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0709.3670

{'0709.3670-1-0-0': 'The structures for the TRI[MATH]P facility have been completed and commissioned.', '0709.3670-1-0-1': 'At the facility radioactive nuclides will be produced to study fundamental interactions and symmetries.', '0709.3670-1-0-2': 'An important feature is the possibility to trap radioactive atom in order to obtain and hold a pure substrate-free sample that will allow precision measurements.', '0709.3670-1-0-3': 'The TRI[MATH]P facility consists of a chain of several separate units: A production target followed by a magnetic separator produces fast radioactive isotopes.', '0709.3670-1-0-4': 'The separator is commissioned and is operating.', '0709.3670-1-0-5': 'Various modes to produce radioactive particles have been tested for optimal production.', '0709.3670-1-0-6': 'A thermal ionizer stops the fast products and transports the nuclides as low energy singly charged ions into a radio frequency quadrupole cooler and buncher.', '0709.3670-1-0-7': 'This allows one to collect and transport the ions via a drift tube and a low energy beam line into a magneto-optical trap (MOT).', '0709.3670-1-0-8': 'The ions are neutralized in the MOT chamber and will be transported to a second trap for [MATH]-decay studies.', '0709.3670-1-0-9': 'Radioactive beam has been transported up to the MOT chamber.', '0709.3670-1-0-10': 'The magneto-optical traps are currently characterized using stable Na beams.', '0709.3670-1-1-0': '# Introduction', '0709.3670-1-2-0': 'Precision measurements of parameters describing [MATH]-decays and searches for permanent electric dipole moments (EDMs) in atomic systems are among the main objectives of the TRI[MATH]P physics programme [CITATION].', '0709.3670-1-2-1': 'Studies of [MATH]-[MATH] correlations in nuclear [MATH] decays require measurement of very low energy recoiling ions.', '0709.3670-1-2-2': 'From these correlations deviations from the V-A structure of the Weak interaction can be searched for.', '0709.3670-1-2-3': 'Another line of research is to improve on the limit set for a permanent EDMs, which are time-reversal violating moments and to improve on the value set for parity non-conservation measuring the weak charge.', '0709.3670-1-2-4': 'For this research advantage can be taken of special high sensitivity to new physics in certain heavy radioactive atoms [CITATION].', '0709.3670-1-3-0': 'The experiments will be performed with samples of radioactive atoms stored in magneto-optical traps (MOT).', '0709.3670-1-3-1': 'Atomic trapping allows to store radioactive atoms without substrates and thereby boosts the performance of high precision experiments.', '0709.3670-1-3-2': 'The samples are confined and localized in space (typically in a volume [MATH] 1 mm[MATH]) at very low temperatures (in the sub-mK range).', '0709.3670-1-3-3': 'Additionally the highly reduced background radioactivity and the exclusive isotopic selectivity further advantages.', '0709.3670-1-4-0': 'At the Kernfysisch Versneller Instituut (KVI) a complex facility was built both to achieve the goals of the TRI[MATH]P group and for experiments with radioactive nuclides by external groups.', '0709.3670-1-4-1': 'The facility employs the in-flight method for production and separation of radioactive isotopes utilizing a dual-mode magnetic separator.', '0709.3670-1-4-2': 'The separator is commissioned and operating [CITATION].', '0709.3670-1-4-3': 'Various modes to produce radioactive particles have been tested for optimal production [CITATION].', '0709.3670-1-4-4': 'A thermal ionizer [CITATION] stops the fast products and transports the nuclides as low-energy singly charged ions into a radio frequency quadrupole (RFQ) cooler and buncher [CITATION].', '0709.3670-1-4-5': 'This allows one to collect and transport the ions towards the optical traps.', '0709.3670-1-4-6': 'The optical trapping of Na atoms is done in two separate MOT stages.', '0709.3670-1-4-7': 'The first MOT is used for neutralization and accumulation of the Na atoms.', '0709.3670-1-4-8': 'It has been commissioned with stable [MATH]Na as described in [CITATION].', '0709.3670-1-4-9': 'The atoms are then transported into a second MOT chamber where they can be trapped again and their decays can be studied in a background-free environment [CITATION].', '0709.3670-1-4-10': 'The second MOT is equipped with [MATH]-detectors and a reaction microscope [CITATION] for detecting of the recoils from the decay.', '0709.3670-1-5-0': '# Production and separation of radioactive nuclides', '0709.3670-1-6-0': 'The TRI[MATH]P magnetic separator is used for in-flight production and separation of a large variety of beam-like radioactive isotopes.', '0709.3670-1-6-1': 'Nuclides are produced mainly using transfer reactions in inverse kinematics (heavy projectile and light target) which provides a secondary radioactive beam that can be matched to the angular and momentum acceptance of the separator.', '0709.3670-1-7-0': 'The production target [CITATION] is filled with a light gas, usually hydrogen, deuterium, or helium.', '0709.3670-1-7-1': 'The length of the gas volume is 10 cm.', '0709.3670-1-7-2': 'The gas is kept at high pressure using Havar windows at the two ends of the target.', '0709.3670-1-7-3': 'They are exchangeable and their thickness can be varied depending on the gas pressure.', '0709.3670-1-7-4': 'Windows of 2.5 and 10 [MATH]m thickness allow safe operation at 1 and 10 atm respectively.', '0709.3670-1-7-5': 'The target thickness is increased several times by cooling of the target to liquid N[MATH] temperature.', '0709.3670-1-8-0': 'The production yields of the desired isotopes and the selection in the separator should be considered simultaneously when the aim is to obtain the highest isotope production rate at the end of the separator.', '0709.3670-1-8-1': 'The factors contributing to the momentum distribution of the secondary beam are related to the production mechanism, i.e. the nuclear reactions, and to the angular and energy straggling in the target and other materials in the separator.', '0709.3670-1-8-2': 'The interplay between the cross sections for a specific reaction, the momentum and angular distributions of the products, and the limited acceptances is considered for the choice of the initial energy of the primary beam.', '0709.3670-1-9-0': 'For the production of [MATH]Na various combinations of reactions and beam energies were examined.', '0709.3670-1-9-1': 'We have found that the reaction [MATH]H([MATH]Ne,[MATH]Na)n is the most convenient, with a typical yield of [MATH]/s/particle nA of primary beam at 1 atm.', '0709.3670-1-9-2': 'The magnetic rigidity of [MATH]Na is [MATH] lower than the rigidity of the [MATH]Ne beam which allows full their separation.', '0709.3670-1-9-3': 'When the separator settings are optimized for the rate of [MATH]Na various other reaction products are also present in the spectra (see Fig. [REF]).', '0709.3670-1-9-4': 'Additional purification can be obtained by using an achromatic degrader in the dispersive plane of the separator.', '0709.3670-1-9-5': 'For the planned [MATH]-decay experiments high yield of [MATH]Na is of primary importance and degraders are not used.', '0709.3670-1-9-6': 'Mass and element selectivity is featured in the stages of the facility following the separator.', '0709.3670-1-10-0': 'The [MATH]Na momentum distributions are shown in more detail in Fig. [REF].', '0709.3670-1-10-1': 'Extrapolating to higher pressures, we conclude that a 10 atm target can be used, allowing production of at least 10[MATH]Na particles/s.', '0709.3670-1-10-2': 'This will allow us to measure [MATH]-[MATH] correlations with a precision of 10[MATH] in less than one week of beam time.', '0709.3670-1-10-3': 'This is an order of magnitude improvement over existing measurement [CITATION].', '0709.3670-1-10-4': 'It also requires theoretical work to describe final state interactions [CITATION].', '0709.3670-1-11-0': 'Several experiments have been performed at the TRI[MATH]P facility with the objective to measure properties of various radioactive elements and their decays.', '0709.3670-1-11-1': 'Depending on the goals of the experiment optimization was focused on either element purity, or on maximal production rate.', '0709.3670-1-11-2': 'Table 1 summarizes the isotopes produced to date with the magnetic separator.', '0709.3670-1-12-0': '# Slowing of radioactive ions', '0709.3670-1-13-0': 'The achromatic focus of the separator coincides with the location of a thermal ionizer (TI) which functions as an ion stopper.', '0709.3670-1-13-1': 'The extraction efficiency from a TI can be close to 100% for alkali ions.', '0709.3670-1-13-2': 'The separator decouples the production site from the TI and the experimental setup completely.', '0709.3670-1-13-3': 'The stopping of the products in the TI is accomplished using a stack of thin W foils which are heated up to at least 2800 K.', '0709.3670-1-13-4': 'The total thickness of the foils is chosen to match the maximum of the energy distribution ([MATH]) of the ions at the focal plane of the separator.', '0709.3670-1-13-5': 'The stopping range is adjusted by using a variable thickness degrader upstream the TI.', '0709.3670-1-13-6': 'In the TI only the ionized particles are extracted but they have multiple chances to be ionized inside the TI and to be electrostatically extracted by an electrode at a negative potential (up to -10 kV).', '0709.3670-1-13-7': 'The increase of output of the TI as function of temperature and of Na-isotope lifetime was observed, but has not yet reached the point where the output is saturated.', '0709.3670-1-13-8': 'Maximal efficiency of 48(3) [MATH] was achieved for [MATH]Na.', '0709.3670-1-13-9': 'The thermal ionizer principles, design, and operation are described in more detail in [CITATION].', '0709.3670-1-14-0': 'Together with the radioactive Na isotopes, various stable ions are present in the low energy beam extracted from the TI.', '0709.3670-1-14-1': 'These ions are formed from diffusion of impurities in the W walls and foils of the TI.', '0709.3670-1-14-2': 'In order to reduce space charge effect mass selection is desired.', '0709.3670-1-14-3': 'This is accomplished using a Wien filter downstream the extractor electrode of the TI.', '0709.3670-1-15-0': '# Cooling and transport of low energy ion beams', '0709.3670-1-16-0': 'The [MATH]Na ion beam leaves the thermal ionizer with a large transverse emittance and an energy distribution width defined by the temperature of the W cavity.', '0709.3670-1-16-1': 'The large emittance is often affecting the transmission of the ions in the beam line and depending on the requirements of the experimental setup cooling may be required.', '0709.3670-1-16-2': 'This can be done by the radio frequency quadrupole (RFQ) cooler and buncher system [CITATION] as part of the low energy beam line (see Fig. [REF]).', '0709.3670-1-17-0': 'The cooling technique is based on collisions with a buffer gas in combination with RF electric fields confining the ions in transverse direction.', '0709.3670-1-17-1': 'The buffer gas is composed of light atoms/molecules, typically He at pressures from [MATH] to [MATH] mbar.', '0709.3670-1-17-2': 'The RF electric field originating from 4 axisymmetric cylindrical electrodes (330 mm long rods) creates a pseudo potential which confines the ions between the rods.', '0709.3670-1-17-3': 'Opposite rods are separated radially by [MATH] mm and potentials [MATH] are applied to them.', '0709.3670-1-17-4': 'Ions with charge [MATH] and mass [MATH] are confined when Mathieu parameters [EQUATION] fulfill certain conditions.', '0709.3670-1-17-5': 'For [MATH] and [MATH] 0.9 it serves as an ion guide.', '0709.3670-1-17-6': 'For [MATH] the RFQ structure can also be exploited for mass selectivity.', '0709.3670-1-18-0': 'To transport the ions through the cooler a DC drag potential must be applied in the longitudinal direction.', '0709.3670-1-18-1': 'It is obtained by dividing the RFQ in longitudinal sections (36 segments per rod) allowing different potentials to be applied.', '0709.3670-1-18-2': 'The DC potentials are set on the separated sections, whereas, the RF potentials are applied on the rods which are capacitively coupled with the segments.', '0709.3670-1-18-3': 'The capacitative coupling is achieved using a 20 [MATH]m thin Kapton[MATH] foil between the rods and the segments.', '0709.3670-1-19-0': 'The cooled ions enter the buncher (mechanically identical to the cooler) through a small aperture on an electrode plate separating the cooler and the buncher and allowing differential pumping.', '0709.3670-1-19-1': 'The ion transmission through the aperture is critical because of the damping of the RF fields near the aperture causing a reduction of the radial confinement.', '0709.3670-1-19-2': 'To overcome this a DC longitudinal acceleration is applied before the aperture and a subsequent deceleration in the beginning of the buncher.', '0709.3670-1-20-0': 'The operation pressure in the buncher varies from [MATH] to [MATH] mbar.', '0709.3670-1-20-1': 'This pressure allows to reduce the velocity of the ions further and is sufficient to confine the ions in the longitudinal direction by creating a DC potential well close to the exit of the buncher.', '0709.3670-1-20-2': 'The trapped ions are then accumulated and extracted in bunches by switching the DC potential configuration when the trapping region is filled.', '0709.3670-1-20-3': 'The accumulation time is in the range from several milliseconds to several seconds.', '0709.3670-1-21-0': 'The ion bunches are accelerated to several keV in a drift tube accelerator and guided into an all-electrostatic low energy beam line where they are transported to the experimental sites.', '0709.3670-1-21-1': 'The drift tube accelerator allows to keep the RFQ as well as the low energy beam line at ground potential.', '0709.3670-1-22-0': 'Monte Carlo simulations of the RFQ performance were made for Na ions in order to estimate the effects of the fringe fields near the apertures and to optimize the coupling of the RFQ sections and the subsequent extraction.', '0709.3670-1-22-1': 'The interactions of the ions with the buffer gas atoms were implemented in the simulations by taking into account the diffusion related random force [CITATION].', '0709.3670-1-22-2': 'The amplitude of the random force was normalized using data for ion mobilities in gases.', '0709.3670-1-23-0': 'The RFQ system was commissioned with stable Na ions.', '0709.3670-1-23-1': 'Various measurements were made in order to characterize the system and establish the operation ranges of various parameters, e.g. potentials, pressures, accumulation times.', '0709.3670-1-23-2': 'Transmission was optimized and efficiencies up to [MATH] were measured for the RFQ stages.', '0709.3670-1-23-3': 'Measured storage times in the buncher showed strong dependence of the buncher efficiency on the buffer gas purity and the vacuum system.', '0709.3670-1-24-0': 'A variety of elements at a wide span of masses extracted from the thermal ionizer was used to determine the mass selectivity of the RFQ (Mathieu parameter [MATH]).', '0709.3670-1-24-1': 'Different ion masses were identified using a microchannel plate detector after the drift tube and measuring time-of-flight spectra of the ions.', '0709.3670-1-24-2': 'The optimal selectivity was achieved by applying a potential [MATH] in the buncher (due to lower operation pressures and longer trajectories compared to the cooler).', '0709.3670-1-24-3': 'Selection was achieved for [MATH] u in the mass region of Na.', '0709.3670-1-25-0': '# Neutralization and trapping', '0709.3670-1-26-0': 'To trap atoms the secondary ion beam needs to be neutralized.', '0709.3670-1-26-1': 'This is done inside a small glass cell, the collection MOT [CITATION] , using a heated neutralizer foil in which the ions are implanted with energies up to a few keV.', '0709.3670-1-26-2': 'The neutralizer thus has common issues with the the thermal ionizer, i.e. fast diffusion from a hot foil, surface sticking time and ionization probability.', '0709.3670-1-26-3': 'In both cases diffusion delay is the slowest time component in the process.', '0709.3670-1-26-4': 'Various materials (e.g. Zr, Y, W, and LiF) were used for neutralization of Na ions.', '0709.3670-1-26-5': 'Stable [MATH]Na ions were used for characterization of the neutralizers by observing the number of captured particles in the MOT.', '0709.3670-1-26-6': 'Trapping was maximal for Zr which is considered currently as the optimal neutralizer material.', '0709.3670-1-27-0': 'Once neutral, the particles can be trapped in the MOT (shown in Fig. [REF]).', '0709.3670-1-27-1': 'The atoms in the cell have a thermal velocity distribution (the neutralizer temperature) and only a fraction of them are in the velocity range allowing trapping.', '0709.3670-1-27-2': 'The maximal velocity for which atoms can be trapped can be increased by increasing the red detuning of the lasers but require higher laser power.', '0709.3670-1-27-3': 'The atoms which are not trapped when passing through the trap region collide with the walls of the cell and may be lost due to sticking and re-diffusion in the glass.', '0709.3670-1-27-4': 'The atoms desorbed from the surface have lower velocities than the ones from the neutralizer since their velocity distribution is defined by the temperature of the walls.', '0709.3670-1-27-5': 'In Fig. [REF] are two sets of data.', '0709.3670-1-27-6': 'The small yields correspond to a glass cell without a non-stick coating.', '0709.3670-1-27-7': 'The data obtained with non-stick coating are higher in yield because the atoms can bounce of the walls and make multiple passes through the trapping region.', '0709.3670-1-27-8': 'The number of trapped atoms increases with temperature due to faster diffusion in the neutralizer.', '0709.3670-1-27-9': 'The fact that at higher temperatures the number of trapped atoms decreases again is due to the increase of the pressure with temperature of the neutralizer.', '0709.3670-1-27-10': 'The trap lifetime is inversely proportional to the pressure.', '0709.3670-1-28-0': 'The trapped atoms will be guided using lasers into a second MOT which will be employed for the actual measurements of the [MATH]-decay observables.', '0709.3670-1-28-1': 'In contrast to the first MOT cell, where many atoms can decay without being trapped, in the decay MOT the atoms will be well localized in the trap reducing the uncertainty of the measurements.', '0709.3670-1-28-2': 'This trap is situated in a large vacuum chamber including detectors for the [MATH] particles, a microchannel plate (MCP) detector with position sensitive read-out, and guiding electrodes for the detection of the recoiling [MATH]Ne ions.', '0709.3670-1-28-3': 'The MOT setup for the measurements of the [MATH]-[MATH] correlations is currently being commissioned [CITATION].', '0709.3670-1-29-0': '# Outlook', '0709.3670-1-30-0': 'A secondary beam of [MATH]Na ions has been produced, purified, and transferred successfully to a neutralizer in the glass cell around the collection MOT site.', '0709.3670-1-30-1': 'We have exploited [MATH]Na and [MATH]Na for the optimizing process using pairs of [MATH]-ray detectors at various places along the low energy beam line to observe the annihilation radiation or [MATH]-delayed [MATH] decay ([MATH]Na).', '0709.3670-1-30-2': 'Stable [MATH]Na from the thermal ionizer is used as a pilot beam in the low energy beam line.', '0709.3670-1-30-3': 'We are currently in the process of optimizing trapping and detection efficiencies in preparation of our first physics experiment, a [MATH] correlation measurement in the decay of [MATH]Na.', '0709.3670-1-30-4': 'The successful trapping and detection of [MATH]Na in the MOT, which we expect very soon as the next and remaining last step in a series of successful preparation stages, will be the start of the physics program at the TRI[MATH]P radioactive beam and trapping facility at KVI.'}

{'0709.3670-2-0-0': 'The structures for the TRI[MATH]P facility have been completed and commissioned.', '0709.3670-2-0-1': 'At the facility radioactive nuclides are produced to study fundamental interactions and symmetries.', '0709.3670-2-0-2': 'An important feature is the possibility to trap radioactive atoms in order to obtain and hold a pure substrate-free sample for precision measurements.', '0709.3670-2-0-3': 'In the TRI[MATH]P facility a production target is followed by a magnetic separator, where radioactive isotopes are produced in inverse reaction kinematics.', '0709.3670-2-0-4': 'Separation up to 99.95[MATH] could be achieved for [MATH]Na.', '0709.3670-2-0-5': 'A novel transmitting thermal ionizing device was developed to stop the energetic isotopes.', '0709.3670-2-0-6': 'Some 50[MATH] of stopped [MATH]Na could be extracted and transported as low energy singly charged ions into a radio frequency quadrupole cooler and buncher with 35[MATH] transmission efficiency.', '0709.3670-2-0-7': 'The ions are transported lossless via a drift tube and a low energy electrostatic beam line into the experimental setup.', '0709.3670-2-0-8': 'Such ions can be neutralized on hot metal foils and the resulting atoms can be stored in a magneto-optical trap.', '0709.3670-2-0-9': 'The functioning of that principle was demonstrated with stable Na extracted from the thermal ionizer, radioactive beams will follow next.', '0709.3670-2-1-0': '# Introduction', '0709.3670-2-2-0': 'Precision measurements of parameters describing [MATH]-decays and searches for permanent electric dipole moments (EDMs) in atomic systems are among the main objectives of the TRI[MATH]P physics programme [CITATION].', '0709.3670-2-2-1': 'Studies of [MATH]-[MATH] correlations in nuclear [MATH]-decays require the detection of very low energy recoiling ions.', '0709.3670-2-2-2': 'From these correlations, deviations from the V-A structure of the weak interaction can be searched for.', '0709.3670-2-2-3': 'Another line of research is to improve the limits for permanent EDMs, which are time-reversal violating moments and to improve the uncertainty of the values for parity non-conservation measuring the weak charge.', '0709.3670-2-2-4': 'For this research, advantage can be taken of special high sensitivity to new physics in certain heavy radioactive atoms [CITATION].', '0709.3670-2-3-0': 'The experiments will be performed with samples of radioactive atoms stored in magneto-optical traps (MOT).', '0709.3670-2-3-1': 'Atomic trapping allows the storage of radioactive atoms without substrates and thereby boosts the performance of high precision experiments.', '0709.3670-2-3-2': 'Other advantages of atomic trapping include confinement and localization of the samples in space (typically in a volume [MATH] 1 mm[MATH]) at very low temperatures (in the sub-mK range), exclusive isotopic selectivity, and highly reduced background radioactivity.', '0709.3670-2-4-0': 'At the Kernfysisch Versneller Instituut (KVI) a complex facility was built both to achieve the goals of the TRI[MATH]P group and for experiments with radioactive nuclides by external groups.', '0709.3670-2-4-1': 'The facility employs the in-flight method for production and separation of radioactive isotopes utilizing a dual-mode magnetic separator.', '0709.3670-2-4-2': 'The separator is commissioned and operating [CITATION].', '0709.3670-2-4-3': 'Various modes to produce radioactive particles have been tested for optimal production [CITATION].', '0709.3670-2-4-4': 'A thermal ionizer [CITATION] stops the fast products and transports the nuclides as low-energy singly charged ions into a radio frequency quadrupole (RFQ) cooler and buncher [CITATION].', '0709.3670-2-4-5': 'This allows the collection and transport of ions towards the optical traps.', '0709.3670-2-4-6': 'The optical trapping of Na atoms is done in two separate MOT stages.', '0709.3670-2-4-7': 'The first MOT is used for neutralization and accumulation of the Na atoms.', '0709.3670-2-4-8': 'It has been commissioned with stable [MATH]Na as described in [CITATION].', '0709.3670-2-4-9': 'The atoms are then transported into a second MOT chamber where they can be trapped again and their decays can be studied in a background-free environment [CITATION].', '0709.3670-2-4-10': 'The second MOT is equipped with [MATH]-detectors and a reaction microscope [CITATION] for detecting of the recoils from the decay.', '0709.3670-2-5-0': '# Production and separation of radioactive nuclides', '0709.3670-2-6-0': 'The TRI[MATH]P magnetic separator is used for in-flight production and separation of a large variety of beam-like radioactive isotopes.', '0709.3670-2-6-1': 'The nuclides are produced mainly using transfer reactions in inverse kinematics (heavy projectile and light target) which provides a secondary radioactive beam that can be matched to the angular and the momentum acceptance of the separator.', '0709.3670-2-7-0': 'The production target [CITATION] is filled with a light gas, usually hydrogen, deuterium, or helium.', '0709.3670-2-7-1': 'The length of the gas volume is 10 cm.', '0709.3670-2-7-2': 'The gas is kept at high pressure using Havar windows at the two ends of the target.', '0709.3670-2-7-3': 'The windows are exchangeable and their thickness can be varied depending on the gas pressure.', '0709.3670-2-7-4': 'Windows of 2.5 and 10 [MATH]m thickness allow safe operation at 1 and 10 atm respectively.', '0709.3670-2-7-5': 'The target thickness is increased several times by cooling of the target to liquid N[MATH] temperature.', '0709.3670-2-8-0': 'The production yield of the desired isotopes in the target should be considered together with the selection in the separator when the aim is to obtain the highest isotope production rate at the end of the separator.', '0709.3670-2-8-1': 'The factors contributing to the momentum distribution of the secondary beam are related to the production mechanism, i.e. the nuclear reactions, and to the angular and energy straggling in the target and other materials in the separator.', '0709.3670-2-8-2': 'The interplay between the cross sections for a specific reaction, the momentum and angular distributions of the products, and the separator acceptances is considered for the choice of the initial energy of the primary beam.', '0709.3670-2-8-3': 'For the production of [MATH]Na various combinations of reactions and beam energies were examined.', '0709.3670-2-8-4': 'We have found that the reaction [MATH]H([MATH]Ne,[MATH]Na)n is the most convenient, with a typical yield of [MATH]/s/particle nA of primary beam at 1 atm.', '0709.3670-2-8-5': 'The magnetic rigidity of [MATH]Na is [MATH] lower than the rigidity of the [MATH]Ne beam which allows their complete separation.', '0709.3670-2-8-6': 'When the separator settings are optimized for the rate of [MATH]Na various other reaction products are also present in the spectra (see Fig. [REF]).', '0709.3670-2-8-7': 'Additional purification (up to 99.95[MATH] for [MATH]Na) is obtained by using an achromatic degrader in the dispersive plane of the separator.', '0709.3670-2-8-8': 'For the planned [MATH]-decay experiments the high yield of [MATH]Na is of primary importance and degraders are not used.', '0709.3670-2-8-9': 'Mass and element selectivity is featured in the stages of the facility following the separator.', '0709.3670-2-9-0': 'The [MATH]Na momentum distributions are shown in more detail in Fig. [REF].', '0709.3670-2-9-1': 'Extrapolating to higher pressures, we conclude that a 10 atm target can be used, allowing the production of at least 10[MATH]Na particles/s.', '0709.3670-2-9-2': 'This will allow us to measure [MATH]-[MATH] correlations with a precision of 10[MATH] in less than one week of beam time.', '0709.3670-2-9-3': 'This is an order of magnitude improvement over existing measurement [CITATION].', '0709.3670-2-9-4': 'It also requires theoretical work to describe final state interactions [CITATION].', '0709.3670-2-10-0': 'Several experiments have been performed at the TRI[MATH]P facility with the objective to measure properties of various radioactive elements and their decays.', '0709.3670-2-10-1': 'Depending on the goals of the experiment, the optimization was focused on either element purity, or on maximal production rate.', '0709.3670-2-10-2': 'Table 1 summarizes the isotopes produced to date with the magnetic separator.', '0709.3670-2-11-0': '# Slowing of radioactive ions', '0709.3670-2-12-0': 'The achromatic focus of the separator coincides with the location of a thermal ionizer (TI) which functions as an ion stopper.', '0709.3670-2-12-1': 'The extraction efficiency from a TI can be close to 100% for alkali ions.', '0709.3670-2-12-2': 'The separator completely decouples the production site from the TI and the experimental setup.', '0709.3670-2-12-3': 'The stopping of the products in the TI is accomplished using a stack of thin W foils which are heated up to at least 2800 K.', '0709.3670-2-12-4': 'The total thickness of the foils is chosen to match the maximum of the energy distribution ([MATH]) of the ions at the focal plane of the separator.', '0709.3670-2-12-5': 'The stopping range is adjusted by using a rotatable degrader upstream the TI.', '0709.3670-2-12-6': 'In the TI, only the ionized particles are extracted but they have multiple chances to be ionized inside the TI and to be electrostatically extracted by an electrode at a negative potential (up to -10 kV).', '0709.3670-2-12-7': 'A yield increase from the TI as function of temperature and of Na-isotope lifetime was observed, but has not yet reached the point where the output is saturated.', '0709.3670-2-12-8': 'A maximal efficiency of 48(3)[MATH] was achieved for [MATH]Na.', '0709.3670-2-12-9': 'The thermal ionizer principles, design, and operation are described in more detail in [CITATION].', '0709.3670-2-13-0': 'Together with the radioactive Na isotopes, various stable ions are present in the low energy beam extracted from the TI.', '0709.3670-2-13-1': 'These ions are formed from diffusion of impurities in the W walls and foils of the TI.', '0709.3670-2-13-2': 'In order to reduce space charge effects, mass selection is desired.', '0709.3670-2-13-3': 'This is accomplished using a Wien filter downstream the extractor electrode of the TI.', '0709.3670-2-14-0': '# Cooling and transport of low energy ion beams', '0709.3670-2-15-0': 'The [MATH]Na ion beam leaves the thermal ionizer with a large transverse emittance and an energy distribution width defined by the temperature of the W cavity.', '0709.3670-2-15-1': 'The large emittance is often affecting the transmission of the ions in the beam line and, depending on the requirements of the experimental setup, cooling may be required.', '0709.3670-2-15-2': 'This can be done by the radio frequency quadrupole (RFQ) cooler and buncher system [CITATION] as part of the low energy beam line (see Fig. [REF]).', '0709.3670-2-16-0': 'The cooling technique is based on collisions with a buffer gas in combination with RF electric fields confining the ions in transverse direction [CITATION].', '0709.3670-2-16-1': 'The buffer gas is composed of light atoms/molecules, typically helium at pressures from [MATH] to [MATH] mbar.', '0709.3670-2-16-2': 'The RF electric field originating from 4 axisymmetric cylindrical electrodes (330 mm long rods) creates a pseudo potential which confines the ions between the rods.', '0709.3670-2-16-3': 'Opposite rods are separated radially by [MATH] mm and potentials [MATH] are applied to them.', '0709.3670-2-16-4': 'To transport the ions through the cooler a DC drag potential is applied in the longitudinal direction.', '0709.3670-2-16-5': 'It is obtained by dividing the RFQ in longitudinal sections (36 segments per rod) allowing different potentials to be applied.', '0709.3670-2-16-6': 'The DC potentials are set on the separated sections whereas the RF potentials are applied on the rods which are capacitively coupled with the segments.', '0709.3670-2-16-7': 'The capacitative coupling is achieved using a 20 [MATH]m thin Kapton[MATH] foil between the rods and the segments.', '0709.3670-2-17-0': 'The cooled ions enter the buncher (mechanically identical to the cooler) through a small aperture on an electrode plate separating the cooler and the buncher and allowing differential pumping (Fig. [REF]).', '0709.3670-2-17-1': 'The ion transmission through the aperture is critical because of the damping of the RF fields near the aperture causing a reduction of the radial confinement.', '0709.3670-2-17-2': 'To overcome this, a DC longitudinal acceleration is applied before the aperture and a subsequent deceleration in the beginning of the buncher.', '0709.3670-2-18-0': 'The operation pressure in the buncher varies from [MATH] to [MATH] mbar.', '0709.3670-2-18-1': 'This pressure allows the further reduction of the ion velocity and is sufficient to confine the ions in the longitudinal direction by creating a DC potential well close to the exit of the buncher.', '0709.3670-2-18-2': 'The trapped ions are then accumulated and extracted in bunches by switching the DC potential configuration when the trapping region is filled.', '0709.3670-2-18-3': 'The accumulation time is in the range from several milliseconds to several seconds.', '0709.3670-2-19-0': 'The ion bunches are accelerated to several keV in a drift tube accelerator and guided into an all-electrostatic low energy beam line where they are transported to the experimental sites.', '0709.3670-2-19-1': 'With the drift tube installed, both the RFQ and the low energy beam line can be kept on ground potential.', '0709.3670-2-20-0': 'Monte Carlo simulations of the RFQ performance were made for Na ions in order to estimate the effects of the fringe fields near the apertures and to optimize the coupling of the RFQ sections and the subsequent extraction.', '0709.3670-2-20-1': 'The interactions of the ions with the buffer gas atoms were implemented in the simulations by taking into account the diffusion related random force [CITATION].', '0709.3670-2-20-2': 'The amplitude of the random force was normalized using data for ion mobilities in gases.', '0709.3670-2-21-0': 'The RFQ system was commissioned with stable Na ions.', '0709.3670-2-21-1': 'Various measurements were made in order to characterize the system and establish the operation ranges of various parameters, e.g. potentials, pressures, accumulation times.', '0709.3670-2-21-2': 'Transmission was optimized and efficiencies up to [MATH] were measured for each of the RFQ stages.', '0709.3670-2-21-3': 'Measured storage times in the buncher showed strong dependence of the buncher efficiency on the buffer gas purity and the vacuum system.', '0709.3670-2-22-0': 'The RFQ structures can also be exploited for mass selectivity.', '0709.3670-2-22-1': 'The variety of elements at a wide span of masses extracted from the thermal ionizer was used to determine the mass selectivity of the RFQ (Mathieu parameter [MATH]).', '0709.3670-2-22-2': 'Different ion masses were identified using a microchannel plate detector after the drift tube and measuring time-of-flight spectra of the ions.', '0709.3670-2-22-3': 'The optimal selectivity was achieved by applying a potential [MATH] in the buncher (due to lower operation pressures and longer trajectories compared to the cooler).', '0709.3670-2-22-4': 'Selection was achieved for [MATH] u in the mass region of Na.', '0709.3670-2-23-0': '# Neutralization and trapping', '0709.3670-2-24-0': 'To trap atoms, the secondary ion beam needs to be neutralized.', '0709.3670-2-24-1': 'This is done inside a small glass cell, the collection MOT [CITATION] , using a heated neutralizer foil in which the ions are implanted with energies up to a few keV.', '0709.3670-2-24-2': 'The neutralizer has thus common issues with the the thermal ionizer, i.e. fast diffusion from a hot foil, surface sticking time and ionization probability.', '0709.3670-2-24-3': 'In both cases diffusion delay is the slowest time component in the process.', '0709.3670-2-24-4': 'Various materials (e.g. Zr, Y, W, and LiF) were used for neutralization of Na ions.', '0709.3670-2-24-5': 'Stable [MATH]Na ions were used for characterization of the neutralizers by observing the number of captured particles in the MOT.', '0709.3670-2-24-6': 'Trapping was maximal for Zr which is considered currently as the optimal neutralizer material.', '0709.3670-2-25-0': 'Once neutral, the particles can be trapped in the MOT (shown in Fig. [REF]).', '0709.3670-2-25-1': 'The atoms in the cell have a thermal velocity distribution (the neutralizer temperature) and only a fraction of them are in the velocity range allowing trapping.', '0709.3670-2-25-2': 'The maximal velocity for which atoms can be trapped can be increased by increasing the red detuning of the lasers but require higher laser power.', '0709.3670-2-25-3': 'The atoms which are not trapped when passing through the trap region collide with the walls of the cell and may be lost due to sticking and re-diffusion in the glass.', '0709.3670-2-25-4': 'The atoms desorbed from the surface have lower velocities than the ones from the neutralizer since their velocity distribution is defined by the temperature of the walls.', '0709.3670-2-25-5': 'Figure [REF] shows two sets of data: from a glass cell without ([MATH]) and with ([MATH]) a non-stick coating.', '0709.3670-2-25-6': 'The data obtained using the coated cell are higher in yield because the atoms can bounce of the walls and make multiple passes through the trapping region.', '0709.3670-2-25-7': 'The number of trapped atoms increases with temperature due to faster diffusion in the neutralizer.', '0709.3670-2-25-8': 'The fact that at higher temperatures the number of trapped atoms decreases again is due to the increase of the pressure with temperature of the neutralizer.', '0709.3670-2-25-9': 'The trap lifetime is inversely proportional to the pressure.', '0709.3670-2-26-0': 'The trapped atoms will be guided using lasers into a second MOT which will be employed for the actual measurements of the [MATH]-decay observables.', '0709.3670-2-26-1': 'In contrast to the first MOT cell, where many atoms can decay without being trapped, in the decay MOT the atoms will be well localized in the trap reducing the uncertainty of the measurements.', '0709.3670-2-26-2': 'This trap is situated in a large vacuum chamber including detectors for the [MATH] particles, a microchannel plate (MCP) detector with position sensitive read-out, and guiding electrodes for the detection of the recoiling [MATH]Ne ions.', '0709.3670-2-26-3': 'The MOT setup for the measurements of the [MATH]-[MATH] correlations is currently being commissioned [CITATION].', '0709.3670-2-27-0': '# Outlook', '0709.3670-2-28-0': 'A secondary beam of [MATH]Na ions has been produced, purified, and transferred successfully to a neutralizer in the glass cell around the collection MOT site.', '0709.3670-2-28-1': 'We have exploited [MATH]Na and [MATH]Na for the optimizing process using pairs of [MATH]-ray detectors at various places along the low energy beam line to observe the annihilation radiation or [MATH]-delayed [MATH] decay ([MATH]Na).', '0709.3670-2-28-2': 'Stable [MATH]Na from the thermal ionizer is used as a pilot beam in the low energy beam line.', '0709.3670-2-28-3': 'We are currently in the process of optimizing the trapping and detection efficiencies in preparation of our first physics experiment, a [MATH] correlation measurement in the decay of [MATH]Na.', '0709.3670-2-28-4': 'The successful trapping and detection of [MATH]Na in the MOT, which we expect very soon as the next and remaining last step in a series of successful preparation stages, will be the start of the physics program at the TRI[MATH]P radioactive beam and trapping facility at KVI.'}

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[]

[['0709.3670-1-27-6', '0709.3670-2-25-5'], ['0709.3670-1-13-2', '0709.3670-2-12-2'], ['0709.3670-1-13-8', '0709.3670-2-12-8'], ['0709.3670-1-21-1', '0709.3670-2-19-1'], ['0709.3670-1-0-3', '0709.3670-2-0-3'], ['0709.3670-1-0-6', '0709.3670-2-0-6'], ['0709.3670-1-0-7', '0709.3670-2-0-7'], ['0709.3670-1-3-2', '0709.3670-2-3-2'], ['0709.3670-1-4-5', '0709.3670-2-4-5']]

[]

[]

{'1': 'http://arxiv.org/licenses/assumed-1991-2003/', '2': 'http://arxiv.org/licenses/assumed-1991-2003/'}

https://arxiv.org/abs/0709.3670

1605.07145

{'1605.07145-1-0-0': 'Training non-linear neural networks is a challenging task, but over the years, various approaches coming from different perspectives have been proposed to improve performance.', '1605.07145-1-0-1': 'However, insights into what fundamentally constitutes optimal network parameters remains obscure.', '1605.07145-1-0-2': 'Similarly, given what properties of data can we hope for a non-linear network to learn is also not well studied.', '1605.07145-1-0-3': 'In order to address these challenges, we take a novel approach by analysing neural network from a data generating perspective, where we assume hidden layers generate the observed data.', '1605.07145-1-0-4': 'This perspective allows us to connect seemingly disparate approaches explored independently in the machine learning community such as batch normalization, Independent Component Analysis, orthogonal weight initialization, etc, as parts of a bigger picture and provide insights into non-linear networks in terms of properties of parameter and data that lead to better performance.', '1605.07145-1-1-0': '# Introduction', '1605.07145-1-2-0': 'Deep networks when trained under different optimization conditions may lead to drastically different results, both in terms of performance and the properties of the network parameters learned.', '1605.07145-1-2-1': 'While many heuristic methods for optimizing parameters exist, it is not well understood what parameter properties lead to good performance.', '1605.07145-1-2-2': 'Therefore our goal is to discover conditions on parameter properties that ensure near optimal performance.', '1605.07145-1-3-0': 'In practice, seemingly unrelated components (e.g. Rectified Linear units [CITATION], orthogonal weight initialization [CITATION], Batch Normalization [CITATION], e.t.c.) lead to performance improvement while resulting in different learned parameters.', '1605.07145-1-3-1': 'This raises the question of whether there exists in general some optimal properties of network parameters.', '1605.07145-1-3-2': 'In order to answer this question we look at neural networks from the perspective of a data generating model where hidden layer representations give rise to the observed data (input layer).', '1605.07145-1-3-3': 'Then, we ask the question: what attributes of network parameters ensure accurate recovery of hidden representation given the data when we forward propagate?', '1605.07145-1-3-4': 'We find the answer to our question lies with a surprisingly unrelated entity-the Auto-Encoder.', '1605.07145-1-4-0': 'Auto-Encoders (AE) [CITATION] are commonly used for unsupervised representation learning.', '1605.07145-1-4-1': 'AEs focus on learning a mapping [MATH], where the reconstructed vector [MATH] is desired to be as close to [MATH] as possible for the entire data distribution.', '1605.07145-1-4-2': 'What we show in this paper is that if we consider [MATH] is actually generated from [MATH] by some process (discussed later), then switching our perspective to analyze [MATH] yields unexpected useful insights into the optimality of model parameters of non-linear networks in terms of signal recovery.', '1605.07145-1-4-3': 'In other words, this perspective lets us look at a neural network layer from a signal recovery point of view where forward propagating [MATH] recovers the true signal [MATH].', '1605.07145-1-4-4': 'In order to do so, we analyze the conditions under which the encoder part of an AE recovers the true [MATH] from [MATH], while the decoder part acts as the data generation process.', '1605.07145-1-5-0': 'In this paper, our main result shows that the true signal [MATH] can be approximately recovered by the encoder part of an AE with high probability under certain conditions on the weight matrix, bias vectors and the distribution of the hidden signal.', '1605.07145-1-5-1': 'We discover that these required conditions, and the properties resulting from them, connect a wide variety of ideas that have been independently explored so far in the machine learning community as pieces of a bigger picture.', '1605.07145-1-5-2': 'These ideas include Batch Normalization [CITATION], Normalization Propagation [CITATION], De-noising Auto-Encoder [CITATION], Independent Component Analysis [CITATION] (ICA), Sparse Coding, orthogonal weight initialization [CITATION], [MATH]-sparse auto-encoders[CITATION], data whitening, Rectified Linear Activation [CITATION] and Sigmoid activation (see Section [REF] for details).', '1605.07145-1-5-3': 'While we provide insights for single layer non-linear network parameters, we believe our approach of analyzing neural networks from a data generation perspective can reveal insights into the workings of deep non-linear networks as well.', '1605.07145-1-6-0': '# Sparse Signal Recovery Point of View', '1605.07145-1-7-0': 'While it is known both empirically and theoretically, that useful features learned by AEs are usually sparse [CITATION].', '1605.07145-1-7-1': "An important question that hasn't been answered yet is whether AEs in general are capable of recovering sparse signals (generated as describe in the next section) in the first place.", '1605.07145-1-7-2': 'This is an important question from Sparse Coding point of view- because it entails recovering the sparsest [MATH] that approximately satisfies [MATH], for any given data vector [MATH] and overcomplete weight matrix [MATH].', '1605.07145-1-7-3': 'However, since this problem is NP complete [CITATION], it is usually relaxed to solving an expensive optimization problem [CITATION], [EQUATION] where [MATH] is a fixed overcomplete ([MATH]) dictionary, [MATH] is the regularization coefficient, [MATH] is the observed data and [MATH] is the signal we want to recover.', '1605.07145-1-7-4': 'For this special case, [CITATION] analyzes the condition under which linear AEs can recover the support of the hidden signal.', '1605.07145-1-8-0': 'The general AE objective, on the other hand, minimizes the expected reconstruction cost [EQUATION] for some encoding and decoding activation function [MATH] and [MATH], and bias vectors [MATH] and [MATH].', '1605.07145-1-8-1': 'In this paper we consider linear activation [MATH] because we are interested in sparse signal recovery analysis.', '1605.07145-1-8-2': 'Notice however, in the case of auto-encoders, the activation functions can be non-linear in general, in contrast to the sparse coding objective.', '1605.07145-1-8-3': 'In addition, we do not have a separate parameter [MATH] for the hidden representation corresponding to every data sample [MATH] individually in case of AEs.', '1605.07145-1-8-4': 'Instead, the hidden representation for every sample is a parametric function of the sample itself.', '1605.07145-1-8-5': 'This is an important distinction between the optimization in equation [REF] and our problem- the identity of [MATH] in equation [REF] is only well defined in the presence of [MATH] regularization due to the overcompleteness of the dictionary; however in our problem, we assume a true signal [MATH] generates the observed data [MATH] as [MATH] where the dictionary [MATH] and bias vector [MATH] are fixed.', '1605.07145-1-8-6': 'Hence, what we mean by recovery of sparse signals in an AE framework is that if we generate data using the above generation process, then Can the estimate [MATH] indeed recover the true [MATH] for some activation functions [MATH], and bias vector [MATH]?', '1605.07145-1-8-7': 'And if so, what properties of [MATH] and [MATH] in general lead to good recovery?', '1605.07145-1-8-8': 'However, when given an [MATH] and the true overcomplete [MATH], the solution [MATH] to [MATH] is not unique.', '1605.07145-1-8-9': 'Then the question arises about the possibility of recovering such an [MATH].', '1605.07145-1-8-10': 'However, as we show, recovery using the AE mechanism is strongest when the signal [MATH] is the sparsest possible one, which from compressed sensing theory, guarantees uniqueness of [MATH] if [MATH] is sufficiently coherent .', '1605.07145-1-9-0': 'We would like to point out that while we analyze the AE recovery mechanism for recovering [MATH], our goal is not data reconstruction and our results hold for layers in a general neural network.', '1605.07145-1-10-0': '# Data Generation Process', '1605.07145-1-11-0': 'We consider the following data generation process: [EQUATION] where [MATH] is the observed data, [MATH] is a bias vector, [MATH] is a noise vector, [MATH] is the weight matrix and [MATH] is the hidden representation (signal) that we want to recover given the observed data.', '1605.07145-1-11-1': 'Through out our analysis, we assume that the signal [MATH] belongs to the following class of distribution,', '1605.07145-1-12-0': 'Bounded Independent Non-negative Sparse (BINS)-Every hidden unit [MATH] is an independent random variable with the following density function: [EQUATION] where [MATH] can be any arbitrary normalized distribution bounded in the interval [MATH], mean [MATH], and [MATH] is the Dirac Delta function at zero.', '1605.07145-1-12-1': 'As a short hand, we say that [MATH] follows the distribution BINS([MATH]).', '1605.07145-1-12-2': 'Notice [MATH].', '1605.07145-1-13-0': 'The above continuous distribution assumption is justified because of the following intuition: In deep networks with ReLU activations, hidden unit pre-activations have a Gaussian like symmetric distribution [CITATION].', '1605.07145-1-13-1': "If we assume these distributions are mean centered, then on the application of ReLU on pre-activation values, hidden units' distribution has a large mass at [MATH] while the rest of the mass concentrates in [MATH] for some finite positive [MATH] since the pre-activation concentrates symmetrically around zero.", '1605.07145-1-13-2': 'As we show in the next section, ReLU is indeed capable of recovering such signals.', '1605.07145-1-13-3': 'On a side note, the distribution from the above assumption can take shapes similar to that of Exponential or Rectified Gaussian distribution (which are generally used for modelling biological neurons) but is simpler to analyse.', '1605.07145-1-13-4': 'This is because our definition is more general in a sense since we allow [MATH] to be any arbitrary normalized distribution.', '1605.07145-1-13-5': 'The only restriction assumption [REF] has is that to be bounded; but this does not change the representative power of this distribution significantly because: a) the distributions used for modelling neurons have very small tail mass; b) in practice, we are generally interested in signals with upper bounded values.', '1605.07145-1-14-0': 'The above data generation process (equation [REF] and the assumptions above) as a whole is justified because of the following reasons:', '1605.07145-1-15-0': '1.', '1605.07145-1-15-1': 'The above data generation model finds applications in a number of areas [CITATION].', '1605.07145-1-15-2': 'Notice that while [MATH] is the measurement vector (observed data), which can in general be noisy, [MATH] denotes the actual signal (internal representation) because it reflects the combination of dictionary ([MATH]) atoms involved in generating the observed samples and hence serves as the true identity of the data.', '1605.07145-1-16-0': '2.', '1605.07145-1-16-1': 'Sparse distributed representation [CITATION] is both observed and desired in hidden representations.', '1605.07145-1-16-2': 'It has been empirically shown that representations that are truly sparse and distributed (large number of hard zeros) usually yield better linear separability and performance [CITATION].', '1605.07145-1-17-0': 'Decoding bias ([MATH]): Consider the data generation process (exclude noise for now) [MATH].', '1605.07145-1-17-1': 'Here [MATH] is a bias vector which can take any arbitrary value but similar to [MATH], it is fixed for any particular data generation process.', '1605.07145-1-17-2': 'However, the following proposition shows that if an AE can recover the sparse code ([MATH]) from a data sample generated as [MATH], then it is also capable of recovering the sparse code from the data generated as [MATH] and vice versa.', '1605.07145-1-18-0': 'Let [MATH] where [MATH], [MATH] and [MATH].', '1605.07145-1-18-1': 'Let [MATH] where [MATH] is a fixed vector.', '1605.07145-1-18-2': 'Let [MATH] and [MATH].', '1605.07145-1-18-3': 'Then [MATH] iff [MATH].', '1605.07145-1-19-0': 'Thus without any loss of generality, we will assume our data is generated by the process [MATH].', '1605.07145-1-20-0': '# Signal Recovery Analysis and Optimal Properties of Network Parameters', '1605.07145-1-21-0': 'Even though auto-encoders themselves are no longer heavily used for parameter initialization since many supervised training methods [CITATION] directly lead to the state-of-the-art results, our analysis of the recovery mechanism [MATH] involved in auto-encoders nonetheless leads to useful insights about the optimization involved in training supervised networks.', '1605.07145-1-21-1': 'This is because we focus on analysing the recovery bound for the hidden representation [MATH] given the corresponding data sample [MATH] and the properties of the weight matrix and encoding bias that lead to good recovery.', '1605.07145-1-21-2': 'Thus instead of serving for simply analysing the data reconstruction bounds, our analysis tells more about what happens when data is forward propagated through a non-linear neural network in terms of the representation achieved at the hidden layer being the true data generator.', '1605.07145-1-21-3': 'Hence, we define the notion of Auto-Encoder signal recovery mechanism that we will analyse though out this paper,', '1605.07145-1-22-0': 'Let a data sample [MATH] be generated by the process [MATH] where [MATH] is a fixed matrix, [MATH] is noise and [MATH].', '1605.07145-1-22-1': 'Then we define the Auto-Encoder signal recovery mechanism as [MATH] that recovers the estimate [MATH] where [MATH] is an activation function.', '1605.07145-1-23-0': '## Recovery Analysis', '1605.07145-1-24-0': 'We analyse two separate class of signals in this category- binary sparse, and continuous sparse signals that follow BINS.', '1605.07145-1-24-1': 'For notational convenience, we will drop the subscript of [MATH] and simply refer this parameter as [MATH] as it is the only bias vector (we are not considering the other bias [MATH] due to proposition [REF]).', '1605.07145-1-24-2': 'Due to space limitations, we have moved the analysis for binary sparse signals to appendix.', '1605.07145-1-24-3': 'In general we found similar conclusions hold for both the binary and continuous signal case.', '1605.07145-1-25-0': '(Noiseless Continuous Signal Recovery): Let each element of [MATH] follow BINS([MATH]) distribution and let [MATH] be an auto-encoder signal recovery mechanism with Rectified Linear activation function (ReLU) and bias [MATH] for a measurement vector [MATH] such that [MATH].', '1605.07145-1-25-1': 'If we set [MATH], then [MATH], [EQUATION] where [MATH]s are vectors such that [EQUATION] [MATH] is the [MATH] row of the matrix [MATH] cast as a column vector.', '1605.07145-1-26-0': 'Analysis: We first analyze the properties of the weight matrix that results in strong recovery bound.', '1605.07145-1-26-1': 'We find that for strong recovery, the terms [MATH] and [MATH] should be as large as possible, while simultaneously, the term [MATH] needs to be as close to zero as possible.', '1605.07145-1-26-2': 'First notice the term [MATH].', '1605.07145-1-26-3': 'Since [MATH] by definition, we have that both terms containing [MATH] are always positive and contributes towards stronger recovery if [MATH] is less than [MATH] (sparse), and becomes stronger as the signal becomes sparser (smaller [MATH]).', '1605.07145-1-27-0': 'Now if we assume the rows of the weight matrix [MATH] are highly incoherent and that each row of [MATH] has unit [MATH] length, then it is safe to assume each [MATH]) is close to [MATH] from the definition of [MATH] and properties of [MATH] we have assumed.', '1605.07145-1-27-1': 'Then for any small positive value of [MATH], we can approximately say [MATH] where each [MATH] is very close to zero.', '1605.07145-1-27-2': 'The same argument holds similarly for the term [MATH].', '1605.07145-1-27-3': 'Thus we find that we get a strong signal recovery bound if the weight matrix is highly incoherent and all hidden weight lengths are set to [MATH].', '1605.07145-1-28-0': 'In the case of bias, we have set each element of the bias [MATH].', '1605.07145-1-28-1': 'Notice from the definition of BINS, [MATH].', '1605.07145-1-28-2': 'Thus in essence, [MATH].', '1605.07145-1-28-3': 'Expanding [MATH], we get, [MATH].', '1605.07145-1-29-0': 'The recovery bound is strong for continuous signals when the recovery mechanism is set to [EQUATION] and the rows of [MATH] are highly incoherent and each hidden weight has length ones ([MATH]).', '1605.07145-1-30-0': 'Now we state the recovery bound for the noisy data generation scenario.', '1605.07145-1-31-0': '(Noisy Continuous Signal Recovery): Let each element of [MATH] follow BINS([MATH]) distribution and let [MATH] be an auto-encoder signal recovery mechanism with Rectified Linear activation function (ReLU) and bias [MATH] for a measurement vector [MATH] such that [MATH] where [MATH] is any noise random vector independent of [MATH].', '1605.07145-1-31-1': 'If we set [MATH], then [MATH], [EQUATION] where [MATH]s are vectors such that [EQUATION] [MATH] is the [MATH] row of the matrix [MATH] cast as a column vector.', '1605.07145-1-32-0': 'Notice we have not assumed any distribution on the noise random variable [MATH].', '1605.07145-1-32-1': 'Also, this term has no effect on recovery (compared to the noiseless case) if the noise distribution is orthogonal to the hidden weight vectors.', '1605.07145-1-32-2': 'On the other hand, the same properties of [MATH] lead to better recovery as in the noiseless case.', '1605.07145-1-32-3': 'However, in the case of bias, we have set each element of the bias [MATH].', '1605.07145-1-32-4': 'Notice from the definition of BINS, [MATH].', '1605.07145-1-32-5': 'Thus in essence, [MATH].', '1605.07145-1-32-6': 'Expanding [MATH], we get, [MATH].', '1605.07145-1-32-7': 'Thus the expression of bias is unaffected by error statistics as long as we can compute the data mean.', '1605.07145-1-33-0': '## Properties of Generated Data', '1605.07145-1-34-0': 'Since the data we observe results from the hidden signal given by [MATH], it would be interesting to analyze the distribution of the generated data.', '1605.07145-1-34-1': 'This would help us answer what kind of pre-processing would ensure stronger signal recovery.', '1605.07145-1-35-0': '(Uncorrelated Distribution Bound): If data is generated as [MATH] where [MATH] has covariance matrix [MATH], ([MATH]) and [MATH]) is such that each row of [MATH] has unit length and the rows of [MATH] are maximally incoherent, then the covariance matrix of the generated data is approximately spherical (uncorrelated) satisfying, [EQUATION] where [MATH] is the covariance matrix of the generated data.', '1605.07145-1-36-0': 'Analysis: Notice, for any vector [MATH], [MATH], and the equality holds when each element of the vector [MATH] is identical.', '1605.07145-1-37-0': 'Data [MATH] generated using a maximally incoherent dictionary [MATH] (with unit [MATH] row length) as [MATH] guarantees [MATH] is highly uncorrelated if [MATH] is uncorrelated with near identity covariance.', '1605.07145-1-37-1': 'This would ensure the hidden units at the following layer are also uncorrelated during training.', '1605.07145-1-37-2': 'Further the covariance matrix of [MATH] is identity if all hidden units have equal variance.', '1605.07145-1-38-0': 'This analysis acts as a justification for data whitening where data is processed to have zero mean and identity covariance matrix.', '1605.07145-1-38-1': 'Notice although the generated data does not have zero mean, the recovery process (equation [REF]) subtracts data mean and hence it does not affect recovery.', '1605.07145-1-39-0': '## Connections with existing work', '1605.07145-1-40-0': 'Auto-Encoders (AE): Our analysis reveals the conditions on weights and bias (section [REF]) in an AE that lead to strong signal recovery (for both continuous and binary signals), which ultimately implies low data reconstruction error.', '1605.07145-1-41-0': 'However, the above arguments hold on AEs from a recovery point of view.', '1605.07145-1-41-1': 'Training a basic AE on data may lead to learning of the identity function.', '1605.07145-1-41-2': 'Thus usually AEs are trained along with a bottle-neck to make the learned representation useful.', '1605.07145-1-41-3': 'One such bottle-neck is the De-noising criteria given by, [EQUATION] where [MATH] is the activation function and [MATH] is a corrupted version of [MATH].', '1605.07145-1-41-4': "It has been shown that the Taylor's expansion of DAE (Theorem 3 of [CITATION]) has the term [MATH].", '1605.07145-1-41-5': 'If we constrain the lengths of the weight vectors to have fixed length, then this regularization term minimizes a weighted sum of cosine of the angle between every pair of weight vectors.', '1605.07145-1-41-6': 'As a result, the weight vectors become increasingly incoherent.', '1605.07145-1-41-7': 'Hence we achieve both our goals by adding one additional constraint to DAE- constraining weight vectors to have unit length.', '1605.07145-1-41-8': 'Even if we do not apply an explicit constraint, we can expect the weight lengths to be upper bounded from the basic AE objective itself, which would explain the learning of incoherent weights due to the DAE regularization (we experimentally confirmed this to be true).', '1605.07145-1-41-9': 'On a side note, our analysis also justifies the use of tied weights in auto-encoders.', '1605.07145-1-42-0': 'Sparse Coding (SC): SC involves minimizing [MATH] using the sparsest possible [MATH].', '1605.07145-1-42-1': 'The analysis after theorem [REF] shows signal recovery using the AE mechanism becomes stronger for sparser signals (as also confirmed experimentally in the next section).', '1605.07145-1-42-2': 'In other words, for any given data sample and weight matrix, the AE recovery mechanism recovers the sparsest possible signal; which justifies using auto-encoders for recovering sparse codes (see [CITATION] for work along this line) as long as the conditions on the weight matrix and bias are met.', '1605.07145-1-43-0': 'Batch Normalization (BN) and Normalization Propagation (NormProp): BN [CITATION] identifies the problem of Internal Covariate Shift in deep network.', '1605.07145-1-43-1': 'This refers to shifting distribution of hidden layer inputs during training since network parameters get updated after each iteration.', '1605.07145-1-43-2': "In order to address this problem, they suggest to make every hidden layer's pre-activation normalized to have Normal distribution.", '1605.07145-1-43-3': 'While BN achieves this by computing a running average of input using training mini-batches for each hidden layer and subtracting it, NormProp parametrically computes this mean and subtracts it at every layer.', '1605.07145-1-43-4': 'As shown by our analysis (equation [REF]), setting bias vector to the negative of the expected pre-activation leads to strong signal recovery.', '1605.07145-1-43-5': 'This is an interesting coincidence since we arrive at this parameterization of the bias vector in order to achieve strong signal recovery bound.', '1605.07145-1-43-6': 'On the other hand, if the generating signal has identity covariance, the generated data is also approximately uncorrelated with equal variance (as shown in theorem [REF]).', '1605.07145-1-43-7': 'This is achieved by BN/NormProp by dividing each pre-activation by the standard deviation of its input.', '1605.07145-1-43-8': 'However, our analysis only holds for the first layer of a deep network, but since BN/NormProp are used for higher layers as well, this suggests a similar signal recovery analysis can be done for modeling higher hidden layers as well.', '1605.07145-1-44-0': 'Independent Component Analysis [CITATION] (ICA): ICA assumes we observe data generated by the process [MATH] where all elements of the signal [MATH] are independent and [MATH] is a mixing matrix.', '1605.07145-1-44-1': 'The task of ICA is to recover both [MATH] and [MATH] given data.', '1605.07145-1-44-2': 'This data generating process is precisely what we assume in section [REF].', '1605.07145-1-44-3': 'Based on this assumption, our results show 1) the properties of [MATH] that can recover such independent signals [MATH]; 2) auto-encoders can be used for recovering such signals and weight matrices [MATH].', '1605.07145-1-45-0': 'Orthogonal weight initialization [CITATION]: The authors of [CITATION] show that orthogonal weight initialization is optimal for training deep linear networks.', '1605.07145-1-45-1': 'We on the other hand, conclude from our analysis that signal recovery is strong when the filters in a weight matrix are highly incoherent.', '1605.07145-1-45-2': 'This condition is achieved trivially by orthogonal weight matrices when feature dimensions are larger than the number of filters (undercomplete).', '1605.07145-1-45-3': 'However, even in the overcomplete weight matrix scenario, we empirically show (next section) that if the transpose of the weight is orthogonalized, it results in highly incoherent filters compared with other methods for generating random weight matrices.', '1605.07145-1-45-4': 'Thus our findings support the orthogonal weight initialization of [CITATION].', '1605.07145-1-45-5': 'However, we do not make any claims for the weight matrices of higher layers of a deep non-linear network.', '1605.07145-1-46-0': 'k-Sparse AEs [CITATION]: The authors of [CITATION] propose to zero out all the values of hidden units smaller than the top-k values for each sample during training.', '1605.07145-1-46-1': 'This is done to achieve sparsity in the learned hidden representation.', '1605.07145-1-46-2': 'This strategy is justified from the perspective of our analysis as well.', '1605.07145-1-46-3': 'This is because the PAC bound (theorem [REF]) derived for signal recovery using the AE signal recovery mechanism shows we recover a noisy version of the true sparse signal.', '1605.07145-1-46-4': 'Since, noise in each recovered signal unit is roughly proportional to the original value, de-noising such recovered signals can be achieved by thresholding the hidden unit values (exploiting the fact that the signal is sparse).', '1605.07145-1-46-5': 'This can be done either by using a fixed threshold or picking the top k values.', '1605.07145-1-47-0': 'Data Whitening: Theorem [REF] shows data generated from BINS and incoherent weight matrices are roughly uncorrelated.', '1605.07145-1-47-1': 'Thus recovering back such signals would be easier, and properties of weights and bias predicted by our analysis would be applicable if we pre-process the sampled data to have uncorrelated dimensions (also suggested by Lecun et al. [CITATION]); a condition achieved by whitening.', '1605.07145-1-48-0': '# Empirical Verification', '1605.07145-1-49-0': 'We empirically verify the fundamental predictions made in section [REF] which both serve to justify the assumptions we have made, as well as confirm our results.', '1605.07145-1-49-1': 'We verify the following: a) the optimality of the rows of a weight matrix [MATH] to have unit length and being highly incoherent for the single hidden layer case; b) effect of sparsity on signal recovery for the single hidden layer case.', '1605.07145-1-50-0': '## Optimal Properties of Weight and Bias', '1605.07145-1-51-0': 'Our analysis on signal recovery in section [REF] (equation [REF]) shows signal recovery bound is strong when a) the data generating weight matrix [MATH] has rows of unit [MATH] length; b) the rows of [MATH] are highly incoherent; c) each bias vector element is set to the negative expectation of the pre-activation; d) signal [MATH] has each dimension independent.', '1605.07145-1-51-1': 'In order to verify this, we generate [MATH] signals [MATH] from BINS([MATH],[MATH]uniform, [MATH],[MATH]) with [MATH] set to uniform distribution for simplicity.', '1605.07145-1-51-2': 'We then generate the corresponding [MATH] data sample [MATH] in [MATH] using an incoherent weight matrix [MATH] (each element sampled from zero mean Gaussian, the columns are orthogonalized, and [MATH] length of each row rescaled to [MATH]; notice the rows cannot be orthogonal).', '1605.07145-1-51-3': 'We then recover each signal using, [EQUATION] where [MATH] and [MATH] are scalars that we vary between [MATH] and [MATH] respectively.', '1605.07145-1-51-4': 'For the recovered signals, we calculate the Average Percentage Recovery Error (APRE) as, [EQUATION] where we set [MATH], [MATH] is the indicator operator, [MATH] denotes the [MATH] dimension of the recovered signal corresponding to the [MATH] true signal and, [EQUATION] r0.3', '1605.07145-1-52-0': 'Coherence of orthogonal and Gaussian weight matrix with varying dimensions.', '1605.07145-1-53-0': 'The error is weighted with [MATH] so that the recovery error for both zero and non-zero [MATH]s are penalized equally.', '1605.07145-1-53-1': 'This is specially needed in this case because [MATH] is sparse and a low error can also be achieved by trivially setting all the recovered [MATH]s to zero.', '1605.07145-1-53-2': 'Along with the incoherent weight matrix, we also generate data separately using a highly coherent weight matrix that we get by sampling each element randomly from a uniform distribution on [MATH] and scaling each row to unit length.', '1605.07145-1-53-3': 'According to our analysis, we should get least error for [MATH] and [MATH] for the incoherent matrix while the coherent matrix should yield both higher recovery error and a different choice of [MATH] and [MATH].', '1605.07145-1-53-4': 'The error heat maps are shown in figure [REF].', '1605.07145-1-53-5': 'For the incoherent weight matrix, we see that the empirical optimal is precisely [MATH] and [MATH] (which is exactly as predicted) with Avg.', '1605.07145-1-53-6': '[MATH] Recovery Error =[MATH] even though the weight matrix is only approximately incoherent (not maximally).', '1605.07145-1-53-7': 'For the coherent weight matrix on the other hand, we get the optimal values at [MATH] and [MATH] with Avg.', '1605.07145-1-53-8': '[MATH] Recovery Error =[MATH].', '1605.07145-1-53-9': 'This clearly shows our predictions indeed hold in practice even under approximate conditions.', '1605.07145-1-54-0': '## Effect of Sparsity on Signal Recovery', '1605.07145-1-55-0': 'We analyze the effect of sparsity of signals on their recovery using the mechanism shown in section [REF].', '1605.07145-1-55-1': 'In order to do so, we generate incoherent matrices using three different methods- Gaussian, Xavier [CITATION] and orthogonal [CITATION].', '1605.07145-1-55-2': 'However, all the generated weight matrices are normalized to have unit [MATH] row length.', '1605.07145-1-55-3': 'Additionally, we sample signals and generate data using the same configurations as mentioned in section [REF]; only this time, we fix [MATH] and [MATH], vary hidden unit activation probability [MATH] in [MATH], and duplicate the generated data while adding noise to the copy, which we sample from a Gaussian distribution with mean [MATH] and standard deviation [MATH].', '1605.07145-1-55-4': "According to our analysis, noise mean should have no effect on recovery so the mean value of [MATH] shouldn't have any effect; only standard deviation affects recovery.", '1605.07145-1-55-5': 'The plot of Avg. [MATH] Recovery Error vs. unit activation probability is shown in figure [REF].', '1605.07145-1-55-6': 'We find for all weight matrices, recovery error reduces with increasing sparsity (decreasing [MATH]).', '1605.07145-1-55-7': "Additionally, we find the recovery error trend is significantly lower for orthogonal weight matrices while it's identical for Gaussian and Xavier weights.", '1605.07145-1-55-8': 'Recall theorem [REF] suggests stronger recovery for more incoherent matrices.', '1605.07145-1-55-9': 'So we plot the row coherence of [MATH] sampled from Gaussian and Orthogonal methods with [MATH] and varying [MATH].', '1605.07145-1-55-10': 'The plots are shown in figure [REF].', '1605.07145-1-55-11': 'Clearly orthogonal matrices have significantly lower coherence even though the orthogonalization is done column-wise.', '1605.07145-1-55-12': 'This explains significantly lower recovery error for orthogonal matrices in figure [REF].', '1605.07145-1-56-0': '# Conclusion and Future Work', '1605.07145-1-57-0': 'Our main contribution is to provide a novel perspective of looking at non-linear neural networks as a generative process.', '1605.07145-1-57-1': 'Specifically, if we assume that observed data is generated by hidden layer signals, then the true hidden representation can be accurately recovered if the weight matrices are highly incoherent with unit [MATH] length filters and bias vectors as described in equation [REF] (theorem [REF]).', '1605.07145-1-57-2': 'Additionally, recovery becomes increasingly accurate with increasing sparsity in hidden signals.', '1605.07145-1-57-3': 'Finally, data generated from such signals (assumption [REF]) have the property of being roughly uncorrelated (theorem [REF]).', '1605.07145-1-57-4': 'As a result of these insights, our analysis brings together a number of independently explored approaches in the machine learning community like- Batch Normalization, Sparse Coding, data whitening, e.t.c- showing them as parts of a bigger picture.', '1605.07145-1-58-0': 'However, our analysis only models the first layer of any non-linear neural network where the input data is Gaussian like while the hidden representation is non-negative sparse (due to the activation function).', '1605.07145-1-58-1': 'It would be interesting to analyze the conditions needed for higher layers of such networks in terms of signal recovery where the input and output are both non-negative sparse (for most activation functions, e.g., ReLU, Sigmoid, e.t.c.).', '1605.07145-1-58-2': 'Our study serves as a necessary step towards this goal.', '1605.07145-1-59-0': '# Binary Sparse Signal Analysis', '1605.07145-1-60-0': 'We consider data generated from binary sparse signals [MATH], i.e., [MATH] that follows BINS([MATH]), where [MATH] (Dirac Delta function with value [MATH] at 1 and 0 elsewhere).', '1605.07145-1-60-1': 'Restricted Boltzmann Machines (RBM) are parametric models widely used for modelling the distribution of any given binary valued data.', '1605.07145-1-60-2': 'The model essentially consists of a bipartite set of nodes, one for modeling the visible units (data) while the other for modeling hidden units (unseen part of data).', '1605.07145-1-60-3': 'These units are interconnected using weights and the goal of RBMs is to maximize the marginal probability of the data with respect to model parameters.', '1605.07145-1-60-4': 'The hidden units in this model are usually modeled as binary valued stochastic variables.', '1605.07145-1-60-5': 'Recently, [CITATION] showed the free energy of an RBM can be used to derive an auto-encoder objective.', '1605.07145-1-60-6': 'In other words, the hidden units of an RBM correspond to the hidden units of AEs.', '1605.07145-1-60-7': 'This connection motivates us to investigate whether the AE recovery mechanism is capable of recovering binary sparse signals.', '1605.07145-1-61-0': 'First we consider the noiseless case of data generation,', '1605.07145-1-62-0': '(Noiseless Binary Signal Recovery): Let each element of [MATH] follow BINS([MATH]) and let [MATH] be an auto-encoder signal recovery mechanism with Sigmoid activation function and bias [MATH] for a measurement vector [MATH] such that [MATH].', '1605.07145-1-62-1': 'If we set [MATH], then [MATH], [EQUATION] where [MATH], [MATH] and [MATH] is the [MATH] row of the matrix [MATH] cast as a column vector.', '1605.07145-1-63-0': 'Analysis: We first analyse the properties of the weight matrix [MATH] that results in strong recovery bound.', '1605.07145-1-63-1': 'Notice the terms [MATH] and [MATH] need to be as large as possible, while simultaneously, the term [MATH] needs to be as close to zero as possible.', '1605.07145-1-63-2': 'For the sake of analysis, lets set [MATH] (achieved when [MATH]).', '1605.07145-1-63-3': 'Then our problem gets reduced to maximizing the ratio [MATH], where [MATH] is the angle between [MATH] and [MATH].', '1605.07145-1-63-4': 'From the property of coherence, if the rows of the weight matrix is highly incoherent, then [MATH] is close to [MATH].', '1605.07145-1-63-5': 'Again, for the ease of analysis, lets replace each [MATH] with a small positive number [MATH].', '1605.07145-1-63-6': 'Then [MATH].', '1605.07145-1-63-7': 'Finally, since we would want this term to be maximized for each hidden unit [MATH] equally, the obvious choice for each weight length [MATH]) is to set it to [MATH].', '1605.07145-1-64-0': 'Finally, lets analyse the bias vector.', '1605.07145-1-64-1': 'Notice we have instantiated each element of the encoding bias [MATH] to take value [MATH].', '1605.07145-1-64-2': 'Since [MATH] is essentially the mean of each binary hidden unit [MATH], we can say that [MATH].', '1605.07145-1-65-0': 'Signal recovery is strong for binary signals when the recovery mechanism is given by [EQUATION] where the rows of [MATH] are highly incoherent and each hidden weight has length ones ([MATH]), and each dimension of data [MATH] is approximately uncorrelated (see theorem [REF]).', '1605.07145-1-66-0': 'Now we state the recovery bound for the noisy data generation scenario.', '1605.07145-1-67-0': '(Noisy Binary Signal Recovery): Let each element of [MATH] follow BINS([MATH]) and let [MATH] be an auto-encoder signal recovery mechanism with Sigmoid activation function and bias [MATH] for a measurement vector [MATH] where [MATH] is any noise vector independent of [MATH].', '1605.07145-1-67-1': 'If we set [MATH], then [MATH], [EQUATION] where [MATH], [MATH] and [MATH] is the [MATH] row of the matrix [MATH] cast as a column vector.', '1605.07145-1-68-0': 'Similar to the continuous signal recovery case, we have not assumed any distribution on the noise random variable [MATH] and this term has no effect on recovery (compared to the noiseless case) if the noise distribution is orthogonal to the hidden weight vectors.', '1605.07145-1-68-1': 'Again, the same properties of [MATH] lead to better recovery as in the noiseless case.', '1605.07145-1-68-2': 'In the case of bias, we have set each element of the bias [MATH].', '1605.07145-1-68-3': 'Notice from the definition of BINS, [MATH].', '1605.07145-1-68-4': 'Thus in essence, [MATH].', '1605.07145-1-68-5': 'Expanding [MATH], we get, [MATH].', '1605.07145-1-68-6': 'Thus the expression of bias is unaffected by error statistics as long as we can compute the data mean.', '1605.07145-1-69-0': '# Proofs', '1605.07145-1-70-0': 'Let [MATH] where [MATH], [MATH] and [MATH].', '1605.07145-1-70-1': 'Let [MATH] where [MATH] is a fixed vector.', '1605.07145-1-70-2': 'Let [MATH] and [MATH].', '1605.07145-1-70-3': 'Then [MATH] iff [MATH].', '1605.07145-1-71-0': 'Proof: Let [MATH].', '1605.07145-1-71-1': 'Thus [MATH].', '1605.07145-1-71-2': 'On the other hand, [MATH].', '1605.07145-1-71-3': 'The other direction can be proved similarly.', '1605.07145-1-72-0': 'Let each element of [MATH] follow BINS([MATH]) distribution and let [MATH] be an auto-encoder signal recovery mechanism with Rectified Linear activation function (ReLU) and bias [MATH] for a measurement vector [MATH] such that [MATH].', '1605.07145-1-72-1': 'If we set [MATH], then [MATH], [EQUATION] where [MATH]s are vectors such that [EQUATION] [MATH] is the [MATH] row of the matrix [MATH] cast as a column vector.', '1605.07145-1-73-0': 'From definition [REF] and the definition of [MATH] above, [EQUATION]', '1605.07145-1-73-1': 'Let [MATH].', '1605.07145-1-73-2': 'Thus, [MATH].', '1605.07145-1-73-3': 'Then, conditioning upon [MATH], [EQUATION]', '1605.07145-1-73-4': 'Since [MATH], we have, [EQUATION]', '1605.07145-1-73-5': 'The above inequality is obtained by ignoring the positive terms that depend on the condition [MATH] and marginalizing over [MATH].', '1605.07145-1-73-6': "For any [MATH], using Chernoff's inequality, [EQUATION]", '1605.07145-1-73-7': 'Setting [MATH], where [MATH], [EQUATION]', '1605.07145-1-73-8': 'Let [MATH].', '1605.07145-1-73-9': 'Then, [EQUATION] where [MATH] denotes any arbitrary distribution in the interval [MATH] with mean [MATH].', '1605.07145-1-73-10': 'If [MATH], let [MATH] and [MATH] which essentially denote the lower and upper bound of [MATH].', '1605.07145-1-73-11': 'Then, [EQUATION] where the first inequality in the above equation is from the property of a convex function.', '1605.07145-1-73-12': 'Define [MATH], [MATH].', '1605.07145-1-73-13': 'Then, [EQUATION]', '1605.07145-1-73-14': 'Define [MATH] and let [MATH].', '1605.07145-1-73-15': 'Then, [EQUATION]', '1605.07145-1-73-16': 'Thus, for getting a maxima for [MATH], we set [MATH] which implies [MATH], or, [MATH].', '1605.07145-1-73-17': 'Substituting this [MATH] in [MATH].', '1605.07145-1-73-18': "By Taylor's theorem, [MATH] such that, [EQUATION]", '1605.07145-1-73-19': 'Thus we can upper bound [MATH] as, [EQUATION]', '1605.07145-1-73-20': 'Substituting for [MATH], we get, [EQUATION]', '1605.07145-1-73-21': 'On the other hand, if [MATH], then we can set [MATH] and [MATH] and proceeding similar to equation [REF], we get, [EQUATION]', '1605.07145-1-73-22': 'Then, collectively, we can write [MATH] as [EQUATION]', '1605.07145-1-73-23': "We similarly bound [MATH] by effectively flipping the sign of [MATH]'s, [EQUATION]", '1605.07145-1-73-24': 'Minimizing both [REF] and [REF] with respect to [MATH] and applying union bound, we get, [EQUATION]', '1605.07145-1-73-25': 'Since the above bound holds for all [MATH], applying union bound on all the units yields the desired result.', '1605.07145-1-74-0': 'Let each element of [MATH] follow BINS([MATH]) distribution and let [MATH] be an auto-encoder signal recovery mechanism with Rectified Linear activation function and bias [MATH] for a measurement vector [MATH] such that [MATH] where [MATH] is any noise random vector independent of [MATH].', '1605.07145-1-74-1': 'If we set [MATH], then [MATH], [EQUATION] where [MATH]s are vectors such that [EQUATION] [MATH] is the [MATH] row of the matrix [MATH] cast as a column vector.', '1605.07145-1-75-0': 'Recall that, [EQUATION]', '1605.07145-1-75-1': 'Let [MATH].', '1605.07145-1-75-2': 'Then, similar to theorem [REF], conditioning upon [MATH], [EQUATION]', '1605.07145-1-75-3': 'Since [MATH], we have, [EQUATION]', '1605.07145-1-75-4': "For any [MATH], using Chernoff's inequality for the random variable [MATH], [EQUATION]", '1605.07145-1-75-5': 'Setting [MATH], where [MATH], [EQUATION]', '1605.07145-1-75-6': 'Setting [MATH], we can rewrite the above inequality as [EQUATION]', '1605.07145-1-75-7': 'Since the above inequality becomes identical to equation [REF], the rest of the proof is similar to theorem [REF].', '1605.07145-1-76-0': '(Uncorrelated Distribution Bound): If data is generated as [MATH] where [MATH] has covariance matrix [MATH], ([MATH]) and [MATH]) is such that each row of [MATH] has unit length and the rows of [MATH] are maximally incoherent, then the covariance matrix of the generated data is approximately spherical (uncorrelated) satisfying, [EQUATION] where [MATH] is the covariance matrix of the generated data.', '1605.07145-1-77-0': 'Notice that, [EQUATION]', '1605.07145-1-77-1': 'Thus, [EQUATION]', '1605.07145-1-77-2': 'Substituting the covariance of [MATH] as [MATH], [EQUATION]', '1605.07145-1-77-3': 'Thus, [EQUATION]', '1605.07145-1-77-4': 'Using the cyclic property of trace, [EQUATION]', '1605.07145-1-77-5': 'Finally minimizing w.r.t [MATH], we get [MATH].', '1605.07145-1-77-6': 'Substituting this into the above inequality, we get,', '1605.07145-1-78-0': '[EQUATION]', '1605.07145-1-79-0': 'Since the weight matrix is maximally incoherent, using Welch bound, we have that, [MATH].', '1605.07145-1-79-1': 'Plugging the lower bound of [MATH] (maximal incoherence) for any fixed [MATH] and [MATH] into the above bound yields, [EQUATION]', '1605.07145-1-80-0': 'Let each element of [MATH] follow BINS([MATH]) and let [MATH] be an auto-encoder signal recovery mechanism with Sigmoid activation function and bias [MATH] for a measurement vector [MATH] such that [MATH].', '1605.07145-1-80-1': 'If we set [MATH], then [MATH], [EQUATION] where [MATH], [MATH] and [MATH] is the [MATH] row of the matrix [MATH] cast as a column vector.', '1605.07145-1-81-0': 'Notice that, [EQUATION] and from definition [REF], [EQUATION]', '1605.07145-1-81-1': 'Thus, [EQUATION]', '1605.07145-1-81-2': 'Notice that [MATH].', '1605.07145-1-81-3': 'Let [MATH] and [MATH].', '1605.07145-1-81-4': "Then, setting [MATH], using Chernoff's inequality, for any [MATH], [EQUATION]", '1605.07145-1-81-5': 'Let [MATH].', '1605.07145-1-81-6': 'Then, [EQUATION]', '1605.07145-1-81-7': 'Let [MATH], thus, [EQUATION]', '1605.07145-1-81-8': 'Setting [MATH], we get [MATH].', '1605.07145-1-81-9': 'Thus, [MATH].', '1605.07145-1-81-10': "By Taylor's theorem, [MATH] s.t., [EQUATION]", '1605.07145-1-81-11': 'Thus we can upper bound [MATH] as, [EQUATION]', '1605.07145-1-81-12': 'Hence we can write [MATH] as [EQUATION]', '1605.07145-1-81-13': 'On the other hand, notice [MATH].', '1605.07145-1-82-0': '[EQUATION]', '1605.07145-1-83-0': 'Let [MATH].', '1605.07145-1-83-1': "Then we can similarly bound [MATH] by effectively flipping the sign of [MATH]'s in the previous derivation, [EQUATION]", '1605.07145-1-83-2': 'Minimizing both [REF] and [REF] with respect to [MATH] and applying union bound, we get, [EQUATION]', '1605.07145-1-83-3': 'Since the above bound holds for all [MATH], applying union bound on all the units yields the desired result.', '1605.07145-1-84-0': 'Let each element of [MATH] follow BINS([MATH]) and let [MATH] be an auto-encoder signal recovery mechanism with Sigmoid activation function and bias [MATH] for a measurement vector [MATH] where [MATH] is any noise vector independent of [MATH].', '1605.07145-1-84-1': 'If we set [MATH], then [MATH], [EQUATION] where [MATH], [MATH] and [MATH] is the [MATH] row of the matrix [MATH] cast as a column vector.', '1605.07145-1-85-0': 'Notice that, [EQUATION] and from definition [REF], [EQUATION]', '1605.07145-1-85-1': 'Thus, [EQUATION]', '1605.07145-1-85-2': 'Notice that [MATH].', '1605.07145-1-85-3': 'Let [MATH] and [MATH].', '1605.07145-1-85-4': "Then, setting [MATH], using Chernoff's inequality on random variable [MATH], for any [MATH], [EQUATION]", '1605.07145-1-85-5': 'Setting [MATH], we can rewrite the above inequality as [EQUATION]', '1605.07145-1-85-6': 'Since the above inequality becomes identical to equation [REF], the rest of the proof is similar to theorem [REF].'}

{'1605.07145-2-0-0': 'Auto-Encoders are unsupervised models that aim to learn patterns from observed data by minimizing a reconstruction cost.', '1605.07145-2-0-1': 'The useful representations learned are often found to be sparse and distributed.', '1605.07145-2-0-2': 'On the other hand, compressed sensing and sparse coding assume a data generating process, where the observed data is generated from some true latent signal source, and try to recover the corresponding signal from measurements.', '1605.07145-2-0-3': 'Looking at auto-encoders from this signal recovery perspective enables us to have a more coherent view of these techniques.', '1605.07145-2-0-4': 'In this paper, in particular, we show that the true hidden representation can be approximately recovered if the weight matrices are highly incoherent with unit [MATH] row length and the bias vectors takes the value (approximately) equal to the negative of the data mean.', '1605.07145-2-0-5': 'The recovery also becomes more and more accurate as the sparsity in hidden signals increases.', '1605.07145-2-0-6': 'Additionally, we empirically also demonstrate that auto-encoders are capable of recovering the data generating dictionary when only data samples are given.', '1605.07145-2-1-0': '# Introduction', '1605.07145-2-2-0': '-1 Recovering hidden signal from measurement vectors (observations) is a long studied problem in compressed sensing and sparse coding with a lot of successful applications.', '1605.07145-2-2-1': 'On the other hand, auto-encoders (AEs) are useful for unsupervised representation learning for uncovering patterns in data.', '1605.07145-2-2-2': 'AEs focus on learning a mapping [MATH], where the reconstructed vector [MATH] is desired to be as close to [MATH] as possible for the entire data distribution.', '1605.07145-2-2-3': 'What we show in this paper is that if we consider [MATH] is actually generated from some true sparse signal [MATH] by some process (see section [REF]), then switching our perspective on AE to analyze [MATH] shows that AE is capable of recovering the true signal that generated the data and yields useful insights into the optimality of model parameters of auto-encoders in terms of signal recovery.', '1605.07145-2-2-4': 'In other words, this perspective lets us look at AEs from a signal recovery point of view where forward propagating [MATH] recovers the true signal [MATH].', '1605.07145-2-2-5': 'We analyze the conditions under which the encoder part of an AE recovers the true [MATH] from [MATH], while the decoder part acts as the data generating process.', '1605.07145-2-2-6': 'Our main result shows that the true sparse signal [MATH] (with mild distribution assumptions) can be approximately recovered by the encoder of an AE with high probability under certain conditions on the weight matrix and bias vectors.', '1605.07145-2-2-7': 'Additionally, we empirically show that in a practical setting when only data is observed, optimizing the AE objective leads to the recovery of both the data generating dictionary [MATH] and the true sparse signal [MATH], which together is not well studied in the auto-encoder framework, to the best of our knowledge.', '1605.07145-2-3-0': '# Sparse Signal Recovery Perspective', '1605.07145-2-4-0': 'While it is known both empirically and theoretically, that useful features learned by AEs are usually sparse .', '1605.07145-2-4-1': "An important question that hasn't been answered yet is whether AEs are capable of recovering sparse signals, in general.", '1605.07145-2-4-2': 'This is an important question for Sparse Coding, which entails recovering the sparsest [MATH] that approximately satisfies [MATH], for any given data vector [MATH] and overcomplete weight matrix [MATH].', '1605.07145-2-4-3': 'However, since this problem is NP complete , it is usually relaxed to solving an expensive optimization problem , [EQUATION] where [MATH] is a fixed overcomplete ([MATH]) dictionary, [MATH] is the regularization coefficient, [MATH] is the data and [MATH] is the signal to recover.', '1605.07145-2-4-4': 'For this special case, [CITATION] analyzed the condition under which linear AEs can recover the support of the hidden signal.', '1605.07145-2-5-0': 'The general AE objective, on the other hand, minimizes the expected reconstruction cost [EQUATION] for some reconstruction cost [MATH], encoding and decoding activation function [MATH] and [MATH], and bias vectors [MATH] and [MATH].', '1605.07145-2-5-1': 'In this paper we consider linear activation [MATH] because it is a more general case.', '1605.07145-2-5-2': 'Notice however, in the case of auto-encoders, the activation functions can be non-linear in general, in contrast to the sparse coding objective.', '1605.07145-2-5-3': 'In addition, in case of AEs we do not have a separate parameter [MATH] for the hidden representation corresponding to every data sample [MATH] individually.', '1605.07145-2-5-4': 'Instead, the hidden representation for every sample is a parametric function of the sample itself.', '1605.07145-2-5-5': 'This is an important distinction between the optimization in eq. [REF] and our problem - the identity of [MATH] in eq. [REF] is only well defined in the presence of [MATH] regularization due to the overcompleteness of the dictionary.', '1605.07145-2-5-6': 'However, in our problem, we assume a true signal [MATH] generates the observed data [MATH] as [MATH], where the dictionary [MATH] and bias vector [MATH] are fixed.', '1605.07145-2-5-7': 'Hence, what we mean by recovery of sparse signals in an AE framework is that if we generate data using the above generation process, then can the estimate [MATH] indeed recover the true [MATH] for some activation functions [MATH], and bias vector [MATH]?', '1605.07145-2-5-8': 'And if so, what properties of [MATH] and [MATH] lead to good recovery?', '1605.07145-2-5-9': 'However, when given an [MATH] and the true overcomplete [MATH], the solution [MATH] to [MATH] is not unique.', '1605.07145-2-5-10': 'Then the question arises about the possibility of recovering such an [MATH].', '1605.07145-2-5-11': 'However, as we show, recovery using the AE mechanism is strongest when the signal [MATH] is the sparsest possible one, which from compressed sensing theory, guarantees uniqueness of [MATH] if [MATH] is sufficiently incoherent .', '1605.07145-2-6-0': '# Data Generation Process', '1605.07145-2-7-0': 'We consider the following data generation process: [EQUATION] where [MATH] is the observed data, [MATH] is a bias vector, [MATH] is a noise vector, [MATH] is the weight matrix and [MATH] is the true hidden representation (signal) that we want to recover.', '1605.07145-2-7-1': 'Throughout our analysis, we assume that the signal [MATH] belongs to the following class of distribution,', '1605.07145-2-8-0': 'Bounded Independent Non-negative Sparse (BINS): Every hidden unit [MATH] is an independent random variable with the following density function: [EQUATION] where [MATH] can be any arbitrary normalized distribution bounded in the interval [MATH], mean [MATH], and [MATH] is the Dirac Delta function at zero.', '1605.07145-2-8-1': 'As a short hand, we say that [MATH] follows the distribution BINS([MATH]).', '1605.07145-2-8-2': 'Notice that [MATH].', '1605.07145-2-9-0': 'The above distribution assumption fits naturally with sparse coding, when the intended signal is non-negative sparse.', '1605.07145-2-9-1': 'From the AE perspective, it is also justified based on the following observation.', '1605.07145-2-9-2': 'In neural networks with ReLU activations, hidden unit pre-activations have a Gaussian like symmetric distribution .', '1605.07145-2-9-3': "If we assume these distributions are mean centered, then the hidden units' distribution after ReLU has a large mass at [MATH] while the rest of the mass concentrates in [MATH] for some finite positive [MATH], because the pre-activations concentrate symmetrically around zero.", '1605.07145-2-9-4': 'As we show in the next section, ReLU is indeed capable of recovering such signals.', '1605.07145-2-9-5': 'On a side note, the distribution from assumption [REF] can take shapes similar to that of Exponential or Rectified Gaussian distribution (which are generally used for modeling biological neurons) but is simpler to analyze.', '1605.07145-2-9-6': 'This is because we allow [MATH] to be any arbitrary normalized distribution.', '1605.07145-2-9-7': 'The only restriction assumption [REF] has is that to be bounded.', '1605.07145-2-9-8': 'However, this does not change the representative power of this distribution significantly because: a) the distributions used for modeling neurons have very small tail mass; b) in practice, we are generally interested in signals with upper bounded values.', '1605.07145-2-10-0': 'The generation process considered in this section (i.e.eq. [REF] and assumptions [REF]) is justified because:', '1605.07145-2-11-0': '1.', '1605.07145-2-11-1': 'This data generation model finds applications in a number of areas .', '1605.07145-2-11-2': 'Notice that while [MATH] is the measurement vector (observed data), which can in general be noisy, [MATH] denotes the actual signal (internal representation) because it reflects the combination of dictionary ([MATH]) atoms involved in generating the observed samples and hence serves as the true identity of the data.', '1605.07145-2-12-0': '2.', '1605.07145-2-12-1': 'Sparse distributed representation is both observed and desired in hidden representations.', '1605.07145-2-12-2': 'It has been empirically shown that representations that are truly sparse (i.e.large number of true zeros) and distributed usually yield better linear separability and performance .', '1605.07145-2-13-0': 'Decoding bias ([MATH]): Consider the data generation process (exclude noise for now) [MATH].', '1605.07145-2-13-1': 'Here [MATH] is a bias vector which can take any arbitrary value but similar to [MATH], it is fixed for any particular data generation process.', '1605.07145-2-13-2': 'However, the following remark shows that if an AE can recover the sparse code ([MATH]) from a data sample generated as [MATH], then it is also capable of recovering the sparse code from the data generated as [MATH] and vice versa.', '1605.07145-2-14-0': 'Let [MATH] where [MATH], [MATH] and [MATH].', '1605.07145-2-14-1': 'Let [MATH] where [MATH] is a fixed vector.', '1605.07145-2-14-2': 'Let [MATH] and [MATH].', '1605.07145-2-14-3': 'Then [MATH] iff [MATH].', '1605.07145-2-15-0': 'Thus without any loss of generality, we will assume our data is generated from [MATH].', '1605.07145-2-16-0': '# Signal Recovery Analysis', '1605.07145-2-17-0': 'We analyse two separate class of signals in this category- continuous sparse, and binary sparse signals that follow BINS.', '1605.07145-2-17-1': 'For notational convenience, we will drop the subscript of [MATH] and simply refer this parameter as [MATH] since it is the only bias vector (we are not considering the other bias [MATH] due to remark [REF]).', '1605.07145-2-17-2': 'The Auto-Encoder signal recovery mechanism that we analyze throughout this paper is defined as,', '1605.07145-2-18-0': 'Let a data sample [MATH] be generated by the process [MATH] where [MATH] is a fixed matrix, [MATH] is noise and [MATH].', '1605.07145-2-18-1': 'Then we define the Auto-Encoder signal recovery mechanism as [MATH] that recovers the estimate [MATH] where [MATH] is an activation function.', '1605.07145-2-19-0': '## Binary Sparse Signal Analysis', '1605.07145-2-20-0': 'First we consider the noiseless case of data generation,', '1605.07145-2-21-0': '(Noiseless Binary Signal Recovery): Let each element of [MATH] follow BINS([MATH]) and let [MATH] be an auto-encoder signal recovery mechanism with Sigmoid activation function and bias [MATH] for a measurement vector [MATH] such that [MATH].', '1605.07145-2-21-1': 'If we set [MATH], then [MATH], [EQUATION] where [MATH], [MATH] and [MATH] is the [MATH] row of the matrix [MATH] cast as a column vector.', '1605.07145-2-22-0': 'Analysis: We first analyse the properties of the weight matrix [MATH] that results in strong recovery bound.', '1605.07145-2-22-1': 'Notice the terms [MATH] and [MATH] need to be as large as possible, while simultaneously, the term [MATH] needs to be as close to zero as possible.', '1605.07145-2-22-2': 'For the sake of analysis, lets set [MATH] (achieved when [MATH]).', '1605.07145-2-22-3': 'Then our problem gets reduced to maximizing the ratio [MATH], where [MATH] is the angle between [MATH] and [MATH].', '1605.07145-2-22-4': 'From the property of coherence, if the rows of the weight matrix is highly incoherent, then [MATH] is close to [MATH].', '1605.07145-2-22-5': 'Again, for the ease of analysis, lets replace each [MATH] with a small positive number [MATH].', '1605.07145-2-22-6': 'Then [MATH].', '1605.07145-2-22-7': 'Finally, since we would want this term to be maximized for each hidden unit [MATH] equally, the obvious choice for each weight length [MATH]) is to set it to [MATH].', '1605.07145-2-23-0': 'Finally, lets analyse the bias vector.', '1605.07145-2-23-1': 'Notice we have instantiated each element of the encoding bias [MATH] to take value [MATH].', '1605.07145-2-23-2': 'Since [MATH] is essentially the mean of each binary hidden unit [MATH], we can say that [MATH].', '1605.07145-2-24-0': 'Signal recovery is strong for binary signals when the recovery mechanism is given by [EQUATION] where the rows of [MATH] are highly incoherent and each hidden weight has length ones ([MATH]), and each dimension of data [MATH] is approximately uncorrelated (see theorem [REF]).', '1605.07145-2-25-0': 'Now we state the recovery bound for the noisy data generation scenario.', '1605.07145-2-26-0': '(Noisy Binary Signal Recovery): Let each element of [MATH] follow BINS([MATH]) and let [MATH] be an auto-encoder signal recovery mechanism with Sigmoid activation function and bias [MATH] for a measurement vector [MATH] where [MATH] is any noise vector independent of [MATH].', '1605.07145-2-26-1': 'If we set [MATH], then [MATH], [EQUATION] where [MATH], [MATH] and [MATH] is the [MATH] row of the matrix [MATH] cast as a column vector.', '1605.07145-2-27-0': 'We have not assumed any distribution on the noise random variable [MATH] and this term has no effect on recovery (compared to the noiseless case) if the noise distribution is orthogonal to the hidden weight vectors.', '1605.07145-2-27-1': 'Again, the same properties of [MATH] lead to better recovery as in the noiseless case.', '1605.07145-2-27-2': 'In the case of bias, we have set each element of the bias [MATH].', '1605.07145-2-27-3': 'Notice from the definition of BINS, [MATH].', '1605.07145-2-27-4': 'Thus in essence, [MATH].', '1605.07145-2-27-5': 'Expanding [MATH], we get, [MATH].', '1605.07145-2-27-6': 'Thus the expression of bias is unaffected by error statistics as long as we can compute the data mean.', '1605.07145-2-28-0': 'In this section, we will first consider the case when data ([MATH]) is generated by linear process [MATH], and if [MATH] and encoding bias [MATH] have certain properties, then the signal recovery bound ([MATH]) is strong.', '1605.07145-2-28-1': 'We will then consider the case when data generated by a non-linear process [MATH] (for certain class of functions [MATH]) can be recovered as well by the same mechanism.', '1605.07145-2-28-2': 'For deep non-linear networks, this means that forward propagating data to hidden layers, such that the network parameters satisfy the required conditions, implies each hidden layer recovers the true signal that generated the corresponding data.', '1605.07145-2-28-3': 'We have moved all the proofs to appendix for better readability.', '1605.07145-2-29-0': '## Continuous Sparse Signal Recovery', '1605.07145-2-30-0': '(Noiseless Continuous Signal Recovery): Let each element of [MATH] follow BINS([MATH]) distribution and let [MATH] be an auto-encoder signal recovery mechanism with Rectified Linear activation function (ReLU) and bias [MATH] for a measurement vector [MATH] such that [MATH].', '1605.07145-2-30-1': 'If we set [MATH], then [MATH], [EQUATION] where [MATH]s are vectors such that [EQUATION] [MATH] is the [MATH] row of the matrix [MATH] cast as a column vector.', '1605.07145-2-31-0': 'Analysis: We first analyze the properties of the weight matrix that results in strong recovery bound.', '1605.07145-2-31-1': 'We find that for strong recovery, the terms [MATH] and [MATH] should be as large as possible, while simultaneously, the term [MATH] needs to be as close to zero as possible.', '1605.07145-2-31-2': 'First, notice the term [MATH].', '1605.07145-2-31-3': 'Since [MATH] by definition, we have that both terms containing [MATH] are always positive and contributes towards stronger recovery if [MATH] is less than [MATH] (sparse), and becomes stronger as the signal becomes sparser (smaller [MATH]).', '1605.07145-2-32-0': 'Now if we assume the rows of the weight matrix [MATH] are highly incoherent and that each row of [MATH] has unit [MATH] length, then it is safe to assume each [MATH]) is close to [MATH] from the definition of [MATH] and properties of [MATH] we have assumed.', '1605.07145-2-32-1': 'Then for any small positive value of [MATH], we can approximately say [MATH] where each [MATH] is very close to zero.', '1605.07145-2-32-2': 'The same argument holds similarly for the other term.', '1605.07145-2-32-3': 'Thus we find that a strong signal recovery bound would be obtained if the weight matrix is highly incoherent and all hidden vectors are of unit length.', '1605.07145-2-33-0': 'In the case of bias, we have set each element of the bias [MATH].', '1605.07145-2-33-1': 'Notice from the definition of BINS, [MATH].', '1605.07145-2-33-2': 'Thus in essence, [MATH].', '1605.07145-2-33-3': 'Expanding [MATH], we get [MATH].', '1605.07145-2-34-0': 'The recovery bound is strong for continuous signals when the recovery mechanism is set to [EQUATION] and the rows of [MATH] are highly incoherent and each hidden weight has length ones ([MATH]).', '1605.07145-2-35-0': 'Now we state the recovery bound for the noisy data generation scenario.', '1605.07145-2-36-0': '(Noisy Continuous Signal Recovery): Let each element of [MATH] follow BINS([MATH]) distribution and let [MATH] be an auto-encoder signal recovery mechanism with Rectified Linear activation function (ReLU) and bias [MATH] for a measurement vector [MATH] such that [MATH] where [MATH] is any noise random vector independent of [MATH].', '1605.07145-2-36-1': 'If we set [MATH], then [MATH], [EQUATION] where [MATH]s are vectors such that [EQUATION] [MATH] is the [MATH] row of the matrix [MATH] cast as a column vector.', '1605.07145-2-37-0': 'Notice that we have not assumed any distribution on variable [MATH], which denotes the noise.', '1605.07145-2-37-1': 'Also, this term has no effect on recovery (compared to the noiseless case) if the noise distribution is orthogonal to the hidden weight vectors.', '1605.07145-2-37-2': 'On the other hand, the same properties of [MATH] lead to better recovery as in the noiseless case.', '1605.07145-2-37-3': 'However, in the case of bias, we have set each element of the bias [MATH].', '1605.07145-2-37-4': 'From the definition of BINS, [MATH].', '1605.07145-2-37-5': 'Thus [MATH].', '1605.07145-2-37-6': 'Expanding [MATH], we get, [MATH].', '1605.07145-2-37-7': 'Thus the expression of bias is unaffected by error statistics as long as we can compute the data mean (i.e.the recovery is the same as shown in eq. [REF]).', '1605.07145-2-38-0': '## Properties of Generated Data', '1605.07145-2-39-0': 'Since the data we observe results from the hidden signal given by [MATH], it would be interesting to analyze the distribution of the generated data.', '1605.07145-2-39-1': 'This would provide us more insight into what kind of pre-processing would ensure stronger signal recovery.', '1605.07145-2-40-0': '(Uncorrelated Distribution Bound): If data is generated as [MATH] where [MATH] has covariance matrix [MATH], ([MATH]) and [MATH]) is such that each row of [MATH] has unit length and the rows of [MATH] are maximally incoherent, then the covariance matrix of the generated data is approximately spherical (uncorrelated) satisfying, [EQUATION] where [MATH] is the covariance matrix of the generated data.', '1605.07145-2-41-0': 'Analysis: Notice that for any vector [MATH], [MATH], and the equality holds when each element of the vector [MATH] is identical.', '1605.07145-2-42-0': 'Data [MATH] generated using a maximally incoherent dictionary [MATH] (with unit [MATH] row length) as [MATH] guarantees [MATH] is highly uncorrelated if [MATH] is uncorrelated with near identity covariance.', '1605.07145-2-42-1': 'This would ensure the hidden units at the following layer are also uncorrelated during training.', '1605.07145-2-42-2': 'Further the covariance matrix of [MATH] is identity, if all hidden units have equal variance.', '1605.07145-2-43-0': 'This analysis acts as a justification for data whitening where data is processed to have zero mean and identity covariance matrix.', '1605.07145-2-43-1': 'Notice that although the generated data does not have zero mean, the recovery process (eq. [REF]) subtracts data mean and hence it does not affect recovery.', '1605.07145-2-44-0': '## Connections with existing work', '1605.07145-2-45-0': 'Auto-Encoders (AE): Our analysis reveals the conditions on parameters of an AE that lead to strong recovery of [MATH] (for both continuous and binary case), which ultimately implies low data reconstruction error.', '1605.07145-2-46-0': 'However, the above arguments hold for AEs from a recovery point of view.', '1605.07145-2-46-1': 'Training an AE on data may lead to learning of the identity function.', '1605.07145-2-46-2': 'Thus usually AEs are trained along with a bottle-neck to make the learned representation useful.', '1605.07145-2-46-3': 'One such bottle-neck is the De-noising criteria given by, [EQUATION] where [MATH] is the activation function and [MATH] is a corrupted version of [MATH].', '1605.07145-2-46-4': "It has been shown that the Taylor's expansion of DAE has the term [MATH].", '1605.07145-2-46-5': 'If we constrain the lengths of the weight vectors to have fixed length, then this regularization term minimizes a weighted sum of cosine of the angle between every pair of weight vectors.', '1605.07145-2-46-6': 'As a result, the weight vectors become increasingly incoherent.', '1605.07145-2-46-7': 'Hence we achieve both our goals by adding one additional constraint to DAE- constraining weight vectors to have unit length.', '1605.07145-2-46-8': 'Even if we do not apply an explicit constraint, we can expect the weight lengths to be upper bounded from the basic AE objective itself, which would explain the learning of incoherent weights due to the DAE regularization.', '1605.07145-2-46-9': 'On a side note, our analysis also justifies the use of tied weights in auto-encoders.', '1605.07145-2-47-0': 'Sparse Coding (SC): SC involves minimizing [MATH] using the sparsest possible [MATH].', '1605.07145-2-47-1': 'The analysis after theorem [REF] shows signal recovery using the AE mechanism becomes stronger for sparser signals (as also confirmed experimentally in section [REF]).', '1605.07145-2-47-2': 'In other words, for any given data sample and weight matrix, as long as the conditions on the weight matrix and bias are met, the AE recovery mechanism recovers the sparsest possible signal; which justifies using auto-encoders for recovering sparse codes .', '1605.07145-2-48-0': 'Independent Component Analysis (ICA): ICA assumes we observe data generated by the process [MATH] where all elements of the [MATH] are independent and [MATH] is a mixing matrix.', '1605.07145-2-48-1': 'The task of ICA is to recover both [MATH] and [MATH] given data.', '1605.07145-2-48-2': 'This data generating process is precisely what we assumed in section [REF].', '1605.07145-2-48-3': 'Based on this assumption, our results show that 1) the properties of [MATH] that can recover such independent signals [MATH]; and 2) auto-encoders can be used for recovering such signals and weight matrix [MATH].', '1605.07145-2-49-0': 'k-Sparse AEs : [CITATION] propose to zero out all the values of hidden units smaller than the top-k values for each sample during training.', '1605.07145-2-49-1': 'This is done to achieve sparsity in the learned hidden representation.', '1605.07145-2-49-2': 'This strategy is justified from the perspective of our analysis as well.', '1605.07145-2-49-3': 'This is because the PAC bound (theorem [REF]) derived for signal recovery using the AE signal recovery mechanism shows we recover a noisy version of the true sparse signal.', '1605.07145-2-49-4': 'Since the noise in each recovered signal unit is roughly proportional to the original value, de-noising such recovered signals can be achieved by thresholding the hidden unit values (exploiting the fact that the signal is sparse).', '1605.07145-2-49-5': 'This can be done either by using a fixed threshold or picking the top k values.', '1605.07145-2-50-0': 'Data Whitening: Theorem [REF] shows that data generated from BINS and incoherent weight matrices are roughly uncorrelated.', '1605.07145-2-50-1': 'Thus recovering back such signals using auto-encoders would be easier if we pre-process the data to have uncorrelated dimensions.', '1605.07145-2-51-0': '# Empirical Verification', '1605.07145-2-52-0': 'We empirically verify the fundamental predictions made in section [REF] which both serve to justify the assumptions we have made, as well as confirm our results.', '1605.07145-2-52-1': 'We verify the following: a) the optimality of the rows of a weight matrix [MATH] to have unit length and being highly incoherent for AE signal recovery; b) effect of sparsity on AE signal recovery; and c) in practice, AE can recover not only the true sparse signal [MATH], but also the dictionary [MATH] that used to generate the data.', '1605.07145-2-53-0': '## Optimal Properties of Weights and Bias', '1605.07145-2-54-0': 'Our analysis on signal recovery in section [REF] (eq. [REF]) shows signal recovery bound is strong when a) the data generating weight matrix [MATH] has rows of unit [MATH] length; b) the rows of [MATH] are highly incoherent; c) each bias vector element is set to the negative expectation of the pre-activation; d) signal [MATH] has each dimension independent.', '1605.07145-2-54-1': 'In order to verify this, we generate [MATH] signals [MATH] from BINS([MATH],[MATH]=uniform, [MATH],[MATH]) with [MATH] set to uniform distribution for simplicity.', '1605.07145-2-54-2': 'We then generate the corresponding [MATH] data sample [MATH] using an incoherent weight matrix [MATH] (each element sampled from zero mean Gaussian, the columns are then orthogonalized, and [MATH] length of each row rescaled to [MATH]; notice the rows cannot be orthogonal).', '1605.07145-2-54-3': 'We then recover each signal using, [EQUATION] where [MATH] and [MATH] are scalars that we vary between [MATH] and [MATH] respectively.', '1605.07145-2-54-4': 'We also generate [MATH] signals [MATH] from BINS([MATH],[MATH], [MATH],[MATH]) with [MATH] set to Dirac delta function at 1.', '1605.07145-2-54-5': 'We then generate the corresponding [MATH] data sample [MATH] following the same procedure as for the continuous signal case.', '1605.07145-2-54-6': 'The signal is recovered using [EQUATION] where [MATH] is the sigmoid function.', '1605.07145-2-54-7': 'For the recovered signals, we calculate the Average Percentage Recovery Error (APRE) as, [EQUATION] where we set [MATH] to [MATH] for continuous signals and [MATH] for binary case, [MATH] is the indicator operator, [MATH] denotes the [MATH] dimension of the recovered signal corresponding to the [MATH] true signal and, [EQUATION]', '1605.07145-2-54-8': 'The error is weighted with [MATH] so that the recovery error for both zero and non-zero [MATH]s are penalized equally.', '1605.07145-2-54-9': 'This is specially needed in this case, because [MATH] is sparse and a low error can also be achieved by trivially setting all the recovered [MATH]s to zero.', '1605.07145-2-54-10': 'Along with the incoherent weight matrix, we also generate data separately using a highly coherent weight matrix that we get by sampling each element randomly from a uniform distribution on [MATH] and scaling each row to unit length.', '1605.07145-2-54-11': 'According to our analysis, we should get least error for [MATH] and [MATH] for the incoherent matrix while the coherent matrix should yield both higher recovery error and a different choice of [MATH] and [MATH] (which is unpredictable).', '1605.07145-2-54-12': 'The error heat maps for both continuous and binary recovery are shown in fig. [REF].', '1605.07145-2-54-13': 'For the incoherent weight matrix, we see that the empirical optimal is precisely [MATH] and [MATH] (which is exactly as predicted) with 0.21 and 0.0 APRE for continuous and binary recovery, respectively.', '1605.07145-2-54-14': 'It is interesting to note that the binary recovery is quite robust with the choice of [MATH] and [MATH], which is because 1) the recovery is denoised through thresholding, and 2) the binary signal inherently contains less information and thus is easier to recover.', '1605.07145-2-54-15': 'For the coherent weight matrix, we get 45.75 and 32.63 APRE instead (see fig. [REF]).', '1605.07145-2-55-0': 'We also experiment on the noisy recovery case, where we generate the data using incoherent weight matrix with [MATH] and [MATH].', '1605.07145-2-55-1': 'For each data dimension we add independent Gaussian noise with mean 100 with standard deviation varying from [MATH] to [MATH].', '1605.07145-2-55-2': 'Both signal recovery schemes are quite robust against noise (see fig. [REF]).', '1605.07145-2-55-3': 'In particular, the binary signal recovery is very robust, which conforms with our previous observation.', '1605.07145-2-56-0': '## Effect of Sparsity on Signal Recovery', '1605.07145-2-57-0': 'We analyze the effect of sparsity of signals on their recovery using the mechanism shown in section [REF].', '1605.07145-2-57-1': 'In order to do so, we generate incoherent matrices using two different methods- Gaussian and orthogonal .', '1605.07145-2-57-2': 'In addition, all the generated weight matrices are normalized to have unit [MATH] row length.', '1605.07145-2-57-3': 'We then sample signals and generate data using the same configurations as mentioned in section [REF]; only this time, we fix [MATH] and [MATH], vary hidden unit activation probability [MATH] in [MATH], and duplicate the generated data while adding noise to the copy, which we sample from a Gaussian distribution with mean [MATH] and standard deviation [MATH].', '1605.07145-2-57-4': "According to our analysis, noise mean should have no effect on recovery so the mean value of [MATH] shouldn't have any effect; only standard deviation affects recovery.", '1605.07145-2-57-5': 'We find for all weight matrices, recovery error reduces with increasing sparsity (decreasing [MATH], see fig. [REF]).', '1605.07145-2-57-6': 'Additionally, we find that both recovery are robust against noise.', '1605.07145-2-57-7': 'We also find the recovery error trend is almost always lower for orthogonal weight matrices, especially when the signal is sparse.', '1605.07145-2-57-8': 'Recall theorem [REF] suggests stronger recovery for more incoherent matrices.', '1605.07145-2-57-9': 'So we look into the row coherence of [MATH] sampled from Gaussian and Orthogonal methods with [MATH] and varying [MATH].', '1605.07145-2-57-10': 'We found that orthogonal initialized matrices have significantly lower coherence even though the orthogonalization is done column-wise (see fig. [REF].)', '1605.07145-2-57-11': 'This explains significantly lower recovery error for orthogonal matrices in figure [REF].', '1605.07145-2-58-0': '## Recovery of Data Dictionary', '1605.07145-2-59-0': 'We showed the conditions on [MATH] and [MATH] for good recovery of sparse signal [MATH].', '1605.07145-2-59-1': 'In practice, however, one does not have access to [MATH], in general.', '1605.07145-2-59-2': 'Therefore, in this section, we empirically demonstrate that AE can indeed recover both [MATH] and [MATH] through optimizing the AE objective.', '1605.07145-2-59-3': 'We generate [MATH] signals [MATH] with the same BINS distribution as in section [REF].', '1605.07145-2-59-4': 'The data are then generate as [MATH] using an incoherent weight matrix [MATH] (same as in section [REF]).', '1605.07145-2-59-5': 'We then recover the data dictionary [MATH] by: [EQUATION]', '1605.07145-2-59-6': 'Notice that although given sparse signal [MATH] the data dictionary [MATH] is unique , there are [MATH] number of equivalent solutions for [MATH], since we can permute dimension of [MATH] in AE.', '1605.07145-2-59-7': 'To check if the original data dictionary is recovered, we therefore pair up the rows of [MATH] and [MATH] by greedily select the pairs that result in the highest dot product value.', '1605.07145-2-59-8': 'We then measure the goodness of the recovery by looking at the values of all the paired dot products.', '1605.07145-2-59-9': 'In addition, since we know the pairing, we can calculate APRE to evaluate the quality of recovered hidden signal.', '1605.07145-2-59-10': 'As can be observed from fig. [REF], by optimizing the AE objective we can recover the the original data dictionary [MATH] (almost all of the cosine distances are 1).', '1605.07145-2-59-11': 'The final achieved [MATH] and [MATH] APRE for continuous and binary signal recovery, which is a bit less than what we achieved in section [REF].', '1605.07145-2-59-12': 'However, one should note that for this set of experiments we only observed data [MATH] and no other information regarding [MATH] is exposed.', '1605.07145-2-59-13': 'Not surprisingly, we again observed that the binary signal recovery is more robust as compared to the continuous counterpart, which may attribute to its lower information content.', '1605.07145-2-59-14': 'We also did experiments on noisy data and achieved similar performance as in section [REF] when the noise is less significant (see supplementary materials for more details).', '1605.07145-2-59-15': 'These results strongly suggests that AEs are capable of recovering the true hidden signal in practice.', '1605.07145-2-60-0': '# Conclusion', '1605.07145-2-61-0': 'In this paper we looked at the sparse signal recovery problem from the Auto-Encoder perspective and provide novel insights into conditions under which AEs can recover such signals.', '1605.07145-2-61-1': 'In particular, 1) from the signal recovery stand point, if we assume that the observed data is generated from some sparse hidden signals according to the assumed data generating process, then, the true hidden representation can be approximately recovered if a) the weight matrices are highly incoherent with unit [MATH] row length, and b) the bias vectors are as described in equation [REF] (theorem [REF]).', '1605.07145-2-61-2': 'The recovery also becomes more and more accurate with increasing sparsity in hidden signals.', '1605.07145-2-61-3': '2) From the data generation perspective, we found that data generated from such signals (assumption [REF]) have the property of being roughly uncorrelated (theorem [REF]), and thus pre-process the data to have uncorrelated dimensions may encourage stronger signal recovery.', '1605.07145-2-61-4': '3) Given only measurement data, we empirically show that the AE reconstruction objective recovers the data generating dictionary, and hence the true signal [MATH].', '1605.07145-2-61-5': '4) These conditions and observations allow us to view various existing techniques, such as data whitening, independent component analysis, etc., in a more coherent picture when considering signal recovery.', '1605.07145-2-62-0': 'Appendix: On Optimality Conditions for Auto-Encoder Signal Recovery', '1605.07145-2-63-0': '# Proofs', '1605.07145-2-64-0': 'Let [MATH] where [MATH], [MATH] and [MATH].', '1605.07145-2-64-1': 'Let [MATH] where [MATH] is a fixed vector.', '1605.07145-2-64-2': 'Let [MATH] and [MATH].', '1605.07145-2-64-3': 'Then [MATH] iff [MATH].', '1605.07145-2-65-0': 'Proof: Let [MATH].', '1605.07145-2-65-1': 'Thus [MATH].', '1605.07145-2-65-2': 'On the other hand, [MATH].', '1605.07145-2-65-3': 'The other direction can be proved similarly.', '1605.07145-2-66-0': 'Let each element of [MATH] follow BINS([MATH]) and let [MATH] be an auto-encoder signal recovery mechanism with Sigmoid activation function and bias [MATH] for a measurement vector [MATH] such that [MATH].', '1605.07145-2-66-1': 'If we set [MATH], then [MATH], [EQUATION] where [MATH], [MATH] and [MATH] is the [MATH] row of the matrix [MATH] cast as a column vector.', '1605.07145-2-67-0': 'Notice that, [EQUATION] and from definition [REF], [EQUATION]', '1605.07145-2-67-1': 'Thus, [EQUATION]', '1605.07145-2-67-2': 'Notice that [MATH].', '1605.07145-2-67-3': 'Let [MATH] and [MATH].', '1605.07145-2-67-4': "Then, setting [MATH], using Chernoff's inequality, for any [MATH], [EQUATION]", '1605.07145-2-67-5': 'Let [MATH].', '1605.07145-2-67-6': 'Then, [EQUATION]', '1605.07145-2-67-7': 'Let [MATH], thus, [EQUATION]', '1605.07145-2-67-8': 'Setting [MATH], we get [MATH].', '1605.07145-2-67-9': 'Thus, [MATH].', '1605.07145-2-67-10': "By Taylor's theorem, [MATH] s.t., [EQUATION]", '1605.07145-2-67-11': 'Thus we can upper bound [MATH] as, [EQUATION]', '1605.07145-2-67-12': 'Hence we can write [MATH] as [EQUATION]', '1605.07145-2-67-13': 'On the other hand, notice [MATH].', '1605.07145-2-68-0': '[EQUATION]', '1605.07145-2-69-0': 'Let [MATH].', '1605.07145-2-69-1': "Then we can similarly bound [MATH] by effectively flipping the sign of [MATH]'s in the previous derivation, [EQUATION]", '1605.07145-2-69-2': 'Minimizing both [REF] and [REF] with respect to [MATH] and applying union bound, we get, [EQUATION]', '1605.07145-2-69-3': 'Since the above bound holds for all [MATH], applying union bound on all the units yields the desired result.', '1605.07145-2-70-0': 'Let each element of [MATH] follow BINS([MATH]) and let [MATH] be an auto-encoder signal recovery mechanism with Sigmoid activation function and bias [MATH] for a measurement vector [MATH] where [MATH] is any noise vector independent of [MATH].', '1605.07145-2-70-1': 'If we set [MATH], then [MATH], [EQUATION] where [MATH], [MATH] and [MATH] is the [MATH] row of the matrix [MATH] cast as a column vector.', '1605.07145-2-71-0': 'Notice that, [EQUATION] and from definition [REF], [EQUATION]', '1605.07145-2-71-1': 'Thus, [EQUATION]', '1605.07145-2-71-2': 'Notice that [MATH].', '1605.07145-2-71-3': 'Let [MATH] and [MATH].', '1605.07145-2-71-4': "Then, setting [MATH], using Chernoff's inequality on random variable [MATH], for any [MATH], [EQUATION]", '1605.07145-2-71-5': 'Setting [MATH], we can rewrite the above inequality as [EQUATION]', '1605.07145-2-71-6': 'Since the above inequality becomes identical to equation [REF], the rest of the proof is similar to theorem [REF].', '1605.07145-2-72-0': 'Let each element of [MATH] follow BINS([MATH]) distribution and let [MATH] be an auto-encoder signal recovery mechanism with Rectified Linear activation function (ReLU) and bias [MATH] for a measurement vector [MATH] such that [MATH].', '1605.07145-2-72-1': 'If we set [MATH], then [MATH], [EQUATION] where [MATH]s are vectors such that [EQUATION] [MATH] is the [MATH] row of the matrix [MATH] cast as a column vector.', '1605.07145-2-73-0': 'From definition [REF] and the definition of [MATH] above, [EQUATION]', '1605.07145-2-73-1': 'Let [MATH].', '1605.07145-2-73-2': 'Thus, [MATH].', '1605.07145-2-73-3': 'Then, conditioning upon [MATH], [EQUATION]', '1605.07145-2-73-4': 'Since [MATH], we have, [EQUATION]', '1605.07145-2-73-5': 'The above inequality is obtained by ignoring the positive terms that depend on the condition [MATH] and marginalizing over [MATH].', '1605.07145-2-73-6': "For any [MATH], using Chernoff's inequality, [EQUATION]", '1605.07145-2-73-7': 'Setting [MATH], where [MATH], [EQUATION]', '1605.07145-2-73-8': 'Let [MATH].', '1605.07145-2-73-9': 'Then, [EQUATION] where [MATH] denotes any arbitrary distribution in the interval [MATH] with mean [MATH].', '1605.07145-2-73-10': 'If [MATH], let [MATH] and [MATH] which essentially denote the lower and upper bound of [MATH].', '1605.07145-2-73-11': 'Then, [EQUATION] where the first inequality in the above equation is from the property of a convex function.', '1605.07145-2-73-12': 'Define [MATH], [MATH].', '1605.07145-2-73-13': 'Then, [EQUATION]', '1605.07145-2-73-14': 'Define [MATH] and let [MATH].', '1605.07145-2-73-15': 'Then, [EQUATION]', '1605.07145-2-73-16': 'Thus, for getting a maxima for [MATH], we set [MATH] which implies [MATH], or, [MATH].', '1605.07145-2-73-17': 'Substituting this [MATH] in [MATH].', '1605.07145-2-73-18': "By Taylor's theorem, [MATH] such that, [EQUATION]", '1605.07145-2-73-19': 'Thus we can upper bound [MATH] as, [EQUATION]', '1605.07145-2-73-20': 'Substituting for [MATH], we get, [EQUATION]', '1605.07145-2-73-21': 'On the other hand, if [MATH], then we can set [MATH] and [MATH] and proceeding similar to equation [REF], we get, [EQUATION]', '1605.07145-2-73-22': 'Then, collectively, we can write [MATH] as [EQUATION]', '1605.07145-2-73-23': "We similarly bound [MATH] by effectively flipping the sign of [MATH]'s, [EQUATION]", '1605.07145-2-73-24': 'Minimizing both [REF] and [REF] with respect to [MATH] and applying union bound, we get, [EQUATION]', '1605.07145-2-73-25': 'Since the above bound holds for all [MATH], applying union bound on all the units yields the desired result.', '1605.07145-2-74-0': 'Let each element of [MATH] follow BINS([MATH]) distribution and let [MATH] be an auto-encoder signal recovery mechanism with Rectified Linear activation function and bias [MATH] for a measurement vector [MATH] such that [MATH] where [MATH] is any noise random vector independent of [MATH].', '1605.07145-2-74-1': 'If we set [MATH], then [MATH], [EQUATION] where [MATH]s are vectors such that [EQUATION] [MATH] is the [MATH] row of the matrix [MATH] cast as a column vector.', '1605.07145-2-75-0': 'Recall that, [EQUATION]', '1605.07145-2-75-1': 'Let [MATH].', '1605.07145-2-75-2': 'Then, similar to theorem [REF], conditioning upon [MATH], [EQUATION]', '1605.07145-2-75-3': 'Since [MATH], we have, [EQUATION]', '1605.07145-2-75-4': "For any [MATH], using Chernoff's inequality for the random variable [MATH], [EQUATION]", '1605.07145-2-75-5': 'Setting [MATH], where [MATH], [EQUATION]', '1605.07145-2-75-6': 'Setting [MATH], we can rewrite the above inequality as [EQUATION]', '1605.07145-2-75-7': 'Since the above inequality becomes identical to equation [REF], the rest of the proof is similar to theorem [REF].', '1605.07145-2-76-0': '(Uncorrelated Distribution Bound): If data is generated as [MATH] where [MATH] has covariance matrix [MATH], ([MATH]) and [MATH]) is such that each row of [MATH] has unit length and the rows of [MATH] are maximally incoherent, then the covariance matrix of the generated data is approximately spherical (uncorrelated) satisfying, [EQUATION] where [MATH] is the covariance matrix of the generated data.', '1605.07145-2-77-0': 'Notice that, [EQUATION]', '1605.07145-2-77-1': 'Thus, [EQUATION]', '1605.07145-2-77-2': 'Substituting the covariance of [MATH] as [MATH], [EQUATION]', '1605.07145-2-77-3': 'Thus, [EQUATION]', '1605.07145-2-77-4': 'Using the cyclic property of trace, [EQUATION]', '1605.07145-2-77-5': 'Finally minimizing w.r.t [MATH], we get [MATH].', '1605.07145-2-77-6': 'Substituting this into the above inequality, we get,', '1605.07145-2-78-0': '[EQUATION]', '1605.07145-2-79-0': 'Since the weight matrix is maximally incoherent, using Welch bound, we have that, [MATH].', '1605.07145-2-79-1': 'Plugging the lower bound of [MATH] (maximal incoherence) for any fixed [MATH] and [MATH] into the above bound yields, [EQUATION]', '1605.07145-2-80-0': '# Supplementary Experiments', '1605.07145-2-81-0': '## Supplementary Experiments for Section [REF]', '1605.07145-2-82-0': 'Here we show the recovery error (APRE) for signals generated with coherent weight matrix, and as expected the recovery result is poor and the values of [MATH] and [MATH] are unpredictable.', '1605.07145-2-82-1': 'The minimum average percentage recovery error we got for continous signal is 45.75, and for binary signal is 32.63.', '1605.07145-2-83-0': '## Supplementary Experiments for Section [REF]', '1605.07145-2-84-0': 'Fig. [REF] shows that the coherence of orthogonal initialized weight matrix is more incoherent as compared to the ones that using Gaussian based initialization.', '1605.07145-2-85-0': '## Supplementary Experiments for Section [REF]', '1605.07145-2-86-0': 'For noisy signal recovery we add independent Gaussian noise to data with mean [MATH] and standard deviation ranging from [MATH] to [MATH].', '1605.07145-2-86-1': 'Note that the data is normally within the range of[MATH], so the noise is quite significant when we have a standard deviation [MATH].', '1605.07145-2-86-2': 'It is clear that even in noisy case AE can recover the dictionary (see fig. [REF]).', '1605.07145-2-86-3': 'However, the recovery is not very strong when the noise is large [MATH] for continous signals, which is because 1) the precise value in this case is continous and thus is more influenced by the noise, 2) the dictionary recovery is poor, which result poor signal recovery.', '1605.07145-2-86-4': 'On the other hand, the recovery is robust in case of recovering binary signals.', '1605.07145-2-86-5': 'Similar results were found on the APRE of recovered hidden signals.', '1605.07145-2-86-6': 'The reason for more robust recovery for binary signal is that 1) the information content is lower and 2) we binarize the recovered hidden signal by thresholding it, which further denoised the recovery.', '1605.07145-2-86-7': 'When optimizing the AE objective for binary singal recovery case, we did a small trick to simulate the binarization of the signal.', '1605.07145-2-86-8': 'From our analysis (see Theorem [REF]), a recovery error [MATH] is reasonable as we can binarize the recovery using some threshold.', '1605.07145-2-86-9': 'However, when optimizing the AE using gradient based method we are unable to do this.', '1605.07145-2-86-10': 'To simulate this effect, we offset the pre-activation by a constant [MATH] and multiply the pre-activation by a constant [MATH], so that it signifies the input and push the post-activation values towards [MATH] and [MATH].', '1605.07145-2-86-11': 'In other words, we optimize the following objective when doing binary signal recovery: [EQUATION] where [MATH] is the sigmoid function.', '1605.07145-2-86-12': 'We find set [MATH] and [MATH] is sufficient to saturate the sigmoid and simulate the binarization of hidden signals.'}

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['1605.07145-1-62-1', '1605.07145-2-21-1'], ['1605.07145-1-51-3', '1605.07145-2-54-3'], ['1605.07145-1-53-0', '1605.07145-2-54-8'], ['1605.07145-1-53-2', '1605.07145-2-54-10'], ['1605.07145-1-28-0', '1605.07145-2-33-0'], ['1605.07145-1-28-1', '1605.07145-2-33-1'], ['1605.07145-1-28-2', '1605.07145-2-33-2'], ['1605.07145-1-68-1', '1605.07145-2-27-1'], ['1605.07145-1-68-2', '1605.07145-2-27-2'], ['1605.07145-1-68-3', '1605.07145-2-27-3'], ['1605.07145-1-68-4', '1605.07145-2-27-4'], ['1605.07145-1-68-5', '1605.07145-2-27-5'], ['1605.07145-1-68-6', '1605.07145-2-27-6'], ['1605.07145-1-32-4', '1605.07145-2-27-3'], ['1605.07145-1-32-5', '1605.07145-2-27-4'], ['1605.07145-1-32-6', '1605.07145-2-27-5'], ['1605.07145-1-32-7', '1605.07145-2-27-6'], ['1605.07145-1-32-1', '1605.07145-2-37-1'], ['1605.07145-1-32-2', '1605.07145-2-37-2'], ['1605.07145-1-32-3', '1605.07145-2-37-3'], ['1605.07145-1-32-6', '1605.07145-2-37-6'], ['1605.07145-1-28-0', '1605.07145-2-27-2'], ['1605.07145-1-28-1', '1605.07145-2-27-3'], ['1605.07145-1-28-2', '1605.07145-2-27-4'], ['1605.07145-1-28-3', '1605.07145-2-27-5'], ['1605.07145-1-30-0', '1605.07145-2-35-0'], ['1605.07145-1-66-0', '1605.07145-2-35-0'], ['1605.07145-1-30-0', '1605.07145-2-25-0'], ['1605.07145-1-11-0', '1605.07145-2-7-0'], ['1605.07145-1-36-0', '1605.07145-2-41-0'], ['1605.07145-1-19-0', '1605.07145-2-15-0'], ['1605.07145-1-42-1', '1605.07145-2-47-1'], ['1605.07145-1-42-2', '1605.07145-2-47-2'], ['1605.07145-1-24-0', '1605.07145-2-17-0'], ['1605.07145-1-24-1', '1605.07145-2-17-1'], ['1605.07145-1-12-0', '1605.07145-2-8-0'], ['1605.07145-1-47-0', '1605.07145-2-50-0'], ['1605.07145-1-13-1', '1605.07145-2-9-3'], ['1605.07145-1-13-3', '1605.07145-2-9-5'], ['1605.07145-1-13-5', '1605.07145-2-9-8'], ['1605.07145-1-4-3', '1605.07145-2-2-4'], ['1605.07145-1-4-4', '1605.07145-2-2-5'], ['1605.07145-1-40-0', '1605.07145-2-45-0'], ['1605.07145-1-38-1', '1605.07145-2-43-1'], ['1605.07145-1-55-2', '1605.07145-2-57-2'], ['1605.07145-1-55-3', '1605.07145-2-57-3'], ['1605.07145-1-55-6', '1605.07145-2-57-5'], ['1605.07145-1-55-9', '1605.07145-2-57-9'], ['1605.07145-1-17-2', '1605.07145-2-13-2'], ['1605.07145-1-41-0', '1605.07145-2-46-0'], ['1605.07145-1-41-1', '1605.07145-2-46-1'], ['1605.07145-1-41-4', '1605.07145-2-46-4'], ['1605.07145-1-41-8', '1605.07145-2-46-8'], ['1605.07145-1-7-0', '1605.07145-2-4-0'], ['1605.07145-1-7-2', '1605.07145-2-4-2'], ['1605.07145-1-7-3', '1605.07145-2-4-3'], ['1605.07145-1-7-4', '1605.07145-2-4-4'], ['1605.07145-1-37-2', '1605.07145-2-42-2'], ['1605.07145-1-44-0', '1605.07145-2-48-0'], ['1605.07145-1-44-2', '1605.07145-2-48-2'], ['1605.07145-1-44-3', '1605.07145-2-48-3'], ['1605.07145-1-8-0', '1605.07145-2-5-0'], ['1605.07145-1-8-3', '1605.07145-2-5-3'], ['1605.07145-1-8-6', '1605.07145-2-5-7'], ['1605.07145-1-8-7', '1605.07145-2-5-8'], ['1605.07145-1-8-10', '1605.07145-2-5-11'], ['1605.07145-1-16-1', '1605.07145-2-12-1'], ['1605.07145-1-16-2', '1605.07145-2-12-2'], 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[]

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[['1605.07145-1-5-0', '1605.07145-2-2-6']]

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{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1605.07145

1508.04375

{'1508.04375-1-0-0': 'We put forward a variation of a recently proposed method to overcome the signal-to-noise problem found in Monte Carlo calculations of the entanglement entropy of interacting fermions.', '1508.04375-1-0-1': 'The present method takes advantage of the approximate lognormal distributions that characterize the signal-to-noise properties of other approaches.', '1508.04375-1-0-2': 'We test our technique by implementing it in combination with the hybrid Monte Carlo algorithm and calculating the second Renyi entanglement entropy of the 1D Hubbard model.', '1508.04375-1-1-0': '# Introduction', '1508.04375-1-2-0': 'Recently [CITATION], we proposed an algorithm to compute the Renyi entanglement entropy of interacting fermions.', '1508.04375-1-2-1': 'Many algorithms have been proposed to this effect in the last few years [CITATION].', '1508.04375-1-2-2': 'Our proposal overcomes the signal-to-noise problem of naive approaches and is compatible with the hybrid Monte Carlo (HMC) method [CITATION] widely used in the context of lattice quantum chromodynamics.', '1508.04375-1-3-0': 'In this work we describe and implement a variation on that Monte Carlo (MC) algorithm which, while sharing the properties mentioned above, also takes advantage of the approximate lognormal shape of the underlying statistical distributions.', '1508.04375-1-3-1': 'Below we review the basic formalism, briefly present the evidence of approximate lognormal distributions, and then explain our method.', '1508.04375-1-3-2': 'After that, we present test results based on the 1D attractive Hubbard model.', '1508.04375-1-4-0': '# Formalism', '1508.04375-1-5-0': 'As in our previous work, we set the stage by briefly presenting the formalism of Ref. [CITATION].', '1508.04375-1-5-1': 'The [MATH]-th Renyi entanglement entropy [MATH] of a sub-system [MATH] of a given quantum system is defined by [EQUATION] where [MATH] is the reduced density matrix of sub-system [MATH] (i.e. the degrees of freedom of the rest of the system are traced over).', '1508.04375-1-5-2': 'An auxiliary-field path-integral form for [MATH], from which [MATH] can be computed using MC methods for a wide variety of systems, was presented in Ref. [CITATION].', '1508.04375-1-5-3': 'In particular, for [MATH]-component fermions, we have [EQUATION] where the field integration measure, given by [EQUATION] is over the [MATH] "replicas" [MATH] of the Hubbard-Stratonovich auxiliary field, and the normalization [EQUATION] was included in the integration measure.', '1508.04375-1-5-4': 'It is worth noting that, by separating a factor of [MATH] in the denominator of Eq. ([REF]), an explicit form can be identified in the numerator as in the replica trick [CITATION], which corresponds to a partition function for [MATH] copies of the system, "glued" together in the region [MATH].', '1508.04375-1-6-0': 'The naive probability measure, given by [EQUATION] factorizes entirely across replicas, and is therefore insensitive to entanglement.', '1508.04375-1-6-1': 'This factorization is the main reason why using [MATH] as a MC probability leads to (seemingly) insurmountable signal-to-noise issues, as shown in Ref. [CITATION].', '1508.04375-1-6-2': 'In Eq. ([REF]), [MATH] is a matrix which encodes the dynamics of the [MATH]-th component in the system, namely the kinetic energy and the form of the interaction after a Hubbard-Stratonovich transformation; it also encodes the form of the trial state [MATH] in ground-state approaches (see e.g. Ref. [CITATION]), which we use here.', '1508.04375-1-6-3': 'We will take [MATH] to be a Slater determinant.', '1508.04375-1-6-4': 'In finite-temperature approaches, [MATH] is obtained by evolving a complete set of single-particle states in imaginary time.', '1508.04375-1-7-0': 'The quantity that contains the essential contributions to entanglement is [EQUATION] where [EQUATION]', '1508.04375-1-7-1': 'The product [MATH] was identified as playing the role of an observable in Ref. [CITATION], which is a natural interpretation in light of Eq. ([REF]), but which we will interpret differently below.', '1508.04375-1-7-2': "Aside from the field replicas, the new element in the determination of [MATH] is the restricted Green's function [MATH].", '1508.04375-1-7-3': "The latter corresponds to a noninteracting single-particle Green's function [MATH] of the [MATH]-th fermion component in the background field [MATH], but such that the arguments [MATH] only take values in the region [MATH] (see Ref. [CITATION] and also Ref. [CITATION], where expressions were first derived for the reduced density matrix of noninteracting systems, based on reduced Green's functions).", '1508.04375-1-7-4': 'Note that, for [MATH], no matrix inversion is required in the calculation of [MATH]; for higher [MATH], however, there is no obvious way to avoid the inversion of [MATH] (see Ref. [CITATION]).', '1508.04375-1-8-0': '# A statistical observation', '1508.04375-1-9-0': 'In Ref. [CITATION], we presented examples of the approximate log-normal distributions obeyed by [MATH] when sampled according to [MATH].', '1508.04375-1-9-1': 'One such example is reproduced here for reference in Fig. [REF].', '1508.04375-1-10-0': 'The fact that such distributions are approximately log-normal, at least visually, suggests that one may use the cumulant expansion to determine [MATH].', '1508.04375-1-10-1': 'Indeed, in general, [EQUATION] where [MATH] is the [MATH]-th cumulant of [MATH].', '1508.04375-1-10-2': 'If the distribution of [MATH] were truly gaussian, the above series would terminate after the first 2 terms, which would provide us with an efficient way to bypass signal-to-noise issues in the determination of [MATH] with stochastic methods [CITATION].', '1508.04375-1-10-3': 'Unfortunately, the distribution is not exactly gaussian.', '1508.04375-1-10-4': 'Moreover, the cumulants beyond [MATH] are often extremely sensitive to the details of the distribution (i.e. they can fluctuate wildly), they are hard to determine stochastically (the signal-to-noise problem re-emerges), and there is no easy way (that we know of) to obtain analytic insight into the large-[MATH] behavior of [MATH].', '1508.04375-1-10-5': 'However, this approximate log-normality does provide a path forward, as it indicates that we may still evaluate [MATH] with good precision with MC methods.', '1508.04375-1-10-6': 'As we will see in the next sections, this is enough to determine [MATH] if we are willing to pay the price of a one-dimensional integration between 0 and 1.', '1508.04375-1-11-0': 'Although (approximate) lognormality in the [MATH] determinant seems very difficult to prove analytically in the present case, evidence of its appearance has been found in systems as different as ultracold atoms and relativistic gauge theories [CITATION].', '1508.04375-1-11-1': 'The underlying reason for this distribution appears to be connected to a similarity between the motion of electrons in disordered media and lattice fermions in the external auxiliary (gauge) field in MC calculations.', '1508.04375-1-12-0': '# Proposed method', '1508.04375-1-13-0': 'Starting from the right-hand side of Eq. ([REF]), we introduce an auxiliary parameter [MATH] and define a function [MATH] via [EQUATION]', '1508.04375-1-13-1': 'At [MATH], [EQUATION] while for [MATH], [MATH] yields the entanglement entropy: [EQUATION]', '1508.04375-1-13-2': 'Using Eq. ([REF]), [EQUATION] where [EQUATION]', '1508.04375-1-13-3': 'When an even number [MATH] of flavors is considered, and the interactions are attractive, [MATH] and [MATH] are real and positive semidefinite for all [MATH], which means that there is no sign problem and [MATH] above is a well-defined, normalized probability measure.', '1508.04375-1-14-0': 'As in our previously proposed method, we can then calculate [MATH] by taking the [MATH] point as a reference and computing [MATH] using [EQUATION] where [EQUATION]', '1508.04375-1-14-1': 'We thus obtain an integral form of the interacting Renyi entropy that can be computed using any MC method (see e.g. [CITATION]), in particular HMC [CITATION].', '1508.04375-1-15-0': 'As in our previous work, we note that the above expectation values are taken with respect to the probability measure [MATH], which incorporates correlations that account for entanglement.', '1508.04375-1-15-1': 'In stark contrast to the naive MC probability [MATH], which corresponds to statistically independent copies of the Hubbard-Stratonovich field, this distribution does not display the decoupling responsible for the signal-to-noise problem mentioned above.', '1508.04375-1-16-0': 'In practice, using Eq. ([REF]) requires MC calculations to evaluate [MATH] as a function of [MATH], followed by numerical integration over [MATH].', '1508.04375-1-16-1': 'We find that [MATH] is an essentially linear function of [MATH].', '1508.04375-1-16-2': 'We therefore carry out the numerical integration using a uniform grid.', '1508.04375-1-16-3': 'It should also be pointed out that the stochastic evaluation of [MATH], for fixed subregion [MATH], can be expected to feature roughly symmetric fluctuations about the mean.', '1508.04375-1-16-4': 'Therefore, after integrating over [MATH], the statistical effects on the entropy are reduced (as we show empirically in the Results section).', '1508.04375-1-17-0': "Finally, we note an interesting application of Jensen's inequality at [MATH].", '1508.04375-1-17-1': 'At that point [EQUATION] which must be satisfied by our calculations.', '1508.04375-1-17-2': 'Our Monte Carlo results at [MATH] indeed satisfy this bound.', '1508.04375-1-18-0': '# Results', '1508.04375-1-19-0': 'We test our algorithm by computing the second Renyi entropy [MATH] for one-dimensional, half-filled Hubbard models with periodic boundary conditions.', '1508.04375-1-19-1': 'The Hamiltonian we used is [EQUATION] where the first sum includes two fermion flavors [MATH] and nearest-neighbor pairs.', '1508.04375-1-19-2': 'To carry out our tests, we implemented a symmetric Trotter-Suzuki decomposition of the Boltzmann weight, with an imaginary-time discretization parameter [MATH] (in lattice units).', '1508.04375-1-19-3': 'The full length of the imaginary-time direction was at most [MATH] (i.e. we used 100 imaginary-time lattice points).', '1508.04375-1-19-4': 'The interaction factor in the Trotter-Suzuki decomposition was addressed, as anticipated in a previous section, by introducing a replica auxiliary field [MATH] for each power of the reduced density matrix.', '1508.04375-1-19-5': 'This insertion was accomplished via a Hubbard-Stratonovich transformation, which we chose to be of a continuous but compact form (see Ref. [CITATION]).', '1508.04375-1-20-0': 'We present plots for [MATH] in Fig. [REF].', '1508.04375-1-20-1': 'In contrast to the results obtained in Ref. [CITATION] and as mentioned above, the resulting expectation demonstrates surprisingly little curvature as the region size [MATH] is varied and is stunningly linear as a function of the auxiliary parameter [MATH].', '1508.04375-1-20-2': 'Even after twice doubling the strength of the interaction, the curvature of constant-subsystem-size slices is increased only marginally.', '1508.04375-1-20-3': 'We note that if one assumes such benign curvature is a somewhat universal feature, at least for weakly-coupled systems, our method provides a means by which to rapidly estimate the entanglement entropy for a large portion of parameter space at the very least yielding a qualitative picture of its behavior as a function of the physically relevant input parameters.', '1508.04375-1-21-0': 'As shown in Fig. [REF], resulting values display almost no features for smaller subregions, and for the weakly-coupled cases, we observe that the surface displays almost no torsion, its dominant features being those present in the noninteracting case i.e. an alternating shell-like structure.', '1508.04375-1-21-1': 'Toward larger region sizes, we observe a combination of twisting and translation culminating in the required, and somewhat delicate, cancellation upon reaching the full system size.', '1508.04375-1-21-2': 'Presented with this relatively forgiving geometry, we performed the required integration via cubic-spline interpolation.', '1508.04375-1-21-3': 'Using a uniformly spaced lattice of size [MATH] points, we determine the desired entropy to a precision limited by statistical rather than systematic considerations.', '1508.04375-1-22-0': '## Comparison to exact diagonalization', '1508.04375-1-23-0': 'Shown in Fig. [REF] are results for a system of size [MATH] with a number of sites [MATH].', '1508.04375-1-23-1': 'For couplings [MATH] and [MATH] and region sizes [MATH], we find solid agreement with previous calculations in Refs. [CITATION], and as in the former, we observe convergence rather quickly with only [MATH] decorrelated samples as can be seen in Fig. [REF].', '1508.04375-1-23-2': 'Further, for large sample sizes [MATH], we observe (see Fig. [REF]) that the standard error in the entropy [MATH], computed from the envelope defined by the MC uncertainty in the source [MATH] at each value in [MATH]-space, scales asymptotically as [MATH] up to minute corrections.', '1508.04375-1-24-0': '# Summary and Conclusions', '1508.04375-1-25-0': 'We have presented a method to compute the entanglement entropy of interacting fermions which takes advantage of an approximate log-normality property.', '1508.04375-1-25-1': 'The resulting approach overcomes the signal-to-noise problem of naive methods, and is very close in principle to another method we proposed recently: both methods involve defining a new parameter, differentiating, and then integrating back after a MC calculation.', '1508.04375-1-25-2': 'As a test of our algorithm, we have presented results for the half-filled 1D Hubbard model with periodic boundary conditions.', '1508.04375-1-26-0': 'This material is based upon work supported by the National Science Foundation under Grants No. PHY1306520 (Nuclear Theory program) and No. PHY1452635 (Computational Physics program).'}

{'1508.04375-2-0-0': 'We put forward a variation of a recently proposed method to overcome the signal-to-noise problem found in Monte Carlo calculations of the entanglement entropy of interacting fermions.', '1508.04375-2-0-1': 'The present method takes advantage of the approximate lognormal distributions that characterize the signal-to-noise properties of other approaches.', '1508.04375-2-0-2': 'We test our technique by implementing it in combination with the hybrid Monte Carlo algorithm and calculating the second Renyi entanglement entropy of the 1D Hubbard model.', '1508.04375-2-1-0': '# Introduction', '1508.04375-2-2-0': 'Recently [CITATION], we proposed an algorithm to compute the Renyi entanglement entropy of interacting fermions.', '1508.04375-2-2-1': 'Many algorithms have been proposed to this effect in the last few years [CITATION].', '1508.04375-2-2-2': 'Our proposal overcomes the signal-to-noise problem of naive approaches and is compatible with the hybrid Monte Carlo (HMC) method [CITATION] widely used in the context of lattice quantum chromodynamics.', '1508.04375-2-3-0': 'In this work we describe and implement a variation on that Monte Carlo (MC) algorithm which, while sharing the properties mentioned above, also takes advantage of the approximate lognormal shape of the underlying statistical distributions.', '1508.04375-2-3-1': 'Below we review the basic formalism, briefly present the evidence of approximate lognormal distributions, and then explain our method.', '1508.04375-2-3-2': 'After that, we present test results based on the 1D attractive Hubbard model.', '1508.04375-2-4-0': '# Formalism', '1508.04375-2-5-0': 'As in our previous work, we set the stage by briefly presenting the formalism of Ref. [CITATION].', '1508.04375-2-5-1': 'The [MATH]-th Renyi entropy [MATH] of a sub-system [MATH] of a given system is [EQUATION] where [MATH] is the reduced density matrix of sub-system [MATH].', '1508.04375-2-5-2': 'An auxiliary-field path-integral form for [MATH], from which [MATH] can be computed using MC methods for a wide variety of systems, was presented in Ref. [CITATION].', '1508.04375-2-5-3': 'For [MATH]-component fermions, we have [EQUATION] where the field integration measure, given by [EQUATION] is over the [MATH] "replicas" [MATH] of the Hubbard-Stratonovich field, and the normalization [EQUATION] was included in the measure.', '1508.04375-2-5-4': 'It is worth noting that, by separating a factor of [MATH] in the denominator of Eq. ([REF]), an explicit form can be identified in the numerator as in the replica trick [CITATION], which corresponds to a partition function for [MATH] copies of the system, "glued" together in the region [MATH].', '1508.04375-2-6-0': 'The naive probability measure, namely [EQUATION] factorizes across replicas, which makes it insensitive to entanglement.', '1508.04375-2-6-1': 'This factorization is the main reason why using [MATH] as a MC probability leads to signal-to-noise issues (see Ref. [CITATION]).', '1508.04375-2-6-2': 'In Eq. ([REF]), [MATH] encodes the dynamics of the [MATH]-th fermion component, including the kinetic energy and the form of the interaction after a Hubbard-Stratonovich transformation.', '1508.04375-2-6-3': 'That matrix also encodes the form of the trial state [MATH] in ground-state approaches (see e.g. Ref. [CITATION]), which we use here, and we have taken [MATH] to be a Slater determinant.', '1508.04375-2-6-4': 'In finite-temperature approaches, [MATH] is obtained by evolving a complete set of single-particle states in imaginary time.', '1508.04375-2-7-0': 'The quantity that contains the pivotal contributions to entanglement is [EQUATION] where [EQUATION]', '1508.04375-2-7-1': 'The product [MATH] played the role of an observable in Ref. [CITATION], which is a natural interpretation given Eq. ([REF]).', '1508.04375-2-7-2': 'However, we will interpret this differently below.', '1508.04375-2-7-3': "Other than the field replicas, the new ingredient in the determination of [MATH] is the restricted Green's function [MATH].", '1508.04375-2-7-4': "This is the same as the noninteracting single-particle Green's function [MATH] of the [MATH]-th fermion component in the background field [MATH], but the arguments [MATH] are restricted to the region [MATH] (see Ref. [CITATION] and also Ref. [CITATION], where expressions were originally derived for the reduced density matrix of noninteracting systems, based on reduced Green's functions).", '1508.04375-2-7-5': 'As was noted in Ref. [CITATION], for [MATH], no matrix inversion is required in the calculation of [MATH], but for higher [MATH] there is no clear way to avoid the inversion of [MATH].', '1508.04375-2-8-0': '# A statistical observation', '1508.04375-2-9-0': 'In Ref. [CITATION], we presented examples of the approximate log-normal distributions obeyed by [MATH] when sampled according to [MATH].', '1508.04375-2-9-1': 'One such example is reproduced here for reference in Fig. [REF].', '1508.04375-2-10-0': 'The fact that such distributions are approximately log-normal, at least visually, suggests that one may use the cumulant expansion to determine [MATH].', '1508.04375-2-10-1': 'Indeed, in general, [EQUATION] where [MATH] is the [MATH]-th cumulant of [MATH].', '1508.04375-2-10-2': 'If the distribution of [MATH] were truly gaussian, the above series would terminate after the first 2 terms, which would provide us with an efficient way to bypass signal-to-noise issues in the determination of [MATH] with stochastic methods [CITATION].', '1508.04375-2-10-3': 'Unfortunately, the distribution is not exactly gaussian.', '1508.04375-2-10-4': 'Moreover, the cumulants beyond [MATH] are often extremely sensitive to the details of the distribution (i.e. they can fluctuate wildly), they are hard to determine stochastically (the signal-to-noise problem re-emerges), and there is no easy way (that we know of) to obtain analytic insight into the large-[MATH] behavior of [MATH].', '1508.04375-2-10-5': 'However, this approximate log-normality does provide a path forward, as it indicates that we may still evaluate [MATH] with good precision with MC methods.', '1508.04375-2-10-6': 'As we will see in the next sections, this is enough to determine [MATH] if we are willing to pay the price of a one-dimensional integration between 0 and 1.', '1508.04375-2-11-0': 'Although (approximate) lognormality in the [MATH] determinant seems very difficult to prove analytically in the present case, evidence of its appearance has been found in systems as different as ultracold atoms and relativistic gauge theories [CITATION].', '1508.04375-2-11-1': 'The underlying reason for this distribution appears to be connected to a similarity between the motion of electrons in disordered media and lattice fermions in the external auxiliary (gauge) field in MC calculations.', '1508.04375-2-12-0': '# Proposed method', '1508.04375-2-13-0': 'Starting from the right-hand side of Eq. ([REF]), we introduce an auxiliary parameter [MATH] and define a function [MATH] via [EQUATION]', '1508.04375-2-13-1': 'At [MATH], [EQUATION] while for [MATH], [MATH] yields the entanglement entropy: [EQUATION]', '1508.04375-2-13-2': 'Using Eq. ([REF]), [EQUATION] where [EQUATION]', '1508.04375-2-13-3': 'In the presence of an even number of flavors [MATH] and attractive interactions, [MATH] and [MATH] are real and non-negative for all [MATH], such that there is no sign problem and [MATH] above is a well-defined, normalized probability measure.', '1508.04375-2-14-0': 'As in our previously proposed method, we can then calculate [MATH] by taking the [MATH] point as a reference and computing [MATH] using [EQUATION] where [EQUATION]', '1508.04375-2-14-1': 'We thus obtain an integral form of the interacting Renyi entropy that can be computed using any MC method (see e.g. [CITATION]), in particular HMC [CITATION].', '1508.04375-2-15-0': 'As in our previous work, we note that the above expectation values are determined with respect to the probability measure [MATH], which communicates correlations responsible for entanglement.', '1508.04375-2-15-1': 'In contrast to the canonical MC probability [MATH], which corresponds to statistically independent copies of the Hubbard-Stratonovich field, this admittedly more complicated distribution does not exhibit the factorization to blame for the signal-to-noise problems present in the approach as originally formulated.', '1508.04375-2-16-0': 'Using Eq. ([REF]) requires Monte Carlo methods to evaluate [MATH] as a function of [MATH], followed by integration over [MATH].', '1508.04375-2-16-1': 'As in our previous method, we find here that [MATH] is smooth as a function of [MATH] and in particular that it is essentially linear in the present case.', '1508.04375-2-16-2': 'It is therefore sufficient to perform the numerical integration using a uniform grid.', '1508.04375-2-16-3': 'The stochastic evaluation of [MATH], for fixed subregion [MATH], can be expected to feature roughly symmetric fluctuations about the mean.', '1508.04375-2-16-4': 'As a consequence, the statistical effects on the entropy are reduced after integrating over [MATH].', '1508.04375-2-17-0': "Finally, we note an interesting application of Jensen's inequality at [MATH].", '1508.04375-2-17-1': 'At that point [EQUATION] which must be satisfied by our calculations.', '1508.04375-2-17-2': 'Our Monte Carlo results at [MATH] indeed satisfy this bound.', '1508.04375-2-18-0': '# Results', '1508.04375-2-19-0': 'As a test of our algorithm and in efforts to make contact with previous work [CITATION], we calculate the second Renyi entropy [MATH] for the one-dimensional Hubbard chain with periodic boundary conditions at half filling, whose Hamiltonian is [EQUATION] where the first sum includes [MATH] and pairs of adjacent sites.', '1508.04375-2-19-1': 'We implemented a symmetric Trotter-Suzuki decomposition of the Boltzmann weight, with an imaginary-time discretization of [MATH] (in lattice units).', '1508.04375-2-19-2': 'As mentioned earlier, the many-body factor in the Trotter-Suzuki approximation was treated by introducing a replica auxiliary field [MATH] for each power of the reduced density matrix.', '1508.04375-2-19-3': 'As in our previous work, we implemented a Hubbard-Stratonovich transformation of a compact continuous form [CITATION].', '1508.04375-2-20-0': 'We present plots for [MATH] in Fig. [REF].', '1508.04375-2-20-1': 'In contrast to the results obtained in Ref. [CITATION] and as mentioned above, the resulting expectation demonstrates surprisingly little curvature as the region size [MATH] is varied and is stunningly linear as a function of the auxiliary parameter [MATH].', '1508.04375-2-20-2': 'Even after twice doubling the strength of the interaction, the curvature of constant-subsystem-size slices is increased only marginally.', '1508.04375-2-20-3': 'We note that if one assumes such benign curvature is a somewhat universal feature, at least for weakly-coupled systems, our method provides a means by which to rapidly estimate the entanglement entropy for a large portion of parameter space at the very least yielding a qualitative picture of its behavior as a function of the physically relevant input parameters.', '1508.04375-2-21-0': 'As shown in Fig. [REF], resulting values display almost no features for smaller subregions, and for the weakly-coupled cases, we observe that the surface displays almost no torsion, its dominant features being those present in the noninteracting case i.e. an alternating shell-like structure.', '1508.04375-2-21-1': '(See Fig. [REF] for a more explicit comparison of weakly and strongly coupled cases.)', '1508.04375-2-21-2': 'Toward larger region sizes, we observe a combination of twisting and translation culminating in the required, and somewhat delicate, cancellation upon reaching the full system size.', '1508.04375-2-21-3': 'Presented with this relatively forgiving geometry, we performed the required integration via cubic-spline interpolation.', '1508.04375-2-21-4': 'Using a uniformly spaced lattice of size [MATH] points, we determine the desired entropy to a precision limited by statistical rather than systematic considerations.', '1508.04375-2-22-0': '## Comparison to exact diagonalization', '1508.04375-2-23-0': 'Shown in Fig. [REF] are results for a system of size [MATH] with a number of sites [MATH].', '1508.04375-2-23-1': 'For couplings [MATH] and [MATH] and region sizes [MATH], we find solid agreement with previous calculations in Refs. [CITATION], and as in the former, we observe convergence rather quickly with only [MATH] decorrelated samples as can be seen in Fig. [REF].', '1508.04375-2-23-2': 'Further, for large sample sizes [MATH], we observe (see Fig. [REF]) that the standard error in the entropy [MATH], computed from the envelope defined by the MC uncertainty in the source [MATH] at each value in [MATH]-space, scales asymptotically as [MATH] up to minute corrections.', '1508.04375-2-24-0': '# Summary and Conclusions', '1508.04375-2-25-0': 'We have presented a method to compute the entanglement entropy of interacting fermions which takes advantage of an approximate log-normality property.', '1508.04375-2-25-1': 'The resulting approach overcomes the signal-to-noise problem of naive methods, and is very close in principle to another method we proposed recently: both methods involve defining a new parameter, differentiating, and then integrating back after a MC calculation.', '1508.04375-2-25-2': 'As a test of our algorithm, we have presented results for the half-filled 1D Hubbard model with periodic boundary conditions.', '1508.04375-2-26-0': 'This material is based upon work supported by the National Science Foundation under Grants No. PHY1306520 (Nuclear Theory program) and No. PHY1452635 (Computational Physics program).'}

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[]

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[]

['1508.04375-1-13-2', '1508.04375-1-26-0', '1508.04375-2-13-2', '1508.04375-2-26-0', '1508.04375-3-22-2', '1508.04375-3-45-0']

{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '3': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1508.04375

{'1508.04375-3-0-0': 'We put forward a simpler and improved variation of a recently proposed method to overcome the signal-to-noise problem found in Monte Carlo calculations of the entanglement entropy of interacting fermions.', '1508.04375-3-0-1': 'The present method takes advantage of the approximate lognormal distributions that characterize the signal-to-noise properties of other approaches.', '1508.04375-3-0-2': 'In addition, we show that a simple rewriting of the formalism allows circumvention of the inversion of the restricted one-body density matrix in the calculation of the [MATH]-th Renyi entanglement entropy for [MATH].', '1508.04375-3-0-3': 'We test our technique by implementing it in combination with the hybrid Monte Carlo algorithm and calculating the [MATH] Renyi entropies of the 1D attractive Hubbard model.', '1508.04375-3-0-4': 'We use that data to extrapolate to the von Neumann ([MATH]) and [MATH] cases.', '1508.04375-3-1-0': '# Introduction', '1508.04375-3-2-0': 'Recently [CITATION], we proposed an algorithm to compute the Renyi entanglement entropy [MATH] of interacting fermions.', '1508.04375-3-2-1': 'Many algorithms have been proposed to this effect in the last few years [CITATION].', '1508.04375-3-2-2': 'Our proposal, based on the free-fermion decomposition approach of Ref. [CITATION], overcomes the signal-to-noise problem present in that approach and is compatible with the hybrid Monte Carlo (HMC) method [CITATION] widely used in the context of lattice quantum chromodynamics.', '1508.04375-3-2-3': 'The core idea of our method is that, by differentiating with respect to an auxiliary parameter [MATH], one may carry out a Monte Carlo (MC) calculation of [MATH] with a probability measure that includes entanglement properties explicitly.', '1508.04375-3-2-4': '[This was not the case in the approach of Ref. [CITATION], where the probability measure factored across auxiliary field replicas; we identified this as the cause of the signal-to-noise problem (see below)].', '1508.04375-3-2-5': 'Once the MC calculation is done, integration with respect to [MATH] returns the desired entanglement entropy relative to that of a noninteracting system (which is easily computed separately).', '1508.04375-3-3-0': 'In this work, we describe and implement a variation on that Monte Carlo algorithm which, while sharing the properties and core idea mentioned above, differs from it in two important ways; the new method, in fact, is different enough that we advocate its use over our original proposal.', '1508.04375-3-3-1': 'First, the new method takes advantage of the approximate lognormal shape of the underlying statistical distributions of the fermion determinants, which we already noted in Ref. [CITATION] and which we explain in detail below.', '1508.04375-3-3-2': 'Second, and more importantly, the present method is simpler than our original proposal: whereas in the latter the parameter [MATH] multiplied the coupling constant [MATH] (thus generating a rather involved set of terms upon differentiation of the fermion determinant), here [MATH] is coupled to the number of fermion species [MATH].', '1508.04375-3-3-3': 'As we show below, this choice not only simplifies the implementation, but also exposes the central role of the logarithm of the fermion determinant in our calculation of [MATH], and thus brings to bear the approximate lognormality property mentioned above.', '1508.04375-3-4-0': 'Below, we present the basic formalism, review the evidence for approximate lognormal distributions, and explain our method.', '1508.04375-3-4-1': 'Besides the points mentioned above, in our calculations we have found the present method to be more numerically stable than its predecessor.', '1508.04375-3-4-2': 'We explain this in detail in our Results section.', '1508.04375-3-5-0': 'In addition to the new method, we show that it is possible to rewrite part of the formalism in order to bypass the calculation of inverses of the restricted density matrix (see e.g. [CITATION]) in the determination of Renyi entropies of order [MATH].', '1508.04375-3-5-1': 'To test our method, we computed the [MATH] Renyi entropy of the 1D attractive Hubbard model using the previous as well as the new formalism, and checked that we obtained identical results.', '1508.04375-3-5-2': 'Going beyond the [MATH] case, we present results for the [MATH] Renyi entropies and find that higher Renyi entropies display lower statistical uncertainty in MC calculations.', '1508.04375-3-6-0': '# Basic formalism', '1508.04375-3-7-0': 'As in our previous work, we set the stage by briefly presenting the formalism of Ref. [CITATION].', '1508.04375-3-7-1': 'The [MATH]-th Renyi entropy [MATH] of a sub-system [MATH] of a given system is [EQUATION] where [MATH] is the reduced density matrix of sub-system [MATH].', '1508.04375-3-7-2': 'For a system with density matrix [MATH], the reduced density matrix is defined via a partial trace over the Hilbert space corresponding to the complement [MATH] of our sub-system: [EQUATION]', '1508.04375-3-7-3': 'An auxiliary-field path-integral form for [MATH], from which [MATH] can be computed using MC methods for a wide variety of systems, was presented in Ref. [CITATION], which we briefly review next.', '1508.04375-3-8-0': 'As is well known from conventional many-body formalism, the full density matrix [MATH] can be written as a path integral by means of a Hubbard-Stratonovich auxiliary-field transformation: [EQUATION] for some normalized probability measure [MATH] determined by the details of the underlying Hamiltonian (for more detail, see below and also Ref. [CITATION]).', '1508.04375-3-8-1': 'Here, [MATH] is the partition function, and [MATH] is the density matrix of noninteracting particles in the external auxiliary field [MATH].', '1508.04375-3-8-2': 'One of the main contributions of Ref. [CITATION] was to show that the above decomposition determines not only the full density matrix but also the restricted one.', '1508.04375-3-8-3': 'Indeed, Ref. [CITATION] shows that [EQUATION] where [MATH] is the same probability used in Eq. ([REF]), [EQUATION] and [EQUATION]', '1508.04375-3-8-4': "Here, [MATH] is the restricted Green's function of the noninteracting system in the external field [MATH] (see below), and [MATH], [MATH] are the fermion creation and annihilation operators.", '1508.04375-3-8-5': 'The sums in the exponent of Eq. ([REF]) go over those points in the system that belong to the subsystem [MATH].', '1508.04375-3-9-0': 'Using the above formalism for the case of [MATH]-component fermions, the entanglement entropy (c.f. Eq. [REF]) takes the form [EQUATION] where the field integration measure, given by [EQUATION] is over the [MATH] "replicas" [MATH] of the Hubbard-Stratonovich field (which result from taking the [MATH]-th power of the path integral representation of [MATH] shown above), and the normalization [EQUATION] was included in the measure.', '1508.04375-3-9-1': 'It is worth noting that, by separating a factor of [MATH] in the denominator of Eq. ([REF]), an explicit form can be identified in the numerator as in the replica trick [CITATION], which corresponds to a partition function for [MATH] copies of the system, "glued" together in the region [MATH].', '1508.04375-3-10-0': 'The naive probability measure, namely [EQUATION] factorizes across replicas, which makes it insensitive to entanglement.', '1508.04375-3-10-1': 'This factorization is the main reason why using [MATH] as a MC probability leads to signal-to-noise issues (see Ref. [CITATION]).', '1508.04375-3-10-2': 'In Eq. ([REF]), [MATH] encodes the dynamics of the [MATH]-th fermion component, including the kinetic energy and the form of the interaction after a Hubbard-Stratonovich transformation.', '1508.04375-3-10-3': 'That matrix also encodes the form of the trial state [MATH] in ground-state approaches (see e.g. Ref. [CITATION]), which we use here; we have taken [MATH] to be a Slater determinant.', '1508.04375-3-10-4': 'In finite-temperature approaches, [MATH] is obtained by evolving a complete set of single-particle states in imaginary time.', '1508.04375-3-11-0': 'The quantity that contains the pivotal contributions to entanglement is [EQUATION] which we refer to below as the "entanglement determinant," and where [EQUATION]', '1508.04375-3-11-1': 'The product [MATH] played the role of an observable in Ref. [CITATION], which is a natural interpretation given Eq. ([REF]).', '1508.04375-3-11-2': 'However, we will interpret this differently below.', '1508.04375-3-11-3': "Other than the field replicas, the new ingredient in the determination of [MATH] is the restricted Green's function [MATH].", '1508.04375-3-11-4': "This is the same as the noninteracting one-body density matrix [MATH] of the [MATH]-th fermion component in the background field [MATH], but the arguments [MATH] are restricted to the region [MATH] (see Ref. [CITATION] and also Ref. [CITATION], where expressions were originally derived for the reduced density matrix of noninteracting systems, based on reduced Green's functions).", '1508.04375-3-12-0': "# Avoiding inversion of the reduced Green's function for [MATH]", '1508.04375-3-13-0': 'As noted in Ref. [CITATION], for [MATH], no inversion of [MATH] is actually required in the calculation of the entanglement determinant [MATH], as the equations clearly simplify in that case.', '1508.04375-3-13-1': 'However, for higher [MATH] it is not obvious how to avoid such an inversion.', '1508.04375-3-13-2': 'Here, however, we show that this calculation can indeed be accomplished without inversion.', '1508.04375-3-13-3': 'We begin by noting that [EQUATION] where [MATH] is a block diagonal matrix (one block per replica [MATH]): [EQUATION] and [EQUATION] with [EQUATION]', '1508.04375-3-13-4': 'The equivalence of the determinants in Eq. ([REF]) can be shown in a straightforward fashion: the [MATH] factor is easily understood, as that matrix is block diagonal and therefore its determinant reproduces the first r.h.s. factor in the first line of Eq. ([REF]); the remaining factor relies on the identity [EQUATION] which is valid for arbitrary block matrices [MATH], is a standard result often used in many-body physics (especially when implementing a Hubbard-Stratonovich transformation), and can be shown using so-called elementary operations on rows and columns.', '1508.04375-3-14-0': 'Within the determinant of Eq. ([REF]), we may of course multiply [MATH] and [MATH]: [EQUATION] where [MATH] is a block diagonal matrix defined by [EQUATION] and [EQUATION]', '1508.04375-3-14-1': 'Equation ([REF]) shows our claim, as we may use [MATH] in our calculations instead of [MATH], and the former contains no inverses of [MATH].', '1508.04375-3-15-0': 'Summarizing, a class of approaches to calculating [MATH] for [MATH], based on the Hubbard-Stratonovich representation of [MATH] (also known as free-fermion decomposition), requires computing [MATH], which in turn requires inverting [MATH] per Eq. ([REF]).', '1508.04375-3-15-1': 'By arriving at Eq. ([REF]), and given that [EQUATION] [Eq. ([REF]) and beyond] we have shown that no inversions are actually required, as [MATH] contains no inverses.', '1508.04375-3-15-2': 'While this is a desirable feature from a numerical point of view, it should be mentioned that, from a computational-cost point of view, the price of not inverting [MATH] reappears in the fact that [MATH], though sparse, scales linearly with [MATH] in size.', '1508.04375-3-16-0': "For the remaining of this work, calculations carried out at [MATH] use the [MATH] approach, which is based on Eq. ([REF]) and the 'proposed method' described below.", '1508.04375-3-16-1': 'We reproduced those results by switching to the [MATH] approach, which uses Eq. ([REF]) (as well as the method described below), and then proceeded to higher [MATH] with the latter.', '1508.04375-3-17-0': '# A statistical observation: lognormal distribution of the entanglement determinant', '1508.04375-3-18-0': 'In Ref. [CITATION], we presented examples of the approximate log-normal distributions obeyed by [MATH] when sampled according to [MATH].', '1508.04375-3-18-1': 'One such example is reproduced here for reference in Fig. [REF].', '1508.04375-3-19-0': 'The fact that such distributions are approximately log-normal, at least visually, suggests that one may use the cumulant expansion to determine [MATH].', '1508.04375-3-19-1': 'Indeed, in general, [EQUATION] where [MATH] is the [MATH]-th cumulant of [MATH], and the first two nonzero cumulants are given by [EQUATION] and [EQUATION] for a functional [MATH], and where the expectation value [MATH] here taken with respect to the produce measure [MATH].', '1508.04375-3-19-2': 'If the distribution of [MATH] were truly gaussian, the above series would terminate after the first two terms, which would provide us with an efficient way to bypass signal-to-noise issues in the determination of [MATH] with stochastic methods [CITATION].', '1508.04375-3-19-3': 'Unfortunately, the distribution is not exactly gaussian.', '1508.04375-3-19-4': 'Moreover, the cumulants beyond [MATH] are often extremely sensitive to the details of the distribution (i.e. they can fluctuate wildly), they are hard to determine stochastically (the signal-to-noise problem re-emerges), and there is no easy way (that we know of) to obtain analytic insight into the large-[MATH] behavior of [MATH].', '1508.04375-3-19-5': 'However, this approximate log-normality does provide a path forward, as it indicates that we may still evaluate [MATH] with good precision with MC methods.', '1508.04375-3-19-6': 'As we will see in the next sections, this is enough to determine [MATH] if we are willing to pay the price of a one-dimensional integration on a compact domain.', '1508.04375-3-20-0': 'Although (approximate) lognormality in the entanglement determinant seems very difficult to prove analytically in the present case, evidence of its appearance has been found in systems as different as ultracold atoms and relativistic gauge theories [CITATION].', '1508.04375-3-20-1': 'The underlying reason for this distribution appears to be connected to a similarity between the motion of electrons in disordered media and lattice fermions in the external auxiliary (gauge) field in MC calculations.', '1508.04375-3-21-0': '# Proposed method', '1508.04375-3-22-0': 'Starting from the right-hand side of Eq. ([REF]), we introduce an auxiliary parameter [MATH] and define a function [MATH] via [EQUATION]', '1508.04375-3-22-1': 'At [MATH], [EQUATION] while for [MATH], [MATH] yields the entanglement entropy: [EQUATION]', '1508.04375-3-22-2': 'Using Eq. ([REF]), [EQUATION] where [EQUATION]', '1508.04375-3-22-3': 'In the presence of an even number of flavors [MATH] and attractive interactions, [MATH] and [MATH] are real and non-negative for all [MATH], such that there is no sign problem and [MATH] above is a well-defined, normalized probability measure.', '1508.04375-3-23-0': 'As in our previously proposed method, we can then calculate [MATH] by taking the [MATH] point as a reference and computing [MATH] using [EQUATION] where [EQUATION]', '1508.04375-3-23-1': 'We thus obtain an integral form of the interacting Renyi entropy that can be computed using any MC method (see e.g. [CITATION]), in particular HMC [CITATION].', '1508.04375-3-24-0': 'As in our previous work, we note that the above expectation values are determined with respect to the probability measure [MATH], which communicates correlations responsible for entanglement.', '1508.04375-3-24-1': 'In contrast to the canonical MC probability [MATH], which corresponds to statistically independent copies of the Hubbard-Stratonovich field, this admittedly more complicated distribution does not exhibit the factorization to blame for the signal-to-noise problems present in the approach as originally formulated.', '1508.04375-3-25-0': 'Using Eq. ([REF]) requires Monte Carlo methods to evaluate [MATH] as a function of [MATH], followed by integration over [MATH].', '1508.04375-3-25-1': 'As in our previous method, we find here that [MATH] is a smooth function of [MATH], which is essentially linear in the present case.', '1508.04375-3-25-2': 'It is therefore sufficient to perform the numerical integration using a uniform grid.', '1508.04375-3-25-3': 'The stochastic evaluation of [MATH], for fixed subregion [MATH], can be expected to feature roughly symmetric fluctuations about the mean.', '1508.04375-3-25-4': 'As a consequence, the statistical effects on the entropy are reduced after integrating over [MATH].', '1508.04375-3-26-0': "Finally, we note an interesting application of Jensen's inequality at [MATH].", '1508.04375-3-26-1': 'At that point [EQUATION] which must be satisfied by our calculations.', '1508.04375-3-26-2': 'Our Monte Carlo results at [MATH] indeed satisfy this bound.', '1508.04375-3-27-0': '# Results', '1508.04375-3-28-0': '## Second Renyi entropy', '1508.04375-3-29-0': 'As a first test of our algorithm and in efforts to make contact with previous work [CITATION], we begin by showing results for the second Renyi entropy [MATH] for the one-dimensional Hubbard chain with periodic boundary conditions at half filling, whose Hamiltonian is [EQUATION] where the first sum includes [MATH] and pairs of adjacent sites.', '1508.04375-3-29-1': 'We implemented a symmetric Trotter-Suzuki decomposition of the Boltzmann weight, with an imaginary-time discretization of [MATH] (in lattice units).', '1508.04375-3-29-2': 'As mentioned earlier, the many-body factor in the Trotter-Suzuki approximation was treated by introducing a replica auxiliary field [MATH] for each power of the reduced density matrix.', '1508.04375-3-29-3': 'As in our previous work, we implemented a Hubbard-Stratonovich transformation of a compact continuous form [CITATION].', '1508.04375-3-30-0': 'We present plots for [MATH] with [MATH] in Fig. [REF].', '1508.04375-3-30-1': 'In contrast to the results obtained in Ref. [CITATION] and as mentioned above, the resulting expectation demonstrates surprisingly little curvature as the region size [MATH] is varied and is stunningly linear as a function of the auxiliary parameter [MATH].', '1508.04375-3-30-2': 'Even after twice doubling the strength of the interaction, the curvature of constant-subsystem-size slices is increased only marginally.', '1508.04375-3-30-3': 'We note that if one assumes such benign curvature is a somewhat universal feature, at least for weakly-coupled systems, our method provides a means by which to rapidly estimate the entanglement entropy for a large portion of parameter space at the very least yielding a qualitative picture of its behavior as a function of the physically relevant input parameters.', '1508.04375-3-31-0': 'We observe that this surface displays almost no torsion, its dominant features being those present in the noninteracting case i.e. an alternating shell-like structure.', '1508.04375-3-31-1': 'Toward larger region sizes, we observe a combination of twisting and translation culminating in the required, and somewhat delicate, cancellation upon reaching the full system size.', '1508.04375-3-31-2': 'Presented with this relatively forgiving geometry, we performed the required integration via cubic-spline interpolation.', '1508.04375-3-31-3': 'Using a uniformly spaced lattice of size [MATH] points, we determine the desired entropy to a precision limited by statistical rather than systematic considerations.', '1508.04375-3-32-0': '## Comparison to exact diagonalization', '1508.04375-3-33-0': 'Shown in Fig. [REF] are results for a system of size [MATH] with a number of sites [MATH].', '1508.04375-3-33-1': 'For couplings [MATH] and [MATH] and region sizes [MATH], we find solid agreement with previous calculations in Refs. [CITATION], and as in the former, we observe convergence rather quickly with only [MATH] decorrelated samples as can be seen in Fig. [REF].', '1508.04375-3-33-2': 'Further, for large sample sizes [MATH], we observe that the standard error in the entropy [MATH], computed from the envelope defined by the MC uncertainty in the source [MATH] for at each value in [MATH]-space, scales asymptotically as [MATH] up to minute corrections.', '1508.04375-3-34-0': '## Results for [MATH]', '1508.04375-3-35-0': 'In this section, we extend the results presented above to [MATH].', '1508.04375-3-35-1': 'In order to highlight the differences between [MATH] and [MATH], we show in Fig. [REF] the Renyi entropies [MATH] for [MATH] (top to bottom) of the 1D attractive Hubbard model, as obtained with our method and the reformulation of the fermion determinant shown in Eq. ([REF]).', '1508.04375-3-36-0': 'As evident from the figure, increasing [MATH] leads to lower values of [MATH] at fixed subsystem size [MATH] consistent with knowledge that the Renyi entropy is a nonincreasing function of its order.', '1508.04375-3-36-1': 'However, increasing [MATH] also amplifies the fluctuations as a function of [MATH].', '1508.04375-3-36-2': 'Interestingly, the approach of our system to the large-[MATH] regime is quite rapid, and after only the first few orders, the difference between consecutive entropies is only marginal, most obviously so at weak coupling.', '1508.04375-3-36-3': 'We also observe that, as [MATH] is increased, the statistical fluctuations that define the error bars appear to be progressively more suppressed, which is particularly evident for the strongest coupling we studied, namely [MATH].', '1508.04375-3-37-0': 'At the level of the auxiliary function [MATH], we again see very predictable changes in the geometry of this surface as a function both arguments as shown in Fig. [REF].', '1508.04375-3-37-1': "With fixed coupling and particle content, increasing the Renyi order results in a tilting effect reminiscent of that seen previously with increasing coupling, but rather than being localized away from vanishing subsystem size, the change is much more global, affecting all subsystems in a qualitatively similar fashion and leaving each surface's characteristic quasi-linearity in [MATH] intact.", '1508.04375-3-37-2': "Although the shell-like structure present in this function's [MATH] dependence is amplified, this increased fluctuation affects the quality of the results negligibly at most, as again, the geometry remains amenable to fairly naive quadratures.", '1508.04375-3-38-0': 'With the data presented above, we would be remiss if we did not attempt an extrapolation not only to the limit of infinite Renyi order [MATH], but also to the von Neumann entropy, despite knowledge of the formidable challenges presented by these extrapolations, particularly in the case of the latter.', '1508.04375-3-38-1': 'The former limit provides a lower bound on all finite-order entropies, whereas the latter is of interest to a variety of disciplines and has proven difficult to study.', '1508.04375-3-38-2': 'At fixed coupling and with the knowledge that the Renyi entropy is nonincreasing in the order, we found that our results at each fixed region size and at every studied coupling were well-characterized by exponential decays.', '1508.04375-3-39-0': 'Interestingly, the relative speed of this decay oscillates as a function of the region size as can be seen in Fig. [REF].', '1508.04375-3-39-1': 'Regions corresponding to an even number of lattice sites demonstrate a much more sudden initial decay than do those regions comprised of an odd number of sites.', '1508.04375-3-39-2': 'This peculiar oscillation results in an inverted shell structure for the extrapolation to [MATH], in contrast to the case where [MATH] in which this feature is preserved.', '1508.04375-3-39-3': 'A representative example of this procedure is shown in Fig. [REF].', '1508.04375-3-40-0': '# Summary and Conclusions', '1508.04375-3-41-0': 'We have presented a method to compute the entanglement entropy of interacting fermions which takes advantage of an approximate log-normality property of the distribution of fermion determinants.', '1508.04375-3-41-1': 'The resulting approach overcomes the signal-to-noise problem of naive methods, and is very close in its core idea to another method we proposed recently: both methods involve defining an auxiliary parameter [MATH], differentiating, and then integrating to recover [MATH] after a MC calculation.', '1508.04375-3-41-2': 'The order of the steps is important, as the differentiation with respect to [MATH] induces the appearance of entanglement-sensitive contributions in the MC probability measure.', '1508.04375-3-41-3': 'Beyond those similarities, the present method has the distinct advantages of being simultaneously simpler to formulate (algebraically as well as computationally) and of explicitly using the approximate log-normality property.', '1508.04375-3-41-4': 'Moreover, we have found that the [MATH] integration step displays clearly more stable numerical behavior in the present approach than in its predecessor: it is approximately linear in the present case and markedly not so in the original incarnation.', '1508.04375-3-41-5': 'We therefore strongly advocate using the present algorithm over the former.', '1508.04375-3-42-0': "In addition to presenting an improved method, we have put forward a straightforward algebraic reformulation of the equations which, while exactly equivalent to the original formalism, avoids the numerical burden of computing inverses of restricted Green's functions in the calculation of [MATH]-th order Renyi entropies for [MATH].", '1508.04375-3-42-1': 'This issue had been pointed out by us and others (see e.g. Ref. [CITATION]) as an inconvenience, as it is perfectly possible for those matrices to be singular.', '1508.04375-3-43-0': 'As a test of our algorithm, we have presented results for the Renyi entropy [MATH] of the half-filled 1D Hubbard model with periodic boundary conditions.', '1508.04375-3-43-1': 'The present and old formalisms were used for calculations at [MATH], which matched exactly.', '1508.04375-3-43-2': 'The rewritten form based on Eq. ([REF]) was then used to extend our computations to [MATH], allowing us to attempt extrapolations in the Renyi order in both directions.', '1508.04375-3-44-0': 'Our results show that, with increasing Renyi order [MATH], the value of [MATH] decreases for all [MATH], and the fluctuations as a function of [MATH] become more pronounced.', '1508.04375-3-44-1': 'Remarkably, the statistical MC fluctuations decrease as [MATH] is increased.', '1508.04375-3-44-2': 'Since the problem we set out to solve was in fact statistical in nature, our observations indicate that calculations for large systems and in higher dimensions will benefit from pursuing orders [MATH].', '1508.04375-3-45-0': 'This material is based upon work supported by the National Science Foundation under Grants No. PHY1306520 (Nuclear Theory program) and No. PHY1452635 (Computational Physics program).'}

hep-ph-0410170

{'hep-ph-0410170-1-0-0': 'REDUCING THE ERROR ON [MATH] in [MATH] Michael Gronau Physics Department, Technion - Israel Institute of Technology 32000 Haifa, Israel Enrico Lunghi and Daniel Wyler Institut fur Theoretische Physik, Universitat Zurich, Winterthurerstrasse 190 CH-8057, Zurich, Switzerland', 'hep-ph-0410170-1-1-0': 'Theoretical errors in the extraction of [MATH] from [MATH] decays are usually given in terms of upper bounds on [MATH] obtained from isospin or from SU(3) relations, where [MATH] is measured through CP asymmetries.', 'hep-ph-0410170-1-1-1': 'We show that mild assumptions about magnitudes and strong phases of penguin and tree amplitudes ([MATH] and [MATH]) in [MATH] and [MATH], imply [MATH], thus reducing by a factor two the error in [MATH].', 'hep-ph-0410170-1-1-2': 'Similarly, the assumptions [MATH] in [MATH] lead to a cancellation between two terms in [MATH].', 'hep-ph-0410170-1-1-3': 'Current data support these conditions.', 'hep-ph-0410170-1-2-0': 'PACS codes: 12.15.', 'hep-ph-0410170-1-2-1': 'Hh, 12.15.', 'hep-ph-0410170-1-2-2': 'Ji, 13.25.', 'hep-ph-0410170-1-2-3': 'Hw, 14.40.', 'hep-ph-0410170-1-2-4': 'Nd', 'hep-ph-0410170-1-3-0': 'Direct extraction of the Cabibbo-Kobayashi-Maskawa (CKM) phase [MATH] from the time-dependent CP asymmetry in [MATH] is obstructed by the penguin amplitude [CITATION].', 'hep-ph-0410170-1-3-1': 'This obstacle may be overcome using isospin symmetry [CITATION], which incorporates electroweak penguin contributions at a percent level [CITATION] but does not include small isospin breaking effects.', 'hep-ph-0410170-1-3-2': 'This method requires separate rate measurements of [MATH].', 'hep-ph-0410170-1-3-3': 'As long as low statistics does not permit these separate measurements, one can use combined [MATH] and [MATH] decay rates into the three [MATH] channels to obtain upper bounds on [MATH] [CITATION].', 'hep-ph-0410170-1-3-4': 'The angle [MATH], which equals [MATH] in the limit of a vanishing penguin amplitude, is given up to a discrete ambiguity by the time-dependent CP asymmetry in [MATH].', 'hep-ph-0410170-1-3-5': 'The upper bound on [MATH] (improved by information from an upper limit on the asymmetry in [MATH]) remains an intrinsic theoretical uncertainty in [MATH].', 'hep-ph-0410170-1-3-6': 'Note that [MATH] may be either positive or negative.', 'hep-ph-0410170-1-4-0': 'Similar considerations can also be applied to [MATH] and lead to analogous bounds on [MATH].', 'hep-ph-0410170-1-4-1': 'Because each of the two [MATH] mesons carries a unit spin, one must distinguish between decays to even-CP and odd-CP final states corresponding to definite polarizations.', 'hep-ph-0410170-1-5-0': 'On the other hand, a complete isospin analysis of the processes [MATH] is complicated by the existence of five different [MATH] charge states in [MATH] and [MATH] decays [CITATION].', 'hep-ph-0410170-1-5-1': 'Thus, it was proposed to measure [MATH] through the time-dependent Dalitz distribution in [MATH] [CITATION], which provides information about interference of amplitudes for [MATH] and [MATH].', 'hep-ph-0410170-1-5-2': 'Alternatively, one may measure an angle [MATH] in quasi two- body decays [MATH], and use broken flavor SU(3) to obtain an upper bound on [MATH] [CITATION].', 'hep-ph-0410170-1-6-0': 'The experimental progress during the past year has been impressive in this class of measurements.', 'hep-ph-0410170-1-6-1': 'The situation in early July [CITATION] was updated in late August [CITATION], following new measurements reported at the International Conference on High Energy Physics in Beijing [CITATION].', 'hep-ph-0410170-1-6-2': 'The overall range of [MATH] determined in [MATH] [CITATION], [MATH], overlaps with and begins to be narrower than the following bounds obtained indirectly in an independent global CKM fit [CITATION], [EQUATION] where a 95[MATH] confidence level (CL) is implied.', 'hep-ph-0410170-1-6-3': 'The error in [MATH] from a time-dependent Dalitz plot analysis of [MATH] [CITATION], [MATH], is statistics-dominated.', 'hep-ph-0410170-1-6-4': 'On the other hand, the theoretical errors in determining [MATH], using isospin in [MATH] and applying broken SU(3) to [MATH], are at least as large as the corresponding statistical errors.', 'hep-ph-0410170-1-6-5': 'The 90[MATH] CL upper bounds on [MATH] in these three cases are [CITATION] [MATH] and [MATH], respectively.', 'hep-ph-0410170-1-6-6': 'It would be very useful to reduce these intrinsic theoretical uncertainties using present data.', 'hep-ph-0410170-1-7-0': 'In this letter we point out that the above theoretical errors in [MATH] may be reduced by about a factor two under very mild and reasonable assumptions.', 'hep-ph-0410170-1-7-1': 'We study the dependence of [MATH] on the ratio of penguin and tree amplitudes, [MATH], and on their relative strong phase, [MATH].', 'hep-ph-0410170-1-7-2': 'We show that the conditions [MATH] and [MATH] predict a positive sign for [MATH] in [MATH] and [MATH].', 'hep-ph-0410170-1-7-3': 'Two phases [MATH] lying in opposite hemispheres tend to suppress [MATH] in [MATH] due to a cancellation between two terms.', 'hep-ph-0410170-1-7-4': 'We propose to include, in future studies of [MATH], the explicit dependence of [MATH] on [MATH], [MATH], [MATH] and [MATH], together with the dependence of the CP asymmetries on these variables.', 'hep-ph-0410170-1-7-5': 'In the argumentation below we assume [MATH] as given in ([REF]).', 'hep-ph-0410170-1-8-0': 'To prove our point, we consider first [MATH].', 'hep-ph-0410170-1-8-1': 'We use the [MATH]-convention defined in [CITATION], in which the top-quark has been integrated out in the [MATH] penguin transition and unitarity of the CKM matrix has been used.', 'hep-ph-0410170-1-8-2': 'Absorbing a [MATH] term in [MATH], the decay amplitude may be written in the following general form, [EQUATION] where by convention [MATH].', 'hep-ph-0410170-1-8-3': 'The phase [MATH] is extracted up to a discrete ambiguity from the two asymmetries [MATH] and [MATH] in [MATH] [CITATION]: [EQUATION] where [EQUATION]', 'hep-ph-0410170-1-8-4': 'Using the definitions [EQUATION] one has [EQUATION]', 'hep-ph-0410170-1-8-5': 'Two very reasonable assumptions, [MATH] and [MATH], imply [MATH].', 'hep-ph-0410170-1-8-6': 'The second condition may be replaced by a stronger one, [MATH]; it is stronger since we are assuming [MATH].', 'hep-ph-0410170-1-8-7': 'The result [MATH] demonstrates the central point of this note.', 'hep-ph-0410170-1-9-0': 'Rewriting Eq. ([REF]), [EQUATION] we see that the above assumptions imply [MATH], or [MATH].', 'hep-ph-0410170-1-9-1': 'The prediction of the sign of [MATH] is based purely on definitions and on the assumptions on [MATH] and [MATH].', 'hep-ph-0410170-1-9-2': 'It implies a considerably narrower range for [MATH] than the bound [MATH] [CITATION], obtained from the [MATH] and [MATH] isospin triangles, with the inclusion of some information about the asymmetry in [MATH].', 'hep-ph-0410170-1-10-0': 'It now remains to justify our two assumptions, [MATH] and [MATH].', 'hep-ph-0410170-1-10-1': 'The ratio [MATH] has a very long history, starting in the late eighties when the penguin amplitude in [MATH] was estimated to be small but non-negligible [CITATION].', 'hep-ph-0410170-1-10-2': 'First measurements of [MATH] and [MATH] decay rates, performed several years later by the CLEO collaboration [CITATION], were analyzed within flavor SU(3) indicating that [MATH] [CITATION].', 'hep-ph-0410170-1-10-3': 'A recent global SU(3) fit to all [MATH] and [MATH] decays obtained an unexpected large value [CITATION] [MATH].', 'hep-ph-0410170-1-10-4': 'The large value of [MATH] is driven partly but not only [CITATION] by a large input value for [MATH] [CITATION].', 'hep-ph-0410170-1-10-5': 'Theoretical calculations based on QCD and a heavy quark expansion [CITATION] find somewhat smaller values for [MATH], all lying comfortably in the range [MATH].', 'hep-ph-0410170-1-10-6': 'All these calculations support strongly the assumption [MATH].', 'hep-ph-0410170-1-11-0': 'Very early theoretical QCD arguments favoring a small value of [MATH] were proposed in [CITATION].', 'hep-ph-0410170-1-11-1': 'Although different calculations of [MATH] [CITATION], based on QCD and a heavy quark expansion, do not always agree in detail, all these computed values of [MATH] are considerably smaller than [MATH].', 'hep-ph-0410170-1-11-2': 'Constraints on CKM parameters based on the assumption [MATH] and on a given range for [MATH] were studied in [CITATION].', 'hep-ph-0410170-1-11-3': 'Long distance [MATH] penguin contributions [CITATION], or equivalently final state rescattering [CITATION], may spoil the QCD calculations [CITATION].', 'hep-ph-0410170-1-11-4': 'These effects cannot be calculated in a model-independent way.', 'hep-ph-0410170-1-11-5': 'A global SU(3) fit to [MATH] and [MATH] decays, which effectively includes these rescattering effects while assuming SU(3) invariant strong phases, obtains [CITATION] [MATH].', 'hep-ph-0410170-1-11-6': 'Large SU(3) breaking effects in strong phases could possibly lead to values of [MATH] outside the range [MATH].', 'hep-ph-0410170-1-12-0': 'A study of the two hadronic parameters, [MATH] and [MATH], was performed in [CITATION], adding measurements of [MATH] and an upper bound on [MATH] available before this summer to all other CKM constraints [CITATION].', 'hep-ph-0410170-1-12-1': 'Values were found in an overall minimum [MATH] fit, [MATH] and [MATH], consistent with [MATH] but possibly violating somewhat the bound [MATH].', 'hep-ph-0410170-1-12-2': 'A recent update [CITATION], using newer measurements, also favors parameters in the range [MATH].', 'hep-ph-0410170-1-13-0': 'Similarly, we attempt a direct experimental proof of [MATH] and [MATH] using current data [CITATION]: [EQUATION]', 'hep-ph-0410170-1-13-1': 'The world averaged asymmetries are based on most recent measurements by the Belle collaboration [CITATION] ([MATH]) and by the BaBar collaboration [CITATION] ([MATH]).', 'hep-ph-0410170-1-13-2': 'Since these two measurements are not in good agreement with each other, an error rescaling factor of 2.39 [CITATION] (determining errors in parentheses) may be used to achieve a conservative confidence level.', 'hep-ph-0410170-1-13-3': 'Expressions for [MATH], and [MATH], given in Eqs. ([REF])([REF]) and ([REF]), and the values in Eq. ([REF]) can be used to constrain [MATH] and [MATH].', 'hep-ph-0410170-1-14-0': 'We performed a separate minimum-[MATH] fit for the parameters [MATH] and [MATH].', 'hep-ph-0410170-1-14-1': 'The [MATH] contains two contributions from [MATH] and [MATH] and the usual terms corresponding to the standard analysis of the unitarity triangle.', 'hep-ph-0410170-1-14-2': 'In the minimization we reject points which yield [MATH].', 'hep-ph-0410170-1-14-3': 'Fig. [REF]a shows the resulting 90[MATH] CL bounds obtained using the unscaled (dark area) and rescaled (dashed line) errors in Eq. ([REF]).', 'hep-ph-0410170-1-14-4': 'The bounds can be summarized by [EQUATION] where the numbers in parentheses correspond to the rescaled errors.', 'hep-ph-0410170-1-14-5': 'Note that the unscaled errors exclude [MATH] at 90[MATH] CL, favoring [MATH].', 'hep-ph-0410170-1-14-6': 'Although they do not exclude completely [MATH], we find that the lowest allowed value of [MATH] is [MATH], quite close to zero.', 'hep-ph-0410170-1-14-7': 'The rescaled errors imply less restrictive bounds.', 'hep-ph-0410170-1-14-8': 'Exclusion of [MATH] would imply [MATH] for the central values given in Eq. ([REF]).', 'hep-ph-0410170-1-14-9': 'In any case, using the dependence of [MATH], [MATH] and [MATH] on [MATH] and [MATH] is expected to reduce the theoretical uncertainty in [MATH] below the upper limit on [MATH].', 'hep-ph-0410170-1-15-0': 'One may study directly the dependence of the sign of [MATH] on the values of [MATH] and [MATH].', 'hep-ph-0410170-1-15-1': 'Assuming that the discrepancy between the BaBar and Belle results disappears and taking errors [MATH], we calculate central values in the ([MATH]) plane for which [MATH] at 90[MATH] CL.', 'hep-ph-0410170-1-15-2': 'An important input is our prior knowledge of [MATH].', 'hep-ph-0410170-1-15-3': 'For this we use three different analyses of the unitarity triangle described in Refs. [CITATION], [CITATION] and [CITATION].', 'hep-ph-0410170-1-15-4': 'Corresponding regions in the ([MATH]) plane, outside of which [MATH], are described in Fig. [REF] by the shaded area and by the dashed and dotted lines, respectively.', 'hep-ph-0410170-1-15-5': 'The heavy dot, denoting the current central values of [MATH] and [MATH], is seen to lie close to the bounds but still inside the most conservative region [CITATION].', 'hep-ph-0410170-1-16-0': 'Let us now turn to discuss briefly [MATH] in [MATH].', 'hep-ph-0410170-1-16-1': 'Using the approximately pure longitudinal polarization of the final states [CITATION], the isospin analysis of the decays [MATH] is greatly simplified, becoming almost identical to that of [MATH] (neglecting the small [MATH]-width effect [CITATION]).', 'hep-ph-0410170-1-16-2': 'The small branching ratio for [MATH] leads to the rather tight 90[MATH] CL upper bound [CITATION], [MATH].', 'hep-ph-0410170-1-16-3': 'The phase difference [MATH] may be written as in Eq. ([REF]) and becomes positive under similar conditions for [MATH] and [MATH].', 'hep-ph-0410170-1-16-4': 'That is, our generic assumptions, [MATH] and [MATH], imply [MATH].', 'hep-ph-0410170-1-17-0': 'Although these assumptions about [MATH] and [MATH] seem reasonable, one may attempt to verify them experimentally, as demonstrated above for [MATH].', 'hep-ph-0410170-1-17-1': 'Current measurements of [MATH] are consistent with a zero CP asymmetry [CITATION], [EQUATION]', 'hep-ph-0410170-1-17-2': 'The central values of [MATH] and [MATH] imply [MATH].', 'hep-ph-0410170-1-17-3': 'An analysis, based on expressions for [MATH] and [MATH] analogous to Eqs. ([REF]) ([REF]) and ([REF]), in terms [MATH], and the weak phase [MATH] or [MATH], proceeds as in [MATH].', 'hep-ph-0410170-1-17-4': 'The rather low upper bound, [MATH], implies a small value for [MATH], as can be seen in Eq. ([REF]) where [MATH] vanishes in the limit [MATH].', 'hep-ph-0410170-1-17-5': 'A small value of [MATH] means a small penguin amplitude and a low sensitivity to the value of [MATH].', 'hep-ph-0410170-1-18-0': 'We have verified the above statement by performing a [MATH] fit based on Eq. ([REF]).', 'hep-ph-0410170-1-18-1': 'In Fig. [REF]b, we show the 90[MATH] CL for the parameters [MATH] and [MATH].', 'hep-ph-0410170-1-18-2': 'We see that while [MATH] is excluded ([MATH]), values of [MATH] are permitted in the entire range [MATH].', 'hep-ph-0410170-1-18-3': 'This situation is unlikely to change with a reduction of errors in [MATH] and [MATH] if [MATH] is much smaller than one.', 'hep-ph-0410170-1-19-0': 'The situation in [MATH] is somewhat more complicated than the one in [MATH] and [MATH] because the final states [MATH] are not CP-eigenstates.', 'hep-ph-0410170-1-19-1': 'Time-dependent decay rates for initially [MATH] decaying into [MATH] are given by [CITATION]: [EQUATION]', 'hep-ph-0410170-1-19-2': 'For initially [MATH] decays the [MATH] and [MATH] terms have opposite signs.', 'hep-ph-0410170-1-19-3': 'One defines a measurable phase [MATH] [CITATION], which equals [MATH] in the limit of vanishing penguin amplitudes, [EQUATION]', 'hep-ph-0410170-1-19-4': 'The phase [MATH] can be expressed in terms of parameters defining decay amplitudes.', 'hep-ph-0410170-1-19-5': 'One denotes two distinct amplitudes for [MATH] and [MATH] by the charge of the [MATH] meson, [EQUATION] where [MATH] are two ratios of penguin and tree amplitudes, and [MATH] are the corresponding relative strong phases.', 'hep-ph-0410170-1-19-6': 'Continuing the analogy with [MATH] by defining [EQUATION] one has [EQUATION]', 'hep-ph-0410170-1-19-7': 'The two angle differences [MATH] have expressions similar to Eq. ([REF]): [EQUATION]', 'hep-ph-0410170-1-19-8': 'In the limit [MATH] one obviously obtains [MATH].', 'hep-ph-0410170-1-20-0': 'One may use flavor SU(3) to obtain two upper bounds at 90[MATH] CL [CITATION]: [MATH].', 'hep-ph-0410170-1-20-1': 'The algebraic average of these bounds provides an upper limit on [MATH], [EQUATION]', 'hep-ph-0410170-1-20-2': 'This bound, which may be modified by SU(3) breaking, does not assume knowledge of the signs of [MATH].', 'hep-ph-0410170-1-20-3': 'It becomes stronger when [MATH] and [MATH] have opposite signs, in which case they cancel each other in [MATH].', 'hep-ph-0410170-1-20-4': 'This possibility will be discussed now.', 'hep-ph-0410170-1-21-0': 'The sings of [MATH] depend on values of [MATH] and [MATH].', 'hep-ph-0410170-1-21-1': 'As we argued in the case of [MATH], both [MATH] and [MATH] would be positive if [MATH] and [MATH].', 'hep-ph-0410170-1-21-2': 'Arguments for [MATH] are very strong.', 'hep-ph-0410170-1-21-3': 'Studies based on flavor SU(3) [CITATION] and a calculation using QCD factorization [CITATION] obtain rather small values [MATH], lying very comfortably in the range [MATH].', 'hep-ph-0410170-1-21-4': 'Both studies obtain values for [MATH] and [MATH] lying in opposite hemispheres.', 'hep-ph-0410170-1-21-5': 'The output of a global SU(3) fit is [CITATION] [MATH] and [MATH].', 'hep-ph-0410170-1-21-6': 'Similar values are obtained in [CITATION], where it is being argued that these values do not differ much from the naive factorization predictions: [MATH] and [MATH] [CITATION].', 'hep-ph-0410170-1-21-7': 'Using the above values for [MATH] and [MATH] we see that [MATH] while [MATH].', 'hep-ph-0410170-1-21-8': 'This implies that [MATH] but [MATH], which means that the upper bound ([REF]) is replaced by a stronger bound, [EQUATION]', 'hep-ph-0410170-1-21-9': 'That is, given [MATH], and assuming that the phases [MATH] and [MATH] lie in opposite hemispheres, improves the upper bound ([REF]) by about a factor two.', 'hep-ph-0410170-1-21-10': 'The actual upper and lower bounds in ([REF]) may be larger by 30[MATH] because of SU(3) breaking corrections some of which have not been taken into account.', 'hep-ph-0410170-1-22-0': 'A direct verification of our assumptions about [MATH] in [MATH] relies on interference between tree and penguin amplitudes in these processes.', 'hep-ph-0410170-1-22-1': 'A possible evidence for such interference is the measured CP asymmetry in [MATH] [CITATION], [EQUATION]', 'hep-ph-0410170-1-22-2': 'This asymmetry provides one equation for [MATH] and [MATH] [CITATION] favoring negative values of [MATH].', 'hep-ph-0410170-1-22-3': 'Similarly, the CP asymmetry in [MATH] [CITATION] [EQUATION] provides an equation for [MATH] and [MATH].', 'hep-ph-0410170-1-22-4': 'Two other observables depending on these four hadronic parameters and on [MATH] are [MATH] and [MATH], related to [MATH] and [MATH] by a simple transformation [CITATION].', 'hep-ph-0410170-1-22-5': 'They obtain values [CITATION] [EQUATION]', 'hep-ph-0410170-1-22-6': 'The error in parentheses includes a rescaling factor of 1.7.', 'hep-ph-0410170-1-22-7': 'The dependence of [MATH] and [MATH] on a small relative phase [MATH] [CITATION] between the two tree amplitudes [MATH] and [MATH] will be neglected.', 'hep-ph-0410170-1-22-8': 'A fifth constraint on [MATH] and [MATH] is provided by the upper bound ([REF]), where [MATH] is given in Eq. ([REF]).', 'hep-ph-0410170-1-22-9': 'This completes a system of five equations for five parameters.', 'hep-ph-0410170-1-22-10': 'An important question is whether these constraints imply that [MATH] and [MATH] lie in opposite hemispheres, thereby leading to Eq. ([REF]).', 'hep-ph-0410170-1-22-11': 'In any case, the inclusion of these five relations is expected to reduce the uncertainty in [MATH] below the upper bound ([REF]).', 'hep-ph-0410170-1-23-0': 'In conclusion, we have shown that the generic assumptions, [MATH] and [MATH], predict [MATH] in [MATH] and [MATH], thereby reducing the theoretical uncertainty in [MATH] by a factor two.', 'hep-ph-0410170-1-23-1': 'In the decay processes [MATH], where two pairs of tree and penguin amplitudes occur, the assumption that the two relative phases between tree and penguin amplitudes lie in opposite hemispheres results in a suppression of about a factor two of the bound on [MATH] due to cancellation between two terms.', 'hep-ph-0410170-1-23-2': 'Measured CP asymmetries and [MATH] were studied in terms of Penguin-to-Tree ratios and their relative strong phases.', 'hep-ph-0410170-1-23-3': 'We presented evidence (although not yet conclusive) that the various hadronic parameters do lie in ranges that allow a reduction of the error on [MATH].', 'hep-ph-0410170-1-24-0': 'We wish to thank Ahmed Ali, Andreas Hoecker and Dan Pirjol for helpful discussions.', 'hep-ph-0410170-1-24-1': 'M. G. is grateful to the Institute for Theoretical Physics at the University of Zurich for its kind hospitality.', 'hep-ph-0410170-1-24-2': 'This work is supported in part by the Schweizerischer Nationalfonds and by the Israel Science Foundation founded by the Israel Academy of Sciences and Humanities, Grant No. 1052/04.', 'hep-ph-0410170-1-25-0': 'def 89CP Violation, edited by C. Jarlskog (World Scientific,Singapore, 1989)'}

{'hep-ph-0410170-2-0-0': 'REDUCING THE ERROR ON [MATH] in [MATH] Michael Gronau Physics Department, Technion - Israel Institute of Technology 32000 Haifa, Israel Enrico Lunghi and Daniel Wyler Institut fur Theoretische Physik, Universitat Zurich, Winterthurerstrasse 190 CH-8057, Zurich, Switzerland', 'hep-ph-0410170-2-1-0': 'Theoretical errors in the extraction of [MATH] from [MATH] decays are usually given in terms of upper bounds on [MATH] obtained from isospin or from SU(3) relations, where [MATH] is measured through CP asymmetries.', 'hep-ph-0410170-2-1-1': 'We show that mild assumptions about magnitudes and strong phases of penguin and tree amplitudes ([MATH] and [MATH]) in [MATH] and [MATH], imply [MATH], thus reducing by a factor two the error in [MATH].', 'hep-ph-0410170-2-1-2': 'Similarly, the assumptions [MATH] in [MATH] lead to a cancellation between two terms in [MATH].', 'hep-ph-0410170-2-1-3': 'Current data support these conditions.', 'hep-ph-0410170-2-2-0': 'PACS codes: 12.15.', 'hep-ph-0410170-2-2-1': 'Hh, 12.15.', 'hep-ph-0410170-2-2-2': 'Ji, 13.25.', 'hep-ph-0410170-2-2-3': 'Hw, 14.40.', 'hep-ph-0410170-2-2-4': 'Nd', 'hep-ph-0410170-2-3-0': 'Direct extraction of the Cabibbo-Kobayashi-Maskawa (CKM) phase [MATH] from the time-dependent CP asymmetry in [MATH] is obstructed by the penguin amplitude [CITATION].', 'hep-ph-0410170-2-3-1': 'This obstacle may be overcome using isospin symmetry [CITATION], which incorporates electroweak penguin contributions at a percent level [CITATION] but does not include small isospin breaking effects.', 'hep-ph-0410170-2-3-2': 'This method requires separate rate measurements of [MATH].', 'hep-ph-0410170-2-3-3': 'As long as low statistics does not permit these separate measurements, one can use combined [MATH] and [MATH] decay rates into the three [MATH] channels to obtain upper bounds on [MATH] [CITATION].', 'hep-ph-0410170-2-3-4': 'The angle [MATH], which equals [MATH] in the limit of a vanishing penguin amplitude, is given up to a discrete ambiguity by the time-dependent CP asymmetry in [MATH].', 'hep-ph-0410170-2-3-5': 'The upper bound on [MATH] (improved by information from an upper limit on the asymmetry in [MATH]) remains an intrinsic theoretical uncertainty in [MATH].', 'hep-ph-0410170-2-3-6': 'Note that [MATH] may be either positive or negative.', 'hep-ph-0410170-2-4-0': 'Similar considerations can also be applied to [MATH] and lead to analogous bounds on [MATH].', 'hep-ph-0410170-2-4-1': 'Because each of the two [MATH] mesons carries a unit spin, one must distinguish between decays to even-CP and odd-CP final states corresponding to definite polarizations.', 'hep-ph-0410170-2-5-0': 'On the other hand, a complete isospin analysis of the processes [MATH] is complicated by the existence of five different [MATH] charge states in [MATH] and [MATH] decays [CITATION].', 'hep-ph-0410170-2-5-1': 'Thus, it was proposed to measure [MATH] through the time-dependent Dalitz distribution in [MATH] [CITATION], which provides information about interference of amplitudes for [MATH] and [MATH].', 'hep-ph-0410170-2-5-2': 'Alternatively, one may measure an angle [MATH] in quasi two- body decays [MATH], and use broken flavor SU(3) to obtain an upper bound on [MATH] [CITATION].', 'hep-ph-0410170-2-6-0': 'The experimental progress during the past year has been impressive in this class of measurements.', 'hep-ph-0410170-2-6-1': 'The situation in early July [CITATION] was updated in late August [CITATION], following new measurements reported at the International Conference on High Energy Physics in Beijing [CITATION].', 'hep-ph-0410170-2-6-2': 'The overall range of [MATH] determined in [MATH] [CITATION], [MATH], overlaps with and begins to be narrower than the following bounds obtained indirectly in an independent global CKM fit [CITATION], [EQUATION] where a 95[MATH] confidence level (CL) is implied.', 'hep-ph-0410170-2-6-3': 'The error in [MATH] from a time-dependent Dalitz plot analysis of [MATH] [CITATION], [MATH], is statistics-dominated.', 'hep-ph-0410170-2-6-4': 'On the other hand, the theoretical errors in determining [MATH], using isospin in [MATH] and applying broken SU(3) to [MATH], are at least as large as the corresponding statistical errors.', 'hep-ph-0410170-2-6-5': 'The 90[MATH] CL upper bounds on [MATH] in these three cases are [CITATION] [MATH] and [MATH], respectively.', 'hep-ph-0410170-2-6-6': 'It would be very useful to reduce these intrinsic theoretical uncertainties using present data.', 'hep-ph-0410170-2-7-0': 'In this letter we point out that the above theoretical errors in [MATH] may be reduced by about a factor two under very mild and reasonable assumptions.', 'hep-ph-0410170-2-7-1': 'We study the dependence of [MATH] on the ratio of penguin and tree amplitudes, [MATH], and on their relative strong phase, [MATH].', 'hep-ph-0410170-2-7-2': 'We show that the conditions [MATH] and [MATH] predict a positive sign for [MATH] in [MATH] and [MATH].', 'hep-ph-0410170-2-7-3': 'Two phases [MATH] lying in opposite hemispheres tend to suppress [MATH] in [MATH] due to a cancellation between two terms.', 'hep-ph-0410170-2-7-4': 'We propose to include, in future studies of [MATH], the explicit dependence of [MATH] on [MATH], [MATH], [MATH] and [MATH], together with the dependence of the CP asymmetries on these variables.', 'hep-ph-0410170-2-7-5': 'In the argumentation below we assume [MATH] as given in ([REF]).', 'hep-ph-0410170-2-8-0': 'To prove our point, we consider first [MATH].', 'hep-ph-0410170-2-8-1': 'We use the [MATH]-convention defined in [CITATION], in which the top-quark has been integrated out in the [MATH] penguin transition and unitarity of the CKM matrix has been used.', 'hep-ph-0410170-2-8-2': 'Absorbing a [MATH] term in [MATH], the decay amplitude may be written in the following general form, [EQUATION] where by convention [MATH].', 'hep-ph-0410170-2-8-3': 'The phase [MATH] is extracted up to a discrete ambiguity from the two asymmetries [MATH] and [MATH] in [MATH] [CITATION]: [EQUATION] where [EQUATION]', 'hep-ph-0410170-2-8-4': 'Using the definitions [EQUATION] one has [EQUATION]', 'hep-ph-0410170-2-8-5': 'Two very reasonable assumptions, [MATH] and [MATH], imply [MATH].', 'hep-ph-0410170-2-8-6': 'The second condition may be replaced by a stronger one, [MATH]; it is stronger since we are assuming [MATH].', 'hep-ph-0410170-2-8-7': 'The result [MATH] demonstrates the central point of this note.', 'hep-ph-0410170-2-9-0': 'Rewriting Eq. ([REF]), [EQUATION] we see that the above assumptions imply [MATH], or [MATH].', 'hep-ph-0410170-2-9-1': 'The prediction of the sign of [MATH] is based purely on definitions and on the assumptions on [MATH] and [MATH].', 'hep-ph-0410170-2-9-2': 'It implies a considerably narrower range for [MATH] than the bound [MATH] [CITATION], obtained from the [MATH] and [MATH] isospin triangles, with the inclusion of some information about the asymmetry in [MATH].', 'hep-ph-0410170-2-10-0': 'It now remains to justify our two assumptions, [MATH] and [MATH].', 'hep-ph-0410170-2-10-1': 'The ratio [MATH] has a very long history, starting in the late eighties when the penguin amplitude in [MATH] was estimated to be small but non-negligible [CITATION].', 'hep-ph-0410170-2-10-2': 'First measurements of [MATH] and [MATH] decay rates, performed several years later by the CLEO collaboration [CITATION], were analyzed within flavor SU(3) indicating that [MATH] [CITATION].', 'hep-ph-0410170-2-10-3': 'A recent global SU(3) fit to all [MATH] and [MATH] decays obtained an unexpected large value [CITATION] [MATH].', 'hep-ph-0410170-2-10-4': 'The large value of [MATH] is driven partly but not only [CITATION] by a large input value for [MATH] [CITATION].', 'hep-ph-0410170-2-10-5': 'Theoretical calculations based on QCD and a heavy quark expansion [CITATION] find somewhat smaller values for [MATH], all lying comfortably in the range [MATH].', 'hep-ph-0410170-2-10-6': 'All these calculations support strongly the assumption [MATH].', 'hep-ph-0410170-2-11-0': 'Very early theoretical QCD arguments favoring a small value of [MATH] were proposed in [CITATION].', 'hep-ph-0410170-2-11-1': 'Although different calculations of [MATH] [CITATION], based on QCD and a heavy quark expansion, do not always agree in detail, all these computed values of [MATH] are considerably smaller than [MATH].', 'hep-ph-0410170-2-11-2': 'Constraints on CKM parameters based on the assumption [MATH] and on a given range for [MATH] were studied in [CITATION].', 'hep-ph-0410170-2-11-3': 'Long distance [MATH] penguin contributions [CITATION], or equivalently final state rescattering [CITATION], may spoil the QCD calculations [CITATION].', 'hep-ph-0410170-2-11-4': 'These effects cannot be calculated in a model-independent way.', 'hep-ph-0410170-2-11-5': 'A global SU(3) fit to [MATH] and [MATH] decays, which effectively includes these rescattering effects while assuming SU(3) invariant strong phases, obtains [CITATION] [MATH].', 'hep-ph-0410170-2-11-6': 'Large SU(3) breaking effects in strong phases could possibly lead to values of [MATH] outside the range [MATH].', 'hep-ph-0410170-2-12-0': 'A study of the two hadronic parameters, [MATH] and [MATH], was performed in [CITATION], adding measurements of [MATH] and an upper bound on [MATH] available before this summer to all other CKM constraints [CITATION].', 'hep-ph-0410170-2-12-1': 'Values were found in an overall minimum [MATH] fit, [MATH] and [MATH], consistent with [MATH] but possibly violating somewhat the bound [MATH].', 'hep-ph-0410170-2-12-2': 'A recent update [CITATION], using newer measurements, also favors parameters in the range [MATH].', 'hep-ph-0410170-2-13-0': 'Similarly, we attempt a direct experimental proof of [MATH] and [MATH] using current data [CITATION]: [EQUATION]', 'hep-ph-0410170-2-13-1': 'The world averaged asymmetries are based on most recent measurements by the Belle collaboration [CITATION] ([MATH]) and by the BaBar collaboration [CITATION] ([MATH]).', 'hep-ph-0410170-2-13-2': 'Since these two measurements are not in good agreement with each other, an error rescaling factor of 2.39 [CITATION] (determining errors in parentheses) may be used to achieve a conservative confidence level.', 'hep-ph-0410170-2-13-3': 'Expressions for [MATH], and [MATH], given in Eqs. ([REF])([REF]) and ([REF]), and the values in Eq. ([REF]) can be used to constrain [MATH] and [MATH].', 'hep-ph-0410170-2-14-0': 'We performed a separate minimum-[MATH] fit for the parameters [MATH] and [MATH].', 'hep-ph-0410170-2-14-1': 'The [MATH] contains two contributions from [MATH] and [MATH] and the usual terms corresponding to the standard analysis of the unitarity triangle.', 'hep-ph-0410170-2-14-2': 'In the minimization we reject points which yield [MATH].', 'hep-ph-0410170-2-14-3': 'Fig. [REF]a shows the resulting 90[MATH] CL bounds obtained using the unscaled (dark area) and rescaled (dashed line) errors in Eq. ([REF]).', 'hep-ph-0410170-2-14-4': 'The bounds can be summarized by [EQUATION] where the numbers in parentheses correspond to the rescaled errors.', 'hep-ph-0410170-2-14-5': 'Note that the unscaled errors exclude [MATH] at 90[MATH] CL, favoring [MATH].', 'hep-ph-0410170-2-14-6': 'Although they do not exclude completely [MATH], we find that the lowest allowed value of [MATH] is [MATH], quite close to zero.', 'hep-ph-0410170-2-14-7': 'The rescaled errors imply less restrictive bounds.', 'hep-ph-0410170-2-14-8': 'Exclusion of [MATH] would imply [MATH] for the central values given in Eq. ([REF]).', 'hep-ph-0410170-2-14-9': 'In any case, using the dependence of [MATH], [MATH] and [MATH] on [MATH] and [MATH] is expected to reduce the theoretical uncertainty in [MATH] below the upper limit on [MATH].', 'hep-ph-0410170-2-15-0': 'One may study directly the dependence of the sign of [MATH] on the values of [MATH] and [MATH].', 'hep-ph-0410170-2-15-1': 'Assuming that the discrepancy between the BaBar and Belle results disappears and taking errors [MATH], we calculate central values in the ([MATH]) plane for which [MATH] at 90[MATH] CL.', 'hep-ph-0410170-2-15-2': 'An important input is our prior knowledge of [MATH].', 'hep-ph-0410170-2-15-3': 'For this we use three different analyses of the unitarity triangle described in Refs. [CITATION], [CITATION] and [CITATION].', 'hep-ph-0410170-2-15-4': 'Corresponding regions in the ([MATH]) plane, outside of which [MATH], are described in Fig. [REF] by the shaded area and by the dashed and dotted lines, respectively.', 'hep-ph-0410170-2-15-5': 'The heavy dot, denoting the current central values of [MATH] and [MATH], is seen to lie close to the bounds but still inside the most conservative region [CITATION].', 'hep-ph-0410170-2-16-0': 'Let us now turn to discuss briefly [MATH] in [MATH].', 'hep-ph-0410170-2-16-1': 'Using the approximately pure longitudinal polarization of the final states [CITATION], the isospin analysis of the decays [MATH] is greatly simplified, becoming almost identical to that of [MATH] (neglecting the small [MATH]-width effect [CITATION]).', 'hep-ph-0410170-2-16-2': 'The small branching ratio for [MATH] [CITATION] leads to the rather tight 90[MATH] CL upper bound [CITATION], [MATH].', 'hep-ph-0410170-2-16-3': 'The phase difference [MATH] may be written as in Eq. ([REF]) and becomes positive under similar conditions for [MATH] and [MATH].', 'hep-ph-0410170-2-16-4': 'That is, our generic assumptions, [MATH] and [MATH], imply [MATH].', 'hep-ph-0410170-2-17-0': 'Although these assumptions about [MATH] and [MATH] seem reasonable, one may attempt to verify them experimentally, as demonstrated above for [MATH].', 'hep-ph-0410170-2-17-1': 'Current measurements of [MATH] are consistent with a zero CP asymmetry [CITATION], [EQUATION]', 'hep-ph-0410170-2-17-2': 'The central values of [MATH] and [MATH] imply [MATH].', 'hep-ph-0410170-2-17-3': 'An analysis, based on expressions for [MATH] and [MATH] analogous to Eqs. ([REF]) ([REF]) and ([REF]), in terms [MATH], and [MATH], proceeds as in [MATH].', 'hep-ph-0410170-2-17-4': 'The rather low upper bound, [MATH], implies a small value for [MATH], as can be seen in Eq. ([REF]) where [MATH] vanishes in the limit [MATH].', 'hep-ph-0410170-2-17-5': 'A small value of [MATH] means a small penguin amplitude and a low sensitivity to the value of [MATH].', 'hep-ph-0410170-2-18-0': 'We have verified the above statement by performing a [MATH] fit based on Eq. ([REF]).', 'hep-ph-0410170-2-18-1': 'In Fig. [REF]b, we show the 90[MATH] CL for the parameters [MATH] and [MATH].', 'hep-ph-0410170-2-18-2': 'We see that while [MATH] is excluded ([MATH]), values of [MATH] are permitted in the entire range [MATH].', 'hep-ph-0410170-2-18-3': 'This situation is unlikely to change with a reduction of errors in [MATH] and [MATH] if [MATH] is much smaller than one.', 'hep-ph-0410170-2-19-0': 'The situation in [MATH] is somewhat more complicated than the one in [MATH] and [MATH] because the final states [MATH] are not CP-eigenstates.', 'hep-ph-0410170-2-19-1': 'Time-dependent decay rates for initially [MATH] decaying into [MATH] are given by [CITATION]: [EQUATION]', 'hep-ph-0410170-2-19-2': 'For initially [MATH] decays the [MATH] and [MATH] terms have opposite signs.', 'hep-ph-0410170-2-19-3': 'One defines a measurable phase [MATH] [CITATION], which equals [MATH] in the limit of vanishing penguin amplitudes, [EQUATION]', 'hep-ph-0410170-2-19-4': 'The phase [MATH] can be expressed in terms of parameters defining decay amplitudes.', 'hep-ph-0410170-2-19-5': 'One denotes two distinct amplitudes for [MATH] and [MATH] by the charge of the [MATH] meson, [EQUATION] where [MATH] are two ratios of penguin and tree amplitudes, and [MATH] are the corresponding relative strong phases.', 'hep-ph-0410170-2-19-6': 'Continuing the analogy with [MATH] by defining [EQUATION] one has [EQUATION]', 'hep-ph-0410170-2-19-7': 'The two angle differences [MATH] have expressions similar to Eq. ([REF]): [EQUATION]', 'hep-ph-0410170-2-19-8': 'In the limit [MATH] one obviously obtains [MATH].', 'hep-ph-0410170-2-20-0': 'Flavor SU(3) relates tree and penguin amplitudes in [MATH] to corresponding contributions in [MATH] and [MATH].', 'hep-ph-0410170-2-20-1': 'Introducing SU(3) breaking in tree amplitudes in terms of ratios of suitable decay constants, and using measured decay rates for the above processes, the following two upper bounds were obtained at 90[MATH] CL [CITATION]: [MATH].', 'hep-ph-0410170-2-20-2': 'The algebraic average of these bounds provides an upper limit on [MATH], [EQUATION]', 'hep-ph-0410170-2-20-3': 'This bound, which may be modified by SU(3) breaking, does not assume knowledge of the signs of [MATH].', 'hep-ph-0410170-2-20-4': 'It becomes stronger when [MATH] and [MATH] have opposite signs, in which case they cancel each other in [MATH].', 'hep-ph-0410170-2-20-5': 'This possibility will be discussed now.', 'hep-ph-0410170-2-21-0': 'The sings of [MATH] depend on values of [MATH] and [MATH].', 'hep-ph-0410170-2-21-1': 'As we argued in the case of [MATH], both [MATH] and [MATH] would be positive if [MATH] and [MATH].', 'hep-ph-0410170-2-21-2': 'Arguments for [MATH] are very strong.', 'hep-ph-0410170-2-21-3': 'Studies based on flavor SU(3) [CITATION] and a calculation using QCD factorization [CITATION] obtain rather small values [MATH], lying very comfortably in the range [MATH].', 'hep-ph-0410170-2-21-4': 'Both studies obtain values for [MATH] and [MATH] lying in opposite hemispheres.', 'hep-ph-0410170-2-21-5': 'The output of a global SU(3) fit is [CITATION] [MATH] and [MATH].', 'hep-ph-0410170-2-21-6': 'Similar values are obtained in [CITATION], where it is being argued that these values do not differ much from the naive factorization predictions: [MATH] and [MATH] [CITATION].', 'hep-ph-0410170-2-21-7': 'Using the above values for [MATH] and [MATH] we see that [MATH] while [MATH].', 'hep-ph-0410170-2-21-8': 'This implies that [MATH] but [MATH], which means that the upper bound ([REF]) is replaced by a stronger bound, [EQUATION]', 'hep-ph-0410170-2-21-9': 'That is, given [MATH], and assuming that the phases [MATH] and [MATH] lie in opposite hemispheres, improves the upper bound ([REF]) by about a factor two.', 'hep-ph-0410170-2-21-10': 'The actual upper and lower bounds in ([REF]) may be larger by 30[MATH] because of SU(3) breaking corrections some of which have not been taken into account.', 'hep-ph-0410170-2-22-0': 'A direct verification of our assumptions about [MATH] in [MATH] relies on interference between tree and penguin amplitudes in these processes.', 'hep-ph-0410170-2-22-1': 'A possible evidence for such interference is the measured CP asymmetry in [MATH] [CITATION], [EQUATION]', 'hep-ph-0410170-2-22-2': 'This asymmetry provides one equation for [MATH] and [MATH] [CITATION] favoring negative values of [MATH].', 'hep-ph-0410170-2-22-3': 'Similarly, the CP asymmetry in [MATH] [CITATION] [EQUATION] provides an equation for [MATH] and [MATH].', 'hep-ph-0410170-2-22-4': 'Two other observables depending on these four hadronic parameters and on [MATH] are [MATH] and [MATH], related to [MATH] and [MATH] by a simple transformation [CITATION].', 'hep-ph-0410170-2-22-5': 'They obtain values [CITATION] [EQUATION]', 'hep-ph-0410170-2-22-6': 'The error in parentheses includes a rescaling factor of 1.7.', 'hep-ph-0410170-2-22-7': 'The dependence of [MATH] and [MATH] on a small relative phase [MATH] [CITATION] between the two tree amplitudes [MATH] and [MATH] will be neglected.', 'hep-ph-0410170-2-22-8': 'A fifth constraint on [MATH] and [MATH] is provided by the upper bound ([REF]), where [MATH] are given in Eq. ([REF]).', 'hep-ph-0410170-2-22-9': 'This completes a system of four equations and one inequality for five parameters.', 'hep-ph-0410170-2-22-10': 'An important question is whether these constraints imply that [MATH] and [MATH] lie in opposite hemispheres, thereby leading to Eq. ([REF]).', 'hep-ph-0410170-2-22-11': 'In any case, the inclusion of these five relations s expected to reduce the uncertainty in [MATH] below the upper bound ([REF]).', 'hep-ph-0410170-2-23-0': 'In conclusion, we have shown that the generic assumptions, [MATH] and [MATH], predict [MATH] in [MATH] and [MATH], thereby reducing the theoretical uncertainty in [MATH] by a factor two.', 'hep-ph-0410170-2-23-1': 'In the decay processes [MATH], where two pairs of tree and penguin amplitudes occur, the assumption that the two relative phases between tree and penguin amplitudes lie in opposite hemispheres results in a suppression of about a factor two of the bound on [MATH] due to cancellation between two terms.', 'hep-ph-0410170-2-23-2': 'Measured CP asymmetries and [MATH] were studied in terms of Penguin-to-Tree ratios and their relative strong phases.', 'hep-ph-0410170-2-23-3': 'We presented evidence (although not yet conclusive) that the various hadronic parameters do lie in ranges that allow a reduction of the error on [MATH].', 'hep-ph-0410170-2-24-0': 'We wish to thank Ahmed Ali, Andreas Hoecker and Dan Pirjol for helpful discussions.', 'hep-ph-0410170-2-24-1': 'M. G. is grateful to the Institute for Theoretical Physics at the University of Zurich for its kind hospitality.', 'hep-ph-0410170-2-24-2': 'This work is supported in part by the Schweizerischer Nationalfonds and by the Israel Science Foundation founded by the Israel Academy of Sciences and Humanities, Grant No. 1052/04.', 'hep-ph-0410170-2-25-0': 'def 89CP Violation, edited by C. Jarlskog (World Scientific,Singapore, 1989)'}

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[]

[['hep-ph-0410170-1-22-9', 'hep-ph-0410170-2-22-9'], ['hep-ph-0410170-2-7-3', 'hep-ph-0410170-3-7-3'], ['hep-ph-0410170-2-16-4', 'hep-ph-0410170-3-16-4'], ['hep-ph-0410170-2-1-3', 'hep-ph-0410170-3-1-3'], ['hep-ph-0410170-2-20-3', 'hep-ph-0410170-3-20-3'], ['hep-ph-0410170-2-22-9', 'hep-ph-0410170-3-23-3']]

[]

['hep-ph-0410170-1-2-0', 'hep-ph-0410170-1-2-1', 'hep-ph-0410170-1-2-2', 'hep-ph-0410170-1-2-3', 'hep-ph-0410170-1-2-4', 'hep-ph-0410170-1-25-0', 'hep-ph-0410170-2-2-0', 'hep-ph-0410170-2-2-1', 'hep-ph-0410170-2-2-2', 'hep-ph-0410170-2-2-3', 'hep-ph-0410170-2-2-4', 'hep-ph-0410170-2-25-0', 'hep-ph-0410170-3-2-0', 'hep-ph-0410170-3-2-1', 'hep-ph-0410170-3-2-2', 'hep-ph-0410170-3-2-3', 'hep-ph-0410170-3-2-4', 'hep-ph-0410170-3-26-0']

{'1': 'http://arxiv.org/licenses/assumed-1991-2003/', '2': 'http://arxiv.org/licenses/assumed-1991-2003/', '3': 'http://arxiv.org/licenses/assumed-1991-2003/'}

https://arxiv.org/abs/hep-ph/0410170

{'hep-ph-0410170-3-0-0': 'REDUCING THE ERROR ON [MATH] in [MATH] Michael Gronau Physics Department, Technion - Israel Institute of Technology 32000 Haifa, Israel Enrico Lunghi and Daniel Wyler Institut fur Theoretische Physik, Universitat Zurich, Winterthurerstrasse 190 CH-8057, Zurich, Switzerland', 'hep-ph-0410170-3-1-0': 'Theoretical errors in the extraction of [MATH] from [MATH] decays are usually given in terms of upper bounds on [MATH] obtained from isospin or from SU(3) relations, where [MATH] is measured through CP asymmetries.', 'hep-ph-0410170-3-1-1': 'We show that mild assumptions about magnitudes and strong phases of penguin and tree amplitudes ([MATH] and [MATH]) in [MATH] and [MATH], imply [MATH], thus reducing by a factor two the error in [MATH].', 'hep-ph-0410170-3-1-2': 'Similarly, the assumptions [MATH] in [MATH] lead to a cancellation between two terms in [MATH].', 'hep-ph-0410170-3-1-3': 'Current data support these conditions, which are justified by both QCD-factorization and flavor SU(3).', 'hep-ph-0410170-3-2-0': 'PACS codes: 12.15.', 'hep-ph-0410170-3-2-1': 'Hh, 12.15.', 'hep-ph-0410170-3-2-2': 'Ji, 13.25.', 'hep-ph-0410170-3-2-3': 'Hw, 14.40.', 'hep-ph-0410170-3-2-4': 'Nd', 'hep-ph-0410170-3-3-0': 'Direct extraction of the Cabibbo-Kobayashi-Maskawa (CKM) phase [MATH] from the time-dependent CP asymmetry in [MATH] is obstructed by the penguin amplitude [CITATION].', 'hep-ph-0410170-3-3-1': 'This obstacle may be overcome using isospin symmetry [CITATION], which incorporates electroweak penguin contributions at a percent level [CITATION] but does not include small isospin breaking effects [CITATION].', 'hep-ph-0410170-3-3-2': 'This method requires separate rate measurements of [MATH].', 'hep-ph-0410170-3-3-3': 'As long as low statistics does not permit these separate measurements, one can use combined [MATH] and [MATH] decay rates into the three [MATH] channels to obtain upper bounds on [MATH] [CITATION].', 'hep-ph-0410170-3-3-4': 'The angle [MATH], which equals [MATH] in the limit of a vanishing penguin amplitude, is given up to a discrete ambiguity by the time-dependent CP asymmetry in [MATH].', 'hep-ph-0410170-3-3-5': 'The upper bound on [MATH], improved by information from an upper limit on the asymmetry in [MATH], remains an intrinsic theoretical uncertainty in [MATH].', 'hep-ph-0410170-3-3-6': 'Note that [MATH] may be either positive or negative.', 'hep-ph-0410170-3-4-0': 'Similar considerations can also be applied to [MATH] and lead to analogous bounds on [MATH].', 'hep-ph-0410170-3-4-1': 'Because each of the two [MATH] mesons carries a unit spin, one must distinguish between decays to even-CP and odd-CP final states corresponding to definite polarizations [CITATION].', 'hep-ph-0410170-3-5-0': 'A complete isospin analysis of the processes [MATH] is complicated by the existence of five different [MATH] charge states in [MATH] and [MATH] decays [CITATION].', 'hep-ph-0410170-3-5-1': 'Thus, it was proposed to measure [MATH] through the time-dependent Dalitz distribution in [MATH] [CITATION], which provides information about interference of amplitudes for [MATH] and [MATH].', 'hep-ph-0410170-3-5-2': 'Alternatively, one may measure an angle [MATH] in quasi two- body decays [MATH], and use broken flavor SU(3) to obtain an upper bound on [MATH] [CITATION].', 'hep-ph-0410170-3-6-0': 'The experimental progress during the past year has been impressive in this class of measurements.', 'hep-ph-0410170-3-6-1': 'The situation in early July [CITATION] was updated in late August [CITATION], following new measurements reported at the International Conference on High Energy Physics in Beijing [CITATION].', 'hep-ph-0410170-3-6-2': 'The overall range of [MATH] determined in [MATH] [CITATION], [MATH], overlaps with and begins to be narrower than the following bounds obtained indirectly in an independent global CKM fit [CITATION], [EQUATION] where a 95[MATH] confidence level (CL) is implied.', 'hep-ph-0410170-3-6-3': 'The error in [MATH] from a time-dependent Dalitz plot analysis of [MATH] [CITATION], [MATH], is statistics-dominated.', 'hep-ph-0410170-3-6-4': 'On the other hand, the theoretical errors in determining [MATH], using isospin in [MATH] and applying broken SU(3) to [MATH], are at least as large as the corresponding statistical errors.', 'hep-ph-0410170-3-6-5': 'The 90[MATH] CL upper bounds on [MATH] in these three cases are [CITATION] [MATH] and [MATH], respectively.', 'hep-ph-0410170-3-6-6': 'It would be very useful to reduce these intrinsic theoretical uncertainties using present data.', 'hep-ph-0410170-3-7-0': 'In this letter we point out that the above theoretical errors in [MATH] may be reduced by about a factor two under very mild and reasonable assumptions, which are justified by both QCD-factorization and flavor SU(3).', 'hep-ph-0410170-3-7-1': 'In [MATH] and [MATH] we study the dependence of [MATH] on the ratio of penguin and tree amplitudes, [MATH], and on their relative strong phase, [MATH].', 'hep-ph-0410170-3-7-2': 'We show that the conditions [MATH] and [MATH] predict a positive sign for [MATH].', 'hep-ph-0410170-3-7-3': 'In [MATH] two small ratios of penguin and tree amplitudes, [MATH], and two phases [MATH] lying in opposite hemispheres, tend to suppress [MATH] due to a cancellation between two terms.', 'hep-ph-0410170-3-7-4': 'We propose to include, in future studies of [MATH], the explicit dependence of [MATH] on [MATH], [MATH], [MATH] and [MATH], together with the dependence of the CP asymmetries on these variables.', 'hep-ph-0410170-3-7-5': 'In the argumentation below we assume [MATH] as given in ([REF]).', 'hep-ph-0410170-3-8-0': 'To prove our point, we consider first [MATH].', 'hep-ph-0410170-3-8-1': 'We use the [MATH]-convention defined in [CITATION], in which the top-quark has been integrated out in the [MATH] penguin transition and unitarity of the CKM matrix has been used.', 'hep-ph-0410170-3-8-2': 'Absorbing a [MATH] term in [MATH], the decay amplitude may be written in the following general form, [EQUATION] where by convention [MATH].', 'hep-ph-0410170-3-8-3': 'The phase [MATH] is extracted up to a discrete ambiguity from the two asymmetries [MATH] and [MATH] in [MATH] [CITATION]: [EQUATION] where [EQUATION]', 'hep-ph-0410170-3-8-4': 'Using the definitions [EQUATION] one has [EQUATION]', 'hep-ph-0410170-3-8-5': 'Two very reasonable assumptions, [MATH] and [MATH], imply [MATH].', 'hep-ph-0410170-3-8-6': 'The second condition may be replaced by a stronger one, [MATH]; it is stronger since we are assuming [MATH].', 'hep-ph-0410170-3-8-7': 'The result [MATH] demonstrates the central point of this note.', 'hep-ph-0410170-3-9-0': 'Rewriting Eq. ([REF]), [EQUATION] we see that the above assumptions imply [MATH], or [MATH].', 'hep-ph-0410170-3-9-1': 'The prediction of the sign of [MATH] is based purely on definitions and on the assumptions on [MATH] and [MATH].', 'hep-ph-0410170-3-9-2': 'It implies a considerably narrower range for [MATH] than the bound [MATH] [CITATION], obtained from the [MATH] and [MATH] isospin triangles, with the inclusion of some information about the asymmetry in [MATH].', 'hep-ph-0410170-3-10-0': 'It now remains to justify our two assumptions, [MATH] and [MATH].', 'hep-ph-0410170-3-10-1': 'The ratio [MATH] has a very long history, starting in the late eighties when the penguin amplitude in [MATH] was estimated to be small but non-negligible [CITATION].', 'hep-ph-0410170-3-10-2': 'First measurements of [MATH] and [MATH] decay rates, performed several years later by the CLEO collaboration [CITATION], were analyzed within flavor SU(3) indicating that [MATH] [CITATION].', 'hep-ph-0410170-3-10-3': 'A recent global SU(3) fit to all [MATH] and [MATH] decays obtained an unexpected large value [CITATION] [MATH].', 'hep-ph-0410170-3-10-4': 'The large value of [MATH] is driven partly but not only [CITATION] by a large input value for [MATH] [CITATION].', 'hep-ph-0410170-3-10-5': 'Theoretical calculations based on QCD and a heavy quark expansion [CITATION] find somewhat smaller values for [MATH], all lying comfortably in the range [MATH].', 'hep-ph-0410170-3-10-6': 'All these calculations support strongly the assumption [MATH].', 'hep-ph-0410170-3-11-0': 'Very early theoretical QCD arguments favoring a small value of [MATH] were proposed in [CITATION].', 'hep-ph-0410170-3-11-1': 'Although different calculations of [MATH] [CITATION], based on QCD and a heavy quark expansion, do not always agree in detail, all these computed values of [MATH] are considerably smaller than [MATH].', 'hep-ph-0410170-3-11-2': 'Constraints on CKM parameters based on the assumption [MATH] and on a given range for [MATH] were studied in [CITATION].', 'hep-ph-0410170-3-11-3': 'Long distance [MATH] penguin contributions [CITATION], or equivalently final state rescattering [CITATION], may spoil the QCD calculations [CITATION].', 'hep-ph-0410170-3-11-4': 'These effects cannot be calculated in a model-independent way.', 'hep-ph-0410170-3-11-5': 'A global SU(3) fit to [MATH] and [MATH] decays, which effectively includes these rescattering effects while assuming SU(3) invariant strong phases, obtains [CITATION] [MATH].', 'hep-ph-0410170-3-11-6': 'Large SU(3) breaking effects in strong phases could possibly lead to values of [MATH] outside the range [MATH].', 'hep-ph-0410170-3-12-0': 'A study of the two hadronic parameters, [MATH] and [MATH], was performed in [CITATION], adding measurements of [MATH] and an upper bound on [MATH] available before this summer to all other CKM constraints [CITATION].', 'hep-ph-0410170-3-12-1': 'Values were found in an overall minimum [MATH] fit, [MATH] and [MATH], consistent with [MATH] but possibly violating somewhat the bound [MATH].', 'hep-ph-0410170-3-12-2': 'A recent update [CITATION], using newer measurements, also favors parameters in the range [MATH].', 'hep-ph-0410170-3-13-0': 'Similarly, we attempt a direct experimental proof of [MATH] and [MATH] using current data [CITATION]: [EQUATION]', 'hep-ph-0410170-3-13-1': 'The world averaged asymmetries are based on most recent measurements by the Belle collaboration [CITATION] ([MATH]) and by the BaBar collaboration [CITATION] ([MATH]).', 'hep-ph-0410170-3-13-2': 'Since these two measurements are not in good agreement with each other, an error rescaling factor of 2.39 [CITATION] (determining errors in parentheses) may be used to achieve a conservative confidence level.', 'hep-ph-0410170-3-13-3': 'Expressions for [MATH], and [MATH], given in Eqs. ([REF])([REF]) and ([REF]), and the values in Eq. ([REF]) can be used to constrain [MATH] and [MATH].', 'hep-ph-0410170-3-14-0': 'We performed a separate minimum-[MATH] fit for the parameters [MATH] and [MATH].', 'hep-ph-0410170-3-14-1': 'The [MATH] contains two contributions from [MATH] and [MATH] and the usual terms corresponding to the standard analysis of the unitarity triangle.', 'hep-ph-0410170-3-14-2': 'In the minimization we reject points which yield [MATH].', 'hep-ph-0410170-3-14-3': 'Fig. [REF]a shows the resulting 90[MATH] CL bounds obtained using the unscaled (dark area) and rescaled (dashed line) errors in Eq. ([REF]).', 'hep-ph-0410170-3-14-4': 'The bounds can be summarized by [EQUATION] where the numbers in parentheses correspond to the rescaled errors.', 'hep-ph-0410170-3-14-5': 'Note that the unscaled errors exclude [MATH] at 90[MATH] CL, favoring [MATH].', 'hep-ph-0410170-3-14-6': 'Although they do not exclude completely [MATH], we find that the lowest allowed value of [MATH] is [MATH], quite close to zero.', 'hep-ph-0410170-3-14-7': 'The rescaled errors imply less restrictive bounds.', 'hep-ph-0410170-3-14-8': 'Exclusion of [MATH] would imply [MATH] for the central values given in Eq. ([REF]).', 'hep-ph-0410170-3-14-9': 'In any case, using the dependence of [MATH], [MATH] and [MATH] on [MATH] and [MATH] is expected to reduce the theoretical uncertainty in [MATH] below the upper limit on [MATH].', 'hep-ph-0410170-3-15-0': 'One may study directly the dependence of the sign of [MATH] on the values of [MATH] and [MATH].', 'hep-ph-0410170-3-15-1': 'Assuming that the discrepancy between the BaBar and Belle results disappears and taking errors [MATH], we calculate central values in the ([MATH]) plane for which [MATH] at 90[MATH] CL.', 'hep-ph-0410170-3-15-2': 'An important input is our prior knowledge of [MATH].', 'hep-ph-0410170-3-15-3': 'For this we use three different analyses of the unitarity triangle described in Refs. [CITATION], [CITATION] and [CITATION].', 'hep-ph-0410170-3-15-4': 'Corresponding regions in the ([MATH]) plane, outside of which [MATH], are described in Fig. [REF] by the shaded area and by the dashed and dotted lines, respectively.', 'hep-ph-0410170-3-15-5': 'The heavy dot, denoting the current central values of [MATH] and [MATH], is seen to lie close to the bounds but still inside the most conservative region [CITATION].', 'hep-ph-0410170-3-16-0': 'Let us now turn to discuss briefly [MATH] in [MATH].', 'hep-ph-0410170-3-16-1': 'Using the approximately pure longitudinal polarization of the final states [CITATION], the isospin analysis of the decays [MATH] is greatly simplified, becoming almost identical to that of [MATH] (neglecting the small [MATH]-width effect [CITATION]).', 'hep-ph-0410170-3-16-2': 'The small branching ratio for [MATH] [CITATION] leads to the rather tight 90[MATH] CL upper bound [CITATION], [MATH].', 'hep-ph-0410170-3-16-3': 'The phase difference [MATH] may be written as in Eq. ([REF]) and becomes positive under similar conditions for [MATH] and [MATH].', 'hep-ph-0410170-3-16-4': 'That is, our generic assumptions, [MATH] and [MATH] for longitudinally polarized [MATH] mesons, imply [MATH].', 'hep-ph-0410170-3-17-0': 'Although these assumptions about [MATH] and [MATH] seem reasonable [CITATION], one may attempt to verify them experimentally, as demonstrated above for [MATH].', 'hep-ph-0410170-3-17-1': 'Current measurements of [MATH] are consistent with a zero CP asymmetry [CITATION], [EQUATION]', 'hep-ph-0410170-3-17-2': 'The central values of [MATH] and [MATH] imply [MATH].', 'hep-ph-0410170-3-17-3': 'An analysis, based on expressions for [MATH] and [MATH] analogous to Eqs. ([REF]) ([REF]) and ([REF]), in terms [MATH], and [MATH], proceeds as in [MATH].', 'hep-ph-0410170-3-17-4': 'The rather low upper bound, [MATH], implies a small value for [MATH], as can be seen in Eq. ([REF]) where [MATH] vanishes in the limit [MATH].', 'hep-ph-0410170-3-17-5': 'A small value of [MATH] means a small penguin amplitude and a low sensitivity to the value of [MATH].', 'hep-ph-0410170-3-18-0': 'We have verified the above statement by performing a [MATH] fit based on Eq. ([REF]).', 'hep-ph-0410170-3-18-1': 'In Fig. [REF]b, we show the 90[MATH] CL for the parameters [MATH] and [MATH].', 'hep-ph-0410170-3-18-2': 'We see that while [MATH] is excluded ([MATH]), values of [MATH] are permitted in the entire range [MATH].', 'hep-ph-0410170-3-18-3': 'This situation is unlikely to change with a reduction of errors in [MATH] and [MATH] if [MATH] is much smaller than one as calculated in naive factorization [CITATION].', 'hep-ph-0410170-3-19-0': 'The situation in [MATH] is somewhat more complicated than the one in [MATH] and [MATH] because the final states [MATH] are not CP-eigenstates.', 'hep-ph-0410170-3-19-1': 'Time-dependent decay rates for initially [MATH] decaying into [MATH] are given by [CITATION]: [EQUATION]', 'hep-ph-0410170-3-19-2': 'For initially [MATH] decays the [MATH] and [MATH] terms have opposite signs.', 'hep-ph-0410170-3-19-3': 'One defines a measurable phase [MATH] [CITATION], which equals [MATH] in the limit of vanishing penguin amplitudes, [EQUATION]', 'hep-ph-0410170-3-19-4': 'The phase [MATH] can be expressed in terms of parameters defining decay amplitudes.', 'hep-ph-0410170-3-19-5': 'One denotes two distinct amplitudes for [MATH] and [MATH] by the charge of the [MATH] meson, [EQUATION] where [MATH] are two ratios of penguin and tree amplitudes, and [MATH] are the corresponding relative strong phases.', 'hep-ph-0410170-3-19-6': 'Continuing the analogy with [MATH] by defining [EQUATION] one has [EQUATION]', 'hep-ph-0410170-3-19-7': 'The two angle differences [MATH] have expressions similar to Eq. ([REF]): [EQUATION]', 'hep-ph-0410170-3-19-8': 'In the limit [MATH] one obviously obtains [MATH].', 'hep-ph-0410170-3-20-0': 'Flavor SU(3) relates tree and penguin amplitudes in [MATH] to corresponding contributions in [MATH] and [MATH].', 'hep-ph-0410170-3-20-1': 'Introducing SU(3) breaking in tree amplitudes in terms of ratios of suitable decay constants, and using measured decay rates for the above processes, the following two upper bounds were obtained at 90[MATH] CL [CITATION]: [MATH].', 'hep-ph-0410170-3-20-2': 'The algebraic average of these bounds provides an upper limit on [MATH], [EQUATION]', 'hep-ph-0410170-3-20-3': 'This bound, which may be modified to [MATH] by SU(3) breaking other than in tree amplitudes and by small annihilation amplitudes which have been neglected, does not assume knowledge of the signs of [MATH].', 'hep-ph-0410170-3-20-4': 'It becomes stronger when [MATH] and [MATH] have opposite signs, in which case they cancel each other in [MATH].', 'hep-ph-0410170-3-20-5': 'This possibility will be discussed now.', 'hep-ph-0410170-3-21-0': 'The sings of [MATH] depend on values of [MATH] and [MATH].', 'hep-ph-0410170-3-21-1': 'As we argued in the case of [MATH], both [MATH] and [MATH] would be positive if [MATH] and [MATH].', 'hep-ph-0410170-3-21-2': 'Arguments for [MATH] are very strong.', 'hep-ph-0410170-3-21-3': 'Studies based on flavor SU(3) [CITATION] and a calculation using QCD-factorization [CITATION] obtain rather small values [MATH], lying very comfortably in the range [MATH].', 'hep-ph-0410170-3-21-4': 'Both studies obtain values for [MATH] and [MATH] lying in opposite hemispheres.', 'hep-ph-0410170-3-21-5': 'The output of a global SU(3) fit is [CITATION] [MATH] and [MATH].', 'hep-ph-0410170-3-21-6': 'Similar values are obtained in [CITATION], where it is being argued that these values do not differ much from the naive factorization predictions: [MATH] and [MATH] [CITATION].', 'hep-ph-0410170-3-21-7': 'Using the above values for [MATH] and [MATH] we see that [MATH] while [MATH].', 'hep-ph-0410170-3-21-8': 'This implies that [MATH] but [MATH], which means that the upper bound ([REF]) is replaced by a stronger bound, [EQUATION]', 'hep-ph-0410170-3-21-9': 'That is, given [MATH], and assuming that the phases [MATH] and [MATH] lie in opposite hemispheres, improves the upper bound ([REF]) by about a factor two.', 'hep-ph-0410170-3-21-10': 'The actual upper and lower bounds in ([REF]) may be larger by about 30[MATH] because of possible SU(3) breaking corrections in penguin amplitudes and small annihilation amplitudes which have not been taken into account.', 'hep-ph-0410170-3-22-0': 'A direct verification of our assumptions about [MATH] in [MATH] relies on interference between tree and penguin amplitudes in these processes.', 'hep-ph-0410170-3-22-1': 'A possible evidence for such interference is the measured direct CP asymmetry in [MATH] [CITATION], [EQUATION]', 'hep-ph-0410170-3-22-2': 'This asymmetry provides one equation for [MATH] and [MATH] [CITATION] favoring negative values of [MATH].', 'hep-ph-0410170-3-22-3': 'Similarly, the direct CP asymmetry in [MATH] [CITATION] [EQUATION] provides an equation for [MATH] and [MATH].', 'hep-ph-0410170-3-22-4': 'Two other observables depending on these four hadronic parameters and on [MATH] are [MATH] and [MATH], related to [MATH] and [MATH] by a simple transformation [CITATION].', 'hep-ph-0410170-3-22-5': 'They obtain values [CITATION] [EQUATION]', 'hep-ph-0410170-3-22-6': 'The error in parentheses includes a rescaling factor of 1.7.', 'hep-ph-0410170-3-23-0': 'The dependence of [MATH] and [MATH] on a small relative phase [MATH] [CITATION] between the two tree amplitudes [MATH] and [MATH] can be neglected.', 'hep-ph-0410170-3-23-1': 'Alternatively, one may use a constraint from the phase [MATH] measured through the interference of the [MATH] and [MATH] overlapping resonances in the [MATH] Dalitz plot [CITATION].', 'hep-ph-0410170-3-23-2': 'Another constraint on [MATH] and [MATH] is provided by the upper bound ([REF]), where [MATH] are given in Eq. ([REF]).', 'hep-ph-0410170-3-23-3': 'This completes a system of four (or five) equations and one inequality for five parameters (or six, if we include [MATH]).', 'hep-ph-0410170-3-23-4': 'An important question is whether these constraints imply that [MATH] and [MATH] lie in opposite hemispheres, thereby leading to Eq. ([REF]).', 'hep-ph-0410170-3-23-5': 'In any case, the inclusion of these five or six relations is expected to reduce the uncertainty in [MATH] below the upper bound ([REF]).', 'hep-ph-0410170-3-24-0': 'In conclusion, we have shown that the generic assumptions, [MATH] and [MATH], predict [MATH] in [MATH] and [MATH], thereby reducing the theoretical uncertainty in [MATH] by a factor two.', 'hep-ph-0410170-3-24-1': 'In the decay processes [MATH], where two pairs of tree and penguin amplitudes occur, the assumption that the two relative phases between tree and penguin amplitudes lie in opposite hemispheres results in a suppression of about a factor two of the bound on [MATH] due to cancellation between two terms.', 'hep-ph-0410170-3-24-2': 'Measured CP asymmetries and [MATH] were studied in terms of Penguin-to-Tree ratios and their relative strong phases.', 'hep-ph-0410170-3-24-3': 'We presented theoretical arguments, based on QCD-factorization and on flavor SU(3), and experimental evidence (although not yet conclusive) that the various hadronic parameters do lie in ranges that allow a reduction of the error on [MATH].', 'hep-ph-0410170-3-25-0': 'We wish to thank Ahmed Ali, Andreas Hoecker and Dan Pirjol for helpful discussions.', 'hep-ph-0410170-3-25-1': 'M. G. is grateful to the Institute for Theoretical Physics at the University of Zurich for its kind hospitality.', 'hep-ph-0410170-3-25-2': 'This work is supported in part by the Schweizerischer Nationalfonds and by the Israel Science Foundation founded by the Israel Academy of Sciences and Humanities, Grant No. 1052/04.', 'hep-ph-0410170-3-26-0': 'def 89CP Violation, edited by C. Jarlskog (World Scientific,Singapore, 1989)'}

1401.6643

{'1401.6643-1-0-0': 'The recent discovery of a topological phase transition in IV-VI narrow-gap semiconductors has revitalized the decades-old interest in the bulk band inversion occurring in these materials.', '1401.6643-1-0-1': 'Here we systematically study the (001) surface states of Pb[MATH]"2081"208B"2093 1-xSn[MATH]"2093xSemixed crystals by means of angle-resolved photoelectron spectroscopy in the parameter space [MATH] "0030"00A0"2264"00A0"0078"00A0 "2264"00A0"0030"002E"0033"00370 <= x <= 0.37 and [MATH].', '1401.6643-1-0-2': 'Using the surface-state observations, we monitor directly the topological phase transition in this solid solution and gain valuable information on the evolution of the underlying fundamental band gap of the system.', '1401.6643-1-0-3': 'In contrast to common model expectations, the band-gap evolution appears to be nonlinear as a function of the studied parameters, resulting in the measuring of a discontinuous band inversion process.', '1401.6643-1-0-4': 'This finding signifies that the anticipated gapless bulk state is in fact not a stable configuration and that the topological phase transition therefore exhibits features akin to a first-order transition.', '1401.6643-1-1-0': '# Introduction', '1401.6643-1-2-0': 'Lead chalcogenides and related compounds have been studied intensely already in the second half of the last century [CITATION].', '1401.6643-1-2-1': 'These narrow-gap semiconductors are used for applications in infrared lasers [CITATION] and detectors [CITATION] as well as in thermoelectric devices [CITATION].', '1401.6643-1-2-2': 'They also exhibit a large range of peculiar fundamental properties, such as positive temperature and negative pressure coefficients of the energy gap [CITATION] and nonparabolic band dispersions [CITATION].', '1401.6643-1-2-3': 'Moreover, for suitable compounds and parameter ranges, the band gap undergoes an inversion (the parities at the top of the valence and the bottom of the conduction bands swap) as a function of temperature, pressure [CITATION], and composition [CITATION].', '1401.6643-1-2-4': 'At the interface between inverted and noninverted insulating layers gapless states were predicted to form [CITATION].', '1401.6643-1-2-5': 'The interest in this class of materials was very recently renewed by the investigation of the so-called topological-crystalline-insulator (TCI) state in SnTe [CITATION] as well as the solid solutions (Pb,Sn)Te [CITATION] and (Pb,Sn)Se [CITATION].', '1401.6643-1-2-6': 'In this state of matter, the mirror symmetry present in the rock-salt structure ensures degenerate energy eigenvalues along mirror lines, and hence metallic surface states on certain high-symmetry surfaces [CITATION], when the band gap is inverted.', '1401.6643-1-2-7': 'Additionally, these topologically protected surface states exhibit Dirac-like dispersions and they are spin-momentum-locked [CITATION].', '1401.6643-1-3-0': 'Although the tunable band inversion in the lead and tin chalcogenides has been established for decades, the details of the gap closing remain in the dark.', '1401.6643-1-3-1': 'It is commonly expected (cf. e.g. most of the aforementioned references) that the size of the band gap goes to zero when a critical composition, temperature or pressure is reached.', '1401.6643-1-3-2': 'Yet, to the best of our knowledge , a completely closed band gap (within the resolution of the respective measurement) has never been observed experimentally.', '1401.6643-1-3-3': 'The lowest confirmed gap values are in the region of a few tens of millielectronvolts.', '1401.6643-1-3-4': 'For instance, in infrared absorption studies, the long-wavelength limit was not accessible [CITATION] and very-low-energy laser emission seems to be hindered by plasmon-phonon excitations [CITATION].', '1401.6643-1-4-0': 'The recent discovery of the TCI phase and the appertaining surface states has opened a new route to study the details of the band inversion in this class of materials by high-resolution photoelectron spectroscopy.', '1401.6643-1-4-1': 'The (001) surface states at the surface high-symmetry point [MATH] lie within the bulk band gap just beyond the valence- and conduction-band edges projected from the bulk [MATH] points [CITATION].', '1401.6643-1-4-2': 'The same is true for the corresponding high-symmetry points of the (110) surface [CITATION].', '1401.6643-1-4-3': 'Hence, studying the surface-state evolution across a parameter range for which the band inversion occurs, can give a reliable estimate of the band gap at [MATH], and thus further elucidate the process of the inversion.', '1401.6643-1-4-4': 'In this article, we report on a systematic angle-resolved photoelectron spectroscopy (ARPES) study of (001)-oriented (Pb,Sn)Se mixed crystals.', '1401.6643-1-4-5': 'The topological phase boundary is established by the investigation of the evolution of the surface states around [MATH] as a function of composition and temperature.', '1401.6643-1-4-6': 'Moreover, contrary to the common theoretical expectations and the conclusions from previous experimental work, our observations point to an unstable zero-gap state in the bulk material.', '1401.6643-1-4-7': 'We compare our results with prior studies and discuss the implications for the band-inversion models of the IV-VI narrow-gap semiconductors.', '1401.6643-1-5-0': '# Experimental details and results', '1401.6643-1-6-0': 'The [MATH]-type Pb[MATH]"2081"208B"2093 1-xSn[MATH]"2093xSe([MATH] "0030"00A0"2264"00A0"0078"00A0 "2264"00A0"0030"002E"0033"00370 <= x <= 0.37) single crystals used in this study have been grown by the self-selecting vapor-growth method [CITATION].', '1401.6643-1-6-1': 'Their composition has been determined by means of X-ray diffraction as well as energy-dispersive X-ray spectroscopy.', '1401.6643-1-6-2': 'Powder X-ray diffraction confirmed the rock-salt structure [space group [MATH] (225)] both at room temperature and [MATH] K.', '1401.6643-1-6-3': 'The ARPES measurements in the temperature range [MATH] have been conducted on samples cleaved along a (001) surface in ultra-high vacuum ([MATH] mbar before cleaving, [MATH] mbar after cleaving).', '1401.6643-1-6-4': 'The experiments have been performed using the laser-based ARPES set-up baltazar equipped with a time-of-flight electron analyzer [CITATION].', '1401.6643-1-6-5': 'Linearly polarized light with a photon energy [MATH] eV was used to excite the electrons.', '1401.6643-1-6-6': 'The total energy and crystal-momentum resolution was about [MATH] meV and better than [MATH], respectively.', '1401.6643-1-7-0': 'The obtained ARPES spectra are shown in Fig. [REF].', '1401.6643-1-7-1': 'The data in the vicinity of [MATH] are plotted along the high-symmetry lines of the surface Brillouin zone [[MATH]-[MATH]-[MATH], cf. Fig. [REF]].', '1401.6643-1-7-2': 'In all spectra surface states are discernible.', '1401.6643-1-7-3': 'While the samples at high temperature ([MATH] K) exhibit gapped states for all [MATH], at lower temperatures the spectra show a change from gapped surface states at low [MATH] to metallic states crossing on the [MATH]-[MATH] line at high [MATH]-the hallmark of the transition from a usual band insulator to a TCI.', '1401.6643-1-7-4': 'A closer inspection of the spectra shows that for [MATH], the energy gap at [MATH]) decreases when the temperature is lowered.', '1401.6643-1-7-5': 'The gap [MATH] also shrinks with increasing tin content when the samples are kept at room temperature.', '1401.6643-1-7-6': 'The opposite behavior is found for samples with high tin content at low temperatures.', '1401.6643-1-8-0': 'As described in the introduction above, the surface-state dispersion can be viewed as an envelope of the projected bulk bands [cf. Fig. [REF]].', '1401.6643-1-8-1': 'Hence, [MATH] provides an estimate of the bulk band gap at [MATH] [CITATION] and thus, the observed qualitative trends are generally anticipated.', '1401.6643-1-8-2': 'They can be explained with the negative pressure coefficient of the energy gap (the lattice constant decreases with temperature as well as with increasing tin content) together with the phonon-related changes (cf. Ref. [CITATION] and references therein).', '1401.6643-1-8-3': 'Also in line with the expectations is that for PbSe the top of the valence band changes more strongly than the bottom of the conduction band across the range of parameters-a direct consequence of the band repulsion due to the occupied Pb [MATH] level in the valence band [CITATION].', '1401.6643-1-9-0': '# Analysis and discussion', '1401.6643-1-10-0': 'To gain a more quantitative insight in the evolution of the energy gap, in particular in the region where the gap is small, we plot the values of [MATH] in Fig. [REF].', '1401.6643-1-10-1': 'The shown values represent the difference in the surface-state positions at the band edges determined by the local extrema in the dispersion along the high-symmetry directions.', '1401.6643-1-10-2': 'The individual positions are exemplarily marked in Fig. [REF].', '1401.6643-1-10-3': 'We correlate the formation of a metallic surface state with an inverted band structure which, following the usual convention, has a negative energy gap.', '1401.6643-1-10-4': 'The resulting temperature and tin-content dependencies are shown in Figs. [REF] and [REF], respectively.', '1401.6643-1-10-5': 'Concentrating first on the low-[MATH] samples ([MATH]), we see that the band gap evolves smoothly as expected and commonly accepted.', '1401.6643-1-10-6': 'The data are very well described by the phenomenological model [CITATION] [EQUATION]', '1401.6643-1-10-7': 'The values of the parameters are given in Tab. [REF] and they are overall compatible with those found in literature.', '1401.6643-1-10-8': 'However, when turning to the high-[MATH] samples, it is apparent immediately that Eq. [REF] does not hold anymore when the band gap becomes small and inverts.', '1401.6643-1-10-9': 'The curves for [MATH] all show a discontinuous band inversion, irrespective of the critical temperature [Fig. [REF]] or the critical composition [Fig. [REF]].', '1401.6643-1-10-10': 'Moreover, the minimal observed absolute value of [MATH] is about [MATH] meV and therefore of the same order of magnitude as the smallest obtained laser energy [CITATION].', '1401.6643-1-11-0': 'To compare our findings with reference values found in literature, we plot the infrared-absorption data of Ref. [CITATION] in Figs. [REF] and [REF] together with the proposed gap dependencies of Refs. [CITATION] (cf. Tab. [REF]).', '1401.6643-1-11-1': 'Also here, clearly, the data for [MATH] are much better described by the phenomenological curves than the data for [MATH].', '1401.6643-1-11-2': 'For instance, in Fig. [REF] the high-[MATH] slope appears to be distinctively smaller than the low-[MATH] slope (for both, positive and negative gaps), therefore naturally being inconsistent with the linear [MATH] dependency in Eq. [REF].', '1401.6643-1-11-3': 'Even though such deviations were pointed out early on [CITATION], models of type [REF] are still widely in use and also stable gapless bulk states are yet postulated [CITATION].', '1401.6643-1-12-0': 'Before discussing our observation of the discontinuous inversion further, we would like to point out that the determination of the position of the gapped surface states in the normal-insulator case is hindered by the very low spectral weight at [MATH], where these states merge into the projected bulk states (cf. Fig. [REF] and Ref. [CITATION]).', '1401.6643-1-12-1': 'While this creates some uncertainty in the [MATH] values, our estimates are overall conservative.', '1401.6643-1-12-2': 'Additionally, it appears as if the bottom of the conduction-band surface state is shifted away from [MATH].', '1401.6643-1-12-3': 'This shift is quantified in Figs. [REF] and [REF], where it is plotted together with the Dirac-point position in the TCI state.', '1401.6643-1-12-4': 'While the gapped-state shift is essentially constant for all studied samples and temperatures, the Dirac points move away further from [MATH] in the direction of [MATH] when the samples "advance deeper" into the TCI state, consistent with observations made previously for (Pb,Sn)Te [CITATION].', '1401.6643-1-12-5': 'Although, we cannot exclude entirely, that the shift in the normal band-insulator phase is merely an artifact of the missing spectral weight at [MATH], the rather clearly visible dispersion in the [MATH] high-temperature data indicates a real effect that so far cannot be reconciled with model calculations [CITATION].', '1401.6643-1-12-6': 'Notably, this observation is reminiscent of the normal-state behavior close to the topological phase transition found in the tunable topological-insulator system BiTl(S[MATH]Se[MATH] [CITATION].', '1401.6643-1-12-7': 'The latter effect also complicates the exact classification of metallic or gapped surface states in the transition region.', '1401.6643-1-12-8': 'For instance the ([MATH], [MATH] K) data set could be classified both ways-without any influence on the following discussion.', '1401.6643-1-13-0': 'Despite the increased uncertainty from the above considerations, the observation of a nonlinear band evolution and the discontinuous inversion remains intact.', '1401.6643-1-13-1': 'It is consistently found across the whole studied parameter range.', '1401.6643-1-13-2': 'Specifically, a linear gap evolution would suggest the high-[MATH] mixed crystals to be found in the TCI phase at room temperature.', '1401.6643-1-13-3': 'Yet, overall gapped surface states are observed, underlining once more the deviation from Eq. [REF].', '1401.6643-1-13-4': 'In addition, also published ARPES data on (Pb,Sn)Te suggest a similar discontinuous band inversion [cf. Fig. 3(d) of Ref. [CITATION]], yet, this option was not further discussed at the time.', '1401.6643-1-13-5': 'So what is responsible for observing an open bulk band gap throughout all measurements?', '1401.6643-1-13-6': 'It seems that pure lattice dilatation effects play no role here, since computational studies of the pressure and strain dependence of the band gap and the topological transition in binary compounds yield a continuous basically linear inversion with a zero crossing of the gap [CITATION].', '1401.6643-1-13-7': 'However, when changing the temperature, the lattice dilatation constitutes only about half of the gap variation in the lead salts [CITATION].', '1401.6643-1-13-8': 'The other half is attributed to electron-phonon interactions [CITATION].', '1401.6643-1-13-9': 'Hence, lattice vibrations (and also mixed plasmon-phonon modes) are generally a grave factor when determining the accurate electronic structure in the studied materials and inter-band scattering becomes potentially important for small gap sizes.', '1401.6643-1-13-10': 'For degenerate semiconductors, as they are studied here, also carrier-carrier scattering plays a role in the transport properties [CITATION].', '1401.6643-1-13-11': 'Therefore, interaction effects might not be entirely negligible.', '1401.6643-1-13-12': 'Finally, we emphasize that the nature of the band inversion is studied here in a solid solution.', '1401.6643-1-13-13': 'In such compounds disorder is always present and influences the electronic properties, although it does not hinder the transition into the TCI phase [CITATION].', '1401.6643-1-13-14': 'For example, "alloy scattering" is known to reduce the overall carrier mobility in mixed crystals compared to pure binary materials [CITATION].', '1401.6643-1-13-15': 'Only recently, an effort has been made to study the band inversion in (Pb,Sn)Te solid solutions by ab initio methods [CITATION].', '1401.6643-1-13-16': 'It was found that fully ordered structures exhibit semimetallic behavior, but that introducing short-range disorder leads to the formation of semiconductor band gaps, although still smaller than experimentally measured.', '1401.6643-1-13-17': 'Going beyond short-range disorder is, however, very computationally challenging.', '1401.6643-1-14-0': 'Here, we propose a scenario for a band-inversion process in which the zero-gap state is essentially an unstable configuration for the system.', '1401.6643-1-14-1': 'The zero-gap state can generally be described by a band degeneracy forming a Dirac point.', '1401.6643-1-14-2': 'It is a general property of a Dirac point that, in the absence of symmetries protecting the Dirac point, the spectrum easily becomes gapped [CITATION].', '1401.6643-1-14-3': 'In the bulk of Pb[MATH]"2081"208B"2093 1-xSn[MATH]"2093xSethere are no symmetries disallowing having a finite bulk gap at any value of the external parameters, such as [MATH] or [MATH].', '1401.6643-1-14-4': 'Thus nothing prevents the creation of an apparent avoided band crossing even around the topological phase transition.', '1401.6643-1-14-5': '(This lack of symmetry protection for the bulk band degeneracy point in the zero-gap state should not be confused with the symmetry-protected surface-state Dirac spectra, which always appear in the presence of mirror-plane symmetry in the inverted band phase.)', '1401.6643-1-14-6': 'As a consequence, although the bulk band gap necessarily has to be zero at some point, when evolving from a positive (topologically trivial state) to a negative (topologically non-trivial state) band gap, the zero-gap state itself can be essentially unstable and thus effectively not detectable in any measurement, even though the system clearly changes its topological state during the process.', '1401.6643-1-15-0': 'To demonstrate the instability of the zero-energy state within a simple model we use [MATH] theory near the bulk band gap minima at any of the [MATH] points.', '1401.6643-1-15-1': 'The low-energy bulk Hamiltonian can there generally be written as [CITATION]: [EQUATION]', '1401.6643-1-15-2': 'Here [MATH] and [MATH] are Pauli matrices, with the eigenvalues of [MATH]) labeling the cation (Pb or Sn) or the anion (Se) [MATH] orbitals, whereas the spectrum of [MATH]) encodes for the Kramers (total angular momentum) degeneracy.', '1401.6643-1-15-3': 'The orthogonal coordinate system for the [MATH] momentum vector has [MATH] along [MATH]-[MATH] and [MATH] aligned with the [MATH] direction perpendicular to the mirror plane.', '1401.6643-1-15-4': 'Moreover, the sign of the mass term [MATH] determines the topological nature of the material.', '1401.6643-1-15-5': 'The normal phase has [MATH], whereas for [MATH] the band structure is inverted and the material is in the nontrivial topological crystalline phase (found at large [MATH] / small [MATH]).', '1401.6643-1-15-6': 'Note that there cannot be any symmetry protecting a zero-gap ([MATH]) state, since the [MATH] mass term is always present in the Hamiltonian, on both sides of the topological phase transition.', '1401.6643-1-15-7': 'The energy bands of the low-energy Hamitonian Eq. [REF] are doubly degenerate and given by [MATH], which forms an anisotropic Dirac spectrum with an energy gap equal to the mass [MATH].', '1401.6643-1-15-8': 'Experimental evidence for massive bulk Dirac fermions in the TCI phase has been reported by recent transport measurements [CITATION].', '1401.6643-1-15-9': 'The electronic free energy as a function of the mass [MATH] of the band structure of Eq. [REF] is reduced for a finite [MATH] compared to the zero-gap [MATH] spectrum.', '1401.6643-1-15-10': 'Thus, with no symmetries disallowing a finite [MATH] and based on the electronic free energy, a finite energy gap is in fact both allowed and energetically favorable, even around the putative zero-gap phase transition point.', '1401.6643-1-15-11': 'The Hamiltonian Eq. [REF] assumes an intrinsic system with the chemical potential in the middle of the bulk band gap.', '1401.6643-1-15-12': 'However, adding a finite [MATH]-type doping, as evident in the experimental system, does not qualitatively change the results.', '1401.6643-1-16-0': 'Thus, unlike in "true" zero-gap semiconductors [CITATION], a perturbation of the system, like potentially the aforementioned disorder, phonon and interaction contributions, can lead to the stabilization of a fully gapped bulk state.', '1401.6643-1-16-1': 'This scenario implies a very sharp, or possibly even first-order, topological phase transition, in which a finite energy gap is present in the bulk everywhere except in a very narrow region, possibly point, where the gap quickly changes sign.', '1401.6643-1-16-2': 'In fact, the characteristic "flattening out" of a first-order transition of the energy gap as function of the tuning parameter near the topological phase transition is visible in the ARPES data in Figs. [REF] through [REF].', '1401.6643-1-16-3': 'First-order transitions between two topologically distinct phases have previously been theoretically predicted between a topological insulator and a Mott insulating phase in the presence of strong electron-electron interactions [CITATION].', '1401.6643-1-16-4': 'Our results suggest that sharp, and possibly even first-order, topological phase transitions might not just be limited to strongly correlated electron systems.', '1401.6643-1-16-5': 'Thus, even if simple theoretical models, such as tight-binding approaches using the virtual crystal approximation (VCA) for solid solutions, seem to capture the coarse features of the band inversion and the topological phase transition well [CITATION], a more accurate description of the electronic state of the IV-VI narrow-gap semiconductors in the region where the fundamental band gap is of the order of a few tens of millielectronvolts is clearly needed.', '1401.6643-1-16-6': 'Such models would need to go beyond a static treatment of the crystal lattice and the VCA for solid solutions.', '1401.6643-1-17-0': '# Summary', '1401.6643-1-18-0': 'To summarize, we have systematically studied the (001) surface states of (Pb,Sn)Se single crystals in the full rock-salt substitution range and for temperatures between [MATH] K and [MATH] K by means of ARPES.', '1401.6643-1-18-1': 'The presented data establish the phase transition into the TCI state over an unprecedented parameter range.', '1401.6643-1-18-2': 'Furthermore, the evolution of the fundamental band gap of this semiconductor system has been investigated.', '1401.6643-1-18-3': 'Contrary to the expectations, our results provide indications that the anticipated gapless bulk state is essentially unstable during the band-inversion process.', '1401.6643-1-18-4': 'Instead, massive bulk states appear to be formed across the whole studied temperature and composition range.', '1401.6643-1-19-0': 'We thank M. Sahlberg (Uppsala University) and M. Hudl (KTH Royal Institute of Technology) for supporting us with low-temperature powder X-ray diffraction measurements and A. V. Balatsky for stimulating discussions.', '1401.6643-1-19-1': 'This work was made possible through support from the Knut and Alice Wallenberg Foundation, the Swedish Research Council, the European Commission Network SemiSpinNet (PITN-GA-2008-215368), the European Regional Development Fund through the Innovative Economy grant (POIG.01.01.02-00-108/09), and the Polish National Science Centre (NCN) Grant No. 2011/03/B/ST3/02659.', '1401.6643-1-19-2': 'P. D. and B. J. K. acknowledge the support from the Baltic Science Link project coordinated by the Swedish Research Council, VR.'}

{'1401.6643-2-0-0': 'The recent discovery of a topological phase transition in IV-VI narrow-gap semiconductors has revitalized the decades-old interest in the bulk band inversion occurring in these materials.', '1401.6643-2-0-1': 'Here we systematically study the (001) surface states of Pb[MATH]"2081"208B"2093 1-xSn[MATH]"2093xSemixed crystals by means of angle-resolved photoelectron spectroscopy in the parameter space [MATH] "0030"00A0"2264"00A0"0078"00A0 "2264"00A0"0030"002E"0033"00370 <= x <= 0.37 and [MATH].', '1401.6643-2-0-2': 'Using the surface-state observations, we monitor directly the topological phase transition in this solid solution and gain valuable information on the evolution of the underlying fundamental band gap of the system.', '1401.6643-2-0-3': 'In contrast to common model expectations, the band-gap evolution appears to be nonlinear as a function of the studied parameters, resulting in the measuring of a discontinuous band inversion process.', '1401.6643-2-0-4': 'This finding signifies that the anticipated gapless bulk state is in fact not a stable configuration and that the topological phase transition therefore exhibits features akin to a first-order transition.', '1401.6643-2-1-0': '# Introduction', '1401.6643-2-2-0': 'Lead chalcogenides and related compounds have been studied intensely already throughout the last century [CITATION].', '1401.6643-2-2-1': 'These narrow-gap semiconductors are used for applications in infrared lasers [CITATION] and detectors [CITATION] as well as in thermoelectric devices [CITATION].', '1401.6643-2-2-2': 'They also exhibit a large range of peculiar fundamental properties, such as positive temperature and negative pressure coefficients of the energy gap [CITATION] and nonparabolic band dispersions [CITATION].', '1401.6643-2-2-3': 'Moreover, for suitable compounds and parameter ranges, the band gap undergoes an inversion as a function of temperature, pressure [CITATION], and composition [CITATION].', '1401.6643-2-2-4': 'At the interface between inverted and noninverted insulating layers gapless states were predicted to form [CITATION].', '1401.6643-2-2-5': 'The interest in this class of materials was very recently renewed by the investigation of the so-called topological-crystalline-insulator (TCI) state in SnTe [CITATION] as well as the solid solutions (Pb,Sn)Te [CITATION] and (Pb,Sn)Se [CITATION].', '1401.6643-2-2-6': 'In this state of matter, the mirror symmetry present in the rock-salt structure ensures degenerate energy eigenvalues along mirror lines, and hence metallic surface states on certain high-symmetry surfaces [CITATION], when the band gap is inverted.', '1401.6643-2-2-7': 'Additionally, these topologically protected surface states exhibit Dirac-like dispersions and they are spin-momentum-locked [CITATION].', '1401.6643-2-3-0': 'Although the tunable band inversion in the lead and tin chalcogenides has been established for decades, the details of the gap closing remain in the dark.', '1401.6643-2-3-1': 'It is commonly expected (cf. e.g. most of the aforementioned references) that the size of the band gap goes to zero when a critical composition, temperature or pressure is reached.', '1401.6643-2-3-2': 'Yet, a completely closed band gap is naturally hard to observe experimentally.', '1401.6643-2-3-3': 'The lowest confirmed gap values are in the region of a few tens of millielectronvolts.', '1401.6643-2-3-4': 'For instance, in infrared absorption studies, the long-wavelength limit was not accessible [CITATION] and very-low-energy laser emission seems to be hindered by plasmon-phonon excitations [CITATION].', '1401.6643-2-4-0': 'The recent discovery of the TCI phase and the appertaining surface states has opened a new route to study the details of the band inversion in this class of materials by high-resolution photoelectron spectroscopy.', '1401.6643-2-4-1': 'The (001) surface states at the surface high-symmetry point [MATH] lie within the bulk band gap just beyond the valence- and conduction-band edges projected from the bulk [MATH] points [CITATION].', '1401.6643-2-4-2': 'The same is true for the corresponding high-symmetry points of the (110) surface [CITATION].', '1401.6643-2-4-3': 'Hence, studying the surface-state evolution across a parameter range for which the band inversion occurs, can give a reliable estimate of the band gap at [MATH] and thus further elucidate the process of the inversion.', '1401.6643-2-5-0': 'In this article, we report on a systematic angle-resolved photoelectron spectroscopy (ARPES) study of (001)-oriented (Pb,Sn)Se mixed crystals.', '1401.6643-2-5-1': 'The topological phase boundary is established by the investigation of the evolution of the surface states around [MATH] as a function of composition and temperature.', '1401.6643-2-5-2': 'Moreover, contrary to the common theoretical expectations and the conclusions from previous experimental work, our observations point to an unstable zero-gap state (ZGS) in the bulk material.', '1401.6643-2-6-0': '# Experimental details and results', '1401.6643-2-7-0': 'The Pb[MATH]"2081"208B"2093 1-xSn[MATH]"2093xSe([MATH] "0030"00A0"2264"00A0"0078"00A0 "2264"00A0"0030"002E"0033"00370 <= x <= 0.37) single crystals used in this study have been grown by the self-selecting vapor-growth method [CITATION].', '1401.6643-2-7-1': 'Their composition has been determined by means of X-ray diffraction as well as energy-dispersive X-ray spectroscopy.', '1401.6643-2-7-2': 'Powder X-ray diffraction confirmed the rock-salt structure [space group [MATH] (225)] both at room temperature and [MATH] K. To avoid the necessity of extrinsic surface doping [CITATION] which might alter the intrinsic electronic structure, bulk [MATH]-type crystals are used in this study.', '1401.6643-2-7-3': 'The ARPES measurements in the temperature range [MATH] have been conducted on samples cleaved along a (001) surface in ultra-high vacuum.', '1401.6643-2-7-4': 'The experiments have been performed using the laser-based ARPES set-up baltazar equipped with a time-of-flight electron analyzer [CITATION].', '1401.6643-2-7-5': 'Linearly polarized light with a photon energy [MATH] eV was used to excite the electrons.', '1401.6643-2-7-6': 'The total energy and crystal-momentum resolution was about [MATH] meV and better than [MATH], respectively.', '1401.6643-2-8-0': 'Characteristic ARPES spectra are shown in Fig. [REF].', '1401.6643-2-8-1': 'The data in the vicinity of [MATH] are plotted along the high-symmetry lines of the surface Brillouin zone ([MATH]-[MATH]-[MATH]).', '1401.6643-2-8-2': 'In all spectra surface states are discernible.', '1401.6643-2-8-3': 'While the samples at high temperature ([MATH] K) exhibit gapped states for all [MATH] [CITATION], at lower temperatures the spectra show a change from gapped surface states at low [MATH] to metallic (gapless) states crossing on the [MATH]-[MATH] line at high [MATH]-the hallmark of the transition from a usual band insulator to a TCI [CITATION].', '1401.6643-2-8-4': 'A closer inspection of the spectra shows that for [MATH], the energy gap at [MATH]) decreases when the temperature is lowered.', '1401.6643-2-8-5': 'The gap [MATH] also shrinks with increasing [MATH] when the samples are kept at room temperature.', '1401.6643-2-8-6': 'The opposite behavior is found for samples with high [MATH] at low temperatures.', '1401.6643-2-9-0': 'As described in the introduction above, the surface-state dispersion can be viewed as an envelope of the projected bulk bands.', '1401.6643-2-9-1': 'Hence, [MATH] provides an estimate of the bulk band gap at [MATH] [CITATION] and thus, the observed qualitative trends are generally anticipated.', '1401.6643-2-9-2': 'They can be explained with the negative pressure coefficient of the energy gap (the lattice constant decreases with temperature as well as with increasing [MATH]) together with the phonon-related changes (cf. Ref. [CITATION] and references therein).', '1401.6643-2-9-3': 'Also in line with the expectations is that for PbSe the top of the valence band changes more strongly than the bottom of the conduction band across the range of parameters-a direct consequence of the band repulsion due to the occupied Pb [MATH] level in the valence band [CITATION].', '1401.6643-2-10-0': '# Analysis and discussion', '1401.6643-2-11-0': 'To gain a more quantitative insight in the evolution of the energy gap, in particular in the region where the gap is small, we plot the values of [MATH] in Fig. [REF].', '1401.6643-2-11-1': 'The shown values represent the difference in the surface-state positions at the band edges determined by the local extrema in the dispersion along the high-symmetry directions.', '1401.6643-2-11-2': 'The determined band-edge positions are marked in Fig. [REF] [CITATION].', '1401.6643-2-11-3': 'We correlate the formation of a metallic surface state with an inverted band structure which, following the usual convention, has a negative energy gap.', '1401.6643-2-11-4': 'The resulting temperature and tin-content dependencies are shown in Figs. [REF] and [REF], respectively.', '1401.6643-2-11-5': 'Concentrating first on the low-[MATH] samples ([MATH]), we see that the band gap evolves nearly linearly as expected and commonly accepted.', '1401.6643-2-11-6': 'The data are very well described by the phenomenological model [CITATION] [EQUATION]', '1401.6643-2-11-7': 'The values of the parameters are given in Tab. [REF] and they are overall compatible with those found in literature.', '1401.6643-2-11-8': 'However, when turning to the high-[MATH] samples, it is apparent immediately that Eq. [REF] does not hold anymore when the band gap becomes small and inverts.', '1401.6643-2-11-9': 'The curves for [MATH] all show a discontinuous band inversion, irrespective of the critical temperature [Fig. [REF]] or the critical composition [Fig. [REF]].', '1401.6643-2-11-10': 'Moreover, the minimal observed absolute value of [MATH] is about ([MATH] to [MATH]) meV and therefore slightly less but of the same order of magnitude as the smallest obtained laser energy [CITATION].', '1401.6643-2-11-11': 'Even though deviations from the approximately linear gap evolution were pointed out early on [CITATION], models of type [REF] are still widely in use and also stable gapless bulk states are yet postulated [CITATION].', '1401.6643-2-12-0': 'Before discussing the observation of the discontinuous inversion further, we would like to point out that the determination of the position of the gapped surface states in the normal-insulator case is somewhat hindered by the very low spectral weight at [MATH], where these states merge into the projected bulk states (cf. Fig. [REF] and Ref. [CITATION]).', '1401.6643-2-12-1': 'While this creates some uncertainty in the [MATH] values in the normal state, our estimates-based on the continuous bands having a finite curvature-are overall conservative.', '1401.6643-2-12-2': 'Additionally, it appears as if occasionally the bottom of the conduction-band surface state is shifted away from [MATH] reminiscent of a "Rashba-like-split" surface state.', '1401.6643-2-12-3': 'Although, we cannot exclude entirely, that the shift in the normal band-insulator phase is merely an artifact of the missing spectral weight at [MATH], the rather clearly visible dispersion in the [MATH] high-temperature data indicates a real effect that so far cannot be reconciled with model calculations [CITATION].', '1401.6643-2-12-4': 'It is worth noting that for (Pb,Sn)Te recent calculations indicate a substantial influence of a finite surface potential gradient on the (111) surface states both in the TCI and the normal state [CITATION].', '1401.6643-2-12-5': 'Until similar calculations become available for the (001) surface we only further quantify the separation of the Dirac points on the mirror line [MATH]-[MATH] in the TCI state of (Pb,Sn)Se.', '1401.6643-2-12-6': 'As shown in Figs. [REF] and [REF] the Dirac points move away farther from [MATH] in the direction of [MATH] when the samples "advance deeper" into the TCI state, consistent with observations made previously for (Pb,Sn)Te [CITATION].', '1401.6643-2-13-0': 'Despite the above considerations, the observation of a nonlinear band evolution and the discontinuous inversion across the whole studied parameter range remains intact.', '1401.6643-2-13-1': 'Specifically, a linear gap evolution would suggest the high-[MATH] mixed crystals to be found in the TCI phase at room temperature.', '1401.6643-2-13-2': 'Yet, overall gapped surface states are observed.', '1401.6643-2-13-3': 'In addition, also published ARPES data on (Pb,Sn)Te suggest a similar discontinuous band inversion [cf. Fig. 3(d) of Ref. [CITATION]].', '1401.6643-2-13-4': 'So what is responsible for observing an open bulk band gap throughout all measurements?', '1401.6643-2-13-5': 'It seems that pure lattice dilatation effects play no role here, since computational studies of the pressure and strain dependence of the band gap and the topological transition in binary compounds yield a continuous basically linear inversion with a zero crossing of the gap [CITATION].', '1401.6643-2-13-6': 'However, when changing the temperature, the lattice dilatation constitutes only about half of the gap variation in the lead salts [CITATION].', '1401.6643-2-13-7': 'The other half is attributed to electron-phonon interactions [CITATION].', '1401.6643-2-13-8': 'Hence, lattice vibrations (and also mixed plasmon-phonon modes) are generally a grave factor when determining the accurate electronic structure in the studied materials and inter-band scattering becomes potentially important for small gap sizes.', '1401.6643-2-13-9': 'For degenerate semiconductors, as they are studied here, also carrier-carrier scattering plays a role in the transport properties [CITATION].', '1401.6643-2-13-10': 'Therefore, interaction effects might not be entirely negligible.', '1401.6643-2-13-11': 'Finally, we emphasize that the nature of the band inversion is studied here in a solid solution.', '1401.6643-2-13-12': 'In such compounds disorder is always present and influences the electronic properties, although it does not hinder the transition into the TCI phase [CITATION].', '1401.6643-2-13-13': 'For example, "alloy scattering" is known to reduce the overall carrier mobility in mixed crystals compared to pure binary materials [CITATION].', '1401.6643-2-13-14': 'Only recently, an effort has been made to study the band inversion in (Pb,Sn)Te solid solutions by ab initio methods [CITATION].', '1401.6643-2-13-15': 'It was found that fully ordered structures exhibit semimetallic behavior, but that introducing short-range disorder leads to the formation of semiconductor band gaps, although still smaller than experimentally measured.', '1401.6643-2-13-16': 'Going beyond short-range disorder is, however, very computationally challenging.', '1401.6643-2-14-0': 'Here, we propose a scenario for a band-inversion process in which the ZGS is essentially an unstable configuration for the system.', '1401.6643-2-14-1': 'The ZGS can generally be described by a band degeneracy forming a Dirac point.', '1401.6643-2-14-2': 'It is a general property of a Dirac point that, in the absence of symmetries protecting it, the spectrum easily becomes gapped [CITATION].', '1401.6643-2-14-3': 'In the bulk of Pb[MATH]"2081"208B"2093 1-xSn[MATH]"2093xSethere are no symmetries disallowing having a finite bulk gap at any value of the external parameters, such as [MATH] or [MATH], even around the topological phase transition (TPT).', '1401.6643-2-14-4': 'As a consequence, although the bulk band gap necessarily has to be zero at some point, when evolving from a positive (topologically trivial state) to a negative (topologically non-trivial state) band gap, the ZGS itself can be essentially unstable and thus effectively not detectable in any measurement.', '1401.6643-2-15-0': 'To demonstrate the instability of the ZGS within a simple model we use [MATH] theory near the bulk band gap minima at any of the [MATH] points.', '1401.6643-2-15-1': 'The low-energy bulk Hamiltonian can there generally be written as [CITATION]: [EQUATION]', '1401.6643-2-15-2': 'Here [MATH] and [MATH] are Pauli matrices, with the eigenvalues of [MATH]) labeling the cation (Pb or Sn) or the anion (Se) [MATH] orbitals, whereas the spectrum of [MATH]) encodes for the Kramers (total angular momentum) degeneracy.', '1401.6643-2-15-3': 'The orthogonal coordinate system for the [MATH] momentum vector has [MATH] along [MATH]-[MATH] and [MATH] aligned with the [MATH] direction perpendicular to the mirror plane.', '1401.6643-2-15-4': 'Moreover, the sign of the mass term [MATH] determines the topological nature of the material.', '1401.6643-2-15-5': 'The normal phase has [MATH], whereas for [MATH] the band structure is inverted and the material is in the nontrivial TCI phase.', '1401.6643-2-15-6': 'Note that there cannot be any symmetry protecting a ZGS ([MATH]), since the [MATH] mass term is always present in the Hamiltonian, on both sides of the TPT.', '1401.6643-2-15-7': 'The energy bands of the low-energy Hamitonian Eq. [REF] are doubly degenerate and given by [MATH], which forms an anisotropic Dirac spectrum with an energy gap equal to the mass [MATH].', '1401.6643-2-15-8': 'Experimental evidence for massive bulk Dirac fermions in the TCI phase has been reported by recent transport measurements [CITATION].', '1401.6643-2-15-9': 'Using the band structure we calculate the electronic free energy as a function of the mass [MATH] for the Hamiltonian in Eq. [REF].', '1401.6643-2-15-10': 'We find that the free energy is reduced for a finite [MATH] compared to the [MATH] spectrum.', '1401.6643-2-15-11': 'Thus, with no symmetries disallowing a finite [MATH] and based on the electronic free energy, a finite energy gap is both allowed and energetically favorable.', '1401.6643-2-15-12': 'Equation [REF] assumes an intrinsic system with the chemical potential in the middle of the bulk band gap.', '1401.6643-2-15-13': 'However, we find that adding a finite [MATH]-type doping, as evident in the experimental system, does not qualitatively change the free energy preference of the gapped system.', '1401.6643-2-16-0': 'Thus, unlike in "true" zero-gap semiconductors [CITATION], a perturbation of the system, like potentially the aforementioned disorder, phonon and interaction contributions, can lead to the stabilization of a fully gapped bulk state.', '1401.6643-2-16-1': 'This scenario implies a very sharp, or possibly even first-order, TPT, in which a finite energy gap is present in the bulk everywhere except in a very narrow region, where the gap quickly changes sign.', '1401.6643-2-16-2': 'In fact, the characteristic "flattening out" of a first-order transition of the energy gap as function of the tuning parameter near the TPT is visible in the ARPES data in Figs. [REF] and [REF].', '1401.6643-2-16-3': 'First-order transitions between two topologically distinct phases have previously been predicted between a topological insulator and a Mott insulating phase in the presence of strong electron-electron interactions [CITATION].', '1401.6643-2-16-4': 'Our results suggest that sharp, and possibly even first-order, TPTs might not just be limited to strongly correlated electron systems.', '1401.6643-2-16-5': 'Thus, even if simple theoretical models, such as tight-binding approaches using the virtual crystal approximation (VCA) for solid solutions, seem to capture the coarse features of the band inversion and the TPT well [CITATION], a more accurate description of the electronic state of the IV-VI narrow-gap semiconductors in the region where the fundamental band gap is of the order of a few tens of millielectronvolts is clearly needed.', '1401.6643-2-16-6': 'Such models would need to go beyond a static treatment of the crystal lattice and the VCA for solid solutions.', '1401.6643-2-17-0': 'We thank M. Sahlberg (Uppsala University) and M. Hudl (KTH Royal Institute of Technology) for supporting us with low-temperature powder X-ray diffraction measurements and A. V. Balatsky for stimulating discussions.', '1401.6643-2-17-1': 'This work was made possible through support from the Knut and Alice Wallenberg Foundation, the Swedish Research Council, the European Commission Network SemiSpinNet (PITN-GA-2008-215368), the European Regional Development Fund through the Innovative Economy grant (POIG.01.01.02-00-108/09), and the Polish National Science Centre (NCN) Grant No. 2011/03/B/ST3/02659.', '1401.6643-2-17-2': 'P. D. and B. J. K. acknowledge the support from the Baltic Science Link project coordinated by the Swedish Research Council, VR.', '1401.6643-2-18-0': '1.5', '1401.6643-2-19-0': '# Supplemental material', '1401.6643-2-20-0': 'Figures [REF], [REF], and [REF] included in this document show ARPES spectra along the high-symmetry lines in the vicinity of [MATH] of the (001) surface of Pb[MATH]"2081"208B"2093 1-xSn[MATH]"2093xSe.', '1401.6643-2-20-1': 'While Fig. [REF] in the main text shows a representative subset of data, here all analyzed spectra contributing to Fig. [REF] in the main text are depicted, both as series of energy distribution curves (EDCs) and as color plots.', '1401.6643-2-20-2': 'The EDC intensity is shown on a linear scale and the spacing between neighboring EDCs is roughly given by [MATH], where [MATH] is the bulk lattice constant.', '1401.6643-2-20-3': 'The intensity scale of the corresponding color plots is linear as well.', '1401.6643-2-20-4': 'In each figure, the solid (open) circles mark the determined EDC peak (band edge) positions.', '1401.6643-2-20-5': 'The different colors are merely used to visualize the derived [MATH] and [MATH] character of the valence- and conduction-band edges.', '1401.6643-2-20-6': 'The data shown in the panels marked with an asterisk [[MATH], ([MATH] K, [MATH])] have been acquired in a separate measurement on a different sample than used for the other temperature points at [MATH].', '1401.6643-2-20-7': 'The spectra in the panels marked with a dagger [[MATH], ([MATH] K, [MATH])] indicate the formation of a gap in the surface state at low temperatures as previously observed in Ref. [CITATION].', '1401.6643-2-20-8': 'Further details on the latter shall be discussed elsewhere.', '1401.6643-2-21-0': 'Eventually, Fig. [REF] depicts the absolute value of the determined energy gap of Pb[MATH]"2081"208B"2093 1-xSn[MATH]"2093xSeat [MATH] as a function of temperature and tin content.', '1401.6643-2-21-1': 'The data are identical to those shown in Fig. [REF] of the main text, however, without following the convention of choosing a negative value for the gap in the band-inverted TCI state.', '1401.6643-2-22-0': '1.0'}

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[]

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[]

['1401.6643-1-0-1', '1401.6643-1-6-0', '1401.6643-2-0-1', '1401.6643-2-7-0', '1401.6643-2-18-0', '1401.6643-2-22-0']

{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1401.6643

cond-mat-0506072

{'cond-mat-0506072-1-0-0': 'Results of magnetization, magnetotransport and Mossbauer spectroscopy measurements of sequentially evaporated Fe-Ag granular composites are presented.', 'cond-mat-0506072-1-0-1': 'The strong magnetic scattering of the conduction electrons is reflected in the sublinear temperature dependence of the resistance and in the large negative magnetoresistance.', 'cond-mat-0506072-1-0-2': 'The simultaneous analysis of the magnetic properties and the transport behavior suggests a bimodal grain size distribution.', 'cond-mat-0506072-1-0-3': 'A detailed quantitative description of the unusual features observed in the transport properties is given.', 'cond-mat-0506072-1-1-0': '# INTRODUCTION', 'cond-mat-0506072-1-2-0': 'As promising candidates for magnetic recording and sensor applications heterogeneous magnetic materials, i. e. multilayer structures [CITATION] of alternating ferromagnetic and nonmagnetic layers and granular composites [CITATION] have been studied widely in the last two decades.', 'cond-mat-0506072-1-2-1': 'The giant magnetoresistance (GMR) in these systems have been explained by elastic scattering of the conduction electrons on magnetic moments of differently aligned magnetic entities.', 'cond-mat-0506072-1-2-2': 'Gittleman et al. have shown that in superparamagnetic granular alloys this consideration leads to a magnetoresistance proportional to the square of the magnetization.[', 'cond-mat-0506072-1-2-3': '[CITATION] Deviations from this relation has been attributed to the size distribution of the magnetic scatterers.[', 'cond-mat-0506072-1-2-4': '[CITATION] and interactions between these scatterers [CITATION] Various assumptions on the form of the size distribution have been made in order to obtain a phenomenological description of the GMR phenomena in different granular systems.', 'cond-mat-0506072-1-3-0': 'In this paper we present a systematical study of the magnetic and magnetotransport properties of vacuum evaporated granular Fe-Ag structures.', 'cond-mat-0506072-1-3-1': 'The observed large, negative non-saturating magnetic field dependence and the unusual sublinear temperature dependence ([MATH]) of the resistivity have been analyzed simultaneously.', 'cond-mat-0506072-1-3-2': 'This allowed the separation of the various scattering processes and the identification of two characteristic grain size determining the macroscopic magnetic and transport properties without making any assumption on the grain size distribution.', 'cond-mat-0506072-1-4-0': '# EXPERIMENTAL', 'cond-mat-0506072-1-5-0': 'The Fe-Ag multilayer samples were prepared by sequential vacuum evaporation in a base pressure of [MATH] Pa onto Si(111) single crystal substrates at room temperature.', 'cond-mat-0506072-1-5-1': 'The mass of the deposited material was measured by a quartz oscillator and the nominal layer thickness was calculated using the bulk density of Fe and Ag.', 'cond-mat-0506072-1-5-2': 'In this paper specimens prepared with the following sequences are discussed:', 'cond-mat-0506072-1-6-0': '(A) [Ag(2.6nm)/Fe(0.2nm)][MATH]/Ag(2.6nm)', 'cond-mat-0506072-1-7-0': '(B) [Ag(1.3nm)/Fe(0.2nm)][MATH]/Ag(1.3nm)', 'cond-mat-0506072-1-8-0': '(C) [Ag(0.8nm)/Fe(0.2nm)][MATH]/Ag(0.8nm)', 'cond-mat-0506072-1-9-0': 'Structural characterization of the samples by X-ray diffraction indicated a nanometer scale grain size of the constituents [for details see Ref. 13], however, due to the strong overlap between the diffraction lines of bcc-Fe and fcc-Ag, a quantitative evaluation of the size of the magnetic Fe grains was not possible.', 'cond-mat-0506072-1-9-1': 'The absence of peaks of the X-ray reflectivity[CITATION] when the Fe layer thickness is less than [MATH] nm is attributed to discontinuities of the Fe layers.', 'cond-mat-0506072-1-9-2': 'This limit of the continuous layer regime is quite similar to that observed in other multilayer systems [CITATION] composed of transition metals with immiscible nonmagnetic elements and these kind of discontinuous multilayers are often referred to as granular multilayers [CITATION].', 'cond-mat-0506072-1-9-3': 'The relation between the size of the grains in granular multilayers and the nominal layer thickness is generally determined by a three-dimensional growth process [CITATION]and the average diameter of the grains can be much larger than the nominal layer thickness.', 'cond-mat-0506072-1-9-4': 'It depends on the material parameters (e.g. lattice parameter mismatch, surface energy, etc.), as well as various parameters (deposition rate, substrate temperature, etc.) of the deposition technique.', 'cond-mat-0506072-1-9-5': 'The decrease of the spacer layer thickness was found [CITATION] to increase the magnetic grain size in an other series of Fe/Ag granular multilayer samples.', 'cond-mat-0506072-1-10-0': 'The magnetic structure of the samples was examined by a superconducting quantum interference device (SQUID) and by transmission Mossbauer spectroscopy.', 'cond-mat-0506072-1-10-1': 'For the latter purpose the samples have been removed from the substrate and folded up to achieve an appropriate thickness for transmission measurements.', 'cond-mat-0506072-1-11-0': 'The largest GMR and the strongest magnetic scattering of the conduction electrons were observed in sample A, as it will be shown later.', 'cond-mat-0506072-1-11-1': 'The [MATH]Fe Mossbauer spectra of this sample taken at several temperatures are shown in Fig. [REF].', 'cond-mat-0506072-1-11-2': 'The room temperature spectrum contains a paramagnetic doublet with a large isomer shift relative to [MATH]-Fe (0.18 mm/s) and a quadrupole splitting (0.45 mm/s) characteristic to a system of small Fe clusters embedded in an Ag matrix [CITATION].', 'cond-mat-0506072-1-11-3': 'The low temperature spectra show that the sample is superparamagnetic (SPM) and as the magnetic clusters gradually freeze below the blocking temperature (around 50K), the six-line pattern characteristic to the Zeeman splitting of the nuclear levels of [MATH]Fe appears.', 'cond-mat-0506072-1-11-4': 'At [MATH]K the paramagnetic fraction is absent, but the spectral lines are much broader and the hyperfine parameters are different than those of bulk bcc-Fe.', 'cond-mat-0506072-1-11-5': 'The 4.2K spectrum could be fitted with a distribution of hyperfine fields[CITATION] with an average value, [MATH]T and standard deviation [MATH]2.4T.', 'cond-mat-0506072-1-12-0': 'It is worth noting that the intensity ratios of the six lines indicate a significant spontaneous alignment of the magnetic moments.', 'cond-mat-0506072-1-12-1': 'The intensity of the six lines of a sextet is distributed as [MATH], where [MATH] is the intensity of the [MATH] transitions, and [MATH] is the angle between the direction of the gamma-ray (perpendicular to the sample plane) and the magnetic moment.', 'cond-mat-0506072-1-12-2': 'In case of a random distribution of the magnetization directions [MATH].', 'cond-mat-0506072-1-12-3': 'The observed small intensity, [MATH], indicates a close-to-perpendicular alignment of the magnetic moments with respect to the sample plane.', 'cond-mat-0506072-1-12-4': 'Applying a magnetic field perpendicular to the sample plane could fully align the moments parallel to the field, as it is indicated by the [MATH] intensity.', 'cond-mat-0506072-1-13-0': 'The spectra measured in external magnetic field were also fitted with a distribution of the hyperfine fields and the parameters obtained are: [MATH] and 27.8 and [MATH] and 2.4T for [MATH] and 7T.', 'cond-mat-0506072-1-13-1': 'The external magnetic field does not affect the width of the distribution indicating that it is not due to relaxation of the magnetic moments.', 'cond-mat-0506072-1-13-2': 'The broad spectral lines of the observed sextet result from a distribution of the Fe neighborhoods, which can be due both to the large number of surface atoms in small grains and to non-equilibrium mixing of the elements [CITATION] during the growth process.', 'cond-mat-0506072-1-13-3': 'The field-independent width and the decrease of the observed average hyperfine field in external field indicate the ferromagnetic alignment of the magnetic grains.', 'cond-mat-0506072-1-13-4': 'The hyperfine field of a ferromagnet is decreased by the applied field, since it is oriented antiparallel to the magnetic moment.', 'cond-mat-0506072-1-13-5': 'We note that at 4.2K the statistical errors allow an SPM fraction containing less than 2 atomic % of the Fe atoms.', 'cond-mat-0506072-1-14-0': 'The freezing of the superparamagnetic moments - seen in the Mossbauer spectra - also appears in the temperature dependence of the low field susceptibility measured by the SQUID.', 'cond-mat-0506072-1-14-1': 'Figure [REF] shows the results for the 3 samples, after cooling them either in zero or in 1 mT permanent magnetic field.', 'cond-mat-0506072-1-14-2': 'The blocking temperatures of about 40, 150 and 300K for samples A, B and C, respectively can be deduced from this experiment.', 'cond-mat-0506072-1-14-3': 'Sample A ([MATH]K) exhibits a textbook example for the superparamagnetic behavior.', 'cond-mat-0506072-1-14-4': 'The maximum of the ZFC curve is more smeared out for samples B and C and in case of sample B the irreversibility temperature, where the FC curve starts to deviate from the ZFC one, is much larger than the [MATH] defined by the ZFC maximum.', 'cond-mat-0506072-1-14-5': 'Such features are generally explained[CITATION] as the effect of a grain size distribution and interaction among the magnetic particles, which obviously play a role as the average grain size and the concentration of the magnetic component increase.', 'cond-mat-0506072-1-15-0': 'The magnetization versus applied magnetic field curves of sample A are shown in Fig. [REF] up to [MATH]T at four different temperatures.', 'cond-mat-0506072-1-15-1': 'Above the blocking temperature the magnetization can quite well be described[CITATION] by a single Langevin function, as expected for a superparamegnetic system of uniform grains.', 'cond-mat-0506072-1-15-2': 'The average magnetic moment of the SPM grains deduced from the fit is [MATH].', 'cond-mat-0506072-1-15-3': 'It indicates that the bulk magnetic properties are mainly determined by the average size grains (about 1.8nm) and the effect of the size distribution is negligible.', 'cond-mat-0506072-1-16-0': 'The magnetotransport measurements have been performed in the current in plane arrangement by four contact method.', 'cond-mat-0506072-1-16-1': 'The coincidence of the parallel ([MATH]) and transversal ([MATH]) magnetoresistance and the absence of any anisotropic component is characteristic to the GMR phenomenon in granular systems[CITATION].', 'cond-mat-0506072-1-16-2': 'The temperature dependence of the resistivity of samples A-C in zero and 12T magnetic field normalized to their [MATH]K values are displayed in Fig. [REF].', 'cond-mat-0506072-1-16-3': 'Contrary to ordinary metallic systems, the resistivity is sublinear above 40 K for each sample.', 'cond-mat-0506072-1-16-4': 'Similar observation was reported by Milner et al. for granular systems,[CITATION] but only in presence of a high magnetic field.', 'cond-mat-0506072-1-16-5': 'As discussed later in detail, the above qualitative feature of the zero field temperature dependencies signifies the presence of an extremely strong magnetic scattering in our samples.', 'cond-mat-0506072-1-17-0': 'The anomalous character of the [MATH] curves is the most dominant in case of sample A. Simultaneously, this sample exhibits the largest magnetoresistance at low temperature, as shown in the bottom panel of Fig. [REF].', 'cond-mat-0506072-1-17-1': 'The magnitude of the GMR measured at [MATH]K in a field of [MATH]T is 26%, 18% and 16%, for samples A, B and C, respectively.', 'cond-mat-0506072-1-17-2': 'Note that at room temperature this order is reversed.', 'cond-mat-0506072-1-18-0': 'The magnetoresistance curves of the three samples have a common feature, they do not saturate even in high magnetic fields and at low temperatures, where the magnetization seems already to be saturated.', 'cond-mat-0506072-1-18-1': 'This indicates that significant magnetic scattering takes place at magnetic entities much smaller than the typical grain size which determines the macroscopic magnetization.', 'cond-mat-0506072-1-18-2': 'It is well known that the scattering amplitude of ferromagnetic grains embedded in a nonmagnetic metallic matrix is size dependent and the contribution of the smaller clusters is strongly enhanced.', 'cond-mat-0506072-1-18-3': '[CITATION] The magnetic moments of these smaller clusters are harder to rotate by an applied magnetic field thus the saturation of the magnetoresistance is slower than that of the net magnetization arising dominantly from the larger grains.', 'cond-mat-0506072-1-18-4': 'Note that similar behavior was found found in Fe-Ag co-deposited granular films [CITATION] as well as in many other systems (for a review see [CITATION]).', 'cond-mat-0506072-1-18-5': 'In the next section we give a more detailed analysis of the above qualitative picture.', 'cond-mat-0506072-1-19-0': '# ANALYSIS', 'cond-mat-0506072-1-20-0': 'In order to identify the resistivity contribution of the magnetic scattering process we assume that the Matthiessen-rule can be applied, i.e the resistivity is composed of 3 terms: [EQUATION]', 'cond-mat-0506072-1-20-1': 'Here [MATH] denotes the residual resistivity, [MATH] is the contribution arising from the phonon scattering, and the remaining part is attributed to the magnetic scattering.', 'cond-mat-0506072-1-20-2': 'The separation of the latter term requires further assumptions.', 'cond-mat-0506072-1-21-0': 'Instead of making assumption on the shape of the grain size distribution function[CITATION] we use physical considerations to get insight into the size distribution of the grains determining the magnetic scattering.', 'cond-mat-0506072-1-21-1': 'It is based on the analysis of the temperature dependence of the resistivity measured in zero and high magnetic field, as shown in Figs. [REF] and [REF].', 'cond-mat-0506072-1-21-2': 'In these limits the large grains - which dominate the bulk magnetic properties - do not give temperature dependent contribution to the magnetic scattering, as the system is either nonmagnetic (zero field limit, above the blocking temperature) or fully polarized ferromagnetically (high field limit).', 'cond-mat-0506072-1-21-3': 'We will see that the small clusters can also be characterized by a single average size, and in high fields the temperature dependence of the magnetic scattering is determined solely by this characteristic size: the magnetic moment of these small clusters is the only fitting parameter for the calculated curves describing the shapes of [MATH] shown in Fig. [REF].', 'cond-mat-0506072-1-21-4': 'Finally, the consistency of our analysis will also be demonstrated by evaluating the magnetic field dependence of the resistivity in the whole magnetic field range (Fig. [REF]) using the characteristic sizes determined independently from the magnetization data and the temperature dependence of the resistivity in the zero and high field limits.', 'cond-mat-0506072-1-22-0': 'In a granular system the magnetic scattering depends on the correlation between the localized magnetic moments of the grains, [MATH], while a conduction electron is scattered from [MATH] to [MATH] within its spin-diffusion length[CITATION].', 'cond-mat-0506072-1-22-1': 'In zero magnetic field, well above the blocking temperature the magnetic moments of the grains (including the large grains) are fully disordered.', 'cond-mat-0506072-1-22-2': 'In these circumstances no temperature dependence is expected from the magnetic scattering, at least until the spin diffusion length is large enough.', 'cond-mat-0506072-1-22-3': 'We assume that at high temperatures the temperature dependence of the resistivity arise solely from the phonon contribution, [EQUATION]', 'cond-mat-0506072-1-22-4': 'As the the phonon term is linear above the Debye temperature ([MATH] K)[CITATION], the strength of phonon scattering, [MATH], can be determined from the high temperature slope of the zero field resistivity curves.', 'cond-mat-0506072-1-22-5': 'The calculated [MATH] curves are shown in Fig. [REF] by dashed lines for the 3 samples.', 'cond-mat-0506072-1-22-6': 'The difference of the total resistivity and its phonon related part is attributed to the magnetic scattering, and [MATH] is displayed in Fig. [REF] by dotted lines for each sample.', 'cond-mat-0506072-1-22-7': 'As it was expected, the magnetic scattering is temperature independent at high temperatures and the lower the blocking temperature the wider is the flat part of the separated magnetic contribution curve.', 'cond-mat-0506072-1-22-8': 'It is also seen that the magnetic scattering gradually decreases as the blocking temperature is approached from above.', 'cond-mat-0506072-1-22-9': 'Note, however, that at [MATH]K there is still a considerable contribution from it, i.e. [MATH] as it can be seen from Fig. [REF].', 'cond-mat-0506072-1-23-0': 'Since the phonon term is magnetic field independent, the [MATH] curves determined from the zero field temperature dependencies can be used to separate the magnetic scattering contribution in the high field measurements.', 'cond-mat-0506072-1-23-1': 'This is shown in Fig. [REF] for the [MATH]T measurements.', 'cond-mat-0506072-1-23-2': 'A comparison of the related panels of Fig. [REF] and Fig. [REF] reveals that sample A has the strongest negative curvature of the resistivity and the biggest change in [MATH].', 'cond-mat-0506072-1-23-3': 'In this sample magnetic scattering at [MATH]T dominates over even the phonon term in a very broad temperature range.', 'cond-mat-0506072-1-24-0': 'In order to describe the evaluated [MATH] curves we assume that in the high field limit the magnetic scattering of the spin-polarized electrons is proportional to the spin disorder of the small clusters.', 'cond-mat-0506072-1-24-1': 'The magnetic moments of the large grains are fully aligned by the applied magnetic field, as it could be deduced from the Mossbauer spectra of Fig. [REF].', 'cond-mat-0506072-1-24-2': 'The spin disorder for a characteristic moment [MATH] is described by the Brillouin-function: [EQUATION]', 'cond-mat-0506072-1-24-3': 'Here, [MATH] and [MATH] are the total spin and its [MATH]-component of the scatterers, respectively.', 'cond-mat-0506072-1-25-0': 'The fitted [MATH] curves are shown in Fig. [REF] by solid lines.', 'cond-mat-0506072-1-25-1': 'Apart from a normalization factor, the only fitting parameter is the magnetic moment characteristic to the small Fe clusters.', 'cond-mat-0506072-1-25-2': 'The good agreement of the experimental and the calculated curves indicates that the size distribution of these clusters is negligible.', 'cond-mat-0506072-1-25-3': 'The fitted values are in the same order of magnitude for all the samples; [MATH][MATH], 17[MATH] and 12.5[MATH] for samples A-C respectively.', 'cond-mat-0506072-1-26-0': 'Next we discuss the magnetic field dependence of the resistivity of sample A, which exhibits the strongest magnetic scattering in the superparamagnetic state, where the process of magnetic saturation is well understood.', 'cond-mat-0506072-1-26-1': 'For a numerical analysis we use the two characteristic sizes determined from the previous experiments.', 'cond-mat-0506072-1-26-2': 'The Langevin fit to the magnetization experiments performed in the superparamagnetic temperature range has shown the presence of large grains with [MATH][MATH] (Fig. [REF]), while the temperature dependence of the resistivity in high magnetic field indicated the presence of small clusters with [MATH][MATH].', 'cond-mat-0506072-1-26-3': "Following Gittleman's model [CITATION] we describe the magnetoresistance by the field dependence of the correlation between the localized magnetic moments responsible for an initial and a final magnetic scattering process: [EQUATION]", 'cond-mat-0506072-1-26-4': 'In case of the observed two largely different grain sizes this can be expressed as [EQUATION] where [MATH] is the Brillouin-function defined under Eq. [REF] for the small iron clusters and [MATH] is the Langevin-function, i.e. the classical limit of [MATH] for the large Fe grains.', 'cond-mat-0506072-1-26-5': 'The parameters [MATH], [MATH] and [MATH] represent the relative weights of scattering from grain to grain, between a grain and a cluster and from cluster to cluster.', 'cond-mat-0506072-1-27-0': 'Figure [REF] shows magnetoresistance for sample A in the superparamagnetic phase and the expected variation calculated by Eq. [REF].', 'cond-mat-0506072-1-27-1': 'The two characteristic magnetic moments ([MATH] and [MATH] ) are determined in the previous analysis from independent experiments: the field and temperature dependence of the bulk magnetization, and the temperature dependence of the resistivity.', 'cond-mat-0506072-1-27-2': 'The relative weights of the various processes were used as fitting parameters: [MATH], [MATH] and [MATH].', 'cond-mat-0506072-1-27-3': 'The value of [MATH], i.e the relatively weight for the grain[MATH]grain scattering process is larger than [MATH], even though the amplitude of the grain scattering is small.', 'cond-mat-0506072-1-27-4': '[CITATION] It reflects the large probability of scattering from grain to grain due to the large volume fraction of this type of magnetic scatterer.', 'cond-mat-0506072-1-27-5': 'The direct interplay between the clusters is negligible, as expected for a small fraction (less than 2%).', 'cond-mat-0506072-1-28-0': 'In Fig. [REF] the dashed and the dashed-dotted lines represents the contribution of the two dominant scattering processes.', 'cond-mat-0506072-1-28-1': 'The grain[MATH]grain scattering process is the leading term in Eq. [REF] up to 4T magnetic field.', 'cond-mat-0506072-1-28-2': 'The dashed line corresponds to the square of the magnetization, and if this would be the only scattering process the simple relation[CITATION] of [MATH] would hold.', 'cond-mat-0506072-1-28-3': 'Above [MATH]T, however, the second term of Eq. [REF] dominates and the observed behavior can be attributed to the effect of the small clusters, not seen in the magnetization curves.', 'cond-mat-0506072-1-28-4': 'The good description of the measured magnetization and resistance curves by two characteristic sizes is the consequence of the narrow size distributions of the grains and the clusters.', 'cond-mat-0506072-1-29-0': 'Bimodal distribution of the grain size has already been observed in granular systems prepared by co-deposition [CITATION], rapid quenching from the melt, [CITATION], or layered growth [CITATION] of the constituents.', 'cond-mat-0506072-1-29-1': 'Since for our samples a non-saturating magnetoresistance indicates the presence of small clusters even in case of 25 nm thick continuous Fe layers [CITATION], we associate the large grains and the small culsters to Fe rich grains of the granular layers and small Fe clusters trapped inside the Ag layers, respectively.', 'cond-mat-0506072-1-29-2': 'Intermixing of the layers can occur during the sample growth even when the heat of mixing is positive [CITATION], like in the case of Fe and Ag.', 'cond-mat-0506072-1-29-3': 'On the other hand, the tendency for non-equilibrium mixing does not seem to depend on the sample preparation method, since the magnetoresistance and the magnetic properties of our granular multilayers are very similar to those observed in co-depositied [CITATION] samples.', 'cond-mat-0506072-1-30-0': 'In our case for sample A the magnetic moments of the large grains and the small clusters differ more than an order of magnitude; [MATH], and [MATH].', 'cond-mat-0506072-1-30-1': 'According to the Mossbauer spectroscopy measurements the small clusters contain only a small fraction (below 2%) of the magnetic atoms.', 'cond-mat-0506072-1-30-2': 'This explains why the bulk magnetization is determined by the properties of the large grains at all temperatures, while in the transport properties the magnetic scattering of small clusters also play an important role.', 'cond-mat-0506072-1-31-0': 'In case of sample B and C a broader grain size distribution is indicated by the smeared out ZFC curves and to describe the magnetic field dependence of the resistance would require further parameters.', 'cond-mat-0506072-1-31-1': 'Interactions between the grains are also likely to play a role [CITATION] as the Fe concentration increases.', 'cond-mat-0506072-1-31-2': 'However, the temperature dependence of the resistivity measured in 12T magnetic field is quite similar to that observed in sample A (see Fig. 4) and the analysis of the magnetic scattering contribution (Fig. 8) undoubtedly indicates that there is a significant contribution from small clusters in these samples, as well.', 'cond-mat-0506072-1-32-0': '# SUMMARY', 'cond-mat-0506072-1-33-0': 'In conclusion, we investigated the magnetic scattering processes in sequentially evaporated granular Fe-Ag films.', 'cond-mat-0506072-1-33-1': 'Unusual magnetotransport features - like sublinear temperature dependence of the resistivity over a wide temperature range both in zero and 12T magnetic fields and large, non saturating GMR - were found experimentally.', 'cond-mat-0506072-1-33-2': 'The contribution of the magnetic scattering was separated and analyzed.', 'cond-mat-0506072-1-33-3': 'The quantitative description suggests a granular system with bimodal size distribution of the magnetic components: coexisting large grains and small clusters.', 'cond-mat-0506072-1-33-4': 'A detailed numerical analysis was given to determine characteristic grain- and cluster-moments, and their influence on both the scattering processes and on the macroscopic magnetization.', 'cond-mat-0506072-1-33-5': 'The analysis reveals that scattering on the small clusters plays a dominant role in high magnetic fields over a wide temperature range.', 'cond-mat-0506072-1-34-0': '# ACKNOWLEDGEMENTS', 'cond-mat-0506072-1-35-0': 'Financial support of the Hungarian Research Founds OTKA TS049881, T048965 and T038383 are acknowledged.', 'cond-mat-0506072-1-36-0': 'Baibich1988 M. 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{'cond-mat-0506072-2-0-0': 'Results of magnetization, magnetotransport and Mossbauer spectroscopy measurements of sequentially evaporated Fe-Ag granular composites are presented.', 'cond-mat-0506072-2-0-1': 'The strong magnetic scattering of the conduction electrons is reflected in the sublinear temperature dependence of the resistance and in the large negative magnetoresistance.', 'cond-mat-0506072-2-0-2': 'The simultaneous analysis of the magnetic properties and the transport behavior suggests a bimodal grain size distribution.', 'cond-mat-0506072-2-0-3': 'A detailed quantitative description of the unusual features observed in the transport properties is given.', 'cond-mat-0506072-2-1-0': '# INTRODUCTION', 'cond-mat-0506072-2-2-0': 'As promising candidates for magnetic recording and sensor applications heterogeneous magnetic materials, i. e. multilayer structures [CITATION] of alternating ferromagnetic and nonmagnetic layers and granular composites [CITATION] have been studied widely in the last two decades.', 'cond-mat-0506072-2-2-1': 'The giant magnetoresistance (GMR) in these systems have been explained by elastic scattering of the conduction electrons on magnetic moments of differently aligned magnetic entities.', 'cond-mat-0506072-2-2-2': 'Gittleman et al. have shown that in superparamagnetic granular alloys this consideration leads to a magnetoresistance proportional to the square of the magnetization.[', 'cond-mat-0506072-2-2-3': '[CITATION] Deviations from this relation has been attributed to the size distribution of the magnetic scatterers.[', 'cond-mat-0506072-2-2-4': '[CITATION] and interactions between these scatterers [CITATION] Various assumptions on the form of the size distribution have been made in order to obtain a phenomenological description of the GMR phenomena in different granular systems.', 'cond-mat-0506072-2-3-0': 'In this paper we present a systematical study of the magnetic and magnetotransport properties of vacuum evaporated granular Fe-Ag structures.', 'cond-mat-0506072-2-3-1': 'The observed large, negative non-saturating magnetic field dependence and the unusual sublinear temperature dependence ([MATH]) of the resistivity have been analyzed simultaneously.', 'cond-mat-0506072-2-3-2': 'This allowed the separation of the various scattering processes and the identification of two characteristic grain size determining the macroscopic magnetic and transport properties without making any assumption on the grain size distribution.', 'cond-mat-0506072-2-4-0': '# EXPERIMENTAL', 'cond-mat-0506072-2-5-0': 'The Fe-Ag multilayer samples were prepared by sequential vacuum evaporation in a base pressure of [MATH] Pa onto Si(111) single crystal substrates at room temperature.', 'cond-mat-0506072-2-5-1': 'The mass of the deposited material was measured by a quartz oscillator and the nominal layer thickness was calculated using the bulk density of Fe and Ag.', 'cond-mat-0506072-2-5-2': 'In this paper specimens prepared with the following sequences are discussed:', 'cond-mat-0506072-2-6-0': '(A) [Ag(2.6nm)/Fe(0.2nm)][MATH]/Ag(2.6nm)', 'cond-mat-0506072-2-7-0': '(B) [Ag(1.3nm)/Fe(0.2nm)][MATH]/Ag(1.3nm)', 'cond-mat-0506072-2-8-0': '(C) [Ag(0.8nm)/Fe(0.2nm)][MATH]/Ag(0.8nm)', 'cond-mat-0506072-2-9-0': 'Structural characterization of the samples by X-ray diffraction indicated a nanometer scale grain size of the constituents [for details see Ref. 13], however, due to the strong overlap between the diffraction lines of bcc-Fe and fcc-Ag, a quantitative evaluation of the size of the magnetic Fe grains was not possible.', 'cond-mat-0506072-2-9-1': 'The absence of peaks of the X-ray reflectivity[CITATION] when the Fe layer thickness is less than [MATH] nm is attributed to discontinuities of the Fe layers.', 'cond-mat-0506072-2-9-2': 'This limit of the continuous layer regime is quite similar to that observed in other multilayer systems [CITATION] composed of transition metals with immiscible nonmagnetic elements and these kind of discontinuous multilayers are often referred to as granular multilayers [CITATION].', 'cond-mat-0506072-2-9-3': 'The relation between the size of the grains in granular multilayers and the nominal layer thickness is generally determined by a three-dimensional growth process [CITATION]and the average diameter of the grains can be much larger than the nominal layer thickness.', 'cond-mat-0506072-2-9-4': 'It depends on the material parameters (e.g. lattice parameter mismatch, surface energy, etc.), as well as various parameters (deposition rate, substrate temperature, etc.) of the deposition technique.', 'cond-mat-0506072-2-9-5': 'The decrease of the spacer layer thickness was found [CITATION] to increase the magnetic grain size in an other series of Fe/Ag granular multilayer samples.', 'cond-mat-0506072-2-10-0': 'The magnetic structure of the samples was examined by a superconducting quantum interference device (SQUID) and by transmission Mossbauer spectroscopy.', 'cond-mat-0506072-2-10-1': 'For the latter purpose the samples have been removed from the substrate and folded up to achieve an appropriate thickness for transmission measurements.', 'cond-mat-0506072-2-11-0': 'The largest GMR and the strongest magnetic scattering of the conduction electrons were observed in sample A, as it will be shown later.', 'cond-mat-0506072-2-11-1': 'The [MATH]Fe Mossbauer spectra of this sample taken at several temperatures are shown in Fig. [REF].', 'cond-mat-0506072-2-11-2': 'The room temperature spectrum contains a paramagnetic doublet with a large isomer shift relative to [MATH]-Fe (0.18 mm/s) and a quadrupole splitting (0.45 mm/s) characteristic to a system of small Fe clusters embedded in an Ag matrix [CITATION].', 'cond-mat-0506072-2-11-3': 'The low temperature spectra show that the sample is superparamagnetic (SPM) and as the magnetic clusters gradually freeze below the blocking temperature (around 50K), the six-line pattern characteristic to the Zeeman splitting of the nuclear levels of [MATH]Fe appears.', 'cond-mat-0506072-2-11-4': 'At [MATH]K the paramagnetic fraction is absent, but the spectral lines are much broader and the hyperfine parameters are different than those of bulk bcc-Fe.', 'cond-mat-0506072-2-11-5': 'The 4.2K spectrum could be fitted with a distribution of hyperfine fields[CITATION] with an average value, [MATH]T and standard deviation [MATH]2.4T.', 'cond-mat-0506072-2-12-0': 'It is worth noting that the intensity ratios of the six lines indicate a significant spontaneous alignment of the magnetic moments.', 'cond-mat-0506072-2-12-1': 'The intensity of the six lines of a sextet is distributed as [MATH], where [MATH] is the intensity of the [MATH] transitions, and [MATH] is the angle between the direction of the gamma-ray (perpendicular to the sample plane) and the magnetic moment.', 'cond-mat-0506072-2-12-2': 'In case of a random distribution of the magnetization directions [MATH].', 'cond-mat-0506072-2-12-3': 'The observed small intensity, [MATH], indicates a close-to-perpendicular alignment of the magnetic moments with respect to the sample plane.', 'cond-mat-0506072-2-12-4': 'Applying a magnetic field perpendicular to the sample plane could fully align the moments parallel to the field, as it is indicated by the [MATH] intensity.', 'cond-mat-0506072-2-13-0': 'The spectra measured in external magnetic field were also fitted with a distribution of the hyperfine fields and the parameters obtained are: [MATH] and 27.8 and [MATH] and 2.4T for [MATH] and 7T.', 'cond-mat-0506072-2-13-1': 'The external magnetic field does not affect the width of the distribution indicating that it is not due to relaxation of the magnetic moments.', 'cond-mat-0506072-2-13-2': 'The broad spectral lines of the observed sextet result from a distribution of the Fe neighborhoods, which can be due both to the large number of surface atoms in small grains and to non-equilibrium mixing of the elements [CITATION] during the growth process.', 'cond-mat-0506072-2-13-3': 'The field-independent width and the decrease of the observed average hyperfine field in external field indicate the ferromagnetic alignment of the magnetic grains.', 'cond-mat-0506072-2-13-4': 'The hyperfine field of a ferromagnet is decreased by the applied field, since it is oriented antiparallel to the magnetic moment.', 'cond-mat-0506072-2-13-5': 'We note that at 4.2K the statistical errors allow an SPM fraction containing less than 2 atomic % of the Fe atoms.', 'cond-mat-0506072-2-14-0': 'The freezing of the superparamagnetic moments - seen in the Mossbauer spectra - also appears in the temperature dependence of the low field susceptibility measured by the SQUID.', 'cond-mat-0506072-2-14-1': 'Figure [REF] shows the results for the 3 samples, after cooling them either in zero or in 1 mT permanent magnetic field.', 'cond-mat-0506072-2-14-2': 'The blocking temperatures of about 40, 150 and 300K for samples A, B and C, respectively can be deduced from this experiment.', 'cond-mat-0506072-2-14-3': 'Sample A ([MATH]K) exhibits a textbook example for the superparamagnetic behavior.', 'cond-mat-0506072-2-14-4': 'The maximum of the ZFC curve is more smeared out for samples B and C and in case of sample B the irreversibility temperature, where the FC curve starts to deviate from the ZFC one, is much larger than the [MATH] defined by the ZFC maximum.', 'cond-mat-0506072-2-14-5': 'Such features are generally explained[CITATION] as the effect of a grain size distribution and interaction among the magnetic particles, which obviously play a role as the average grain size and the concentration of the magnetic component increase.', 'cond-mat-0506072-2-15-0': 'The magnetization versus applied magnetic field curves of sample A are shown in Fig. [REF] up to [MATH]T at four different temperatures.', 'cond-mat-0506072-2-15-1': 'Above the blocking temperature the magnetization can quite well be described[CITATION] by a single Langevin function, as expected for a superparamegnetic system of uniform grains.', 'cond-mat-0506072-2-15-2': 'The average magnetic moment of the SPM grains deduced from the fit is [MATH].', 'cond-mat-0506072-2-15-3': 'It indicates that the bulk magnetic properties are mainly determined by the average size grains (about 1.8nm) and the effect of the size distribution is negligible.', 'cond-mat-0506072-2-16-0': 'The magnetotransport measurements have been performed in the current in plane arrangement by four contact method.', 'cond-mat-0506072-2-16-1': 'The coincidence of the parallel ([MATH]) and transversal ([MATH]) magnetoresistance and the absence of any anisotropic component is characteristic to the GMR phenomenon in granular systems[CITATION].', 'cond-mat-0506072-2-16-2': 'The temperature dependence of the resistivity of samples A-C in zero and 12T magnetic field normalized to their [MATH]K values are displayed in Fig. [REF].', 'cond-mat-0506072-2-16-3': 'Contrary to ordinary metallic systems, the resistivity is sublinear above 40 K for each sample.', 'cond-mat-0506072-2-16-4': 'Similar observation was reported by Milner et al. for granular systems,[CITATION] but only in presence of a high magnetic field.', 'cond-mat-0506072-2-16-5': 'As discussed later in detail, the above qualitative feature of the zero field temperature dependencies signifies the presence of an extremely strong magnetic scattering in our samples.', 'cond-mat-0506072-2-17-0': 'The anomalous character of the [MATH] curves is the most dominant in case of sample A. Simultaneously, this sample exhibits the largest magnetoresistance at low temperature, as shown in the bottom panel of Fig. [REF].', 'cond-mat-0506072-2-17-1': 'The magnitude of the GMR measured at [MATH]K in a field of [MATH]T is 26%, 18% and 16%, for samples A, B and C, respectively.', 'cond-mat-0506072-2-17-2': 'Note that at room temperature this order is reversed.', 'cond-mat-0506072-2-18-0': 'The magnetoresistance curves of the three samples have a common feature, they do not saturate even in high magnetic fields and at low temperatures, where the magnetization seems already to be saturated.', 'cond-mat-0506072-2-18-1': 'This indicates that significant magnetic scattering takes place at magnetic entities much smaller than the typical grain size which determines the macroscopic magnetization.', 'cond-mat-0506072-2-18-2': 'It is well known that the scattering amplitude of ferromagnetic grains embedded in a nonmagnetic metallic matrix is size dependent and the contribution of the smaller clusters is strongly enhanced.', 'cond-mat-0506072-2-18-3': '[CITATION] The magnetic moments of these smaller clusters are harder to rotate by an applied magnetic field thus the saturation of the magnetoresistance is slower than that of the net magnetization arising dominantly from the larger grains.', 'cond-mat-0506072-2-18-4': 'Note that similar behavior was found found in Fe-Ag co-deposited granular films [CITATION] as well as in many other systems (for a review see [CITATION]).', 'cond-mat-0506072-2-18-5': 'In the next section we give a more detailed analysis of the above qualitative picture.', 'cond-mat-0506072-2-19-0': '# ANALYSIS', 'cond-mat-0506072-2-20-0': 'In order to identify the resistivity contribution of the magnetic scattering process we assume that the Matthiessen-rule can be applied, i.e the resistivity is composed of 3 terms: [EQUATION]', 'cond-mat-0506072-2-20-1': 'Here [MATH] denotes the residual resistivity, [MATH] is the contribution arising from the phonon scattering, and the remaining part is attributed to the magnetic scattering.', 'cond-mat-0506072-2-20-2': 'The separation of the latter term requires further assumptions.', 'cond-mat-0506072-2-21-0': 'Instead of making assumption on the shape of the grain size distribution function[CITATION] we use physical considerations to get insight into the size distribution of the grains determining the magnetic scattering.', 'cond-mat-0506072-2-21-1': 'It is based on the analysis of the temperature dependence of the resistivity measured in zero and high magnetic field, as shown in Figs. [REF] and [REF].', 'cond-mat-0506072-2-21-2': 'In these limits the large grains - which dominate the bulk magnetic properties - do not give temperature dependent contribution to the magnetic scattering, as the system is either nonmagnetic (zero field limit, above the blocking temperature) or fully polarized ferromagnetically (high field limit).', 'cond-mat-0506072-2-21-3': 'We will see that the small clusters can also be characterized by a single average size, and in high fields the temperature dependence of the magnetic scattering is determined solely by this characteristic size: the magnetic moment of these small clusters is the only fitting parameter for the calculated curves describing the shapes of [MATH] shown in Fig. [REF].', 'cond-mat-0506072-2-21-4': 'Finally, the consistency of our analysis will also be demonstrated by evaluating the magnetic field dependence of the resistivity in the whole magnetic field range (Fig. [REF]) using the characteristic sizes determined independently from the magnetization data and the temperature dependence of the resistivity in the zero and high field limits.', 'cond-mat-0506072-2-22-0': 'In a granular system the magnetic scattering depends on the correlation between the localized magnetic moments of the grains, [MATH], while a conduction electron is scattered from [MATH] to [MATH] within its spin-diffusion length[CITATION].', 'cond-mat-0506072-2-22-1': 'In zero magnetic field, well above the blocking temperature the magnetic moments of the grains (including the large grains) are fully disordered.', 'cond-mat-0506072-2-22-2': 'In these circumstances no temperature dependence is expected from the magnetic scattering, at least until the spin diffusion length is large enough.', 'cond-mat-0506072-2-22-3': 'We assume that at high temperatures the temperature dependence of the resistivity arise solely from the phonon contribution, [EQUATION]', 'cond-mat-0506072-2-22-4': 'As the the phonon term is linear above the Debye temperature ([MATH] K)[CITATION], the strength of phonon scattering, [MATH], can be determined from the high temperature slope of the zero field resistivity curves.', 'cond-mat-0506072-2-22-5': 'The calculated [MATH] curves are shown in Fig. [REF] by dashed lines for the 3 samples.', 'cond-mat-0506072-2-22-6': 'The difference of the total resistivity and its phonon related part is attributed to the magnetic scattering, and [MATH] is displayed in Fig. [REF] by dotted lines for each sample.', 'cond-mat-0506072-2-22-7': 'As it was expected, the magnetic scattering is temperature independent at high temperatures and the lower the blocking temperature the wider is the flat part of the separated magnetic contribution curve.', 'cond-mat-0506072-2-22-8': 'It is also seen that the magnetic scattering gradually decreases as the blocking temperature is approached from above.', 'cond-mat-0506072-2-22-9': 'Note, however, that at [MATH]K there is still a considerable contribution from it, i.e. [MATH] as it can be seen from Fig. [REF].', 'cond-mat-0506072-2-23-0': 'Since the phonon term is magnetic field independent, the [MATH] curves determined from the zero field temperature dependencies can be used to separate the magnetic scattering contribution in the high field measurements.', 'cond-mat-0506072-2-23-1': 'This is shown in Fig. [REF] for the [MATH]T measurements.', 'cond-mat-0506072-2-23-2': 'A comparison of the related panels of Fig. [REF] and Fig. [REF] reveals that sample A has the strongest negative curvature of the resistivity and the biggest change in [MATH].', 'cond-mat-0506072-2-23-3': 'In this sample magnetic scattering at [MATH]T dominates over even the phonon term in a very broad temperature range.', 'cond-mat-0506072-2-24-0': 'In order to describe the evaluated [MATH] curves we assume that in the high field limit the magnetic scattering of the spin-polarized electrons is proportional to the spin disorder of the small clusters.', 'cond-mat-0506072-2-24-1': 'The magnetic moments of the large grains are fully aligned by the applied magnetic field, as it could be deduced from the Mossbauer spectra of Fig. [REF].', 'cond-mat-0506072-2-24-2': 'The spin disorder for a characteristic moment [MATH] is described by the Brillouin-function: [EQUATION]', 'cond-mat-0506072-2-24-3': 'Here, [MATH] and [MATH] are the total spin and its [MATH]-component of the scatterers, respectively.', 'cond-mat-0506072-2-25-0': 'The fitted [MATH] curves are shown in Fig. [REF] by solid lines.', 'cond-mat-0506072-2-25-1': 'Apart from a normalization factor, the only fitting parameter is the magnetic moment characteristic to the small Fe clusters.', 'cond-mat-0506072-2-25-2': 'The good agreement of the experimental and the calculated curves indicates that the size distribution of these clusters is negligible.', 'cond-mat-0506072-2-25-3': 'The fitted values are in the same order of magnitude for all the samples; [MATH][MATH], 17[MATH] and 12.5[MATH] for samples A-C respectively.', 'cond-mat-0506072-2-26-0': 'Next we discuss the magnetic field dependence of the resistivity of sample A, which exhibits the strongest magnetic scattering in the superparamagnetic state, where the process of magnetic saturation is well understood.', 'cond-mat-0506072-2-26-1': 'For a numerical analysis we use the two characteristic sizes determined from the previous experiments.', 'cond-mat-0506072-2-26-2': 'The Langevin fit to the magnetization experiments performed in the superparamagnetic temperature range has shown the presence of large grains with [MATH][MATH] (Fig. [REF]), while the temperature dependence of the resistivity in high magnetic field indicated the presence of small clusters with [MATH][MATH].', 'cond-mat-0506072-2-26-3': "Following Gittleman's model [CITATION] we describe the magnetoresistance by the field dependence of the correlation between the localized magnetic moments responsible for an initial and a final magnetic scattering process: [EQUATION]", 'cond-mat-0506072-2-26-4': 'In case of the observed two largely different grain sizes this can be expressed as [EQUATION] where [MATH] is the Brillouin-function defined under Eq. [REF] for the small iron clusters and [MATH] is the Langevin-function, i.e. the classical limit of [MATH] for the large Fe grains.', 'cond-mat-0506072-2-26-5': 'The parameters [MATH], [MATH] and [MATH] represent the relative weights of scattering from grain to grain, between a grain and a cluster and from cluster to cluster.', 'cond-mat-0506072-2-27-0': 'Figure [REF] shows magnetoresistance for sample A in the superparamagnetic phase and the expected variation calculated by Eq. [REF].', 'cond-mat-0506072-2-27-1': 'The two characteristic magnetic moments ([MATH] and [MATH] ) are determined in the previous analysis from independent experiments: the field and temperature dependence of the bulk magnetization, and the temperature dependence of the resistivity.', 'cond-mat-0506072-2-27-2': 'The relative weights of the various processes were used as fitting parameters: [MATH], [MATH] and [MATH].', 'cond-mat-0506072-2-27-3': 'The value of [MATH], i.e the relatively weight for the grain[MATH]grain scattering process is larger than [MATH], even though the amplitude of the grain scattering is small.', 'cond-mat-0506072-2-27-4': '[CITATION] It reflects the large probability of scattering from grain to grain due to the large volume fraction of this type of magnetic scatterer.', 'cond-mat-0506072-2-27-5': 'The direct interplay between the clusters is negligible, as expected for a small fraction (less than 2%).', 'cond-mat-0506072-2-28-0': 'In Fig. [REF] the dashed and the dashed-dotted lines represents the contribution of the two dominant scattering processes.', 'cond-mat-0506072-2-28-1': 'The grain[MATH]grain scattering process is the leading term in Eq. [REF] up to 4T magnetic field.', 'cond-mat-0506072-2-28-2': 'The dashed line corresponds to the square of the magnetization, and if this would be the only scattering process the simple relation[CITATION] of [MATH] would hold.', 'cond-mat-0506072-2-28-3': 'Above [MATH]T, however, the second term of Eq. [REF] dominates and the observed behavior can be attributed to the effect of the small clusters, not seen in the magnetization curves.', 'cond-mat-0506072-2-28-4': 'The good description of the measured magnetization and resistance curves by two characteristic sizes is the consequence of the narrow size distributions of the grains and the clusters.', 'cond-mat-0506072-2-29-0': 'Bimodal distribution of the grain size has already been observed in granular systems prepared by co-deposition [CITATION], rapid quenching from the melt, [CITATION], or layered growth [CITATION] of the constituents.', 'cond-mat-0506072-2-29-1': 'Since for our samples a non-saturating magnetoresistance indicates the presence of small clusters even in case of 25 nm thick continuous Fe layers [CITATION], we associate the large grains and the small culsters to Fe rich grains of the granular layers and small Fe clusters trapped inside the Ag layers, respectively.', 'cond-mat-0506072-2-29-2': 'Intermixing of the layers can occur during the sample growth even when the heat of mixing is positive [CITATION], like in the case of Fe and Ag.', 'cond-mat-0506072-2-29-3': 'On the other hand, the tendency for non-equilibrium mixing does not seem to depend on the sample preparation method, since the magnetoresistance and the magnetic properties of our granular multilayers are very similar to those observed in co-depositied [CITATION] samples.', 'cond-mat-0506072-2-30-0': 'In our case for sample A the magnetic moments of the large grains and the small clusters differ more than an order of magnitude; [MATH], and [MATH].', 'cond-mat-0506072-2-30-1': 'According to the Mossbauer spectroscopy measurements the small clusters contain only a small fraction (below 2%) of the magnetic atoms.', 'cond-mat-0506072-2-30-2': 'This explains why the bulk magnetization is determined by the properties of the large grains at all temperatures, while in the transport properties the magnetic scattering of small clusters also play an important role.', 'cond-mat-0506072-2-31-0': 'In case of sample B and C a broader grain size distribution is indicated by the smeared out ZFC curves and to describe the magnetic field dependence of the resistance would require further parameters.', 'cond-mat-0506072-2-31-1': 'Interactions between the grains are also likely to play a role [CITATION] as the Fe concentration increases.', 'cond-mat-0506072-2-31-2': 'However, the temperature dependence of the resistivity measured in 12T magnetic field is quite similar to that observed in sample A (see Fig. 4) and the analysis of the magnetic scattering contribution (Fig. 8) undoubtedly indicates that there is a significant contribution from small clusters in these samples, as well.', 'cond-mat-0506072-2-32-0': '# SUMMARY', 'cond-mat-0506072-2-33-0': 'In conclusion, we investigated the magnetic scattering processes in sequentially evaporated granular Fe-Ag films.', 'cond-mat-0506072-2-33-1': 'Unusual magnetotransport features - like sublinear temperature dependence of the resistivity over a wide temperature range both in zero and 12T magnetic fields and large, non saturating GMR - were found experimentally.', 'cond-mat-0506072-2-33-2': 'The contribution of the magnetic scattering was separated and analyzed.', 'cond-mat-0506072-2-33-3': 'The quantitative description suggests a granular system with bimodal size distribution of the magnetic components: coexisting large grains and small clusters.', 'cond-mat-0506072-2-33-4': 'A detailed numerical analysis was given to determine characteristic grain- and cluster-moments, and their influence on both the scattering processes and on the macroscopic magnetization.', 'cond-mat-0506072-2-33-5': 'The analysis reveals that scattering on the small clusters plays a dominant role in high magnetic fields over a wide temperature range.', 'cond-mat-0506072-2-34-0': '# ACKNOWLEDGEMENTS', 'cond-mat-0506072-2-35-0': 'Financial support of the Hungarian Research Founds OTKA TS049881, T034602 and T038383 are acknowledged.', 'cond-mat-0506072-2-36-0': 'Baibich1988 M. 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'cond-mat-0506072-2-29-0'], ['cond-mat-0506072-1-29-1', 'cond-mat-0506072-2-29-1'], ['cond-mat-0506072-1-29-2', 'cond-mat-0506072-2-29-2'], ['cond-mat-0506072-1-29-3', 'cond-mat-0506072-2-29-3'], ['cond-mat-0506072-1-40-0', 'cond-mat-0506072-2-40-0']]

[['cond-mat-0506072-1-35-0', 'cond-mat-0506072-2-35-0']]

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['cond-mat-0506072-1-5-2', 'cond-mat-0506072-1-6-0', 'cond-mat-0506072-1-7-0', 'cond-mat-0506072-1-8-0', 'cond-mat-0506072-2-5-2', 'cond-mat-0506072-2-6-0', 'cond-mat-0506072-2-7-0', 'cond-mat-0506072-2-8-0']

{'1': 'http://arxiv.org/licenses/assumed-1991-2003/', '2': 'http://arxiv.org/licenses/assumed-1991-2003/'}

https://arxiv.org/abs/cond-mat/0506072

1708.04260

{'1708.04260-1-0-0': 'Particle transport in Markov mixtures can be addressed by the so-called Chord Length Sampling (CLS) methods, a family of Monte Carlo algorithms taking into account the effects of stochastic media on particle propagation by generating on-the-fly the material interfaces crossed by the random walkers during their trajectories.', '1708.04260-1-0-1': 'Such methods enable a significant reduction of computational resources as opposed to reference solutions obtained by solving the Boltzmann equation for a large number of realizations of random media.', '1708.04260-1-0-2': 'CLS solutions, which neglect correlations induced by the spatial disorder, are faster albeit approximate, and might thus show discrepancies with respect to reference solutions.', '1708.04260-1-0-3': "In this work we propose a new family of algorithms (called 'Poisson Box Sampling', PBS) aimed at improving the accuracy of the CLS approach for transport in [MATH]-dimensional binary Markov mixtures.", '1708.04260-1-0-4': 'In order to probe the features of PBS methods, we will focus on three-dimensional Markov media and revisit the benchmark problem originally proposed by Adams, Larsen and Pomraning [CITATION] and extended by Brantley [CITATION]: for these configurations we will compare reference solutions, standard CLS solutions and the new PBS solutions for scalar particle flux, transmission and reflection coefficients.', '1708.04260-1-0-5': 'PBS will be shown to perform better than CLS at the expense of a reasonable increase in computational time.', '1708.04260-1-1-0': '# Introduction', '1708.04260-1-2-0': 'Linear particle transport theory in random media is key to several applications in nuclear science and engineering, such as neutron diffusion in pebble-bed reactors or randomly mixed water-vapour phases in boiling water reactors [CITATION], and inertial confinement fusion [CITATION].', '1708.04260-1-2-1': 'Material and life sciences as well as radiative transport also often involve particle propagation in random media [CITATION].', '1708.04260-1-3-0': 'In this context, the material cross sections composing the traversed medium and the particle sources are distributed according to some statistical laws, and the physical observable of interest is typically the ensemble-averaged angular particle flux [MATH], namely, [EQUATION] where [MATH] satisfies the linear Boltzmann equation corresponding to a single realization [MATH], and [MATH] is the stationary probability of observing the state [MATH] for the material cross sections and/or the sources [CITATION].', '1708.04260-1-3-1': 'In the following, we consider linear particle transport in binary stochastic mixing composed of two immiscible random media (say [MATH] and [MATH]).', '1708.04260-1-4-0': 'Exact solutions for [MATH], or more generally for some ensemble-averaged functional [MATH] of the particle flux, can be obtained using a so-called quenched disorder approach: an ensemble of medium realizations are first sampled from the underlying mixing statistics; then, the linear transport equation is solved for each realization by either deterministic or Monte Carlo methods, and the physical observables of interest [MATH] are determined; ensemble averages are finally computed.', '1708.04260-1-4-1': 'In a series of recent papers, we have provided reference solutions for particle transport in [MATH]-dimensional random media with Markov statistics [CITATION], where the spatial disorder has been generated by means of homogeneous and isotropic [MATH]-dimensional Poisson tessellations [CITATION].', '1708.04260-1-5-0': 'Rreference solutions for particle transport in stochastic media are computationally expensive, so faster but approximate methods have been therefore proposed.', '1708.04260-1-5-1': 'A first approximate approach consists in deriving an expression for the ensemble-averaged flux [MATH] in each material: this generally leads to an infinite hierarchy of equations, which ultimately requires a closure formula, such as in the celebrated Levermore-Pomraning model [CITATION].', '1708.04260-1-5-2': 'A second approach is based on Monte Carlo algorithms that reproduce the ensemble-averaged solutions to various degrees of accuracy by modifying the displacement laws of the simulated particles in order to take into account the effects of spatial disorder [CITATION].', '1708.04260-1-5-3': 'The Chord Length Sampling (CLS) algorithm is perhaps the most representative and best-known example of such algorithms: the basic idea behind CLS is that the interfaces between the constituents of the stochastic medium are sampled on-the-fly during the particle displacements by drawing the distances to the following material boundaries from a distribution depending on the mixing statistics.', '1708.04260-1-5-4': 'It has been shown that the CLS algorithm formally solves the Levermore-Pomraning model for Markovian binary mixing [CITATION].', '1708.04260-1-5-5': 'The free parameters of the CLS model are the average chord length [MATH] through each material, and the volume fraction [MATH].', '1708.04260-1-5-6': 'Since the spatial configuration seen by each particle is regenerated at each particle flight, the CLS corresponds to an annealed disorder model, as opposed to the quenched disorder of the reference solutions, where the spatial configuration is frozen for all the traversing particles.', '1708.04260-1-5-7': 'This means that the correlations on particle trajectories induced by the spatial disorder are neglected in the standard implementation of CLS.', '1708.04260-1-5-8': 'Generalization of these Monte Carlo algorithms including partial memory effects due to correlations for particles crossing back and forth the same materials have been also proposed [CITATION].', '1708.04260-1-6-0': 'CLS, which had been originally formulated for Markov statistics, has been extensively applied also to randomly dispersed spherical inclusions into background matrices, with application to pebble-bed and very high temperature gas-cooled reactors [CITATION].', '1708.04260-1-6-1': 'In order to quantify the accuracy of CLS with respect to reference solutions for spherical inclusions, several comparisons have been proposed in two and three dimensions [CITATION].', '1708.04260-1-6-2': 'For Markov mixing specifically, a number of benchmark problems comparing CLS and reference solutions have been proposed in the literature so far [CITATION] with focus on [MATH]-geometries (either of the rod or slab type); flat [MATH] geometries have been considered in [CITATION].', '1708.04260-1-6-3': 'These benchmark comparisons have been recently extended to [MATH]-dimensional Markov geometries, for [MATH] (extruded) and [MATH] [CITATION].', '1708.04260-1-7-0': 'Not surprisingly, CLS solutions may display discrepancies as compared to reference solutions, whose relevance varies strongly with the system dimensionality, the average chord length and the material volume fraction [CITATION].', '1708.04260-1-7-1': 'For the case of [MATH] slab geometries with Markov mixing, possible improvements to the standard CLS algorithm accounting for partial memory effects for particle trajectories have been detailed [CITATION], and numerical tests have revealed that these corrections contribute to palliating the discrepancies [CITATION], although a generalization to higher dimensions seems hardly feasible with reasonable computational burden [CITATION].', '1708.04260-1-8-0': 'In this work we propose a new family of Monte Carlo algorithms aimed at improving the standard CLS for [MATH]-dimensional Markov media, yet keeping the increase in algorithmic complexity to a minimum.', '1708.04260-1-8-1': 'Inspiration comes from the observation that the physical observables related to particle transport through quasi-isotropic Poisson tessellations based on Cartesian boxes are almost identical to those computed for isotropic Poisson tessellations, for any dimension [MATH] [CITATION], which confirms the considerations in [CITATION].', '1708.04260-1-8-2': 'This quite remarkable property suggests that the standard CLS algorithm can be extended by replacing the memoryless sampling of material interfaces by the sampling of [MATH]-dimensional Cartesian boxes sharing the statistical features of quasi-isotropic Poisson tessellations, so as to mimic the spatial correlations that would be induced by isotropic Poisson tessellations.', '1708.04260-1-8-3': 'We will call this class of algorithms Poisson Box Sampling (PBS).', '1708.04260-1-9-0': 'In order to illustrate the behaviour of the PBS with respect reference solutions and to CLS, we will revisit the classical benchmark problem for transport in Markov binary mixtures proposed by Adams, Larsen and Pomraning [CITATION] and revisited by Brantley [CITATION].', '1708.04260-1-9-1': 'The physical observables of interest will be the particle flux [MATH], the transmission coefficient [MATH] and the reflection coefficient [MATH], for incident flux conditions and for uniform interior sources.', '1708.04260-1-10-0': 'This paper is organized as follows: in Sec. [REF] we will recall the benchmark specifications that will be used for our analysis in dimension [MATH].', '1708.04260-1-10-1': 'In Sec. [REF] we will illustrate the reference solutions for the benchmark problem obtained by using isotropic and quasi-isotropic Poisson tessellations: this preliminary investigation will allow establishing that quasi-isotropic tessellations yield results very close to those of isotropic tessellations, as expected based on previous investigations.', '1708.04260-1-10-2': 'Then, in Sec. [REF] we will describe in detail the PBS algorithms, compare these methods to the reference solutions and to the standard CLS approach, and discuss their respective merits and drawbacks.', '1708.04260-1-10-3': 'Conclusions will be finally drawn in Sec. [REF].', '1708.04260-1-11-0': '# Benchmark specifications', '1708.04260-1-12-0': 'In order for this paper to be self-contained, we briefly recall here the benchmark specifications that have been selected for this work, which are essentially drawn from those originally proposed in [CITATION] and [CITATION], and later extended in [CITATION].', '1708.04260-1-13-0': 'We consider mono-kinetic linear particle transport through a stochastic binary medium with homogeneous and isotropic Markov mixing.', '1708.04260-1-13-1': 'The medium is non-multiplying, with isotropic scattering.', '1708.04260-1-13-2': 'The geometry consists of a cubic box of side [MATH] (in arbitrary units), with reflective boundary conditions on all sides of the box except two opposite faces (say those perpendicular to the [MATH] axis), where leakage boundary conditions are imposed.', '1708.04260-1-13-3': 'Two kinds of sources will be considered: either an imposed normalized incident angular flux on the leakage surface at [MATH] (with zero interior sources), or a distributed homogeneous and isotropic normalized interior source (with zero incident angular flux on the leakage surfaces).', '1708.04260-1-13-4': 'The benchmark configurations pertaining to the former kind of source will be called suite I, whereas those pertaining to the latter will be called suite II [CITATION].', '1708.04260-1-13-5': 'Markov mixing statistics are entirely defined by assigning the average chord length for each material [MATH], namely [MATH].', '1708.04260-1-13-6': 'The (homogeneous) probability [MATH] of finding material [MATH] at an arbitrary location within the box follows from [EQUATION]', '1708.04260-1-13-7': 'By definition, the material probability [MATH] yields the volume fraction for material [MATH].', '1708.04260-1-13-8': 'The cross sections for each material will be denoted as customary [MATH] for the total cross section and [MATH] for the scattering cross section.', '1708.04260-1-13-9': 'The average number of particles surviving a collision in material [MATH] will be denoted by [MATH].', '1708.04260-1-13-10': 'The physical parameters for the benchmark configurations are recalled in Tabs.', '1708.04260-1-13-11': '[REF] and [REF]: the benchmark specifications include three cases (numbered [MATH], [MATH] and [MATH], corresponding to different materials), and three sub-cases (noted [MATH], [MATH] and [MATH], corresponding to different [MATH] for a given material) for each case [CITATION].', '1708.04260-1-14-0': 'Following [CITATION], the physical observables of interest for the benchmark will be the ensemble-averaged outgoing particle currents [MATH] on the two surfaces with leakage boundary conditions, the ensemble-averaged scalar particle flux [MATH] along [MATH], and the total scalar flux [MATH].', '1708.04260-1-14-1': 'For the suite I configurations, the outgoing particle current on the side opposite to the imposed current source represents the ensemble-averaged transmission coefficient, namely, [MATH], whereas the outgoing particle current on the side of the current source represents the ensemble-averaged reflection coefficient, namely, [MATH].', '1708.04260-1-14-2': 'For the suite II configurations, the outgoing currents on opposite faces are expected to be equal (within statistical fluctuations), for symmetry reasons.', '1708.04260-1-14-3': 'In this case, we also introduce the average leakage current [MATH].', '1708.04260-1-15-0': '# Reference solutions', '1708.04260-1-16-0': 'In view of computing reference solutions for particle transport in three-dimensional quenched disorder, the generation of Markov mixing statistics will be based on random tessellations, which are stochastic aggregates of disjoint and space-filling cells obeying a given probability distribution [CITATION].', '1708.04260-1-16-1': 'In the following we will describe the two kinds of stochastic geometries that will be used for our analysis, namely isotropic Poisson tessellations and quasi-isotropic Poisson tessellations.', '1708.04260-1-16-2': 'Reference solutions for a [MATH]-dimensional generalization of the Adams, Larsen and Pomraning benchmark with homogeneous and isotropic Markov mixing have been thoroughly described in [CITATION], where the ensemble-averaged scalar particle flux [MATH] and the currents [MATH] and [MATH] have been determined.', '1708.04260-1-16-3': 'In this section we recall the methods and the key results, and detail the changes and the additions that have been made with respect to our previous work.', '1708.04260-1-17-0': '## Isotropic Poisson tessellations', '1708.04260-1-18-0': 'Three-dimensional homogeneous and isotropic Poisson tessellations are obtained by partitioning an arbitrary domain with random planes sampled from an auxiliary Poisson process [CITATION].', '1708.04260-1-18-1': 'A single free parameter [MATH] (which is called the tessellation density) is required, which formally corresponds to the average number of planes of the tessellation that would be intersected by an arbitrary segment of unit length.', '1708.04260-1-18-2': 'An explicit construction amenable to Monte Carlo realizations for [MATH] geometries of finite size had been established in [CITATION] (for a numerical investigation see, e.g., [CITATION]), and recently generalized to [MATH]-dimensional domains [CITATION].', '1708.04260-1-18-3': 'The algorithm for [MATH] tessellations of a cube of side [MATH] has been detailed in [CITATION].', '1708.04260-1-18-4': 'An example of realization of homogeneous and isotropic Poisson tessellation is provided in Fig. [REF].', '1708.04260-1-19-0': 'Isotropic Poisson geometries satisfy a Markov property: for domains of infinite size, arbitrary lines drawn through the tessellation will be cut by the surfaces of the polyhedra into segments whose lengths [MATH] are exponentially distributed, with average chord length [MATH] [CITATION].', '1708.04260-1-19-1': 'The quantity [MATH] intuitively defines the correlation length of the Poisson geometry, i.e, the typical linear size of a polyhedron composing the random tessellation [CITATION].', '1708.04260-1-20-0': '## Colored stochastic geometries', '1708.04260-1-21-0': 'Homogeneous and isotropic binary Markov mixtures required for the reference solutions corresponding to the benchmark specifications are obtained as follows: first, an isotropic Poisson tessellation is constructed as described above.', '1708.04260-1-21-1': "Then, each polyhedron of the geometry is assigned a material composition by formally attributing a distinct 'color' [MATH] or [MATH], with associated complementary probabilities [MATH] and [MATH] [CITATION].", '1708.04260-1-21-2': 'This gives rise to (generally) non-convex [MATH] and [MATH] clusters, each composed of a random number of convex polyhedra.', '1708.04260-1-21-3': 'An example of realization for a colored Poisson tessellation is shown in Fig. [REF].', '1708.04260-1-22-0': 'The average chord length [MATH] through clusters with composition [MATH] is related to the correlation length [MATH] of the geometry via [MATH], and for [MATH] we similarly have [MATH].', '1708.04260-1-22-1': 'This yields [MATH], and we recover [EQUATION]', '1708.04260-1-22-2': 'Based on the formulas above, and using [MATH], the parameters of the colored Poisson geometries corresponding to the benchmark specifications provided in Tab. [REF] are easily derived.', '1708.04260-1-23-0': '## Poisson-Box tessellations', '1708.04260-1-24-0': 'Box tessellations refer to a class of anisotropic stochastic geometries composed of Cartesian parallelepipeds with random sides [CITATION].', '1708.04260-1-24-1': 'The special case of Poisson-Box tessellations was proposed by [CITATION]: a domain is partitioned by randomly generated planes orthogonal to the three axes [MATH], [MATH] and [MATH] through a Poisson process of intensity [MATH], [MATH] and [MATH], respectively.', '1708.04260-1-24-2': 'We will assume that the three parameters are equal, namely, [MATH], which leads to homogeneous quasi-isotropic Poisson tessellations of density [MATH].', '1708.04260-1-25-0': 'The explicit construction for Poisson-Box tessellations restricted to a cubic box of side [MATH] has been provided in [CITATION].', '1708.04260-1-25-1': 'An example of realization of Poisson-Box tessellation is illustrated in Fig. [REF].', '1708.04260-1-25-2': 'The chord distribution through Poisson-Box tessellations is not exponential; its average can be computed exactly, and yields [MATH] [CITATION].', '1708.04260-1-25-3': 'The coloring procedure is identical to that of isotropic Poisson tessellations, and the properties of the average chord lengths through colored clusters carry over as they stand: an example of colored realization is shown in Fig. [REF].', '1708.04260-1-25-4': 'In order to avoid confusion with the Poisson tessellations described above, we will refer to Poisson-Box geometries simply as Box tessellations in the following.', '1708.04260-1-26-0': '## Comparing Poisson and Box tessellations', '1708.04260-1-27-0': 'In a series of benchmark calculations for multiplying and non-multiplying systems we have shown by Monte Carlo simulations that Box tessellations yield physical observables related to particle transport that are very close to those computed for isotropic Poisson tessellations [CITATION], which confirms the findings in [CITATION].', '1708.04260-1-27-1': 'Since this property represents a crucial step towards the construction of the PBS algorithms that will be presented in Sec. [REF], we would like to preliminarily verify that this peculiar feature carries over to the Adam, Larsen and Pomraning benchmark configurations.', '1708.04260-1-28-0': 'The key point is that both tessellations depend on a single free parameter, namely the average chord length [MATH], in addition to the coloring probability [MATH].', '1708.04260-1-28-1': 'For isotropic Markovian binary mixtures, the average chord length of the Poisson tessellation and the coloring probability are chosen so that the resulting average chord lengths in the colored clusters, namely [MATH] and [MATH], match the correlation lengths in the random media [CITATION].', '1708.04260-1-29-0': "A natural choice is therefore to set [MATH] and [MATH] to be equal for Poisson and Box tessellations, which ensures that the two colored geometries are 'statistically equivalent'.", '1708.04260-1-29-1': 'As shown in [CITATION], this can be achieved by choosing the same parameters [MATH] and [MATH] for the two tessellations.', '1708.04260-1-29-2': 'Correspondingly, we have a constraint on the tessellation densities [MATH] and [MATH], which must now satisfy [EQUATION]', '1708.04260-1-29-3': 'Numerical simulations show that by imposing Eq. [REF] the chord length distributions for the two tessellations are barely distinguishable [CITATION], which is of utmost importance since the properties of particle transport through random media mostly depend on the shape of the chord length distribution [CITATION].', '1708.04260-1-30-0': 'The similarity of the chord length distributions is all the more striking when considering that other geometrical features do not share comparable affinities.', '1708.04260-1-30-1': 'For instance, if we set the average chord length [MATH] to be equal for the two tessellations as in Eq. [REF], the average volumes of a typical polyhedron read [EQUATION] respectively, i.e., the average volume of the Box tessellations is much larger than that of the Poisson tessellations [CITATION].', '1708.04260-1-31-0': '## Simulation results for reference solutions', '1708.04260-1-32-0': 'For each benchmark configuration, a large number [MATH] of geometries has been generated, and the material properties have been attributed to each volume as described in [CITATION].', '1708.04260-1-32-1': 'Then, for each realization [MATH] of the ensemble, linear particle transport has been simulated by using the production Monte Carlo code Tripoli-4 , developed at CEA [CITATION].', '1708.04260-1-32-2': 'Tripoli-4 is a general-purpose stochastic transport code capable of simulating the propagation of neutral and charged particles with continuous-energy cross sections in arbitrary geometries.', '1708.04260-1-32-3': 'In order to comply with the benchmark specifications, constant cross sections adapted to mono-energetic transport and isotropic angular scattering have been prepared.', '1708.04260-1-32-4': 'The number of simulated particle histories per configuration is [MATH].', '1708.04260-1-32-5': 'For a given physical observable [MATH], the benchmark solution is obtained as the ensemble average [EQUATION] where [MATH] is the Monte Carlo estimate for the observable [MATH] obtained for the [MATH]-th realization.', '1708.04260-1-32-6': 'Specifically, currents [MATH] and [MATH] at a given surface are estimated by summing the statistical weights of the particles crossing that surface.', '1708.04260-1-32-7': 'Scalar fluxes [MATH] have been tallied using the standard track length estimator over a pre-defined spatial grid containing [MATH] uniformly spaced meshes along the [MATH] axis.', '1708.04260-1-33-0': 'The error affecting the average observable [MATH] results from two separate contributions, the dispersion [EQUATION] of the observables exclusively due to the stochastic nature of the geometries and of the material compositions, and [EQUATION] which is an estimate of the variance due to the stochastic nature of the Monte Carlo method for particle transport, [MATH] being the dispersion of a single calculation [CITATION].', '1708.04260-1-33-1': 'The statistical error on [MATH] is then estimated as [EQUATION]', '1708.04260-1-33-2': 'The reference solutions corresponding to isotropic Poisson tessellations have been presented in [CITATION] with [MATH] realizations for every benchmark configuration.', '1708.04260-1-33-3': 'For this work, in order to improve the statistical uncertainty affecting the simulation results we have re-run the less fragmented configurations by increasing the number of realizations: specifically, we have now [MATH] realizations for case [MATH], and [MATH] realizations for case [MATH].', '1708.04260-1-33-4': 'For some sub-cases of suite II having poor convergence, we have further increased the number of realizations: [MATH] for sub-case [MATH], and [MATH] for [MATH].', '1708.04260-1-33-5': 'Additionally, reference solutions corresponding to Box tessellations have been computed for each benchmark configuration by following the same procedure as above, and the number of realizations has been set equal to that of the corresponding Poisson tessellations.', '1708.04260-1-34-0': 'Particle transport calculations have been run on a cluster based at CEA, with Intel Xeon E5-2680 V2 2.8 GHz processors.', '1708.04260-1-34-1': 'For the simulations discussed here considerable speed-ups have been obtained for the most fragmented geometries thanks to the possibility of reading pre-computed connectivity maps for the volumes composing the geometry, which largely increases the performances of particle tracking.', '1708.04260-1-35-0': 'Reference solutions for both tessellations are provided in Tabs.', '1708.04260-1-35-1': '[REF] to [REF] for the benchmark cases corresponding to suite I, and in Tabs.', '1708.04260-1-35-2': '[REF] to [REF] for the benchmark cases corresponding to suite II, respectively: the ensemble-averaged total scalar flux [MATH], transmission coefficient [MATH], and reflection coefficient [MATH] are displayed for Poisson and Box tessellations.', '1708.04260-1-35-3': 'The respective computer times are also provided in the same tables.', '1708.04260-1-35-4': 'The ensemble-averaged spatial flux [MATH] is illustrated in Figs. [REF] to [REF].', '1708.04260-1-35-5': 'As mentioned above, for case [MATH] the reference solutions for Poisson geometries are taken from reference [CITATION]; for cases [MATH] and [MATH], the results have been updated with better statistics.', '1708.04260-1-35-6': 'The reference solutions for Box tessellations have never been presented before.', '1708.04260-1-36-0': 'Simulation results for the Adam, Larsen and Pomraning benchmark configurations basically confirm our previous findings: the physical observables related to particle transport through Box tessellations are very close to those of isotropic Poisson tessellations, which was expected based on their respective chord length distributions being very similar.', '1708.04260-1-36-1': 'The agreement between the two sets of results increases by decreasing the average chord length (i.e., for more fragmented tessellations).', '1708.04260-1-36-2': 'An exception must be remarked for sub-case [MATH] of suite I, in particular for the transmission coefficient [MATH], despite this configuration being highly fragmented.', '1708.04260-1-36-3': 'Since this sub-case is composed of absorbing chunks dispersed in a scattering background, the observed discrepancy might be attributed to the effects induced by the shape of the chunks on particle transport (which are different for the two tessellations, as noticed above).', '1708.04260-1-36-4': 'For the spatial flux profiles, slight differences emerge for the less fragmented configurations, e.g., sub-cases [MATH], [MATH] and [MATH] of suite I.', '1708.04260-1-37-0': 'The computer time required for the reference solutions (as shown in Tabs.', '1708.04260-1-37-1': '[REF] to [REF]) depends on the material compositions and increases with the complexity of the configurations, i.e., with the number of polyhedra composing the tessellation.', '1708.04260-1-37-2': 'For a given average chord length [MATH], the average number of volumes is smaller in Box geometries than in Poisson geometries, which follows from the expressions of the typical volumes in Eq. [REF].', '1708.04260-1-37-3': 'Transport simulations in Box tessellations are faster than in Poisson tessellations for configurations composed of a large number of polyhedra, such as those of case [MATH]; for cases [MATH] and [MATH], finite-size effects due to [MATH] being comparable to [MATH] come into play, and computer times become almost identical for Poisson and Box geometries.', '1708.04260-1-38-0': '# Approximate solutions: from CLS to PBS', '1708.04260-1-39-0': 'Reference solutions based on the quenched disorder approach are very demanding in terms of computational resources, so that intensive research efforts have been devoted to the development of Monte Carlo-based annealed disorder models capable of approximating the effects of spatial disorder on-the-fly during particle trajectories, i.e., within a single transport simulation.', '1708.04260-1-39-1': 'In this section we first briefly recall the standard CLS algorithm, for the sake of completeness, and then introduce a new class of Monte Carlo methods, called Poisson Box Sampling (PBS), combining the principles of CLS with the generation of material volumes inspired by the findings concerning Box tessellations.', '1708.04260-1-39-2': 'Simulation results of the PBS for the benchmark configurations will be compared to those of CLS and to the reference solutions obtained above.', '1708.04260-1-40-0': '## Chord Length Sampling (CLS)', '1708.04260-1-41-0': 'The annealed disorder algorithms initially developed by Zimmerman and Adams go now under the name of Chord Length Sampling methods [CITATION].', '1708.04260-1-41-1': 'The standard form of CLS (Algorithm A in [CITATION]) formally solves the Levermore-Pomraning equations corresponding to Markov mixing with the approximation that memory of the crossed material interfaces is lost at each particle flight [CITATION].', '1708.04260-1-41-2': 'Algorithm A has the following structure [CITATION]:', '1708.04260-1-42-0': 'The particle will ultimately either get absorbed in one of the chunks or leak out of the boundaries of the random medium.', '1708.04260-1-42-1': 'As observed above, Algorithm A assumes that the particle has no memory of its past history, and in particular the crossed interfaces are immediately forgotten (which is consistent with the closure formula of the Levermore-Pomraning model).', '1708.04260-1-42-2': 'In this respect, CLS is an approximation of the exact treatment of disorder-induced spatial correlations.', '1708.04260-1-42-3': 'In particular, CLS is expected to be less accurate in the presence of strong scatterers with optically thick mean material chunk length [CITATION].', '1708.04260-1-43-0': '## Poisson Box Sampling (PBS)', '1708.04260-1-44-0': 'For the case of [MATH] slab geometries, two improved versions of CLS Algorithm A have been proposed in the literature, by partially taking into account the memory effects induced by the spatial correlations [CITATION]: in the former, called Algorithm B, instead of sampling the material interfaces one at a time a full random slab is generated, and particles do not switch material properties until either the forward or the backward surfaces of the slab are crossed; in the latter, called Algorithm C, a slab is generated as in Algorithm B, and the slab traversed before entering the current one is also kept in memory.', '1708.04260-1-44-1': 'The basic idea behind Algorithms B and C is to preserve the shape and the position of the material chunks (thus partially restoring spatial correlations) by generating an additional typical random slab whenever particles cross the material surfaces of the current volume.', '1708.04260-1-45-0': 'As expected, Algorithms B and C have been shown to approximate the reference solutions for Markov mixing in [MATH] more accurately than Algorithm A, at the expense of an increased computational cost [CITATION].', '1708.04260-1-45-1': 'Algorithm B in particular has been extensively tested for the Adams, Larsen and Pomraning benchmark in slab geometries, and performs better than Algorithm A for all configurations [CITATION].', '1708.04260-1-45-2': 'As observed in [CITATION], it is not trivial to extend Algorithms B and C to higher dimensions: this can be immediately understood by remarking that randomly generating a typical material chunk in dimension three with Markov mixing would correspond to sampling a typical polyhedral cell of the isotropic Poisson tessellations, whose exact distributions for the volume, surface, number of faces, etc., are unfortunately unknown to this day [CITATION].', '1708.04260-1-45-3': 'In dimension one the typical chunk is a slab of exponentially distributed width, which considerably simplifies the computational burden.', '1708.04260-1-46-0': 'A possible way to overcome this issue and improve Algorithm A in higher dimensions is however suggested by the numerical findings concerning Box tessellations.', '1708.04260-1-46-1': 'Since the chord length distribution of Box tessellations is very close to that of Poisson tessellations, it seems reasonable to extend Algorithm B by generating on-the-fly the typical cells of Box tessellations, i.e., Cartesian boxes with exponentially distributed side lengths.', '1708.04260-1-46-2': 'The generalization of Algorithm C would immediately follow by keeping memory of the last visited box.', '1708.04260-1-46-3': 'We will call this new class of Monte Carlo algorithms Poisson Box Sampling (PBS), and we will denote by PBS-1 the former (inspired by Algorithm B) and by PBS the latter (inspired by Algorithm C).', '1708.04260-1-46-4': 'In view of the aforementioned similarity between quasi-isotropic and isotropic Poisson tessellations, intuitively we expect that PBS methods will preserve the increased accuracy of Algorithms B and C over Algorithm A, yet allowing for a relatively straightforward construction and a fairly minor additional computational burden.', '1708.04260-1-47-0': 'By adapting the strategy of CLS, the algorithm for PBS-1 proceeds as follows:', '1708.04260-1-48-0': 'For the sampling of a new box at Step [MATH], we begin by drawing a random spacing [MATH] from an exponential distribution with average [MATH].', '1708.04260-1-48-1': 'Without loss of generality, if the interface of the current box hit by the particle is perpendicular to the [MATH]-axis, we set the following values for the side [MATH] of the new box and the position [MATH] of its center: [MATH], [MATH], where [MATH] is the particle direction along the [MATH]-axis.', '1708.04260-1-48-2': 'The other features of the current box, namely, [MATH], [MATH], [MATH] and [MATH], are left unchanged for the new box (as suggested by the construction of Box tessellations).', '1708.04260-1-48-3': 'We would proceed in the same way for the [MATH]- and [MATH]-axis.', '1708.04260-1-48-4': 'Finally, the label of the new box is randomly sampled according to the coloring probability [MATH].', '1708.04260-1-49-0': 'Contrary to Algorithm A, the correlations induced by spatial disorder are partially preserved by the PBS-1 algorithm: indeed, each particle will see the same material properties until the current box is left.', '1708.04260-1-49-1': 'Moreover, when a new box is created, its features strongly depend on those of the previous box.', '1708.04260-1-49-2': 'This should globally improve the accuracy of PBS-1 with respect to CLS in reproducing the reference solutions for the benchmark.', '1708.04260-1-49-3': 'Long-range correlations spanning more than a box (i.e., a linear size of the order of [MATH]) are nonetheless suppressed, so that we still expect some discrepancies between PBS-1 solutions and those obtained by the quenched disorder approach for either Poisson or Box tessellations.', '1708.04260-1-50-0': 'In order to further improve the accuracy of the PBS methods, we propose a second method, inspired by Algorithm C, that will be denoted PBS-2.', '1708.04260-1-50-1': 'The strategy is exactly as in the PBS-1 algorithm, the only difference being in the fact that, once a new box has been sampled, the old box is not deleted but is kept in memory (size, position and material label) until a new material interface is selected.', '1708.04260-1-50-2': 'If the particle leaves the new box by another interface, the old box is definitively deleted, another box is sampled and the new box becomes the old box.', '1708.04260-1-50-3': 'If the selected interface is the one that has been kept in memory, the new box will simply be the old box, and the roles are reversed.', '1708.04260-1-50-4': 'This implementation intuitively extends the range of spatial correlations, and is thus supposed to correspondingly enhance the accuracy with respect to reference solutions, at the expense of increasing the computational burden, too.', '1708.04260-1-51-0': '## Simulation results', '1708.04260-1-52-0': 'The simulation results corresponding to CLS and PBS for the total scalar flux [MATH], the transmission coefficient [MATH] and the reflection coefficient [MATH] are provided in Tabs.', '1708.04260-1-52-1': '[REF] to [REF] for the benchmark cases corresponding to suite I, and in Tabs.', '1708.04260-1-52-2': '[REF] to [REF] for the benchmark cases corresponding to suite II, respectively.', '1708.04260-1-52-3': 'The spatial flux [MATH] is illustrated in Figs. [REF] to [REF].', '1708.04260-1-52-4': 'For the CLS and PBS simulations of the benchmark configurations we have used 10[MATH] particles (10[MATH] replicas with 10[MATH] particles per replica), with resulting statistical uncertainties associated to each physical observable [MATH] denoted by [MATH] and [MATH], respectively.', '1708.04260-1-53-0': 'Generally speaking, the solutions computed with PBS-1 show a better agreement with respect to the reference solutions based on Poisson tessellations than those computed with CLS, and overall remarkably well approximate the benchmark observables.', '1708.04260-1-53-1': 'Moreover, as expected from the previous considerations, PBS-2 shows a further enhanced accuracy with respect to PBS-1.', '1708.04260-1-53-2': 'A single exception has been detected for sub-case [MATH] of suite I, as reported in Tab. [REF] and in Fig. [REF].', '1708.04260-1-53-3': 'For this configuration, the results of the Box tessellations are slightly different from those of Poisson tessellations, as observed above, for the spatial flux and the transmission coefficient.', '1708.04260-1-53-4': 'It turns out that both PBS algorithms provide results that are in excellent agreement with the reference solutions for the Box tessellation, which is consistent with their implementation.', '1708.04260-1-53-5': 'However, because of the observed discrepancy between Box and Poisson tessellations for sub-case [MATH], PBS show a small bias with respect to Poisson reference solutions.', '1708.04260-1-53-6': 'For the same case, CLS displays a better accuracy as compared to Poisson solutions, and this is most probably due to the fact that this algorithm exactly preserves isotropy.', '1708.04260-1-54-0': 'The analysis of the approximate solutions suggests that the accuracy of CLS globally improves when decreasing the average chord length [MATH]: configurations pertaining to case [MATH] globally show a better agreement than those of case [MATH], and those of case [MATH] show a better agreement than those of case [MATH], as pointed out in [CITATION].', '1708.04260-1-54-1': 'The improved PBS methods are less sensitive to the average chord length [MATH] and show a satisfactory agreement for all benchmark configurations.', '1708.04260-1-55-0': 'Computer times for the CLS and PBS solutions are also provided in Tabs.', '1708.04260-1-55-1': '[REF] to [REF]: not surprisingly, the approaches based on annealed disorder are much faster than the reference methods, since a single transport simulation is needed.', '1708.04260-1-55-2': 'PBS methods, while still much faster than reference solutions, for most configurations take sensibly longer than CLS: this is partly due to the increased complexity of the algorithms, and partly due to the fact that CLS is based on the sampling of the colored chord lengths (corresponding to clusters of polyhedra sharing all the same material label), whereas PBS require the sampling of un-colored boxes one at a time.', '1708.04260-1-55-3': 'Nonetheless, keeping in memory a further box amounts to an almost negligible additional computational burden for PBS-2 as opposed to PBS-1.', '1708.04260-1-56-0': '# Conclusions', '1708.04260-1-57-0': 'In this paper we have proposed a new family of Monte Carlo methods aimed at approximating ensemble-averaged observables for particle transport in Markov binary mixtures, where reference results are obtained by sampling medium realizations from homogeneous and isotropic Poisson tessellations.', '1708.04260-1-57-1': 'The so-called Algorithm A of Chord Length Sampling method is perhaps the most widely adopted simulation tool to provide such approximate solutions, based on the Levermore-Pomraning model.', '1708.04260-1-57-2': 'Several numerical investigations have shown that Algorithm A works reasonably well in most cases, yet discrepancies between CLS and reference solutions may appear due to the fact that Algorithm A neglects the correlations induced by spatial disorder.', '1708.04260-1-57-3': 'For the case of one-dimensional slab geometries, two variants of the standard CLS method, namely Algorithm B and Algorithm C, have been proposed by partially including spatial correlations and memory effects.', '1708.04260-1-57-4': 'These algorithms provide an increased accuracy with respect to Algorithm A thanks to the on-the-fly generation of typical slabs during the particle displacements, but their generalization to higher dimensions appears to be non-trivial.', '1708.04260-1-57-5': 'A rigorous generalization in dimension three would for instance demand sampling on-the-fly typical polyhedra from homogeneous and isotropic Poisson tessellations, whose exact statistical distribution are unfortunately unknown.', '1708.04260-1-58-0': 'In order to overcome these issues and derive CLS-like methods capable of taking into account spatial correlations for [MATH]-dimensional configurations, we have resorted to the key observation that quasi-isotropic Poisson tessellations (also called Box tessellations) based on Cartesian boxes yield chord length distributions and transport-related physical observables that in most cases are barely distinguishable from those coming from isotropic Poisson tessellations.', '1708.04260-1-58-1': 'This remarkable feature has inspired a generalization of CLS Algorithms B and C based on sampling on-the-fly random boxes obeying the same statistical properties as for Box tessellations.', '1708.04260-1-58-2': 'We have called these family of algorithms Poisson Box Sampling, or PBS.', '1708.04260-1-59-0': 'We have proposed two variants of PBS: in PBS-1 we generate random [MATH]-dimensional boxes, similarly as done in Algorithm B of CLS, and in PBS-2 we additionally keep memory of the last generated box, in full analogy with Algorithm C of CLS.', '1708.04260-1-59-1': 'In order to test the performances of these new methods, we have compared PBS simulation results to the reference solutions and CLS solutions for the classical benchmark problem proposed by Adams, Larsen and Pomraning for particle propagation in stochastic media with binary Markov mixing.', '1708.04260-1-59-2': 'In particular, we have examined the evolution of the transmission coefficient, the reflection coefficient and the particle flux for the benchmark configurations in dimension [MATH].', '1708.04260-1-60-0': 'A preliminary investigation has shown that Poisson and Box tessellations lead to very similar results for all the benchmark configurations, as expected on the basis of previous works, which substantiates our motivation for PBS methods.', '1708.04260-1-60-1': 'Overall, the PBS-1 algorithm reproduces reference solutions based on Poisson tessellations more accurately that Algorithm A of CLS, at the expense of an increased computational cost.', '1708.04260-1-60-2': 'PBS-2 further increases the accuracy of PBS-1 by including memory effects and thus enhancing the range of spatial correlations that are correctly captured by the algorithm; the additional computational burden required by PBS-2 is almost negligible.', '1708.04260-1-61-0': 'Future research work will be aimed at testing the performances of PBS methods as applied to other benchmark configurations with Markov mixtures, such as diffusing matrices with void or absorbing chunks [CITATION], or multiplying systems [CITATION].'}

{'1708.04260-2-0-0': 'Particle transport in Markov mixtures can be addressed by the so-called Chord Length Sampling (CLS) methods, a family of Monte Carlo algorithms taking into account the effects of stochastic media on particle propagation by generating on-the-fly the material interfaces crossed by the random walkers during their trajectories.', '1708.04260-2-0-1': 'Such methods enable a significant reduction of computational resources as opposed to reference solutions obtained by solving the Boltzmann equation for a large number of realizations of random media.', '1708.04260-2-0-2': 'CLS solutions, which neglect correlations induced by the spatial disorder, are faster albeit approximate, and might thus show discrepancies with respect to reference solutions.', '1708.04260-2-0-3': "In this work we propose a new family of algorithms (called 'Poisson Box Sampling', PBS) aimed at improving the accuracy of the CLS approach for transport in [MATH]-dimensional binary Markov mixtures.", '1708.04260-2-0-4': 'In order to probe the features of PBS methods, we will focus on three-dimensional Markov media and revisit the benchmark problem originally proposed by Adams, Larsen and Pomraning [CITATION] and extended by Brantley [CITATION]: for these configurations we will compare reference solutions, standard CLS solutions and the new PBS solutions for scalar particle flux, transmission and reflection coefficients.', '1708.04260-2-0-5': 'PBS will be shown to perform better than CLS at the expense of a reasonable increase in computational time.', '1708.04260-2-1-0': '# Introduction', '1708.04260-2-2-0': 'Linear particle transport theory in random media is key to several applications in nuclear science and engineering, such as neutron diffusion in pebble-bed reactors or randomly mixed water-vapour phases in boiling water reactors [CITATION], and inertial confinement fusion [CITATION].', '1708.04260-2-2-1': 'Material and life sciences as well as radiative transport also often involve particle propagation in random media [CITATION].', '1708.04260-2-3-0': 'In this context, the material cross sections composing the traversed medium and the particle sources are distributed according to some statistical laws, and the physical observable of interest is typically the ensemble-averaged angular particle flux [MATH], namely, [EQUATION] where [MATH] satisfies the linear Boltzmann equation corresponding to a single realization [MATH], and [MATH] is the stationary probability of observing the state [MATH] for the material cross sections and/or the sources [CITATION].', '1708.04260-2-3-1': 'In the following, we consider linear particle transport in binary stochastic mixing composed of two immiscible random media (say [MATH] and [MATH]).', '1708.04260-2-4-0': 'Exact solutions for [MATH], or more generally for some ensemble-averaged functional [MATH] of the particle flux, can be obtained using a so-called quenched disorder approach: an ensemble of medium realizations are first sampled from the underlying mixing statistics; then, the linear transport equation is solved for each realization by either deterministic or Monte Carlo methods, and the physical observables of interest [MATH] are determined; ensemble averages are finally computed.', '1708.04260-2-4-1': 'In a series of recent papers, we have provided reference solutions for particle transport in [MATH]-dimensional random media with Markov statistics [CITATION], where the spatial disorder has been generated by means of homogeneous and isotropic [MATH]-dimensional Poisson tessellations [CITATION].', '1708.04260-2-5-0': 'Reference solutions for particle transport in stochastic media are computationally expensive, so faster but approximate methods have been therefore proposed.', '1708.04260-2-5-1': 'A first approximate approach consists in deriving an expression for the ensemble-averaged flux [MATH] in each material: this generally leads to an infinite hierarchy of equations, which ultimately requires a closure formula, such as in the celebrated Levermore-Pomraning model [CITATION].', '1708.04260-2-5-2': 'A second approach is based on Monte Carlo algorithms that reproduce the ensemble-averaged solutions to various degrees of accuracy by modifying the displacement laws of the simulated particles in order to take into account the effects of spatial disorder [CITATION].', '1708.04260-2-5-3': 'The Chord Length Sampling (CLS) algorithm is perhaps the most representative and best-known example of such algorithms: the basic idea behind CLS is that the interfaces between the constituents of the stochastic medium are sampled on-the-fly during the particle displacements by drawing the distances to the following material boundaries from a distribution depending on the mixing statistics.', '1708.04260-2-5-4': 'It has been shown that the CLS algorithm formally solves the Levermore-Pomraning model for Markovian binary mixing [CITATION].', '1708.04260-2-5-5': 'The free parameters of the CLS model are the average chord length [MATH] through each material, and the volume fraction [MATH].', '1708.04260-2-5-6': 'Since the spatial configuration seen by each particle is regenerated at each particle flight, the CLS corresponds to an annealed disorder model, as opposed to the quenched disorder of the reference solutions, where the spatial configuration is frozen for all the traversing particles.', '1708.04260-2-5-7': 'This means that the correlations on particle trajectories induced by the spatial disorder are neglected in the standard implementation of CLS.', '1708.04260-2-5-8': 'Generalization of these Monte Carlo algorithms including partial memory effects due to correlations for particles crossing back and forth the same materials have been also proposed [CITATION].', '1708.04260-2-6-0': 'CLS, which had been originally formulated for Markov statistics, has been extensively applied also to randomly dispersed spherical inclusions into background matrices, with application to pebble-bed and very high temperature gas-cooled reactors [CITATION].', '1708.04260-2-6-1': 'In order to quantify the accuracy of CLS with respect to reference solutions for spherical inclusions, several comparisons have been proposed in two and three dimensions [CITATION].', '1708.04260-2-6-2': 'Some methods to mitigate the errors between CLS and the reference solutions have been presented in the context of eigenvalue calculations, e.g., in [CITATION].', '1708.04260-2-6-3': 'For Markov mixing specifically, a number of benchmark problems comparing CLS and reference solutions have been proposed in the literature so far [CITATION] with focus on [MATH]-geometries (either of the rod or slab type); flat [MATH] geometries have been considered in [CITATION].', '1708.04260-2-6-4': 'These benchmark comparisons have been recently extended to [MATH]-dimensional Markov geometries, for [MATH] (extruded) and [MATH] [CITATION].', '1708.04260-2-7-0': 'Not surprisingly, CLS solutions may display discrepancies as compared to reference solutions, whose relevance varies strongly with the system dimensionality, the average chord length and the material volume fraction [CITATION].', '1708.04260-2-7-1': 'For the case of [MATH] slab geometries with Markov mixing, possible improvements to the standard CLS algorithm accounting for partial memory effects for particle trajectories have been detailed [CITATION], and numerical tests have revealed that these corrections contribute to palliating the discrepancies [CITATION], although a generalization to higher dimensions seems hardly feasible with reasonable computational burden [CITATION].', '1708.04260-2-8-0': 'In this work we propose a new family of Monte Carlo algorithms aimed at improving the standard CLS for [MATH]-dimensional Markov media, yet keeping the increase in algorithmic complexity to a minimum.', '1708.04260-2-8-1': 'Inspiration comes from the observation that the physical observables related to particle transport through quasi-isotropic Poisson tessellations based on Cartesian boxes are almost identical to those computed for isotropic Poisson tessellations, for any dimension [MATH] [CITATION], which confirms the considerations in [CITATION].', '1708.04260-2-8-2': 'This quite remarkable property suggests that the standard CLS algorithm can be extended by replacing the memoryless sampling of material interfaces by the sampling of [MATH]-dimensional Cartesian boxes sharing the statistical features of quasi-isotropic Poisson tessellations, so as to mimic the spatial correlations that would be induced by isotropic Poisson tessellations.', '1708.04260-2-8-3': 'We will call this class of algorithms Poisson Box Sampling (PBS).', '1708.04260-2-9-0': 'In order to illustrate the behaviour of the PBS with respect reference solutions and to CLS, we will revisit the classical benchmark problem for transport in Markov binary mixtures proposed by Adams, Larsen and Pomraning [CITATION] and revisited by Brantley [CITATION].', '1708.04260-2-9-1': 'The physical observables of interest will be the particle flux [MATH], the transmission coefficient [MATH] and the reflection coefficient [MATH], for incident flux conditions and for uniform interior sources.', '1708.04260-2-10-0': 'This paper is organized as follows: in Sec. [REF] we will recall the benchmark specifications that will be used for our analysis in dimension [MATH].', '1708.04260-2-10-1': 'In Sec. [REF] we will illustrate the reference solutions for the benchmark problem obtained by using isotropic and quasi-isotropic Poisson tessellations: this preliminary investigation will allow establishing that quasi-isotropic tessellations yield results very close to those of isotropic tessellations, as expected based on previous investigations.', '1708.04260-2-10-2': 'Then, in Sec. [REF] we will describe in detail the PBS algorithms, compare these methods to the reference solutions and to the standard CLS approach, and discuss their respective merits and drawbacks.', '1708.04260-2-10-3': 'Conclusions will be finally drawn in Sec. [REF].', '1708.04260-2-11-0': '# Benchmark specifications', '1708.04260-2-12-0': 'In order for this paper to be self-contained, we briefly recall here the benchmark specifications that have been selected for this work, which are essentially drawn from those originally proposed in [CITATION] and [CITATION], and later extended in [CITATION].', '1708.04260-2-13-0': 'We consider mono-kinetic linear particle transport through a stochastic binary medium with homogeneous and isotropic Markov mixing.', '1708.04260-2-13-1': 'The medium is non-multiplying, with isotropic scattering.', '1708.04260-2-13-2': 'The geometry consists of a cubic box of side [MATH] (in arbitrary units), with reflective boundary conditions on all sides of the box except two opposite faces (say those perpendicular to the [MATH] axis), where leakage boundary conditions are imposed.', '1708.04260-2-13-3': 'Two kinds of sources will be considered: either an imposed normalized incident angular flux on the leakage surface at [MATH] (with zero interior sources), or a distributed homogeneous and isotropic normalized interior source (with zero incident angular flux on the leakage surfaces).', '1708.04260-2-13-4': 'The benchmark configurations pertaining to the former kind of source will be called suite I, whereas those pertaining to the latter will be called suite II [CITATION].', '1708.04260-2-13-5': 'Markov mixing statistics are entirely defined by assigning the average chord length for each material [MATH], namely [MATH].', '1708.04260-2-13-6': 'The (homogeneous) probability [MATH] of finding material [MATH] at an arbitrary location within the box follows from [EQUATION]', '1708.04260-2-13-7': 'By definition, the material probability [MATH] yields the volume fraction for material [MATH].', '1708.04260-2-13-8': 'The cross sections for each material will be denoted as customary [MATH] for the total cross section and [MATH] for the scattering cross section.', '1708.04260-2-13-9': 'The average number of particles surviving a collision in material [MATH] will be denoted by [MATH].', '1708.04260-2-13-10': 'The physical parameters for the benchmark configurations are recalled in Tabs.', '1708.04260-2-13-11': '[REF] and [REF]: the benchmark specifications include three cases (numbered [MATH], [MATH] and [MATH], corresponding to different materials), and three sub-cases (noted [MATH], [MATH] and [MATH], corresponding to different [MATH] for a given material) for each case [CITATION].', '1708.04260-2-14-0': 'Following [CITATION], the physical observables of interest for the benchmark will be the ensemble-averaged outgoing particle currents [MATH] on the two surfaces with leakage boundary conditions, the ensemble-averaged scalar particle flux [MATH] along [MATH], and the total scalar flux [MATH].', '1708.04260-2-14-1': 'For the suite I configurations, the outgoing particle current on the side opposite to the imposed current source represents the ensemble-averaged transmission coefficient, namely, [MATH], whereas the outgoing particle current on the side of the current source represents the ensemble-averaged reflection coefficient, namely, [MATH].', '1708.04260-2-14-2': 'For the suite II configurations, the outgoing currents on opposite faces are expected to be equal (within statistical fluctuations), for symmetry reasons.', '1708.04260-2-14-3': 'In this case, we also introduce the average leakage current [MATH].', '1708.04260-2-15-0': '# Reference solutions', '1708.04260-2-16-0': 'In view of computing reference solutions for particle transport in three-dimensional quenched disorder, the generation of Markov mixing statistics will be based on random tessellations, which are stochastic aggregates of disjoint and space-filling cells obeying a given probability distribution [CITATION].', '1708.04260-2-16-1': 'In the following we will describe the two kinds of stochastic geometries that will be used for our analysis, namely isotropic Poisson tessellations and quasi-isotropic Poisson tessellations.', '1708.04260-2-16-2': 'Reference solutions for a [MATH]-dimensional generalization of the Adams, Larsen and Pomraning benchmark with homogeneous and isotropic Markov mixing have been thoroughly described in [CITATION], where the ensemble-averaged scalar particle flux [MATH] and the currents [MATH] and [MATH] have been determined.', '1708.04260-2-16-3': 'In this section we recall the methods and the key results, and detail the changes and the additions that have been made with respect to our previous work.', '1708.04260-2-17-0': '## Isotropic Poisson tessellations', '1708.04260-2-18-0': 'Three-dimensional homogeneous and isotropic Poisson tessellations are obtained by partitioning an arbitrary domain with random planes sampled from an auxiliary Poisson process [CITATION].', '1708.04260-2-18-1': 'A single free parameter [MATH] (which is called the tessellation density) is required, which formally corresponds to the average number of planes of the tessellation that would be intersected by an arbitrary segment of unit length.', '1708.04260-2-18-2': 'An explicit construction amenable to Monte Carlo realizations for [MATH] geometries of finite size had been established in [CITATION] (for a numerical investigation see, e.g., [CITATION]), and recently generalized to [MATH]-dimensional domains [CITATION].', '1708.04260-2-18-3': 'The algorithm for [MATH] tessellations of a cube of side [MATH] has been detailed in [CITATION].', '1708.04260-2-18-4': 'An example of realization of homogeneous and isotropic Poisson tessellation is provided in Fig. [REF].', '1708.04260-2-19-0': 'Isotropic Poisson geometries satisfy a Markov property: for domains of infinite size, arbitrary lines drawn through the tessellation will be cut by the surfaces of the polyhedra into segments whose lengths [MATH] are exponentially distributed, with average chord length [MATH] [CITATION].', '1708.04260-2-19-1': 'The quantity [MATH] intuitively defines the correlation length of the Poisson geometry, i.e, the typical linear size of a polyhedron composing the random tessellation [CITATION].', '1708.04260-2-20-0': '## Colored stochastic geometries', '1708.04260-2-21-0': 'Homogeneous and isotropic binary Markov mixtures required for the reference solutions corresponding to the benchmark specifications are obtained as follows: first, an isotropic Poisson tessellation is constructed as described above.', '1708.04260-2-21-1': "Then, each polyhedron of the geometry is assigned a material composition by formally attributing a distinct 'color' [MATH] or [MATH], with associated complementary probabilities [MATH] and [MATH] [CITATION].", '1708.04260-2-21-2': 'This gives rise to (generally) non-convex [MATH] and [MATH] clusters, each composed of a random number of convex polyhedra.', '1708.04260-2-21-3': 'An example of realization for a colored Poisson tessellation is shown in Fig. [REF].', '1708.04260-2-22-0': 'The average chord length [MATH] through clusters with composition [MATH] is related to the correlation length [MATH] of the geometry via [MATH], and for [MATH] we similarly have [MATH].', '1708.04260-2-22-1': 'This yields [MATH], and we recover [EQUATION]', '1708.04260-2-22-2': 'Based on the formulas above, and using [MATH], the parameters of the colored Poisson geometries corresponding to the benchmark specifications provided in Tab. [REF] are easily derived.', '1708.04260-2-23-0': '## Poisson-Box tessellations', '1708.04260-2-24-0': 'Box tessellations refer to a class of anisotropic stochastic geometries composed of Cartesian parallelepipeds with random sides [CITATION].', '1708.04260-2-24-1': 'The special case of Poisson-Box tessellations was proposed in [CITATION]: a domain is partitioned by randomly generated planes orthogonal to the three axes [MATH], [MATH] and [MATH] through a Poisson process of intensity [MATH], [MATH] and [MATH], respectively.', '1708.04260-2-24-2': 'We will assume that the three parameters are equal, namely, [MATH], which leads to homogeneous quasi-isotropic Poisson tessellations of density [MATH].', '1708.04260-2-25-0': 'The explicit construction for Poisson-Box tessellations restricted to a cubic box of side [MATH] has been provided in [CITATION].', '1708.04260-2-25-1': 'An example of realization of Poisson-Box tessellation is illustrated in Fig. [REF].', '1708.04260-2-25-2': 'The chord distribution through Poisson-Box tessellations is not exponential; its average can be computed exactly, and yields [MATH] [CITATION].', '1708.04260-2-25-3': 'The coloring procedure is identical to that of isotropic Poisson tessellations, and the properties of the average chord lengths through colored clusters carry over as they stand: an example of colored realization is shown in Fig. [REF].', '1708.04260-2-25-4': 'In order to avoid confusion with the Poisson tessellations described above, we will refer to Poisson-Box geometries simply as Box tessellations in the following.', '1708.04260-2-26-0': '## Comparing Poisson and Box tessellations', '1708.04260-2-27-0': 'In a series of benchmark calculations for multiplying and non-multiplying systems we have shown by Monte Carlo simulations that Box tessellations yield physical observables related to particle transport that are very close to those computed for isotropic Poisson tessellations [CITATION], which confirms the findings in [CITATION].', '1708.04260-2-27-1': 'Since this property represents a crucial step towards the construction of the PBS algorithms that will be presented in Sec. [REF], we would like to preliminarily verify that this peculiar feature carries over to the Adam, Larsen and Pomraning benchmark configurations.', '1708.04260-2-28-0': 'The key point is that both tessellations depend on a single free parameter, namely the average chord length [MATH], in addition to the coloring probability [MATH].', '1708.04260-2-28-1': 'For isotropic Markovian binary mixtures, the average chord length of the Poisson tessellation and the coloring probability are chosen so that the resulting average chord lengths in the colored clusters, namely [MATH] and [MATH], match the correlation lengths in the random media [CITATION].', '1708.04260-2-29-0': "A natural choice is therefore to set [MATH] and [MATH] to be equal for Poisson and Box tessellations, which ensures that the two colored geometries are 'statistically equivalent'.", '1708.04260-2-29-1': 'As shown in [CITATION], this can be achieved by choosing the same parameters [MATH] and [MATH] for the two tessellations.', '1708.04260-2-29-2': 'Correspondingly, we have a constraint on the tessellation densities [MATH] and [MATH], which must now satisfy [EQUATION]', '1708.04260-2-29-3': 'Numerical simulations show that by imposing Eq. [REF] the chord length distributions for the two tessellations are barely distinguishable [CITATION], which is of utmost importance since the properties of particle transport through random media mostly depend on the shape of the chord length distribution [CITATION].', '1708.04260-2-30-0': 'The similarity of the chord length distributions is all the more striking when considering that other geometrical features do not share comparable affinities.', '1708.04260-2-30-1': 'For instance, if we set the average chord length [MATH] to be equal for the two tessellations as in Eq. [REF], the average volumes of a typical polyhedron read [EQUATION] respectively, i.e., the average volume of the Box tessellations is much larger than that of the Poisson tessellations [CITATION].', '1708.04260-2-31-0': '## Simulation results for reference solutions', '1708.04260-2-32-0': 'For each benchmark configuration, a large number [MATH] of geometries has been generated, and the material properties have been attributed to each volume as described in [CITATION].', '1708.04260-2-32-1': 'Then, for each realization [MATH] of the ensemble, linear particle transport has been simulated by using the production Monte Carlo code Tripoli-4 , developed at CEA [CITATION].', '1708.04260-2-32-2': 'Tripoli-4 is a general-purpose stochastic transport code capable of simulating the propagation of neutral and charged particles with continuous-energy cross sections in arbitrary geometries.', '1708.04260-2-32-3': 'In order to comply with the benchmark specifications, constant cross sections adapted to mono-energetic transport and isotropic angular scattering have been prepared.', '1708.04260-2-32-4': 'The number of simulated particle histories per configuration is [MATH].', '1708.04260-2-32-5': 'For a given physical observable [MATH], the benchmark solution is obtained as the ensemble average [EQUATION] where [MATH] is the Monte Carlo estimate for the observable [MATH] obtained for the [MATH]-th realization.', '1708.04260-2-32-6': 'Specifically, currents [MATH] and [MATH] at a given surface are estimated by summing the statistical weights of the particles crossing that surface.', '1708.04260-2-32-7': 'Scalar fluxes [MATH] have been tallied using the standard track length estimator over a pre-defined spatial grid containing [MATH] uniformly spaced meshes along the [MATH] axis.', '1708.04260-2-33-0': 'The error affecting the average observable [MATH] results from two separate contributions, the dispersion [EQUATION] of the observables exclusively due to the stochastic nature of the geometries and of the material compositions, and [EQUATION] which is an estimate of the variance due to the stochastic nature of the Monte Carlo method for particle transport, [MATH] being the dispersion of a single calculation [CITATION].', '1708.04260-2-33-1': 'The statistical error on [MATH] is then estimated as [EQUATION]', '1708.04260-2-33-2': 'The reference solutions corresponding to isotropic Poisson tessellations have been first presented in [CITATION] with [MATH] realizations for every benchmark configuration.', '1708.04260-2-33-3': 'In order to reduce the dispersion of the observables due to the statistical nature of the geometries, a new set of reference solutions has been computed in [CITATION] by increasing the number of realizations for the benchmark configurations displaying larger correlation lengths (i.e., larger material chunks).', '1708.04260-2-33-4': 'The data for reference solutions presented here are taken from [CITATION]: we have set [MATH] for the sub-case [MATH] of the suite II; [MATH] for all the other sub-cases of case [MATH]; [MATH] for the sub-case [MATH] of the suite II; and [MATH] for all the other sub-cases of case [MATH].', '1708.04260-2-33-5': 'For all remaining cases and sub-cases, we have used the same number of realizations as in [CITATION], namely, [MATH].', '1708.04260-2-33-6': 'Additionally, reference solutions corresponding to Box tessellations have been computed for each benchmark configuration by following the same procedure as above, and the number of realizations has been set equal to that of the corresponding Poisson tessellations.', '1708.04260-2-34-0': 'Particle transport calculations have been run on a cluster based at CEA, with Intel Xeon E5-2680 V2 2.8 GHz processors.', '1708.04260-2-34-1': 'For the simulations discussed here considerable speed-ups have been obtained for the most fragmented geometries thanks to the possibility of reading pre-computed connectivity maps for the volumes composing the geometry, which largely increases the performances of particle tracking.', '1708.04260-2-35-0': 'Reference solutions for both tessellations are provided in Tabs.', '1708.04260-2-35-1': '[REF] to [REF] for the benchmark cases corresponding to suite I, and in Tabs.', '1708.04260-2-35-2': '[REF] to [REF] for the benchmark cases corresponding to suite II, respectively: the ensemble-averaged total scalar flux [MATH], transmission coefficient [MATH], and reflection coefficient [MATH] are displayed for Poisson and Box tessellations.', '1708.04260-2-35-3': 'The respective computer times are also provided in the same tables.', '1708.04260-2-35-4': 'The ensemble-averaged spatial flux [MATH] is illustrated in Figs. [REF] to [REF].', '1708.04260-2-35-5': 'As mentioned above, the reference solutions for Poisson geometries are taken from reference [CITATION]; the reference solutions for Box tessellations have never been presented before.', '1708.04260-2-36-0': 'Simulation results for the Adam, Larsen and Pomraning benchmark configurations basically confirm our previous findings: the physical observables related to particle transport through Box tessellations are very close to those of isotropic Poisson tessellations, which was expected based on their respective chord length distributions being very similar.', '1708.04260-2-36-1': 'The agreement between the two sets of results increases by decreasing the average chord length (i.e., for more fragmented tessellations).', '1708.04260-2-36-2': 'An exception must be remarked for sub-case [MATH] of suite I, in particular for the transmission coefficient [MATH], despite this configuration being highly fragmented.', '1708.04260-2-36-3': 'Since this sub-case is composed of absorbing chunks dispersed in a scattering background, the observed discrepancy might be attributed to the effects induced by the shape of the chunks on particle transport (which are different for the two tessellations, as noticed above).', '1708.04260-2-36-4': 'For the spatial flux profiles, slight differences emerge for the less fragmented configurations, e.g., sub-cases [MATH], [MATH] and [MATH] of suite I.', '1708.04260-2-37-0': 'The computer time required for the reference solutions (as shown in Tabs.', '1708.04260-2-37-1': '[REF] to [REF]) depends on the material compositions and increases with the complexity of the configurations, i.e., with the number of polyhedra composing the tessellation.', '1708.04260-2-37-2': 'For a given average chord length [MATH], the average number of volumes is smaller in Box geometries than in Poisson geometries, which follows from the expressions of the typical volumes in Eq. [REF].', '1708.04260-2-37-3': 'Transport simulations in Box tessellations are faster than in Poisson tessellations for configurations composed of a large number of polyhedra, such as those of case [MATH]; for cases [MATH] and [MATH], finite-size effects due to [MATH] being comparable to [MATH] come into play, and computer times become almost identical for Poisson and Box geometries.', '1708.04260-2-38-0': '# Approximate solutions: from CLS to PBS', '1708.04260-2-39-0': 'Reference solutions based on the quenched disorder approach are very demanding in terms of computational resources, so that intensive research efforts have been devoted to the development of Monte Carlo-based annealed disorder models capable of approximating the effects of spatial disorder on-the-fly during particle trajectories, i.e., within a single transport simulation.', '1708.04260-2-39-1': 'In this section we first briefly recall the standard CLS algorithm, for the sake of completeness, and then introduce a new class of Monte Carlo methods, called Poisson Box Sampling (PBS), combining the principles of CLS with the generation of material volumes inspired by the findings concerning Box tessellations.', '1708.04260-2-39-2': 'Simulation results of the PBS for the benchmark configurations will be compared to those of CLS and to the reference solutions obtained above.', '1708.04260-2-40-0': '## Chord Length Sampling (CLS)', '1708.04260-2-41-0': 'The annealed disorder algorithms initially developed by Zimmerman and Adams go now under the name of Chord Length Sampling methods [CITATION].', '1708.04260-2-41-1': 'The standard form of CLS (Algorithm A in [CITATION]) formally solves the Levermore-Pomraning equations corresponding to Markov mixing with the approximation that memory of the crossed material interfaces is lost at each particle flight [CITATION].', '1708.04260-2-41-2': 'Algorithm A has the following structure [CITATION]:', '1708.04260-2-42-0': 'The particle will ultimately either get absorbed in one of the chunks or leak out of the boundaries of the random medium.', '1708.04260-2-42-1': 'As observed above, Algorithm A assumes that the particle has no memory of its past history, and in particular the crossed interfaces are immediately forgotten (which is consistent with the closure formula of the Levermore-Pomraning model).', '1708.04260-2-42-2': 'In this respect, CLS is an approximation of the exact treatment of disorder-induced spatial correlations.', '1708.04260-2-42-3': 'In particular, CLS is expected to be less accurate in the presence of strong scatterers with optically thick mean material chunk length [CITATION].', '1708.04260-2-42-4': 'A thorough discussion of the shortcomings of the CLS approach for [MATH] can be found, e.g., in [CITATION].', '1708.04260-2-43-0': '## Poisson Box Sampling (PBS)', '1708.04260-2-44-0': 'For the case of [MATH] slab geometries, two improved versions of CLS Algorithm A have been proposed in the literature, by partially taking into account the memory effects induced by the spatial correlations [CITATION]: in the former, called Algorithm B, instead of sampling the material interfaces one at a time a full random slab is generated, and particles do not switch material properties until either the forward or the backward surfaces of the slab are crossed; in the latter, called Algorithm C, a slab is generated as in Algorithm B, and the slab traversed before entering the current one is also kept in memory.', '1708.04260-2-44-1': 'The basic idea behind Algorithms B and C is to preserve the shape and the position of the material chunks (thus partially restoring spatial correlations) by generating an additional typical random slab whenever particles cross the material surfaces of the current volume.', '1708.04260-2-45-0': 'As expected, Algorithms B and C have been shown to approximate the reference solutions for Markov mixing in [MATH] more accurately than Algorithm A, at the expense of an increased computational cost [CITATION].', '1708.04260-2-45-1': 'Algorithm B in particular has been extensively tested for the Adams, Larsen and Pomraning benchmark in slab geometries, and performs better than Algorithm A for all configurations [CITATION].', '1708.04260-2-45-2': 'As observed in [CITATION], it is not trivial to extend Algorithms B and C to higher dimensions: this can be immediately understood by remarking that randomly generating a typical material chunk in dimension three with Markov mixing would correspond to sampling a typical polyhedral cell of the isotropic Poisson tessellations, whose exact distributions for the volume, surface, number of faces, etc., are unfortunately unknown to this day [CITATION].', '1708.04260-2-45-3': 'In dimension one the typical chunk is a slab of exponentially distributed width, which considerably simplifies the computational burden.', '1708.04260-2-46-0': 'A possible way to overcome this issue and improve Algorithm A in higher dimensions is however suggested by the numerical findings concerning Box tessellations.', '1708.04260-2-46-1': 'Since the chord length distribution of Box tessellations is very close to that of Poisson tessellations, it seems reasonable to extend Algorithm B by generating on-the-fly the typical cells of Box tessellations, i.e., Cartesian boxes with exponentially distributed side lengths.', '1708.04260-2-46-2': 'The generalization of Algorithm C would immediately follow by keeping memory of the last visited box.', '1708.04260-2-46-3': 'We will call this new class of Monte Carlo algorithms Poisson Box Sampling (PBS), and we will denote by PBS-1 the former (inspired by Algorithm B) and by PBS the latter (inspired by Algorithm C).', '1708.04260-2-46-4': 'In view of the aforementioned similarity between quasi-isotropic and isotropic Poisson tessellations, intuitively we expect that PBS methods will preserve the increased accuracy of Algorithms B and C over Algorithm A, yet allowing for a relatively straightforward construction and a fairly minor additional computational burden.', '1708.04260-2-47-0': 'By adapting the strategy of CLS, the algorithm for PBS-1 proceeds as follows:', '1708.04260-2-48-0': 'For the sampling of a new box at Step [MATH], we begin by drawing a random spacing [MATH] from an exponential distribution with average [MATH].', '1708.04260-2-48-1': 'Without loss of generality, if the interface of the current box hit by the particle is perpendicular to the [MATH]-axis, we set the following values for the side [MATH] of the new box and the position [MATH] of its center: [MATH], [MATH], where [MATH] is the particle direction along the [MATH]-axis.', '1708.04260-2-48-2': 'The other features of the current box, namely, [MATH], [MATH], [MATH] and [MATH], are left unchanged for the new box (as suggested by the construction of Box tessellations).', '1708.04260-2-48-3': 'We would proceed in the same way for the [MATH]- and [MATH]-axis.', '1708.04260-2-48-4': 'Finally, the label of the new box is randomly sampled according to the coloring probability [MATH].', '1708.04260-2-49-0': 'Contrary to Algorithm A, the correlations induced by spatial disorder are partially preserved by the PBS-1 algorithm: indeed, each particle will see the same material properties until the current box is left.', '1708.04260-2-49-1': 'Moreover, when a new box is created, its features strongly depend on those of the previous box.', '1708.04260-2-49-2': 'This should globally improve the accuracy of PBS-1 with respect to CLS in reproducing the reference solutions for the benchmark.', '1708.04260-2-49-3': 'Long-range correlations spanning more than a box (i.e., a linear size of the order of [MATH]) are nonetheless suppressed, so that we still expect some discrepancies between PBS-1 solutions and those obtained by the quenched disorder approach for either Poisson or Box tessellations.', '1708.04260-2-50-0': 'In order to further improve the accuracy of the PBS methods, we propose a second method, inspired by Algorithm C, that will be denoted PBS-2.', '1708.04260-2-50-1': 'The strategy is exactly as in the PBS-1 algorithm, the only difference being in the fact that, once a new box has been sampled, the old box is not deleted but is kept in memory (size, position and material label) until a new material interface is selected.', '1708.04260-2-50-2': 'If the particle leaves the new box by another interface, the old box is definitively deleted, another box is sampled and the new box becomes the old box.', '1708.04260-2-50-3': 'If the selected interface is the one that has been kept in memory, the new box will simply be the old box, and the roles are reversed.', '1708.04260-2-50-4': 'This implementation intuitively extends the range of spatial correlations, and is thus supposed to correspondingly enhance the accuracy with respect to reference solutions, at the expense of increasing the computational burden, too.', '1708.04260-2-51-0': '## Simulation results', '1708.04260-2-52-0': 'The simulation results corresponding to CLS and PBS for the total scalar flux [MATH], the transmission coefficient [MATH] and the reflection coefficient [MATH] are provided in Tabs.', '1708.04260-2-52-1': '[REF] to [REF] for the benchmark cases corresponding to suite I, and in Tabs.', '1708.04260-2-52-2': '[REF] to [REF] for the benchmark cases corresponding to suite II, respectively.', '1708.04260-2-52-3': 'The spatial flux [MATH] is illustrated in Figs. [REF] to [REF].', '1708.04260-2-52-4': 'For the CLS and PBS simulations of the benchmark configurations we have used 10[MATH] particles (10[MATH] replicas with 10[MATH] particles per replica), with resulting statistical uncertainties associated to each physical observable [MATH] denoted by [MATH] and [MATH], respectively.', '1708.04260-2-53-0': 'Generally speaking, the solutions computed with PBS-1 show a better agreement with respect to the reference solutions based on Poisson tessellations than those computed with CLS, and overall remarkably well approximate the benchmark observables.', '1708.04260-2-53-1': 'Moreover, as expected from the previous considerations, PBS-2 shows a further enhanced accuracy with respect to PBS-1.', '1708.04260-2-53-2': 'A single exception has been detected for sub-case [MATH] of suite I, as reported in Tab. [REF] and in Fig. [REF].', '1708.04260-2-53-3': 'For this configuration, the results of the Box tessellations are slightly different from those of Poisson tessellations, as observed above, for the spatial flux and the transmission coefficient.', '1708.04260-2-53-4': 'It turns out that both PBS algorithms provide results that are in excellent agreement with the reference solutions for the Box tessellation, which is consistent with their implementation.', '1708.04260-2-53-5': 'However, because of the observed discrepancy between Box and Poisson tessellations for sub-case [MATH], PBS show a small bias with respect to Poisson reference solutions.', '1708.04260-2-53-6': 'For the same case, CLS displays a better accuracy as compared to Poisson solutions, and this is most probably due to the fact that this algorithm exactly preserves isotropy.', '1708.04260-2-54-0': 'The analysis of the approximate solutions suggests that the accuracy of CLS globally improves when decreasing the average chord length [MATH]: configurations pertaining to case [MATH] globally show a better agreement than those of case [MATH], and those of case [MATH] show a better agreement than those of case [MATH], as pointed out in [CITATION].', '1708.04260-2-54-1': 'The improved PBS methods are less sensitive to the average chord length [MATH] and show a satisfactory agreement for all benchmark configurations.', '1708.04260-2-55-0': 'Computer times for the CLS and PBS solutions are also provided in Tabs.', '1708.04260-2-55-1': '[REF] to [REF]: not surprisingly, the approaches based on annealed disorder are much faster than the reference methods, since a single transport simulation is needed.', '1708.04260-2-55-2': 'PBS methods, while still much faster than reference solutions, for most configurations take sensibly longer than CLS: this is partly due to the increased complexity of the algorithms, and partly due to the fact that CLS is based on the sampling of the colored chord lengths (corresponding to clusters of polyhedra sharing all the same material label), whereas PBS require the sampling of un-colored boxes one at a time.', '1708.04260-2-55-3': 'Nonetheless, keeping in memory a further box amounts to an almost negligible additional computational burden for PBS-2 as opposed to PBS-1.', '1708.04260-2-56-0': '# Conclusions', '1708.04260-2-57-0': 'In this paper we have proposed a new family of Monte Carlo methods aimed at approximating ensemble-averaged observables for particle transport in Markov binary mixtures, where reference results are obtained by sampling medium realizations from homogeneous and isotropic Poisson tessellations.', '1708.04260-2-57-1': 'The so-called Algorithm A of Chord Length Sampling method is perhaps the most widely adopted simulation tool to provide such approximate solutions, based on the Levermore-Pomraning model.', '1708.04260-2-57-2': 'Several numerical investigations have shown that Algorithm A works reasonably well in most cases, yet discrepancies between CLS and reference solutions may appear due to the fact that Algorithm A neglects the correlations induced by spatial disorder.', '1708.04260-2-57-3': 'For the case of one-dimensional slab geometries, two variants of the standard CLS method, namely Algorithm B and Algorithm C, have been proposed by partially including spatial correlations and memory effects.', '1708.04260-2-57-4': 'These algorithms provide an increased accuracy with respect to Algorithm A thanks to the on-the-fly generation of typical slabs during the particle displacements, but their generalization to higher dimensions appears to be non-trivial.', '1708.04260-2-57-5': 'A rigorous generalization in dimension three would for instance demand sampling on-the-fly typical polyhedra from homogeneous and isotropic Poisson tessellations, whose exact statistical distribution are unfortunately unknown.', '1708.04260-2-58-0': 'In order to overcome these issues and derive CLS-like methods capable of taking into account spatial correlations for [MATH]-dimensional configurations, we have resorted to the key observation that quasi-isotropic Poisson tessellations (also called Box tessellations) based on Cartesian boxes yield chord length distributions and transport-related physical observables that in most cases are barely distinguishable from those coming from isotropic Poisson tessellations.', '1708.04260-2-58-1': 'This remarkable feature has inspired a generalization of CLS Algorithms B and C based on sampling on-the-fly random boxes obeying the same statistical properties as for Box tessellations.', '1708.04260-2-58-2': 'We have called these family of algorithms Poisson Box Sampling, or PBS.', '1708.04260-2-59-0': 'We have proposed two variants of PBS: in PBS-1 we generate random [MATH]-dimensional boxes, similarly as done in Algorithm B of CLS, and in PBS-2 we additionally keep memory of the last generated box, in full analogy with Algorithm C of CLS.', '1708.04260-2-59-1': 'In order to test the performances of these new methods, we have compared PBS simulation results to the reference solutions and CLS solutions for the classical benchmark problem proposed by Adams, Larsen and Pomraning for particle propagation in stochastic media with binary Markov mixing.', '1708.04260-2-59-2': 'In particular, we have examined the evolution of the transmission coefficient, the reflection coefficient and the particle flux for the benchmark configurations in dimension [MATH].', '1708.04260-2-60-0': 'A preliminary investigation has shown that Poisson and Box tessellations lead to very similar results for all the benchmark configurations, as expected on the basis of previous works, which substantiates our motivation for PBS methods.', '1708.04260-2-60-1': 'Overall, the PBS-1 algorithm reproduces reference solutions based on Poisson tessellations more accurately that Algorithm A of CLS, at the expense of an increased computational cost.', '1708.04260-2-60-2': 'PBS-2 further increases the accuracy of PBS-1 by including memory effects and thus enhancing the range of spatial correlations that are correctly captured by the algorithm; the additional computational burden required by PBS-2 is almost negligible.', '1708.04260-2-61-0': 'A local realization preserving (LRP) algorithm that extends the standard CLS in a way similar to PBS (i.e., by preserving information about the shape of the traversed polyhedra) has been independently developed at LLNL and tested against reference solutions and CLS Algorithm A [CITATION]: in the future, it will be interesting to compare PBS to LRP.', '1708.04260-2-61-1': 'Moreover, future research work will be aimed at testing the performances of PBS methods as applied to other benchmark configurations with Markov mixtures, such as diffusing matrices with void or absorbing chunks [CITATION], or multiplying systems [CITATION].'}

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['1708.04260-1-54-0', '1708.04260-2-54-0'], ['1708.04260-1-54-1', '1708.04260-2-54-1'], ['1708.04260-1-58-0', '1708.04260-2-58-0'], ['1708.04260-1-58-1', '1708.04260-2-58-1'], ['1708.04260-1-58-2', '1708.04260-2-58-2'], ['1708.04260-1-59-0', '1708.04260-2-59-0'], ['1708.04260-1-59-1', '1708.04260-2-59-1'], ['1708.04260-1-59-2', '1708.04260-2-59-2'], ['1708.04260-1-21-0', '1708.04260-2-21-0'], ['1708.04260-1-21-1', '1708.04260-2-21-1'], ['1708.04260-1-21-2', '1708.04260-2-21-2'], ['1708.04260-1-21-3', '1708.04260-2-21-3'], ['1708.04260-1-28-0', '1708.04260-2-28-0'], ['1708.04260-1-28-1', '1708.04260-2-28-1'], ['1708.04260-1-9-0', '1708.04260-2-9-0'], ['1708.04260-1-9-1', '1708.04260-2-9-1'], ['1708.04260-1-25-0', '1708.04260-2-25-0'], ['1708.04260-1-25-1', '1708.04260-2-25-1'], ['1708.04260-1-25-2', '1708.04260-2-25-2'], ['1708.04260-1-25-3', '1708.04260-2-25-3'], ['1708.04260-1-25-4', '1708.04260-2-25-4'], ['1708.04260-1-3-0', '1708.04260-2-3-0'], ['1708.04260-1-3-1', '1708.04260-2-3-1']]

[['1708.04260-1-5-0', '1708.04260-2-5-0'], ['1708.04260-1-61-0', '1708.04260-2-61-1'], ['1708.04260-1-33-2', '1708.04260-2-33-2'], ['1708.04260-1-24-1', '1708.04260-2-24-1']]

[]

[['1708.04260-1-35-5', '1708.04260-2-35-5'], ['1708.04260-1-35-6', '1708.04260-2-35-5'], ['1708.04260-1-33-4', '1708.04260-2-33-4']]

[]

['1708.04260-1-41-2', '1708.04260-1-47-0', '1708.04260-2-41-2', '1708.04260-2-47-0']

{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1708.04260

1301.7014

{'1301.7014-1-0-0': 'We study the electromagnetic Bethe-Heitler type contribution to neutrino-induced deeply virtual meson production ([MATH]DVMP).', '1301.7014-1-0-1': 'Such [MATH]-corrections decrease with [MATH] in the Bjorken regime less steeply than the standard [MATH]DVMP handbag contribution.', '1301.7014-1-0-2': 'Therefore, they are relatively enhanced at high [MATH].', '1301.7014-1-0-3': 'The Bethe-Heitler terms give rise to an angular correlation between the lepton and hadron scattering planes with harmonics sensitive to the real and imaginary parts of the DVMP amplitude.', '1301.7014-1-0-4': 'These corrections constitute a few percent effect in the kinematics of the forthcoming Minerva experiment at Fermilab and should be taken into account in precision tests of GPD parametrizations.', '1301.7014-1-0-5': 'For virtualities [MATH] GeV[MATH] these corrections become on a par with DVMP handbag contributions.', '1301.7014-1-0-6': 'A computational code, which can be used for the evaluation of these corrections employing various GPD models is provided.', '1301.7014-1-1-0': '# Introduction', '1301.7014-1-2-0': 'Generalized parton distributions (GPD) allow evaluation of cross-sections for a wide class of processes, where the collinear factorization is applicable [CITATION].', '1301.7014-1-2-1': 'The main source of experimental information on GPDs has been so far the electron(positron)-proton measurements performed at JLAB and HERA, in particular deeply virtual Compton scattering (DVCS) and deeply virtual meson production (DVMP) [CITATION].', '1301.7014-1-2-2': 'The 12 GeV upgrade at Jefferson lab will open new opportunities for further improvement of our knowledge of the GPDs [CITATION].', '1301.7014-1-3-0': 'However, the practical realization of this program suffers from large uncertainties.', '1301.7014-1-3-1': 'For instance, the results at moderately high [MATH] can be affected by poorly known higher-twist components of GPDs and distribution amplitudes (DA) of the produced mesons [CITATION].', '1301.7014-1-4-0': 'Neutrino experiments provide a powerful tool for consistency checks for the extraction of GPD from JLAB data, especially of their flavor structure.', '1301.7014-1-4-1': 'The study of various processes in the Bjorken regime may be done with the high-intensity NuMI beam at Fermilab, which will switch soon to the so-called middle-energy (ME) regime with an average neutrino energy of about [MATH] GeV, and potentially may reach energies up to 20 GeV, without essential loss of luminosity.', '1301.7014-1-4-2': 'In this setup the Minerva experiment [CITATION] should be able to probe the quark flavor structure of the targets.', '1301.7014-1-4-3': 'Even higher luminosities in multi-GeV regime can be achieved at the planned Muon Collider/Neutrino Factory [CITATION].', '1301.7014-1-5-0': 'Certain information on the GPD flavor structure can be extracted from comparison of analogous processes in neutrino- and electro-induced processes employing the difference of flavor structures of electromagnetic and weak neutral currents.', '1301.7014-1-5-1': 'An example is the weak DVCS [CITATION], which alone, however, is not sufficient to constrain the flavor structure.', '1301.7014-1-6-0': 'Recently we discussed the possibility of GPD extraction from deeply virtual neutrino-production of the pseudo-Goldstone mesons ([MATH]) [CITATION].', '1301.7014-1-6-1': 'The [MATH]DVMP measurements with neutrino and antineutrino beams are complementary to the electromagnetic DVMP.', '1301.7014-1-6-2': 'The octet of pseudo-Goldstone bosons, originating from the chiral symmetry breaking, acts in the axial current as a natural probe for the flavor content.', '1301.7014-1-6-3': 'Due to the [MATH] structure of the charged current, in [MATH]DVMP one can access simultaneously the unpolarized GPDs, [MATH] and [MATH], and the helicity flip GPDs, [MATH] and [MATH].', '1301.7014-1-6-4': 'We expect the contributions of the GPDs [MATH] and [MATH], which are controlled by the poorly known twist-3 pion DA [MATH], to be negligible.', '1301.7014-1-6-5': 'Besides, important information on the flavor structure can be obtained by studying the transitional GPDs in the processes with nucleon to hyperon transitions.', '1301.7014-1-6-6': 'As was discussed in [CITATION], assuming [MATH] flavor symmetry one can relate these GPDs to the ordinary diagonal ones in the proton.', '1301.7014-1-7-0': 'In this paper we study the Bethe-Heitler (BH) type radiative corrections to the diffractive neutrino-production of charged pseudo-Golstone mesons with the target remained intact, related to meson emission from lepton line with subsequent electromagnetic interaction with the target.', '1301.7014-1-7-1': 'Although such processes are formally suppressed by [MATH], at large [MATH] they fall off less steeply than the DVMP cross-section.', '1301.7014-1-7-2': 'While at virtualities [MATH] GeV[MATH], relevant for modern neutrino experiments, this is a few percent correction, already at [MATH] GeV[MATH] it becomes comparable with the DVMP cross section.', '1301.7014-1-7-3': 'Such corrections are relevant only in case of the [MATH]DVMP: for the electron-induced DVMP [MATH] they are suppressed by factor [MATH], where [MATH] is the Fermi coupling, and are negligible unless we consider extremely high [MATH].', '1301.7014-1-7-4': 'Existence of such diagrams opens a possibility to probe separately the real and imaginary parts of the DVMP amplitude (not only the total cross-sections), in close analogy to DVCS studies [CITATION].', '1301.7014-1-8-0': 'The paper is organized as follows.', '1301.7014-1-8-1': 'In Section [REF] we present the results for the DVMP and BH contributions to the experimentally measurable total [MATH]DVMP cross-section (technical details of evaluation may be found in Appendix [REF]).', '1301.7014-1-8-2': 'Also, at the end of Section [REF] we construct two asymmetries which have particular sensitivity to the real and imaginary parts of the DVMP amplitude.', '1301.7014-1-8-3': 'In Section [REF], for the sake of completeness we discuss the features of the GPD parametrization used in calculations.', '1301.7014-1-8-4': 'In Section [REF] we present the numerical results and make conclusions.', '1301.7014-1-9-0': '# Cross-section of the [MATH]DVMP and BH processes', '1301.7014-1-10-0': 'Exclusive neutrino-production of pions is presented by the diagram (a) in the Figure [REF].', '1301.7014-1-10-1': 'Production of pions by the vector current was studied in [CITATION], and was recently extended to neutrino interactions in [CITATION].', '1301.7014-1-11-0': 'As was shown in [CITATION], at large [MATH] where the collinear factorization is applicable, the amplitude of this process is suppressed due to hard gluon exchange in the coefficient function, and is small.', '1301.7014-1-11-1': 'This raises a question, how important could be the [MATH] corrections ?', '1301.7014-1-11-2': 'While a systematic study of radiative corrections is beyond the scope of the present paper, we would like to consider in detail the contributions which decrease with [MATH] less rapidly than the diagram (a).', '1301.7014-1-11-3': 'In the leading order in [MATH] and [MATH] there are two such diagrams (b,c) shown in the Figure [REF].', '1301.7014-1-11-4': 'These diagrams are enhanced by a factor [MATH], where [MATH] and [MATH] are the lepton and proton 4-momentum transfers squared, respectively.', '1301.7014-1-11-5': 'This factor is large in the Bjorken regime of [MATH], and despite the formal suppression by [MATH], the diagrams (b,c) are numerically comparable to the diagram (a).', '1301.7014-1-11-6': 'Notice that such corrections are sizable only in the case of neutrino-production of mesons.', '1301.7014-1-11-7': 'In case of electroproduction the corresponding corrections are suppressed by [MATH], where [MATH] is the Fermi constant, and are negligible, unless we go to extremely large [MATH] .', '1301.7014-1-11-8': 'In what follows we evaluate the contribution of each diagram in Figure [REF].', '1301.7014-1-12-0': 'The cross-section of pion production can be presented as a sum of contributions of DVMP (diagram (a)), BH mechanism (diagrams (b,c)) and their interference, [EQUATION] where [MATH], and [MATH] is the angle between the lepton and hadron scattering planes.', '1301.7014-1-13-0': 'Evaluation of the diagram (a) is straightforward and yields [CITATION] [EQUATION] where [MATH] is the fractional loss of lepton energy defined as [MATH].', '1301.7014-1-13-1': 'Notice that the DVMP cross section turns out to be independent of the angle [MATH] between lepton and hadron planes.', '1301.7014-1-13-2': 'This happens because the momentum [MATH] does not have transverse components in the Bjorken reference frame.', '1301.7014-1-14-0': 'For unpolarized target, the matrix element squared [MATH] in Eqn. ([REF]) can be simplified as, [EQUATION] where we introduced a shorthand notation, [EQUATION] and the script letters [MATH] signify convolution of the GPDs [MATH] with corresponding coefficient functions given in Table [REF].', '1301.7014-1-15-0': 'Comparing different elements in Table [REF], one gets relations, [EQUATION] which are just a manifestation of the isospin symmetry.', '1301.7014-1-15-1': 'As shown below, these relations are broken by the BH corrections.', '1301.7014-1-16-0': 'In the leading order in [MATH] both BH diagrams, Fig. [REF] (b) and (c), acquire dominant contribution from longitudinally polarized photons.', '1301.7014-1-16-1': 'However, as discussed below, certain angular harmonics are suppressed by [MATH] and get similar contributions from transverse and longitudinal photons.', '1301.7014-1-16-2': 'For this reason, for the form factors we include both the longitudinal and transverse components and evaluate the BH diagrams exactly, and only after that we make an expansion in [MATH].', '1301.7014-1-16-3': 'The diagram (b) contains the matrix element [EQUATION] where [MATH] and [MATH] are the vector and axial vector isovector currents.', '1301.7014-1-16-4': 'The correlator ([REF]) can be evaluated in pQCD in the collinear approximation, because [MATH] is large, and we assume that the dominant contribution comes from the leading twist-2 pion DA.', '1301.7014-1-16-5': 'Notice that the amplitude [MATH] cannot be interpreted as the pion form factor because (i) the virtuality of [MATH] is large, (ii) the insertion of the pion state between [MATH] and [MATH] leads to [MATH], which gives zero acting on the transverse on-shell lepton current.', '1301.7014-1-16-6': 'The details of the calculations are moved to Appendix [REF], and we present here the final result, which reads, [EQUATION] where [EQUATION]', '1301.7014-1-16-7': 'Here, following Refs. [CITATION] we introduced the notations, [EQUATION]', '1301.7014-1-16-8': 'Notice that in the leading order in [MATH] the contributions of diagrams (b) and (c) to the terms [MATH] and [MATH] exactly cancel each other, so that the coefficients [MATH] and [MATH] acquire an extra suppression factor [MATH].', '1301.7014-1-16-9': 'This can be understood as a screening of the opposite charges of the pion and muon, when both of them move in the forward direction in the limit of massless leptons.', '1301.7014-1-16-10': 'As we can see, the BH cross-section is symmetric under the [MATH] transformation.', '1301.7014-1-16-11': 'For asymptotically large [MATH], the [MATH] harmonic is suppressed by [MATH], whereas [MATH], so the distribution is symmetric relative to the replacement [MATH].', '1301.7014-1-16-12': 'Note also that [MATH] and vanishes when [MATH].', '1301.7014-1-17-0': 'The interference term in ([REF]) has a form (see details in Appendix [REF]) [EQUATION] where [EQUATION]', '1301.7014-1-17-1': 'As one can see from ([REF]), the angular dependence of the interference term has a [MATH] term which is absent both in the BH and DVMP taken alone.', '1301.7014-1-17-2': 'This term stems from the interference of the vector and axial vector current in lepton part of the diagram.', '1301.7014-1-17-3': 'It has the same sign for neutrino and antineutrino beams (in contrast, all the other terms in ([REF]) change sign).', '1301.7014-1-17-4': 'Such terms are absent in case of DVMP and BH since the interference is asymmetric w.r.t. polarization vectors of the emitted boson.', '1301.7014-1-17-5': 'Due to presence of [MATH] harmonics, the antisymmetrized cross-section directly probes the imaginary part of the DVMP amplitude as [EQUATION]', '1301.7014-1-17-6': 'Another way to access the interference term is based on the isospin symmetry for the pure DVMP cross-sections ([REF],[REF]).', '1301.7014-1-17-7': 'Since the BH correction nearly vanishes on a neutron target, and that BH cross-sections may be easily calculated, one may directly probe the interference term and extract the real and imaginary parts of the DVMP amplitude as, [EQUATION]', '1301.7014-1-18-0': '# GPD and DA parametrizations', '1301.7014-1-19-0': 'As was mentioned in the introduction, an essential uncertainty in the calculations of DVMP originate from the poorly known DAs of the produced mesons.', '1301.7014-1-19-1': 'Only the DAs of pions and [MATH]-mesons have been tested experimentally, and even in this case the situation remains rather controversial.', '1301.7014-1-19-2': 'The early experiments CELLO and CLEO [CITATION], which studied the small-[MATH] behavior of the form factor [MATH], found it to be consistent with the asymptotic form, [MATH].', '1301.7014-1-19-3': 'Later the BABAR collaboration [CITATION] found a steep rise with [MATH] of the form factor [MATH] in the large-[MATH] regime.', '1301.7014-1-19-4': 'This observation gave rise to speculations that the pion DA might have a [MATH]-dependence quite different from the asymptotic form [CITATION] (see also the recent review by Brodsky et.', '1301.7014-1-19-5': 'al. in [CITATION]).', '1301.7014-1-19-6': 'However, the most recent mesurements by the BELLE collaboration [CITATION] did not confirm the rapid growth with [MATH] observed in the BABAR experiment.', '1301.7014-1-19-7': 'As was found in [CITATION] based on the fits to BELLE, CLEO and CELLO data, the Gegenbauer expansion coefficients of the pion DA [MATH] are small and give at most [MATH] correction for the minus-first moment [MATH] defined in ([REF]).', '1301.7014-1-19-8': 'Although there are no direct measurements of the kaon DAs, it is expected that their deviations from the pion DAs are parametrically suppressed by the quark mass [MATH].', '1301.7014-1-19-9': 'Numerically this corresponds to a 10-20% deviation.', '1301.7014-1-20-0': 'For this reason in what follows we assume all the Goldstone DAs to have the asymptotic form, [EQUATION]', '1301.7014-1-20-1': 'For the decay constants we use the standard values [MATH] MeV, [MATH] MeV, and [MATH].', '1301.7014-1-21-0': 'For GPDs more than a dozen different parametrizations have been proposed so far [CITATION].', '1301.7014-1-21-1': 'While we neither endorse nor refute any of them, for the sake of concreteness we select the parametrization [CITATION], which succeeded to describe HERA [CITATION] and JLAB [CITATION] data on electro- and photoproduction of different mesons, so it might provide a reasonable description of [MATH]DVMP.', '1301.7014-1-21-2': "This parametrization is based on the Radyushkin's double distribution ansatz.", '1301.7014-1-21-3': 'It assumes additivity of the valence and sea parts of the GPDs, [EQUATION] which are defined as [EQUATION] and [MATH] and [MATH] are the ordinary valence and sea components of PDFs.', '1301.7014-1-21-4': 'The coefficients [MATH], [MATH], as well as the parametrization of the input PDFs [MATH] and pseudo-PDFs [MATH] (corresponding to the forward limit of the GPDs [MATH]), are discussed in [CITATION].', '1301.7014-1-21-5': 'The unpolarized PDFs [MATH] within the range of [MATH] GeV[MATH] roughly coincide with the CTEQ PDFs.', '1301.7014-1-21-6': 'Notice that in this model the sea is flavor symmetric for asymptotically large [MATH], [EQUATION] where [EQUATION]', '1301.7014-1-21-7': 'The equality of the sea components for [MATH] and [MATH] quarks in ([REF]) should be considered as a rough approximation, since in the forward limit [MATH] was firmly established by the E866/NuSea experiment [CITATION].', '1301.7014-1-21-8': 'For this reason the predictions made with this parametrization of GPDs for the [MATH] transitions in the region [MATH] might slightly underestimate the data.', '1301.7014-1-22-0': 'The Dirac and Pauli form factors [MATH] are extracted from GPDs in the standard way, [EQUATION]', '1301.7014-1-23-0': '# Numerical results and discussion', '1301.7014-1-24-0': 'In this section we perform numerical analysis of the electromagnetic corrections to the processes listed in Table [REF], relying on the GPDs described in the previous section.', '1301.7014-1-24-1': 'At small [MATH] the angular harmonics are small and the cross-section is dominated by the angular-independent DVMP contribution.', '1301.7014-1-24-2': 'Therefore, it is convenient to normalize all the coefficients to DVMP cross-section, [EQUATION]', '1301.7014-1-24-3': 'Notice that in the limit [MATH], no BH corrections are possible, the coefficient [MATH], and all other coefficients vanish.', '1301.7014-1-25-0': 'The results for the [MATH]-dependence of the relative BH corrections to the neutrino-DVMP cross section for pions and kaons are presented in Figure [REF].', '1301.7014-1-26-0': 'We see that the isotropic part of the BH correction [MATH] steeply rises in all channels from few percent or less at [MATH] up to few tens of percent at [MATH].', '1301.7014-1-26-1': 'It behaves like [MATH] modulo logarithmic corrections.', '1301.7014-1-26-2': 'As a result, the cross section is reduced about twice compared to the DVMP contribution.', '1301.7014-1-26-3': 'The asymmetry [MATH] also rises with [MATH] and reaches about [MATH] at [MATH] GeV[MATH].', '1301.7014-1-26-4': 'Notice that some of the coefficients (e.g. [MATH], [MATH]) have nodes for [MATH] production, while they are absent for [MATH].', '1301.7014-1-26-5': 'The reason is purely algebraic: as one can see from the Table [REF], for [MATH] the large [MATH]-channel coefficient function [MATH] is convoluted with the small [MATH] -quark GPD, whereas the small [MATH]-channel coefficient function [MATH] is convoluted with the large [MATH]-quark GPD.', '1301.7014-1-26-6': 'This produces a node in the real part of the DVMP amplitude, because the real parts of the two contributions have opposite signs.', '1301.7014-1-26-7': 'Such a node is absent for negatively charged mesons, because the "large" [MATH]-quark GPD is convoluted with the "large" [MATH].', '1301.7014-1-26-8': 'The full DVMP cross section has no nodes, because it gets a large contribution from the imaginary part, which homogeneously depends on [MATH] (the coefficient [MATH], which probes the imaginary part, has no nodes).', '1301.7014-1-26-9': 'The difference between the Cabibbo suppressed and allowed processes comes from the sensitivity to different flavor combinations of GPDs in the corresponding DVMP amplitude.', '1301.7014-1-27-0': 'The terms [MATH] and [MATH] in Eqn. ([REF]) are dominated by the interference of the DVMP and BH amplitudes, therefore they have different signs for [MATH] and [MATH] (and [MATH] and [MATH]).', '1301.7014-1-27-1': 'The term [MATH] gets contribution only from BH process, so it always has the same sign.', '1301.7014-1-27-2': 'The [MATH]-term does not change its sign under the [MATH]-conjugation in the lepton part, because it originates from the [MATH]-odd interference between the vector and axial vector currents.', '1301.7014-1-28-0': 'The results for [MATH]-dependence of the BH corrections are depicted in Figure [REF].', '1301.7014-1-29-0': 'It shows that there is a qualitative difference between [MATH] and other angular harmonics.', '1301.7014-1-29-1': 'The coefficient [MATH] reaches its maximum at [MATH] due to the [MATH] behavior of the BH cross section.', '1301.7014-1-29-2': 'In contrast, the angular harmonics [MATH] vanish at small [MATH] due the [MATH]-factors in front of them.', '1301.7014-1-29-3': 'As a consequence, the harmonics reach their maxima at [MATH] GeV.', '1301.7014-1-30-0': 'The results for the angular harmonic coefficients vs the elasticity parameter [MATH] are presented in Figure [REF].', '1301.7014-1-31-0': 'The coefficient [MATH] does not depend on [MATH] at all due to exact cancellation of the pre-factors [MATH] in [MATH] and the pre-factor in the DVMP cross-section Eqn. ([REF]).', '1301.7014-1-31-1': 'The harmonics [MATH] and [MATH] have a mild dependence on [MATH], except in the region [MATH], where they blow up, because the DVMP cross-section ([REF]) is suppressed there by the factor [MATH], whereas the harmonics in ([REF],[REF]) are suppressed at most as [MATH].', '1301.7014-1-31-2': 'The harmonic [MATH] in accordance with Eqn. ([REF]) in the kinematics [MATH] is proportional to [MATH] and has a node near [MATH].', '1301.7014-1-32-0': '# Summary', '1301.7014-1-33-0': 'In this paper we studied the electromagnetic Bethe-Heitler corrections to neutrino-induced deeply virtual meson production.', '1301.7014-1-33-1': 'We found these corrections to fall with [MATH] less steeply compared with the [MATH]DVMP cross section, so they tend to become a dominant mechanism in the Bjorken limit of [MATH].', '1301.7014-1-33-2': 'Besides, they are enhanced at small-[MATH] due to the [MATH]-channel Coulomb pole [MATH].', '1301.7014-1-33-3': 'Remarkably, these corrections generate an angular correlation between the lepton and hadron scattering planes.', '1301.7014-1-33-4': 'Similar to the BH corrections in DVCS, some angular harmonics are sensitive to the real or imaginary parts of the DVMP amplitude (see ([REF]-[REF])).', '1301.7014-1-33-5': 'Notice that the appearance of such angular dependence was also predicted in [CITATION], however there it appears due to interference of the longitudinally and transversely polarized charged bosons, which is a twist-three effect.', '1301.7014-1-33-6': 'In contrast, our result is a twist-two effect, and as one can see from Figure [REF], it is not suppressed for asymptotically large [MATH].', '1301.7014-1-34-0': 'Numerically, the BH corrections are subject to the interplay between the suppression factor [MATH] and the relative enhancement, which is as large as [MATH] for some harmonics.', '1301.7014-1-34-1': 'In the kinematics of the Minerva experiment, the BH contribution for the proton target represents a few percent correction and thus is important for precision tests of the GPD parametrizations.', '1301.7014-1-34-2': 'At [MATH] GeV[MATH], which can be accessed in future neutrino experiments, these corrections are expected to become on par with the DVMP contribution.', '1301.7014-1-34-3': 'For a neutron target, these corrections are two orders of magnitude smaller than for a proton and can be neglected up to very high [MATH].', '1301.7014-1-34-4': 'Combining this fact with isospin symmetry of the DVMP amplitude, we construct combinations of the cross-sections Eqs. ([REF],[REF]), which are sensitive only to the interference term.', '1301.7014-1-35-0': 'The electromagnetic corrections discussed in this paper are important only in neutrino-induced DVMP: in the case of electron-induced processes [MATH] the BH corrections are suppressed by the factor [MATH], and are negligibly small.', '1301.7014-1-35-1': 'We provide a computational code, which can be used for evaluation of the cross-sections relying on different GPD models.', '1301.7014-1-36-0': '# Evaluation of the [MATH]DVMP and BH cross-sections', '1301.7014-1-37-0': 'In this section we present some technical details of the evaluation of diagrams (a-c) in the Figure [REF].', '1301.7014-1-37-1': 'The calculation of the diagram (a) in Figure [REF] is rather straightforward and yields for the amplitude of the process [CITATION] [EQUATION] where [MATH] and [MATH] are the spinors of the final muon and initial neutrino; [MATH] is the polarization vector of the photon; [MATH] are the spinors of the initial/final state baryons; [MATH] is the DA of the produced meson; [MATH] is the decay constant of the meson [MATH]; subscript index for each momentum in Eqn. ([REF]) and in what follows shows to which particle it corresponds; [MATH] is a symbolic notation for summation of all the leading twist GPDs contributions (defined below); and [MATH] are the convolutions of the GPDs [MATH] of the target with the proper coefficient function.', '1301.7014-1-37-2': 'Currently, the amplitude of the DVMP is known up to the NLO accuracy [CITATION].', '1301.7014-1-37-3': 'Extension of the analysis of [CITATION] to neutrinos is straightforward.', '1301.7014-1-37-4': 'In contrast to electro-production, due to the [MATH] structure, the amplitudes acquire contributions from both the unpolarized and helicity flip GPDs.', '1301.7014-1-38-0': 'Four GPDs, [MATH] and [MATH] contribute to this process in the leading twist.', '1301.7014-1-38-1': 'They are defined as [EQUATION] where [MATH], [MATH] and [MATH] (see e.g. [CITATION] for the details of kinematics).', '1301.7014-1-38-2': 'In what follows we assume that the target [MATH] is either a proton or a neutron.', '1301.7014-1-38-3': 'Since in neutrino experiments the target cannot be polarized due to its large size, it makes no sense to discuss the transversity GPDs [MATH].', '1301.7014-1-38-4': 'We also ignore the contributions of gluons in this paper because in the current neutrino experiments the region of small [MATH] but very high [MATH], is hardly accessible, so the amplitude ([REF]) simplifies to [EQUATION]', '1301.7014-1-38-5': 'In Table [REF] the corresponding amplitudes are listed for each final state [MATH].', '1301.7014-1-38-6': 'The DVMP part of the corresponding neutrino cross-section for charged currents is given by ([REF]).', '1301.7014-1-39-0': 'In the leading order in [MATH] both BH diagrams Figure [REF] (b,c) are dominated by longitudinally polarized photons.', '1301.7014-1-39-1': 'Nevertheless, as was mentioned in Section [REF], we evaluate the BH contribution exactly, because various angular harmonics, suppressed by [MATH], get contribution from the transverse components, which is of the same order as the longitudinal result.', '1301.7014-1-39-2': 'Only after that we make expansion in [MATH].', '1301.7014-1-40-0': 'The dipole scattering amplitude, which contributes to the diagram Figure [REF] (b), has the form [EQUATION] where [MATH] and [MATH] are the vector and axial-vector isovector currents.', '1301.7014-1-40-1': 'Notice that the amplitude [MATH] should not be interpreted as a pion form factor, because: (i) the virtuality is large; (ii) the insertion of the pion state between [MATH] and [MATH] leads to [MATH], which vanishes when is multiplied by an on-shell lepton current.', '1301.7014-1-41-0': 'We evaluated ([REF]) in pQCD in the collinear approximation.', '1301.7014-1-41-1': 'This is justified in Bjorken kinematics by the high virtuality of the charged boson, so we assume that the dominant contribution comes from the leading twist-2 pion DA.', '1301.7014-1-41-2': 'The result reads, [EQUATION]', '1301.7014-1-41-3': 'Here [MATH] and [MATH] are the positive and the negative direction light-cone vectors respectively.', '1301.7014-1-41-4': 'The plus-components of [MATH] and [MATH] have the form, [EQUATION] where [MATH].', '1301.7014-1-41-5': 'In what follows we encounter the combination [MATH], for which we need to make expansion up to [MATH].', '1301.7014-1-41-6': 'While separately the series expansion coefficients for each factor [MATH] have non-integrable singularities [MATH], which signal a sensitivity to the transverse degrees of freedom and presence of the non-analytic terms [MATH], in the above-mentioned combination, these terms cancel each other resulting in [EQUATION]', '1301.7014-1-41-7': 'The amplitude of the axial current transition into an on-shell pion in the diagram Figure [REF] (c) according to PCAC has the form, [EQUATION]', '1301.7014-1-41-8': 'In order to simplify the calculation of the leptonic part of the diagram (c), we employ the chain of identities [EQUATION] where in ([REF]) we make use of the fact that the initial state neutrino is on-shell, [MATH].', '1301.7014-1-41-9': 'Actually, the simplification of ([REF]) is a manifestation of the Ward-Takahashi-Slavnov-Taylor identity for the charged lepton current, [EQUATION]', '1301.7014-1-41-10': 'This simplification explains why there is no harmonics in the denominator of the BH and interference terms, and it is valid in the limit of massless leptons.', '1301.7014-1-41-11': 'The diagrams (b, c) yield for the amplitude (sign corresponds to [MATH]) [EQUATION]', '1301.7014-1-41-12': 'Further evaluation of Bethe-Heitler ([REF]) and interference ([REF]) terms requires some trivial but tedious Dirac algebra, which was done with FeynCalc [CITATION].'}

{'1301.7014-2-0-0': 'We study the electromagnetic Bethe-Heitler type contribution to neutrino-induced deeply virtual meson production ([MATH]DVMP).', '1301.7014-2-0-1': 'Such [MATH]-corrections decrease with [MATH] in the Bjorken regime less steeply than the standard [MATH]DVMP handbag contribution.', '1301.7014-2-0-2': 'Therefore, they are relatively enhanced at high [MATH].', '1301.7014-2-0-3': 'The Bethe-Heitler terms give rise to an angular correlation between the lepton and hadron scattering planes with harmonics sensitive to the real and imaginary parts of the DVMP amplitude.', '1301.7014-2-0-4': 'These corrections constitute a few percent effect in the kinematics of the forthcoming Minerva experiment at Fermilab and should be taken into account in precision tests of GPD parametrizations.', '1301.7014-2-0-5': 'For virtualities [MATH] GeV[MATH] these corrections become on a par with DVMP handbag contributions.', '1301.7014-2-0-6': 'A computational code, which can be used for the evaluation of these corrections employing various GPD models is provided.', '1301.7014-2-1-0': '# Introduction', '1301.7014-2-2-0': 'Generalized parton distributions (GPD) allow evaluation of cross-sections for a wide class of processes, where the collinear factorization is applicable [CITATION].', '1301.7014-2-2-1': 'The main source of experimental information on GPDs has been so far the electron(positron)-proton measurements performed at JLAB and HERA, in particular deeply virtual Compton scattering (DVCS) and deeply virtual meson production (DVMP) [CITATION].', '1301.7014-2-2-2': 'The 12 GeV upgrade at Jefferson lab will open new opportunities for further improvement of our knowledge of the GPDs [CITATION].', '1301.7014-2-3-0': 'However, the practical realization of this program suffers from large uncertainties.', '1301.7014-2-3-1': 'For instance, the results at moderately high [MATH] can be affected by poorly known higher-twist components of GPDs and distribution amplitudes (DA) of the produced mesons [CITATION].', '1301.7014-2-4-0': 'Neutrino experiments provide a powerful tool for consistency checks for the extraction of GPD from JLAB data, especially of their flavor structure.', '1301.7014-2-4-1': 'The study of various processes in the Bjorken regime may be done with the high-intensity NuMI beam at Fermilab, which will switch soon to the so-called middle-energy (ME) regime with an average neutrino energy of about [MATH] GeV, and potentially may reach energies up to 20 GeV, without essential loss of luminosity.', '1301.7014-2-4-2': 'In this setup the Minerva experiment [CITATION] should be able to probe the quark flavor structure of the targets.', '1301.7014-2-4-3': 'Even higher luminosities in multi-GeV regime can be achieved at the planned Muon Collider/Neutrino Factory [CITATION].', '1301.7014-2-5-0': 'Certain information on the GPD flavor structure can be extracted from comparison of analogous processes in neutrino- and electro-induced processes employing the difference of flavor structures of electromagnetic and weak neutral currents.', '1301.7014-2-5-1': 'An example is the weak DVCS [CITATION], which alone, however, is not sufficient to constrain the flavor structure.', '1301.7014-2-6-0': 'Recently we discussed the possibility of GPD extraction from deeply virtual neutrino-production of the pseudo-Goldstone mesons ([MATH]) [CITATION].', '1301.7014-2-6-1': 'The [MATH]DVMP measurements with neutrino and antineutrino beams are complementary to the electromagnetic DVMP.', '1301.7014-2-6-2': 'The octet of pseudo-Goldstone bosons, originating from the chiral symmetry breaking, acts in the axial current as a natural probe for the flavor content.', '1301.7014-2-6-3': 'Due to the [MATH] structure of the charged current, in [MATH]DVMP one can access simultaneously the unpolarized GPDs, [MATH] and [MATH], and the helicity flip GPDs, [MATH] and [MATH].', '1301.7014-2-6-4': 'We expect the contributions of the GPDs [MATH] and [MATH], which are controlled by the poorly known twist-3 pion DA [MATH], to be negligible.', '1301.7014-2-6-5': 'Besides, important information on the flavor structure can be obtained by studying the transitional GPDs in the processes with nucleon to hyperon transitions.', '1301.7014-2-6-6': 'As was discussed in [CITATION], assuming [MATH] flavor symmetry one can relate these GPDs to the ordinary diagonal ones in the proton.', '1301.7014-2-7-0': 'In this paper we study the Bethe-Heitler (BH) type radiative corrections to the diffractive neutrino-production of charged pseudo-Golstone mesons with the target remained intact, related to meson emission from lepton line with subsequent electromagnetic interaction with the target.', '1301.7014-2-7-1': 'Although such processes are formally suppressed by [MATH], at large [MATH] they fall off less steeply than the DVMP cross-section.', '1301.7014-2-7-2': 'While at virtualities [MATH] GeV[MATH], relevant for modern neutrino experiments, this is a few percent correction, already at [MATH] GeV[MATH] it becomes comparable with the DVMP cross section.', '1301.7014-2-7-3': 'Such corrections are relevant only in case of the [MATH]DVMP: for the electron-induced DVMP [MATH] they are suppressed by factor [MATH], where [MATH] is the Fermi coupling, and are negligible unless we consider extremely high [MATH].', '1301.7014-2-7-4': 'Existence of such diagrams opens a possibility to probe separately the real and imaginary parts of the DVMP amplitude (not only the total cross-sections), in close analogy to DVCS studies [CITATION].', '1301.7014-2-8-0': 'The paper is organized as follows.', '1301.7014-2-8-1': 'In Section [REF] we present the results for the DVMP and BH contributions to the experimentally measurable total [MATH]DVMP cross-section (technical details of evaluation may be found in Appendix [REF]).', '1301.7014-2-8-2': 'Also, at the end of Section [REF] we construct two asymmetries which have particular sensitivity to the real and imaginary parts of the DVMP amplitude.', '1301.7014-2-8-3': 'In Section [REF], for the sake of completeness we discuss the features of the GPD parametrization used in calculations.', '1301.7014-2-8-4': 'In Section [REF] we present the numerical results and make conclusions.', '1301.7014-2-9-0': '# Cross-section of the [MATH]DVMP and BH processes', '1301.7014-2-10-0': 'Exclusive neutrino-production of pions is presented by the diagram (a) in the Figure [REF].', '1301.7014-2-10-1': 'Production of pions by the vector current was studied in [CITATION], and was recently extended to neutrino interactions in [CITATION].', '1301.7014-2-11-0': 'As was shown in [CITATION], at large [MATH] where the collinear factorization is applicable, the amplitude of this process is suppressed due to hard gluon exchange in the coefficient function, and is small.', '1301.7014-2-11-1': 'This raises a question, how important could be the [MATH] corrections ?', '1301.7014-2-11-2': 'While a systematic study of radiative corrections is beyond the scope of the present paper, we would like to consider in detail the contributions which decrease with [MATH] less rapidly than the diagram (a).', '1301.7014-2-11-3': 'In the leading order in [MATH] and [MATH] there are two such diagrams (b,c) shown in the Figure [REF].', '1301.7014-2-11-4': 'These diagrams are enhanced by a factor [MATH], where [MATH] and [MATH] are the lepton and proton 4-momentum transfers squared, respectively.', '1301.7014-2-11-5': 'This factor is large in the Bjorken regime of [MATH], and despite the formal suppression by [MATH], the diagrams (b,c) are numerically comparable to the diagram (a).', '1301.7014-2-11-6': 'Notice that such corrections are sizable only in the case of neutrino-production of mesons.', '1301.7014-2-11-7': 'In case of electroproduction the corresponding corrections are suppressed by [MATH], where [MATH] is the Fermi constant, and are negligible, unless we go to extremely large [MATH] .', '1301.7014-2-11-8': 'In what follows we evaluate the contribution of each diagram in Figure [REF].', '1301.7014-2-12-0': 'The cross-section of pion production can be presented as a sum of contributions of DVMP (diagram (a)), BH mechanism (diagrams (b,c)) and their interference, [EQUATION] where [MATH], and [MATH] is the angle between the lepton and hadron scattering planes.', '1301.7014-2-13-0': 'Evaluation of the diagram (a) is straightforward and yields [CITATION] [EQUATION] where [MATH] is the fractional loss of lepton energy defined as [MATH].', '1301.7014-2-13-1': 'Notice that the DVMP cross section turns out to be independent of the angle [MATH] between lepton and hadron planes.', '1301.7014-2-13-2': 'This happens because the momentum [MATH] does not have transverse components in the Bjorken reference frame.', '1301.7014-2-14-0': 'For unpolarized target, the matrix element squared [MATH] in Eqn. ([REF]) can be simplified as, [EQUATION] where we introduced a shorthand notation, [EQUATION] and the script letters [MATH] signify convolution of the GPDs [MATH] with corresponding coefficient functions given in Table [REF].', '1301.7014-2-15-0': 'Comparing different elements in Table [REF], one gets relations, [EQUATION] which are just a manifestation of the isospin symmetry.', '1301.7014-2-15-1': 'As shown below, these relations are broken by the BH corrections.', '1301.7014-2-16-0': 'In the leading order in [MATH] both BH diagrams, Fig. [REF] (b) and (c), acquire dominant contribution from longitudinally polarized photons.', '1301.7014-2-16-1': 'However, as discussed below, certain angular harmonics are suppressed by [MATH] and get similar contributions from transverse and longitudinal photons.', '1301.7014-2-16-2': 'For this reason, for the form factors we include both the longitudinal and transverse components and evaluate the BH diagrams exactly, and only after that we make an expansion in [MATH].', '1301.7014-2-16-3': 'The diagram (b) contains the matrix element [EQUATION] where [MATH] and [MATH] are the vector and axial vector isovector currents.', '1301.7014-2-16-4': 'The correlator ([REF]) can be evaluated in pQCD in the collinear approximation, because [MATH] is large, and we assume that the dominant contribution comes from the leading twist-2 pion DA.', '1301.7014-2-16-5': 'Notice that the amplitude [MATH] cannot be interpreted as the pion form factor because (i) the virtuality of [MATH] is large, (ii) the insertion of the pion state between [MATH] and [MATH] leads to [MATH], which gives zero acting on the transverse on-shell lepton current.', '1301.7014-2-16-6': 'The details of the calculations are moved to Appendix [REF], and we present here the final result, which reads, [EQUATION] where [EQUATION]', '1301.7014-2-16-7': 'Here, following Refs. [CITATION] we introduced the notations, [EQUATION]', '1301.7014-2-16-8': 'Notice that in the leading order in [MATH] the contributions of diagrams (b) and (c) to the terms [MATH] and [MATH] exactly cancel each other, so that the coefficients [MATH] and [MATH] acquire an extra suppression factor [MATH].', '1301.7014-2-16-9': 'This can be understood as a screening of the opposite charges of the pion and muon, when both of them move in the forward direction in the limit of massless leptons.', '1301.7014-2-16-10': 'As we can see, the BH cross-section is symmetric under the [MATH] transformation.', '1301.7014-2-16-11': 'For asymptotically large [MATH], the [MATH] harmonic is suppressed by [MATH], whereas [MATH], so the distribution is symmetric relative to the replacement [MATH].', '1301.7014-2-16-12': 'Note also that [MATH] and vanishes when [MATH].', '1301.7014-2-17-0': 'The interference term in ([REF]) has a form (see details in Appendix [REF]) [EQUATION] where [EQUATION]', '1301.7014-2-17-1': 'As one can see from ([REF]), the angular dependence of the interference term has a [MATH] term which is absent both in the BH and DVMP taken alone.', '1301.7014-2-17-2': 'This term stems from the interference of the vector and axial vector current in lepton part of the diagram.', '1301.7014-2-17-3': 'It has the same sign for neutrino and antineutrino beams (in contrast, all the other terms in ([REF]) change sign).', '1301.7014-2-17-4': 'Such terms are absent in case of DVMP and BH since the interference is asymmetric w.r.t. polarization vectors of the emitted boson.', '1301.7014-2-17-5': 'Due to presence of [MATH] harmonics, the antisymmetrized cross-section directly probes the imaginary part of the DVMP amplitude as [EQUATION]', '1301.7014-2-17-6': 'Another way to access the interference term is based on the isospin symmetry for the pure DVMP cross-sections ([REF],[REF]).', '1301.7014-2-17-7': 'Since the BH correction nearly vanishes on a neutron target, and that BH cross-sections may be easily calculated, one may directly probe the interference term and extract the real and imaginary parts of the DVMP amplitude as, [EQUATION]', '1301.7014-2-18-0': '# GPD and DA parametrizations', '1301.7014-2-19-0': 'As was mentioned in the introduction, an essential uncertainty in the calculations of DVMP originate from the poorly known DAs of the produced mesons.', '1301.7014-2-19-1': 'Only the DAs of pions and [MATH]-mesons have been tested experimentally, and even in this case the situation remains rather controversial.', '1301.7014-2-19-2': 'The early experiments CELLO and CLEO [CITATION], which studied the small-[MATH] behavior of the form factor [MATH], found it to be consistent with the asymptotic form, [MATH].', '1301.7014-2-19-3': 'Later the BABAR collaboration [CITATION] found a steep rise with [MATH] of the form factor [MATH] in the large-[MATH] regime.', '1301.7014-2-19-4': 'This observation gave rise to speculations that the pion DA might have a [MATH]-dependence quite different from the asymptotic form [CITATION] (see also the recent review by Brodsky et.', '1301.7014-2-19-5': 'al. in [CITATION]).', '1301.7014-2-19-6': 'However, the most recent mesurements by the BELLE collaboration [CITATION] did not confirm the rapid growth with [MATH] observed in the BABAR experiment.', '1301.7014-2-19-7': 'As was found in [CITATION] based on the fits to BELLE, CLEO and CELLO data, the Gegenbauer expansion coefficients of the pion DA [MATH] are small and give at most [MATH] correction for the minus-first moment [MATH] defined in ([REF]).', '1301.7014-2-19-8': 'Although there are no direct measurements of the kaon DAs, it is expected that their deviations from the pion DAs are parametrically suppressed by the quark mass [MATH].', '1301.7014-2-19-9': 'Numerically this corresponds to a 10-20% deviation.', '1301.7014-2-20-0': 'For this reason in what follows we assume all the Goldstone DAs to have the asymptotic form, [EQUATION]', '1301.7014-2-20-1': 'For the decay constants we use the standard values [MATH] MeV, [MATH] MeV, and [MATH].', '1301.7014-2-21-0': 'For GPDs more than a dozen different parametrizations have been proposed so far [CITATION].', '1301.7014-2-21-1': 'While we neither endorse nor refute any of them, for the sake of concreteness we select the parametrization [CITATION], which succeeded to describe HERA [CITATION] and JLAB [CITATION] data on electro- and photoproduction of different mesons, so it might provide a reasonable description of [MATH]DVMP.', '1301.7014-2-21-2': "This parametrization is based on the Radyushkin's double distribution ansatz.", '1301.7014-2-21-3': 'It assumes additivity of the valence and sea parts of the GPDs, [EQUATION] which are defined as [EQUATION] and [MATH] and [MATH] are the ordinary valence and sea components of PDFs.', '1301.7014-2-21-4': 'The coefficients [MATH], [MATH], as well as the parametrization of the input PDFs [MATH] and pseudo-PDFs [MATH] (corresponding to the forward limit of the GPDs [MATH]), are discussed in [CITATION].', '1301.7014-2-21-5': 'The unpolarized PDFs [MATH] within the range of [MATH] GeV[MATH] roughly coincide with the CTEQ PDFs.', '1301.7014-2-21-6': 'Notice that in this model the sea is flavor symmetric for asymptotically large [MATH], [EQUATION] where [EQUATION]', '1301.7014-2-21-7': 'The equality of the sea components for [MATH] and [MATH] quarks in ([REF]) should be considered as a rough approximation, since in the forward limit [MATH] was firmly established by the E866/NuSea experiment [CITATION].', '1301.7014-2-21-8': 'For this reason the predictions made with this parametrization of GPDs for the [MATH] transitions in the region [MATH] might slightly underestimate the data.', '1301.7014-2-22-0': 'The Dirac and Pauli form factors [MATH] are extracted from GPDs in the standard way, [EQUATION]', '1301.7014-2-23-0': '# Numerical results and discussion', '1301.7014-2-24-0': 'In this section we perform numerical analysis of the electromagnetic corrections to the processes listed in Table [REF], relying on the GPDs described in the previous section.', '1301.7014-2-24-1': 'At small [MATH] the angular harmonics are small and the cross-section is dominated by the angular-independent DVMP contribution.', '1301.7014-2-24-2': 'Therefore, it is convenient to normalize all the coefficients to DVMP cross-section, [EQUATION]', '1301.7014-2-24-3': 'Notice that in the limit [MATH], no BH corrections are possible, the coefficient [MATH], and all other coefficients vanish.', '1301.7014-2-25-0': 'The results for the [MATH]-dependence of the relative BH corrections to the neutrino-DVMP cross section for pions and kaons are presented in Figure [REF].', '1301.7014-2-26-0': 'We see that the isotropic part of the BH correction [MATH] steeply rises in all channels from few percent or less at [MATH] up to few tens of percent at [MATH].', '1301.7014-2-26-1': 'It behaves like [MATH] modulo logarithmic corrections.', '1301.7014-2-26-2': 'As a result, the cross section is reduced about twice compared to the DVMP contribution.', '1301.7014-2-26-3': 'The asymmetry [MATH] also rises with [MATH] and reaches about [MATH] at [MATH] GeV[MATH].', '1301.7014-2-26-4': 'Notice that some of the coefficients (e.g. [MATH], [MATH]) have nodes for [MATH] production, while they are absent for [MATH].', '1301.7014-2-26-5': 'The reason is purely algebraic: as one can see from the Table [REF], for [MATH] the large [MATH]-channel coefficient function [MATH] is convoluted with the small [MATH] -quark GPD, whereas the small [MATH]-channel coefficient function [MATH] is convoluted with the large [MATH]-quark GPD.', '1301.7014-2-26-6': 'This produces a node in the real part of the DVMP amplitude, because the real parts of the two contributions have opposite signs.', '1301.7014-2-26-7': 'Such a node is absent for negatively charged mesons, because the "large" [MATH]-quark GPD is convoluted with the "large" [MATH].', '1301.7014-2-26-8': 'The full DVMP cross section has no nodes, because it gets a large contribution from the imaginary part, which homogeneously depends on [MATH] (the coefficient [MATH], which probes the imaginary part, has no nodes).', '1301.7014-2-26-9': 'The difference between the Cabibbo suppressed and allowed processes comes from the sensitivity to different flavor combinations of GPDs in the corresponding DVMP amplitude.', '1301.7014-2-27-0': 'The terms [MATH] and [MATH] in Eqn. ([REF]) are dominated by the interference of the DVMP and BH amplitudes, therefore they have different signs for [MATH] and [MATH] (and [MATH] and [MATH]).', '1301.7014-2-27-1': 'The term [MATH] gets contribution only from BH process, so it always has the same sign.', '1301.7014-2-27-2': 'The [MATH]-term does not change its sign under the [MATH]-conjugation in the lepton part, because it originates from the [MATH]-odd interference between the vector and axial vector currents.', '1301.7014-2-28-0': 'The results for [MATH]-dependence of the BH corrections are depicted in Figure [REF].', '1301.7014-2-29-0': 'It shows that there is a qualitative difference between [MATH] and other angular harmonics.', '1301.7014-2-29-1': 'The coefficient [MATH] reaches its maximum at [MATH] due to the [MATH] behavior of the BH cross section.', '1301.7014-2-29-2': 'In contrast, the angular harmonics [MATH] vanish at small [MATH] due the [MATH]-factors in front of them.', '1301.7014-2-29-3': 'As a consequence, the harmonics reach their maxima at [MATH] GeV.', '1301.7014-2-30-0': 'The results for the angular harmonic coefficients vs the elasticity parameter [MATH] are presented in Figure [REF].', '1301.7014-2-31-0': 'The coefficient [MATH] does not depend on [MATH] at all due to exact cancellation of the pre-factors [MATH] in [MATH] and the pre-factor in the DVMP cross-section Eqn. ([REF]).', '1301.7014-2-31-1': 'The harmonics [MATH] and [MATH] have a mild dependence on [MATH], except in the region [MATH], where they blow up, because the DVMP cross-section ([REF]) is suppressed there by the factor [MATH], whereas the harmonics in ([REF],[REF]) are suppressed at most as [MATH].', '1301.7014-2-31-2': 'The harmonic [MATH] in accordance with Eqn. ([REF]) in the kinematics [MATH] is proportional to [MATH] and has a node near [MATH].', '1301.7014-2-32-0': '# Summary', '1301.7014-2-33-0': 'In this paper we studied the electromagnetic Bethe-Heitler corrections to neutrino-induced deeply virtual meson production.', '1301.7014-2-33-1': 'We found these corrections to fall with [MATH] less steeply compared with the [MATH]DVMP cross section, so they tend to become a dominant mechanism in the Bjorken limit of [MATH].', '1301.7014-2-33-2': 'Besides, they are enhanced at small-[MATH] due to the [MATH]-channel Coulomb pole [MATH].', '1301.7014-2-33-3': 'Remarkably, these corrections generate an angular correlation between the lepton and hadron scattering planes.', '1301.7014-2-33-4': 'Similar to the BH corrections in DVCS, some angular harmonics are sensitive to the real or imaginary parts of the DVMP amplitude (see ([REF]-[REF])).', '1301.7014-2-33-5': 'Notice that the appearance of such angular dependence was also predicted in [CITATION], however there it appears due to interference of the longitudinally and transversely polarized charged bosons, which is a twist-three effect.', '1301.7014-2-33-6': 'In contrast, our result is a twist-two effect, and as one can see from Figure [REF], it is not suppressed for asymptotically large [MATH].', '1301.7014-2-34-0': 'Numerically, the BH corrections are subject to the interplay between the suppression factor [MATH] and the relative enhancement, which is as large as [MATH] for some harmonics.', '1301.7014-2-34-1': 'In the kinematics of the Minerva experiment, the BH contribution for the proton target represents a few percent correction and thus is important for precision tests of the GPD parametrizations.', '1301.7014-2-34-2': 'At [MATH] GeV[MATH], which can be accessed in future neutrino experiments, these corrections are expected to become on par with the DVMP contribution.', '1301.7014-2-34-3': 'For a neutron target, these corrections are two orders of magnitude smaller than for a proton and can be neglected up to very high [MATH].', '1301.7014-2-34-4': 'Combining this fact with isospin symmetry of the DVMP amplitude, we construct combinations of the cross-sections Eqs. ([REF],[REF]), which are sensitive only to the interference term.', '1301.7014-2-35-0': 'The electromagnetic corrections discussed in this paper are important only in neutrino-induced DVMP: in the case of electron-induced processes [MATH] the BH corrections are suppressed by the factor [MATH], and are negligibly small.', '1301.7014-2-35-1': 'We provide a computational code, which can be used for evaluation of the cross-sections relying on different GPD models.', '1301.7014-2-36-0': '# Evaluation of the [MATH]DVMP and BH cross-sections', '1301.7014-2-37-0': 'In this section we present some technical details of the evaluation of diagrams (a-c) in the Figure [REF].', '1301.7014-2-37-1': 'The calculation of the diagram (a) in Figure [REF] is rather straightforward and yields for the amplitude of the process [CITATION] [EQUATION] where [MATH] and [MATH] are the spinors of the final muon and initial neutrino; [MATH] is the polarization vector of the photon; [MATH] are the spinors of the initial/final state baryons; [MATH] is the DA of the produced meson; [MATH] is the decay constant of the meson [MATH]; subscript index for each momentum in Eqn. ([REF]) and in what follows shows to which particle it corresponds; [MATH] is a symbolic notation for summation of all the leading twist GPDs contributions (defined below); and [MATH] are the convolutions of the GPDs [MATH] of the target with the proper coefficient function.', '1301.7014-2-37-2': 'Currently, the amplitude of the DVMP is known up to the NLO accuracy [CITATION].', '1301.7014-2-37-3': 'Extension of the analysis of [CITATION] to neutrinos is straightforward.', '1301.7014-2-37-4': 'In contrast to electro-production, due to the [MATH] structure, the amplitudes acquire contributions from both the unpolarized and helicity flip GPDs.', '1301.7014-2-38-0': 'Four GPDs, [MATH] and [MATH] contribute to this process in the leading twist.', '1301.7014-2-38-1': 'They are defined as [EQUATION] where [MATH], [MATH] and [MATH] (see e.g. [CITATION] for the details of kinematics).', '1301.7014-2-38-2': 'In what follows we assume that the target [MATH] is either a proton or a neutron.', '1301.7014-2-38-3': 'Since in neutrino experiments the target cannot be polarized due to its large size, it makes no sense to discuss the transversity GPDs [MATH].', '1301.7014-2-38-4': 'We also ignore the contributions of gluons in this paper because in the current neutrino experiments the region of small [MATH] but very high [MATH], is hardly accessible, so the amplitude ([REF]) simplifies to [EQUATION]', '1301.7014-2-38-5': 'In Table [REF] the corresponding amplitudes are listed for each final state [MATH].', '1301.7014-2-38-6': 'The DVMP part of the corresponding neutrino cross-section for charged currents is given by ([REF]).', '1301.7014-2-39-0': 'In the leading order in [MATH] both BH diagrams Figure [REF] (b,c) are dominated by longitudinally polarized photons.', '1301.7014-2-39-1': 'Nevertheless, as was mentioned in Section [REF], we evaluate the BH contribution exactly, because various angular harmonics, suppressed by [MATH], get contribution from the transverse components, which is of the same order as the longitudinal result.', '1301.7014-2-39-2': 'Only after that we make expansion in [MATH].', '1301.7014-2-40-0': 'The dipole scattering amplitude, which contributes to the diagram Figure [REF] (b), has the form [EQUATION] where [MATH] and [MATH] are the vector and axial-vector isovector currents.', '1301.7014-2-40-1': 'Notice that the amplitude [MATH] should not be interpreted as a pion form factor, because: (i) the virtuality is large; (ii) the insertion of the pion state between [MATH] and [MATH] leads to [MATH], which vanishes when is multiplied by an on-shell lepton current.', '1301.7014-2-41-0': 'We evaluated ([REF]) in pQCD in the collinear approximation.', '1301.7014-2-41-1': 'This is justified in Bjorken kinematics by the high virtuality of the charged boson, so we assume that the dominant contribution comes from the leading twist-2 pion DA.', '1301.7014-2-41-2': 'The result reads, [EQUATION]', '1301.7014-2-41-3': 'Here [MATH] and [MATH] are the positive and the negative direction light-cone vectors respectively.', '1301.7014-2-41-4': 'The plus-components of [MATH] and [MATH] have the form, [EQUATION] where [MATH].', '1301.7014-2-41-5': 'In what follows we encounter the combination [MATH], for which we need to make expansion up to [MATH].', '1301.7014-2-41-6': 'While separately the series expansion coefficients for each factor [MATH] have non-integrable singularities [MATH], which signal a sensitivity to the transverse degrees of freedom and presence of the non-analytic terms [MATH], in the above-mentioned combination, these terms cancel each other resulting in [EQUATION]', '1301.7014-2-41-7': 'The amplitude of the axial current transition into an on-shell pion in the diagram Figure [REF] (c) according to PCAC has the form, [EQUATION]', '1301.7014-2-41-8': 'In order to simplify the calculation of the leptonic part of the diagram (c), we employ the chain of identities [EQUATION] where in ([REF]) we make use of the fact that the initial state neutrino is on-shell, [MATH].', '1301.7014-2-41-9': 'Actually, the simplification of ([REF]) is a manifestation of the Ward-Takahashi-Slavnov-Taylor identity for the charged lepton current, [EQUATION]', '1301.7014-2-41-10': 'This simplification explains why there is no harmonics in the denominator of the BH and interference terms, and it is valid in the limit of massless leptons.', '1301.7014-2-41-11': 'The diagrams (b, c) yield for the amplitude (sign corresponds to [MATH]) [EQUATION]', '1301.7014-2-41-12': 'Further evaluation of Bethe-Heitler ([REF]) and interference ([REF]) terms requires some trivial but tedious Dirac algebra, which was done with FeynCalc [CITATION].'}

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'1301.7014-2-21-4'], ['1301.7014-1-21-5', '1301.7014-2-21-5'], ['1301.7014-1-21-6', '1301.7014-2-21-6'], ['1301.7014-1-21-7', '1301.7014-2-21-7'], ['1301.7014-1-21-8', '1301.7014-2-21-8'], ['1301.7014-1-24-0', '1301.7014-2-24-0'], ['1301.7014-1-24-1', '1301.7014-2-24-1'], ['1301.7014-1-24-2', '1301.7014-2-24-2'], ['1301.7014-1-24-3', '1301.7014-2-24-3'], ['1301.7014-1-0-0', '1301.7014-2-0-0'], ['1301.7014-1-0-1', '1301.7014-2-0-1'], ['1301.7014-1-0-2', '1301.7014-2-0-2'], ['1301.7014-1-0-3', '1301.7014-2-0-3'], ['1301.7014-1-0-4', '1301.7014-2-0-4'], ['1301.7014-1-0-5', '1301.7014-2-0-5'], ['1301.7014-1-0-6', '1301.7014-2-0-6'], ['1301.7014-1-34-0', '1301.7014-2-34-0'], ['1301.7014-1-34-1', '1301.7014-2-34-1'], ['1301.7014-1-34-2', '1301.7014-2-34-2'], ['1301.7014-1-34-3', '1301.7014-2-34-3'], ['1301.7014-1-34-4', '1301.7014-2-34-4'], ['1301.7014-1-40-0', '1301.7014-2-40-0'], ['1301.7014-1-40-1', '1301.7014-2-40-1'], ['1301.7014-1-5-0', '1301.7014-2-5-0'], ['1301.7014-1-5-1', '1301.7014-2-5-1'], ['1301.7014-1-6-0', '1301.7014-2-6-0'], ['1301.7014-1-6-1', '1301.7014-2-6-1'], ['1301.7014-1-6-2', '1301.7014-2-6-2'], ['1301.7014-1-6-3', '1301.7014-2-6-3'], ['1301.7014-1-6-4', '1301.7014-2-6-4'], ['1301.7014-1-6-5', '1301.7014-2-6-5'], ['1301.7014-1-6-6', '1301.7014-2-6-6'], ['1301.7014-1-4-0', '1301.7014-2-4-0'], ['1301.7014-1-4-1', '1301.7014-2-4-1'], ['1301.7014-1-4-2', '1301.7014-2-4-2'], ['1301.7014-1-4-3', '1301.7014-2-4-3'], ['1301.7014-1-33-0', '1301.7014-2-33-0'], ['1301.7014-1-33-1', '1301.7014-2-33-1'], ['1301.7014-1-33-2', '1301.7014-2-33-2'], ['1301.7014-1-33-3', '1301.7014-2-33-3'], ['1301.7014-1-33-4', '1301.7014-2-33-4'], ['1301.7014-1-33-5', '1301.7014-2-33-5'], ['1301.7014-1-33-6', '1301.7014-2-33-6'], ['1301.7014-1-35-0', '1301.7014-2-35-0'], ['1301.7014-1-35-1', '1301.7014-2-35-1'], ['1301.7014-1-8-0', '1301.7014-2-8-0'], ['1301.7014-1-8-1', '1301.7014-2-8-1'], ['1301.7014-1-8-2', '1301.7014-2-8-2'], ['1301.7014-1-8-3', '1301.7014-2-8-3'], ['1301.7014-1-8-4', '1301.7014-2-8-4'], ['1301.7014-1-41-0', '1301.7014-2-41-0'], ['1301.7014-1-41-1', '1301.7014-2-41-1'], ['1301.7014-1-41-2', '1301.7014-2-41-2'], ['1301.7014-1-41-3', '1301.7014-2-41-3'], ['1301.7014-1-41-4', '1301.7014-2-41-4'], ['1301.7014-1-41-5', '1301.7014-2-41-5'], ['1301.7014-1-41-6', '1301.7014-2-41-6'], ['1301.7014-1-41-7', '1301.7014-2-41-7'], ['1301.7014-1-41-8', '1301.7014-2-41-8'], ['1301.7014-1-41-9', '1301.7014-2-41-9'], ['1301.7014-1-41-10', '1301.7014-2-41-10'], ['1301.7014-1-41-11', '1301.7014-2-41-11'], ['1301.7014-1-41-12', '1301.7014-2-41-12'], ['1301.7014-1-11-0', '1301.7014-2-11-0'], ['1301.7014-1-11-1', '1301.7014-2-11-1'], ['1301.7014-1-11-2', '1301.7014-2-11-2'], ['1301.7014-1-11-3', '1301.7014-2-11-3'], ['1301.7014-1-11-4', '1301.7014-2-11-4'], ['1301.7014-1-11-5', '1301.7014-2-11-5'], ['1301.7014-1-11-6', '1301.7014-2-11-6'], ['1301.7014-1-11-7', '1301.7014-2-11-7'], ['1301.7014-1-11-8', '1301.7014-2-11-8']]

[]

[]

[]

[]

['1301.7014-1-19-5', '1301.7014-2-19-5']

{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1301.7014

1309.2616

{'1309.2616-1-0-0': 'We demonstrate and test the adiabatic projection method, a general new framework for calculating scattering and reactions on the lattice.', '1309.2616-1-0-1': 'The method is based upon calculating a low-energy effective theory for clusters which becomes exact in the limit of large Euclidean projection time.', '1309.2616-1-0-2': 'As a detailed example we calculate the adiabatic two-body Hamiltonian for elastic fermion-dimer scattering in lattice effective field theory.', '1309.2616-1-0-3': 'Our calculation corresponds to neutron-deuteron scattering in the spin-quartet channel at leading order in pionless effective field theory.', '1309.2616-1-0-4': 'We show that the spectrum of the adiabatic Hamiltonian reproduces the spectrum of the original Hamiltonian below the inelastic threshold to arbitrary accuracy.', '1309.2616-1-0-5': 'We also show that the calculated [MATH]-wave phase shift reproduces the known exact result in the continuum and infinite-volume limits.', '1309.2616-1-0-6': 'When extended to more than one scattering channel, the adiabatic projection method can be used to calculate inelastic reactions on the lattice.', '1309.2616-1-1-0': '# Introduction', '1309.2616-1-2-0': 'There has been recent progress in ab initio calculations of nuclear scattering and reactions.', '1309.2616-1-2-1': "This includes calculations using the no-core shell model and resonating group method [CITATION], fermionic molecular dynamics [CITATION], the coupled-cluster expansion [CITATION], and variational and Green's function Monte Carlo [CITATION].", '1309.2616-1-2-2': 'For calculations using lattice methods there has been progress in using finite periodic volumes to analyze coupled-channel scattering [CITATION] and three-body systems [CITATION].', '1309.2616-1-2-3': 'Also the first steps towards calculating nuclear reactions on the lattice were introduced in Ref. [CITATION] using an adiabatic projection formalism.', '1309.2616-1-2-4': 'The general strategy involves separating the calculation into two parts.', '1309.2616-1-2-5': 'The first part of the method uses Euclidean time projection to determine an adiabatic Hamiltonian for the participating nuclei.', '1309.2616-1-2-6': 'This is done by starting with a set of cluster states [MATH] labeled by their separation vector [MATH], as illustrated in Fig. [REF].', '1309.2616-1-2-7': 'These states are propagated in Euclidean time to form dressed cluster states, [EQUATION]', '1309.2616-1-3-0': 'By evolving in Euclidean time with the full microscopic Hamiltonian, we are in essence cooling the initial cluster states to the correct physical state dynamically with the interaction.', '1309.2616-1-3-1': 'Deformations and polarizations of the interacting clusters are incorporated automatically by means of Euclidean time projection.', '1309.2616-1-3-2': 'In the limit of large Euclidean time, these dressed cluster states span the low-energy subspace of two-body continuum states for our clusters.', '1309.2616-1-3-3': 'We then calculate an adiabatic Hamiltonian matrix defined by the original Hamiltonian restricted to the subspace of dressed cluster states.', '1309.2616-1-3-4': 'The construction is still ab initio though restricted to the description of cluster configurations.', '1309.2616-1-3-5': 'For inelastic processes we construct dressed cluster states for each of the possible scattering channels and calculate matrix elements for all operators relevant to the reaction process.', '1309.2616-1-3-6': 'For example, in the case of radiative capture reactions, we calculate the adiabatic Hamiltonian and matrix elements of one-photon vertex operators.', '1309.2616-1-4-0': 'The second part of the adiabatic projection method entails using the adiabatic Hamiltonian and operator matrix elements for the dressed cluster states to calculate scattering amplitudes.', '1309.2616-1-4-1': 'For elastic phase shifts, we can apply the finite-volume scaling analysis developed by Luscher [CITATION].', '1309.2616-1-4-2': 'For inelastic reactions, additional steps in the calculation are required.', '1309.2616-1-4-3': "Since we have reduced the problem to a few-body system of nuclear clusters, this part of the calculation can be performed using Green's function methods defined in Minkowskian time.", '1309.2616-1-4-4': "In Ref. [CITATION] this is demonstrated using finite-volume Green's functions for radiative neutron-proton capture, [MATH], in pionless effective field theory.", '1309.2616-1-5-0': 'In this paper we study in depth the first part of the adiabatic projection method, the construction of dressed cluster states and the adiabatic Hamiltonian.', '1309.2616-1-5-1': 'We use the example of elastic fermion-dimer scattering for attractive two-component fermions in the limit of zero-range interactions.', '1309.2616-1-5-2': 'This corresponds to deuteron-neutron scattering in the spin-quartet channel at leading order in pionless effective field theory.', '1309.2616-1-5-3': 'One of the key requirements of an ab initio approach is that all errors are under control and can be systematically reduced.', '1309.2616-1-5-4': 'In our analysis we will find that the spectrum of the adiabatic Hamiltonian matches the spectrum of the original microscopic Hamiltonian below the inelastic threshold to arbitrary accuracy.', '1309.2616-1-5-5': 'Furthermore, we reproduce the [MATH]-wave phase shift in good agreement with exact results in the continuum and infinite-volume limits the Skorniakov-Ter-Martirosian (STM) integral equation [CITATION].', '1309.2616-1-6-0': 'The outline of the paper is as follows.', '1309.2616-1-6-1': 'In Section [REF] we introduce the underlying interactions for our effective field theory description of two-component fermions and fermion-dimer scattering on the lattice.', '1309.2616-1-6-2': 'In Section [REF] we apply the adiabatic projection method to this system and calculate the corresponding two-body adiabatic Hamiltonian.', '1309.2616-1-6-3': 'In Section [REF] we compare the spectrum of the two-body adiabatic Hamiltonian with the spectrum of the original microscopic Hamiltonian.', '1309.2616-1-6-4': 'In Section [REF] we compute the elastic [MATH]-wave phase shift for fermion-dimer scattering and compare with exact continuum infinite-volume results.', '1309.2616-1-6-5': 'We then conclude in Section [REF] with a summary and discussion of the results and applications to inelastic processes.', '1309.2616-1-7-0': '# two-component fermions on the lattice', '1309.2616-1-8-0': 'The example we consider in depth is fermion-dimer scattering for two-component fermions.', '1309.2616-1-8-1': 'We call the two components spin-up and spin-down.', '1309.2616-1-8-2': 'The bound dimer state is composed of one spin-up and one spin-down fermion.', '1309.2616-1-8-3': 'The interactions are chosen to be attractive, and we take the limit where the range of the interactions is negligible.', '1309.2616-1-8-4': 'At leading order in pionless effective field theory, neutron-deuteron scattering in the spin-quartet channel is completely equivalent to our fermion-dimer scattering system.', '1309.2616-1-8-5': 'In the neutron-deuteron case the two fermion components correspond to isospin, while all the nucleon intrinsic spins are fully symmetrized into a spin-quartet.', '1309.2616-1-9-0': 'The Hamiltonian for our system can be written as [EQUATION] where we take the zero-range limit [MATH], with coupling constant [MATH].', '1309.2616-1-9-1': 'Here [MATH] and [MATH] are creation and annihilation operators.', '1309.2616-1-9-2': 'Motivated by the neutron-deuteron system, we take the mass of the fermions to be [MATH] MeV and tune the strength of [MATH] to match the binding energy of the deuteron, [MATH] MeV.', '1309.2616-1-9-3': 'For a shallow bound dimer such as this, the fermion-dimer scattering problem is known to be strongly coupled even at rather low momenta.', '1309.2616-1-9-4': 'See, for example, Ref. [CITATION] and references therein.', '1309.2616-1-10-0': 'We will calculate the properties of this system using a lattice Hamiltonian.', '1309.2616-1-10-1': "While we don't need the full computational machinery in this analysis, we should note that the adiabatic projection formalism fits conveniently into the framework of lattice effective field theory.", '1309.2616-1-10-2': 'Lattice effective field theory is a combination of effective field theory and numerical lattice methods that has been used to study nuclei in pionless EFT [CITATION] and chiral EFT [CITATION].', '1309.2616-1-10-3': 'A review of lattice effective field theory calculations can be found in Ref. [CITATION].', '1309.2616-1-11-0': 'We denote the lattice spacing as [MATH].', '1309.2616-1-11-1': 'We write all quantities in lattice units, meaning that we form dimensionless combinations involving the appropriate power of [MATH].', '1309.2616-1-11-2': 'Using the simplest possible lattice action with nearest-neighbor hopping terms, we find that the lattice Hamiltonian has the form [EQUATION] where [MATH] labels the lattice sites, [MATH] is the lattice-regularized coupling, and [MATH] is a lattice unit vector in the [MATH] direction.', '1309.2616-1-11-3': 'We apply cubic periodic boundary conditions, where the physical size of the cube is [MATH] times the lattice spacing [MATH].', '1309.2616-1-12-0': '# Adiabatic projection method', '1309.2616-1-13-0': 'The first step of the adiabatic projection method is to set up the initial cluster states.', '1309.2616-1-13-1': 'Without loss of generality we take the fermion-dimer system to consist of two spin-up fermions and one spin-down fermion.', '1309.2616-1-13-2': 'We will work in the center-of-mass frame and measure particle locations relative to the spin-down fermion.', '1309.2616-1-13-3': 'In our coordinate convention the spin-down fermion is anchored at the origin, [MATH], while the two spin-up fermions locations are unconstrained except for Fermi statistic.', '1309.2616-1-13-4': 'We choose our cluster initial states to have the form [EQUATION] for any lattice separation vector [MATH].', '1309.2616-1-13-5': 'This is illustrated in Fig. [REF].', '1309.2616-1-13-6': 'On our cubic periodic lattice, there are [MATH] possible values for [MATH].', '1309.2616-1-14-0': 'We now evolve the initial states [MATH] in Euclidean time [MATH] with the microscopic Hamiltonian to produce the dressed cluster states, [EQUATION]', '1309.2616-1-14-1': 'We have chosen a simple form for the initial cluster states in Eq. ([REF]) to demonstrate the general properties of the adiabatic projection method as simply as possible.', '1309.2616-1-14-2': 'We can accelerate the convergence of the method by choosing an initial cluster state that better reproduces the dimer wavefunction.', '1309.2616-1-15-0': 'The initial state [MATH] plays a role analogous to an interpolating field.', '1309.2616-1-15-1': 'We start with a configuration which roughly approximates the desired continuum state.', '1309.2616-1-15-2': 'The Euclidean time projection then systematically improves the approximation while accounting for all possible deformations and polarizations due to the interacting bodies.', '1309.2616-1-15-3': 'In the limit of large projection time [MATH], the set of dressed cluster states [MATH] will span the low-energy spectrum of the original Hamiltonian [MATH].', '1309.2616-1-16-0': 'The technique of generating cluster scattering states using Euclidean time projection is motivated by recent studies of alpha-particle clusters in the carbon-12 nucleus [CITATION].', '1309.2616-1-16-1': 'In those investigations, two different characteristic time scales are apparent from the projection Monte Carlo simulations.', '1309.2616-1-16-2': 'The first is a fast time scale associated with the formation of alpha clusters.', '1309.2616-1-16-3': 'Starting from any initial state of carbon-12, individual clusters emerge quickly as a function of projection time [MATH].', '1309.2616-1-16-4': 'However the overall structure of the alpha clusters relative to each another develops only much later in projection time [MATH].', '1309.2616-1-16-5': 'The underlying physics is related to the original motivation of Wheeler when he first introduced the resonating group method to describe the structure of compound nuclei [CITATION].', '1309.2616-1-17-0': 'The same separation of time scales can be seen in the Euclidean time projection of continuum states.', '1309.2616-1-17-1': 'The formation time for individual clusters is fast while the physics of inter-cluster interactions develops more slowly.', '1309.2616-1-17-2': 'The adiabatic projection formalism uses this separation of time scales to represent the low-energy continuum states efficiently as superpositions of dressed cluster states.', '1309.2616-1-18-0': 'The dressed cluster states [MATH] are in general not orthogonal.', '1309.2616-1-18-1': 'Therefore it is convenient to define the dual vector [MATH] as a linear functional, [EQUATION] where [MATH] is the inverse of the norm matrix [MATH] with components given by the inner product [EQUATION]', '1309.2616-1-18-2': 'The dual vector [MATH] will annihilate any vector which is orthogonal to all dressed cluster states: [EQUATION]', '1309.2616-1-18-3': 'It also serves as a dual basis within the linear subspace of dressed cluster states, [EQUATION]', '1309.2616-1-18-4': 'Let [MATH] be the matrix representation of the Hamiltonian operator [MATH] projected onto the set of dressed cluster states, [EQUATION]', '1309.2616-1-18-5': 'We will call [MATH] the two-body adiabatic Hamiltonian, an effective two-body matrix Hamiltonian describing the fermion-dimer system.', '1309.2616-1-18-6': 'From Eq. ([REF]) we can write the adiabatic Hamiltonian as [EQUATION]', '1309.2616-1-18-7': 'If we can apply a similarity transform involving the inverse square root of the norm matrix, [MATH], then in the new basis the adiabatic Hamiltonian is Hermitian, [EQUATION]', '1309.2616-1-18-8': 'The structure of this Hermitian adiabatic Hamiltonian is similar to the Hamiltonian matrix used in recent calculations of the no-core shell model together with the resonating group method [CITATION].', '1309.2616-1-19-0': 'We should mention that it is neither necessary nor advantageous to include all possible lattice separation vectors [MATH] in the set of initial cluster states [MATH].', '1309.2616-1-19-1': 'It is sufficient and often more efficient to keep a smaller set of vectors.', '1309.2616-1-19-2': 'The only requirement is that the dressed cluster states [MATH] are overlapping in their spatial wavefunctions and cover the full spatial range of allowed cluster separation vectors.', '1309.2616-1-19-3': 'This technique is useful as it significantly reduces the numerical task of computing of [MATH] and avoids numerical stability problems produced by large ill-conditioned norm matrices [MATH].', '1309.2616-1-20-0': '# Results for the Finite-Volume Spectrum', '1309.2616-1-21-0': 'Let the low-energy spectrum of the microscopic [MATH] be denoted [EQUATION]', '1309.2616-1-21-1': 'Suppose now we construct an adiabatic Hamiltonian [MATH] defined in the subspace that is spanned by [MATH] cluster separation states [MATH].', '1309.2616-1-21-2': 'The number of initial states that we use in the calculation can be smaller than [MATH] as discussed in Section [REF] above.', '1309.2616-1-21-3': 'In the asymptotic limit [MATH], it is straightforward to prove that the spectrum of [MATH] will match the low-energy spectrum [MATH] with an error [EQUATION]', '1309.2616-1-21-4': 'In most practical applications, however, we cannot go to extremely large values for [MATH].', '1309.2616-1-21-5': 'Therefore we actually see a more complicated dependence on [MATH] associated with higher-body continuum states.', '1309.2616-1-21-6': 'These effects arise because our initial two-body cluster states [MATH] have very little overlap with three-body continuum states.', '1309.2616-1-21-7': 'Therefore we will not reproduce the three-body continuum spectrum of [MATH] for any computationally practical value of [MATH].', '1309.2616-1-21-8': 'Furthermore, the three-body continuum states introduce a small additional error to the calculation of the two-body energy levels.', '1309.2616-1-21-9': 'The error estimate in Eq. ([REF]) is modified by extra terms due to these three-body states, [EQUATION]', '1309.2616-1-21-10': 'Here [MATH] denotes an energy-dependent spectral function which characterizes the small overlap between the three-body states and our two-body cluster states [MATH].', '1309.2616-1-22-0': 'In Fig. [REF] we plot the low-lying energy levels of [MATH] as a function of Euclidean time [MATH].', '1309.2616-1-22-1': 'We show results for lattice spacing [MATH] MeV[MATH] and cubic periodic box size [MATH].', '1309.2616-1-22-2': 'This corresponds with a physical length of [MATH].', '1309.2616-1-22-3': 'The energy levels of the microscopic Hamiltonian [MATH] are indicated by the horizontal lines.', '1309.2616-1-22-4': 'As the system evolves in Euclidean time, the lowest ten energy levels of the adiabatic Hamiltonian fall onto the lowest ten energy levels of [MATH] with exponential convergence.', '1309.2616-1-22-5': 'The degeneracies of these levels are not shown in Fig. [REF], but these first ten eigenvalues correspond with five unique energies.', '1309.2616-1-22-6': 'For this chosen periodic box volume, the spectrum of predominantly three-body continuum states starts at [MATH], the topmost horizontal line shown in Fig. [REF].', '1309.2616-1-22-7': 'We identify the different continuum states by measuring spatial correlations among the three fermions.', '1309.2616-1-22-8': 'In particular, the fermion-dimer states are easily distinguished due to their significant probability for a spin-up and spin-down fermion occupying the same lattice site.', '1309.2616-1-23-0': 'As expected, the three-body continuum states are not accurately reproduced for the values of [MATH] shown in Fig. [REF].', '1309.2616-1-23-1': 'However, the two-body continuum states below the three-body threshold are produced with rapid exponential convergence.', '1309.2616-1-23-2': 'The exponential dependence of the errors are consistent with spectral functions as defined in Eq. ([REF]) that are peaked at energies slightly above the three-body threshold at [MATH].', '1309.2616-1-23-3': 'As the box size increases, the number of two-body continuum states below the three-body threshold increases.', '1309.2616-1-23-4': 'Therefore we will reproduce more two-body continuum states using the two-body adiabatic Hamiltonian at larger volumes.', '1309.2616-1-24-0': '# Adiabatic Projection Calculation of s-wave Phase Shift', '1309.2616-1-25-0': 'We now use adiabatic projection to calculate the fermion-dimer elastic [MATH]-wave phase shift [MATH].', '1309.2616-1-25-1': "We will extract the phase shift at finite volume using Luscher's finite-volume method [CITATION].", '1309.2616-1-25-2': 'We consider three lattice spacings [MATH] MeV[MATH] , 1/150 MeV[MATH], 1/200 MeV[MATH] and a range of lattice box sizes [MATH].', '1309.2616-1-25-3': 'This corresponds to maximum physical box sizes of [MATH] fm, 21 fm and [MATH] fm, respectively.', '1309.2616-1-26-0': 'We will work in the center-of-mass frame of the fermion-dimer system.', '1309.2616-1-26-1': 'The phase shift [MATH] is calculated from the low-energy spectrum in the cubic periodic box using the relation [EQUATION] where [MATH] is the relative momentum between the two scattering bodies as deduced from the finite-volume energy and [MATH] is the three-dimensional zeta function, [EQUATION]', '1309.2616-1-26-2': 'All quantities are written in terms of lattice units.', '1309.2616-1-26-3': 'The physics of the scattering process is encoded in the discrete values of the momentum [MATH] set by the energy levels in the periodic cube.', '1309.2616-1-27-0': 'Let [MATH] be the finite-volume energy of our fermion-dimer system with relative momentum [MATH].', '1309.2616-1-27-1': 'In the large volume limit, the fermion-dimer energy [MATH] and relative momentum [MATH] are related by [EQUATION] where [MATH] is the dimer effective mass, the binding momentum [MATH] is [MATH] in the non-relativistic continuum limit, and the binding energy is [MATH] MeV.', '1309.2616-1-27-2': 'The exponentially small finite-volume corrections in Eq. ([REF]) are due to the dimer wavefunction wrapping once around the periodic boundary [CITATION].', '1309.2616-1-28-0': 'In order to reduce systematic errors in our lattice calculation, we take into account the renormalization of the dimer effective mass [MATH] at nonzero lattice spacing [MATH].', '1309.2616-1-28-1': 'We determine the dimer effective mass by numerically calculating the dispersion relation of the dimer on the lattice using a very large volume, [MATH], in order to eliminate any finite-volume effects.', '1309.2616-1-28-2': 'In the continuum limit, of course, we find that [MATH] equals [MATH].', '1309.2616-1-29-0': 'In order to reduce systematic errors further, we also consider the exponentially small finite-volume corrections in Eq. ([REF]) associated with modifications to the dimer binding energy.', '1309.2616-1-29-1': 'Let [MATH] be the finite-volume energy of the dimer with momentum [MATH].', '1309.2616-1-29-2': 'We write the difference between the finite-volume dimer energy and the infinite-volume dimer energy as [MATH], [EQUATION]', '1309.2616-1-29-3': 'As shown in Ref. [CITATION], the boundary conditions for the dimer wavefunction must satisfy different twisted boundary conditions as a function of the dimer momentum.', '1309.2616-1-29-4': 'The momentum dependence of this finite-volume correction is [EQUATION] where [EQUATION]', '1309.2616-1-29-5': 'The shift in the energy of the fermion-dimer [MATH]-wave scattering state can then be written as [CITATION] [EQUATION] where the topological volume factor [MATH] is given by [EQUATION]', '1309.2616-1-29-6': 'Taking into account this topological finite-volume correction to the dimer binding energy, we can write the fermion-dimer finite-volume energy as [EQUATION]', '1309.2616-1-29-7': "The ellipses denote other exponentially small corrections which we don't consider here arising from higher-order winding effects and finite-volume corrections to the fermion-dimer interactions.", '1309.2616-1-29-8': 'We solve Eq. ([REF]) iteratively for the relative momentum [MATH], and this in turn determines the phase shift at momentum [MATH] from Eq. ([REF]).', '1309.2616-1-30-0': 'We now compare the lattice calculation of the fermion-dimer [MATH]-wave phase shift to the exact result in the continuum and infinite volume limits solved using the STM integral equation.', '1309.2616-1-30-1': 'For the [MATH]-wave half-off-shell [MATH]-matrix we find [CITATION] [EQUATION] with dimer binding momentum [MATH] and total energy [MATH].', '1309.2616-1-30-2': 'We determine the phase shift from the on-shell [MATH]-matrix, [EQUATION]', '1309.2616-1-30-3': 'In Fig. [REF] we compare the lattice results and STM equation results for the [MATH]-wave phase shift.', '1309.2616-1-30-4': 'We show data for three different lattice spacings [MATH] MeV[MATH], [MATH] MeV[MATH] and [MATH] MeV[MATH].', '1309.2616-1-30-5': 'At low momentum the lattice results are expected to be accurate, and this is evident in the plotted results.', '1309.2616-1-30-6': 'There is only a small deviation starting near the dimer breakup momentum 52.7 MeV.', '1309.2616-1-30-7': 'The three-body breakup amplitude can also be calculated using the adiabatic projection formalism.', '1309.2616-1-30-8': 'However this requires the inclusion of low-lying three-body states and is beyond the scope of our analysis here.', '1309.2616-1-30-9': 'Investigations on this topic are planned in future work.', '1309.2616-1-30-10': 'The fermion-dimer breakup amplitude happens to be numerically small at low momenta.', '1309.2616-1-30-11': 'Consequently we see reasonable agreement between the lattice and STM results for the elastic phase shift even above the breakup momentum.', '1309.2616-1-31-0': '# Summary and Discussion', '1309.2616-1-32-0': 'In this paper we have demonstrated and tested the adiabatic projection method, a general framework for calculating scattering and reactions on the lattice.', '1309.2616-1-32-1': 'The adiabatic projection method is based upon computing a low-energy effective theory for clusters.', '1309.2616-1-32-2': 'In our analysis we calculated the adiabatic two-body Hamiltonian for elastic fermion-dimer scattering with zero-range attractive two-component fermions.', '1309.2616-1-32-3': 'This system corresponds to neutron-deuteron scattering in the quartet channel at leading order in pionless effective field theory.', '1309.2616-1-33-0': 'We found that the spectrum of the two-body adiabatic Hamiltonian matches the low-energy spectrum of the fermion-dimer system below the three-body continuum threshold.', '1309.2616-1-33-1': 'In the limit of large projection time, the adiabatic Hamiltonian estimates for the energy levels become exact, and the errors are exponentially small in the projection time [MATH].', '1309.2616-1-33-2': "Using Luscher's finite-volume method, we found good agreement between lattice results and continuum STM equation results for the [MATH]-wave phase shift up to the dimer breakup threshold.", '1309.2616-1-33-3': 'While we did not employ Monte Carlo methods in our calculation here, the adiabatic projection method uses Euclidean time projection and is therefore compatible with large-scale projection Monte Carlo codes being used in lattice effective field theory calculations.', '1309.2616-1-34-0': 'In order to calculate two-body scattering processes with charge or mass transfer, we generalize the formalism to include an additional scattering channel.', '1309.2616-1-34-1': 'This is done by using two sets of initial cluster states [MATH] and [MATH].', '1309.2616-1-34-2': 'We then consider the dressed cluster states for each channel and their mixing to compute the multi-channel norm matrix, [EQUATION] and multi-channel adiabatic Hamiltonian, [EQUATION]', '1309.2616-1-34-3': 'These inelastic scattering processes will be investigated in future studies.', '1309.2616-1-35-0': 'Another interesting and important application of the adiabatic projection method is radiative capture.', '1309.2616-1-35-1': 'Radiative capture reactions have great relevance to understanding hydrogen and helium burning in stars.', '1309.2616-1-35-2': 'The determination of astrophysical [MATH]-factors and asymptotic normalization coefficients are important areas where more theoretical input is needed.', '1309.2616-1-35-3': 'For example, model-independent analyses of [MATH] and [MATH] at low energies indicate that the strong nuclear interaction component of this process is sensitive to the elastic [MATH]-[MATH]Li and [MATH]-[MATH]Be scattering parameters at leading order and these are not well constrained experimentally [CITATION].', '1309.2616-1-36-0': 'In order to carry out radiative capture reactions in the adiabatic projection formalism, we need to add a one-body cluster state [MATH] to the set of initial cluster states.', '1309.2616-1-36-1': 'This state will correspond to the outgoing nucleus after capture.', '1309.2616-1-36-2': 'We again compute the corresponding multi-channel norm matrix and multi-channel adiabatic Hamiltonian.', '1309.2616-1-36-3': 'In this case, though, we also need to compute one-photon transition matrix elements between the dressed cluster states [EQUATION]', '1309.2616-1-36-4': 'After computing all of the quantities involving the dressed cluster states, we have now reduced the problem to radiative capture involving only two incoming bodies.', '1309.2616-1-36-5': 'Hence the capture amplitude can be calculated in the same manner as calculating radiative neutron-proton capture, [MATH].', '1309.2616-1-36-6': "This is demonstrated in detail on the lattice in Ref. [CITATION] using infrared-regulated Green's function methods.", '1309.2616-1-37-0': 'Computing support was provided by the HPCC at MSU.', '1309.2616-1-37-1': 'Part of this work was completed at the Institute for Nuclear Theory.', '1309.2616-1-37-2': 'Partial support provided by the U.S. Department of Energy grant DE-FG02-03ER41260 (D.L. and M.P.), U.S. Department of Education GAANN Fellowship (M.P.), and the U.S. National Science Foundation grant No. PHY-0969378 (G.R.)'}

{'1309.2616-2-0-0': 'We demonstrate and test the adiabatic projection method, a general new framework for calculating scattering and reactions on the lattice.', '1309.2616-2-0-1': 'The method is based upon calculating a low-energy effective theory for clusters which becomes exact in the limit of large Euclidean projection time.', '1309.2616-2-0-2': 'As a detailed example we calculate the adiabatic two-body Hamiltonian for elastic fermion-dimer scattering in lattice effective field theory.', '1309.2616-2-0-3': 'Our calculation corresponds to neutron-deuteron scattering in the spin-quartet channel at leading order in pionless effective field theory.', '1309.2616-2-0-4': 'We show that the spectrum of the adiabatic Hamiltonian reproduces the spectrum of the original Hamiltonian below the inelastic threshold to arbitrary accuracy.', '1309.2616-2-0-5': 'We also show that the calculated [MATH]-wave phase shift reproduces the known exact result in the continuum and infinite-volume limits.', '1309.2616-2-0-6': 'When extended to more than one scattering channel, the adiabatic projection method can be used to calculate inelastic reactions on the lattice in future work.', '1309.2616-2-1-0': '# Introduction', '1309.2616-2-2-0': 'There has been recent progress in ab initio calculations of nuclear scattering and reactions.', '1309.2616-2-2-1': "This includes calculations using the no-core shell model and resonating group method [CITATION], fermionic molecular dynamics [CITATION], the coupled-cluster expansion [CITATION], and variational and Green's function Monte Carlo [CITATION].", '1309.2616-2-2-2': 'For calculations using lattice methods there has been progress in using finite periodic volumes to analyze coupled-channel scattering [CITATION] and three-body systems [CITATION].', '1309.2616-2-2-3': 'Also the first steps towards calculating nuclear reactions on the lattice were introduced in Ref. [CITATION] using an adiabatic projection formalism.', '1309.2616-2-2-4': 'The general strategy in the adiabatic projection formalism involves separating the calculation into two parts.', '1309.2616-2-2-5': 'The first part of the method uses Euclidean time projection to determine an adiabatic Hamiltonian for the participating nuclei.', '1309.2616-2-2-6': 'This is done by starting with a set of cluster states [MATH] labeled by their separation vector [MATH], as illustrated in Fig. [REF].', '1309.2616-2-2-7': 'These states are propagated in Euclidean time to form dressed cluster states, [EQUATION]', '1309.2616-2-3-0': 'By evolving in Euclidean time with the full microscopic Hamiltonian, we are in essence cooling the initial cluster states to the correct physical state dynamically with the interaction.', '1309.2616-2-3-1': 'Deformations and polarizations of the interacting clusters are incorporated automatically by means of Euclidean time projection.', '1309.2616-2-3-2': 'In the limit of large Euclidean time, these dressed cluster states span the low-energy subspace of two-body continuum states for our clusters.', '1309.2616-2-3-3': 'We then calculate an adiabatic Hamiltonian matrix defined by the original Hamiltonian restricted to the subspace of dressed cluster states.', '1309.2616-2-3-4': 'The construction is still ab initio though restricted to the description of cluster configurations.', '1309.2616-2-3-5': 'For inelastic processes we construct dressed cluster states for each of the possible scattering channels and calculate matrix elements for all operators relevant to the reaction process.', '1309.2616-2-3-6': 'For example, in the case of radiative capture reactions, we calculate the adiabatic Hamiltonian and matrix elements of one-photon vertex operators.', '1309.2616-2-4-0': 'The second part of the adiabatic projection method entails using the adiabatic Hamiltonian and operator matrix elements for the dressed cluster states to calculate scattering amplitudes.', '1309.2616-2-4-1': 'For elastic phase shifts, we can apply the finite-volume scaling analysis developed by Luscher [CITATION].', '1309.2616-2-4-2': 'For inelastic reactions, additional steps in the calculation are required as follows.', '1309.2616-2-4-3': "Since we have reduced the problem to a few-body system of nuclear clusters, this part of the calculation can be performed using Green's function methods defined in Minkowskian time.", '1309.2616-2-4-4': "In Ref. [CITATION] this is demonstrated using finite-volume Green's functions for radiative neutron-proton capture, [MATH], in pionless effective field theory.", '1309.2616-2-4-5': 'See Section [REF] for more details.', '1309.2616-2-5-0': 'In this paper we study in depth the first part of the adiabatic projection method, the construction of dressed cluster states and the adiabatic Hamiltonian.', '1309.2616-2-5-1': 'We use the example of elastic fermion-dimer scattering for attractive two-component fermions in the limit of zero-range interactions.', '1309.2616-2-5-2': 'This corresponds to deuteron-neutron scattering in the spin-quartet channel at leading order in pionless effective field theory.', '1309.2616-2-5-3': 'One of the key requirements of an ab initio approach is that all errors are under control and can be systematically reduced.', '1309.2616-2-5-4': 'In our analysis we will find that the spectrum of the adiabatic Hamiltonian matches the spectrum of the original microscopic Hamiltonian below the inelastic threshold to arbitrary accuracy.', '1309.2616-2-5-5': 'Furthermore, we reproduce the [MATH]-wave phase shift in good agreement with exact results in the continuum and infinite-volume limits from the Skorniakov-Ter-Martirosian (STM) integral equation [CITATION].', '1309.2616-2-6-0': 'The outline of the paper is as follows.', '1309.2616-2-6-1': 'In Section [REF] we introduce the underlying interactions for our effective field theory description of two-component fermions and fermion-dimer scattering on the lattice.', '1309.2616-2-6-2': 'In Section [REF] we apply the adiabatic projection method to this system and calculate the corresponding two-body adiabatic Hamiltonian.', '1309.2616-2-6-3': 'In Section [REF] we compare the spectrum of the two-body adiabatic Hamiltonian with the spectrum of the original microscopic Hamiltonian.', '1309.2616-2-6-4': 'In Section [REF] we compute the elastic [MATH]-wave phase shift for fermion-dimer scattering and compare with exact continuum infinite-volume results.', '1309.2616-2-6-5': 'Applications to inelastic processes in future work is discussed in Section [REF].', '1309.2616-2-6-6': 'We then conclude in Section [REF] with a summary and outlook.', '1309.2616-2-7-0': '# Two-component fermions on the lattice', '1309.2616-2-8-0': 'The example we consider in depth is fermion-dimer scattering for two-component fermions.', '1309.2616-2-8-1': 'We call the two components spin-up and spin-down.', '1309.2616-2-8-2': 'The bound dimer state is composed of one spin-up and one spin-down fermion.', '1309.2616-2-8-3': 'The interactions are chosen to be attractive, and we take the limit where the range of the interactions is negligible.', '1309.2616-2-8-4': 'At leading order in pionless effective field theory, neutron-deuteron scattering in the spin-quartet channel is completely equivalent to our fermion-dimer scattering system.', '1309.2616-2-8-5': 'In the neutron-deuteron case the two fermion components correspond to isospin, while all the nucleon intrinsic spins are fully symmetrized into a spin-quartet.', '1309.2616-2-9-0': 'The Hamiltonian for our system can be written as [EQUATION] where we take the zero-range limit [MATH], with coupling constant [MATH].', '1309.2616-2-9-1': 'Here [MATH] and [MATH] are creation and annihilation operators.', '1309.2616-2-9-2': 'Motivated by the neutron-deuteron system, we take the mass of the fermions to be [MATH] MeV and tune the strength of [MATH] to match the binding energy of the deuteron, [MATH] MeV.', '1309.2616-2-9-3': 'For a shallow bound dimer such as this, the fermion-dimer scattering problem is known to be strongly coupled even at rather low momenta.', '1309.2616-2-9-4': 'See, for example, Ref. [CITATION] and references therein.', '1309.2616-2-10-0': 'We will calculate the properties of this system using a lattice Hamiltonian.', '1309.2616-2-10-1': "While we don't need the full computational machinery of Monte Carlo simulations in this analysis, we should note that the adiabatic projection formalism fits conveniently into the framework of lattice effective field theory.", '1309.2616-2-10-2': 'Lattice effective field theory is a combination of effective field theory and numerical lattice methods that has been used to study nuclei in pionless EFT [CITATION] and chiral EFT [CITATION].', '1309.2616-2-10-3': 'A review of lattice effective field theory calculations can be found in Ref. [CITATION].', '1309.2616-2-11-0': 'We denote the lattice spacing as [MATH].', '1309.2616-2-11-1': 'We write all quantities in lattice units, meaning that we form dimensionless combinations involving the appropriate power of [MATH].', '1309.2616-2-11-2': 'Using the simplest possible lattice action with nearest-neighbor hopping terms, we find that the lattice Hamiltonian has the form [EQUATION] where [MATH] labels the lattice sites, [MATH] is the lattice-regularized coupling, and [MATH] is a lattice unit vector in the [MATH] direction.', '1309.2616-2-11-3': 'We apply cubic periodic boundary conditions, where the physical size of the cube is [MATH] times the lattice spacing [MATH].', '1309.2616-2-12-0': '# Adiabatic projection method', '1309.2616-2-13-0': 'The first step of the adiabatic projection method is to set up the initial cluster states.', '1309.2616-2-13-1': 'Without loss of generality we take the fermion-dimer system to consist of two spin-up fermions and one spin-down fermion.', '1309.2616-2-13-2': 'We will work in the center-of-mass frame and measure particle locations relative to the spin-down fermion.', '1309.2616-2-13-3': 'In our coordinate convention the spin-down fermion is anchored at the origin, [MATH], while the two spin-up fermion locations are unconstrained except for Fermi statistics.', '1309.2616-2-13-4': 'We choose our cluster initial states to have the form [EQUATION] for any lattice separation vector [MATH].', '1309.2616-2-13-5': 'This is illustrated in Fig. [REF].', '1309.2616-2-13-6': 'In the actual code we use a Slater determinant to construct the fermionic state.', '1309.2616-2-13-7': 'On our cubic periodic lattice, there are [MATH] possible values for [MATH].', '1309.2616-2-14-0': 'We now evolve the initial states [MATH] in Euclidean time [MATH] with the microscopic Hamiltonian to produce the dressed cluster states, [EQUATION]', '1309.2616-2-14-1': 'We have chosen a simple form for the initial cluster states in Eq. ([REF]) to demonstrate the general properties of the adiabatic projection method as simply as possible.', '1309.2616-2-14-2': 'We can accelerate the convergence of the method by choosing an initial cluster state that better reproduces the dimer wavefunction.', '1309.2616-2-14-3': 'The Euclidean time evolution is done by exact matrix multiplication using the Trotter approximation [EQUATION] for large number of time steps [MATH].', '1309.2616-2-15-0': 'The initial state [MATH] plays a role analogous to an interpolating field.', '1309.2616-2-15-1': 'We start with a configuration which roughly approximates the desired continuum state.', '1309.2616-2-15-2': 'The Euclidean time projection then systematically improves the approximation while accounting for all possible deformations and polarizations due to the interacting bodies.', '1309.2616-2-15-3': 'In the limit of large projection time [MATH], the set of dressed cluster states [MATH] will span the low-energy spectrum of the original Hamiltonian [MATH].', '1309.2616-2-16-0': 'The technique of generating cluster scattering states using Euclidean time projection is motivated by recent studies of alpha-particle clusters in the carbon-12 nucleus [CITATION].', '1309.2616-2-16-1': 'In those investigations, two different characteristic time scales are apparent from the projection Monte Carlo simulations.', '1309.2616-2-16-2': 'The first is a fast time scale associated with the formation of alpha clusters.', '1309.2616-2-16-3': 'Starting from any initial state of carbon-12, individual clusters emerge quickly as a function of projection time [MATH].', '1309.2616-2-16-4': 'However the overall structure of the alpha clusters relative to each other develops only much later in projection time [MATH].', '1309.2616-2-16-5': 'The underlying physics is related to the original motivation of Wheeler when he first introduced the resonating group method to describe the structure of compound nuclei [CITATION].', '1309.2616-2-17-0': 'The same separation of time scales can be seen in the Euclidean time projection of continuum states.', '1309.2616-2-17-1': 'The formation time for individual clusters is fast while the physics of inter-cluster interactions develops more slowly.', '1309.2616-2-17-2': 'The adiabatic projection formalism uses this separation of time scales to represent the low-energy continuum states efficiently as superpositions of dressed cluster states.', '1309.2616-2-18-0': 'The dressed cluster states [MATH] are in general not orthogonal.', '1309.2616-2-18-1': 'Therefore it is convenient to define the dual vector [MATH] as a linear functional, [EQUATION] where [MATH] is the inverse of the norm matrix [MATH] with components given by the inner product [EQUATION]', '1309.2616-2-18-2': 'The dual vector [MATH] will annihilate any vector which is orthogonal to all dressed cluster states: [EQUATION]', '1309.2616-2-18-3': 'It also serves as a dual basis within the linear subspace of dressed cluster states, [EQUATION]', '1309.2616-2-18-4': 'Let [MATH] be the matrix representation of the Hamiltonian operator [MATH] projected onto the set of dressed cluster states, [EQUATION]', '1309.2616-2-18-5': 'We will call [MATH] the two-body adiabatic Hamiltonian, an effective two-body matrix Hamiltonian describing the fermion-dimer system.', '1309.2616-2-18-6': 'From Eq. ([REF]) we can write the adiabatic Hamiltonian as [EQUATION]', '1309.2616-2-18-7': 'If we apply a similarity transform involving the inverse square root of the norm matrix, [MATH], then in the new basis the adiabatic Hamiltonian is Hermitian, [EQUATION]', '1309.2616-2-18-8': 'The structure of this Hermitian adiabatic Hamiltonian is similar to the Hamiltonian matrix used in recent calculations of the no-core shell model together with the resonating group method [CITATION].', '1309.2616-2-19-0': '# Results for the Finite-Volume Spectrum', '1309.2616-2-20-0': 'Let the low-energy spectrum of the microscopic [MATH] be denoted [EQUATION]', '1309.2616-2-20-1': 'Let [MATH] be the number of initial cluster states that we use in our calculation.', '1309.2616-2-20-2': 'It is not necessary to use every possible cluster state on the lattice, and we will discuss the choice of [MATH] a bit later in our discussion.', '1309.2616-2-20-3': 'Suppose now we construct an adiabatic Hamiltonian [MATH] defined in the subspace that is spanned by [MATH] cluster separation states [MATH].', '1309.2616-2-20-4': 'In the asymptotic limit [MATH], it is straightforward to prove that the spectrum of [MATH] will match the low-energy spectrum [MATH] with an error [EQUATION]', '1309.2616-2-20-5': 'In most practical applications, however, we cannot go to extremely large values for [MATH].', '1309.2616-2-20-6': 'Therefore we actually see a more complicated dependence on [MATH] associated with higher-body continuum states.', '1309.2616-2-21-0': 'At finite volume and above the threshold for three-body states, the eigenstates of [MATH] will in general be a mixture of two-body and three-body states.', '1309.2616-2-21-1': 'However at large volumes we can still classify which energy eigenstates are predominantly two-body or predominantly three-body.', '1309.2616-2-21-2': 'Our initial two-body cluster states [MATH] will have only a very small overlap with the three-body continuum states.', '1309.2616-2-21-3': 'In the adiabatic projection method we would need to include initial states that have better overlap with three-body states in order to reproduce the three-body continuum spectrum of [MATH].', '1309.2616-2-22-0': 'Consider for the moment the idealized case where our initial states [MATH] are completely orthogonal to all three-body and higher-body states.', '1309.2616-2-22-1': 'Let [MATH] be the energy-ordered spectrum of [MATH] including up to at most two-body states.', '1309.2616-2-22-2': 'We then reproduce the energy level [MATH] using the adiabatic projection method with error [EQUATION]', '1309.2616-2-22-3': 'Now let us return to the actual situation where there is some small but nonzero overlap between our initial states [MATH] and higher-body states.', '1309.2616-2-22-4': 'These higher-body states introduce a small additional error to the calculation of the two-body energy levels, [EQUATION]', '1309.2616-2-22-5': 'Here [MATH] denotes an energy-dependent spectral function which characterizes the small overlap between the higher-body states and our two-body cluster states [MATH].', '1309.2616-2-23-0': 'We should mention that it is neither necessary nor advantageous to include all possible lattice separation vectors [MATH] in the set of initial cluster states [MATH].', '1309.2616-2-23-1': 'It is sufficient and often more efficient to keep a smaller set of vectors.', '1309.2616-2-23-2': 'This technique is useful as it significantly reduces the numerical task of computing [MATH] and avoids numerical stability problems produced by large ill-conditioned norm matrices [MATH].', '1309.2616-2-23-3': 'Skipping states [MATH] will reduce [MATH] and this in turn will decrease the energy level [MATH].', '1309.2616-2-23-4': 'The effect on the convergence of the method as a function of projection time [MATH] can be understood from the error estimate in Eq. ([REF]).', '1309.2616-2-23-5': 'Roughly speaking it is most efficient to choose a value for [MATH] which is somewhat larger than the number of scattering states we would like to compute.', '1309.2616-2-23-6': 'The strategy then is to choose [MATH] sufficiently large so that the desired accuracy goal is achieved.', '1309.2616-2-24-0': 'In Fig. [REF] we plot the low-lying energy levels of [MATH] as a function of Euclidean time [MATH].', '1309.2616-2-24-1': 'We show results for lattice spacing [MATH] MeV[MATH] and cubic periodic box size [MATH].', '1309.2616-2-24-2': 'This corresponds with a physical length of [MATH].', '1309.2616-2-24-3': 'For these calculations we use initial cluster states [MATH] which are separated by at least two lattice sites in each direction.', '1309.2616-2-24-4': 'The energy levels of the microscopic Hamiltonian [MATH] are indicated by the horizontal lines.', '1309.2616-2-24-5': 'As the system evolves in Euclidean time, the lowest ten energy levels of the adiabatic Hamiltonian fall onto the lowest ten energy levels of [MATH] with exponential convergence.', '1309.2616-2-24-6': 'The degeneracies of these levels are not shown in Fig. [REF], but these first ten eigenvalues correspond to five energy levels.', '1309.2616-2-24-7': 'For this chosen periodic box volume, the spectrum of predominantly three-body continuum states starts at [MATH], the topmost horizontal line shown in Fig. [REF].', '1309.2616-2-24-8': 'We identify the different continuum states by measuring spatial correlations among the three fermions.', '1309.2616-2-24-9': 'In particular, the fermion-dimer states are easily distinguished due to their significant probability for a spin-up and spin-down fermion occupying the same lattice site.', '1309.2616-2-25-0': 'As expected, the three-body continuum states are not accurately reproduced for the values of [MATH] shown in Fig. [REF].', '1309.2616-2-25-1': 'However, the two-body continuum states below the three-body threshold are produced with rapid exponential convergence.', '1309.2616-2-25-2': 'The exponential dependence of the errors are consistent with spectral functions as defined in Eq. ([REF]) that are peaked at energies slightly above the three-body threshold at [MATH].', '1309.2616-2-25-3': 'As the box size increases, the number of two-body continuum states below the three-body threshold increases.', '1309.2616-2-25-4': 'Therefore we will reproduce more two-body continuum states using the two-body adiabatic Hamiltonian at larger volumes.', '1309.2616-2-26-0': '# Adiabatic Projection Calculation of s-wave Phase Shift', '1309.2616-2-27-0': 'We now use adiabatic projection to calculate the fermion-dimer elastic [MATH]-wave phase shift [MATH].', '1309.2616-2-27-1': "We will extract the phase shift at finite volume using Luscher's finite-volume method [CITATION].", '1309.2616-2-27-2': 'We consider three lattice spacings [MATH] MeV[MATH] , 1/150 MeV[MATH], 1/200 MeV[MATH] and a range of lattice box sizes [MATH].', '1309.2616-2-27-3': 'This corresponds to maximum physical box sizes of [MATH] fm, 21 fm and [MATH] fm, respectively.', '1309.2616-2-28-0': 'We will work in the center-of-mass frame of the fermion-dimer system.', '1309.2616-2-28-1': 'The phase shift [MATH] is calculated from the low-energy spectrum in the cubic periodic box using the relation [EQUATION] where [MATH] is the relative momentum between the two scattering bodies as deduced from the finite-volume fermion-dimer energy [MATH], and [MATH] is the three-dimensional zeta function, [EQUATION]', '1309.2616-2-28-2': 'All quantities are written in terms of lattice units.', '1309.2616-2-28-3': 'The physics of the scattering process is encoded in the discrete values of the momentum [MATH] set by the energy levels in the periodic cube.', '1309.2616-2-29-0': 'The fermion-dimer energy in the periodic box of size [MATH] for [MATH]-wave scattering states can be written as [CITATION] [EQUATION]', '1309.2616-2-29-1': 'The lattice momentum [MATH] in Eq. ([REF]) is self-consistently derived from the above equation using the various expressions as described below.', '1309.2616-2-30-0': 'The first term on the right hand side of Eq. ([REF]) is the infinite-volume fermion-dimer energy given by [EQUATION] with [MATH] the dimer effective mass, and [MATH] MeV the infinite-volume dimer binding energy.', '1309.2616-2-30-1': 'In the non-relativistic continuum limit, the dimer mass [MATH] equals [MATH].', '1309.2616-2-30-2': 'However, in order to reduce systematic errors in our lattice calculation, we take into account the renormalization of the dimer effective mass [MATH] at nonzero lattice spacing [MATH].', '1309.2616-2-30-3': 'We determine the dimer effective mass by numerically calculating the dispersion relation of the dimer on the lattice using a very large volume, [MATH], in order to eliminate any finite-volume effects.', '1309.2616-2-31-0': 'The second term [MATH] in Eq. ([REF]) encapsulates the finite-volume corrections to the fermion-dimer energy due to the dimer wavefunction wrapping around the periodic boundary [CITATION], where [MATH] is the finite-volume energy shift for the bound dimer state in the two-body center-of-mass frame.', '1309.2616-2-31-1': 'The topological factor [MATH] for the [MATH]-wave dimer wavefunction wrapping around the periodic boundary once is given by [CITATION] [EQUATION] where [EQUATION] accounts for the finite-volume energy shift of a dimer moving with center-of-mass momentum [MATH] in the fermion-dimer system [EQUATION]', '1309.2616-2-31-2': 'The expression for the topological factor [MATH] in Eq. ([REF]) ignores higher-order volume corrections associated with the dimer wavefunction winding around the boundary more than once and finite-volume corrections to the fermion-dimer interactions.', '1309.2616-2-32-0': 'From low energy spectrum in the lattice calculation, we directly determine [MATH] and [MATH].', '1309.2616-2-32-1': 'Then the lattice momentum [MATH] (and [MATH]) is the one that satisfied Eq. ([REF]) with these energies.', '1309.2616-2-32-2': 'We solve for [MATH] iteratively by setting [MATH] at the start, and then recursively solving for [MATH] and [MATH] until they convergence.', '1309.2616-2-32-3': 'Note [MATH] gives [MATH] that corresponds to [EQUATION] a naive but reasonable initial guess for the lattice momentum [MATH] given [MATH] and [MATH] from the lattice calculation.', '1309.2616-2-32-4': 'In our calculations, we find convergence within approximately 10-15 iterations [CITATION].', '1309.2616-2-32-5': 'Having determined [MATH], the phase shift is calculated from Eq. ([REF]).', '1309.2616-2-33-0': 'We now compare the lattice calculation of the fermion-dimer [MATH]-wave phase shift to the exact result in the continuum and infinite volume limits obtained from the STM integral equation.', '1309.2616-2-33-1': 'For the [MATH]-wave half-off-shell fermion-dimer scattering [MATH]-matrix, we find [CITATION] [EQUATION] with dimer binding momentum [MATH] and total energy [MATH].', '1309.2616-2-33-2': 'We determine the phase shift from the on-shell [MATH]-matrix, [EQUATION]', '1309.2616-2-33-3': 'In Fig. [REF] we compare the lattice results and STM equation results for the [MATH]-wave phase shift.', '1309.2616-2-33-4': 'We show data for three different lattice spacings [MATH] MeV[MATH], [MATH] MeV[MATH] and [MATH] MeV[MATH].', '1309.2616-2-33-5': 'For each of these adiabatic projection calculations we use about thirty initial cluster states, [MATH].', '1309.2616-2-33-6': 'At low momentum the lattice results are expected to be accurate, and this is evident in the plotted results.', '1309.2616-2-33-7': 'There is only a small deviation starting near the dimer breakup momentum 52.7 MeV.', '1309.2616-2-33-8': 'The three-body breakup amplitude can also be calculated using the adiabatic projection formalism.', '1309.2616-2-33-9': 'However this requires the inclusion of low-lying three-body states and is beyond the scope of our analysis here.', '1309.2616-2-33-10': 'Investigations on this topic are planned in future work.', '1309.2616-2-33-11': 'The fermion-dimer breakup amplitude happens to be numerically small at low momenta.', '1309.2616-2-33-12': 'Consequently we see reasonable agreement between the lattice and STM results for the elastic phase shift even above the breakup momentum.', '1309.2616-2-34-0': '# Future Directions and Extensions', '1309.2616-2-35-0': 'In order to calculate two-body scattering processes with charge or mass transfer, we can generalize the formalism to include an additional scattering channel.', '1309.2616-2-35-1': 'This is done by using two sets of initial cluster states [MATH] and [MATH].', '1309.2616-2-35-2': 'We then consider the dressed cluster states for each channel and their mixing to compute the multi-channel norm matrix, [EQUATION] and multi-channel adiabatic Hamiltonian, [EQUATION]', '1309.2616-2-35-3': 'These inelastic scattering processes will be investigated in future studies.', '1309.2616-2-36-0': 'Another interesting and important application of the adiabatic projection method is radiative capture.', '1309.2616-2-36-1': 'Radiative capture reactions have great relevance to understanding hydrogen and helium burning in stars.', '1309.2616-2-36-2': 'The determination of astrophysical [MATH]-factors and asymptotic normalization coefficients are important areas where more theoretical input is needed.', '1309.2616-2-36-3': 'For example, model-independent analyses of [MATH] and [MATH] at low energies indicate that the strong nuclear interaction component of this process is sensitive to the elastic [MATH]-[MATH]Li and [MATH]-[MATH]Be scattering parameters at leading order and these are not well constrained experimentally [CITATION].', '1309.2616-2-37-0': 'To calculate radiative capture reactions in the adiabatic projection formalism, we need to add a one-body cluster state [MATH] to the set of initial cluster states.', '1309.2616-2-37-1': 'This state will correspond to the outgoing nucleus after capture.', '1309.2616-2-37-2': 'We again compute the corresponding multi-channel norm matrix and multi-channel adiabatic Hamiltonian.', '1309.2616-2-37-3': 'In this case, though, we also need to compute one-photon transition matrix elements between the dressed cluster states [EQUATION]', '1309.2616-2-37-4': 'After computing all of the quantities involving the dressed cluster states, we have now reduced the problem to radiative capture involving only two incoming bodies.', '1309.2616-2-37-5': 'Hence the capture amplitude can be calculated in the same manner as calculating radiative neutron-proton capture, [MATH].', '1309.2616-2-37-6': "This is demonstrated in detail on the lattice in Ref. [CITATION] using infrared-regulated Green's function methods.", '1309.2616-2-38-0': '# Summary and Outlook', '1309.2616-2-39-0': 'In this paper we have demonstrated and tested the adiabatic projection method, a general framework for calculating scattering and reactions on the lattice.', '1309.2616-2-39-1': 'The adiabatic projection method is based upon computing a low-energy effective theory for clusters.', '1309.2616-2-39-2': 'In our analysis we calculated the adiabatic two-body Hamiltonian for elastic fermion-dimer scattering with zero-range attractive two-component fermions.', '1309.2616-2-39-3': 'This system corresponds to neutron-deuteron scattering in the quartet channel at leading order in pionless effective field theory.', '1309.2616-2-39-4': 'Future work should include higher order corrections in the effective field theory.', '1309.2616-2-40-0': 'We found that the spectrum of the two-body adiabatic Hamiltonian matches the low-energy spectrum of the fermion-dimer system below the three-body continuum threshold.', '1309.2616-2-40-1': 'In the limit of large projection time, the adiabatic Hamiltonian estimates for the energy levels become exact, and the errors are exponentially small in the projection time [MATH].', '1309.2616-2-40-2': "Using Luscher's finite-volume method, we found good agreement between lattice results and continuum STM equation results for the [MATH]-wave phase shift up to the dimer breakup threshold.", '1309.2616-2-41-0': 'While we did not employ Monte Carlo methods in our calculation here, the adiabatic projection method uses Euclidean time projection and is therefore compatible with large-scale projection Monte Carlo codes being used in lattice effective field theory calculations.', '1309.2616-2-41-1': 'In particular, the Euclidean time projection can be performed using the same auxiliary-field projection Monte Carlo method used in previous calculations.', '1309.2616-2-41-2': 'These have the advantage of very favorable scaling with particle number when the sign problem is not severe.', '1309.2616-2-41-3': 'The computational effort scales with nucleon number roughly as [MATH] for [MATH] in the range of about twenty nucleons.', '1309.2616-2-42-0': 'The scattering calculations, however, require more work than simple bound state calculations.', '1309.2616-2-42-1': 'If we use [MATH] initial cluster states, then the calculation is at least a factor of [MATH] times more in computational effort.', '1309.2616-2-42-2': 'In addition to this, we would also need to explore volumes larger than the box sizes used in bound state calculations.', '1309.2616-2-42-3': 'However the overall difficulty of each individual Monte Carlo calculation is not significantly greater than that for bound state calculations.', '1309.2616-2-42-4': "We are currently working on the application of Green's function methods to calculate scattering amplitudes for elastic and inelastic processes by computing [MATH]-matrix elements.", '1309.2616-2-42-5': "This should eliminate problems associated with Luscher's finite-volume method and the need for high-accuracy calculations of finite-volume energy levels.", '1309.2616-2-42-6': 'Benchmark calculations of the adiabatic projection method using Monte Carlo simulations will be presented in several publications in the near future.', '1309.2616-2-43-0': 'The authors thank U.-G. Meissner for valuable comments on the manuscript.', '1309.2616-2-43-1': 'Computing support was provided by the HPCC at MSU.', '1309.2616-2-43-2': 'Part of this work was completed at the Institute for Nuclear Theory, Seattle.', '1309.2616-2-43-3': 'The authors thank E. Epelbaum and H. Krebs for kind hospitality at Ruhr-Universitat, Bochum.', '1309.2616-2-43-4': 'Partial support provided by the U.S. Department of Energy grant DE-FG02-03ER41260 (D.L. and M.P.), U.S. Department of Education GAANN Fellowship (M.P.), and the U.S. National Science Foundation grant No. PHY-0969378 (G.R.)'}

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['1309.2616-1-22-1', '1309.2616-2-24-1'], ['1309.2616-1-22-2', '1309.2616-2-24-2'], ['1309.2616-1-22-3', '1309.2616-2-24-4'], ['1309.2616-1-22-4', '1309.2616-2-24-5'], ['1309.2616-1-22-6', '1309.2616-2-24-7'], ['1309.2616-1-22-7', '1309.2616-2-24-8'], ['1309.2616-1-22-8', '1309.2616-2-24-9'], ['1309.2616-1-16-0', '1309.2616-2-16-0'], ['1309.2616-1-16-1', '1309.2616-2-16-1'], ['1309.2616-1-16-2', '1309.2616-2-16-2'], ['1309.2616-1-16-3', '1309.2616-2-16-3'], ['1309.2616-1-16-5', '1309.2616-2-16-5'], ['1309.2616-1-33-0', '1309.2616-2-40-0'], ['1309.2616-1-33-1', '1309.2616-2-40-1'], ['1309.2616-1-33-2', '1309.2616-2-40-2'], ['1309.2616-1-33-3', '1309.2616-2-41-0'], ['1309.2616-1-21-0', '1309.2616-2-20-0'], ['1309.2616-1-21-1', '1309.2616-2-20-3'], ['1309.2616-1-21-3', '1309.2616-2-20-4'], ['1309.2616-1-21-4', '1309.2616-2-20-5'], ['1309.2616-1-21-5', '1309.2616-2-20-6']]

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[]

[['1309.2616-1-6-5', '1309.2616-2-6-6'], ['1309.2616-1-2-4', '1309.2616-2-2-4'], ['1309.2616-1-27-0', '1309.2616-2-31-1'], ['1309.2616-1-27-2', '1309.2616-2-31-0'], ['1309.2616-1-27-2', '1309.2616-2-31-1'], ['1309.2616-1-27-2', '1309.2616-2-31-2'], ['1309.2616-1-4-2', '1309.2616-2-4-2'], ['1309.2616-1-28-2', '1309.2616-2-30-1'], ['1309.2616-1-21-2', '1309.2616-2-20-1'], ['1309.2616-1-21-6', '1309.2616-2-21-2'], ['1309.2616-1-21-6', '1309.2616-2-22-0'], ['1309.2616-1-21-8', '1309.2616-2-21-2'], ['1309.2616-1-21-8', '1309.2616-2-22-4']]

[]

['1309.2616-1-37-2', '1309.2616-2-43-0', '1309.2616-2-43-4']

{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1309.2616

cond-mat-0509616

{'cond-mat-0509616-1-0-0': 'We investigate, for two water models displaying a liquid-liquid critical point, the relation between changes in dynamic and thermodynamic anomalies arising from the presence of the liquid-liquid critical point.', 'cond-mat-0509616-1-0-1': 'We find a correlation between the dynamic fragility transition and the locus of specific heat maxima [MATH] ("Widom line") emanating from the critical point.', 'cond-mat-0509616-1-0-2': 'Our findings are consistent with a possible relation between the previously hypothesized liquid-liquid phase transition and the transition in the dynamics recently observed in neutron scattering experiments on confined water.', 'cond-mat-0509616-1-0-3': 'More generally, we argue that this connection between [MATH] and dynamic crossover is not limited to the case of water, a hydrogen bond network forming liquid, but is a more general feature of crossing the Widom line.', 'cond-mat-0509616-1-0-4': 'Specifically, we also study the Jagla potential, a spherically-symmetric two-scale potential known to possess a liquid-liquid critical point, in which the competition between two liquid structures is generated by repulsive and attractive ramp interactions.', 'cond-mat-0509616-1-1-0': '# Introduction', 'cond-mat-0509616-1-2-0': 'By definition, in a first order phase transition, thermodynamic state functions such as density [MATH] and enthalpy [MATH] change discontinuously as we cool the system along a path crossing the equilibrium coexistence line [Fig. [REF](a), path [MATH]].', 'cond-mat-0509616-1-2-1': 'However in a real experiment, this discontinuous change may not occur at the coexistence line since a substance can remain in a supercooled metastable phase until a limit of stability (a spinodal) is reached [CITATION] [Fig. [REF](b), path [MATH]].', 'cond-mat-0509616-1-3-0': 'If the system is cooled isobarically along a path above the critical pressure [MATH] [Fig. [REF](b), path [MATH]], the state functions continuously change from the values characteristic of a high temperature phase (gas) to those characteristic of a low temperature phase (liquid).', 'cond-mat-0509616-1-3-1': 'The thermodynamic response functions which are the derivatives of the state functions with respect to temperature [e.g., isobaric heat capacity [MATH]] have maxima at temperatures denoted [MATH].', 'cond-mat-0509616-1-3-2': 'Remarkably these maxima are still prominent far above the critical pressure [CITATION], and the values of the response functions at [MATH] (e.g., [MATH]) diverge as the critical point is approached.', 'cond-mat-0509616-1-3-3': 'The lines of the maxima for different response functions asymptotically approach one another as the critical point is approached, since all response functions become expressible in terms of the correlation length.', 'cond-mat-0509616-1-3-4': 'This asymptotic line is sometimes called the Widom line, and is often regarded as an extension of the coexistence line into the "one-phase regime."', 'cond-mat-0509616-1-4-0': 'If the system is cooled at constant pressure [MATH], and [MATH] is not too far from the critical pressure [MATH], then there are two classes of behavior possible.', 'cond-mat-0509616-1-4-1': '(i) If [MATH] (path [MATH]), then experimentally-measured quantities will change dramatically but continuously in the vicinity of the Widom line (with huge fluctuations as measured by, e.g., [MATH]).', 'cond-mat-0509616-1-4-2': '(ii) If [MATH] (path [MATH]), experimentally-measured quantities will change discontinuously if the coexistence line is actually seen.', 'cond-mat-0509616-1-4-3': 'However the coexistence line can be difficult to detect in a pure system due to metastability, and changes will occur only when the spinodal is approached where the gas phase is no longer stable.', 'cond-mat-0509616-1-4-4': 'The changes in behavior may include not only static quantities like response functions [CITATION] but also dynamic quantities like diffusivity.', 'cond-mat-0509616-1-5-0': 'In the case of water-the most important solvent for biological function [CITATION]-a significant change in dynamical properties has been suggested to take place in deeply supercooled states [CITATION].', 'cond-mat-0509616-1-5-1': 'Unlike other network forming materials [CITATION], water behaves as a fragile liquid in the experimentally accessible window [CITATION].', 'cond-mat-0509616-1-5-2': 'Based on analogies with other network forming liquids and with the thermodynamic properties of the amorphous forms of water, it has been suggested that, at ambient pressure, liquid water should show a crossover between fragile behavior at high T to strong behavior at low T [CITATION] in the deep supercooled region of the phase diagram below the homogeneous nucleation line.', 'cond-mat-0509616-1-5-3': 'This region may contain the hypothesized liquid-liquid critical point [CITATION], the terminal point of a line of first order liquid-liquid phase transitions.', 'cond-mat-0509616-1-5-4': "According to one current hypothesis, the liquid-liquid critical point is the thermodynamic source of all water's anomalies [CITATION].", 'cond-mat-0509616-1-5-5': 'This region has been called the "no-man\'s land" because to date no experiments have been able to make direct measurements on the bulk liquid phase [CITATION].', 'cond-mat-0509616-1-5-6': 'Recently the fragility transition in confined water was studied experimentally [CITATION] since nucleation can be avoided in confined geometries.', 'cond-mat-0509616-1-5-7': 'Also, a dynamic crossover has been associated with the liquid-liquid phase transition in silicon and silica [CITATION].', 'cond-mat-0509616-1-5-8': 'In this work, we offer an interpretation of the fragility transition in water as arising from crossing the Widom line emanating from the hypothesized liquid-liquid critical point [CITATION] [Fig. [REF](c), path [MATH]].', 'cond-mat-0509616-1-6-0': '# Methods', 'cond-mat-0509616-1-7-0': 'Using molecular dynamic (MD) simulations [CITATION], we study three models, each of which has a liquid-liquid critical point.', 'cond-mat-0509616-1-7-1': 'Two of the models, (the TIP5P [CITATION] and the ST2 [CITATION]) treat the water molecular as a multiple site rigid body, interacting via electrostatic site-site interactions complemented by a Lennard-Jones potential.', 'cond-mat-0509616-1-7-2': 'The third model is the spherical "two-scale" Jagla potential with attractive and repulsive ramps [Fig. [REF]] which has been studied in the context of liquid-liquid phase transitions and liquid anomalies [CITATION].', 'cond-mat-0509616-1-7-3': 'For all three models, we evaluate the loci of maxima of the relevant response functions, compressibility and specific heat, which coincide close to the critical point and give rise to the Widom line.', 'cond-mat-0509616-1-7-4': 'We provide evidence that, for all three potentials, a dynamic crossover occurs when the Widom line is crossed.', 'cond-mat-0509616-1-8-0': 'Our results for the TIP5P potential are based on MD simulations of a system of [MATH] molecules, carried out both in the NPT and NVT ensembles using the techniques described in [CITATION].', 'cond-mat-0509616-1-8-1': 'For ST2 simulations [MATH] molecules are used and all the simulations are carried out in NVT ensemble.', 'cond-mat-0509616-1-8-2': 'For the Jagla potential, discrete molecular dynamics simulation [CITATION] implemented for [MATH] particles interacting with step potentials [CITATION] is used in both NVT and NVE ensembles.', 'cond-mat-0509616-1-9-0': '# Results', 'cond-mat-0509616-1-10-0': 'Fig. [REF](a) shows for TIP5P the relevant portion of the [MATH] phase diagram.', 'cond-mat-0509616-1-10-1': 'A liquid-liquid critical point is observed [CITATION], from which the Widom line develops.', 'cond-mat-0509616-1-10-2': 'The coexistence curve is negatively sloped, so the Clapeyron equation implies that the high-temperature phase is a high-density liquid (HDL) and the low-temperature phase is a low-density liquid (LDL).', 'cond-mat-0509616-1-10-3': 'Fig. [REF](b) shows the [MATH] dependence of the diffusion coefficient [MATH], evaluated from the long time limit of the mean square displacement along isobars.', 'cond-mat-0509616-1-10-4': 'The isobars crossing the Widom line [Fig. [REF](c), path [MATH]] show a clear crossover (i) from a non-Arrhenius behavior at high [MATH] [which can be well fit by a power law function [MATH]], consistent with the mode coupling theory predictions [CITATION]), (ii) to an Arrhenius behavior at low [MATH] [which can be described by [MATH]].', 'cond-mat-0509616-1-10-5': 'The crossover between these two functional forms takes place when crossing the Widom line.', 'cond-mat-0509616-1-11-0': 'For paths [MATH] [Fig. [REF](d)], crystallization occurs in TIP5P [CITATION], so the hypothesis that there is no fragility transition cannot be checked at low temperature.', 'cond-mat-0509616-1-11-1': 'Hence we consider a related potential, ST2, for which crystallization is absent within the time scale of the simulation.', 'cond-mat-0509616-1-11-2': 'Simulation details are described in [CITATION].', 'cond-mat-0509616-1-11-3': 'This potential also displays a liquid-liquid critical point [CITATION], as seen in the phase diagram of Fig. [REF](a).', 'cond-mat-0509616-1-11-4': 'The analog of Fig. [REF](b) is shown in Fig. [REF](b).', 'cond-mat-0509616-1-11-5': 'We confirm that along paths [MATH] a fragility transition takes place [Fig. [REF](c)].', 'cond-mat-0509616-1-11-6': 'Moreover, along paths [MATH] the [MATH] dependence of [MATH] does not show any sign of crossover to Arrhenius behavior and the fragile behavior is retained down to the lowest studied temperature (note that [MATH] extends to [MATH].', 'cond-mat-0509616-1-11-7': 'Indeed, for paths [MATH], the entire [MATH] dependence can be fit by a power law [MATH] [Fig. [REF](c)].', 'cond-mat-0509616-1-12-0': 'Thus we see that the simulations for both TIP5P and ST2 water models support the connection between the Widom line and the dynamic fragility transition.', 'cond-mat-0509616-1-12-1': "It is natural to ask which features of the water molecular potential are responsible for the properties of water discussed here, especially because water's unusual properties are shared by several other liquids whose inter-molecular potential has two energy (length) scales [CITATION].", 'cond-mat-0509616-1-12-2': 'We next investigate the two-scale spherically symmetric Jagla potential.', 'cond-mat-0509616-1-12-3': 'The Jagla model displays-without the need to supercool-a liquid-liquid coexistence line which, unlike water, has a positive slope, implying that the Widom line is now crossed along [MATH] paths with [MATH] [Figs. [REF](d) and [REF](a)].', 'cond-mat-0509616-1-12-4': 'There is a crossover in the behavior of [MATH] when the [MATH] line is crossed [Figs. [REF](b) and [REF](c)].', 'cond-mat-0509616-1-12-5': 'At high temperature, [MATH] exhibits an Arrhenius behavior [Figs. [REF](b) and [REF](c)], while at low temperature it follows a non-Arrhenius behavior, consistent with a power law.', 'cond-mat-0509616-1-12-6': 'Along a [MATH] path ([MATH]), [MATH] follows the Arrhenius behavior over the entire studied temperature range [Figs. [REF](b) and [REF](d)].', 'cond-mat-0509616-1-12-7': 'Thus, the fragility transition coincides with the location of the [MATH] line, extending the conclusion of the TIP5P and ST2 potentials to a general two-scale spherically symmetric potential.', 'cond-mat-0509616-1-13-0': '# Discussion and Summary', 'cond-mat-0509616-1-14-0': 'Before concluding, we note that our findings are consistent with the possibility that the observed fragility transition along path [MATH] is related to the behavior of [MATH], suggesting that enthalpy or entropy fluctuations may have a strong influence on the dynamic properties.', 'cond-mat-0509616-1-14-1': 'The role of [MATH] is consistent with expectations based on the Adam-Gibbs [CITATION] interpretation of the water dynamics [CITATION] and of the fragility transition [CITATION].', 'cond-mat-0509616-1-15-0': 'For both water and the Jagla model, crossing the Widom line is associated with a change in the T-dependence of the dynamics.', 'cond-mat-0509616-1-15-1': 'In the case of water, [MATH] changes from non-Arrhenius ("fragile") to Arrhenius ("strong") behavior, while the structural and thermodynamic properties change from those of HDL to those of LDL.', 'cond-mat-0509616-1-15-2': 'For the Jagla potential, due to the positive slope of the Widom line, [MATH] changes from Arrhenius to non-Arrhenius while the structural and thermodynamic properties change from those of LDL to those of HDL.', 'cond-mat-0509616-1-16-0': 'In summary, our results for water are consistent with the experimental observation in confined water of (i) a fragility transition for [MATH] [CITATION], and (ii) a peak in [MATH] upon cooling water at atmospheric pressure [CITATION].', 'cond-mat-0509616-1-16-1': "Thus our work offers a plausible interpretation of the results of Ref. [CITATION] consistent with the existence of a liquid-liquid critical point located in the no-man's land.", 'cond-mat-0509616-1-17-0': 'We thank C. A. Angell, G. Franzese, J. M. H. Levelt Sengers, L. Liu, M. Mazza, S. Sastry, F. W. Starr, B. Widom, and Z. Yan for helpful discussions and NSF grant CHE 0096892 for support.', 'cond-mat-0509616-1-17-1': 'We also thank the Boston University Computation Center, Yeshiva University, and StFX hpcLAB for allocation of CPU time.'}

{'cond-mat-0509616-2-0-0': 'We investigate, for two water models displaying a liquid-liquid critical point, the relation between changes in dynamic and thermodynamic anomalies arising from the presence of the liquid-liquid critical point.', 'cond-mat-0509616-2-0-1': 'We find a correlation between the dynamic fragility transition and the locus of specific heat maxima [MATH] ("Widom line") emanating from the critical point.', 'cond-mat-0509616-2-0-2': 'Our findings are consistent with a possible relation between the previously hypothesized liquid-liquid phase transition and the transition in the dynamics recently observed in neutron scattering experiments on confined water.', 'cond-mat-0509616-2-0-3': 'More generally, we argue that this connection between [MATH] and dynamic crossover is not limited to the case of water, a hydrogen bond network forming liquid, but is a more general feature of crossing the Widom line.', 'cond-mat-0509616-2-0-4': 'Specifically, we also study the Jagla potential, a spherically-symmetric two-scale potential known to possess a liquid-liquid critical point, in which the competition between two liquid structures is generated by repulsive and attractive ramp interactions.', 'cond-mat-0509616-2-1-0': '# Introduction', 'cond-mat-0509616-2-2-0': 'By definition, in a first order phase transition, thermodynamic state functions such as density [MATH] and enthalpy [MATH] change discontinuously as we cool the system along a path crossing the equilibrium coexistence line [Fig. [REF](a), path [MATH]].', 'cond-mat-0509616-2-2-1': 'However in a real experiment, this discontinuous change may not occur at the coexistence line since a substance can remain in a supercooled metastable phase until a limit of stability (a spinodal) is reached [CITATION] [Fig. [REF](b), path [MATH]].', 'cond-mat-0509616-2-3-0': 'If the system is cooled isobarically along a path above the critical pressure [MATH] [Fig. [REF](b), path [MATH]], the state functions continuously change from the values characteristic of a high temperature phase (gas) to those characteristic of a low temperature phase (liquid).', 'cond-mat-0509616-2-3-1': 'The thermodynamic response functions which are the derivatives of the state functions with respect to temperature [e.g., isobaric heat capacity [MATH]] have maxima at temperatures denoted [MATH].', 'cond-mat-0509616-2-3-2': 'Remarkably these maxima are still prominent far above the critical pressure [CITATION], and the values of the response functions at [MATH] (e.g., [MATH]) diverge as the critical point is approached.', 'cond-mat-0509616-2-3-3': 'The lines of the maxima for different response functions asymptotically approach one another as the critical point is approached, since all response functions become expressible in terms of the correlation length.', 'cond-mat-0509616-2-3-4': 'This asymptotic line is sometimes called the Widom line, and is often regarded as an extension of the coexistence line into the "one-phase region."', 'cond-mat-0509616-2-4-0': 'If the system is cooled at constant pressure [MATH], and [MATH] is not too far from the critical pressure [MATH], then there are two classes of behavior possible.', 'cond-mat-0509616-2-4-1': '(i) If [MATH] (path [MATH]), then experimentally-measured quantities will change dramatically but continuously in the vicinity of the Widom line (with huge fluctuations as measured by, e.g., [MATH]).', 'cond-mat-0509616-2-4-2': '(ii) If [MATH] (path [MATH]), experimentally-measured quantities will change discontinuously if the coexistence line is actually seen.', 'cond-mat-0509616-2-4-3': 'However the coexistence line can be difficult to detect in a pure system due to metastability, and changes will occur only when the spinodal is approached where the gas phase is no longer stable.', 'cond-mat-0509616-2-4-4': 'The changes in behavior may include not only static quantities like response functions [CITATION] but also dynamic quantities like diffusivity.', 'cond-mat-0509616-2-5-0': 'In the case of water-the most important solvent for biological function [CITATION]-a significant change in dynamical properties has been suggested to take place in deeply supercooled states [CITATION].', 'cond-mat-0509616-2-5-1': 'Unlike other network forming materials [CITATION], water behaves as a fragile liquid in the experimentally accessible window [CITATION].', 'cond-mat-0509616-2-5-2': 'Based on analogies with other network forming liquids and with the thermodynamic properties of the amorphous forms of water, it has been suggested that, at ambient pressure, liquid water should show a crossover between fragile behavior at high T to strong behavior at low T [CITATION] in the deep supercooled region of the phase diagram below the homogeneous nucleation line.', 'cond-mat-0509616-2-5-3': 'This region may contain the hypothesized liquid-liquid critical point [CITATION], the terminal point of a line of first order liquid-liquid phase transitions.', 'cond-mat-0509616-2-5-4': "According to one current hypothesis, the liquid-liquid critical point is the thermodynamic source of all water's anomalies [CITATION].", 'cond-mat-0509616-2-5-5': 'This region has been called the "no-man\'s land" because to date no experiments have been able to make direct measurements on the bulk liquid phase [CITATION].', 'cond-mat-0509616-2-5-6': 'Recently the fragility transition in confined water was studied experimentally [CITATION] since nucleation can be avoided in confined geometries.', 'cond-mat-0509616-2-5-7': 'Also, a dynamic crossover has been associated with the liquid-liquid phase transition in silicon and silica [CITATION].', 'cond-mat-0509616-2-5-8': 'In this work, we offer an interpretation of the dynamic crossover (called a fragility transition or fragile-strong transition by many authors) in water as arising from crossing the Widom line emanating from the hypothesized liquid-liquid critical point [CITATION] [Fig. [REF](c), path [MATH]].', 'cond-mat-0509616-2-5-9': 'Our thermodynamic and structural interpretation of the dynamic crossover may not hold for liquids for which the fragile-strong dynamic crossover can be caused by other mechanisms, as discussed in [CITATION].', 'cond-mat-0509616-2-6-0': '# Methods', 'cond-mat-0509616-2-7-0': 'Using molecular dynamic (MD) simulations, we study three models, each of which has a liquid-liquid critical point.', 'cond-mat-0509616-2-7-1': 'Two of the models, (the TIP5P [CITATION] and the ST2 [CITATION]) treat the water molecule as a multiple-site rigid body, interacting via electrostatic site-site interactions complemented by a Lennard-Jones potential.', 'cond-mat-0509616-2-7-2': 'The third model is the spherical "two-scale" Jagla potential with attractive and repulsive ramps [Fig. [REF]] which has been studied in the context of liquid-liquid phase transitions and liquid anomalies [CITATION].', 'cond-mat-0509616-2-7-3': 'For all three models, we evaluate the loci of maxima of the relevant response functions, compressibility and specific heat, which coincide close to the critical point and give rise to the Widom line.', 'cond-mat-0509616-2-7-4': 'We provide evidence that, for all three potentials, a dynamic crossover occurs when the Widom line is crossed.', 'cond-mat-0509616-2-8-0': 'Our results for the TIP5P potential are based on MD simulations of a system of [MATH] molecules, carried out both in the NPT and NVT ensembles using the techniques described in [CITATION].', 'cond-mat-0509616-2-8-1': 'For ST2 simulations [MATH] molecules are used and all the simulations are carried out in NVT ensemble.', 'cond-mat-0509616-2-8-2': 'For the Jagla potential, discrete molecular dynamics simulation [CITATION] implemented for [MATH] particles interacting with step potentials [CITATION] is used in both NVT and NVE ensembles.', 'cond-mat-0509616-2-9-0': '# Results', 'cond-mat-0509616-2-10-0': 'Fig. [REF](a) shows for TIP5P the relevant portion of the [MATH] phase diagram.', 'cond-mat-0509616-2-10-1': 'A liquid-liquid critical point is observed [CITATION], from which the Widom line develops.', 'cond-mat-0509616-2-10-2': 'The coexistence curve is negatively sloped, so the Clapeyron equation implies that the high-temperature phase is a high-density liquid (HDL) and the low-temperature phase is a low-density liquid (LDL).', 'cond-mat-0509616-2-10-3': 'Fig. [REF](b) shows the [MATH] dependence of the diffusion coefficient [MATH], evaluated from the long time limit of the mean square displacement along isobars.', 'cond-mat-0509616-2-10-4': 'The isobars crossing the Widom line [Fig. [REF](c), path [MATH]] show a clear crossover from a non-Arrhenius behavior at high [MATH] [which can be well fit by a power law function [MATH]], consistent with the mode coupling theory predictions [CITATION]), to an Arrhenius behavior at low [MATH] [which can be described by [MATH]].', 'cond-mat-0509616-2-10-5': 'The crossover between these two functional forms takes place when crossing the Widom line.', 'cond-mat-0509616-2-11-0': 'For paths [MATH] [Fig. [REF](d)], crystallization occurs in TIP5P [CITATION], so the hypothesis that there is no fragility transition cannot be checked at low temperature.', 'cond-mat-0509616-2-11-1': 'Hence we consider a related potential, ST2, for which crystallization is absent within the time scale of the simulation.', 'cond-mat-0509616-2-11-2': 'Simulation details are described in [CITATION].', 'cond-mat-0509616-2-11-3': 'This potential also displays a liquid-liquid critical point [CITATION], as seen in the phase diagram of Fig. [REF](a).', 'cond-mat-0509616-2-11-4': 'The analog of Fig. [REF](b) is shown in Fig. [REF](b).', 'cond-mat-0509616-2-11-5': 'We confirm that along paths [MATH] a fragility transition takes place [Fig. [REF](c)].', 'cond-mat-0509616-2-11-6': 'Moreover, along paths [MATH] the [MATH] dependence of [MATH] does not show any sign of crossover to Arrhenius behavior and the fragile behavior is retained down to the lowest studied temperature (note that [MATH] extends to [MATH].', 'cond-mat-0509616-2-11-7': 'Indeed, for paths [MATH], the entire [MATH] dependence can be fit by a power law [MATH] [Fig. [REF](c)].', 'cond-mat-0509616-2-12-0': 'Thus we see that the simulations for both TIP5P and ST2 water models support the connection between the Widom line and the dynamic fragility transition.', 'cond-mat-0509616-2-12-1': "It is natural to ask which features of the water molecular potential are responsible for the properties of water discussed here, especially because water's unusual properties are shared by several other liquids whose inter-molecular potential has two energy (length) scales [CITATION].", 'cond-mat-0509616-2-12-2': 'We next investigate the two-scale spherically symmetric Jagla potential.', 'cond-mat-0509616-2-12-3': 'The Jagla model displays-without the need to supercool-a liquid-liquid coexistence line which, unlike water, has a positive slope, implying that the Widom line is now crossed along [MATH] paths with [MATH] [Figs. [REF](d) and [REF](a)].', 'cond-mat-0509616-2-12-4': 'There is a crossover in the behavior of [MATH] when the [MATH] line is crossed [Figs. [REF](b) and [REF](c)].', 'cond-mat-0509616-2-12-5': 'At high temperature, [MATH] exhibits an Arrhenius behavior [Figs. [REF](b) and [REF](c)], while at low temperature it follows a non-Arrhenius behavior, consistent with a power law.', 'cond-mat-0509616-2-12-6': 'Along a [MATH] path ([MATH]), [MATH] follows the Arrhenius behavior over the entire studied temperature range [Figs. [REF](b) and [REF](d)].', 'cond-mat-0509616-2-12-7': 'Thus, the dynamic crossover coincides with the location of the [MATH] line, extending the conclusion of the TIP5P and ST2 potentials to a general two-scale spherically symmetric potential.', 'cond-mat-0509616-2-13-0': '# Discussion and Summary', 'cond-mat-0509616-2-14-0': 'Before concluding, we note that our findings are consistent with the possibility that the observed dynamic crossover along path [MATH] is related to the behavior of [MATH], suggesting that enthalpy or entropy fluctuations may have a strong influence on the dynamic properties.', 'cond-mat-0509616-2-14-1': 'The role of [MATH] is consistent with expectations based on the Adam-Gibbs [CITATION] interpretation of the water dynamics [CITATION] and of the fragility transition [CITATION].', 'cond-mat-0509616-2-15-0': 'For both water and the Jagla model, crossing the Widom line is associated with a change in the T-dependence of the dynamics.', 'cond-mat-0509616-2-15-1': 'In the case of water, [MATH] changes from non-Arrhenius ("fragile") to Arrhenius ("strong") behavior, while the structural and thermodynamic properties change from those of HDL to those of LDL.', 'cond-mat-0509616-2-15-2': 'For the Jagla potential, due to the positive slope of the Widom line, [MATH] changes from Arrhenius to non-Arrhenius while the structural and thermodynamic properties change from those of LDL to those of HDL.', 'cond-mat-0509616-2-16-0': 'In summary, our results for water are consistent with the experimental observation in confined water of (i) a fragility transition for [MATH] [CITATION], and (ii) a peak in [MATH] upon cooling water at atmospheric pressure [CITATION].', 'cond-mat-0509616-2-16-1': "Thus our work offers a plausible interpretation of the results of Ref. [CITATION] consistent with the existence of a liquid-liquid critical point located in the no-man's land.", 'cond-mat-0509616-2-17-0': 'We thank C. A. Angell, G. Franzese, J. M. H. Levelt Sengers, L. Liu, M. Mazza, S. Sastry, F. W. Starr, B. Widom, and Z. Yan for helpful discussions and NSF grant CHE 0096892 and MIUR-FIRB for support.', 'cond-mat-0509616-2-17-1': 'We thank D. Chandler and J. P. Garrahan for helpful criticisms on the manuscript.', 'cond-mat-0509616-2-17-2': 'We also thank the Boston University Computation Center, Yeshiva University, and StFX hpcLAB (high performance computing laboratory) for allocation of CPU time.'}

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{'1': 'http://arxiv.org/licenses/assumed-1991-2003/', '2': 'http://arxiv.org/licenses/assumed-1991-2003/'}

https://arxiv.org/abs/cond-mat/0509616

1906.01277

{'1906.01277-1-0-0': 'Graph kernels are an instance of the class of [MATH]-Convolution kernels, which measure the similarity of objects by comparing their substructures.', '1906.01277-1-0-1': 'Despite their empirical success, most graph kernels use a naive aggregation of the final set of substructures, usually a sum or average, thereby potentially discarding valuable information about the distribution of individual components.', '1906.01277-1-0-2': 'Furthermore, only a limited instance of these approaches can be extended to continuously attributed graphs.', '1906.01277-1-0-3': 'We propose a novel method that relies on the Wasserstein distance between the node feature vector distributions of two graphs, which allows to find subtler differences in data sets by considering graphs as high-dimensional objects, rather than simple means.', '1906.01277-1-0-4': 'We further propose a Weisfeiler-Lehman inspired embedding scheme for graphs with continuous node attributes and weighted edges, enhance it with the computed Wasserstein distance, and thus improve the state-of-the-art prediction performance on several graph classification tasks.', '1906.01277-1-1-0': '# Introduction', '1906.01277-1-2-0': 'Graph-structured data have become ubiquitous across domains over the last decades, with examples ranging from social and sensor networks, to chemo- and bioinformatics.', '1906.01277-1-2-1': 'Graph kernels have been very successful in dealing with the complexity of graphs and have shown good predictive performance on a variety of classification problems .', '1906.01277-1-2-2': 'Most graph kernels rely on the [MATH]-Convolution framework , which decomposes structured objects into substructures to compute local similarities that are then aggregated.', '1906.01277-1-2-3': 'While being generally successful for several applications, [MATH]-Convolution kernels on graphs are not without limitations: [(1)] the simplicity of the way in which the similarities between substructures are aggregated might limit their ability to capture complex nonlinear structural characteristics of the graph; most proposed variants do not generalise to graphs with high-dimensional continuous node attributes, and extensions are far from being straightforward.', '1906.01277-1-3-0': 'Various solutions have been proposed in order to address point (1).', '1906.01277-1-3-1': 'For example, [CITATION] introduced kernels based on optimal assignment of node labels for molecular graphs, even though these kernels are not positive definite .', '1906.01277-1-3-2': 'More recently, another approach was proposed by [CITATION] which set a new state-of-the-art performance in graph classification on categorical node labels.', '1906.01277-1-3-3': 'However, this formulation cannot handle continuous nodes attributes, leaving point (2) as an open problem.', '1906.01277-1-4-0': 'To overcome both limitations, we propose a method that combines the most successful vectorial graph-representations derived from the graph kernel literature with ideas from optimal transport theory, which have recently gained considerable attention.', '1906.01277-1-4-1': 'In particular, improvements of the computational strategies to efficiently obtain Wasserstein distances have led to many applications in machine learning that use it for various purposes, ranging from generative models to new loss functions .', '1906.01277-1-4-2': 'For applications to graphs, notions from optimal transport were used to tackle the graph alignment problem .', '1906.01277-1-4-3': 'In this work, we provide the theoretical foundations of our method, then define a new graph kernel formulation and finally present successful experimental results.', '1906.01277-1-4-4': 'More precisely, our main contributions can be summarised as follows:', '1906.01277-1-5-0': 'We present the graph Wasserstein distance, a new distance between graphs based on their node feature representations and we discuss how kernels can be derived from it; We introduce a Weisfeiler-Lehman inspired embedding scheme that works for both categorically labelled and continuously attributed graphs and couple it with our graph Wasserstein distance; We establish a new state of the art for graph kernels on traditional graph classification benchmarks with continuous attributes.', '1906.01277-1-6-0': '# Background: Graph Kernels and Wasserstein Distance', '1906.01277-1-7-0': 'In this section, we introduce the notation that will be used throughout the manuscript and provide the necessary background on graph kernel methods and on the Wasserstein distance.', '1906.01277-1-8-0': '## Graph kernels', '1906.01277-1-9-0': 'Kernels are a class of similarity functions that present attractive properties to be used in learning algorithms .', '1906.01277-1-9-1': 'Let [MATH] be a set and [MATH] be a function associated with a Hilbert space [MATH], such that there exists a map [MATH] with [MATH].', '1906.01277-1-9-2': 'Then, [MATH] is a reproducing kernel Hilbert space (RKHS) and [MATH] is said to be a positive definite kernel.', '1906.01277-1-9-3': 'A positive definite kernel can be interpreted as a dot product in a high-dimensional space, thus permitting its use in any learning algorithm that relies on dot products, such as Support Vector Machines (SVM), by virtue of the kernel trick .', '1906.01277-1-9-4': 'Because ensuring positive definiteness is not always feasible, many learning algorithms were recently proposed to extend SVMs to indefinite kernels .', '1906.01277-1-10-0': 'We define a graph as a tuple [MATH], where [MATH] and [MATH] denote the set of nodes and edges, respectively; we further assume that the edges are undirected.', '1906.01277-1-10-1': 'Moreover, we denote the cardinality of nodes and edges for [MATH] as [MATH] and [MATH].', '1906.01277-1-10-2': 'For a node [MATH], we write [MATH] and [MATH] to denote its first-order neighbourhood.', '1906.01277-1-10-3': 'We say that a graph is labelled if its nodes have categorical labels.', '1906.01277-1-10-4': 'A label on the nodes is a function [MATH] that assigns to each node [MATH] in [MATH] a value [MATH] from a finite label alphabet [MATH].', '1906.01277-1-10-5': 'Additionally, we say that a graph is attributed if for each node [MATH] there exists an associated vector [MATH].', '1906.01277-1-10-6': 'In the following, we refer to [MATH] as the node attributes and to [MATH] as the categorical node labels of node [MATH].', '1906.01277-1-10-7': 'In particular, the node attributes are high-dimensional continuous vectors, while the categorical node labels are assumed to be integer numbers, encoding either an ordered discrete value or a category.', '1906.01277-1-10-8': 'With the term node labels, we will implicitly refer to categorical node labels.', '1906.01277-1-10-9': 'Finally, a graph can have weighted edges and the function [MATH] defines the weight [MATH] of an edge [MATH].', '1906.01277-1-11-0': 'Kernels on graphs are generally defined using the [MATH]-Convolution framework by [CITATION].', '1906.01277-1-11-1': 'The main idea is to decompose the graph [MATH] into substructures and to define a kernel value [MATH] as a combination of substructure similarities.', '1906.01277-1-11-2': 'For instance, the shortest path kernel computes each kernel value [MATH] as a sum of the similarities between each shortest path in [MATH] and each shortest path in [MATH].', '1906.01277-1-11-3': 'Despite the practical success of [MATH]-Convolution kernels, they often rely on aggregation strategies that ignore valuable information, such as the distribution of the substructures.', '1906.01277-1-11-4': 'An example is the Weisfeiler-Lehman (WL) subtree kernel , which generates graph-level features by summing the contribution of the node representations.', '1906.01277-1-11-5': 'In order to avoid these simplifications, we want to use concepts from optimal transport theory such as the Wasserstein distance; they can be used to better capture the similarities between graphs.', '1906.01277-1-12-0': '## Wasserstein distance', '1906.01277-1-13-0': 'The Wasserstein distance is a distance function between probability distributions defined on a given metric space.', '1906.01277-1-13-1': 'Let [MATH] and [MATH] be two probability distributions on a metric space [MATH] equipped with a ground distance [MATH], such as the Euclidean distance.', '1906.01277-1-14-0': 'The [MATH]-Wasserstein distance for [MATH] is defined as [EQUATION] where [MATH] is the set of all transportation plans [MATH] over [MATH] with marginals [MATH] and [MATH] on the first and second factors respectively.', '1906.01277-1-15-0': 'The Wasserstein distance satisfies the axioms of a metric, provided that [MATH] is a metric (see the monograph of [CITATION], chapter 6, for a proof).', '1906.01277-1-15-1': 'Throughout the paper, we will focus on the distance for [MATH] and we will refer to the [MATH]-Wasserstein distance when mentioning the Wasserstein distance, unless noted otherwise.', '1906.01277-1-16-0': 'The Wasserstein distance is linked to the optimal transport problem , where the aim is to find the most "inexpensive" way, in terms of the ground distance, to transport all the probability mass from the distribution [MATH] so as to match the distribution [MATH] .', '1906.01277-1-16-1': 'An intuitive illustration can be made for the [MATH]-dimensional case, where the two probability distributions can be imagined as piles of dirt or sand.', '1906.01277-1-16-2': "The Wasserstein distance, sometimes also referred to as the Earth Mover's Distance , can be interpreted as the minimum effort required to move the content of the first pile to reproduce the second pile.", '1906.01277-1-17-0': 'In this paper, we deal with finite sets of node embeddings and not with continuous probability distributions.', '1906.01277-1-17-1': 'We can therefore reformulate the Wasserstein distance as a sum rather than an integral, and use the matrix notation commonly encountered in the optimal transport literature to represent the transportation plan.', '1906.01277-1-17-2': 'Given two sets of vectors [MATH] and [MATH], we can equivalently define the Wasserstein distance between them as [EQUATION]', '1906.01277-1-17-3': 'Here, [MATH] is the distance matrix containing the distances [MATH] between each element [MATH] of [MATH] and [MATH] of [MATH], [MATH] is a transport matrix (or joint probability), and [MATH] is the Frobenius dot product.', '1906.01277-1-17-4': 'The transport matrix [MATH] contains the fractions that indicate how to transport the values from [MATH] to [MATH] with the minimal total transport effort.', '1906.01277-1-17-5': 'Since we assume that the total mass to be transported equals [MATH] and is evenly distributed across the elements of [MATH] and [MATH], the row and column values of [MATH] must sum up to [MATH] and [MATH], respectively.', '1906.01277-1-18-0': '# Wasserstein distance on graphs', '1906.01277-1-19-0': 'The unsatisfactory nature of the aggregation step of current [MATH]-Convolution graph kernels, which may mask important substructure differences by averaging, motivated us to have a finer distance measure between structures and their components.', '1906.01277-1-19-1': 'In parallel, recent advances in optimisation solutions for faster computation of the optimal transport problem inspired us to consider this framework for the problem of graph classification.', '1906.01277-1-19-2': 'Our method relies on the following steps: [(1)] transform each graph into a set of node embeddings, measure the Wasserstein distance between each pair of graphs, and compute a similarity matrix to be used in the learning algorithm.', '1906.01277-1-20-0': 'Figure [REF] provides an illustration of the first two steps, while Algorithm [REF] summarises the whole procedure.', '1906.01277-1-20-1': 'We start by defining an embedding scheme and illustrate how we integrate embeddings in the Wasserstein distance.', '1906.01277-1-21-0': '[Graph Embedding Scheme]', '1906.01277-1-22-0': 'Given a graph [MATH], a graph embedding scheme [MATH] is a function that outputs a fixed-size vectorial representation for each node in the graph.', '1906.01277-1-22-1': 'For each [MATH], the [MATH]-th row of [MATH] is called the node embedding of [MATH].', '1906.01277-1-23-0': 'Note that Definition [REF] permits treating node labels, which are categorical attributes, as one-dimensional attributes with [MATH].', '1906.01277-1-24-0': '[Graph Wasserstein Distance]', '1906.01277-1-25-0': 'Given two graphs [MATH] and [MATH], a graph embedding scheme [MATH] and a ground distance [MATH], we define the Graph Wasserstein Distance (GWD) as [EQUATION]', '1906.01277-1-25-1': 'We will now propose a graph embedding scheme inspired by the WL kernel on categorically labeled graphs, extend it to continuously attributed graphs with weighted edges, and show how to integrate it with the GWD presented in Definition [REF].', '1906.01277-1-26-0': '## Generating node embeddings', '1906.01277-1-27-0': 'The Weisfeiler-Lehman scheme.', '1906.01277-1-27-1': 'The Weisfeiler-Lehman subtree kernel , designed for labelled non-attributed graphs, looks at similarities among subtree patterns, defined by a propagation scheme on the graphs that iteratively compares labels on the nodes and their neighbours.', '1906.01277-1-27-2': 'This is achieved by creating a sequence of ordered strings through the aggregation of the labels of a node and its neighbours; those strings are subsequently hashed to create updated compressed node labels.', '1906.01277-1-27-3': 'With increasing iterations of the algorithm, these labels represent increasingly larger neighbourhoods of each node, allowing to compare more extended substructures.', '1906.01277-1-28-0': 'More precisely, consider a graph [MATH], let [MATH] be the initial node label of [MATH] for each [MATH], and let [MATH] be the number of WL iterations.', '1906.01277-1-28-1': 'Then, we can define a recursive scheme to compute [MATH] for [MATH] by looking at the ordered set of neighbours [MATH], as [EQUATION]', '1906.01277-1-28-2': 'We call this procedure the WL labelling scheme.', '1906.01277-1-28-3': 'As in the original publication , we use perfect hashing for the [MATH] function, so nodes at iteration [MATH] will have the same label if and only if their label and those of their neighbours are identical at iteration [MATH].', '1906.01277-1-29-0': 'Extension to continuous attributes.', '1906.01277-1-29-1': 'For graphs with continuous attributes [MATH], we need to improve the WL refinement step, whose original definition prohibits handling the continuous case.', '1906.01277-1-29-2': 'The key idea is to create an explicit propagation scheme that leverages and updates the current node features by averaging over the neighbourhoods.', '1906.01277-1-29-3': 'While similar approaches have been implicitly investigated for computing node-level kernel similarities , they rely on additional hashing steps on the continuous features.', '1906.01277-1-29-4': 'Moreover, we can easily account for edge weights by considering them in the average calculation of each neighbourhood.', '1906.01277-1-29-5': 'Suppose we have a continuous attribute [MATH] for each node [MATH], then we recursively define [EQUATION]', '1906.01277-1-29-6': 'When edge weights are not available, we set [MATH].', '1906.01277-1-29-7': 'We consider the weighted average of the neighbourhood attribute values instead of a sum and add the [MATH] factor because we want to ensure a similar scale of the features across iterations; in fact, we concatenate such features for building our proposed kernel (see Definition [REF] for more details) and observe better empirical results with similarly-scaled features.', '1906.01277-1-29-8': 'While not constituting a test of graph isomorphism, this refinement step can be seen as an intuitive extension for continuous attributes of the one used by the WL subtree kernel on categorical node labels, a widely successful baseline.', '1906.01277-1-29-9': 'Moreover, it resembles the propagation scheme used in many graph neural networks, which have proven to be very successful for node classification on large data sets .', '1906.01277-1-29-10': 'Finally, its ability to account for edge weights makes it applicable to all types of graphs without having to perform an hashing step .', '1906.01277-1-30-0': 'Graph embedding scheme.', '1906.01277-1-30-1': 'Using the recursive procedure described above, we propose a WL-based graph embedding scheme that generates node embeddings from the node labels or attributes of the graphs.', '1906.01277-1-30-2': 'In the following, we use [MATH] to denote the dimensionality of the node attributes (thus, [MATH] for the categorical labels).', '1906.01277-1-31-0': '[WL Features]', '1906.01277-1-32-0': 'Let [MATH] and let [MATH] be the number of WL iterations.', '1906.01277-1-32-1': 'Then, for every [MATH], we define the WL features as [EQUATION] where [MATH] for categorically labelled graphs and [MATH] for continuously attributed graphs.', '1906.01277-1-32-2': 'We refer to [MATH] as the node features of graph [MATH] at iteration [MATH].', '1906.01277-1-32-3': 'Then, the node embeddings of graph [MATH] at iteration [MATH] are defined as: [EQUATION]', '1906.01277-1-32-4': 'We observe that a graph can be both categorically labelled and continuously attributed, and one could extend the above scheme by jointly considering this information, for instance by concatenating the node features.', '1906.01277-1-32-5': 'However, we will leave this scenario as an extension for future work, and we thus avoid having to define an appropriate distance measure between categorical and continuous data, as this is a long-standing issue .', '1906.01277-1-33-0': '## Computing the Wasserstein distance', '1906.01277-1-34-0': 'Once the node embeddings are generated by the graph embedding scheme, we evaluate the pairwise Wasserstein distance between graphs.', '1906.01277-1-34-1': 'We start by computing the ground distances between each pair of nodes.', '1906.01277-1-34-2': 'For categorical node features, we use the normalised Hamming distance, i.e.: [EQUATION]', '1906.01277-1-34-3': 'The Hamming distance can be pictured as the normalised sum of the discrete metric [MATH] on each of the features.', '1906.01277-1-34-4': 'The Hamming distance equals [MATH] when two vectors have no features in common while it is [MATH] when the vectors are identical.', '1906.01277-1-34-5': 'We use the Hamming distance as, in this case, the Weisfeiler-Lehman features are indeed categorical and values carry no meaning.', '1906.01277-1-34-6': 'For continuous node features, on the other hand, we employ the Euclidean distance [EQUATION]', '1906.01277-1-34-7': 'We then plug the ground distance into the equation of Definition [REF] and compute the Wasserstein distance using a network simplex method .', '1906.01277-1-35-0': 'Computational complexity.', '1906.01277-1-35-1': 'Naively, the computation of the Wasserstein Distance has a complexity of [MATH], with [MATH] being the cardinality of the indexed set of node embeddings, i.e. the number of nodes in the two graphs.', '1906.01277-1-35-2': 'Nevertheless, efficient speedup tricks can be employed.', '1906.01277-1-35-3': 'In particular, approximations relying on Sinkhorn regularisation have been proposed , some of which reduce the computational burden to near-linear time while preserving accuracy .', '1906.01277-1-35-4': 'Such speed-up strategies become incredibly useful for larger data sets, i.e. graphs with thousands of nodes, and can be easily integrated in our method.', '1906.01277-1-35-5': 'See Appendix [REF] for a practical discussion.', '1906.01277-1-36-0': '[tb] Compute Wasserstein Graph Kernel', '1906.01277-1-37-0': 'Input: Two graphs [MATH], [MATH]; graph embedding scheme [MATH]; ground distance [MATH]; [MATH].', '1906.01277-1-37-1': 'Output: kernel value [MATH].', '1906.01277-1-38-0': '[MATH] Generate node embeddings [MATH] Compute the ground distance between each pair of nodes [MATH] Compute the Wasserstein distance', '1906.01277-1-39-0': '[MATH]', '1906.01277-1-40-0': '# From Wasserstein distance to kernels', '1906.01277-1-41-0': 'From the graph Wasserstein distance, one can construct a similarity measure to be used in a learning algorithm.', '1906.01277-1-41-1': 'In this section, we propose a new graph kernel, state some claims about its (in)definiteness, and elaborate on how to use it for classifying graphs with continuous and categorical node labels.', '1906.01277-1-42-0': '[Wasserstein Weisfeiler-Lehman]', '1906.01277-1-43-0': 'Given a set of graphs [MATH] and the GWD defined for each pair of graph on their WL embeddings, we define the Wasserstein Weisfeiler-Lehman (WWL) kernel as: [EQUATION]', '1906.01277-1-43-1': 'This is an instance of a Laplacian kernel, which was shown to offer favourable conditions for positive definiteness in case of non-Euclidean distances .', '1906.01277-1-43-2': 'Obtaining the WWL kernel concludes the procedure described in Algorithm [REF].', '1906.01277-1-43-3': 'In the remainder of this section, we distinguish between the categorical WWL kernel, obtained on graphs with categorical labels, and the continuous WWL kernel, obtained on continuously attributed graphs via the graph embedding schemes described in Section [REF].', '1906.01277-1-44-0': 'For Euclidean spaces, obtaining positive definite kernels from distance functions is a well-studied topic .', '1906.01277-1-44-1': 'However, the Wasserstein distance in its general form is not isometric, i.e. there is no metric-preserving mapping, to an [MATH]-norm as the metric space it induces strongly depends on the chosen ground distance .', '1906.01277-1-44-2': 'Therefore, despite being a metric, it is not necessarily possible to derive a positive definite kernel from the Wasserstein distance in its general formulation, because the classical approaches cannot be applied here.', '1906.01277-1-44-3': 'Nevertheless, as a consequence of using the Laplacian kernel , we can show that, in the setting of categorical node labels, the obtained kernel is positive definite.', '1906.01277-1-45-0': 'The categorical WWL kernel is positive definite for all [MATH].', '1906.01277-1-46-0': 'For a proof, see Sections [REF] and [REF] in the Appendix.', '1906.01277-1-46-1': 'By contrast, for the continuous case, establishing the definiteness of the obtained kernel remains an open problem.', '1906.01277-1-46-2': 'We refer the reader to Section [REF] in the supplementary materials for further discussions and conjectures.', '1906.01277-1-47-0': 'To ensure the theoretical and practical correctness of our results, we employ recently-developed methods for learning with indefinite kernels.', '1906.01277-1-47-1': 'More precisely, we utilise learning methods for Krein spaces, which have been specifically designed to work with indefinite kernels ; in general, kernels that are not positive definite induce Reproducible Kernel Krein Spaces (RKKS).', '1906.01277-1-47-2': 'These spaces can be seen as a generalisation of Reproducible Kernel Hilbert Spaces (RKHS), with which they share similar mathematical properties, making them amenable to machine learning techniques.', '1906.01277-1-47-3': 'Recent algorithms are capable of solving learning problems in RKKS; their results indicate that there are clear benefits (in terms of classification performance, for example) of learning in such spaces.', '1906.01277-1-47-4': 'Therefore, when evaluating WWL, we will use a Krein SVM (KSVM, ) as a classifier for the case of continuous attributes.', '1906.01277-1-48-0': '# Experimental evaluation', '1906.01277-1-49-0': 'In this section, we analyse how the performance of WWL compares with state-of-the-art graph kernels.', '1906.01277-1-49-1': 'In particular, we empirically observe that WWL [(1)] is competitive with the best graph kernel for categorically labelled data, and outperforms all the state-of-the-art graph kernels for attributed graphs.', '1906.01277-1-50-0': '## Data sets', '1906.01277-1-51-0': 'We report results on real-world data sets from multiple sources and use either their continuous attributes or categorical labels for evaluation.', '1906.01277-1-51-1': 'In particular, MUTAG, PTC-MR, NCI1 and DD are equipped with categorical node labels only; ENZYMES, PROTEINS have both categorical labels and continuous attributes; IMDB-B, BZR and COX2 only contain continuous attributes; finally, BZR-MD and COX2-MD have both continuous node attributes and edge weights.', '1906.01277-1-51-2': 'Further information on the data sets are available in Supplementary Table [REF].', '1906.01277-1-51-3': 'Additionally, we also report results on synthetic data (Synthie and SYNTHETIC-new) in Appendix [REF].', '1906.01277-1-51-4': 'All the data sets have been downloaded from [CITATION].', '1906.01277-1-52-0': '## Experimental setup', '1906.01277-1-53-0': 'We compare WWL with state-of-the-art graph kernel methods from the literature as well as relevant baselines, which we trained ourselves on the same splits (see below).', '1906.01277-1-53-1': 'In particular, for the categorical case we compare with WL and WL-OA as well as with the vertex (V) and edge (E) histograms.', '1906.01277-1-53-2': 'For the continuously attributed data sets, we compare with two instances of the Hash Graph Kernel (HGK-SP; HGK-WL) , and with the Graph Hopper (GH) .', '1906.01277-1-53-3': 'As a further comparison partner, we use a continuous vertex histogram (VH-C), which is defined as an RBF kernel between the sum of the graph node embeddings.', '1906.01277-1-53-4': 'Furthermore, to highlight the benefits of using the Wasserstein distance in our method, we replace it with an RBF kernel.', '1906.01277-1-53-5': 'More precisely, given two graphs [MATH] and [MATH], with [MATH] and [MATH], we first compute the Gaussian kernel between each pair of the node embeddings obtained in the same fashion as for WWL, therefore obtaining a kernel matrix between node embeddings [MATH]; then, we sum up the values [MATH] and set [MATH].', '1906.01277-1-53-6': 'This procedure is repeated for each pair of graphs to obtain the final graph kernel matrix.', '1906.01277-1-53-7': 'We refer to this baseline as RBF-WL.', '1906.01277-1-54-0': 'As a classifier, we use an SVM (or a KSVM in the case of WWL) and 10-fold cross-validation, selecting the parameters on the training set only.', '1906.01277-1-54-1': 'We repeat each cross-validation split 10 times and report the average accuracy.', '1906.01277-1-54-2': 'We employ the same split for each evaluated method, thus guaranteeing a fully comparable setup among all evaluated methods.', '1906.01277-1-54-3': 'Please refer to Appendix [REF] for details on the hyperparameter selection.', '1906.01277-1-55-0': 'Implementation Available Python implementations can be used to compute the WL kernel and the Wasserstein distance .', '1906.01277-1-55-1': 'We leverage these resources to develop our method and make our code publicly available.', '1906.01277-1-55-2': 'We use the original implementations provided by the respective authors to compute the WL-OA, the HGK, and the GH methods.', '1906.01277-1-56-0': '## Results and discussion', '1906.01277-1-57-0': 'The results are evaluated via classification accuracy and summarised in Table [REF] and Table [REF] for the categorical labels and continuous attributes, respectively.', '1906.01277-1-58-0': '### Categorical labels', '1906.01277-1-59-0': 'On the categorical data sets, WWL is comparable to the WL-OA kernel; it improves over the classical WL, though.', '1906.01277-1-59-1': 'In particular, WWL largely improves over WL-OA in PTC-MR and is slightly better on DD, while WL-OA is better on NCI1 and PROTEINS.', '1906.01277-1-60-0': 'Unsurprisingly, our approach is comparable to the WL-OA, whose main idea is to solve the optimal assignment problem by defining Dirac kernels on histograms of node labels, using multiple iterations of WL.', '1906.01277-1-60-1': 'This formulation is similar to the one we provide for categorical data, but relies on optimal assignment rather than optimal transport and therefore requires one-to-one mappings instead of continuous transport maps.', '1906.01277-1-60-2': 'Beside, we solve the optimal transport problem on the concatenated embeddings, hereby jointly exploiting representations at multiple WL iterations.', '1906.01277-1-60-3': 'Contrarily, the WL-OA performs optimal assignment at each iteration of WL and only combines them in a second stage.', '1906.01277-1-60-4': 'However, the key advantage of WWL over WL-OA is its capacity to account for continuous attributes.', '1906.01277-1-61-0': '### Continuous attributes', '1906.01277-1-62-0': 'In this setting, WWL significantly outperforms the other methods on [MATH] out of [MATH] data sets, is better on another one, and is on a par on the remaining [MATH].', '1906.01277-1-62-1': 'We further compute the average rank of each method in the continuous setting, with WWL scoring as first.', '1906.01277-1-62-2': 'The ranks calculated from Table [REF] are: WWL = [MATH]; HGK-WL = [MATH]; RBF-WL = [MATH]; HGK-SP = [MATH]; VH-C = [MATH].', '1906.01277-1-62-3': 'This is a remarkable improvement over the current state of the art, and it indeed establishes a new one.', '1906.01277-1-62-4': 'When looking at the average rank of the method, WWL always scores first.', '1906.01277-1-62-5': 'We thus raise the bar in kernel graph classification for attributed graphs.', '1906.01277-1-62-6': 'As mentioned in Section [REF], the kernel obtained from continuous attributes is not necessarily positive definite.', '1906.01277-1-62-7': 'However, we empirically observe the kernel matrices to be positive definite (up to a numerical error), further supporting our theoretical considerations (see Appendix [REF]).', '1906.01277-1-62-8': 'In practice, the difference between the results obtained from classical SVMs in RKHS and the results obtained with the KSVM approach are negligible.', '1906.01277-1-63-0': 'Comparison with Hash Graph Kernels.', '1906.01277-1-63-1': 'The Hash Graph Kernel (HGK) approach is somewhat related to our propagation scheme.', '1906.01277-1-63-2': 'By using multiple hashing functions, the HGK method is capable to extend certain existing graph kernels to the continuous setting.', '1906.01277-1-63-3': 'This helps avoid the limitations of perfect hashing, which cannot express small differences in continuous attributes.', '1906.01277-1-63-4': 'A drawback of the random hashing performed by HGK is that it requires additional parameters and introduces a stochastic element to the kernel matrix computation.', '1906.01277-1-63-5': 'By contrast, our propagation scheme is fully continuous and uses the Wasserstein distance to capture small differences in distributions of continuous node attributes.', '1906.01277-1-63-6': 'Moreover, the observed performance gap suggests that an entirely continuous representation of the graphs provides clear benefits over the hashing.', '1906.01277-1-64-0': '# Conclusion', '1906.01277-1-65-0': 'In this paper, we present a new family of graph kernels, the Wasserstein Weisfeiler-Lehman (WWL) graph kernels.', '1906.01277-1-65-1': 'Our experiments show that WWL graph kernels constitute the new state of the art for graph classification in the scenario of continuous node attributes, while matching the state of the art in the categorical setting.', '1906.01277-1-65-2': 'As a line of research for future work, we see a great potential in the runtime improvement, thus allowing applications of our method on regimes and data sets with larger graphs.', '1906.01277-1-65-3': 'In fact, preliminary experiments (see Section [REF] as well as Figure [REF] in the Appendix) already confirm the benefit of Sinkhorn regularisation, when the average number of nodes in the graph increases.', '1906.01277-1-65-4': 'In parallel, it would be beneficial to derive approximations of the explicit feature representations in the RKKS, as this would also provide a consistent speedup.', '1906.01277-1-65-5': 'We further envision that major theoretical contributions could be made by defining theoretical bounds to ensure the positive definiteness of the WWL kernel in the case of continuous node attributes.', '1906.01277-1-65-6': 'Finally, optimisation objectives based on optimal transport could be employed to develop new algorithms based on graph neural networks .', '1906.01277-1-65-7': 'On a more general level, our proposed method provides a solid foundation and highlight the large potential of optimal transport theory for machine learning.', '1906.01277-1-66-0': 'figuresection tablesection', '1906.01277-1-67-0': '# Appendix', '1906.01277-1-68-0': '## Extended considerations on WWL definiteness', '1906.01277-1-69-0': 'We will now discuss on the positive definite nature of our WWL kernel.', '1906.01277-1-70-0': 'In general, whether distances obtained from optimal transport problems can be used to create positive definite kernels remains an open research question.', '1906.01277-1-70-1': 'Several attempts to draw general conclusions on the definiteness of the Wasserstein distance were unsuccessful, but insightful results on particular cases were obtained along the way.', '1906.01277-1-70-2': 'Here, we first collect some of these contributions and use them to prove that our WWL Kernel for categorical embeddings is positive definite.', '1906.01277-1-70-3': 'We then elaborate further on the continuous embeddings case, for which we provide conjectures on practical conditions to obtain a pd kernel.', '1906.01277-1-71-0': 'Before proceeding, let us remind some useful notions.', '1906.01277-1-71-1': '[CITATION] A symmetric function [MATH] is called a positive definite (pd) kernel if it satisfies the condition [EQUATION] for every [MATH], [MATH] and [MATH].', '1906.01277-1-72-0': 'The matrix of kernel values [MATH] with entries [MATH] is called the Gram matrix of [MATH] with respect to [MATH].', '1906.01277-1-72-1': 'A conditional positive definite (cpd) kernel is a function that satisfies Equation [REF] for all [MATH] with [MATH], whereas a conditional negative definite kernel is a function that satisfies [MATH] for all [MATH] with [MATH].', '1906.01277-1-73-0': 'For Euclidean spaces, obtaining kernels from distance functions is a well-studied topic.', '1906.01277-1-74-0': 'Let [MATH] be a symmetric, non-negative distance function with [MATH].', '1906.01277-1-74-1': 'If [MATH] is isometric to an [MATH]-norm, then [EQUATION] is a valid cpd kernel.', '1906.01277-1-75-0': 'However, the Wasserstein distance in its general form is not isometric to an [MATH]-norm as the metric space it induces strongly depends on the chosen ground distance .', '1906.01277-1-75-1': 'More recently, [CITATION] argued that many types of data, including probability distributions, do not always reside in Euclidean spaces.', '1906.01277-1-75-2': 'Therefore, they define the family of exponential kernels relying on a non-Euclidean distance [MATH], as follows [EQUATION] and, based on earlier considerations from [CITATION], show that, under certain conditions, the Laplacian kernel ([MATH] in Equation [REF]) is positive definite.', '1906.01277-1-76-0': 'The geodesic Laplacian kernel is positive definite for all [MATH] if and only if the geodesic distance [MATH] is conditional negative definite.', '1906.01277-1-77-0': 'Once again, considerations on the negative definiteness of Wasserstein distance functions cannot be made on the general level.', '1906.01277-1-77-1': 'Certain ground distances, however, guarantee the negative definiteness of the resulting Wasserstein distance.', '1906.01277-1-77-2': 'In particular, the Wasserstein distance with the discrete metric (i.e. [MATH] in Equation [REF]) as the ground distance, was proved to be conditional negative definite .', '1906.01277-1-78-0': 'We will now leverage these results to prove that the Wasserstein distance equipped with the Hamming ground distance is conditional negative definite, therefore it yields positive definite kernels for the categorical WL embeddings.', '1906.01277-1-79-0': '### The case of categorical embeddings', '1906.01277-1-80-0': 'When generating node embeddings using the Weisfeiler-Lehman labelling scheme with a shared dictionary across all the graphs, the solutions to the optimal transport problem are also shared across iterations.', '1906.01277-1-80-1': 'We denote the Weisfeiler-Lehman embedding scheme as defined in Definition [REF] as [MATH], and let [MATH] be the corresponding GWD on a set of graphs [MATH] with categorical labels.', '1906.01277-1-80-2': 'Let [MATH] of Equation [REF] be the ground distance of [MATH].', '1906.01277-1-80-3': 'Then, the following useful results hold.', '1906.01277-1-81-0': 'If a transportation plan [MATH] with transport matrix [MATH] is optimal in the sense of Definition [REF] for distances [MATH] between embeddings obtained with [MATH], then it is also optimal for the discrete distances [MATH] between the [MATH]-th iteration values obtained with the Weisfeiler-Lehman procedure.', '1906.01277-1-82-0': 'Proof.', '1906.01277-1-82-1': 'See Appendix [REF]', '1906.01277-1-83-0': 'If a transportation plan [MATH] with transport matrix [MATH] is optimal in the sense of Definition [REF] for distances [MATH] between embeddings obtained with [MATH], then it is also optimal for distances [MATH] between embeddings obtained with [MATH].', '1906.01277-1-84-0': 'Proof.', '1906.01277-1-84-1': 'See Appendix [REF]', '1906.01277-1-85-0': 'We therefore postulate that the Wasserstein distance between categorical WL node embeddings is a conditional negative definite function.', '1906.01277-1-86-0': '[MATH] is a conditional negative definite function.', '1906.01277-1-87-0': 'Proof.', '1906.01277-1-87-1': 'See Appendix [REF]', '1906.01277-1-88-0': 'Proof of Theorem [REF].', '1906.01277-1-88-1': 'Theorem [REF] in light of Proposition [REF] implies that the WWL kernel of Definition [REF] is positive definite for all [MATH].', '1906.01277-1-88-2': '[MATH]', '1906.01277-1-89-0': 'We will now consider the case of the definiteness of kernels in the continuous setting.', '1906.01277-1-90-0': '### The case of continuous embeddings', '1906.01277-1-91-0': 'On one hand, in the categorical case, we proved the positive definiteness of our kernel.', '1906.01277-1-91-1': 'On the other hand, the continuous case is considerably harder to tackle.', '1906.01277-1-91-2': 'We conjecture that, under certain conditions, the same might hold for continuous features.', '1906.01277-1-91-3': 'While we do not have a formal proof yet, in what follows, we discuss arguments to support this conjecture which seems to agree with our empirical findings.', '1906.01277-1-92-0': 'The curvature of the metric space induced by the Wasserstein metric for a given ground distance plays an important role here.', '1906.01277-1-92-1': 'We first need to define Alexandrov spaces.', '1906.01277-1-92-2': '[Alexandrov space] Given a metric space and a real number [MATH], the space is called an Alexandrov space if its sectional curvature is [MATH].', '1906.01277-1-93-0': 'Roughly speaking, the curvature indicates to what extent a geodesic triangle will be deformed in the space.', '1906.01277-1-93-1': 'The case of [MATH] is special as no distortion is happening here-hence, spaces that satisfy this property are called flat.', '1906.01277-1-93-2': 'The concept of Alexandrov spaces is required in the following proposition, taken from a theorem by [CITATION], which shows the relationship between a kernel and its underlying metric space.', '1906.01277-1-94-0': 'The geodesic Gaussian kernel (i.e. [MATH] in Equation [REF]) is positive definite for all [MATH] if and only if the underlying metric space [MATH] is flat in the sense of Alexandrov, i.e. if any geodesic triangle in [MATH] can be isometrically embedded in a Euclidean space.', '1906.01277-1-95-0': 'However, it is unlikely that the space induced by the Wasserstein distance is locally flat, as not even the geodesics (i.e. a generalization of a shortest path to arbitrary metric spaces) between graph embeddings are necessarily unique, as we shall show below.', '1906.01277-1-95-1': 'Hence, we use the geodesic Laplacian kernel instead of the Gaussian one because it poses less strict requirements on the induced space, as stated in Proposition [REF].', '1906.01277-1-95-2': 'More precisely, the metric used in the kernel function needs to be cnd.', '1906.01277-1-95-3': 'We cannot directly prove this yet, but we can provide a proof that the converse is not true.', '1906.01277-1-95-4': 'To this end, we first notice that the metric space induced by the GWD, which we shall here refer to as [MATH], does not have a curvature that is bounded from above.', '1906.01277-1-96-0': 'A metric space [MATH] is said to be [MATH] if its curvature is bounded by some real number [MATH] from above.', '1906.01277-1-96-1': 'This can also be seen a "relaxed" definition, or generalisation, of a Riemannian manifold.', '1906.01277-1-97-0': '[MATH] is not in [MATH] for any [MATH], meaning that its curvature is not bounded by any [MATH] from above.', '1906.01277-1-98-0': 'This follows from a similar argument as presented by [CITATION].', '1906.01277-1-98-1': 'We briefly sketch the argument.', '1906.01277-1-98-2': 'Let [MATH] and [MATH] be two graphs.', '1906.01277-1-98-3': 'Assume that [MATH] is a [MATH] space for some [MATH].', '1906.01277-1-98-4': 'Then it follows that if [MATH], there is a unique geodesic between them.', '1906.01277-1-98-5': 'However, we can construct a family of graph embeddings for which this is not the case.', '1906.01277-1-98-6': 'To this end, let [MATH] and [MATH] and [MATH] be two graph embeddings with node embeddings [MATH], [MATH] as well as [MATH] and [MATH], respectively.', '1906.01277-1-98-7': 'Since we use the Euclidean distance as a ground distance, there will be two optimal transport plans: the first maps [MATH] to [MATH] and [MATH] to [MATH], while the second maps [MATH] to [MATH] and [MATH] to [MATH].', '1906.01277-1-98-8': 'Hence, we have found two geodesics that connect [MATH] and [MATH].', '1906.01277-1-98-9': 'Since we may choose [MATH] to be arbitrarily small, the space cannot be [MATH] for [MATH].', '1906.01277-1-99-0': 'While this means that we do not have a simple upper bound on the curvature, we have the following conjecture.', '1906.01277-1-100-0': '[MATH] is an Alexandrov space with curvature bounded from below by zero.', '1906.01277-1-101-0': 'For a proof idea, we refer to [CITATION]; the main argument involves characterizing the distance between triples of graph embeddings.', '1906.01277-1-101-1': 'This conjecture is helpful insofar as being a non-negatively-curved Alexandrov space is a necessary prerequisite for [MATH] to be a Hilbert space .', '1906.01277-1-101-2': 'In turn, [CITATION] shows that cnd metrics and Hilbert spaces are intricately linked.', '1906.01277-1-101-3': 'We thus have some hope in obtaining a cnd metric, even though we lack a proof as of now.', '1906.01277-1-101-4': 'Our empirical results, however, indicate that it is possible to turn the GWD into a cnd metric, provided a proper normalisation takes place.', '1906.01277-1-101-5': 'Intuitively, for high dimensional input spaces, standardisation of input features changes the curvature of the induced space by making it locally (almost) flat.', '1906.01277-1-102-0': 'To support this argumentation, we refer to an existing way to ensure positive definiteness.', '1906.01277-1-102-1': 'One can use an alternative to the classical Wasserstein distance denoted as the sliced Wasserstein .', '1906.01277-1-102-2': 'The idea is to project high dimensional ditributions into one-dimensional spaces, hereby calculating the Wasserstein distance as a combination of one dimensional representations.', '1906.01277-1-102-3': '[CITATION] showed that each of the one-dimensional Wasserstein distances are conditional negative definite.', '1906.01277-1-102-4': 'The kernel on high-dimensional representations is then defined as a combination of the one-dimensional positive definite counterparts.', '1906.01277-1-103-0': '## Proof of Lemma [REF]', '1906.01277-1-104-0': 'We recall the matrix notation introduced in Equation [REF] of the main paper, where [MATH] is the cost or distance matrix, [MATH] is a transport matrix (or joint probability), and [MATH] is the Frobenius dot product.', '1906.01277-1-104-1': 'Since we give equal weight (i.e. equal probability mass) to each of the vectors in each set, [MATH] contains all nonnegative [MATH] matrices [MATH] with [EQUATION]', '1906.01277-1-104-2': 'For notation simplicity, let us denote by [MATH] the Hamming matrix [MATH], where the [MATH]-th entry is given by the Hamming distance between the embedding of the [MATH]-th node of graph [MATH] and the embedding of the [MATH]-th node of graph [MATH] at iteration [MATH].', '1906.01277-1-104-3': 'Similarly, we define [MATH] to be the discrete metric distance matrix, where the [MATH]-th entry is given by the discrete distance between feature [MATH] of node embedding [MATH] of graph [MATH] and feature [MATH] of node embedding [MATH] of graph [MATH].', '1906.01277-1-104-4': 'It is easy to see that [MATH] while [MATH] and, by definition, [EQUATION].', '1906.01277-1-105-0': 'Moreover, due to the WL procedure, two labels that are different at iteration [MATH] will also be different at iteration [MATH].', '1906.01277-1-105-1': 'Hence, the following identity holds [EQUATION]', '1906.01277-1-105-2': 'Which implies that [MATH].', '1906.01277-1-105-3': 'An optimal transportation plan [MATH] for [MATH] embeddings satisfies [EQUATION]', '1906.01277-1-105-4': 'Assuming that [MATH] is not optimal for [MATH], we can define [MATH] such that [EQUATION]', '1906.01277-1-105-5': 'Since the entries of [MATH] are either [MATH] or [MATH], we can define the set of indices tuples [MATH] and rewrite the inequality as [EQUATION]', '1906.01277-1-105-6': 'Due to the constraints on the entries of [MATH] and [MATH], namely [MATH], this implies that, by rearranging the transport map, there is more mass that could be transported at [MATH] cost.', '1906.01277-1-105-7': 'In our formalism: [EQUATION]', '1906.01277-1-105-8': 'However, as stated before, entries of [MATH] that are [MATH] are also [MATH] in [MATH] and a better transport plan [MATH] would therefore also be optimal for [MATH]: [EQUATION]', '1906.01277-1-105-9': 'Which contradicts the optimality assumption above.', '1906.01277-1-105-10': 'Hence, [MATH] is also optimal for [MATH].', '1906.01277-1-106-0': '## Proof of Lemma [REF]', '1906.01277-1-107-0': 'Intuitively, the transportation plan at iteration [MATH] is a "refinement" of the transportation plan at iteration [MATH], where only a subset of the optimal transportation plans remain optimal for the new cost matrix [MATH].', '1906.01277-1-107-1': 'Using the same notation as for the Proof in Appendix [REF], and due to the WL procedure, two labels that are different at iteration [MATH] will also be different at iteration [MATH].', '1906.01277-1-107-2': 'Hence, the following identities hold [EQUATION]', '1906.01277-1-107-3': 'An optimal transportation plan [MATH] for [MATH] embeddings satisfies [EQUATION]', '1906.01277-1-107-4': 'Which can also be written as [EQUATION]', '1906.01277-1-107-5': 'The values of [MATH] increase in a step-wise fashion for increasing [MATH] and their ordering remains constant except for entries that were [MATH] at iteration [MATH] and become [MATH] at iteration [MATH].', '1906.01277-1-107-6': 'Hence, since our metric distance matrices satisfy monotonicity conditions and because [MATH] is optimal for [MATH] according to Lemma [REF], it follows that [EQUATION]', '1906.01277-1-107-7': 'Hence, [MATH] is also optimal for [MATH] embeddings.', '1906.01277-1-108-0': '## Proof of Theorem [REF]', '1906.01277-1-109-0': 'Using the same notation as for the Proof in Appendix [REF] and the formulation in Equation [REF], we can write [EQUATION]', '1906.01277-1-109-1': 'Let [MATH] be an optimal solution for iteration [MATH] then, from Lemmas [REF] and [REF], it is also an optimal solution for [MATH] as well as for all [MATH] and we can rewrite the equation as a sum of optimal transport problems [EQUATION]', '1906.01277-1-109-2': 'This corresponds to a sum of 1-dimensional optimal transport problems relying on the discrete metric, which were shown to be conditional negative functions .', '1906.01277-1-109-3': 'The final sum is therefore conditional negative definite as well.', '1906.01277-1-110-0': '## Data sets and additional results', '1906.01277-1-111-0': 'We report additional information on the data sets used in our experimental comparison in Supplementary Table [REF]', '1906.01277-1-112-0': 'Our data sets belong to multiple chemoinformatics domains, including small molecules (MUTAG, PTC-MR, NCI1), macromolecules (ENZYMES, PROTEINS, DD) and chemical compounds (BZR, COX2).', '1906.01277-1-112-1': 'We further consider a movie collaboration dataset (IMDB, see for a desription) and two synthetic data sets Synthie and Synthetic-new, created by [CITATION] and [CITATION], respectively.', '1906.01277-1-112-2': 'The BZR-MD and COX2-MD data sets do not have node attributes but contain the atomic distance between each connected atom as an edge weight.', '1906.01277-1-112-3': 'We do not consider distances between non-connected nodes and we equip the node with one-hot-encoding categorical attribtues representing the atom type, i.e. what is originally intended as a categorical node label.', '1906.01277-1-112-4': 'On IMDB-B, IMDB-BINARY was used with the node degree as a (semi-)continuous feature for each node .', '1906.01277-1-112-5': 'For all the other data sets, we use the off-the-shelf version provided by [CITATION].', '1906.01277-1-113-0': 'Results on synthetic data sets are provided in Table [REF].', '1906.01277-1-113-1': 'We decided not to include those in the main manuscript because of the very unstable and unreliable performances we obtained.', '1906.01277-1-113-2': 'In particular, for both data sets there is a very high variation among the different methods.', '1906.01277-1-113-3': 'Furthermore, we experimentally observed that even a slight modification of the node features (e.g. normalisation or scaling of the embedding scheme) resulted in large change in performances (up to [MATH]).', '1906.01277-1-113-4': 'Additionally, it has been previously reported that on Synthetic-new, a WL with degree treated as categorical node label outperforms the competitors, suggesting that the continuous attribtues are indeed not very informative.', '1906.01277-1-113-5': 'Therefore, we excluded these data sets from the main manuscript, as we concluded that they are unsuitable to provide a fair assessment of methods quality.', '1906.01277-1-114-0': '## Details on hyperparameter selection', '1906.01277-1-115-0': 'The following ranges are used for the hyperparameter selection: the parameter of the SVM [MATH] (for continuous attributes) and [MATH] (for categorical attributes); the WL number of iterations [MATH]; the [MATH] parameter of the WWL [MATH].', '1906.01277-1-115-1': 'For the RBF-WL and VH-C we use default [MATH] parameter for the Gaussian kernel, i.e [MATH], where [MATH] is the size of node attributes.', '1906.01277-1-115-2': 'For the GH kernel, we also fix the [MATH] parameter to [MATH], and for HGK we fix the number of iterations to [MATH] for each data set except for SYNTHETICnew, where we use [MATH] (these setups were suggested by the respective authors .', '1906.01277-1-115-3': 'Furthermore, since HGK is a randomised method, we compute each kernel matrix [MATH] times and average the results.', '1906.01277-1-115-4': 'When the dimensionality of the continuous attributes [MATH], these are normalised to ensure comparability among the different feature scales, in each data set but BZR and COX2, due to the meaning of the node attributes being location coordinates.', '1906.01277-1-116-0': '## Runtime comparison', '1906.01277-1-117-0': 'Overall, we note that WL and WL-OA scale linearly with the number of nodes, and will therefore be faster than our approach.', '1906.01277-1-117-1': 'Due to the differences in programming language implementations of the different methods, it is hard to provide an accurate runtime comparison.', '1906.01277-1-117-2': 'However, we empirically observe that the Wasserstein Graph Kernels are still competitive and a kernel matrix can be computed in a median time of [MATH]s, depending on the size and number of graphs (see Figure [REF]).', '1906.01277-1-117-3': 'For the continuous attributes, our approach has a runtime in the same order of magnitude as the GH.', '1906.01277-1-117-4': 'However, while our approach can benefit from significant speed-up (see discussion below and Section [REF]), the GH was shown to empirically scale quadratically with the number of nodes in the graphs .', '1906.01277-1-117-5': 'The HGK, on the other hand, is considerably slower given the number of iterations and multiple repetitions needed while taking into account the randomisation.', '1906.01277-1-118-0': 'To evaluate our approach with respect to the size of the graphs and recalling that computing the Wasserstein distance has complexity [MATH], we simulated a fixed number of graphs with a varying average number of nodes per graph.', '1906.01277-1-118-1': 'We generated random node embeddings for [MATH] where the number of nodes is taken from a normal distribution centered around the average number of nodes.', '1906.01277-1-118-2': 'We then computed the kernel matrix on each set of graphs to compare the runtime of regular Wasserstein with the Sinkhorn regularized optimization.', '1906.01277-1-118-3': 'As shown in Supplementary Figure [REF], the speedup starts to become beneficial at around 100 nodes per graph on average, which is larger than the average number of nodes in the benchmark data sets we used.', '1906.01277-1-119-0': 'In order to ensure that good performance is maintained when using the Sinkhorn approximation, we evaluate the obtained accuracy of the model.', '1906.01277-1-119-1': 'Recalling that the Sinkhorn method solves the following entropic regularization problem: [EQUATION] we further need to select [MATH].', '1906.01277-1-119-2': 'Therefore, on top of the cross-validation scheme described above, we further cross-validate over the regularization parameter values of [MATH] for the Enzymes data set and obtain an accuracy of [MATH], which remains above the current state-of-the-art.', '1906.01277-1-119-3': "The [MATH]'s selected most of the time are [MATH], [MATH] and [MATH]."}

{'1906.01277-2-0-0': 'Most graph kernels are an instance of the class of [MATH]-Convolution kernels, which measure the similarity of objects by comparing their substructures.', '1906.01277-2-0-1': 'Despite their empirical success, most graph kernels use a naive aggregation of the final set of substructures, usually a sum or average, thereby potentially discarding valuable information about the distribution of individual components.', '1906.01277-2-0-2': 'Furthermore, only a limited instance of these approaches can be extended to continuously attributed graphs.', '1906.01277-2-0-3': 'We propose a novel method that relies on the Wasserstein distance between the node feature vector distributions of two graphs, which allows finding subtler differences in data sets by considering graphs as high-dimensional objects rather than simple means.', '1906.01277-2-0-4': 'We further propose a Weisfeiler-Lehman-inspired embedding scheme for graphs with continuous node attributes and weighted edges, enhance it with the computed Wasserstein distance, and thereby improve the state-of-the-art prediction performance on several graph classification tasks.', '1906.01277-2-1-0': '# Introduction', '1906.01277-2-2-0': 'Graph-structured data have become ubiquitous across domains over the last decades, with examples ranging from social and sensor networks to chemo- and bioinformatics.', '1906.01277-2-2-1': 'Graph kernels have been highly successful in dealing with the complexity of graphs and have shown good predictive performance on a variety of classification problems .', '1906.01277-2-2-2': 'Most graph kernels rely on the [MATH]-Convolution framework , which decomposes structured objects into substructures to compute local similarities that are then aggregated.', '1906.01277-2-2-3': 'Although being successful in several applications, [MATH]-Convolution kernels on graphs have limitations: [(1)] the simplicity of the way in which the similarities between substructures are aggregated might limit their ability to capture complex characteristics of the graph; most proposed variants do not generalise to graphs with high-dimensional continuous node attributes, and extensions are far from being straightforward.', '1906.01277-2-3-0': 'Various solutions have been proposed to address point (1).', '1906.01277-2-3-1': 'For example, [CITATION] introduced kernels based on the optimal assignment of node labels for molecular graphs, although these kernels are not positive definite .', '1906.01277-2-3-2': 'Recently, another approach was proposed by [CITATION], which employs a Weisfeiler-Lehman based colour refinement scheme and uses an optimal assignment of the nodes to compute the kernel.', '1906.01277-2-3-3': 'However, this method cannot handle continuous node attributes, leaving point (2) as an open problem.', '1906.01277-2-4-0': 'To overcome both limitations, we propose a method that combines the most successful vectorial graph representations derived from the graph kernel literature with ideas from optimal transport theory, which have recently gained considerable attention.', '1906.01277-2-4-1': 'In particular, improvements of the computational strategies to efficiently obtain Wasserstein distances have led to many applications in machine learning that use it for various purposes, ranging from generative models to new loss functions .', '1906.01277-2-4-2': 'In graph applications, notions from optimal transport were used to tackle the graph alignment problem .', '1906.01277-2-4-3': 'In this paper, we provide the theoretical foundations of our method, define a new graph kernel formulation, and present successful experimental results.', '1906.01277-2-4-4': 'Specifically, our main contributions can be summarised as follows:', '1906.01277-2-5-0': 'We present the graph Wasserstein distance, a new distance between graphs based on their node feature representations, and we discuss how kernels can be derived from it.', '1906.01277-2-5-1': 'We introduce a Weisfeiler-Lehman-inspired embedding scheme that works for both categorically labelled and continuously attributed graphs, and we couple it with our graph Wasserstein distance; We outperform the state of the art for graph kernels on traditional graph classification benchmarks with continuous attributes.', '1906.01277-2-6-0': '# Background: graph kernels and Wasserstein distance', '1906.01277-2-7-0': 'In this section, we introduce the notation that will be used throughout the manuscript.', '1906.01277-2-7-1': 'Moreover, we provide the necessary background on graph kernel methods and the Wasserstein distance.', '1906.01277-2-8-0': '## Graph kernels', '1906.01277-2-9-0': 'Kernels are a class of similarity functions that present attractive properties to be used in learning algorithms .', '1906.01277-2-9-1': 'Let [MATH] be a set and [MATH] be a function associated with a Hilbert space [MATH], such that there exists a map [MATH] with [MATH].', '1906.01277-2-9-2': 'Then, [MATH] is a reproducing kernel Hilbert space (RKHS) and [MATH] is said to be a positive definite kernel.', '1906.01277-2-9-3': 'A positive definite kernel can be interpreted as a dot product in a high-dimensional space, thereby permitting its use in any learning algorithm that relies on dot products, such as support vector machines (SVMs), by virtue of the kernel trick .', '1906.01277-2-9-4': 'Because ensuring positive definiteness is not always feasible, many learning algorithms were recently proposed to extend SVMs to indefinite kernels .', '1906.01277-2-10-0': 'We define a graph as a tuple [MATH], where [MATH] and [MATH] denote the set of nodes and edges, respectively; we further assume that the edges are undirected.', '1906.01277-2-10-1': 'Moreover, we denote the cardinality of nodes and edges for [MATH] as [MATH] and [MATH].', '1906.01277-2-10-2': 'For a node [MATH], we write [MATH] and [MATH] to denote its first-order neighbourhood.', '1906.01277-2-10-3': 'We say that a graph is labelled if its nodes have categorical labels.', '1906.01277-2-10-4': 'A label on the nodes is a function [MATH] that assigns to each node [MATH] in [MATH] a value [MATH] from a finite label alphabet [MATH].', '1906.01277-2-10-5': 'Additionally, we say that a graph is attributed if for each node [MATH] there exists an associated vector [MATH].', '1906.01277-2-10-6': 'In this paper, [MATH] are the node attributes and [MATH] are the categorical node labels of node [MATH].', '1906.01277-2-10-7': 'In particular, the node attributes are high-dimensional continuous vectors, whereas the categorical node labels are assumed to be integer numbers (encoding either an ordered discrete value or a category).', '1906.01277-2-10-8': 'With the term "node labels", we will implicitly refer to categorical node labels.', '1906.01277-2-10-9': 'Finally, a graph can have weighted edges, and the function [MATH] defines the weight [MATH] of an edge [MATH].', '1906.01277-2-11-0': 'Kernels on graphs are generally defined using the [MATH]-Convolution framework by [CITATION].', '1906.01277-2-11-1': 'The main idea is to decompose graph [MATH] into substructures and to define a kernel value [MATH] as a combination of substructure similarities.', '1906.01277-2-11-2': 'A pioneer kernel on graphs was presented by , where node and edge attributes are exploited for label sequence generation using a random walk scheme.', '1906.01277-2-11-3': 'Successively, a more efficient approach based on shortest paths was proposed, which computes each kernel value [MATH] as a sum of the similarities between each shortest path in [MATH] and each shortest path in [MATH].', '1906.01277-2-11-4': 'Despite the practical success of [MATH]-Convolution kernels, they often rely on aggregation strategies that ignore valuable information, such as the distribution of the substructures.', '1906.01277-2-11-5': 'An example is the Weisfeiler-Lehman (WL) subtree kernel or one of its variants , which generates graph-level features by summing the contribution of the node representations.', '1906.01277-2-11-6': 'To avoid these simplifications, we want to use concepts from optimal transport theory, such as the Wasserstein distance, which can help to better capture the similarities between graphs.', '1906.01277-2-12-0': '## Wasserstein distance', '1906.01277-2-13-0': 'The Wasserstein distance is a distance function between probability distributions defined on a given metric space.', '1906.01277-2-13-1': 'Let [MATH] and [MATH] be two probability distributions on a metric space [MATH] equipped with a ground distance [MATH], such as the Euclidean distance.', '1906.01277-2-14-0': 'The [MATH]-Wasserstein distance for [MATH] is defined as [EQUATION] where [MATH] is the set of all transportation plans [MATH] over [MATH] with marginals [MATH] and [MATH] on the first and second factors, respectively.', '1906.01277-2-15-0': 'The Wasserstein distance satisfies the axioms of a metric, provided that [MATH] is a metric (see the monograph of [CITATION], chapter 6, for a proof).', '1906.01277-2-15-1': 'Throughout the paper, we will focus on the distance for [MATH] and we will refer to the [MATH]-Wasserstein distance when mentioning the Wasserstein distance, unless noted otherwise.', '1906.01277-2-16-0': 'The Wasserstein distance is linked to the optimal transport problem , where the aim is to find the most "inexpensive" way, in terms of the ground distance, to transport all the probability mass from distribution [MATH] to match distribution [MATH].', '1906.01277-2-16-1': 'An intuitive illustration can be made for the [MATH]-dimensional case, where the two probability distributions can be imagined as piles of dirt or sand.', '1906.01277-2-16-2': "The Wasserstein distance, sometimes also referred to as the earth mover's distance , can be interpreted as the minimum effort required to move the content of the first pile to reproduce the second pile.", '1906.01277-2-17-0': 'In this paper, we deal with finite sets of node embeddings and not with continuous probability distributions.', '1906.01277-2-17-1': 'Therefore, we can reformulate the Wasserstein distance as a sum rather than an integral, and use the matrix notation commonly encountered in the optimal transport literature to represent the transportation plan.', '1906.01277-2-17-2': 'Given two sets of vectors [MATH] and [MATH], we can equivalently define the Wasserstein distance between them as [EQUATION]', '1906.01277-2-17-3': 'Here, [MATH] is the distance matrix containing the distances [MATH] between each element [MATH] of [MATH] and [MATH] of [MATH], [MATH] is a transport matrix (or joint probability), and [MATH] is the Frobenius dot product.', '1906.01277-2-17-4': 'The transport matrix [MATH] contains the fractions that indicate how to transport the values from [MATH] to [MATH] with the minimal total transport effort.', '1906.01277-2-17-5': 'Because we assume that the total mass to be transported equals [MATH] and is evenly distributed across the elements of [MATH] and [MATH], the row and column values of [MATH] must sum up to [MATH] and [MATH], respectively.', '1906.01277-2-18-0': '# Wasserstein distance on graphs', '1906.01277-2-19-0': 'The unsatisfactory nature of the aggregation step of current [MATH]-Convolution graph kernels, which may mask important substructure differences by averaging, motivated us to have a finer distance measure between structures and their components.', '1906.01277-2-19-1': 'In parallel, recent advances in optimisation solutions for faster computation of the optimal transport problem inspired us to consider this framework for the problem of graph classification.', '1906.01277-2-19-2': 'Our method relies on the following steps: [(1)] transform each graph into a set of node embeddings, measure the Wasserstein distance between each pair of graphs, and compute a similarity matrix to be used in the learning algorithm.', '1906.01277-2-20-0': 'Figure [REF] illustrates the first two steps, and Algorithm [REF] summarises the whole procedure.', '1906.01277-2-20-1': 'We start by defining an embedding scheme and illustrate how we integrate embeddings in the Wasserstein distance.', '1906.01277-2-21-0': '[Graph Embedding Scheme]', '1906.01277-2-22-0': 'Given a graph [MATH], a graph embedding scheme [MATH] is a function that outputs a fixed-size vectorial representation for each node in the graph.', '1906.01277-2-22-1': 'For each [MATH], the [MATH]-th row of [MATH] is called the node embedding of [MATH].', '1906.01277-2-23-0': 'Note that Definition [REF] permits treating node labels, which are categorical attributes, as one-dimensional attributes with [MATH].', '1906.01277-2-24-0': '[Graph Wasserstein Distance]', '1906.01277-2-25-0': 'Given two graphs [MATH] and [MATH], a graph embedding scheme [MATH] and a ground distance [MATH], we define the Graph Wasserstein Distance (GWD) as [EQUATION]', '1906.01277-2-25-1': 'We will now propose a graph embedding scheme inspired by the WL kernel on categorically labeled graphs, extend it to continuously attributed graphs with weighted edges, and show how to integrate it with the GWD presented in Definition [REF].', '1906.01277-2-26-0': '## Generating node embeddings', '1906.01277-2-27-0': 'The Weisfeiler-Lehman scheme.', '1906.01277-2-27-1': 'The Weisfeiler-Lehman subtree kernel , designed for labelled non-attributed graphs, looks at similarities among subtree patterns, defined by a propagation scheme on the graphs that iteratively compares labels on the nodes and their neighbours.', '1906.01277-2-27-2': 'This is achieved by creating a sequence of ordered strings through the aggregation of the labels of a node and its neighbours; those strings are subsequently hashed to create updated compressed node labels.', '1906.01277-2-27-3': 'With increasing iterations of the algorithm, these labels represent increasingly larger neighbourhoods of each node, allowing to compare more extended substructures.', '1906.01277-2-28-0': 'Specifically, consider a graph [MATH], let [MATH] be the initial node label of [MATH] for each [MATH], and let [MATH] be the number of WL iterations.', '1906.01277-2-28-1': 'Then, we can define a recursive scheme to compute [MATH] for [MATH] by looking at the ordered set of neighbours labels [MATH] as [EQUATION]', '1906.01277-2-28-2': 'We call this procedure the WL labelling scheme.', '1906.01277-2-28-3': 'As in the original publication , we use perfect hashing for the [MATH] function, so nodes at iteration [MATH] will have the same label if and only if their label and those of their neighbours are identical at iteration [MATH].', '1906.01277-2-29-0': 'Extension to continuous attributes.', '1906.01277-2-29-1': 'For graphs with continuous attributes [MATH], we need to improve the WL refinement step, whose original definition prohibits handling the continuous case.', '1906.01277-2-29-2': 'The key idea is to create an explicit propagation scheme that leverages and updates the current node features by averaging over the neighbourhoods.', '1906.01277-2-29-3': 'Although similar approaches have been implicitly investigated for computing node-level kernel similarities , they rely on additional hashing steps for the continuous features.', '1906.01277-2-29-4': 'Moreover, we can easily account for edge weights by considering them in the average calculation of each neighbourhood.', '1906.01277-2-29-5': 'Suppose we have a continuous attribute [MATH] for each node [MATH].', '1906.01277-2-29-6': 'Then, we recursively define [EQUATION]', '1906.01277-2-29-7': 'When edge weights are not available, we set [MATH].', '1906.01277-2-29-8': 'We consider the weighted average of the neighbourhood attribute values instead of a sum and add the [MATH] factor because we want to ensure a similar scale of the features across iterations; in fact, we concatenate such features for building our proposed kernel (see Definition [REF] for more details) and observe better empirical results with similarly scaled features.', '1906.01277-2-29-9': 'Although this is not a test of graph isomorphism, this refinement step can be seen as an intuitive extension for continuous attributes of the one used by the WL subtree kernel on categorical node labels, a widely successful baseline.', '1906.01277-2-29-10': 'Moreover, it resembles the propagation scheme used in many graph neural networks, which have proven to be successful for node classification on large data sets .', '1906.01277-2-29-11': 'Finally, its ability to account for edge weights makes it applicable to all types of graphs without having to perform a hashing step .', '1906.01277-2-29-12': 'Further extensions of the refinement step to account for high-dimensional edge attributes are left for future work.', '1906.01277-2-29-13': 'A straightforward example would be to also apply the scheme on the dual graph (where each edge is represented as a node, and connectivity is established if two edges in the primal graph share the same node) to then combine the obtained kernel with the kernel obtained on primal graphs via appropriate weighting.', '1906.01277-2-30-0': 'Graph embedding scheme.', '1906.01277-2-30-1': 'Using the recursive procedure described above, we propose a WL-based graph embedding scheme that generates node embeddings from the node labels or attributes of the graphs.', '1906.01277-2-30-2': 'In the following, we use [MATH] to denote the dimensionality of the node attributes ([MATH] for the categorical labels).', '1906.01277-2-31-0': '[WL features]', '1906.01277-2-32-0': 'Let [MATH] and let [MATH] be the number of WL iterations.', '1906.01277-2-32-1': 'Then, for every [MATH], we define the WL features as [EQUATION] where [MATH] for categorically labelled graphs and [MATH] for continuously attributed graphs.', '1906.01277-2-32-2': 'We refer to [MATH] as the node features of graph [MATH] at iteration [MATH].', '1906.01277-2-32-3': 'Then, the node embeddings of graph [MATH] at iteration [MATH] are defined as [EQUATION]', '1906.01277-2-32-4': 'We observe that a graph can be both categorically labelled and continuously attributed, and one could extend the above scheme by jointly considering this information (for instance, by concatenating the node features).', '1906.01277-2-32-5': 'However, we will leave this scenario as an extension for future work; thereby, we avoid having to define an appropriate distance measure between categorical and continuous data, as this is a long-standing issue .', '1906.01277-2-33-0': '## Computing the Wasserstein distance', '1906.01277-2-34-0': 'Once the node embeddings are generated by the graph embedding scheme, we evaluate the pairwise Wasserstein distance between graphs.', '1906.01277-2-34-1': 'We start by computing the ground distances between each pair of nodes.', '1906.01277-2-34-2': 'For categorical node features, we use the normalised Hamming distance: [EQUATION]', '1906.01277-2-34-3': 'The Hamming distance can be pictured as the normalised sum of discrete metric [MATH] on each of the features.', '1906.01277-2-34-4': 'The Hamming distance equals [MATH] when two vectors have no features in common and [MATH] when the vectors are identical.', '1906.01277-2-34-5': 'We use the Hamming distance as, in this case, the Weisfeiler-Lehman features are indeed categorical, and values carry no meaning.', '1906.01277-2-34-6': 'For continuous node features, on the other hand, we employ the Euclidean distance: [EQUATION]', '1906.01277-2-34-7': 'Next, we substitute the ground distance into the equation of Definition [REF] and compute the Wasserstein distance using a network simplex method .', '1906.01277-2-35-0': 'Computational complexity.', '1906.01277-2-35-1': 'Naively, the computation of the Wasserstein Distance has a complexity of [MATH], with [MATH] being the cardinality of the indexed set of node embeddings, i.e., the number of nodes in the two graphs.', '1906.01277-2-35-2': 'Nevertheless, efficient speedup tricks can be employed.', '1906.01277-2-35-3': 'In particular, approximations relying on Sinkhorn regularisation have been proposed , some of which reduce the computational burden to near-linear time while preserving accuracy .', '1906.01277-2-35-4': 'Such speedup strategies become incredibly useful for larger data sets, i.e., graphs with thousands of nodes, and can be easily integrated into our method.', '1906.01277-2-35-5': 'See Appendix [REF] for a practical discussion.', '1906.01277-2-36-0': '[tb] Compute Wasserstein graph kernel', '1906.01277-2-37-0': 'Input: Two graphs [MATH], [MATH]; graph embedding scheme [MATH]; ground distance [MATH]; [MATH].', '1906.01277-2-37-1': 'Output: kernel value [MATH].', '1906.01277-2-38-0': '[MATH] Generate node embeddings [MATH] Compute the ground distance between each pair of nodes [MATH] Compute the Wasserstein distance', '1906.01277-2-39-0': '[MATH]', '1906.01277-2-40-0': '# From Wasserstein distance to kernels', '1906.01277-2-41-0': 'From the graph Wasserstein distance, one can construct a similarity measure to be used in a learning algorithm.', '1906.01277-2-41-1': 'In this section, we propose a new graph kernel, state some claims about its (in)definiteness, and elaborate on how to use it for classifying graphs with continuous and categorical node labels.', '1906.01277-2-42-0': '[Wasserstein Weisfeiler-Lehman]', '1906.01277-2-43-0': 'Given a set of graphs [MATH] and the GWD defined for each pair of graph on their WL embeddings, we define the Wasserstein Weisfeiler-Lehman (WWL) kernel as [EQUATION]', '1906.01277-2-43-1': 'This is an instance of a Laplacian kernel, which was shown to offer favourable conditions for positive definiteness in the case of non-Euclidean distances .', '1906.01277-2-43-2': 'Obtaining the WWL kernel concludes the procedure described in Algorithm [REF].', '1906.01277-2-43-3': 'In the remainder of this section, we distinguish between the categorical WWL kernel, obtained on graphs with categorical labels, and the continuous WWL kernel, obtained on continuously attributed graphs via the graph embedding schemes described in Section [REF].', '1906.01277-2-44-0': 'For Euclidean spaces, obtaining positive definite kernels from distance functions is a well-studied topic .', '1906.01277-2-44-1': 'However, the Wasserstein distance in its general form is not isometric, i.e., there is no metric-preserving mapping to an [MATH]-norm, as the metric space it induces strongly depends on the chosen ground distance .', '1906.01277-2-44-2': 'Therefore, despite being a metric, it is not necessarily possible to derive a positive definite kernel from the Wasserstein distance in its general formulation, because the classical approaches cannot be applied here.', '1906.01277-2-44-3': 'Nevertheless, as a consequence of using the Laplacian kernel , we can show that, in the setting of categorical node labels, the obtained kernel is positive definite.', '1906.01277-2-45-0': 'The categorical WWL kernel is positive definite for all [MATH].', '1906.01277-2-46-0': 'For a proof, see Sections [REF] and [REF] in the Appendix.', '1906.01277-2-46-1': 'By contrast, for the continuous case, establishing the definiteness of the obtained kernel remains an open problem.', '1906.01277-2-46-2': 'We refer the reader to Section [REF] in the supplementary materials for further discussions and conjectures.', '1906.01277-2-47-0': 'Therefore, to ensure the theoretical and practical correctness of our results in the continuous case, we employ recently developed methods for learning with indefinite kernels.', '1906.01277-2-47-1': 'Specifically, we use learning methods for Krein spaces, which have been specifically designed to work with indefinite kernels ; in general, kernels that are not positive definite induce reproducing kernel Krein spaces (RKKS).', '1906.01277-2-47-2': 'These spaces can be seen as a generalisation of reproducing kernel Hilbert spaces, with which they share similar mathematical properties, making them amenable to machine learning techniques.', '1906.01277-2-47-3': 'Recent algorithms are capable of solving learning problems in RKKS; their results indicate that there are clear benefits (in terms of classification performance, for example) of learning in such spaces.', '1906.01277-2-47-4': 'Therefore, when evaluating WWL, we will use a Krein SVM (KSVM, ) as a classifier for the case of continuous attributes.', '1906.01277-2-48-0': '# Experimental evaluation', '1906.01277-2-49-0': 'In this section, we analyse how the performance of WWL compares with state-of-the-art graph kernels.', '1906.01277-2-49-1': 'In particular, we empirically observe that WWL [(1)] is competitive with the best graph kernel for categorically labelled data, and outperforms all the state-of-the-art graph kernels for attributed graphs.', '1906.01277-2-50-0': '## Data sets', '1906.01277-2-51-0': 'We report results on real-world data sets from multiple sources and use either their continuous attributes or categorical labels for evaluation.', '1906.01277-2-51-1': 'In particular, MUTAG, PTC-MR, NCI1, and DD are equipped with categorical node labels only; ENZYMES and PROTEINS have both categorical labels and continuous attributes; IMDB-B, BZR, and COX2 only contain continuous attributes; finally, BZR-MD and COX2-MD have both continuous node attributes and edge weights.', '1906.01277-2-51-2': 'Further information on the data sets is available in Supplementary Table [REF].', '1906.01277-2-51-3': 'Additionally, we report results on synthetic data (Synthie and SYNTHETIC-new) in Appendix [REF].', '1906.01277-2-51-4': 'All the data sets have been downloaded from [CITATION].', '1906.01277-2-52-0': '## Experimental setup', '1906.01277-2-53-0': 'We compare WWL with state-of-the-art graph kernel methods from the literature and relevant baselines, which we trained ourselves on the same splits (see below).', '1906.01277-2-53-1': 'In particular, for the categorical case, we compare with WL and WL-OA as well as with the vertex (V) and edge (E) histograms.', '1906.01277-2-53-2': 'Because already showed that the WL-OA is superior to previous approaches, we do not include the whole set of kernels in our comparison.', '1906.01277-2-53-3': 'For the continuously attributed data sets, we compare with two instances of the hash graph kernel (HGK-SP; HGK-WL) and with the GraphHopper (GH) .', '1906.01277-2-53-4': 'For comparison, we additionally use a continuous vertex histogram (VH-C), which is defined as a radial basis function (RBF) kernel between the sum of the graph node embeddings.', '1906.01277-2-53-5': 'Furthermore, to highlight the benefits of using the Wasserstein distance in our method, we replace it with an RBF kernel.', '1906.01277-2-53-6': 'Specifically, given two graphs [MATH] and [MATH], with [MATH] and [MATH], we first compute the Gaussian kernel between each pair of the node embeddings obtained in the same fashion as for WWL; therefore, we obtain a kernel matrix between node embeddings [MATH].', '1906.01277-2-53-7': 'Next, we sum up the values [MATH] and set [MATH].', '1906.01277-2-53-8': 'This procedure is repeated for each pair of graphs to obtain the final graph kernel matrix.', '1906.01277-2-53-9': 'We refer to this baseline as RBF-WL.', '1906.01277-2-54-0': 'As a classifier, we use an SVM (or a KSVM in the case of WWL) and 10-fold cross-validation, selecting the parameters on the training set only.', '1906.01277-2-54-1': 'We repeat each cross-validation split 10 times and report the average accuracy.', '1906.01277-2-54-2': 'We employ the same split for each evaluated method, thereby guaranteeing a fully comparable setup among all evaluated methods.', '1906.01277-2-54-3': 'Please refer to Appendix [REF] for details on the hyperparameter selection.', '1906.01277-2-55-0': 'Implementation and computing infrastructure Available Python implementations can be used to compute the WL kernel and the Wasserstein distance .', '1906.01277-2-55-1': 'We leverage these resources and make our code publicly available.', '1906.01277-2-55-2': 'We use the original implementations provided by the respective authors to compute the WL-OA, HGK, and GH methods.', '1906.01277-2-55-3': 'All our analyses were performed on a shared server running Ubuntu 14.04.5 LTS, with 4 CPUs (Intel Xeon E7-4860 v2 @ 2.60GHz) each with 12 cores and 24 threads, and 512 GB of RAM.', '1906.01277-2-56-0': '## Results and discussion', '1906.01277-2-57-0': 'The results are evaluated by classification accuracy and summarised in Table [REF] and Table [REF] for the categorical labels and continuous attributes, respectively.', '1906.01277-2-58-0': '### Categorical labels', '1906.01277-2-59-0': 'On the categorical data sets, WWL is comparable to the WL-OA kernel; however, it improves over the classical WL.', '1906.01277-2-59-1': 'In particular, WWL largely improves over WL-OA in PTC-MR and is slightly better on DD, whereas WL-OA is better on NCI1 and PROTEINS.', '1906.01277-2-60-0': 'Unsurprisingly, our approach is comparable to the WL-OA, whose main idea is to solve the optimal assignment problem by defining Dirac kernels on histograms of node labels, using multiple iterations of WL.', '1906.01277-2-60-1': 'This formulation is similar to the one we provide for categorical data, but it relies on the optimal assignment rather than the optimal transport; therefore, it requires one-to-one mappings instead of continuous transport maps.', '1906.01277-2-60-2': 'Besides, we solve the optimal transport problem on the concatenated embeddings, hereby jointly exploiting representations at multiple WL iterations.', '1906.01277-2-60-3': 'Contrarily, the WL-OA performs an optimal assignment at each iteration of WL and only combines them in the second stage.', '1906.01277-2-60-4': 'However, the key advantage of WWL over WL-OA is its capacity to account for continuous attributes.', '1906.01277-2-61-0': '### Continuous attributes', '1906.01277-2-62-0': 'In this setting, WWL significantly outperforms the other methods on [MATH] out of [MATH] data sets, is better on another one, and is on a par on the remaining [MATH].', '1906.01277-2-62-1': 'We further compute the average rank of each method in the continuous setting, with WWL scoring as first.', '1906.01277-2-62-2': 'The ranks calculated from Table [REF] are WWL = [MATH], HGK-WL = [MATH], RBF-WL = [MATH], HGK-SP = [MATH], and VH-C = [MATH].', '1906.01277-2-62-3': 'This is a remarkable improvement over the current state of the art, and it indeed establishes a new one.', '1906.01277-2-62-4': 'When looking at the average rank of the method, WWL always scores first.', '1906.01277-2-62-5': 'Therefore, we raise the bar in kernel graph classification for attributed graphs.', '1906.01277-2-62-6': 'As mentioned in Section [REF], the kernel obtained from continuous attributes is not necessarily positive definite.', '1906.01277-2-62-7': 'However, we empirically observe the kernel matrices to be positive definite (up to a numerical error), further supporting our theoretical considerations (see Appendix [REF]).', '1906.01277-2-62-8': 'In practice, the difference between the results obtained from classical SVMs in RKHS and the results obtained with the KSVM approach is negligible.', '1906.01277-2-63-0': 'Comparison with hash graph kernels The hash graph kernel (HGK) approach is somewhat related to our propagation scheme.', '1906.01277-2-63-1': 'By using multiple hashing functions, the HGK method is capable of extending certain existing graph kernels to the continuous setting.', '1906.01277-2-63-2': 'This helps to avoid the limitations of perfect hashing, which cannot express small differences in continuous attributes.', '1906.01277-2-63-3': 'A drawback of the random hashing performed by HGK is that it requires additional parameters and introduces a stochastic element to the kernel matrix computation.', '1906.01277-2-63-4': 'By contrast, our propagation scheme is fully continuous and uses the Wasserstein distance to capture small differences in distributions of continuous node attributes.', '1906.01277-2-63-5': 'Moreover, the observed performance gap suggests that an entirely continuous representation of the graphs provides clear benefits over the hashing.', '1906.01277-2-64-0': '# Conclusion', '1906.01277-2-65-0': 'In this paper, we present a new family of graph kernels, the Wasserstein Weisfeiler-Lehman (WWL) graph kernels.', '1906.01277-2-65-1': 'Our experiments show that WWL graph kernels outperform the state of the art for graph classification in the scenario of continuous node attributes, while matching the state of the art in the categorical setting.', '1906.01277-2-65-2': 'As a line of research for future work, we see great potential in the runtime improvement, thus, allowing applications of our method on regimes and data sets with larger graphs.', '1906.01277-2-65-3': 'In fact, preliminary experiments (see Section [REF] as well as Figure [REF] in the Appendix) already confirm the benefit of Sinkhorn regularisation when the average number of nodes in the graph increases.', '1906.01277-2-65-4': 'In parallel, it would be beneficial to derive approximations of the explicit feature representations in the RKKS, as this would also provide a consistent speedup.', '1906.01277-2-65-5': 'We further envision that major theoretical contributions could be made by defining theoretical bounds to ensure the positive definiteness of the WWL kernel in the case of continuous node attributes.', '1906.01277-2-65-6': 'Finally, optimisation objectives based on optimal transport could be employed to develop new algorithms based on graph neural networks .', '1906.01277-2-65-7': 'On a more general level, our proposed method provides a solid foundation of the use of optimal transport theory for kernel methods and highlights the large potential of optimal transport for machine learning.'}

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[]

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[]

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{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1906.01277

1010.3722

{'1010.3722-1-0-0': '# Motivation', '1010.3722-1-1-0': 'Non-Gaussianity that originates from the inflationary epoch leaves distinct signatures in present-day astrophysical measurements, and therefore provides a unique link to the early universe.', '1010.3722-1-1-1': 'Interactions of the field(s) sourcing the primordial curvature fluctuations introduce non-Gaussian imprints in the statistics of the temperature fluctuations in the Cosmic Microwave Background (CMB) and of the density fluctuations that collapse into bound objects.', '1010.3722-1-1-2': 'These effects give us many independent probes of the signals of inflationary physics at different redshifts, sensitive to a range of scales.', '1010.3722-1-1-3': 'While current measurements from the CMB confirm that the spectrum of primordial fluctuations is Gaussian to a remarkable part in [MATH], that bound is still four orders of magnitude away from testing primordial non-Gaussianity at the level predicted by slow-roll inflation and more than one order of magnitude above the level expected from non-linear post-inflationary processing of the fluctuations (see eg [CITATION] for a recent calculation).', '1010.3722-1-1-4': 'In addition, any deviation from the simplest single field slow-roll inflationary scenario, including multiple fields, derivative interactions, features in the potential, or non-Bunch-Davies initial conditions (see [CITATION] for a summary) can lead to observable non-Gaussianity at levels within current constraints but well above the slow-roll prediction.', '1010.3722-1-1-5': 'Upcoming data from the Planck satellite [CITATION] and a variety of galaxy surveys [CITATION] have the potential to achieve accuracy on non-Gaussianity at the level expected from non-linear evolution alone.', '1010.3722-1-1-6': 'For recent reviews, see [CITATION].', '1010.3722-1-2-0': 'Primordial non-Gaussianity is most effectively constrained by complementary measurements from the CMB and Large Scale Structure (LSS).', '1010.3722-1-2-1': 'The galaxy power spectrum and bispectrum and cluster number counts provide independent statistics with different systematics, sensitive to different qualitative features of the primordial non-Gausianity.', '1010.3722-1-2-2': 'Combined with the CMB, these observations will constrain a wide range of qualitative features of any observed non-Gaussianity (including amplitude, shape, sign and scale dependence) which can rule out large classes of inflationary models.', '1010.3722-1-3-0': 'Non-Gaussianity of the local type (with bispectrum maximum in the squeezed limit; [MATH]) has recently generated a good deal of interest in part because it will be especially well-constrained by LSS observations [CITATION].', '1010.3722-1-3-1': 'Even in the case of Gaussian fluctuations, the statistics of collapsed objects are different from those of the underlying density field, and the ratio of the clustering of the two is known as the halo bias [CITATION].', '1010.3722-1-3-2': 'The particular coupling of long and short wavelength modes in local non-Gaussianity introduces an additional, distinctive correction (proportional to [MATH]) in the power spectrum of collapsed objects which will allow strong observational constraints on local non-Gaussianity [CITATION].', '1010.3722-1-3-3': 'From a theoretical point of view, observably large primordial non-Gaussianity of this type requires at least two fields to contribute to the scenario - single field inflation alone can only generate a bispectrum of the local shape with an extremely small amplitude (of order the spectral index of the primordial power spectrum [CITATION]).', '1010.3722-1-3-4': 'The complete phenomenology of multi-field models is rich but we will show here that there are qualitative differences that are observationally distinguishable in the halo bias.', '1010.3722-1-3-5': 'We propose here a generalization of the local ansatz that is phenomenologically useful and captures the physics of many possible multi-field models.', '1010.3722-1-3-6': 'The generalized ansatz allows for different types of scale-dependent amplitude [MATH] along with the standard local shape.', '1010.3722-1-4-0': 'While signatures of primordial non-Gaussianity in LSS can often be predicted analytically, accurate comparisons of observables with theoretical predictions require the intermediate step of numerical simulations to validate or correct any the analytical relations.', '1010.3722-1-4-1': 'In this paper we build on previous work of Dalal et al. [CITATION] to numerically investigate the effect of scale-dependent, local non-Gaussian initial conditions.', '1010.3722-1-4-2': 'Interestingly, we find theoretically and numerically that the halo bias is sensitive to two different types of scale dependence that can constrain and distinguish between inflationary models.', '1010.3722-1-4-3': 'However unlike in the constant [MATH] case, the simplest theoretical prediction for the bias in models with scale-dependent non-Gaussianity does not fully agree with our numerical results.', '1010.3722-1-4-4': 'In this paper we will motivate our new non-Gaussian ansatz, present the analytic predictions from that model and the associated simulations.', '1010.3722-1-4-5': 'We will discuss a possible explanation for the discrepancy, which indicates that this problem constitutes an interesting test for our understanding of structure growth, although we postpone a detailed analysis for future work.', '1010.3722-1-5-0': 'The paper is organized as follows.', '1010.3722-1-5-1': 'In Section [REF] we discuss in more detail the motivation from inflationary theory.', '1010.3722-1-5-2': 'A self-contained and purely phenomenological discussion starts in Section [REF], where we use the peak-background split method to demonstrate the qualitative ways scale-dependent non-Gaussianity may be observable.', '1010.3722-1-5-3': 'We also present forecasts for differentiating the bispectra based on the analytic predictions.', '1010.3722-1-5-4': 'In Section [REF] we show the results of numerical simulations, which demonstrate a stronger signal than the analytic prediction, and so are encouraging for the observational prospects.', '1010.3722-1-5-5': 'We speculate on a possible explanation for the discrepancy between theory and simulation and then conclude in Section [REF].', '1010.3722-1-6-0': '# A bigger family for the local ansatz', '1010.3722-1-7-0': 'The original "local ansatz" to add non-Gaussianity to the primordial perturbations is [CITATION]: [EQUATION] where [MATH] is (minus) the gravitational potential, [MATH] is a Gaussian random field and the degree of non-Gaussianity is parameterized by (typically constant) [MATH].', '1010.3722-1-7-1': 'Here a positive [MATH] leads to a positive skewness in the density perturbations, and so more very large objects, in the same sign convention as WMAP [CITATION].', '1010.3722-1-8-0': 'In many scenarios the primary effect of the non-Gaussian correction appears as a non-zero bispectrum, defined as [EQUATION]', '1010.3722-1-8-1': 'For the local ansatz above the bispectrum is [EQUATION] where as usual we define [EQUATION]', '1010.3722-1-8-2': 'The subscript [MATH] distinguishes the Dirac delta function from the density perturbation.', '1010.3722-1-8-3': 'Assuming the spectral index, [MATH], has no significant [MATH]-dependence, the dimensionless power spectrum is given by [MATH].', '1010.3722-1-9-0': 'CMB data (WMAP7) already constrain [MATH] at 95 confidence ([CITATION]; see also [CITATION]) and could potentially achieve [MATH] few from the Planck satellite [CITATION].', '1010.3722-1-9-1': 'The best current constraint from LSS comes from the scale-dependent bias induced in the galaxy power spectrum, giving [MATH] at 95 CL ([CITATION]; see also [CITATION]).', '1010.3722-1-10-0': 'While the local ansatz is a useful phenomenological tool, it is only a first step toward modeling and constraining primordial non-Gaussianity motivated by the fundamental physics of inflation.', '1010.3722-1-10-1': 'Specifically, the local ansatz resembles the first term in a series that arises from the transfer of isocurvature to curvature fluctuations during or at the end of inflation.', '1010.3722-1-10-2': 'Such a transfer may be due to additional scalar fields during inflation (multi-field [CITATION]), or after (the curvaton scenario [CITATION]), or inhomogeneous reheating [CITATION].', '1010.3722-1-11-0': 'There are at least four possible sources of non-Gaussianity that generate bispectra that are largely well-captured by the local ansatz shape in the final curvature perturbations.', '1010.3722-1-11-1': 'First, a spectator field during inflation is not constrained to have a flat potential, so there may be intrinsic non-Gaussianity in that field that is not tightly constrained by the slow-roll conditions and which can be transferred to non-Gaussianity in the curvature.', '1010.3722-1-11-2': 'Second, in multi-field models non-linear evolution of curvature modes outside the horizon will generate non-Gaussianity in the observed curvature perturbations even if the field(s) themselves have no interactions other than gravitational [CITATION].', '1010.3722-1-11-3': 'Third, the conversion of curvaton isocurvature fluctuations to curvature after inflation depends on the energy density in the curvaton field, which is at least quadratic in the fluctuations and so introduces non-Gaussianity of the local type [CITATION].', '1010.3722-1-11-4': 'Finally, loop corrections may, in special cases, generate a scale-dependent non-Gaussianity [CITATION].', '1010.3722-1-12-0': 'Phenomenologically, we can write a more general ansatz for the bispectrum of (minus) the gravitational potential that is factorizable and symmetric in momentum by introducing two functions, [MATH] and [MATH]: [EQUATION]', '1010.3722-1-12-1': 'This ansatz captures a wide range of physically motivated and perturbatively controlled models, where the functions [MATH] are at most weak functions of scale.', '1010.3722-1-12-2': 'The notation refers to the physical origin of the two functions in inflationary scenarios: [MATH], with s for single field, is different from one if one of the fields has non-trivial self interactions or nonlinearly sources curvature perturbations; [MATH], with m for multi-field, is different from one when two or more fields both contribute to the power in curvature fluctuations.', '1010.3722-1-12-3': 'We will discuss several illustrative examples next.', '1010.3722-1-13-0': '## Two field inflation', '1010.3722-1-14-0': 'First, we consider a two field inflation scenario where running non-Gaussianity can be obtained, following [CITATION].', '1010.3722-1-14-1': 'In the [MATH] formalism [CITATION], one uses the dependence on the number of e-folds of inflation on the fields present to relate the curvature fluctuations to the scalar field fluctuations.', '1010.3722-1-14-2': 'Even if a field does not source the inflationary Hubble parameter [MATH], the point where inflation ends (and so the number of e-folds, [MATH]) can still depend on the position of the field.', '1010.3722-1-14-3': 'Then, we can express the curvature perturbation resulting from fluctuations of two fields [MATH] and [MATH] (up to second order) as [EQUATION] where for simplicity we have assumed one of the non-Gaussian terms ([MATH]) dominates the other ([MATH]) and [MATH].', '1010.3722-1-14-4': 'All quantities are evaluated at horizon crossing for the mode [MATH], and [MATH] is the derivative of the number of e-folds with respect to the field [MATH].', '1010.3722-1-14-5': 'To gain some intuition about the pattern of multiplications and convolutions in this expression, recall that in single field inflation the running of [MATH] contributes the term proportional to [MATH] to the spectral index.', '1010.3722-1-14-6': 'In addition, the form of the quadratic term generates the standard result that the bispectrum in the squeezed limit goes like the spectral index, [MATH], evaluated at horizon crossing of the short wavelength (large [MATH]) modes [CITATION] (although one must take into account pre-horizon crossing non-Gaussianity generated in the statistics of the field [MATH] to get the complete bispectrum correct).', '1010.3722-1-14-7': 'In the single field case, the amplitude of the non-Gaussianity in the curvature perturbation is small, but it does run if the spectral index runs and the dominant term has scale-dependent [MATH] evaluated at the scale of the short modes.', '1010.3722-1-15-0': 'In a multi-field scenario, the modes for each individual field have fluctuations of order [MATH], so that [EQUATION] where the asterisk is a reminder that [MATH] is evaluated at horizon-crossing for each wavenumber [MATH].', '1010.3722-1-15-1': 'Then the total curvature power spectrum can be written [EQUATION]', '1010.3722-1-15-2': 'The tree-level bispectrum, assuming [MATH], is [EQUATION] where [MATH] depends on non-trivial self (and gravitational) interaction terms of just the field [MATH], so we relabel it [MATH], with [MATH] for single field.', '1010.3722-1-15-3': 'The fraction of power in the [MATH] field is different from one only if both fields contribute significantly to the power in fluctuations so we have labeled this function with an [MATH] for multi-field.', '1010.3722-1-15-4': '(Otherwise, the bispectrum would reduce to the usual single-field expression, where [MATH] must be of order slow-roll - that is, the term quadratic in [MATH] would be most important, giving a bispectrum with the same form as the first line of Eq. (2.9) but with the coefficient of the power spectrum terms [MATH].)', '1010.3722-1-15-5': 'Assuming that the potential [MATH] is defined in the matter era, the precise relationship between the first and second lines above is [EQUATION]', '1010.3722-1-15-6': 'We note that quite generally all the coefficients [MATH], [MATH], [MATH], etc will be scale-dependent as the potentials for the fields are not exactly flat.', '1010.3722-1-15-7': 'In that sense, in any two-field scenario with large local non-Gaussianity, running of the amplitude through the function [MATH] is as natural as running of the spectral index.', '1010.3722-1-15-8': 'It may be somewhat fine-tuned to have two fields contribute to the amplitude of fluctuations (although this is hard to say in the absence of compelling particle physics realizations of inflation), but if they do it is likely natural for their potentials to be slightly different so that [MATH] is scale-dependent.', '1010.3722-1-15-9': 'We will parametrize this scale dependence by writing [EQUATION] where [MATH] is a (theoretically irrelevant) pivot point.', '1010.3722-1-16-0': '## Mixed curvaton/inflaton scenario', '1010.3722-1-17-0': "Now suppose the curvature perturbation comes partly from a Gaussian inflaton field ([MATH]) and partly from a 'curvaton' field ([MATH]) which was a spectator during inflation but contributes to the curvature perturbation afterwards [CITATION].", '1010.3722-1-17-1': 'The curvaton naturally has a contribution that is quadratic in real space since it contributes proportionally to the energy density in its fluctuations.', '1010.3722-1-17-2': 'Assuming a purely quadratic potential for the curvaton gives [EQUATION]', '1010.3722-1-17-3': 'The field fluctuations are still generated during inflation, with amplitude [MATH].', '1010.3722-1-17-4': 'The quadratic term means that the curvaton can contribute a local-type non-Gaussianity with [MATH] constant and determined by the proportion of energy in the curvaton at the time the field decays.', '1010.3722-1-18-0': 'Then we can write the total curvature field as a sum of contributions from the inflaton and curvaton: [EQUATION] where the factor of [MATH] enters since [MATH] is conventionally defined for the matter era potential (Eq. ([REF])) rather than the primordial curvature.', '1010.3722-1-19-0': "Assuming the fields don't couple, the bispectrum takes the familiar local form, but now in terms of [MATH]: [EQUATION]", '1010.3722-1-19-1': 'If we define the ratio of power contributed by the curvaton [EQUATION] we can write [EQUATION] where we have absorbed [MATH] into [MATH], that is [MATH].', '1010.3722-1-19-2': 'Again, we parametrize the function [MATH] as a simple power law, [MATH].', '1010.3722-1-20-0': '## Curvaton alone', '1010.3722-1-21-0': 'If non-Gaussianity comes from the curvaton alone, and a potential other than quadratic is considered, the bispectrum can again take the form [CITATION] [EQUATION] where [MATH] can be parametrized as a power law, at least for some potentials, and [MATH] apparently can have either sign.', '1010.3722-1-21-1': 'Inhomogeneous reheating may similarly generate this bispectrum [CITATION].', '1010.3722-1-22-0': '## Relation to the spectral index', '1010.3722-1-23-0': 'The running of [MATH] is an important physical feature of either type of two-field model: it is the evolution of the relative power in the two fields during inflation.', '1010.3722-1-23-1': 'Just as the spectral index measures the variation of the overall amplitude of fluctuations during inflation, for two-field models the bispectral index [MATH] can provide complementary information about how the contribution from each field evolves.', '1010.3722-1-23-2': 'For some curvaton scenarios there would be a link between running non-Gaussianity and large scale power asymmetry in the CMB [CITATION].', '1010.3722-1-24-0': 'There is a precise relationship between the spectral index and the bispectral index [MATH]: [EQUATION]', '1010.3722-1-24-1': 'Although the running of the bispectrum may have either sign, models with a red tilt for the field [MATH] are anecdotally more common and in that case we have [EQUATION]', '1010.3722-1-24-2': 'Notice that some of the literature (e.g. [CITATION]) defines [MATH] and so quotes [MATH].', '1010.3722-1-24-3': 'Here however, we will see that there are two different shifts in the non-Gaussian bias, each dependent on one factor of [MATH], so we define [MATH] as the running in that function.', '1010.3722-1-24-4': 'Finally, notice that the spectral index of the observed curvature perturbation depends on the running of both fields [MATH] and [MATH].', '1010.3722-1-24-5': 'If the running of the fields is large enough, it will change which field dominates the curvature statistics.', '1010.3722-1-25-0': 'Whatever the origin of the running in either function [MATH], it parametrizes the deviation from exactly quadratic potentials in either field and so is expected to be generically on the order of slow-roll parameters (and should be to avoid substantial corrections to this parameterization).', '1010.3722-1-25-1': 'We will use somewhat large values of the running to confirm the behavior of this type of model in our simulations, but the observational goal should be to measure [MATH].', '1010.3722-1-25-2': 'We discuss the potential of future surveys to reach this goal in Section 5.', '1010.3722-1-25-3': 'For the standard quadratic curvaton case, [MATH] seems more natural (that is, non-Gaussianity increases on small scales) while Byrnes et al. [CITATION] found [MATH] in a survey of multi-field hybrid inflation models.', '1010.3722-1-25-4': 'The sign can be understood if the non-Gaussianity is entirely due to non-linear evolution outside the horizon.', '1010.3722-1-25-5': 'Then large scale modes (which exit earlier) will to be more non-Gaussian than smal scale modes.', '1010.3722-1-26-0': '## A comment on naturalness and completeness', '1010.3722-1-27-0': 'Given that it is already difficult to convincingly explain one field with a very flat potential, we may reasonably ask if the scenarios we are considering are even less likely than the usual single-field inflation.', '1010.3722-1-27-1': 'It is very hard to answer that question without better fundamental models - it may be that where there is one inflaton-like field, there are naturally several (especially in higher dimensional models), or not.', '1010.3722-1-27-2': 'In inflation, there is a very compelling reason why the spectral index should be slightly different from one: old inflation models with exact de Sitter space are difficult to connect to the early, hot universe after inflation, while slow-roll with the Hubble parameter not exactly constant can have a natural end to inflation and a period of reheating.', '1010.3722-1-27-3': 'If the slow-roll scenario is right and if two fields are present and relevant during inflation, it may be reasonable to expect that they both have nearly flat and yet not identical potentials.', '1010.3722-1-27-4': 'If one accepts that local type non-Gaussianity is natural (or more compellingly, if it is observed), scale dependence is also natural.', '1010.3722-1-27-5': 'In the absence of a range of compelling high energy models, it is hard to quantify the likelihood of any of these scenarios.', '1010.3722-1-28-0': 'However, from a phenomenological point of view, considering a generalized local ansatz is helpful in two ways: first, it argues for a careful analysis of different mass tracers in any test for primordial local type non-Gaussianity and second, it provides a test of our understanding of structure formation.', '1010.3722-1-28-1': 'As we will see, the existing expressions for halo bias do not give particularly satisfactory agreement with our simulations.', '1010.3722-1-29-0': 'The generalized local ansatz above is useful to uncover new observational signatures, and it would be interesting to investigate to what extent it holds in more complicated models with more (and coupled) fields.', '1010.3722-1-29-1': 'However, even with this ansatz we are still far from considering all possible known effects.', '1010.3722-1-29-2': 'In most two field models, we expect higher order terms (like a [MATH] contribution) to be present in the expression for the non-Gaussian curvature.', '1010.3722-1-29-3': 'Those corrections are also important for comparison of observation with realistic models and have been considered in [CITATION].', '1010.3722-1-29-4': 'In addition, there are other possibilities that require something even more general than the symmetric, factorizable form.', '1010.3722-1-29-5': 'For example, non-gaussianity generated by loop effects can sometimes be large and goes like [CITATION] [EQUATION]', '1010.3722-1-29-6': 'In addition, the power-law behavior of our ansatz is a poor model for scenarios with a feature at some particular scale, such as [CITATION] (features in the potential) or [CITATION] where non-Gaussianity effectively switches on at some scale where a spectator field becomes light.', '1010.3722-1-30-0': 'Finally, we note that scale-dependent non-Gaussianity may also arise in other ways and for other bispectra, but most other examples in standard inflationary models are less divergent in the squeezed limit than the local shape is and so have a weaker signal in the bias.', '1010.3722-1-30-1': 'However, there is a small region of parameter space in ekpyrotic models that seems to generate bispectra with scale-dependent amplitudes consistent with current observations, and more divergent than the local ansatz [CITATION].', '1010.3722-1-31-0': '# Generalized local ansatz and large scale structure statistics', '1010.3722-1-32-0': 'Our ansatz for the factorizable, symmetric, scale-dependent local bispectrum is [EQUATION] where we parametrize the [MATH]-dependence of the amplitude as [EQUATION] with [MATH].', '1010.3722-1-32-1': 'Ideally, the pivot scale [MATH] can be chosen at a point where the amplitude and running of the shape are as close to uncorrelated as possible.', '1010.3722-1-32-2': 'We adopt [MATH] based on analysis for the CMB in [CITATION], although they used a slightly different ansatz for the scale dependence.', '1010.3722-1-32-3': 'Note, however, that the constraints on [MATH] will be entirely independent of the chosen value of [MATH].', '1010.3722-1-33-0': 'Although generically we might expect both functions [MATH] and [MATH] to be present, we can consider the two functions separately for simplicity.', '1010.3722-1-33-1': 'For that reason, we will compare the following two bispectra in what follows: [EQUATION]', '1010.3722-1-33-2': 'The first line applies to a model where only one field contributes to the curvature perturbations (and the inflationary background is sourced by something else).', '1010.3722-1-33-3': 'For example, it is generated by a curvaton model where the potential has terms other than the mass term, Eq. ([REF]), or from a simplified version of the [MATH] case from Section 2 (where we take [MATH], the ratio of power in the two fields, to be constant).', '1010.3722-1-33-4': 'Since the curvature perturbations come only from one field, we label the new function [MATH] with [MATH] for single field.', '1010.3722-1-33-5': 'Scale-dependence in this function indicates the presence of non-trivial self-interactions (eg, deviation of the curvaton potential from exactly quadratic).', '1010.3722-1-33-6': 'The second line corresponds to a scenario where (at least) two fields contribute to the curvature perturbations, but the relevant self-interactions are purely quadratic.', '1010.3722-1-33-7': 'For example, this is the mixed inflaton/curvaton model of Eq. ([REF]), where the curvaton has only a quadratic potential.', '1010.3722-1-33-8': 'The label [MATH] on [MATH] indicates that multiple fields contribute to the curvature perturbations.', '1010.3722-1-33-9': 'Scale-dependence in [MATH] shows how much the potentials for the fields differ.', '1010.3722-1-34-0': 'Notice that the first model in Eq. ([REF]) has a form that is equivalent to the bispectrum one would get from generalizing the local ansatz by [EQUATION]', '1010.3722-1-34-1': 'This also justifies the single-field label.', '1010.3722-1-34-2': 'Scale dependence of this type was studied recently in Ref. [CITATION].', '1010.3722-1-35-0': '## Scale-independent non-Gaussianity and bias', '1010.3722-1-36-0': 'In this section we will work out a prediction for the possible signatures of our generalized local ansatz in the halo power spectrum.', '1010.3722-1-36-1': 'We will only be concerned with the behavior of the power spectrum at very small [MATH], where the deviation from the Gaussian case is largest.', '1010.3722-1-36-2': 'The matter perturbations [MATH] at redshift [MATH] are related to the perturbations in the early matter era potential [MATH] by [EQUATION] so that [MATH].', '1010.3722-1-36-3': 'Here [MATH] is the matter density relative to critical, [MATH] is the Hubble constant, [MATH] is the linear growth function at redshift [MATH] normalized to one today, and the growth suppression factor is [MATH] in the best-fit [MATH]CDM model.', '1010.3722-1-36-4': 'We use the Eisenstein Hu [CITATION] fit to the transfer function [MATH].', '1010.3722-1-36-5': 'The variance of density fluctuations at redshift [MATH] smoothed on a scale [MATH] associated to mass [MATH] is [MATH], defined by [EQUATION] where the power spectrum of the primordial curvature perturbations is given by Eq. ([REF]) and [MATH] is the Fourier transform of the top-hat window function.', '1010.3722-1-36-6': 'The spatial smoothing scale [MATH] is related to the smoothing mass scale [MATH] via [EQUATION] where [MATH] is the matter energy density today.', '1010.3722-1-36-7': 'We write the combination [MATH].', '1010.3722-1-37-0': '## Peak-background split and halo bias', '1010.3722-1-38-0': 'Halos in N-body simulations are associated with peaks of the initial, linear density field [MATH], whose heights exceed some threshold [CITATION].', '1010.3722-1-38-1': 'The basic idea of the peak-background split [CITATION] is to compute the effect of long-wavelength background modes on the heights of small-scale peaks, and thereby estimate the large-scale clustering of halos.', '1010.3722-1-38-2': 'The procedure used in the peak-background split is to perturb a single background mode [MATH] and propagate the effect of this perturbation to the height of a peak near threshold.', '1010.3722-1-38-3': 'In Gaussian cosmologies, where there is no mode coupling, the heights of peaks are simply boosted by the density associated with the background mode, [MATH].', '1010.3722-1-38-4': 'With non-Gaussianity, however, there is mode coupling, so we have to compute how this background mode affects shorter-wavelength modes [MATH] as well.', '1010.3722-1-38-5': 'This clearly involves looking at the bispectrum in the squeezed limit [MATH] where [MATH].', '1010.3722-1-39-0': 'Using the argument above, we can predict the consequences of modifying the local ansatz to include some form of scale dependence; our discussion here is similar to that in [CITATION] and [CITATION].', '1010.3722-1-39-1': 'To get a feel for the effect of the scale-dependent non-Gaussianity on the bias, notice that we can rewrite the expression for the [MATH] type non-Gaussian field (the first line of Eq. ([REF])) in Fourier space as a sum of Gaussian modes [MATH] and a non-Gaussian piece [MATH] designed to recover the single-field model bispectrum: [EQUATION] where we have dropped the [MATH] term which is not important for this discussion.', '1010.3722-1-39-2': 'Now we can use Eq. ([REF]) to consider the effect of some long-wavelength perturbation [MATH].', '1010.3722-1-39-3': 'Considering [MATH] and [MATH] in that expression we see that [EQUATION] where in the latter equation, we have used the Poisson equation.', '1010.3722-1-39-4': 'We then sum over all the short wavelengths to get the total boost in peak height, which then translates into the halo excess and halo bias.', '1010.3722-1-39-5': 'Since short wavelength modes are essentially modes with length scales up to the scale of the object (there is a window function in the integration), we see that the presence of the scale-dependent function [MATH] implies that collapsed objects have a shift in bias with amplitude given by an effective [MATH] roughly on the scale of the object.', '1010.3722-1-39-6': 'That is, if [MATH] (which in this simple case is like [MATH] as in Eq. ([REF])) increases on small scales, smaller mass objects will have a larger non-Gaussian correction than very massive objects.', '1010.3722-1-40-0': 'For the curvaton type model, the second line of Eq. ([REF]), the effect above comes along with an additional [MATH]-dependence in the bias: [EQUATION]', '1010.3722-1-40-1': 'In this case, different mass tracers have a non-Gaussian shift in bias, [MATH], with an amplitude proportional to an "effective [MATH]" set by their mass (a consequence of [MATH]) and a scale dependence that goes as [MATH] (a consequence of [MATH]).', '1010.3722-1-41-0': '## Alternative derivation of scale-dependent effects', '1010.3722-1-42-0': 'The intuitive procedure above leads to essentially the same result as the procedure outlined by Grinstein and Wise [CITATION] and further developed by Matarrese et al. [CITATION].', '1010.3722-1-42-1': 'For a generic non-Gaussian distribution, Refs. [CITATION] found a way to express the two-point function of peaks in terms of a series expansion in correlation functions.', '1010.3722-1-42-2': 'Motivated by the results of [CITATION], this expression was recently applied to the case of local non-Gaussianity by Matarrese and Verde [CITATION].', '1010.3722-1-42-3': 'We can use the same starting point to consider the effects of our generalized local ansatz, and express the two-point function for halos of mass [MATH]) in terms of the [MATH]-point functions of the density field smoothed on the associated scale [MATH]) [EQUATION] where the dots represent higher order terms (both higher correlation functions and higher powers of the two- and three-point function).', '1010.3722-1-42-4': 'The collapse threshold [MATH] is contained in [MATH].', '1010.3722-1-42-5': 'Now we can use the Fourier transform of the halo auto-correlation, Eq. ([REF]), to compute the bias: [EQUATION] where [MATH] and the redshift independent integral is [EQUATION]', '1010.3722-1-42-6': 'The second expression is in the small [MATH] limit, so [MATH].', '1010.3722-1-42-7': 'In that limit, [MATH] is a constant that depends on the smoothing scale.', '1010.3722-1-42-8': 'When [MATH], this expression reduces to that of [CITATION] and is identically one in the small [MATH] limit.', '1010.3722-1-43-0': 'The Lagrangian halo bias [MATH] for halos of mass M is defined by [EQUATION] where in the second equality the [MATH] contribution [MATH] has been explicitly factored out.', '1010.3722-1-43-1': 'Then from Eq. ([REF]) the change in the bias relative to the Gaussian value is [EQUATION]', '1010.3722-1-43-2': 'In using the definition of [MATH] above to derive Eq. ([REF]) from Eq. ([REF]), we have expanded the square-root which is not always strictly valid.', '1010.3722-1-43-3': 'However, the resulting expression agrees with the peak background split (and we find that keeping the square-root yields worse agreement between theory and simulation).', '1010.3722-1-43-4': 'The results for the constant, small [MATH] part of the integral in Eq. ([REF]), [MATH] for the representative two-parameter cases are plotted as a function of mass (related to smoothing scale [MATH] following Eq. ([REF])) in Fig. [REF].', '1010.3722-1-43-5': 'The functions [MATH] are normalized to [MATH] so that the left panel shows an effective [MATH] generated by the scale-dependence for each scenario.', '1010.3722-1-43-6': 'The right panel compares the prediction for the non-Gaussian correction to the (Lagrangian) bias for the single-field and multi-field scenarios.', '1010.3722-1-43-7': '(This label indicates how many fields contribute to the curvature perturbations - the inflaton itself may be separate).', '1010.3722-1-44-0': '## Summary of analytic results', '1010.3722-1-45-0': 'We have arrived at the same prediction in both of the previous subsections: a general factorizable and symmetric extension of the local ansatz leads to two possible modifications of the non-Gaussian halo bias.', '1010.3722-1-45-1': 'First, different mass objects may see a different non-Gaussian correction that goes roughly like the amplitude of the non-Gaussianity on the scale of the object.', '1010.3722-1-45-2': 'Second, the power of [MATH] appearing in the scale-dependent correction can be shifted away from the standard [MATH] result when there are two fields contributing to the curvature power and their relative importance is a function of scale.', '1010.3722-1-45-3': 'Either the first effect alone or a combination of both may be found, depending on the origin of the scale dependence.', '1010.3722-1-45-4': 'The most general case has two parameters to characterize the running, and one to characterize the amplitude.', '1010.3722-1-45-5': 'From the point of view of measurements of bias, these combine into the mass-dependent coefficient of the scale-dependent term, [MATH], and the power of [MATH] that appears in the denominator.', '1010.3722-1-45-6': 'In other words, phenomenologically we have (in the small [MATH] limit) [EQUATION]', '1010.3722-1-45-7': 'More precisely (and in terms of the Gaussian Eulerian bias [MATH]) [EQUATION] where [EQUATION]', '1010.3722-1-45-8': 'There is some suggestion, both from simulations and from analytic considerations, that there is an additional factor multiplying the expression above for [MATH] even in the case of constant local non-Gaussianity.', '1010.3722-1-45-9': 'For example, Giannantonio and Porciani [CITATION] have suggested a multiplication by a factor [EQUATION] where [MATH] is a second order non-Gaussian correction that can be calculated from some choice of non-Gaussian mass function (and the subscript [MATH] indicates that it is scale-independent).', '1010.3722-1-45-10': 'Although it is reasonably well motivated, we do not find that such a correction alone substantially improves the fit to our simulations (especially for negative [MATH]), so we remain agnostic about the analytic form of any additional corrections and instead focus on the effects unique to the generalized local form, especially [MATH].', '1010.3722-1-45-11': 'From a practical perspective, the coefficient above can be fit from simulation and will not affect our conclusions.', '1010.3722-1-46-0': '## Forecasts based on the analytic prediction', '1010.3722-1-47-0': 'We now estimate the ability of future observations to detect slow-roll values of the running parameter.', '1010.3722-1-47-1': 'Here we present Fisher matrix forecasts based on the analytic predictions above.', '1010.3722-1-47-2': 'This analysis complements earlier forecasts made for both the scale-independent [CITATION], and also scale-dependent [CITATION] models of non-Gaussianity.', '1010.3722-1-48-0': 'We first consider the two simpler scenarios, given in Eq. ([REF]), that isolate the single field or multi-field effects and each have only two parameters.', '1010.3722-1-48-1': 'These can be obtained by setting either of the functions [MATH] or [MATH] to one in the general expressions above (and so either [MATH] or [MATH] is set to zero).', '1010.3722-1-48-2': 'The explicit expressions are [EQUATION]', '1010.3722-1-48-3': 'We have used only the small [MATH] portion of the integral expression from Eq. ([REF]) since this corresponds to the prediction from the peak-background split, and since the integrand at high wavenumbers (e.g. [MATH]) depends on the explicit form of the window function.', '1010.3722-1-48-4': 'We report constraints on the momentum dependent functions that contribute to the integral in [MATH]: [EQUATION] where the last two lines specialize to the simpler cases of considering only the single-field or multi-field effects.', '1010.3722-1-48-5': 'Conceptually, [MATH] and its Fourier-space analogue, [MATH] capture the high frequency scale-dependence of non-Gaussianity, [MATH].', '1010.3722-1-48-6': 'In addition to this, the bias has low-frequency scale-dependence [MATH] for [MATH].', '1010.3722-1-48-7': 'This is why our expressions for [MATH] and [MATH] contain only one power of [MATH], even though the bispectrum has two [MATH] terms.', '1010.3722-1-48-8': 'Fiducial values adopted were [MATH] and [MATH] while [MATH] as before.', '1010.3722-1-48-9': 'We will see in a moment that the true best-measured scale from the large-scale clustering of galaxies and clusters is somewhat smaller.', '1010.3722-1-49-0': 'Suppose that we have measurements of the power spectrum using objects (galaxies and clusters of galaxies) that have been separated in several mass bins.', '1010.3722-1-49-1': 'We assume that the covariance matrix of measured Fourier-space overdensities in a given redshift bin centered at [MATH] is given by [EQUATION] where the labels [MATH] and [MATH] refer to mass bins.', '1010.3722-1-49-2': 'This equation encodes how to combine observations from different mass bins, and also straightforwardly specifies dependence on the parameters of interest [MATH] and [MATH] via Eqs. ([REF]) and ([REF]).', '1010.3722-1-50-0': 'The Fisher matrix can now be evaluated in the FKP approximation [CITATION], where information is summed over the redshift bins and wavenumber shells.', '1010.3722-1-50-1': 'We have [EQUATION] where [MATH] and [MATH] are the solid angle and maximum redshift in the survey respectively, [MATH] is volume, commas denote derivatives with respect to the non-Gaussian parameters, and we have suppressed the dependencies of [MATH] on wavenumber and redshift.', '1010.3722-1-50-2': 'In practice we replace integrals with sums to evaluate this expressions.', '1010.3722-1-50-3': 'We neglect the effect of redshift uncertainties, but assume thick redshift bins with [MATH].', '1010.3722-1-51-0': 'For definiteness, we assume a dataset of the quality expected from the Dark Energy Survey (DES; [CITATION]), with [MATH] and covering 5000 square degrees; the total volume in this survey is about [MATH].', '1010.3722-1-51-1': 'We assume [MATH] and [MATH]; the latter ensures that all information safely comes from the linear regime.', '1010.3722-1-51-2': 'Finally, we choose the number density of sources above some mass to correspond exactly to the expectation from the Jenkins mass function [CITATION].', '1010.3722-1-51-3': 'Therefore, [MATH], where [MATH] and [MATH] and the boundaries of the [MATH]-th mass bin.', '1010.3722-1-51-4': 'The total number density of sources at [MATH] and above [MATH] is [MATH].', '1010.3722-1-51-5': 'We assume a large number of mass bins (forty) in [MATH], uniformly distributed in [MATH] from [MATH] to [MATH].', '1010.3722-1-52-0': 'First considering the single-field case, the error in [MATH] at any [MATH] is given by a simple propagation of errors [EQUATION] where [MATH] is the covariance matrix of the two non-Gaussian parameters that we consider.', '1010.3722-1-52-1': 'The errors in [MATH] are shown in the left panel of Figure [REF].', '1010.3722-1-53-0': 'We are also interested in finding the best-constrained wavenumber, [MATH].', '1010.3722-1-53-1': 'When [MATH], then the errors on the parameters [MATH] and [MATH] are uncorrelated .', '1010.3722-1-53-2': 'While this best-constrained wavenumber can obviously be read off from Fig. [REF], it can also be calculated analytically as [EQUATION] where [MATH] is the arbitrary pivot in Eq. ([REF]).', '1010.3722-1-53-3': "The way that [MATH] 'runs' with changing mass illustrates the point we made in Sec. [REF] that different mass halos probe scale-dependent NG on scales corresponding to those masses.", '1010.3722-1-54-0': 'We find that the best-constrained wavenumber of our survey, for the single-field model and assuming DES-quality data, is [MATH], and the corresponding parameter errors at [MATH] are [EQUATION]', '1010.3722-1-54-1': 'We also find that the error in [MATH] is largely insensitive to the fiducial value of [MATH], while the error in the spectral index [MATH] becomes larger for a smaller fiducial [MATH] (which is expected, since a larger fiducial non-Gaussianity increases the absolute change in [MATH] for a fixed change in [MATH]).', '1010.3722-1-55-0': 'We repeated the same exercise assuming data of the quality expected from the Large Synoptic Survey Telescope (LSST; [CITATION]), with [MATH] and covering 20,000 square degrees; the constraints became [EQUATION]', '1010.3722-1-55-1': 'Next we consider the multi-field model; see the right panel of Fig. [REF].', '1010.3722-1-55-2': 'As expected, the numerical constraints on the amplitude are comparable to the single-field case, however the constraints on the running improve, and the best-determined scale moves to a slightly lower [MATH]: [EQUATION]', '1010.3722-1-55-3': 'The colored contours in Fig. [REF] show the individual constraints from each of the six narrow mass bins uniformly distributed in [MATH] from [MATH] to [MATH] (this is for the DES survey scenario and the single-field inflaton model).', '1010.3722-1-55-4': 'The thick black curve in either panel shows the combined constraint.', '1010.3722-1-55-5': 'Note that the combined constraint contains the information from the individual bins and the correlations between them (corresponding to [MATH] in Eq. ([REF])).', '1010.3722-1-55-6': 'A particularly interesting feature of testing these models with primordial non-Gaussianity is that halos of different mass complement in producing the overall constraint.', '1010.3722-1-55-7': 'For example, inspection of Eqs. ([REF]) and ([REF]) shows that, with a single mass measurement, the normalization and slope of the single-field model, [MATH] and [MATH], are completely degenerate, and only [MATH] at a single [MATH] value is measured.', '1010.3722-1-55-8': 'By adding a wide range of masses, this degeneracy is broken.', '1010.3722-1-55-9': 'This is shown in the left panel of Fig. [REF], where narrow mass bins only constrain this function near a single [MATH] value.', '1010.3722-1-55-10': 'In contrast, the right panel shows that for the multi-field scenario the degeneracy is broken even with halos of a fixed mass, as expected from Eq. ([REF]).', '1010.3722-1-56-0': 'Similarly, Fig. [REF] contains more visual information on how the degeneracy is broken with multiple mass measurements.', '1010.3722-1-56-1': 'The left panel shows the constraint in the [MATH]-[MATH] plane for the DES survey, with lines showing degeneracy directions that each of the six individual mass bins suffers (these mass bins correspond to colored curves in Fig. [REF]).', '1010.3722-1-56-2': 'The right panel shows the constraint in the [MATH]-[MATH] plane, marginalized over the amplitude [MATH], for both DES and LSST surveys.', '1010.3722-1-57-0': 'Clearly, even the information from large-scale structure alone offers the possibility of distinguishing the origin of primordial non-Gaussianity by constraining both single-field and multi-field model parameters simultaneously, but the most interesting level to probe is running of order the spectral index, [MATH].', '1010.3722-1-57-1': 'It is not completely clear if we can reach that level, and there are several factors that could push the predictions above in either direction.', '1010.3722-1-57-2': 'First, the forecasts presented here are in some sense a best-case scenario, given that for simplicity we did not marginalize over the standard cosmological parameters, and we assumed no systematic errors in recovering the power spectra of halos, only taking into account the statistical uncertainties.', '1010.3722-1-57-3': 'In particular, measurements of mass of clusters of galaxies suffer from statistical and systematic errors that are currently at least at the [MATH] level per cluster.', '1010.3722-1-57-4': 'On the other hand, constraints presented might be reached in the near future because we do not expect that serious degeneracies exist between the non-Gaussian and other cosmological parameters [CITATION].', '1010.3722-1-57-5': 'One exception might be the Gaussian bias [MATH], which will need to be measured concurrently rather than predicted by theory as we assumed here.', '1010.3722-1-58-0': 'In addition, we caution that our simulation results do not agree quantitatively with the analytic prediction (see the next section).', '1010.3722-1-58-1': 'In fact, the simulations find a substantially stronger dependence on mass that what is predicted.', '1010.3722-1-58-2': 'If the simulations are proven correct, then the effects shown above will be easier to distinguish, which is very encouraging for distinguishing between different mechanisms that may lead to large local non-Gaussianity.', '1010.3722-1-59-0': 'Finally, the constraints above are from bias alone, but the Planck satellite will considerably improve constraints on [MATH] from the CMB through measurements of the bispectrum itself, so that any scale dependence may also be constrained [CITATION].', '1010.3722-1-59-1': 'Whereas current constraints come from [MATH], Planck constraints will extend to higher [MATH] and should overlap with constraints from the bias.', '1010.3722-1-59-2': 'Extant CMB analyses [CITATION] have used a different parametrization of possible scale-dependence, and so it would be interesting to repeat this with our ansatz.', '1010.3722-1-59-3': 'As pointed out by [CITATION] and explored in detail by [CITATION], even more information can be extracted from joint constraints on models where two fields contribute to the curvature fluctuation.', '1010.3722-1-59-4': 'Multiple observations can separate the inherent size of the non-Gaussian interaction in one field ([MATH]) from the rescaling by the fraction of power from the non-Gaussian field, [MATH], which combine as shown in Eq. ([REF]) to give the amplitude of the non-Gaussian term in the bias, [MATH], that we have constrained here.', '1010.3722-1-60-0': '# Simulation results', '1010.3722-1-61-0': 'To check the dependence of the effective [MATH] on the tracer mass, we generated initial conditions with a non-zero bispectrum of the form shown in the first line of Eq. ([REF]) (the [MATH] model) with scale-dependent amplitude as defined in Eq. ([REF]).', '1010.3722-1-61-1': 'This is equivalent to Eq. ([REF]), so our function [MATH] corresponds to a commonly used definition for [MATH] (see also Eq. ([REF])).', '1010.3722-1-61-2': 'This is the simplest possible model, but serves to check the predictions for how the effective coefficient of the non-Gaussian bias ("[MATH]") varies with the mass of the halo.', '1010.3722-1-61-3': 'We have also performed a small number of simulations using the bispectrum form in the second line of Eq. ([REF]), to verify that the bias has the expected scale-dependence.', '1010.3722-1-61-4': 'These simulations confirm that in such models [MATH] no longer simply scales as [MATH] on large scales, but has [MATH] dependence as well.', '1010.3722-1-61-5': 'However, at large enough values of the running to be distinguished by our simulations, second order effects are significant.', '1010.3722-1-61-6': 'For now we focus on the first scenario which is simpler and already uncovers a disagreement between the analytic predictions and the numerical results.', '1010.3722-1-62-0': 'To perform these non-Gaussian simulations, we first generated a realization of a Gaussian random field [MATH] with amplitude chosen so that [MATH] and with spectral index [MATH].', '1010.3722-1-62-1': 'Then we squared the field, Fourier transformed and multiplied by the scale-dependent [MATH] shown in the second line of Eq. ([REF]).', '1010.3722-1-62-2': 'Finally, we transformed this component back to real space and added it to the Gaussian piece.', '1010.3722-1-62-3': 'We evolved the resulting non-Gaussian field forward from the scale factor of [MATH] using a flat cosmology consistent to WMAP7 best fit values ([MATH], [MATH]).', '1010.3722-1-63-0': 'We ran 8 realizations each of Gaussian and non-Gaussian initial conditions, including [MATH] with [MATH], [MATH] with [MATH], and [MATH] with [MATH].', '1010.3722-1-63-1': 'All cases had pivot point [MATH].', '1010.3722-1-63-2': 'The box size was 2400 [MATH] Mpc, with [MATH] particles, giving a mass per particle of [MATH].', '1010.3722-1-63-3': 'Although our low-mass halos do not have many particles in these simulations, we used a few [MATH]Mpc simulations (with [MATH]) where these halos were well-resolved to verify our results.', '1010.3722-1-63-4': 'These simulations were performed on the SciNet machines, where each run took about 3.5 hours on 16 nodes.', '1010.3722-1-64-0': 'We find that the simulations with constant [MATH] are offset from the analytic expectation at small [MATH] by a factor that is nearly constant with mass and is less than one for both positive and negative [MATH].', '1010.3722-1-64-1': 'This is consistent with findings by other simulations, and the behavior of the offset was studied in detail by Pillepich et al. [CITATION] and Giannantonio and Porciani [CITATION].', '1010.3722-1-64-2': 'However, as discussed above, we will effectively fit this offset out and examine only the difference in behavior between our [MATH] constant simulations and those with running.', '1010.3722-1-65-0': 'From the simulations with scale-dependent non-Gaussianity, we find that different mass objects are indeed sensitive to an effective [MATH] that depends on the scale of the object and which increases (decreases) for positive (negative) running as the mass and size of the object decreases.', '1010.3722-1-65-1': 'Figure [REF] illustrates this effect: the non-Gaussian term in the bias for small mass objects has a smaller (larger) amplitude for positive (negative) running than for constant [MATH] (left hand panel).', '1010.3722-1-65-2': 'The curves converge for larger mass objects (right hand panel).', '1010.3722-1-65-3': 'The bias correction [MATH] is calculated from the difference between the matter-halo cross correlation in a Gaussian simulation and the non-Gaussian case built from the same Gaussian realization, then averaged over realizations.', '1010.3722-1-66-0': 'The qualitative effect we expected is present, but for some halos the magnitude of the effect is not well predicted by the analytic expressions from Section 3.', '1010.3722-1-66-1': 'Figure [REF] shows the deviation between simulation and prediction for [MATH], [MATH] (the same generic trend was seen in all parameter sets).', '1010.3722-1-66-2': 'We plot the ratio of the non-Gaussian correction with running to the non-Gaussian correction for constant [MATH]: [EQUATION]', '1010.3722-1-66-3': 'This is compared with the theoretical expectation calculated from the small [MATH] limit of Eq. ([REF]).', '1010.3722-1-66-4': 'The curves are plotted as a function of [MATH].', '1010.3722-1-66-5': 'As the figure demonstrates, the simulation results agree well with our analytic model in the high-mass limit [MATH], but towards lower masses (e.g. [MATH]) the simulations produce a stronger effect than Eq. ([REF]) would predict.', '1010.3722-1-66-6': 'Note that the discrepancy does not appear at a fixed mass, but rather at a fixed [MATH].', '1010.3722-1-66-7': 'This is illustrated in the right panel of Figure [REF], which is identical to the left panel of Figure except that now mass [MATH] is the abscissa.', '1010.3722-1-66-8': 'The figure shows that for fixed mass [MATH], the simulations agree with Eq. ([REF]) at high redshift, but begin to disagree at low redshift as [MATH] grows.', '1010.3722-1-67-0': 'One very plausible explanation for this discrepancy at low mass ([MATH]) is that the profiles of the peaks that produce halos begin to change as [MATH] increases.', '1010.3722-1-67-1': 'As we have argued, the non-Gaussian bias of halos of mass [MATH] is sensitive to the value of [MATH] at some effective [MATH].', '1010.3722-1-67-2': 'Implicit in this scaling is the assumption that the profiles of peaks that collapse into halos are similar at different masses, just rescaled in size.', '1010.3722-1-67-3': 'However, we know that this assumption is incorrect.', '1010.3722-1-67-4': 'Bardeen et al. [CITATION] argued from Gaussian statistics that as [MATH] increases, the peaks that collapse into halos generally become steeper.', '1010.3722-1-67-5': 'N-body simulations confirm the presence of this effect, but show that it is much stronger in magnitude than predicted by Bardeen et al., apparently due to environmental effects during halo formation [CITATION].', '1010.3722-1-67-6': 'Because peaks at high [MATH] are much steeper than rare peaks at low [MATH], they are sensitive to non-Gaussianity at higher wavenumbers, even at the same peak size [MATH].', '1010.3722-1-67-7': 'For scale-independent [MATH], this change in peak profile has no effect, but for nonzero [MATH] it can dramatically enhance the mass dependence of non-Gaussianity, as our simulations show.', '1010.3722-1-67-8': 'It remains to be seen whether the magnitude of the change in peak profile can account for the discrepancy between our simulations and Eq. ([REF]); this is work in progress.', '1010.3722-1-68-0': '# Conclusions', '1010.3722-1-69-0': 'In summary, we have introduced a generalization of the local ansatz, Eq. ([REF]), that is a symmetric, factorizable function of the momenta and includes scale-dependent non-Gaussianity.', '1010.3722-1-69-1': 'This more general expression is motivated by natural features of models that give an observably large amplitude for local type non-Gaussianity, and distinguishes between non-Gaussian curvature fluctuations generated by a single field and multiple fields.', '1010.3722-1-69-2': 'If only one field contributes to the curvature fluctuation (and is different from the inflaton so that the non-Gaussianity may be large), the scale-dependence of the non-Gaussianity characterizes the self-interactions of the field.', '1010.3722-1-69-3': 'If two fields contribute to the curvature fluctuations, scale-dependence indicates how the ratio of power in the fields changes, which is a function of how different the potentials are.', '1010.3722-1-69-4': 'If local non-Gaussianity is large enough to be observed, such scale-dependence is as natural as running of the power spectrum.', '1010.3722-1-70-0': 'Models with scale-dependent local non-Gaussianity can generate two signatures in the non-Gaussian contribution to the halo bias.', '1010.3722-1-70-1': 'First, the non-Gaussian term may be proportional to an effective [MATH] related to the amplitude of the bispectrum on the scale of the object so that different mass objects have a different amplitude correction.', '1010.3722-1-70-2': 'Second, the [MATH] behavior of the non-Gaussian bias can be modified to [MATH] (where [MATH]), and one should expect the first effect to accompany this one.', '1010.3722-1-71-0': 'We have used N-body simulations to verify that different mass objects do indeed have a non-Gaussian bias proportional to an effective [MATH] that varies with the mass of the object.', '1010.3722-1-71-1': 'It is interesting that the simulations show that scale dependance for with [MATH] can erase the scale dependent effect on the bias for some range of masses, highlighting the need for analysis using multiple tracers of different mass.', '1010.3722-1-71-2': 'However, the quantitative result for halos at large [MATH] is not well predicted by our analytic expressions.', '1010.3722-1-71-3': 'We have speculated that the origin of this discrepancy may be related to differences in the initial peak profiles of the halos, but leave a detailed investigation for a later work.', '1010.3722-1-72-0': 'Future surveys are sure to bring interesting results.', '1010.3722-1-72-1': 'Using the analytic predictions, we find that they may be able to distinguish the different pieces of our generalized local ansatz, and so different origins of local non-Gaussianity, especially if the running is somewhat large ([MATH]).', '1010.3722-1-72-2': 'However, the existing analytic expressions predict a weaker effect than we see in the simulations, and our forecasts only account for constraints from massive groups and clusters of galaxies, neglecting the (potentially) greater sensitivity to running possible when galaxy correlations are included as well.', '1010.3722-1-72-3': 'Our forecasts for future surveys should therefore be taken as a lower limit on the potential to observationally distinguish these features.'}

{'1010.3722-2-0-0': '# Motivation', '1010.3722-2-1-0': 'Non-Gaussianity that originates from the inflationary epoch leaves distinct signatures in present-day astrophysical measurements, and therefore provides a unique link to the early universe.', '1010.3722-2-1-1': 'Interactions of the field(s) sourcing the primordial curvature fluctuations introduce non-Gaussian imprints in the statistics of the temperature fluctuations in the Cosmic Microwave Background (CMB) and of the density fluctuations that collapse into bound objects.', '1010.3722-2-1-2': 'These effects give us many independent probes of the signals of inflationary physics at different redshifts, sensitive to a range of scales.', '1010.3722-2-1-3': 'While current measurements from the CMB confirm that the spectrum of primordial fluctuations is Gaussian to a remarkable part in [MATH], that bound is still four orders of magnitude away from testing primordial non-Gaussianity at the level predicted by slow-roll inflation and more than one order of magnitude above the level expected from non-linear post-inflationary processing of the fluctuations (see eg [CITATION] for a recent calculation).', '1010.3722-2-1-4': 'In addition, any deviation from the simplest single field slow-roll inflationary scenario, including multiple fields, derivative interactions, features in the potential, or non-Bunch-Davies initial conditions (see [CITATION] for a summary) can lead to observable non-Gaussianity at levels within current constraints but well above the slow-roll prediction.', '1010.3722-2-1-5': 'Upcoming data from the Planck satellite [CITATION] and a variety of galaxy surveys [CITATION] have the potential to achieve accuracy on non-Gaussianity at the level expected from non-linear evolution alone.', '1010.3722-2-1-6': 'For recent reviews, see [CITATION].', '1010.3722-2-2-0': 'Primordial non-Gaussianity is most effectively constrained by complementary measurements from the CMB and Large Scale Structure (LSS).', '1010.3722-2-2-1': 'The galaxy power spectrum and bispectrum and cluster number counts provide independent statistics with different systematics, sensitive to different qualitative features of the primordial non-Gausianity.', '1010.3722-2-2-2': 'Combined with the CMB, these observations will constrain a wide range of qualitative features of any observed non-Gaussianity (including amplitude, shape, sign and scale dependence) which can rule out large classes of inflationary models.', '1010.3722-2-3-0': 'Non-Gaussianity of the local type (with bispectrum maximum in the squeezed limit; [MATH]) has recently generated a good deal of interest in part because it will be especially well-constrained by LSS observations [CITATION].', '1010.3722-2-3-1': 'Even in the case of Gaussian fluctuations, the statistics of collapsed objects are different from those of the underlying density field, and the ratio of the clustering of the two is known as the halo bias [CITATION].', '1010.3722-2-3-2': 'The particular coupling of long and short wavelength modes in local non-Gaussianity introduces an additional, distinctive correction (proportional to [MATH]) in the power spectrum of collapsed objects which will allow strong observational constraints on local non-Gaussianity [CITATION].', '1010.3722-2-3-3': 'From a theoretical point of view, observably large primordial non-Gaussianity of this type requires at least two fields to contribute to the scenario - single field inflation alone can only generate a bispectrum of the local shape with an extremely small amplitude (of order the spectral index of the primordial power spectrum [CITATION]).', '1010.3722-2-3-4': 'The complete phenomenology of multi-field models is rich but we will show here that there are qualitative differences that are observationally distinguishable in the halo bias.', '1010.3722-2-3-5': 'We propose here a generalization of the local ansatz that is phenomenologically useful and captures the physics of many possible multi-field models.', '1010.3722-2-3-6': 'The generalized ansatz allows for different types of scale-dependent amplitude [MATH] along with the standard local shape.', '1010.3722-2-4-0': 'While signatures of primordial non-Gaussianity in LSS can often be predicted analytically, accurate comparisons of observables with theoretical predictions require the intermediate step of numerical simulations to validate or correct any the analytical relations.', '1010.3722-2-4-1': 'In this paper we build on previous work of Dalal et al. [CITATION] to numerically investigate the effect of scale-dependent, local non-Gaussian initial conditions.', '1010.3722-2-4-2': 'Interestingly, we find theoretically and numerically that the halo bias is sensitive to two different types of scale dependence that can constrain and distinguish between inflationary models.', '1010.3722-2-4-3': 'However unlike in the constant [MATH] case, the simplest theoretical prediction for the bias in models with scale-dependent non-Gaussianity does not fully agree with our numerical results.', '1010.3722-2-4-4': 'In this paper we will motivate our new non-Gaussian ansatz, present the analytic predictions from that model and the associated simulations.', '1010.3722-2-4-5': 'We will discuss a possible explanation for the discrepancy, which indicates that this problem constitutes an interesting test for our understanding of structure growth, although we postpone a detailed analysis for future work.', '1010.3722-2-5-0': 'The paper is organized as follows.', '1010.3722-2-5-1': 'In Section [REF] we discuss in more detail the motivation from inflationary theory.', '1010.3722-2-5-2': 'A self-contained and purely phenomenological discussion starts in Section [REF], where we use the peak-background split method to demonstrate the qualitative ways scale-dependent non-Gaussianity may be observable.', '1010.3722-2-5-3': 'We also present forecasts for differentiating the bispectra based on the analytic predictions.', '1010.3722-2-5-4': 'In Section [REF] we show the results of numerical simulations, which demonstrate a stronger signal than the analytic prediction, and so are encouraging for the observational prospects.', '1010.3722-2-5-5': 'We speculate on a possible explanation for the discrepancy between theory and simulation and then conclude in Section [REF].', '1010.3722-2-6-0': '# A bigger family for the local ansatz', '1010.3722-2-7-0': 'The original "local ansatz" to add non-Gaussianity to the primordial perturbations is [CITATION]: [EQUATION] where [MATH] is (minus) the gravitational potential, [MATH] is a Gaussian random field and the degree of non-Gaussianity is parameterized by (typically constant) [MATH].', '1010.3722-2-7-1': 'Here a positive [MATH] leads to a positive skewness in the density perturbations, and so more very large objects, in the same sign convention as WMAP [CITATION].', '1010.3722-2-8-0': 'In many scenarios the primary effect of the non-Gaussian correction appears as a non-zero bispectrum, defined as [EQUATION]', '1010.3722-2-8-1': 'For the local ansatz above the bispectrum is [EQUATION] where as usual we define [EQUATION]', '1010.3722-2-8-2': 'The subscript [MATH] distinguishes the Dirac delta function from the density perturbation.', '1010.3722-2-8-3': 'Assuming the spectral index, [MATH], has no significant [MATH]-dependence, the dimensionless power spectrum is given by [MATH].', '1010.3722-2-9-0': 'CMB data (WMAP7) already constrain [MATH] at 95 confidence ([CITATION]; see also [CITATION]) and could potentially achieve [MATH] few from the Planck satellite [CITATION].', '1010.3722-2-9-1': 'The best current constraint from LSS comes from the scale-dependent bias induced in the galaxy power spectrum, giving [MATH] at 95 CL ([CITATION]; see also [CITATION]).', '1010.3722-2-10-0': 'While the local ansatz is a useful phenomenological tool, it is only a first step toward modeling and constraining primordial non-Gaussianity motivated by the fundamental physics of inflation.', '1010.3722-2-10-1': 'Specifically, the local ansatz resembles the first term in a series that arises from the transfer of isocurvature to curvature fluctuations during or at the end of inflation.', '1010.3722-2-10-2': 'Such a transfer may be due to additional scalar fields during inflation (multi-field [CITATION]), or after (the curvaton scenario [CITATION]), or inhomogeneous reheating [CITATION].', '1010.3722-2-11-0': 'There are at least four possible sources of non-Gaussianity that generate bispectra that are largely well-captured by the local ansatz shape in the final curvature perturbations.', '1010.3722-2-11-1': 'First, a spectator field during inflation is not constrained to have a flat potential, so there may be intrinsic non-Gaussianity in that field that is not tightly constrained by the slow-roll conditions and which can be transferred to non-Gaussianity in the curvature.', '1010.3722-2-11-2': 'Second, in multi-field models non-linear evolution of curvature modes outside the horizon will generate non-Gaussianity in the observed curvature perturbations even if the field(s) themselves have no interactions other than gravitational [CITATION].', '1010.3722-2-11-3': 'Third, the conversion of curvaton isocurvature fluctuations to curvature after inflation depends on the energy density in the curvaton field, which is at least quadratic in the fluctuations and so introduces non-Gaussianity of the local type [CITATION].', '1010.3722-2-11-4': 'Finally, loop corrections may, in special cases, generate a scale-dependent non-Gaussianity [CITATION].', '1010.3722-2-12-0': 'Phenomenologically, we can write a more general ansatz for the bispectrum of (minus) the gravitational potential that is factorizable and symmetric in momentum by introducing two functions, [MATH] and [MATH]: [EQUATION]', '1010.3722-2-12-1': 'This ansatz captures a wide range of physically motivated and perturbatively controlled models, where the functions [MATH] are at most weak functions of scale.', '1010.3722-2-12-2': 'The notation refers to the physical origin of the two functions in inflationary scenarios: [MATH], with s for single field, is different from one if one of the fields has non-trivial self interactions or nonlinearly sources curvature perturbations; [MATH], with m for multi-field, is different from one when two or more fields both contribute to the power in curvature fluctuations.', '1010.3722-2-12-3': 'We will discuss several illustrative examples next.', '1010.3722-2-13-0': '## Two field inflation', '1010.3722-2-14-0': 'First, we consider a two field inflation scenario where running non-Gaussianity can be obtained, following [CITATION].', '1010.3722-2-14-1': 'In the [MATH] formalism [CITATION], one uses the dependence on the number of e-folds of inflation on the fields present to relate the curvature fluctuations to the scalar field fluctuations.', '1010.3722-2-14-2': 'Even if a field does not source the inflationary Hubble parameter [MATH], the point where inflation ends (and so the number of e-folds, [MATH]) can still depend on the position of the field.', '1010.3722-2-14-3': 'Then, we can express the curvature perturbation resulting from fluctuations of two fields [MATH] and [MATH] (up to second order) as [EQUATION] where for simplicity we have assumed one of the non-Gaussian terms ([MATH]) dominates the other ([MATH]) and [MATH].', '1010.3722-2-14-4': 'All quantities are evaluated at horizon crossing for the mode [MATH], and [MATH] is the derivative of the number of e-folds with respect to the field [MATH].', '1010.3722-2-14-5': 'To gain some intuition about the pattern of multiplications and convolutions in this expression, recall that in single field inflation the running of [MATH] contributes the term proportional to [MATH] to the spectral index.', '1010.3722-2-14-6': 'In addition, the form of the quadratic term generates the standard result that the bispectrum in the squeezed limit goes like the spectral index, [MATH], evaluated at horizon crossing of the short wavelength (large [MATH]) modes [CITATION] (although one must take into account pre-horizon crossing non-Gaussianity generated in the statistics of the field [MATH] to get the complete bispectrum correct).', '1010.3722-2-14-7': 'In the single field case, the amplitude of the non-Gaussianity in the curvature perturbation is small, but it does run if the spectral index runs and the dominant term has scale-dependent [MATH] evaluated at the scale of the short modes.', '1010.3722-2-15-0': 'In a multi-field scenario, the modes for each individual field have fluctuations of order [MATH], so that [EQUATION] where the asterisk is a reminder that [MATH] is evaluated at horizon-crossing for each wavenumber [MATH].', '1010.3722-2-15-1': 'Then the total curvature power spectrum can be written [EQUATION]', '1010.3722-2-15-2': 'The tree-level bispectrum, assuming [MATH], is [EQUATION] where [MATH] depends on non-trivial self (and gravitational) interaction terms of just the field [MATH], so we relabel it [MATH], with [MATH] for single field.', '1010.3722-2-15-3': 'The fraction of power in the [MATH] field is different from one only if both fields contribute significantly to the power in fluctuations so we have labeled this function with an [MATH] for multi-field.', '1010.3722-2-15-4': '(Otherwise, the bispectrum would reduce to the usual single-field expression, where [MATH] must be of order slow-roll - that is, the term quadratic in [MATH] would be most important, giving a bispectrum with the same form as the first line of Eq. (2.9) but with the coefficient of the power spectrum terms [MATH].)', '1010.3722-2-15-5': 'Assuming that the potential [MATH] is defined in the matter era, the precise relationship between the first and second lines above is [EQUATION]', '1010.3722-2-15-6': 'We note that quite generally all the coefficients [MATH], [MATH], [MATH], etc will be scale-dependent as the potentials for the fields are not exactly flat.', '1010.3722-2-15-7': 'In that sense, in any two-field scenario with large local non-Gaussianity, running of the amplitude through the function [MATH] is as natural as running of the spectral index.', '1010.3722-2-15-8': 'It may be somewhat fine-tuned to have two fields contribute to the amplitude of fluctuations (although this is hard to say in the absence of compelling particle physics realizations of inflation), but if they do it is likely natural for their potentials to be slightly different so that [MATH] is scale-dependent.', '1010.3722-2-15-9': 'We will parametrize this scale dependence by writing [EQUATION] where [MATH] is a (theoretically irrelevant) pivot point.', '1010.3722-2-16-0': '## Mixed curvaton/inflaton scenario', '1010.3722-2-17-0': "Now suppose the curvature perturbation comes partly from a Gaussian inflaton field ([MATH]) and partly from a 'curvaton' field ([MATH]) which was a spectator during inflation but contributes to the curvature perturbation afterwards [CITATION].", '1010.3722-2-17-1': 'The curvaton naturally has a contribution that is quadratic in real space since it contributes proportionally to the energy density in its fluctuations.', '1010.3722-2-17-2': 'Assuming a purely quadratic potential for the curvaton gives [EQUATION]', '1010.3722-2-17-3': 'The field fluctuations are still generated during inflation, with amplitude [MATH].', '1010.3722-2-17-4': 'The quadratic term means that the curvaton can contribute a local-type non-Gaussianity with [MATH] constant and determined by the proportion of energy in the curvaton at the time the field decays.', '1010.3722-2-18-0': 'Then we can write the total curvature field as a sum of contributions from the inflaton and curvaton: [EQUATION] where the factor of [MATH] enters since [MATH] is conventionally defined for the matter era potential (Eq. ([REF])) rather than the primordial curvature.', '1010.3722-2-19-0': "Assuming the fields don't couple, the bispectrum takes the familiar local form, but now in terms of [MATH]: [EQUATION]", '1010.3722-2-19-1': 'If we define the ratio of power contributed by the curvaton [EQUATION] we can write [EQUATION] where we have absorbed [MATH] into [MATH], that is [MATH].', '1010.3722-2-19-2': 'Again, we parametrize the function [MATH] as a simple power law, [MATH].', '1010.3722-2-20-0': '## Curvaton alone', '1010.3722-2-21-0': 'If non-Gaussianity comes from the curvaton alone, and a potential other than quadratic is considered, the bispectrum can again take the form [CITATION] [EQUATION] where [MATH] can be parametrized as a power law, at least for some potentials, and [MATH] apparently can have either sign.', '1010.3722-2-21-1': 'Inhomogeneous reheating may similarly generate this bispectrum [CITATION].', '1010.3722-2-22-0': '## Relation to the spectral index', '1010.3722-2-23-0': 'The running of [MATH] is an important physical feature of either type of two-field model: it is the evolution of the relative power in the two fields during inflation.', '1010.3722-2-23-1': 'Just as the spectral index measures the variation of the overall amplitude of fluctuations during inflation, for two-field models the bispectral index [MATH] can provide complementary information about how the contribution from each field evolves.', '1010.3722-2-23-2': 'For some curvaton scenarios there would be a link between running non-Gaussianity and large scale power asymmetry in the CMB [CITATION].', '1010.3722-2-24-0': 'There is a precise relationship between the spectral index and the bispectral index [MATH]: [EQUATION]', '1010.3722-2-24-1': 'Although the running of the bispectrum may have either sign, models with a red tilt for the field [MATH] are anecdotally more common and in that case we have [EQUATION]', '1010.3722-2-24-2': 'Notice that some of the literature (e.g. [CITATION]) defines [MATH] and so quotes [MATH].', '1010.3722-2-24-3': 'Here however, we will see that there are two different shifts in the non-Gaussian bias, each dependent on one factor of [MATH], so we define [MATH] as the running in that function.', '1010.3722-2-24-4': 'Finally, notice that the spectral index of the observed curvature perturbation depends on the running of both fields [MATH] and [MATH].', '1010.3722-2-24-5': 'If the running of the fields is large enough, it will change which field dominates the curvature statistics.', '1010.3722-2-25-0': 'Whatever the origin of the running in either function [MATH], it parametrizes the deviation from exactly quadratic potentials in either field and so is expected to be generically on the order of slow-roll parameters (and should be to avoid substantial corrections to this parameterization).', '1010.3722-2-25-1': 'We will use somewhat large values of the running to confirm the behavior of this type of model in our simulations, but the observational goal should be to measure [MATH].', '1010.3722-2-25-2': 'We discuss the potential of future surveys to reach this goal in Section 5.', '1010.3722-2-25-3': 'For the standard quadratic curvaton case, [MATH] seems more natural (that is, non-Gaussianity increases on small scales) while Byrnes et al. [CITATION] found [MATH] in a survey of multi-field hybrid inflation models.', '1010.3722-2-25-4': 'The sign can be understood if the non-Gaussianity is entirely due to non-linear evolution outside the horizon.', '1010.3722-2-25-5': 'Then large scale modes (which exit earlier) will to be more non-Gaussian than smal scale modes.', '1010.3722-2-26-0': '## A comment on naturalness and completeness', '1010.3722-2-27-0': 'Given that it is already difficult to convincingly explain one field with a very flat potential, we may reasonably ask if the scenarios we are considering are even less likely than the usual single-field inflation.', '1010.3722-2-27-1': 'It is very hard to answer that question without better fundamental models - it may be that where there is one inflaton-like field, there are naturally several (especially in higher dimensional models), or not.', '1010.3722-2-27-2': 'In inflation, there is a very compelling reason why the spectral index should be slightly different from one: old inflation models with exact de Sitter space are difficult to connect to the early, hot universe after inflation, while slow-roll with the Hubble parameter not exactly constant can have a natural end to inflation and a period of reheating.', '1010.3722-2-27-3': 'If the slow-roll scenario is right and if two fields are present and relevant during inflation, it may be reasonable to expect that they both have nearly flat and yet not identical potentials.', '1010.3722-2-27-4': 'If one accepts that local type non-Gaussianity is natural (or more compellingly, if it is observed), scale dependence is also natural.', '1010.3722-2-27-5': 'In the absence of a range of compelling high energy models, it is hard to quantify the likelihood of any of these scenarios.', '1010.3722-2-28-0': 'However, from a phenomenological point of view, considering a generalized local ansatz is helpful in two ways: first, it argues for a careful analysis of different mass tracers in any test for primordial local type non-Gaussianity and second, it provides a test of our understanding of structure formation.', '1010.3722-2-28-1': 'As we will see, the existing expressions for halo bias do not give particularly satisfactory agreement with our simulations.', '1010.3722-2-29-0': 'The generalized local ansatz above is useful to uncover new observational signatures, and it would be interesting to investigate to what extent it holds in more complicated models with more (and coupled) fields.', '1010.3722-2-29-1': 'However, even with this ansatz we are still far from considering all possible known effects.', '1010.3722-2-29-2': 'In most two field models, we expect higher order terms (like a [MATH] contribution) to be present in the expression for the non-Gaussian curvature.', '1010.3722-2-29-3': 'Those corrections are also important for comparison of observation with realistic models and have been considered in [CITATION].', '1010.3722-2-29-4': 'In addition, there are other possibilities that require something even more general than the symmetric, factorizable form.', '1010.3722-2-29-5': 'For example, non-gaussianity generated by loop effects can sometimes be large and goes like [CITATION] [EQUATION]', '1010.3722-2-29-6': 'In addition, the power-law behavior of our ansatz is a poor model for scenarios with a feature at some particular scale, such as [CITATION] (features in the potential) or [CITATION] where non-Gaussianity effectively switches on at some scale where a spectator field becomes light.', '1010.3722-2-30-0': 'Finally, we note that scale-dependent non-Gaussianity may also arise in other ways and for other bispectra, but most other examples in standard inflationary models are less divergent in the squeezed limit than the local shape is and so have a weaker signal in the bias.', '1010.3722-2-30-1': 'However, there is a small region of parameter space in ekpyrotic models that seems to generate bispectra with scale-dependent amplitudes consistent with current observations, and more divergent than the local ansatz [CITATION].', '1010.3722-2-31-0': '# Generalized local ansatz and large scale structure statistics', '1010.3722-2-32-0': 'Our ansatz for the factorizable, symmetric, scale-dependent local bispectrum is [EQUATION] where we parametrize the [MATH]-dependence of the amplitude as [EQUATION] with [MATH].', '1010.3722-2-32-1': 'Ideally, the pivot scale [MATH] can be chosen at a point where the amplitude and running of the shape are as close to uncorrelated as possible.', '1010.3722-2-32-2': 'We adopt [MATH] based on analysis for the CMB in [CITATION], although they used a slightly different ansatz for the scale dependence.', '1010.3722-2-32-3': 'Note, however, that the constraints on [MATH] will be entirely independent of the chosen value of [MATH].', '1010.3722-2-33-0': 'Although generically we might expect both functions [MATH] and [MATH] to be present, we can consider the two functions separately for simplicity.', '1010.3722-2-33-1': 'For that reason, we will compare the following two bispectra in what follows: [EQUATION]', '1010.3722-2-33-2': 'The first line applies to a model where only one field contributes to the curvature perturbations (and the inflationary background is sourced by something else).', '1010.3722-2-33-3': 'For example, it is generated by a curvaton model where the potential has terms other than the mass term, Eq. ([REF]), or from a simplified version of the [MATH] case from Section 2 (where we take [MATH], the ratio of power in the two fields, to be constant).', '1010.3722-2-33-4': 'Since the curvature perturbations come only from one field, we label the new function [MATH] with [MATH] for single field.', '1010.3722-2-33-5': 'Scale-dependence in this function indicates the presence of non-trivial self-interactions (eg, deviation of the curvaton potential from exactly quadratic).', '1010.3722-2-33-6': 'The second line corresponds to a scenario where (at least) two fields contribute to the curvature perturbations, but the relevant self-interactions are purely quadratic.', '1010.3722-2-33-7': 'For example, this is the mixed inflaton/curvaton model of Eq. ([REF]), where the curvaton has only a quadratic potential.', '1010.3722-2-33-8': 'The label [MATH] on [MATH] indicates that multiple fields contribute to the curvature perturbations.', '1010.3722-2-33-9': 'Scale-dependence in [MATH] shows how much the potentials for the fields differ.', '1010.3722-2-34-0': 'Notice that the first model in Eq. ([REF]) has a form that is equivalent to the bispectrum one would get from generalizing the local ansatz by [EQUATION]', '1010.3722-2-34-1': 'This also justifies the single-field label.', '1010.3722-2-34-2': 'Scale dependence of this type was studied recently in Ref. [CITATION].', '1010.3722-2-35-0': '## Scale-independent non-Gaussianity and bias', '1010.3722-2-36-0': 'In this section we will work out a prediction for the possible signatures of our generalized local ansatz in the halo power spectrum.', '1010.3722-2-36-1': 'We will only be concerned with the behavior of the power spectrum at very small [MATH], where the deviation from the Gaussian case is largest.', '1010.3722-2-36-2': 'The matter perturbations [MATH] at redshift [MATH] are related to the perturbations in the early matter era potential [MATH] by [EQUATION] so that [MATH].', '1010.3722-2-36-3': 'Here [MATH] is the matter density relative to critical, [MATH] is the Hubble constant, [MATH] is the linear growth function at redshift [MATH] normalized to one today, and the growth suppression factor is [MATH] in the best-fit [MATH]CDM model.', '1010.3722-2-36-4': 'We use the Eisenstein Hu [CITATION] fit to the transfer function [MATH].', '1010.3722-2-36-5': 'The variance of density fluctuations at redshift [MATH] smoothed on a scale [MATH] associated to mass [MATH] is [MATH], defined by [EQUATION] where the power spectrum of the primordial curvature perturbations is given by Eq. ([REF]) and [MATH] is the Fourier transform of the top-hat window function.', '1010.3722-2-36-6': 'The spatial smoothing scale [MATH] is related to the smoothing mass scale [MATH] via [EQUATION] where [MATH] is the matter energy density today.', '1010.3722-2-36-7': 'We write the combination [MATH].', '1010.3722-2-37-0': '## Peak-background split and halo bias', '1010.3722-2-38-0': 'Halos in N-body simulations are associated with peaks of the initial, linear density field [MATH], whose heights exceed some threshold [CITATION].', '1010.3722-2-38-1': 'The basic idea of the peak-background split [CITATION] is to compute the effect of long-wavelength background modes on the heights of small-scale peaks, and thereby estimate the large-scale clustering of halos.', '1010.3722-2-38-2': 'The procedure used in the peak-background split is to perturb a single background mode [MATH] and propagate the effect of this perturbation to the height of a peak near threshold.', '1010.3722-2-38-3': 'In Gaussian cosmologies, where there is no mode coupling, the heights of peaks are simply boosted by the density associated with the background mode, [MATH].', '1010.3722-2-38-4': 'With non-Gaussianity, however, there is mode coupling, so we have to compute how this background mode affects shorter-wavelength modes [MATH] as well.', '1010.3722-2-38-5': 'This clearly involves looking at the bispectrum in the squeezed limit [MATH] where [MATH].', '1010.3722-2-39-0': 'Using the argument above, we can predict the consequences of modifying the local ansatz to include some form of scale dependence; our discussion here is similar to that in [CITATION].', '1010.3722-2-39-1': 'To get a feel for the effect of the scale-dependent non-Gaussianity on the bias, notice that we can rewrite the expression for the [MATH] type non-Gaussian field (the first line of Eq. ([REF])) in Fourier space as a sum of Gaussian modes [MATH] and a non-Gaussian piece [MATH] designed to recover the single-field model bispectrum: [EQUATION] where we have dropped the [MATH] term which is not important for this discussion.', '1010.3722-2-39-2': 'Now we can use Eq. ([REF]) to consider the effect of some long-wavelength perturbation [MATH].', '1010.3722-2-39-3': 'Considering [MATH] and [MATH] in that expression we see that [EQUATION] where in the latter equation, we have used the Poisson equation.', '1010.3722-2-39-4': 'We then sum over all the short wavelengths to get the total boost in peak height, which then translates into the halo excess and halo bias.', '1010.3722-2-39-5': 'Since short wavelength modes are essentially modes with length scales up to the scale of the object (there is a window function in the integration), we see that the presence of the scale-dependent function [MATH] implies that collapsed objects have a shift in bias with amplitude given by an effective [MATH] roughly on the scale of the object.', '1010.3722-2-39-6': 'That is, if [MATH] (which in this simple case is like [MATH] as in Eq. ([REF])) increases on small scales, smaller mass objects will have a larger non-Gaussian correction than very massive objects.', '1010.3722-2-40-0': 'For the curvaton type model, the second line of Eq. ([REF]), the effect above comes along with an additional [MATH]-dependence in the bias: [EQUATION]', '1010.3722-2-40-1': 'In this case, different mass tracers have a non-Gaussian shift in bias, [MATH], with an amplitude proportional to an "effective [MATH]" set by their mass (a consequence of [MATH]) and a scale dependence that goes as [MATH] (a consequence of [MATH]).', '1010.3722-2-41-0': '## Alternative derivation of scale-dependent effects', '1010.3722-2-42-0': 'The intuitive procedure above leads to essentially the same result as the procedure outlined by Grinstein and Wise [CITATION] and further developed by Matarrese et al. [CITATION].', '1010.3722-2-42-1': 'For a generic non-Gaussian distribution, Refs. [CITATION] found a way to express the two-point function of peaks in terms of a series expansion in correlation functions.', '1010.3722-2-42-2': 'Motivated by the results of [CITATION], this expression was recently applied to the case of local non-Gaussianity by Matarrese and Verde [CITATION].', '1010.3722-2-42-3': 'We can use the same starting point to consider the effects of our generalized local ansatz, and express the two-point function for halos of mass [MATH]) in terms of the [MATH]-point functions of the density field smoothed on the associated scale [MATH]) [EQUATION] where the dots represent higher order terms (both higher correlation functions and higher powers of the two- and three-point function).', '1010.3722-2-42-4': 'The collapse threshold [MATH] is contained in [MATH].', '1010.3722-2-42-5': 'Now we can use the Fourier transform of the halo auto-correlation, Eq. ([REF]), to compute the bias: [EQUATION] where [MATH] and the redshift independent integral is [EQUATION]', '1010.3722-2-42-6': 'The second expression is in the small [MATH] limit, so [MATH].', '1010.3722-2-42-7': 'In that limit, [MATH] is a constant that depends on the smoothing scale.', '1010.3722-2-42-8': 'When [MATH], this expression reduces to that of [CITATION] and is identically one in the small [MATH] limit.', '1010.3722-2-43-0': 'The Lagrangian halo bias [MATH] for halos of mass M is defined by [EQUATION] where in the second equality the [MATH] contribution [MATH] has been explicitly factored out.', '1010.3722-2-43-1': 'Then from Eq. ([REF]) the change in the bias relative to the Gaussian value is [EQUATION]', '1010.3722-2-43-2': 'In using the definition of [MATH] above to derive Eq. ([REF]) from Eq. ([REF]), we have expanded the square-root which is not always strictly valid.', '1010.3722-2-43-3': 'However, the resulting expression agrees with the peak background split (and we find that keeping the square-root yields worse agreement between theory and simulation).', '1010.3722-2-43-4': 'The results for the constant, small [MATH] part of the integral in Eq. ([REF]), [MATH] for the representative two-parameter cases are plotted as a function of mass (related to smoothing scale [MATH] following Eq. ([REF])) in Fig. [REF].', '1010.3722-2-43-5': 'The functions [MATH] are normalized to [MATH] so that the left panel shows an effective [MATH] generated by the scale-dependence for each scenario.', '1010.3722-2-43-6': 'The right panel compares the prediction for the non-Gaussian correction to the (Lagrangian) bias for the single-field and multi-field scenarios.', '1010.3722-2-43-7': '(This label indicates how many fields contribute to the curvature perturbations - the inflaton itself may be separate).', '1010.3722-2-44-0': '## Summary of analytic results', '1010.3722-2-45-0': 'We have arrived at the same prediction in both of the previous subsections: a general factorizable and symmetric extension of the local ansatz leads to two possible modifications of the non-Gaussian halo bias.', '1010.3722-2-45-1': 'First, different mass objects may see a different non-Gaussian correction that goes roughly like the amplitude of the non-Gaussianity on the scale of the object.', '1010.3722-2-45-2': 'Second, the power of [MATH] appearing in the scale-dependent correction can be shifted away from the standard [MATH] result when there are two fields contributing to the curvature power and their relative importance is a function of scale.', '1010.3722-2-45-3': 'Either the first effect alone or a combination of both may be found, depending on the origin of the scale dependence.', '1010.3722-2-45-4': 'The most general case has two parameters to characterize the running, and one to characterize the amplitude.', '1010.3722-2-45-5': 'From the point of view of measurements of bias, these combine into the mass-dependent coefficient of the scale-dependent term, [MATH], and the power of [MATH] that appears in the denominator.', '1010.3722-2-45-6': 'In other words, phenomenologically we have (in the small [MATH] limit) [EQUATION]', '1010.3722-2-45-7': 'More precisely (and in terms of the Gaussian Eulerian bias [MATH]) [EQUATION] where [EQUATION]', '1010.3722-2-45-8': 'There is some suggestion, both from simulations and from analytic considerations, that there is an additional factor multiplying the expression above for [MATH] even in the case of constant local non-Gaussianity.', '1010.3722-2-45-9': 'For example, Giannantonio and Porciani [CITATION] have suggested a multiplication by a factor [EQUATION] where [MATH] is a second order non-Gaussian correction that can be calculated from some choice of non-Gaussian mass function (and the subscript [MATH] indicates that it is scale-independent).', '1010.3722-2-45-10': 'Although it is reasonably well motivated, we do not find that such a correction alone substantially improves the fit to our simulations (especially for negative [MATH]), so we remain agnostic about the analytic form of any additional corrections and instead focus on the effects unique to the generalized local form, especially [MATH].', '1010.3722-2-45-11': 'From a practical perspective, the coefficient above can be fit from simulation and will not affect our conclusions.', '1010.3722-2-46-0': '## Forecasts based on the analytic prediction', '1010.3722-2-47-0': 'We now estimate the ability of future observations to detect slow-roll values of the running parameter.', '1010.3722-2-47-1': 'Here we present Fisher matrix forecasts based on the analytic predictions above.', '1010.3722-2-47-2': 'This analysis complements earlier forecasts made for both the scale-independent [CITATION], and also scale-dependent [CITATION] models of non-Gaussianity.', '1010.3722-2-48-0': 'We first consider the two simpler scenarios, given in Eq. ([REF]), that isolate the single field or multi-field effects and each have only two parameters.', '1010.3722-2-48-1': 'These can be obtained by setting either of the functions [MATH] or [MATH] to one in the general expressions above (and so either [MATH] or [MATH] is set to zero).', '1010.3722-2-48-2': 'The explicit expressions are [EQUATION]', '1010.3722-2-48-3': 'We have used only the small [MATH] portion of the integral expression from Eq. ([REF]) since this corresponds to the prediction from the peak-background split, and since the integrand at high wavenumbers (e.g. [MATH]) depends on the explicit form of the window function.', '1010.3722-2-48-4': 'We report constraints on the momentum dependent functions that contribute to the integral in [MATH]: [EQUATION] where the last two lines specialize to the simpler cases of considering only the single-field or multi-field effects.', '1010.3722-2-48-5': 'Conceptually, [MATH] and its Fourier-space analogue, [MATH] capture the high frequency scale-dependence of non-Gaussianity, [MATH].', '1010.3722-2-48-6': 'In addition to this, the bias has low-frequency scale-dependence [MATH] for [MATH].', '1010.3722-2-48-7': 'This is why our expressions for [MATH] and [MATH] contain only one power of [MATH], even though the bispectrum has two [MATH] terms.', '1010.3722-2-48-8': 'Fiducial values adopted were [MATH] and [MATH] while [MATH] as before.', '1010.3722-2-48-9': 'We will see in a moment that the true best-measured scale from the large-scale clustering of galaxies and clusters is somewhat smaller.', '1010.3722-2-49-0': 'Suppose that we have measurements of the power spectrum using objects (galaxies and clusters of galaxies) that have been separated in several mass bins.', '1010.3722-2-49-1': 'We assume that the covariance matrix of measured Fourier-space overdensities in a given redshift bin centered at [MATH] is given by [EQUATION] where the labels [MATH] and [MATH] refer to mass bins.', '1010.3722-2-49-2': 'This equation encodes how to combine observations from different mass bins, and also straightforwardly specifies the dependence on the parameters of interest [MATH] and [MATH] via Eqs. ([REF]) and ([REF]).', '1010.3722-2-50-0': 'The Fisher matrix can now be evaluated in the FKP approximation [CITATION], where information is summed over the redshift bins and wavenumber shells.', '1010.3722-2-50-1': 'We have [EQUATION] where [MATH] and [MATH] are the solid angle and maximum redshift in the survey respectively, [MATH] is volume, commas denote derivatives with respect to the non-Gaussian parameters, and we have suppressed the dependencies of [MATH] on wavenumber and redshift.', '1010.3722-2-50-2': 'In practice we replace integrals with sums to evaluate this expressions.', '1010.3722-2-50-3': 'We neglect the effect of redshift uncertainties, but assume thick redshift bins with [MATH].', '1010.3722-2-51-0': 'For definiteness, we assume a dataset of the quality expected from the Dark Energy Survey (DES; [CITATION]), with [MATH] and covering 5000 square degrees; the total volume in this survey is about [MATH].', '1010.3722-2-51-1': 'We assume [MATH] and [MATH]; the latter ensures that all information safely comes from the linear regime.', '1010.3722-2-51-2': 'Finally, we choose the number density of sources above some mass to correspond exactly to the expectation from the Jenkins mass function [CITATION].', '1010.3722-2-51-3': 'Therefore, [MATH], where [MATH] and [MATH] and the boundaries of the [MATH]-th mass bin.', '1010.3722-2-51-4': 'The total number density of sources at [MATH] and above [MATH] is [MATH].', '1010.3722-2-51-5': 'We assume a large number of mass bins (forty) in [MATH], uniformly distributed in [MATH] from [MATH] to [MATH].', '1010.3722-2-52-0': 'First considering the single-field case, the error in [MATH] at any [MATH] is given by a simple propagation of errors [EQUATION] where [MATH] is the covariance matrix of the two non-Gaussian parameters that we consider.', '1010.3722-2-52-1': 'The errors in [MATH] are shown in the left panel of Figure [REF].', '1010.3722-2-53-0': 'We are also interested in finding the best-constrained wavenumber, [MATH].', '1010.3722-2-53-1': 'When [MATH], then the errors on the parameters [MATH] and [MATH] are uncorrelated .', '1010.3722-2-53-2': 'While this best-constrained wavenumber can obviously be read off from Fig. [REF], it can also be calculated analytically as [EQUATION] where [MATH] is the arbitrary pivot in Eq. ([REF]).', '1010.3722-2-53-3': "The way that [MATH] 'runs' with changing mass illustrates the point we made in Sec. [REF] that different mass halos probe scale-dependent NG on scales corresponding to those masses.", '1010.3722-2-54-0': 'We find that the best-constrained wavenumber of our survey, for the single-field model and assuming DES-quality data, is [MATH], and the corresponding parameter errors at [MATH] are [EQUATION]', '1010.3722-2-54-1': 'We also find that the error in [MATH] is largely insensitive to the fiducial value of [MATH], while the error in the spectral index [MATH] becomes larger for a smaller fiducial [MATH] (which is expected, since a larger fiducial non-Gaussianity increases the absolute change in [MATH] for a fixed change in [MATH]).', '1010.3722-2-55-0': 'We repeated the same exercise assuming data of the quality expected from the Large Synoptic Survey Telescope (LSST; [CITATION]), with [MATH] and covering 20,000 square degrees; the constraints became [EQUATION]', '1010.3722-2-55-1': 'Next we consider the multi-field model; see the right panel of Fig. [REF].', '1010.3722-2-55-2': 'As expected, the numerical constraints on the amplitude are comparable to the single-field case, however the constraints on the running improve, and the best-determined scale moves to a slightly lower [MATH]: [EQUATION]', '1010.3722-2-55-3': 'The colored contours in Fig. [REF] show the individual constraints from each of the six narrow mass bins uniformly distributed in [MATH] from [MATH] to [MATH] (this is for the DES survey scenario and the single-field inflaton model).', '1010.3722-2-55-4': 'The thick black curve in either panel shows the combined constraint.', '1010.3722-2-55-5': 'Note that the combined constraint contains the information from the individual bins and the correlations between them (corresponding to [MATH] in Eq. ([REF])).', '1010.3722-2-55-6': 'A particularly interesting feature of testing these models with primordial non-Gaussianity is that halos of different mass complement in producing the overall constraint.', '1010.3722-2-55-7': 'For example, inspection of Eqs. ([REF]) and ([REF]) shows that, with a single mass measurement, the normalization and slope of the single-field model, [MATH] and [MATH], are completely degenerate, and only [MATH] at a single [MATH] value is measured.', '1010.3722-2-55-8': 'By adding a wide range of masses, this degeneracy is broken.', '1010.3722-2-55-9': 'This is shown in the left panel of Fig. [REF], where narrow mass bins only constrain this function near a single [MATH] value.', '1010.3722-2-55-10': 'In contrast, the right panel shows that for the multi-field scenario the degeneracy is broken even with halos of a fixed mass, as expected from Eq. ([REF]).', '1010.3722-2-56-0': 'Similarly, Fig. [REF] contains more visual information on how the degeneracy is broken with multiple mass measurements.', '1010.3722-2-56-1': 'The left panel shows the constraint in the [MATH]-[MATH] plane for the DES survey, with lines showing degeneracy directions that each of the six individual mass bins suffers (these mass bins correspond to colored curves in Fig. [REF]).', '1010.3722-2-56-2': 'The right panel shows the constraint in the [MATH]-[MATH] plane, marginalized over the amplitude [MATH], for both DES and LSST surveys.', '1010.3722-2-57-0': 'Clearly, even the information from large-scale structure alone offers the possibility of distinguishing the origin of primordial non-Gaussianity by constraining both single-field and multi-field model parameters simultaneously, but the most interesting level to probe is running of order the spectral index, [MATH].', '1010.3722-2-57-1': 'It is not completely clear if we can reach that level, and there are several factors that could push the predictions above in either direction.', '1010.3722-2-57-2': 'First, the forecasts presented here are in some sense a best-case scenario, given that for simplicity we did not marginalize over the standard cosmological parameters, and we assumed no systematic errors in recovering the power spectra of halos, only taking into account the statistical uncertainties.', '1010.3722-2-57-3': 'In particular, measurements of mass of clusters of galaxies suffer from statistical and systematic errors that are currently at least at the [MATH] level per cluster.', '1010.3722-2-57-4': 'On the other hand, constraints presented might be reached in the near future because we do not expect that serious degeneracies exist between the non-Gaussian and other cosmological parameters [CITATION].', '1010.3722-2-57-5': 'One exception might be the Gaussian bias [MATH], which will need to be measured concurrently rather than predicted by theory as we assumed here.', '1010.3722-2-58-0': 'In addition, we caution that our simulation results do not agree quantitatively with the analytic prediction (see the next section).', '1010.3722-2-58-1': 'In fact, the simulations find a substantially stronger dependence on mass that what is predicted.', '1010.3722-2-58-2': 'If the simulations are proven correct, then the effects shown above will be easier to distinguish, which is very encouraging for distinguishing between different mechanisms that may lead to large local non-Gaussianity.', '1010.3722-2-59-0': 'Finally, the constraints above are from bias alone, but the Planck satellite will considerably improve constraints on [MATH] from the CMB through measurements of the bispectrum itself, so that any scale dependence may also be constrained [CITATION].', '1010.3722-2-59-1': 'Whereas current constraints come from [MATH], Planck constraints will extend to higher [MATH] and should overlap with constraints from the bias.', '1010.3722-2-59-2': 'Extant CMB analyses [CITATION] have used a different parametrization of possible scale-dependence, and so it would be interesting to repeat this with our ansatz.', '1010.3722-2-59-3': 'As pointed out by [CITATION] and explored in detail by [CITATION], even more information can be extracted from joint constraints on models where two fields contribute to the curvature fluctuation.', '1010.3722-2-59-4': 'Multiple observations can separate the inherent size of the non-Gaussian interaction in one field ([MATH]) from the rescaling by the fraction of power from the non-Gaussian field, [MATH], which combine as shown in Eq. ([REF]) to give the amplitude of the non-Gaussian term in the bias, [MATH], that we have constrained here.', '1010.3722-2-60-0': '# Simulation results', '1010.3722-2-61-0': 'To check the dependence of the effective [MATH] on the tracer mass, we generated initial conditions with a non-zero bispectrum of the form shown in the first line of Eq. ([REF]) (the single field model) with scale-dependent amplitude as defined in Eq. ([REF]).', '1010.3722-2-61-1': 'This is equivalent to Eq. ([REF]), so our function [MATH] corresponds to a commonly used definition for [MATH] (see also Eq. ([REF])).', '1010.3722-2-61-2': 'This is the simplest possible model, but serves to check the predictions for how the effective coefficient of the non-Gaussian bias ("[MATH]") varies with the mass of the halo.', '1010.3722-2-61-3': 'We have also performed a small number of simulations using the bispectrum form in the second line of Eq. ([REF]), to verify that the bias has the expected scale-dependence.', '1010.3722-2-61-4': 'These simulations confirm that in such models [MATH] no longer simply scales as [MATH] on large scales, but has [MATH] dependence as well.', '1010.3722-2-61-5': 'However, at large enough values of the running to be distinguished by our simulations, second order effects are significant.', '1010.3722-2-61-6': 'For now we focus on the first scenario which is simpler and already uncovers a disagreement between the analytic predictions and the numerical results.', '1010.3722-2-62-0': 'To perform these non-Gaussian simulations, we first generated a realization of a Gaussian random field [MATH] with amplitude chosen so that [MATH] and with spectral index [MATH].', '1010.3722-2-62-1': 'Then we squared the field, Fourier transformed and multiplied by the scale-dependent [MATH] shown in the second line of Eq. ([REF]).', '1010.3722-2-62-2': 'Finally, we transformed this component back to real space and added it to the Gaussian piece.', '1010.3722-2-62-3': 'We evolved the resulting non-Gaussian field forward from the scale factor of [MATH] using a flat cosmology consistent to WMAP7 best fit values ([MATH], [MATH]).', '1010.3722-2-63-0': 'We ran 8 realizations each of Gaussian and non-Gaussian initial conditions, including [MATH] with [MATH], [MATH] with [MATH], and [MATH] with [MATH].', '1010.3722-2-63-1': 'All cases had pivot point [MATH].', '1010.3722-2-63-2': 'The box size was 2400 [MATH] Mpc, with [MATH] particles, giving a mass per particle of [MATH].', '1010.3722-2-63-3': 'Although our low-mass halos do not have many particles in these simulations, we used a few [MATH]Mpc simulations (with [MATH]) where these halos were well-resolved to verify our results.', '1010.3722-2-63-4': 'These simulations were performed on the SciNet machines, where each run took about 3.5 hours on 16 nodes.', '1010.3722-2-64-0': 'We find that the simulations with constant [MATH] are offset from the analytic expectation at small [MATH] by a factor that is nearly constant with mass and is less than one for both positive and negative [MATH].', '1010.3722-2-64-1': 'This is consistent with findings by other simulations, and the behavior of the offset was studied in detail by Pillepich et al. [CITATION] and Giannantonio and Porciani [CITATION].', '1010.3722-2-64-2': 'However, as discussed above, we will effectively fit this offset out and examine only the difference in behavior between our [MATH] constant simulations and those with running.', '1010.3722-2-65-0': 'From the simulations with scale-dependent non-Gaussianity, we find that different mass objects are indeed sensitive to an effective [MATH] that depends on the scale of the object and which increases (decreases) for positive (negative) running as the mass and size of the object decreases.', '1010.3722-2-65-1': 'Figure [REF] illustrates this effect: the non-Gaussian term in the bias for small mass objects has a smaller (larger) amplitude for positive (negative) running than for constant [MATH] (left hand panel).', '1010.3722-2-65-2': 'The curves converge for larger mass objects (right hand panel).', '1010.3722-2-65-3': 'The bias correction [MATH] is calculated from the difference between the matter-halo cross correlation in a Gaussian simulation and the non-Gaussian case built from the same Gaussian realization, then averaged over realizations.', '1010.3722-2-66-0': 'The qualitative effect we expected is present, but for some halos the magnitude of the effect is not well predicted by the analytic expressions from Section 3.', '1010.3722-2-66-1': 'Figure [REF] shows the deviation between simulation and prediction for [MATH], [MATH] (the same generic trend was seen in all parameter sets).', '1010.3722-2-66-2': 'We plot the ratio of the non-Gaussian correction with running to the non-Gaussian correction for constant [MATH]: [EQUATION]', '1010.3722-2-66-3': 'This is compared with the theoretical expectation calculated from the small [MATH] limit of Eq. ([REF]).', '1010.3722-2-66-4': 'The curves are plotted as a function of [MATH].', '1010.3722-2-66-5': 'As the figure demonstrates, the simulation results agree well with our analytic model in the high-mass limit [MATH], but towards lower masses (e.g. [MATH]) the simulations produce a stronger effect than Eq. ([REF]) would predict.', '1010.3722-2-66-6': 'Note that the discrepancy does not appear at a fixed mass, but rather at a fixed [MATH].', '1010.3722-2-66-7': 'This is illustrated in the right panel of Figure [REF], which is identical to the left panel of Figure except that now mass [MATH] is the abscissa.', '1010.3722-2-66-8': 'The figure shows that for fixed mass [MATH], the simulations agree with Eq. ([REF]) at high redshift, but begin to disagree at low redshift as [MATH] grows.', '1010.3722-2-67-0': 'One very plausible explanation for this discrepancy at low mass ([MATH]) is that the profiles of the peaks that produce halos begin to change as [MATH] increases.', '1010.3722-2-67-1': 'As we have argued, the non-Gaussian bias of halos of mass [MATH] is sensitive to the value of [MATH] at some effective [MATH].', '1010.3722-2-67-2': 'Implicit in this scaling is the assumption that the profiles of peaks that collapse into halos are similar at different masses, just rescaled in size.', '1010.3722-2-67-3': 'However, we know that this assumption is incorrect.', '1010.3722-2-67-4': 'Bardeen et al. [CITATION] argued from Gaussian statistics that as [MATH] increases, the peaks that collapse into halos generally become steeper.', '1010.3722-2-67-5': 'N-body simulations confirm the presence of this effect, but show that it is much stronger in magnitude than predicted by Bardeen et al., apparently due to environmental effects during halo formation [CITATION].', '1010.3722-2-67-6': 'Because peaks at high [MATH] are much steeper than rare peaks at low [MATH], they are sensitive to non-Gaussianity at higher wavenumbers, even at the same peak size [MATH].', '1010.3722-2-67-7': 'For scale-independent [MATH], this change in peak profile has no effect, but for nonzero [MATH] it can dramatically enhance the mass dependence of non-Gaussianity, as our simulations show.', '1010.3722-2-67-8': 'It remains to be seen whether the magnitude of the change in peak profile can account for the discrepancy between our simulations and Eq. ([REF]); this is work in progress.', '1010.3722-2-68-0': '# Conclusions', '1010.3722-2-69-0': 'In summary, we have introduced a generalization of the local ansatz, Eq. ([REF]), that is a symmetric, factorizable function of the momenta and includes scale-dependent non-Gaussianity.', '1010.3722-2-69-1': 'This more general expression is motivated by natural features of models that give an observably large amplitude for local type non-Gaussianity, and distinguishes between non-Gaussian curvature fluctuations generated by a single field and multiple fields.', '1010.3722-2-69-2': 'If only one field contributes to the curvature fluctuation (and is different from the inflaton so that the non-Gaussianity may be large), the scale-dependence of the non-Gaussianity characterizes the self-interactions of the field.', '1010.3722-2-69-3': 'If two fields contribute to the curvature fluctuations, scale-dependence indicates how the ratio of power in the fields changes, which is a function of how different the potentials are.', '1010.3722-2-69-4': 'If local non-Gaussianity is large enough to be observed, such scale-dependence is as natural as running of the power spectrum.', '1010.3722-2-70-0': 'Models with scale-dependent local non-Gaussianity can generate two signatures in the non-Gaussian contribution to the halo bias.', '1010.3722-2-70-1': 'First, the non-Gaussian term may be proportional to an effective [MATH] related to the amplitude of the bispectrum on the scale of the object so that different mass objects have a different amplitude correction.', '1010.3722-2-70-2': 'Second, the [MATH] behavior of the non-Gaussian bias can be modified to [MATH] (where [MATH]), and one should expect the first effect to accompany this one.', '1010.3722-2-71-0': 'We have used N-body simulations to verify that different mass objects do indeed have a non-Gaussian bias proportional to an effective [MATH] that varies with the mass of the object.', '1010.3722-2-71-1': 'It is interesting that the simulations show that scale dependance with [MATH] can erase the scale dependent effect on the bias for some range of masses, highlighting the need for analysis using multiple tracers of different mass.', '1010.3722-2-71-2': 'However, the quantitative result for halos at large [MATH] is not well predicted by our analytic expressions.', '1010.3722-2-71-3': 'We have speculated that the origin of this discrepancy may be related to differences in the initial peak profiles of the halos, but leave a detailed investigation for a later work.', '1010.3722-2-72-0': 'Future surveys are sure to bring interesting results.', '1010.3722-2-72-1': 'Using the analytic predictions, we find that they may be able to distinguish the different pieces of our generalized local ansatz, and so different origins of local non-Gaussianity, especially if the running is somewhat large ([MATH]).', '1010.3722-2-72-2': 'However, the existing analytic expressions predict a weaker effect than we see in the simulations, and our forecasts only account for constraints from massive groups and clusters of galaxies, neglecting the (potentially) greater sensitivity to running possible when galaxy correlations are included as well.', '1010.3722-2-72-3': 'Our forecasts for future surveys should therefore be taken as a lower limit on the potential to observationally distinguish these features.'}

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'1010.3722-2-15-4'], ['1010.3722-1-15-5', '1010.3722-2-15-5'], ['1010.3722-1-15-6', '1010.3722-2-15-6'], ['1010.3722-1-15-7', '1010.3722-2-15-7'], ['1010.3722-1-15-8', '1010.3722-2-15-8'], ['1010.3722-1-15-9', '1010.3722-2-15-9'], ['1010.3722-1-56-0', '1010.3722-2-56-0'], ['1010.3722-1-56-1', '1010.3722-2-56-1'], ['1010.3722-1-56-2', '1010.3722-2-56-2'], ['1010.3722-1-61-1', '1010.3722-2-61-1'], ['1010.3722-1-61-2', '1010.3722-2-61-2'], ['1010.3722-1-61-3', '1010.3722-2-61-3'], ['1010.3722-1-61-4', '1010.3722-2-61-4'], ['1010.3722-1-61-5', '1010.3722-2-61-5'], ['1010.3722-1-61-6', '1010.3722-2-61-6'], ['1010.3722-1-59-0', '1010.3722-2-59-0'], ['1010.3722-1-59-1', '1010.3722-2-59-1'], ['1010.3722-1-59-2', '1010.3722-2-59-2'], ['1010.3722-1-59-3', '1010.3722-2-59-3'], ['1010.3722-1-59-4', '1010.3722-2-59-4'], ['1010.3722-1-19-0', '1010.3722-2-19-0'], ['1010.3722-1-19-1', '1010.3722-2-19-1'], ['1010.3722-1-19-2', '1010.3722-2-19-2'], ['1010.3722-1-52-0', '1010.3722-2-52-0'], ['1010.3722-1-52-1', '1010.3722-2-52-1'], ['1010.3722-1-11-0', '1010.3722-2-11-0'], ['1010.3722-1-11-1', '1010.3722-2-11-1'], ['1010.3722-1-11-2', '1010.3722-2-11-2'], ['1010.3722-1-11-3', '1010.3722-2-11-3'], ['1010.3722-1-11-4', '1010.3722-2-11-4'], ['1010.3722-1-29-0', '1010.3722-2-29-0'], ['1010.3722-1-29-1', '1010.3722-2-29-1'], ['1010.3722-1-29-2', '1010.3722-2-29-2'], ['1010.3722-1-29-3', '1010.3722-2-29-3'], ['1010.3722-1-29-4', '1010.3722-2-29-4'], ['1010.3722-1-29-5', '1010.3722-2-29-5'], ['1010.3722-1-29-6', '1010.3722-2-29-6'], ['1010.3722-1-67-0', '1010.3722-2-67-0'], ['1010.3722-1-67-1', '1010.3722-2-67-1'], ['1010.3722-1-67-2', '1010.3722-2-67-2'], ['1010.3722-1-67-3', '1010.3722-2-67-3'], ['1010.3722-1-67-4', '1010.3722-2-67-4'], ['1010.3722-1-67-5', '1010.3722-2-67-5'], ['1010.3722-1-67-6', '1010.3722-2-67-6'], ['1010.3722-1-67-7', '1010.3722-2-67-7'], ['1010.3722-1-67-8', '1010.3722-2-67-8'], ['1010.3722-1-33-0', '1010.3722-2-33-0'], ['1010.3722-1-33-1', '1010.3722-2-33-1'], ['1010.3722-1-33-2', '1010.3722-2-33-2'], ['1010.3722-1-33-3', '1010.3722-2-33-3'], ['1010.3722-1-33-4', '1010.3722-2-33-4'], ['1010.3722-1-33-5', '1010.3722-2-33-5'], ['1010.3722-1-33-6', '1010.3722-2-33-6'], ['1010.3722-1-33-7', '1010.3722-2-33-7'], ['1010.3722-1-33-8', '1010.3722-2-33-8'], ['1010.3722-1-33-9', '1010.3722-2-33-9'], ['1010.3722-1-54-0', '1010.3722-2-54-0'], ['1010.3722-1-54-1', '1010.3722-2-54-1'], ['1010.3722-1-70-0', '1010.3722-2-70-0'], ['1010.3722-1-70-1', '1010.3722-2-70-1'], ['1010.3722-1-70-2', '1010.3722-2-70-2'], ['1010.3722-1-1-0', '1010.3722-2-1-0'], ['1010.3722-1-1-1', '1010.3722-2-1-1'], ['1010.3722-1-1-2', '1010.3722-2-1-2'], ['1010.3722-1-1-3', '1010.3722-2-1-3'], ['1010.3722-1-1-4', '1010.3722-2-1-4'], ['1010.3722-1-1-5', '1010.3722-2-1-5'], ['1010.3722-1-1-6', '1010.3722-2-1-6'], ['1010.3722-1-58-0', '1010.3722-2-58-0'], ['1010.3722-1-58-1', '1010.3722-2-58-1'], ['1010.3722-1-58-2', '1010.3722-2-58-2'], ['1010.3722-1-12-0', '1010.3722-2-12-0'], ['1010.3722-1-12-1', '1010.3722-2-12-1'], ['1010.3722-1-12-2', '1010.3722-2-12-2'], ['1010.3722-1-12-3', '1010.3722-2-12-3'], ['1010.3722-1-25-0', '1010.3722-2-25-0'], ['1010.3722-1-25-1', '1010.3722-2-25-1'], ['1010.3722-1-25-2', '1010.3722-2-25-2'], ['1010.3722-1-25-3', '1010.3722-2-25-3'], ['1010.3722-1-25-4', '1010.3722-2-25-4'], ['1010.3722-1-25-5', '1010.3722-2-25-5']]

[['1010.3722-1-39-0', '1010.3722-2-39-0'], ['1010.3722-1-49-2', '1010.3722-2-49-2'], ['1010.3722-1-71-1', '1010.3722-2-71-1'], ['1010.3722-1-61-0', '1010.3722-2-61-0']]

[]

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{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1010.3722

1306.3888

{'1306.3888-1-0-0': 'This article is an overview of the SP theory of intelligence, which aims to simplify and integrate concepts across artificial intelligence, mainstream computing and human perception and cognition, with information compression as a unifying theme.', '1306.3888-1-0-1': "It is conceived as a brain-like system that receives 'New' information and stores some or all of it in compressed form as 'Old' information; and it is realised in the form of a computer model, a first version of the SP machine.", '1306.3888-1-0-2': 'The matching and unification of patterns and the concept of multiple alignment are central ideas.', '1306.3888-1-0-3': "Using heuristic techniques, the system builds multiple alignments that are 'good' in terms of information compression.", '1306.3888-1-0-4': 'For each multiple alignment, probabilities may be calculated for associated inferences.', '1306.3888-1-0-5': "Unsupervised learning is done by deriving new structures from partial matches between patterns and via heuristic search for sets of structures that are 'good' in terms of information compression.", '1306.3888-1-0-6': "These are normally ones that people judge to be 'natural', in accordance with the 'DONSVIC' principle-the discovery of natural structures via information compression.", '1306.3888-1-0-7': "The SP theory provides an interpretation for concepts and phenomena in several other areas including 'computing', aspects of mathematics and logic, the representation of knowledge, natural language processing, pattern recognition, several kinds of reasoning, information storage and retrieval, planning and problem solving, information compression, neuroscience, and human perception and cognition.", '1306.3888-1-0-8': "Examples include the parsing and production of language with discontinuous dependencies in syntax, pattern recognition at multiple levels of abstraction and its integration with part-whole relations, nonmonotonic reasoning and reasoning with default values, reasoning in Bayesian networks including 'explaining away', causal diagnosis, and the solving of a geometric analogy problem.", '1306.3888-1-1-0': 'Keywords: information compression, artificial intelligence, multiple alignment, computing, representation of knowledge, natural language processing, pattern recognition, information retrieval, probabilistic reasoning, planning, problem solving, unsupervised learning.', '1306.3888-1-2-0': '# Introduction', '1306.3888-1-3-0': 'The SP theory of intelligence, which has been under development since about 1987, aims to simplify and integrate concepts across artificial intelligence, mainstream computing and human perception and cognition, with information compression as a unifying theme.', '1306.3888-1-4-0': "The name 'SP' is short for Simplicity and Power, because compression of any given body of information, [MATH], may be seen as a process of reducing informational 'redundancy' in [MATH] and thus increasing its 'simplicity', whilst retaining as much as possible of its non-redundant expressive 'power'.", '1306.3888-1-4-1': "Likewise with Occam's Razor (Section [REF], below).", '1306.3888-1-5-0': 'Aspects of the theory, as it has developed, have been described in several peer-reviewed articles.', '1306.3888-1-5-1': 'The most comprehensive description of the theory as it stands now, with many examples, is in [CITATION].', '1306.3888-1-6-0': 'But this book, with more than 450 pages, it is too long to serve as an introduction to the theory.', '1306.3888-1-6-1': 'This article aims to meet that need, with a fairly full description of the theory and a selection of examples.', '1306.3888-1-6-2': "For the sake of brevity, the book will be referred to as 'B' (for 'book').", '1306.3888-1-7-0': 'The next section describes the origins and motivation for the SP theory, Section [REF] introduces the theory, Sections [REF] and [REF] fill in a lot of the details, while the following sections describe aspects of the theory and what it can do.', '1306.3888-1-8-0': '# Origins and motivation', '1306.3888-1-9-0': 'The following subsections outline the origins of the SP theory and how it relates to some other research.', '1306.3888-1-10-0': '## Information compression', '1306.3888-1-11-0': 'Much of the inspiration for the SP theory is a body of research, pioneered by Fred Attneave , Horace Barlow , and others, showing that several aspects of the workings of brains and nervous systems may be understood in terms of information compression.', '1306.3888-1-11-1': 'For example, when we view a scene with two eyes, the image on the retina of the left eye is almost exactly the same as the image on the retina of right eye, but our brains merge the two images into a single percept, and thus compress the information .', '1306.3888-1-12-0': 'More immediately, the theory has grown out of my own research, developing models of the unsupervised learning of a first language, where the importance of information compression became increasingly clear .', '1306.3888-1-13-0': "The theory also draws on principles of 'minimum length encoding' pioneered by [CITATION], and others.", '1306.3888-1-13-1': 'And it has become apparent that several aspects of computing, mathematics, and logic may be understood in terms of information compression (B, Chapters 2 and 10).', '1306.3888-1-14-0': 'At an abstract level, information compression can bring two main benefits:', '1306.3888-1-15-0': 'In animals, we would expect these things to have been favoured by natural selection because of the competitive advantage they can bring.', '1306.3888-1-15-1': "Notwithstanding the 'QWERTY' phenomenon, there is reason to believe that information compression, properly applied, may yield comparable advantages in artificial systems.", '1306.3888-1-16-0': '## The matching and unification of patterns', '1306.3888-1-17-0': 'In the SP theory, the matching and unification of patterns is seen as being closer to the bedrock of information compression than more mathematical techniques such as wavelets or arithmetic coding, and closer to the bedrock of information processing and intelligence than, say, concepts of probability.', '1306.3888-1-17-1': "A working hypothesis in this programme of research is that, by staying close to relatively simple, 'primitive', concepts of matching patterns and unifying them, there is a better chance of cutting through unnecessary complexity, and in gaining new insights and better solutions to problems.", '1306.3888-1-17-2': 'The mathematical basis of wavelets, arithmetic coding, and probabilities, may itself be founded on the matching and unification of patterns (B, Chapter 10).', '1306.3888-1-18-0': '## Simplification and integration of concepts', '1306.3888-1-19-0': "In accordance with Occam's Razor, the SP system aims to combine conceptual simplicity with descriptive and explanatory power.", '1306.3888-1-19-1': "Apart from this widely-accepted principle, the drive for simplification and integration of concepts in this research programme has been motivated in part by Allen Newell's critique of some kinds of research in cognitive science , and in part by the apparent fragmentation of research in artificial intelligence and mainstream computing, with their myriad of concepts and many specialisms.", '1306.3888-1-20-0': 'In attempting to simplify and integrate ideas, the SP theory belongs in the same tradition as unified theories of cognition such as Soar and ACT-R .', '1306.3888-1-20-1': "And it chimes with the resurgence of interest in understanding artificial intelligence as a whole and with research on 'natural computation' .", '1306.3888-1-21-0': "Although the SP programme shares some objectives with projects such as the Godel Machine , and 'universal artificial intelligence' , the approach is very different.", '1306.3888-1-22-0': '## Transparency in the representation of knowledge', '1306.3888-1-23-0': "In this research, it is assumed that knowledge in the SP system should normally be transparent or comprehensible, much as in the 'symbolic' tradition in artificial intelligence (see also Section [REF]), and distinct from the kind of 'sub-symbolic' representation of knowledge that is the rule in, for example, 'neural networks' as they are generally conceived in computer science.", '1306.3888-1-24-0': 'As we shall see in Section [REF] and elsewhere in this article, SP patterns in the multiple alignment framework may serve to represent a variety of kinds of knowledge, in symbolic forms.', '1306.3888-1-25-0': '# Introduction to the SP theory', '1306.3888-1-26-0': 'The main elements of the SP theory are:', '1306.3888-1-27-0': '## The SP computer model', '1306.3888-1-28-0': 'The SP theory is realised most fully in the SP70 computer model, with capabilities in the building of multiple alignments and in unsupervised learning.', '1306.3888-1-28-1': 'This will be referred to as the SP model, although in some cases examples are from a subset of the model or slightly earlier precursors of it.', '1306.3888-1-29-0': 'The SP model and its precursors have played a key part in the development of the theory:', '1306.3888-1-30-0': 'The workings of the SP model is described in some detail in B (Sections 3.9, 3.10, and 9.2) and more briefly in Sections [REF] and [REF], below.', '1306.3888-1-30-1': "The source code for the models, with associated documents and files, may be downloaded via links under the heading 'SOURCE CODE' at the bottom of the page on http://bit.ly/WtXa3gbit.ly/WtXa3g.", '1306.3888-1-31-0': 'The two main elements of the models, described in the following two sections, are the building of multiple alignments and the unsupervised learning of Old patterns.', '1306.3888-1-32-0': '## The SP machine', '1306.3888-1-33-0': 'The SP model may be regarded as a first version of the SP machine, an expression of the SP theory and a means for it to be applied.', '1306.3888-1-34-0': 'A useful step forward in the development of the SP theory would be the creation of a high-parallel, open-source version of the SP machine, accessible via the web, and with a good user interface.', '1306.3888-1-34-1': 'This would provide a means for researchers to explore what can be done with the system and to improve it.', '1306.3888-1-34-2': 'How things may develop is shown schematically in Figure [REF].', '1306.3888-1-35-0': 'The high-parallel search mechanisms in any of the existing internet search engines would probably provide a good foundation for the proposed development.', '1306.3888-1-36-0': 'Further ahead, there may be a case for the creation of new kinds of hardware, dedicated to the building of multiple alignments and other processes in the SP framework .', '1306.3888-1-37-0': '## Unfinished business', '1306.3888-1-38-0': 'Like most theories, the SP theory has shortcomings, but it appears that they may be overcome.', '1306.3888-1-38-1': 'At present, the most immediate problems are:', '1306.3888-1-39-0': '# The multiple alignment concept', '1306.3888-1-40-0': 'The concept of multiple alignment in the SP theory has been adapted from a similar concept in bioinformatics, where it means a process of arranging, in rows or columns, two or more DNA sequences or amino-acid sequences so that matching symbols-as many as possible-are aligned orthogonally in columns or rows.', '1306.3888-1-41-0': 'Multiple alignments like these are normally used in the computational analysis of (symbolic representations of) sequences of DNA bases or sequences of amino acid residues as part of the process of elucidating the structure, functions or evolution of the corresponding molecules.', '1306.3888-1-41-1': 'An example of this kind of multiple alignment is shown in Figure [REF].', '1306.3888-1-42-0': 'As in bioinformatics, a multiple alignment in the SP system is an arrangement of two or more patterns in rows (or columns), with one pattern in each row (or column).', '1306.3888-1-42-1': 'The main difference between the two concepts is that, in bioinformatics, all sequences have the same status, whereas in the SP theory, the system attempts to create a multiple alignment which enables one New pattern (sometimes more) to be encoded economically in terms of one or more Old patterns.', '1306.3888-1-42-2': 'Other differences are described in B (Section 3.4.1).', '1306.3888-1-43-0': "In Figure [REF], row 0 contains a New pattern representing a sentence: 't h i s b o y l o v e s t h a t g i r l', while each of rows 1 to 8 contains an Old pattern representing a grammatical rule or a word with grammatical markers.", '1306.3888-1-43-1': 'This multiple alignment, which achieves the effect of parsing the sentence in terms of grammatical structures, is the best of several built by the model when it is supplied with the New pattern and a set of Old patterns that includes those shown in the figure and several others as well.', '1306.3888-1-44-0': "In this example, and others in this article, 'best' means that the multiple alignment in the figure is the one that enables the New pattern to be encoded most economically in terms of the Old patterns, as described in Section [REF], below.", '1306.3888-1-45-0': '## Coding and the evaluation of an alignment in terms of compression', '1306.3888-1-46-0': 'This section describes in outline how multiple alignments are evaluated in the SP model.', '1306.3888-1-46-1': 'More detail may be found in B (Section 3.5).', '1306.3888-1-47-0': "Each Old pattern in the SP system contains one or more 'identification' symbols or ID-symbols which, as their name suggests, serve to identify the pattern.", '1306.3888-1-47-1': "Examples of ID-symbols in Figure [REF] are 'D' and '0' at the beginning of 'D 0 t h i s D' (row 6), and 'N' and '1' at the beginning of 'N 1 b o y N' (row 8).", '1306.3888-1-48-0': "Associated with each type of symbol (where a 'type' of symbol is any one of a set of symbols that match each other exactly) is a notional code or bit pattern that serves to distinguish the given type from all the others.", '1306.3888-1-48-1': 'This is only notional because the bit patterns are not actually constructed.', '1306.3888-1-48-2': 'All that is needed for the purpose of evaluating multiple alignments is the size of the notional bit pattern associated with each type.', '1306.3888-1-48-3': 'This is calculated via the Shannon-Fano-Elias coding scheme (described by [CITATION]), using information about the frequency of occurence of each Old pattern, so that the shortest codes represent the most frequent symbol types and vice versa.', '1306.3888-1-48-4': 'Notice that these bit patterns and their sizes are totally independent of the names for symbols that are used in written accounts like this one: names that are chosen purely for their mnemonic value.', '1306.3888-1-49-0': 'Given a multiple alignment like the one shown in Figure [REF], one can derive a code pattern from the multiple alignment in the following way:', '1306.3888-1-50-0': 'Scan the multiple alignment from left to right looking for columns that contain an ID-symbol by itself, not aligned with any other symbol.', '1306.3888-1-51-0': 'Copy these symbols into a code pattern in the same order that they appear in the multiple alignment.', '1306.3888-1-52-0': "The code pattern derived in this way from the multiple alignment shown in Figure [REF] is 'S 0 1 0 1 0 S'.", '1306.3888-1-52-1': 'This is, in effect, a compressed representation of those symbols in the New pattern that are aligned with Old symbols in the multiple alignment.', '1306.3888-1-52-2': 'In this case, the code pattern is a compressed representation of all the symbols in the New pattern but it often happens that some of the symbols in the New pattern are not matched with any Old symbols and then the code pattern will represent only those New symbols that are aligned with Old symbols.', '1306.3888-1-53-0': "In the context of natural language processing, it perhaps more plausible to suppose that the encoding of a sentence is some kind of representation of the meaning of the sentence, instead of a pattern like 'S 0 1 0 1 0 S'.", '1306.3888-1-53-1': 'How a meaning may be derived from a sentence via multiple alignment is described in B (Section 5.7).', '1306.3888-1-54-0': '### Compression difference and compression ratio', '1306.3888-1-55-0': "Given a code pattern like 'S 0 1 0 1 0 S', we may calculate a 'compression difference'CD as: [EQUATION] or a 'compression ratio'CR as: [EQUATION] where [MATH] is the total number of bits in those symbols in the New pattern that are aligned with Old symbols in the alignment and [MATH] is the total number of bits in the symbols in the code pattern, and the number of bits for each symbol is calculated via the Shannon-Fano-Elias scheme as mentioned above.", '1306.3888-1-56-0': '[MATH] and [MATH] are each an indication of how effectively the New pattern (or those parts of the New pattern that are aligned with symbols within Old patterns in the alignment) may be compressed in terms of the Old patterns that appear in the given multiple alignment.', '1306.3888-1-56-1': 'The [MATH] of a multiple alignment-which has been found to be more useful than [MATH]-may be referred to as the compression score of the multiple alignment.', '1306.3888-1-57-0': 'In each of these equations, [MATH] is calculated as: [EQUATION] where [MATH] is the size of the code for [MATH]th symbol in a sequence, [MATH], comprising those symbols within the New pattern that are aligned with Old symbols within the multiple alignment.', '1306.3888-1-58-0': '[MATH] is calculated as: [EQUATION] where [MATH] is the size of the code for [MATH]th symbol in the sequence of [MATH] symbols in the code pattern derived from the multiple alignment.', '1306.3888-1-59-0': '## The building of multiple alignments', '1306.3888-1-60-0': 'This section describes in outline how the SP model builds multiple alignments.', '1306.3888-1-60-1': 'More detail may be found in B (Section 3.10).', '1306.3888-1-61-0': 'Multiple alignments are built in stages, with pairwise matching and alignment of patterns.', '1306.3888-1-61-1': 'At each stage, any partially-constructed multiple alignment may be processed as if it was a basic pattern and carried forward to later stages.', '1306.3888-1-61-2': 'This is broadly similar to some programs for the creation of multiple alignments in bioinformatics.', '1306.3888-1-61-3': 'At all stages, the aim is to encode New information economically in terms of Old information and to weed out multiple alignments that score poorly in that regard.', '1306.3888-1-62-0': 'The model may create Old patterns for itself, as described in Section [REF], but when the formation of multiple alignments is the focus of interest, Old patterns may be supplied by the user.', '1306.3888-1-62-1': 'In all cases, New patterns must be supplied by the user.', '1306.3888-1-63-0': 'At each stage of building multiple alignments, the operations are as follows:', '1306.3888-1-64-0': "Identify a set of 'driving' patterns and a set of 'target' patterns.", '1306.3888-1-64-1': 'At the beginning, the New pattern is the sole driving pattern and the Old patterns are the target patterns.', '1306.3888-1-64-2': 'In all subsequent stages, the best of the multiple alignments formed so far (in terms of their [MATH] scores) are chosen to be driving patterns and the target patterns are the Old patterns together with a selection of the best multiple alignments formed so far, including all of those that are driving patterns.', '1306.3888-1-65-0': 'Compare each driving pattern with each of the target patterns to find full matches and good partial matches between patterns.', '1306.3888-1-65-1': "This is done with a process that is essentially a form of 'dynamic programming' , somewhat like the WinMerge utility for finding similarities and differences between files.", '1306.3888-1-65-2': 'The process is described quite fully in B (Appendix A) and outlined in Section [REF], below.', '1306.3888-1-65-3': "The main difference between the SP process and others, is that the former can deliver several alternative matches between a pair of patterns, while WinMerge and standard methods for finding alignments deliver one 'best' result.", '1306.3888-1-66-0': 'From the best of the matches found in the current stage, create corresponding multiple alignments and add them to the repository of multiple alignments created by the program.', '1306.3888-1-67-0': 'This process of matching driving patterns against target patterns and building multiple alignments is repeated until no more multiple alignments can be found.', '1306.3888-1-67-1': 'For the best of the multiple alignments created since the start of processing, probabilities are calculated, as described in Section [REF].', '1306.3888-1-68-0': '### Finding good matches between patterns', '1306.3888-1-69-0': "Figure [REF] shows with a simple example how the SP model finds good full and partial matches between a 'query' string of atomic symbols (alphabetic characters in this example) and a 'database' string:", '1306.3888-1-70-0': 'The query is processed left to right, one symbol at a time.', '1306.3888-1-71-0': 'Each symbol in the query is, in effect, broadcast to every symbol in the database to make a yes/no match in each case.', '1306.3888-1-72-0': 'Every positive match (hit) between a symbol from the query and a symbol in the database is recorded in a hit structure, illustrated in the figure.', '1306.3888-1-73-0': 'If the memory space allocated to the hit structure is exhausted at any time then the hit structure is purged: the leaf nodes of the tree are sorted in reverse order of their probability values and each leaf node in the bottom half of the set is extracted from the hit structure, together with all nodes on its path which are not shared with any other path.', '1306.3888-1-73-1': 'After the hit structure has been purged, the recording of hits may continue using the space which has been released.', '1306.3888-1-74-0': '### Noisy data', '1306.3888-1-75-0': 'Because of the way each model searches for a global optimum in the building of multiple alignments, it does not depend on the presence or absence of any particular feature or combination of features.', '1306.3888-1-75-1': 'Up to a point, plausible results may be obtained in the face of errors of omission, commission and substitution in the data.', '1306.3888-1-75-2': "This is illustrated in the two multiple alignments in Figure [REF] where the New pattern in row 0 of (b) is the same sentence as in (a) ('t w o k i t t e n s p l a y') but with the omission of the 'w' in 't w o', the substitution of 'm' for ''n' in 'k i t t e n s', and the addition of 'x' within the word 'p l a y'.", '1306.3888-1-75-3': "Despite these errors, the best multiple alignment created by the SP model is, as shown in (b), the one that we judge intuitively to be 'correct'.", '1306.3888-1-76-0': 'This kind of ability to cope gracefully with noisy data is very much in keeping with our ability to understand speech in noisy surroundings, to understand written language despite errors, and to recognise people, trees, houses, and the like, despite fog, snow, falling leaves, or other things that may obstruct our view.', '1306.3888-1-76-1': 'In a similar way, it is likely to prove useful in artificial systems for such applications as the processing of natural language and the recognition of patterns.', '1306.3888-1-77-0': '## Computational complexity', '1306.3888-1-78-0': 'In considering the matching and unification of patterns, it not hard to see that, for any body of information [MATH], except very small examples, there is a huge number of alternative ways in which patterns may be matched against each other, there will normally be many alternative ways in which patterns may be unified, and exhaustive search is not tractable (B, Section 2.2.8.4).', '1306.3888-1-79-0': 'However, with the kinds of heuristic techniques that are familiar in other AI applications-reducing the size of the search space by pruning the search tree at appropriate points, and being content with approximate solutions which are not necessarily perfect-this kind of matching becomes quite practical.', '1306.3888-1-79-1': 'Much the same can be said about the heuristic techniques used for the building of multiple alignments (Section [REF]) and for unsupervised learning (Section [REF]).', '1306.3888-1-80-0': 'For the process of building multiple alignments in the SP model, the time complexity in a serial processing environment, with conservative assumptions, has been estimated to be O[MATH], where [MATH] is the size of the pattern from New (in bits) and [MATH] is the sum of the lengths of the patterns in Old (in bits).', '1306.3888-1-80-1': 'In a parallel processing environment, the time complexity may approach O[MATH], depending on how well the parallel processing is applied.', '1306.3888-1-80-2': 'In serial and parallel environments, the space complexity has been estimated to be O[MATH].', '1306.3888-1-81-0': 'There is reason to be confident that the models can be scaled up to deal with large data sets because the kind of flexible matching of patterns which is at the heart of the SP model is done very fast and with huge volumes of data by all the leading internet search engines.', '1306.3888-1-81-1': 'As was suggested in Section [REF], the relevant processes in any one of those search engines would probably provide a good basis for the creation of a high-parallel version of the SP machine.', '1306.3888-1-82-0': '## Calculation of probabilities associated with multiple alignments', '1306.3888-1-83-0': 'As described in B (Chapter 7), the formation of multiple alignments in the SP framework supports several kinds of probabilistic reasoning.', '1306.3888-1-83-1': 'The core idea is that any Old symbol in a multiple alignment that is not aligned with a New symbol represents an inference that may be drawn from the multiple alignment.', '1306.3888-1-83-2': 'This section outlines how probabilities for such inferences may be calculated.', '1306.3888-1-83-3': 'There is more detail in B (Section 3.7).', '1306.3888-1-84-0': '### Absolute probabilities', '1306.3888-1-85-0': 'Any sequence of [MATH] symbols, drawn from an alphabet of [MATH] alphabetic types, represents one point in a set of [MATH] points where [MATH] is calculated as: [EQUATION]', '1306.3888-1-85-1': 'If we assume that the sequence is random or nearly so, which means that the [MATH] points are equi-probable or nearly so, the probability of any one point (which represents a sequence of length [MATH]) is close to: [EQUATION]', '1306.3888-1-85-2': 'This equation may be used to calculate the absolute probability of the code pattern that may be derived from any given multiple alignment (as described in Section [REF]).', '1306.3888-1-85-3': 'That number may also be regarded as the absolute probability of any inferences that may be drawn from the multiple alignment.', '1306.3888-1-85-4': 'In this calculation, [MATH] is the sum of all the bits in the symbols of the code pattern and [MATH] is 2.', '1306.3888-1-86-0': 'As we shall see (Section [REF]), Equation [REF] may, with advantage, be generalised by replacing [MATH] with a value, [MATH], calculated in a slightly different way.', '1306.3888-1-87-0': '### Relative probabilities', '1306.3888-1-88-0': 'The absolute probabilities of multiple alignments, calculated as described in the last subsection, are normally very small and not very interesting in themselves.', '1306.3888-1-88-1': 'From the standpoint of practical applications, we are normally interested in the relative values of probabilities, calculated as follows.', '1306.3888-1-89-0': 'For the multiple alignment which has the highest [MATH] (which we shall call the reference multiple alignment), identify the reference set of symbols in New, meaning the symbols from New which are encoded by the multiple alignment.', '1306.3888-1-90-0': 'Compile a reference set of multiple alignments which includes the reference multiple alignment and all other multiple alignments (if any) which encode exactly the reference set of symbols from New, neither more nor less.', '1306.3888-1-91-0': 'Calculate the sum of the values for [MATH] in the reference set of multiple alignments: [EQUATION] where [MATH] is the size of the reference set of multiple alignments and [MATH] is the value of [MATH] for the [MATH]th multiple alignment in the reference set.', '1306.3888-1-92-0': 'For each multiple alignment in the reference set, calculate its relative probability as: [EQUATION]', '1306.3888-1-92-1': 'The values of [MATH], calculated as just described, provide an effective means of comparing the multiple alignments in the reference set.', '1306.3888-1-93-0': '### A generalisation of the method for calculating absolute and relative probabilities', '1306.3888-1-94-0': "The value of [MATH], calculated as described in Section [REF], may be regarded as the informational 'cost' of encoding the New symbol or symbols that appear in the multiple alignment, excluding those New symbols that have not appeared in the multiple alignment.", '1306.3888-1-95-0': 'This is OK but it is somewhat restrictive because it means that if we want to calculate relative probabilities for two or more multiple alignments they must all encode the same symbol or symbols from New.', '1306.3888-1-95-1': 'We cannot easily compare multiple alignments that encode different New symbols.', '1306.3888-1-96-0': 'The generalisation proposed here is that, in the calculation of absolute probabilities, a new value, [MATH], would be used instead of [MATH].', '1306.3888-1-96-1': 'This would be calculated as: [EQUATION] where [MATH] is the total number of bits in the symbols in the code patterns (as in Section [REF]) and [MATH] is the total number of bits in the New symbols that have not appeared in the multiple alignment.', '1306.3888-1-97-0': 'The rationale is that, to encode all the symbols in New, we can use the code pattern to encode those New symbols that do appear in the multiple alignment and, for each of the remaining New symbols, we can simply use its code.', '1306.3888-1-97-1': 'The advantage of this scheme is that we can compare any two or more multiple alignments, regardless of the number of New symbols that appear in the multiple alignment.', '1306.3888-1-98-0': '### Relative probabilities of patterns and symbols', '1306.3888-1-99-0': 'It often happens that a given pattern from Old, or a given symbol type within patterns from Old, appears in more than one of the multiple alignments in the reference set.', '1306.3888-1-99-1': 'In cases like these, one would expect the relative probability of the pattern or symbol type to be higher than if it appeared in only one multiple alignment.', '1306.3888-1-99-2': 'To take account of this kind of situation, the SP model calculates relative probabilities for individual patterns and symbol types in the following way:', '1306.3888-1-100-0': 'Compile a set of patterns from Old, each of which appears at least once in the reference set of multiple alignments.', '1306.3888-1-100-1': 'No single pattern from Old should appear more than once in the set.', '1306.3888-1-101-0': 'For each pattern, calculate a value for its relative probability as the sum of the [MATH] values for the multiple alignments in which it appears.', '1306.3888-1-101-1': 'If a pattern appears more than once in a multiple alignment, it is only counted once for that multiple alignment.', '1306.3888-1-102-0': 'Compile a set of symbol types which appear anywhere in the patterns identified in step 2.', '1306.3888-1-103-0': 'For each alphabetic symbol type identified in step 3, calculate its relative probability as the sum of the relative probabilities of the patterns in which it appears.', '1306.3888-1-103-1': 'If it appears more than once in a given pattern, it is only counted once.', '1306.3888-1-104-0': 'The foregoing applies only to symbol types which do not appear in New.', '1306.3888-1-104-1': 'Any symbol type that appears in New necessarily has a probability of [MATH]-because it has been observed, not inferred.', '1306.3888-1-105-0': '## One system for both the analysis and the production of information', '1306.3888-1-106-0': 'A potentially useful feature of the SP system is that the processes which serve to analyse or parse a New pattern in terms of Old patterns, and to create an economical encoding of the New pattern, may also work in reverse, to recreate the New pattern from its encoding.', '1306.3888-1-106-1': "This is the 'output' perspective, mentioned in Section [REF].", '1306.3888-1-107-0': "If the New pattern is the code sequence 'S 0 1 0 1 0 S' (as described in Section [REF]), and if the Old patterns are the same as were used to create the multiple alignment shown in Figure [REF], then the best multiple alignment found by the system is the one shown in Figure [REF].", '1306.3888-1-107-1': "This multiple alignment contains the same words as the original sentence ('t h i s b o y l o v e s t h a t g i r l'), in the same order as the original.", '1306.3888-1-107-2': "Readers who are familiar with Prolog, will recognise that this process of recreating the original sentence from its encoding is similar in some respects to the way in which an appropriately-constructed Prolog program may be run 'backwards', deriving 'data' from 'results'.", '1306.3888-1-108-0': 'How is it possible to decompress the compressed code for the original sentence by using information compression?', '1306.3888-1-108-1': "This apparent paradox-decompression by compression-may be resolved by ensuring that, when a code pattern like 'S 0 1 0 1 0 S' is used to recreate the original data, each symbol is treated, at least notionally, as if contained a few more bits of information than is strictly necessary.", '1306.3888-1-108-2': 'That residual redundancy allows the system to recreate the original sentence by the same process of compression as was used to create the original parsing and encoding.', '1306.3888-1-109-0': 'This process of creating a relatively large pattern from a relatively small encoding provides a model for the creation of sentences by a person or an artificial system.', '1306.3888-1-109-1': "But instead of the New pattern being a rather dry code, like 'S 0 1 0 1 0 S', it would be more plausible if it were some kind of representation of the meaning of the sentence, like that mentioned in Section [REF].", '1306.3888-1-109-2': 'How a sentence may be generated from a representation of meaning is outlined in B (Section 5.7.1).', '1306.3888-1-110-0': "Similar principles may apply to other kinds of 'output', such as planning an outing, cooking a meal, and so on.", '1306.3888-1-111-0': '# Unsupervised learning', '1306.3888-1-112-0': 'As was mentioned in Section [REF], part of the inspiration for the SP theory has been a programme of research developing models of the unsupervised learning of language.', '1306.3888-1-112-1': 'But although the SNPR model is quite successful in deriving plausible grammars from samples of English-like artificial language, it has proved to be quite unsuitable as a basis for the SP theory.', '1306.3888-1-112-2': 'In order to accommodate other aspects of intelligence, such as pattern recognition, reasoning, and problem solving, it has been necessary to develop an entirely new conceptual framework, with multiple alignment at centre stage.', '1306.3888-1-113-0': 'So there is now the curious paradox that, while the SP theory is rooted in work on unsupervised learning, and that kind of learning has a central role in the theory, the SP model does much the same things as the earlier model, and with similar limitations (Sections [REF] and [REF]).', '1306.3888-1-113-1': 'But I believe that the new conceptual framework has many advantages, that it provides a much sounder footing for further developments, and that with some reorganisation of the learning processes in the SP model, its current weaknesses may be overcome (Section [REF]).', '1306.3888-1-114-0': '## Outline of unsupervised learning in the SP model', '1306.3888-1-115-0': 'The outline of the SP model in this section aims to provide sufficient detail for a good intuitive grasp of how it works.', '1306.3888-1-115-1': 'A lot more detail may be found in B (Chapter 9).', '1306.3888-1-116-0': 'In addition to the processes for building multiple alignments, the SP model has processes for deriving Old patterns from multiple alignments, evaluating sets of newly-created Old patterns in terms of their effectiveness for the economical encoding of the New information, and the weeding out low-scoring sets.', '1306.3888-1-116-1': 'The system does not merely record statistical information, it uses that information to learn new structures.', '1306.3888-1-117-0': '### Deriving Old patterns from multiple alignments', '1306.3888-1-118-0': 'The process of deriving Old patterns from multiple alignments is illustrated schematically in Figure [REF].', '1306.3888-1-118-1': "As was mentioned in Section [REF], the SP system is conceived as an abstract brain-like system that, in 'input' mode, may receive 'New' information via its senses and store some or all of it as 'Old' information.", '1306.3888-1-118-2': 'Here, we may think of it as the brain of a baby who is listening to what people are saying.', '1306.3888-1-118-3': "Let's imagine that he or she hears someone say 't h a t b o y r u n s'.", '1306.3888-1-118-4': 'If the baby has never heard anything similar, then, if it is stored at all, that New information may be stored as a relatively straightforward copy, something like the Old pattern shown in row 1 of the multiple alignment in part (a) of the figure.', '1306.3888-1-119-0': "Now let us imagine that the information has been stored and that, at some later stage, the baby hears someone say 't h a t g i r l r u n s'.", '1306.3888-1-119-1': 'Then, from that New information and the previously-stored Old pattern, a multiple alignment may be created like the one shown in part (a) of Figure [REF].', '1306.3888-1-119-2': "And, by picking out coherent sequences that are either fully matched or not matched at all, four putative words may be extracted: 't h a t', 'g i r l', 'b o y', and 'r u n s', as shown in the first four patterns in part (b) of the figure.", '1306.3888-1-119-3': "In each newly-created Old pattern there are additional symbols such as 'B', '2', and 'B' that serve to identify the pattern, to mark its boundaries, and to mark its grammatical category or categories.", '1306.3888-1-120-0': "In addition to these four patterns, a fifth pattern is created, 'E 6 B B C C D D E', as shown in the figure, that records the sequence 't h a t ... r u n s', with the category 'C C' in the middle representing a choice between 'b o y' and 'g i r l'.", '1306.3888-1-120-1': 'Part (b) in the figure is the beginnings of a grammar to describe that kind of phrase.', '1306.3888-1-121-0': '### Evaluating and selecting sets of newly-created Old patterns', '1306.3888-1-122-0': 'The example just described shows how Old patterns may be derived from a multiple alignment but it gives a highly misleading impression of how the SP model actually works.', '1306.3888-1-122-1': "In practice, the program forms many multiple alignments that are much less tidy than the one shown and it creates many Old patterns that are clearly 'wrong'.", '1306.3888-1-122-2': "However, the program contains procedures for evaluating candidate sets of patterns ('grammars') and weeding out those that score badly in terms of their effectiveness for encoding the New information economically.", '1306.3888-1-122-3': "Out of all the muddle, it can normally abstract one or two 'best' grammars and these are normally ones that appear intuitively to be 'correct', or nearly so.", '1306.3888-1-122-4': 'In general, the program can abstract one or more plausible grammars from a sample of English-like artificial language, including words, grammatical categories of words, and sentence structure.', '1306.3888-1-123-0': 'In accordance with the principles of minimum length encoding , the aim of these processes of sifting and sorting is to minimise [MATH], where [MATH] is the size (in bits) of the grammar that is under development and [MATH] is the size (in bits) of the New patterns when they have been encoded in terms of the grammar.', '1306.3888-1-124-0': 'For a given grammar comprising patterns [MATH], the value of [MATH] is calculated as: [EQUATION] where [MATH] is the number of symbols in the [MATH]th pattern and [MATH] is the encoding cost of the [MATH]th symbol in that pattern.', '1306.3888-1-125-0': 'Given that each grammar is derived from a set [MATH] of multiple alignments (one multiple alignment for each pattern from New), the value of [MATH] for the grammar is calculated as: [EQUATION] where [MATH] is the size, in bits, of the code string derived from the [MATH]th multiple alignment (Section [REF]).', '1306.3888-1-126-0': 'For a given set of patterns from New, a tree of alternative grammars is created with branching occurring wherever there are two or more alternative multiple alignments for a given pattern from New.', '1306.3888-1-126-1': 'The tree is grown in stages and pruned periodically to keep it within reasonable bounds.', '1306.3888-1-126-2': 'At each stage, grammars with high values for [MATH] (which will be referred to as [MATH]) are eliminated.', '1306.3888-1-127-0': '### Plotting values for [MATH], [MATH] and [MATH]', '1306.3888-1-128-0': 'Figure [REF] shows cumulative values for [MATH], [MATH] and [MATH] as the SP model searches for good grammars for a succession of 8 New patterns, each of which represents a sentence.', '1306.3888-1-128-1': 'Each point on each of the lower three graphs represents the relevant value (on the scale at the left) from the best grammar found after a given pattern from New has been processed.', '1306.3888-1-128-2': "The graph labelled '[MATH]' shows cumulative values on the scale at the left for the succession of New patterns.", '1306.3888-1-128-3': "The graph labelled '[MATH]' shows the amount of compression achieved (on the scale to the right).", '1306.3888-1-129-0': '### Limitations in the SP model and how they may be overcome', '1306.3888-1-130-0': 'As mentioned before (Section [REF]), there are two main weaknesses in the processes for unsupervised learning in the SP model as it is now: the model does not learn intermediate levels in a grammar (phrases or clauses) or discontinuous dependencies of the kind described in Sections [REF] to [REF].', '1306.3888-1-131-0': 'It appears that some reorganisation of the learning processes in the model would solve both problems.', '1306.3888-1-131-1': "What seems to be needed is a tighter focus on the principle that, with appropriately-constructed Old patterns, multiple alignments may be created without the kind of mis-match between patterns that may be seen in Figure [REF] (a) ('g i r l' and 'b o y' do not match each other), and that any such multiple alignment may be treated as if it was a simple pattern.", '1306.3888-1-131-2': 'That reform should facilitate the discovery of structures at multiple levels and the discovery of structures that are discontinuous in the sense that they can bridge intervening structures.', '1306.3888-1-132-0': '### Computational complexity', '1306.3888-1-133-0': 'As with the building of multiple alignments (Section [REF]), the computational complexity of learning in the SP model is kept under control by pruning the search tree at appropriate points, aiming to discover grammars that are reasonably good and not necessarily perfect.', '1306.3888-1-134-0': 'In a serial processing environment, the time complexity of learning in the SP model has been estimated to be O[MATH] where [MATH] is the number of patterns in New.', '1306.3888-1-134-1': 'In a parallel processing environment, the time complexity may approach O[MATH], depending on how well the parallel processing is applied.', '1306.3888-1-134-2': 'In serial or parallel environments, the space complexity has been estimated to be O[MATH].', '1306.3888-1-135-0': '## The discovery of natural structures via information compression (DONSVIC)', '1306.3888-1-136-0': "In our dealings with the world, certain kinds of structures appear to be more prominent and useful than others: in natural languages, there are words, phrase and sentences; we understand the visual and tactile worlds to be composed of discrete 'objects'; and conceptually, we recognise classes of things like 'person', 'house', 'tree', and so on.", '1306.3888-1-137-0': "It appears that these 'natural' kinds of structure are significant in our thinking because they provide a means of compressing sensory information, and that compression of information provides the key to their learning or discovery.", '1306.3888-1-137-1': 'At first sight, this looks like nonsense because popular programs for compression of information, such as those based on the LZW algorithm, or programs for JPEG compression of images, seem not to recognise anything resembling words, objects, or classes.', '1306.3888-1-137-2': 'But those programs are designed to work fast on low-powered computers.', '1306.3888-1-137-3': 'With other programs that are designed to be relatively thorough in their compression of information, natural structures can be revealed:', '1306.3888-1-138-0': 'It seems likely that the principles that have been outlined in this subsection may be applied not only to the discovery of words, phrases and grammars in language-like data but also to such things as the discovery of objects in images , and classes of entity in all kinds of data.', '1306.3888-1-138-1': "These principles may be characterised as the discovery of natural structures via information compression, or 'DONSVIC' for short.", '1306.3888-1-139-0': '## Generalisation, the correction of overgeneralisations, and learning from noisy data', '1306.3888-1-140-0': 'Issues that arise in the learning of a first language and, probably, in other kinds of learning, are illustrated in Figure [REF]:', '1306.3888-1-141-0': 'One possible answer is that mistakes are corrected by parents, teachers, and others.', '1306.3888-1-141-1': 'But the weight of evidence is that children can learn their first language without that kind of assistance.', '1306.3888-1-142-0': 'A better answer is the principle of minimum length encoding (described in its essentials in Section [REF]):', '1306.3888-1-143-0': '# Computing, mathematics, and logic', '1306.3888-1-144-0': 'Drawing mainly on B (Chapters 4 to 11), this and the following sections describe, with a selection of examples, how the SP theory relates to several areas in artificial intelligence, mainstream computing, and human perception and cognition.', '1306.3888-1-145-0': 'In B (Chapter 4), I have argued that the SP system is equivalent to a universal Turing machine , in the sense that anything that may be computed with a Turing machine may, in principle, also be computed with an SP machine.', '1306.3888-1-145-1': "The 'in principle' qualification is necessary because the SP theory is still not fully mature and there are still some weaknesses in the SP computer models.", '1306.3888-1-145-2': 'The gist of the argument is that the operation of a Post canonical system may be understood in terms of the SP theory and, since it is accepted that the Post canonical system is equivalent to the Turing machine (as a computational system), the Turing machine may also be understood in terms of the SP theory.', '1306.3888-1-146-0': "The key differences between the SP theory and earlier theories of computing are that the SP theory has a lot more to say about the nature of intelligence than earlier theories, that the theory is founded on principles of information compression via the matching and unification of patterns ('computing as compression'), and that it includes mechanisms for building multiple alignments and for heuristic search that are not present in earlier models.", '1306.3888-1-147-0': '## Conventional computing systems', '1306.3888-1-148-0': 'In conventional computing systems, compression of information may be seen in the matching of patterns with at least implicit unification of patterns that match each other-processes that appear in a variety of guises (B, Chapter 2).', '1306.3888-1-148-1': 'And three basic techniques for the compression of information-chunking-with-codes, schema-plus-correction, and run-length coding-may be seen in various forms in the organisation of computer programs (ibid.)', '1306.3888-1-149-0': '## Mathematics and logic', '1306.3888-1-150-0': 'In a similar way, several structures and processes in mathematics and logic may be interpreted in terms of information compression via the matching and unification of patterns, and the compression techniques just mentioned (B, Chapter 10).', '1306.3888-1-150-1': 'For example, multiplication (as repeated addition) and exponentiation (as repeated multiplication) may be seen as examples of run-length coding; a function with parameters may be seen as an example of schema-plus-correction; the chunking-with-codes technique may be seen in the organisation of number systems; and so on.', '1306.3888-1-151-0': '## Computing and probabilities', '1306.3888-1-152-0': 'As we have seen, the SP system is fundamentally probabilistic.', '1306.3888-1-152-1': "If it is indeed Turing-equivalent, as suggested above, and if the Turing machine is regarded as a definition of 'computing', then we may conclude that computing is fundamentally probabilistic.", '1306.3888-1-152-2': 'That may seem like a strange conclusion in view of the clockwork certainties that we associate with the operation of ordinary computers and the workings of mathematics and logic.', '1306.3888-1-152-3': 'There are at least three answers to that apparent contradiction:', '1306.3888-1-153-0': '# Representation of knowledge', '1306.3888-1-154-0': "Within the multiple alignment framework (Section [REF]), SP patterns may serve to represent several kinds of knowledge, including grammars for natural languages, ontologies, class hierarchies with inheritance of attributes, including cross-classification or multiple inheritance, part-whole hierarchies and their integration with class-inclusion hierarchies, decision networks and trees, relational tuples, if-then rules, associations of medical signs and symptoms, causal relations, and concepts in mathematics and logic such as 'function', 'variable', 'value', 'set', and 'type definition'.", '1306.3888-1-155-0': 'The use of one simple format for the representation of knowledge facilitates the seamless integration of different kinds of knowledge.', '1306.3888-1-156-0': '# Natural language processing', '1306.3888-1-157-0': 'One of the main strengths of the SP system is in natural language processing (B, Chapter 5):', '1306.3888-1-158-0': '## Discontinuous dependencies in syntax', '1306.3888-1-159-0': 'The way in which the SP system can record discontinuous dependencies in syntax may be seen in both of the two parsings in Figure [REF].', '1306.3888-1-159-1': "The pattern in row 8 of each multiple alignment records the syntactic dependency between the plural noun phrase ('t w o k i t t e n s') which is the subject of the sentence-marked with 'Np'-and the plural verb phrase ('p l a y')-marked with 'Vp'-which belongs with it.", '1306.3888-1-160-0': "This kind of dependency is discontinuous because it can bridge arbitrarily large amounts of intervening structure such as, for example, 'from the West' in a sentence like 'Winds from the West are strong'.", '1306.3888-1-161-0': 'This method of marking discontinuous dependencies can accommodate overlapping dependencies such as number dependencies and gender dependencies in languages like French (B, Section 5.4).', '1306.3888-1-161-1': 'It also provides a means of encoding the interesting system of overlapping and interlocking dependencies in English auxiliary verbs, described by Noam Chomsky in Syntactic Structuresyearparchomsky_1957.', '1306.3888-1-162-0': "In that book, the structure of English auxiliary verbs is part of Chomsky's evidence in support of Transformational Grammar.", '1306.3888-1-162-1': 'Despite the elegance and persuasiveness of his arguments, it turns out that the structure of English auxiliary verbs may be described with non-transformational rules in, for example, Definite Clause Grammars , and also in the SP system, as outlined in the subsections that follow.', '1306.3888-1-163-0': '## Two quasi-independent patterns of constraint in English auxiliary verbs', '1306.3888-1-164-0': 'In English, the syntax for main verbs and the auxiliary verbs which may accompany them follows two quasi-independent patterns of constraint which interact in an interesting way.', '1306.3888-1-165-0': 'The primary constraints may be expressed with this sequence of symbols,', '1306.3888-1-166-0': 'M H B B V,', '1306.3888-1-167-0': 'which should be interpreted in the following way:', '1306.3888-1-168-0': 'The words occur in the order shown but any of the words may be omitted.', '1306.3888-1-169-0': "Questions of 'standard' form follow exactly the same pattern as statements except that the first verb, whatever it happens to be ('M', 'H', the first 'B', the second 'B' or 'V'), precedes the subject noun phrase instead of following it.", '1306.3888-1-170-0': 'Here are two examples of the primary pattern with all of the words included:', '1306.3888-1-171-0': 'It will have been being washed M H B B V', '1306.3888-1-172-0': 'Will it have been being washed?', '1306.3888-1-172-1': 'M H B B V', '1306.3888-1-173-0': 'The secondary constraints are these:', '1306.3888-1-174-0': 'Figure [REF] shows a selection of examples with the dependencies marked.', '1306.3888-1-175-0': '## Multiple alignments and English auxiliary verbs', '1306.3888-1-176-0': 'Without reproducing all the detail in B (Section 5.5), we can see from Figures [REF] and [REF] how the primary and secondary constraints may be applied in the multiple alignment framework.', '1306.3888-1-177-0': 'In each figure, the sentence to be analysed is shown as a New pattern in column 0.', '1306.3888-1-177-1': "The primary constraints are applied via the matching of symbols in Old patterns in the remaining columns, with a consequent interlocking of the patterns so that they recognise sentences of the form 'M H B B V', with options as described above.", '1306.3888-1-178-0': 'In Figure [REF], the secondary constraints apply as follows:', '1306.3888-1-179-0': 'In Figure [REF], the secondary constraints apply like this:', '1306.3888-1-180-0': '# Pattern recognition', '1306.3888-1-181-0': 'The system also has some useful features as a framework for pattern recognition (B, (Chapter 6):', '1306.3888-1-182-0': 'One area of application is medical diagnosis , viewed as pattern recognition.', '1306.3888-1-182-1': 'There is also potential to assist in the understanding of natural vision and in the development of computer vision, as discussed in [CITATION].', '1306.3888-1-183-0': '## Part-whole hierarchies, class hierarchies, and their integration', '1306.3888-1-184-0': 'A strength of the multiple alignment concept is that it provides a simple but effective vehicle for the representation and processing of class-inclusion hierarchies, part-whole hierarchies, and their integration.', '1306.3888-1-185-0': "Figure [REF] shows the best multiple alignment found by the SP model with the New pattern 'white-bib eats furry purrs' (column 0) representing some features of an unknown creature, and with a set of Old patterns representing different classes of animal, at varying levels of abstraction.", '1306.3888-1-185-1': "From this multiple alignment, we may conclude that the unknown entity is an animal (column 1), a mammal (column 2), a cat (column 3) and the specific individual 'Tibs' (column 4).", '1306.3888-1-186-0': "The framework also provides for the representation of heterarchies or cross classification: a given entity, such as 'Jane' (or a class of entities), may belong in two or more higher-level classes that are not themselves hierarchically related, such as 'woman' and 'doctor'.", '1306.3888-1-187-0': 'The way that class-inclusion relations and part-whole relations may be combined in one multiple alignment is illustrated in Figure [REF].', '1306.3888-1-187-1': 'Here, some features of an unknown plant are expressed as a set of New patterns, shown in column 0: the plant has chlorophyll, the stem is hairy, it has yellow petals, and so on.', '1306.3888-1-188-0': 'From this multiple alignment, we can see that the unknown plant is most likely to be the Meadow Buttercup, Ranunculus acris, as shown in column 1.', '1306.3888-1-188-1': 'As such, it belongs in the genus Ranunculus (column 6), the family Ranunculaceae (column 5), the order Ranunculales (column 4), the class Angiospermae (column 3), and the phylum Plants (column 2).', '1306.3888-1-189-0': 'Each of these higher-level classifications contributes information about attributes of the plant and its division into parts and sub-parts.', '1306.3888-1-189-1': "For example, as a member of the class Angiospermae (column 3), the plant has a shoot and roots, with the shoot divided into stem, leaves, and flowers; as a member of the family Ranunculaceae (column 5), the plant has flowers that are 'regular', with all parts 'free'; as a member of the phylum Plants (column 2), the buttercup has chlorophyll and creates its own food by photosynthesis; and so on.", '1306.3888-1-190-0': '## Inference and inheritance', '1306.3888-1-191-0': 'In the example just described, we can infer from the multiple alignment, very directly, that the plant which has been provisionally identified as the Meadow Buttercup performs photosynthesis (column 2), has five petals (column 6), is poisonous (column 5), and so on.', '1306.3888-1-191-1': "As in other object-oriented systems, the first of these attributes has been 'inherited' from the class 'Plants', the second from the class Ranunculus, and the third from the class Ranunculaceae.", '1306.3888-1-191-2': 'These kinds of inference illustrate the close connection, often remarked, between pattern recognition and inferential reasoning .', '1306.3888-1-192-0': '# Probabilistic reasoning', '1306.3888-1-193-0': "The SP system can model several kinds of reasoning including inheritance of attributes (as just described), one-step 'deductive' reasoning, abductive reasoning, reasoning with probabilistic decision networks and decision trees, reasoning with 'rules', nonmonotonic reasoning and reasoning with default values, reasoning in Bayesian networks (including 'explaining away'), causal diagnosis, and reasoning which is not supported by evidence (B, Chapter 7).", '1306.3888-1-194-0': 'Since these several kinds of reasoning all flow from one computational framework (multiple alignment), they may be seen as aspects of one process, working individually or together without awkward boundaries.', '1306.3888-1-195-0': 'Plausible lines of reasoning may be achieved, even when relevant information is incomplete.', '1306.3888-1-196-0': "Probabilities of inferences may be calculated, including extreme values (0 or 1) in the case of logic-like 'deductions'.", '1306.3888-1-197-0': 'A selection of examples is described in the following subsections.', '1306.3888-1-198-0': '## Nonmonotonic reasoning and reasoning with default values', '1306.3888-1-199-0': 'Conventional deductive reasoning is monotonic because deductions made on the strength of current knowledge cannot be invalidated by new knowledge: the conclusion that "Socrates is mortal", deduced from "All humans are mortal" and "Socrates is human", remains true for all time, regardless of anything we learn later.', '1306.3888-1-199-1': 'By contrast, the inference that "Tweety can probably fly" from the propositions that "Most birds fly" and "Tweety is a bird" is nonmonotonic because it may be changed if, for example, we learn that Tweety is a penguin.', '1306.3888-1-200-0': 'This section presents some examples which show how the SP system can accommodate nonmonotonic reasoning.', '1306.3888-1-201-0': '### Typically, birds fly', '1306.3888-1-202-0': "The idea that (all) birds can fly may be expressed with an SP pattern like 'Bd bird name name canfly warm-blooded wings feathers ... Bd'.", '1306.3888-1-202-1': 'This, of course, is an oversimplification of the real-world facts because, while it true that the majority of birds fly, we know that there are also flightless birds like ostriches, penguins and kiwis.', '1306.3888-1-203-0': "In order to model these facts more closely, we need to modify the pattern that describes birds to be something like this: 'Bd bird name name f f warm-blooded wings feathers ... Bd.", '1306.3888-1-203-1': 'And, to our database of Old patterns, we need to add patterns like this:', '1306.3888-1-204-0': 'Default Bd f canfly #f #Bd #Default P penguin Bd f cannotfly #f #Bd ... #P O ostrich Bd f cannotfly #f #Bd ... #O.', '1306.3888-1-205-0': "Now, the pair of symbols 'f f' in 'Bd bird name name f f warm-blooded wings feathers ... Bd' functions like a 'variable' that may take the value 'canfly' if a given class of birds can fly and 'cannotfly' when a type of bird cannot fly.", '1306.3888-1-205-1': "The pattern 'P penguin Bd f cannotfly f Bd ... P' shows that penguins cannot fly and, likewise, the pattern 'O ostrich Bd f cannotfly f Bd ... O' shows that ostriches cannot fly.", '1306.3888-1-205-2': "The pattern 'Default Bd f canfly f Bd Default', which has a substantially higher frequency than the other two patterns, represents the default value for the variable which is 'canfly'.", '1306.3888-1-205-3': "Notice that all three of these patterns contains the pair of symbols 'Bd ... Bd' showing that the corresponding classes are all subclasses of birds.", '1306.3888-1-206-0': '### Tweety is a bird so, probably, Tweety can fly', '1306.3888-1-207-0': "When the SP model is run with 'bird Tweety' in New and the same patterns in Old as before, modified as just described, the three best multiple alignments found are those shown in Figures [REF], [REF] and [REF].", '1306.3888-1-208-0': 'The first multiple alignment tells us that, with a relative probability of 0.66, Tweety may be the typical kind of bird that can fly.', '1306.3888-1-208-1': 'The second multiple alignment tells us that, with a relative probability of 0.22, Tweety might be an ostrich and, as such, he or she would not be able to fly.', '1306.3888-1-208-2': 'Likewise, the third multiple alignment tells us that, with a relative probability of 0.12, Tweety might be a penguin and would not be able to fly.', '1306.3888-1-208-3': 'The values for probabilities in this simple example are derived from guestimated frequencies that are, almost certainly, not ornithologically correct.', '1306.3888-1-209-0': '### Tweety is a penguin, so Tweety cannot fly', '1306.3888-1-210-0': "Figure [REF] shows the best multiple alignment found by the SP model when it is run again, with 'penguin Tweety' in New instead of 'bird Tweety'.", '1306.3888-1-210-1': 'This time, there is only one multiple alignment in the reference set and its relative probability is 1.0.', '1306.3888-1-210-2': 'Correspondingly, all inferences that we can draw from this multiple alignment have a probability of 1.0.', '1306.3888-1-210-3': 'In particular, we can be confident, within the limits of the available knowledge, that Tweety cannot fly.', '1306.3888-1-211-0': 'In a similar way, if Tweety were an ostrich, we would be able to say with confidence (p = 1.0) that he or she would not be able to fly.', '1306.3888-1-212-0': "## Reasoning in Bayesian networks, including 'explaining away'", '1306.3888-1-213-0': "A Bayesian network is a directed, acyclic graph like the one shown in Figure [REF], below, where each node has zero or more 'inputs' (connections with nodes that can influence the given node) and one or more 'outputs' (connections to other nodes that the given node can influence).", '1306.3888-1-214-0': 'Each node contains a set of conditional probability values, each one the probability of a given output value for a given input value or combination of input values.', '1306.3888-1-214-1': 'With this information, conditional probabilities of alternative outputs for any node may be computed for any given combination of inputs.', '1306.3888-1-214-2': "By combining these calculations for sequences of nodes, probabilities may be propagated through the network from one or more 'start' nodes to one or more 'finishing' nodes.", '1306.3888-1-215-0': 'This section describes how the SP system may perform that kind of probabilistic reasoning, and some advantages compared with Bayesian networks.', '1306.3888-1-216-0': 'Judea Pearlyearpar[p. 7]pearl_1997 describes the phenomenon of \'explaining away\' like this: "If A implies B, C implies B, and B is true, then finding that C is true makes A less credible.', '1306.3888-1-216-1': 'In other words, finding a second explanation for an item of data makes the first explanation less credible.', '1306.3888-1-216-2': '" (his italics).', '1306.3888-1-216-3': 'Here is an example:', '1306.3888-1-217-0': 'Normally an alarm sound alerts us to the possibility of a burglary.', '1306.3888-1-217-1': 'If somebody calls you at the office and tells you that your alarm went off, you will surely rush home in a hurry, even though there could be other causes for the alarm sound.', '1306.3888-1-217-2': 'If you hear a radio announcement that there was an earthquake nearby, and if the last false alarm you recall was triggered by an earthquake, then your certainty of a burglary will diminish.', '1306.3888-1-218-0': 'The causal relationships in the example just described may be captured in a Bayesian network like the one shown in Figure [REF].', '1306.3888-1-219-0': 'Pearl argues that, with appropriate values for conditional probabilities, the phenomenon of "explaining away" can be explained in terms of this network (representing the case where there is a radio announcement of an earthquake) compared with the same network without the node for "radio announcement" (representing the situation where there is no radio announcement of an earthquake).', '1306.3888-1-220-0': '### Representing contingencies with patterns and frequencies', '1306.3888-1-221-0': 'To see how this phenomenon may be understood in terms of the SP theory, consider, first, the set of patterns shown in Figure [REF], which are to be stored in Old.', '1306.3888-1-221-1': "The patterns in the figure show events which occur together in some notional sample of the 'World' together with their frequencies of occurrence in the sample.", '1306.3888-1-222-0': "As with other knowledge-based systems, we shall assume that the 'closed-world' assumption applies so that the absence of any pattern may be taken to mean that the corresponding combination of events did not occur in the period when observations were made.", '1306.3888-1-223-0': "The first pattern ('burglary alarm') shows that there were 1000 occasions when there was a burglary and the alarm went off and the second pattern ('earthquake alarm') shows just 20 occasions when there was an earthquake and the alarm went off (presumably triggered by the earthquake).", '1306.3888-1-223-1': 'Thus we have assumed that, as triggers for the alarm, burglaries are much more common than earthquakes.', '1306.3888-1-224-0': 'Since there is no pattern showing that the alarm sounded when there was a burglary and an earthquake at the same time, we may assume, via the closed-world assumption, that nothing like that happened during the sampling period.', '1306.3888-1-225-0': "The third pattern ('alarm phonealarmcall') shows that, out of the 1020 cases when the alarm went off, there were 980 cases where a phone call about the alarm was made.", '1306.3888-1-225-1': 'Since there is no pattern showing phone calls about the alarm in any other context, the closed-world assumption allows us to assume that there were no false positives (eg., hoaxes)-phone calls about the alarm when no alarm had sounded.', '1306.3888-1-226-0': "The fourth pattern ('earthquake radioearthquakeannouncement') shows that, in the sampling period, there were 40 occasions when there was an earthquake with an announcement about it on the radio.", '1306.3888-1-226-1': "And the fifth pattern ('e1 earthquake e2') shows that an earthquake has occurred on 40 occasions in contexts where the alarm did not ring and there was no radio announcement.", '1306.3888-1-227-0': "As before, the absence of patterns like 'earthquake alarm radioearthquakeannouncement' representing cases where an earthquake triggers the alarm and also leads to a radio announcement, allows us to assume via the closed-world assumption that cases of that kind have not occurred in the sampling period.", '1306.3888-1-228-0': '### Approximating the temporal order of events', '1306.3888-1-229-0': 'In these patterns and in the multiple alignments shown below, the left-to-right order of symbols may be regarded as an approximation to the order of events in time.', '1306.3888-1-229-1': 'Thus in the first two patterns, events that can trigger an alarm precede the sounding of the alarm.', '1306.3888-1-229-2': "Likewise, in the third pattern, 'alarm' (meaning that the alarm has sounded) precedes 'phonealarmcall' (a phone call to say the alarm has sounded).", '1306.3888-1-229-3': 'A single dimension can only approximate the order of events in time because it cannot represent events which overlap in time or which occur simultaneously.', '1306.3888-1-229-4': 'However, this kind of approximation has little or no bearing on the points to be illustrated here.', '1306.3888-1-230-0': '### Other considerations', '1306.3888-1-231-0': 'Other points relating to the patterns shown in Figure [REF] include:', '1306.3888-1-232-0': '### Formation of alignments: the burglar alarm has sounded', '1306.3888-1-233-0': "Receiving a phone call to say that the burglar alarm at one's house has gone off may be represented by placing the symbol 'phonealarmcall' in New.", '1306.3888-1-233-1': 'Figure [REF] shows, at the top, the best multiple alignment formed by the SP model in this case, with the patterns from Figure [REF] in Old.', '1306.3888-1-233-2': 'The other two multiple alignments in the reference set are shown below the best multiple alignment, in order of CD value and relative probability.', '1306.3888-1-233-3': 'The actual values for [MATH] and relative probability are given in the caption to Figure [REF].', '1306.3888-1-234-0': 'The unmatched Old symbols in these multiple alignments represent inferences made by the system.', '1306.3888-1-234-1': 'The probabilities for these inferences which are calculated by the SP model (as outlined in Section [REF]) are shown in Table [REF].', '1306.3888-1-234-2': 'These probabilities do not add up to 1 and we should not expect them to because any given multiple alignment may contain two or more of these symbols.', '1306.3888-1-235-0': 'The most probable inference is the rather trivial inference that the alarm has indeed sounded.', '1306.3888-1-235-1': 'This reflects the fact that there is no pattern in Figure [REF] representing false positives for telephone calls about the alarm.', '1306.3888-1-235-2': 'Apart from the inference that the alarm has sounded, the most probable inference (p = 0.328) is that there has been a burglary.', '1306.3888-1-235-3': 'However, there is a distinct possibility that there has been an earthquake-but the probability in this case (p = 0.016) is much lower than the probability of a burglary.', '1306.3888-1-236-0': "These inferences and their relative probabilities seem to accord quite well with what one would naturally think following a telephone call to say that the burglar alarm at one's house has gone off (given that one was living in a part of the world where earthquakes were not vanishingly rare).", '1306.3888-1-237-0': '## Formation of alignments: the burglar alarm has sounded and there is a radio announcement of an earthquake', '1306.3888-1-238-0': "In this example, the phenomenon of 'explaining away' occurs when you learn not only that the burglar alarm has sounded but that there has been an announcement on the radio that there has been an earthquake.", '1306.3888-1-238-1': 'In terms of the SP model, the two events (the phone call about the alarm and the announcement about the earthquake) can be represented in New by a pattern like this:', '1306.3888-1-239-0': 'phone_alarm_call radio_earthquake_announcement', '1306.3888-1-240-0': "or 'radioearthquakeannouncement phonealarmcall'.", '1306.3888-1-240-1': 'The order of the two symbols makes no difference to the result.', '1306.3888-1-241-0': "In this case, there is only one multiple alignment (shown at the top of Figure [REF]) that can 'explain' all the information in New.", '1306.3888-1-241-1': "Since there is only this one multiple alignment in the reference set for the best multiple alignment, the associated probabilities of the inferences that can be read from the multiple alignment ('alarm' and 'earthquake') are 1.0: it was an earthquake that caused the alarm to go off (and led to the phone call), and not a burglary.", '1306.3888-1-242-0': "These results show how 'explaining away' may be explained in terms of the SP theory.", '1306.3888-1-242-1': "The main point is that the multiple alignment or multiple alignments that provide the best 'explanation' of a telephone call to say that one's burglar alarm has sounded is different from the multiple alignment or multiple alignments that best explain the same telephone call coupled with an announcement on the radio that there has been an earthquake.", '1306.3888-1-242-2': 'In the latter case, the best explanation is that the earthquake triggered the alarm.', '1306.3888-1-242-3': 'Other possible explanations have lower probabilities.', '1306.3888-1-243-0': '### Other possibilities', '1306.3888-1-244-0': 'As mentioned above, the closed-world assumption allows us to rule out possibilities such as:', '1306.3888-1-245-0': 'Nevertheless, we may consider possibilities of that kind by combining multiple alignments as described in B (Section 7.8.7).', '1306.3888-1-245-1': 'But as a general rule, that kind of further analysis makes no difference to the original conclusion: the multiple alignment which was originally judged to represent the best interpretation of the available facts has not been dislodged from this position.', '1306.3888-1-245-2': 'This is in keeping with way we normally concentrate on the most likely explanations of events and ignore the many conceivable but unlikely alternatives.', '1306.3888-1-246-0': '## The SP framework and Bayesian networks', '1306.3888-1-247-0': 'The foregoing examples show that the SP framework is a viable alternative to Bayesian networks, at least in the kinds of situation that have been described.', '1306.3888-1-247-1': 'This subsection makes some general observations about the relative merits of the two frameworks for probabilistic reasoning where the events of interest are subject to multiple influences or chains of influence or both those things.', '1306.3888-1-248-0': "Without in any way diminishing Thomas Bayes' achievement, his theorem appears to have shortcomings as the basis for theorising about perception and cognition:", '1306.3888-1-249-0': '## Causal diagnosis', '1306.3888-1-250-0': 'In this section, we consider a simple example of fault diagnosis in an electronic circuit-described by [CITATION].', '1306.3888-1-250-1': 'Figure [REF] shows the circuit with inputs on the left, outputs on the right and, in between, three multipliers ([MATH], [MATH], and [MATH]) and two adders ([MATH] and [MATH]).', '1306.3888-1-250-2': 'For the given inputs on the left, it is clear that output F is false and output G is correct.', '1306.3888-1-251-0': 'Figure [REF] shows a causal network derived from the electronic circuit in Figure [REF] .', '1306.3888-1-251-1': 'In this diagram, [MATH], [MATH], [MATH], [MATH] and [MATH] represent the outputs of components [MATH], [MATH], [MATH], [MATH] and [MATH], respectively.', '1306.3888-1-251-2': "In each case, there are three causal influences on the output: the two inputs to the component and the state of the component which may be 'good' or 'bad'.", '1306.3888-1-251-3': 'These influences are shown in Figure [REF] by lines with arrows connecting the source of the influence to the target node.', '1306.3888-1-251-4': 'Thus, for example, the two inputs of component [MATH] are represented by [MATH] and [MATH], the good or bad state of component [MATH] is represented by the node labelled [MATH], and their causal influences on node [MATH] are shown by the three arrows pointing at that node.', '1306.3888-1-252-0': 'Given a causal analysis like this, and given appropriate information about conditional probabilities, it is possible to derive one or more alternative diagnoses of which components are good and which are bad.', '1306.3888-1-252-1': "In Pearl's example, it is assumed that components of the same type have the same prior probability of failure and that the probability of failure of multipliers is greater than for adders.", '1306.3888-1-252-2': 'Given these assumptions and some others together with the inputs and outputs shown in Figure [REF] (but not the intermediate values), the best diagnosis derived from the causal network is that the [MATH] component is bad and the second best diagnosis is that [MATH] is bad.', '1306.3888-1-252-3': 'Pearl indicates that some third-best interpretations may be retrievable (e.g., [MATH] and [MATH] are bad) "... but in general, it is not guaranteed that interpretations beyond the second-best will be retrievable."', '1306.3888-1-252-4': '(p. 272).', '1306.3888-1-253-0': '## An SP approach to causal diagnosis', '1306.3888-1-254-0': 'The main elements of the SP analysis presented here are as follows:', '1306.3888-1-255-0': 'Figure [REF] shows a set of patterns for the circuit shown in Figure [REF].', '1306.3888-1-255-1': "In the figure, the patterns that start with the symbol 'M1' represent I/O relations for component [MATH], those that start with 'M2' represent I/O relations for the [MATH] component and likewise for the other patterns except the last one (starting with the symbol 'frame') which is the framework pattern mentioned above.", '1306.3888-1-255-2': "For each initial symbol there is a corresponding 'terminating' symbol with an initial '' character.", '1306.3888-1-255-3': "For reasons explained shortly, there may be other symbols following the 'terminating' symbol.", '1306.3888-1-256-0': "Let us now consider the first pattern in the figure ('M1 M1GOOD TM1I1 TM1I2 TM1O M1 TM4I2') representing I/O relations for component [MATH] when that component is good, as indicated by the symbol 'M1GOOD'.", '1306.3888-1-256-1': "In this pattern, the symbols 'TM1I1', 'TM1I2' and 'TM1O' represent the two inputs and the output of the component, 'M1' is the terminating symbol, and 'TM4I2' serves to transfer the output of [MATH] to the second input of component [MATH] as will be explained.", '1306.3888-1-256-2': "In a symbol like 'TM1I1', 'T' indicates that the input is true, 'M1' identifies the component, and 'I1' indicates that this is the first input of the component.", '1306.3888-1-256-3': 'Other symbols may be interpreted in a similar way, following the key given in the caption of Figure [REF].', '1306.3888-1-256-4': 'In effect, this pattern says that, when the component is working correctly, true inputs yield a true output.', '1306.3888-1-256-5': 'The pattern has a relatively high frequency of occurrence (500,000) reflecting the idea that the component will normally work correctly.', '1306.3888-1-257-0': "The other two patterns for component [MATH] ('M1 M1BAD TM1I1 TM1I2 TM1O M1 TM4I2' and 'M1 M1BAD TM1I1 TM1I2 FM1O M1 FM4I2') describe I/O relations when the component is bad.", '1306.3888-1-257-1': 'The first one describes the situation where true inputs to a faulty component yield a true result, a possibility noted by Pearl (ibid.', '1306.3888-1-257-2': 'p. 265).', '1306.3888-1-257-3': 'The second pattern-with a higher frequency-describes the more usual situation where true inputs to a faulty component yield a false result.', '1306.3888-1-257-4': 'Both these bad patterns have much lower frequencies than the good pattern.', '1306.3888-1-258-0': 'The other patterns in Figure [REF] may be interpreted in a similar way.', '1306.3888-1-258-1': 'Components [MATH], [MATH] and [MATH] have only three patterns each because it is assumed that inputs to the circuit will always be true so it is not necessary to include patterns describing what happens when one or both of the inputs are false.', '1306.3888-1-258-2': 'By contrast, there are 4 good patterns and 8 bad patterns for each of [MATH] and [MATH] because either of these components may receive faulty input.', '1306.3888-1-259-0': 'For each of the five components, the frequencies of the bad patterns sum to 100.', '1306.3888-1-259-1': 'However, for each of the components [MATH], [MATH], and [MATH], the total frequency of the good patterns is 500,000 compared with 1,000,000 for the set of good patterns associated with each of the component [MATH] and [MATH].', '1306.3888-1-259-2': "These figures accord with the assumptions in Pearl's example that components of the same type have the same probability of failure and that the probability of failure of multipliers ([MATH], [MATH], and [MATH]) is greater than the probability of failure of adders ([MATH] and [MATH]).", '1306.3888-1-260-0': '## Multiple alignments in causal diagnosis', '1306.3888-1-261-0': 'Given appropriate patterns, the SP model constructs multiple alignments from which diagnoses may be obtained.', '1306.3888-1-261-1': "Figure [REF] shows the best multiple alignment created by the SP model with the Old patterns shown in Figure [REF] and 'TM1I1 TM1I2 TM2I1 TM2I2 TM3I1 TM3I2 FM4O TM5O' as the New pattern.", '1306.3888-1-261-2': 'The first six symbols in this pattern express the idea that all the inputs for components [MATH], [MATH] and [MATH] are true.', '1306.3888-1-261-3': "The penultimate symbol ('FM4O') shows that the output of [MATH] is false and the last symbol ('TM5O') shows that the output of [MATH] is true-in accordance with the outputs shown in Figure [REF].", '1306.3888-1-262-0': 'From the multiple alignment in Figure [REF] it can be inferred that component [MATH] is bad and all the other components are good.', '1306.3888-1-262-1': 'A total of seven alternative diagnoses can be derived from those multiple alignments created by the SP model that encode all the symbols in New.', '1306.3888-1-262-2': 'These diagnoses are shown in Table [REF], each with its relative probability.', '1306.3888-1-263-0': "It is interesting to see that the best diagnosis derived by the SP model ([MATH] is bad) and the second best diagnosis ([MATH] is bad) are in accordance with first two diagnoses obtained by Pearl's method.", '1306.3888-1-263-1': "The remaining five diagnoses derived by the SP model are different from the one obtained by Pearl's method ([MATH] and [MATH] are bad) but this is not altogether surprising because detailed frequencies or probabilities are different from Pearl's example and there are differences in assumptions that have been made.", '1306.3888-1-264-0': '# Information storage and retrieval', '1306.3888-1-265-0': "The SP theory provides a versatile model for database systems, with the ability to accommodate object-oriented structures, as well as relational 'tuples', and network and tree models of data .", '1306.3888-1-265-1': 'It lends itself most directly to information retrieval in the manner of query-by-example but it appears to have potential to support the use of natural language or query languages such as SQL.', '1306.3888-1-266-0': 'Unlike some ordinary database systems:', '1306.3888-1-267-0': '# Planning and problem solving', '1306.3888-1-268-0': 'The SP framework provides a means of planning a route between two places, and, with the translation of geometric patterns into textual form, it can solve the kind of geometric analogy problem that may be seen in some puzzle books and IQ tests (B, Chapter 8).', '1306.3888-1-269-0': 'Figure [REF] shows an example of the latter kind of problem.', '1306.3888-1-269-1': 'The task is to complete the relationship "A is to B as C is to ?"', '1306.3888-1-269-2': "using one of the figures 'D', 'E', 'F' or 'G' in the position marked with '?", '1306.3888-1-269-3': "'.", '1306.3888-1-269-4': "For this example, the 'correct' answer is clearly 'E'.", '1306.3888-1-269-5': "Quote marks have been used for the word 'correct' because in many problems of this type, there may be two or even more alternative answers for which cases can be made and there is a corresponding uncertainty about which answer is the right one.", '1306.3888-1-270-0': "Computer-based methods for solving this kind of problem have existed for some time (e.g., Evans'yearparevans_1968 well-known heuristic algorithm).", '1306.3888-1-270-1': 'In more recent work , minimum length encoding principles have been applied to good effect.', '1306.3888-1-270-2': 'This kind of problem may also be understood in terms of the SP concepts.', '1306.3888-1-271-0': 'As in most previous work, the proposed solution assumes that some mechanism is available which can translate the geometric forms in each problem into patterns of text symbols like other patterns in this article.', '1306.3888-1-271-1': "For example, item 'A' in Figure [REF] may be described as 'small circle inside large triangle'.", '1306.3888-1-272-0': 'How this kind of translation may be done is not part of the present proposals (one such translation mechanism is described in [CITATION]).', '1306.3888-1-272-1': 'As noted elsewhere , successful solutions for this kind of problem require consistency in the way the translation is done.', '1306.3888-1-272-2': "For this example, it would be unhelpful if item 'A' in Figure [REF] were described as 'large triangle outside small circle' while item 'C' were described as 'small square inside large ellipse'.", '1306.3888-1-272-3': "For any one puzzle, the description needs to stick to one or other of 'X outside Y' or 'Y inside X'-and likewise for 'above/below' and 'left-of/right-of'.", '1306.3888-1-273-0': 'Given that the diagrammatic form of the problem has been translated into patterns as just described, this kind of problem can be cast as a problem of partial matching, well within the scope of the SP model.', '1306.3888-1-273-1': 'To do this, symbolic representations of item A and item B in Figure [REF] are treated as a single pattern, thus:', '1306.3888-1-274-0': 'small circle inside large triangle ; large circle above small triangle', '1306.3888-1-275-0': 'and this pattern is placed in New.', '1306.3888-1-275-1': 'Four other patterns are constructed by pairing a symbolic representation of item C (on the left) with symbolic representations of each of D, E, F and G (on the right), thus:', '1306.3888-1-276-0': 'C1 small square inside large ellipse ; D small square inside large circle #C1 C2 small square inside large ellipse ; E large square above small ellipse #C2 C3 small square inside large ellipse ; F small circle left-of large square #C3 C4 small square inside large ellipse ; G small ellipse above large rectangle #C4.', '1306.3888-1-277-0': 'These four patterns are placed in Old, each with an arbitrary frequency value of 1.', '1306.3888-1-278-0': 'Figure [REF] shows the best multiple alignment found by the SP model with New and Old as just described.', '1306.3888-1-278-1': 'The multiple alignment is a partial match between the New pattern (in column 0) and the second of the four patterns in Old (in column 1).', '1306.3888-1-278-2': "This corresponds with the 'correct' result (item E) as noted above.", '1306.3888-1-279-0': '# Compression of information', '1306.3888-1-280-0': 'Since information compression is central to the workings of the SP system, it is natural to consider whether the system might provide useful insights in that area.', '1306.3888-1-280-1': 'In that connection, the most promising aspects of the SP system appear to be:', '1306.3888-1-281-0': "In terms of the trade-off that exists between computational resources that are required and the level of compression that can be achieved, it is intended that the system will operate towards the 'up market' end of the spectrum-by contrast with LZW algorithms and the like, which have been designed to be 'quick-and-dirty', sacrificing performance for speed on low-powered computers.", '1306.3888-1-282-0': '# Human perception and cognition, and neuroscience', '1306.3888-1-283-0': 'Since much of the inspiration for the SP theory has come from evidence, mentioned in Section [REF], that, to a large extent, the workings of brains and nervous systems may be understood in terms of information compression, the theory is about perception and cognition as well as artificial intelligence and mainstream computing (B, Chapters 11 and 12).', '1306.3888-1-284-0': 'That said, the main elements of the theory-the multiple alignment concept in particular-are theoretical constructs derived from what appears to be necessary to model, in an economical way, such things as pattern recognition, reasoning, and so on.', '1306.3888-1-284-1': 'In B (Chapter 11), I have described in outline, and tentatively, how such things as SP patterns and multiple alignments may be realised with neurons and connections between them.', '1306.3888-1-285-0': 'The cortex of the brains of mammals-which is, topologically, a two-dimensional sheet-may be, in some respects, like a sheet of paper on which pattern assemblies (neural analogues of SP patterns) may be written-as shown schematically in Figure [REF].', '1306.3888-1-285-1': 'Unlike information written on a sheet of paper, there are neural connections between patterns-as shown in the figure-and communications amongst them.', '1306.3888-1-286-0': "These proposals are very different from how 'neural networks' are generally conceived in computer science.", '1306.3888-1-287-0': '# Conclusion', '1306.3888-1-288-0': 'The SP theory aims to simplify and integrate concepts across artificial intelligence, mainstream computing and human perception and cognition, with information compression as a unifying theme.', '1306.3888-1-288-1': 'The matching and unification of patterns and the concept of multiple alignment are central ideas.', '1306.3888-1-289-0': "In accordance with Occam's Razor, the SP system combines conceptual simplicity with descriptive and explanatory power.", '1306.3888-1-289-1': "A relatively simple mechanism provides an interpretation for a range of concepts and phenomena in several areas including conepts of 'computing', aspects of mathematics and logic, representation of knowledge, natural language processing, pattern recognition, several kinds of probabilistic reasoning, information storage and retrieval, planning and problem solving, information compression, neuroscience, and human perception and cognition.", '1306.3888-1-290-0': 'As suggested in Section [REF], an aid to further research would be the creation of a high-parallel, open-source version of the SP machine, that may be accessed via the web.'}

{'1306.3888-2-0-0': 'This article is an overview of the SP theory of intelligence, which aims to simplify and integrate concepts across artificial intelligence, mainstream computing and human perception and cognition, with information compression as a unifying theme.', '1306.3888-2-0-1': "It is conceived as a brain-like system that receives 'New' information and stores some or all of it in compressed form as 'Old' information; and it is realised in the form of a computer model, a first version of the SP machine.", '1306.3888-2-0-2': 'The matching and unification of patterns and the concept of multiple alignment are central ideas.', '1306.3888-2-0-3': "Using heuristic techniques, the system builds multiple alignments that are 'good' in terms of information compression.", '1306.3888-2-0-4': 'For each multiple alignment, probabilities may be calculated for associated inferences.', '1306.3888-2-0-5': "Unsupervised learning is done by deriving new structures from partial matches between patterns and via heuristic search for sets of structures that are 'good' in terms of information compression.", '1306.3888-2-0-6': "These are normally ones that people judge to be 'natural', in accordance with the 'DONSVIC' principle-the discovery of natural structures via information compression.", '1306.3888-2-0-7': "The SP theory provides an interpretation for concepts and phenomena in several other areas including 'computing', aspects of mathematics and logic, the representation of knowledge, natural language processing, pattern recognition, several kinds of reasoning, information storage and retrieval, planning and problem solving, information compression, neuroscience, and human perception and cognition.", '1306.3888-2-0-8': "Examples include the parsing and production of language with discontinuous dependencies in syntax, pattern recognition at multiple levels of abstraction and its integration with part-whole relations, nonmonotonic reasoning and reasoning with default values, reasoning in Bayesian networks including 'explaining away', causal diagnosis, and the solving of a geometric analogy problem.", '1306.3888-2-1-0': 'Keywords: information compression, artificial intelligence, multiple alignment, computing, representation of knowledge, natural language processing, pattern recognition, information retrieval, probabilistic reasoning, planning, problem solving, unsupervised learning.', '1306.3888-2-2-0': '# Introduction', '1306.3888-2-3-0': 'The SP theory of intelligence, which has been under development since about 1987, aims to simplify and integrate concepts across artificial intelligence, mainstream computing and human perception and cognition, with information compression as a unifying theme.', '1306.3888-2-4-0': "The name 'SP' is short for Simplicity and Power, because compression of any given body of information, [MATH], may be seen as a process of reducing informational 'redundancy' in [MATH] and thus increasing its 'simplicity', whilst retaining as much as possible of its non-redundant expressive 'power'.", '1306.3888-2-4-1': "Likewise with Occam's Razor (Section [REF], below).", '1306.3888-2-5-0': 'Aspects of the theory, as it has developed, have been described in several peer-reviewed articles.', '1306.3888-2-5-1': 'The most comprehensive description of the theory as it stands now, with many examples, is in [CITATION].', '1306.3888-2-6-0': 'But this book, with more than 450 pages, it is too long to serve as an introduction to the theory.', '1306.3888-2-6-1': 'This article aims to meet that need, with a fairly full description of the theory and a selection of examples.', '1306.3888-2-6-2': "For the sake of brevity, the book will be referred to as 'BK'.", '1306.3888-2-7-0': 'The next section describes the origins and motivation for the SP theory, Section [REF] introduces the theory, Sections [REF] and [REF] fill in a lot of the details, while the following sections describe aspects of the theory and what it can do.', '1306.3888-2-8-0': '# Origins and motivation', '1306.3888-2-9-0': 'The following subsections outline the origins of the SP theory, how it relates to some other research, and how it has developed.', '1306.3888-2-10-0': '## Information compression', '1306.3888-2-11-0': 'Much of the inspiration for the SP theory is a body of research, pioneered by Fred Attneaveyearparattneave_1954, Horace Barlowyearparbarlow_1959,barlow_1969, and others, showing that several aspects of the workings of brains and nervous systems may be understood in terms of information compression.', '1306.3888-2-11-1': 'For example, when we view a scene with two eyes, the image on the retina of the left eye is almost exactly the same as the image on the retina of right eye, but our brains merge the two images into a single percept, and thus compress the information .', '1306.3888-2-12-0': 'More immediately, the theory has grown out of my own research, developing models of the unsupervised learning of a first language, where the importance of information compression became increasingly clear .', '1306.3888-2-13-0': "The theory also draws on principles of 'minimum length encoding' pioneered by [CITATION], and others.", '1306.3888-2-13-1': 'And it has become apparent that several aspects of computing, mathematics, and logic may be understood in terms of information compression (BK, Chapters 2 and 10).', '1306.3888-2-14-0': 'At an abstract level, information compression can bring two main benefits:', '1306.3888-2-15-0': 'In animals, we would expect these things to have been favoured by natural selection because of the competitive advantage they can bring.', '1306.3888-2-15-1': "Notwithstanding the 'QWERTY' phenomenon, there is reason to believe that information compression, properly applied, may yield comparable advantages in artificial systems.", '1306.3888-2-16-0': '## The matching and unification of patterns', '1306.3888-2-17-0': 'In the SP theory, the matching and unification of patterns is seen as being closer to the bedrock of information compression than more mathematical techniques such as wavelets or arithmetic coding, and closer to the bedrock of information processing and intelligence than, say, concepts of probability.', '1306.3888-2-17-1': "A working hypothesis in this programme of research is that, by staying close to relatively simple, 'primitive', concepts of matching patterns and unifying them, there is a better chance of cutting through unnecessary complexity, and in gaining new insights and better solutions to problems.", '1306.3888-2-17-2': 'The mathematical basis of wavelets, arithmetic coding, and probabilities, may itself be founded on the matching and unification of patterns (BK, Chapter 10).', '1306.3888-2-18-0': '## Simplification and integration of concepts', '1306.3888-2-19-0': "In accordance with Occam's Razor, the SP system aims to combine conceptual simplicity with descriptive and explanatory power.", '1306.3888-2-19-1': "Apart from this widely-accepted principle, the drive for simplification and integration of concepts in this research programme has been motivated in part by Allen Newell's critique of some kinds of research in cognitive science , and in part by the apparent fragmentation of research in artificial intelligence and mainstream computing, with their myriad of concepts and many specialisms.", '1306.3888-2-20-0': 'In attempting to simplify and integrate ideas, the SP theory belongs in the same tradition as unified theories of cognition such as Soar and ACT-R .', '1306.3888-2-20-1': "And it chimes with the resurgence of interest in understanding artificial intelligence as a whole and with research on 'natural computation' .", '1306.3888-2-21-0': "Although the SP programme shares some objectives with projects such as the Godel Machine , and 'universal artificial intelligence' , the approach is very different.", '1306.3888-2-22-0': '## Transparency in the representation of knowledge', '1306.3888-2-23-0': "In this research, it is assumed that knowledge in the SP system should normally be transparent or comprehensible, much as in the 'symbolic' tradition in artificial intelligence (see also Section [REF]), and distinct from the kind of 'sub-symbolic' representation of knowledge that is the rule in, for example, 'neural networks' as they are generally conceived in computer science.", '1306.3888-2-24-0': 'As we shall see in Section [REF] and elsewhere in this article, SP patterns in the multiple alignment framework may serve to represent a variety of kinds of knowledge, in symbolic forms.', '1306.3888-2-25-0': '## Development of the theory', '1306.3888-2-26-0': 'In developing the theory, it was apparent at an early stage that existing systems-such my models of language learning and systems like Prolog-would need radical re-thinking to meet the goal of simplifying and integrating ideas across a wide area.', '1306.3888-2-27-0': "The first published version of the SP theory [CITATION]) described 'some unifying ideas in computing'.", '1306.3888-2-27-1': "Early work on the SP computer model concentrated on developing an improved version of the 'dynamic programming' technique for the alignment of two sequences (see BK, Appendix A) as a possible route to modelling human-like flexibility in pattern recognition, analysis of language, and the like.", '1306.3888-2-28-0': "About 1992, it became apparent that the explanatory range of the theory could be greatly expanded by forming alignments of 2, 3, or more sequences, much as in the 'multiple alignment' concept of bioinformatics.", '1306.3888-2-28-1': 'That idea was developed and adapted in new versions of the SP model, and incorporated in new procedures for unsupervised learning.', '1306.3888-2-29-0': 'Aspects of the theory, with many examples, have been developed in [CITATION].', '1306.3888-2-30-0': '# Introduction to the SP theory', '1306.3888-2-31-0': 'The main elements of the SP theory are:', '1306.3888-2-32-0': '## The SP computer model', '1306.3888-2-33-0': 'The SP theory is realised most fully in the SP70 computer model, with capabilities in the building of multiple alignments and in unsupervised learning.', '1306.3888-2-33-1': 'This will be referred to as the SP model, although in some cases examples are from a subset of the model or slightly earlier precursors of it.', '1306.3888-2-34-0': 'The SP model and its precursors have played a key part in the development of the theory:', '1306.3888-2-35-0': 'The workings of the SP model is described in some detail in BK (Sections 3.9, 3.10, and 9.2) and more briefly in Sections [REF] and [REF], below.', '1306.3888-2-35-1': "The source code for the models, with associated documents and files, may be downloaded via links under the heading 'SOURCE CODE' at the bottom of the page on http://bit.ly/WtXa3gbit.ly/WtXa3g.", '1306.3888-2-36-0': 'The two main elements of the models, described in the following two sections, are the building of multiple alignments and the unsupervised learning of Old patterns.', '1306.3888-2-37-0': '## The SP machine', '1306.3888-2-38-0': 'The SP model may be regarded as a first version of the SP machine, an expression of the SP theory and a means for it to be applied.', '1306.3888-2-39-0': 'A useful step forward in the development of the SP theory would be the creation of a high-parallel, open-source version of the SP machine, accessible via the web, and with a good user interface.', '1306.3888-2-39-1': 'This would provide a means for researchers to explore what can be done with the system and to improve it.', '1306.3888-2-39-2': 'How things may develop is shown schematically in Figure [REF].', '1306.3888-2-40-0': 'The high-parallel search mechanisms in any of the existing internet search engines would probably provide a good foundation for the proposed development.', '1306.3888-2-41-0': 'Further ahead, there may be a case for the creation of new kinds of hardware, dedicated to the building of multiple alignments and other processes in the SP framework .', '1306.3888-2-42-0': '## Unfinished business', '1306.3888-2-43-0': 'Like most theories, the SP theory has shortcomings, but it appears that they may be overcome.', '1306.3888-2-43-1': 'At present, the most immediate problems are:', '1306.3888-2-44-0': '# The multiple alignment concept', '1306.3888-2-45-0': 'The concept of multiple alignment in the SP theory has been adapted from a similar concept in bioinformatics, where it means a process of arranging, in rows or columns, two or more DNA sequences or amino-acid sequences so that matching symbols-as many as possible-are aligned orthogonally in columns or rows.', '1306.3888-2-46-0': 'Multiple alignments like these are normally used in the computational analysis of (symbolic representations of) sequences of DNA bases or sequences of amino acid residues as part of the process of elucidating the structure, functions or evolution of the corresponding molecules.', '1306.3888-2-46-1': 'An example of this kind of multiple alignment is shown in Figure [REF].', '1306.3888-2-47-0': 'As in bioinformatics, a multiple alignment in the SP system is an arrangement of two or more patterns in rows (or columns), with one pattern in each row (or column).', '1306.3888-2-47-1': 'The main difference between the two concepts is that, in bioinformatics, all sequences have the same status, whereas in the SP theory, the system attempts to create a multiple alignment which enables one New pattern (sometimes more) to be encoded economically in terms of one or more Old patterns.', '1306.3888-2-47-2': 'Other differences are described in BK (Section 3.4.1).', '1306.3888-2-48-0': "In Figure [REF], row 0 contains a New pattern representing a sentence: 't h i s b o y l o v e s t h a t g i r l', while each of rows 1 to 8 contains an Old pattern representing a grammatical rule or a word with grammatical markers.", '1306.3888-2-48-1': 'This multiple alignment, which achieves the effect of parsing the sentence in terms of grammatical structures, is the best of several built by the model when it is supplied with the New pattern and a set of Old patterns that includes those shown in the figure and several others as well.', '1306.3888-2-49-0': "In this example, and others in this article, 'best' means that the multiple alignment in the figure is the one that enables the New pattern to be encoded most economically in terms of the Old patterns, as described in Section [REF], below.", '1306.3888-2-50-0': '## Coding and the evaluation of an alignment in terms of compression', '1306.3888-2-51-0': 'This section describes in outline how multiple alignments are evaluated in the SP model.', '1306.3888-2-51-1': 'More detail may be found in BK (Section 3.5).', '1306.3888-2-52-0': "Each Old pattern in the SP system contains one or more 'identification' symbols or ID-symbols which, as their name suggests, serve to identify the pattern.", '1306.3888-2-52-1': "Examples of ID-symbols in Figure [REF] are 'D' and '0' at the beginning of 'D 0 t h i s D' (row 6), and 'N' and '1' at the beginning of 'N 1 b o y N' (row 8).", '1306.3888-2-53-0': "Associated with each type of symbol (where a 'type' of symbol is any one of a set of symbols that match each other exactly) is a notional code or bit pattern that serves to distinguish the given type from all the others.", '1306.3888-2-53-1': 'This is only notional because the bit patterns are not actually constructed.', '1306.3888-2-53-2': 'All that is needed for the purpose of evaluating multiple alignments is the size of the notional bit pattern associated with each type.', '1306.3888-2-53-3': 'This is calculated via the Shannon-Fano-Elias coding scheme (described in [CITATION]), using information about the frequency of occurrence of each Old pattern, so that the shortest codes represent the most frequent symbol types and vice versa.', '1306.3888-2-53-4': 'Notice that these bit patterns and their sizes are totally independent of the names for symbols that are used in written accounts like this one: names that are chosen purely for their mnemonic value.', '1306.3888-2-54-0': 'Given a multiple alignment like the one shown in Figure [REF], one can derive a code pattern from the multiple alignment in the following way:', '1306.3888-2-55-0': 'Scan the multiple alignment from left to right looking for columns that contain an ID-symbol by itself, not aligned with any other symbol.', '1306.3888-2-56-0': 'Copy these symbols into a code pattern in the same order that they appear in the multiple alignment.', '1306.3888-2-57-0': "The code pattern derived in this way from the multiple alignment shown in Figure [REF] is 'S 0 1 0 1 0 S'.", '1306.3888-2-57-1': 'This is, in effect, a compressed representation of those symbols in the New pattern that are aligned with Old symbols in the multiple alignment.', '1306.3888-2-57-2': 'In this case, the code pattern is a compressed representation of all the symbols in the New pattern but it often happens that some of the symbols in the New pattern are not matched with any Old symbols and then the code pattern will represent only those New symbols that are aligned with Old symbols.', '1306.3888-2-58-0': "In the context of natural language processing, it perhaps more plausible to suppose that the encoding of a sentence is some kind of representation of the meaning of the sentence, instead of a pattern like 'S 0 1 0 1 0 S'.", '1306.3888-2-58-1': 'How a meaning may be derived from a sentence via multiple alignment is described in BK (Section 5.7).', '1306.3888-2-59-0': '### Compression difference and compression ratio', '1306.3888-2-60-0': "Given a code pattern like 'S 0 1 0 1 0 S', we may calculate a 'compression difference'CD as: [EQUATION] or a 'compression ratio'CR as: [EQUATION] where [MATH] is the total number of bits in those symbols in the New pattern that are aligned with Old symbols in the alignment and [MATH] is the total number of bits in the symbols in the code pattern, and the number of bits for each symbol is calculated via the Shannon-Fano-Elias scheme as mentioned above.", '1306.3888-2-61-0': '[MATH] and [MATH] are each an indication of how effectively the New pattern (or those parts of the New pattern that are aligned with symbols within Old patterns in the alignment) may be compressed in terms of the Old patterns that appear in the given multiple alignment.', '1306.3888-2-61-1': 'The [MATH] of a multiple alignment-which has been found to be more useful than [MATH]-may be referred to as the compression score of the multiple alignment.', '1306.3888-2-62-0': 'In each of these equations, [MATH] is calculated as: [EQUATION] where [MATH] is the size of the code for [MATH]th symbol in a sequence, [MATH], comprising those symbols within the New pattern that are aligned with Old symbols within the multiple alignment.', '1306.3888-2-63-0': '[MATH] is calculated as: [EQUATION] where [MATH] is the size of the code for [MATH]th symbol in the sequence of [MATH] symbols in the code pattern derived from the multiple alignment.', '1306.3888-2-64-0': '## The building of multiple alignments', '1306.3888-2-65-0': 'This section describes in outline how the SP model builds multiple alignments.', '1306.3888-2-65-1': 'More detail may be found in BK (Section 3.10).', '1306.3888-2-66-0': 'Multiple alignments are built in stages, with pairwise matching and alignment of patterns.', '1306.3888-2-66-1': 'At each stage, any partially-constructed multiple alignment may be processed as if it was a basic pattern and carried forward to later stages.', '1306.3888-2-66-2': 'This is broadly similar to some programs for the creation of multiple alignments in bioinformatics.', '1306.3888-2-66-3': 'At all stages, the aim is to encode New information economically in terms of Old information and to weed out multiple alignments that score poorly in that regard.', '1306.3888-2-67-0': 'The model may create Old patterns for itself, as described in Section [REF], but when the formation of multiple alignments is the focus of interest, Old patterns may be supplied by the user.', '1306.3888-2-67-1': 'In all cases, New patterns must be supplied by the user.', '1306.3888-2-68-0': 'At each stage of building multiple alignments, the operations are as follows:', '1306.3888-2-69-0': "Identify a set of 'driving' patterns and a set of 'target' patterns.", '1306.3888-2-69-1': 'At the beginning, the New pattern is the sole driving pattern and the Old patterns are the target patterns.', '1306.3888-2-69-2': 'In all subsequent stages, the best of the multiple alignments formed so far (in terms of their [MATH] scores) are chosen to be driving patterns and the target patterns are the Old patterns together with a selection of the best multiple alignments formed so far, including all of those that are driving patterns.', '1306.3888-2-70-0': 'Compare each driving pattern with each of the target patterns to find full matches and good partial matches between patterns.', '1306.3888-2-70-1': "This is done with a process that is essentially a form of 'dynamic programming' , somewhat like the WinMerge utility for finding similarities and differences between files.", '1306.3888-2-70-2': 'The process is described quite fully in BK (Appendix A) and outlined in Section [REF], below.', '1306.3888-2-70-3': "The main difference between the SP process and others, is that the former can deliver several alternative matches between a pair of patterns, while WinMerge and standard methods for finding alignments deliver one 'best' result.", '1306.3888-2-71-0': 'From the best of the matches found in the current stage, create corresponding multiple alignments and add them to the repository of multiple alignments created by the program.', '1306.3888-2-72-0': 'This process of matching driving patterns against target patterns and building multiple alignments is repeated until no more multiple alignments can be found.', '1306.3888-2-72-1': 'For the best of the multiple alignments created since the start of processing, probabilities are calculated, as described in Section [REF].', '1306.3888-2-73-0': '### Finding good matches between patterns', '1306.3888-2-74-0': "Figure [REF] shows with a simple example how the SP model finds good full and partial matches between a 'query' string of atomic symbols (alphabetic characters in this example) and a 'database' string:", '1306.3888-2-75-0': 'The query is processed left to right, one symbol at a time.', '1306.3888-2-76-0': 'Each symbol in the query is, in effect, broadcast to every symbol in the database to make a yes/no match in each case.', '1306.3888-2-77-0': 'Every positive match (hit) between a symbol from the query and a symbol in the database is recorded in a hit structure, illustrated in the figure.', '1306.3888-2-78-0': 'If the memory space allocated to the hit structure is exhausted at any time then the hit structure is purged: the leaf nodes of the tree are sorted in reverse order of their probability values and each leaf node in the bottom half of the set is extracted from the hit structure, together with all nodes on its path which are not shared with any other path.', '1306.3888-2-78-1': 'After the hit structure has been purged, the recording of hits may continue using the space which has been released.', '1306.3888-2-79-0': '### Noisy data', '1306.3888-2-80-0': 'Because of the way each model searches for a global optimum in the building of multiple alignments, it does not depend on the presence or absence of any particular feature or combination of features.', '1306.3888-2-80-1': 'Up to a point, plausible results may be obtained in the face of errors of omission, commission and substitution in the data.', '1306.3888-2-80-2': "This is illustrated in the two multiple alignments in Figure [REF] where the New pattern in row 0 of (b) is the same sentence as in (a) ('t w o k i t t e n s p l a y') but with the omission of the 'w' in 't w o', the substitution of 'm' for ''n' in 'k i t t e n s', and the addition of 'x' within the word 'p l a y'.", '1306.3888-2-80-3': "Despite these errors, the best multiple alignment created by the SP model is, as shown in (b), the one that we judge intuitively to be 'correct'.", '1306.3888-2-81-0': 'This kind of ability to cope gracefully with noisy data is very much in keeping with our ability to understand speech in noisy surroundings, to understand written language despite errors, and to recognise people, trees, houses, and the like, despite fog, snow, falling leaves, or other things that may obstruct our view.', '1306.3888-2-81-1': 'In a similar way, it is likely to prove useful in artificial systems for such applications as the processing of natural language and the recognition of patterns.', '1306.3888-2-82-0': '## Computational complexity', '1306.3888-2-83-0': 'In considering the matching and unification of patterns, it not hard to see that, for any body of information [MATH], except very small examples, there is a huge number of alternative ways in which patterns may be matched against each other, there will normally be many alternative ways in which patterns may be unified, and exhaustive search is not tractable (BK, Section 2.2.8.4).', '1306.3888-2-84-0': 'However, with the kinds of heuristic techniques that are familiar in other AI applications-reducing the size of the search space by pruning the search tree at appropriate points, and being content with approximate solutions which are not necessarily perfect-this kind of matching becomes quite practical.', '1306.3888-2-84-1': 'Much the same can be said about the heuristic techniques used for the building of multiple alignments (Section [REF]) and for unsupervised learning (Section [REF]).', '1306.3888-2-85-0': 'For the process of building multiple alignments in the SP model, the time complexity in a serial processing environment, with conservative assumptions, has been estimated to be O[MATH], where [MATH] is the size of the pattern from New (in bits) and [MATH] is the sum of the lengths of the patterns in Old (in bits).', '1306.3888-2-85-1': 'In a parallel processing environment, the time complexity may approach O[MATH], depending on how well the parallel processing is applied.', '1306.3888-2-85-2': 'In serial and parallel environments, the space complexity has been estimated to be O[MATH].', '1306.3888-2-86-0': 'Although the data sets used with the current SP model have generally been small, there is reason to be confident that the models can be scaled up to deal with large data sets because the kind of flexible matching of patterns which is at the heart of the SP model is done very fast and with huge volumes of data by all the leading internet search engines.', '1306.3888-2-86-1': 'As was suggested in Section [REF], the relevant processes in any one of those search engines would probably provide a good basis for the creation of a high-parallel version of the SP machine.', '1306.3888-2-87-0': '## Calculation of probabilities associated with multiple alignments', '1306.3888-2-88-0': 'As described in BK (Chapter 7), the formation of multiple alignments in the SP framework supports several kinds of probabilistic reasoning.', '1306.3888-2-88-1': 'The core idea is that any Old symbol in a multiple alignment that is not aligned with a New symbol represents an inference that may be drawn from the multiple alignment.', '1306.3888-2-88-2': 'This section outlines how probabilities for such inferences may be calculated.', '1306.3888-2-88-3': 'There is more detail in BK (Section 3.7).', '1306.3888-2-89-0': '### Absolute probabilities', '1306.3888-2-90-0': 'Any sequence of [MATH] symbols, drawn from an alphabet of [MATH] alphabetic types, represents one point in a set of [MATH] points where [MATH] is calculated as: [EQUATION]', '1306.3888-2-90-1': 'If we assume that the sequence is random or nearly so, which means that the [MATH] points are equi-probable or nearly so, the probability of any one point (which represents a sequence of length [MATH]) is close to: [EQUATION]', '1306.3888-2-90-2': 'This equation may be used to calculate the absolute probability of the code pattern that may be derived from any given multiple alignment (as described in Section [REF]).', '1306.3888-2-90-3': 'That number may also be regarded as the absolute probability of any inferences that may be drawn from the multiple alignment.', '1306.3888-2-90-4': 'In this calculation, [MATH] is the sum of all the bits in the symbols of the code pattern and [MATH] is 2.', '1306.3888-2-91-0': 'As we shall see (Section [REF]), Equation [REF] may, with advantage, be generalised by replacing [MATH] with a value, [MATH], calculated in a slightly different way.', '1306.3888-2-92-0': '### Relative probabilities', '1306.3888-2-93-0': 'The absolute probabilities of multiple alignments, calculated as described in the last subsection, are normally very small and not very interesting in themselves.', '1306.3888-2-93-1': 'From the standpoint of practical applications, we are normally interested in the relative values of probabilities, calculated as follows.', '1306.3888-2-94-0': 'For the multiple alignment which has the highest [MATH] (which we shall call the reference multiple alignment), identify the reference set of symbols in New, meaning the symbols from New which are encoded by the multiple alignment.', '1306.3888-2-95-0': 'Compile a reference set of multiple alignments which includes the reference multiple alignment and all other multiple alignments (if any) which encode exactly the reference set of symbols from New, neither more nor less.', '1306.3888-2-96-0': 'Calculate the sum of the values for [MATH] in the reference set of multiple alignments: [EQUATION] where [MATH] is the size of the reference set of multiple alignments and [MATH] is the value of [MATH] for the [MATH]th multiple alignment in the reference set.', '1306.3888-2-97-0': 'For each multiple alignment in the reference set, calculate its relative probability as: [EQUATION]', '1306.3888-2-97-1': 'The values of [MATH], calculated as just described, provide an effective means of comparing the multiple alignments in the reference set.', '1306.3888-2-98-0': '### A generalisation of the method for calculating absolute and relative probabilities', '1306.3888-2-99-0': "The value of [MATH], calculated as described in Section [REF], may be regarded as the informational 'cost' of encoding the New symbol or symbols that appear in the multiple alignment, excluding those New symbols that have not appeared in the multiple alignment.", '1306.3888-2-100-0': 'This is OK but it is somewhat restrictive because it means that if we want to calculate relative probabilities for two or more multiple alignments they must all encode the same symbol or symbols from New.', '1306.3888-2-100-1': 'We cannot easily compare multiple alignments that encode different New symbols.', '1306.3888-2-101-0': 'The generalisation proposed here is that, in the calculation of absolute probabilities, a new value, [MATH], would be used instead of [MATH].', '1306.3888-2-101-1': 'This would be calculated as: [EQUATION] where [MATH] is the total number of bits in the symbols in the code patterns (as in Section [REF]) and [MATH] is the total number of bits in the New symbols that have not appeared in the multiple alignment.', '1306.3888-2-102-0': 'The rationale is that, to encode all the symbols in New, we can use the code pattern to encode those New symbols that do appear in the multiple alignment and, for each of the remaining New symbols, we can simply use its code.', '1306.3888-2-102-1': 'The advantage of this scheme is that we can compare any two or more multiple alignments, regardless of the number of New symbols that appear in the multiple alignment.', '1306.3888-2-103-0': '### Relative probabilities of patterns and symbols', '1306.3888-2-104-0': 'It often happens that a given pattern from Old, or a given symbol type within patterns from Old, appears in more than one of the multiple alignments in the reference set.', '1306.3888-2-104-1': 'In cases like these, one would expect the relative probability of the pattern or symbol type to be higher than if it appeared in only one multiple alignment.', '1306.3888-2-104-2': 'To take account of this kind of situation, the SP model calculates relative probabilities for individual patterns and symbol types in the following way:', '1306.3888-2-105-0': 'Compile a set of patterns from Old, each of which appears at least once in the reference set of multiple alignments.', '1306.3888-2-105-1': 'No single pattern from Old should appear more than once in the set.', '1306.3888-2-106-0': 'For each pattern, calculate a value for its relative probability as the sum of the [MATH] values for the multiple alignments in which it appears.', '1306.3888-2-106-1': 'If a pattern appears more than once in a multiple alignment, it is only counted once for that multiple alignment.', '1306.3888-2-107-0': 'Compile a set of symbol types which appear anywhere in the patterns identified in step 2.', '1306.3888-2-108-0': 'For each alphabetic symbol type identified in step 3, calculate its relative probability as the sum of the relative probabilities of the patterns in which it appears.', '1306.3888-2-108-1': 'If it appears more than once in a given pattern, it is only counted once.', '1306.3888-2-109-0': 'The foregoing applies only to symbol types which do not appear in New.', '1306.3888-2-109-1': 'Any symbol type that appears in New necessarily has a probability of [MATH]-because it has been observed, not inferred.', '1306.3888-2-110-0': '## One system for both the analysis and the production of information', '1306.3888-2-111-0': 'A potentially useful feature of the SP system is that the processes which serve to analyse or parse a New pattern in terms of Old patterns, and to create an economical encoding of the New pattern, may also work in reverse, to recreate the New pattern from its encoding.', '1306.3888-2-111-1': "This is the 'output' perspective, mentioned in Section [REF].", '1306.3888-2-112-0': "If the New pattern is the code sequence 'S 0 1 0 1 0 S' (as described in Section [REF]), and if the Old patterns are the same as were used to create the multiple alignment shown in Figure [REF], then the best multiple alignment found by the system is the one shown in Figure [REF].", '1306.3888-2-112-1': "This multiple alignment contains the same words as the original sentence ('t h i s b o y l o v e s t h a t g i r l'), in the same order as the original.", '1306.3888-2-112-2': "Readers who are familiar with Prolog, will recognise that this process of recreating the original sentence from its encoding is similar in some respects to the way in which an appropriately-constructed Prolog program may be run 'backwards', deriving 'data' from 'results'.", '1306.3888-2-113-0': 'How is it possible to decompress the compressed code for the original sentence by using information compression?', '1306.3888-2-113-1': "This apparent paradox-decompression by compression-may be resolved by ensuring that, when a code pattern like 'S 0 1 0 1 0 S' is used to recreate the original data, each symbol is treated, at least notionally, as if contained a few more bits of information than is strictly necessary.", '1306.3888-2-113-2': 'That residual redundancy allows the system to recreate the original sentence by the same process of compression as was used to create the original parsing and encoding.', '1306.3888-2-114-0': 'This process of creating a relatively large pattern from a relatively small encoding provides a model for the creation of sentences by a person or an artificial system.', '1306.3888-2-114-1': "But instead of the New pattern being a rather dry code, like 'S 0 1 0 1 0 S', it would be more plausible if it were some kind of representation of the meaning of the sentence, like that mentioned in Section [REF].", '1306.3888-2-114-2': 'How a sentence may be generated from a representation of meaning is outlined in BK (Section 5.7.1).', '1306.3888-2-115-0': "Similar principles may apply to other kinds of 'output', such as planning an outing, cooking a meal, and so on.", '1306.3888-2-116-0': '# Unsupervised learning', '1306.3888-2-117-0': 'As was mentioned in Section [REF], part of the inspiration for the SP theory has been a programme of research developing models of the unsupervised learning of language.', '1306.3888-2-117-1': 'But although the SNPR model is quite successful in deriving plausible grammars from samples of English-like artificial language, it has proved to be quite unsuitable as a basis for the SP theory.', '1306.3888-2-117-2': 'In order to accommodate other aspects of intelligence, such as pattern recognition, reasoning, and problem solving, it has been necessary to develop an entirely new conceptual framework, with multiple alignment at centre stage.', '1306.3888-2-118-0': 'So there is now the curious paradox that, while the SP theory is rooted in work on unsupervised learning, and that kind of learning has a central role in the theory, the SP model does much the same things as the earlier model, and with similar limitations (Sections [REF] and [REF]).', '1306.3888-2-118-1': 'But I believe that the new conceptual framework has many advantages, that it provides a much sounder footing for further developments, and that with some reorganisation of the learning processes in the SP computer model, its current weaknesses may be overcome (Section [REF]).', '1306.3888-2-119-0': '## Outline of unsupervised learning in the SP model', '1306.3888-2-120-0': 'The outline of the SP model in this section aims to provide sufficient detail for a good intuitive grasp of how it works.', '1306.3888-2-120-1': 'A lot more detail may be found in BK (Chapter 9).', '1306.3888-2-121-0': 'In addition to the processes for building multiple alignments, the SP model has processes for deriving Old patterns from multiple alignments, evaluating sets of newly-created Old patterns in terms of their effectiveness for the economical encoding of the New information, and the weeding out low-scoring sets.', '1306.3888-2-121-1': 'The system does not merely record statistical information, it uses that information to learn new structures.', '1306.3888-2-122-0': '### Deriving Old patterns from multiple alignments', '1306.3888-2-123-0': 'The process of deriving Old patterns from multiple alignments is illustrated schematically in Figure [REF].', '1306.3888-2-123-1': "As was mentioned in Section [REF], the SP system is conceived as an abstract brain-like system that, in 'input' mode, may receive 'New' information via its senses and store some or all of it as 'Old' information.", '1306.3888-2-123-2': 'Here, we may think of it as the brain of a baby who is listening to what people are saying.', '1306.3888-2-123-3': "Let's imagine that he or she hears someone say 't h a t b o y r u n s'.", '1306.3888-2-123-4': 'If the baby has never heard anything similar, then, if it is stored at all, that New information may be stored as a relatively straightforward copy, something like the Old pattern shown in row 1 of the multiple alignment in part (a) of the figure.', '1306.3888-2-124-0': "Now let us imagine that the information has been stored and that, at some later stage, the baby hears someone say 't h a t g i r l r u n s'.", '1306.3888-2-124-1': 'Then, from that New information and the previously-stored Old pattern, a multiple alignment may be created like the one shown in part (a) of Figure [REF].', '1306.3888-2-124-2': "And, by picking out coherent sequences that are either fully matched or not matched at all, four putative words may be extracted: 't h a t', 'g i r l', 'b o y', and 'r u n s', as shown in the first four patterns in part (b) of the figure.", '1306.3888-2-124-3': "In each newly-created Old pattern there are additional symbols such as 'B', '2', and 'B' that are added by the system, and which serve to identify the pattern, to mark its boundaries, and to mark its grammatical category or categories.", '1306.3888-2-125-0': "In addition to these four patterns, a fifth pattern is created, 'E 6 B B C C D D E', as shown in the figure, that records the sequence 't h a t ... r u n s', with the category 'C C' in the middle representing a choice between 'b o y' and 'g i r l'.", '1306.3888-2-125-1': 'Part (b) in the figure is the beginnings of a grammar to describe that kind of phrase.', '1306.3888-2-126-0': '### Evaluating and selecting sets of newly-created Old patterns', '1306.3888-2-127-0': 'The example just described shows how Old patterns may be derived from a multiple alignment but it gives a highly misleading impression of how the SP model actually works.', '1306.3888-2-127-1': "In practice, the program forms many multiple alignments that are much less tidy than the one shown and it creates many Old patterns that are clearly 'wrong'.", '1306.3888-2-127-2': "However, the program contains procedures for evaluating candidate sets of patterns ('grammars') and weeding out those that score badly in terms of their effectiveness for encoding the New information economically.", '1306.3888-2-127-3': "Out of all the muddle, it can normally abstract one or two 'best' grammars and these are normally ones that appear intuitively to be 'correct', or nearly so.", '1306.3888-2-127-4': 'In general, the program can abstract one or more plausible grammars from a sample of English-like artificial language, including words, grammatical categories of words, and sentence structure.', '1306.3888-2-128-0': 'In accordance with the principles of minimum length encoding , the aim of these processes of sifting and sorting is to minimise [MATH], where [MATH] is the size (in bits) of the grammar that is under development and [MATH] is the size (in bits) of the New patterns when they have been encoded in terms of the grammar.', '1306.3888-2-129-0': 'For a given grammar comprising patterns [MATH], the value of [MATH] is calculated as: [EQUATION] where [MATH] is the number of symbols in the [MATH]th pattern and [MATH] is the encoding cost of the [MATH]th symbol in that pattern.', '1306.3888-2-130-0': 'Given that each grammar is derived from a set [MATH] of multiple alignments (one multiple alignment for each pattern from New), the value of [MATH] for the grammar is calculated as: [EQUATION] where [MATH] is the size, in bits, of the code string derived from the [MATH]th multiple alignment (Section [REF]).', '1306.3888-2-131-0': 'For a given set of patterns from New, a tree of alternative grammars is created with branching occurring wherever there are two or more alternative multiple alignments for a given pattern from New.', '1306.3888-2-131-1': 'The tree is grown in stages and pruned periodically to keep it within reasonable bounds.', '1306.3888-2-131-2': 'At each stage, grammars with high values for [MATH] (which will be referred to as [MATH]) are eliminated.', '1306.3888-2-132-0': '### Plotting values for [MATH], [MATH] and [MATH]', '1306.3888-2-133-0': 'Figure [REF] shows cumulative values for [MATH], [MATH] and [MATH] as the SP model searches for good grammars for a succession of 8 New patterns, each of which represents a sentence.', '1306.3888-2-133-1': 'Each point on each of the lower three graphs represents the relevant value (on the scale at the left) from the best grammar found after a given pattern from New has been processed.', '1306.3888-2-133-2': "The graph labelled '[MATH]' shows cumulative values on the scale at the left for the succession of New patterns.", '1306.3888-2-133-3': "The graph labelled '[MATH]' shows the amount of compression achieved (on the scale to the right).", '1306.3888-2-134-0': '### Limitations in the SP model and how they may be overcome', '1306.3888-2-135-0': 'As mentioned before (Section [REF]), there are two main weaknesses in the processes for unsupervised learning in the SP model as it is now: the model does not learn intermediate levels in a grammar (phrases or clauses) or discontinuous dependencies of the kind described in Sections [REF] to [REF].', '1306.3888-2-136-0': 'It appears that some reorganisation of the learning processes in the model would solve both problems.', '1306.3888-2-136-1': "What seems to be needed is a tighter focus on the principle that, with appropriately-constructed Old patterns, multiple alignments may be created without the kind of mis-match between patterns that may be seen in Figure [REF] (a) ('g i r l' and 'b o y' do not match each other), and that any such multiple alignment may be treated as if it was a simple pattern.", '1306.3888-2-136-2': 'That reform should facilitate the discovery of structures at multiple levels and the discovery of structures that are discontinuous in the sense that they can bridge intervening structures.', '1306.3888-2-137-0': '### Computational complexity', '1306.3888-2-138-0': 'As with the building of multiple alignments (Section [REF]), the computational complexity of learning in the SP model is kept under control by pruning the search tree at appropriate points, aiming to discover grammars that are reasonably good and not necessarily perfect.', '1306.3888-2-139-0': 'In a serial processing environment, the time complexity of learning in the SP model has been estimated to be O[MATH] where [MATH] is the number of patterns in New.', '1306.3888-2-139-1': 'In a parallel processing environment, the time complexity may approach O[MATH], depending on how well the parallel processing is applied.', '1306.3888-2-139-2': 'In serial or parallel environments, the space complexity has been estimated to be O[MATH].', '1306.3888-2-140-0': '## The discovery of natural structures via information compression (DONSVIC)', '1306.3888-2-141-0': "In our dealings with the world, certain kinds of structures appear to be more prominent and useful than others: in natural languages, there are words, phrase and sentences; we understand the visual and tactile worlds to be composed of discrete 'objects'; and conceptually, we recognise classes of things like 'person', 'house', 'tree', and so on.", '1306.3888-2-142-0': "It appears that these 'natural' kinds of structure are significant in our thinking because they provide a means of compressing sensory information, and that compression of information provides the key to their learning or discovery.", '1306.3888-2-142-1': 'At first sight, this looks like nonsense because popular programs for compression of information, such as those based on the LZW algorithm, or programs for JPEG compression of images, seem not to recognise anything resembling words, objects, or classes.', '1306.3888-2-142-2': 'But those programs are designed to work fast on low-powered computers.', '1306.3888-2-142-3': 'With other programs that are designed to be relatively thorough in their compression of information, natural structures can be revealed:', '1306.3888-2-143-0': 'It seems likely that the principles that have been outlined in this subsection may be applied not only to the discovery of words, phrases and grammars in language-like data but also to such things as the discovery of objects in images , and classes of entity in all kinds of data.', '1306.3888-2-143-1': "These principles may be characterised as the discovery of natural structures via information compression, or 'DONSVIC' for short.", '1306.3888-2-144-0': '## Generalisation, the correction of overgeneralisations, and learning from noisy data', '1306.3888-2-145-0': 'Issues that arise in the learning of a first language and, probably, in other kinds of learning, are illustrated in Figure [REF]:', '1306.3888-2-146-0': 'One possible answer is that mistakes are corrected by parents, teachers, and others.', '1306.3888-2-146-1': 'But the weight of evidence is that children can learn their first language without that kind of assistance.', '1306.3888-2-147-0': 'A better answer is the principle of minimum length encoding (described in its essentials in Section [REF]):', '1306.3888-2-148-0': '## One-trial learning and its implications', '1306.3888-2-149-0': 'In many theories of learning, the process is seen as gradual: behaviour is progressively shaped by rewards or punishments or other kinds of experience.', '1306.3888-2-150-0': 'But any theory of learning in which the process is necessarily gradual is out of step with our ordinary experience that we can and do learn things from a single experience, especially if that single experience is very significant for us (BK, Section 11.4.4.1).', '1306.3888-2-151-0': 'In the SP theory, one-trial learning is accommodated in the way the system can store New information directly.', '1306.3888-2-151-1': 'And the gradual nature of, for example, language learning, may be explained by the complexity of the process of sifting and sorting the many alternative sets of candidate patterns to find one or more sets that are good in terms of information compression (BK, Section 11.4.4.2).', '1306.3888-2-152-0': '# Computing, mathematics, and logic', '1306.3888-2-153-0': 'Drawing mainly on BK (Chapters 4 to 11), this and the following sections describe, with a selection of examples, how the SP theory relates to several areas in artificial intelligence, mainstream computing, and human perception and cognition.', '1306.3888-2-154-0': 'In BK (Chapter 4), I have argued that the SP system is equivalent to a universal Turing machine , in the sense that anything that may be computed with a Turing machine may, in principle, also be computed with an SP machine.', '1306.3888-2-154-1': "The 'in principle' qualification is necessary because the SP theory is still not fully mature and there are still some weaknesses in the SP computer models.", '1306.3888-2-154-2': 'The gist of the argument is that the operation of a Post canonical system may be understood in terms of the SP theory and, since it is accepted that the Post canonical system is equivalent to the Turing machine (as a computational system), the Turing machine may also be understood in terms of the SP theory.', '1306.3888-2-155-0': "The key differences between the SP theory and earlier theories of computing are that the SP theory has a lot more to say about the nature of intelligence than earlier theories, that the theory is founded on principles of information compression via the matching and unification of patterns ('computing as compression'), and that it includes mechanisms for building multiple alignments and for heuristic search that are not present in earlier models.", '1306.3888-2-156-0': '## Conventional computing systems', '1306.3888-2-157-0': 'In conventional computing systems, compression of information may be seen in the matching of patterns with at least implicit unification of patterns that match each other-processes that appear in a variety of guises (BK, Chapter 2).', '1306.3888-2-157-1': 'And three basic techniques for the compression of information-chunking-with-codes, schema-plus-correction, and run-length coding-may be seen in various forms in the organisation of computer programs (ibid.)', '1306.3888-2-158-0': '## Mathematics and logic', '1306.3888-2-159-0': 'In a similar way, several structures and processes in mathematics and logic may be interpreted in terms of information compression via the matching and unification of patterns, and the compression techniques just mentioned (BK, Chapter 10).', '1306.3888-2-159-1': 'For example, multiplication (as repeated addition) and exponentiation (as repeated multiplication) may be seen as examples of run-length coding; a function with parameters may be seen as an example of schema-plus-correction; the chunking-with-codes technique may be seen in the organisation of number systems; and so on.', '1306.3888-2-160-0': '## Computing and probabilities', '1306.3888-2-161-0': 'As we have seen, the SP system is fundamentally probabilistic.', '1306.3888-2-161-1': "If it is indeed Turing-equivalent, as suggested above, and if the Turing machine is regarded as a definition of 'computing', then we may conclude that computing is fundamentally probabilistic.", '1306.3888-2-161-2': 'That may seem like a strange conclusion in view of the clockwork certainties that we associate with the operation of ordinary computers and the workings of mathematics and logic.', '1306.3888-2-161-3': 'There are at least three answers to that apparent contradiction:', '1306.3888-2-162-0': '# Representation of knowledge', '1306.3888-2-163-0': "Within the multiple alignment framework (Section [REF]), SP patterns may serve to represent several kinds of knowledge, including grammars for natural languages, ontologies, class hierarchies with inheritance of attributes, including cross-classification or multiple inheritance, part-whole hierarchies and their integration with class-inclusion hierarchies, decision networks and trees, relational tuples, if-then rules, associations of medical signs and symptoms, causal relations, and concepts in mathematics and logic such as 'function', 'variable', 'value', 'set', and 'type definition'.", '1306.3888-2-164-0': 'The use of one simple format for the representation of knowledge facilitates the seamless integration of different kinds of knowledge.', '1306.3888-2-165-0': '# Natural language processing', '1306.3888-2-166-0': 'One of the main strengths of the SP system is in natural language processing (BK, Chapter 5):', '1306.3888-2-167-0': '## Discontinuous dependencies in syntax', '1306.3888-2-168-0': 'The way in which the SP system can record discontinuous dependencies in syntax may be seen in both of the two parsings in Figure [REF].', '1306.3888-2-168-1': "The pattern in row 8 of each multiple alignment records the syntactic dependency between the plural noun phrase ('t w o k i t t e n s') which is the subject of the sentence-marked with 'Np'-and the plural verb phrase ('p l a y')-marked with 'Vp'-which belongs with it.", '1306.3888-2-169-0': "This kind of dependency is discontinuous because it can bridge arbitrarily large amounts of intervening structure such as, for example, 'from the West' in a sentence like 'Winds from the West are strong'.", '1306.3888-2-170-0': 'This method of marking discontinuous dependencies can accommodate overlapping dependencies such as number dependencies and gender dependencies in languages like French (BK, Section 5.4).', '1306.3888-2-170-1': 'It also provides a means of encoding the interesting system of overlapping and interlocking dependencies in English auxiliary verbs, described by Noam Chomsky in Syntactic Structuresyearparchomsky_1957.', '1306.3888-2-171-0': "In that book, the structure of English auxiliary verbs is part of Chomsky's evidence in support of Transformational Grammar.", '1306.3888-2-171-1': 'Despite the elegance and persuasiveness of his arguments, it turns out that the structure of English auxiliary verbs may be described with non-transformational rules in, for example, Definite Clause Grammars , and also in the SP system, as outlined in the subsections that follow.', '1306.3888-2-172-0': '## Two quasi-independent patterns of constraint in English auxiliary verbs', '1306.3888-2-173-0': 'In English, the syntax for main verbs and the auxiliary verbs which may accompany them follows two quasi-independent patterns of constraint which interact in an interesting way.', '1306.3888-2-174-0': 'The primary constraints may be expressed with this sequence of symbols,', '1306.3888-2-175-0': 'M H B B V,', '1306.3888-2-176-0': 'which should be interpreted in the following way:', '1306.3888-2-177-0': 'The words occur in the order shown but any of the words may be omitted.', '1306.3888-2-178-0': "Questions of 'standard' form follow exactly the same pattern as statements except that the first verb, whatever it happens to be ('M', 'H', the first 'B', the second 'B' or 'V'), precedes the subject noun phrase instead of following it.", '1306.3888-2-179-0': 'Here are two examples of the primary pattern with all of the words included:', '1306.3888-2-180-0': 'It will have been being washed M H B B V', '1306.3888-2-181-0': 'Will it have been being washed?', '1306.3888-2-181-1': 'M H B B V', '1306.3888-2-182-0': 'The secondary constraints are these:', '1306.3888-2-183-0': 'Figure [REF] shows a selection of examples with the dependencies marked.', '1306.3888-2-184-0': '## Multiple alignments and English auxiliary verbs', '1306.3888-2-185-0': 'Without reproducing all the detail in BK (Section 5.5), we can see from Figures [REF] and [REF] how the primary and secondary constraints may be applied in the multiple alignment framework.', '1306.3888-2-186-0': 'In each figure, the sentence to be analysed is shown as a New pattern in column 0.', '1306.3888-2-186-1': "The primary constraints are applied via the matching of symbols in Old patterns in the remaining columns, with a consequent interlocking of the patterns so that they recognise sentences of the form 'M H B B V', with options as described above.", '1306.3888-2-187-0': 'In Figure [REF], the secondary constraints apply as follows:', '1306.3888-2-188-0': 'In Figure [REF], the secondary constraints apply like this:', '1306.3888-2-189-0': '# Pattern recognition', '1306.3888-2-190-0': 'The system also has some useful features as a framework for pattern recognition (BK, (Chapter 6):', '1306.3888-2-191-0': 'One area of application is medical diagnosis , viewed as pattern recognition.', '1306.3888-2-191-1': 'There is also potential to assist in the understanding of natural vision and in the development of computer vision, as discussed in [CITATION].', '1306.3888-2-192-0': '## Part-whole hierarchies, class hierarchies, and their integration', '1306.3888-2-193-0': 'A strength of the multiple alignment concept is that it provides a simple but effective vehicle for the representation and processing of class-inclusion hierarchies, part-whole hierarchies, and their integration.', '1306.3888-2-194-0': "Figure [REF] shows the best multiple alignment found by the SP model with the New pattern 'white-bib eats furry purrs' (column 0) representing some features of an unknown creature, and with a set of Old patterns representing different classes of animal, at varying levels of abstraction.", '1306.3888-2-194-1': "From this multiple alignment, we may conclude that the unknown entity is an animal (column 1), a mammal (column 2), a cat (column 3) and the specific individual 'Tibs' (column 4).", '1306.3888-2-195-0': "The framework also provides for the representation of heterarchies or cross classification: a given entity, such as 'Jane' (or a class of entities), may belong in two or more higher-level classes that are not themselves hierarchically related, such as 'woman' and 'doctor'.", '1306.3888-2-196-0': 'The way that class-inclusion relations and part-whole relations may be combined in one multiple alignment is illustrated in Figure [REF].', '1306.3888-2-196-1': 'Here, some features of an unknown plant are expressed as a set of New patterns, shown in column 0: the plant has chlorophyll, the stem is hairy, it has yellow petals, and so on.', '1306.3888-2-197-0': 'From this multiple alignment, we can see that the unknown plant is most likely to be the Meadow Buttercup, Ranunculus acris, as shown in column 1.', '1306.3888-2-197-1': 'As such, it belongs in the genus Ranunculus (column 6), the family Ranunculaceae (column 5), the order Ranunculales (column 4), the class Angiospermae (column 3), and the phylum Plants (column 2).', '1306.3888-2-198-0': 'Each of these higher-level classifications contributes information about attributes of the plant and its division into parts and sub-parts.', '1306.3888-2-198-1': "For example, as a member of the class Angiospermae (column 3), the plant has a shoot and roots, with the shoot divided into stem, leaves, and flowers; as a member of the family Ranunculaceae (column 5), the plant has flowers that are 'regular', with all parts 'free'; as a member of the phylum Plants (column 2), the buttercup has chlorophyll and creates its own food by photosynthesis; and so on.", '1306.3888-2-199-0': '## Inference and inheritance', '1306.3888-2-200-0': 'In the example just described, we can infer from the multiple alignment, very directly, that the plant which has been provisionally identified as the Meadow Buttercup performs photosynthesis (column 2), has five petals (column 6), is poisonous (column 5), and so on.', '1306.3888-2-200-1': "As in other object-oriented systems, the first of these attributes has been 'inherited' from the class 'Plants', the second from the class Ranunculus, and the third from the class Ranunculaceae.", '1306.3888-2-200-2': 'These kinds of inference illustrate the close connection, often remarked, between pattern recognition and inferential reasoning .', '1306.3888-2-201-0': '# Probabilistic reasoning', '1306.3888-2-202-0': "The SP system can model several kinds of reasoning including inheritance of attributes (as just described), one-step 'deductive' reasoning, abductive reasoning, reasoning with probabilistic decision networks and decision trees, reasoning with 'rules', nonmonotonic reasoning and reasoning with default values, reasoning in Bayesian networks (including 'explaining away'), causal diagnosis, and reasoning which is not supported by evidence (BK, Chapter 7).", '1306.3888-2-203-0': 'Since these several kinds of reasoning all flow from one computational framework (multiple alignment), they may be seen as aspects of one process, working individually or together without awkward boundaries.', '1306.3888-2-204-0': 'Plausible lines of reasoning may be achieved, even when relevant information is incomplete.', '1306.3888-2-205-0': "Probabilities of inferences may be calculated, including extreme values (0 or 1) in the case of logic-like 'deductions'.", '1306.3888-2-206-0': 'A selection of examples is described in the following subsections.', '1306.3888-2-207-0': '## Nonmonotonic reasoning and reasoning with default values', '1306.3888-2-208-0': 'Conventional deductive reasoning is monotonic because deductions made on the strength of current knowledge cannot be invalidated by new knowledge: the conclusion that "Socrates is mortal", deduced from "All humans are mortal" and "Socrates is human", remains true for all time, regardless of anything we learn later.', '1306.3888-2-208-1': 'By contrast, the inference that "Tweety can probably fly" from the propositions that "Most birds fly" and "Tweety is a bird" is nonmonotonic because it may be changed if, for example, we learn that Tweety is a penguin.', '1306.3888-2-209-0': 'This section presents some examples which show how the SP system can accommodate nonmonotonic reasoning.', '1306.3888-2-210-0': '### Typically, birds fly', '1306.3888-2-211-0': "The idea that (all) birds can fly may be expressed with an SP pattern like 'Bd bird name name canfly warm-blooded wings feathers ... Bd'.", '1306.3888-2-211-1': 'This, of course, is an oversimplification of the real-world facts because, while it true that the majority of birds fly, we know that there are also flightless birds like ostriches, penguins and kiwis.', '1306.3888-2-212-0': "In order to model these facts more closely, we need to modify the pattern that describes birds to be something like this: 'Bd bird name name f f warm-blooded wings feathers ... Bd.", '1306.3888-2-212-1': 'And, to our database of Old patterns, we need to add patterns like this:', '1306.3888-2-213-0': 'Default Bd f canfly #f #Bd #Default P penguin Bd f cannotfly #f #Bd ... #P O ostrich Bd f cannotfly #f #Bd ... #O.', '1306.3888-2-214-0': "Now, the pair of symbols 'f f' in 'Bd bird name name f f warm-blooded wings feathers ... Bd' functions like a 'variable' that may take the value 'canfly' if a given class of birds can fly and 'cannotfly' when a type of bird cannot fly.", '1306.3888-2-214-1': "The pattern 'P penguin Bd f cannotfly f Bd ... P' shows that penguins cannot fly and, likewise, the pattern 'O ostrich Bd f cannotfly f Bd ... O' shows that ostriches cannot fly.", '1306.3888-2-214-2': "The pattern 'Default Bd f canfly f Bd Default', which has a substantially higher frequency than the other two patterns, represents the default value for the variable which is 'canfly'.", '1306.3888-2-214-3': "Notice that all three of these patterns contains the pair of symbols 'Bd ... Bd' showing that the corresponding classes are all subclasses of birds.", '1306.3888-2-215-0': '### Tweety is a bird so, probably, Tweety can fly', '1306.3888-2-216-0': "When the SP model is run with 'bird Tweety' in New and the same patterns in Old as before, modified as just described, the three best multiple alignments found are those shown in Figures [REF], [REF] and [REF].", '1306.3888-2-217-0': 'The first multiple alignment tells us that, with a relative probability of 0.66, Tweety may be the typical kind of bird that can fly.', '1306.3888-2-217-1': 'The second multiple alignment tells us that, with a relative probability of 0.22, Tweety might be an ostrich and, as such, he or she would not be able to fly.', '1306.3888-2-217-2': 'Likewise, the third multiple alignment tells us that, with a relative probability of 0.12, Tweety might be a penguin and would not be able to fly.', '1306.3888-2-217-3': 'The values for probabilities in this simple example are derived from guestimated frequencies that are, almost certainly, not ornithologically correct.', '1306.3888-2-218-0': '### Tweety is a penguin, so Tweety cannot fly', '1306.3888-2-219-0': "Figure [REF] shows the best multiple alignment found by the SP model when it is run again, with 'penguin Tweety' in New instead of 'bird Tweety'.", '1306.3888-2-219-1': 'This time, there is only one multiple alignment in the reference set and its relative probability is 1.0.', '1306.3888-2-219-2': 'Correspondingly, all inferences that we can draw from this multiple alignment have a probability of 1.0.', '1306.3888-2-219-3': 'In particular, we can be confident, within the limits of the available knowledge, that Tweety cannot fly.', '1306.3888-2-220-0': 'In a similar way, if Tweety were an ostrich, we would be able to say with confidence (p = 1.0) that he or she would not be able to fly.', '1306.3888-2-221-0': "## Reasoning in Bayesian networks, including 'explaining away'", '1306.3888-2-222-0': "A Bayesian network is a directed, acyclic graph like the one shown in Figure [REF], below, where each node has zero or more 'inputs' (connections with nodes that can influence the given node) and one or more 'outputs' (connections to other nodes that the given node can influence).", '1306.3888-2-223-0': 'Each node contains a set of conditional probability values, each one the probability of a given output value for a given input value or combination of input values.', '1306.3888-2-223-1': 'With this information, conditional probabilities of alternative outputs for any node may be computed for any given combination of inputs.', '1306.3888-2-223-2': "By combining these calculations for sequences of nodes, probabilities may be propagated through the network from one or more 'start' nodes to one or more 'finishing' nodes.", '1306.3888-2-224-0': 'This section describes how the SP system may perform that kind of probabilistic reasoning, and some advantages compared with Bayesian networks.', '1306.3888-2-225-0': 'Judea Pearlyearpar[p. 7]pearl_1997 describes the phenomenon of \'explaining away\' like this: "If A implies B, C implies B, and B is true, then finding that C is true makes A less credible.', '1306.3888-2-225-1': 'In other words, finding a second explanation for an item of data makes the first explanation less credible.', '1306.3888-2-225-2': '" (his italics).', '1306.3888-2-225-3': 'Here is an example:', '1306.3888-2-226-0': 'Normally an alarm sound alerts us to the possibility of a burglary.', '1306.3888-2-226-1': 'If somebody calls you at the office and tells you that your alarm went off, you will surely rush home in a hurry, even though there could be other causes for the alarm sound.', '1306.3888-2-226-2': 'If you hear a radio announcement that there was an earthquake nearby, and if the last false alarm you recall was triggered by an earthquake, then your certainty of a burglary will diminish.', '1306.3888-2-227-0': 'The causal relationships in the example just described may be captured in a Bayesian network like the one shown in Figure [REF].', '1306.3888-2-228-0': 'Pearl argues that, with appropriate values for conditional probabilities, the phenomenon of "explaining away" can be explained in terms of this network (representing the case where there is a radio announcement of an earthquake) compared with the same network without the node for "radio announcement" (representing the situation where there is no radio announcement of an earthquake).', '1306.3888-2-229-0': '### Representing contingencies with patterns and frequencies', '1306.3888-2-230-0': 'To see how this phenomenon may be understood in terms of the SP theory, consider, first, the set of patterns shown in Figure [REF], which are to be stored in Old.', '1306.3888-2-230-1': "The patterns in the figure show events which occur together in some notional sample of the 'World' together with their frequencies of occurrence in the sample.", '1306.3888-2-231-0': "As with other knowledge-based systems, we shall assume that the 'closed-world' assumption applies so that the absence of any pattern may be taken to mean that the corresponding combination of events did not occur in the period when observations were made.", '1306.3888-2-232-0': "The first pattern ('burglary alarm') shows that there were 1000 occasions when there was a burglary and the alarm went off and the second pattern ('earthquake alarm') shows just 20 occasions when there was an earthquake and the alarm went off (presumably triggered by the earthquake).", '1306.3888-2-232-1': 'Thus we have assumed that, as triggers for the alarm, burglaries are much more common than earthquakes.', '1306.3888-2-233-0': 'Since there is no pattern showing that the alarm sounded when there was a burglary and an earthquake at the same time, we may assume, via the closed-world assumption, that nothing like that happened during the sampling period.', '1306.3888-2-234-0': "The third pattern ('alarm phonealarmcall') shows that, out of the 1020 cases when the alarm went off, there were 980 cases where a phone call about the alarm was made.", '1306.3888-2-234-1': 'Since there is no pattern showing phone calls about the alarm in any other context, the closed-world assumption allows us to assume that there were no false positives (eg., hoaxes)-phone calls about the alarm when no alarm had sounded.', '1306.3888-2-235-0': "The fourth pattern ('earthquake radioearthquakeannouncement') shows that, in the sampling period, there were 40 occasions when there was an earthquake with an announcement about it on the radio.", '1306.3888-2-235-1': "And the fifth pattern ('e1 earthquake e2') shows that an earthquake has occurred on 40 occasions in contexts where the alarm did not ring and there was no radio announcement.", '1306.3888-2-236-0': "As before, the absence of patterns like 'earthquake alarm radioearthquakeannouncement' representing cases where an earthquake triggers the alarm and also leads to a radio announcement, allows us to assume via the closed-world assumption that cases of that kind have not occurred in the sampling period.", '1306.3888-2-237-0': '### Approximating the temporal order of events', '1306.3888-2-238-0': 'In these patterns and in the multiple alignments shown below, the left-to-right order of symbols may be regarded as an approximation to the order of events in time.', '1306.3888-2-238-1': 'Thus in the first two patterns, events that can trigger an alarm precede the sounding of the alarm.', '1306.3888-2-238-2': "Likewise, in the third pattern, 'alarm' (meaning that the alarm has sounded) precedes 'phonealarmcall' (a phone call to say the alarm has sounded).", '1306.3888-2-238-3': 'A single dimension can only approximate the order of events in time because it cannot represent events which overlap in time or which occur simultaneously.', '1306.3888-2-238-4': 'However, this kind of approximation has little or no bearing on the points to be illustrated here.', '1306.3888-2-239-0': '### Other considerations', '1306.3888-2-240-0': 'Other points relating to the patterns shown in Figure [REF] include:', '1306.3888-2-241-0': '### Formation of alignments: the burglar alarm has sounded', '1306.3888-2-242-0': "Receiving a phone call to say that the burglar alarm at one's house has gone off may be represented by placing the symbol 'phonealarmcall' in New.", '1306.3888-2-242-1': 'Figure [REF] shows, at the top, the best multiple alignment formed by the SP model in this case, with the patterns from Figure [REF] in Old.', '1306.3888-2-242-2': 'The other two multiple alignments in the reference set are shown below the best multiple alignment, in order of CD value and relative probability.', '1306.3888-2-242-3': 'The actual values for [MATH] and relative probability are given in the caption to Figure [REF].', '1306.3888-2-243-0': 'The unmatched Old symbols in these multiple alignments represent inferences made by the system.', '1306.3888-2-243-1': 'The probabilities for these inferences which are calculated by the SP model (as outlined in Section [REF]) are shown in Table [REF].', '1306.3888-2-243-2': 'These probabilities do not add up to 1 and we should not expect them to because any given multiple alignment may contain two or more of these symbols.', '1306.3888-2-244-0': 'The most probable inference is the rather trivial inference that the alarm has indeed sounded.', '1306.3888-2-244-1': 'This reflects the fact that there is no pattern in Figure [REF] representing false positives for telephone calls about the alarm.', '1306.3888-2-244-2': 'Apart from the inference that the alarm has sounded, the most probable inference (p = 0.328) is that there has been a burglary.', '1306.3888-2-244-3': 'However, there is a distinct possibility that there has been an earthquake-but the probability in this case (p = 0.016) is much lower than the probability of a burglary.', '1306.3888-2-245-0': "These inferences and their relative probabilities seem to accord quite well with what one would naturally think following a telephone call to say that the burglar alarm at one's house has gone off (given that one was living in a part of the world where earthquakes were not vanishingly rare).", '1306.3888-2-246-0': '## Formation of alignments: the burglar alarm has sounded and there is a radio announcement of an earthquake', '1306.3888-2-247-0': "In this example, the phenomenon of 'explaining away' occurs when you learn not only that the burglar alarm has sounded but that there has been an announcement on the radio that there has been an earthquake.", '1306.3888-2-247-1': 'In terms of the SP model, the two events (the phone call about the alarm and the announcement about the earthquake) can be represented in New by a pattern like this:', '1306.3888-2-248-0': 'phone_alarm_call radio_earthquake_announcement', '1306.3888-2-249-0': "or 'radioearthquakeannouncement phonealarmcall'.", '1306.3888-2-249-1': 'The order of the two symbols makes no difference to the result.', '1306.3888-2-250-0': "In this case, there is only one multiple alignment (shown at the top of Figure [REF]) that can 'explain' all the information in New.", '1306.3888-2-250-1': "Since there is only this one multiple alignment in the reference set for the best multiple alignment, the associated probabilities of the inferences that can be read from the multiple alignment ('alarm' and 'earthquake') are 1.0: it was an earthquake that caused the alarm to go off (and led to the phone call), and not a burglary.", '1306.3888-2-251-0': "These results show how 'explaining away' may be explained in terms of the SP theory.", '1306.3888-2-251-1': "The main point is that the multiple alignment or multiple alignments that provide the best 'explanation' of a telephone call to say that one's burglar alarm has sounded is different from the multiple alignment or multiple alignments that best explain the same telephone call coupled with an announcement on the radio that there has been an earthquake.", '1306.3888-2-251-2': 'In the latter case, the best explanation is that the earthquake triggered the alarm.', '1306.3888-2-251-3': 'Other possible explanations have lower probabilities.', '1306.3888-2-252-0': '### Other possibilities', '1306.3888-2-253-0': 'As mentioned above, the closed-world assumption allows us to rule out possibilities such as:', '1306.3888-2-254-0': 'Nevertheless, we may consider possibilities of that kind by combining multiple alignments as described in BK (Section 7.8.7).', '1306.3888-2-254-1': 'But as a general rule, that kind of further analysis makes no difference to the original conclusion: the multiple alignment which was originally judged to represent the best interpretation of the available facts has not been dislodged from this position.', '1306.3888-2-254-2': 'This is in keeping with way we normally concentrate on the most likely explanations of events and ignore the many conceivable but unlikely alternatives.', '1306.3888-2-255-0': '## The SP framework and Bayesian networks', '1306.3888-2-256-0': 'The foregoing examples show that the SP framework is a viable alternative to Bayesian networks, at least in the kinds of situation that have been described.', '1306.3888-2-256-1': 'This subsection makes some general observations about the relative merits of the two frameworks for probabilistic reasoning where the events of interest are subject to multiple influences or chains of influence or both those things.', '1306.3888-2-257-0': "Without in any way diminishing Thomas Bayes' achievement, his theorem appears to have shortcomings as the basis for theorising about perception and cognition:", '1306.3888-2-258-0': '## Causal diagnosis', '1306.3888-2-259-0': 'In this section, we consider a simple example of fault diagnosis in an electronic circuit-described by [CITATION].', '1306.3888-2-259-1': 'Figure [REF] shows the circuit with inputs on the left, outputs on the right and, in between, three multipliers ([MATH], [MATH], and [MATH]) and two adders ([MATH] and [MATH]).', '1306.3888-2-259-2': 'For the given inputs on the left, it is clear that output F is false and output G is correct.', '1306.3888-2-260-0': 'Figure [REF] shows a causal network derived from the electronic circuit in Figure [REF] .', '1306.3888-2-260-1': 'In this diagram, [MATH], [MATH], [MATH], [MATH] and [MATH] represent the outputs of components [MATH], [MATH], [MATH], [MATH] and [MATH], respectively.', '1306.3888-2-260-2': "In each case, there are three causal influences on the output: the two inputs to the component and the state of the component which may be 'good' or 'bad'.", '1306.3888-2-260-3': 'These influences are shown in Figure [REF] by lines with arrows connecting the source of the influence to the target node.', '1306.3888-2-260-4': 'Thus, for example, the two inputs of component [MATH] are represented by [MATH] and [MATH], the good or bad state of component [MATH] is represented by the node labelled [MATH], and their causal influences on node [MATH] are shown by the three arrows pointing at that node.', '1306.3888-2-261-0': 'Given a causal analysis like this, and given appropriate information about conditional probabilities, it is possible to derive one or more alternative diagnoses of which components are good and which are bad.', '1306.3888-2-261-1': "In Pearl's example, it is assumed that components of the same type have the same prior probability of failure and that the probability of failure of multipliers is greater than for adders.", '1306.3888-2-261-2': 'Given these assumptions and some others together with the inputs and outputs shown in Figure [REF] (but not the intermediate values), the best diagnosis derived from the causal network is that the [MATH] component is bad and the second best diagnosis is that [MATH] is bad.', '1306.3888-2-261-3': 'Pearl indicates that some third-best interpretations may be retrievable (e.g., [MATH] and [MATH] are bad) "... but in general, it is not guaranteed that interpretations beyond the second-best will be retrievable."', '1306.3888-2-261-4': '(p. 272).', '1306.3888-2-262-0': '## An SP approach to causal diagnosis', '1306.3888-2-263-0': 'The main elements of the SP analysis presented here are as follows:', '1306.3888-2-264-0': 'Figure [REF] shows a set of patterns for the circuit shown in Figure [REF].', '1306.3888-2-264-1': "In the figure, the patterns that start with the symbol 'M1' represent I/O relations for component [MATH], those that start with 'M2' represent I/O relations for the [MATH] component and likewise for the other patterns except the last one (starting with the symbol 'frame') which is the framework pattern mentioned above.", '1306.3888-2-264-2': "For each initial symbol there is a corresponding 'terminating' symbol with an initial '' character.", '1306.3888-2-264-3': "For reasons explained shortly, there may be other symbols following the 'terminating' symbol.", '1306.3888-2-265-0': "Let us now consider the first pattern in the figure ('M1 M1GOOD TM1I1 TM1I2 TM1O M1 TM4I2') representing I/O relations for component [MATH] when that component is good, as indicated by the symbol 'M1GOOD'.", '1306.3888-2-265-1': "In this pattern, the symbols 'TM1I1', 'TM1I2' and 'TM1O' represent the two inputs and the output of the component, 'M1' is the terminating symbol, and 'TM4I2' serves to transfer the output of [MATH] to the second input of component [MATH] as will be explained.", '1306.3888-2-265-2': "In a symbol like 'TM1I1', 'T' indicates that the input is true, 'M1' identifies the component, and 'I1' indicates that this is the first input of the component.", '1306.3888-2-265-3': 'Other symbols may be interpreted in a similar way, following the key given in the caption of Figure [REF].', '1306.3888-2-265-4': 'In effect, this pattern says that, when the component is working correctly, true inputs yield a true output.', '1306.3888-2-265-5': 'The pattern has a relatively high frequency of occurrence (500,000) reflecting the idea that the component will normally work correctly.', '1306.3888-2-266-0': "The other two patterns for component [MATH] ('M1 M1BAD TM1I1 TM1I2 TM1O M1 TM4I2' and 'M1 M1BAD TM1I1 TM1I2 FM1O M1 FM4I2') describe I/O relations when the component is bad.", '1306.3888-2-266-1': 'The first one describes the situation where true inputs to a faulty component yield a true result, a possibility noted by Pearl (ibid.', '1306.3888-2-266-2': 'p. 265).', '1306.3888-2-266-3': 'The second pattern-with a higher frequency-describes the more usual situation where true inputs to a faulty component yield a false result.', '1306.3888-2-266-4': 'Both these bad patterns have much lower frequencies than the good pattern.', '1306.3888-2-267-0': 'The other patterns in Figure [REF] may be interpreted in a similar way.', '1306.3888-2-267-1': 'Components [MATH], [MATH] and [MATH] have only three patterns each because it is assumed that inputs to the circuit will always be true so it is not necessary to include patterns describing what happens when one or both of the inputs are false.', '1306.3888-2-267-2': 'By contrast, there are 4 good patterns and 8 bad patterns for each of [MATH] and [MATH] because either of these components may receive faulty input.', '1306.3888-2-268-0': 'For each of the five components, the frequencies of the bad patterns sum to 100.', '1306.3888-2-268-1': 'However, for each of the components [MATH], [MATH], and [MATH], the total frequency of the good patterns is 500,000 compared with 1,000,000 for the set of good patterns associated with each of the component [MATH] and [MATH].', '1306.3888-2-268-2': "These figures accord with the assumptions in Pearl's example that components of the same type have the same probability of failure and that the probability of failure of multipliers ([MATH], [MATH], and [MATH]) is greater than the probability of failure of adders ([MATH] and [MATH]).", '1306.3888-2-269-0': '## Multiple alignments in causal diagnosis', '1306.3888-2-270-0': 'Given appropriate patterns, the SP model constructs multiple alignments from which diagnoses may be obtained.', '1306.3888-2-270-1': "Figure [REF] shows the best multiple alignment created by the SP model with the Old patterns shown in Figure [REF] and 'TM1I1 TM1I2 TM2I1 TM2I2 TM3I1 TM3I2 FM4O TM5O' as the New pattern.", '1306.3888-2-270-2': 'The first six symbols in this pattern express the idea that all the inputs for components [MATH], [MATH] and [MATH] are true.', '1306.3888-2-270-3': "The penultimate symbol ('FM4O') shows that the output of [MATH] is false and the last symbol ('TM5O') shows that the output of [MATH] is true-in accordance with the outputs shown in Figure [REF].", '1306.3888-2-271-0': 'From the multiple alignment in Figure [REF] it can be inferred that component [MATH] is bad and all the other components are good.', '1306.3888-2-271-1': 'A total of seven alternative diagnoses can be derived from those multiple alignments created by the SP model that encode all the symbols in New.', '1306.3888-2-271-2': 'These diagnoses are shown in Table [REF], each with its relative probability.', '1306.3888-2-272-0': "It is interesting to see that the best diagnosis derived by the SP model ([MATH] is bad) and the second best diagnosis ([MATH] is bad) are in accordance with first two diagnoses obtained by Pearl's method.", '1306.3888-2-272-1': "The remaining five diagnoses derived by the SP model are different from the one obtained by Pearl's method ([MATH] and [MATH] are bad) but this is not altogether surprising because detailed frequencies or probabilities are different from Pearl's example and there are differences in assumptions that have been made.", '1306.3888-2-273-0': '# Information storage and retrieval', '1306.3888-2-274-0': "The SP theory provides a versatile model for database systems, with the ability to accommodate object-oriented structures, as well as relational 'tuples', and network and tree models of data .", '1306.3888-2-274-1': 'It lends itself most directly to information retrieval in the manner of query-by-example but it appears to have potential to support the use of natural language or query languages such as SQL.', '1306.3888-2-275-0': 'Unlike some ordinary database systems:', '1306.3888-2-276-0': '# Planning and problem solving', '1306.3888-2-277-0': 'The SP framework provides a means of planning a route between two places, and, with the translation of geometric patterns into textual form, it can solve the kind of geometric analogy problem that may be seen in some puzzle books and IQ tests (BK, Chapter 8).', '1306.3888-2-278-0': 'Figure [REF] shows an example of the latter kind of problem.', '1306.3888-2-278-1': 'The task is to complete the relationship "A is to B as C is to ?"', '1306.3888-2-278-2': "using one of the figures 'D', 'E', 'F' or 'G' in the position marked with '?", '1306.3888-2-278-3': "'.", '1306.3888-2-278-4': "For this example, the 'correct' answer is clearly 'E'.", '1306.3888-2-278-5': "Quote marks have been used for the word 'correct' because in many problems of this type, there may be two or even more alternative answers for which cases can be made and there is a corresponding uncertainty about which answer is the right one.", '1306.3888-2-279-0': "Computer-based methods for solving this kind of problem have existed for some time (e.g., Evans'yearparevans_1968 well-known heuristic algorithm).", '1306.3888-2-279-1': 'In more recent work , minimum length encoding principles have been applied to good effect.', '1306.3888-2-279-2': 'This kind of problem may also be understood in terms of the SP concepts.', '1306.3888-2-280-0': 'As in most previous work, the proposed solution assumes that some mechanism is available which can translate the geometric forms in each problem into patterns of text symbols like other patterns in this article.', '1306.3888-2-280-1': "For example, item 'A' in Figure [REF] may be described as 'small circle inside large triangle'.", '1306.3888-2-281-0': 'How this kind of translation may be done is not part of the present proposals (one such translation mechanism is described in [CITATION]).', '1306.3888-2-281-1': 'As noted elsewhere , successful solutions for this kind of problem require consistency in the way the translation is done.', '1306.3888-2-281-2': "For this example, it would be unhelpful if item 'A' in Figure [REF] were described as 'large triangle outside small circle' while item 'C' were described as 'small square inside large ellipse'.", '1306.3888-2-281-3': "For any one puzzle, the description needs to stick to one or other of 'X outside Y' or 'Y inside X'-and likewise for 'above/below' and 'left-of/right-of'.", '1306.3888-2-282-0': 'Given that the diagrammatic form of the problem has been translated into patterns as just described, this kind of problem can be cast as a problem of partial matching, well within the scope of the SP model.', '1306.3888-2-282-1': 'To do this, symbolic representations of item A and item B in Figure [REF] are treated as a single pattern, thus:', '1306.3888-2-283-0': 'small circle inside large triangle ; large circle above small triangle', '1306.3888-2-284-0': 'and this pattern is placed in New.', '1306.3888-2-284-1': 'Four other patterns are constructed by pairing a symbolic representation of item C (on the left) with symbolic representations of each of D, E, F and G (on the right), thus:', '1306.3888-2-285-0': 'C1 small square inside large ellipse ; D small square inside large circle #C1 C2 small square inside large ellipse ; E large square above small ellipse #C2 C3 small square inside large ellipse ; F small circle left-of large square #C3 C4 small square inside large ellipse ; G small ellipse above large rectangle #C4.', '1306.3888-2-286-0': 'These four patterns are placed in Old, each with an arbitrary frequency value of 1.', '1306.3888-2-287-0': 'Figure [REF] shows the best multiple alignment found by the SP model with New and Old as just described.', '1306.3888-2-287-1': 'The multiple alignment is a partial match between the New pattern (in column 0) and the second of the four patterns in Old (in column 1).', '1306.3888-2-287-2': "This corresponds with the 'correct' result (item E) as noted above.", '1306.3888-2-288-0': '# Compression of information', '1306.3888-2-289-0': 'Since information compression is central to the workings of the SP system, it is natural to consider whether the system might provide useful insights in that area.', '1306.3888-2-289-1': 'In that connection, the most promising aspects of the SP system appear to be:', '1306.3888-2-290-0': "In terms of the trade-off that exists between computational resources that are required and the level of compression that can be achieved, it is intended that the system will operate towards the 'up market' end of the spectrum-by contrast with LZW algorithms and the like, which have been designed to be 'quick-and-dirty', sacrificing performance for speed on low-powered computers.", '1306.3888-2-291-0': '# Human perception, cognition and neuroscience', '1306.3888-2-292-0': 'Since much of the inspiration for the SP theory has come from evidence, mentioned in Section [REF], that, to a large extent, the workings of brains and nervous systems may be understood in terms of information compression, the theory is about perception and cognition as well as artificial intelligence and mainstream computing.', '1306.3888-2-293-0': 'That said, the main elements of the theory-the multiple alignment concept in particular-are theoretical constructs derived from what appears to be necessary to model, in an economical way, such things as pattern recognition, reasoning, and so on.', '1306.3888-2-293-1': 'In BK (Chapter 12), there is some discussion of how the SP concepts relate to a selection of issues in human perception and cognition.', '1306.3888-2-293-2': "A particular interest at the time of writing (after that chapter was written) is the way that the SP theory may provide an alternative to quantum probability as an explanation of phenomena such as the 'conjunction fallacy' [CITATION].", '1306.3888-2-294-0': 'In BK (Chapter 11), I have described in outline, and tentatively, how such things as SP patterns and multiple alignments may be realised with neurons and connections between them.', '1306.3888-2-294-1': 'The cortex of the brains of mammals-which is, topologically, a two-dimensional sheet-may be, in some respects, like a sheet of paper on which pattern assemblies may be written.', '1306.3888-2-294-2': 'These are neural analogues of SP patterns, shown schematically in Figure [REF].', '1306.3888-2-294-3': 'Unlike information written on a sheet of paper, there are neural connections between patterns-as shown in the figure-and communications amongst them.', '1306.3888-2-295-0': "These proposals, which are adapted with modifications from Hebb'syearparhebb_1949 concept of a 'cell assembly', are very different from how artificial 'neural networks' are generally conceived in computer science.", '1306.3888-2-295-1': 'As noted in Section [REF], learning in the SP system is very different from learning in that kind of network-or Hebbian learning.', '1306.3888-2-296-0': '# Conclusion', '1306.3888-2-297-0': 'The SP theory aims to simplify and integrate concepts across artificial intelligence, mainstream computing and human perception and cognition, with information compression as a unifying theme.', '1306.3888-2-297-1': 'The matching and unification of patterns and the concept of multiple alignment are central ideas.', '1306.3888-2-298-0': "In accordance with Occam's Razor, the SP system combines conceptual simplicity with descriptive and explanatory power.", '1306.3888-2-298-1': "A relatively simple mechanism provides an interpretation for a range of concepts and phenomena in several areas including conepts of 'computing', aspects of mathematics and logic, representation of knowledge, natural language processing, pattern recognition, several kinds of probabilistic reasoning, information storage and retrieval, planning and problem solving, information compression, neuroscience, and human perception and cognition.", '1306.3888-2-299-0': 'As suggested in Section [REF], an aid to further research would be the creation of a high-parallel, open-source version of the SP machine, that may be accessed via the web.'}

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[]

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{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '3': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '4': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1306.3888

{'1306.3888-3-0-0': 'This article is an overview of the SP theory of intelligence, which aims to simplify and integrate concepts across artificial intelligence, mainstream computing and human perception and cognition, with information compression as a unifying theme.', '1306.3888-3-0-1': "It is conceived as a brain-like system that receives 'New' information and stores some or all of it in compressed form as 'Old' information; and it is realised in the form of a computer model, a first version of the SP machine.", '1306.3888-3-0-2': 'The matching and unification of patterns and the concept of multiple alignment are central ideas.', '1306.3888-3-0-3': "Using heuristic techniques, the system builds multiple alignments that are 'good' in terms of information compression.", '1306.3888-3-0-4': 'For each multiple alignment, probabilities may be calculated for associated inferences.', '1306.3888-3-0-5': "Unsupervised learning is done by deriving new structures from partial matches between patterns and via heuristic search for sets of structures that are 'good' in terms of information compression.", '1306.3888-3-0-6': "These are normally ones that people judge to be 'natural', in accordance with the 'DONSVIC' principle-the discovery of natural structures via information compression.", '1306.3888-3-0-7': "The SP theory provides an interpretation for concepts and phenomena in several other areas including 'computing', aspects of mathematics and logic, the representation of knowledge, natural language processing, pattern recognition, several kinds of reasoning, information storage and retrieval, planning and problem solving, information compression, neuroscience, and human perception and cognition.", '1306.3888-3-0-8': "Examples include the parsing and production of language with discontinuous dependencies in syntax, pattern recognition at multiple levels of abstraction and its integration with part-whole relations, nonmonotonic reasoning and reasoning with default values, reasoning in Bayesian networks including 'explaining away', causal diagnosis, and the solving of a geometric analogy problem.", '1306.3888-3-1-0': 'Keywords: information compression, artificial intelligence, multiple alignment, computing, representation of knowledge, natural language processing, pattern recognition, information retrieval, probabilistic reasoning, planning, problem solving, unsupervised learning.', '1306.3888-3-2-0': '# Introduction', '1306.3888-3-3-0': 'The SP theory of intelligence, which has been under development since about 1987, aims to simplify and integrate concepts across artificial intelligence, mainstream computing and human perception and cognition, with information compression as a unifying theme.', '1306.3888-3-4-0': "The name 'SP' is short for Simplicity and Power, because compression of any given body of information, [MATH], may be seen as a process of reducing informational 'redundancy' in [MATH] and thus increasing its 'simplicity', whilst retaining as much as possible of its non-redundant expressive 'power'.", '1306.3888-3-4-1': "Likewise with Occam's Razor (Section [REF], below).", '1306.3888-3-5-0': 'Aspects of the theory, as it has developed, have been described in several peer-reviewed articles.', '1306.3888-3-5-1': 'The most comprehensive description of the theory as it stands now, with many examples, is in [CITATION].', '1306.3888-3-6-0': 'But this book, with more than 450 pages, is too long to serve as an introduction to the theory.', '1306.3888-3-6-1': 'This article aims to meet that need, with a fairly full description of the theory and a selection of examples.', '1306.3888-3-6-2': "For the sake of brevity, the book will be referred to as 'BK'.", '1306.3888-3-7-0': 'The next section describes the origins and motivation for the SP theory, Section [REF] introduces the theory, Sections [REF] and [REF] fill in a lot of the details, while the following sections describe aspects of the theory and what it can do.', '1306.3888-3-8-0': '# Origins and motivation', '1306.3888-3-9-0': 'The following subsections outline the origins of the SP theory, how it relates to some other research, and how it has developed.', '1306.3888-3-10-0': '## Information compression', '1306.3888-3-11-0': 'Much of the inspiration for the SP theory is a body of research, pioneered by Fred Attneaveyearparattneave_1954, Horace Barlowyearparbarlow_1959,barlow_1969, and others, showing that several aspects of the workings of brains and nervous systems may be understood in terms of information compression.', '1306.3888-3-11-1': 'For example, when we view a scene with two eyes, the image on the retina of the left eye is almost exactly the same as the image on the retina of right eye, but our brains merge the two images into a single percept, and thus compress the information .', '1306.3888-3-12-0': 'More immediately, the theory has grown out of my own research, developing models of the unsupervised learning of a first language, where the importance of information compression became increasingly clear .', '1306.3888-3-13-0': "The theory also draws on principles of 'minimum length encoding' pioneered by [CITATION], and others.", '1306.3888-3-13-1': 'And it has become apparent that several aspects of computing, mathematics, and logic may be understood in terms of information compression (BK, Chapters 2 and 10).', '1306.3888-3-14-0': 'At an abstract level, information compression can bring two main benefits:', '1306.3888-3-15-0': 'In animals, we would expect these things to have been favoured by natural selection because of the competitive advantage they can bring.', '1306.3888-3-15-1': "Notwithstanding the 'QWERTY' phenomenon, there is reason to believe that information compression, properly applied, may yield comparable advantages in artificial systems.", '1306.3888-3-16-0': '## The matching and unification of patterns', '1306.3888-3-17-0': 'In the SP theory, the matching and unification of patterns is seen as being closer to the bedrock of information compression than more mathematical techniques such as wavelets or arithmetic coding, and closer to the bedrock of information processing and intelligence than, say, concepts of probability.', '1306.3888-3-17-1': "A working hypothesis in this programme of research is that, by staying close to relatively simple, 'primitive', concepts of matching patterns and unifying them, there is a better chance of cutting through unnecessary complexity, and in gaining new insights and better solutions to problems.", '1306.3888-3-17-2': 'The mathematical basis of wavelets, arithmetic coding, and probabilities, may itself be founded on the matching and unification of patterns (BK, Chapter 10).', '1306.3888-3-18-0': '## Simplification and integration of concepts', '1306.3888-3-19-0': "In accordance with Occam's Razor, the SP system aims to combine conceptual simplicity with descriptive and explanatory power.", '1306.3888-3-19-1': "Apart from this widely-accepted principle, the drive for simplification and integration of concepts in this research programme has been motivated in part by Allen Newell's critique of some kinds of research in cognitive science , and in part by the apparent fragmentation of research in artificial intelligence and mainstream computing, with their myriad of concepts and many specialisms.", '1306.3888-3-20-0': 'In attempting to simplify and integrate ideas, the SP theory belongs in the same tradition as unified theories of cognition such as Soar and ACT-R .', '1306.3888-3-20-1': "And it chimes with the resurgence of interest in understanding artificial intelligence as a whole and with research on 'natural computation' .", '1306.3888-3-21-0': "Although the SP programme shares some objectives with projects such as the Godel Machine , and 'universal artificial intelligence' , the approach is very different.", '1306.3888-3-22-0': '## Transparency in the representation of knowledge', '1306.3888-3-23-0': "In this research, it is assumed that knowledge in the SP system should normally be transparent or comprehensible, much as in the 'symbolic' tradition in artificial intelligence (see also Section [REF]), and distinct from the kind of 'sub-symbolic' representation of knowledge that is the rule in, for example, 'neural networks' as they are generally conceived in computer science.", '1306.3888-3-24-0': 'As we shall see in Section [REF] and elsewhere in this article, SP patterns in the multiple alignment framework may serve to represent a variety of kinds of knowledge, in symbolic forms.', '1306.3888-3-25-0': '## Development of the theory', '1306.3888-3-26-0': 'In developing the theory, it was apparent at an early stage that existing systems-such my models of language learning and systems like Prolog-would need radical re-thinking to meet the goal of simplifying and integrating ideas across a wide area.', '1306.3888-3-27-0': "The first published version of the SP theory described 'some unifying ideas in computing'.", '1306.3888-3-27-1': "Early work on the SP computer model concentrated on developing an improved version of the 'dynamic programming' technique for the alignment of two sequences (see BK, Appendix A) as a possible route to modelling human-like flexibility in pattern recognition, analysis of language, and the like.", '1306.3888-3-28-0': "About 1992, it became apparent that the explanatory range of the theory could be greatly expanded by forming alignments of 2, 3, or more sequences, much as in the 'multiple alignment' concept of bioinformatics.", '1306.3888-3-28-1': 'That idea was developed and adapted in new versions of the SP model, and incorporated in new procedures for unsupervised learning.', '1306.3888-3-29-0': 'Aspects of the theory, with many examples, have been developed in [CITATION].', '1306.3888-3-30-0': '# Introduction to the SP theory', '1306.3888-3-31-0': 'The main elements of the SP theory are:', '1306.3888-3-32-0': '## The SP computer model', '1306.3888-3-33-0': 'The SP theory is realised most fully in the SP70 computer model, with capabilities in the building of multiple alignments and in unsupervised learning.', '1306.3888-3-33-1': 'This will be referred to as the SP model, although in some cases examples are from a subset of the model or slightly earlier precursors of it.', '1306.3888-3-34-0': 'The SP model and its precursors have played a key part in the development of the theory:', '1306.3888-3-35-0': 'The workings of the SP model is described in some detail in BK (Sections 3.9, 3.10, and 9.2) and more briefly in Sections [REF] and [REF], below.', '1306.3888-3-35-1': "The source code for the models, with associated documents and files, may be downloaded via links under the heading 'SOURCE CODE' at the bottom of the page on http://bit.ly/WtXa3gbit.ly/WtXa3g.", '1306.3888-3-36-0': 'The two main elements of the models, described in the following two sections, are the building of multiple alignments and the unsupervised learning of Old patterns.', '1306.3888-3-37-0': '## The SP machine', '1306.3888-3-38-0': 'The SP model may be regarded as a first version of the SP machine, an expression of the SP theory and a means for it to be applied.', '1306.3888-3-39-0': 'A useful step forward in the development of the SP theory would be the creation of a high-parallel, open-source version of the SP machine, accessible via the web, and with a good user interface.', '1306.3888-3-39-1': 'This would provide a means for researchers to explore what can be done with the system and to improve it.', '1306.3888-3-39-2': 'How things may develop is shown schematically in Figure [REF].', '1306.3888-3-40-0': 'The high-parallel search mechanisms in any of the existing internet search engines would probably provide a good foundation for the proposed development.', '1306.3888-3-41-0': 'Further ahead, there may be a case for the creation of new kinds of hardware, dedicated to the building of multiple alignments and other processes in the SP framework .', '1306.3888-3-42-0': '## Unfinished business', '1306.3888-3-43-0': 'Like most theories, the SP theory has shortcomings, but it appears that they may be overcome.', '1306.3888-3-43-1': 'At present, the most immediate problems are:', '1306.3888-3-44-0': '# The multiple alignment concept', '1306.3888-3-45-0': 'The concept of multiple alignment in the SP theory has been adapted from a similar concept in bioinformatics, where it means a process of arranging, in rows or columns, two or more DNA sequences or amino-acid sequences so that matching symbols-as many as possible-are aligned orthogonally in columns or rows.', '1306.3888-3-46-0': 'Multiple alignments like these are normally used in the computational analysis of (symbolic representations of) sequences of DNA bases or sequences of amino acid residues as part of the process of elucidating the structure, functions or evolution of the corresponding molecules.', '1306.3888-3-46-1': 'An example of this kind of multiple alignment is shown in Figure [REF].', '1306.3888-3-47-0': 'As in bioinformatics, a multiple alignment in the SP system is an arrangement of two or more patterns in rows (or columns), with one pattern in each row (or column).', '1306.3888-3-47-1': 'The main difference between the two concepts is that, in bioinformatics, all sequences have the same status, whereas in the SP theory, the system attempts to create a multiple alignment which enables one New pattern (sometimes more) to be encoded economically in terms of one or more Old patterns.', '1306.3888-3-47-2': 'Other differences are described in BK (Section 3.4.1).', '1306.3888-3-48-0': "In Figure [REF], row 0 contains a New pattern representing a sentence: 't h i s b o y l o v e s t h a t g i r l', while each of rows 1 to 8 contains an Old pattern representing a grammatical rule or a word with grammatical markers.", '1306.3888-3-48-1': 'This multiple alignment, which achieves the effect of parsing the sentence in terms of grammatical structures, is the best of several built by the model when it is supplied with the New pattern and a set of Old patterns that includes those shown in the figure and several others as well.', '1306.3888-3-49-0': "In this example, and others in this article, 'best' means that the multiple alignment in the figure is the one that enables the New pattern to be encoded most economically in terms of the Old patterns, as described in Section [REF], below.", '1306.3888-3-50-0': '## Coding and the evaluation of an alignment in terms of compression', '1306.3888-3-51-0': 'This section describes in outline how multiple alignments are evaluated in the SP model.', '1306.3888-3-51-1': 'More detail may be found in BK (Section 3.5).', '1306.3888-3-52-0': "Each Old pattern in the SP system contains one or more 'identification' symbols or ID-symbols which, as their name suggests, serve to identify the pattern.", '1306.3888-3-52-1': "Examples of ID-symbols in Figure [REF] are 'D' and '0' at the beginning of 'D 0 t h i s D' (row 6), and 'N' and '1' at the beginning of 'N 1 b o y N' (row 8).", '1306.3888-3-53-0': "Associated with each type of symbol (where a 'type' of symbol is any one of a set of symbols that match each other exactly) is a notional code or bit pattern that serves to distinguish the given type from all the others.", '1306.3888-3-53-1': 'This is only notional because the bit patterns are not actually constructed.', '1306.3888-3-53-2': 'All that is needed for the purpose of evaluating multiple alignments is the size of the notional bit pattern associated with each type.', '1306.3888-3-53-3': 'This is calculated via the Shannon-Fano-Elias coding scheme (described in [CITATION]), using information about the frequency of occurrence of each Old pattern, so that the shortest codes represent the most frequent symbol types and vice versa.', '1306.3888-3-53-4': 'Notice that these bit patterns and their sizes are totally independent of the names for symbols that are used in written accounts like this one: names that are chosen purely for their mnemonic value.', '1306.3888-3-54-0': 'Given a multiple alignment like the one shown in Figure [REF], one can derive a code pattern from the multiple alignment in the following way:', '1306.3888-3-55-0': 'Scan the multiple alignment from left to right looking for columns that contain an ID-symbol by itself, not aligned with any other symbol.', '1306.3888-3-56-0': 'Copy these symbols into a code pattern in the same order that they appear in the multiple alignment.', '1306.3888-3-57-0': "The code pattern derived in this way from the multiple alignment shown in Figure [REF] is 'S 0 1 0 1 0 S'.", '1306.3888-3-57-1': 'This is, in effect, a compressed representation of those symbols in the New pattern that are aligned with Old symbols in the multiple alignment.', '1306.3888-3-57-2': 'In this case, the code pattern is a compressed representation of all the symbols in the New pattern but it often happens that some of the symbols in the New pattern are not matched with any Old symbols and then the code pattern will represent only those New symbols that are aligned with Old symbols.', '1306.3888-3-58-0': "In the context of natural language processing, it perhaps more plausible to suppose that the encoding of a sentence is some kind of representation of the meaning of the sentence, instead of a pattern like 'S 0 1 0 1 0 S'.", '1306.3888-3-58-1': 'How a meaning may be derived from a sentence via multiple alignment is described in BK (Section 5.7).', '1306.3888-3-59-0': '### Compression difference and compression ratio', '1306.3888-3-60-0': "Given a code pattern like 'S 0 1 0 1 0 S', we may calculate a 'compression difference'CD as: [EQUATION] or a 'compression ratio'CR as: [EQUATION] where [MATH] is the total number of bits in those symbols in the New pattern that are aligned with Old symbols in the alignment and [MATH] is the total number of bits in the symbols in the code pattern, and the number of bits for each symbol is calculated via the Shannon-Fano-Elias scheme as mentioned above.", '1306.3888-3-61-0': '[MATH] and [MATH] are each an indication of how effectively the New pattern (or those parts of the New pattern that are aligned with symbols within Old patterns in the alignment) may be compressed in terms of the Old patterns that appear in the given multiple alignment.', '1306.3888-3-61-1': 'The [MATH] of a multiple alignment-which has been found to be more useful than [MATH]-may be referred to as the compression score of the multiple alignment.', '1306.3888-3-62-0': 'In each of these equations, [MATH] is calculated as: [EQUATION] where [MATH] is the size of the code for [MATH]th symbol in a sequence, [MATH], comprising those symbols within the New pattern that are aligned with Old symbols within the multiple alignment.', '1306.3888-3-63-0': '[MATH] is calculated as: [EQUATION] where [MATH] is the size of the code for [MATH]th symbol in the sequence of [MATH] symbols in the code pattern derived from the multiple alignment.', '1306.3888-3-64-0': '## The building of multiple alignments', '1306.3888-3-65-0': 'This section describes in outline how the SP model builds multiple alignments.', '1306.3888-3-65-1': 'More detail may be found in BK (Section 3.10).', '1306.3888-3-66-0': 'Multiple alignments are built in stages, with pairwise matching and alignment of patterns.', '1306.3888-3-66-1': 'At each stage, any partially-constructed multiple alignment may be processed as if it was a basic pattern and carried forward to later stages.', '1306.3888-3-66-2': 'This is broadly similar to some programs for the creation of multiple alignments in bioinformatics.', '1306.3888-3-66-3': 'At all stages, the aim is to encode New information economically in terms of Old information and to weed out multiple alignments that score poorly in that regard.', '1306.3888-3-67-0': 'The model may create Old patterns for itself, as described in Section [REF], but when the formation of multiple alignments is the focus of interest, Old patterns may be supplied by the user.', '1306.3888-3-67-1': 'In all cases, New patterns must be supplied by the user.', '1306.3888-3-68-0': 'At each stage of building multiple alignments, the operations are as follows:', '1306.3888-3-69-0': "Identify a set of 'driving' patterns and a set of 'target' patterns.", '1306.3888-3-69-1': 'At the beginning, the New pattern is the sole driving pattern and the Old patterns are the target patterns.', '1306.3888-3-69-2': 'In all subsequent stages, the best of the multiple alignments formed so far (in terms of their [MATH] scores) are chosen to be driving patterns and the target patterns are the Old patterns together with a selection of the best multiple alignments formed so far, including all of those that are driving patterns.', '1306.3888-3-70-0': 'Compare each driving pattern with each of the target patterns to find full matches and good partial matches between patterns.', '1306.3888-3-70-1': "This is done with a process that is essentially a form of 'dynamic programming' , somewhat like the WinMerge utility for finding similarities and differences between files.", '1306.3888-3-70-2': 'The process is described quite fully in BK (Appendix A) and outlined in Section [REF], below.', '1306.3888-3-70-3': "The main difference between the SP process and others, is that the former can deliver several alternative matches between a pair of patterns, while WinMerge and standard methods for finding alignments deliver one 'best' result.", '1306.3888-3-71-0': 'From the best of the matches found in the current stage, create corresponding multiple alignments and add them to the repository of multiple alignments created by the program.', '1306.3888-3-72-0': 'This process of matching driving patterns against target patterns and building multiple alignments is repeated until no more multiple alignments can be found.', '1306.3888-3-72-1': 'For the best of the multiple alignments created since the start of processing, probabilities are calculated, as described in Section [REF].', '1306.3888-3-73-0': '### Finding good matches between patterns', '1306.3888-3-74-0': "Figure [REF] shows with a simple example how the SP model finds good full and partial matches between a 'query' string of atomic symbols (alphabetic characters in this example) and a 'database' string:", '1306.3888-3-75-0': 'The query is processed left to right, one symbol at a time.', '1306.3888-3-76-0': 'Each symbol in the query is, in effect, broadcast to every symbol in the database to make a yes/no match in each case.', '1306.3888-3-77-0': 'Every positive match (hit) between a symbol from the query and a symbol in the database is recorded in a hit structure, illustrated in the figure.', '1306.3888-3-78-0': 'If the memory space allocated to the hit structure is exhausted at any time then the hit structure is purged: the leaf nodes of the tree are sorted in reverse order of their probability values and each leaf node in the bottom half of the set is extracted from the hit structure, together with all nodes on its path which are not shared with any other path.', '1306.3888-3-78-1': 'After the hit structure has been purged, the recording of hits may continue using the space which has been released.', '1306.3888-3-79-0': '### Noisy data', '1306.3888-3-80-0': 'Because of the way each model searches for a global optimum in the building of multiple alignments, it does not depend on the presence or absence of any particular feature or combination of features.', '1306.3888-3-80-1': 'Up to a point, plausible results may be obtained in the face of errors of omission, commission and substitution in the data.', '1306.3888-3-80-2': "This is illustrated in the two multiple alignments in Figure [REF] where the New pattern in row 0 of (b) is the same sentence as in (a) ('t w o k i t t e n s p l a y') but with the omission of the 'w' in 't w o', the substitution of 'm' for 'n' in 'k i t t e n s', and the addition of 'x' within the word 'p l a y'.", '1306.3888-3-80-3': "Despite these errors, the best multiple alignment created by the SP model is, as shown in (b), the one that we judge intuitively to be 'correct'.", '1306.3888-3-81-0': 'This kind of ability to cope gracefully with noisy data is very much in keeping with our ability to understand speech in noisy surroundings, to understand written language despite errors, and to recognise people, trees, houses, and the like, despite fog, snow, falling leaves, or other things that may obstruct our view.', '1306.3888-3-81-1': 'In a similar way, it is likely to prove useful in artificial systems for such applications as the processing of natural language and the recognition of patterns.', '1306.3888-3-82-0': '## Computational complexity', '1306.3888-3-83-0': 'In considering the matching and unification of patterns, it not hard to see that, for any body of information [MATH], except very small examples, there is a huge number of alternative ways in which patterns may be matched against each other, there will normally be many alternative ways in which patterns may be unified, and exhaustive search is not tractable (BK, Section 2.2.8.4).', '1306.3888-3-84-0': 'However, with the kinds of heuristic techniques that are familiar in other AI applications-reducing the size of the search space by pruning the search tree at appropriate points, and being content with approximate solutions which are not necessarily perfect-this kind of matching becomes quite practical.', '1306.3888-3-84-1': 'Much the same can be said about the heuristic techniques used for the building of multiple alignments (Section [REF]) and for unsupervised learning (Section [REF]).', '1306.3888-3-85-0': 'For the process of building multiple alignments in the SP model, the time complexity in a serial processing environment, with conservative assumptions, has been estimated to be O[MATH], where [MATH] is the size of the pattern from New (in bits) and [MATH] is the sum of the lengths of the patterns in Old (in bits).', '1306.3888-3-85-1': 'In a parallel processing environment, the time complexity may approach O[MATH], depending on how well the parallel processing is applied.', '1306.3888-3-85-2': 'In serial and parallel environments, the space complexity has been estimated to be O[MATH].', '1306.3888-3-86-0': 'Although the data sets used with the current SP model have generally been small, there is reason to be confident that the models can be scaled up to deal with large data sets because the kind of flexible matching of patterns which is at the heart of the SP model is done very fast and with huge volumes of data by all the leading internet search engines.', '1306.3888-3-86-1': 'As was suggested in Section [REF], the relevant processes in any one of those search engines would probably provide a good basis for the creation of a high-parallel version of the SP machine.', '1306.3888-3-87-0': '## Calculation of probabilities associated with multiple alignments', '1306.3888-3-88-0': 'As described in BK (Chapter 7), the formation of multiple alignments in the SP framework supports several kinds of probabilistic reasoning.', '1306.3888-3-88-1': 'The core idea is that any Old symbol in a multiple alignment that is not aligned with a New symbol represents an inference that may be drawn from the multiple alignment.', '1306.3888-3-88-2': 'This section outlines how probabilities for such inferences may be calculated.', '1306.3888-3-88-3': 'There is more detail in BK (Section 3.7).', '1306.3888-3-89-0': '### Absolute probabilities', '1306.3888-3-90-0': 'Any sequence of [MATH] symbols, drawn from an alphabet of [MATH] alphabetic types, represents one point in a set of [MATH] points where [MATH] is calculated as: [EQUATION]', '1306.3888-3-90-1': 'If we assume that the sequence is random or nearly so, which means that the [MATH] points are equi-probable or nearly so, the probability of any one point (which represents a sequence of length [MATH]) is close to: [EQUATION]', '1306.3888-3-90-2': 'This equation may be used to calculate the absolute probability of the code pattern that may be derived from any given multiple alignment (as described in Section [REF]).', '1306.3888-3-90-3': 'That number may also be regarded as the absolute probability of any inferences that may be drawn from the multiple alignment.', '1306.3888-3-90-4': 'In this calculation, [MATH] is the sum of all the bits in the symbols of the code pattern and [MATH] is 2.', '1306.3888-3-91-0': 'As we shall see (Section [REF]), Equation [REF] may, with advantage, be generalised by replacing [MATH] with a value, [MATH], calculated in a slightly different way.', '1306.3888-3-92-0': '### Relative probabilities', '1306.3888-3-93-0': 'The absolute probabilities of multiple alignments, calculated as described in the last subsection, are normally very small and not very interesting in themselves.', '1306.3888-3-93-1': 'From the standpoint of practical applications, we are normally interested in the relative values of probabilities, calculated as follows.', '1306.3888-3-94-0': 'For the multiple alignment which has the highest [MATH] (which we shall call the reference multiple alignment), identify the reference set of symbols in New, meaning the symbols from New which are encoded by the multiple alignment.', '1306.3888-3-95-0': 'Compile a reference set of multiple alignments which includes the reference multiple alignment and all other multiple alignments (if any) which encode exactly the reference set of symbols from New, neither more nor less.', '1306.3888-3-96-0': 'Calculate the sum of the values for [MATH] in the reference set of multiple alignments: [EQUATION] where [MATH] is the size of the reference set of multiple alignments and [MATH] is the value of [MATH] for the [MATH]th multiple alignment in the reference set.', '1306.3888-3-97-0': 'For each multiple alignment in the reference set, calculate its relative probability as: [EQUATION]', '1306.3888-3-97-1': 'The values of [MATH], calculated as just described, provide an effective means of comparing the multiple alignments in the reference set.', '1306.3888-3-98-0': '### A generalisation of the method for calculating absolute and relative probabilities', '1306.3888-3-99-0': "The value of [MATH], calculated as described in Section [REF], may be regarded as the informational 'cost' of encoding the New symbol or symbols that appear in the multiple alignment, excluding those New symbols that have not appeared in the multiple alignment.", '1306.3888-3-100-0': 'This is OK but it is somewhat restrictive because it means that if we want to calculate relative probabilities for two or more multiple alignments they must all encode the same symbol or symbols from New.', '1306.3888-3-100-1': 'We cannot easily compare multiple alignments that encode different New symbols.', '1306.3888-3-101-0': 'The generalisation proposed here is that, in the calculation of absolute probabilities, a new value, [MATH], would be used instead of [MATH].', '1306.3888-3-101-1': 'This would be calculated as: [EQUATION] where [MATH] is the total number of bits in the symbols in the code patterns (as in Section [REF]) and [MATH] is the total number of bits in the New symbols that have not appeared in the multiple alignment.', '1306.3888-3-102-0': 'The rationale is that, to encode all the symbols in New, we can use the code pattern to encode those New symbols that do appear in the multiple alignment and, for each of the remaining New symbols, we can simply use its code.', '1306.3888-3-102-1': 'The advantage of this scheme is that we can compare any two or more multiple alignments, regardless of the number of New symbols that appear in the multiple alignment.', '1306.3888-3-103-0': '### Relative probabilities of patterns and symbols', '1306.3888-3-104-0': 'It often happens that a given pattern from Old, or a given symbol type within patterns from Old, appears in more than one of the multiple alignments in the reference set.', '1306.3888-3-104-1': 'In cases like these, one would expect the relative probability of the pattern or symbol type to be higher than if it appeared in only one multiple alignment.', '1306.3888-3-104-2': 'To take account of this kind of situation, the SP model calculates relative probabilities for individual patterns and symbol types in the following way:', '1306.3888-3-105-0': 'Compile a set of patterns from Old, each of which appears at least once in the reference set of multiple alignments.', '1306.3888-3-105-1': 'No single pattern from Old should appear more than once in the set.', '1306.3888-3-106-0': 'For each pattern, calculate a value for its relative probability as the sum of the [MATH] values for the multiple alignments in which it appears.', '1306.3888-3-106-1': 'If a pattern appears more than once in a multiple alignment, it is only counted once for that multiple alignment.', '1306.3888-3-107-0': 'Compile a set of symbol types which appear anywhere in the patterns identified in step 2.', '1306.3888-3-108-0': 'For each alphabetic symbol type identified in step 3, calculate its relative probability as the sum of the relative probabilities of the patterns in which it appears.', '1306.3888-3-108-1': 'If it appears more than once in a given pattern, it is only counted once.', '1306.3888-3-109-0': 'The foregoing applies only to symbol types which do not appear in New.', '1306.3888-3-109-1': 'Any symbol type that appears in New necessarily has a probability of [MATH]-because it has been observed, not inferred.', '1306.3888-3-110-0': '## One system for both the analysis and the production of information', '1306.3888-3-111-0': 'A potentially useful feature of the SP system is that the processes which serve to analyse or parse a New pattern in terms of Old patterns, and to create an economical encoding of the New pattern, may also work in reverse, to recreate the New pattern from its encoding.', '1306.3888-3-111-1': "This is the 'output' perspective, mentioned in Section [REF].", '1306.3888-3-112-0': "If the New pattern is the code sequence 'S 0 1 0 1 0 S' (as described in Section [REF]), and if the Old patterns are the same as were used to create the multiple alignment shown in Figure [REF], then the best multiple alignment found by the system is the one shown in Figure [REF].", '1306.3888-3-112-1': "This multiple alignment contains the same words as the original sentence ('t h i s b o y l o v e s t h a t g i r l'), in the same order as the original.", '1306.3888-3-112-2': "Readers who are familiar with Prolog, will recognise that this process of recreating the original sentence from its encoding is similar in some respects to the way in which an appropriately-constructed Prolog program may be run 'backwards', deriving 'data' from 'results'.", '1306.3888-3-113-0': 'How is it possible to decompress the compressed code for the original sentence by using information compression?', '1306.3888-3-113-1': "This apparent paradox-decompression by compression-may be resolved by ensuring that, when a code pattern like 'S 0 1 0 1 0 S' is used to recreate the original data, each symbol is treated, at least notionally, as if contained a few more bits of information than is strictly necessary.", '1306.3888-3-113-2': 'That residual redundancy allows the system to recreate the original sentence by the same process of compression as was used to create the original parsing and encoding.', '1306.3888-3-114-0': 'This process of creating a relatively large pattern from a relatively small encoding provides a model for the creation of sentences by a person or an artificial system.', '1306.3888-3-114-1': "But instead of the New pattern being a rather dry code, like 'S 0 1 0 1 0 S', it would be more plausible if it were some kind of representation of the meaning of the sentence, like that mentioned in Section [REF].", '1306.3888-3-114-2': 'How a sentence may be generated from a representation of meaning is outlined in BK (Section 5.7.1).', '1306.3888-3-115-0': "Similar principles may apply to other kinds of 'output', such as planning an outing, cooking a meal, and so on.", '1306.3888-3-116-0': '# Unsupervised learning', '1306.3888-3-117-0': 'As was mentioned in Section [REF], part of the inspiration for the SP theory has been a programme of research developing models of the unsupervised learning of language.', '1306.3888-3-117-1': 'But although the SNPR model is quite successful in deriving plausible grammars from samples of English-like artificial language, it has proved to be quite unsuitable as a basis for the SP theory.', '1306.3888-3-117-2': 'In order to accommodate other aspects of intelligence, such as pattern recognition, reasoning, and problem solving, it has been necessary to develop an entirely new conceptual framework, with multiple alignment at centre stage.', '1306.3888-3-118-0': 'So there is now the curious paradox that, while the SP theory is rooted in work on unsupervised learning, and that kind of learning has a central role in the theory, the SP model does much the same things as the earlier model, and with similar limitations (Sections [REF] and [REF]).', '1306.3888-3-118-1': 'But I believe that the new conceptual framework has many advantages, that it provides a much sounder footing for further developments, and that with some reorganisation of the learning processes in the SP computer model, its current weaknesses may be overcome (Section [REF]).', '1306.3888-3-119-0': '## Outline of unsupervised learning in the SP model', '1306.3888-3-120-0': 'The outline of the SP model in this section aims to provide sufficient detail for a good intuitive grasp of how it works.', '1306.3888-3-120-1': 'A lot more detail may be found in BK (Chapter 9).', '1306.3888-3-121-0': 'In addition to the processes for building multiple alignments, the SP model has processes for deriving Old patterns from multiple alignments, evaluating sets of newly-created Old patterns in terms of their effectiveness for the economical encoding of the New information, and the weeding out low-scoring sets.', '1306.3888-3-121-1': 'The system does not merely record statistical information, it uses that information to learn new structures.', '1306.3888-3-122-0': '### Deriving Old patterns from multiple alignments', '1306.3888-3-123-0': 'The process of deriving Old patterns from multiple alignments is illustrated schematically in Figure [REF].', '1306.3888-3-123-1': "As was mentioned in Section [REF], the SP system is conceived as an abstract brain-like system that, in 'input' mode, may receive 'New' information via its senses and store some or all of it as 'Old' information.", '1306.3888-3-123-2': 'Here, we may think of it as the brain of a baby who is listening to what people are saying.', '1306.3888-3-123-3': "Let's imagine that he or she hears someone say 't h a t b o y r u n s'.", '1306.3888-3-123-4': 'If the baby has never heard anything similar, then, if it is stored at all, that New information may be stored as a relatively straightforward copy, something like the Old pattern shown in row 1 of the multiple alignment in part (a) of the figure.', '1306.3888-3-124-0': "Now let us imagine that the information has been stored and that, at some later stage, the baby hears someone say 't h a t g i r l r u n s'.", '1306.3888-3-124-1': 'Then, from that New information and the previously-stored Old pattern, a multiple alignment may be created like the one shown in part (a) of Figure [REF].', '1306.3888-3-124-2': "And, by picking out coherent sequences that are either fully matched or not matched at all, four putative words may be extracted: 't h a t', 'g i r l', 'b o y', and 'r u n s', as shown in the first four patterns in part (b) of the figure.", '1306.3888-3-124-3': "In each newly-created Old pattern there are additional symbols such as 'B', '2', and 'B' that are added by the system, and which serve to identify the pattern, to mark its boundaries, and to mark its grammatical category or categories.", '1306.3888-3-125-0': "In addition to these four patterns, a fifth pattern is created, 'E 6 B B C C D D E', as shown in the figure, that records the sequence 't h a t ... r u n s', with the category 'C C' in the middle representing a choice between 'b o y' and 'g i r l'.", '1306.3888-3-125-1': 'Part (b) in the figure is the beginnings of a grammar to describe that kind of phrase.', '1306.3888-3-126-0': '### Evaluating and selecting sets of newly-created Old patterns', '1306.3888-3-127-0': 'The example just described shows how Old patterns may be derived from a multiple alignment but it gives a highly misleading impression of how the SP model actually works.', '1306.3888-3-127-1': "In practice, the program forms many multiple alignments that are much less tidy than the one shown and it creates many Old patterns that are clearly 'wrong'.", '1306.3888-3-127-2': "However, the program contains procedures for evaluating candidate sets of patterns ('grammars') and weeding out those that score badly in terms of their effectiveness for encoding the New information economically.", '1306.3888-3-127-3': "Out of all the muddle, it can normally abstract one or two 'best' grammars and these are normally ones that appear intuitively to be 'correct', or nearly so.", '1306.3888-3-127-4': 'In general, the program can abstract one or more plausible grammars from a sample of English-like artificial language, including words, grammatical categories of words, and sentence structure.', '1306.3888-3-128-0': 'In accordance with the principles of minimum length encoding , the aim of these processes of sifting and sorting is to minimise [MATH], where [MATH] is the size (in bits) of the grammar that is under development and [MATH] is the size (in bits) of the New patterns when they have been encoded in terms of the grammar.', '1306.3888-3-129-0': 'For a given grammar comprising patterns [MATH], the value of [MATH] is calculated as: [EQUATION] where [MATH] is the number of symbols in the [MATH]th pattern and [MATH] is the encoding cost of the [MATH]th symbol in that pattern.', '1306.3888-3-130-0': 'Given that each grammar is derived from a set [MATH] of multiple alignments (one multiple alignment for each pattern from New), the value of [MATH] for the grammar is calculated as: [EQUATION] where [MATH] is the size, in bits, of the code string derived from the [MATH]th multiple alignment (Section [REF]).', '1306.3888-3-131-0': 'For a given set of patterns from New, a tree of alternative grammars is created with branching occurring wherever there are two or more alternative multiple alignments for a given pattern from New.', '1306.3888-3-131-1': 'The tree is grown in stages and pruned periodically to keep it within reasonable bounds.', '1306.3888-3-131-2': 'At each stage, grammars with high values for [MATH] (which will be referred to as [MATH]) are eliminated.', '1306.3888-3-132-0': '### Plotting values for [MATH], [MATH] and [MATH]', '1306.3888-3-133-0': 'Figure [REF] shows cumulative values for [MATH], [MATH] and [MATH] as the SP model searches for good grammars for a succession of 8 New patterns, each of which represents a sentence.', '1306.3888-3-133-1': 'Each point on each of the lower three graphs represents the relevant value (on the scale at the left) from the best grammar found after a given pattern from New has been processed.', '1306.3888-3-133-2': "The graph labelled '[MATH]' shows cumulative values on the scale at the left for the succession of New patterns.", '1306.3888-3-133-3': "The graph labelled '[MATH]' shows the amount of compression achieved (on the scale to the right).", '1306.3888-3-134-0': '### Limitations in the SP model and how they may be overcome', '1306.3888-3-135-0': 'As mentioned before (Section [REF]), there are two main weaknesses in the processes for unsupervised learning in the SP model as it is now: the model does not learn intermediate levels in a grammar (phrases or clauses) or discontinuous dependencies of the kind described in Sections [REF] to [REF].', '1306.3888-3-136-0': 'It appears that some reorganisation of the learning processes in the model would solve both problems.', '1306.3888-3-136-1': "What seems to be needed is a tighter focus on the principle that, with appropriately-constructed Old patterns, multiple alignments may be created without the kind of mis-match between patterns that may be seen in Figure [REF] (a) ('g i r l' and 'b o y' do not match each other), and that any such multiple alignment may be treated as if it was a simple pattern.", '1306.3888-3-136-2': 'That reform should facilitate the discovery of structures at multiple levels and the discovery of structures that are discontinuous in the sense that they can bridge intervening structures.', '1306.3888-3-137-0': '### Computational complexity', '1306.3888-3-138-0': 'As with the building of multiple alignments (Section [REF]), the computational complexity of learning in the SP model is kept under control by pruning the search tree at appropriate points, aiming to discover grammars that are reasonably good and not necessarily perfect.', '1306.3888-3-139-0': 'In a serial processing environment, the time complexity of learning in the SP model has been estimated to be O[MATH] where [MATH] is the number of patterns in New.', '1306.3888-3-139-1': 'In a parallel processing environment, the time complexity may approach O[MATH], depending on how well the parallel processing is applied.', '1306.3888-3-139-2': 'In serial or parallel environments, the space complexity has been estimated to be O[MATH].', '1306.3888-3-140-0': '## The discovery of natural structures via information compression (DONSVIC)', '1306.3888-3-141-0': "In our dealings with the world, certain kinds of structures appear to be more prominent and useful than others: in natural languages, there are words, phrase and sentences; we understand the visual and tactile worlds to be composed of discrete 'objects'; and conceptually, we recognise classes of things like 'person', 'house', 'tree', and so on.", '1306.3888-3-142-0': "It appears that these 'natural' kinds of structure are significant in our thinking because they provide a means of compressing sensory information, and that compression of information provides the key to their learning or discovery.", '1306.3888-3-142-1': 'At first sight, this looks like nonsense because popular programs for compression of information, such as those based on the LZW algorithm, or programs for JPEG compression of images, seem not to recognise anything resembling words, objects, or classes.', '1306.3888-3-142-2': 'But those programs are designed to work fast on low-powered computers.', '1306.3888-3-142-3': 'With other programs that are designed to be relatively thorough in their compression of information, natural structures can be revealed:', '1306.3888-3-143-0': 'It seems likely that the principles that have been outlined in this subsection may be applied not only to the discovery of words, phrases and grammars in language-like data but also to such things as the discovery of objects in images , and classes of entity in all kinds of data.', '1306.3888-3-143-1': "These principles may be characterised as the discovery of natural structures via information compression, or 'DONSVIC' for short.", '1306.3888-3-144-0': '## Generalisation, the correction of overgeneralisations, and learning from noisy data', '1306.3888-3-145-0': 'Issues that arise in the learning of a first language and, probably, in other kinds of learning, are illustrated in Figure [REF]:', '1306.3888-3-146-0': 'One possible answer is that mistakes are corrected by parents, teachers, and others.', '1306.3888-3-146-1': 'But the weight of evidence is that children can learn their first language without that kind of assistance.', '1306.3888-3-147-0': 'A better answer is the principle of minimum length encoding (described in its essentials in Section [REF]):', '1306.3888-3-148-0': '## One-trial learning and its implications', '1306.3888-3-149-0': 'In many theories of learning, the process is seen as gradual: behaviour is progressively shaped by rewards or punishments or other kinds of experience.', '1306.3888-3-150-0': 'But any theory of learning in which the process is necessarily gradual is out of step with our ordinary experience that we can and do learn things from a single experience, especially if that single experience is very significant for us (BK, Section 11.4.4.1).', '1306.3888-3-151-0': 'In the SP theory, one-trial learning is accommodated in the way the system can store New information directly.', '1306.3888-3-151-1': 'And the gradual nature of, for example, language learning, may be explained by the complexity of the process of sifting and sorting the many alternative sets of candidate patterns to find one or more sets that are good in terms of information compression (BK, Section 11.4.4.2).', '1306.3888-3-152-0': '# Computing, mathematics, and logic', '1306.3888-3-153-0': 'Drawing mainly on BK (Chapters 4 to 11), this and the following sections describe, with a selection of examples, how the SP theory relates to several areas in artificial intelligence, mainstream computing, and human perception and cognition.', '1306.3888-3-154-0': 'In BK (Chapter 4), I have argued that the SP system is equivalent to a universal Turing machine , in the sense that anything that may be computed with a Turing machine may, in principle, also be computed with an SP machine.', '1306.3888-3-154-1': "The 'in principle' qualification is necessary because the SP theory is still not fully mature and there are still some weaknesses in the SP computer models.", '1306.3888-3-154-2': 'The gist of the argument is that the operation of a Post canonical system may be understood in terms of the SP theory and, since it is accepted that the Post canonical system is equivalent to the Turing machine (as a computational system), the Turing machine may also be understood in terms of the SP theory.', '1306.3888-3-155-0': "The key differences between the SP theory and earlier theories of computing are that the SP theory has a lot more to say about the nature of intelligence than earlier theories, that the theory is founded on principles of information compression via the matching and unification of patterns ('computing as compression'), and that it includes mechanisms for building multiple alignments and for heuristic search that are not present in earlier models.", '1306.3888-3-156-0': '## Conventional computing systems', '1306.3888-3-157-0': 'In conventional computing systems, compression of information may be seen in the matching of patterns with at least implicit unification of patterns that match each other-processes that appear in a variety of guises (BK, Chapter 2).', '1306.3888-3-157-1': 'And three basic techniques for the compression of information-chunking-with-codes, schema-plus-correction, and run-length coding-may be seen in various forms in the organisation of computer programs (ibid.)', '1306.3888-3-158-0': '## Mathematics and logic', '1306.3888-3-159-0': 'In a similar way, several structures and processes in mathematics and logic may be interpreted in terms of information compression via the matching and unification of patterns, and the compression techniques just mentioned (BK, Chapter 10).', '1306.3888-3-159-1': 'For example, multiplication (as repeated addition) and exponentiation (as repeated multiplication) may be seen as examples of run-length coding; a function with parameters may be seen as an example of schema-plus-correction; the chunking-with-codes technique may be seen in the organisation of number systems; and so on.', '1306.3888-3-160-0': '## Computing and probabilities', '1306.3888-3-161-0': 'As we have seen, the SP system is fundamentally probabilistic.', '1306.3888-3-161-1': "If it is indeed Turing-equivalent, as suggested above, and if the Turing machine is regarded as a definition of 'computing', then we may conclude that computing is fundamentally probabilistic.", '1306.3888-3-161-2': 'That may seem like a strange conclusion in view of the clockwork certainties that we associate with the operation of ordinary computers and the workings of mathematics and logic.', '1306.3888-3-161-3': 'There are at least three answers to that apparent contradiction:', '1306.3888-3-162-0': '# Representation of knowledge', '1306.3888-3-163-0': "Within the multiple alignment framework (Section [REF]), SP patterns may serve to represent several kinds of knowledge, including grammars for natural languages, ontologies, class hierarchies with inheritance of attributes, including cross-classification or multiple inheritance, part-whole hierarchies and their integration with class-inclusion hierarchies, decision networks and trees, relational tuples, if-then rules, associations of medical signs and symptoms, causal relations, and concepts in mathematics and logic such as 'function', 'variable', 'value', 'set', and 'type definition'.", '1306.3888-3-164-0': 'The use of one simple format for the representation of knowledge facilitates the seamless integration of different kinds of knowledge.', '1306.3888-3-165-0': '# Natural language processing', '1306.3888-3-166-0': 'One of the main strengths of the SP system is in natural language processing (BK, Chapter 5):', '1306.3888-3-167-0': '## Discontinuous dependencies in syntax', '1306.3888-3-168-0': 'The way in which the SP system can record discontinuous dependencies in syntax may be seen in both of the two parsings in Figure [REF].', '1306.3888-3-168-1': "The pattern in row 8 of each multiple alignment records the syntactic dependency between the plural noun phrase ('t w o k i t t e n s') which is the subject of the sentence-marked with 'Np'-and the plural verb phrase ('p l a y')-marked with 'Vp'-which belongs with it.", '1306.3888-3-169-0': "This kind of dependency is discontinuous because it can bridge arbitrarily large amounts of intervening structure such as, for example, 'from the West' in a sentence like 'Winds from the West are strong'.", '1306.3888-3-170-0': 'This method of marking discontinuous dependencies can accommodate overlapping dependencies such as number dependencies and gender dependencies in languages like French (BK, Section 5.4).', '1306.3888-3-170-1': 'It also provides a means of encoding the interesting system of overlapping and interlocking dependencies in English auxiliary verbs, described by Noam Chomsky in Syntactic Structuresyearparchomsky_1957.', '1306.3888-3-171-0': "In that book, the structure of English auxiliary verbs is part of Chomsky's evidence in support of Transformational Grammar.", '1306.3888-3-171-1': 'Despite the elegance and persuasiveness of his arguments, it turns out that the structure of English auxiliary verbs may be described with non-transformational rules in, for example, Definite Clause Grammars , and also in the SP system, as outlined in the subsections that follow.', '1306.3888-3-172-0': '## Two quasi-independent patterns of constraint in English auxiliary verbs', '1306.3888-3-173-0': 'In English, the syntax for main verbs and the auxiliary verbs which may accompany them follows two quasi-independent patterns of constraint which interact in an interesting way.', '1306.3888-3-174-0': 'The primary constraints may be expressed with this sequence of symbols,', '1306.3888-3-175-0': 'M H B B V,', '1306.3888-3-176-0': 'which should be interpreted in the following way:', '1306.3888-3-177-0': 'The words occur in the order shown but any of the words may be omitted.', '1306.3888-3-178-0': "Questions of 'standard' form follow exactly the same pattern as statements except that the first verb, whatever it happens to be ('M', 'H', the first 'B', the second 'B' or 'V'), precedes the subject noun phrase instead of following it.", '1306.3888-3-179-0': 'Here are two examples of the primary pattern with all of the words included:', '1306.3888-3-180-0': 'It will have been being washed M H B B V', '1306.3888-3-181-0': 'Will it have been being washed?', '1306.3888-3-181-1': 'M H B B V', '1306.3888-3-182-0': 'The secondary constraints are these:', '1306.3888-3-183-0': 'Figure [REF] shows a selection of examples with the dependencies marked.', '1306.3888-3-184-0': '## Multiple alignments and English auxiliary verbs', '1306.3888-3-185-0': 'Without reproducing all the detail in BK (Section 5.5), we can see from Figures [REF] and [REF] how the primary and secondary constraints may be applied in the multiple alignment framework.', '1306.3888-3-186-0': 'In each figure, the sentence to be analysed is shown as a New pattern in column 0.', '1306.3888-3-186-1': "The primary constraints are applied via the matching of symbols in Old patterns in the remaining columns, with a consequent interlocking of the patterns so that they recognise sentences of the form 'M H B B V', with options as described above.", '1306.3888-3-187-0': 'In Figure [REF], the secondary constraints apply as follows:', '1306.3888-3-188-0': 'In Figure [REF], the secondary constraints apply like this:', '1306.3888-3-189-0': '# Pattern recognition', '1306.3888-3-190-0': 'The system also has some useful features as a framework for pattern recognition (BK, (Chapter 6):', '1306.3888-3-191-0': 'One area of application is medical diagnosis , viewed as pattern recognition.', '1306.3888-3-191-1': 'There is also potential to assist in the understanding of natural vision and in the development of computer vision, as discussed in [CITATION].', '1306.3888-3-192-0': '## Part-whole hierarchies, class hierarchies, and their integration', '1306.3888-3-193-0': 'A strength of the multiple alignment concept is that it provides a simple but effective vehicle for the representation and processing of class-inclusion hierarchies, part-whole hierarchies, and their integration.', '1306.3888-3-194-0': "Figure [REF] shows the best multiple alignment found by the SP model with the New pattern 'white-bib eats furry purrs' (column 0) representing some features of an unknown creature, and with a set of Old patterns representing different classes of animal, at varying levels of abstraction.", '1306.3888-3-194-1': "From this multiple alignment, we may conclude that the unknown entity is an animal (column 1), a mammal (column 2), a cat (column 3) and the specific individual 'Tibs' (column 4).", '1306.3888-3-195-0': "The framework also provides for the representation of heterarchies or cross classification: a given entity, such as 'Jane' (or a class of entities), may belong in two or more higher-level classes that are not themselves hierarchically related, such as 'woman' and 'doctor'.", '1306.3888-3-196-0': 'The way that class-inclusion relations and part-whole relations may be combined in one multiple alignment is illustrated in Figure [REF].', '1306.3888-3-196-1': 'Here, some features of an unknown plant are expressed as a set of New patterns, shown in column 0: the plant has chlorophyll, the stem is hairy, it has yellow petals, and so on.', '1306.3888-3-197-0': 'From this multiple alignment, we can see that the unknown plant is most likely to be the Meadow Buttercup, Ranunculus acris, as shown in column 1.', '1306.3888-3-197-1': 'As such, it belongs in the genus Ranunculus (column 6), the family Ranunculaceae (column 5), the order Ranunculales (column 4), the class Angiospermae (column 3), and the phylum Plants (column 2).', '1306.3888-3-198-0': 'Each of these higher-level classifications contributes information about attributes of the plant and its division into parts and sub-parts.', '1306.3888-3-198-1': "For example, as a member of the class Angiospermae (column 3), the plant has a shoot and roots, with the shoot divided into stem, leaves, and flowers; as a member of the family Ranunculaceae (column 5), the plant has flowers that are 'regular', with all parts 'free'; as a member of the phylum Plants (column 2), the buttercup has chlorophyll and creates its own food by photosynthesis; and so on.", '1306.3888-3-199-0': '## Inference and inheritance', '1306.3888-3-200-0': 'In the example just described, we can infer from the multiple alignment, very directly, that the plant which has been provisionally identified as the Meadow Buttercup performs photosynthesis (column 2), has five petals (column 6), is poisonous (column 5), and so on.', '1306.3888-3-200-1': "As in other object-oriented systems, the first of these attributes has been 'inherited' from the class 'Plants', the second from the class Ranunculus, and the third from the class Ranunculaceae.", '1306.3888-3-200-2': 'These kinds of inference illustrate the close connection, often remarked, between pattern recognition and inferential reasoning .', '1306.3888-3-201-0': '# Probabilistic reasoning', '1306.3888-3-202-0': "The SP system can model several kinds of reasoning including inheritance of attributes (as just described), one-step 'deductive' reasoning, abductive reasoning, reasoning with probabilistic decision networks and decision trees, reasoning with 'rules', nonmonotonic reasoning and reasoning with default values, reasoning in Bayesian networks (including 'explaining away'), causal diagnosis, and reasoning which is not supported by evidence (BK, Chapter 7).", '1306.3888-3-203-0': 'Since these several kinds of reasoning all flow from one computational framework (multiple alignment), they may be seen as aspects of one process, working individually or together without awkward boundaries.', '1306.3888-3-204-0': 'Plausible lines of reasoning may be achieved, even when relevant information is incomplete.', '1306.3888-3-205-0': "Probabilities of inferences may be calculated, including extreme values (0 or 1) in the case of logic-like 'deductions'.", '1306.3888-3-206-0': 'A selection of examples is described in the following subsections.', '1306.3888-3-207-0': '## Nonmonotonic reasoning and reasoning with default values', '1306.3888-3-208-0': 'Conventional deductive reasoning is monotonic because deductions made on the strength of current knowledge cannot be invalidated by new knowledge: the conclusion that "Socrates is mortal", deduced from "All humans are mortal" and "Socrates is human", remains true for all time, regardless of anything we learn later.', '1306.3888-3-208-1': 'By contrast, the inference that "Tweety can probably fly" from the propositions that "Most birds fly" and "Tweety is a bird" is nonmonotonic because it may be changed if, for example, we learn that Tweety is a penguin.', '1306.3888-3-209-0': 'This section presents some examples which show how the SP system can accommodate nonmonotonic reasoning.', '1306.3888-3-210-0': '### Typically, birds fly', '1306.3888-3-211-0': "The idea that (all) birds can fly may be expressed with an SP pattern like 'Bd bird name name canfly warm-blooded wings feathers ... Bd'.", '1306.3888-3-211-1': 'This, of course, is an oversimplification of the real-world facts because, while it true that the majority of birds fly, we know that there are also flightless birds like ostriches, penguins and kiwis.', '1306.3888-3-212-0': "In order to model these facts more closely, we need to modify the pattern that describes birds to be something like this: 'Bd bird name name f f warm-blooded wings feathers ... Bd.", '1306.3888-3-212-1': 'And, to our database of Old patterns, we need to add patterns like this:', '1306.3888-3-213-0': 'Default Bd f canfly #f #Bd #Default P penguin Bd f cannotfly #f #Bd ... #P O ostrich Bd f cannotfly #f #Bd ... #O.', '1306.3888-3-214-0': "Now, the pair of symbols 'f f' in 'Bd bird name name f f warm-blooded wings feathers ... Bd' functions like a 'variable' that may take the value 'canfly' if a given class of birds can fly and 'cannotfly' when a type of bird cannot fly.", '1306.3888-3-214-1': "The pattern 'P penguin Bd f cannotfly f Bd ... P' shows that penguins cannot fly and, likewise, the pattern 'O ostrich Bd f cannotfly f Bd ... O' shows that ostriches cannot fly.", '1306.3888-3-214-2': "The pattern 'Default Bd f canfly f Bd Default', which has a substantially higher frequency than the other two patterns, represents the default value for the variable which is 'canfly'.", '1306.3888-3-214-3': "Notice that all three of these patterns contains the pair of symbols 'Bd ... Bd' showing that the corresponding classes are all subclasses of birds.", '1306.3888-3-215-0': '### Tweety is a bird so, probably, Tweety can fly', '1306.3888-3-216-0': "When the SP model is run with 'bird Tweety' in New and the same patterns in Old as before, modified as just described, the three best multiple alignments found are those shown in Figures [REF], [REF] and [REF].", '1306.3888-3-217-0': 'The first multiple alignment tells us that, with a relative probability of 0.66, Tweety may be the typical kind of bird that can fly.', '1306.3888-3-217-1': 'The second multiple alignment tells us that, with a relative probability of 0.22, Tweety might be an ostrich and, as such, he or she would not be able to fly.', '1306.3888-3-217-2': 'Likewise, the third multiple alignment tells us that, with a relative probability of 0.12, Tweety might be a penguin and would not be able to fly.', '1306.3888-3-217-3': 'The values for probabilities in this simple example are derived from guestimated frequencies that are, almost certainly, not ornithologically correct.', '1306.3888-3-218-0': '### Tweety is a penguin, so Tweety cannot fly', '1306.3888-3-219-0': "Figure [REF] shows the best multiple alignment found by the SP model when it is run again, with 'penguin Tweety' in New instead of 'bird Tweety'.", '1306.3888-3-219-1': 'This time, there is only one multiple alignment in the reference set and its relative probability is 1.0.', '1306.3888-3-219-2': 'Correspondingly, all inferences that we can draw from this multiple alignment have a probability of 1.0.', '1306.3888-3-219-3': 'In particular, we can be confident, within the limits of the available knowledge, that Tweety cannot fly.', '1306.3888-3-220-0': 'In a similar way, if Tweety were an ostrich, we would be able to say with confidence (p = 1.0) that he or she would not be able to fly.', '1306.3888-3-221-0': "## Reasoning in Bayesian networks, including 'explaining away'", '1306.3888-3-222-0': "A Bayesian network is a directed, acyclic graph like the one shown in Figure [REF], below, where each node has zero or more 'inputs' (connections with nodes that can influence the given node) and one or more 'outputs' (connections to other nodes that the given node can influence).", '1306.3888-3-223-0': 'Each node contains a set of conditional probability values, each one the probability of a given output value for a given input value or combination of input values.', '1306.3888-3-223-1': 'With this information, conditional probabilities of alternative outputs for any node may be computed for any given combination of inputs.', '1306.3888-3-223-2': "By combining these calculations for sequences of nodes, probabilities may be propagated through the network from one or more 'start' nodes to one or more 'finishing' nodes.", '1306.3888-3-224-0': 'This section describes how the SP system may perform that kind of probabilistic reasoning, and some advantages compared with Bayesian networks.', '1306.3888-3-225-0': 'Judea Pearlyearpar[p. 7]pearl_1997 describes the phenomenon of \'explaining away\' like this: "If A implies B, C implies B, and B is true, then finding that C is true makes A less credible.', '1306.3888-3-225-1': 'In other words, finding a second explanation for an item of data makes the first explanation less credible.', '1306.3888-3-225-2': '" (his italics).', '1306.3888-3-225-3': 'Here is an example:', '1306.3888-3-226-0': 'Normally an alarm sound alerts us to the possibility of a burglary.', '1306.3888-3-226-1': 'If somebody calls you at the office and tells you that your alarm went off, you will surely rush home in a hurry, even though there could be other causes for the alarm sound.', '1306.3888-3-226-2': 'If you hear a radio announcement that there was an earthquake nearby, and if the last false alarm you recall was triggered by an earthquake, then your certainty of a burglary will diminish.', '1306.3888-3-227-0': 'The causal relationships in the example just described may be captured in a Bayesian network like the one shown in Figure [REF].', '1306.3888-3-228-0': 'Pearl argues that, with appropriate values for conditional probabilities, the phenomenon of "explaining away" can be explained in terms of this network (representing the case where there is a radio announcement of an earthquake) compared with the same network without the node for "radio announcement" (representing the situation where there is no radio announcement of an earthquake).', '1306.3888-3-229-0': '### Representing contingencies with patterns and frequencies', '1306.3888-3-230-0': 'To see how this phenomenon may be understood in terms of the SP theory, consider, first, the set of patterns shown in Figure [REF], which are to be stored in Old.', '1306.3888-3-230-1': "The patterns in the figure show events which occur together in some notional sample of the 'World' together with their frequencies of occurrence in the sample.", '1306.3888-3-231-0': "As with other knowledge-based systems, we shall assume that the 'closed-world' assumption applies so that the absence of any pattern may be taken to mean that the corresponding combination of events did not occur in the period when observations were made.", '1306.3888-3-232-0': "The first pattern ('burglary alarm') shows that there were 1000 occasions when there was a burglary and the alarm went off and the second pattern ('earthquake alarm') shows just 20 occasions when there was an earthquake and the alarm went off (presumably triggered by the earthquake).", '1306.3888-3-232-1': 'Thus we have assumed that, as triggers for the alarm, burglaries are much more common than earthquakes.', '1306.3888-3-233-0': 'Since there is no pattern showing that the alarm sounded when there was a burglary and an earthquake at the same time, we may assume, via the closed-world assumption, that nothing like that happened during the sampling period.', '1306.3888-3-234-0': "The third pattern ('alarm phonealarmcall') shows that, out of the 1020 cases when the alarm went off, there were 980 cases where a phone call about the alarm was made.", '1306.3888-3-234-1': 'Since there is no pattern showing phone calls about the alarm in any other context, the closed-world assumption allows us to assume that there were no false positives (eg., hoaxes)-phone calls about the alarm when no alarm had sounded.', '1306.3888-3-235-0': "The fourth pattern ('earthquake radioearthquakeannouncement') shows that, in the sampling period, there were 40 occasions when there was an earthquake with an announcement about it on the radio.", '1306.3888-3-235-1': "And the fifth pattern ('e1 earthquake e2') shows that an earthquake has occurred on 40 occasions in contexts where the alarm did not ring and there was no radio announcement.", '1306.3888-3-236-0': "As before, the absence of patterns like 'earthquake alarm radioearthquakeannouncement' representing cases where an earthquake triggers the alarm and also leads to a radio announcement, allows us to assume via the closed-world assumption that cases of that kind have not occurred in the sampling period.", '1306.3888-3-237-0': '### Approximating the temporal order of events', '1306.3888-3-238-0': 'In these patterns and in the multiple alignments shown below, the left-to-right order of symbols may be regarded as an approximation to the order of events in time.', '1306.3888-3-238-1': 'Thus in the first two patterns, events that can trigger an alarm precede the sounding of the alarm.', '1306.3888-3-238-2': "Likewise, in the third pattern, 'alarm' (meaning that the alarm has sounded) precedes 'phonealarmcall' (a phone call to say the alarm has sounded).", '1306.3888-3-238-3': 'A single dimension can only approximate the order of events in time because it cannot represent events which overlap in time or which occur simultaneously.', '1306.3888-3-238-4': 'However, this kind of approximation has little or no bearing on the points to be illustrated here.', '1306.3888-3-239-0': '### Other considerations', '1306.3888-3-240-0': 'Other points relating to the patterns shown in Figure [REF] include:', '1306.3888-3-241-0': '### Formation of alignments: the burglar alarm has sounded', '1306.3888-3-242-0': "Receiving a phone call to say that the burglar alarm at one's house has gone off may be represented by placing the symbol 'phonealarmcall' in New.", '1306.3888-3-242-1': 'Figure [REF] shows, at the top, the best multiple alignment formed by the SP model in this case, with the patterns from Figure [REF] in Old.', '1306.3888-3-242-2': 'The other two multiple alignments in the reference set are shown below the best multiple alignment, in order of CD value and relative probability.', '1306.3888-3-242-3': 'The actual values for [MATH] and relative probability are given in the caption to Figure [REF].', '1306.3888-3-243-0': 'The unmatched Old symbols in these multiple alignments represent inferences made by the system.', '1306.3888-3-243-1': 'The probabilities for these inferences which are calculated by the SP model (as outlined in Section [REF]) are shown in Table [REF].', '1306.3888-3-243-2': 'These probabilities do not add up to 1 and we should not expect them to because any given multiple alignment may contain two or more of these symbols.', '1306.3888-3-244-0': 'The most probable inference is the rather trivial inference that the alarm has indeed sounded.', '1306.3888-3-244-1': 'This reflects the fact that there is no pattern in Figure [REF] representing false positives for telephone calls about the alarm.', '1306.3888-3-244-2': 'Apart from the inference that the alarm has sounded, the most probable inference (p = 0.328) is that there has been a burglary.', '1306.3888-3-244-3': 'However, there is a distinct possibility that there has been an earthquake-but the probability in this case (p = 0.016) is much lower than the probability of a burglary.', '1306.3888-3-245-0': "These inferences and their relative probabilities seem to accord quite well with what one would naturally think following a telephone call to say that the burglar alarm at one's house has gone off (given that one was living in a part of the world where earthquakes were not vanishingly rare).", '1306.3888-3-246-0': '## Formation of alignments: the burglar alarm has sounded and there is a radio announcement of an earthquake', '1306.3888-3-247-0': "In this example, the phenomenon of 'explaining away' occurs when you learn not only that the burglar alarm has sounded but that there has been an announcement on the radio that there has been an earthquake.", '1306.3888-3-247-1': 'In terms of the SP model, the two events (the phone call about the alarm and the announcement about the earthquake) can be represented in New by a pattern like this:', '1306.3888-3-248-0': 'phone_alarm_call radio_earthquake_announcement', '1306.3888-3-249-0': "or 'radioearthquakeannouncement phonealarmcall'.", '1306.3888-3-249-1': 'The order of the two symbols makes no difference to the result.', '1306.3888-3-250-0': "In this case, there is only one multiple alignment (shown at the top of Figure [REF]) that can 'explain' all the information in New.", '1306.3888-3-250-1': "Since there is only this one multiple alignment in the reference set for the best multiple alignment, the associated probabilities of the inferences that can be read from the multiple alignment ('alarm' and 'earthquake') are 1.0: it was an earthquake that caused the alarm to go off (and led to the phone call), and not a burglary.", '1306.3888-3-251-0': "These results show how 'explaining away' may be explained in terms of the SP theory.", '1306.3888-3-251-1': "The main point is that the multiple alignment or multiple alignments that provide the best 'explanation' of a telephone call to say that one's burglar alarm has sounded is different from the multiple alignment or multiple alignments that best explain the same telephone call coupled with an announcement on the radio that there has been an earthquake.", '1306.3888-3-251-2': 'In the latter case, the best explanation is that the earthquake triggered the alarm.', '1306.3888-3-251-3': 'Other possible explanations have lower probabilities.', '1306.3888-3-252-0': '### Other possibilities', '1306.3888-3-253-0': 'As mentioned above, the closed-world assumption allows us to rule out possibilities such as:', '1306.3888-3-254-0': 'Nevertheless, we may consider possibilities of that kind by combining multiple alignments as described in BK (Section 7.8.7).', '1306.3888-3-254-1': 'But as a general rule, that kind of further analysis makes no difference to the original conclusion: the multiple alignment which was originally judged to represent the best interpretation of the available facts has not been dislodged from this position.', '1306.3888-3-254-2': 'This is in keeping with way we normally concentrate on the most likely explanations of events and ignore the many conceivable but unlikely alternatives.', '1306.3888-3-255-0': '## The SP framework and Bayesian networks', '1306.3888-3-256-0': 'The foregoing examples show that the SP framework is a viable alternative to Bayesian networks, at least in the kinds of situation that have been described.', '1306.3888-3-256-1': 'This subsection makes some general observations about the relative merits of the two frameworks for probabilistic reasoning where the events of interest are subject to multiple influences or chains of influence or both those things.', '1306.3888-3-257-0': "Without in any way diminishing Thomas Bayes' achievement, his theorem appears to have shortcomings as the basis for theorising about perception and cognition:", '1306.3888-3-258-0': '## Causal diagnosis', '1306.3888-3-259-0': 'In this section, we consider a simple example of fault diagnosis in an electronic circuit-described by [CITATION].', '1306.3888-3-259-1': 'Figure [REF] shows the circuit with inputs on the left, outputs on the right and, in between, three multipliers ([MATH], [MATH], and [MATH]) and two adders ([MATH] and [MATH]).', '1306.3888-3-259-2': 'For the given inputs on the left, it is clear that output F is false and output G is correct.', '1306.3888-3-260-0': 'Figure [REF] shows a causal network derived from the electronic circuit in Figure [REF] .', '1306.3888-3-260-1': 'In this diagram, [MATH], [MATH], [MATH], [MATH] and [MATH] represent the outputs of components [MATH], [MATH], [MATH], [MATH] and [MATH], respectively.', '1306.3888-3-260-2': "In each case, there are three causal influences on the output: the two inputs to the component and the state of the component which may be 'good' or 'bad'.", '1306.3888-3-260-3': 'These influences are shown in Figure [REF] by lines with arrows connecting the source of the influence to the target node.', '1306.3888-3-260-4': 'Thus, for example, the two inputs of component [MATH] are represented by [MATH] and [MATH], the good or bad state of component [MATH] is represented by the node labelled [MATH], and their causal influences on node [MATH] are shown by the three arrows pointing at that node.', '1306.3888-3-261-0': 'Given a causal analysis like this, and given appropriate information about conditional probabilities, it is possible to derive one or more alternative diagnoses of which components are good and which are bad.', '1306.3888-3-261-1': "In Pearl's example, it is assumed that components of the same type have the same prior probability of failure and that the probability of failure of multipliers is greater than for adders.", '1306.3888-3-261-2': 'Given these assumptions and some others together with the inputs and outputs shown in Figure [REF] (but not the intermediate values), the best diagnosis derived from the causal network is that the [MATH] component is bad and the second best diagnosis is that [MATH] is bad.', '1306.3888-3-261-3': 'Pearl indicates that some third-best interpretations may be retrievable (e.g., [MATH] and [MATH] are bad) "... but in general, it is not guaranteed that interpretations beyond the second-best will be retrievable."', '1306.3888-3-261-4': '(p. 272).', '1306.3888-3-262-0': '## An SP approach to causal diagnosis', '1306.3888-3-263-0': 'The main elements of the SP analysis presented here are as follows:', '1306.3888-3-264-0': 'Figure [REF] shows a set of patterns for the circuit shown in Figure [REF].', '1306.3888-3-264-1': "In the figure, the patterns that start with the symbol 'M1' represent input-output relations for component [MATH], those that start with 'M2' represent input-output relations for the [MATH] component and likewise for the other patterns except the last one (starting with the symbol 'frame') which is the framework pattern mentioned above.", '1306.3888-3-264-2': "For each initial symbol there is a corresponding 'terminating' symbol with an initial '' character.", '1306.3888-3-264-3': "For reasons explained shortly, there may be other symbols following the 'terminating' symbol.", '1306.3888-3-265-0': "Let us now consider the first pattern in the figure ('M1 M1GOOD TM1I1 TM1I2 TM1O M1 TM4I2') representing I/O relations for component [MATH] when that component is good, as indicated by the symbol 'M1GOOD'.", '1306.3888-3-265-1': "In this pattern, the symbols 'TM1I1', 'TM1I2' and 'TM1O' represent the two inputs and the output of the component, 'M1' is the terminating symbol, and 'TM4I2' serves to transfer the output of [MATH] to the second input of component [MATH] as will be explained.", '1306.3888-3-265-2': "In a symbol like 'TM1I1', 'T' indicates that the input is true, 'M1' identifies the component, and 'I1' indicates that this is the first input of the component.", '1306.3888-3-265-3': 'Other symbols may be interpreted in a similar way, following the key given in the caption of Figure [REF].', '1306.3888-3-265-4': 'In effect, this pattern says that, when the component is working correctly, true inputs yield a true output.', '1306.3888-3-265-5': 'The pattern has a relatively high frequency of occurrence (500,000) reflecting the idea that the component will normally work correctly.', '1306.3888-3-266-0': "The other two patterns for component [MATH] ('M1 M1BAD TM1I1 TM1I2 TM1O M1 TM4I2' and 'M1 M1BAD TM1I1 TM1I2 FM1O M1 FM4I2') describe I/O relations when the component is bad.", '1306.3888-3-266-1': 'The first one describes the situation where true inputs to a faulty component yield a true result, a possibility noted by Pearl (ibid.', '1306.3888-3-266-2': 'p. 265).', '1306.3888-3-266-3': 'The second pattern-with a higher frequency-describes the more usual situation where true inputs to a faulty component yield a false result.', '1306.3888-3-266-4': 'Both these bad patterns have much lower frequencies than the good pattern.', '1306.3888-3-267-0': 'The other patterns in Figure [REF] may be interpreted in a similar way.', '1306.3888-3-267-1': 'Components [MATH], [MATH] and [MATH] have only three patterns each because it is assumed that inputs to the circuit will always be true so it is not necessary to include patterns describing what happens when one or both of the inputs are false.', '1306.3888-3-267-2': 'By contrast, there are 4 good patterns and 8 bad patterns for each of [MATH] and [MATH] because either of these components may receive faulty input.', '1306.3888-3-268-0': 'For each of the five components, the frequencies of the bad patterns sum to 100.', '1306.3888-3-268-1': 'However, for each of the components [MATH], [MATH], and [MATH], the total frequency of the good patterns is 500,000 compared with 1,000,000 for the set of good patterns associated with each of the component [MATH] and [MATH].', '1306.3888-3-268-2': "These figures accord with the assumptions in Pearl's example that components of the same type have the same probability of failure and that the probability of failure of multipliers ([MATH], [MATH], and [MATH]) is greater than the probability of failure of adders ([MATH] and [MATH]).", '1306.3888-3-269-0': '## Multiple alignments in causal diagnosis', '1306.3888-3-270-0': 'Given appropriate patterns, the SP model constructs multiple alignments from which diagnoses may be obtained.', '1306.3888-3-270-1': "Figure [REF] shows the best multiple alignment created by the SP model with the Old patterns shown in Figure [REF] and 'TM1I1 TM1I2 TM2I1 TM2I2 TM3I1 TM3I2 FM4O TM5O' as the New pattern.", '1306.3888-3-270-2': 'The first six symbols in this pattern express the idea that all the inputs for components [MATH], [MATH] and [MATH] are true.', '1306.3888-3-270-3': "The penultimate symbol ('FM4O') shows that the output of [MATH] is false and the last symbol ('TM5O') shows that the output of [MATH] is true-in accordance with the outputs shown in Figure [REF].", '1306.3888-3-271-0': 'From the multiple alignment in Figure [REF] it can be inferred that component [MATH] is bad and all the other components are good.', '1306.3888-3-271-1': 'A total of seven alternative diagnoses can be derived from those multiple alignments created by the SP model that encode all the symbols in New.', '1306.3888-3-271-2': 'These diagnoses are shown in Table [REF], each with its relative probability.', '1306.3888-3-272-0': "It is interesting to see that the best diagnosis derived by the SP model ([MATH] is bad) and the second best diagnosis ([MATH] is bad) are in accordance with first two diagnoses obtained by Pearl's method.", '1306.3888-3-272-1': "The remaining five diagnoses derived by the SP model are different from the one obtained by Pearl's method ([MATH] and [MATH] are bad) but this is not altogether surprising because detailed frequencies or probabilities are different from Pearl's example and there are differences in assumptions that have been made.", '1306.3888-3-273-0': '# Information storage and retrieval', '1306.3888-3-274-0': "The SP theory provides a versatile model for database systems, with the ability to accommodate object-oriented structures, as well as relational 'tuples', and network and tree models of data .", '1306.3888-3-274-1': 'It lends itself most directly to information retrieval in the manner of query-by-example but it appears to have potential to support the use of natural language or query languages such as SQL.', '1306.3888-3-275-0': 'Unlike some ordinary database systems:', '1306.3888-3-276-0': '# Planning and problem solving', '1306.3888-3-277-0': 'The SP framework provides a means of planning a route between two places, and, with the translation of geometric patterns into textual form, it can solve the kind of geometric analogy problem that may be seen in some puzzle books and IQ tests (BK, Chapter 8).', '1306.3888-3-278-0': 'Figure [REF] shows an example of the latter kind of problem.', '1306.3888-3-278-1': 'The task is to complete the relationship "A is to B as C is to ?"', '1306.3888-3-278-2': "using one of the figures 'D', 'E', 'F' or 'G' in the position marked with '?", '1306.3888-3-278-3': "'.", '1306.3888-3-278-4': "For this example, the 'correct' answer is clearly 'E'.", '1306.3888-3-278-5': "Quote marks have been used for the word 'correct' because in many problems of this type, there may be two or even more alternative answers for which cases can be made and there is a corresponding uncertainty about which answer is the right one.", '1306.3888-3-279-0': "Computer-based methods for solving this kind of problem have existed for some time (e.g., Evans'yearparevans_1968 well-known heuristic algorithm).", '1306.3888-3-279-1': 'In more recent work , minimum length encoding principles have been applied to good effect.', '1306.3888-3-279-2': 'This kind of problem may also be understood in terms of the SP concepts.', '1306.3888-3-280-0': 'As in most previous work, the proposed solution assumes that some mechanism is available which can translate the geometric forms in each problem into patterns of text symbols like other patterns in this article.', '1306.3888-3-280-1': "For example, item 'A' in Figure [REF] may be described as 'small circle inside large triangle'.", '1306.3888-3-281-0': 'How this kind of translation may be done is not part of the present proposals (one such translation mechanism is described in [CITATION]).', '1306.3888-3-281-1': 'As noted elsewhere , successful solutions for this kind of problem require consistency in the way the translation is done.', '1306.3888-3-281-2': "For this example, it would be unhelpful if item 'A' in Figure [REF] were described as 'large triangle outside small circle' while item 'C' were described as 'small square inside large ellipse'.", '1306.3888-3-281-3': "For any one puzzle, the description needs to stick to one or other of 'X outside Y' or 'Y inside X'-and likewise for 'above/below' and 'left-of/right-of'.", '1306.3888-3-282-0': 'Given that the diagrammatic form of the problem has been translated into patterns as just described, this kind of problem can be cast as a problem of partial matching, well within the scope of the SP model.', '1306.3888-3-282-1': 'To do this, symbolic representations of item A and item B in Figure [REF] are treated as a single pattern, thus:', '1306.3888-3-283-0': 'small circle inside large triangle ; large circle above small triangle', '1306.3888-3-284-0': 'and this pattern is placed in New.', '1306.3888-3-284-1': 'Four other patterns are constructed by pairing a symbolic representation of item C (on the left) with symbolic representations of each of D, E, F and G (on the right), thus:', '1306.3888-3-285-0': 'C1 small square inside large ellipse ; D small square inside large circle #C1 C2 small square inside large ellipse ; E large square above small ellipse #C2 C3 small square inside large ellipse ; F small circle left-of large square #C3 C4 small square inside large ellipse ; G small ellipse above large rectangle #C4.', '1306.3888-3-286-0': 'These four patterns are placed in Old, each with an arbitrary frequency value of 1.', '1306.3888-3-287-0': 'Figure [REF] shows the best multiple alignment found by the SP model with New and Old as just described.', '1306.3888-3-287-1': 'The multiple alignment is a partial match between the New pattern (in column 0) and the second of the four patterns in Old (in column 1).', '1306.3888-3-287-2': "This corresponds with the 'correct' result (item E) as noted above.", '1306.3888-3-288-0': '# Compression of information', '1306.3888-3-289-0': 'Since information compression is central to the workings of the SP system, it is natural to consider whether the system might provide useful insights in that area.', '1306.3888-3-289-1': 'In that connection, the most promising aspects of the SP system appear to be:', '1306.3888-3-290-0': "In terms of the trade-off that exists between computational resources that are required and the level of compression that can be achieved, it is intended that the system will operate towards the 'up market' end of the spectrum-by contrast with LZW algorithms and the like, which have been designed to be 'quick-and-dirty', sacrificing performance for speed on low-powered computers.", '1306.3888-3-291-0': '# Perception, cognition and neuroscience', '1306.3888-3-292-0': 'Since much of the inspiration for the SP theory has come from evidence, mentioned in Section [REF], that, to a large extent, the workings of brains and nervous systems may be understood in terms of information compression, the theory is about perception and cognition as well as artificial intelligence and mainstream computing.', '1306.3888-3-293-0': 'That said, the main elements of the theory-the multiple alignment concept in particular-are theoretical constructs derived from what appears to be necessary to model, in an economical way, such things as pattern recognition, reasoning, and so on.', '1306.3888-3-293-1': 'In BK (Chapter 12), there is some discussion of how the SP concepts relate to a selection of issues in human perception and cognition.', '1306.3888-3-293-2': "A particular interest at the time of writing (after that chapter was written) is the way that the SP theory may provide an alternative to quantum probability as an explanation of phenomena such as the 'conjunction fallacy' [CITATION].", '1306.3888-3-294-0': 'In BK (Chapter 11), I have described in outline, and tentatively, how such things as SP patterns and multiple alignments may be realised with neurons and connections between them.', '1306.3888-3-294-1': 'The cortex of the brains of mammals-which is, topologically, a two-dimensional sheet-may be, in some respects, like a sheet of paper on which pattern assemblies may be written.', '1306.3888-3-294-2': 'These are neural analogues of SP patterns, shown schematically in Figure [REF].', '1306.3888-3-294-3': 'Unlike information written on a sheet of paper, there are neural connections between patterns-as shown in the figure-and communications amongst them.', '1306.3888-3-295-0': "These proposals, which are adapted with modifications from Hebb'syearparhebb_1949 concept of a 'cell assembly', are very different from how artificial 'neural networks' are generally conceived in computer science.", '1306.3888-3-295-1': 'As noted in Section [REF], learning in the SP system is very different from learning in that kind of network-or Hebbian learning.', '1306.3888-3-296-0': '# Conclusion', '1306.3888-3-297-0': 'The SP theory aims to simplify and integrate concepts across artificial intelligence, mainstream computing and human perception and cognition, with information compression as a unifying theme.', '1306.3888-3-297-1': 'The matching and unification of patterns and the concept of multiple alignment are central ideas.', '1306.3888-3-298-0': "In accordance with Occam's Razor, the SP system combines conceptual simplicity with descriptive and explanatory power.", '1306.3888-3-298-1': "A relatively simple mechanism provides an interpretation for a range of concepts and phenomena in several areas including conepts of 'computing', aspects of mathematics and logic, representation of knowledge, natural language processing, pattern recognition, several kinds of probabilistic reasoning, information storage and retrieval, planning and problem solving, information compression, neuroscience, and human perception and cognition.", '1306.3888-3-299-0': 'As suggested in Section [REF], an aid to further research would be the creation of a high-parallel, open-source version of the SP machine, that may be accessed via the web.'}

{'1306.3888-4-0-0': 'This article is an overview of the SP theory of intelligence, which aims to simplify and integrate concepts across artificial intelligence, mainstream computing and human perception and cognition, with information compression as a unifying theme.', '1306.3888-4-0-1': "It is conceived as a brain-like system that receives 'New' information and stores some or all of it in compressed form as 'Old' information; and it is realised in the form of a computer model, a first version of the SP machine.", '1306.3888-4-0-2': 'The matching and unification of patterns and the concept of multiple alignment are central ideas.', '1306.3888-4-0-3': "Using heuristic techniques, the system builds multiple alignments that are 'good' in terms of information compression.", '1306.3888-4-0-4': 'For each multiple alignment, probabilities may be calculated for associated inferences.', '1306.3888-4-0-5': "Unsupervised learning is done by deriving new structures from partial matches between patterns and via heuristic search for sets of structures that are 'good' in terms of information compression.", '1306.3888-4-0-6': "These are normally ones that people judge to be 'natural', in accordance with the 'DONSVIC' principle-the discovery of natural structures via information compression.", '1306.3888-4-0-7': "The SP theory provides an interpretation for concepts and phenomena in several other areas including 'computing', aspects of mathematics and logic, the representation of knowledge, natural language processing, pattern recognition, several kinds of reasoning, information storage and retrieval, planning and problem solving, information compression, neuroscience, and human perception and cognition.", '1306.3888-4-0-8': "Examples include the parsing and production of language with discontinuous dependencies in syntax, pattern recognition at multiple levels of abstraction and its integration with part-whole relations, nonmonotonic reasoning and reasoning with default values, reasoning in Bayesian networks including 'explaining away', causal diagnosis, and the solving of a geometric analogy problem.", '1306.3888-4-1-0': 'Keywords: information compression, artificial intelligence, multiple alignment, computing, representation of knowledge, natural language processing, pattern recognition, information retrieval, probabilistic reasoning, planning, problem solving, unsupervised learning.', '1306.3888-4-2-0': '# Introduction', '1306.3888-4-3-0': 'The SP theory of intelligence, which has been under development since about 1987, aims to simplify and integrate concepts across artificial intelligence, mainstream computing and human perception and cognition, with information compression as a unifying theme.', '1306.3888-4-4-0': "The name 'SP' is short for Simplicity and Power, because compression of any given body of information, [MATH], may be seen as a process of reducing informational 'redundancy' in [MATH] and thus increasing its 'simplicity', whilst retaining as much as possible of its non-redundant expressive 'power'.", '1306.3888-4-4-1': "Likewise with Occam's Razor (Section [REF], below).", '1306.3888-4-5-0': 'Aspects of the theory, as it has developed, have been described in several peer-reviewed articles.', '1306.3888-4-5-1': 'The most comprehensive description of the theory as it stands now, with many examples, is in [CITATION].', '1306.3888-4-6-0': 'But this book, with more than 450 pages, is too long to serve as an introduction to the theory.', '1306.3888-4-6-1': 'This article aims to meet that need, with a fairly full description of the theory and a selection of examples.', '1306.3888-4-6-2': "For the sake of brevity, the book will be referred to as 'BK'.", '1306.3888-4-7-0': 'The next section describes the origins and motivation for the SP theory, Section [REF] introduces the theory, Sections [REF] and [REF] fill in a lot of the details, while the following sections describe aspects of the theory and what it can do.', '1306.3888-4-8-0': '# Origins and motivation', '1306.3888-4-9-0': 'The following subsections outline the origins of the SP theory, how it relates to some other research, and how it has developed.', '1306.3888-4-10-0': '## Information compression', '1306.3888-4-11-0': 'Much of the inspiration for the SP theory is a body of research, pioneered by Fred Attneaveyearparattneave_1954, Horace Barlowyearparbarlow_1959,barlow_1969, and others, showing that several aspects of the workings of brains and nervous systems may be understood in terms of information compression.', '1306.3888-4-11-1': 'For example, when we view a scene with two eyes, the image on the retina of the left eye is almost exactly the same as the image on the retina of right eye, but our brains merge the two images into a single percept, and thus compress the information .', '1306.3888-4-12-0': 'More immediately, the theory has grown out of my own research, developing models of the unsupervised learning of a first language, where the importance of information compression became increasingly clear .', '1306.3888-4-13-0': "The theory also draws on principles of 'minimum length encoding' pioneered by [CITATION], and others.", '1306.3888-4-13-1': 'And it has become apparent that several aspects of computing, mathematics, and logic may be understood in terms of information compression (BK, Chapters 2 and 10).', '1306.3888-4-14-0': 'At an abstract level, information compression can bring two main benefits:', '1306.3888-4-15-0': 'In animals, we would expect these things to have been favoured by natural selection because of the competitive advantage they can bring.', '1306.3888-4-15-1': "Notwithstanding the 'QWERTY' phenomenon, there is reason to believe that information compression, properly applied, may yield comparable advantages in artificial systems.", '1306.3888-4-16-0': '## The matching and unification of patterns', '1306.3888-4-17-0': 'In the SP theory, the matching and unification of patterns is seen as being closer to the bedrock of information compression than more mathematical techniques such as wavelets or arithmetic coding, and closer to the bedrock of information processing and intelligence than, say, concepts of probability.', '1306.3888-4-17-1': "A working hypothesis in this programme of research is that, by staying close to relatively simple, 'primitive', concepts of matching patterns and unifying them, there is a better chance of cutting through unnecessary complexity, and in gaining new insights and better solutions to problems.", '1306.3888-4-17-2': 'The mathematical basis of wavelets, arithmetic coding, and probabilities, may itself be founded on the matching and unification of patterns (BK, Chapter 10).', '1306.3888-4-18-0': '## Simplification and integration of concepts', '1306.3888-4-19-0': "In accordance with Occam's Razor, the SP system aims to combine conceptual simplicity with descriptive and explanatory power.", '1306.3888-4-19-1': "Apart from this widely-accepted principle, the drive for simplification and integration of concepts in this research programme has been motivated in part by Allen Newell's critique of some kinds of research in cognitive science , and in part by the apparent fragmentation of research in artificial intelligence and mainstream computing, with their myriad of concepts and many specialisms.", '1306.3888-4-20-0': 'In attempting to simplify and integrate ideas, the SP theory belongs in the same tradition as unified theories of cognition such as Soar and ACT-R .', '1306.3888-4-20-1': "And it chimes with the resurgence of interest in understanding artificial intelligence as a whole and with research on 'natural computation' .", '1306.3888-4-21-0': "Although the SP programme shares some objectives with projects such as the Godel Machine , and 'universal artificial intelligence' , the approach is very different.", '1306.3888-4-22-0': '## Transparency in the representation of knowledge', '1306.3888-4-23-0': "In this research, it is assumed that knowledge in the SP system should normally be transparent or comprehensible, much as in the 'symbolic' tradition in artificial intelligence (see also Section [REF]), and distinct from the kind of 'sub-symbolic' representation of knowledge that is the rule in, for example, 'neural networks' as they are generally conceived in computer science.", '1306.3888-4-24-0': 'As we shall see in Section [REF] and elsewhere in this article, SP patterns in the multiple alignment framework may serve to represent a variety of kinds of knowledge, in symbolic forms.', '1306.3888-4-25-0': '## Development of the theory', '1306.3888-4-26-0': 'In developing the theory, it was apparent at an early stage that existing systems-such my models of language learning and systems like Prolog-would need radical re-thinking to meet the goal of simplifying and integrating ideas across a wide area.', '1306.3888-4-27-0': "The first published version of the SP theory described 'some unifying ideas in computing'.", '1306.3888-4-27-1': "Early work on the SP computer model concentrated on developing an improved version of the 'dynamic programming' technique for the alignment of two sequences (see BK, Appendix A) as a possible route to modelling human-like flexibility in pattern recognition, analysis of language, and the like.", '1306.3888-4-28-0': "About 1992, it became apparent that the explanatory range of the theory could be greatly expanded by forming alignments of 2, 3, or more sequences, much as in the 'multiple alignment' concept of bioinformatics.", '1306.3888-4-28-1': 'That idea was developed and adapted in new versions of the SP model, and incorporated in new procedures for unsupervised learning.', '1306.3888-4-29-0': 'Aspects of the theory, with many examples, have been developed in [CITATION].', '1306.3888-4-30-0': '# Introduction to the SP theory', '1306.3888-4-31-0': 'The main elements of the SP theory are:', '1306.3888-4-32-0': '## The SP computer model', '1306.3888-4-33-0': 'The SP theory is realised most fully in the SP70 computer model, with capabilities in the building of multiple alignments and in unsupervised learning.', '1306.3888-4-33-1': 'This will be referred to as the SP model, although in some cases examples are from a subset of the model or slightly earlier precursors of it.', '1306.3888-4-34-0': 'The SP model and its precursors have played a key part in the development of the theory:', '1306.3888-4-35-0': 'The workings of the SP model is described in some detail in BK (Sections 3.9, 3.10, and 9.2) and more briefly in Sections [REF] and [REF], below.', '1306.3888-4-35-1': "The source code for the models, with associated documents and files, may be downloaded via links under the heading 'SOURCE CODE' at the bottom of the page on http://bit.ly/WtXa3gbit.ly/WtXa3g.", '1306.3888-4-36-0': 'The two main elements of the models, described in the following two sections, are the building of multiple alignments and the unsupervised learning of Old patterns.', '1306.3888-4-37-0': '## The SP machine', '1306.3888-4-38-0': 'The SP model may be regarded as a first version of the SP machine, an expression of the SP theory and a means for it to be applied.', '1306.3888-4-39-0': 'A useful step forward in the development of the SP theory would be the creation of a high-parallel, open-source version of the SP machine, accessible via the web, and with a good user interface.', '1306.3888-4-39-1': 'This would provide a means for researchers to explore what can be done with the system and to improve it.', '1306.3888-4-39-2': 'How things may develop is shown schematically in Figure [REF].', '1306.3888-4-40-0': 'The high-parallel search mechanisms in any of the existing internet search engines would probably provide a good foundation for the proposed development.', '1306.3888-4-41-0': 'Further ahead, there may be a case for the creation of new kinds of hardware, dedicated to the building of multiple alignments and other processes in the SP framework .', '1306.3888-4-42-0': '## Unfinished business', '1306.3888-4-43-0': 'Like most theories, the SP theory has shortcomings, but it appears that they may be overcome.', '1306.3888-4-43-1': 'At present, the most immediate problems are:', '1306.3888-4-44-0': '# The multiple alignment concept', '1306.3888-4-45-0': 'The concept of multiple alignment in the SP theory has been adapted from a similar concept in bioinformatics, where it means a process of arranging, in rows or columns, two or more DNA sequences or amino-acid sequences so that matching symbols-as many as possible-are aligned orthogonally in columns or rows.', '1306.3888-4-46-0': 'Multiple alignments like these are normally used in the computational analysis of (symbolic representations of) sequences of DNA bases or sequences of amino acid residues as part of the process of elucidating the structure, functions or evolution of the corresponding molecules.', '1306.3888-4-46-1': 'An example of this kind of multiple alignment is shown in Figure [REF].', '1306.3888-4-47-0': 'As in bioinformatics, a multiple alignment in the SP system is an arrangement of two or more patterns in rows (or columns), with one pattern in each row (or column).', '1306.3888-4-47-1': 'The main difference between the two concepts is that, in bioinformatics, all sequences have the same status, whereas in the SP theory, the system attempts to create a multiple alignment which enables one New pattern (sometimes more) to be encoded economically in terms of one or more Old patterns.', '1306.3888-4-47-2': 'Other differences are described in BK (Section 3.4.1).', '1306.3888-4-48-0': "In Figure [REF], row 0 contains a New pattern representing a sentence: 't h i s b o y l o v e s t h a t g i r l', while each of rows 1 to 8 contains an Old pattern representing a grammatical rule or a word with grammatical markers.", '1306.3888-4-48-1': 'This multiple alignment, which achieves the effect of parsing the sentence in terms of grammatical structures, is the best of several built by the model when it is supplied with the New pattern and a set of Old patterns that includes those shown in the figure and several others as well.', '1306.3888-4-49-0': "In this example, and others in this article, 'best' means that the multiple alignment in the figure is the one that enables the New pattern to be encoded most economically in terms of the Old patterns, as described in Section [REF], below.", '1306.3888-4-50-0': '## Coding and the evaluation of an alignment in terms of compression', '1306.3888-4-51-0': 'This section describes in outline how multiple alignments are evaluated in the SP model.', '1306.3888-4-51-1': 'More detail may be found in BK (Section 3.5).', '1306.3888-4-52-0': "Each Old pattern in the SP system contains one or more 'identification' symbols or ID-symbols which, as their name suggests, serve to identify the pattern.", '1306.3888-4-52-1': "Examples of ID-symbols in Figure [REF] are 'D' and '0' at the beginning of 'D 0 t h i s D' (row 6), and 'N' and '1' at the beginning of 'N 1 b o y N' (row 8).", '1306.3888-4-53-0': "Associated with each type of symbol (where a 'type' of symbol is any one of a set of symbols that match each other exactly) is a notional code or bit pattern that serves to distinguish the given type from all the others.", '1306.3888-4-53-1': 'This is only notional because the bit patterns are not actually constructed.', '1306.3888-4-53-2': 'All that is needed for the purpose of evaluating multiple alignments is the size of the notional bit pattern associated with each type.', '1306.3888-4-53-3': 'This is calculated via the Shannon-Fano-Elias coding scheme (described in [CITATION]), using information about the frequency of occurrence of each Old pattern, so that the shortest codes represent the most frequent symbol types and vice versa.', '1306.3888-4-53-4': 'Notice that these bit patterns and their sizes are totally independent of the names for symbols that are used in written accounts like this one: names that are chosen purely for their mnemonic value.', '1306.3888-4-54-0': 'Given a multiple alignment like the one shown in Figure [REF], one can derive a code pattern from the multiple alignment in the following way:', '1306.3888-4-55-0': 'Scan the multiple alignment from left to right looking for columns that contain an ID-symbol by itself, not aligned with any other symbol.', '1306.3888-4-56-0': 'Copy these symbols into a code pattern in the same order that they appear in the multiple alignment.', '1306.3888-4-57-0': "The code pattern derived in this way from the multiple alignment shown in Figure [REF] is 'S 0 1 0 1 0 S'.", '1306.3888-4-57-1': 'This is, in effect, a compressed representation of those symbols in the New pattern that are aligned with Old symbols in the multiple alignment.', '1306.3888-4-57-2': 'In this case, the code pattern is a compressed representation of all the symbols in the New pattern but it often happens that some of the symbols in the New pattern are not matched with any Old symbols and then the code pattern will represent only those New symbols that are aligned with Old symbols.', '1306.3888-4-58-0': "In the context of natural language processing, it perhaps more plausible to suppose that the encoding of a sentence is some kind of representation of the meaning of the sentence, instead of a pattern like 'S 0 1 0 1 0 S'.", '1306.3888-4-58-1': 'How a meaning may be derived from a sentence via multiple alignment is described in BK (Section 5.7).', '1306.3888-4-59-0': '### Compression difference and compression ratio', '1306.3888-4-60-0': "Given a code pattern like 'S 0 1 0 1 0 S', we may calculate a 'compression difference'CD as: [EQUATION] or a 'compression ratio'CR as: [EQUATION] where [MATH] is the total number of bits in those symbols in the New pattern that are aligned with Old symbols in the alignment and [MATH] is the total number of bits in the symbols in the code pattern, and the number of bits for each symbol is calculated via the Shannon-Fano-Elias scheme as mentioned above.", '1306.3888-4-61-0': '[MATH] and [MATH] are each an indication of how effectively the New pattern (or those parts of the New pattern that are aligned with symbols within Old patterns in the alignment) may be compressed in terms of the Old patterns that appear in the given multiple alignment.', '1306.3888-4-61-1': 'The [MATH] of a multiple alignment-which has been found to be more useful than [MATH]-may be referred to as the compression score of the multiple alignment.', '1306.3888-4-62-0': 'In each of these equations, [MATH] is calculated as: [EQUATION] where [MATH] is the size of the code for [MATH]th symbol in a sequence, [MATH], comprising those symbols within the New pattern that are aligned with Old symbols within the multiple alignment.', '1306.3888-4-63-0': '[MATH] is calculated as: [EQUATION] where [MATH] is the size of the code for [MATH]th symbol in the sequence of [MATH] symbols in the code pattern derived from the multiple alignment.', '1306.3888-4-64-0': '## The building of multiple alignments', '1306.3888-4-65-0': 'This section describes in outline how the SP model builds multiple alignments.', '1306.3888-4-65-1': 'More detail may be found in BK (Section 3.10).', '1306.3888-4-66-0': 'Multiple alignments are built in stages, with pairwise matching and alignment of patterns.', '1306.3888-4-66-1': 'At each stage, any partially-constructed multiple alignment may be processed as if it was a basic pattern and carried forward to later stages.', '1306.3888-4-66-2': 'This is broadly similar to some programs for the creation of multiple alignments in bioinformatics.', '1306.3888-4-66-3': 'At all stages, the aim is to encode New information economically in terms of Old information and to weed out multiple alignments that score poorly in that regard.', '1306.3888-4-67-0': 'The model may create Old patterns for itself, as described in Section [REF], but when the formation of multiple alignments is the focus of interest, Old patterns may be supplied by the user.', '1306.3888-4-67-1': 'In all cases, New patterns must be supplied by the user.', '1306.3888-4-68-0': 'At each stage of building multiple alignments, the operations are as follows:', '1306.3888-4-69-0': "Identify a set of 'driving' patterns and a set of 'target' patterns.", '1306.3888-4-69-1': 'At the beginning, the New pattern is the sole driving pattern and the Old patterns are the target patterns.', '1306.3888-4-69-2': 'In all subsequent stages, the best of the multiple alignments formed so far (in terms of their [MATH] scores) are chosen to be driving patterns and the target patterns are the Old patterns together with a selection of the best multiple alignments formed so far, including all of those that are driving patterns.', '1306.3888-4-70-0': 'Compare each driving pattern with each of the target patterns to find full matches and good partial matches between patterns.', '1306.3888-4-70-1': "This is done with a process that is essentially a form of 'dynamic programming' , somewhat like the WinMerge utility for finding similarities and differences between files.", '1306.3888-4-70-2': 'The process is described quite fully in BK (Appendix A) and outlined in Section [REF], below.', '1306.3888-4-70-3': "The main difference between the SP process and others, is that the former can deliver several alternative matches between a pair of patterns, while WinMerge and standard methods for finding alignments deliver one 'best' result.", '1306.3888-4-71-0': 'From the best of the matches found in the current stage, create corresponding multiple alignments and add them to the repository of multiple alignments created by the program.', '1306.3888-4-72-0': 'This process of matching driving patterns against target patterns and building multiple alignments is repeated until no more multiple alignments can be found.', '1306.3888-4-72-1': 'For the best of the multiple alignments created since the start of processing, probabilities are calculated, as described in Section [REF].', '1306.3888-4-73-0': '### Finding good matches between patterns', '1306.3888-4-74-0': "Figure [REF] shows with a simple example how the SP model finds good full and partial matches between a 'query' string of atomic symbols (alphabetic characters in this example) and a 'database' string:", '1306.3888-4-75-0': 'The query is processed left to right, one symbol at a time.', '1306.3888-4-76-0': 'Each symbol in the query is, in effect, broadcast to every symbol in the database to make a yes/no match in each case.', '1306.3888-4-77-0': 'Every positive match (hit) between a symbol from the query and a symbol in the database is recorded in a hit structure, illustrated in the figure.', '1306.3888-4-78-0': 'If the memory space allocated to the hit structure is exhausted at any time then the hit structure is purged: the leaf nodes of the tree are sorted in reverse order of their probability values and each leaf node in the bottom half of the set is extracted from the hit structure, together with all nodes on its path which are not shared with any other path.', '1306.3888-4-78-1': 'After the hit structure has been purged, the recording of hits may continue using the space which has been released.', '1306.3888-4-79-0': '### Noisy data', '1306.3888-4-80-0': 'Because of the way each model searches for a global optimum in the building of multiple alignments, it does not depend on the presence or absence of any particular feature or combination of features.', '1306.3888-4-80-1': 'Up to a point, plausible results may be obtained in the face of errors of omission, commission and substitution in the data.', '1306.3888-4-80-2': "This is illustrated in the two multiple alignments in Figure [REF] where the New pattern in row 0 of (b) is the same sentence as in (a) ('t w o k i t t e n s p l a y') but with the omission of the 'w' in 't w o', the substitution of 'm' for 'n' in 'k i t t e n s', and the addition of 'x' within the word 'p l a y'.", '1306.3888-4-80-3': "Despite these errors, the best multiple alignment created by the SP model is, as shown in (b), the one that we judge intuitively to be 'correct'.", '1306.3888-4-81-0': 'This kind of ability to cope gracefully with noisy data is very much in keeping with our ability to understand speech in noisy surroundings, to understand written language despite errors, and to recognise people, trees, houses, and the like, despite fog, snow, falling leaves, or other things that may obstruct our view.', '1306.3888-4-81-1': 'In a similar way, it is likely to prove useful in artificial systems for such applications as the processing of natural language and the recognition of patterns.', '1306.3888-4-82-0': '## Computational complexity', '1306.3888-4-83-0': 'In considering the matching and unification of patterns, it not hard to see that, for any body of information [MATH], except very small examples, there is a huge number of alternative ways in which patterns may be matched against each other, there will normally be many alternative ways in which patterns may be unified, and exhaustive search is not tractable (BK, Section 2.2.8.4).', '1306.3888-4-84-0': 'However, with the kinds of heuristic techniques that are familiar in other AI applications-reducing the size of the search space by pruning the search tree at appropriate points, and being content with approximate solutions which are not necessarily perfect-this kind of matching becomes quite practical.', '1306.3888-4-84-1': 'Much the same can be said about the heuristic techniques used for the building of multiple alignments (Section [REF]) and for unsupervised learning (Section [REF]).', '1306.3888-4-85-0': 'For the process of building multiple alignments in the SP model, the time complexity in a serial processing environment, with conservative assumptions, has been estimated to be O[MATH], where [MATH] is the size of the pattern from New (in bits) and [MATH] is the sum of the lengths of the patterns in Old (in bits).', '1306.3888-4-85-1': 'In a parallel processing environment, the time complexity may approach O[MATH], depending on how well the parallel processing is applied.', '1306.3888-4-85-2': 'In serial and parallel environments, the space complexity has been estimated to be O[MATH].', '1306.3888-4-86-0': 'Although the data sets used with the current SP model have generally been small, there is reason to be confident that the models can be scaled up to deal with large data sets because the kind of flexible matching of patterns which is at the heart of the SP model is done very fast and with huge volumes of data by all the leading internet search engines.', '1306.3888-4-86-1': 'As was suggested in Section [REF], the relevant processes in any one of those search engines would probably provide a good basis for the creation of a high-parallel version of the SP machine.', '1306.3888-4-87-0': '## Calculation of probabilities associated with multiple alignments', '1306.3888-4-88-0': 'As described in BK (Chapter 7), the formation of multiple alignments in the SP framework supports several kinds of probabilistic reasoning.', '1306.3888-4-88-1': 'The core idea is that any Old symbol in a multiple alignment that is not aligned with a New symbol represents an inference that may be drawn from the multiple alignment.', '1306.3888-4-88-2': 'This section outlines how probabilities for such inferences may be calculated.', '1306.3888-4-88-3': 'There is more detail in BK (Section 3.7).', '1306.3888-4-89-0': '### Absolute probabilities', '1306.3888-4-90-0': 'Any sequence of [MATH] symbols, drawn from an alphabet of [MATH] alphabetic types, represents one point in a set of [MATH] points where [MATH] is calculated as: [EQUATION]', '1306.3888-4-90-1': 'If we assume that the sequence is random or nearly so, which means that the [MATH] points are equi-probable or nearly so, the probability of any one point (which represents a sequence of length [MATH]) is close to: [EQUATION]', '1306.3888-4-90-2': 'This equation may be used to calculate the absolute probability of the code pattern that may be derived from any given multiple alignment (as described in Section [REF]).', '1306.3888-4-90-3': 'That number may also be regarded as the absolute probability of any inferences that may be drawn from the multiple alignment.', '1306.3888-4-90-4': 'In this calculation, [MATH] is the sum of all the bits in the symbols of the code pattern and [MATH] is 2.', '1306.3888-4-91-0': 'As we shall see (Section [REF]), Equation [REF] may, with advantage, be generalised by replacing [MATH] with a value, [MATH], calculated in a slightly different way.', '1306.3888-4-92-0': '### Relative probabilities', '1306.3888-4-93-0': 'The absolute probabilities of multiple alignments, calculated as described in the last subsection, are normally very small and not very interesting in themselves.', '1306.3888-4-93-1': 'From the standpoint of practical applications, we are normally interested in the relative values of probabilities, calculated as follows.', '1306.3888-4-94-0': 'For the multiple alignment which has the highest [MATH] (which we shall call the reference multiple alignment), identify the reference set of symbols in New, meaning the symbols from New which are encoded by the multiple alignment.', '1306.3888-4-95-0': 'Compile a reference set of multiple alignments which includes the reference multiple alignment and all other multiple alignments (if any) which encode exactly the reference set of symbols from New, neither more nor less.', '1306.3888-4-96-0': 'Calculate the sum of the values for [MATH] in the reference set of multiple alignments: [EQUATION] where [MATH] is the size of the reference set of multiple alignments and [MATH] is the value of [MATH] for the [MATH]th multiple alignment in the reference set.', '1306.3888-4-97-0': 'For each multiple alignment in the reference set, calculate its relative probability as: [EQUATION]', '1306.3888-4-97-1': 'The values of [MATH], calculated as just described, provide an effective means of comparing the multiple alignments in the reference set.', '1306.3888-4-98-0': '### A generalisation of the method for calculating absolute and relative probabilities', '1306.3888-4-99-0': "The value of [MATH], calculated as described in Section [REF], may be regarded as the informational 'cost' of encoding the New symbol or symbols that appear in the multiple alignment, excluding those New symbols that have not appeared in the multiple alignment.", '1306.3888-4-100-0': 'This is OK but it is somewhat restrictive because it means that if we want to calculate relative probabilities for two or more multiple alignments they must all encode the same symbol or symbols from New.', '1306.3888-4-100-1': 'We cannot easily compare multiple alignments that encode different New symbols.', '1306.3888-4-101-0': 'The generalisation proposed here is that, in the calculation of absolute probabilities, a new value, [MATH], would be used instead of [MATH].', '1306.3888-4-101-1': 'This would be calculated as: [EQUATION] where [MATH] is the total number of bits in the symbols in the code patterns (as in Section [REF]) and [MATH] is the total number of bits in the New symbols that have not appeared in the multiple alignment.', '1306.3888-4-102-0': 'The rationale is that, to encode all the symbols in New, we can use the code pattern to encode those New symbols that do appear in the multiple alignment and, for each of the remaining New symbols, we can simply use its code.', '1306.3888-4-102-1': 'The advantage of this scheme is that we can compare any two or more multiple alignments, regardless of the number of New symbols that appear in the multiple alignment.', '1306.3888-4-103-0': '### Relative probabilities of patterns and symbols', '1306.3888-4-104-0': 'It often happens that a given pattern from Old, or a given symbol type within patterns from Old, appears in more than one of the multiple alignments in the reference set.', '1306.3888-4-104-1': 'In cases like these, one would expect the relative probability of the pattern or symbol type to be higher than if it appeared in only one multiple alignment.', '1306.3888-4-104-2': 'To take account of this kind of situation, the SP model calculates relative probabilities for individual patterns and symbol types in the following way:', '1306.3888-4-105-0': 'Compile a set of patterns from Old, each of which appears at least once in the reference set of multiple alignments.', '1306.3888-4-105-1': 'No single pattern from Old should appear more than once in the set.', '1306.3888-4-106-0': 'For each pattern, calculate a value for its relative probability as the sum of the [MATH] values for the multiple alignments in which it appears.', '1306.3888-4-106-1': 'If a pattern appears more than once in a multiple alignment, it is only counted once for that multiple alignment.', '1306.3888-4-107-0': 'Compile a set of symbol types which appear anywhere in the patterns identified in step 2.', '1306.3888-4-108-0': 'For each alphabetic symbol type identified in step 3, calculate its relative probability as the sum of the relative probabilities of the patterns in which it appears.', '1306.3888-4-108-1': 'If it appears more than once in a given pattern, it is only counted once.', '1306.3888-4-109-0': 'The foregoing applies only to symbol types which do not appear in New.', '1306.3888-4-109-1': 'Any symbol type that appears in New necessarily has a probability of [MATH]-because it has been observed, not inferred.', '1306.3888-4-110-0': '## One system for both the analysis and the production of information', '1306.3888-4-111-0': 'A potentially useful feature of the SP system is that the processes which serve to analyse or parse a New pattern in terms of Old patterns, and to create an economical encoding of the New pattern, may also work in reverse, to recreate the New pattern from its encoding.', '1306.3888-4-111-1': "This is the 'output' perspective, mentioned in Section [REF].", '1306.3888-4-112-0': "If the New pattern is the code sequence 'S 0 1 0 1 0 S' (as described in Section [REF]), and if the Old patterns are the same as were used to create the multiple alignment shown in Figure [REF], then the best multiple alignment found by the system is the one shown in Figure [REF].", '1306.3888-4-112-1': "This multiple alignment contains the same words as the original sentence ('t h i s b o y l o v e s t h a t g i r l'), in the same order as the original.", '1306.3888-4-112-2': "Readers who are familiar with Prolog, will recognise that this process of recreating the original sentence from its encoding is similar in some respects to the way in which an appropriately-constructed Prolog program may be run 'backwards', deriving 'data' from 'results'.", '1306.3888-4-113-0': 'How is it possible to decompress the compressed code for the original sentence by using information compression?', '1306.3888-4-113-1': "This apparent paradox-decompression by compression-may be resolved by ensuring that, when a code pattern like 'S 0 1 0 1 0 S' is used to recreate the original data, each symbol is treated, at least notionally, as if contained a few more bits of information than is strictly necessary.", '1306.3888-4-113-2': 'That residual redundancy allows the system to recreate the original sentence by the same process of compression as was used to create the original parsing and encoding.', '1306.3888-4-114-0': 'This process of creating a relatively large pattern from a relatively small encoding provides a model for the creation of sentences by a person or an artificial system.', '1306.3888-4-114-1': "But instead of the New pattern being a rather dry code, like 'S 0 1 0 1 0 S', it would be more plausible if it were some kind of representation of the meaning of the sentence, like that mentioned in Section [REF].", '1306.3888-4-114-2': 'How a sentence may be generated from a representation of meaning is outlined in BK (Section 5.7.1).', '1306.3888-4-115-0': "Similar principles may apply to other kinds of 'output', such as planning an outing, cooking a meal, and so on.", '1306.3888-4-116-0': '# Unsupervised learning', '1306.3888-4-117-0': 'As was mentioned in Section [REF], part of the inspiration for the SP theory has been a programme of research developing models of the unsupervised learning of language.', '1306.3888-4-117-1': 'But although the SNPR model is quite successful in deriving plausible grammars from samples of English-like artificial language, it has proved to be quite unsuitable as a basis for the SP theory.', '1306.3888-4-117-2': 'In order to accommodate other aspects of intelligence, such as pattern recognition, reasoning, and problem solving, it has been necessary to develop an entirely new conceptual framework, with multiple alignment at centre stage.', '1306.3888-4-118-0': 'So there is now the curious paradox that, while the SP theory is rooted in work on unsupervised learning, and that kind of learning has a central role in the theory, the SP model does much the same things as the earlier model, and with similar limitations (Sections [REF] and [REF]).', '1306.3888-4-118-1': 'But I believe that the new conceptual framework has many advantages, that it provides a much sounder footing for further developments, and that with some reorganisation of the learning processes in the SP computer model, its current weaknesses may be overcome (Section [REF]).', '1306.3888-4-119-0': '## Outline of unsupervised learning in the SP model', '1306.3888-4-120-0': 'The outline of the SP model in this section aims to provide sufficient detail for a good intuitive grasp of how it works.', '1306.3888-4-120-1': 'A lot more detail may be found in BK (Chapter 9).', '1306.3888-4-121-0': 'In addition to the processes for building multiple alignments, the SP model has processes for deriving Old patterns from multiple alignments, evaluating sets of newly-created Old patterns in terms of their effectiveness for the economical encoding of the New information, and the weeding out low-scoring sets.', '1306.3888-4-121-1': 'The system does not merely record statistical information, it uses that information to learn new structures.', '1306.3888-4-122-0': '### Deriving Old patterns from multiple alignments', '1306.3888-4-123-0': 'The process of deriving Old patterns from multiple alignments is illustrated schematically in Figure [REF].', '1306.3888-4-123-1': "As was mentioned in Section [REF], the SP system is conceived as an abstract brain-like system that, in 'input' mode, may receive 'New' information via its senses and store some or all of it as 'Old' information.", '1306.3888-4-123-2': 'Here, we may think of it as the brain of a baby who is listening to what people are saying.', '1306.3888-4-123-3': "Let's imagine that he or she hears someone say 't h a t b o y r u n s'.", '1306.3888-4-123-4': 'If the baby has never heard anything similar, then, if it is stored at all, that New information may be stored as a relatively straightforward copy, something like the Old pattern shown in row 1 of the multiple alignment in part (a) of the figure.', '1306.3888-4-124-0': "Now let us imagine that the information has been stored and that, at some later stage, the baby hears someone say 't h a t g i r l r u n s'.", '1306.3888-4-124-1': 'Then, from that New information and the previously-stored Old pattern, a multiple alignment may be created like the one shown in part (a) of Figure [REF].', '1306.3888-4-124-2': "And, by picking out coherent sequences that are either fully matched or not matched at all, four putative words may be extracted: 't h a t', 'g i r l', 'b o y', and 'r u n s', as shown in the first four patterns in part (b) of the figure.", '1306.3888-4-124-3': "In each newly-created Old pattern there are additional symbols such as 'B', '2', and 'B' that are added by the system, and which serve to identify the pattern, to mark its boundaries, and to mark its grammatical category or categories.", '1306.3888-4-125-0': "In addition to these four patterns, a fifth pattern is created, 'E 6 B B C C D D E', as shown in the figure, that records the sequence 't h a t ... r u n s', with the category 'C C' in the middle representing a choice between 'b o y' and 'g i r l'.", '1306.3888-4-125-1': 'Part (b) in the figure is the beginnings of a grammar to describe that kind of phrase.', '1306.3888-4-126-0': '### Evaluating and selecting sets of newly-created Old patterns', '1306.3888-4-127-0': 'The example just described shows how Old patterns may be derived from a multiple alignment but it gives a highly misleading impression of how the SP model actually works.', '1306.3888-4-127-1': "In practice, the program forms many multiple alignments that are much less tidy than the one shown and it creates many Old patterns that are clearly 'wrong'.", '1306.3888-4-127-2': "However, the program contains procedures for evaluating candidate sets of patterns ('grammars') and weeding out those that score badly in terms of their effectiveness for encoding the New information economically.", '1306.3888-4-127-3': "Out of all the muddle, it can normally abstract one or two 'best' grammars and these are normally ones that appear intuitively to be 'correct', or nearly so.", '1306.3888-4-127-4': 'In general, the program can abstract one or more plausible grammars from a sample of English-like artificial language, including words, grammatical categories of words, and sentence structure.', '1306.3888-4-128-0': 'In accordance with the principles of minimum length encoding , the aim of these processes of sifting and sorting is to minimise [MATH], where [MATH] is the size (in bits) of the grammar that is under development and [MATH] is the size (in bits) of the New patterns when they have been encoded in terms of the grammar.', '1306.3888-4-129-0': 'For a given grammar comprising patterns [MATH], the value of [MATH] is calculated as: [EQUATION] where [MATH] is the number of symbols in the [MATH]th pattern and [MATH] is the encoding cost of the [MATH]th symbol in that pattern.', '1306.3888-4-130-0': 'Given that each grammar is derived from a set [MATH] of multiple alignments (one multiple alignment for each pattern from New), the value of [MATH] for the grammar is calculated as: [EQUATION] where [MATH] is the size, in bits, of the code string derived from the [MATH]th multiple alignment (Section [REF]).', '1306.3888-4-131-0': 'For a given set of patterns from New, a tree of alternative grammars is created with branching occurring wherever there are two or more alternative multiple alignments for a given pattern from New.', '1306.3888-4-131-1': 'The tree is grown in stages and pruned periodically to keep it within reasonable bounds.', '1306.3888-4-131-2': 'At each stage, grammars with high values for [MATH] (which will be referred to as [MATH]) are eliminated.', '1306.3888-4-132-0': '### Plotting values for [MATH], [MATH] and [MATH]', '1306.3888-4-133-0': 'Figure [REF] shows cumulative values for [MATH], [MATH] and [MATH] as the SP model searches for good grammars for a succession of 8 New patterns, each of which represents a sentence.', '1306.3888-4-133-1': 'Each point on each of the lower three graphs represents the relevant value (on the scale at the left) from the best grammar found after a given pattern from New has been processed.', '1306.3888-4-133-2': "The graph labelled '[MATH]' shows cumulative values on the scale at the left for the succession of New patterns.", '1306.3888-4-133-3': "The graph labelled '[MATH]' shows the amount of compression achieved (on the scale to the right).", '1306.3888-4-134-0': '### Limitations in the SP model and how they may be overcome', '1306.3888-4-135-0': 'As mentioned before (Section [REF]), there are two main weaknesses in the processes for unsupervised learning in the SP model as it is now: the model does not learn intermediate levels in a grammar (phrases or clauses) or discontinuous dependencies of the kind described in Sections [REF] to [REF].', '1306.3888-4-136-0': 'It appears that some reorganisation of the learning processes in the model would solve both problems.', '1306.3888-4-136-1': "What seems to be needed is a tighter focus on the principle that, with appropriately-constructed Old patterns, multiple alignments may be created without the kind of mis-match between patterns that may be seen in Figure [REF] (a) ('g i r l' and 'b o y' do not match each other), and that any such multiple alignment may be treated as if it was a simple pattern.", '1306.3888-4-136-2': 'That reform should facilitate the discovery of structures at multiple levels and the discovery of structures that are discontinuous in the sense that they can bridge intervening structures.', '1306.3888-4-137-0': '### Computational complexity', '1306.3888-4-138-0': 'As with the building of multiple alignments (Section [REF]), the computational complexity of learning in the SP model is kept under control by pruning the search tree at appropriate points, aiming to discover grammars that are reasonably good and not necessarily perfect.', '1306.3888-4-139-0': 'In a serial processing environment, the time complexity of learning in the SP model has been estimated to be O[MATH] where [MATH] is the number of patterns in New.', '1306.3888-4-139-1': 'In a parallel processing environment, the time complexity may approach O[MATH], depending on how well the parallel processing is applied.', '1306.3888-4-139-2': 'In serial or parallel environments, the space complexity has been estimated to be O[MATH].', '1306.3888-4-140-0': '## The discovery of natural structures via information compression (DONSVIC)', '1306.3888-4-141-0': "In our dealings with the world, certain kinds of structures appear to be more prominent and useful than others: in natural languages, there are words, phrase and sentences; we understand the visual and tactile worlds to be composed of discrete 'objects'; and conceptually, we recognise classes of things like 'person', 'house', 'tree', and so on.", '1306.3888-4-142-0': "It appears that these 'natural' kinds of structure are significant in our thinking because they provide a means of compressing sensory information, and that compression of information provides the key to their learning or discovery.", '1306.3888-4-142-1': 'At first sight, this looks like nonsense because popular programs for compression of information, such as those based on the LZW algorithm, or programs for JPEG compression of images, seem not to recognise anything resembling words, objects, or classes.', '1306.3888-4-142-2': 'But those programs are designed to work fast on low-powered computers.', '1306.3888-4-142-3': 'With other programs that are designed to be relatively thorough in their compression of information, natural structures can be revealed:', '1306.3888-4-143-0': 'It seems likely that the principles that have been outlined in this subsection may be applied not only to the discovery of words, phrases and grammars in language-like data but also to such things as the discovery of objects in images , and classes of entity in all kinds of data.', '1306.3888-4-143-1': "These principles may be characterised as the discovery of natural structures via information compression, or 'DONSVIC' for short.", '1306.3888-4-144-0': '## Generalisation, the correction of overgeneralisations, and learning from noisy data', '1306.3888-4-145-0': 'Issues that arise in the learning of a first language and, probably, in other kinds of learning, are illustrated in Figure [REF]:', '1306.3888-4-146-0': 'One possible answer is that mistakes are corrected by parents, teachers, and others.', '1306.3888-4-146-1': 'But the weight of evidence is that children can learn their first language without that kind of assistance.', '1306.3888-4-147-0': 'A better answer is the principle of minimum length encoding (described in its essentials in Section [REF]):', '1306.3888-4-148-0': '## One-trial learning and its implications', '1306.3888-4-149-0': 'In many theories of learning, the process is seen as gradual: behaviour is progressively shaped by rewards or punishments or other kinds of experience.', '1306.3888-4-150-0': 'But any theory of learning in which the process is necessarily gradual is out of step with our ordinary experience that we can and do learn things from a single experience, especially if that single experience is very significant for us (BK, Section 11.4.4.1).', '1306.3888-4-151-0': 'In the SP theory, one-trial learning is accommodated in the way the system can store New information directly.', '1306.3888-4-151-1': 'And the gradual nature of, for example, language learning, may be explained by the complexity of the process of sifting and sorting the many alternative sets of candidate patterns to find one or more sets that are good in terms of information compression (BK, Section 11.4.4.2).', '1306.3888-4-152-0': '# Computing, mathematics, and logic', '1306.3888-4-153-0': 'Drawing mainly on BK (Chapters 4 to 11), this and the following sections describe, with a selection of examples, how the SP theory relates to several areas in artificial intelligence, mainstream computing, and human perception and cognition.', '1306.3888-4-154-0': 'In BK (Chapter 4), I have argued that the SP system is equivalent to a universal Turing machine , in the sense that anything that may be computed with a Turing machine may, in principle, also be computed with an SP machine.', '1306.3888-4-154-1': "The 'in principle' qualification is necessary because the SP theory is still not fully mature and there are still some weaknesses in the SP computer models.", '1306.3888-4-154-2': 'The gist of the argument is that the operation of a Post canonical system may be understood in terms of the SP theory and, since it is accepted that the Post canonical system is equivalent to the Turing machine (as a computational system), the Turing machine may also be understood in terms of the SP theory.', '1306.3888-4-155-0': "The key differences between the SP theory and earlier theories of computing are that the SP theory has a lot more to say about the nature of intelligence than earlier theories, that the theory is founded on principles of information compression via the matching and unification of patterns ('computing as compression'), and that it includes mechanisms for building multiple alignments and for heuristic search that are not present in earlier models.", '1306.3888-4-156-0': '## Conventional computing systems', '1306.3888-4-157-0': 'In conventional computing systems, compression of information may be seen in the matching of patterns with at least implicit unification of patterns that match each other-processes that appear in a variety of guises (BK, Chapter 2).', '1306.3888-4-157-1': 'And three basic techniques for the compression of information-chunking-with-codes, schema-plus-correction, and run-length coding-may be seen in various forms in the organisation of computer programs (ibid.)', '1306.3888-4-158-0': '## Mathematics and logic', '1306.3888-4-159-0': 'In a similar way, several structures and processes in mathematics and logic may be interpreted in terms of information compression via the matching and unification of patterns, and the compression techniques just mentioned (BK, Chapter 10).', '1306.3888-4-159-1': 'For example, multiplication (as repeated addition) and exponentiation (as repeated multiplication) may be seen as examples of run-length coding; a function with parameters may be seen as an example of schema-plus-correction; the chunking-with-codes technique may be seen in the organisation of number systems; and so on.', '1306.3888-4-160-0': '## Computing and probabilities', '1306.3888-4-161-0': 'As we have seen, the SP system is fundamentally probabilistic.', '1306.3888-4-161-1': "If it is indeed Turing-equivalent, as suggested above, and if the Turing machine is regarded as a definition of 'computing', then we may conclude that computing is fundamentally probabilistic.", '1306.3888-4-161-2': 'That may seem like a strange conclusion in view of the clockwork certainties that we associate with the operation of ordinary computers and the workings of mathematics and logic.', '1306.3888-4-161-3': 'There are at least three answers to that apparent contradiction:', '1306.3888-4-162-0': '# Representation of knowledge', '1306.3888-4-163-0': "Within the multiple alignment framework (Section [REF]), SP patterns may serve to represent several kinds of knowledge, including grammars for natural languages, ontologies, class hierarchies with inheritance of attributes, including cross-classification or multiple inheritance, part-whole hierarchies and their integration with class-inclusion hierarchies, decision networks and trees, relational tuples, if-then rules, associations of medical signs and symptoms, causal relations, and concepts in mathematics and logic such as 'function', 'variable', 'value', 'set', and 'type definition'.", '1306.3888-4-164-0': 'The use of one simple format for the representation of knowledge facilitates the seamless integration of different kinds of knowledge.', '1306.3888-4-165-0': '# Natural language processing', '1306.3888-4-166-0': 'One of the main strengths of the SP system is in natural language processing (BK, Chapter 5):', '1306.3888-4-167-0': '## Discontinuous dependencies in syntax', '1306.3888-4-168-0': 'The way in which the SP system can record discontinuous dependencies in syntax may be seen in both of the two parsings in Figure [REF].', '1306.3888-4-168-1': "The pattern in row 8 of each multiple alignment records the syntactic dependency between the plural noun phrase ('t w o k i t t e n s') which is the subject of the sentence-marked with 'Np'-and the plural verb phrase ('p l a y')-marked with 'Vp'-which belongs with it.", '1306.3888-4-169-0': "This kind of dependency is discontinuous because it can bridge arbitrarily large amounts of intervening structure such as, for example, 'from the West' in a sentence like 'Winds from the West are strong'.", '1306.3888-4-170-0': 'This method of marking discontinuous dependencies can accommodate overlapping dependencies such as number dependencies and gender dependencies in languages like French (BK, Section 5.4).', '1306.3888-4-170-1': 'It also provides a means of encoding the interesting system of overlapping and interlocking dependencies in English auxiliary verbs, described by Noam Chomsky in Syntactic Structuresyearparchomsky_1957.', '1306.3888-4-171-0': "In that book, the structure of English auxiliary verbs is part of Chomsky's evidence in support of Transformational Grammar.", '1306.3888-4-171-1': 'Despite the elegance and persuasiveness of his arguments, it turns out that the structure of English auxiliary verbs may be described with non-transformational rules in, for example, Definite Clause Grammars , and also in the SP system, as outlined in the subsections that follow.', '1306.3888-4-172-0': '## Two quasi-independent patterns of constraint in English auxiliary verbs', '1306.3888-4-173-0': 'In English, the syntax for main verbs and the auxiliary verbs which may accompany them follows two quasi-independent patterns of constraint which interact in an interesting way.', '1306.3888-4-174-0': 'The primary constraints may be expressed with this sequence of symbols,', '1306.3888-4-175-0': 'M H B B V,', '1306.3888-4-176-0': 'which should be interpreted in the following way:', '1306.3888-4-177-0': 'The words occur in the order shown but any of the words may be omitted.', '1306.3888-4-178-0': "Questions of 'standard' form follow exactly the same pattern as statements except that the first verb, whatever it happens to be ('M', 'H', the first 'B', the second 'B' or 'V'), precedes the subject noun phrase instead of following it.", '1306.3888-4-179-0': 'Here are two examples of the primary pattern with all of the words included:', '1306.3888-4-180-0': 'It will have been being washed M H B B V', '1306.3888-4-181-0': 'Will it have been being washed?', '1306.3888-4-181-1': 'M H B B V', '1306.3888-4-182-0': 'The secondary constraints are these:', '1306.3888-4-183-0': 'Figure [REF] shows a selection of examples with the dependencies marked.', '1306.3888-4-184-0': '## Multiple alignments and English auxiliary verbs', '1306.3888-4-185-0': 'Without reproducing all the detail in BK (Section 5.5), we can see from Figures [REF] and [REF] how the primary and secondary constraints may be applied in the multiple alignment framework.', '1306.3888-4-186-0': 'In each figure, the sentence to be analysed is shown as a New pattern in column 0.', '1306.3888-4-186-1': "The primary constraints are applied via the matching of symbols in Old patterns in the remaining columns, with a consequent interlocking of the patterns so that they recognise sentences of the form 'M H B B V', with options as described above.", '1306.3888-4-187-0': 'In Figure [REF], the secondary constraints apply as follows:', '1306.3888-4-188-0': 'In Figure [REF], the secondary constraints apply like this:', '1306.3888-4-189-0': '# Pattern recognition', '1306.3888-4-190-0': 'The system also has some useful features as a framework for pattern recognition (BK, (Chapter 6):', '1306.3888-4-191-0': 'One area of application is medical diagnosis , viewed as pattern recognition.', '1306.3888-4-191-1': 'There is also potential to assist in the understanding of natural vision and in the development of computer vision, as discussed in [CITATION].', '1306.3888-4-192-0': '## Part-whole hierarchies, class hierarchies, and their integration', '1306.3888-4-193-0': 'A strength of the multiple alignment concept is that it provides a simple but effective vehicle for the representation and processing of class-inclusion hierarchies, part-whole hierarchies, and their integration.', '1306.3888-4-194-0': "Figure [REF] shows the best multiple alignment found by the SP model with the New pattern 'white-bib eats furry purrs' (column 0) representing some features of an unknown creature, and with a set of Old patterns representing different classes of animal, at varying levels of abstraction.", '1306.3888-4-194-1': "From this multiple alignment, we may conclude that the unknown entity is an animal (column 1), a mammal (column 2), a cat (column 3) and the specific individual 'Tibs' (column 4).", '1306.3888-4-195-0': "The framework also provides for the representation of heterarchies or cross classification: a given entity, such as 'Jane' (or a class of entities), may belong in two or more higher-level classes that are not themselves hierarchically related, such as 'woman' and 'doctor'.", '1306.3888-4-196-0': 'The way that class-inclusion relations and part-whole relations may be combined in one multiple alignment is illustrated in Figure [REF].', '1306.3888-4-196-1': 'Here, some features of an unknown plant are expressed as a set of New patterns, shown in column 0: the plant has chlorophyll, the stem is hairy, it has yellow petals, and so on.', '1306.3888-4-197-0': 'From this multiple alignment, we can see that the unknown plant is most likely to be the Meadow Buttercup, Ranunculus acris, as shown in column 1.', '1306.3888-4-197-1': 'As such, it belongs in the genus Ranunculus (column 6), the family Ranunculaceae (column 5), the order Ranunculales (column 4), the class Angiospermae (column 3), and the phylum Plants (column 2).', '1306.3888-4-198-0': 'Each of these higher-level classifications contributes information about attributes of the plant and its division into parts and sub-parts.', '1306.3888-4-198-1': "For example, as a member of the class Angiospermae (column 3), the plant has a shoot and roots, with the shoot divided into stem, leaves, and flowers; as a member of the family Ranunculaceae (column 5), the plant has flowers that are 'regular', with all parts 'free'; as a member of the phylum Plants (column 2), the buttercup has chlorophyll and creates its own food by photosynthesis; and so on.", '1306.3888-4-199-0': '## Inference and inheritance', '1306.3888-4-200-0': 'In the example just described, we can infer from the multiple alignment, very directly, that the plant which has been provisionally identified as the Meadow Buttercup performs photosynthesis (column 2), has five petals (column 6), is poisonous (column 5), and so on.', '1306.3888-4-200-1': "As in other object-oriented systems, the first of these attributes has been 'inherited' from the class 'Plants', the second from the class Ranunculus, and the third from the class Ranunculaceae.", '1306.3888-4-200-2': 'These kinds of inference illustrate the close connection, often remarked, between pattern recognition and inferential reasoning .', '1306.3888-4-201-0': '# Probabilistic reasoning', '1306.3888-4-202-0': "The SP system can model several kinds of reasoning including inheritance of attributes (as just described), one-step 'deductive' reasoning, abductive reasoning, reasoning with probabilistic decision networks and decision trees, reasoning with 'rules', nonmonotonic reasoning and reasoning with default values, reasoning in Bayesian networks (including 'explaining away'), causal diagnosis, and reasoning which is not supported by evidence (BK, Chapter 7).", '1306.3888-4-203-0': 'Since these several kinds of reasoning all flow from one computational framework (multiple alignment), they may be seen as aspects of one process, working individually or together without awkward boundaries.', '1306.3888-4-204-0': 'Plausible lines of reasoning may be achieved, even when relevant information is incomplete.', '1306.3888-4-205-0': "Probabilities of inferences may be calculated, including extreme values (0 or 1) in the case of logic-like 'deductions'.", '1306.3888-4-206-0': 'A selection of examples is described in the following subsections.', '1306.3888-4-207-0': '## Nonmonotonic reasoning and reasoning with default values', '1306.3888-4-208-0': 'Conventional deductive reasoning is monotonic because deductions made on the strength of current knowledge cannot be invalidated by new knowledge: the conclusion that "Socrates is mortal", deduced from "All humans are mortal" and "Socrates is human", remains true for all time, regardless of anything we learn later.', '1306.3888-4-208-1': 'By contrast, the inference that "Tweety can probably fly" from the propositions that "Most birds fly" and "Tweety is a bird" is nonmonotonic because it may be changed if, for example, we learn that Tweety is a penguin.', '1306.3888-4-209-0': 'This section presents some examples which show how the SP system can accommodate nonmonotonic reasoning.', '1306.3888-4-210-0': '### Typically, birds fly', '1306.3888-4-211-0': "The idea that (all) birds can fly may be expressed with an SP pattern like 'Bd bird name name canfly warm-blooded wings feathers ... Bd'.", '1306.3888-4-211-1': 'This, of course, is an oversimplification of the real-world facts because, while it true that the majority of birds fly, we know that there are also flightless birds like ostriches, penguins and kiwis.', '1306.3888-4-212-0': "In order to model these facts more closely, we need to modify the pattern that describes birds to be something like this: 'Bd bird name name f f warm-blooded wings feathers ... Bd.", '1306.3888-4-212-1': 'And, to our database of Old patterns, we need to add patterns like this:', '1306.3888-4-213-0': 'Default Bd f canfly #f #Bd #Default P penguin Bd f cannotfly #f #Bd ... #P O ostrich Bd f cannotfly #f #Bd ... #O.', '1306.3888-4-214-0': "Now, the pair of symbols 'f f' in 'Bd bird name name f f warm-blooded wings feathers ... Bd' functions like a 'variable' that may take the value 'canfly' if a given class of birds can fly and 'cannotfly' when a type of bird cannot fly.", '1306.3888-4-214-1': "The pattern 'P penguin Bd f cannotfly f Bd ... P' shows that penguins cannot fly and, likewise, the pattern 'O ostrich Bd f cannotfly f Bd ... O' shows that ostriches cannot fly.", '1306.3888-4-214-2': "The pattern 'Default Bd f canfly f Bd Default', which has a substantially higher frequency than the other two patterns, represents the default value for the variable which is 'canfly'.", '1306.3888-4-214-3': "Notice that all three of these patterns contains the pair of symbols 'Bd ... Bd' showing that the corresponding classes are all subclasses of birds.", '1306.3888-4-215-0': '### Tweety is a bird so, probably, Tweety can fly', '1306.3888-4-216-0': "When the SP model is run with 'bird Tweety' in New and the same patterns in Old as before, modified as just described, the three best multiple alignments found are those shown in Figures [REF], [REF] and [REF].", '1306.3888-4-217-0': 'The first multiple alignment tells us that, with a relative probability of 0.66, Tweety may be the typical kind of bird that can fly.', '1306.3888-4-217-1': 'The second multiple alignment tells us that, with a relative probability of 0.22, Tweety might be an ostrich and, as such, he or she would not be able to fly.', '1306.3888-4-217-2': 'Likewise, the third multiple alignment tells us that, with a relative probability of 0.12, Tweety might be a penguin and would not be able to fly.', '1306.3888-4-217-3': 'The values for probabilities in this simple example are derived from guestimated frequencies that are, almost certainly, not ornithologically correct.', '1306.3888-4-218-0': '### Tweety is a penguin, so Tweety cannot fly', '1306.3888-4-219-0': "Figure [REF] shows the best multiple alignment found by the SP model when it is run again, with 'penguin Tweety' in New instead of 'bird Tweety'.", '1306.3888-4-219-1': 'This time, there is only one multiple alignment in the reference set and its relative probability is 1.0.', '1306.3888-4-219-2': 'Correspondingly, all inferences that we can draw from this multiple alignment have a probability of 1.0.', '1306.3888-4-219-3': 'In particular, we can be confident, within the limits of the available knowledge, that Tweety cannot fly.', '1306.3888-4-220-0': 'In a similar way, if Tweety were an ostrich, we would be able to say with confidence (p = 1.0) that he or she would not be able to fly.', '1306.3888-4-221-0': "## Reasoning in Bayesian networks, including 'explaining away'", '1306.3888-4-222-0': "A Bayesian network is a directed, acyclic graph like the one shown in Figure [REF], below, where each node has zero or more 'inputs' (connections with nodes that can influence the given node) and one or more 'outputs' (connections to other nodes that the given node can influence).", '1306.3888-4-223-0': 'Each node contains a set of conditional probability values, each one the probability of a given output value for a given input value or combination of input values.', '1306.3888-4-223-1': 'With this information, conditional probabilities of alternative outputs for any node may be computed for any given combination of inputs.', '1306.3888-4-223-2': "By combining these calculations for sequences of nodes, probabilities may be propagated through the network from one or more 'start' nodes to one or more 'finishing' nodes.", '1306.3888-4-224-0': 'This section describes how the SP system may perform that kind of probabilistic reasoning, and some advantages compared with Bayesian networks.', '1306.3888-4-225-0': 'Judea Pearlyearpar[p. 7]pearl_1997 describes the phenomenon of \'explaining away\' like this: "If A implies B, C implies B, and B is true, then finding that C is true makes A less credible.', '1306.3888-4-225-1': 'In other words, finding a second explanation for an item of data makes the first explanation less credible.', '1306.3888-4-225-2': '" (his italics).', '1306.3888-4-225-3': 'Here is an example:', '1306.3888-4-226-0': 'Normally an alarm sound alerts us to the possibility of a burglary.', '1306.3888-4-226-1': 'If somebody calls you at the office and tells you that your alarm went off, you will surely rush home in a hurry, even though there could be other causes for the alarm sound.', '1306.3888-4-226-2': 'If you hear a radio announcement that there was an earthquake nearby, and if the last false alarm you recall was triggered by an earthquake, then your certainty of a burglary will diminish.', '1306.3888-4-227-0': 'The causal relationships in the example just described may be captured in a Bayesian network like the one shown in Figure [REF].', '1306.3888-4-228-0': 'Pearl argues that, with appropriate values for conditional probabilities, the phenomenon of "explaining away" can be explained in terms of this network (representing the case where there is a radio announcement of an earthquake) compared with the same network without the node for "radio announcement" (representing the situation where there is no radio announcement of an earthquake).', '1306.3888-4-229-0': '### Representing contingencies with patterns and frequencies', '1306.3888-4-230-0': 'To see how this phenomenon may be understood in terms of the SP theory, consider, first, the set of patterns shown in Figure [REF], which are to be stored in Old.', '1306.3888-4-230-1': "The patterns in the figure show events which occur together in some notional sample of the 'World' together with their frequencies of occurrence in the sample.", '1306.3888-4-231-0': "As with other knowledge-based systems, we shall assume that the 'closed-world' assumption applies so that the absence of any pattern may be taken to mean that the corresponding combination of events did not occur in the period when observations were made.", '1306.3888-4-232-0': "The first pattern ('burglary alarm') shows that there were 1000 occasions when there was a burglary and the alarm went off and the second pattern ('earthquake alarm') shows just 20 occasions when there was an earthquake and the alarm went off (presumably triggered by the earthquake).", '1306.3888-4-232-1': 'Thus we have assumed that, as triggers for the alarm, burglaries are much more common than earthquakes.', '1306.3888-4-233-0': 'Since there is no pattern showing that the alarm sounded when there was a burglary and an earthquake at the same time, we may assume, via the closed-world assumption, that nothing like that happened during the sampling period.', '1306.3888-4-234-0': "The third pattern ('alarm phonealarmcall') shows that, out of the 1020 cases when the alarm went off, there were 980 cases where a phone call about the alarm was made.", '1306.3888-4-234-1': 'Since there is no pattern showing phone calls about the alarm in any other context, the closed-world assumption allows us to assume that there were no false positives (eg., hoaxes)-phone calls about the alarm when no alarm had sounded.', '1306.3888-4-235-0': "The fourth pattern ('earthquake radioearthquakeannouncement') shows that, in the sampling period, there were 40 occasions when there was an earthquake with an announcement about it on the radio.", '1306.3888-4-235-1': "And the fifth pattern ('e1 earthquake e2') shows that an earthquake has occurred on 40 occasions in contexts where the alarm did not ring and there was no radio announcement.", '1306.3888-4-236-0': "As before, the absence of patterns like 'earthquake alarm radioearthquakeannouncement' representing cases where an earthquake triggers the alarm and also leads to a radio announcement, allows us to assume via the closed-world assumption that cases of that kind have not occurred in the sampling period.", '1306.3888-4-237-0': '### Approximating the temporal order of events', '1306.3888-4-238-0': 'In these patterns and in the multiple alignments shown below, the left-to-right order of symbols may be regarded as an approximation to the order of events in time.', '1306.3888-4-238-1': 'Thus in the first two patterns, events that can trigger an alarm precede the sounding of the alarm.', '1306.3888-4-238-2': "Likewise, in the third pattern, 'alarm' (meaning that the alarm has sounded) precedes 'phonealarmcall' (a phone call to say the alarm has sounded).", '1306.3888-4-238-3': 'A single dimension can only approximate the order of events in time because it cannot represent events which overlap in time or which occur simultaneously.', '1306.3888-4-238-4': 'However, this kind of approximation has little or no bearing on the points to be illustrated here.', '1306.3888-4-239-0': '### Other considerations', '1306.3888-4-240-0': 'Other points relating to the patterns shown in Figure [REF] include:', '1306.3888-4-241-0': '### Formation of alignments: the burglar alarm has sounded', '1306.3888-4-242-0': "Receiving a phone call to say that the burglar alarm at one's house has gone off may be represented by placing the symbol 'phonealarmcall' in New.", '1306.3888-4-242-1': 'Figure [REF] shows, at the top, the best multiple alignment formed by the SP model in this case, with the patterns from Figure [REF] in Old.', '1306.3888-4-242-2': 'The other two multiple alignments in the reference set are shown below the best multiple alignment, in order of CD value and relative probability.', '1306.3888-4-242-3': 'The actual values for [MATH] and relative probability are given in the caption to Figure [REF].', '1306.3888-4-243-0': 'The unmatched Old symbols in these multiple alignments represent inferences made by the system.', '1306.3888-4-243-1': 'The probabilities for these inferences which are calculated by the SP model (as outlined in Section [REF]) are shown in Table [REF].', '1306.3888-4-243-2': 'These probabilities do not add up to 1 and we should not expect them to because any given multiple alignment may contain two or more of these symbols.', '1306.3888-4-244-0': 'The most probable inference is the rather trivial inference that the alarm has indeed sounded.', '1306.3888-4-244-1': 'This reflects the fact that there is no pattern in Figure [REF] representing false positives for telephone calls about the alarm.', '1306.3888-4-244-2': 'Apart from the inference that the alarm has sounded, the most probable inference (p = 0.328) is that there has been a burglary.', '1306.3888-4-244-3': 'However, there is a distinct possibility that there has been an earthquake-but the probability in this case (p = 0.016) is much lower than the probability of a burglary.', '1306.3888-4-245-0': "These inferences and their relative probabilities seem to accord quite well with what one would naturally think following a telephone call to say that the burglar alarm at one's house has gone off (given that one was living in a part of the world where earthquakes were not vanishingly rare).", '1306.3888-4-246-0': '## Formation of alignments: the burglar alarm has sounded and there is a radio announcement of an earthquake', '1306.3888-4-247-0': "In this example, the phenomenon of 'explaining away' occurs when you learn not only that the burglar alarm has sounded but that there has been an announcement on the radio that there has been an earthquake.", '1306.3888-4-247-1': 'In terms of the SP model, the two events (the phone call about the alarm and the announcement about the earthquake) can be represented in New by a pattern like this:', '1306.3888-4-248-0': 'phone_alarm_call radio_earthquake_announcement', '1306.3888-4-249-0': "or 'radioearthquakeannouncement phonealarmcall'.", '1306.3888-4-249-1': 'The order of the two symbols makes no difference to the result.', '1306.3888-4-250-0': "In this case, there is only one multiple alignment (shown at the top of Figure [REF]) that can 'explain' all the information in New.", '1306.3888-4-250-1': "Since there is only this one multiple alignment in the reference set for the best multiple alignment, the associated probabilities of the inferences that can be read from the multiple alignment ('alarm' and 'earthquake') are 1.0: it was an earthquake that caused the alarm to go off (and led to the phone call), and not a burglary.", '1306.3888-4-251-0': "These results show how 'explaining away' may be explained in terms of the SP theory.", '1306.3888-4-251-1': "The main point is that the multiple alignment or multiple alignments that provide the best 'explanation' of a telephone call to say that one's burglar alarm has sounded is different from the multiple alignment or multiple alignments that best explain the same telephone call coupled with an announcement on the radio that there has been an earthquake.", '1306.3888-4-251-2': 'In the latter case, the best explanation is that the earthquake triggered the alarm.', '1306.3888-4-251-3': 'Other possible explanations have lower probabilities.', '1306.3888-4-252-0': '### Other possibilities', '1306.3888-4-253-0': 'As mentioned above, the closed-world assumption allows us to rule out possibilities such as:', '1306.3888-4-254-0': 'Nevertheless, we may consider possibilities of that kind by combining multiple alignments as described in BK (Section 7.8.7).', '1306.3888-4-254-1': 'But as a general rule, that kind of further analysis makes no difference to the original conclusion: the multiple alignment which was originally judged to represent the best interpretation of the available facts has not been dislodged from this position.', '1306.3888-4-254-2': 'This is in keeping with way we normally concentrate on the most likely explanations of events and ignore the many conceivable but unlikely alternatives.', '1306.3888-4-255-0': '## The SP framework and Bayesian networks', '1306.3888-4-256-0': 'The foregoing examples show that the SP framework is a viable alternative to Bayesian networks, at least in the kinds of situation that have been described.', '1306.3888-4-256-1': 'This subsection makes some general observations about the relative merits of the two frameworks for probabilistic reasoning where the events of interest are subject to multiple influences or chains of influence or both those things.', '1306.3888-4-257-0': "Without in any way diminishing Thomas Bayes' achievement, his theorem appears to have shortcomings as the basis for theorising about perception and cognition:", '1306.3888-4-258-0': '## Causal diagnosis', '1306.3888-4-259-0': 'In this section, we consider a simple example of fault diagnosis in an electronic circuit-described by [CITATION].', '1306.3888-4-259-1': 'Figure [REF] shows the circuit with inputs on the left, outputs on the right and, in between, three multipliers ([MATH], [MATH], and [MATH]) and two adders ([MATH] and [MATH]).', '1306.3888-4-259-2': 'For the given inputs on the left, it is clear that output F is false and output G is correct.', '1306.3888-4-260-0': 'Figure [REF] shows a causal network derived from the electronic circuit in Figure [REF] .', '1306.3888-4-260-1': 'In this diagram, [MATH], [MATH], [MATH], [MATH] and [MATH] represent the outputs of components [MATH], [MATH], [MATH], [MATH] and [MATH], respectively.', '1306.3888-4-260-2': "In each case, there are three causal influences on the output: the two inputs to the component and the state of the component which may be 'good' or 'bad'.", '1306.3888-4-260-3': 'These influences are shown in Figure [REF] by lines with arrows connecting the source of the influence to the target node.', '1306.3888-4-260-4': 'Thus, for example, the two inputs of component [MATH] are represented by [MATH] and [MATH], the good or bad state of component [MATH] is represented by the node labelled [MATH], and their causal influences on node [MATH] are shown by the three arrows pointing at that node.', '1306.3888-4-261-0': 'Given a causal analysis like this, and given appropriate information about conditional probabilities, it is possible to derive one or more alternative diagnoses of which components are good and which are bad.', '1306.3888-4-261-1': "In Pearl's example, it is assumed that components of the same type have the same prior probability of failure and that the probability of failure of multipliers is greater than for adders.", '1306.3888-4-261-2': 'Given these assumptions and some others together with the inputs and outputs shown in Figure [REF] (but not the intermediate values), the best diagnosis derived from the causal network is that the [MATH] component is bad and the second best diagnosis is that [MATH] is bad.', '1306.3888-4-261-3': 'Pearl indicates that some third-best interpretations may be retrievable (e.g., [MATH] and [MATH] are bad) "... but in general, it is not guaranteed that interpretations beyond the second-best will be retrievable."', '1306.3888-4-261-4': '(p. 272).', '1306.3888-4-262-0': '## An SP approach to causal diagnosis', '1306.3888-4-263-0': 'The main elements of the SP analysis presented here are as follows:', '1306.3888-4-264-0': 'Figure [REF] shows a set of patterns for the circuit shown in Figure [REF].', '1306.3888-4-264-1': "In the figure, the patterns that start with the symbol 'M1' represent input-output relations for component [MATH], those that start with 'M2' represent input-output relations for the [MATH] component and likewise for the other patterns except the last one (starting with the symbol 'frame') which is the framework pattern mentioned above.", '1306.3888-4-264-2': "For each initial symbol there is a corresponding 'terminating' symbol with an initial '' character.", '1306.3888-4-264-3': "For reasons explained shortly, there may be other symbols following the 'terminating' symbol.", '1306.3888-4-265-0': "Let us now consider the first pattern in the figure ('M1 M1GOOD TM1I1 TM1I2 TM1O M1 TM4I2') representing I/O relations for component [MATH] when that component is good, as indicated by the symbol 'M1GOOD'.", '1306.3888-4-265-1': "In this pattern, the symbols 'TM1I1', 'TM1I2' and 'TM1O' represent the two inputs and the output of the component, 'M1' is the terminating symbol, and 'TM4I2' serves to transfer the output of [MATH] to the second input of component [MATH] as will be explained.", '1306.3888-4-265-2': "In a symbol like 'TM1I1', 'T' indicates that the input is true, 'M1' identifies the component, and 'I1' indicates that this is the first input of the component.", '1306.3888-4-265-3': 'Other symbols may be interpreted in a similar way, following the key given in the caption of Figure [REF].', '1306.3888-4-265-4': 'In effect, this pattern says that, when the component is working correctly, true inputs yield a true output.', '1306.3888-4-265-5': 'The pattern has a relatively high frequency of occurrence (500,000) reflecting the idea that the component will normally work correctly.', '1306.3888-4-266-0': "The other two patterns for component [MATH] ('M1 M1BAD TM1I1 TM1I2 TM1O M1 TM4I2' and 'M1 M1BAD TM1I1 TM1I2 FM1O M1 FM4I2') describe I/O relations when the component is bad.", '1306.3888-4-266-1': 'The first one describes the situation where true inputs to a faulty component yield a true result, a possibility noted by Pearl (ibid.', '1306.3888-4-266-2': 'p. 265).', '1306.3888-4-266-3': 'The second pattern-with a higher frequency-describes the more usual situation where true inputs to a faulty component yield a false result.', '1306.3888-4-266-4': 'Both these bad patterns have much lower frequencies than the good pattern.', '1306.3888-4-267-0': 'The other patterns in Figure [REF] may be interpreted in a similar way.', '1306.3888-4-267-1': 'Components [MATH], [MATH] and [MATH] have only three patterns each because it is assumed that inputs to the circuit will always be true so it is not necessary to include patterns describing what happens when one or both of the inputs are false.', '1306.3888-4-267-2': 'By contrast, there are 4 good patterns and 8 bad patterns for each of [MATH] and [MATH] because either of these components may receive faulty input.', '1306.3888-4-268-0': 'For each of the five components, the frequencies of the bad patterns sum to 100.', '1306.3888-4-268-1': 'However, for each of the components [MATH], [MATH], and [MATH], the total frequency of the good patterns is 500,000 compared with 1,000,000 for the set of good patterns associated with each of the component [MATH] and [MATH].', '1306.3888-4-268-2': "These figures accord with the assumptions in Pearl's example that components of the same type have the same probability of failure and that the probability of failure of multipliers ([MATH], [MATH], and [MATH]) is greater than the probability of failure of adders ([MATH] and [MATH]).", '1306.3888-4-269-0': '## Multiple alignments in causal diagnosis', '1306.3888-4-270-0': 'Given appropriate patterns, the SP model constructs multiple alignments from which diagnoses may be obtained.', '1306.3888-4-270-1': "Figure [REF] shows the best multiple alignment created by the SP model with the Old patterns shown in Figure [REF] and 'TM1I1 TM1I2 TM2I1 TM2I2 TM3I1 TM3I2 FM4O TM5O' as the New pattern.", '1306.3888-4-270-2': 'The first six symbols in this pattern express the idea that all the inputs for components [MATH], [MATH] and [MATH] are true.', '1306.3888-4-270-3': "The penultimate symbol ('FM4O') shows that the output of [MATH] is false and the last symbol ('TM5O') shows that the output of [MATH] is true-in accordance with the outputs shown in Figure [REF].", '1306.3888-4-271-0': 'From the multiple alignment in Figure [REF] it can be inferred that component [MATH] is bad and all the other components are good.', '1306.3888-4-271-1': 'A total of seven alternative diagnoses can be derived from those multiple alignments created by the SP model that encode all the symbols in New.', '1306.3888-4-271-2': 'These diagnoses are shown in Table [REF], each with its relative probability.', '1306.3888-4-272-0': "It is interesting to see that the best diagnosis derived by the SP model ([MATH] is bad) and the second best diagnosis ([MATH] is bad) are in accordance with first two diagnoses obtained by Pearl's method.", '1306.3888-4-272-1': "The remaining five diagnoses derived by the SP model are different from the one obtained by Pearl's method ([MATH] and [MATH] are bad) but this is not altogether surprising because detailed frequencies or probabilities are different from Pearl's example and there are differences in assumptions that have been made.", '1306.3888-4-273-0': '# Information storage and retrieval', '1306.3888-4-274-0': "The SP theory provides a versatile model for database systems, with the ability to accommodate object-oriented structures, as well as relational 'tuples', and network and tree models of data .", '1306.3888-4-274-1': 'It lends itself most directly to information retrieval in the manner of query-by-example but it appears to have potential to support the use of natural language or query languages such as SQL.', '1306.3888-4-275-0': 'Unlike some ordinary database systems:', '1306.3888-4-276-0': '# Planning and problem solving', '1306.3888-4-277-0': 'The SP framework provides a means of planning a route between two places, and, with the translation of geometric patterns into textual form, it can solve the kind of geometric analogy problem that may be seen in some puzzle books and IQ tests (BK, Chapter 8).', '1306.3888-4-278-0': 'Figure [REF] shows an example of the latter kind of problem.', '1306.3888-4-278-1': 'The task is to complete the relationship "A is to B as C is to ?"', '1306.3888-4-278-2': "using one of the figures 'D', 'E', 'F' or 'G' in the position marked with '?", '1306.3888-4-278-3': "'.", '1306.3888-4-278-4': "For this example, the 'correct' answer is clearly 'E'.", '1306.3888-4-278-5': "Quote marks have been used for the word 'correct' because in many problems of this type, there may be two or even more alternative answers for which cases can be made and there is a corresponding uncertainty about which answer is the right one.", '1306.3888-4-279-0': "Computer-based methods for solving this kind of problem have existed for some time (e.g., Evans'yearparevans_1968 well-known heuristic algorithm).", '1306.3888-4-279-1': 'In more recent work , minimum length encoding principles have been applied to good effect.', '1306.3888-4-279-2': 'This kind of problem may also be understood in terms of the SP concepts.', '1306.3888-4-280-0': 'As in most previous work, the proposed solution assumes that some mechanism is available which can translate the geometric forms in each problem into patterns of text symbols like other patterns in this article.', '1306.3888-4-280-1': "For example, item 'A' in Figure [REF] may be described as 'small circle inside large triangle'.", '1306.3888-4-281-0': 'How this kind of translation may be done is not part of the present proposals (one such translation mechanism is described in [CITATION]).', '1306.3888-4-281-1': 'As noted elsewhere , successful solutions for this kind of problem require consistency in the way the translation is done.', '1306.3888-4-281-2': "For this example, it would be unhelpful if item 'A' in Figure [REF] were described as 'large triangle outside small circle' while item 'C' were described as 'small square inside large ellipse'.", '1306.3888-4-281-3': "For any one puzzle, the description needs to stick to one or other of 'X outside Y' or 'Y inside X'-and likewise for 'above/below' and 'left-of/right-of'.", '1306.3888-4-282-0': 'Given that the diagrammatic form of the problem has been translated into patterns as just described, this kind of problem can be cast as a problem of partial matching, well within the scope of the SP model.', '1306.3888-4-282-1': 'To do this, symbolic representations of item A and item B in Figure [REF] are treated as a single pattern, thus:', '1306.3888-4-283-0': 'small circle inside large triangle ; large circle above small triangle', '1306.3888-4-284-0': 'and this pattern is placed in New.', '1306.3888-4-284-1': 'Four other patterns are constructed by pairing a symbolic representation of item C (on the left) with symbolic representations of each of D, E, F and G (on the right), thus:', '1306.3888-4-285-0': 'C1 small square inside large ellipse ; D small square inside large circle #C1 C2 small square inside large ellipse ; E large square above small ellipse #C2 C3 small square inside large ellipse ; F small circle left-of large square #C3 C4 small square inside large ellipse ; G small ellipse above large rectangle #C4.', '1306.3888-4-286-0': 'These four patterns are placed in Old, each with an arbitrary frequency value of 1.', '1306.3888-4-287-0': 'Figure [REF] shows the best multiple alignment found by the SP model with New and Old as just described.', '1306.3888-4-287-1': 'The multiple alignment is a partial match between the New pattern (in column 0) and the second of the four patterns in Old (in column 1).', '1306.3888-4-287-2': "This corresponds with the 'correct' result (item E) as noted above.", '1306.3888-4-288-0': '# Compression of information', '1306.3888-4-289-0': 'Since information compression is central to the workings of the SP system, it is natural to consider whether the system might provide useful insights in that area.', '1306.3888-4-289-1': 'In that connection, the most promising aspects of the SP system appear to be:', '1306.3888-4-290-0': "In terms of the trade-off that exists between computational resources that are required and the level of compression that can be achieved, it is intended that the system will operate towards the 'up market' end of the spectrum-by contrast with LZW algorithms and the like, which have been designed to be 'quick-and-dirty', sacrificing performance for speed on low-powered computers.", '1306.3888-4-291-0': '# Perception, cognition and neuroscience', '1306.3888-4-292-0': 'Since much of the inspiration for the SP theory has come from evidence, mentioned in Section [REF], that, to a large extent, the workings of brains and nervous systems may be understood in terms of information compression, the theory is about perception and cognition as well as artificial intelligence and mainstream computing.', '1306.3888-4-293-0': 'That said, the main elements of the theory-the multiple alignment concept in particular-are theoretical constructs derived from what appears to be necessary to model, in an economical way, such things as pattern recognition, reasoning, and so on.', '1306.3888-4-293-1': 'In BK (Chapter 12), there is some discussion of how the SP concepts relate to a selection of issues in human perception and cognition.', '1306.3888-4-293-2': "A particular interest at the time of writing (after that chapter was written) is the way that the SP theory may provide an alternative to quantum probability as an explanation of phenomena such as the 'conjunction fallacy' [CITATION].", '1306.3888-4-294-0': 'In BK (Chapter 11), I have described in outline, and tentatively, how such things as SP patterns and multiple alignments may be realised with neurons and connections between them.', '1306.3888-4-294-1': 'The cortex of the brains of mammals-which is, topologically, a two-dimensional sheet-may be, in some respects, like a sheet of paper on which pattern assemblies may be written.', '1306.3888-4-294-2': 'These are neural analogues of SP patterns, shown schematically in Figure [REF].', '1306.3888-4-294-3': 'Unlike information written on a sheet of paper, there are neural connections between patterns-as shown in the figure-and communications amongst them.', '1306.3888-4-295-0': "These proposals, which are adapted with modifications from Hebb'syearparhebb_1949 concept of a 'cell assembly', are very different from how artificial 'neural networks' are generally conceived in computer science.", '1306.3888-4-295-1': 'As noted in Section [REF], learning in the SP system is very different from learning in that kind of network-or Hebbian learning.', '1306.3888-4-296-0': '# Conclusion', '1306.3888-4-297-0': 'The SP theory aims to simplify and integrate concepts across artificial intelligence, mainstream computing and human perception and cognition, with information compression as a unifying theme.', '1306.3888-4-297-1': 'The matching and unification of patterns and the concept of multiple alignment are central ideas.', '1306.3888-4-298-0': "In accordance with Occam's Razor, the SP system combines conceptual simplicity with descriptive and explanatory power.", '1306.3888-4-298-1': "A relatively simple mechanism provides an interpretation for a range of concepts and phenomena in several areas including conepts of 'computing', aspects of mathematics and logic, representation of knowledge, natural language processing, pattern recognition, several kinds of probabilistic reasoning, information storage and retrieval, planning and problem solving, information compression, neuroscience, and human perception and cognition.", '1306.3888-4-299-0': 'As suggested in Section [REF], an aid to further research would be the creation of a high-parallel, open-source version of the SP machine, that may be accessed via the web.'}

1008.3198

{'1008.3198-1-0-0': 'Two-field slow-roll inflation is the most conservative modification of the single-field model.', '1008.3198-1-0-1': 'The main motivations to study it are its entropic mode and non-Gaussianity.', '1008.3198-1-0-2': 'Several years ago, for a two-field model with additive separable potentials, Vernizzi and Wands invented an analytic method to estimate its non-Gaussianities.', '1008.3198-1-0-3': 'Later on, Choi et al. applied this method to the model with multiplicative separable potentials.', '1008.3198-1-0-4': 'In this note, we design a larger class of models whose non-Gaussianity can be estimated by the same method.', '1008.3198-1-0-5': 'Under some simplistic assumptions, generally these models is unlikely able to generate a large non-Gaussianity.', '1008.3198-1-0-6': 'But for some specific models of this class, after scanning the full parameter space, we dig out large non-Gaussianities, whose signature could be positive or negative.', '1008.3198-1-0-7': 'These models and scanning techniques would be useful for future model hunt if observational evidence shows up for two-field inflation.', '1008.3198-1-1-0': '# Introduction', '1008.3198-1-2-0': 'Cosmic inflation [CITATION] is a great idea to solve some cosmological problems and to predict the fine fluctuations of cosmic microwave background (CMB).', '1008.3198-1-2-1': 'Hitherto the survived and most economical model of inflation is a single scalar field slowly rolling down its effective potential [CITATION], with a canonical kinetic term and minimally coupled to the Einstein gravity.', '1008.3198-1-2-2': 'We will call it the simplest single-field inflation, although there are still freedom to design its exact potential.', '1008.3198-1-2-3': 'The single-field inflation passed the latest observational test [CITATION] successfully, even with the simplest quadratic potential.', '1008.3198-1-3-0': 'Nevertheless there are perpetual attempts to modify the simplest single-field inflation.', '1008.3198-1-3-1': 'Some of them are motivated by incorporating inflation model into certain theoretical frameworks, such as standard model of particle physics [CITATION] or string theory [CITATION].', '1008.3198-1-3-2': 'Some others put their stake on signatures that cannot appear in the simplest single-field model, such as a large deviation from the Gaussian distribution in the CMB temperature fluctuations.', '1008.3198-1-4-0': 'Among these modifications, the two-field slow-roll inflation is the most conservative one, at least in my personal point of view.', '1008.3198-1-4-1': 'It introduces another scalar field rather than a non-conventional Lagrangian such as non-canonical kinetic terms or modifications of gravity.', '1008.3198-1-4-2': 'It also retains the slow-roll condition, which makes the model simple and consistent with the observed CMB power spectrum.', '1008.3198-1-4-3': 'If both conventional Lagrangian and non-conventional Lagrangian are adaptable to the observational data, then the model with conventional Lagrangian would be more acceptable, unless there are better and solid theoretical motivations for non-conventional Lagrangian.', '1008.3198-1-5-0': 'On the observational side, two new features appear in two-field model.', '1008.3198-1-5-1': 'First, the model is able to leave a residual entropic perturbation between the fluctuations of dark matter and CMB [CITATION].', '1008.3198-1-5-2': 'Second, in a simple model with quadratic potential, numerical computations [CITATION] found that the non-Gaussianity can be temporarily large at the turn of inflation trajectory in field space.', '1008.3198-1-5-3': 'The same phenomenon was discovered recently by [CITATION] in many other potentials.', '1008.3198-1-6-0': 'Compared with the simplest one-field inflation, the field space becomes two-dimensional in a two-field model.', '1008.3198-1-6-1': 'When the inflation trajectory is curved in field space, the entropic perturbation will be coupled to the adiabatic perturbation.', '1008.3198-1-6-2': 'So there are more uncertainties in calculate cosmological observables, such as power spectra of CMB and their indices.', '1008.3198-1-6-3': 'It would be more complicated to honestly compute the bispectra and non-linear parameters, which reflect the non-Gaussianity of the primordial fluctuations.', '1008.3198-1-7-0': 'Fortunately, base on the extended [MATH]-formalism [CITATION], Vernizzi and Wands [CITATION] invented an analytic method to estimate its non-Gaussianities.', '1008.3198-1-7-1': 'They demonstrate the power of this method in a two-field model with additive separable potentials.', '1008.3198-1-7-2': 'Later on, this method was applied by Choi et al. [CITATION] to a model with multiplicative separable potentials.', '1008.3198-1-8-0': 'Encouraged by this method, we tried to improve it for the two-field slow-roll model with generic potentials but failed.', '1008.3198-1-8-1': 'Finally, we only designed a larger class of model whose non-Gaussianity can be estimated by the method of Vernizzi and Wands.', '1008.3198-1-8-2': 'This is a class of model whose potential takes the form [MATH] with [MATH] or [MATH].', '1008.3198-1-8-3': 'Here [MATH], [MATH] and [MATH] are arbitrary functions of the indicated variables as long as the slow-roll condition is satisfied.', '1008.3198-1-8-4': 'Two scalar fields [MATH] and [MATH] are inflatons.', '1008.3198-1-9-0': 'This paper will be organized as follows.', '1008.3198-1-9-1': 'In our convention of notations, we will concisely prepare some well-known but necessary knowledge in section [REF].', '1008.3198-1-9-2': 'In section [REF], we will present the exact form of our models, whose non-linear parameters will be worked out in sections [REF] and [REF].', '1008.3198-1-9-3': 'Some specific examples are investigated in section [REF].', '1008.3198-1-9-4': 'We summarize the main results of this paper in the final section.', '1008.3198-1-10-0': 'This is a note when the author read references [CITATION].', '1008.3198-1-10-1': 'Some of our techniques stem from these references or slightly generalize theirs.', '1008.3198-1-10-2': 'Sometimes we employ the techniques with few explanation as long as the mathematical development is smooth.', '1008.3198-1-10-3': 'To better understand them, the readers are strongly recommended to review the relevant parts of [CITATION].', '1008.3198-1-11-0': '# Non-Gaussianities in Two-field Inflation', '1008.3198-1-12-0': 'We are interested in inflation models described by the following action [CITATION] [EQUATION]', '1008.3198-1-12-1': 'Due to appearance of [MATH], the field [MATH] has a non-standard kinetic term.', '1008.3198-1-12-2': 'Following the notation of slow-roll parameters defined in [CITATION] [EQUATION] the slow-roll condition can be expressed as [MATH], [MATH] with [MATH].', '1008.3198-1-13-0': 'As an aside, we mention that model [REF] is equivalent to the [MATH] generalized gravity [CITATION] when [MATH].', '1008.3198-1-13-1': 'But then we find [MATH], which violates the the slow-roll condition.', '1008.3198-1-13-2': 'This is a pitfall in treating generalized gravity as a two-field model.', '1008.3198-1-13-3': 'A method circumventing the pitfall was employed in [CITATION].', '1008.3198-1-14-0': 'Under the slow-roll condition, the background equations of motion are very simple [EQUATION]', '1008.3198-1-14-1': 'Using them one may directly demonstrate [EQUATION]', '1008.3198-1-14-2': 'Observationally, the most promising probe of primordial non-Gaussianities comes from the bispectrum of CMB fluctuations, which is characterized by the non-linear parameter [MATH].', '1008.3198-1-14-3': 'If [MATH], it would be detectable by ongoing or planned satellite experiments [CITATION].', '1008.3198-1-15-0': 'It has been shown in [CITATION] that the non-linear parameter in two-field inflation models can be separated into a momentum dependent term and a momentum independent term [EQUATION]', '1008.3198-1-15-1': 'It is also proved in [CITATION] that the first term is always suppressed by the tensor-to-scalar ratio, resulting in [MATH].', '1008.3198-1-15-2': 'Hence this term is negligible in observation.', '1008.3198-1-15-3': 'For action [REF], the second term [EQUATION] may be large and deserves a closer look.', '1008.3198-1-15-4': 'Here [MATH] is the [MATH]-folding number from the initial flat hypersurface [MATH] to the final comoving hypersurface [MATH].', '1008.3198-1-15-5': 'To evaluate [REF], we will work out the derivatives of [MATH] with respect to [MATH] and [MATH] in the next section, focusing on a class of analytically solvable models.', '1008.3198-1-16-0': '# Hunting for Analytically Solvable Models', '1008.3198-1-17-0': 'Making use of background equations [REF], the [MATH]-folding number can be cast as [EQUATION]', '1008.3198-1-17-1': 'Here [MATH] is an arbitrary function of [MATH] and [MATH] in principle.', '1008.3198-1-17-2': 'However, for a given [MATH], we have to choose a suitable form of [MATH] so that the integrations can be performed.', '1008.3198-1-17-3': 'It is straightforward to obtain the first order partial derivatives [EQUATION]', '1008.3198-1-17-4': 'Akin to [CITATION], we define an integral of motion [MATH] along the trajectory of inflation [EQUATION] where the explicit form of [MATH] is determined by scalar potential [MATH].', '1008.3198-1-17-5': 'Later on we will give the expression of [MATH] for a class of [MATH].', '1008.3198-1-17-6': 'In order to know [MATH], [MATH], [MATH], [MATH] in [REF], we should calculate the first order derivatives of [MATH] on the initial flat hypersurface [MATH], [EQUATION]', '1008.3198-1-17-7': 'Differentiating [REF] with respect to [MATH], it gives [EQUATION]', '1008.3198-1-17-8': 'On large scales, the comoving hypersurface [MATH] coincides with the uniform density hypersurface.', '1008.3198-1-17-9': 'This implies under the slow-roll condition [EQUATION] whose differentiation with respect to [MATH] is [EQUATION]', '1008.3198-1-17-10': 'Combined with [REF] on the final comoving surface [MATH], it could give the solution for [MATH] and [MATH].', '1008.3198-1-17-11': 'This is in general difficult analytically.', '1008.3198-1-17-12': 'To overcome the difficulty, we introduce an ansatz: [EQUATION]', '1008.3198-1-17-13': 'Although we are free to design the function [MATH], this condition is not always satisfiable.', '1008.3198-1-17-14': 'We have hunted for analytical models meeting the above condition, and found it is achievable if [MATH] with [MATH] or [MATH].', '1008.3198-1-17-15': 'Here [MATH], [MATH] and [MATH] are arbitrary functions of the indicated variables as long as the slow-roll condition is satisfied.', '1008.3198-1-17-16': 'In this paper, we will pay our attention on this situation.', '1008.3198-1-17-17': 'But it is never excluded that there might be other situations in which [MATH] and [MATH] are solvable from [REF] and [REF], even if ansatz [REF] is violated.', '1008.3198-1-18-0': 'Ansatz [REF] simplifies our discussion significantly.', '1008.3198-1-18-1': 'When it holds, equations [REF] and [REF] lead to [EQUATION] while [REF] is reduced as [EQUATION]', '1008.3198-1-18-2': 'As a result, the partial derivatives of [MATH] take the form [EQUATION]', '1008.3198-1-18-3': 'In these equations, we have adopted the notations [EQUATION]', '1008.3198-1-18-4': 'In the above, the expression of [MATH] and its derivatives involves nuisance integrals.', '1008.3198-1-18-5': 'To further simplify our study, we utilize one more ansatz [EQUATION]', '1008.3198-1-18-6': 'In favor of this ansatz, we have [MATH] and so do its derivatives.', '1008.3198-1-19-0': 'As was mentioned, ansatz [REF] can be satisfied by special forms of potential [MATH].', '1008.3198-1-19-1': 'Now ansatz [REF] will further constrain the form of [MATH] and [MATH].', '1008.3198-1-19-2': 'Let us discuss it in details case by case.', '1008.3198-1-20-0': '## Case I: [MATH], [MATH]', '1008.3198-1-21-0': 'For this class of model, according to [REF], we set [EQUATION] while condition [REF] is met by [EQUATION] or [EQUATION]', '1008.3198-1-21-1': 'Hereafter, as free parameters in our models, [MATH], [MATH], [MATH], [MATH], [MATH] and [MATH] are arbitrary real constants.', '1008.3198-1-21-2': 'The normalization of [MATH] is fixed for simplicity.', '1008.3198-1-21-3': 'This is always realizable by rescaling the field [MATH].', '1008.3198-1-22-0': 'Taking [MATH], this model recovers the well-studied sum potential [CITATION], to which we will return in subsection [REF].', '1008.3198-1-22-1': 'In subsection [REF], we will study a specific example of non-separable potential that corresponds to [MATH] in [REF].', '1008.3198-1-23-0': 'As will be discussed in subsection [REF], there is an equivalence relation between case I in this subsection and case II in the next subsection.', '1008.3198-1-23-1': 'Models in class I can be transformed to those in class II, and vice versa.', '1008.3198-1-23-2': 'We will translate model [REF] to a nicer form [REF] in subsection [REF] and explore it.', '1008.3198-1-24-0': '## Case II: [MATH], [MATH]', '1008.3198-1-25-0': 'For this class of model, we take [EQUATION] then condition [REF] is satisfied.', '1008.3198-1-25-1': 'Condition [REF] can be met by [EQUATION] or [EQUATION]', '1008.3198-1-25-2': 'We observed that [REF], [REF] and [REF] can be obtained from [REF], [REF] and [REF] perfectly by the following replacement: [EQUATION]', '1008.3198-1-25-3': 'In fact, there is a general equivalence relation between case I and case II, as will be elaborated in subsection [REF].', '1008.3198-1-26-0': 'Equation [REF] dictates [MATH] implicitly as a differential equation.', '1008.3198-1-26-1': 'To obtain the explicit form of [MATH], we should solve the equation.', '1008.3198-1-26-2': 'This could be done analytically in some corners of the parameter space.', '1008.3198-1-26-3': 'For instance, setting [MATH], equation [REF] gives [EQUATION]', '1008.3198-1-26-4': 'However, if [MATH], it leads to a larger class of model [EQUATION] leaving [MATH] as an arbitrary function of [MATH].', '1008.3198-1-26-5': 'Model [REF] or [REF] is separable and can be seen as the well-studied product potential [CITATION].', '1008.3198-1-26-6': 'More discussion on models with product potential will be given in subsection [REF].', '1008.3198-1-26-7': 'In the case that [MATH] and [MATH], we find another model [EQUATION]', '1008.3198-1-26-8': 'In subsection [REF], we will study an example of non-separable potential which corresponds to [MATH] in [REF].', '1008.3198-1-26-9': 'Since [MATH] is an arbitrary real constant, equation [REF] can generate many other forms of potential [MATH].', '1008.3198-1-26-10': 'For example, when [MATH] and [MATH], we get a model [EQUATION]', '1008.3198-1-27-0': '## Equivalence between Case I and Case II', '1008.3198-1-28-0': 'We have classified our models into two categories, corresponding to subsections [REF] and [REF].', '1008.3198-1-28-1': 'In case I, the potential [MATH] is a function of sum [MATH].', '1008.3198-1-28-2': 'In case II, the potential [MATH] is a function of product [MATH].', '1008.3198-1-28-3': 'After the non-dimensionalization, case I can be translated to case II by the transformation [EQUATION]', '1008.3198-1-28-4': 'The last relation in [REF] is a corollary of the former ones because [MATH].', '1008.3198-1-28-5': 'On the other hand, via transformation [REF], an arbitrary potential of case I can be transformed to that of case II.', '1008.3198-1-28-6': 'So the two "cases\'\'are just two different formalisms for studying the same models.', '1008.3198-1-28-7': 'They are equivalent to each other.', '1008.3198-1-28-8': 'We are free to study a model in either formalism contingent on the convenience.', '1008.3198-1-29-0': '# Model I: [MATH], [MATH], [MATH]', '1008.3198-1-30-0': 'This model is given by [REF].', '1008.3198-1-30-1': 'Corresponding to this model, the number of [MATH]-foldings and the integral constant along the inflation trajectory are [EQUATION]', '1008.3198-1-30-2': 'We have defined the slow-roll parameters in [REF].', '1008.3198-1-30-3': 'In the present case, they are of the form [EQUATION]', '1008.3198-1-30-4': 'Now equations [REF] and [REF] become [EQUATION] while the function [MATH] defined by [REF] takes the form [EQUATION]', '1008.3198-1-30-5': 'Then we get the partial derivatives of [MATH] with respect to [MATH] and [MATH], [EQUATION] in terms of [EQUATION]', '1008.3198-1-30-6': 'With the above result at hand, it is straightforward to calculate [EQUATION] where for convenience we used notations [EQUATION]', '1008.3198-1-30-7': 'For these notations, the relation [MATH] holds.', '1008.3198-1-30-8': 'In the next section, the definitions of [MATH] and [MATH] are different, but the same relation also holds.', '1008.3198-1-31-0': 'As a result, using formula [REF] we get the main part of non-linear parameter in this model [EQUATION]', '1008.3198-1-31-1': 'The non-linear parameter [REF] depends on the exponent [MATH] in a complicated manner.', '1008.3198-1-31-2': 'For the purpose of rough estimation, we assume both [MATH] and [MATH] are of order unity.', '1008.3198-1-31-3': 'This assumption is reasonable if [MATH], [MATH] and [MATH] are of the same order.', '1008.3198-1-31-4': 'It is also consistent with the relation [MATH].', '1008.3198-1-31-5': 'Furthermore, motivated by the slow-roll condition and the observational constraint on spectral indices, we assume the slow-roll parameters are of order [MATH].', '1008.3198-1-31-6': 'In saying this we mean all of the slow-roll parameters are of the same order, which is a strong but still allowable assumption.', '1008.3198-1-31-7': 'After making these assumptions, we can estimate the magnitude of [REF] in three regions according to the value of [MATH].', '1008.3198-1-32-0': 'Firstly, in the limit [MATH], we have [MATH].', '1008.3198-1-32-1': 'So the third term in curly brackets of [REF] is of order [MATH], while the other two terms are of order [MATH].', '1008.3198-1-32-2': 'Consequently, we can estimate [MATH].', '1008.3198-1-32-3': 'Hence it seems that a small value of [MATH] could give rise to a large non-linear parameter.', '1008.3198-1-32-4': 'Specifically, under our assumptions above, if [MATH], then the non-linear parameter [MATH].', '1008.3198-1-32-5': 'However, this limit violates our assumptions.', '1008.3198-1-32-6': 'On the one hand, we have assumed [MATH].', '1008.3198-1-32-7': 'On the other hand, equations [REF] tell us [MATH], which apparently violates our assumption in the limit [MATH].', '1008.3198-1-32-8': 'So we cannot use the oversimplified assumptions to estimate the non-linear parameter in this limit.', '1008.3198-1-33-0': 'Secondly, for [MATH], we would have [MATH].', '1008.3198-1-33-1': 'Then the last term in braces of [REF] is of order [MATH].', '1008.3198-1-33-2': 'The other terms can be of order [MATH].', '1008.3198-1-33-3': 'After cancelation with the prefactor, it leads the estimation [MATH].', '1008.3198-1-33-4': 'That is to say, in this limit, the non-linear parameter is independent of [MATH] in the leading order and suppressed by the slow-roll parameters.', '1008.3198-1-34-0': 'The third region is [MATH].', '1008.3198-1-34-1': 'In this region, the non-linear parameter is still suppressed, [MATH].', '1008.3198-1-35-0': 'Our conclusion is somewhat unexciting.', '1008.3198-1-35-1': 'This model could not generate large non-Gaussianities under our simplistic assumptions.', '1008.3198-1-35-2': 'However, one should be warned that our estimation above relies on two assumptions: [MATH] and [MATH].', '1008.3198-1-35-3': 'Although these assumptions are reasonable, they may be avoided in very special circumstances.', '1008.3198-1-35-4': 'To get a large non-Gaussianity with our formula [REF], one should give up these assumptions and carefully scan the whole parameter space in a consistent way.', '1008.3198-1-35-5': 'Generally that is an ambitious task if not impossible.', '1008.3198-1-35-6': 'But for a specific model of this type, We will perform such a scanning in subsection [REF].', '1008.3198-1-36-0': '# Model II: [MATH], [MATH], [MATH]', '1008.3198-1-37-0': 'As we have discussed, models [REF] and [REF] are equivalent.', '1008.3198-1-37-1': 'Thus it is enough to study them in the relatively simpler form, namely in the form [REF].', '1008.3198-1-37-2': 'For this model, we calculated the number of [MATH]-foldings and the integral constant along the inflation trajectory [EQUATION]', '1008.3198-1-37-3': 'Parallel to section [REF], we also calculated the slow-roll parameters in this model, [EQUATION]', '1008.3198-1-37-4': 'Subsequently, after obtaining the equations [EQUATION] and [EQUATION] we find by tedious computation [EQUATION]', '1008.3198-1-37-5': 'Here notation [MATH] is different from the one in the previous section, [EQUATION]', '1008.3198-1-37-6': 'In terms of [EQUATION] and the relation [MATH], once again straightforward calculation gives [EQUATION]', '1008.3198-1-37-7': 'Therefore, the non-linear parameter in this model is [EQUATION]', '1008.3198-1-37-8': 'Similar to the previous section, we can estimate [MATH] by assuming [MATH] and [MATH].', '1008.3198-1-37-9': 'Under these assumptions, the only possibility to generate a large non-linear parameter is in the limit [MATH].', '1008.3198-1-37-10': 'Unfortunately, careful analysis ruled out this possibility.', '1008.3198-1-37-11': 'Because the assumption [MATH] implies [MATH], we find the non-linear parameter is not enhanced by [MATH] but is suppressed by the slow-roll parameters, [MATH].', '1008.3198-1-37-12': 'The same suppression applies if [MATH] lies in other regions.', '1008.3198-1-37-13': 'So we conclude that it is hopeless to generate large non-Gaussianities in this model unless one goes beyond the assumptions we made.', '1008.3198-1-37-14': 'A careful scan of parameter space will be done in subsection [REF] for a specific model.', '1008.3198-1-38-0': '# Examples', '1008.3198-1-39-0': 'In sections above, we have generalized the method of [CITATION] to a larger class of model, which is summarized by equations [REF] and [REF], whose non-linear parameters are given by [REF] and [REF] generally.', '1008.3198-1-39-1': 'To check our general formulae, we will reduce [REF] and [REF] to previously known limit in subsections [REF] and [REF].', '1008.3198-1-39-2': 'The reduced expressions are consistent with the results of [CITATION].', '1008.3198-1-39-3': 'In subsections [REF], [REF] and [REF], we will apply our formulae to non-separable examples and scan the full parameter spaces.', '1008.3198-1-40-0': '## Additive Potential: [MATH], [MATH], [MATH]', '1008.3198-1-41-0': 'This potential is obtained from [REF] by setting [MATH].', '1008.3198-1-41-1': 'The condition [MATH] is necessary to guarantee [REF].', '1008.3198-1-41-2': 'After taking [MATH], the result in section [REF] matches with that in [CITATION] obviously.', '1008.3198-1-42-0': '## Multiplicative Potential: [MATH], [MATH]', '1008.3198-1-43-0': 'Like equation [REF], we leave [MATH] as an arbitrary function of [MATH], as long as the slow-roll parameters [REF] are small.', '1008.3198-1-43-1': 'This is a special limit of section [REF].', '1008.3198-1-44-0': 'Using relations [EQUATION] we get the reduced form of non-linear parameter [EQUATION] where we have made use of the fact that [MATH] as well as the following notations [EQUATION]', '1008.3198-1-44-1': 'One may compare this formula with [CITATION].', '1008.3198-1-44-2': 'Note that their definitions of [MATH], [MATH] and [MATH] are slightly different from ours by some factors.', '1008.3198-1-44-3': 'Taking these factors into account, the result here is in accordance with [CITATION].', '1008.3198-1-45-0': '## Non-separable Potential I: [MATH], [MATH]', '1008.3198-1-46-0': 'We list this model as an independent subsection not because of its non-Gaussianity, but because it has an elegant relation between the [MATH]-foling number and the angle variable of fields.', '1008.3198-1-46-1': 'For this model, the number of [MATH]-foldings from time [MATH] during the inflation stage to the end of inflation is [EQUATION]', '1008.3198-1-46-2': 'Note that [MATH] can be regarded as sum of squares.', '1008.3198-1-46-3': 'Its time derivative gives the Hubble parameter [MATH].', '1008.3198-1-46-4': 'So we can follow the standard treatment to parameterize the scalars in polar coordinates [EQUATION]', '1008.3198-1-46-5': 'Rewriting the equations of motion [REF] in terms of the polar coordinates, we obtain a differential relation between [MATH] and [MATH] for the present model, [EQUATION] with [MATH].', '1008.3198-1-46-6': 'It can be solved out to give [EQUATION]', '1008.3198-1-46-7': 'If the scalars have rolled down to the bottom of potential at the end of inflation, one may simply set [MATH].', '1008.3198-1-47-0': "Relation [REF] is a trivial but probably useful generalization of Polarski and Starobinsky's relation [CITATION].", '1008.3198-1-47-1': "Recall that Polarski and Starobinsky's relation has been widely used for the inflation model with two massive scalar fields, which corresponds to exponent [MATH] in the model of this subsection.", '1008.3198-1-47-2': 'The simple demonstration above generalized the relation to arbitrary [MATH].', '1008.3198-1-48-0': '## Non-separable Potential II: [MATH], [MATH]', '1008.3198-1-49-0': 'Our interest in this and the next subsections is to hunt for large non-Gaussianities by parameter scanning.', '1008.3198-1-49-1': 'Two common assumptions will be used: the [MATH]-folding number is fixed to be [MATH] and the inflation is supposed to conclude at the point [MATH].', '1008.3198-1-50-0': 'Using the latter assumptions and the equations in section [REF], we find all of the relevant quantities can be expressed by [MATH], [MATH] and [MATH]: [EQUATION]', '1008.3198-1-50-1': 'Here we defined [MATH] like the previous subsection.', '1008.3198-1-50-2': 'If [MATH], it can be proved that [MATH].', '1008.3198-1-50-3': 'Without loss of generality, we will consider the parameter region [MATH].', '1008.3198-1-51-0': 'In the above expressions, there are five parameters: [MATH], [MATH], [MATH], [MATH] and [MATH].', '1008.3198-1-51-1': 'The number can be reduced by the assumptions we made at the beginning of this section.', '1008.3198-1-51-2': 'Firstly, [MATH] and [MATH] can be traded to each other with the relation [MATH].', '1008.3198-1-51-3': 'Secondly, when [MATH], we can solve equation [REF], [EQUATION] [MATH] is defined by the first term of [REF].', '1008.3198-1-51-4': 'It is not hard to check that [MATH] using [REF].', '1008.3198-1-51-5': 'One can also check that [MATH] is obeyed by both roots as long as [MATH], so both roots are reasonable.', '1008.3198-1-51-6': 'Now we see only three parameters are independent: [MATH], [MATH] and [MATH].', '1008.3198-1-52-0': 'The inequality [MATH] impose a constraint on [MATH] and [MATH]: [EQUATION]', '1008.3198-1-52-1': 'This constraint should be obeyed, otherwise our assumptions will break down.', '1008.3198-1-52-2': 'When scanning the parameter space, we will take it into consideration.', '1008.3198-1-53-0': 'We scanned the parameter space continuously in [MATH] and [MATH] for discrete values of [MATH].', '1008.3198-1-53-1': 'Typical results are illustrated in figures [REF] and [REF], corresponding to two roots of [MATH] respectively.', '1008.3198-1-53-2': 'To our surprise, in some corners of the parameter space, the non-linear parameter [MATH] could be large, and its signature could be positive or negative.', '1008.3198-1-54-0': 'As figures [REF] and [REF] indicate, large values of [MATH] appear generally at small [MATH], namely in the vicinity of [MATH].', '1008.3198-1-54-1': 'This corner is of special interest.', '1008.3198-1-54-2': 'Obviously more exploration is demanded here.', '1008.3198-1-54-3': 'In the limit [MATH], equations [REF] and [REF] are greatly simplified, [EQUATION] with [MATH] and [MATH].', '1008.3198-1-54-4': 'We have plotted these functions in figure [REF] by scanning [MATH] and [MATH].', '1008.3198-1-54-5': 'According to the figure, large non-Gaussianities show up in the following limits: [MATH], [MATH], [MATH] or [MATH].', '1008.3198-1-54-6': 'It can be confirmed analytically by [REF] in these limits.', '1008.3198-1-55-0': '## Non-separable Potential III: [MATH], [MATH]', '1008.3198-1-56-0': 'This is a special model of [REF] with [MATH], [MATH], [MATH].', '1008.3198-1-56-1': 'As in the previous subsection, we assume [MATH] and [MATH].', '1008.3198-1-56-2': 'Then from section [REF] we get the relations [EQUATION]', '1008.3198-1-56-3': 'If we introduce the notation [MATH], then combining it with the condition [MATH], we can express [MATH] and [MATH] in terms of [MATH], [MATH] and [MATH], [EQUATION]', '1008.3198-1-56-4': 'From equation [REF], we can see [MATH] be positive and suppressed by slow-roll parameters.', '1008.3198-1-56-5': 'Thus we are focus on the region [MATH].', '1008.3198-1-56-6': 'The parameter scanning is illustrated in figure [REF].', '1008.3198-1-56-7': 'Although [MATH] changes as the parameters vary, the non-Gaussianity is small in the entire parameter space.', '1008.3198-1-57-0': '# Summary', '1008.3198-1-58-0': 'In this paper, we investigated a class of two-field slow-roll inflation model whose non-linear parameter is analytically calculable.', '1008.3198-1-59-0': 'In our convention of notations, we collected some well-known but necessary knowledge in section [REF].', '1008.3198-1-59-1': 'Slightly generalizing the method of [CITATION], we showed in section [REF] how their method could be utilized in a larger class of model satisfying two ansatzes, namely [REF] and [REF].', '1008.3198-1-59-2': 'In subsections [REF] and [REF] we proposed models meeting these ansatzes.', '1008.3198-1-59-3': 'We put our models in the form of [MATH] with [MATH] in subsection [REF] and with [MATH] in subsection [REF].', '1008.3198-1-59-4': 'At first glance, these are two different classes of models.', '1008.3198-1-59-5': 'But in fact they are two dual forms of the same class of model, just as proved in subsection [REF].', '1008.3198-1-59-6': 'In a succinct form, our models can be summarized by equations [REF] and [REF], whose non-linear parameters were worked out in sections [REF] and [REF] respectively, see equations [REF] and [REF].', '1008.3198-1-59-7': 'Under simplistic assumptions, we found no large non-Gaussianity in these models.', '1008.3198-1-60-0': 'As a double check, we reduced the expression [REF] for non-linear parameter to the additive potential in subsection [REF], and [REF] to multiplicative potential in subsection [REF].', '1008.3198-1-60-1': 'The resulting non-linear parameters match with [CITATION], confirming our calculations.', '1008.3198-1-60-2': "In subsection [REF], for a special class of model, we generalized Polarski and Starobinsky's relation [REF].", '1008.3198-1-60-3': 'For more specific models, we scanned the parameter space to evaluate the non-linear parameter, as shown by figures in subsections [REF] and [REF].', '1008.3198-1-60-4': 'In the scanning, we assumed the [MATH]-folding number [MATH] and the inflation terminates at [MATH].', '1008.3198-1-60-5': 'For the model studied in subsection [REF], the non-linear parameter [MATH] can be large and its signature can be positive or negative.', '1008.3198-1-60-6': 'For the model studied in [REF], [MATH] always takes a small positive value.'}

{'1008.3198-2-0-0': 'Two-field slow-roll inflation is the most conservative modification of a single-field model.', '1008.3198-2-0-1': 'The main motivations to study it are its entropic mode and non-Gaussianity.', '1008.3198-2-0-2': 'Several years ago, for a two-field model with additive separable potentials, Vernizzi and Wands invented an analytic method to estimate its non-Gaussianities.', '1008.3198-2-0-3': 'Later on, Choi et al. applied this method to the model with multiplicative separable potentials.', '1008.3198-2-0-4': 'In this note, we design a larger class of models whose non-Gaussianity can be estimated by the same method.', '1008.3198-2-0-5': 'Under some simplistic assumptions, roughly these models are unlikely able to generate a large non-Gaussianity.', '1008.3198-2-0-6': 'We look over some specific models of this class by scanning the full parameter space, but still no large non-Gaussianity appears in the slow-roll region.', '1008.3198-2-0-7': 'These models and scanning techniques would be useful for future model hunt if observational evidence shows up for two-field inflation.', '1008.3198-2-1-0': '# Introduction', '1008.3198-2-2-0': 'Cosmic inflation [CITATION] is a great idea to solve some cosmological problems and to predict the fine fluctuations of cosmic microwave background (CMB).', '1008.3198-2-2-1': 'Hitherto the surviving and most economical model of inflation involves a single scalar field slowly rolling down its effective potential [CITATION], with a canonical kinetic term and minimally coupled to the Einstein gravity.', '1008.3198-2-2-2': 'We will call it the simplest single-field inflation, although there is still freedom to design its exact potential.', '1008.3198-2-2-3': 'The single-field inflation passed the latest observational test [CITATION] successfully, even with the simplest quadratic potential.', '1008.3198-2-3-0': 'Nevertheless there are perpetual attempts to modify the simplest single-field inflation.', '1008.3198-2-3-1': 'Some of them are motivated by incorporating inflation model into certain theoretical frameworks, such as the standard model of particle physics [CITATION] or string theory [CITATION].', '1008.3198-2-3-2': 'Some others put their stake on signatures that cannot appear in the simplest single-field model, such as a large deviation from the Gaussian distribution in the CMB temperature fluctuations.', '1008.3198-2-4-0': 'Among these modifications, the two-field slow-roll inflation is the most conservative one, at least in my personal point of view.', '1008.3198-2-4-1': 'It introduces another scalar field rather than a non-conventional Lagrangian such as non-canonical kinetic terms or modifications of gravity.', '1008.3198-2-4-2': 'It also retains the slow-roll condition, which makes the model simple and consistent with the observed CMB power spectrum.', '1008.3198-2-4-3': 'If both conventional Lagrangian and non-conventional Lagrangian are adaptable to the observational data, then the model with conventional Lagrangian would be more acceptable, unless there are better and solid theoretical motivations for non-conventional Lagrangian.', '1008.3198-2-5-0': 'On the observational side, two new features arise in two-field model.', '1008.3198-2-5-1': 'First, the model is able to leave a residual entropic perturbation between the fluctuations of dark matter and CMB [CITATION].', '1008.3198-2-5-2': 'Second, in a simple model with quadratic potential, numerical computations [CITATION] found that the non-Gaussianity can be temporarily large at the turn of inflation trajectory in field space.', '1008.3198-2-5-3': 'Longer-lived large non-Gaussianities were discovered recently by [CITATION] in many other two-field models.', '1008.3198-2-6-0': 'Compared with the simplest one-field inflation, the field space becomes two-dimensional in a two-field model.', '1008.3198-2-6-1': 'When the inflation trajectory is curved in field space, the entropic perturbation will be coupled to the adiabatic perturbation.', '1008.3198-2-6-2': 'So there are more uncertainties in calculation of cosmological observables, such as power spectra of CMB and their indices.', '1008.3198-2-6-3': 'It would be more complicated to honestly compute the bispectra and non-linear parameters, which reflect the non-Gaussianity of the primordial fluctuations.', '1008.3198-2-7-0': 'Fortunately, based on the extended [MATH]-formalism [CITATION], Vernizzi and Wands [CITATION] invented an analytic method to estimate such non-Gaussianities.', '1008.3198-2-7-1': 'They demonstrated the power of this method in a two-field model with additive separable potentials.', '1008.3198-2-7-2': 'This method was later applied by Choi et al. [CITATION] to a model with multiplicative separable potentials.', '1008.3198-2-8-0': 'Encouraged by the method of Vernizzi and Wands, we tried to improve it for the two-field slow-roll model with generic potentials but failed.', '1008.3198-2-8-1': 'Finally, we only designed a larger class of models whose non-Gaussianity can be estimated by this method.', '1008.3198-2-8-2': 'It is a class of models whose potential take the form [MATH] with [MATH] or [MATH].', '1008.3198-2-8-3': 'Here [MATH], [MATH] and [MATH] are arbitrary functions of the indicated variables as long as the slow-roll condition is satisfied.', '1008.3198-2-8-4': 'Scalar fields [MATH] and [MATH] are inflatons.', '1008.3198-2-9-0': 'The outline of this paper is as follows.', '1008.3198-2-9-1': 'In our convention of notations, we will prepare some well-known but necessary knowledge in section [REF] concisely.', '1008.3198-2-9-2': 'In section [REF], we will present the exact form of our models, whose non-linear parameters will be worked out in sections [REF] and [REF].', '1008.3198-2-9-3': 'Some specific examples are investigated in section [REF].', '1008.3198-2-9-4': 'We summarize the main results of this paper in the final section.', '1008.3198-2-10-0': 'This is a note concerning references [CITATION].', '1008.3198-2-10-1': 'Some of our techniques stem from these references or slightly generalize theirs.', '1008.3198-2-10-2': 'Sometimes we employ the techniques with few explanation if the mathematical development is smooth.', '1008.3198-2-10-3': 'To better understand them, the readers are strongly recommended to review the relevant parts of [CITATION].', '1008.3198-2-11-0': '# Non-Gaussianities in Two-field Inflation', '1008.3198-2-12-0': 'We are interested in inflation models described by the following action [CITATION] [EQUATION]', '1008.3198-2-12-1': 'Because of the appearance of [MATH], the field [MATH] has a non-standard kinetic term.', '1008.3198-2-12-2': 'Following the notation of slow-roll parameters defined in [CITATION] [EQUATION] the slow-roll condition can be expressed as [MATH], [MATH], [MATH] with [MATH].', '1008.3198-2-13-0': 'As an aside, we mention that model [REF] is equivalent to the [MATH] generalized gravity [CITATION] when [MATH].', '1008.3198-2-13-1': 'But then we find [MATH], which violates the the slow-roll condition.', '1008.3198-2-13-2': 'This is a pitfall in treating generalized gravity as a two-field model.', '1008.3198-2-13-3': 'This pitfall can be circumvented by the scheme in [CITATION].', '1008.3198-2-14-0': 'Under the slow-roll condition, the background equations of motion are very simple [EQUATION]', '1008.3198-2-14-1': 'Using them one may directly demonstrate [EQUATION]', '1008.3198-2-14-2': 'Observationally, the most promising probe of primordial non-Gaussianities comes from the bispectrum of CMB fluctuations, which is characterized by the non-linear parameter [MATH].', '1008.3198-2-14-3': 'If [MATH], it would be detectable by ongoing or planned satellite experiments [CITATION].', '1008.3198-2-15-0': 'It has been shown in [CITATION] that the non-linear parameter in two-field inflation models can be separated into a momentum dependent term and a momentum independent term [EQUATION]', '1008.3198-2-15-1': 'It is also proved in [CITATION] that the first term is always suppressed by the tensor-to-scalar ratio, leading to [MATH].', '1008.3198-2-15-2': 'Hence this term is negligible in observation.', '1008.3198-2-15-3': 'For action [REF], the second term [EQUATION] may be large and deserves a closer look.', '1008.3198-2-15-4': 'Here [MATH] is the [MATH]-folding number from the initial flat hypersurface [MATH] to the final comoving hypersurface [MATH].', '1008.3198-2-15-5': 'To evaluate [REF], we will work out the derivatives of [MATH] with respect to [MATH] and [MATH] in the next section, focusing on a class of analytically solvable models.', '1008.3198-2-16-0': '# Hunting for Analytically Solvable Models', '1008.3198-2-17-0': 'Making use of equations [REF], the [MATH]-folding number can be cast as [EQUATION]', '1008.3198-2-17-1': 'Hence [MATH] is an arbitrary function of [MATH] and [MATH] in principle, because [MATH] along any classical trajectory under the slow-roll condition.', '1008.3198-2-17-2': 'However, for a given [MATH], we have to choose a suitable form of [MATH] so that the integrations defined by [MATH] in [REF] can be performed.', '1008.3198-2-17-3': 'Later on we will fix [MATH] to meet the ansatz [REF] for simplicity.', '1008.3198-2-17-4': 'But for the moment let us leave it as an arbitrary function of [MATH] and [MATH].', '1008.3198-2-17-5': 'It is straightforward to obtain the first order partial derivatives [EQUATION]', '1008.3198-2-17-6': 'Akin to [CITATION], we define an integral of motion [MATH] along the trajectory of inflation [EQUATION]', '1008.3198-2-17-7': 'Here the explicit form of [MATH] is determined by scalar potential [MATH].', '1008.3198-2-17-8': 'We will give the expression of [MATH] for some types of potential in this section.', '1008.3198-2-17-9': 'If we fix the limits of integration to run from [MATH] to [MATH], then due to the background equations [REF], [EQUATION] along classical trajectories under the slow-roll approximation.', '1008.3198-2-17-10': 'So the constant [MATH] parameterizes the motion off classical trajectories.', '1008.3198-2-17-11': 'In order to know [MATH], [MATH], [MATH], [MATH] in [REF], we should calculate the first order derivatives of [MATH] on the initial flat hypersurface [MATH], [EQUATION]', '1008.3198-2-17-12': 'Differentiating [REF] with respect to [MATH], it gives [EQUATION]', '1008.3198-2-17-13': 'On large scales, the comoving hypersurface [MATH] coincides with the uniform density hypersurface.', '1008.3198-2-17-14': 'This implies under the slow-roll condition [EQUATION] whose differentiation with respect to [MATH] is [EQUATION]', '1008.3198-2-17-15': 'Combined with [REF] on the final comoving surface [MATH], it could give the solution for [MATH] and [MATH].', '1008.3198-2-17-16': 'This is in general difficult analytically.', '1008.3198-2-17-17': 'To overcome the difficulty, we introduce an ansatz: [EQUATION]', '1008.3198-2-17-18': 'Although we are free to design the function [MATH], the above condition is not always satisfiable.', '1008.3198-2-17-19': 'We have hunted for analytical models meeting this condition, and found it is achievable if [MATH] with [MATH] or [MATH].', '1008.3198-2-17-20': 'Here [MATH], [MATH] and [MATH] are arbitrary functions of the indicated variables as long as the slow-roll condition is satisfied.', '1008.3198-2-17-21': 'In this paper, we will pay attention to this situation.', '1008.3198-2-17-22': 'But it is never excluded that there might be other situations in which [MATH] and [MATH] are solvable from [REF] and [REF], even if ansatz [REF] is violated.', '1008.3198-2-18-0': 'Ansatz [REF] simplifies our discussion significantly.', '1008.3198-2-18-1': 'Once it holds, equations [REF] and [REF] lead to [EQUATION] while [REF] is reduced as [EQUATION]', '1008.3198-2-18-2': 'As a result, the partial derivatives of [MATH] take the form [EQUATION]', '1008.3198-2-18-3': 'In these equations, we have adopted the notations [EQUATION]', '1008.3198-2-18-4': 'In the above, the expression of [MATH] and its derivatives involve nuisance integrals.', '1008.3198-2-18-5': 'To further simplify our study, we utilize one more ansatz [EQUATION]', '1008.3198-2-18-6': 'In favor of this ansatz, we have [MATH] and so do its derivatives.', '1008.3198-2-19-0': 'As was mentioned, ansatz [REF] can be satisfied by special forms of potential [MATH].', '1008.3198-2-19-1': 'Now ansatz [REF] further constrains the form of [MATH] and [MATH].', '1008.3198-2-19-2': 'Let us discuss it in details case by case.', '1008.3198-2-20-0': '## Case I: [MATH], [MATH]', '1008.3198-2-21-0': 'For this class of models, according to [REF], we set [EQUATION] while condition [REF] is met by [EQUATION] or [EQUATION]', '1008.3198-2-21-1': 'Hereafter, as free parameters in our models, [MATH], [MATH], [MATH], [MATH], [MATH] and [MATH] are arbitrary real constants.', '1008.3198-2-21-2': 'The normalization of [MATH] is fixed for simplicity.', '1008.3198-2-21-3': 'This is always realizable by rescaling the field [MATH].', '1008.3198-2-22-0': 'Taking [MATH], model [REF] recovers the well-studied sum potential [CITATION], to which we will return in subsection [REF].', '1008.3198-2-22-1': 'In subsection [REF], we will study a specific example of non-separable potential that corresponds to [MATH] in [REF].', '1008.3198-2-23-0': 'As will be discussed in subsection [REF], there is an equivalence relation between case I in this subsection and case II in the next subsection.', '1008.3198-2-23-1': 'Models in class I can be transformed to those in class II, and vice versa.', '1008.3198-2-23-2': 'We will translate model [REF] to a nicer form [REF] and explore it.', '1008.3198-2-24-0': '## Case II: [MATH], [MATH]', '1008.3198-2-25-0': 'For this class of models, we take [EQUATION] then condition [REF] is satisfied.', '1008.3198-2-25-1': 'Condition [REF] can be met by [EQUATION] or [EQUATION]', '1008.3198-2-25-2': 'We observed that [REF], [REF] and [REF] can be obtained from [REF], [REF] and [REF] perfectly by the following replacement: [EQUATION]', '1008.3198-2-25-3': 'In fact, there is a general equivalence relation between case I and case II, on which will be elaborated in subsection [REF].', '1008.3198-2-26-0': 'Equation [REF] dictates [MATH] implicitly as a differential equation.', '1008.3198-2-26-1': 'To obtain the explicit form of [MATH], one should solve the equation.', '1008.3198-2-26-2': 'This could be done analytically in some corners of the parameter space.', '1008.3198-2-26-3': 'For instance, setting [MATH], equation [REF] gives [EQUATION]', '1008.3198-2-26-4': 'However, if [MATH], it leads to a larger class of model [EQUATION] leaving [MATH] as an arbitrary function of [MATH].', '1008.3198-2-26-5': 'Model [REF] or [REF] is separable and can be seen as the well-studied product potential [CITATION].', '1008.3198-2-26-6': 'More discussion on models with product potential will be given in subsection [REF].', '1008.3198-2-26-7': 'In the case that [MATH] and [MATH], we find another model [EQUATION]', '1008.3198-2-26-8': 'In subsection [REF], we will study an example of non-separable potential which corresponds to [MATH] in [REF].', '1008.3198-2-26-9': 'Since [MATH] is an arbitrary real constant, equation [REF] can generate many other forms of potential [MATH].', '1008.3198-2-26-10': 'For example, when [MATH] and [MATH], we get a model [EQUATION]', '1008.3198-2-27-0': '## Equivalence between Case I and Case II', '1008.3198-2-28-0': 'We have classified our models into two categories, corresponding to subsections [REF] and [REF].', '1008.3198-2-28-1': 'In case I, the potential [MATH] is a function of sum [MATH].', '1008.3198-2-28-2': 'In case II, the potential [MATH] is a function of product [MATH].', '1008.3198-2-28-3': 'After the non-dimensionalization, case I can be translated to case II by the transformation [EQUATION]', '1008.3198-2-28-4': 'The last relation in [REF] is a corollary of the former ones because [MATH].', '1008.3198-2-28-5': 'On the other hand, via transformation [REF], an arbitrary potential of case I can be transformed to that of case II.', '1008.3198-2-28-6': 'So the two "cases\'\'are just two different formalisms for studying the same models.', '1008.3198-2-28-7': 'They are equivalent to each other.', '1008.3198-2-28-8': 'We are free to study a model in either formalism contingent on the convenience.', '1008.3198-2-29-0': 'For instance, using the formulae in this section, a model with potential [MATH] and prefactor [MATH] can be studied in two different formalisms:', '1008.3198-2-30-0': 'But apparently, for this model the calculation will be easier in formalism II.', '1008.3198-2-30-1': 'Because the dependence of [MATH] and [MATH] on [MATH] and [MATH] is unaltered, the quantization of perturbations is not affected by the choice of formalism.', '1008.3198-2-30-2': 'For the same reason, the exact dependence of [MATH] on [MATH] and [MATH] is the same in both formalisms.', '1008.3198-2-31-0': '# Model I: [MATH], [MATH], [MATH]', '1008.3198-2-32-0': 'This model is given by [REF], which is equivalent to model [REF].', '1008.3198-2-32-1': 'Corresponding to this model, the number of [MATH]-foldings and the integral constant along the inflation trajectory are [EQUATION]', '1008.3198-2-32-2': 'We have defined the slow-roll parameters in [REF].', '1008.3198-2-32-3': 'In the present case, they are of the form [EQUATION]', '1008.3198-2-32-4': 'Now equations [REF] and [REF] become [EQUATION] while the function [MATH] defined by [REF] takes the form [EQUATION]', '1008.3198-2-32-5': 'Then we get the partial derivatives of [MATH] with respect to [MATH] and [MATH], [EQUATION] in terms of [EQUATION]', '1008.3198-2-32-6': 'With the above result at hand, it is straightforward to calculate [EQUATION] where for convenience we used notations [EQUATION]', '1008.3198-2-32-7': 'For these notations, the relation [MATH] holds.', '1008.3198-2-32-8': 'In the next section, the definitions of [MATH] and [MATH] are different, but the same relation also holds.', '1008.3198-2-33-0': 'As a result, using formula [REF] we get the main part of non-linear parameter in this model [EQUATION]', '1008.3198-2-33-1': 'The non-linear parameter [REF] depends on the exponent [MATH] in a complicated manner.', '1008.3198-2-33-2': 'For the purpose of rough estimation, we assume both [MATH] and [MATH] are of order unity.', '1008.3198-2-33-3': 'This assumption is reasonable if [MATH], [MATH] and [MATH] are of the same order.', '1008.3198-2-33-4': 'It is also consistent with the relation [MATH].', '1008.3198-2-33-5': 'Furthermore, motivated by the slow-roll condition and the observational constraint on spectral indices, we assume the slow-roll parameters are of order [MATH].', '1008.3198-2-33-6': 'In saying this we mean all of the slow-roll parameters are of the same order, which is a strong but still allowable assumption.', '1008.3198-2-33-7': 'After making these assumptions, we can estimate the magnitude of [REF] in three regions according to the value of [MATH].', '1008.3198-2-34-0': 'Firstly, in the limit [MATH], we have [MATH].', '1008.3198-2-34-1': 'So the third term in curly brackets of [REF] is of order [MATH], while the other two terms are of order [MATH].', '1008.3198-2-34-2': 'Consequently, we can estimate [MATH].', '1008.3198-2-34-3': 'It seems that a small value of [MATH] could give rise to a large non-linear parameter.', '1008.3198-2-34-4': 'Specifically, under our assumptions above, if [MATH], then the non-linear parameter [MATH].', '1008.3198-2-34-5': 'However, this limit violates our assumptions.', '1008.3198-2-34-6': 'On the one hand, we have assumed [MATH].', '1008.3198-2-34-7': 'On the other hand, equations [REF] tell us [MATH], which apparently violates our assumption in the limit [MATH].', '1008.3198-2-34-8': 'So we cannot use the oversimplified assumptions to estimate the non-linear parameter in this limit.', '1008.3198-2-35-0': 'Secondly, for [MATH], we would have [MATH].', '1008.3198-2-35-1': 'Then the last term in braces of [REF] is of order [MATH].', '1008.3198-2-35-2': 'The other terms can be of order [MATH].', '1008.3198-2-35-3': 'After cancelation with the prefactor, it leads to the estimation [MATH].', '1008.3198-2-35-4': 'That is to say, in this limit, the non-linear parameter is independent of [MATH] in the leading order and suppressed by the slow-roll parameters.', '1008.3198-2-36-0': 'The third region is [MATH].', '1008.3198-2-36-1': 'In this region, the non-linear parameter is still suppressed, [MATH].', '1008.3198-2-37-0': 'Our conclusion is somewhat unexciting.', '1008.3198-2-37-1': 'This model could not generate large non-Gaussianities under our simplistic assumptions.', '1008.3198-2-37-2': 'However, one should be warned that our estimation above relies on two assumptions: [MATH] and [MATH].', '1008.3198-2-37-3': 'Although these assumptions are reasonable, they may be avoided in very special circumstances.', '1008.3198-2-37-4': 'To further look for a large non-Gaussianity with our formula [REF], one should give up these assumptions and carefully scan the whole parameter space in a consistent way.', '1008.3198-2-37-5': 'Generally that is an ambitious task if not impossible.', '1008.3198-2-37-6': 'But for a specific model of this type, we will perform such a scanning in subsection [REF].', '1008.3198-2-38-0': '# Model II: [MATH], [MATH], [MATH]', '1008.3198-2-39-0': 'As we have discussed, model [REF] and model [REF] are equivalent.', '1008.3198-2-39-1': 'Thus it is enough to study them in the relatively simpler form, namely in the form [REF].', '1008.3198-2-39-2': 'For this model, we calculated the number of [MATH]-foldings and the integral constant along the inflation trajectory [EQUATION]', '1008.3198-2-39-3': 'Parallel to section [REF], we also calculated the slow-roll parameters in this model, [EQUATION]', '1008.3198-2-39-4': 'Subsequently, after obtaining the equations [EQUATION] and [EQUATION] we find by a little computation [EQUATION]', '1008.3198-2-39-5': 'Here notation [MATH] is different from the one in the previous section, [EQUATION]', '1008.3198-2-39-6': 'In terms of [EQUATION] and the relation [MATH], once again straightforward calculation gives [EQUATION]', '1008.3198-2-39-7': 'Therefore, the non-linear parameter in this model is [EQUATION]', '1008.3198-2-39-8': 'Similar to the previous section, we can estimate [MATH] by assuming [MATH] and [MATH].', '1008.3198-2-39-9': 'Under these assumptions, the only possibility to generate a large non-linear parameter is in the limit [MATH].', '1008.3198-2-39-10': 'Unfortunately, careful analysis ruled out this possibility.', '1008.3198-2-39-11': 'Because the assumption [MATH] implies [MATH], we find the non-linear parameter is not enhanced by [MATH] but is suppressed by the slow-roll parameters, [MATH].', '1008.3198-2-39-12': 'The same suppression applies if [MATH] lies in other regions.', '1008.3198-2-39-13': 'So we conclude that it is hopeless to generate large non-Gaussianities in this model unless one goes beyond the assumptions we made.', '1008.3198-2-39-14': 'A careful scan of parameter space will be done in subsection [REF] for a specific model.', '1008.3198-2-40-0': '# Examples', '1008.3198-2-41-0': 'In sections above, we have generalized the method of [CITATION] and applied it to a larger class of models.', '1008.3198-2-41-1': 'These models are summarized by equations [REF] and [REF], whose non-linear parameters are given by [REF] and [REF] generally.', '1008.3198-2-41-2': 'To check our general formulae, we will reduce [REF] and [REF] to previously known limit in subsections [REF] and [REF].', '1008.3198-2-41-3': 'The reduced expressions are consistent with the results of [CITATION].', '1008.3198-2-41-4': 'In subsections [REF], [REF] and [REF], we will apply our formulae to non-separable examples and scan the full parameter spaces.', '1008.3198-2-42-0': 'We should stress that all results in this paper are reliable only in the slow-roll region, that means at the least [MATH], [MATH], [MATH] with [MATH].', '1008.3198-2-42-1': 'A method free of slow-roll condition for some special models has been explored in reference [CITATION].', '1008.3198-2-43-0': '## Additive Potential: [MATH], [MATH], [MATH]', '1008.3198-2-44-0': 'This potential is obtained from [REF] by setting [MATH].', '1008.3198-2-44-1': 'The condition [MATH] is necessary to guarantee [REF].', '1008.3198-2-44-2': 'After taking [MATH], the result in section [REF] matches with that in [CITATION] obviously.', '1008.3198-2-45-0': '## Multiplicative Potential: [MATH], [MATH]', '1008.3198-2-46-0': 'Like equation [REF], we leave [MATH] as an arbitrary function of [MATH], as long as the slow-roll parameters [REF] are small.', '1008.3198-2-46-1': 'This is a special limit of section [REF].', '1008.3198-2-47-0': 'Using relations [EQUATION] we get the reduced form of non-linear parameter [EQUATION] where we have made use of the fact that [MATH] as well as the following notations [EQUATION]', '1008.3198-2-47-1': 'One may compare this formula with [CITATION].', '1008.3198-2-47-2': 'Note that their definitions of [MATH], [MATH] and [MATH] are slightly different from ours by some factors.', '1008.3198-2-47-3': 'Taking these factors into account, the result here is in accordance with [CITATION].', '1008.3198-2-48-0': '## Non-separable Potential I: [MATH], [MATH]', '1008.3198-2-49-0': 'We spend an independent subsection on this model not because of its non-Gaussianity, but because it has an elegant relation between the [MATH]-folding number and the angle variable of fields.', '1008.3198-2-49-1': 'For this model, the number of [MATH]-foldings from time [MATH] during the inflation stage to the end of inflation is [EQUATION]', '1008.3198-2-49-2': 'Note that [MATH] can be regarded as sum of squares.', '1008.3198-2-49-3': 'Its time derivative gives the Hubble parameter [MATH].', '1008.3198-2-49-4': 'So we can follow the standard treatment to parameterize the scalars in polar coordinates [EQUATION]', '1008.3198-2-49-5': 'Rewriting the equations of motion [REF] in terms of the polar coordinates, we obtain a differential relation between [MATH] and [MATH] for the present model, [EQUATION] with [MATH].', '1008.3198-2-49-6': 'It can be solved out to give [EQUATION]', '1008.3198-2-49-7': 'At the end of inflation, if the scalars arrive at the bottom of potential, one may simply set [MATH].', '1008.3198-2-50-0': "Relation [REF] is a trivial but useful generalization of Polarski and Starobinsky's relation [CITATION].", '1008.3198-2-50-1': "Recall that Polarski and Starobinsky's relation has been widely used for the inflation model with two massive scalar fields, which corresponds to exponent [MATH] in the model of this subsection.", '1008.3198-2-50-2': 'The simple demonstration above generalized the relation to arbitrary [MATH].', '1008.3198-2-51-0': 'As an application, we evaluate [REF] on the initial flat hypersurface [MATH] and then on the final comoving hypersurface [MATH], getting the ratio [EQUATION] which reduces to [EQUATION]', '1008.3198-2-51-1': 'This result can be also achieved from [REF] directly.', '1008.3198-2-52-0': '## Non-separable Potential II: [MATH], [MATH]', '1008.3198-2-53-0': 'Our purpose in this and the next subsections is to examine non-Gaussianities by parameter scanning.', '1008.3198-2-53-1': 'Two common assumptions will be used: the [MATH]-folding number is fixed to be [MATH] and the inflation is supposed to conclude at the point [MATH].', '1008.3198-2-54-0': 'Using the latter assumption and the general formulae in section [REF], we find all of the relevant quantities can be expressed by [MATH], [MATH] and [MATH]: [EQUATION]', '1008.3198-2-54-1': 'Here we defined [MATH] like the previous subsection.', '1008.3198-2-54-2': 'If [MATH], it can be proved that [MATH].', '1008.3198-2-54-3': 'Without loss of generality, we will consider the parameter region [MATH].', '1008.3198-2-54-4': 'As has been mentioned, from [REF] or [REF], one can get relation [REF].', '1008.3198-2-54-5': 'This relation is equivalent to [EQUATION]', '1008.3198-2-54-6': 'If [MATH], it gives [MATH] and thus [MATH].', '1008.3198-2-55-0': 'In the above expressions, there are five parameters: [MATH], [MATH], [MATH], [MATH] and [MATH].', '1008.3198-2-55-1': 'The number can be reduced by the assumptions we made at the beginning of this section.', '1008.3198-2-55-2': 'Firstly, [MATH] and [MATH] can be traded to each other with the relation [MATH].', '1008.3198-2-55-3': 'Secondly, since we have assumed [MATH], equations [REF] and [REF] can be used to eliminate two degrees of freedom further.', '1008.3198-2-55-4': 'Now we see only two parameters are independent, and we choose them to be [MATH] and [MATH] in the analysis below.', '1008.3198-2-55-5': 'The number counting in this way agrees with the fact that [REF] is a first order system under the slow-roll approximation.', '1008.3198-2-56-0': 'As a useful trick, we introduce a dimensionless notation [MATH], then equations [REF] and [REF] can be reformed as [MATH] and [EQUATION]', '1008.3198-2-56-1': 'Usually the second equation has no analytical expression for the root [MATH], but one may still find the root numerically.', '1008.3198-2-56-2': 'In the region [MATH], both [MATH] and [MATH] increase monotonically from zero to infinity, so this equation with respect to [MATH] has exactly one positive real root if the right hand side is finite.', '1008.3198-2-56-3': 'In terms of [MATH], [MATH] and [MATH], this equation is of the form [EQUATION]', '1008.3198-2-56-4': 'Fixing [MATH], the recipe of our numerical simulation is as follows:', '1008.3198-2-57-0': 'Given the values of [MATH] and [MATH] in parameter space [MATH], [MATH], numerically find the root [MATH] of equation [REF], where [MATH].', '1008.3198-2-57-1': 'Compute [MATH], [MATH], [MATH] and [MATH] according to [EQUATION] and equations [REF].', '1008.3198-2-57-2': 'Evaluate [MATH] with the formula [EQUATION]', '1008.3198-2-57-3': 'Repeat the above steps to scan the entire parameter space of [MATH] and [MATH].', '1008.3198-2-57-4': 'Due to the violation of slow-roll condition, the vicinity of [MATH] should be skipped to avoid numerical singularities (see spikes in figure [REF]).', '1008.3198-2-58-0': 'In a practical simulation, we scan the region [MATH], [MATH] on a uniform grid with [MATH] points.', '1008.3198-2-58-1': 'Some simulation results are illustrated in figure [REF].', '1008.3198-2-58-2': 'When drawing the figure, we have imposed the slow-roll condition [MATH], [MATH], [MATH], [MATH].', '1008.3198-2-58-3': 'In the limit [MATH], they are in agreement with the analytical results [MATH], [MATH].', '1008.3198-2-58-4': 'One may also check the results in other limits analytically, such as [MATH] or [MATH].', '1008.3198-2-58-5': 'Theoretically, [MATH] should correspond to an inflation model driven by one field [MATH].', '1008.3198-2-58-6': 'But our method does not apply to that limit, because it would violate the slow-roll condition for [MATH].', '1008.3198-2-59-0': 'From figure [REF],we can see the non-linear parameter [MATH] is suppressed by slow-roll parameters.', '1008.3198-2-59-1': 'Especially, in the neighborhood of [MATH], the spikes of [MATH] are located at the same positions as the spikes of [MATH].', '1008.3198-2-59-2': 'Such a coincidence continues to exist even if one relaxes the slow-roll condition.', '1008.3198-2-59-3': 'But there is no spike in similar graphs for [MATH], [MATH] and [MATH].', '1008.3198-2-59-4': 'Actually, these spikes are mainly attributed to the enhancement of [MATH] and [MATH] by [MATH] in the small [MATH] limit.', '1008.3198-2-59-5': 'After the parameter scanning and the numerical simulation, our lesson is that this model cannot generate a large non-Gaussianity unless the slow-roll condition breaks down.', '1008.3198-2-60-0': '## Non-separable Potential III: [MATH], [MATH]', '1008.3198-2-61-0': 'This is a special model of [REF] with [MATH], [MATH], [MATH].', '1008.3198-2-61-1': 'As in the previous subsection, we assume [MATH] and [MATH].', '1008.3198-2-61-2': 'Then from section [REF] we get the relations [EQUATION]', '1008.3198-2-61-3': 'For the present model, equation [REF] gives [EQUATION] that is [EQUATION]', '1008.3198-2-61-4': 'If we introduce the notations [MATH], then combining it with equation [REF] and the condition [MATH], we can express [MATH], [MATH] and [MATH] in terms of [MATH], [MATH] and [MATH], [EQUATION]', '1008.3198-2-61-5': 'On the basis of equation [REF], we deduce that [MATH] should be positive and suppressed by slow-roll parameters.', '1008.3198-2-61-6': 'In particular, [EQUATION]', '1008.3198-2-61-7': 'Thus we focus on the region [MATH].', '1008.3198-2-62-0': 'As indicated by the above analysis, if we are interested only in the non-linear parameter and slow-roll parameters, this model has two free parameters after using our assumptions and equations of motion.', '1008.3198-2-62-1': 'They will be chosen as [MATH] and [MATH] in our simulation, just like in the previous subsection.', '1008.3198-2-62-2': 'But we should warn that, compared with the previous subsection, the notation [MATH] has a distinct meaning in the current subsection.', '1008.3198-2-63-0': 'The parameter scanning is illustrated by figures [REF] and [REF].', '1008.3198-2-63-1': 'In figure [REF], parameter [MATH] decreases exponentially from 1 to [MATH].', '1008.3198-2-63-2': 'In this process, the non-linear parameter grows roughly proportional to [MATH] while the slow-roll condition [MATH] is violated gradually.', '1008.3198-2-63-3': 'This phenomenon agrees with equations [REF] and [REF], both of whose amplitude are enhanced by the factor [MATH] when [MATH] is small.', '1008.3198-2-63-4': 'In figure [REF],we find a sharp spike for the non-linear parameter in the corner [MATH], [MATH].', '1008.3198-2-63-5': 'Figure [REF] is drawn to zoom in this corner, with [MATH] scaled linearly.', '1008.3198-2-63-6': 'As shown by this figure, the spike dwells in a position violating the slow-roll condition [MATH].', '1008.3198-2-63-7': 'Therefore, the non-linear parameter in this model must be small once the slow-roll condition [MATH], [MATH], [MATH]) is imposed.', '1008.3198-2-64-0': '# Summary', '1008.3198-2-65-0': 'In this paper, we investigated a class of two-field slow-roll inflation models whose non-linear parameter is analytically calculable.', '1008.3198-2-66-0': 'In our convention of notations, we collected some well-known but necessary knowledge in section [REF].', '1008.3198-2-66-1': 'Slightly generalizing the method of [CITATION], we showed in section [REF] how their method could be utilized in a larger class of models satisfying two ansatzes, namely [REF] and [REF].', '1008.3198-2-66-2': 'In subsections [REF] and [REF] we proposed models meeting these ansatzes.', '1008.3198-2-66-3': 'We put our models in the form of [MATH] with [MATH] in subsection [REF] and with [MATH] in subsection [REF].', '1008.3198-2-66-4': 'At first glance, these are two different classes of models.', '1008.3198-2-66-5': 'But in fact they are two dual forms of the same class of models, just as proved in subsection [REF].', '1008.3198-2-66-6': 'In a succinct form, our models can be summarized by equations [REF] and [REF], whose non-linear parameters were worked out in sections [REF] and [REF] respectively, see equations [REF] and [REF].', '1008.3198-2-66-7': 'Under simplistic assumptions, we found no large non-Gaussianity in these models.', '1008.3198-2-67-0': 'As a double check, we reduced the expression [REF] for non-linear parameter to the additive potential in subsection [REF], and [REF] to multiplicative potential in subsection [REF].', '1008.3198-2-67-1': 'The resulting non-linear parameters match with [CITATION], confirming our calculations.', '1008.3198-2-67-2': "In subsection [REF], for a special class of models, we generalized Polarski and Starobinsky's relation [REF].", '1008.3198-2-67-3': 'For more specific models, we scanned the parameter space to evaluate the non-linear parameter, as shown by figures in subsections [REF] and [REF].', '1008.3198-2-67-4': 'In the scanning, we assumed the [MATH]-folding number [MATH] and the inflation terminates at [MATH].', '1008.3198-2-67-5': 'For the models we studied in subsections [REF] and [REF], the non-linear parameter [MATH] always takes a small positive value under the slow-roll approximation.', '1008.3198-2-68-0': 'The author would like to thank Christian T. Byrnes for private communications and helpful comments .'}

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[]

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[]

['1008.3198-2-29-0', '1008.3198-2-56-4', '1008.3198-2-61-6']

{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1008.3198

cond-mat-0105145

{'cond-mat-0105145-1-0-0': 'The spin-dependent mobility for the lateral transport of the hole gas in a GaMnAs/GaAs heterostructure containing several metallic-like ferromagnetic layers is calculated.', 'cond-mat-0105145-1-0-1': 'The electronic structure is obtained self-consistently taking into account the direct Coulomb Hartree and exchange-correlation terms, besides the sp-d exchange interaction with the Mn magnetic moments.', 'cond-mat-0105145-1-1-0': 'Recent advances on growing [MATH] multilayers opened a wide range of interest in heterostructures of that Diluted Magnetic Semiconductor (DMS) [CITATION], specially in spin-polarized transport due to potential applications in the area of quantum computers.', 'cond-mat-0105145-1-1-1': 'Besides a magnetic impurity, [MATH] in this alloy is a strong [MATH] dopant.', 'cond-mat-0105145-1-1-2': 'However, the density of carriers, as observed in measurements of the (anomalous) Hall resistance, is only a small fraction (of the order of 10%) of the [MATH] concentration, indicating that a major part of them remains neutral.', 'cond-mat-0105145-1-1-3': 'The sp-d exchange interaction of Ruderman-Kittel-Kasuya-Yosida (RKKY) type has been recognized as the main origin of the observed ferromagnetism in the metallic phase of III-V based DMS [CITATION], and the possible magnetic ordering occurring in [MATH] heterostructures have been studied via Monte Carlo simulations on metallic samples [CITATION] .', 'cond-mat-0105145-1-1-4': 'Recently, a two component model has been suggested to contempt to the iinterplay of neutral and ionized [MATH] atoms in the magnetic interactions [CITATION].', 'cond-mat-0105145-1-1-5': 'Chiba et al [CITATION] investigated a trilayer structure and observed a ferromagnetic interaction, although weak, between two [MATH] ferromagnetic layers.', 'cond-mat-0105145-1-1-6': 'More recently, ferromagnetic arrangements of [MATH] multilayers have been observed, forming a ferromagnetic superlattice [CITATION].', 'cond-mat-0105145-1-1-7': 'Carriers, according to their polarization, are attracted or repelled by the magnetic layer.', 'cond-mat-0105145-1-1-8': 'Therefore, the transport mean free path is expected to be different for each polarization.', 'cond-mat-0105145-1-1-9': 'The aim of this work is to obtain the spin-polarized charge density, and to understand its effect on the spin-polarized transport in a model structure consisting of a sequence of [MATH] layers grown inside a thick non-magnetic [MATH] layer.', 'cond-mat-0105145-1-2-0': 'The spin-polarized electronic structure for holes is obtained self-consistently in the reciprocal space [CITATION], taking into account the hole-hole interaction as well as the hole interaction with the magnetic impurities through the contact potential: [EQUATION] where [MATH] is the [MATH] exchange coupling constant, [MATH] denotes the positions of the [MATH] impurities [MATH] , [MATH] is the (classical) spin of the impurity, and [MATH] is the spin of the hole.', 'cond-mat-0105145-1-2-1': 'We assume the many layer magnetizations to be oriented in a single direction, each of them in their metallic and ferromagnetic phase.', 'cond-mat-0105145-1-2-2': 'Thus, the spin of the hole is well defined in that direction, being polarized either up (parallel) or down(anti-parallel).', 'cond-mat-0105145-1-2-3': 'When integrating the magnetic term in the Hamiltonian over [MATH], the magnetic impurities are assumed to be uniformly distributed in the [MATH] DMS layers, all of them in each magnetic layer [MATH] having the same magnetization, namely the thermal average magnetization [MATH].', 'cond-mat-0105145-1-2-4': 'Therefore, [EQUATION] where [MATH] for the hole spin orientation up (parallel) or down (anti-parallel) and [MATH] the two-dimensional [MATH]-function in the [MATH]-plane.', 'cond-mat-0105145-1-2-5': 'As usual, [MATH], where [MATH] is the volume of the Mn[MATH] ion.', 'cond-mat-0105145-1-2-6': '[MATH] is the integral performed on the [MATH]-coordinate across the [MATH]-th DMS layer: [EQUATION]', 'cond-mat-0105145-1-2-7': 'A net [MATH] magnetization [MATH] polarizes the hole gas by introducing additional effective confining potentials given by [EQUATION] where [MATH] if [MATH] lies inside the [MATH]-layer, and [MATH] otherwise.', 'cond-mat-0105145-1-2-8': 'With the use of [EQUATION] where [MATH] and [MATH] are, respectively, the confining and the hole-hole interaction potentials, one gets the one-dimensional secular equation in the [MATH]-direction: [EQUATION]', 'cond-mat-0105145-1-2-9': 'Each DMS layer works effectively as a barrier or a well for spins parallel or anti-parallel to the local average magnetization, depending on the sign of [MATH].', 'cond-mat-0105145-1-2-10': 'For [MATH], and for [MATH] [CITATION] this corresponds to band offsets of [MATH].', 'cond-mat-0105145-1-2-11': 'Throughout the calculation we made the hole density [MATH], a fraction of roughly [MATH]% of the [MATH] concentration, T=0[MATH], and [MATH].', 'cond-mat-0105145-1-2-12': 'Due to the high carrier density, several subbands are occupied.', 'cond-mat-0105145-1-3-0': 'The spin-polarized charge density distributions are shown in Fig. [REF] for (a) one, (b) five, (c) eleven, and (d) seventeen DMS layers.', 'cond-mat-0105145-1-3-1': 'We considered symmetric structures containing a number of [MATH] layers of width [MATH]AA, separated [MATH]by [MATH] width [MATH] layers, inside a [MATH] QW of width [MATH]AA.', 'cond-mat-0105145-1-3-2': 'A single [MATH] layer occupying partially a [MATH] quantum well can be ferromagnetic for widths smaller than the case where the DMS occupy completely the quantum well, as it has been shown by Monte Carlo simulations [CITATION].', 'cond-mat-0105145-1-3-3': 'However, a single layer of [MATH] is probably much too thin to be ferromagnetic.', 'cond-mat-0105145-1-3-4': 'As the number of layers increases, the spin-polarized charge density distribution approaches that of a semiconductor superlattice with a band offset of [MATH].', 'cond-mat-0105145-1-3-5': 'Holes with the spin polarized parallel to the average magnetization occupy the interstitial regions between the magnetic layers, being repelled by the [MATH] layers due to the negative value of [MATH] .', 'cond-mat-0105145-1-3-6': 'Holes polarized anti-parallel to the average magnetization, on the ccontrary, are mostly located inside the DMS layers.', 'cond-mat-0105145-1-3-7': 'Their density, however, is much lower than that of the anti-parallel spin holes, because the lowest occupied sub-bands are for this spin orientation.', 'cond-mat-0105145-1-3-8': 'We expect the concentration of charge in the magnetic region to favor the ferromagnetic order of the system as compared to the single-layered structure, a fact that may be related with the recent experiments of Sadowski et al. [CITATION]', 'cond-mat-0105145-1-4-0': 'Spin motion for carriers polarized parallel and anti-parallel to the equilibrium polarization in otherwise non-magnetic GaAs samples has been predicted [CITATION] to show a difference of an order of magnitude between the two speeds.', 'cond-mat-0105145-1-4-1': 'The hole gas in these structures, when the spin-flip scattering is neglected, can be modeled by a two-carrier system.', 'cond-mat-0105145-1-4-2': 'The corresponding Hamiltonian, with an electric field [MATH] applied in a direction parallel to the interfaces, is separated into center of mass (CM) and relative motions, in this case for each spin sub-system, each one with the instantaneous position of its CM, [MATH], and its drift velocity expressed in terms of the spin-dependent mobility [MATH], [MATH].', 'cond-mat-0105145-1-4-3': 'The bare inter (and intra) sub-band impurity scattering potential [MATH] is: [EQUATION] where [MATH] is the hole wavefunction at the bottom of the sub-band [MATH], which corresponds to a specific spin polarization.', 'cond-mat-0105145-1-4-4': 'The frictional force due to the scattering by impurities balances the net effect of the applied electric field on the CM [CITATION]: [EQUATION]', 'cond-mat-0105145-1-4-5': 'Here, the screening is included through the imaginary part of the polarization function [MATH], with the frequency argument [MATH], which can be approximated, in this simple case, by the unperturbed polarization [CITATION].', 'cond-mat-0105145-1-5-0': 'Fig. ([REF]) shows the spin polarized mobilities for a single [MATH] layer in a [MATH] QW of width [MATH] as a function of the DMS layer width.', 'cond-mat-0105145-1-5-1': 'As the width increases, corresponding to the increase of the effective magnetic well (barrier) width, holes with spins parallel (anti-parallel) are repelled from (attracted to) the region where the scatterers are located.', 'cond-mat-0105145-1-5-2': 'An increasing number of occupied sub-bands corresponds to the anti-parallel polarization, reducing the parallel spin hole charge as compared to the anti-parallel one.', 'cond-mat-0105145-1-5-3': 'It follows a decrease of the parallel polarization mobility, which is much more pronounced than that of the anti-parallel one.', 'cond-mat-0105145-1-5-4': 'The mobility for a paramagnetic sample (at T=100K) is also included.', 'cond-mat-0105145-1-5-5': 'It decreases as a function of the layer width almost in the same way as for the majority spins.', 'cond-mat-0105145-1-5-6': 'Fig. ([REF]) shows the mobilities for a series of structures in which a variable number of [MATH] layers of width [MATH]AA, separated by [MATH] width [MATH] layers, are symmetrically placed inside a [MATH] QW, as in the calculation of the electronic structure of Fig. ([REF]).', 'cond-mat-0105145-1-5-7': 'We observe an oscillation of the mobilities for small number of layers, but the general trend is that above a certain number of layers the mobility of the anti-parallel polarization increase monotonically, while that of the parallel polarization decreases, reaching a plateau, the former being a factor of eight higher than the latter.', 'cond-mat-0105145-1-5-8': 'Note that the paramagnetic mobility reaches a plateau just above a few layers.', 'cond-mat-0105145-1-6-0': 'We obtained a strongly spin-dependent mobility, with drift velocities differing by almost one order of magnitude, in agreement with other results [CITATION].', 'cond-mat-0105145-1-6-1': 'This has an additional interest in what concerns the hole gas properties under an external magnetic field.', 'cond-mat-0105145-1-6-2': 'On the other hand, from the theoretical point of view of non-linear transport, one could expect that a two-gas model presenting such a difference in the drift velocities, as shown in this calculation, will generate higher order processes like a mutual Coulomb drag, and other non-equilibrium effects in the two spin-polarized electron gases.', 'cond-mat-0105145-1-7-0': 'This work was partially supported by CNPq, CAPES, FAPERJ and CENAPAD/UNICAMP-FINEP in Brazil.', 'cond-mat-0105145-1-8-0': 'matsu F. Matsukura, H. Ohno, A. Shen, and Y. Sugawara, Phys.', 'cond-mat-0105145-1-9-0': 'utiv U. Gummich and I.C. da Cunha Lima, Solid State Commun.', 'cond-mat-0105145-1-10-0': 'boselli M.A. Boselli, A. Ghazali, and I.C. da Cunha Lima, Phys.', 'cond-mat-0105145-1-11-0': 'chiba D. Chiba, N. Akiba, F. Matsukura, Y. Ohno, and H. Ohno, Appl.', 'cond-mat-0105145-1-12-0': 'upsalla S. Sadowski, K. Switek, R. Mathieu, P. Svedlindh, M. Karlsteen, J. Kansik, and L. Ilver (unpublished).', 'cond-mat-0105145-1-13-0': 'ghaz A. Ghazali, I.C. da Cunha Lima, and M.A. Boselli, Phys.', 'cond-mat-0105145-1-14-0': 'okaba J. Okabayashi, A. Kimura, O. Rader, T. Mizokawa, A. Fujimori, T. Hayashi, and M. Tanaka, Phys.', 'cond-mat-0105145-1-15-0': 'jap M.A. Boselli, I.C. da Cunha Lima, and A. Ghazali, J. App.'}

{'cond-mat-0105145-2-0-0': 'The spin-dependent mobility for the lateral transport of the hole gas in a GaMnAs/GaAs heterostructure containing several metallic-like ferromagnetic layers is calculated.', 'cond-mat-0105145-2-0-1': 'The electronic structure is obtained self-consistently taking into account the direct Coulomb Hartree and exchange-correlation terms, besides the sp-d exchange interaction with the Mn magnetic moments.', 'cond-mat-0105145-2-1-0': 'Recent advances on growing [MATH] multilayers opened a wide range of interest in heterostructures of that Diluted Magnetic Semiconductor (DMS) [CITATION], specially in spin-polarized transport due to potential applications in the area of quantum computers.', 'cond-mat-0105145-2-1-1': 'Besides a magnetic impurity, [MATH] in this alloy is a strong [MATH] dopant.', 'cond-mat-0105145-2-1-2': 'However, the density of carriers, as observed in measurements of the (anomalous) Hall resistance, is only a small fraction (of the order of 10%) of the [MATH] concentration, indicating that a major part of them remains neutral.', 'cond-mat-0105145-2-1-3': 'The sp-d exchange interaction of Ruderman-Kittel-Kasuya-Yosida (RKKY) type has been recognized as the main origin of the observed ferromagnetism in the metallic phase of III-V based DMS [CITATION], and the possible magnetic ordering occurring in [MATH] heterostructures have been studied via Monte Carlo simulations on metallic samples [CITATION] .', 'cond-mat-0105145-2-1-4': 'Recently, a two component model has been suggested to contempt to the iinterplay of neutral and ionized [MATH] atoms in the magnetic interactions [CITATION].', 'cond-mat-0105145-2-1-5': 'Chiba et al [CITATION] investigated a trilayer structure and observed a ferromagnetic interaction, although weak, between two [MATH] ferromagnetic layers.', 'cond-mat-0105145-2-1-6': 'More recently, ferromagnetic arrangements of [MATH] multilayers have been observed, forming a ferromagnetic superlattice [CITATION].', 'cond-mat-0105145-2-1-7': 'Carriers, according to their polarization, are attracted or repelled by the magnetic layer.', 'cond-mat-0105145-2-1-8': 'Therefore, the transport mean free path is expected to be different for each polarization.', 'cond-mat-0105145-2-1-9': 'The aim of this work is to obtain the spin-polarized charge density, and to understand its effect on the spin-polarized transport in a model structure consisting of a sequence of [MATH] layers grown inside a thick non-magnetic [MATH] layer.', 'cond-mat-0105145-2-2-0': 'The spin-polarized electronic structure for holes is obtained self-consistently in the reciprocal space [CITATION], taking into account the hole-hole interaction as well as the hole interaction with the magnetic impurities through the contact potential: [EQUATION] where [MATH] is the [MATH] exchange coupling constant, [MATH] denotes the positions of the [MATH] impurities [MATH] , [MATH] is the (classical) spin of the impurity, and [MATH] is the spin of the hole.', 'cond-mat-0105145-2-2-1': 'We assume the many layer magnetizations to be oriented in a single direction, each of them in their metallic and ferromagnetic phase.', 'cond-mat-0105145-2-2-2': 'Thus, the spin of the hole is well defined in that direction, being polarized either up (parallel) or down(anti-parallel).', 'cond-mat-0105145-2-2-3': 'When integrating the magnetic term in the Hamiltonian over [MATH], the magnetic impurities are assumed to be uniformly distributed in the [MATH] DMS layers, all of them in each magnetic layer [MATH] having the same magnetization, namely the thermal average magnetization [MATH].', 'cond-mat-0105145-2-2-4': 'Therefore, [EQUATION] where [MATH] for the hole spin orientation up (parallel) or down (anti-parallel) and [MATH] the two-dimensional [MATH]-function in the [MATH]-plane.', 'cond-mat-0105145-2-2-5': 'As usual, [MATH], where [MATH] is the volume of the Mn[MATH] ion.', 'cond-mat-0105145-2-2-6': '[MATH] is the integral performed on the [MATH]-coordinate across the [MATH]-th DMS layer: [EQUATION]', 'cond-mat-0105145-2-2-7': 'A net [MATH] magnetization [MATH] polarizes the hole gas by introducing additional effective confining potentials given by [EQUATION] where [MATH] if [MATH] lies inside the [MATH]-layer, and [MATH] otherwise.', 'cond-mat-0105145-2-2-8': 'With the use of [EQUATION] where [MATH] and [MATH] are, respectively, the confining and the hole-hole interaction potentials, one gets the one-dimensional secular equation in the [MATH]-direction: [EQUATION]', 'cond-mat-0105145-2-2-9': 'Each DMS layer works effectively as a barrier or a well for spins parallel or anti-parallel to the local average magnetization, depending on the sign of [MATH].', 'cond-mat-0105145-2-2-10': 'For [MATH], and for [MATH] [CITATION] this corresponds to band offsets of [MATH].', 'cond-mat-0105145-2-2-11': 'Throughout the calculation we made the hole density [MATH], a fraction of roughly [MATH]% of the [MATH] concentration, T=0[MATH], and [MATH].', 'cond-mat-0105145-2-2-12': 'Due to the high carrier density, several subbands are occupied.', 'cond-mat-0105145-2-3-0': 'The spin-polarized charge density distributions are shown in Fig. [REF] for (a) one, (b) five, (c) eleven, and (d) seventeen DMS layers.', 'cond-mat-0105145-2-3-1': 'We considered symmetric structures containing a number of [MATH] layers of width [MATH]AA, separated [MATH]by [MATH] width [MATH] layers, inside a [MATH] QW of width [MATH]AA.', 'cond-mat-0105145-2-3-2': 'A single [MATH] layer occupying partially a [MATH] quantum well can be ferromagnetic for widths smaller than the case where the DMS occupy completely the quantum well, as it has been shown by Monte Carlo simulations [CITATION].', 'cond-mat-0105145-2-3-3': 'However, a single layer of [MATH] is probably much too thin to be ferromagnetic.', 'cond-mat-0105145-2-3-4': 'As the number of layers increases, the spin-polarized charge density distribution approaches that of a semiconductor superlattice with a band offset of [MATH].', 'cond-mat-0105145-2-3-5': 'Holes with the spin polarized parallel to the average magnetization occupy the interstitial regions between the magnetic layers, being repelled by the [MATH] layers due to the negative value of [MATH] .', 'cond-mat-0105145-2-3-6': 'Holes polarized anti-parallel to the average magnetization, on the ccontrary, are mostly located inside the DMS layers.', 'cond-mat-0105145-2-3-7': 'Their density, however, is much lower than that of the anti-parallel spin holes, because the lowest occupied sub-bands are for this spin orientation.', 'cond-mat-0105145-2-3-8': 'We expect the concentration of charge in the magnetic region to favor the ferromagnetic order of the system as compared to the single-layered structure, a fact that may be related with the recent experiments of Sadowski et al. [CITATION]', 'cond-mat-0105145-2-4-0': 'Spin motion for carriers polarized parallel and anti-parallel to the equilibrium polarization in otherwise non-magnetic GaAs samples has been predicted [CITATION] to show a difference of an order of magnitude between the two speeds.', 'cond-mat-0105145-2-4-1': 'The hole gas in these structures, when the spin-flip scattering is neglected, can be modeled by a two-carrier system.', 'cond-mat-0105145-2-4-2': 'The corresponding Hamiltonian, with an electric field [MATH] applied in a direction parallel to the interfaces, is separated into center of mass (CM) and relative motions, in this case for each spin sub-system, each one with the instantaneous position of its CM, [MATH], and its drift velocity expressed in terms of the spin-dependent mobility [MATH], [MATH].', 'cond-mat-0105145-2-4-3': 'The bare inter (and intra) sub-band impurity scattering potential [MATH] is: [EQUATION] where [MATH] is the hole wavefunction at the bottom of the sub-band [MATH], which corresponds to a specific spin polarization.', 'cond-mat-0105145-2-4-4': 'The frictional force due to the scattering by impurities balances the net effect of the applied electric field on the CM [CITATION]: [EQUATION]', 'cond-mat-0105145-2-4-5': 'Here, the screening is included through the imaginary part of the polarization function [MATH], with the frequency argument [MATH], which can be approximated, in this simple case, by the unperturbed polarization [CITATION].', 'cond-mat-0105145-2-5-0': 'Fig. ([REF]) shows the spin polarized mobilities for a single [MATH] layer in a [MATH] QW of width [MATH] as a function of the DMS layer width.', 'cond-mat-0105145-2-5-1': 'As the width increases, corresponding to the increase of the effective magnetic well (barrier) width, holes with spins parallel (anti-parallel) are repelled from (attracted to) the region where the scatterers are located.', 'cond-mat-0105145-2-5-2': 'An increasing number of occupied sub-bands corresponds to the anti-parallel polarization, reducing the parallel spin hole charge as compared to the anti-parallel one.', 'cond-mat-0105145-2-5-3': 'It follows a decrease of the parallel polarization mobility, which is much more pronounced than that of the anti-parallel one.', 'cond-mat-0105145-2-5-4': 'The mobility for a paramagnetic sample (at T=100K) is also included.', 'cond-mat-0105145-2-5-5': 'It decreases as a function of the layer width almost in the same way as for the majority spins.', 'cond-mat-0105145-2-5-6': 'Fig. ([REF]) shows the mobilities for a series of structures in which a variable number of [MATH] layers of width [MATH]AA, separated by [MATH] width [MATH] layers, are symmetrically placed inside a [MATH] QW, as in the calculation of the electronic structure of Fig. ([REF]).', 'cond-mat-0105145-2-5-7': 'We observe an oscillation of the mobilities for small number of layers, but the general trend is that above a certain number of layers the mobility of the anti-parallel polarization increase monotonically, while that of the parallel polarization decreases, reaching a plateau, the former being a factor of eight higher than the latter.', 'cond-mat-0105145-2-5-8': 'Note that the paramagnetic mobility reaches a plateau just above a few layers.', 'cond-mat-0105145-2-6-0': 'We obtained a strongly spin-dependent mobility, with drift velocities differing by almost one order of magnitude, in agreement with other results [CITATION].', 'cond-mat-0105145-2-6-1': 'This has an additional interest in what concerns the hole gas properties under an external magnetic field.', 'cond-mat-0105145-2-6-2': 'On the other hand, from the theoretical point of view of non-linear transport, one could expect that a two-gas model presenting such a difference in the drift velocities, as shown in this calculation, will generate higher order processes like a mutual Coulomb drag, and other non-equilibrium effects in the two spin-polarized electron gases.', 'cond-mat-0105145-2-7-0': 'This work was partially supported by CNPq, CAPES, FAPERJ and CENAPAD/UNICAMP-FINEP in Brazil.', 'cond-mat-0105145-2-8-0': 'matsu F. Matsukura, H. Ohno, A. Shen, and Y. Sugawara, Phys.', 'cond-mat-0105145-2-9-0': 'utiv U. Gummich and I.C. da Cunha Lima, Solid State Commun.', 'cond-mat-0105145-2-10-0': 'boselli M.A. Boselli, A. Ghazali, and I.C. da Cunha Lima, Phys.', 'cond-mat-0105145-2-11-0': 'chiba D. Chiba, N. Akiba, F. Matsukura, Y. Ohno, and H. Ohno, Appl.', 'cond-mat-0105145-2-12-0': 'upsalla J. Sadowski, K. Switek, R. Mathieu, P. Svedlindh, M. Karlsteen, J. Kansik, and L. Ilver (unpublished).', 'cond-mat-0105145-2-13-0': 'ghaz A. Ghazali, I.C. da Cunha Lima, and M.A. Boselli, Phys.', 'cond-mat-0105145-2-14-0': 'okaba J. Okabayashi, A. Kimura, O. Rader, T. Mizokawa, A. Fujimori, T. Hayashi, and M. Tanaka, Phys.', 'cond-mat-0105145-2-15-0': 'jap M.A. Boselli, I.C. da Cunha Lima, and A. Ghazali, J. App.'}

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[['cond-mat-0105145-1-12-0', 'cond-mat-0105145-2-12-0']]

[]

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{'1': 'http://arxiv.org/licenses/assumed-1991-2003/', '2': 'http://arxiv.org/licenses/assumed-1991-2003/'}

https://arxiv.org/abs/cond-mat/0105145

1309.6215

{'1309.6215-1-0-0': 'A design study of a horizontal neutron reflectometer adapted to the general baseline of the long pulse European Spallation Source (ESS) is presented.', '1309.6215-1-0-1': 'The instrument layout comprises solutions for the neutron guide, high-resolution pulse shaping and beam bending onto a sample surface being so far unique in the field of reflectometry.', '1309.6215-1-0-2': 'The length of this instrument is roughly 55 m, enabling [MATH] resolutions from 0.5% to 10%.', '1309.6215-1-0-3': 'The incident beam is focussed in horizontal plane to boost measurements of sample sizes of [MATH] cm[MATH] and smaller with potential beam deflection in both downward and upward direction.', '1309.6215-1-0-4': 'The range of neutron wavelengths untilized by the instrument is 2 to 7.1 (12.2, ...) [MATH] , if every (second, ...) neutron source pulse is used.', '1309.6215-1-0-5': 'Angles of incidence can be set between 0[MATH] and 9[MATH] with a total accessible q-range from [MATH] up to 1 [MATH].', '1309.6215-1-0-6': 'The instrument operates both in [MATH] (free liquid surfaces) and [MATH] (solid/liquid, air/solid interfaces) geometry.', '1309.6215-1-0-7': 'The experimental setup will in particular enable direct studies on ultrathin films (d [MATH]) and buried monolayers to multilayered structures of up to 3000 [MATH] total thickness.', '1309.6215-1-0-8': 'The horizontal reflectometer will further foster investigations of hierarchical systems from nm to micrometer length scale (the latter by offspecular scattering), as well as their kinetics and dynamical properties, in particular under load (shear, pressure, external fields).', '1309.6215-1-0-9': 'Polarization and polarization analysis as well as the GISANS option are designed as potential modules to be implemented separately in the generic instrument layout.', '1309.6215-1-0-10': 'The instrument is highly flexible layout and offers a variety of different measurement modes.', '1309.6215-1-0-11': 'With respect to its mechanical components the instrument is exclusively based on current technology.', '1309.6215-1-0-12': 'Risks of failure for the chosen setup are minimum.', '1309.6215-1-1-0': '# Introduction and science case', '1309.6215-1-2-0': 'Soft matter and life science systems investigated by neutron reflectometry (NR) continuously increase in complexity - both in structure as well as in the number and the specific roles of their components.', '1309.6215-1-2-1': 'The same is true for hard matter systems.', '1309.6215-1-2-2': 'This development presents a permanent challenge and demand for continuous improvement of the neutron facilities.', '1309.6215-1-2-3': 'On the other hand there is a constantly increasing request for NR measurements, in particular in soft matter and life sciences, which stems from the unique possibilities of neutrons in probing amphiphilic and self-organizing structures at air-liquid [CITATION], liquid-liquid [CITATION] and solid-liquid (buried) interfaces [CITATION].', '1309.6215-1-2-4': 'NR in combination with off-specular and grazing-incidence small angle neutron scattering (GISANS [CITATION], SESANS [CITATION], SERGIS [CITATION]) provides deep insight in self-assembly- and aggregation processes [CITATION], 1-3d ordered interfacial films [CITATION] and the interplay of length scales in hierarchically structured systems [CITATION].', '1309.6215-1-3-0': 'Current research at the forefront of science with NR covers investigations of protein organization in bio-membranes [CITATION], structure, organization and functioning of living cells at interfaces [CITATION], nano-engineering [CITATION], new materials and interface properties for energy (storage) applications [CITATION], magnetic fluctuations and domain propagation [CITATION] as well as magnetic ordering and the understanding of relevant length scales and energies [CITATION].', '1309.6215-1-4-0': 'Complementary (on-board) in-situ techniques in addition to NR are often needed to elucidate complex interfacial structures and processes that are difficult to assess otherwise.', '1309.6215-1-4-1': 'In particular X-ray reflectometry [CITATION], ellipsometry [CITATION], Brewster angle microscopy [CITATION] and IR spectroscopy [CITATION] are essential complementary tools for achieving integrated insight into soft matter and life science systems.', '1309.6215-1-4-2': 'Soft matter sample sizes required by NR today are typically of the order of 1000 mm[MATH] or larger which hinders studies with components that are not available or cannot be produced in sufficient quantities.', '1309.6215-1-4-3': 'The latter holds in particular, but not exclusively, for biological systems where typically human proteins provide such bottle-neck.', '1309.6215-1-4-4': 'Thus, parametric studies for instance on biological, genetic or pharmaceutical activity are outside the scope of NR measurements today.', '1309.6215-1-4-5': 'Such investigations will become available with the high flux of the ESS source by utilizing much reduced beam cross sections (foot prints) and sample sizes.', '1309.6215-1-5-0': 'Minimized beam dimensions (with not necessarily minimized sample size) will further allow for characterizations of rough and curved interfaces that are common in functional and natural materials like industrial coatings, ball bearings, bone or skin.', '1309.6215-1-5-1': 'In addition, inhomogeneous, patterned surfaces might be scanned for local features with sub-mm beams.', '1309.6215-1-5-2': 'Magnetic reference layers in combination with polarized neutron beams offer additional contrast and sensitivity for samples that do not withstand solution contrast changes [CITATION].', '1309.6215-1-5-3': 'In combination with polarization analysis such experimental set-up will help discriminating the incoherent background in soft matter and life science systems by spin-filtering [CITATION].', '1309.6215-1-6-0': 'The ESS source [CITATION] will enable kinetic studies of systems in non-equilibrium situations on time scales not available today.', '1309.6215-1-6-1': 'Tackling the millisecond range in the future however might still require concurrent sample sizes.', '1309.6215-1-6-2': 'Therefore, the reflectometer at ESS should foresee flexible and adaptable collimation optics.', '1309.6215-1-6-3': 'The latter is also important whenever interfaces are to be assessed from above and below the sample horizon as can be the case for buried interfaces and interfaces under load, e.g. shear forces in combined NR and rheology set-ups [CITATION].', '1309.6215-1-7-0': 'The length scale of interest in soft matter and life science systems of the next generation will cover 1-3000 [MATH], hence an instrument with variable resolution from [MATH] is required.', '1309.6215-1-7-1': 'For the majority of purposes (not including wide angle and diffraction experiments) a q-range of 0.005 [MATH] to 1 [MATH] will suffice.', '1309.6215-1-8-0': '# Layout of the horizontal reflectometer', '1309.6215-1-9-0': 'The scientific motivation described in the last section combined with additional requirements from ESS advisory panels were taken into account when choosing the basic instrument parameters.', '1309.6215-1-9-1': 'As a result, the key strength of this instrument was decided to be the ability to use small samples and to access large momentum transfer values [MATH] reaching up to [MATH] in both downward and upward direction for liquid samples.', '1309.6215-1-9-2': 'In order to match these requirements, the following basic parameters were selected:', '1309.6215-1-10-0': 'In the following, the layout of the instrument based on the key parameters listed above is described in more detail (see also Tab. [REF] for an overview).', '1309.6215-1-10-1': 'Individual beamline components are presented and their characteristics are justified based on results of neutronic MC simulations carried out with the VITESS software package [CITATION].', '1309.6215-1-11-0': '## Moderator and extraction system', '1309.6215-1-12-0': 'The reflectometer is set up for using wavelengths of 2 [MATH] and longer.', '1309.6215-1-12-1': 'Based on the current moderator description provided by the ESS and implemented in the simulation packages like VITESS [CITATION] and McStas [CITATION], the cold moderator with 12 cm edge length, which provides a larger flux for wavelenghts longer than 2.5 [MATH], was selected for the reflectometer.', '1309.6215-1-12-2': 'Neutron optic elements can be placed in the shielding monolith, which is currently 6 m in radius, beginning at 2 m behind the moderator.', '1309.6215-1-12-3': 'Thus beamline components like mirrors or guides can be placed in the monolith to increase the number of neutrons fed into the main guide system outside the first 6 m.', '1309.6215-1-12-4': 'While the horizontal shape is elliptic (see above), three options were tested for the vertical shape of the extraction system, that is a constant height of 2 cm, and a tapered guide with the entry height being 8.67 cm and 12 cm, respectively.', '1309.6215-1-12-5': 'For the latter, the guide entry height matches the size of the moderator.', '1309.6215-1-12-6': 'Simulations reveal that the best beam extraction is provided by the tapered feeding guide with 8.67 cm entry height, though it is noted that the performance of the constant guide is only slightly worse.', '1309.6215-1-12-7': 'The conclusion is that a large part of addional phase space accepted by the tapered guide is not transported by the subsequent beamline to the sample position.', '1309.6215-1-13-0': '## Direct line of sight', '1309.6215-1-14-0': 'A reflectometer aiming for high-q measurements needs to be free of prompt pulse background.', '1309.6215-1-14-1': 'Taking into account the unprecedented intensity of currently 5 MW and the energy of the primary proton beam of [MATH] GeV it is unclear whether a T0-chopper is able to remove all of the fast particle and gamma-ray background.', '1309.6215-1-14-2': 'This is why a T0-chopper is not foreseen and the background from the prompt pulse is avoided by the geometry of the guide system.', '1309.6215-1-14-3': 'The latter has to be designed such that in the field of view of the detector, there are no guide elements that are in the direct flight path from the source.', '1309.6215-1-14-4': 'This is the definition of avoiding the line of sight twice.', '1309.6215-1-14-5': 'Since the horizontal shape of the guide is elliptic and a modification of this shape would interfere with its focusing properties, avoiding the line of sight must be carried out in the vertical plane.', '1309.6215-1-15-0': 'As a sophisticated chopper system will be placed within the first few meters after the shielding monolith, where also one chopper needs to be moved along the beamline, it was decided to leave the in-monolith section horizontally, i.e. not to have any inclination in the guide system there.', '1309.6215-1-15-1': 'On the other hand, the guide section 4, which is upstream from the bending section, should be horizontal, to enable the latter to equally access samples from above and below.', '1309.6215-1-15-2': 'Avoiding the line of sight requires a difference in height between the guide position at the monolith and at the bending section.', '1309.6215-1-15-3': 'Ways to achieve that are to place a double bender (s-bender) or a double kink (z-kink) in between.', '1309.6215-1-15-4': 'Here, a z-kink shows a clearly superiour performance, see Fig. [REF].', '1309.6215-1-15-5': 'The kink angle is fixed at [MATH], which leads to very moderate losses only if compared with a guide system without a kink.', '1309.6215-1-15-6': 'The shift of the maximum in the vertical divergence distribution towards 0.25[MATH] can be reduced by optimizing the vertical shape of the kink guide element.', '1309.6215-1-15-7': 'The latter shows the best performance if the shape is slightly double-elliptic.', '1309.6215-1-15-8': 'The total length of the kink guide section is 11.2 m and the shift in height of the guide position amounts to [MATH] cm.', '1309.6215-1-16-0': 'In accordance with the demand of avoiding the line of sight twice, the z-kink element should be placed such that fast primary particles are unable to illuminate the guide section after the z-kink, independent of whether their flight path would interfere with the detector area.', '1309.6215-1-16-1': 'A simple raytracing study was performed to determine an optimal position of the latter guide element by scanning the neutron parameter space at the source between [MATH] cm and [MATH], where [MATH] is the vertical coordinate of the neutron trajectory at the source and [MATH] is its vertical divergence.', '1309.6215-1-16-2': 'The study revealed that if the z-kink is placed at a distance of 18.7 m away from the source, all fast particles would propagate at least 6 m outside the guide before entering the last guide section.', '1309.6215-1-16-3': 'Since propagation outside the guide can for simplicity be considered as propagation in shielding, it can safely be argued that due to the guide geometry, all fast background will be sufficiently absorbed and the prompt pulse will not contribute to the background at the detector position.', '1309.6215-1-17-0': '## Beam delivery on sample', '1309.6215-1-18-0': 'As stated above, in order to access a momentum transfer [MATH] on a free liquid surface the beam must have an incident angle on a horizontal sample of [MATH], taking into account the shortest utilized wavelength of 2 [MATH].', '1309.6215-1-18-1': 'Short wavelength neutrons require several bounces off the beam bending walls in order to be inclined by such large angle [MATH], since there is a strong limitation on their maximum reflection angle, e.g. [MATH] for m=5 supermirror coatings.', '1309.6215-1-18-2': 'At the same time, the reflectometer is to provide sufficient flexibility concerning the variety of incident angles such that also regions of small [MATH] can be covered.', '1309.6215-1-18-3': 'Thus, the beam is tilted down- or upwards on a horizontal sample surface by means of 5 moveable deflecting elements, of which the top and the bottom surfaces are covered with m = 5 coating.', '1309.6215-1-18-4': 'The length of each element is [MATH] 1.28 m with the height of 2 cm.', '1309.6215-1-18-5': 'When tilted by [MATH] each element bends the full beam of 2 cm height by 1.8[MATH].', '1309.6215-1-18-6': 'The horizontal shape of the beam bending elements is elliptic, since these elements are still part of the main guide ellipse (see Fig. 1).', '1309.6215-1-18-7': 'The maximum bending angle [MATH] guaranties for a maximum [MATH].', '1309.6215-1-18-8': 'Bending guide elements can be replaced upon request by a GISANS guide module providing horizontal collimation.', '1309.6215-1-18-9': 'The beam characteristics after the bending section are shown in Figs. [REF] and [REF].', '1309.6215-1-18-10': 'In principle, it is also possible to have shorter (1.15 m) bending elements that can be tilted by 1[MATH], hence reaching the maximum reflection angle for a 2 [MATH] neutron.', '1309.6215-1-18-11': 'On the other hand, MC simulations show that in this case the intensity at wavelengths around 2 [MATH] is heavily supressed.', '1309.6215-1-19-0': 'The layout of the bending section is optimized for maximum flexibility and high-[MATH] while keeping the complexity of the system as low as possible.', '1309.6215-1-19-1': 'The five bending elements can be arranged to bend the neutron beam down- or upwards by a number of possible angles.', '1309.6215-1-19-2': 'Their work principle is that neutrons bounce off each element once and propagate towards the sample afterwards, see Fig. [REF].', '1309.6215-1-19-3': 'The drawback is that neutrons with a non-zero divergence have a certain chance of hitting the bottom (top) wall of one of the bending elements, if the beam is bent downwards (upwards).', '1309.6215-1-19-4': 'The resulting mis-orientation can be further increased during the propagation of the beam until the neutron is absorbed or leaves the bending section with a high divergence, i.e. does not propagate towards the sample.', '1309.6215-1-19-5': 'This is why the vertical divergence provided by the collimation system does not exceed 0.2[MATH] FWHM (0.3 [MATH] as maximum) for higher bending angles, even though the maximum divergence can in principle achieve maximum [MATH], taking into account the height of the guide of 2 cm and the collimation length of 2 m.', '1309.6215-1-19-6': 'It was investigated whether the divergence can be increased by involving more bending elements that would approximate a curved guide.', '1309.6215-1-19-7': 'This, however, would increase the mechanical complexity of the system, while MC simulations showed that there was no flux gain at the sample position.', '1309.6215-1-20-0': '## Collimation', '1309.6215-1-21-0': 'The collimation of the neutron beam exiting the bending section of the guide is done by means of two diaphragms (slits) located at the guide exit and 2 m downstream from the guide, respectively.', '1309.6215-1-21-1': 'The first slit determines the total vertical divergence on the sample, while the second, so-called footprint slit, is used to reduce the beam size such that it matches the sample dimensions or the requirements for the beam size, e.g., for position sensitive measurements.', '1309.6215-1-21-2': 'The adjusted divergence depends on the desired vertical angular resolution on the sample.', '1309.6215-1-21-3': 'For simplicity one usually uses matching resolution for divergence and wavelength, thus the opening (height) of the first slit depends on the pulse shaping regime used.', '1309.6215-1-21-4': 'The free propagation length of 2 m further allows to install, e.g., a polarisation device between the collimation slits.', '1309.6215-1-22-0': '## Pulse shaping', '1309.6215-1-23-0': 'The reflectometer operates with two chopper setups for basic and high-resolution, respectively.', '1309.6215-1-23-1': 'All choppers are located at the side of the guide to make use of the small beam height and thus reduce the opening/closing time.', '1309.6215-1-23-2': 'The basic setup requires three choppers, which define the main frame and prevent neutrons with other wavelengths from propagating to the detector.', '1309.6215-1-23-3': 'In the basic setup, the wavelength resolution is defined by the total pulse length [MATH] ms. Hende the first chopper does not need to be close to the source, as the pulse length will not be reduced.', '1309.6215-1-23-4': 'Furthermore, the first few meters after the shielding monolith are occupied by pulse shaping choppers belonging to the high-resolution setup (see below).', '1309.6215-1-23-5': 'Thus, the first chopper of the basic setup is placed at 12.5 m, followed by a second chopper at 15 m and a third chopper at 31 m.', '1309.6215-1-23-6': 'While the first two choppers are designed to suppress background from slow neutrons that otherwise could pollute the subsequent frame(s), the third chopper defines the accepted waveband being [MATH], if every source pulse is used.', '1309.6215-1-23-7': 'The waveband can be extended, if every second (third) pulse is used, see Fig. [REF].', '1309.6215-1-23-8': 'This can be achieved by reducing the chopper rotation frequency to [MATH], ...) of the original value.', '1309.6215-1-24-0': 'A dedicated wavelength frame multiplication system (WFM) was developed for the high-resolution work regime.', '1309.6215-1-24-1': 'Its purpose is to provide a high, constant wavelength resolution between [MATH] and [MATH] for the utilized waveband, while removing contaminant neutrons.', '1309.6215-1-24-2': 'This layout is very close to the one described in [CITATION] comprising six choppers in total.', '1309.6215-1-24-3': 'Solely the rotation speed of the first three choppers is increased to 112 Hz to further reduce intensity losses, see Fig. [REF] and Tab. [REF].', '1309.6215-1-24-4': 'It is noted that the intensity loss in subframe overlap regions is intrinsic to the way the WFM system is designed, as the subframes are kept separated in time very strictly.', '1309.6215-1-24-5': 'This is necessary for highly structured reflectivity spectra that are recorded with the high-resolution setup.', '1309.6215-1-24-6': 'In general, the ansatz described in [CITATION], which is very similar to the one used here, leads to less intensity losses in the subframe overlap regions, but at the expense of a somewhat more challenging subframe separation.', '1309.6215-1-25-0': '## Sample environment', '1309.6215-1-26-0': 'In case of free liquid surfaces or situations where the interface of study needs to be fixed in the horizontal plane, samples are put on a moveable sample table equipped with motors for necessary movements in horizontal and in vertical direction.', '1309.6215-1-26-1': 'The positional displacement of the sample from the [MATH] position of the main guide [upstream the bending section] is 84 cm at maximum for the highest possible deflection angle of 9°.', '1309.6215-1-26-2': 'This maximum vertical movement is only 1.7 times the vertical displacement of the sample stage realised at FIGARO at present and thus, will not pose any difficulty in handling on the instrument.', '1309.6215-1-26-3': 'The sample table will allow for accessing the sample by the neutron beam from above and below.', '1309.6215-1-26-4': 'In the case of air-solid and solid-liquid interfaces samples will be tilted against the fixed incident beam in the horizontal at fixed zero height thus operating the instrument in the more simple [MATH] mode.', '1309.6215-1-26-5': 'In addition, there should exist enough space for sample environment controlling temperature, pressure, mechanical load etc.', '1309.6215-1-26-6': 'For the very reason the distance between the sample position and the footprint slit is set to 40 cm.', '1309.6215-1-27-0': '## Detector', '1309.6215-1-28-0': 'It is envisaged that the reflectometer comprises a position sensitive area detector of 50 x 50 cm[MATH], located 2 m away from the sample position, with a pixel size fine enough to match the angular resolution.', '1309.6215-1-28-1': 'The smallest reflection angles are expected to be 0.3[MATH] - 0.4[MATH].', '1309.6215-1-28-2': 'Taking into account the loosest wavelength resolution of 10% the detector resolution should be [MATH].', '1309.6215-1-28-3': 'Using [MATH] mm, where [MATH] is the pixel size and [MATH] the desired resolution (expressed in [MATH] under the presumption that the divergence distribution is Gaussian), it follows that a detector with pixels having 2 mm egde length fulfills the instrument requirements.', '1309.6215-1-28-4': 'Furthermore, it will be advantegeous that the detection process takes place in a single conversion layer, since in this case the detector can be tilted by an angle [MATH] w.r.t the neutron flight path after the sample to improve the angular resolution in vertical direction to [MATH].', '1309.6215-1-28-5': 'Such setup could even match the high wavelength resolution achieved with the WFM chopper setup.', '1309.6215-1-29-0': '## SERGIS add-on', '1309.6215-1-30-0': 'Interfacial systems continuously increase in complexity.', '1309.6215-1-30-1': 'While in the last decade main focus of neutron scattering investigations of interfaces was on 1-dimensional systems, this will change in near future with the development of hierarchical 2D- to 3D-ordered systems on nanoscale.', '1309.6215-1-30-2': 'Typical examples are porous interfacial films for use in catalysis, as scaffolds or templates for nanomaterial synthesis [CITATION], as selective cell culture substrates [CITATION], separation media, energy storage materials [CITATION], spin valves or magnetic random access memory devices.', '1309.6215-1-30-3': 'This development requires continuous improvement of adequate instrumentation for analysis.', '1309.6215-1-30-4': 'Here, SERGIS (Spin-Echo Resolved Grazing Incidence Scattering) [CITATION] as add-on offers a promissing route for the investigations of 2-3D structures on nanoscale.', '1309.6215-1-30-5': 'It utilizes the polarized beam of the instrument, can be mounted without alterations of instrument length into the reflectometry set-up and decouples the intensity of the incident beam from the resolution of the experiment.', '1309.6215-1-30-6': 'Although SERGIS is still not matured, in particular with respect to data interpretation and analysis [CITATION], there is a great potential of this technique.', '1309.6215-1-30-7': 'In particular when used as rather simple add-on, and limited to nanoscale, functional systems, both in soft matter and life sciences [CITATION] as well as in materials science [CITATION], also under non-equilibrium conditions might be explored with high spatial and temporal resolution.', '1309.6215-1-31-0': 'The design of the SERGIS add-on is developed to fit the conditions imposed by the layout of the liquids reflectometer.', '1309.6215-1-31-1': 'For the setup 2 m in front and behind the sample position are available.', '1309.6215-1-31-2': 'Figure (muss ich dir noch schicken) shows the technical design of the complete setup (polariser and analyser not shown).', '1309.6215-1-31-3': 'The installation at the incoming (upstream) side accommodates', '1309.6215-1-32-0': 'polariser (S-bender or polarized [MATH]He gas cell), which is able to polarise a polychromatic beam with high efficiency.', '1309.6215-1-32-1': 'adiabatic gradient field radio frequency spin flipper for calibration and measurement possibility with both polarisations (spin up and spin down) gradient-field coupling device (Forte coil) between the strong-field region (polarizer, flipper with strong permanent magnets) and the low-field region (spin-echo precession devices, i.e. magnetic air-core coils with relatively weak fields) first pair of triangular coils (A) guide field (GF1) second pair of triangular coils (B)', '1309.6215-1-33-0': 'The setup behind the sample (downstream) consists of:', '1309.6215-1-34-0': 'third pair of triangular coils (C) guide field (GF2) fourth pair of triangular coils (D) gradient-field coupling device (Forte coil, (FC)) between the strong-field region (polarizer, flipper with strong permanent magnets) and the low-field region (spin-echo precession devices, i.e. magnetic air-core coils with relatively weak fields) analyser (multi-cavity) (Ana) position sensitive 2D detector (PSD)', '1309.6215-1-35-0': 'The guide field (GF1,2) is 1m in length in order to leave enough room on the incoming side to accommodate all spin manipulation devices.', '1309.6215-1-35-1': 'The angle [MATH] of the triangular coils with respect to the incoming neutron beam can be changed between 30[MATH] and 60[MATH] depending on the orientation of the coils (see Figure ).', '1309.6215-1-35-2': 'Magnetic fields used lie in the range from 1.5 to 15 mT. Above parameters in combination with a wavelength band of 2 - 7.1AA(2 - 12[MATH], every second pulse) result in a spin echo length of 5-662nm (30[MATH]) and 9-1150mm (5-1850nm, 30[MATH] and 90-3200nm, 60[MATH]).', '1309.6215-1-36-0': '## GISANS add-on', '1309.6215-1-37-0': 'As mentioned in the introduction, studies of laterally structured interfacial systems will become ever more important [CITATION].', '1309.6215-1-37-1': 'For this reason the reflectometer is equipped with a conventional GISANS module.', '1309.6215-1-37-2': 'The collimation concept is based on the approach successfully used at REFSANS [CITATION].', '1309.6215-1-37-3': 'In this concept the sample is illuminated by multiple low divergence beams which all converge to a single focus point on the detector (Fig. [REF]).', '1309.6215-1-37-4': 'In this way the best possible use of the available source divergence is realized and a high reciprocal space resolution is maintained.', '1309.6215-1-37-5': 'The radial collimator is replacing guide part 5 (bending section) and collimation slits in front of the sample (see Fig. [REF] (b) and Tab. [REF]).', '1309.6215-1-37-6': 'It consists of elements with non-reflecting side walls.', '1309.6215-1-37-7': 'Top and bottom walls are made from m=5 mirrors in order to provide sufficient beam bending for GISANS experiments.', '1309.6215-1-37-8': 'The collimator is divided laterally in three sub-channels separated by 0.1 mm thick absorbing walls.', '1309.6215-1-37-9': 'The chosen length L of the collimator is 6.4m with [MATH] = 5 mm and [MATH] = 3.3 mm and the detector is placed at 5.7 m downstream from the collimator exit with a sample-to-detector distance, LSD, of 5 m, slightly reducing the usable waveband by [MATH].', '1309.6215-1-37-10': 'In the chosen 3 beams configuration, the intensity distribution is homogeneous horizontally (see Fig B).', '1309.6215-1-37-11': 'The shadows of the collimator walls are small in comparison to the individual beam width.', '1309.6215-1-37-12': 'This ensures a very good sample illumination (>90).', '1309.6215-1-37-13': 'From Fig. [REF] it appears that all three channels deliver the same intensity (resp. 30, 38, 32 of the total), which ensures a homogeneous sampling of the surface.', '1309.6215-1-38-0': 'The long collimation distance produces an intensity distribution that is uniform over the total horizontal divergence (Fig. [REF], left panel), and does not exceed +/-1mrad.', '1309.6215-1-38-1': 'As expected, each of the beams contributes equally to this divergence (Fig. [REF], middle panel).', '1309.6215-1-38-2': 'At the detector position, which was chosen to be 5m from the sample in order not to depart too much from the normal operation distance of the reflectometer, one observes -as desired- a single spot of 4 mm full width at half maximum, matching the targeted in-plane resolution (Fig. [REF], right panel).', '1309.6215-1-38-3': 'In conclusion the radial collimation option offers the possibility to perform GISANS measurements at the liquid reflectometer with an intensity gain of 3 compared to a single pinhole channel with a resolution matching that of a current state-of-the-art SANS instrument.', '1309.6215-1-39-0': '# Performance', '1309.6215-1-40-0': 'The layout of the horizontal reflectometer is optimized for measurements with small samples and high q transfers.', '1309.6215-1-40-1': 'Its performance is demonstrated by carrying out virtual experiments with different samples of 1x1 cm[MATH] area.', '1309.6215-1-40-2': 'Fluxes and count rates are given for different angular and resolution settings, which are summarized in Tab. [REF].', '1309.6215-1-40-3': 'The beam intensity at the sample position is also shown for selected angular settings in Fig. [REF].', '1309.6215-1-40-4': 'The simulations were carried out including the ESS cold moderator characteristics as of May 2013.', '1309.6215-1-41-0': '## Basic setup', '1309.6215-1-42-0': 'As discussed in the previous section, the basic setup of the reflectometer utilizes three choppers that provide the desired waveband with the wavelength resolution mainly determined by the length of the instrument going up to 10 for 2 [MATH] neutrons.', '1309.6215-1-42-1': 'The slit collimation yields a vertical beam divergence that matches the loosest wavelength resolution (but is restricted to max.', '1309.6215-1-42-2': 'Such a setup is most suitable for measuring reflectivity spectra of free liquid surfaces and monolayers on liquid surfaces, like e.g. Langmuir and Gibbs adsorption layer of amphiphiles.', '1309.6215-1-42-3': 'The measurement can be performed using the single waveband and three angular settings or the double waveband and two angular settings (and so on).', '1309.6215-1-42-4': 'In general, for each angular setting the waveband can be freely selected, e.g. for small angles it is advantageous to use a larger waveband, since the reflectivity is high at low q and thus a larger q-region can be covered with a measurement time still being of the order of a few seconds.', '1309.6215-1-42-5': 'The applicability will depend on the time required by the choppers to change settings from single to double (triple,...) waveband regime.', '1309.6215-1-43-0': 'The measured reflectivity curves of an ideal D[MATH]O reflectivity surface and an adsorption monolayer and corresponding count rates are shown in Fig. [REF].', '1309.6215-1-43-1': 'The usage of the double waveband leads to a larger measurement time, but on the other hand the time needed to adjust for an intermediate reflection angle is saved.', '1309.6215-1-43-2': 'The longest measurement time is required, as expected, for highest angular settings.', '1309.6215-1-43-3': 'Nevertheless, it is possible to measure reflectivities up to [MATH] within minutes on a 1x1 cm[MATH] sample, with at least 100 counts for every data point.', '1309.6215-1-43-4': 'The ability of measuring reflectivities at high q and on samples of 1 cm[MATH] size is unprecedented.', '1309.6215-1-43-5': 'For example, the FIGARO reflectometer at ILL is used for measurements up to 0.4 [MATH] with typically much larger sample sizes.', '1309.6215-1-43-6': 'T', '1309.6215-1-44-0': 'It is well known that incoherent background originating from the sample sets a limit on minimum reflectivity that can still be measured with a certain statistical significance.', '1309.6215-1-44-1': 'For the ideal D[MATH]O surface of 1x1 cm[MATH] area and 0.3 mm thickness, the background is of the order of [MATH] using the mean free path of [MATH] cm for D[MATH]O for incoherent scattering.', '1309.6215-1-44-2': 'Neutronic simulations with incoherent background included show that even in that case data acquisition times of 100 s suffice in the range up to 0.6 [MATH], see Fig. [REF].', '1309.6215-1-44-3': 'Since its shape is flat, the background can be fitted and subtracted from the reflectivity spectrum.', '1309.6215-1-44-4': 'The residual spectrum agrees well with the theoretical curve and all data points still have a high enough statistical significance, with 2[MATH] being the significance of the lowest data point.', '1309.6215-1-44-5': 'Thus, if the measurement can be carried out for a sufficiently long time, being only a few minutes for a 1x1 cm[MATH] D[MATH]O sample, reflectivities down to 10[MATH] can be accessed by background recording and subtraction.', '1309.6215-1-44-6': 'In the case of the 1x1 cm[MATH] monolayer sample, the mean free path of the null reflecting water is [MATH] and yields a background level of [MATH].', '1309.6215-1-44-7': 'For such a sample the total counting time of the order of 4 to 5 hours (for the largest angular setting) will suffice to achieve a statistically significant measurement of the monolayer signal after background subtraction.', '1309.6215-1-44-8': 'Naturally, the required sampling time is anti-correlated with the total sample area and can be significantly reduced if larger samples (of the order of 10 cm[MATH]) can be used.', '1309.6215-1-45-0': '## High-resolution setup', '1309.6215-1-46-0': 'The WFM chopper setup is optimized to provide a constant and high wavelength resolution for the single waveband from 2 [MATH] to 7.1 [MATH] neutrons, combined with an adequate collimation before the sample for a high q-resolution.', '1309.6215-1-46-1': 'Due to a substantial loss in flux because of increased resolution as compared to the basic setup, see Fig. [REF], the available q-range for such measurements is most likely reduced to [MATH].', '1309.6215-1-46-2': 'Within the accessible q-range, however, fast measurements of highly structured reflectivity spectra are rendered possible.', '1309.6215-1-46-3': 'In Fig. [REF], the measurement of a NiTi sample ([86 [MATH] Ni/115 [MATH] Ti] glass, total thickness = 2010 [MATH]) is shown in grey for comparison.', '1309.6215-1-46-4': 'is shown.', '1309.6215-1-46-5': 'Its reflectivity spectrum exhibits several main peaks along with Kiessig oscillations over the entire q-range.', '1309.6215-1-46-6': 'The high-resolution setup of the reflectometer proves capable of a precise reconstruction of most of these features.', '1309.6215-1-46-7': 'To achieve at least 100 counts in each data point, 15 min of acquisition time for the highest angular setting is needed.', '1309.6215-1-47-0': '# Concept robustness', '1309.6215-1-48-0': 'The concept of the horizontal reflectometer makes mainly use of established and well known instrument components.', '1309.6215-1-48-1': 'The guide system consists of straight sections of 0.5 m. Solely the z-kink piece might require a somewhat smaller segmentation.', '1309.6215-1-48-2': 'The required coating is m=5 for the top and bottom guide surface, while except for the feeding section, the z-kink and the bending section, m=3 coating can be used everywhere.', '1309.6215-1-48-3': 'The avoidance of the line of sight by the chosen guide geometry is assumed to clear the detector area from background arising from the prompt pulse.', '1309.6215-1-48-4': 'Two guide sections were studied more carefully, being the feeding and the deflection sections.', '1309.6215-1-48-5': 'The impact of various solutions for the feeding section located in the central monolith was studied with respect to the delivered flux on sample.', '1309.6215-1-48-6': 'It was observed that even if the first 2 m of the feeding section cannot be installed due to cooling problems or other technical constraints, the flux on a 1x1 cm[MATH] sample stays without significant changes.', '1309.6215-1-48-7': 'Solely for samples with widths larger than 2 cm a drop of intensity occurs.', '1309.6215-1-48-8': 'The conclusion is that the instrument does not heavily depend on the actual performance of the extraction system.', '1309.6215-1-49-0': 'The required stability of the deflecting guide system was studied with respect to the flux delivered on the sample.', '1309.6215-1-49-1': 'It was found that the misalignment of the individual guide pieces can be as large as 0.01[MATH] without causing noticeable flux losses.', '1309.6215-1-49-2': 'The precision that is routinely reached today is 0.001[MATH].', '1309.6215-1-49-3': 'Thus we conclude that the technical demands for the deflecting section do not pose any risk.', '1309.6215-1-50-0': 'The chopper system is rather complex, but on the other hand the sizes of the chopper discs and their speed are well within what is technically feasible today.', '1309.6215-1-50-1': 'The practical validity of the WFM concept itself was already demonstrated at the BNC reactor in Hungary [CITATION].', '1309.6215-1-50-2': 'Presuming that an adequate detector matching the instrument requirements is provided, the setup for high-resolution measurements does not entail additional risks.', '1309.6215-1-51-0': '# Discussion and conclusions', '1309.6215-1-52-0': 'The design of the horizontal reflectometer presented in this work has been developed with respect to scientific questions that will be relevant at the time when the ESS facility comes into operation.', '1309.6215-1-52-1': 'The horizontal reflectomer proves capable of measurements of high q transfers on small horizontal samples possible within reasonable measurement time, while being very flexible in terms of covered q-range, sample size and beam direction.', '1309.6215-1-52-2': 'A dedicated WFM chopper setup for high-resolution measurements on very thick multistructured samples is currently the first one to be proposed for a reflectometer instrument and its design presented in this work and in [CITATION] shows that the required [MATH] resolution can be achieved without pushing chopper layouts at or beyond current technical limits.', '1309.6215-1-52-3': 'If this setup will be realized at the ESS, areas of parameter space that are currently inaccessible to neutron reflectometry due to limited flux and/or precision are expected to open up for exploration and new insights in materials organization.'}

{'1309.6215-2-0-0': 'A design study of a horizontal neutron reflectometer adapted to the general baseline of the long pulse European Spallation Source (ESS) is presented.', '1309.6215-2-0-1': 'The instrument layout comprises solutions for the neutron guide, high-resolution pulse shaping and beam bending onto a sample surface being so far unique in the field of reflectometry.', '1309.6215-2-0-2': 'The length of this instrument is roughly 55 m, enabling [MATH] resolutions from 0.5% to 10%.', '1309.6215-2-0-3': 'The incident beam is focused in horizontal plane to boost measurements of sample sizes of [MATH] cm[MATH] and smaller with potential beam deflection in both downward and upward direction.', '1309.6215-2-0-4': 'The range of neutron wavelengths utilized by the instrument is 2 to 7.1 (12.2, ...) [MATH] , if every (second, ...) neutron source pulse is used.', '1309.6215-2-0-5': 'Angles of incidence can be set between 0[MATH] and 9[MATH] with a total accessible q-range from [MATH] up to 1 [MATH].', '1309.6215-2-0-6': 'The instrument operates both in [MATH] (free liquid surfaces) and [MATH] (solid/liquid, air/solid interfaces) geometry.', '1309.6215-2-0-7': 'The experimental setup will in particular enable direct studies on ultrathin films (d [MATH]) and buried monolayers to multilayered structures of up to 3000 [MATH] total thickness.', '1309.6215-2-0-8': 'The horizontal reflectometer will further foster investigations of hierarchical systems from nanometer to micrometer length scale (the latter by off-specular scattering), as well as their kinetics and dynamical properties, in particular under load (shear, pressure, external fields).', '1309.6215-2-0-9': 'Polarization and polarization analysis as well as the GISANS option are designed as potential modules to be implemented separately in the generic instrument layout.', '1309.6215-2-0-10': 'The instrument is highly flexible and offers a variety of different measurement modes.', '1309.6215-2-0-11': 'With respect to its mechanical components the instrument is exclusively based on current technology.', '1309.6215-2-0-12': 'Risks of failure for the chosen setup are minimum.', '1309.6215-2-1-0': '# Introduction and science case', '1309.6215-2-2-0': 'Soft matter and life science systems investigated by neutron reflectometry (NR) continuously increase in complexity - both in structure as well as in the number and the specific roles of their components.', '1309.6215-2-2-1': 'The same is true for hard matter systems.', '1309.6215-2-2-2': 'This development presents a permanent challenge and demand for continuous improvement of the neutron facilities.', '1309.6215-2-2-3': 'On the other hand there is a constantly increasing request for NR measurements, in particular in soft matter and life sciences, which stems from the unique possibilities of neutrons in probing amphiphilic and self-organizing structures at air-liquid [CITATION], liquid-liquid [CITATION] and solid-liquid (buried) interfaces [CITATION].', '1309.6215-2-2-4': 'NR in combination with off-specular and grazing-incidence small angle neutron scattering (GISANS [CITATION], SESANS [CITATION], SERGIS [CITATION]) provides deep insight in self-assembly- and aggregation processes [CITATION], 1-3d ordered interfacial films [CITATION] and the interplay of length scales in hierarchically structured systems [CITATION].', '1309.6215-2-3-0': 'Current research at the forefront of science with NR covers investigations of protein organization in bio-membranes [CITATION], structure, organization and functioning of living cells at interfaces [CITATION], nano-engineering [CITATION], new materials and interface properties for energy (storage) applications [CITATION], magnetic fluctuations and domain propagation [CITATION] as well as magnetic ordering and the understanding of relevant length scales and energies [CITATION].', '1309.6215-2-4-0': 'Complementary (on-board) in-situ techniques in addition to NR are often needed to elucidate complex interfacial structures and processes that are difficult to assess otherwise.', '1309.6215-2-4-1': 'In particular X-ray reflectometry [CITATION], ellipsometry [CITATION], Brewster angle microscopy [CITATION] and IR spectroscopy [CITATION] are essential complementary tools for achieving integrated insight into soft matter and life science systems.', '1309.6215-2-4-2': 'Soft matter sample sizes required by NR today are typically of the order of 1000 mm[MATH] or larger which hinders studies with components that are not available or cannot be produced in sufficient quantities.', '1309.6215-2-4-3': 'The latter holds in particular, but not exclusively, for biological systems where typically human proteins provide such bottle-neck.', '1309.6215-2-4-4': 'Thus, parametric studies for instance on biological, genetic or pharmaceutical activity are outside the scope of NR measurements today.', '1309.6215-2-4-5': 'Such investigations will become available with the high flux of the ESS source by utilizing much reduced beam cross sections (foot prints) and sample sizes.', '1309.6215-2-5-0': 'Minimized beam dimensions (with not necessarily minimized sample size) will further allow for characterizations of rough and curved interfaces that are common in functional and natural materials like industrial coatings, ball bearings, bone or skin.', '1309.6215-2-5-1': 'In addition, inhomogeneous, patterned surfaces might be scanned for local features with sub-mm beams.', '1309.6215-2-5-2': 'Magnetic reference layers in combination with polarized neutron beams offer additional contrast and sensitivity for samples that do not withstand solution contrast changes [CITATION].', '1309.6215-2-5-3': 'In combination with polarization analysis such experimental set-up will help discriminating the incoherent background in soft matter and life science systems by spin-filtering [CITATION].', '1309.6215-2-6-0': 'The ESS source [CITATION] will enable kinetic studies of systems in non-equilibrium situations on time scales not available today.', '1309.6215-2-6-1': 'Tackling the millisecond range in the future however might still require concurrent sample sizes.', '1309.6215-2-6-2': 'Therefore, the reflectometer at ESS should foresee flexible and adaptable collimation optics.', '1309.6215-2-6-3': 'The latter is also important whenever interfaces are to be assessed from above and below the sample horizon as can be the case for buried interfaces and interfaces under load, e.g. shear forces in combined NR and rheology set-ups [CITATION].', '1309.6215-2-7-0': 'The length scale of interest in soft matter and life science systems of the next generation will cover 1-3000 [MATH], hence an instrument with variable resolution from [MATH] is required.', '1309.6215-2-7-1': 'For the majority of purposes (not including wide angle and diffraction experiments) a q-range of 0.005 [MATH] to 1 [MATH] will suffice.', '1309.6215-2-8-0': '# Layout of the horizontal reflectometer', '1309.6215-2-9-0': 'The scientific motivation described in the last section combined with additional requirements from ESS advisory panels were taken into account when choosing the basic instrument parameters.', '1309.6215-2-9-1': 'As a result, the key strength of this instrument was decided to be the ability to use small samples and to access large momentum transfer values [MATH] reaching up to [MATH] in both downward and upward direction for liquid samples.', '1309.6215-2-9-2': 'In order to match these requirements, the following basic parameters were selected:', '1309.6215-2-10-0': 'In the following, the layout of the instrument based on the key parameters listed above is described in more detail (see also Tab. [REF] for an overview).', '1309.6215-2-10-1': 'Individual beamline components are presented and their characteristics are justified based on results of neutronic MC simulations carried out with the VITESS software package [CITATION].', '1309.6215-2-11-0': '## Moderator and extraction system', '1309.6215-2-12-0': 'The reflectometer is set up for using wavelengths of 2 [MATH] and longer.', '1309.6215-2-12-1': 'Based on the current moderator description provided by the ESS and implemented in simulation packages like VITESS [CITATION] and McStas [CITATION], the cold moderator with 12 cm edge length, which provides a larger flux for wavelengths longer than 2.5 [MATH], was selected for the reflectometer.', '1309.6215-2-12-2': 'Neutron optic elements can be placed in the shielding monolith, which is currently 6 m in radius, beginning at 2 m behind the moderator.', '1309.6215-2-12-3': 'Thus beamline components like mirrors or guides can be placed in the monolith to increase the number of neutrons fed into the main guide system outside the first 6 m.', '1309.6215-2-12-4': 'While the horizontal shape is elliptic (see above), three options were tested for the vertical shape of the extraction system, that is a constant height of 2 cm, and a tapered guide with the entry height of 8.67 cm and 12 cm, respectively.', '1309.6215-2-12-5': 'For the latter, the guide entry height matches the size of the moderator.', '1309.6215-2-12-6': 'Simulations reveal that the best beam extraction is provided by the tapered feeding guide with 8.67 cm entry height, though it is noted that the performance of the constant guide is only slightly worse.', '1309.6215-2-12-7': 'The conclusion is that a large part of additional phase space accepted by the tapered guide is not transported by the subsequent beamline to the sample position.', '1309.6215-2-13-0': '## Direct line of sight', '1309.6215-2-14-0': 'A reflectometer aiming for high-q measurements needs to be free of prompt pulse background.', '1309.6215-2-14-1': 'Taking into account the unprecedented intensity of currently 5 MW and the energy of the primary proton beam of [MATH] GeV it is unclear whether a T0-chopper is able to remove all of the fast particle and gamma-ray background.', '1309.6215-2-14-2': 'This is why a T0-chopper is not foreseen and the background from the prompt pulse is avoided by the geometry of the guide system.', '1309.6215-2-14-3': 'The latter has to be designed such that in the field of view of the detector, there are no guide elements that lie in the direct flight path from the source.', '1309.6215-2-14-4': 'This is the definition of avoiding the line of sight twice.', '1309.6215-2-14-5': 'Since the horizontal shape of the guide is elliptic and a modification of this shape would interfere with its focusing properties, avoiding the line of sight must be carried out in the vertical plane.', '1309.6215-2-15-0': 'As a sophisticated chopper system will be placed within the first few meters after the shielding monolith, where also one chopper needs to be moved along the beamline, it was decided to leave the in-monolith section horizontally, i.e. not to have any inclination in the guide system there.', '1309.6215-2-15-1': 'On the other hand, the guide section 4, which is upstream from the bending section, should be horizontal, to enable the latter to equally access samples from above and below.', '1309.6215-2-15-2': 'Avoiding the line of sight requires a difference in height between the guide position at the monolith and at the bending section.', '1309.6215-2-15-3': 'Ways to achieve that are to place a double bender (s-bender) or a double kink (z-kink) in between.', '1309.6215-2-15-4': 'Here, a z-kink shows a clearly superior performance, see Fig. [REF].', '1309.6215-2-15-5': 'The kink angle is fixed at [MATH], which leads to very moderate losses only if compared with a guide system without a kink.', '1309.6215-2-15-6': 'The shift of the maximum in the vertical divergence distribution towards 0.25[MATH] can be reduced by optimizing the vertical shape of the kink guide element.', '1309.6215-2-15-7': 'The latter shows the best performance if the shape is slightly double-elliptic.', '1309.6215-2-15-8': 'The total length of the kink guide section is 11.2 m and the shift in height of the guide position amounts to [MATH] cm.', '1309.6215-2-16-0': 'In accordance with the demand of avoiding the line of sight twice, the z-kink element should be placed such that fast primary particles are unable to illuminate the guide section after the z-kink, independent of whether their flight path would interfere with the detector area.', '1309.6215-2-16-1': 'A simple raytracing study was performed to determine an optimal position of the latter guide element by scanning the neutron parameter space at the source between [MATH] cm and [MATH], where [MATH] is the vertical coordinate of the neutron trajectory at the source and [MATH] is its vertical divergence.', '1309.6215-2-16-2': 'The study revealed that if the z-kink is placed at a distance of 18.7 m away from the source, all fast particles would propagate at least 6 m outside the guide before entering the last guide section.', '1309.6215-2-16-3': 'Since propagation outside the guide can for simplicity be considered as propagation in shielding, it can safely be argued that due to the guide geometry, all fast background will be sufficiently absorbed and the prompt pulse will not contribute to the background at the detector position.', '1309.6215-2-17-0': '## Beam delivery on sample', '1309.6215-2-18-0': 'As stated above, in order to access a momentum transfer [MATH] on a free liquid surface the beam must have an incident angle on a horizontal sample of [MATH], taking into account the shortest utilized wavelength of 2 [MATH].', '1309.6215-2-18-1': 'Short wavelength neutrons require several bounces off the beam bending walls in order to be inclined by such large angle [MATH], since there is a strong limitation on their maximum reflection angle, e.g. [MATH] for m=5 supermirror coatings.', '1309.6215-2-18-2': 'At the same time, the reflectometer is to provide sufficient flexibility concerning the variety of incident angles such that also regions of small [MATH] can be covered.', '1309.6215-2-18-3': 'Thus, the beam is tilted down- or upwards on a horizontal sample surface by means of 5 movable deflecting elements, of which the top and the bottom surfaces are covered with m = 5 coating.', '1309.6215-2-18-4': 'The length of each element is [MATH] 1.28 m with the height of 2 cm.', '1309.6215-2-18-5': 'When tilted by [MATH] each element bends the full beam of 2 cm height by 1.8[MATH].', '1309.6215-2-18-6': 'The horizontal shape of the beam bending elements is elliptic, since these elements are still part of the main guide ellipse (see Fig. 1).', '1309.6215-2-18-7': 'The maximum bending angle [MATH] guaranties for a maximum [MATH].', '1309.6215-2-18-8': 'Bending guide elements can be replaced upon request by a GISANS guide module providing horizontal collimation.', '1309.6215-2-18-9': 'The beam characteristics after the bending section are shown in Figs. [REF] and [REF].', '1309.6215-2-18-10': 'In principle, it is also possible to have shorter (1.15 m) bending elements that can be tilted by 1[MATH], hence reaching the maximum reflection angle for a 2 [MATH] neutron.', '1309.6215-2-18-11': 'On the other hand, MC simulations show that in this case the intensity at wavelengths around 2 [MATH] is heavily suppressed.', '1309.6215-2-19-0': 'The layout of the bending section is optimized for maximum flexibility and high-[MATH] while keeping the complexity of the system as low as possible.', '1309.6215-2-19-1': 'The five bending elements can be arranged to bend the neutron beam down- or upwards by a number of possible angles.', '1309.6215-2-19-2': 'Their work principle is that neutrons bounce off each element once and propagate towards the sample afterwards, see Fig. [REF].', '1309.6215-2-19-3': 'The drawback is that neutrons with a non-zero divergence have a certain chance of hitting the bottom (top) wall of one of the bending elements, if the beam is bent downwards (upwards).', '1309.6215-2-19-4': 'The resulting mis-orientation can be further increased during the propagation of the beam until the neutron is absorbed or leaves the bending section with a high divergence, i.e. does not propagate towards the sample.', '1309.6215-2-19-5': 'This is why the vertical divergence provided by the collimation system does not exceed 0.2[MATH] FWHM (0.3[MATH] as maximum) for higher bending angles, even though the maximum divergence can in principle achieve maximum [MATH], taking into account the height of the guide of 2 cm and the collimation length of 2 m.', '1309.6215-2-19-6': 'It was investigated whether the divergence can be increased by involving more bending elements that would approximate a curved guide.', '1309.6215-2-19-7': 'This, however, would increase the mechanical complexity of the system, while MC simulations showed that there was no significant flux gain at the sample position.', '1309.6215-2-20-0': '## Collimation', '1309.6215-2-21-0': 'The collimation of the neutron beam exiting the bending section of the guide is done by means of two diaphragms (slits) located at the guide exit and 2 m downstream from the guide, respectively.', '1309.6215-2-21-1': 'The first slit determines the total vertical divergence on the sample, while the second, so-called footprint slit, is used to reduce the beam size such that it matches the sample dimensions or the requirements for the beam size, e.g., for position sensitive measurements.', '1309.6215-2-21-2': 'The adjusted divergence depends on the desired vertical angular resolution on the sample.', '1309.6215-2-21-3': 'For simplicity one usually uses matching resolution for divergence and wavelength, thus the opening (height) of the first slit depends on the pulse shaping regime used.', '1309.6215-2-21-4': 'The free propagation length of 2 m further allows to install, e.g., a polarization device between the collimation slits.', '1309.6215-2-22-0': '## Pulse shaping', '1309.6215-2-23-0': 'The reflectometer operates with two chopper setups for basic and high-resolution, respectively.', '1309.6215-2-23-1': 'All choppers are located at the side of the guide to make use of the small beam height and thus reduce the opening/closing time.', '1309.6215-2-23-2': 'The basic setup requires three choppers, which define the main frame and prevent neutrons with other wavelengths from propagating to the detector.', '1309.6215-2-23-3': 'In the basic setup, the wavelength resolution is defined by the total pulse length [MATH] ms. Hence the first chopper does not need to be close to the source, as the pulse length will not be reduced.', '1309.6215-2-23-4': 'Furthermore, the first few meters after the shielding monolith are occupied by pulse shaping choppers belonging to the high-resolution setup (see below).', '1309.6215-2-23-5': 'Thus, the first chopper of the basic setup is placed at 12.5 m, followed by a second chopper at 17.5 m and a third chopper at 31 m.', '1309.6215-2-23-6': 'While the first two choppers are designed to suppress background from slow neutrons that otherwise could pollute the subsequent frame(s), the third chopper defines the accepted waveband of [MATH], if every source pulse is used.', '1309.6215-2-23-7': 'The waveband can be extended, if every second (third) pulse is used, see Fig. [REF].', '1309.6215-2-23-8': 'This can be achieved by reducing the chopper rotation frequency to [MATH], ...) of the original value.', '1309.6215-2-24-0': 'A dedicated wavelength frame multiplication system (WFM) was developed for the high-resolution work regime.', '1309.6215-2-24-1': 'Its purpose is to provide a high, constant wavelength resolution between [MATH] and [MATH] for the utilized waveband, while removing contaminant neutrons.', '1309.6215-2-24-2': 'This layout is very close to the one described in [CITATION] comprising six choppers in total.', '1309.6215-2-24-3': 'Solely the rotation speed of the first three choppers is increased to 112 Hz to further reduce intensity losses, see Fig. [REF] and Tab. [REF].', '1309.6215-2-24-4': 'It is noted that the intensity loss in subframe overlap regions is intrinsic to the way the WFM system is designed, as the subframes are kept separated in time very strictly.', '1309.6215-2-24-5': 'This is necessary for highly structured reflectivity spectra that are recorded with the high-resolution setup.', '1309.6215-2-24-6': 'In general, the ansatz described in [CITATION], which is very similar to the one used here, could lead to less intensity losses in the subframe overlap regions, but at the expense of a somewhat more challenging subframe separation.', '1309.6215-2-25-0': '## Sample environment', '1309.6215-2-26-0': 'In case of free liquid surfaces or situations where the interface of study needs to be fixed in the horizontal plane, samples are put on a movable sample table equipped with motors for necessary movements in horizontal and in vertical direction.', '1309.6215-2-26-1': 'The positional displacement of the sample from the [MATH] position of the main guide (upstream the bending section) is 84 cm at maximum for the highest possible deflection angle of 9[MATH].', '1309.6215-2-26-2': 'This maximum vertical movement is only 1.7 times the vertical displacement of the sample stage realized at FIGARO at present and thus, will not pose any difficulty in handling on the instrument.', '1309.6215-2-26-3': 'The sample table will allow for accessing the sample by the neutron beam from above and below.', '1309.6215-2-26-4': 'In the case of air-solid and solid-liquid interfaces samples will be tilted against the fixed incident beam in the horizontal at fixed zero height thus operating the instrument in the more simple [MATH] mode.', '1309.6215-2-26-5': 'In addition, there should exist enough space for sample environment controlling temperature, pressure, mechanical load etc.', '1309.6215-2-26-6': 'For the very reason the distance between the sample position and the footprint slit is set to 40 cm.', '1309.6215-2-27-0': '## Detector', '1309.6215-2-28-0': 'It is envisaged that the reflectometer comprises a position sensitive area detector of 50 x 50 cm[MATH], located 2 m away from the sample position, with a pixel size fine enough to match the angular resolution.', '1309.6215-2-28-1': 'The smallest reflection angles are expected to be 0.3[MATH] - 0.4[MATH].', '1309.6215-2-28-2': 'Taking into account the loosest wavelength resolution of 10% the detector resolution should be [MATH].', '1309.6215-2-28-3': 'Using [MATH] mm, where [MATH] is the pixel size and [MATH] the desired resolution expressed in [MATH] under the presumption that the divergence distribution is Gaussian, it follows that a detector with pixels having 2 mm egde length fulfills the instrument requirements.', '1309.6215-2-28-4': 'Furthermore, it will be advantegeous that the detection process takes place in a single conversion layer, since in this case the detector can be tilted by an angle [MATH] w.r.t the neutron flight path after the sample to improve the angular resolution in vertical direction to [MATH].', '1309.6215-2-28-5': 'Such setup could even match the high wavelength resolution achieved with the WFM chopper setup.', '1309.6215-2-29-0': '## SERGIS add-on', '1309.6215-2-30-0': 'As mentioned in the Introduction, interfacial systems continuously increase in complexity.', '1309.6215-2-30-1': 'While in the last decade main focus of neutron scattering investigations of interfaces was on 1-dimensional systems, this will change in near future with the development of hierarchical 2D- to 3D-ordered systems on nanoscale.', '1309.6215-2-30-2': 'Typical examples are porous interfacial films for use in catalysis, as scaffolds or templates for nanomaterial synthesis [CITATION], as selective cell culture substrates [CITATION], separation media, energy storage materials [CITATION], spin valves or magnetic random access memory devices [CITATION].', '1309.6215-2-30-3': 'This development requires continuous improvement of adequate instrumentation for analysis.', '1309.6215-2-30-4': 'Here, SERGIS (Spin-Echo Resolved Grazing Incidence Scattering) [CITATION] as add-on offers a promissing route for the investigations of 2-3D structures on nanoscale.', '1309.6215-2-30-5': 'It utilizes the polarized beam of the instrument, can be mounted without alterations of instrument length into the reflectometry set-up and decouples the intensity of the incident beam from the resolution of the experiment.', '1309.6215-2-30-6': 'Although SERGIS is still not matured, in particular with respect to data interpretation and analysis [CITATION], there is a great potential of this technique.', '1309.6215-2-30-7': 'In particular when used as rather simple add-on, and limited to nanoscale, functional systems, both in soft matter and life sciences [CITATION] as well as in materials science [CITATION], also under non-equilibrium conditions might be explored with high spatial and temporal resolution.', '1309.6215-2-31-0': 'The design of the SERGIS add-on is developed to fit the conditions imposed by the layout of the liquids reflectometer.', '1309.6215-2-31-1': 'For the setup 2 m in front and behind the sample position are available.', '1309.6215-2-31-2': 'Figure [REF] shows the technical design of the complete setup (polariser and analyser not shown).', '1309.6215-2-31-3': 'The installation at the incoming (upstream) side accommodates', '1309.6215-2-32-0': 'polariser (S-bender or polarized [MATH]He gas cell), which is able to polarise a polychromatic beam with high efficiency.', '1309.6215-2-32-1': 'adiabatic gradient field radio frequency spin flipper for calibration and measurement possibility with both polarisations (spin up and spin down) gradient-field coupling device (Forte coil) between the low field region (spin-echo precession devices, i.e. magnetic air-core coils with relatively weak fields) and the strong field region of the analyser first pair of triangular coils (A) guide field (GF1) second pair of triangular coils (B)', '1309.6215-2-33-0': 'The setup behind the sample (downstream) consists of:', '1309.6215-2-34-0': 'third pair of triangular coils (C) guide field (GF2) fourth pair of triangular coils (D) gradient-field coupling device (Forte coil, (FC)) between the strong-field region (polarizer, flipper with strong permanent magnets) and the low-field region (spin-echo precession devices, i.e. magnetic air-core coils with relatively weak fields) analyser (multi-cavity) (Ana) position sensitive 2D detector (PSD)', '1309.6215-2-35-0': 'The guide field (GF1,2) is 1m in length in order to leave enough room on the incoming side to accommodate all spin manipulation devices.', '1309.6215-2-35-1': 'The angle [MATH] of the triangular coils with respect to the incoming neutron beam can be changed between 30[MATH] and 60[MATH] depending on the orientation of the coils (see Figure [REF]).', '1309.6215-2-35-2': 'Magnetic fields used lie in the range from 1.5 to 15 mT. Above parameters in combination with a wavelength band of 2 - 7.1[MATH] (2 - 12[MATH], every second pulse) result in a spin echo length of 5-662 nm (30[MATH]) and 9-1150 nm (5-1850 nm, 30[MATH] and 90-3200 nm, 60[MATH]).', '1309.6215-2-36-0': '## GISANS add-on', '1309.6215-2-37-0': 'The alternative module for studies of laterally structured interfacial systems is a GISANS add-on.', '1309.6215-2-37-1': 'For this reason the reflectometer can also be equipped with a conventional GISANS module.', '1309.6215-2-37-2': 'The collimation concept of this module is based on the approach successfully used at REFSANS [CITATION].', '1309.6215-2-37-3': 'In this concept the sample is illuminated by multiple low divergence beams which all converge to a single focus point on the detector (Fig. [REF]).', '1309.6215-2-37-4': 'In this way the best possible use of the available source divergence is realized and a high reciprocal space resolution is maintained.', '1309.6215-2-37-5': 'The radial collimator is replacing guide part 5 (bending section) and the collimation slits in front of the sample (see Fig. [REF] (b) and Tab. [REF]).', '1309.6215-2-37-6': 'It consists of elements with non-reflecting side walls.', '1309.6215-2-37-7': 'Top and bottom walls are made from m=5 mirrors in order to provide sufficient beam bending for GISANS experiments.', '1309.6215-2-37-8': 'The collimator is divided laterally in three sub-channels separated by 0.1 mm thick absorbing walls.', '1309.6215-2-37-9': 'The chosen length L of the collimator is 6.4m with [MATH] = 5 mm and [MATH] = 3.3 mm.', '1309.6215-2-37-10': 'The detector is placed at 5.7 m downstream from the collimator exit with a sample-to-detector distance, LSD, of 5 m, slightly reducing the usable waveband by [MATH].', '1309.6215-2-37-11': 'In the chosen 3 beams configuration, the intensity distribution is homogeneous horizontally (see Fig. [REF]).', '1309.6215-2-37-12': 'The shadows of the collimator walls are small in comparison to the individual beam width.', '1309.6215-2-37-13': 'This ensures a very good sample illumination ([MATH]).', '1309.6215-2-37-14': 'From Fig. [REF] a) it appears that all three channels deliver the same intensity (resp. 30%, 38%, 32% of the total), which ensures a homogeneous sampling of the surface.', '1309.6215-2-38-0': 'The long collimation distance produces an intensity distribution that is uniform over the total horizontal divergence (Fig. [REF], left panel), and does not exceed +/-1mrad.', '1309.6215-2-38-1': 'As expected, each of the beams contributes equally to this divergence (Fig. [REF], middle panel).', '1309.6215-2-38-2': 'At the detector position, which was chosen to be 5 m from the sample in order not to depart too much from the normal operation distance of the reflectometer, one observes -as desired- a single spot of 4 mm full width at half maximum, matching the targeted in-plane resolution (Fig. [REF], right panel).', '1309.6215-2-38-3': 'In conclusion the radial collimation option offers the possibility to perform GISANS measurements at the liquid reflectometer with a resolution matching that of a current state-of-the-art SANS instrument.', '1309.6215-2-39-0': '# Performance', '1309.6215-2-40-0': 'The layout of the horizontal reflectometer is optimized for measurements with small samples and high q transfers.', '1309.6215-2-40-1': 'Its performance is demonstrated by carrying out virtual experiments with different samples of 1x1 cm[MATH] area.', '1309.6215-2-40-2': 'Fluxes and count rates are given for different angular and resolution settings, which are summarized in Tab. [REF].', '1309.6215-2-40-3': 'The beam intensity at the sample position is also shown for selected angular settings in Fig. [REF].', '1309.6215-2-40-4': 'The simulations were carried out including the ESS cold moderator characteristics as of May 2013.', '1309.6215-2-41-0': '## Basic setup', '1309.6215-2-42-0': 'As discussed in the previous section, the basic setup of the reflectometer utilizes three choppers that provide the desired waveband with the wavelength resolution mainly determined by the length of the instrument going up to 10 for 2 [MATH] neutrons.', '1309.6215-2-42-1': 'The slit collimation yields a vertical beam divergence that matches the loosest wavelength resolution (but is restricted to max.', '1309.6215-2-42-2': 'Such a setup is most suitable for measuring reflectivity spectra of free liquid surfaces and monolayers on liquid surfaces, like e.g. Langmuir and Gibbs adsorption layer of amphiphiles.', '1309.6215-2-42-3': 'The measurement can be performed using the single waveband and three angular settings or the double waveband and two angular settings (and so on).', '1309.6215-2-42-4': 'In general, for each angular setting the waveband can be freely selected, e.g. for small angles it is advantageous to use a larger waveband, since the reflectivity is high at low q and thus a larger q-region can be covered with a measurement time still being of the order of a few seconds.', '1309.6215-2-42-5': 'The applicability will depend on the time required by the choppers to change settings from single to double (triple,...) waveband regime.', '1309.6215-2-43-0': 'The measured reflectivity curves of an ideal D[MATH]O reflectivity surface and an adsorption monolayer and corresponding count rates are shown in Fig. [REF].', '1309.6215-2-43-1': 'The usage of the double waveband leads to a larger measurement time, but on the other hand the time needed to adjust for an intermediate reflection angle is saved.', '1309.6215-2-43-2': 'The longest measurement time is required, as expected, for highest angular settings.', '1309.6215-2-43-3': 'Nevertheless, it is possible to measure reflectivities up to [MATH] within minutes on a 1x1 cm[MATH] sample, with at least 100 counts for every data point.', '1309.6215-2-43-4': 'The ability of measuring reflectivities at high q and on samples of 1 cm[MATH] size is unprecedented.', '1309.6215-2-43-5': 'For example, the FIGARO reflectometer at ILL is used for measurements up to 0.4 [MATH] with typically much larger sample sizes.', '1309.6215-2-44-0': 'It is well known that incoherent background originating from the sample sets a limit on minimum reflectivity that can still be measured with a certain statistical significance.', '1309.6215-2-44-1': 'For the ideal D[MATH]O surface of 1x1 cm[MATH] area and 0.3 mm thickness, the background is of the order of [MATH] using the mean free path of [MATH] cm for D[MATH]O for incoherent scattering.', '1309.6215-2-44-2': 'Neutronic simulations with incoherent background included show that even in that case data acquisition times of 100 s suffice in the range up to 0.6 [MATH], see Fig. [REF].', '1309.6215-2-44-3': 'Since its shape is flat, the background can be fitted and subtracted from the reflectivity spectrum.', '1309.6215-2-44-4': 'The residual spectrum agrees well with the theoretical curve and all data points still have a high enough statistical significance, with 2[MATH] being the significance of the lowest data point.', '1309.6215-2-44-5': 'Thus, if the measurement can be carried out for a sufficiently long time, being only a few minutes for a 1x1 cm[MATH] D[MATH]O sample, reflectivities down to 10[MATH] can be accessed by background recording and subtraction.', '1309.6215-2-44-6': 'In the case of the 1x1 cm[MATH] monolayer sample, the mean free path of the null reflecting water is [MATH] and yields a background level of [MATH].', '1309.6215-2-44-7': 'For such a sample the total counting time of the order of 4 to 5 hours (for the largest angular setting) will suffice to achieve a statistically significant measurement of the monolayer signal after background subtraction.', '1309.6215-2-44-8': 'Naturally, the required sampling time is inversely proportional to the total sample area and can be significantly reduced if larger samples (of the order of 10 cm[MATH]) are used.', '1309.6215-2-45-0': '## High-resolution setup', '1309.6215-2-46-0': 'The WFM chopper setup is optimized to provide a constant and high wavelength resolution for the single waveband from 2 [MATH] to 7.1 [MATH], combined with an adequate collimation before the sample for a high q-resolution.', '1309.6215-2-46-1': 'Due to a substantial loss in flux because of increased resolution as compared to the basic setup, see Fig. [REF], the available q-range for such measurements is most likely reduced to [MATH].', '1309.6215-2-46-2': 'Within the accessible q-range, however, fast measurements of highly structured reflectivity spectra are rendered possible.', '1309.6215-2-46-3': 'In Fig. [REF], the measurement of a NiTi-multilayer sample ([86 [MATH] Ni/115 [MATH] Ti] on glass, total thickness = 2010 [MATH]) is shown together with the ideal spectrum of this sample in grey for comparison.', '1309.6215-2-46-4': 'Its reflectivity spectrum exhibits several main peaks along with Kiessig oscillations over the entire q-range.', '1309.6215-2-46-5': 'The high-resolution setup of the reflectometer proves capable of a precise reconstruction of most of these features.', '1309.6215-2-46-6': 'To achieve at least 100 counts in each data point, 15 min of acquisition time for the highest angular setting is needed.', '1309.6215-2-47-0': '# Concept robustness', '1309.6215-2-48-0': 'The concept of the horizontal reflectometer makes mainly use of established and well known instrument components.', '1309.6215-2-48-1': 'The guide system consists of straight sections of 0.5 m. Solely the z-kink piece might require a somewhat smaller segmentation.', '1309.6215-2-48-2': 'The required coating is m=5 for the top and bottom guide surface, while except for the feeding section, the z-kink and the bending section, m=3 coating can be used everywhere else.', '1309.6215-2-48-3': 'The avoidance of the line of sight by the chosen guide geometry is assumed to clear the detector area from background arising from the prompt pulse.', '1309.6215-2-48-4': 'Two guide sections were studied more carefully, being the feeding and the deflection sections.', '1309.6215-2-48-5': 'The impact of various solutions for the feeding section located in the central monolith was inspected with respect to the delivered flux on sample.', '1309.6215-2-48-6': 'It was observed that even if the first 2 m of the feeding section cannot be installed due to cooling problems or other technical constraints, the flux on a 1x1 cm[MATH] sample stays without significant changes.', '1309.6215-2-48-7': 'Solely for samples with widths larger than 2 cm a drop of intensity occurs.', '1309.6215-2-48-8': 'The conclusion is that the instrument does not heavily depend on the actual performance of the extraction system.', '1309.6215-2-49-0': 'The required stability of the deflecting guide system was studied with respect to the flux delivered on the sample.', '1309.6215-2-49-1': 'It was found that the misalignment of the individual guide pieces can be as large as 0.01[MATH] without causing noticeable flux losses.', '1309.6215-2-49-2': 'The precision that is routinely reached today is 0.001[MATH].', '1309.6215-2-49-3': 'Thus we conclude that the technical demands for the deflecting section do not pose any risk.', '1309.6215-2-50-0': 'The chopper system is rather complex, but on the other hand the sizes of the chopper discs and their speed are well within what is technically feasible today.', '1309.6215-2-50-1': 'The practical validity of the WFM concept itself was already demonstrated at the BNC reactor in Hungary [CITATION].', '1309.6215-2-50-2': 'Presuming that an adequate detector matching the instrument requirements is provided, the setup for high-resolution measurements does not entail additional risks.', '1309.6215-2-51-0': '# Discussion and conclusions', '1309.6215-2-52-0': 'The design of the horizontal reflectometer presented in this work has been developed with respect to scientific questions that will be relevant at the time when the ESS facility comes into operation.', '1309.6215-2-52-1': 'The horizontal reflectometer proves capable of measurements of high q transfers on small horizontal samples possible within reasonable measurement time, while being very flexible in terms of covered q-range, sample size and beam direction.', '1309.6215-2-52-2': 'A dedicated WFM chopper setup for high-resolution measurements on very thick multi-structured samples is currently the first one to be proposed for a reflectometer instrument and its design presented in this work and in [CITATION] shows that the required [MATH] resolution can be achieved without pushing chopper layouts at or beyond current technical limits.', '1309.6215-2-52-3': 'If this setup will be realized at the ESS, areas of parameter space that are currently inaccessible to neutron reflectometry due to limited flux and/or precision are expected to open up for exploration and new insights in materials organization.'}

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['1309.6215-1-48-8', '1309.6215-2-48-8'], ['1309.6215-1-5-0', '1309.6215-2-5-0'], ['1309.6215-1-5-1', '1309.6215-2-5-1'], ['1309.6215-1-5-2', '1309.6215-2-5-2'], ['1309.6215-1-5-3', '1309.6215-2-5-3'], ['1309.6215-1-3-0', '1309.6215-2-3-0'], ['1309.6215-1-14-0', '1309.6215-2-14-0'], ['1309.6215-1-14-1', '1309.6215-2-14-1'], ['1309.6215-1-14-2', '1309.6215-2-14-2'], ['1309.6215-1-14-4', '1309.6215-2-14-4'], ['1309.6215-1-14-5', '1309.6215-2-14-5'], ['1309.6215-1-42-0', '1309.6215-2-42-0'], ['1309.6215-1-42-1', '1309.6215-2-42-1'], ['1309.6215-1-42-2', '1309.6215-2-42-2'], ['1309.6215-1-42-3', '1309.6215-2-42-3'], ['1309.6215-1-42-4', '1309.6215-2-42-4'], ['1309.6215-1-42-5', '1309.6215-2-42-5'], ['1309.6215-1-50-0', '1309.6215-2-50-0'], ['1309.6215-1-50-1', '1309.6215-2-50-1'], ['1309.6215-1-50-2', '1309.6215-2-50-2'], ['1309.6215-1-7-0', '1309.6215-2-7-0'], ['1309.6215-1-7-1', '1309.6215-2-7-1'], ['1309.6215-1-9-0', '1309.6215-2-9-0'], ['1309.6215-1-9-1', '1309.6215-2-9-1'], ['1309.6215-1-24-0', '1309.6215-2-24-0'], ['1309.6215-1-24-1', '1309.6215-2-24-1'], ['1309.6215-1-24-2', '1309.6215-2-24-2'], ['1309.6215-1-24-3', '1309.6215-2-24-3'], ['1309.6215-1-24-4', '1309.6215-2-24-4'], ['1309.6215-1-24-5', '1309.6215-2-24-5'], ['1309.6215-1-40-0', '1309.6215-2-40-0'], ['1309.6215-1-40-1', '1309.6215-2-40-1'], ['1309.6215-1-40-2', '1309.6215-2-40-2'], ['1309.6215-1-40-3', '1309.6215-2-40-3'], ['1309.6215-1-40-4', '1309.6215-2-40-4'], ['1309.6215-1-26-3', '1309.6215-2-26-3'], ['1309.6215-1-26-4', '1309.6215-2-26-4'], ['1309.6215-1-26-5', '1309.6215-2-26-5'], ['1309.6215-1-26-6', '1309.6215-2-26-6'], ['1309.6215-1-37-3', '1309.6215-2-37-3'], ['1309.6215-1-37-4', '1309.6215-2-37-4'], ['1309.6215-1-37-6', '1309.6215-2-37-6'], ['1309.6215-1-37-7', '1309.6215-2-37-7'], ['1309.6215-1-37-8', '1309.6215-2-37-8'], ['1309.6215-1-37-11', '1309.6215-2-37-12']]

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[]

[['1309.6215-1-30-0', '1309.6215-2-30-0'], ['1309.6215-1-31-2', '1309.6215-2-31-2']]

[]

['1309.6215-1-9-2', '1309.6215-1-33-0', '1309.6215-1-43-6', '1309.6215-1-46-4', '1309.6215-2-9-2', '1309.6215-2-33-0']

{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1309.6215

1808.04597

{'1808.04597-1-0-0': 'Recent observations reveal a bulk flow in the local Universe which is faster and extends to larger scales than is expected around a typical observer in the standard [MATH]CDM cosmology.', '1808.04597-1-0-1': 'The deceleration parameter [MATH] derived from local observations is then expected to show a scale-dependent dipolar modulation.', '1808.04597-1-0-2': 'From a maximum likelihood analysis of the Joint Lightcurve Analysis (JLA) catalogue of Type Ia supernovae we do find such a dipole in [MATH] extending out to [MATH], with a magnitude comparable to its monopole.', '1808.04597-1-0-3': 'Although not statistically significant in current data, such a dipole must be allowed for, especially in analysing surveys with incomplete sky coverage such as JLA and its successor Pantheon; out of 740 (1048) SNe IA in the JLA (Pantheon) catalogue, 632 (890) are in the hemisphere opposite to the direction of bulk flow for which their redshifts have been corrected.', '1808.04597-1-0-4': 'However when we do so, the monopole component of [MATH], which has been widely ascribed to a cosmological constant (dark energy), drops in statistical significance and becomes consistent with zero at [MATH] (95% c.l.).', '1808.04597-1-0-5': 'This suggests that the apparent acceleration of the expansion rate deduced from supernovae may be an artefact of our bulk flow.', '1808.04597-1-1-0': '# Introduction', '1808.04597-1-2-0': "The foundations of the current 'standard model' of cosmology date back to the 1930s when there was essentially no data available.", '1808.04597-1-2-1': 'In particular the Universe was assumed to be exactly isotropic and homogeneous, with space-time described by the maximally symmetric Robertson-Walker metric, and occupied by ideal fluids with purely diagonal energy-momentum tensors [CITATION].', '1808.04597-1-2-2': 'Subsequently it has been recognised that the distribution of galaxies, which is a biased tracer of the underlying distribution of the dominant dark matter, is in fact rather inhomogeneous.', '1808.04597-1-2-3': 'Counts-in-spheres of galaxy catalogues suggest that there is a transition to (statistical) homogeneity on scales exceeding [MATH] Mpc [CITATION] although sufficiently large volumes have not yet been surveyed to establish this definitively.', '1808.04597-1-2-4': 'This is however the standard model expectation if the observed large-scale structure has grown under gravity in the sea of dark matter, starting with an initially gaussian random field of small density perturbations with an approximately scale-invariant spectrum.', '1808.04597-1-2-5': "Detailed observations of the temperature fluctuations in the cosmic microwave background (CMB) have broadly confirmed this picture [CITATION] although several anomalies remain (e.g. the 'hemispherical asymmetry', the quadrupole-octupole alignment, a quadrupolar modulation) suggesting that further revision of the underlying picture may prove necessary.", '1808.04597-1-3-0': "In our real universe there are 'peculiar motions' due to the local inhomogeneity and anisotropy of surrounding structure, i.e. we are not idealised comoving observers but have a 'tilt' velocity relative to the idealised Hubble flow [CITATION].", '1808.04597-1-3-1': 'These are non-negligible, e.g. our Local Group of galaxies moves with respect to the universal expansion at [MATH] towards [MATH], as is inferred from the observed dipolar modulation of the CMB temperature [CITATION].', '1808.04597-1-3-2': 'Moreover diverse observations [CITATION] reaching out as far as [MATH] Mpc have not seen the [MATH] fall-off of the peculiar velocity expected in the standard [MATH]CDM cosmology.', '1808.04597-1-3-3': 'The odds of this happening by chance in that framework can be estimated by querying Hubble volume simulations of large-scale structure formation e.g. Dark Sky [CITATION].', '1808.04597-1-3-4': 'We found [CITATION] that [MATH] of Milky Way-like observers should observe the bulk flow ([MATH] extending to [MATH]) that we do.', '1808.04597-1-3-5': 'The implications of our being such a special observer have been discussed for e.g. measurements of [MATH] [CITATION] but not for the inference of cosmic acceleration.', '1808.04597-1-4-0': "Since cosmological observables are generally formulated in the 'CMB frame' in which the universe is supposedly perfectly isotropic, it is in any case always necessary to correct what we measure from our relative moving frame.", '1808.04597-1-4-1': 'For example the observed redshifts of the Type Ia supernovae (SNe Ia) in catalogues like JLA [CITATION] have been corrected in order to convert from the heliocentric frame to the CMB frame.', '1808.04597-1-4-2': 'The methodology used follows earlier work [CITATION] which used bulk flow observations made back in 2004 [CITATION] and moreover assumed that there is convergence to the CMB frame beyond 150 Mpc.', '1808.04597-1-4-3': 'Since this is not in accordance with subsequent deeper observations, we reverse these corrections in order to examine whether the deceleration parameter measured in our (bulk flow) rest frame can indeed differ from that of comoving observers in the model universe as was suggested by Tsagas [CITATION].', '1808.04597-1-4-4': 'Such theoretical considerations predict [CITATION] that there should be a dipole asymmetry in the derived cosmic deceleration parameter [MATH] towards the bulk flow direction.', '1808.04597-1-4-5': 'We do find indications of such a dipole and, even more importantly, that the significance of the monopole in [MATH] decreases simultaneously.', '1808.04597-1-4-6': 'Hence not only is the indication for acceleration statistically marginal [CITATION], it may arise due to our being located in a bulk flow, rather than being the effect of a cosmological constant or dark energy.', '1808.04597-1-5-0': '# The SDSS-II/SNLS3 Joint Lightcurve Analysis catalogue', '1808.04597-1-6-0': 'We use the most up to date publicly available sample of supernova lightcurve properties and directions: the SDSS-II/SNLS3 Joint Lightcurve Analysis (JLA) catalogue [CITATION].', '1808.04597-1-6-1': "This consists of 740 spectroscopically confirmed SNe Ia, including several low redshift ([MATH]) samples, three seasons of SDSS-II ([MATH]) and three years of SNLS ([MATH] data, all calibrated consistently in the 'Spectral Adaptive Lightcurve Template 2' (SALT2) scheme.", '1808.04597-1-6-2': "This assigns to each supernova 3 parameters: the apparent magnitude [MATH] at maximum in the rest frame '[MATH]-band', and the light curve shape and colour corrections, [MATH] and [MATH].", '1808.04597-1-6-3': "The distance modulus is then given by: [EQUATION] where [MATH] and [MATH] are assumed to be constants, as is [MATH] the absolute SNe Ia magnitude, as befits a 'standard candle'.", '1808.04597-1-6-4': 'In the standard [MATH]CDM cosmological model this is related to the luminosity distance [MATH] as: [EQUATION]', '1808.04597-1-6-5': 'Here [MATH] is the Hubble distance and [MATH] the Hubble parameter ([MATH] being its present value), and [MATH] are the matter, cosmological constant and curvature densities in units of the critical density.', '1808.04597-1-6-6': "In the standard [MATH]CDM model these are related by the 'cosmic sum rule': [MATH].", '1808.04597-1-6-7': 'However we make no such model assumptions and will simply expand the luminosity distance [MATH] in a Taylor series in order to examine its second derivative i.e. the acceleration (see [REF]).', '1808.04597-1-6-8': 'This is because acceleration is a kinematic quantity and can be measured without making any assumptions about the dynamics underlying the universal expansion.', '1808.04597-1-6-9': '(There may be concern that such a Taylor expansion fails at high redshift, however we have verified that [MATH] in the best-fit [MATH]CDM model differs by only 7% even at [MATH] (the highest redshift in the JLA sample) which is much less than the measurement uncertainty.', '1808.04597-1-6-10': 'Indeed as seen in the Tables, the Taylor expansion fits the data just as well as [MATH]CDM.)', '1808.04597-1-7-0': 'Figure [REF] is a Mollewide projection of the directions of the 740 SNe Ia in Galactic coordinates.', '1808.04597-1-7-1': 'Due to the diverse survey strategies of the sub-samples that make up the JLA catalogue, its coverage of the sky is patchy and anisotropic.', '1808.04597-1-7-2': 'While the low redshift objects are spread out unevenly across the sky, the intermediate redshift ones from SDSS are mainly confined to a narrow disk at low declination, while the high redshift ones from SNLS are clustered along four specific directions.', '1808.04597-1-8-0': 'The JLA analysis [CITATION] individually corrects the observed redshifts [MATH] in the heliocentric frame in order to account for perturbations to the cosmological redshifts [MATH] due to peculiar motions in the local Universe.', '1808.04597-1-8-1': 'These corrections are carried over unchanged from an earlier analysis [CITATION], which in turn cites an earlier method [CITATION] and the peculiar velocity model of Hudson et al. [CITATION].', '1808.04597-1-8-2': 'It is stated that the inclusion of these corrections allow SNe Ia with redshifts down to 0.01 to be included in the cosmological analysis, in contrast to earlier analyses [CITATION] which employed only SNe Ia down to [MATH].', '1808.04597-1-9-0': 'In Figure [REF] we examine these corrections by exhibiting the quantity [MATH], defined as [EQUATION] where [MATH] and [MATH] are the heliocentric and CMB rest frame redshifts tabulated by JLA, while [MATH] is given by [EQUATION] where [MATH] is 369 km s[MATH] in the direction of the CMB dipole,[CITATION] and [MATH] is the unit vector in the direction of the supernova.', '1808.04597-1-9-1': 'It can be seen in Fig. [REF] that SNe Ia beyond [MATH] have been assumed to be stationary w.r.t. the CMB rest frame, and corrections applied only to those at lower redshifts.', '1808.04597-1-9-2': 'It is not clear how these corrections have in fact been derived beyond [MATH], which is the extent to which the Streaming Motions of Abell Clusters (SMAC) sample [CITATION] extends.', '1808.04597-1-9-3': 'This has a bulk velocity of [MATH] km s[MATH] towards [MATH] out to [MATH] at 90%C.L., and a bulk velocity of [MATH] km s[MATH] towards [MATH] generated by sources beyond 200 [MATH] Mpc ([MATH]) at 98% C.L.', '1808.04597-1-9-4': 'If the peculiar velocity field is not discontinuous, the SNe Ia immediately outside this volume should have comparable velocities.', '1808.04597-1-9-5': "Fig. [REF] indicates however that the JLA peculiar velocity 'corrections' have assumed that the bulk flow abruptly disappears at this point!", '1808.04597-1-9-6': 'The JLA analysis [CITATION] allows SNe Ia beyond this distance to only have an uncorrelated velocity dispersion of 150 km s[MATH].', '1808.04597-1-9-7': 'In the absence of any evidence of convergence to the CMB rest frame, this assumption is unjustified since it is it quite possible that the observed bulk flow stretches out to much larger scales.', '1808.04597-1-9-8': "In fact there have been persistent claims of a 'dark flow' extending out to several hundreds of Mpc [CITATION], although this is still under debate.", '1808.04597-1-10-0': 'At this point it is worth noting the anisotropy of the JLA catalogue.', '1808.04597-1-10-1': 'Out of the 740 SNe Ia, 551 are in the hemisphere pointing away from the CMB dipole.', '1808.04597-1-10-2': 'With respect to the [MATH] km s[MATH] bulk flow of the model [CITATION] using which the redshifts of the local SNe Ia have been corrected, only 108 are in the upper hemisphere while 632 are in the lower hemisphere.', '1808.04597-1-10-3': 'With respect to the direction of the abnormally high flow reported by 6dFGSv, the largest and most homogeneous peculiar velocity sample of nearly 9000 galaxies [CITATION], 103 SNe Ia are in the upper hemisphere while 637 are in the lower hemisphere.', '1808.04597-1-11-0': 'The subsequent Pantheon catalogue[CITATION], which incorporates additional SNe Ia in addition to JLA, continues to suffer from these problems.', '1808.04597-1-11-1': 'While the flow model [CITATION] using which the redshifts of the Pantheon sample have been corrected go out to [MATH], this model has a residual bulk flow of [MATH] km s[MATH], attributed to sources beyond [MATH], and 890 of the 1048 Pantheon SNe are in the hemisphere opposite to the direction of this flow.', '1808.04597-1-12-0': 'Both JLA and Pantheon include SNe to which anomalously large peculiar velocity corrections have been applied at redshifts far higher than the limit to which the corresponding flow models extend.', '1808.04597-1-12-1': 'SDSS2308 (see Fig. [REF]) in JLA at [MATH] in JLA and SN2246 in Pantheon at [MATH] are two of the many examples.', '1808.04597-1-13-0': 'We use the heliocentric redshifts tabulated by JLA [CITATION] and subtract out the bias corrections applied to [MATH].', '1808.04597-1-13-1': 'For the Pantheon catalogue [CITATION] the [MATH] values and individual contributions to the covariance are not yet public.', '1808.04597-1-14-0': '# Cosmological analysis', '1808.04597-1-15-0': 'We compare the distance modulus ([REF]) obtained from the JLA sample with the apparent magnitude ([REF]) using the Maximum Likelihood Estimator [CITATION].', '1808.04597-1-15-1': "We use the kinematic Taylor series expansion for the luminosity distance up to the third term, given by [CITATION]: [EQUATION] where [MATH] is the cosmic deceleration parameter in the Hubble flow frame, defined in terms of the scale factor of the universe [MATH] and its derivatives w.r.t. proper time, [MATH] is the cosmic 'jerk' [MATH], and [MATH] is in fact [MATH].", '1808.04597-1-15-2': 'Note that since the last two appear together in the coefficient of the [MATH] term, they cannot be determined separately.', '1808.04597-1-15-3': 'In the [MATH]CDM model: [MATH], however we wish to analyse the data without making assumptions about the matter content or the dynamics.', '1808.04597-1-16-0': 'To look for a possible dipole in the deceleration parameter, we allow it to have a direction dependence: [EQUATION] where [MATH] and [MATH] are, respectively, the magnitudes of the dipole and monopole (the latter corresponding to what is usually called [MATH]), while [MATH] is the direction of the dipole and [MATH] describes its scale dependence.', '1808.04597-1-16-1': 'We consider four representative functional forms:', '1808.04597-1-17-0': '(a) No scale dependence: [MATH] independent of [MATH],', '1808.04597-1-18-0': "(b) 'Top hat': [MATH] for [MATH], and [MATH] otherwise,", '1808.04597-1-19-0': '(c) Exponential: [MATH], and', '1808.04597-1-20-0': '(d) Linear: [MATH].', '1808.04597-1-21-0': 'Due to the very anisotropic sky coverage of the dataset, we do not attempt to find [MATH], instead we choose it to be along the CMB dipole direction.', '1808.04597-1-21-1': 'This is reasonable as the directions of the reported bulk flows [CITATION] are all within [MATH] of each other and of the CMB dipole.', '1808.04597-1-22-0': 'Subsequently, we maximise a likelihood as constructed earlier [CITATION], simultaneously w.r.t. the 4 cosmological parameters [MATH] and [MATH], as well as the 8 parameters that go into the standardisation of the SNe Ia candles: [MATH] and [MATH].', '1808.04597-1-22-1': "Ours is a frequentist analysis, however it is equivalent to the 'Bayesian Hierarchical Model' of March et al. [CITATION] which in fact yielded the same result [CITATION] as the frequentist analysis of Nielsen et al. [CITATION] when applied to the JLA catalogue.", '1808.04597-1-23-0': '# Results', '1808.04597-1-24-0': 'The results of the fits described in [REF] are given in Table [REF].', '1808.04597-1-24-1': 'The first row presents the results of a 10-parameter fit without a dipole, using the SN-by-SN corrections to the redshift as included in JLA.', '1808.04597-1-24-2': 'We find the best fit value of [MATH] to be -0.314, and that no acceleration is disfavoured at slightly less than [MATH], completely consistent with previous principled statistical analyses [CITATION].', '1808.04597-1-24-3': 'Similar results were also found by Rubin Hayden [CITATION] (see the left panels of their Fig.2 and 3rd row, Fig.3).', '1808.04597-1-25-0': 'However the quality of fit improves further (-2 log [MATH] decreases) when [MATH] is allowed to have a dipole.', '1808.04597-1-25-1': 'In the best fit where this has an exponentially decaying form [MATH], the dipole [MATH] is larger than the monopole [MATH] and its scale parameter is [MATH] suggesting that the bulk flow extends out to far larger redshift than is expected in a [MATH]CDM universe.', '1808.04597-1-25-2': 'In the presence of this dipole, [MATH] is disfavoured at only about [MATH].', '1808.04597-1-25-3': 'In other words, in an universe where we have theoretical reasons to expect a dipolar modulation in the deceleration parameter in the direction of our motion through the CMB, there is no significant evidence for a non-zero value of its monopole component.', '1808.04597-1-25-4': 'Fig. [REF] shows the 1, 2 and 3 [MATH] contours in the likelihood around the maximum as a function of [MATH] and [MATH], profiling over all other parameters.', '1808.04597-1-26-0': '# Discussion', '1808.04597-1-27-0': 'In this work we have used the statistical approach as well as the treatment of lightcurve parameters espoused by Nielsen et al [CITATION].', '1808.04597-1-27-1': 'These authors were criticised by Rubin Hayden [CITATION] for using redshift-independent distributions for [MATH] and [MATH].', '1808.04597-1-27-2': 'In this respect we note the following:', '1808.04597-1-28-0': 'The fit for determination of cosmological parameters [CITATION] is to determine the relationship between the luminosity distance [MATH] and the redshift.', '1808.04597-1-28-1': 'To inspect a posteriori the distribution of two ([MATH] and [MATH]) out of the three ingredients that go into standardising SNe Ia, and add empirical terms to describe their sample dependence and redshift evolution in the fit, is fundamentally against the principles of blind hypothesis testing, especially since no such dependence had been suggested by the JLA analysis [CITATION].', '1808.04597-1-29-0': 'We nevertheless carry out the same 16- and 22-parameter fits [CITATION] and present the results in Table [REF].', '1808.04597-1-29-1': 'While the log maximum likelihood ratio improves to 12.5 and 16 respectively for these fits, both these parameterisations also allow for a large dipole in [MATH] out to [MATH] and [MATH] respectively, which reduces the likelihood in favour of acceleration.', '1808.04597-1-30-0': 'When progressively adding more parameters to improve the quality of a fit and obtain a desired outcome, the parameters have to be justified by both physical and information theoretic arguments.', '1808.04597-1-30-1': 'The additional parameters of Rubin Hayden [CITATION] can be justified by the Akaike information criterion (AIC) but not by the Bayesian information criterion (BIC).', '1808.04597-1-30-2': 'The same applies to the two additional parameters we introduce ([MATH] and [MATH]) however there is a physical motivation to expect a scale-dependent dipole in an inhomogeneous universe [CITATION].', '1808.04597-1-31-0': 'If the light curve parameters [MATH] and [MATH] are allowed to be sample/redshift dependent one can ask why the absolute magnitude of SNe Ia should also not be sample/redshift-dependent.', '1808.04597-1-31-1': "Allowing this of course undermines their use as 'standard candles' and the data is then unsurprisingly consistent with no acceleration, as can be seen in Table [REF] (see also Tutusaus et al. [CITATION]).", '1808.04597-1-31-2': 'Note that the additional 3 parameters describing the sample dependence of [MATH] are justified by the AIC.', '1808.04597-1-32-0': 'It has been observed, e.g. by Bernal et al. [CITATION] that the deceleration parameter inferred from previous SNe Ia datasets has a redshift, and indeed directional, dependence.', '1808.04597-1-32-1': 'This was interpreted as indicative of local anisotropy in the matter distribution, i.e. our being located in an asymmetric void.', '1808.04597-1-32-2': 'The refinement in the present work is that we consider the most recent comprehensive database of SNe Ia and take into account all systematic effects as encoded in the covariance matrices provided [CITATION].', '1808.04597-1-32-3': 'Moreover we focus on the local velocity rather than the density field as this fully reflects the gravitational dynamics due to inhomogeneities.', '1808.04597-1-32-4': "We can then explore the expected consequences of our being 'tilted' i.e. non-Copernican observers.", '1808.04597-1-32-5': 'Our analysis is guided by the suggestion that we may then infer acceleration even when the overall expansion rate is decelerating - a signature of which would be a dipolar modulation of the inferred [MATH] along the direction of the bulk flow [CITATION].', '1808.04597-1-33-0': 'The effect of peculiar velocities on SNe Ia cosmology has been previously studied [CITATION], however these studies relied on covariances that apply to Copernican observers.', '1808.04597-1-33-1': 'As we show in an accompanying paper [CITATION], although the bulk flow we are embedded in is rare at only about the [MATH] level [CITATION] according to the DarkSky simulation [CITATION], the conditional covariances can be a factor of 10 larger and introduce a preferred direction locally.', '1808.04597-1-33-2': 'This can make a much bigger impact on cosmological inferences than was found in these previous studies.', '1808.04597-1-33-3': 'In particular the JLA analysis [CITATION] of the same dataset concluded that the effect of peculiar velocities is only a tiny ([MATH]) shift in the best-fit cosmological parameters.', '1808.04597-1-34-0': 'In summary, the model-independent evidence for acceleration of the Hubble expansion rate from the largest public catalogue of Type Ia supernovae is only [MATH].', '1808.04597-1-34-1': 'This is in stark contrast to the claim [CITATION] that acceleration is established by SNe Ia at [MATH] in the framework of the [MATH]CDM model.', '1808.04597-1-34-2': 'Moreover there is an indication for a dipole in [MATH] towards the CMB dipole - as is expected if the apparent acceleration is an artefact of our being located in a local bulk flow which extends out far enough to include most of the supernovae studied [CITATION].', '1808.04597-1-34-3': 'Given the observational evidence that there is no convergence to the CMB frame as far out as redshift [MATH] which includes half the known SNe Ia, this possibility must be taken seriously.', '1808.04597-1-35-0': 'It is often stated that regardless of the marginal evidence for cosmic acceleration from the SNe I a Hubble diagram there are other reasons to believe that the expansion rate is accelerating.', '1808.04597-1-35-1': 'It has been shown [CITATION] however that all low redshift probes e.g. measurements of Baryon Acoustic Oscillations, of [MATH] from stellar chronometry, and of the growth of matter perturbations [MATH], are also consistent with no acceleration.', '1808.04597-1-35-2': 'The precision data on CMB anisotropies [CITATION] do not directly probe cosmic acceleration.', '1808.04597-1-35-3': 'Whether the Universe is indeed dominated by dark energy thus remains an open question.'}

{'1808.04597-2-0-0': 'Recent observations reveal a bulk flow in the local Universe which is faster and extends to larger scales than is expected around a typical observer in the standard [MATH]CDM cosmology.', '1808.04597-2-0-1': 'The deceleration parameter [MATH] derived from local observations is then expected to show a scale-dependent dipolar modulation.', '1808.04597-2-0-2': 'From a maximum likelihood analysis of the Joint Lightcurve Analysis (JLA) catalogue of Type Ia supernovae we do find such a dipole in [MATH] extending out to [MATH], with a magnitude comparable to its monopole.', '1808.04597-2-0-3': 'Although not statistically significant in current data, such a dipole must be allowed for, especially in analysing surveys with incomplete sky coverage such as JLA and its successor Pantheon; out of 740 (1048) SNe IA in the JLA (Pantheon) catalogue, 632 (890) are in the hemisphere opposite to the direction of bulk flow for which their redshifts have been corrected.', '1808.04597-2-0-4': 'However when we do so, the monopole component of [MATH], which has been widely ascribed to a cosmological constant (dark energy), drops in statistical significance and becomes consistent with zero at [MATH] (95% c.l.).', '1808.04597-2-0-5': 'This suggests that the apparent acceleration of the expansion rate deduced from supernovae may be an artefact of our bulk flow.', '1808.04597-2-1-0': '# Introduction', '1808.04597-2-2-0': "The foundations of the current 'standard model' of cosmology date back to the 1930s when there was essentially no data available.", '1808.04597-2-2-1': 'In particular the Universe was assumed to be exactly isotropic and homogeneous, with space-time described by the maximally symmetric Robertson-Walker metric, and occupied by ideal fluids with a purely diagonal energy-momentum tensor [CITATION].', '1808.04597-2-2-2': 'Subsequently it has been recognised that the distribution of galaxies, which is a biased tracer of the underlying distribution of the dominant dark matter, is in fact rather inhomogeneous.', '1808.04597-2-2-3': 'Counts-in-spheres of galaxy catalogues suggest that there is a transition to (statistical) homogeneity on scales exceeding [MATH] Mpc [CITATION] although sufficiently large volumes have not yet been surveyed to establish this definitively.', '1808.04597-2-2-4': 'This is however the standard model expectation if the observed large-scale structure has grown under gravity in the sea of dark matter, starting with an initially gaussian random field of small density perturbations with an approximately scale-invariant spectrum.', '1808.04597-2-2-5': "Detailed observations of the temperature fluctuations in the cosmic microwave background (CMB) have broadly confirmed this picture [CITATION] although several anomalies remain (e.g. the 'hemispherical asymmetry', the quadrupole-octupole alignment, a quadrupolar modulation) suggesting that further revision of the underlying picture may prove necessary.", '1808.04597-2-3-0': "In our real universe there are 'peculiar motions' due to the local inhomogeneity and anisotropy of surrounding structure, i.e. we are not idealised comoving observers but have a 'tilt' velocity relative to the idealised Hubble flow [CITATION].", '1808.04597-2-3-1': 'These are non-negligible, e.g. our Local Group of galaxies moves with respect to the universal expansion at [MATH] towards [MATH], as is inferred from the observed dipolar modulation of the CMB temperature [CITATION].', '1808.04597-2-3-2': 'Moreover diverse observations [CITATION] reaching out as far as [MATH] Mpc have not seen the [MATH] fall-off of the peculiar velocity expected in the standard [MATH]CDM cosmology.', '1808.04597-2-3-3': 'The odds of this happening purely by chance can be estimated by querying Hubble volume simulations of large-scale structure formation e.g. Dark Sky [CITATION].', '1808.04597-2-3-4': 'We found [CITATION] that [MATH] of Milky Way-like observers should observe the bulk flow ([MATH] extending to [MATH]) that we do.', '1808.04597-2-3-5': 'The implications of our being such a special observer have been discussed for e.g. measurements of [MATH] [CITATION] but not for the inference of cosmic acceleration.', '1808.04597-2-3-6': 'Since this underpins the standard [MATH]CDM model, we address this important lacuna in this paper.', '1808.04597-2-4-0': "Since cosmological observables are generally formulated in the 'CMB frame' in which the universe is supposedly perfectly isotropic, it is in any case always necessary to correct what we measure from our relative moving frame.", '1808.04597-2-4-1': 'In particular the observed redshifts of the Type Ia supernovae (SNe Ia) in catalogues like JLA [CITATION] have been corrected in order to convert from the heliocentric frame to the CMB frame.', '1808.04597-2-4-2': 'The methodology used follows earlier work [CITATION] which used now outdated bulk flow observations made back in 2004 [CITATION] and moreover assumed without any physical basis that there is (abrupt) convergence to the CMB frame beyond 150 Mpc.', '1808.04597-2-4-3': 'Since this is not in accordance with subsequent deeper observations, we reverse these corrections in order to examine whether the deceleration parameter measured in our (bulk flow) rest frame can indeed differ from that of comoving observers in the isotropic model universe as was suggested by Tsagas [CITATION].', '1808.04597-2-4-4': 'Such theoretical considerations predict [CITATION] that there should be a dipole asymmetry in the derived cosmic deceleration parameter [MATH] towards the bulk flow direction.', '1808.04597-2-4-5': 'We do find indications of such a dipole and, even more importantly, that the significance of the monopole in [MATH] decreases simultaneously.', '1808.04597-2-4-6': 'Hence not only is the indication for acceleration statistically marginal [CITATION], it may well arise due to our being located in a bulk flow, rather than being the effect of a cosmological constant or dark energy.', '1808.04597-2-5-0': '# The SDSS-II/SNLS3 Joint Lightcurve Analysis catalogue', '1808.04597-2-6-0': 'We use the most up to date publicly available sample of supernova lightcurve properties and directions: the SDSS-II/SNLS3 Joint Lightcurve Analysis (JLA) catalogue [CITATION].', '1808.04597-2-6-1': "This consists of 740 spectroscopically confirmed SNe Ia, including several low redshift ([MATH]) samples, three seasons of SDSS-II ([MATH]) and three years of SNLS ([MATH] data, all calibrated consistently in the 'Spectral Adaptive Lightcurve Template 2' (SALT2) scheme.", '1808.04597-2-6-2': "This assigns to each supernova 3 parameters: the apparent magnitude [MATH] at maximum in the rest frame '[MATH]-band', and the light curve shape and colour corrections, [MATH] and [MATH].", '1808.04597-2-6-3': "The distance modulus is then given by: [EQUATION] where [MATH] and [MATH] are assumed to be constants, as is [MATH] the absolute SNe Ia magnitude, as befits a 'standard candle'.", '1808.04597-2-6-4': "In the 'standard' [MATH]CDM cosmological model this is related to the luminosity distance [MATH] as: [EQUATION]", '1808.04597-2-6-5': 'Here [MATH] is the Hubble distance, [MATH] the Hubble parameter ([MATH] being its present value), and [MATH] are the matter, cosmological constant and curvature densities in units of the critical density.', '1808.04597-2-6-6': "In the [MATH]CDM model these are thus related by the 'cosmic sum rule': [EQUATION]", '1808.04597-2-6-7': 'However we make no such model-dependent assumptions and will simply expand the luminosity distance [MATH] in a Taylor series in order to examine its second derivative i.e. the acceleration (see [REF]).', '1808.04597-2-6-8': 'This is because acceleration is a kinematic quantity and can be measured without making any assumptions about the dynamics underlying the universal expansion.', '1808.04597-2-6-9': 'There may be concern that such a Taylor expansion fails at high redshift and instead an expansion in [MATH] has been advocated [CITATION] for better convergence.', '1808.04597-2-6-10': 'However we have verified that [MATH] in the best-fit [MATH]CDM model differs by only 7% even at [MATH] (the highest redshift in the JLA sample) which is much less than the measurement uncertainty.', '1808.04597-2-6-11': 'As seen in Table I, the best-fit kinematic universe has -2 log [MATH] = - 213.84, cf. -214.97 for the best-fit [MATH]CDM model.', '1808.04597-2-6-12': 'Note that [MATH] and [MATH] are degenerate in the fits.', '1808.04597-2-7-0': 'Figure [REF] is a Mollewide projection of the directions of the 740 SNe Ia in Galactic coordinates.', '1808.04597-2-7-1': 'Due to the diverse survey strategies of the sub-samples that make up the JLA catalogue, its coverage of the sky is patchy and anisotropic.', '1808.04597-2-7-2': 'While the low redshift objects are spread out unevenly across the sky, the intermediate redshift ones from SDSS are mainly confined to a narrow disk at low declination, while the high redshift ones from SNLS are clustered along four specific directions.', '1808.04597-2-8-0': 'The JLA analysis [CITATION] individually corrects the observed redshifts [MATH] in the heliocentric frame in order to account for perturbations to the cosmological redshifts [MATH] due to peculiar motions in the local Universe.', '1808.04597-2-8-1': 'These corrections are carried over unchanged from an earlier analysis [CITATION], which in turn cites an earlier method [CITATION] and the peculiar velocity model of Hudson et al. [CITATION].', '1808.04597-2-8-2': 'It is stated that the inclusion of these corrections allow SNe Ia with redshifts down to 0.01 to be included in the cosmological analysis, in contrast to earlier analyses [CITATION] which employed only SNe Ia down to [MATH].', '1808.04597-2-9-0': 'In Figure [REF] we examine these corrections by exhibiting the quantity [MATH], defined as [EQUATION] where [MATH] and [MATH] are the heliocentric and CMB rest frame redshifts tabulated by JLA, while [MATH] is given by [EQUATION] where [MATH] is 369 km s[MATH] in the direction of the CMB dipole,[CITATION] and [MATH] is the unit vector in the direction of the supernova.', '1808.04597-2-9-1': 'It can be seen in Fig. [REF] that SNe Ia beyond [MATH] have been assumed to be stationary w.r.t. the CMB rest frame, and corrections applied only to those at lower redshifts.', '1808.04597-2-9-2': 'It is not clear how these corrections have in fact been derived beyond [MATH], which is the extent to which the Streaming Motions of Abell Clusters (SMAC) sample [CITATION] extends.', '1808.04597-2-9-3': 'This has a bulk velocity of [MATH] km s[MATH] towards [MATH] out to [MATH] at 90%C.L., and a bulk velocity of [MATH] km s[MATH] towards [MATH] generated by sources beyond 200 [MATH] Mpc ([MATH]) at 98% C.L.', '1808.04597-2-9-4': 'If the peculiar velocity field is not discontinuous, the SNe Ia immediately outside this volume should have comparable velocities.', '1808.04597-2-9-5': "Fig. [REF] indicates however that the JLA peculiar velocity 'corrections' have assumed that the bulk flow abruptly disappears at this point!", '1808.04597-2-9-6': 'The JLA analysis [CITATION] allows SNe Ia beyond this distance to only have an uncorrelated velocity dispersion of 150 km s[MATH].', '1808.04597-2-9-7': 'In the absence of any evidence of convergence to the CMB rest frame, this assumption is unjustified since it is it quite possible that the observed bulk flow stretches out to much larger scales.', '1808.04597-2-9-8': "In fact there have been persistent claims of a 'dark flow' extending out to several hundreds of Mpc [CITATION], although this is still under debate.", '1808.04597-2-10-0': 'At this point it is worth noting the anisotropy of the JLA catalogue.', '1808.04597-2-10-1': 'Out of the 740 SNe Ia, 551 are in the hemisphere pointing away from the CMB dipole.', '1808.04597-2-10-2': 'With respect to the [MATH] km s[MATH] bulk flow of the model [CITATION] using which the redshifts of the local SNe Ia have been corrected, only 108 are in the upper hemisphere while 632 are in the lower hemisphere.', '1808.04597-2-10-3': 'With respect to the direction of the abnormally high flow reported by 6dFGSv, the largest and most homogeneous peculiar velocity sample of nearly 9000 galaxies [CITATION], 103 SNe Ia are in the upper hemisphere while 637 are in the lower hemisphere.', '1808.04597-2-11-0': 'The subsequent Pantheon catalogue[CITATION], which incorporates 365 additional SNe Ia from the Pan-STARRS1 survey, continues to suffer from these problems.', '1808.04597-2-11-1': 'While the flow model [CITATION] using which the redshifts of the Pantheon sample have been corrected go out to [MATH], this model has a residual bulk flow of [MATH] km s[MATH], attributed to sources beyond [MATH], and 890 of the 1048 Pantheon SNe are in the hemisphere opposite to the direction of this flow.', '1808.04597-2-12-0': 'Both JLA and Pantheon include SNe to which anomalously large peculiar velocity corrections have been applied at redshifts far higher than the limit to which the corresponding flow models extend.', '1808.04597-2-12-1': 'SDSS2308 (see Fig. [REF]) in JLA at [MATH] in JLA and SN2246 in Pantheon at [MATH] are two of the many examples.', '1808.04597-2-13-0': 'We use the heliocentric redshifts tabulated by JLA [CITATION] and subtract out the bias corrections applied to [MATH].', '1808.04597-2-13-1': 'For the Pantheon catalogue [CITATION] the [MATH] values and individual contributions to the covariance are not yet public.', '1808.04597-2-14-0': '# Cosmological analysis', '1808.04597-2-15-0': 'We compare the distance modulus ([REF]) obtained from the JLA sample with the apparent magnitude ([REF]) using the Maximum Likelihood Estimator [CITATION].', '1808.04597-2-15-1': "We use the kinematic Taylor series expansion for the luminosity distance up to the third term, given by [CITATION]: [EQUATION] where [MATH] is the cosmic deceleration parameter in the Hubble flow frame, defined in terms of the scale factor of the universe [MATH] and its derivatives w.r.t. proper time, [MATH] is the cosmic 'jerk' [MATH], and [MATH] is in fact [MATH].", '1808.04597-2-15-2': 'Note that since the last two appear together in the coefficient of the [MATH] term, they cannot be determined separately.', '1808.04597-2-15-3': 'In the [MATH]CDM model: [MATH], however we wish to analyse the data without making assumptions about the matter content or the dynamics.', '1808.04597-2-16-0': 'To look for a possible dipole in the deceleration parameter, we allow it to have a direction dependence: [EQUATION] where [MATH] and [MATH] are, respectively, the magnitudes of the dipole and monopole (the latter corresponding to what is usually called [MATH]), while [MATH] is the direction of the dipole and [MATH] describes its scale dependence.', '1808.04597-2-16-1': 'We consider four representative functional forms:', '1808.04597-2-17-0': '(a) No scale dependence: [MATH] independent of [MATH],', '1808.04597-2-18-0': "(b) 'Top hat': [MATH] for [MATH], and [MATH] otherwise,", '1808.04597-2-19-0': '(c) Exponential: [MATH], and', '1808.04597-2-20-0': '(d) Linear: [MATH].', '1808.04597-2-21-0': 'Due to the very anisotropic sky coverage of the dataset, we do not attempt to find [MATH], instead we choose it to be along the CMB dipole direction.', '1808.04597-2-21-1': 'This is reasonable as the directions of the reported bulk flows [CITATION] are all within [MATH] of each other and of the CMB dipole.', '1808.04597-2-22-0': 'Subsequently, we maximise a likelihood as constructed earlier [CITATION], simultaneously w.r.t. the 4 cosmological parameters [MATH] and [MATH], as well as the 8 parameters that go into the standardisation of the SNe Ia candles: [MATH] and [MATH].', '1808.04597-2-22-1': "Ours is a frequentist analysis, however it is equivalent to the 'Bayesian Hierarchical Model' of March et al. [CITATION] which in fact yielded the same result [CITATION] as the frequentist analysis of Nielsen et al. [CITATION] when applied to the JLA catalogue.", '1808.04597-2-23-0': '# Results', '1808.04597-2-24-0': 'The results of the fits described in [REF] are given in Table [REF].', '1808.04597-2-24-1': 'The first row presents the results of a 10-parameter fit without a dipole, using the SN-by-SN corrections to the redshift as included in JLA.', '1808.04597-2-24-2': 'We find the best fit value of [MATH] to be -0.314, and that no acceleration is disfavoured at slightly less than [MATH], completely consistent with previous principled statistical analyses [CITATION].', '1808.04597-2-24-3': 'Similar results were also found by Rubin Hayden [CITATION] (see the left panels of their Fig.2 and 3rd row, Fig.3).', '1808.04597-2-25-0': 'However the quality of fit improves further (-2 log [MATH] decreases) when [MATH] is allowed to have a dipole.', '1808.04597-2-25-1': 'In the best fit where this has an exponentially decaying form [MATH], the dipole [MATH] is larger than the monopole [MATH] and its scale parameter is [MATH] suggesting that the bulk flow extends out to far larger redshift than is expected in a [MATH]CDM universe.', '1808.04597-2-25-2': 'In the presence of this dipole, [MATH] is disfavoured at only about [MATH].', '1808.04597-2-25-3': 'In other words, in an universe where we have theoretical reasons to expect a dipolar modulation in the deceleration parameter in the direction of our motion through the CMB, there is no significant evidence for a non-zero value of its monopole component.', '1808.04597-2-25-4': 'Figure [REF] shows the 1, 2 and 3 [MATH] contours in the likelihood around the maximum as a function of [MATH] and [MATH], profiling over all other parameters.', '1808.04597-2-26-0': '# Discussion', '1808.04597-2-27-0': 'In this work we have used the statistical approach as well as the treatment of lightcurve parameters espoused by Nielsen et al [CITATION].', '1808.04597-2-27-1': 'These authors were criticised by Rubin Hayden [CITATION] for using redshift-independent distributions for [MATH] and [MATH].', '1808.04597-2-27-2': 'In this respect we note the following:', '1808.04597-2-28-0': 'The fit for determination of cosmological parameters [CITATION] is to determine the relationship between the luminosity distance [MATH] and the redshift.', '1808.04597-2-28-1': 'To inspect a posteriori the distribution of two ([MATH] and [MATH]) out of the three ingredients that go into standardising SNe Ia, and add empirical terms to describe their sample dependence and redshift evolution in the fit, is fundamentally against the principles of blind hypothesis testing, especially since no such dependence had been suggested by the JLA analysis [CITATION].', '1808.04597-2-29-0': 'We nevertheless carry out the same 16- and 22-parameter fits [CITATION] and present the results in Table [REF].', '1808.04597-2-29-1': 'While the log maximum likelihood ratio improves to 12.5 and 16 respectively for these fits, both these parameterisations also allow for a large dipole in [MATH] out to [MATH] and [MATH] respectively, which reduces the likelihood in favour of acceleration.', '1808.04597-2-30-0': 'When progressively adding more parameters to improve the quality of a fit and obtain a desired outcome, the parameters have to be justified by both physical and information theoretic arguments.', '1808.04597-2-30-1': 'The additional parameters of Rubin Hayden [CITATION] can be justified by the Akaike information criterion (AIC) but not by the Bayesian information criterion (BIC).', '1808.04597-2-30-2': 'The same applies to the two additional parameters we introduce ([MATH] and [MATH]) however there is a physical motivation to expect a scale-dependent dipole in an inhomogeneous universe [CITATION] so the BIC need not apply [CITATION].', '1808.04597-2-31-0': 'If the light curve parameters [MATH] and [MATH] are allowed to be sample/redshift dependent one can ask why the absolute magnitude of SNe Ia should also not be sample/redshift-dependent.', '1808.04597-2-31-1': "Allowing this of course undermines their use as 'standard candles' and the data is then unsurprisingly consistent with no acceleration, as can be seen in Table [REF] (see also Tutusaus et al. [CITATION]).", '1808.04597-2-31-2': 'Note that the additional 3 parameters describing the sample dependence of [MATH] are justified by the AIC.', '1808.04597-2-32-0': 'It has been observed, e.g. by Bernal et al. [CITATION] that the deceleration parameter inferred from previous SNe Ia datasets has a redshift, and indeed directional, dependence.', '1808.04597-2-32-1': 'This was interpreted as indicative of local anisotropy in the matter distribution, i.e. our being located in an asymmetric void.', '1808.04597-2-32-2': 'The refinement in the present work is that we consider the most recent comprehensive database of SNe Ia and take into account all systematic effects as encoded in the covariance matrices provided [CITATION].', '1808.04597-2-32-3': 'Moreover we focus on the local velocity rather than the density field as this fully reflects the gravitational dynamics due to inhomogeneities.', '1808.04597-2-32-4': "We can then explore the expected consequences of our being 'tilted' i.e. non-Copernican observers.", '1808.04597-2-32-5': 'Our analysis is guided by the suggestion that we may then infer acceleration even when the overall expansion rate is decelerating - a signature of which would be a dipolar modulation of the inferred [MATH] along the direction of the bulk flow [CITATION].', '1808.04597-2-33-0': 'The effect of peculiar velocities on SNe Ia cosmology has been previously studied [CITATION], however these studies relied on covariances that apply to Copernican observers.', '1808.04597-2-33-1': 'As we show in an accompanying paper [CITATION], although the bulk flow we are embedded in is rare at only about the [MATH] level [CITATION] according to the DarkSky simulation [CITATION], the conditional covariances can be a factor of 10 larger and introduce a preferred direction locally.', '1808.04597-2-33-2': 'This can make a much bigger impact on cosmological inferences than was found in these previous studies.', '1808.04597-2-33-3': 'In particular the JLA analysis [CITATION] of the same dataset concluded that the effect of peculiar velocities is only a tiny ([MATH]) shift in the best-fit cosmological parameters.', '1808.04597-2-34-0': 'In summary, the model-independent evidence for acceleration of the Hubble expansion rate from the largest public catalogue of Type Ia supernovae is only [MATH].', '1808.04597-2-34-1': 'This is in stark contrast to the claim [CITATION] that acceleration is established by SNe Ia at [MATH] in the framework of the [MATH]CDM model.', '1808.04597-2-34-2': 'Moreover there is an indication for a dipole in [MATH] towards the CMB dipole - as is expected if the apparent acceleration is an artefact of our being located in a local bulk flow which extends out far enough to include most of the supernovae studied [CITATION].', '1808.04597-2-34-3': 'Given the observational evidence that there is no convergence to the CMB frame as far out as redshift [MATH] which includes half the known SNe Ia, this possibility must be taken seriously.', '1808.04597-2-35-0': 'It is often stated that regardless of the marginal evidence for cosmic acceleration from the SNe I a Hubble diagram there are other reasons to believe that the expansion rate is accelerating.', '1808.04597-2-35-1': 'It has been shown [CITATION] however that all low redshift probes e.g. measurements of Baryon Acoustic Oscillations (BAO), of [MATH] from stellar chronometry, and of the growth of matter perturbations [MATH], are also consistent with no acceleration.', '1808.04597-2-35-2': 'The relevant present datasets are simply not large enough and there are significant systematic uncertainties, e.g. to determine the BAO scale requires adopting a fiducial model like [MATH]CDM itself in order to translate angular and redshift separations to comoving distances, and only the location of the BAO feature is allowed to shift relative to the model expectation.', '1808.04597-2-35-3': "Moreover the precision data on CMB anisotropies [CITATION] do not directly probe cosmic acceleration so evidence for [MATH] can only be inferred using the 'cosmic sum rule' ([REF]) for the assumed [MATH]CDM model.", '1808.04597-2-35-4': 'Whether the universe is indeed dominated by dark energy thus remains an open question.', '1808.04597-2-36-0': 'Looking forward, the Large Synoptic Survey Telescope (https://www.lsst.org/) is expected to discover 3-4 million supernovae during its 10 year survey so should convincingly test whether there is indeed a dipole in the deceleration parameter aligned with the CMB dipole.'}

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[]

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[]

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{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '3': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1808.04597

{'1808.04597-3-0-0': "Observations reveal a 'bulk flow' in the local Universe which is faster and extends to much larger scales than is expected around a typical observer in the standard [MATH]CDM cosmology.", '1808.04597-3-0-1': 'This is expected to result in a scale-dependent dipolar modulation of the acceleration of the expansion rate inferred from observations of objects within the bulk flow.', '1808.04597-3-0-2': 'From a maximum-likelihood analysis of the Joint Lightcurve Analysis (JLA) catalogue of Type Ia supernovae we find that the deceleration parameter, in addition to a small monopole, indeed has a much bigger dipole component aligned with the CMB dipole which falls exponentially with redshift [MATH]: [MATH].', '1808.04597-3-0-3': 'The best fit to data yields [MATH] and [MATH], rejecting isotropy ([MATH]) with [MATH] statistical significance, while [MATH] and consistent with no acceleration ([MATH]) at [MATH].', '1808.04597-3-0-4': "Thus the cosmic acceleration deduced from supernovae may be an artefact of our being non-Copernican observers, rather than evidence for a dominant component of 'dark energy' in the Universe.", '1808.04597-3-1-0': '# Introduction', '1808.04597-3-2-0': 'The foundations of the current standard model of cosmology date back nearly a century to when there was essentially no data available.', '1808.04597-3-2-1': 'In particular the Universe was assumed to be exactly isotropic and homogeneous, with space-time described by the maximally symmetric Friedmann-Lemaitre-Robertson-Walker metric, and occupied by ideal fluids with purely diagonal energy-momentum tensors .', '1808.04597-3-2-2': 'Subsequently it has been recognised that the distribution of galaxies, which is a biased tracer of the underlying distribution of the dominant dark matter, is in fact rather inhomogeneous.', '1808.04597-3-2-3': 'Counts-in-spheres of galaxy catalogues suggest that there is a transition to (statistical) homogeneity on scales exceeding [MATH] Mpc although sufficiently large volumes have not yet been surveyed to establish this definitively.', '1808.04597-3-2-4': 'This is however the expectation in the current standard cosmological model if the observed large-scale structure has grown under gravity in the sea of dark matter, starting with an initially gaussian random field of small density perturbations with an approximately scale-invariant spectrum.', '1808.04597-3-2-5': 'Detailed observations of the temperature fluctuations in the cosmic microwave background (CMB) have broadly confirmed this model .', '1808.04597-3-2-6': 'However several anomalies have been noted, e.g. the lack of correlations on large angular scales, the quadrupole-octupole alignment, and the hemispherical power asymmetry - which seem to imply a violation of statistical isotropy and scale-invariance of primordial perturbations - although there is no consensus yet on either their physical nature or their origin .', '1808.04597-3-3-0': "In our real Universe there are 'peculiar motions' due to the local inhomogeneity and anisotropy of surrounding structure.", '1808.04597-3-3-1': 'These are non-negligible, e.g. our Local Group of galaxies moves with respect to the universal expansion at [MATH] towards [MATH], as is inferred from the observed dipolar modulation of the CMB temperature .', '1808.04597-3-3-2': 'Moreover diverse observations e.g. [CITATION], reaching out as far as [MATH] Mpc, have not seen the expected [MATH] fall-off of the peculiar velocity in the standard [MATH]CDM cosmology.', '1808.04597-3-3-3': 'The odds of this happening by chance in that framework can be estimated by querying Hubble volume simulations of large-scale structure formation e.g. Dark Sky .', '1808.04597-3-3-4': 'In fact less than 1% of Milky Way-like observers should observe the bulk flow ([MATH] extending to [MATH]) that we do .', '1808.04597-3-3-5': "Thus we are not comoving observers but are 'tilted' relative to the idealised Hubble flow .", '1808.04597-3-3-6': 'The implications of this have been discussed for measurements of the Hubble parameter [MATH] , but not for the inference of cosmic acceleration.', '1808.04597-3-4-0': "Since cosmological observables are formulated in the 'CMB frame' in which the Universe is supposedly perfectly isotropic, it is in any case always necessary to correct what we measure from our relative moving frame.", '1808.04597-3-4-1': 'For example the observed redshifts of the Type Ia supernovae (SNe Ia) in catalogues like JLA have been corrected in order to convert from the heliocentric frame to the CMB frame.', '1808.04597-3-4-2': 'The methodology used follows earlier work which used bulk flow observations made back in 2004 and moreover assumed that there is convergence to the CMB frame beyond 150 Mpc.', '1808.04597-3-4-3': 'Since this is not in accordance with subsequent deeper observations, we use only the heliocentric redshifts and reverse the corrections applied to the magnitudes in order to examine whether the deceleration parameter measured in our (bulk flow) rest frame can indeed differ from that of comoving observers in the model universe .', '1808.04597-3-4-4': 'Such theoretical considerations imply that there should be a dipole asymmetry in the derived cosmic deceleration parameter [MATH] towards the bulk flow direction.', '1808.04597-3-4-5': 'In this work we do find a significant ([MATH]) indication of such a dipole, and also that the monopole in [MATH] decreases simultaneously in significance (to [MATH]).', '1808.04597-3-4-6': 'Hence not only is the indication for acceleration statistically marginal , it probably arises due to our being tilted observers located in a bulk flow, rather than being the effect of a cosmological constant or dark energy.', '1808.04597-3-5-0': "# The 'Joint Lightcurve Analysis'", '1808.04597-3-6-0': "We use the most up to date publicly available sample of supernova lightcurve properties and directions: the SDSS-II/SNLS3 'Joint Lightcurve Analysis' (JLA) catalogue .", '1808.04597-3-6-1': "This consists of 740 spectroscopically confirmed SNe Ia, including several low redshift ([MATH]) samples, three seasons of SDSS-II ([MATH]) and three years of SNLS ([MATH] data, all calibrated consistently in the 'Spectral Adaptive Lightcurve Template 2' (SALT2) scheme.", '1808.04597-3-6-2': "This assigns to each supernova 3 parameters: the apparent magnitude [MATH] at maximum in the rest frame '[MATH]-band', and the light curve shape and colour corrections, [MATH] and [MATH].", '1808.04597-3-6-3': "The distance modulus is then given by: [EQUATION] where [MATH] and [MATH] are assumed to be constants, as is [MATH] the absolute SNe Ia magnitude, as befits a 'standard candle'.", '1808.04597-3-6-4': 'In the standard [MATH]CDM cosmological model this is related to the luminosity distance [MATH] as: [EQUATION]', '1808.04597-3-6-5': 'Here [MATH] is the Hubble distance and [MATH] the Hubble parameter ([MATH] being its present value), and [MATH] are the matter, cosmological constant and curvature densities in units of the critical density.', '1808.04597-3-6-6': "In the [MATH]CDM model these are related by the 'cosmic sum rule': [MATH].", '1808.04597-3-6-7': 'However we make no such model assumptions and will simply expand the luminosity distance [MATH] in a Taylor series in order to examine its second derivative i.e. the acceleration (see [REF]).', '1808.04597-3-6-8': 'This is because acceleration is a kinematic quantity and can be measured without making any assumptions about the dynamics underlying the universal expansion.', '1808.04597-3-6-9': '(There may be concern that such a Taylor expansion fails at high redshift, however we have verified that [MATH] in the best-fit [MATH]CDM model differs by only 7% even at [MATH] (the highest redshift in the JLA sample), which is much less than the measurement uncertainty.', '1808.04597-3-6-10': 'Indeed the Taylor expansion fits the data just as well as [MATH]CDM.)', '1808.04597-3-7-0': 'Figure [REF] is a Mollewide projection of the directions of the 740 SNe Ia in Galactic coordinates.', '1808.04597-3-7-1': 'Due to the diverse survey strategies of the sub-samples that make up the JLA catalogue, its sky coverage is patchy and anisotropic.', '1808.04597-3-7-2': 'While the low redshift objects are spread out unevenly across the sky, the intermediate redshift ones from SDSS are mainly confined to a narrow disk at low declination, while the high redshift ones from SNLS are clustered along 4 specific directions.', '1808.04597-3-8-0': 'The JLA analysis corrects the observed redshifts in the heliocentric frame, [MATH], in order to obtain the cosmological redshifts, [MATH], after accounting for peculiar motions in the local Universe.', '1808.04597-3-8-1': 'These corrections are carried over unchanged from an earlier analysis , which in turn cites an earlier method and the peculiar velocity model of Hudson et al. .', '1808.04597-3-8-2': 'It is stated that the inclusion of these corrections allow SNe Ia with redshifts down to 0.01 to be included in the cosmological analysis, in contrast to earlier analyses which employed only SNe Ia down to [MATH].', '1808.04597-3-9-0': 'In Figure [REF] we scrutinise these corrections by exhibiting the velocity parameter [MATH], defined as [EQUATION] where [MATH] and [MATH] are as tabulated by JLA, while [MATH] is given by [EQUATION] where [MATH] is 369 km s[MATH] in the direction of the CMB dipole, and [MATH] is the unit vector in the direction of the supernova.', '1808.04597-3-9-1': 'It can be seen in Figure [REF] that SNe Ia beyond [MATH] have been assumed to be stationary w.r.t. the CMB rest frame, and corrections applied only to those at lower redshifts.', '1808.04597-3-9-2': 'It is not clear how these corrections were made beyond [MATH], which is the maximum extent to which the Streaming Motions of Abell Clusters (SMAC) sample extends.', '1808.04597-3-9-3': 'This has a bulk velocity of [MATH] km s[MATH] towards [MATH] out to [MATH] at 90%C.L., and a bulk velocity of [MATH] km s[MATH] towards [MATH] generated by sources beyond 200 [MATH] Mpc ([MATH]) at 98% C.L.', '1808.04597-3-9-4': 'If the peculiar velocity field is not discontinuous, the SNe Ia immediately outside this volume should have comparable velocities.', '1808.04597-3-9-5': "Figure [REF] indicates however that the JLA peculiar velocity 'corrections' have arbitrarily assumed that the bulk flow abruptly disappears at this point!", '1808.04597-3-9-6': 'The JLA analysis allows SNe Ia beyond this distance to only have an uncorrelated velocity dispersion of [MATH] km s[MATH].', '1808.04597-3-9-7': 'In the absence of any evidence of convergence to the CMB rest frame, this assumption is unjustified since it is it quite possible that the observed bulk flow stretches out to much larger scales.', '1808.04597-3-9-8': "In fact there have been persistent claims of a 'dark flow' extending out to several hundreds of Mpc , although this is still under debate.", '1808.04597-3-9-9': 'At any rate the value of [MATH] should be determined by fitting to the data, rather than put in by hand.', '1808.04597-3-10-0': 'At this point it is worth noting the anisotropy of the JLA catalogue.', '1808.04597-3-10-1': 'Out of the 740 SNe Ia, 551 are in the hemisphere pointing away from the CMB dipole.', '1808.04597-3-10-2': 'With respect to the [MATH] km s[MATH] bulk flow of the model using which the redshifts of the local SNe Ia have been corrected, only 108 are in the upper hemisphere while 632 are in the lower hemisphere.', '1808.04597-3-10-3': 'With respect to the direction of the abnormally high flow reported by 6dFGSv, the largest and most homogeneous peculiar velocity sample of nearly 9000 galaxies , 103 SNe Ia are in the upper hemisphere while 637 are in the lower hemisphere.', '1808.04597-3-11-0': 'The subsequent Pantheon catalogue, which incorporates 308 additional SNe Ia (many from the Pan-STARRS survey), continues to suffer from these problems.', '1808.04597-3-11-1': 'While the flow model using which the redshifts of the Pantheon sample have been corrected go out to [MATH], this model has a residual bulk flow of [MATH] km s[MATH] attributed to sources beyond [MATH], and 890 of the 1048 Pantheon SNe are in the hemisphere opposite to the direction of this flow.', '1808.04597-3-12-0': 'Both JLA and Pantheon include SNe to which anomalously large peculiar velocity corrections have been applied at redshifts far higher than the limit to which the corresponding flow models extend.', '1808.04597-3-12-1': 'Two of the many such examples are SDSS2308 in JLA at [MATH] in JLA (the outlier in Figure 2) and SN2246 in Pantheon at [MATH].', '1808.04597-3-13-0': 'We use the heliocentric redshifts tabulated by JLA and subtract out the bias corrections applied to [MATH].', '1808.04597-3-13-1': 'For the Pantheon catalogue the [MATH] values and individual contributions to the covariance are not public, and moreover there are unresolved concerns about the accuracy of the data therein so we cannot use it.', '1808.04597-3-14-0': '# Cosmological analysis', '1808.04597-3-15-0': 'We now compare the distance modulus (equation [REF]) obtained from the JLA sample with the apparent magnitude (equation [REF]) using the Maximum Likelihood Estimator .', '1808.04597-3-15-1': "For the luminosity distance we use its kinematic Taylor series expansion up to the third term since we wish to analyse the data without making assumptions about the matter content or the dynamics: [EQUATION] where [MATH] is the cosmic deceleration parameter in the Hubble flow frame, defined in terms of the scale factor of the universe [MATH] and its derivatives w.r.t. proper time, [MATH] is the cosmic 'jerk' [MATH], and [MATH] is just [MATH].", '1808.04597-3-15-2': 'Note that the last two appear together in the coefficient of the [MATH] term so cannot be determined separately.', '1808.04597-3-15-3': 'In the [MATH]CDM model: [MATH].', '1808.04597-3-16-0': 'To look for a dipole in the deceleration parameter, we allow it to have a direction dependence: [EQUATION] where [MATH] and [MATH] are the monopole and dipole components, while [MATH] is the direction of the dipole and [MATH] describes its scale dependence.', '1808.04597-3-16-1': 'We consider four representative functional forms:', '1808.04597-3-17-0': '(a) Constant: [MATH] independent of [MATH],', '1808.04597-3-18-0': '(b) Top hat: [MATH] for [MATH], and [MATH] otherwise,', '1808.04597-3-19-0': '(c) Linear: [MATH], and', '1808.04597-3-20-0': '(d) Exponential: [MATH].', '1808.04597-3-21-0': 'Due to the anisotropic sky coverage of the dataset, it would be hard to find [MATH] from the data, so we choose it to be along the CMB dipole direction.', '1808.04597-3-21-1': 'This is reasonable as the directions of the reported bulk flows are all within [MATH] of each other and of the CMB dipole.', '1808.04597-3-21-2': '(Later we allow the direction to vary as an a posteriori test to demonstrate that our result is indeed robust.)', '1808.04597-3-22-0': 'We maximise a likelihood constructed earlier , simultaneously with respect to the 4 cosmological parameters [MATH] and [MATH], as well as the 8 parameters that go into the standardisation of the SNe Ia candles: [MATH] and [MATH].', '1808.04597-3-22-1': "While our analysis is frequentist, it is equivalent to the 'Bayesian Hierarchical Model' of March et al. (which indeed yielded the same result as the frequentist analysis of [CITATION] when applied to the JLA catalogue).", '1808.04597-3-23-0': 'In Table [REF] we show how this compares with using the prevalent "constrained [MATH] method" used by e.g. [CITATION], wherein an arbitrary error [MATH] is added to each data point and varied until a good fit (with [MATH]/d.o.f.) is obtained to the assumed model.', '1808.04597-3-23-1': 'This may be appropriate for parameter estimation, but not for model selection.', '1808.04597-3-24-0': 'However as seen in Table [REF] the quality of fit improves further (-2 log [MATH] decreases) when [MATH] is allowed to have a dipole.', '1808.04597-3-24-1': 'In the best fit where this has an exponentially decaying form [MATH], the dipole [MATH] is much larger than the monopole [MATH] and its scale parameter is [MATH] indicating that the impact of the bulk flow dominates over any isotropic acceleration out to [MATH].', '1808.04597-3-24-2': "Since [MATH] between the model with [MATH] and the model with [MATH] is 9.86, this constitutes 'strong' evidence against a universe that is accelerating isotropically.", '1808.04597-3-24-3': 'In the presence of this dipole, [MATH] is disfavoured at only [MATH].', '1808.04597-3-24-4': 'In other words, in a universe where we have theoretical reasons to expect a dipolar modulation in the deceleration parameter in the direction of our motion through the CMB, there is no significant evidence for a non-zero value of its monopole component.', '1808.04597-3-24-5': 'Figure 4 shows the 1, 2 and 3 [MATH] contours in the likelihood around the maximum as a function of [MATH] and [MATH], profiling over all other parameters.', '1808.04597-3-25-0': 'We also study the effect of allowing an additional uncorrelated velocity dispersion [MATH] in the fit - rather than fixing it to be [MATH] km s[MATH] as in the JLA analysis .', '1808.04597-3-25-1': 'As shown in Table [REF] this improves the overall fit even further for [MATH] km s[MATH]; the best-fit dipole drops a little to [MATH], while the monopole is nearly unchanged at [MATH].', '1808.04597-3-25-2': "The [MATH] between the model with [MATH] and the one with [MATH] is 4.91, providing 'positive' evidence against a universe that is accelerating isotropically.", '1808.04597-3-25-3': 'Our main result is thus robust in that the maximum likelihood estimator prefers to interpret the data as evidence of a dipole in the deceleration parameter aligned with the CMB dipole, rather than as an isotropic acceleration of the universe which may indicate the presence of a cosmological constant.', '1808.04597-3-26-0': 'As an a posteriori test, we examine the direction dependence of this scale-dependent dipolar modulation in [MATH], by scanning the direction of [MATH] on a grid corresponding to a HEALpix map of nside=8.', '1808.04597-3-26-1': 'The best fit direction is 23 degrees away from the CMB dipole, where [MATH] increases to -9.851 but -2 log [MATH] improves by only 3.22.', '1808.04597-3-26-2': 'This demonstrates that the direction of the anisotropy we find is also robust.', '1808.04597-3-27-0': '# Discussion', '1808.04597-3-28-0': 'It has been observed that the deceleration parameter inferred from previous SNe Ia datasets has a redshift, and indeed directional, dependence.', '1808.04597-3-28-1': 'This was interpreted as indicative of local anisotropy in the matter distribution, i.e. our being located in an asymmetric void.', '1808.04597-3-28-2': 'The refinement in the present work is that we consider a recent comprehensive database of SNe Ia and take into account all systematic effects as encoded in the covariance matrices provided .', '1808.04597-3-28-3': 'Moreover we focus on the local velocity rather than the density field as this fully reflects the gravitational dynamics due to inhomogeneities.', '1808.04597-3-28-4': "We can then explore the expected consequences of our being 'tilted' i.e. non-Copernican observers.", '1808.04597-3-28-5': 'Our analysis is guided by the suggestion that we may then infer acceleration even when the overall expansion rate is decelerating - a signature of which would be a dipolar modulation of the inferred [MATH] along the direction of the bulk flow .', '1808.04597-3-29-0': 'The effect of peculiar velocities on SNe Ia cosmology has been discussed earlier , however these studies relied on covariances that apply to Copernican observers.', '1808.04597-3-29-1': 'As we show elsewhere , the bulk flow we are embedded in is rare at a level of [MATH] according to the DarkSky simulation , but the conditional covariances can be up to a factor of [MATH] larger and introduce a preferred direction locally.', '1808.04597-3-29-2': 'This can make a much bigger impact on cosmological inferences than was found in previous studies.', '1808.04597-3-29-3': 'In particular the JLA analysis of the same dataset claimed that the effect of peculiar velocities is a tiny ([MATH]) shift in the best-fit cosmological parameters.', '1808.04597-3-30-0': 'In summary, the model-independent evidence for acceleration of the Hubble expansion rate from the largest public catalogue of Type Ia supernovae is only [MATH].', '1808.04597-3-30-1': 'This is in contrast to the claim that acceleration is established by SNe Ia at [MATH] in the framework of the [MATH]CDM model.', '1808.04597-3-30-2': 'Moreover there is a significant ([MATH]) indication for a dipole in [MATH] towards the CMB dipole - as is indeed expected if the apparent acceleration is an artefact of our being located in a local bulk flow which extends out far enough to include most of the supernovae studied .', '1808.04597-3-30-3': 'Given the observational evidence that there is no convergence to the CMB frame as far out as redshift [MATH] which includes half the known SNe Ia, this possibility must be taken seriously.', '1808.04597-3-31-0': 'We thank the JLA collaboration for making all their data public and Dan Scolnic for correspondence concerning the Pantheon catalogue.', '1808.04597-3-31-1': 'We are grateful to Mike Hudson and Christos Tsagas for discussions.', '1808.04597-3-31-2': "MR acknowledges a Carlsberg distinguished postdoctoral fellowship and hospitality at the Institut d'Astrophysique, Paris.", '1808.04597-3-32-0': 'Code availability: The code used here is available at: https://github.com/rameez3333/DipoleJLA.', '1808.04597-3-33-0': '# Redshift-dependence of light curve fitting parameters', '1808.04597-3-34-0': 'In this work we have used the statistical approach as well as the treatment of lightcurve parameters espoused by [CITATION].', '1808.04597-3-34-1': 'These authors were criticised by [CITATION] for using redshift-independent distributions for [MATH] and [MATH].', '1808.04597-3-34-2': 'In this respect we note the following:', '1808.04597-3-35-0': 'The JLA analysis determined the relationship between the luminosity distance and redshift for SNe Ia.', '1808.04597-3-35-1': 'To inspect a posteriori the distribution of two ([MATH] and [MATH]) out of the three ingredients that go into standardising SNe Ia, and then add empirical terms in the fit to describe their sample dependence and redshift evolution, is fundamentally against the principles of blind hypothesis testing, especially since no such dependence had been suggested by [CITATION].', '1808.04597-3-36-0': 'Nevertheless we carry out the same 22-parameter fit for comparison and present the results in Table [REF].', '1808.04597-3-36-1': 'While the log maximum likelihood ratio does improve for these fits, this parameterisation actually increases the significance of the dipole in [MATH] to [MATH] (likelihood ratio of 18.3) and reduces further the significance of a monopole.', '1808.04597-3-37-0': 'The addition of parameters to improve the quality of a fit and obtain a desired outcome have to be justified by physical and/or information theoretic arguments.', '1808.04597-3-37-1': 'The additional parameters of [CITATION] can be justified by the Akaike information criterion but not by the stricter Bayesian information criterion.', '1808.04597-3-37-2': "This also applies to the two additional parameters we introduce ([MATH] and [MATH]) but these are physically motivated for a 'tilted observer' .", '1808.04597-3-38-0': 'If the light curve parameters [MATH] and [MATH] are allowed to be sample/redshift dependent one can ask why the absolute magnitude of SNe Ia should also not be sample/redshift-dependent.', '1808.04597-3-38-1': "Allowing this of course undermines their use as 'standard candles' and the data is then unsurprisingly consistent with no acceleration , as seen in Table [REF].", '1808.04597-3-39-0': '# Uncertainties', '1808.04597-3-40-0': 'The JLA covariance matrix includes uncertainties from the lightcurve template fitting process, calibration uncertainties, dust extinction in the Galaxy etc, together with the expected dispersion due to peculiar velocities (which mainly affects low redshift SNe) and lensing (which mainly affects high redshift SNe Ia), as well as the propagated uncertainties from the flow model using which the SN by SN peculiar velocity corrections are performed.', '1808.04597-3-40-1': 'In addition it is also necessary to fit for a global intrinsic dispersion as in previous analyses .', '1808.04597-3-40-2': 'We use heliocentric redshifts in this analysis so do not include uncertainties related to the peculiar velocity corrections based on the flow model.', '1808.04597-3-40-3': 'The redshift dependence of the dispersions in the fit are shown in Figure [REF].'}

1406.4864

{'1406.4864-1-0-0': 'Barotropic fluids, for which the pressure is only a function of the density, rotate on cylinders in the presence of a gravitational potential, so that the angular frequency of such a disk is independent of height.', '1406.4864-1-0-1': 'Therefore the shearing box framework, representing a small disk volume with height-independent angular frequency, can consistently model barotropic disks.', '1406.4864-1-0-2': 'If the fluid in the disk is baroclinic, the angular frequency does in general depend on height and it is thus necessary to go beyond the standard shearing box approach.', '1406.4864-1-0-3': 'In this paper, we show that given a global disk model, it is possible to develop consistent models that are local in horizontal planes and global in height with shearing-periodic boundary conditions.', '1406.4864-1-0-4': 'These models can be non-axisymmetric for globally barotropic disks but should be axisymmetric for globally baroclinic disks.', '1406.4864-1-0-5': 'We provide explicit equations for this vertically global shearing box which can be implemented in standard magnetohydrodynamic codes by generalizing the shearing-periodic boundary conditions to allow for a height-dependent angular frequency.', '1406.4864-1-0-6': 'We illustrate the potential for this framework by studying a vertical shear instability in disks with cylindrical temperature structure, and examining the modes associated with the magnetorotational instability in a disk model with spherical temperature structure.', '1406.4864-1-0-7': 'The framework provided by the vertically global shearing box will benefit the study of a wide variety of astrophysical phenomena in baroclinic disks; including instabilities, convection, turbulent transport, as well as the structure and dynamics of disk coronae and winds, and the interstellar medium in galactic disks.', '1406.4864-1-1-0': '# Introduction', '1406.4864-1-2-0': 'Astrophysical disks play a crucial role in the formation, evolution, and fate of a wide variety of celestial objects, by mediating the transport of mass, energy, and angular momentum.', '1406.4864-1-2-1': 'Building realistic disk models is of fundamental importance for understanding, for example, protoplanetary disks around young stars, accretion flows onto stellar compact objects and active galactic nuclei, as well as the interstellar medium in galactic disks.', '1406.4864-1-2-2': 'The large dynamic range involved makes it particularly challenging to produce detailed global numerical simulations of these systems.', '1406.4864-1-2-3': 'Moreover, while global models allow us to investigate large-scale phenomena, local models with a hierarchy of increasingly complex microphysics have proved critical to elucidating the processes that are crucial at small scales.', '1406.4864-1-2-4': 'Because of this, several types of local approximations have been employed for studying astrophysical disks.', '1406.4864-1-3-0': '[CITATION] pioneered the use of a local approximation to study the dynamics of particles orbiting a host system and subject to encounters with a perturber, and used it to study the motion of the Moon.', '1406.4864-1-3-1': '[CITATION] and [CITATION] applied these ideas to galactic disks using the concept of a locally shearing coordinate system.', '1406.4864-1-3-2': 'This approach constitutes the basis of the shearing sheet framework which has been widely used to study the dynamics of orbiting particles and planetesimals, as well as local processes in hydrodynamic and magnetohydrodynamic disks.', '1406.4864-1-3-3': 'The implementation of the concept behind the shearing sheet, with an appropriate shear-periodic radial-boundary condition , forms a computational model used for studying local disk dynamics referred to as the shearing box .', '1406.4864-1-4-0': 'Shearing box models solve the equations of motion for the fluid in a local cartesian frame co-rotating with the disk at a fiducial radius.', '1406.4864-1-4-1': 'In the standard framework, the differential rotation of the disk is locally accounted for with a height-independent angular frequency.', '1406.4864-1-4-2': 'This is appropriate for barotropic disks for which the pressure is only a function of density and thus rotate on cylinders.', '1406.4864-1-4-3': 'The standard shearing box framework (SSB) relies on a first order expansion of the steady bulk flow in the radial direction, which is the highest order compatible with shearing periodic boundary conditions.', '1406.4864-1-4-4': 'Depending on whether zeroth or first order expansions are considered for the gravitational field in the direction perpendicular to the disk, usually denoted with the coordinate [MATH], this leads to the so called unstratified or stratified shearing box models.', '1406.4864-1-4-5': 'These approximations are appropriate when the disk is thin and the vertical scales of interest are small compared to the fiducial disk radius.', '1406.4864-1-4-6': 'There have been works that retained the correct expression for the vertical component of the gravitational field , allowing for larger vertical domains to be considered.', '1406.4864-1-4-7': 'However, this generalization does not allow to study baroclinic disks without relaxing the assumption that the angular frequency is height-independent.', '1406.4864-1-4-8': 'In the early formulation of the shearing sheet, [CITATION] avoided making any approximation in height ([MATH]), which is possible when considering barotropic fluids.', '1406.4864-1-5-0': 'Baroclinic astrophysical disks, for which the pressure is not solely a function of density, posses angular frequency profiles that depend in general on height, especially if these are not thin.', '1406.4864-1-5-1': 'Therefore, building a framework to study these disks demands going beyond the standard shearing box, where the assumption that the angular frequency is height-independent is rooted deep.', '1406.4864-1-5-2': 'In this work, we generalize the standard shearing box by considering the full height-dependence of a steady state, axisymmetric bulk flow to leading order in radius, without making any expansion in the vertical coordinate.', '1406.4864-1-5-3': 'We show that given a global disk model, it is possible to develop consistent models that are local in horizontal planes and global in height and are amenable to shearing-periodic boundary conditions.', '1406.4864-1-5-4': 'These models can be non-axisymmetric for global barotropic disks but should be axisymmetric for global baroclinic disks.', '1406.4864-1-5-5': 'We term the resulting framework the vertically global shearing box (VGSB).', '1406.4864-1-5-6': 'This formalism provides a novel framework to investigate dynamical aspects of astrophysical disks that cannot be studied with the standard shearing box and for which full global modeling is too demanding.', '1406.4864-1-6-0': 'The paper is organized as follows.', '1406.4864-1-6-1': 'We derive the equations involved in the VGSB framework in Section [REF], providing some of the algebraic details in Appendix [REF].', '1406.4864-1-6-2': 'We state the final form of the VGSB equations and discuss its novel features in Section [REF].', '1406.4864-1-6-3': 'For convenience, we provide a self-contained summary of the VGSB equations that can be incorporated in magnetohydrodynamic codes in Appendix [REF].', '1406.4864-1-6-4': 'We use this new framework to explore the behavior of two important instabilities in a baroclinic context.', '1406.4864-1-6-5': 'We demonstrate that a linear vertical shear instability, akin to those studied by [CITATION] and [CITATION], appears in the VGSB in Section [REF].', '1406.4864-1-6-6': 'We examine some basic aspects of the magnetorotational instability in the VGSB in Section [REF].', '1406.4864-1-6-7': 'We conclude by briefly discussing several potential applications of the VGSB in Section [REF].', '1406.4864-1-7-0': '# Equations of Motion', '1406.4864-1-8-0': 'We are concerned with the equations of ideal magnetohydrodynamics, in cylindrical coordinates [MATH], in a reference frame rotating with angular frequency [MATH], i.e., [EQUATION]', '1406.4864-1-8-1': 'Here, [MATH] is the mass density, [MATH] is the fluid velocity in the rotating frame, [MATH] is the magnetic field, with [MATH], [MATH] is the internal energy density, [MATH] is the pressure determined through an equation of state, and [MATH] is the gravitational potential, which is assumed to be cylindrically symmetric, but not necessarily spherical.', '1406.4864-1-8-2': 'The current density is [MATH], with [MATH] a constant dependent on the unit system adopted.', '1406.4864-1-9-0': 'Fluid flows described by these equations are subject to conservation laws.', '1406.4864-1-9-1': 'It is thus important to understand under what circumstances these properties are satisfied by the equations describing the local dynamics involving expansions of the original set of equations.', '1406.4864-1-9-2': "It is easy to show that the approximations embodied in the standard isothermal shearing box are such that the vortex lines of an inviscid flow are frozen into the fluid (Kelvin's Circulation Theorem) and that the magnetic flux is also frozen into the fluid in the absence of magnetic dissipation (Alfven's Frozen-in Theorem).", '1406.4864-1-10-0': 'Understanding under what conditions these properties also hold for the equations of motion that result from invoking a local approximation of global disk models which are baroclinic is more subtle.', '1406.4864-1-10-1': 'Here, we state the general versions of the aforementioned theorems in order to prepare the ground to address these issues in subsequent sections.', '1406.4864-1-10-2': 'These conservation theorems can be derived by calculating the Lagrangian derivative of the fluxes associated with the vorticity and the magnetic field.', '1406.4864-1-10-3': 'It is thus useful to recall that, see e.g., , any vector field [MATH] satisfies [EQUATION] where the integral is carried over an open surface advected by the flow with velocity [MATH].', '1406.4864-1-11-0': "## Kelvin's Circulation Theorem", '1406.4864-1-12-0': 'The momentum equation ([REF]) for an an inviscid, unmagnetized, barotropic flow in the rotating frame is given by [EQUATION] where [MATH] is the generalized gravito-thermal potential, where [MATH] is the enthalpy, with [MATH].', '1406.4864-1-12-1': 'The equation governing the evolution of the vorticity is thus [EQUATION] which preserves the solenoidal character of the vorticity.', '1406.4864-1-12-2': 'This implies that, by virtue of Equation ([REF]) with [MATH], vortex lines are frozen into the fluid, i.e., the flow preserves the circulation [MATH] [EQUATION] with [EQUATION] where [MATH] is a closed contour, delimiting the open surface [MATH], advected by the flow with velocity [MATH].', '1406.4864-1-13-0': "## Alfve n's Frozen-in Theorem", '1406.4864-1-14-0': 'The induction equation ([REF]) preserves the divergence of the magnetic field, i.e., [EQUATION]', '1406.4864-1-14-1': 'This implies that, provided that [MATH] at some initial time, the magnetic flux remains frozen into the fluid, i.e., [EQUATION] with [EQUATION]', '1406.4864-1-14-2': 'This follows from Equation ([REF]) with [MATH] and the induction equation ([REF]).', '1406.4864-1-15-0': '# The Vertically Global, Horizontally Local Approximation', '1406.4864-1-16-0': 'We seek to derive a set of equations that describe the local dynamics of the magnetized fluid with respect to a known steady state bulk flow around a point co-rotating with the disk at a distance [MATH].', '1406.4864-1-16-1': 'Here, we outline the steps of the derivation, which is carried out in detail below.', '1406.4864-1-17-0': 'We find a suitable steady flow and background equilibrium, which enables the derivation of exact equations of motion for the departures from this solution.', '1406.4864-1-18-0': 'We transform to a locally cartesian coordinate system.', '1406.4864-1-18-1': 'We expand the bulk flow and background equilibrium to leading order in the radial direction leaving the direction perpendicular to the disk midplane unaltered.', '1406.4864-1-19-0': 'We determine under which circumstances the resulting equations are amenable to being solved with shearing-periodic boundary conditions, which could depend on height for baroclinic disk models.', '1406.4864-1-20-0': 'We show explicitly these steps for the momentum and induction equations, while we state the results for the continuity equation and energy equation that are simpler to work with.', '1406.4864-1-21-0': '## Equations for Departures from Background Equilibrium', '1406.4864-1-22-0': '### Bulk Flow and Background Equilibrium', '1406.4864-1-23-0': 'As in the standard shearing box, we first seek a steady background flow.', '1406.4864-1-23-1': 'In this case, we will not make any a priori assumptions about the [MATH]-dependence of the angular frequency.', '1406.4864-1-24-0': 'We begin by noting that the momentum equation ([REF]) admits a force-free, steady state solution [MATH], with [EQUATION] where the angular frequency is [EQUATION] and the vertical, hydrostatic pressure gradient satisfies [EQUATION]', '1406.4864-1-24-1': 'Here, [MATH] and [MATH] are the mass density and energy density associated with the steady state background flow.', '1406.4864-1-25-0': '### Momentum and Induction Equations', '1406.4864-1-26-0': 'Using Equations ([REF]) and ([REF]) we can recast the gravitational force in the momentum equation ([REF]) in terms of the angular frequency and the pressure gradient, both corresponding to the steady state bulk flow.', '1406.4864-1-26-1': 'We obtain [EQUATION]', '1406.4864-1-26-2': 'The velocity field describing the departure from the bulk flow satisfying Equations ([REF])-([REF]) [EQUATION] evolves according to the momentum equation, [EQUATION]', '1406.4864-1-26-3': 'This equation is exact and it displays the particular feature that the last term on the left hand side resembles the Coriolis acceleration, with one important difference.', '1406.4864-1-26-4': 'The angular frequency involved is not the fixed angular frequency of the rotating frame, [MATH], but rather the angular frequency of the steady state flow, [MATH].', '1406.4864-1-26-5': 'As a quick consistency check, note that if [MATH] and [MATH] then there are no departures from the steady state bulk flow, i.e., [MATH].', '1406.4864-1-27-0': 'Taking the induction equation ([REF]) and replacing the velocity field with [MATH] yields, after some algebra, [EQUATION]', '1406.4864-1-28-0': '### Background-Flow Advection and Shear Rate', '1406.4864-1-29-0': 'The fact that the speed of the background flow [MATH] depends, in general, on height implies that the departures from the bulk flow will be advected and sheared in a height-dependent way.', '1406.4864-1-29-1': 'This motivates the definition of the advection operator [EQUATION] which is defined so that it acts on scalar fields, such as the density [MATH], and on each of the components of a vector field, e.g., [MATH] and [MATH], but not on the unit coordinate-vectors, i.e., [MATH].', '1406.4864-1-29-2': 'It is also convenient to define the shear rate [EQUATION]', '1406.4864-1-29-3': 'Using these definitions, Equations ([REF]) and ([REF]) become, without approximations, [EQUATION]', '1406.4864-1-30-0': '## Local Approximation in Horizontal Planes', '1406.4864-1-31-0': 'We now seek to derive a set of equations of motion which is local in radius and azimut by expanding Equation ([REF]) and ([REF]) around a fiducial point [MATH].', '1406.4864-1-31-1': 'In order to simplify this task, we choose a reference frame that corotates with the bulk flow at radius [MATH], i.e., [EQUATION]', '1406.4864-1-31-2': 'We also adopt a coordinate frame centered at [MATH] with locally cartesian coordinates [MATH], such that [MATH] and [MATH], with [MATH] and [MATH].', '1406.4864-1-31-3': 'In this locally cartesian frame, the differential vector operators are well approximated by their cartesian versions, provided that the radial coordinate versor [MATH] Because of the axisymmetric character of the background flow, in what follows, we choose [MATH] without loss of generality.', '1406.4864-1-32-0': 'In this locally cartesian coordinate system, we can expand to leading order in [MATH] the various functions appearing in the momentum and induction equations, Equation ([REF]) and ([REF]), respectively.', '1406.4864-1-33-0': '### Approximation of the Bulk Flow', '1406.4864-1-34-0': 'The local approximations of the angular frequency, the bulk flow in Equation ([REF]), and the advection operator in Equation ([REF]) yield [EQUATION]', '1406.4864-1-34-1': 'Here, we have defined the local, height-dependent angular frequency, bulk flow, and shear rate, all evaluated at the fiducial radius [MATH], i.e., [EQUATION]', '1406.4864-1-34-2': 'The operator [MATH] is a height-dependent generalization of the shearing sheet advection operator originally introduced in [CITATION].', '1406.4864-1-34-3': 'In order to illustrate the action of the advection operator [MATH], Figure [REF] shows the effects of considering the local bulk flow in Equation ([REF]) that results from expanding Equation ([REF]) to leading order in the radial direction, leaving unaltered the vertical dependence.', '1406.4864-1-34-4': 'In this particular example, we have considered a baroclinic global disk model with a cylindrical temperature structure, which is discussed in detail in Section [REF].', '1406.4864-1-35-0': '### Approximation of the Background Equilibrium', '1406.4864-1-36-0': 'The fourth term on the right hand side of Equation ([REF]) can be dealt with along the lines proposed by [CITATION] for the hydrostatic background quantities [EQUATION] where [MATH] and [MATH] are the leading order terms associated with the mass and internal energy density profiles of the background flow at the fiducial radius .', '1406.4864-1-36-1': 'Within the level of approximation we are working at, we thus have [EQUATION] where the vertical acceleration induced by the background pressure gradient is balanced by gravity at the fiducial radius [EQUATION] with [MATH], the gravitational potential evaluated at the fiducial radius [MATH].', '1406.4864-1-37-0': '### Approximation of Momentum and Induction Equations', '1406.4864-1-38-0': 'Using the approximations above, we arrive at expressions for the momentum and induction equations, Equation ([REF]) and ([REF]), which are correct to leading order in [MATH] and [MATH]: [EQUATION]', '1406.4864-1-38-1': 'Here, all the differential operators are defined in a cartesian coordinate system centered at the fiducial radius [MATH].', '1406.4864-1-39-0': 'Because we have consistently retained the leading order in all the approximations involving the bulk-flow, the two important flow properties discussed in Section [REF] remain unaltered.', '1406.4864-1-39-1': 'In the case of an inviscid, unmagnetized, barotropic flow the momentum equation ([REF]) that results from the local approximation in horizontal planes leads to [EQUATION]', '1406.4864-1-39-2': 'Furthermore, the induction equation ([REF]) that results from the local approximation in horizontal planes preserves the solenoidal character of the the magnetic field, i.e., [EQUATION]', '1406.4864-1-39-3': 'This implies that the magnetic flux is frozen into the fluid flow that results from the local expansion in horizontal planes, i.e., [EQUATION]', '1406.4864-1-39-4': "Therefore, the local approximation in horizontal planes leads to equations that still satisfy Kelvin's circulation theorem and Alfven's frozen in theorem discussed in Section [REF].", '1406.4864-1-40-0': '## Compatibility of the Local Approximation with Shearing-Periodic Boundary Conditions', '1406.4864-1-41-0': '### Shearing-Periodic Boundary Conditions in the SSB', '1406.4864-1-42-0': 'All the explicit coordinate dependences in the equations of motion defining the standard shearing box are contained in the advection operator [EQUATION] which is obtained as a limit of the operator [MATH] introduced in Equation ([REF]).', '1406.4864-1-42-1': 'The explicit dependence on the coordinate [MATH] can be eliminated by the linear transformation [MATH], [MATH], [MATH], and [EQUATION]', '1406.4864-1-42-2': 'In the primed coordinate system, the advection operator simply becomes [MATH], and the equations of motion can be solved by using strictly periodic boundary conditions in horizontal planes, i.e., [EQUATION] and appropriate boundary conditions for the vertical boundaries.', '1406.4864-1-42-3': 'Note that in the original coordinate system, the equations for the departures from the bulk flow satisfy shearing-periodic boundary conditions given by [EQUATION]', '1406.4864-1-43-0': '### Height-dependent Shearing-Periodic Boundary Conditions', '1406.4864-1-44-0': 'Defining a consistent set of equations and boundary conditions for baroclinic disks, for which the angular frequency is in general a function of height, is more subtle.', '1406.4864-1-45-0': 'The coordinate dependence arising through the advection operator [MATH] in Equation ([REF]) can still be removed by defining the linear transformation [MATH], [MATH], [MATH], and [EQUATION]', '1406.4864-1-45-1': 'In this primed coordinate system, the advection operator is coordinate-independent, i.e., [MATH].', '1406.4864-1-45-2': 'Therefore, if this were the only explicit coordinate dependence then, in each horizontal plane, it would be enough to consider the height-dependent, shearing-periodic boundary conditions given by [EQUATION]', '1406.4864-1-45-3': 'However, the coordinate dependence induced by the terms proportional to [EQUATION] on the right hand sides of Equations ([REF]) and ([REF]) cannot be eliminated by the same coordinate transformation that removes the [MATH]-dependence in [MATH].', '1406.4864-1-45-4': 'This prevents Equation ([REF]) and ([REF]) from being solved with the shearing-periodic boundary conditions in Equation ([REF]) in a straightforward way.', '1406.4864-1-46-0': 'In what follows we analyze the consequences of proceeding by neglecting the term proportional to the coordinate [MATH] in Equation ([REF]).', '1406.4864-1-46-1': 'We thus approximate this equation as [EQUATION]', '1406.4864-1-46-2': 'This approximation eliminates the explicit coordinate dependence on the right hand sides of Equations ([REF]) and ([REF]), leading to [EQUATION]', '1406.4864-1-46-3': 'Because the only coordinate dependences arise through the advection operator [MATH], these equations are compatible with the height-dependent, shearing-periodic boundary conditions ([REF]).', '1406.4864-1-46-4': 'However, the approximation embodied in Equation ([REF]) does, in general, affect the validity of the conservation theorems discussed in Section [REF].', '1406.4864-1-46-5': "We conclude this section by showing that Equations ([REF]) and ([REF]) do satisfy Kelvin's circulation theorem and Alfven's frozen-in theorem when the underlying global disk model is barotropic or when we consider axisymmetry.", '1406.4864-1-47-0': '### Consistent Vertically Global Hydrodynamic Disk Models', '1406.4864-1-48-0': 'It can be seen that equation ([REF]) when applied to an inviscid, barotropic, unmagnetized flow, leads to [EQUATION]', '1406.4864-1-48-1': "This means that under the conditions over which Kelvin's circulation theorem is satisfied, the approximation embodied in Equation ([REF]) does not lead to spurious sources of circulation when considering Equation ([REF]) instead of ([REF]).", '1406.4864-1-49-0': 'If the global disk model under consideration is baroclinic, the circulation [MATH] is no longer conserved and thus, for physical reasons, [MATH] is no longer expected to vanish.', '1406.4864-1-49-1': 'However, one should also realize that the approximation invoked in Equation ([REF]) leads to a source term that contributes spuriously to the evolution of the circulation [EQUATION]', '1406.4864-1-49-2': 'Here, the dots represent the physical sources of circulation present in fluids which are either viscous or baroclinic.', '1406.4864-1-49-3': 'The spurious source of circulation in Equation ([REF]) vanishes under axisymmetry.', '1406.4864-1-49-4': 'In order to demonstrate this, let us examine the integral involved.', '1406.4864-1-49-5': "Using Stoke's theorem, it follows that [EQUATION]", '1406.4864-1-49-6': 'In axisymmetry, i.e., [MATH], the problem reduces to understanding the dynamics of the fluid in the [MATH] plane.', '1406.4864-1-49-7': 'Under this condition, the line integral over a closed loop vanishes.', '1406.4864-1-50-0': 'We thus infer that Equation ([REF]) makes it possible to construct models for hydrodynamic disks, whether these are barotropic or baroclinic, that are local in horizontal planes but global in height and are compatible with the shearing-periodic boundaries.', '1406.4864-1-50-1': 'In the former case, the boundary conditions are height-independent, as in the standard shearing box, see Equation ([REF]), while in the latter case the boundary conditions depend on height, see Equation ([REF]).', '1406.4864-1-51-0': '### Consistent Vertically Global MHD Disk Models', '1406.4864-1-52-0': 'Let us now consider the implications for the induction equation.', '1406.4864-1-52-1': 'In general, neglecting the term proportional to the coordinate [MATH] in Equation ([REF]) leads to an approximated induction equation that no longer preserves the solenoidal character of the magnetic field.', '1406.4864-1-52-2': 'More specifically, Equation ([REF]) leads to [EQUATION]', '1406.4864-1-52-3': 'Taken at face value, this implies that Equation ([REF]) induces spurious generation of magnetic monopoles that will break flux freezing even in the absence of dissipation.', '1406.4864-1-52-4': 'Therefore, in the case of a magnetized fluid, while eliminating the explicit [MATH]-dependence that can not be removed by a coordinate transformation allows the induction equation to be solved with shearing-periodic, height-dependent boundary conditions, this approximation, in general, would destroy the solenoidal character of the magnetic field.', '1406.4864-1-53-0': 'However, the spurious evolution of the divergence of the magnetic field that results from the approximation Equation ([REF]) is absent in Equation ([REF]) if the underlying global disk model is barotropic or axisymmetry, i.e., [MATH], is considered.', '1406.4864-1-54-0': 'Therefore, given a global disk model, it is possible to develop consistent models that are local in horizontal planes and global in height that are amenable to shearing-periodic boundary conditions.', '1406.4864-1-54-1': 'These models can be non-axisymmetric for globally barotropic disks but should be axisymmetric for globally baroclinic disks.', '1406.4864-1-54-2': 'Under either of these conditions, the approximation invoked in Equation ([REF]), which is necessary to employ the shearing-periodic boundary conditions given by Equation ([REF]), does not lead to spurious source terms that could affect the evolution of the circulation, magnetic flux freezing, or the solenoidal character of the magnetic field, i.e., [EQUATION] and [EQUATION]', '1406.4864-1-54-3': 'This motivates the development of the vertically global shearing box.', '1406.4864-1-55-0': '# The Vertically Global Shearing Box (VGSB)', '1406.4864-1-56-0': '## Equations of Motion for the VGSB', '1406.4864-1-57-0': 'Following the steps outlined in the previous section, we arrive at the expressions for the continuity, momentum, induction, and energy equations that define the framework of the vertically global shearing box (VGSB) [EQUATION]', '1406.4864-1-57-1': 'Here, all the operators are defined in a cartesian coordinate system centered at the fiducial radius [MATH].', '1406.4864-1-58-0': 'The only explicit coordinate dependence in the VGSB arises through the advection operator [MATH], which is linear in the coordinate [MATH] for all heights [MATH].', '1406.4864-1-58-1': 'This implies that the equations are suitable for being solved with a vertically varying shear-periodic [MATH]-direction boundary condition.', '1406.4864-1-58-2': 'The radial and azimuthal boundary conditions for mapping a field variable [MATH] in a VGSB of size [MATH] are, respectively, [EQUATION]', '1406.4864-1-58-3': 'In order to completely define the problem, appropriate boundary conditions in the vertical direction must be specified.', '1406.4864-1-59-0': 'The set of Equations ([REF])-([REF]), together with the boundary conditions Equations ([REF])-([REF]) lead to consistent vertically global, radially local disk models if the underlying global disk model is barotropic or if we assume axisymmetry.', '1406.4864-1-60-0': 'It is important to emphasize that the velocity [MATH] is the departure from the local approximation of the bulk flow and thus the total fluid velocity [MATH] in the VGSB is [EQUATION] where [EQUATION] and the local shear rate [MATH] is related to the generalization of the height-independent [MATH]-parameter in the standard shearing box: [EQUATION]', '1406.4864-1-60-1': 'At the midplane, i.e., [MATH], the flow velocity [MATH] is the same as the steady state flow velocity in the standard shearing box.', '1406.4864-1-60-2': 'However, in global disk models where the shear rate of the flow decreases to zero at high altitude, i.e., [MATH] and, therefore, [MATH] for [MATH].', '1406.4864-1-60-3': 'This is just the reflex motion induced by the rotating frame.', '1406.4864-1-60-4': 'This shows that in the limit [MATH], Equations ([REF])-([REF]) reduce to the MHD equations in the absence of a gravitational field.', '1406.4864-1-61-0': '## The SSB as a limit of the VGSB', '1406.4864-1-62-0': 'In order to show that the isothermal, stratified standard shearing box is a limiting case of the VGSB, it is useful to recall some results that apply to barotropic fluids.', '1406.4864-1-62-1': 'If the pressure is [MATH] it follows that:', '1406.4864-1-63-0': 'Let us now focus our attention on a global isothermal disk in the gravitational potential of a point source of mass [MATH], i.e., [EQUATION] with [MATH] the gravitational constant.', '1406.4864-1-63-1': 'The pressure is [MATH], with constant sound speed [MATH], and hydrostatic balance in the vertical direction implies that the density is [EQUATION] where the constant [MATH] and we defined the Keplerian speed [EQUATION]', '1406.4864-1-63-2': 'In this case, the enthalpy is simply [MATH], and thus the generalized gravito-thermal potential [MATH], is [EQUATION]', '1406.4864-1-63-3': 'The angular frequency can be obtained from Equation ([REF]) as [EQUATION] where the Keplerian frequency is [EQUATION]', '1406.4864-1-63-4': 'The generalized gravito-thermal potential [MATH] and the angular frequency [MATH] are both independent of height, as expected.', '1406.4864-1-64-0': 'Expanding Equations ([REF]) and ([REF]) in radius around [MATH], we obtain expressions for the local values the angular frequency [MATH] and the shear rate [MATH] that can be used, in the framework of the VGSB, to study isothermal disks which are not necessarily thin compared to the local radius [MATH], i.e., [EQUATION] where we have defined [MATH] and [MATH].', '1406.4864-1-65-0': 'These expressions are useful to show that in the limit of a cold, thin disk, i.e., [MATH], the density profile becomes [MATH], with [MATH], and the angular frequency and shear rate become [MATH], [MATH], respectively, and Equations ([REF])-([REF]) reduce to the equations for the standard shearing box.', '1406.4864-1-66-0': '## New Aspects of the Baroclinic VGSB', '1406.4864-1-67-0': 'The approach that we followed in deriving the equations defining the framework of the vertically global shearing box is similar in spirit to the one employed in the derivation of the standard shearing box.', '1406.4864-1-67-1': 'This consists of expanding the steady sate bulk flow, retaining only the leading order terms in the ratio [MATH].', '1406.4864-1-67-2': 'The significant difference resides in that we have avoided making any expansion in the vertical direction.', '1406.4864-1-67-3': 'This allows us to retain the full height-dependence of the angular frequency.', '1406.4864-1-67-4': 'This is essential when dealing with disks in which the fluid is described by a baroclinic equation of state, since their angular frequency is in general a function of height.', '1406.4864-1-68-0': 'In order to illustrate the new aspects that are open to examination by retaining the full height-dependence in Equations ([REF]), ([REF]), and ([REF]), let us consider a baroclinic global disk model, in which the temperature is a function of the spherical radius, as in [CITATION].', '1406.4864-1-68-1': 'The four panels in Figure [REF] compare the temperature, density, pressure, and angular frequency corresponding to the global equilibrium of the disk model (labeled "Global"), the local equilibrium defining the vertically global shearing box (VGSB), and local equilibrium that would result in the standard shearing box (SSB).', '1406.4864-1-69-0': 'The VGSB provides a local representation of a global baroclinic disk model, capturing effects that the standard shearing box would not be able to account for.', '1406.4864-1-69-1': 'Because the lowest order discarded in Equation ([REF]) is [MATH], the radial dependence of the angular frequency is correct, to linear order in radius, for all heights.', '1406.4864-1-69-2': 'Moreover, the VGSB retains the correct global vertical gradients in temperature, density, and pressure, while neglecting the local radial variation of the isopycnic, isothermal, isobaric surfaces.', '1406.4864-1-69-3': 'This is in sharp contrast with the local equilibrium involved in the standard shearing box framework that is unable to capture the variation of the angular frequency with height, preventing its use when modeling baroclinic disks.', '1406.4864-1-69-4': 'Note also that, for the case under consideration, the equilibrium density and pressure profiles involved in the standard isothermal shearing box are less accurate than the ones associated with the VGSB.', '1406.4864-1-70-0': 'We now consider two families of baroclinic, global disk equilibria that are characterized by angular frequencies and shear rates that depend on height, and are thus impossible to study within the standard shearing box approximation.', '1406.4864-1-70-1': 'The VGSB framework, embodied in Equations ([REF])-([REF]), together with their associated boundary conditions, can be used to produce axisymmetric, vertically global, horizontally local models for these astrophysical disks.', '1406.4864-1-71-0': '## VGSB for Global Disk Models with Cylindrical Temperature Structure', '1406.4864-1-72-0': 'There is a family of disk models which are isothermal in the vertical direction at any fiducial radius.', '1406.4864-1-72-1': 'These have been used, for example, by [CITATION].', '1406.4864-1-72-2': 'The corresponding global hydrostatic equilibrium configuration is given by [EQUATION] where [MATH] stands for the sound speed, assuming an ideal gas, [EQUATION]', '1406.4864-1-72-3': 'The local expansions involved in the VGSB for this global baroclinic disk model are [EQUATION] with [MATH] and [MATH].', '1406.4864-1-73-0': 'Note that all the explicit [MATH]-dependences in [MATH] and [MATH] arise only when [MATH].', '1406.4864-1-73-1': 'This is in agreement with the isothermal case, corresponding to [MATH], in which the angular frequency is independent of height.', '1406.4864-1-73-2': 'It can also be seen from inspection of these equations that in the thin disk limit, i.e., [MATH], the VGSB reduces to the standard shearing box with [MATH], [MATH], and [MATH], provided that [MATH], even if the temperature depends on radius, i.e., [MATH].', '1406.4864-1-73-3': 'This is what ultimately justifies the use of a collection of standard shearing boxes to produce local barotropic models, with height-independent angular frequency at different radii, even though the underlying global disk model is baroclinic.', '1406.4864-1-74-0': '## VGSB for Global Disk Models with Spherical Temperature Structure', '1406.4864-1-75-0': 'A spherical temperature dependence of the form [EQUATION] leads to a global hydrostatic disk configuration given by [EQUATION]', '1406.4864-1-75-1': 'Here, the power-law coefficients [MATH] and [MATH] are related via [EQUATION] and the sound speed is [EQUATION] where [MATH] is the sound speed at the midplane at [MATH].', '1406.4864-1-75-2': 'Such models have been used, for example, by [CITATION].', '1406.4864-1-76-0': 'The expansion of these expressions leads to the VGSB model corresponding to a global disk with a temperature structure varying in spherical shells.', '1406.4864-1-76-1': 'The local angular frequency and shear rate are, respectively, [EQUATION]', '1406.4864-1-76-2': 'In the limiting case in which [MATH]; the angular frequency and the shear rate take the Keplerian values at the midplane, i.e., [MATH] and [MATH], and the parameter [MATH] alone determines the disk thickness.', '1406.4864-1-77-0': 'In the next two sections, we illustrate how the VGSB framework relates to, and extends, previous treatments of two disk instabilities that are relevant for a wide variety of astrophysical disks; namely the vertical shear instability (VSI), also known as the GSF instability , and the magnetorotational instability (MRI) .', '1406.4864-1-77-1': 'These instabilities will be the subject of future, more detailed, work.', '1406.4864-1-78-0': '# Vertical Shear Instability (VSI) in the VGSB', '1406.4864-1-79-0': 'Unmagnetized disks with shear profiles that depend on height have been long suspected to be unstable to various instabilities that feed off this angular frequency gradient.', '1406.4864-1-79-1': 'The pioneering studies of [CITATION] and [CITATION] invoked local approximations in both radius and height, capturing the essence of these instabilities but leaving open questions about their global behavior.', '1406.4864-1-79-2': 'The global, non-linear evolution of these instabilities has ben recently studied in [CITATION] by performing numerical simulations.', '1406.4864-1-79-3': 'This work includes an extension of the original local analysis, by considering the effects of compressibility and also an approximate, vertically global linear mode analysis.', '1406.4864-1-79-4': 'In this section, we show that the VGSB formalism recovers the local dispersion relation found in [CITATION] and can be used to address their vertically global mode analysis without approximations.', '1406.4864-1-80-0': 'There are general considerations which are common to both local and global approaches in height.', '1406.4864-1-80-1': 'Let us first write the VGSB equations for an unmagnetized disk for which the local angular frequency and shear rate derive from a global disk model with temperature [MATH] dependent on the cylindrical radius, as described in Section [REF].', '1406.4864-1-80-2': 'Following [CITATION], we make the change of variables [MATH] so that Equations ([REF])-([REF]) become [EQUATION]', '1406.4864-1-80-3': 'In order to derive the equations for the linear mode analysis that lead to the VSI starting from Equations ([REF]) and ([REF]), we proceed as follows: (i) We write the density variable as [MATH], i.e., the sum of its hydrostatic value and a fluctuation over this background, and the pressure as [MATH].', '1406.4864-1-80-4': '(ii) We use the scales of length and time provided by [MATH] and [MATH] to define the dimensionless variables [MATH] and [MATH], so that [MATH] is the dimensionless shear rate, [MATH] is the dimensionless sound speed, etc. (iii) We assume that all the perturbations are small and axisymmetric.', '1406.4864-1-80-5': '(iv) We focus on radial scales that are small compared to the fiducial radius [MATH] and take the Fourier transform of the set of equations ([REF]) and ([REF]), which reduces to making the substitution [MATH], for all flow variables, [MATH], [MATH], [MATH], and [MATH], where the tilde denotes Fourier amplitudes.', '1406.4864-1-80-6': 'This procedure leads to the following set of dimensionless equations [EQUATION]', '1406.4864-1-81-0': '## Vertically Local Analysis of VSI', '1406.4864-1-82-0': 'Let us first show that the equations for the VGSB lead to the dispersion relation derived by [CITATION], which is a generalization of the dispersion relation derived originally by GSF , who considered an incompressible gas.', '1406.4864-1-82-1': 'In order to do this, we restrict the analysis to the vertical location about the height [MATH], focusing on scales that are small in both the radial and vertical direction.', '1406.4864-1-82-2': 'Note that, in this case, the convenient time scale to define dimensionless variables is [MATH].', '1406.4864-1-82-3': 'Taking the Fourier transform of the set of equations ([REF]) and ([REF]) reduces to making the substitution [MATH], so that [EQUATION] where the background quantities are understood to be evaluated at [MATH] and we have used that [MATH], where [MATH] is the local value of the gravitational accelleration.', '1406.4864-1-83-0': 'The characteristic polynomial of the homogeneous system of equations ([REF])-([REF]) yields the dispersion relation in terms of dimensionless variables [EQUATION]', '1406.4864-1-83-1': 'This result is equivalent to the equation one before Equation (32) in [CITATION].', '1406.4864-1-83-2': 'Therefore, the dispersion relation of the vertical shear instability present in global disk models with height-dependent angular frequencies is correctly obtained by the VGSB framework.', '1406.4864-1-84-0': '## Vertically Global Analysis of VSI', '1406.4864-1-85-0': 'In this section we perform a vertically global analysis of isothermal, axisymmetric unmagnetized perturbations by solving the coupled set of ordinary differential equations ([REF])-([REF]).', '1406.4864-1-85-1': 'For a fixed radial wavenumber [MATH], the solution to the eigenvalue problem defined by these equations yields a set of eigenfunctions [MATH], [MATH], [MATH], and [MATH] associated with the eigenvalues [MATH].', '1406.4864-1-85-2': 'We are particularly interested in finding the modes with growing amplitude, i.e., where the real part of the eigenvalue [MATH].', '1406.4864-1-86-0': 'We illustrate the eigenmodes present in a disk model with cylindrical temperature structure [MATH], as presented in Section [REF], in Figures [REF] and [REF].', '1406.4864-1-86-1': 'We consider as an example a global disk model with [MATH], [MATH], and the finite domain [MATH], where [MATH] and [MATH].', '1406.4864-1-86-2': 'We solve the problem corresponding to a single radial wavenumber [MATH], associated to a wavelength of [MATH], as follows.', '1406.4864-1-86-3': 'We discretize the problem in terms of Chebyshev cardinal functions on the Gauss-Lobatto grid, and the boundary condition [MATH] is enforced by the "boundary bordering" method .', '1406.4864-1-86-4': 'We vary the resolution, using a maximum of 1200 grid points (yielding a 4800 by 4800 matrix) to obtain the converged eigenvalues shown.', '1406.4864-1-86-5': 'We find that the basic pattern of modes in Figure [REF] agrees with the approximate analysis carried out by [CITATION], who solved a second order differential equation that results from combining and approximating Equations ([REF])-([REF]).', '1406.4864-1-86-6': 'In order to facilitate making a connection to their findings, we discuss the modes that we obtained using their terminology.', '1406.4864-1-87-0': 'The eigenvalues have the symmetry in the complex plane [MATH].', '1406.4864-1-87-1': 'The fastest growing modes are a branch of "surface" modes with degenerate eigenvalues.', '1406.4864-1-87-2': 'The top row in Figure [REF] illustrates the fastest growing pair of modes, which share the same structure, mirrored across [MATH].', '1406.4864-1-87-3': 'A branch of eigenvalues proceeding from the origin of the complex plane contains the fundamental body modes, which are associated with "corrugation" and "breathing" modes.', '1406.4864-1-87-4': 'Like in [CITATION], the eigenvalue closest to the origin is associated with the fundamental corrugation mode, shown in the middle row of Figure [REF].', '1406.4864-1-87-5': 'Next on this branch is the fundamental breathing mode, shown in the bottom row of Figure [REF].', '1406.4864-1-87-6': 'Unlike the modes found by [CITATION], these modes have rapid oscillations at high altitudes, however, in that calculation the domain was limited to [MATH].', '1406.4864-1-87-7': 'Finally, a long tail of "body" modes proceeds along the imaginary axis.', '1406.4864-1-87-8': 'It is not clear where, or if, this sequence ends, and higher resolution calculations will be required to determine this.', '1406.4864-1-87-9': 'At much higher frequencies [MATH], a series of converged, very slowly growing modes appears in the calculation.', '1406.4864-1-88-0': 'The prodigious number of fast growing surface modes which appear in this calculation as the vertical size of the domain is increased suggests that magnetic fields will be a significant consideration in the astrophysical context for vertical shear instabilities.', '1406.4864-1-88-1': 'In the low density regime high above the disk midplane the very low density gas will almost certainly be ionized in many types of astrophysical disks; the presence of magnetic fields ought then to have a significant impact on the dynamics of these instabilities.', '1406.4864-1-89-0': '# Magnetorotational Instability in the VGSB', '1406.4864-1-90-0': 'Following, and extending, the general technique of [CITATION], we perform a linear stability analysis of the MRI in the framework of the VGSB.', '1406.4864-1-90-1': '3 In order to derive the equations for the linear mode analysis that lead to the MRI, we assume a homogeneous background magnetic field [MATH] and examine the evolution of the perturbations of the form [MATH] for the velocity and magnetic field components [MATH], [MATH], [MATH], [MATH], considering [MATH].', '1406.4864-1-90-2': 'Upon substitution in Equations ([REF])-([REF]) and linearizing in the perturbation amplitudes, the equations for the perturbation are [EQUATION]', '1406.4864-1-90-3': 'In a similar way as we did for the VSI, it is convenient to use the scales of length and time provided by [MATH] and [MATH] in order to define dimensionless variables, i.e., [MATH] and [MATH], etc.', '1406.4864-1-90-4': 'Note that the time unit of the growth rate [MATH] is [MATH], so its actual value will depend on the parameters of the shearing box model employed.', '1406.4864-1-91-0': 'The set of Equations ([REF])-([REF]) generalize Equations (12)-(15) in [CITATION], to include the effects of height-dependent angular frequency and shear rate that arise in the context of baroclinic disk models .', '1406.4864-1-91-1': 'In order to illustrate how the MRI modes are affected, we examine the spectrum of growing modes present in a disk with spherical temperature structure, [MATH], as discussed in Section [REF], taking as an example the [MATH].', '1406.4864-1-91-2': 'These values corresponding to a scale height of [MATH], and magnetic field strength [MATH].', '1406.4864-1-92-0': 'We solve the eigenproblem posed by Equations ([REF])-([REF]) by discretizing in the infinite domain [MATH] in terms of rational Chebyshev cardinal functions on the "roots", or interior grid.', '1406.4864-1-92-1': 'For the set of chosen parameters, 200 grid points were sufficient to obtain converged results.', '1406.4864-1-92-2': 'We adopted the same set of boundary conditions used in [CITATION], i.e., [MATH] and [MATH].', '1406.4864-1-92-3': 'To make the problem numerically tractable, we limit [MATH] to a minimum value of [MATH], which is equivalent to limiting the Alfven speed at high altitudes.', '1406.4864-1-92-4': 'The VGSB model of this particular disk model has one more unstable eigenmode than the standard shearing box.', '1406.4864-1-92-5': 'Note that our VGSB analysis yields growth rates scaled by [MATH], which is smaller than the Keplerian frequency.', '1406.4864-1-92-6': 'Thus, even though the parameters characterizing the specific VGSB model do not appear explicitly in the linear analysis, the physical values obtained do depend on them.', '1406.4864-1-92-7': 'If we consider [MATH], then in the VGSB model analyzed here [MATH].', '1406.4864-1-92-8': 'This implies that the physical growth rate of the MRI is smaller in the VGSB model of this particular disk than in the standard shearing box.', '1406.4864-1-92-9': 'In addition to solving the eigenproblem for the VGSB, we solve the matching problem for the standard shearing box by using an isothermal disk with the same midplane density and scale height [MATH] as used in the VGSB.', '1406.4864-1-92-10': 'The magnetic field and velocity perturbations of the eigenmodes, shown in Figures [REF] and [REF], are directionally orthogonal in the standard shearing box, whereas in the VGSB this is not the case.', '1406.4864-1-92-11': 'The eigenmodes in the VGSB are generally more complex, and, as it can be seen from these figures, they do not appear to have such a simple ordering in terms of the number of nodes as the standard shearing box eigenmodes do.', '1406.4864-1-93-0': '# Discussion', '1406.4864-1-94-0': 'Because of its inherent local character, the framework provided by the standard shearing box is well suited to study physical processes involving scales that are smaller than the characteristic disk scales.', '1406.4864-1-94-1': 'Global disk simulations, which are becoming ever more accessible , are useful for understanding the large-scale disk dynamics but face the challenge of resolving the physical processes at small scales.', '1406.4864-1-94-2': 'In order to understand how local and global processes interact, it is desirable to devise a framework to bridge local and global approaches.', '1406.4864-1-94-3': 'Hence, there have been several initiatives to relax the local character of the standard shearing box.', '1406.4864-1-95-0': '## Previous Works Beyond the Standard Shearing Box', '1406.4864-1-96-0': 'The periodic "shearing disk" annulus introduced in [CITATION] relaxes the condition of azimuthal locality by building a radially periodic annulus and remapping quantities across the radial background according to an imposed power-law radial disk structure.', '1406.4864-1-96-1': 'This approach involves equations which have explicit coordinate dependence in the direction where the computational domain is shear-periodic.', '1406.4864-1-96-2': 'Therefore, the flow properties present a jump at the shearing boundary, which might lead to unphysical effects.', '1406.4864-1-96-3': 'In this context, the formulations presented in [CITATION] and [CITATION] complement each other in the sense that they retain the terms accounting for finite curvature but discard global gradients and vice-versa, but both lead to similar issues.', '1406.4864-1-97-0': 'There have been several attempts to make the stratified shearing box global in height.', '1406.4864-1-97-1': 'Foremost, [CITATION] treated the case of a polytropic gas, but only made local expansions in the horizontal direction.', '1406.4864-1-97-2': 'An early example of modifying the vertical gravity in a standard shearing box is given in [CITATION], where the full [MATH]-dependence of the vertical component of the gravitational force is considered with an isothermal gas.', '1406.4864-1-97-3': 'Several other authors have used similar approaches, for both galactic and Keplerian disks .', '1406.4864-1-97-4': 'These works assume that all the global effects in the vertical direction can be accounted for by modifying only the momentum equation in the [MATH]-direction and exclusively through the term [MATH].', '1406.4864-1-97-5': 'When the gas in the global equilibrium configuration is strictly baroptropic, it must rotate on cylinders, and the height-independent shear in the standard shearing box is a consistent approximation.', '1406.4864-1-97-6': 'Importantly, in the case that the gas is assumed to be isothermal, the standard shearing box provides a consistent approach if the global disk configuration, at all radii, is isothermal.', '1406.4864-1-97-7': 'If another equation of state is used, or if the global configuration of which the shearing box is a small part has a radial temperature gradient, then the VGSB provides a more consistent approximation, with important modifications beyond the standard shearing box formalism, see Equations ([REF])-([REF]).', '1406.4864-1-98-0': '## New Aspects of the Vertically Global Shearing Box', '1406.4864-1-99-0': 'The framework of the VGSB allows, for the first time, to develop models for astrophysical disks which are local in radius but global in height.', '1406.4864-1-99-1': 'This is critical to study astrophysical disks that cannot be described by a barotropic equation of state, as these, in general, do not rotate on cylinders and are thus not amenable to the standard shearing box framework (see Figure [REF]).', '1406.4864-1-99-2': 'The VGSB naturally accounts for height-dependent radial and vertical flux of fluid and electromagnetic momentum in the azimuthal direction.', '1406.4864-1-99-3': 'This is relevant for the dynamics of baroclinic disks, especially if these cannot be regarded as thin.', '1406.4864-1-99-4': 'Physical domains with a large vertical extent allow for the global magnetic field threading the disk to be efficiently anchored into the surrounding medium.', '1406.4864-1-99-5': 'The coupling between stresses and shear, both radial and vertical, can efficiently transport momentum and energy to/from the disk.', '1406.4864-1-99-6': 'We envision that these effects will have important consequences, for example, in the study of the coupled dynamics between disks and their coronae and winds.', '1406.4864-1-100-0': '## Applications', '1406.4864-1-101-0': 'We anticipate that the VGSB framework, summarized in Appendix [REF], will benefit the modeling of a wide variety of phenomena in astrophysical disks.', '1406.4864-1-101-1': 'The advances made possible with respect to the standard shearing box will depend on the nature of the physical phenomena under study, e.g.:', '1406.4864-1-102-0': 'Hydrodynamic Disk Instabilities The VGSB framework may be used for the local study of vertical shear instabilities, such as Goldreich-Schubert-Fricke (GSF) instability and generalizations .', '1406.4864-1-102-1': 'We have demonstrated in Section [REF] that the VGSB captures the correct local linear behavior.', '1406.4864-1-102-2': 'It should also be useful to analyze the propagation of hydrodynamic waves in disks .', '1406.4864-1-103-0': 'Disk Convection Vertical convective instabilities in disks have been studied with the standard shearing box as an angular momentum transport mechanism.', '1406.4864-1-103-1': 'Though early studies gave negative results, [CITATION] have suggested that earlier models are under-resolved, and outward angular momentum transport is possible.', '1406.4864-1-103-2': 'Given that convection can lead to fluid motions far away from the disk midplane, the VGSB naturally provides a more consistent framework to investigate these issues.', '1406.4864-1-104-0': 'Disk Coronae Previous standard stratified shearing box simulations show the buoyant rise of magnetic field to the upper disk layers , where it is thought to dissipate giving rise to a hot corona .', '1406.4864-1-104-1': 'If the disk is not barotropic, The fact that the shear, i.e., the source of free energy, decreases with height can have important implications for the turbulent disk dynamics and energetics in the disk corona.', '1406.4864-1-104-2': 'This could affect the vertical disk structure and its corona .', '1406.4864-1-104-3': 'These effects could be important for thick disks, such as advection dominated accretion flows (ADAFs) .', '1406.4864-1-105-0': 'Disk Winds Standard shearing boxes extended in height have been used to study the magnetorotational instability (MRI) as a mechanism for launching disk winds though only in the framework provided by modifying the gravitational forces through the term [MATH].', '1406.4864-1-105-1': 'We have shown in Section [REF] that the linear MRI is modified by effects included in the VGSB for baroclinic disks.', '1406.4864-1-105-2': 'Because the wind dynamics are particularly sensitive to the forces acting on the fluid far away from the midplane, the VGSB provides a more consistent framework for local studies of this nature.', '1406.4864-1-106-0': 'Interstellar Medium and Galactic Disks In galactic disks, the VGSB makes it possible to capture critical physics for studying the non-linear evolution of the magnetorotational, Parker, and magneto-Jeans instabilities .', '1406.4864-1-106-1': 'The framework provided by the VGSB also allows for the inclusion of fundamental height-dependent physical effects that can play an important role in the study of star formation, galactic dynamos, and the structure of the interstellar medium .', '1406.4864-1-107-0': 'In future papers, we will present the implementation of the VGSB framework in a magnetohydrodynamic code and investigate these outstanding problems in astrophysical disk dynamics.', '1406.4864-1-108-0': 'We are grateful to Tobias Heinemann, whose valuable insights helped us to better understand the subtleties involved in the derivation of the VGSB and to present it in a transparent way.', '1406.4864-1-108-1': 'We acknowledge thoughtful comments from Scott Tremaine and Eric Blackman, and discussions with Troels Haugbolle, AAke Nordlund, Andrew Jackson, Oliver Gressel, Orkan M. Umurhan, Gopakumar, Henrik Latter, Andrea Mignone, Gianluigi Bodo and Sacha Brun.', '1406.4864-1-108-2': 'We thank an anonymous referee for an enlightening report, which led us to better appreciate how the VGSB connects to, and extends, the standard shearing box framework, and to improve the original manuscript significantly.', '1406.4864-1-108-3': "The research leading to these results has received funding from the People Programme (Marie Curie Actions) of the European Union's Seventh Framework Programme (FP7/2007-2013) under REA grant agreement 327995 (CPM), and the European Research Council under the European Union's Seventh Framework Programme (FP/2007-2013) under ERC grant agreement 306614 (MEP).", '1406.4864-1-108-4': 'MEP also acknowledges support from the Young Investigator Programme of the Villum Foundation.', '1406.4864-1-109-0': '# Derivation of the momentum equation for departures from the background bulk flow', '1406.4864-1-110-0': 'Here, we provide the algebraic details involved in obtaining the momentum equation ([REF]) starting from Equation ([REF]).', '1406.4864-1-111-0': 'Let us first expand the left hand side of Equation ([REF]), LHS for short, after substituting [MATH], we obtain [EQUATION]', '1406.4864-1-111-1': 'We now expand the last three terms, recalling that the differential operators involved are in cylindrical coordinates and using the definition of the bulk flow in terms of the angular frequency [MATH].', '1406.4864-1-111-2': 'This yields [EQUATION]', '1406.4864-1-111-3': 'This completes the expansion of the left hand side in terms of the new velocity variable [MATH].', '1406.4864-1-112-0': 'We now proceed similarly with the right hand side of the momentum equation ([REF]), RHS for short, by substituting [MATH] and expanding [EQUATION]', '1406.4864-1-112-1': 'Equating the left hand side expression (Equation [REF]) and the right hand side expression (Equation [REF]) and canceling out the term [MATH] which appears on both sides, the momentum Equation [REF] becomes, without approximation, [EQUATION] which corresponds to Equation ([REF]).', '1406.4864-1-113-0': 'As stated in Section [REF], a cancellation of terms from the left and right hand sides of the momentum equation results in the term [MATH] that looks similar to the Coriolis acceleration, i.e., [MATH], but with the angular frequency [MATH] taking the place of the constant angular frequency of the rotating frame [MATH].', '1406.4864-1-114-0': '# Summary of the VGSB Equations', '1406.4864-1-115-0': 'Here we provide a self-contained summary of the equations defining the vertically global shearing box (VGSB).', '1406.4864-1-116-0': 'VGSB Equations The continuity, momentum, induction, and energy equations in the VGSB are, respectively, given by [EQUATION]', '1406.4864-1-116-1': 'Here, [MATH] is the mass density, [MATH] is the velocity with respect to the local bulk flow in the disk, [MATH] is the magnetic field, and [MATH] is the internal energy density.', '1406.4864-1-116-2': 'The pressure [MATH] is determined as a function of [MATH] and [MATH], through an appropriate equation of state.', '1406.4864-1-116-3': 'The current density is [MATH], with [MATH] a constant dependent on the unit system adopted.', '1406.4864-1-116-4': 'All the operators in the VGSB equations are defined in a cartesian coordinate system centered at the fiducial radius [MATH].', '1406.4864-1-117-0': 'For barotropic background disc structures, the above equations can be used in fully three dimensions, while for baroclinic background disc structures, these equations must by restricted to axisymmetric solutions ([MATH]-invariant).', '1406.4864-1-118-0': 'VGSB Definitions The equations defining the VGSB framework depend on a number of functions of the coordinate [MATH], which result from local radial expansions of the global disk model around the radius [MATH].', '1406.4864-1-118-1': 'The height-dependent advection by the background shear, [MATH], is given by [EQUATION] where [MATH], [MATH], and [MATH] correspond, respectively, to the local values of the background azimuthal velocity, angular frequency, and background shear flow, which are all derived from the angular frequency of the global disk model [MATH], i.e., [EQUATION]', '1406.4864-1-118-2': 'The gravitational potential in the VGSB [MATH] is given by the value of the global gravitational potential [MATH] at [MATH] [EQUATION]', '1406.4864-1-118-3': 'VGSB Boundary Conditions For a domain with size [MATH] the horizontal boundary conditions are [EQUATION] which correspond to a generalization of the shearing-periodic boundary conditions adopted in the standard shearing box.', '1406.4864-1-119-0': '# Neglecting Curvature Terms', '1406.4864-1-120-0': 'The cylindrical coordinate system that we adopt to describe the global disk model leads naturally to the presence of the quadratic terms [MATH], [MATH], [MATH], and [MATH] in the momentum equation ([REF]).', '1406.4864-1-120-1': 'All of these terms are usually neglected in the standard shearing box.', '1406.4864-1-120-2': 'The terms related to the velocity field can be neglected safely when they are small compared to the corresponding components of the acceleration [MATH] and [MATH].', '1406.4864-1-120-3': 'Both of these conditions lead to the inequality [MATH].', '1406.4864-1-120-4': 'The magnetic terms [MATH] and [MATH] in Equation ([REF]) are usually absent in local studies (see for an exception).', '1406.4864-1-120-5': 'If the magnetic field is sufficiently sub-thermal everywhere in the domain, these terms can be self-consistently dropped .', '1406.4864-1-120-6': 'However, they have been commonly neglected even when this is not the case.', '1406.4864-1-120-7': 'This is perhaps because it is not obvious that retaining them will lead to a physically consistent problem in the framework of the shearing box.', '1406.4864-1-120-8': 'Consider, for example, the case of a shearing box in which a strong net azimuthal field develops as a consequence of the local disk dynamics.', '1406.4864-1-120-9': 'This could lead to the force arising from the term [MATH] to increase with time.', '1406.4864-1-120-10': 'However, this force cannot be balanced by an increase in the centripetal acceleration as the latter is fixed by the choice of the bulk flow .', '1406.4864-1-120-11': 'Note that all of this statements regarding the quadratic terms that reminisce the curvilinear nature of the original cylindrical coordinate system are independent of whether the standard shearing box, or the vertically global version developed in this paper, is considered.'}

{'1406.4864-2-0-0': 'Disks with a barotropic equilibrium structure, for which the pressure is only a function of the density, rotate on cylinders in the presence of a gravitational potential, so that the angular frequency of such a disk is independent of height.', '1406.4864-2-0-1': 'Such disks with barotropic equilibria can be approximately modeled using the shearing box framework, representing a small disk volume with height-independent angular frequency.', '1406.4864-2-0-2': 'If the disk is in baroclinic equilibrium, the angular frequency does in general depend on height and it is thus necessary to go beyond the standard shearing box approach.', '1406.4864-2-0-3': 'In this paper, we show that given a global disk model, it is possible to develop approximate models that are local in horizontal planes without an expansion in height with shearing-periodic boundary conditions.', '1406.4864-2-0-4': 'We refer to the resulting framework as the vertically global shearing box.', '1406.4864-2-0-5': 'These models can be non-axisymmetric for globally barotropic equilibria but should be axisymmetric for globally baroclinic equilibria.', '1406.4864-2-0-6': 'We provide explicit equations for this vertically global shearing box which can be implemented in standard magnetohydrodynamic codes by generalizing the shearing-periodic boundary conditions to allow for a height-dependent angular frequency and shear rate.', '1406.4864-2-0-7': 'We also discuss the limitations that result from the radial approximations that are needed in order to impose height-dependent shearing periodic boundary conditions.', '1406.4864-2-0-8': 'We illustrate the potential of this framework by studying a vertical shear instability and examining the modes associated with the magnetorotational instability.', '1406.4864-2-1-0': '# Introduction', '1406.4864-2-2-0': 'Astrophysical disks play a crucial role in the formation, evolution, and fate of a wide variety of celestial objects, by mediating the transport of mass, energy, and angular momentum.', '1406.4864-2-2-1': 'Building realistic disk models is of fundamental importance for understanding, for example, protoplanetary disks around young stars, accretion flows onto stellar compact objects and active galactic nuclei, as well as the interstellar medium in galactic disks.', '1406.4864-2-2-2': 'The large dynamic range involved makes it particularly challenging to produce detailed global numerical simulations of these systems.', '1406.4864-2-2-3': 'Moreover, while global models allow us to investigate large-scale phenomena, local models with a hierarchy of increasingly complex microphysics have proved critical to elucidating the processes that are crucial at small scales.', '1406.4864-2-2-4': 'Because of this, several types of local approximations have been employed for studying astrophysical disks.', '1406.4864-2-3-0': '[CITATION] pioneered the use of a local approximation to study the dynamics of particles orbiting a host system and subject to encounters with a perturber, and used it to study the motion of the Moon.', '1406.4864-2-3-1': '[CITATION] and [CITATION] applied these ideas to galactic disks using the concept of a locally shearing coordinate system.', '1406.4864-2-3-2': 'This approach constitutes the basis of the shearing sheet framework which has been widely used to study the dynamics of orbiting particles and planetesimals, as well as local processes in hydrodynamic and magnetohydrodynamic disks.', '1406.4864-2-3-3': 'The implementation of the concept behind the shearing sheet, with an appropriate shear-periodic radial-boundary condition , forms a computational model used for studying local disk dynamics referred to as the shearing box .', '1406.4864-2-4-0': 'Shearing box models solve the equations of motion for the fluid in a local cartesian frame co-rotating with the disk at a fiducial radius.', '1406.4864-2-4-1': 'In the standard framework, the differential rotation of the disk is locally accounted for with a height-independent angular frequency.', '1406.4864-2-4-2': 'This is appropriate for disks with a barotropic equilibrium for which the pressure is only a function of density and thus rotate on cylinders.', '1406.4864-2-4-3': 'The shearing box framework relies on a first order expansion of the steady bulk flow in the radial direction, which is the highest order compatible with shearing periodic boundary conditions.', '1406.4864-2-4-4': 'Depending on whether zeroth or first order expansions are considered for the gravitational field in the direction perpendicular to the disk, usually denoted with the coordinate [MATH], this leads to the so called unstratified or stratified shearing box models.', '1406.4864-2-4-5': 'This stratified, compressible shearing box is what we will refer to in this paper as the standard shearing box (SSB).', '1406.4864-2-4-6': 'These approximations are appropriate when the disk is thin and the vertical scales of interest are small compared to the fiducial disk radius.', '1406.4864-2-4-7': 'There have been works that retained the correct expression for the vertical component of the gravitational field , allowing for larger vertical domains to be considered.', '1406.4864-2-4-8': 'However, this generalization does not allow to study of disks with baroclinic equilibria without relaxing the assumption that the angular frequency is height-independent.', '1406.4864-2-4-9': 'In the early formulation of the shearing sheet, [CITATION] avoided making any approximation in height ([MATH]), which is possible when considering barotropic equilibrium structures.', '1406.4864-2-5-0': 'Astrophysical disks with baroclinic equilibrium structure, for which the pressure is not solely a function of density, posses angular frequency profiles that depend in general on height, especially if these are not thin.', '1406.4864-2-5-1': 'Therefore, building a framework to study these disks demands going beyond the standard shearing box, where the assumption that the angular frequency is height-independent is rooted deep.', '1406.4864-2-5-2': 'In this work, we generalize the standard shearing box by considering the full height-dependence of a steady state, axisymmetric bulk flow to leading order in radius, without making any expansion in the vertical coordinate.', '1406.4864-2-5-3': 'We show that given a global disk model, it is possible to develop approximate models that are local in horizontal planes and global in height and are amenable to shearing-periodic boundary conditions.', '1406.4864-2-5-4': 'These models can be non-axisymmetric for disks with a barotropic global equilibrium but should be axisymmetric for disks with a baroclinic global equilibrium.', '1406.4864-2-5-5': 'We term the resulting framework the vertically global shearing box (VGSB).', '1406.4864-2-5-6': "The terminology of 'vertically global' specifically and solely refers to the approach of never making a expansion in the vertical direction in the derivation.", '1406.4864-2-5-7': 'This yields a model which allows us to account for the vertical variation of gravity without approximations and the possible presence of vertical shear.', '1406.4864-2-5-8': 'As we discuss below, the radial expansions that are needed to apply height-dependent shearing periodic boundary conditions do limit the range of vertical scales which can be modeled approximately.', '1406.4864-2-5-9': 'In spite of its limitations, the VGSB formalism provides a novel framework that goes beyond the standard shearing box and promises to provide a bridge between strictly local and fully global approaches to model astrophysical disks.', '1406.4864-2-6-0': 'The paper is organized as follows.', '1406.4864-2-6-1': 'We derive the equations involved in the VGSB framework in Section [REF], providing some of the algebraic details in Appendix [REF].', '1406.4864-2-6-2': 'We state the final form of the VGSB equations and discuss its novel features in Section [REF].', '1406.4864-2-6-3': 'For convenience, we provide a self-contained summary of the VGSB equations that can be incorporated in magnetohydrodynamic codes in Appendix [REF].', '1406.4864-2-6-4': 'We use this new framework to explore the behavior of two important instabilities in a baroclinic context.', '1406.4864-2-6-5': 'We demonstrate that a linear vertical shear instability, akin to those studied by [CITATION] and [CITATION], appears in the VGSB in Section [REF].', '1406.4864-2-6-6': 'We examine some basic aspects of the magnetorotational instability in the VGSB in Section [REF].', '1406.4864-2-6-7': 'We conclude by briefly discussing the limitations and several potential applications of the VGSB in Section [REF].', '1406.4864-2-6-8': 'In addition, in Appendix [REF] we outline some details related to neglecting curvature terms, in Appendix [REF] we analyze the hydrodynamic momentum equation in order to assess under what conditions it is acceptable to discard radial pressure gradients in the SSB and VGSB, and in Appendix [REF] we discuss issues related to potential vorticity in shearing boxes.', '1406.4864-2-7-0': '# Equations of Motion', '1406.4864-2-8-0': 'We are concerned with the equations of ideal magnetohydrodynamics, in cylindrical coordinates [MATH], in a reference frame rotating with angular frequency [MATH], i.e., [EQUATION]', '1406.4864-2-8-1': 'Here, [MATH] is the mass density, [MATH] is the fluid velocity in the rotating frame, [MATH] is the magnetic field, with [MATH], [MATH] is the internal energy density, [MATH] is the pressure determined through an equation of state, and [MATH] is the gravitational potential, which is assumed to be cylindrically symmetric, but not necessarily spherical.', '1406.4864-2-8-2': 'The current density is [MATH], with [MATH] a constant dependent on the unit system adopted.', '1406.4864-2-9-0': 'Fluid flows described by these equations are subject to conservation laws.', '1406.4864-2-9-1': 'It is thus important to understand under what circumstances these properties are satisfied by the equations describing the local dynamics involving expansions of the original set of equations.', '1406.4864-2-9-2': "It is easy to show that the approximations embodied in the standard isothermal shearing box are such that the vortex lines of an inviscid flow are frozen into the fluid (Kelvin's Circulation Theorem) and that the magnetic flux is also frozen into the fluid in the absence of magnetic dissipation (Alfven's Frozen-in Theorem).", '1406.4864-2-10-0': 'Understanding under what conditions these properties also hold for the equations of motion that result from invoking a local approximation of global disk models which have a baroclinic equilibrium structure is more subtle.', '1406.4864-2-10-1': 'Here, we state the general versions of the aforementioned theorems in order to prepare the ground to address these issues in subsequent sections.', '1406.4864-2-10-2': 'These conservation theorems can be derived by calculating the Lagrangian derivative of the fluxes associated with the vorticity and the magnetic field.', '1406.4864-2-10-3': 'It is thus useful to recall that, see e.g., , any vector field [MATH] satisfies [EQUATION] where the integral is carried out over any open surface [MATH] advected by the flow with velocity [MATH].', '1406.4864-2-11-0': "## Kelvin's Circulation Theorem", '1406.4864-2-12-0': 'The momentum equation ([REF]) for an an inviscid, unmagnetized, barotropic flow in the rotating frame is given by [EQUATION] where [MATH] is the generalized gravito-thermal potential, where [MATH] is the enthalpy, with [MATH].', '1406.4864-2-12-1': 'The equation governing the evolution of the vorticity is thus [EQUATION] which preserves the solenoidal character of the vorticity.', '1406.4864-2-12-2': 'This implies that, by virtue of Equation ([REF]) with [MATH], vortex lines are frozen into the fluid, i.e., the flow preserves the circulation [MATH] [EQUATION] with [EQUATION] where [MATH] is a closed contour, delimiting the open surface [MATH], advected by the flow with velocity [MATH].', '1406.4864-2-13-0': "## Alfve n's Frozen-in Theorem", '1406.4864-2-14-0': 'The induction equation ([REF]) preserves the divergence of the magnetic field, i.e., [EQUATION]', '1406.4864-2-14-1': 'This implies that, provided that [MATH] at some initial time, the magnetic flux remains frozen into the fluid, i.e., [EQUATION] with [EQUATION] where the integral is carried out over any open surface [MATH] advected by the flow with velocity [MATH].', '1406.4864-2-14-2': 'This follows from Equation ([REF]) with [MATH] and the induction equation ([REF]).', '1406.4864-2-15-0': '# The Vertically Global, Horizontally Local Approximation', '1406.4864-2-16-0': 'We seek to derive a set of equations that describe the local dynamics of the magnetized fluid with respect to a known steady state bulk flow around a point co-rotating with the disk at a distance [MATH].', '1406.4864-2-16-1': 'Here, we outline the steps of the derivation, which is carried out in detail below.', '1406.4864-2-17-0': 'We find a suitable steady flow and background equilibrium, which enables the derivation of exact equations of motion for the departures from this solution.', '1406.4864-2-18-0': 'We transform to a locally cartesian coordinate system.', '1406.4864-2-18-1': 'We expand the bulk flow and background equilibrium to leading order in the radial direction leaving the direction perpendicular to the disk midplane unaltered.', '1406.4864-2-19-0': 'We determine under which circumstances the resulting equations are amenable to being solved with shearing-periodic boundary conditions, which could depend on height for disk models with baroclinic global equilibria.', '1406.4864-2-20-0': 'We show explicitly these steps for the momentum and induction equations, while we state the results for the continuity equation and energy equation that are simpler to work with.', '1406.4864-2-21-0': '## Equations for Departures from Background Equilibrium', '1406.4864-2-22-0': '### Bulk Flow and Background Equilibrium', '1406.4864-2-23-0': 'As in the standard shearing box, we first seek a steady background flow.', '1406.4864-2-23-1': 'In this case, we will not make any a priori assumptions about the [MATH]-dependence of the angular frequency.', '1406.4864-2-24-0': 'We begin by noting that the momentum equation ([REF]) admits a force-free (magnetic fields playing no role), steady state solution [MATH], with [EQUATION] where the angular frequency is [EQUATION] and the vertical, hydrostatic pressure gradient satisfies [EQUATION]', '1406.4864-2-24-1': 'Here, [MATH] and [MATH] are the mass density and energy density associated with the steady state background flow.', '1406.4864-2-25-0': '### Momentum and Induction Equations', '1406.4864-2-26-0': 'Using Equations ([REF]) and ([REF]) we can recast the gravitational force in the momentum equation ([REF]) in terms of the angular frequency and the pressure gradient, both corresponding to the steady state bulk flow.', '1406.4864-2-26-1': 'We obtain [EQUATION]', '1406.4864-2-26-2': 'The velocity field describing the departure from the bulk flow satisfying Equations ([REF])-([REF]) [EQUATION] evolves according to the momentum equation, [EQUATION]', '1406.4864-2-26-3': 'This equation is exact and it displays the particular feature that the last term on the left hand side resembles the Coriolis acceleration, with one important difference.', '1406.4864-2-26-4': 'The angular frequency involved is not the fixed angular frequency of the rotating frame, [MATH], but rather the angular frequency of the steady state flow, [MATH].', '1406.4864-2-26-5': 'As a quick check, note that if [MATH] and [MATH] then there are no departures from the steady state bulk flow, i.e., [MATH].', '1406.4864-2-27-0': 'Taking the induction equation ([REF]) and replacing the velocity field with [MATH] yields, after some algebra, [EQUATION]', '1406.4864-2-28-0': '### Background-Flow Advection and Shear Rate', '1406.4864-2-29-0': 'The fact that the speed of the background flow [MATH] depends, in general, on height implies that the departures from the bulk flow will be advected and sheared in a height-dependent way.', '1406.4864-2-29-1': 'This motivates the definition of the advection operator [EQUATION] which is defined so that it acts on scalar fields, such as the density [MATH], and on each of the components of a vector field, e.g., [MATH] and [MATH], but not on the unit coordinate-vectors, i.e., [MATH].', '1406.4864-2-29-2': 'It is also convenient to define the shear rate [EQUATION]', '1406.4864-2-29-3': 'Using these definitions, Equations ([REF]) and ([REF]) become, without approximations, [EQUATION]', '1406.4864-2-30-0': '## Local Approximation in Horizontal Planes', '1406.4864-2-31-0': 'We now seek to derive a set of equations of motion which is local in radius and azimuth by expanding Equation ([REF]) and ([REF]) around a fiducial point [MATH].', '1406.4864-2-31-1': 'In order to simplify this task, we choose a reference frame that corotates with the bulk flow at radius [MATH], i.e., [EQUATION]', '1406.4864-2-31-2': 'We also adopt a coordinate frame centered at [MATH] with locally cartesian coordinates [MATH], such that [MATH] and [MATH], with [MATH] and [MATH].', '1406.4864-2-31-3': 'In this locally cartesian frame, the differential vector operators are well approximated by their cartesian versions, provided that the radial coordinate versor [MATH]', '1406.4864-2-32-0': 'Because of the axisymmetric character of the background flow, in what follows, we choose [MATH] without loss of generality.', '1406.4864-2-33-0': 'In this locally cartesian coordinate system, we can expand to leading order in [MATH] the various functions appearing in the momentum and induction equations, Equation ([REF]) and ([REF]), respectively.', '1406.4864-2-34-0': '### Approximation of the Bulk Flow', '1406.4864-2-35-0': 'The local approximations of the angular frequency, the bulk flow in Equation ([REF]), and the advection operator in Equation ([REF]) yield [EQUATION]', '1406.4864-2-35-1': 'Here, we have defined the local, height-dependent angular frequency, bulk flow, and shear rate, all evaluated at the fiducial radius [MATH], i.e., [EQUATION]', '1406.4864-2-35-2': 'The operator [MATH] is a height-dependent generalization of the shearing sheet advection operator originally introduced in [CITATION].', '1406.4864-2-35-3': 'In order to illustrate the action of the advection operator [MATH], Figure [REF] shows the effects of considering the local bulk flow in Equation ([REF]) that results from expanding Equation ([REF]) to leading order in the radial direction, leaving unaltered the vertical dependence.', '1406.4864-2-35-4': 'In this particular example, we have considered a baroclinic equilibrium global disk model with a cylindrical temperature structure, which is discussed in detail in Section [REF].', '1406.4864-2-36-0': '### Approximation of Momentum and Induction Equations', '1406.4864-2-37-0': 'Using the approximations above, we arrive at expressions for the momentum and induction equations, Equation ([REF]) and ([REF]), which are correct to leading order in [MATH] and [MATH]: [EQUATION]', '1406.4864-2-37-1': 'Here, all the differential operators are defined in a cartesian coordinate system centered at the fiducial radius [MATH].', '1406.4864-2-38-0': 'Because we have retained the leading order in all the approximations involving the bulk-flow, the two important flow properties discussed in Section [REF] remain unaltered.', '1406.4864-2-38-1': 'In the case of an inviscid, unmagnetized, barotropic flow the momentum equation ([REF]) that results from the local approximation in horizontal planes leads to [EQUATION]', '1406.4864-2-38-2': 'Furthermore, the induction equation ([REF]) that results from the local approximation in horizontal planes preserves the solenoidal character of the the magnetic field, i.e., [EQUATION]', '1406.4864-2-38-3': 'This implies that the magnetic flux is frozen into the fluid flow that results from the local expansion in horizontal planes, i.e., [EQUATION]', '1406.4864-2-38-4': "Therefore, the local approximation in horizontal planes leads to equations that still satisfy Kelvin's circulation theorem and Alfven's frozen in theorem discussed in Section [REF].", '1406.4864-2-39-0': '## Compatibility of the Local Approximation with Shearing-Periodic Boundary Conditions', '1406.4864-2-40-0': '### Shearing-Periodic Boundary Conditions in the SSB', '1406.4864-2-41-0': 'All the explicit coordinate dependences in the equations of motion defining the standard shearing box are contained in the advection operator [EQUATION] which is obtained as a limit of the operator [MATH] introduced in Equation ([REF]).', '1406.4864-2-41-1': 'The explicit dependence on the coordinate [MATH] can be eliminated by the linear transformation [MATH], [MATH], [MATH], and [EQUATION]', '1406.4864-2-41-2': 'In the primed coordinate system, the advection operator simply becomes [MATH], and the equations of motion can be solved by using strictly periodic boundary conditions in horizontal planes, i.e., [EQUATION] and appropriate boundary conditions for the vertical boundaries.', '1406.4864-2-41-3': 'Note that in the original coordinate system, the equations for the departures from the bulk flow satisfy shearing-periodic boundary conditions given by [EQUATION]', '1406.4864-2-42-0': '### Height-dependent Shearing-Periodic Boundary Conditions', '1406.4864-2-43-0': 'Defining an approximate, radially local set of equations and boundary conditions for disks with baroclinic equilibria, for which the angular frequency is in general a function of height, is more subtle.', '1406.4864-2-44-0': 'The coordinate dependence arising through the advection operator [MATH] in Equation ([REF]) can still be removed by defining the linear transformation [MATH], [MATH], [MATH], and [EQUATION]', '1406.4864-2-44-1': 'In this primed coordinate system, the advection operator is coordinate-independent, i.e., [MATH].', '1406.4864-2-44-2': 'Therefore, if this were the only explicit coordinate dependence then, in each horizontal plane, it would be enough to consider the height-dependent, shearing-periodic boundary conditions given by [EQUATION]', '1406.4864-2-44-3': 'However, the coordinate dependence induced by the terms proportional to [EQUATION] on the right hand sides of Equations ([REF]) and ([REF]) cannot be eliminated by the same coordinate transformation that removes the [MATH]-dependence in [MATH].', '1406.4864-2-44-4': 'This prevents Equation ([REF]) and ([REF]) from being solved with the shearing-periodic boundary conditions in Equation ([REF]) in a straightforward way.', '1406.4864-2-45-0': 'In what follows we analyze the consequences of proceeding by neglecting the term proportional to the coordinate [MATH] in Equation ([REF]).', '1406.4864-2-45-1': 'We thus approximate this equation as [EQUATION]', '1406.4864-2-45-2': 'With the exception of the background hydrostatic profile which will be dealt with in the next section, this approximation eliminates the explicit coordinate dependence on the right hand sides of Equations ([REF]) and ([REF]), leading to [EQUATION]', '1406.4864-2-45-3': 'Because the only coordinate dependences arise through the advection operator [MATH], these equations are compatible with the height-dependent, shearing-periodic boundary conditions Equations ([REF]).', '1406.4864-2-45-4': 'By reducing the coordinate dependence down to the advection operator [MATH], and the background equilibrium [MATH] we have brought these equation closer to compatibility with with the height-dependent, shearing-periodic boundary conditions Equations ([REF]).', '1406.4864-2-45-5': 'The background equilibrium will be treated in Section [REF], but first we deal with issues which arise from this step.', '1406.4864-2-45-6': 'The approximation embodied in Equation ([REF]) does, in general, affect the validity of the conservation theorems discussed in Section [REF].', '1406.4864-2-45-7': "We proceed by showing that Equations ([REF]) and ([REF]) do satisfy Kelvin's circulation theorem and Alfven's frozen-in theorem when the underlying global disk model has a barotropic equilibrium or when we consider axisymmetry.", '1406.4864-2-46-0': '### Radially local, Vertically Global Hydrodynamic Disk Models', '1406.4864-2-47-0': 'It can be seen that equation ([REF]) when applied to an inviscid, barotropic, unmagnetized flow, leads to [EQUATION]', '1406.4864-2-47-1': "This means that under the conditions over which Kelvin's circulation theorem is satisfied, the approximation embodied in Equation ([REF]) does not lead to spurious sources of circulation when considering Equation ([REF]) instead of ([REF]).", '1406.4864-2-48-0': 'If the global disk model under consideration has baroclinic equilibrium, the circulation [MATH] is no longer conserved and thus, for physical reasons, [MATH] is no longer expected to vanish.', '1406.4864-2-48-1': 'However, one should also realize that the approximation invoked in Equation ([REF]) leads to a source term that contributes spuriously to the evolution of the circulation [EQUATION]', '1406.4864-2-48-2': 'Here, the dots represent the physical sources of circulation present in fluids which are either viscous or baroclinic.', '1406.4864-2-48-3': 'The spurious source of circulation in Equation ([REF]) vanishes under axisymmetry.', '1406.4864-2-48-4': 'In order to demonstrate this, let us examine the integral involved.', '1406.4864-2-48-5': "Using Stoke's theorem, it follows that [EQUATION]", '1406.4864-2-48-6': 'In axisymmetry, i.e., [MATH], the problem reduces to understanding the dynamics of the fluid in the [MATH] plane.', '1406.4864-2-48-7': 'Under this condition, the line integral over a closed loop vanishes.', '1406.4864-2-49-0': 'Thus, given a global hydrodynamic disk model, corresponding to either barotropic or baroclinic equilibria, Equation ([REF]) can be used to define an associated disk model that is local in horizontal planes but global in height.', '1406.4864-2-49-1': 'This model can be non-axisymmetric for disks with globally barotropic equilibria but should be axisymmetric for disks with globally baroclinic equilibria.', '1406.4864-2-50-0': "The flow of hydrodynamic fluids is constrained by the evolution of the potential vorticity as governed by Ertel's theorem .", '1406.4864-2-50-1': 'In Appendix [REF], we derive the equations associated with the evolution of potential vorticity in the framework of the VGSB and discuss how these relate to the Kelvin circulation theorem alluded to in this section.', '1406.4864-2-50-2': 'The details of this also depend on the final approximations needed to make the model compatible with shearing periodic radial boundaries in Section [REF].', '1406.4864-2-50-3': 'Before that, we discuss the analogous issue which occurs with magnetic fields in the induction equation.', '1406.4864-2-51-0': '### Radially local, Vertically Global MHD Disk Models', '1406.4864-2-52-0': 'Let us now consider the implications for the induction equation.', '1406.4864-2-52-1': 'In general, neglecting the term proportional to the coordinate [MATH] in Equation ([REF]) leads to an approximated induction equation that no longer preserves the solenoidal character of the magnetic field.', '1406.4864-2-52-2': 'More specifically, Equation ([REF]) leads to [EQUATION]', '1406.4864-2-52-3': 'Taken at face value, this implies that Equation ([REF]) induces spurious generation of magnetic monopoles that will break flux freezing even in the absence of dissipation.', '1406.4864-2-52-4': 'Therefore, in the case of a magnetized fluid, while eliminating the explicit [MATH]-dependence that can not be removed by a coordinate transformation allows the induction equation to be solved with shearing-periodic, height-dependent boundary conditions, this approximation, in general, would destroy the solenoidal character of the magnetic field.', '1406.4864-2-53-0': 'However, the spurious evolution of the divergence of the magnetic field that results from the approximation Equation ([REF]) is absent in Equation ([REF]) if the underlying global disk model is has a barotropic equilibrium or axisymmetry, i.e., [MATH], is considered.', '1406.4864-2-54-0': 'Therefore, given a global disk model, it is possible to develop models which preserve the solenoidal character of the magnetic field that are local in horizontal planes and global in height that are amenable to shearing-periodic boundary conditions.', '1406.4864-2-54-1': 'These models can be non-axisymmetric for disks with globally barotropic equilibria but should be axisymmetric for disks with globally baroclinic equilibria.', '1406.4864-2-54-2': 'Under either of these conditions, the approximation invoked in Equation ([REF]), which is necessary to employ the shearing-periodic boundary conditions given by Equation ([REF]), does not lead to spurious source terms that could affect the evolution of the circulation, magnetic flux freezing, or the solenoidal character of the magnetic field, i.e., [EQUATION] and [EQUATION]', '1406.4864-2-55-0': '### Approximation of the Background Equilibrium', '1406.4864-2-56-0': 'The fourth term on the right hand side of Equation ([REF]) can be dealt with along the lines proposed by [CITATION] for the hydrostatic background quantities [EQUATION] where [MATH] and [MATH] are the leading order terms associated with the mass and internal energy density profiles of the background flow at the fiducial radius .', '1406.4864-2-56-1': 'Within the level of approximation we are working at, we thus have [EQUATION] where the vertical acceleration induced by the background pressure gradient is balanced by gravity at the fiducial radius [EQUATION] with [MATH], the gravitational potential evaluated at the fiducial radius [MATH].', '1406.4864-2-57-0': 'The set of approximations to the background described above can also be considered on more formal grounds by introducing a set of dimensionless parameters that describe the relative scale of the phenomena of interest and the departures from a thin, Keplerian disk.', '1406.4864-2-57-1': 'In Appendix [REF], we provide the details involved in this procedure, emphasizing in particular the handling of the terms related to pressure gradients in the momentum equation.', '1406.4864-2-57-2': 'Our considerations build on, and extend, the analysis carried out in [CITATION].', '1406.4864-2-57-3': 'The analysis suggests the condition [MATH] as a limit on the height of the best modeled part of the domain.', '1406.4864-2-57-4': 'This criterion also applies to the SSB, as we demonstrate in Appendix [REF].', '1406.4864-2-58-0': 'The set of equations that result from considering the approximations described in this section are compatible with the height-dependent, shearing-periodic boundary conditions Equations ([REF]).', '1406.4864-2-58-1': 'This motivates introducing the vertically global shearing box.', '1406.4864-2-59-0': '# The Vertically Global Shearing Box (VGSB)', '1406.4864-2-60-0': '## Equations of Motion for the VGSB', '1406.4864-2-61-0': 'Following the steps outlined in the previous section, we arrive at the expressions for the continuity, momentum, induction, and energy equations that define the framework of the vertically global shearing box (VGSB) [EQUATION]', '1406.4864-2-61-1': 'Here, all the operators are defined in a cartesian coordinate system centered at the fiducial radius [MATH].', '1406.4864-2-62-0': 'The only explicit coordinate dependence in the VGSB arises through the advection operator [MATH], which is linear in the coordinate [MATH] for all heights [MATH].', '1406.4864-2-62-1': 'This implies that the equations are suitable for being solved with a vertically varying shear-periodic [MATH]-direction boundary condition.', '1406.4864-2-62-2': 'The radial and azimuthal boundary conditions for mapping a field variable [MATH] in a VGSB of size [MATH] are, respectively, [EQUATION]', '1406.4864-2-62-3': 'In order to completely define the problem, appropriate boundary conditions in the vertical direction must be specified.', '1406.4864-2-63-0': 'The set of Equations ([REF])-([REF]), together with the boundary conditions Equations ([REF])-([REF]) lead to approximate radially local, vertically global disk models if the underlying global disk model has a barotropic equilibrium or if we assume axisymmetry.', '1406.4864-2-64-0': 'It is important to emphasize that the velocity [MATH] is the departure from the local approximation of the bulk flow and thus the total fluid velocity [MATH] in the VGSB is [EQUATION] where [EQUATION] and the local shear rate [MATH] is related to the generalization of the height-independent [MATH]-parameter in the standard shearing box: [EQUATION]', '1406.4864-2-64-1': 'At the midplane, i.e., [MATH], the flow velocity [MATH] is the same as the steady state flow velocity in the standard shearing box.', '1406.4864-2-64-2': 'However, in global disk models where the shear rate of the flow decreases to zero at high altitude, i.e., [MATH] and, therefore, [MATH] for [MATH].', '1406.4864-2-64-3': 'This is just the reflex motion induced by the rotating frame.', '1406.4864-2-64-4': 'This shows that in the limit [MATH], Equations ([REF])-([REF]) reduce to the MHD equations in the absence of a gravitational field.', '1406.4864-2-65-0': '## Connecting the VGSB to the SSB', '1406.4864-2-66-0': 'In order to connect the SSB to the VGSB, with its generalization to baroclinic disk equilibria, it is useful to recall some results that apply to barotropic hydrostatic equilibria.', '1406.4864-2-66-1': 'If the pressure is [MATH] it follows that:', '1406.4864-2-67-0': 'Let us now focus our attention on a global isothermal disk in the gravitational potential of a point source of mass [MATH], i.e., [EQUATION] with [MATH] the gravitational constant.', '1406.4864-2-67-1': 'The pressure is [MATH], with constant sound speed [MATH].', '1406.4864-2-67-2': 'Adopting a power-law density dependence with radius in the midplane, tha hydrostatic balance in the vertical direction implies that the density is [EQUATION] where the constant [MATH] and we defined the Keplerian speed [EQUATION]', '1406.4864-2-67-3': 'In this case, the enthalpy is simply [MATH], and thus the generalized gravito-thermal potential [MATH], is [EQUATION]', '1406.4864-2-67-4': 'The angular frequency can be obtained from Equation ([REF]) as [EQUATION] where the Keplerian frequency is [EQUATION]', '1406.4864-2-67-5': 'The generalized gravito-thermal potential [MATH] and the angular frequency [MATH] are both independent of height, as expected.', '1406.4864-2-68-0': 'Expanding Equations ([REF]) and ([REF]) in radius around [MATH], we obtain expressions for the local values the angular frequency [MATH] and the shear rate [MATH] that can be used, in the framework of the VGSB, to study isothermal disks which are not necessarily thin compared to the local radius [MATH], i.e., [EQUATION] where we have defined [MATH] and [MATH].', '1406.4864-2-69-0': 'These expressions are useful to show that in the limit of a cold, thin disk, i.e., [MATH], the density profile becomes [MATH], with [MATH], and the angular frequency and shear rate become [MATH], [MATH], respectively, and Equations ([REF])-([REF]) reduce to the equations for the standard shearing box.', '1406.4864-2-70-0': '## Examples of VGSB Models for Global Equilibria', '1406.4864-2-71-0': 'We now consider two families of baroclinic, global disk equilibria that are characterized by angular frequencies and shear rates that depend on height, and are thus impossible to study within the standard shearing box approximation.', '1406.4864-2-71-1': 'The VGSB framework, embodied in Equations ([REF])-([REF]), together with their associated boundary conditions, can be used to produce axisymmetric, vertically global, horizontally local models for these astrophysical disks.', '1406.4864-2-71-2': 'After these examples, we summarize the new aspects of the baroclinic VGSB.', '1406.4864-2-72-0': '### VGSB for Global Disk Models with Cylindrical Temperature Structure', '1406.4864-2-73-0': 'There is a family of disk models which are isothermal in the vertical direction at any fiducial radius.', '1406.4864-2-73-1': 'These have been used, for example, by [CITATION].', '1406.4864-2-73-2': 'The corresponding global hydrostatic equilibrium configuration is given by [EQUATION] where [MATH] stands for the sound speed, assuming an ideal gas, [EQUATION]', '1406.4864-2-73-3': 'The local expansions involved in the VGSB for this global baroclinic equilibrium disk model are [EQUATION] with [MATH] and [MATH].', '1406.4864-2-74-0': 'Note that all the explicit [MATH]-dependences in [MATH] and [MATH] arise only when [MATH].', '1406.4864-2-74-1': 'This is in agreement with the isothermal case, corresponding to [MATH], in which the angular frequency is independent of height.', '1406.4864-2-74-2': 'It can also be seen from inspection of these equations that in the thin disk limit, i.e., [MATH], the VGSB reduces to the standard shearing box with [MATH], [MATH], and [MATH], provided that [MATH], even if the temperature depends on radius, i.e., [MATH].', '1406.4864-2-74-3': 'This is what ultimately justifies the use of a collection of standard shearing boxes to produce local barotropic equilibrium disk models, with height-independent angular frequency at different radii, even though the underlying global disk model has a baroclinic equilibrium structure.', '1406.4864-2-75-0': '### VGSB for Global Disk Models with Spherical Temperature Structure', '1406.4864-2-76-0': 'A spherical temperature dependence of the form [EQUATION] leads to a global hydrostatic disk configuration given by [EQUATION]', '1406.4864-2-76-1': 'Here, the power-law coefficients [MATH] and [MATH] are related via [EQUATION] and the sound speed is [EQUATION] where [MATH] is the sound speed at the midplane at [MATH].', '1406.4864-2-76-2': 'Such models have been used, for example, by [CITATION].', '1406.4864-2-77-0': 'The expansion of these expressions leads to the VGSB model corresponding to a global disk with a temperature structure varying in spherical shells.', '1406.4864-2-77-1': 'The local angular frequency and shear rate are, respectively, [EQUATION]', '1406.4864-2-77-2': 'In the limiting case in which [MATH]; the angular frequency and the shear rate take the Keplerian values at the midplane, i.e., [MATH] and [MATH], and the parameter [MATH] alone determines the disk thickness.', '1406.4864-2-78-0': '### New Aspects of the Vertically Global Shearing Box', '1406.4864-2-79-0': 'The approach that we followed in deriving the equations defining the framework of the vertically global shearing box is similar in spirit to the one employed in the derivation of the standard shearing box.', '1406.4864-2-79-1': 'This consists of expanding the steady sate bulk flow, retaining only the leading order terms in the ratio [MATH].', '1406.4864-2-79-2': 'The significant difference resides in that we have avoided making any expansion in the vertical direction.', '1406.4864-2-79-3': 'This allows us to retain the full height-dependence of the angular frequency.', '1406.4864-2-79-4': 'This is essential when dealing with disks in which the equilibrium structure is described by a baroclinic equation of state, since their angular frequency is in general a function of height.', '1406.4864-2-80-0': 'In order to illustrate the new aspects that are open to examination by retaining the full height-dependence in Equations ([REF]), ([REF]), and ([REF]), let us consider a baroclinic equilibrium global disk model, in which the temperature is a function of the spherical radius, as in [CITATION].', '1406.4864-2-80-1': 'The four panels in Figure [REF] compare the temperature, density, pressure, and angular frequency corresponding to the global equilibrium of the disk model (labeled "Global"), the local equilibrium defining the vertically global shearing box (VGSB), and local equilibrium that would result in the standard shearing box (SSB).', '1406.4864-2-81-0': 'The VGSB provides a local representation of a global baroclinic equilibrium disk model, capturing effects that the standard shearing box would not be able to account for.', '1406.4864-2-81-1': 'Because the lowest order discarded in Equation ([REF]) is [MATH], the radial dependence of the angular frequency is correct, to linear order in radius, for all heights.', '1406.4864-2-81-2': 'Moreover, the VGSB retains the correct global vertical gradients in temperature, density, and pressure, while neglecting the local radial variation of the isopycnic, isothermal, isobaric surfaces.', '1406.4864-2-81-3': 'This is in sharp contrast with the local equilibrium involved in the standard shearing box framework that is unable to capture the variation of the angular frequency with height, preventing its use when modeling disks with a baroclinic equilibrium structure.', '1406.4864-2-81-4': 'Note also that, for the case under consideration, the equilibrium density and pressure profiles involved in the standard isothermal shearing box are less accurate than the ones associated with the VGSB.', '1406.4864-2-82-0': 'In the next two sections, we illustrate how the VGSB framework relates to, and extends, previous treatments of two disk instabilities that are relevant for a wide variety of astrophysical disks; namely the vertical shear instability (VSI), also known as the GSF instability , and the magnetorotational instability (MRI) .', '1406.4864-2-82-1': 'These instabilities will be the subject of future, more detailed, work.', '1406.4864-2-83-0': '# Vertical Shear Instability (VSI) in the VGSB', '1406.4864-2-84-0': 'Unmagnetized disks with shear profiles that depend on height have been long suspected to be unstable to various instabilities that feed off this angular frequency gradient.', '1406.4864-2-84-1': 'The pioneering studies of [CITATION] and [CITATION] invoked local approximations in both radius and height, capturing the essence of these instabilities but leaving open questions about their global behavior.', '1406.4864-2-84-2': 'These instabilities have been studied locally, and in conjunction with magnetorotational instability, in the context of accretion disks.', '1406.4864-2-84-3': 'It has been suggested that they can play a role in the low-conductivity regime characterizing protoplanetary disks .', '1406.4864-2-84-4': 'The global, non-linear evolution of these instabilities has ben recently studied in [CITATION] by performing numerical simulations.', '1406.4864-2-84-5': 'This work includes an extension of the original local analysis, by considering the effects of compressibility and also an approximate, vertically global linear mode analysis.', '1406.4864-2-84-6': 'In this section, we show that the VGSB formalism recovers the local dispersion relation found in [CITATION] and can be used to address their vertically global mode analysis without invoking the approximations related to compressibility considered by the authors .', '1406.4864-2-85-0': 'There are general considerations which are common to both local and global approaches in height.', '1406.4864-2-85-1': 'Let us first write the VGSB equations for an unmagnetized disk for which the local angular frequency and shear rate derive from a global disk model with temperature [MATH] dependent on the cylindrical radius, as described in Section [REF].', '1406.4864-2-85-2': 'Following [CITATION], we make the change of variables [MATH] so that Equations ([REF])-([REF]) become [EQUATION]', '1406.4864-2-85-3': 'In order to derive the equations for the linear mode analysis that lead to the VSI starting from Equations ([REF]) and ([REF]), we proceed as follows: (i) We write the density variable as [MATH], i.e., the sum of its hydrostatic value and a fluctuation over this background, and the pressure as [MATH].', '1406.4864-2-85-4': '(ii) We use the scales of length and time provided by [MATH] and [MATH] to define the dimensionless variables [MATH] and [MATH], so that [MATH] is the dimensionless shear rate, [MATH] is the dimensionless sound speed, etc. (iii) We assume that all the perturbations are small and axisymmetric.', '1406.4864-2-85-5': '(iv) We focus on radial scales that are small compared to the fiducial radius [MATH] and take the Fourier transform of the set of equations ([REF]) and ([REF]), which reduces to making the substitution [MATH], for all flow variables, [MATH], [MATH], [MATH], and [MATH], where the tilde denotes Fourier amplitudes.', '1406.4864-2-85-6': 'This procedure leads to the following set of dimensionless equations [EQUATION]', '1406.4864-2-86-0': '## Vertically Local Analysis of VSI', '1406.4864-2-87-0': 'Let us first show that the equations for the VGSB lead to the dispersion relation derived by [CITATION], which is a generalization of the dispersion relation derived originally by GSF , who considered an incompressible gas.', '1406.4864-2-87-1': 'In order to do this, we restrict the analysis to the vertical location about the height [MATH], focusing on scales that are small in both the radial and vertical direction.', '1406.4864-2-87-2': 'Note that, in this case, the convenient time scale to define dimensionless variables is [MATH].', '1406.4864-2-87-3': 'Taking the Fourier transform of the set of equations ([REF]) and ([REF]) reduces to making the substitution [MATH], so that [EQUATION] where the background quantities are understood to be evaluated at [MATH] and we have used that [MATH], where [MATH] is the local value of the gravitational accelleration.', '1406.4864-2-88-0': 'The characteristic polynomial of the homogeneous system of equations ([REF])-([REF]) yields the dispersion relation in terms of dimensionless variables [EQUATION]', '1406.4864-2-88-1': 'This result is equivalent to the equation one before Equation (32) in [CITATION].', '1406.4864-2-88-2': 'Therefore, the dispersion relation of the vertical shear instability present in global disk models with height-dependent angular frequencies is correctly obtained by the VGSB framework.', '1406.4864-2-89-0': '## Vertically Global Analysis of VSI', '1406.4864-2-90-0': 'In this section we perform a vertically global analysis of isothermal, axisymmetric unmagnetized perturbations by solving the coupled set of ordinary differential equations ([REF])-([REF]).', '1406.4864-2-90-1': 'For a fixed radial wavenumber [MATH], the solution to the eigenvalue problem defined by these equations yields a set of eigenfunctions [MATH], [MATH], [MATH], and [MATH] associated with the eigenvalues [MATH].', '1406.4864-2-90-2': 'We are particularly interested in finding the modes with growing amplitude, i.e., where the real part of the eigenvalue [MATH].', '1406.4864-2-91-0': 'We compare the eigenvalues for a given set of parameters to those found through the approximated eigenproblem derived by [CITATION], which invokes additional approximations.', '1406.4864-2-91-1': 'Reasonable agreement is found, as shown in Figure [REF], with the main difference being that the surface modes show slower growth when full compressibility is retained.', '1406.4864-2-92-0': 'We illustrate the eigenmodes present in a disk model with cylindrical temperature structure [MATH], as presented in Section [REF], in Figures [REF] and [REF].', '1406.4864-2-92-1': 'We consider as an example a global disk model with [MATH], [MATH], and the finite domain [MATH], where [MATH] and [MATH].', '1406.4864-2-92-2': 'These parameters match those presented in [CITATION].', '1406.4864-2-92-3': 'We solve the problem corresponding to a single radial wavenumber [MATH], associated to a wavelength of [MATH], as follows.', '1406.4864-2-92-4': 'We discretize the problem in terms of Chebyshev cardinal functions on the Gauss-Lobatto grid, and the boundary condition [MATH] is enforced by the "boundary bordering" method .', '1406.4864-2-92-5': 'We vary the resolution, using a maximum of 300 grid points (yielding a 1200 by 1200 matrix) to obtain the converged eigenvalues shown.', '1406.4864-2-93-0': 'The basic pattern of modes shown in Figure [REF] agrees well with the approximate analysis carried out by [CITATION], who solved a second order differential equation that results from combining and approximate Equations ([REF])-([REF]) .', '1406.4864-2-93-1': 'The important difference between these two approaches is that [CITATION] make an approximation which removes full compressibility from the problem.', '1406.4864-2-93-2': 'We see that the fastest growing modes are relatively damped when full compressibility is retained in our analysis.', '1406.4864-2-93-3': 'In order to facilitate making a connection to their findings, we discuss the modes that we obtained using their terminology.', '1406.4864-2-94-0': 'The eigenvalues have the symmetry in the complex plane [MATH].', '1406.4864-2-94-1': 'The fastest growing modes are a branch of "surface" modes with degenerate eigenvalues.', '1406.4864-2-94-2': 'In Figure [REF] we show eigenfunctions where all components have been normalized by the complex value [EQUATION] where [MATH] is the maximum [MATH] of the domain.', '1406.4864-2-94-3': 'Because the Fourier amplitudes given by solving the eigenproblem as posed in transformed-[MATH] Fourier space for [MATH] and [MATH] are complex conjugates, they correspond to a single real-valued eigenfunction when the [MATH]-direction Fourier transformation is inverted to bring the eigenfunction into real space.', '1406.4864-2-94-4': 'Therefore, we have plotted this real-valued eigenfunction resulting from the [MATH] and [MATH] pair.', '1406.4864-2-94-5': 'The first and second row in Figure [REF] illustrate the fastest growing pair of modes with degenerate eigenvalue.', '1406.4864-2-94-6': 'A branch of eigenvalues proceeding from the origin of the complex plane contains the fundamental body modes, which are associated with "corrugation" and "breathing" modes.', '1406.4864-2-94-7': 'Like in [CITATION], the eigenvalue closest to the origin is associated with the fundamental corrugation mode, shown in the third row of Figure [REF].', '1406.4864-2-94-8': 'Next on this branch is the fundamental breathing mode, shown in the fourth row of Figure [REF].', '1406.4864-2-95-0': 'Repeating the same calculation with a larger finite domain [MATH], and thicker disk [MATH] yields instead the modes shown in Figure [REF].', '1406.4864-2-95-1': 'The remarkable change is the introduction if high frequency oscillations at high altitudes.', '1406.4864-2-95-2': 'The prodigious number of fast growing surface modes which appear in this calculation as the vertical size of the domain is increased suggests that magnetic fields will be a significant consideration in the astrophysical context for vertical shear instabilities.', '1406.4864-2-95-3': 'In the low density regime high above the disk midplane the gas will almost certainly be ionized in many types of astrophysical disks; the presence of magnetic fields ought then to have a significant impact on the dynamics of these instabilities.', '1406.4864-2-96-0': '# Magnetorotational Instability in the VGSB', '1406.4864-2-97-0': 'Following, and extending, the general technique of [CITATION], we perform a linear stability analysis of the MRI in the framework of the VGSB.', '1406.4864-2-97-1': 'In order to derive the equations for the linear mode analysis that lead to the MRI, we assume a homogeneous background magnetic field [MATH] and examine the evolution of the perturbations of the form [MATH] for the velocity and magnetic field components [MATH], [MATH], [MATH], [MATH], considering [MATH].', '1406.4864-2-97-2': 'Upon substitution in Equations ([REF])-([REF]) and linearizing in the perturbation amplitudes, the equations for the perturbation are [EQUATION]', '1406.4864-2-97-3': 'In a similar way as we did for the VSI, it is convenient to use the scales of length and time provided by [MATH] and [MATH] in order to define dimensionless variables, i.e., [MATH] and [MATH], etc.', '1406.4864-2-97-4': 'Note that the time unit of the growth rate [MATH] is [MATH], so its actual value will depend on the parameters of the shearing box model employed.', '1406.4864-2-98-0': 'The set of equations ([REF])-([REF]) generalize Equations (12)-(15) in [CITATION] .', '1406.4864-2-98-1': 'to include the effects of height-dependent angular frequency and shear rate that arise in the context of baroclinic equilibrium disk models.', '1406.4864-2-98-2': 'For the sake of simplicity, and in order to make a direct connection with the linear mode analysis in [CITATION], we have neglected the term proportional to [MATH] on the right hand side of Equation ([REF]), which contributes to the secular evolution of the magnetic field.', '1406.4864-2-98-3': 'This is a sensible approximation if the associated modes are localized in height in regions where the vertical shear rate [MATH] is smaller than the growth rate of the unstable modes.', '1406.4864-2-98-4': 'Figure [REF] shows the vertical shear rate [MATH] for two spherical temperature structure VGSB models, and the case which we consider below is roughly compatible with this approximation.', '1406.4864-2-98-5': 'However, though for the thin disk case the instability growth is roughly ten times faster than than the vertical shear rate at [MATH], this separation is such that the possible effects warrant further exploration in the future.', '1406.4864-2-99-0': 'In order to illustrate how the MRI modes are affected, we examine the spectrum of growing modes present in a disk with spherical temperature structure, [MATH], as discussed in Section [REF], taking as an example the [MATH].', '1406.4864-2-99-1': 'These values corresponding to a scale height of [MATH], and magnetic field strength [MATH].', '1406.4864-2-100-0': 'We solve the eigenproblem posed by Equations ([REF])-([REF]) by discretizing in the infinite domain [MATH] in terms of rational Chebyshev cardinal functions on the "roots", or interior grid.', '1406.4864-2-100-1': 'For the set of chosen parameters, 200 grid points were sufficient to obtain converged results.', '1406.4864-2-100-2': 'We adopted the same set of boundary conditions used in [CITATION], i.e., [MATH] and [MATH].', '1406.4864-2-100-3': 'To make the problem numerically tractable, we limit [MATH] to a minimum value of [MATH], which is equivalent to limiting the Alfven speed at high altitudes.', '1406.4864-2-100-4': 'The VGSB model of this particular disk model has one more unstable eigenmode than the standard shearing box.', '1406.4864-2-100-5': 'Note that our VGSB analysis yields growth rates scaled by [MATH], which is smaller than the Keplerian frequency.', '1406.4864-2-100-6': 'Thus, even though the parameters characterizing the specific VGSB model do not appear explicitly in the linear analysis, the physical values obtained do depend on them.', '1406.4864-2-100-7': 'If we consider [MATH], then in the VGSB model analyzed here [MATH].', '1406.4864-2-100-8': 'This implies that the physical growth rate of the MRI is smaller in the VGSB model of this particular disk than in the standard shearing box.', '1406.4864-2-100-9': 'In addition to solving the eigenproblem for the VGSB, we solve the matching problem for the standard shearing box by using an isothermal disk with the same midplane density and scale height [MATH] as used in the VGSB.', '1406.4864-2-100-10': 'The magnetic field and velocity perturbations of the eigenmodes, shown in Figures [REF] and [REF], are directionally orthogonal in the standard shearing box, whereas in the VGSB this is not the case.', '1406.4864-2-100-11': 'The eigenmodes in the VGSB are generally more complex, and, as it can be seen from these figures, they do not appear to have such a simple ordering in terms of the number of nodes as the standard shearing box eigenmodes do.', '1406.4864-2-101-0': '# Discussion', '1406.4864-2-102-0': 'The framework of the VGSB allows, for the first time, to develop models for astrophysical disks which are local in radius but are global in height, in that no expansion is made vertically.', '1406.4864-2-102-1': 'This is critical to study astrophysical disks where the equilibrium cannot be described by a barotropic equation of state, as these, in general, do not rotate on cylinders and are thus not amenable to the standard shearing box framework (see Figure [REF]).', '1406.4864-2-102-2': 'The VGSB naturally accounts for height-dependent radial and vertical flux of fluid and electromagnetic momentum in the azimuthal direction.', '1406.4864-2-102-3': 'This is relevant for the dynamics of disks with baroclinic equilibrium structure, especially if these cannot be regarded as thin.', '1406.4864-2-102-4': 'Physical domains with a large vertical extent allow for the global magnetic field threading the disk to be efficiently anchored into the surrounding medium.', '1406.4864-2-102-5': 'The coupling between stresses and shear, both radial and vertical, can efficiently transport momentum and energy to/from the disk.', '1406.4864-2-102-6': 'We envision that these effects will have important consequences, for example, in the study of the coupled dynamics between disks and their coronae and winds.', '1406.4864-2-103-0': 'Because of its inherent local character, the framework provided by the standard shearing box is well suited to study physical processes involving scales that are smaller than the characteristic disk scales.', '1406.4864-2-103-1': 'Global disk simulations, which are becoming ever more accessible , are useful for understanding the large-scale disk dynamics but face the challenge of resolving the physical processes at small scales.', '1406.4864-2-103-2': 'In order to understand how local and global processes interact, it is desirable to devise a framework to bridge local and global approaches.', '1406.4864-2-103-3': 'Hence, there have been several initiatives to relax the local character of the standard shearing box.', '1406.4864-2-104-0': '## Previous Works Beyond the Standard Shearing Box', '1406.4864-2-105-0': 'The periodic "shearing disk" annulus introduced in [CITATION] relaxes the condition of azimuthal locality by building a radially periodic annulus and remapping quantities across the radial background according to an imposed power-law radial disk structure.', '1406.4864-2-105-1': 'This approach involves equations which have explicit coordinate dependence in the direction where the computational domain is shear-periodic.', '1406.4864-2-105-2': 'Therefore, the flow properties present a jump at the shearing boundary, which might lead to unphysical effects.', '1406.4864-2-105-3': 'In this context, the formulations presented in [CITATION] and [CITATION] complement each other in the sense that they retain the terms accounting for finite curvature but discard global gradients and vice-versa, but both lead to similar issues.', '1406.4864-2-106-0': 'There have been several attempts to make the stratified shearing box global in height.', '1406.4864-2-106-1': 'Foremost, [CITATION] treated the case of a polytropic gas, but only made local expansions in the horizontal direction.', '1406.4864-2-106-2': 'An early example of modifying the vertical gravity in a standard shearing box is given in [CITATION], where the full [MATH]-dependence of the vertical component of the gravitational force is considered with an isothermal gas.', '1406.4864-2-106-3': 'Several other authors have used similar approaches, for both galactic and Keplerian disks .', '1406.4864-2-106-4': 'These works assume that all the global effects in the vertical direction can be accounted for by modifying only the momentum equation in the [MATH]-direction and exclusively through the term [MATH].', '1406.4864-2-106-5': 'When the gas in the global equilibrium configuration is strictly baroptropic, it must rotate on cylinders, and the height-independent shear in the standard shearing box is a consistent approximation (given the caveats in Appendix [REF]).', '1406.4864-2-106-6': 'Importantly, in the case that the gas is assumed to be isothermal, the standard shearing box provides a consistent approach if the global disk configuration, at all radii, is isothermal.', '1406.4864-2-106-7': 'If another equation of state for the hydrostatic structure is used, or if the global configuration of which the shearing box is a small part has a radial temperature gradient, then the VGSB provides a viable framework going beyond these previous approaches, with important modifications beyond the standard shearing box formalism, see Equations ([REF])-([REF]).', '1406.4864-2-107-0': '## Limitations of the VGSB', '1406.4864-2-108-0': 'The generalization of the SSB to account for the full vertical variation of gravity and the presence of vertical shear, inherent to global baroclinic disk models, does posses limitations.', '1406.4864-2-108-1': 'Some of these limitations are also characteristic of the SSB, while others are inherent to the VGSB.', '1406.4864-2-109-0': 'It should be clear that the VGSB, as well as the SSB, framework consists of a set of dynamical equations for the perturbations with respect to a time-independent background representing a smaller section of a steady global disk model.', '1406.4864-2-109-1': 'In this approach, by construction, the fluctuations cannot modify the dynamics of the background.', '1406.4864-2-109-2': 'This could be considered a serious limitation, however, this kind of approach as embodied in the SSB has proven exceptionally useful as a workhorse to understand a wide variety of phenomena in disks.', '1406.4864-2-109-3': 'Another limitation of the VGSB, that is shared by the SSB, is that the local approximation in radius prevents the full consideration curvature terms, which are inherent to the cylindrical geometry usually employed to model disks (see Appendix [REF]).', '1406.4864-2-110-0': 'The set of equations that define the SSB can be obtained by expanding in radius and height either the gravitational potential of a point source or the Keplerian background flow that ensues when radial pressure support can be neglected.', '1406.4864-2-110-1': 'In order to derive the equations defining the VGSB we have approximated the background flow itself.', '1406.4864-2-110-2': 'The body and frame forces which appear in this formulation relate to a fluid element in that flow with the given global thermodynamics, not the forces on a test particle.', '1406.4864-2-110-3': 'Thus, these forces are not the same as an expansion of the tidal potential about a Keplerian orbit.', '1406.4864-2-110-4': 'In the special case of a barotropic global equilibrium, the body forces do correspond to a conservative potential, but in general for baroclinic backgrounds this is not the case.', '1406.4864-2-110-5': 'The advantage of expanding the angular frequency is that its equilibrium profile [MATH] is sensitive to the equilibrium pressure gradient, enabling access to rotation laws beyond strictly Keplerian, in particular those with vertical shear.', '1406.4864-2-111-0': 'The most important limitations of the VGSB stem from the fact that we have opted to explore the consequences of imposing height-dependent shearing-periodic boundary conditions (see Section [REF]).', '1406.4864-2-111-1': 'In order to do this, we had to approximate Equation ([REF]) and use Equation ([REF]) instead.', '1406.4864-2-111-2': "Whereas this seems a reasonable approximation, it had in general undesired consequences for some basic conservation laws such as Kelvin's Circulation Theorem and Alfven's Frozen-in Theorem unless axisymmetry is imposed.", '1406.4864-2-111-3': 'In order to consider height-dependent shearing-periodic boundary conditions we also needed to understand under what circumstances it was acceptable to neglect background pressure gradients.', '1406.4864-2-111-4': 'Even though the VGSB can be considered to be global in height because there is no need to expand the background in the [MATH]-direction, the analysis we carried out of the pressure gradients in Appendix [REF] shows that it is necessary to limit the vertical extent of the domain to [MATH].', '1406.4864-2-111-5': 'This is more restrictive that [MATH], but nevertheless can accommodate for many scale-heights with the added value of being able to account for the full expression of the vertical gravity and vertical shear within the domain.', '1406.4864-2-112-0': 'The derivation of the VGSB formalism has allowed us to shed light into several subtle issues that are usually not addressed in the context of the SSB, see e.g., Appendices [REF] and [REF].', '1406.4864-2-112-1': 'In spite of its limitations, the SSB has proven to be a useful tool to learn about local disk dynamics.', '1406.4864-2-112-2': 'We believe that the VGSB will allow the relaxation of some of the assumptions that have been widely adopted by using the SSB.', '1406.4864-2-112-3': 'In particular, it will enable the investigation of some dynamical aspects of astrophysical disks that cannot be studied with the standard shearing box and for which full global modeling is too demanding.', '1406.4864-2-113-0': '## Applications of the VGSB', '1406.4864-2-114-0': 'We anticipate that the VGSB framework, summarized in Appendix [REF], will benefit the modeling of a wide variety of phenomena in astrophysical disks.', '1406.4864-2-114-1': 'The advances made possible with respect to the standard shearing box will depend on the nature of the physical phenomena under study, e.g.:', '1406.4864-2-115-0': 'Hydrodynamic Disk Instabilities The VGSB framework may be used for the local study of vertical shear instabilities, such as Goldreich-Schubert-Fricke (GSF) instability and generalizations .', '1406.4864-2-115-1': 'We have demonstrated in Section [REF] that the VGSB captures the correct local linear behavior.', '1406.4864-2-115-2': 'It should also be useful to analyze the propagation of hydrodynamic waves in disks .', '1406.4864-2-116-0': 'Disk Convection Vertical convective instabilities in disks have been studied with the standard shearing box as an angular momentum transport mechanism.', '1406.4864-2-116-1': 'Though early studies gave negative results, [CITATION] have suggested that earlier models are under-resolved, and outward angular momentum transport is possible.', '1406.4864-2-116-2': 'The VGSB framework enables to study the effects that vertical shear can have on the long-term evolution of convective motions in a way which is not accessible with the standard shearing box.', '1406.4864-2-117-0': 'Disk Coronae Previous standard stratified shearing box simulations show the buoyant rise of magnetic field to the upper disk layers , where it is thought to dissipate giving rise to a hot corona .', '1406.4864-2-117-1': "If the disk's equilibrium structure is not barotropic, the fact that the shear, i.e., the source of free energy, decreases with height can have important implications for the turbulent disk dynamics and energetics in the disk corona.", '1406.4864-2-117-2': 'This could affect the vertical disk structure and its corona .', '1406.4864-2-117-3': 'These effects could be important for thick disks, such as advection dominated accretion flows (ADAFs) .', '1406.4864-2-118-0': 'Disk Winds Standard shearing boxes extended in height have been used to study the magnetorotational instability (MRI) as a mechanism for launching disk winds though only in the framework provided by modifying the gravitational forces through the term [MATH].', '1406.4864-2-118-1': 'We have shown in Section [REF] that the linear MRI is modified by effects included in the VGSB for disks with baroclinic equilibria.', '1406.4864-2-118-2': 'Because the wind dynamics are particularly sensitive to the forces acting on the fluid far away from the midplane, the VGSB provides a more general framework for local studies of this nature.', '1406.4864-2-119-0': 'Interstellar Medium and Galactic Disks In galactic disks, the VGSB makes it possible to capture critical physics for studying the non-linear evolution of the magnetorotational, Parker, and magneto-Jeans instabilities .', '1406.4864-2-119-1': 'The framework provided by the VGSB also allows for the inclusion of fundamental height-dependent physical effects that can play an important role in the study of star formation, galactic dynamos, and the structure of the interstellar medium .', '1406.4864-2-120-0': 'We are grateful to Tobias Heinemann, whose valuable insights helped us to better understand the subtleties involved in the derivation of the VGSB and to present it in a transparent way.', '1406.4864-2-120-1': 'We acknowledge thoughtful comments from Scott Tremaine and Eric Blackman, and discussions with Troels Haugbolle, AAke Nordlund, Andrew Jackson, Oliver Gressel, Orkan M. Umurhan, Gopakumar Mohandas, Henrik Latter, Andrea Mignone, Gianluigi Bodo and Sacha Brun.', '1406.4864-2-120-2': 'We are thankful to the anonymous referee, whose tenacious reports led us to better appreciate how the VGSB connects to, and extends, the standard shearing box framework, and to improve the original manuscript significantly.', '1406.4864-2-120-3': "The research leading to these results has received funding from the People Programme (Marie Curie Actions) of the European Union's Seventh Framework Programme (FP7/2007-2013) under REA grant agreement 327995 (CPM), and the European Research Council under the European Union's Seventh Framework Programme (FP/2007-2013) under ERC grant agreement 306614 (MEP).", '1406.4864-2-120-4': 'MEP also acknowledges support from the Young Investigator Programme of the Villum Foundation.', '1406.4864-2-121-0': '# Derivation of the momentum equation for departures from the background bulk flow', '1406.4864-2-122-0': 'Here, we provide the algebraic details involved in obtaining the momentum equation ([REF]) starting from Equation ([REF]).', '1406.4864-2-123-0': 'Let us first expand the left hand side of Equation ([REF]), LHS for short, after substituting [MATH], we obtain [EQUATION]', '1406.4864-2-123-1': 'We now expand the last three terms, recalling that the differential operators involved are in cylindrical coordinates and using the definition of the bulk flow in terms of the angular frequency [MATH].', '1406.4864-2-123-2': 'This yields [EQUATION]', '1406.4864-2-123-3': 'This completes the expansion of the left hand side in terms of the new velocity variable [MATH].', '1406.4864-2-124-0': 'We now proceed similarly with the right hand side of the momentum equation ([REF]), RHS for short, by substituting [MATH] and expanding [EQUATION]', '1406.4864-2-124-1': 'Equating the left hand side expression (Equation [REF]) and the right hand side expression (Equation [REF]) and canceling out the term [MATH] which appears on both sides, the momentum Equation ([REF]) becomes, without approximation, [EQUATION] which corresponds to Equation ([REF]).', '1406.4864-2-125-0': 'As stated in Section [REF], a cancellation of terms from the left and right hand sides of the momentum equation results in the term [MATH] that looks similar to the Coriolis acceleration, i.e., [MATH], but with the angular frequency [MATH] taking the place of the constant angular frequency of the rotating frame [MATH].', '1406.4864-2-126-0': '# Summary of the VGSB Equations', '1406.4864-2-127-0': 'Here we provide a self-contained summary of the equations defining the vertically global shearing box (VGSB).', '1406.4864-2-128-0': 'VGSB Equations The continuity, momentum, induction, and energy equations in the VGSB are, respectively, given by [EQUATION]', '1406.4864-2-128-1': 'Here, [MATH] is the mass density, [MATH] is the velocity with respect to the local bulk flow in the disk, [MATH] is the magnetic field, and [MATH] is the internal energy density.', '1406.4864-2-128-2': 'The pressure [MATH] is determined as a function of [MATH] and [MATH], through an appropriate equation of state.', '1406.4864-2-128-3': 'The current density is [MATH], with [MATH] a constant dependent on the unit system adopted.', '1406.4864-2-128-4': 'All the operators in the VGSB equations are defined in a cartesian coordinate system centered at the fiducial radius [MATH].', '1406.4864-2-129-0': 'For barotropic background equilibrium disc structures, the above equations can be used in fully three dimensions, while for baroclinic background equilibrium disc structures, these equations must by restricted to axisymmetric solutions ([MATH]-invariant).', '1406.4864-2-130-0': 'VGSB Definitions The equations defining the VGSB framework depend on a number of functions of the coordinate [MATH], which result from local radial expansions of the global disk model around the radius [MATH].', '1406.4864-2-130-1': 'The height-dependent advection by the background shear, [MATH], is given by [EQUATION] where [MATH], [MATH], and [MATH] correspond, respectively, to the local values of the background azimuthal velocity, angular frequency, and background shear flow, which are all derived from the angular frequency of the global disk model [MATH], i.e., [EQUATION]', '1406.4864-2-130-2': 'The gravitational potential in the VGSB [MATH] is given by the value of the global gravitational potential [MATH] at [MATH] [EQUATION]', '1406.4864-2-130-3': 'VGSB Boundary Conditions For a domain with size [MATH] the horizontal boundary conditions are [EQUATION] which correspond to a generalization of the shearing-periodic boundary conditions adopted in the standard shearing box.', '1406.4864-2-131-0': '# Neglecting Curvature Terms', '1406.4864-2-132-0': 'The cylindrical coordinate system that we adopt to describe the global disk model leads naturally to the presence of the quadratic terms [MATH], [MATH], [MATH], and [MATH] in the momentum equation ([REF]).', '1406.4864-2-132-1': 'All of these terms are usually neglected in the standard shearing box.', '1406.4864-2-132-2': 'The terms related to the velocity field can be neglected safely when they are small compared to the corresponding components of the acceleration [MATH] and [MATH].', '1406.4864-2-132-3': 'Both of these conditions lead to the inequality [MATH].', '1406.4864-2-132-4': 'The magnetic terms [MATH] and [MATH] in Equation ([REF]) are usually absent in local studies (see for an exception).', '1406.4864-2-132-5': 'If the magnetic field is sufficiently sub-thermal everywhere in the domain, these terms can be neglected .', '1406.4864-2-132-6': 'However, they have been commonly neglected even when this is not the case.', '1406.4864-2-132-7': 'This is perhaps because it is not obvious that retaining them will lead to a physically consistent problem in the framework of the shearing box.', '1406.4864-2-132-8': 'Consider, for example, the case of a shearing box in which a strong net azimuthal field develops as a consequence of the local disk dynamics.', '1406.4864-2-132-9': 'This could lead to the force arising from the term [MATH] to increase with time.', '1406.4864-2-132-10': 'However, this force cannot be balanced by an increase in the centripetal acceleration as the latter is fixed by the choice of the bulk flow .', '1406.4864-2-132-11': 'Note that all of this statements regarding the quadratic terms that reminisce the curvilinear nature of the original cylindrical coordinate system are independent of whether the standard shearing box, or the vertically global version developed in this paper, is considered.', '1406.4864-2-133-0': '# Nondimensionalization and Ordering of the Hydrodynamic Momentum Equation', '1406.4864-2-134-0': 'A more rigorous description of the approximations that lead to the equations for the VGSB can be made by introducing nondimensional variables in order to expose the hierarchy of the various terms involved.', '1406.4864-2-134-1': 'Here, we only deal with the hydrodynamic part of the momentum equation, for the magnetic components we make the assumption that the magnetic field are weak enough to not effect the global equilibrium background configuration.', '1406.4864-2-134-2': 'Our approach follows and generalizes [CITATION].', '1406.4864-2-135-0': 'The hydrodynamic part of the momentum equation ([REF]) is given by [EQUATION]', '1406.4864-2-135-1': 'We define a horizontal length scale [MATH], a vertical length scale [MATH], time scale [MATH], pressure scale [MATH], and density scale [MATH].', '1406.4864-2-135-2': 'Horizontal velocities are nondimensionalized by [MATH] and vertical velocities by [MATH].', '1406.4864-2-135-3': 'We use primes to denote normalized quantities, like [MATH] the radial component of the fluctuation velocity, and [MATH] the radial partial derivative with respect to the radial nondimensional length, with radial length nondimensionalized as [MATH].', '1406.4864-2-135-4': 'The components of the momentum equation are thus [EQUATION] where we have introduced the dimensionless parameters [EQUATION]', '1406.4864-2-135-5': 'Note that [MATH] is most readily identified as the [MATH] appearing at the location of the dynamics in question, not the Keplerian frequency.', '1406.4864-2-135-6': 'Thus [MATH] is not directly the usual nondimensional scale height.', '1406.4864-2-135-7': 'Taylor expanding in small [MATH] about [MATH], keeping only lowest order terms, and transforming to cartesian coordinates as this allows (Section [REF]) yields [EQUATION]', '1406.4864-2-135-8': 'We now decompose the pressure and density into their hydrostatic part and a their associated fluctuation as [EQUATION]', '1406.4864-2-135-9': 'The components of the momentum equation then become [EQUATION]', '1406.4864-2-135-10': 'The fluctuation pressure terms have the same scaling as the background hydrostatic pressure terms, as in [CITATION].', '1406.4864-2-135-11': 'Using the expression for hydrostatic equilibrium given by Equation ([REF]), we obtain the form of the radial and vertical components of the acceleration as [EQUATION]', '1406.4864-2-135-12': 'The vertical acceleration depends only on the gravitational potential, but the radial acceleration depends on the thermodynamics of the hydrostatic equilibrium though the nondimensionalized rotation [MATH].', '1406.4864-2-136-0': 'In order to illustrate the procedure for determining the appropriate expansion of the radial acceleration, we work with the structure of the hydrostatic background provided by the cylindrical temperature profile [MATH], as in Section [REF].', '1406.4864-2-136-1': 'This thermal structure contains a barotropic equilibrium as a special case.', '1406.4864-2-136-2': 'The nondimensional version of the rotation law given by Equation ([REF]) is [EQUATION]', '1406.4864-2-136-3': 'When [MATH] this structure is isothermal, and barotropic, and when [MATH] the disk is thin, with the rotation close to Keplerian.', '1406.4864-2-137-0': '## Case of SSB and VGSB for a Globally Isothermal Background', '1406.4864-2-138-0': 'In order to connect our analysis with [CITATION], who considered the case with [MATH] and [MATH], we examine explicitly here the globally isothermal case with [MATH].', '1406.4864-2-138-1': 'Even though the equations that we presented for the VGSB do not explicitly involve expansions in the vertical direction, at this point in the analysis we do make such an expansion in order to assess the conditions under which it is acceptable to neglect the radial pressure gradients that need to be discarded in order to impose shearing-periodic boundary conditions, as discussed in Section [REF].', '1406.4864-2-138-2': 'Thus, the results of this section apply equally to the VGSB model for a globally isothermal (barotropic) disk, as well as to the SSB.', '1406.4864-2-139-0': 'Expanding the radial and vertical components of the acceleration about [MATH] and [MATH] we obtain, to leading order, [EQUATION]', '1406.4864-2-139-1': 'It is evident, as any finite domain contains arbitrarily small [MATH], that the only rigorous limit in which we can strictly justify discarding every term on the right hand side of equation ([REF]) is the one corresponding to a disk with a constant midplane density, [MATH], and/or infinitely thin, [MATH], in which the characteristic scales of interest in the vertical direction vanish identically, [MATH].', '1406.4864-2-139-2': 'This is certainly not an interesting limit, especially in the case where [MATH].', '1406.4864-2-139-3': 'Perhaps this is the reason for which these terms are usually discarded in the standard shearing box formalism, even though they cannot be strictly neglected in a disk with a radial density gradient and/or non-vanishing scale height when the vertical extent of the domain does not vanish identically.', '1406.4864-2-140-0': 'Having this caveat in mind, let us examine under what conditions we can argue that the terms proportional to [MATH] and [MATH] are sufficiently small that it is acceptable to neglect them.', '1406.4864-2-140-1': 'This requires understanding when each of the dimensionless terms in question is small compared to unity.', '1406.4864-2-140-2': 'In equation ([REF]) for the radial acceleration, the first term can be argued to be sufficiently small for [MATH] provided that [MATH], which implies [MATH].', '1406.4864-2-140-3': 'This condition can be satisfied for power-law indices [MATH] of order unity.', '1406.4864-2-140-4': 'The second term, of order [MATH], is small based on the same condition as [MATH] is at most of order unity.', '1406.4864-2-140-5': 'The third and fourth terms are clearly small for [MATH].', '1406.4864-2-140-6': 'The fifth term in this equation is more difficult to deal with because it scales with the ratio [MATH].', '1406.4864-2-140-7': 'In the case considered by [CITATION], because [MATH], this term is simply proportional to [MATH], and it can be directly neglected in the limit [MATH], along with the higher order terms in the radial acceleration.', '1406.4864-2-140-8': 'When the assumption [MATH] is relaxed, the second term in the radial acceleration is small when [MATH].', '1406.4864-2-140-9': 'Restoring dimensional quantities, this condition becomes [MATH].', '1406.4864-2-140-10': 'This implies that the fifth term in the equation for the radial acceleration is small provided that the vertical extent of the domain is smaller than the geometric mean between its radial extent and the radial location of the box.', '1406.4864-2-140-11': 'Note that this condition is more restrictive that requesting that [MATH], but far less restrictive than imposing that [MATH] vanishes identically.', '1406.4864-2-140-12': 'The higher order terms, such as the sixth term on the right hand side of equation ([REF]), are negligible provided that [MATH].', '1406.4864-2-140-13': 'The vertical acceleration in equation ([REF]) does not depend on the parameters [MATH] or [MATH].', '1406.4864-2-140-14': 'In this case, it is straightforward to neglect higher order terms proportional to [MATH] and [MATH] with respect to the leading term [MATH].', '1406.4864-2-141-0': '## Case of VGSB for Cylindrical Temperature Profiles', '1406.4864-2-142-0': 'Here, we generalize the above result in order to include a baroclinic background with non-zero radial temperature power law index [MATH].', '1406.4864-2-142-1': 'The generalization of Equation ([REF]) is [EQUATION]', '1406.4864-2-142-2': 'It is clear that there is a one-to-one correspondence between each of the terms present in Equation ([REF]) and the ones appearing in Equation ([REF]).', '1406.4864-2-142-3': 'For any reasonable value of [MATH], the numerical coefficients in both equations are of the same order.', '1406.4864-2-142-4': 'Because of this, the conditions required to neglect each of the terms in Equation ([REF]) and Equation ([REF]) are, within factors of order unity, identical.', '1406.4864-2-142-5': 'The equation for the expansion of the vertical pressure gradient is independent of [MATH] and thus identical to Equation ([REF]).', '1406.4864-2-143-0': 'From the analysis in this appendix, we thus conclude that, within factors of order unity, the requirements to neglect the radial background pressure gradients in the VGSB are expected to be similar to the ones involved in the SSB.', '1406.4864-2-144-0': "# Potential Vorticity and Ertel's Theorem", '1406.4864-2-145-0': "In previous sections of this paper, we have addressed the impact that the approximations embodied in the VGSB framework have on Kelvin's Circulation Theorem and Alfven's Frozen-in Theorem and showed that these are satisfied if the fluid is barotropic or axisymmetric.", '1406.4864-2-145-1': "Another important conservation law is given by Ertel's Theorem, which governs the evolution of the potential vorticity [MATH], where [MATH] is the absolute vorticity and [MATH] is a fluid property advected with the flow, according to [EQUATION]", '1406.4864-2-146-0': "Note that for any conserved scalar field [MATH], Ertel's Theorem leads to a conservation law for the potential vorticity if the flow is barotropic, i.e., [MATH].", '1406.4864-2-146-1': "If [MATH] is taken to be the specific entropy [MATH] in an isentropic flow, i.e. [MATH], Ertel's Theorem provides a conservation law for the potential vorticity even if the flow is baroclinic, because [MATH].", '1406.4864-2-147-0': "In order to shed light into the implications that the approximations embodied in the VGSB have for Ertel's Theorem let us examine the evolution equation for a PV-type quantity defined, in terms of the VGSB background velocity field [MATH] and the velocity fluctuations [MATH], as [EQUATION]", '1406.4864-2-147-1': 'We can assess when such a quantity obeys a conservation law in the form of Equation ([REF]).', '1406.4864-2-147-2': 'Starting from the approximate VGSB momentum equation ([REF]), transforming from the fluctuation velocity [MATH] to the velocity [MATH], and following the usual steps for deriving a potential vorticity evolution equation , we arrive to the following evolution equation [EQUATION]', '1406.4864-2-147-3': 'The first term on the right hand side can be non-zero, so this is not in general a conservation law of the form Equation ([REF]).', '1406.4864-2-147-4': 'This term has three components within it, the first being a nonconservative tidal force, the second being a coriolis-like force which remains from the velocity transformation, and the final term being due to the approximation made in Equation ([REF]).', '1406.4864-2-147-5': 'A coriolis-like force remains from the velocity transform from [MATH] to [MATH] because in Cartesian coordinates it is just a linear velocity boost in [MATH], and the VGSB momentum equation contains a coriolis-like term which is proportional to the [MATH] velocity.', '1406.4864-2-147-6': 'Interestingly, when a velocity transformation is done in cylindrical coordinates with an analogous change in the [MATH] velocity, the matching change in the coriolis term cancels with components of the cylindrical coordinate advection operator on the left hand side, as was seen in Appendix [REF].', '1406.4864-2-147-7': 'In the case of a VGSB model with a height-independent [MATH] or an SSB, the first term on the right hand side of Equation ([REF]) is zero, as the first two components are conservative and curl-free, and the third component is zero.', '1406.4864-2-147-8': 'In those cases, this applies for any choice of [MATH], as [MATH] is constant.', '1406.4864-2-147-9': 'Thus in those cases this PV-type quantity is conserved in the same sense as in Equation ([REF]).', '1406.4864-2-147-10': 'Beyond simply not being in general in the form of a conservation law, Equation ([REF]) also has the property that the Lagrangian derivative term with [MATH] is not zero when the velocity fluctuations [MATH] are zero.', '1406.4864-2-147-11': 'This is because the quantity defined in Equation ([REF]) contains both the background velocity and the fluctuation velocity.', '1406.4864-2-147-12': 'Note that in the absence of fluctuations in an axisymmetric flow, i.e., [MATH], and the first term on the right hand side of Equation ([REF]) does vanish.', '1406.4864-2-148-0': 'Understanding the implications of the source terms that would appear in general on the right hand side of Equation ([REF]) is beyond the scope of this work.', '1406.4864-2-148-1': 'In the reminder of this section we have a more modest goal which consists of assessing how the approximations leading to the VGSB impact the dynamics of a PV-type quantity [MATH] defined solely in terms of the velocity fluctuations, [MATH].', '1406.4864-2-149-0': 'In a way similar to that in the SSB, the VGSB framework consists of a set of equations for the fluctuations with respect to a known local equilibrium background (which corresponds to a local approximation of a global equilibrium).', '1406.4864-2-149-1': 'Starting from the momentum equation in terms of the fluid velocity [MATH] in the frame rotating with [MATH], we have derived the exact Equation ([REF]) for the velocity [MATH] characterizing the departures from the background equilibrium [MATH], Equation ([REF]).', '1406.4864-2-149-2': 'In what follows, we derive an evolution equation for the PV-type quantity associated with the velocity fluctuations in an exact way (Equation [REF]), and then repeat the derivation for the VGSB approximation (Equation [REF]) with the goal of comparing both results.', '1406.4864-2-150-0': 'Starting from the exact momentum equation ([REF]), taking the curl and using the continuity equation, we arrive to [EQUATION] where it has been natural to define the quantity [MATH], i.e., the vorticity associated with the fluctuations.', '1406.4864-2-150-1': 'We can obtain the following relation from the continuity equation ([REF]) [EQUATION] and use it to replace the second term on the left hand side in Equation ([REF]) to obtain [EQUATION]', '1406.4864-2-150-2': 'If we consider a scalar quantity [MATH] which is frozen-in to the flow, i.e., [MATH], we can derive the following identity [EQUATION]', '1406.4864-2-150-3': 'We can combine equations ([REF]) and ([REF]) to obtain an evolution equation for [MATH] as [EQUATION]', '1406.4864-2-150-4': 'This result is exact and it follows directly from Equations ([REF]) and ([REF]) in this paper, provided that [MATH] is frozen-in to the flow.', '1406.4864-2-151-0': 'Following the same procedures in the VGSB context, starting from the approximate VGSB momentum equation ([REF]) and again taking the curl and combining with the continuity equation we arrive at [EQUATION]', '1406.4864-2-151-1': 'Here, it has also been natural to define [MATH], i.e., the voticity associated with the fluctuations with respect to the VGSB background.', '1406.4864-2-151-2': 'This result is the VGSB equivalent of Equation ([REF]).', '1406.4864-2-151-3': 'The notable difference between the VGSB and the exact form is the lack of the [MATH] term.', '1406.4864-2-151-4': 'Considering a scalar quantity frozen-in to the approximate background [MATH], we obtain [EQUATION]', '1406.4864-2-151-5': 'This result is the VGSB version of the exact result in Equation ([REF]).', '1406.4864-2-151-6': 'From this form, we can see that in the shearing box (both SSB and VGSB) the lack of the term [MATH] prevents the thermodynamic driving of vorticity by the background, which would usually lead in a baroclinic disk to phenomena like the Rossby wave instability .', '1406.4864-2-151-7': 'In shearing boxes, this kind of instability can be driven by localized gradients, or for example, by adopting a Boussinesq approximation for the system at the point of Equation ([REF]) and thus retaining the background radial hydrostatic pressure gradient with shear-periodic radial boundary conditions .'}

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[]

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[]

['1406.4864-1-26-1', '1406.4864-1-62-1', '1406.4864-1-101-1', '1406.4864-1-111-2', '1406.4864-2-26-1', '1406.4864-2-66-1', '1406.4864-2-98-0', '1406.4864-2-114-1', '1406.4864-2-123-2', '1406.4864-3-26-1', '1406.4864-3-66-1', '1406.4864-3-98-0', '1406.4864-3-122-2']

{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '3': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1406.4864

{'1406.4864-3-0-0': 'Disks with a barotropic equilibrium structure, for which the pressure is only a function of the density, rotate on cylinders in the presence of a gravitational potential, so that the angular frequency of such a disk is independent of height.', '1406.4864-3-0-1': 'Such disks with barotropic equilibria can be approximately modeled using the shearing box framework, representing a small disk volume with height-independent angular frequency.', '1406.4864-3-0-2': 'If the disk is in baroclinic equilibrium, the angular frequency does in general depend on height and it is thus necessary to go beyond the standard shearing box approach.', '1406.4864-3-0-3': 'In this paper, we show that given a global disk model, it is possible to develop approximate models that are local in horizontal planes without an expansion in height with shearing-periodic boundary conditions.', '1406.4864-3-0-4': 'We refer to the resulting framework as the vertically global shearing box (VGSB).', '1406.4864-3-0-5': 'These models can be non-axisymmetric for globally barotropic equilibria but should be axisymmetric for globally baroclinic equilibria.', '1406.4864-3-0-6': 'We provide explicit equations for this VGSB which can be implemented in standard magnetohydrodynamic codes by generalizing the shearing-periodic boundary conditions to allow for a height-dependent angular frequency and shear rate.', '1406.4864-3-0-7': 'We also discuss the limitations that result from the radial approximations that are needed in order to impose height-dependent shearing periodic boundary conditions.', '1406.4864-3-0-8': 'We illustrate the potential of this framework by studying a vertical shear instability and examining the modes associated with the magnetorotational instability.', '1406.4864-3-1-0': '# Introduction', '1406.4864-3-2-0': 'Astrophysical disks play a crucial role in the formation, evolution, and fate of a wide variety of celestial objects, by mediating the transport of mass, energy, and angular momentum.', '1406.4864-3-2-1': 'Building realistic disk models is of fundamental importance for understanding, for example, protoplanetary disks around young stars, accretion flows onto stellar compact objects and active galactic nuclei, as well as the interstellar medium in galactic disks.', '1406.4864-3-2-2': 'The large dynamic range involved makes it particularly challenging to produce detailed global numerical simulations of these systems.', '1406.4864-3-2-3': 'Moreover, while global models allow us to investigate large-scale phenomena, local models with a hierarchy of increasingly complex microphysics have proved critical to elucidating the processes that are crucial at small scales.', '1406.4864-3-2-4': 'Because of this, several types of local approximations have been employed for studying astrophysical disks.', '1406.4864-3-3-0': '[CITATION] pioneered the use of a local approximation to study the dynamics of particles orbiting a host system and subject to encounters with a perturber, and used it to study the motion of the Moon.', '1406.4864-3-3-1': '[CITATION] and [CITATION] applied these ideas to galactic disks using the concept of a locally shearing coordinate system.', '1406.4864-3-3-2': 'This approach constitutes the basis of the shearing sheet framework which has been widely used to study the dynamics of orbiting particles and planetesimals, as well as local processes in hydrodynamic and magnetohydrodynamic disks.', '1406.4864-3-3-3': 'The implementation of the concept behind the shearing sheet, with an appropriate shear-periodic radial-boundary condition , forms a computational model used for studying local disk dynamics referred to as the shearing box .', '1406.4864-3-4-0': 'Shearing box models solve the equations of motion for the fluid in a local cartesian frame co-rotating with the disk at a fiducial radius.', '1406.4864-3-4-1': 'In the standard framework, the differential rotation of the disk is locally accounted for with a height-independent angular frequency.', '1406.4864-3-4-2': 'This is appropriate for disks with a barotropic equilibrium for which the pressure is only a function of density and thus rotate on cylinders.', '1406.4864-3-4-3': 'The shearing box framework relies on a first order expansion of the steady bulk flow in the radial direction, which is the highest order compatible with shearing periodic boundary conditions.', '1406.4864-3-4-4': 'Depending on whether zeroth or first order expansions are considered for the gravitational field in the direction perpendicular to the disk, usually denoted with the coordinate [MATH], this leads to the so called unstratified or stratified shearing box models.', '1406.4864-3-4-5': 'This stratified, compressible shearing box is what we will refer to in this paper as the standard shearing box (SSB).', '1406.4864-3-4-6': 'These approximations are appropriate when the disk is thin and the vertical scales of interest are small compared to the fiducial disk radius.', '1406.4864-3-4-7': 'There have been works that retained the correct expression for the vertical component of the gravitational field , allowing for larger vertical domains to be considered.', '1406.4864-3-4-8': 'However, this generalization does not allow to study of disks with baroclinic equilibria without relaxing the assumption that the angular frequency is height-independent.', '1406.4864-3-4-9': 'In the early formulation of the shearing sheet, [CITATION] avoided making any approximation in height ([MATH]), which is possible when considering barotropic equilibrium structures.', '1406.4864-3-5-0': 'Astrophysical disks with baroclinic equilibrium structure, for which the pressure is not solely a function of density, posses angular frequency profiles that depend in general on height, especially if these are not thin.', '1406.4864-3-5-1': 'Therefore, building a framework to study these disks demands going beyond the SSB, where the assumption that the angular frequency is height-independent is rooted deep.', '1406.4864-3-5-2': 'In this work, we generalize the SSB by considering the full height-dependence of a steady state, axisymmetric bulk flow to leading order in radius, without making any expansion in the vertical coordinate.', '1406.4864-3-5-3': 'We show that given a global disk model, it is possible to develop approximate models that are local in horizontal planes and global in height and are amenable to shearing-periodic boundary conditions.', '1406.4864-3-5-4': 'These models can be non-axisymmetric for disks with a barotropic global equilibrium but should be axisymmetric for disks with a baroclinic global equilibrium.', '1406.4864-3-5-5': 'We term the resulting framework the vertically global shearing box (VGSB).', '1406.4864-3-5-6': "The terminology of 'vertically global' specifically and solely refers to the approach of never making a expansion in the vertical direction in the derivation.", '1406.4864-3-5-7': 'This yields a model which allows us to account for the vertical variation of gravity without approximations and the possible presence of vertical shear.', '1406.4864-3-5-8': 'As we discuss below, the radial expansions that are needed to apply height-dependent shearing periodic boundary conditions do limit the range of vertical scales which can be modeled approximately.', '1406.4864-3-5-9': 'In spite of its limitations, the VGSB formalism provides a novel framework that goes beyond the SSB and promises to provide a bridge between strictly local and fully global approaches to model astrophysical disks.', '1406.4864-3-6-0': 'The paper is organized as follows.', '1406.4864-3-6-1': 'We derive the equations involved in the VGSB framework in Section [REF], providing some of the algebraic details in Appendix [REF].', '1406.4864-3-6-2': 'We state the final form of the VGSB equations and discuss its novel features in Section [REF].', '1406.4864-3-6-3': 'For convenience, we provide a self-contained summary of the VGSB equations that can be incorporated in magnetohydrodynamic codes in Appendix [REF].', '1406.4864-3-6-4': 'We use this new framework to explore the behavior of two important instabilities in a baroclinic context.', '1406.4864-3-6-5': 'We demonstrate that a linear vertical shear instability (VSI), akin to those studied by [CITATION] and [CITATION], appears in the VGSB in Section [REF].', '1406.4864-3-6-6': 'We examine some basic aspects of the magnetorotational instability (MRI) in the VGSB in Section [REF].', '1406.4864-3-6-7': 'We conclude by briefly discussing the limitations and several potential applications of the VGSB in Section [REF].', '1406.4864-3-6-8': 'In addition, in Appendix [REF] we outline some details related to neglecting curvature terms, in Appendix [REF] we analyze the hydrodynamic momentum equation in order to assess under what conditions it is acceptable to discard radial pressure gradients in the SSB and VGSB, and in Appendix [REF] we discuss issues related to potential vorticity in shearing boxes.', '1406.4864-3-7-0': '# Equations of Motion', '1406.4864-3-8-0': 'We are concerned with the equations of ideal magnetohydrodynamics, in cylindrical coordinates [MATH], in a reference frame rotating with angular frequency [MATH], i.e., [EQUATION]', '1406.4864-3-8-1': 'Here, [MATH] is the mass density, [MATH] is the fluid velocity in the rotating frame, [MATH] is the magnetic field, with [MATH], [MATH] is the internal energy density, [MATH] is the pressure determined through an equation of state, and [MATH] is the gravitational potential, which is assumed to be cylindrically symmetric, but not necessarily spherical.', '1406.4864-3-8-2': 'The current density is [MATH], with [MATH] a constant dependent on the unit system adopted.', '1406.4864-3-9-0': 'Fluid flows described by these equations are subject to conservation laws.', '1406.4864-3-9-1': 'It is thus important to understand under what circumstances these properties are satisfied by the equations describing the local dynamics involving expansions of the original set of equations.', '1406.4864-3-9-2': "It is easy to show that the approximations embodied in the standard isothermal shearing box are such that the vortex lines of an inviscid flow are frozen into the fluid (Kelvin's Circulation Theorem) and that the magnetic flux is also frozen into the fluid in the absence of magnetic dissipation (Alfven's Frozen-in Theorem).", '1406.4864-3-10-0': 'Understanding under what conditions these properties also hold for the equations of motion that result from invoking a local approximation of global disk models which have a baroclinic equilibrium structure is more subtle.', '1406.4864-3-10-1': 'Here, we state the general versions of the aforementioned theorems in order to prepare the ground to address these issues in subsequent sections.', '1406.4864-3-10-2': 'These conservation theorems can be derived by calculating the Lagrangian derivative of the fluxes associated with the vorticity and the magnetic field.', '1406.4864-3-10-3': 'It is thus useful to recall that, see e.g., , any vector field [MATH] satisfies [EQUATION] where the integral is carried out over any open surface [MATH] advected by the flow with velocity [MATH].', '1406.4864-3-11-0': "## Kelvin's Circulation Theorem", '1406.4864-3-12-0': 'The momentum equation ([REF]) for an an inviscid, unmagnetized, barotropic flow in the rotating frame is given by [EQUATION] where [MATH] is the generalized gravito-thermal potential, where [MATH] is the enthalpy, with [MATH].', '1406.4864-3-12-1': 'The equation governing the evolution of the vorticity is thus [EQUATION] which preserves the solenoidal character of the vorticity.', '1406.4864-3-12-2': 'This implies that, by virtue of Equation ([REF]) with [MATH], vortex lines are frozen into the fluid, i.e., the flow preserves the circulation [MATH] [EQUATION] with [EQUATION] where [MATH] is a closed contour, delimiting the open surface [MATH], advected by the flow with velocity [MATH].', '1406.4864-3-13-0': "## Alfve n's Frozen-in Theorem", '1406.4864-3-14-0': 'The induction equation ([REF]) preserves the divergence of the magnetic field, i.e., [EQUATION]', '1406.4864-3-14-1': 'This implies that, provided that [MATH] at some initial time, the magnetic flux remains frozen into the fluid, i.e., [EQUATION] with [EQUATION] where the integral is carried out over any open surface [MATH] advected by the flow with velocity [MATH].', '1406.4864-3-14-2': 'This follows from Equation ([REF]) with [MATH] and the induction equation ([REF]).', '1406.4864-3-15-0': '# The Vertically Global, Horizontally Local Approximation', '1406.4864-3-16-0': 'We seek to derive a set of equations that describe the local dynamics of the magnetized fluid with respect to a known steady state bulk flow around a point co-rotating with the disk at a distance [MATH].', '1406.4864-3-16-1': 'Here, we outline the steps of the derivation, which is carried out in detail below.', '1406.4864-3-17-0': 'We find a suitable steady flow and background equilibrium, which enables the derivation of exact equations of motion for the departures from this solution.', '1406.4864-3-18-0': 'We transform to a locally cartesian coordinate system.', '1406.4864-3-18-1': 'We expand the bulk flow and background equilibrium to leading order in the radial direction leaving the direction perpendicular to the disk midplane unaltered.', '1406.4864-3-19-0': 'We determine under which circumstances the resulting equations are amenable to being solved with shearing-periodic boundary conditions, which could depend on height for disk models with baroclinic global equilibria.', '1406.4864-3-20-0': 'We show explicitly these steps for the momentum and induction equations, while we state the results for the continuity equation and energy equation that are simpler to work with.', '1406.4864-3-21-0': '## Equations for Departures from Background Equilibrium', '1406.4864-3-22-0': '### Bulk Flow and Background Equilibrium', '1406.4864-3-23-0': 'As in the SSB, we first seek a steady background flow.', '1406.4864-3-23-1': 'In this case, we will not make any a priori assumptions about the [MATH]-dependence of the angular frequency.', '1406.4864-3-24-0': 'We begin by noting that the momentum equation ([REF]) admits a force-free (magnetic fields playing no role), steady state solution [MATH], with [EQUATION] where the angular frequency is [EQUATION] and the vertical, hydrostatic pressure gradient satisfies [EQUATION]', '1406.4864-3-24-1': 'Here, [MATH] and [MATH] are the mass density and energy density associated with the steady state background flow.', '1406.4864-3-25-0': '### Momentum and Induction Equations', '1406.4864-3-26-0': 'Using Equations ([REF]) and ([REF]) we can recast the gravitational force in the momentum equation ([REF]) in terms of the angular frequency and the pressure gradient, both corresponding to the steady state bulk flow.', '1406.4864-3-26-1': 'We obtain [EQUATION]', '1406.4864-3-26-2': 'The velocity field describing the departure from the bulk flow satisfying Equations ([REF])-([REF]) [EQUATION] evolves according to the momentum equation, [EQUATION]', '1406.4864-3-26-3': 'This equation is exact and it displays the particular feature that the last term on the left-hand side resembles the Coriolis acceleration, with one important difference.', '1406.4864-3-26-4': 'The angular frequency involved is not the fixed angular frequency of the rotating frame, [MATH], but rather the angular frequency of the steady state flow, [MATH].', '1406.4864-3-26-5': 'As a quick check, note that if [MATH] and [MATH] then there are no departures from the steady state bulk flow, i.e., [MATH].', '1406.4864-3-27-0': 'Taking the induction equation ([REF]) and replacing the velocity field with [MATH] yields, after some algebra, [EQUATION]', '1406.4864-3-28-0': '### Background-flow Advection and Shear Rate', '1406.4864-3-29-0': 'The fact that the speed of the background flow [MATH] depends, in general, on height implies that the departures from the bulk flow will be advected and sheared in a height-dependent way.', '1406.4864-3-29-1': 'This motivates the definition of the advection operator [EQUATION] which is defined so that it acts on scalar fields, such as the density [MATH], and on each of the components of a vector field, e.g., [MATH] and [MATH], but not on the unit coordinate-vectors, i.e., [MATH].', '1406.4864-3-29-2': 'It is also convenient to define the shear rate [EQUATION]', '1406.4864-3-29-3': 'Using these definitions, Equations ([REF]) and ([REF]) become, without approximations, [EQUATION]', '1406.4864-3-30-0': '## Local Approximation in Horizontal Planes', '1406.4864-3-31-0': 'We now seek to derive a set of equations of motion which is local in radius and azimuth by expanding Equation ([REF]) and ([REF]) around a fiducial point [MATH].', '1406.4864-3-31-1': 'In order to simplify this task, we choose a reference frame that corotates with the bulk flow at radius [MATH], i.e., [EQUATION]', '1406.4864-3-31-2': 'We also adopt a coordinate frame centered at [MATH] with locally cartesian coordinates [MATH], such that [MATH] and [MATH], with [MATH] and [MATH].', '1406.4864-3-31-3': 'In this locally cartesian frame, the differential vector operators are well approximated by their cartesian versions, provided that the radial coordinate versor [MATH]', '1406.4864-3-32-0': 'Because of the axisymmetric character of the background flow, in what follows, we choose [MATH] without loss of generality.', '1406.4864-3-33-0': 'In this locally cartesian coordinate system, we can expand to leading order in [MATH] the various functions appearing in the momentum and induction equations, Equation ([REF]) and ([REF]), respectively.', '1406.4864-3-34-0': '### Approximation of the Bulk Flow', '1406.4864-3-35-0': 'The local approximations of the angular frequency, the bulk flow in Equation ([REF]), and the advection operator in Equation ([REF]) yield [EQUATION]', '1406.4864-3-35-1': 'Here, we have defined the local, height-dependent angular frequency, bulk flow, and shear rate, all evaluated at the fiducial radius [MATH], i.e., [EQUATION]', '1406.4864-3-35-2': 'The operator [MATH] is a height-dependent generalization of the shearing sheet advection operator originally introduced in [CITATION].', '1406.4864-3-35-3': 'In order to illustrate the action of the advection operator [MATH], Figure [REF] shows the effects of considering the local bulk flow in Equation ([REF]) that results from expanding Equation ([REF]) to leading order in the radial direction, leaving unaltered the vertical dependence.', '1406.4864-3-35-4': 'In this particular example, we have considered a baroclinic equilibrium global disk model with a cylindrical temperature structure, which is discussed in detail in Section [REF].', '1406.4864-3-36-0': '### Approximation of Momentum and Induction Equations', '1406.4864-3-37-0': 'Using the approximations above, we arrive at expressions for the momentum and induction equations, Equations ([REF]) and ([REF]), which are correct to leading order in [MATH] and [MATH]: [EQUATION]', '1406.4864-3-37-1': 'Here, all the differential operators are defined in a cartesian coordinate system centered at the fiducial radius [MATH].', '1406.4864-3-38-0': 'Because we have retained the leading order in all the approximations involving the bulk-flow, the two important flow properties discussed in Section [REF] remain unaltered.', '1406.4864-3-38-1': 'In the case of an inviscid, unmagnetized, barotropic flow the momentum equation ([REF]) that results from the local approximation in horizontal planes leads to [EQUATION]', '1406.4864-3-38-2': 'Furthermore, the induction equation ([REF]) that results from the local approximation in horizontal planes preserves the solenoidal character of the the magnetic field, i.e., [EQUATION]', '1406.4864-3-38-3': 'This implies that the magnetic flux is frozen into the fluid flow that results from the local expansion in horizontal planes, i.e., [EQUATION]', '1406.4864-3-38-4': "Therefore, the local approximation in horizontal planes leads to equations that still satisfy Kelvin's circulation theorem and Alfven's frozen in theorem discussed in Section [REF].", '1406.4864-3-39-0': '## Compatibility of the Local Approximation with Shearing-periodic Boundary Conditions', '1406.4864-3-40-0': '### Shearing-periodic Boundary Conditions in the SSB', '1406.4864-3-41-0': 'All the explicit coordinate dependences in the equations of motion defining the SSB are contained in the advection operator [EQUATION] which is obtained as a limit of the operator [MATH] introduced in Equation ([REF]).', '1406.4864-3-41-1': 'The explicit dependence on the coordinate [MATH] can be eliminated by the linear transformation [MATH], [MATH], [MATH], and [EQUATION]', '1406.4864-3-41-2': 'In the primed coordinate system, the advection operator simply becomes [MATH], and the equations of motion can be solved by using strictly periodic boundary conditions in horizontal planes, i.e., [EQUATION] and appropriate boundary conditions for the vertical boundaries.', '1406.4864-3-41-3': 'Note that in the original coordinate system, the equations for the departures from the bulk flow satisfy shearing-periodic boundary conditions given by [EQUATION]', '1406.4864-3-42-0': '### Height-dependent Shearing-periodic Boundary Conditions', '1406.4864-3-43-0': 'Defining an approximate, radially local set of equations and boundary conditions for disks with baroclinic equilibria, for which the angular frequency is in general a function of height, is more subtle.', '1406.4864-3-44-0': 'The coordinate dependence arising through the advection operator [MATH] in Equation ([REF]) can still be removed by defining the linear transformation [MATH], [MATH], [MATH], and [EQUATION]', '1406.4864-3-44-1': 'In this primed coordinate system, the advection operator is coordinate-independent, i.e., [MATH].', '1406.4864-3-44-2': 'Therefore, if this were the only explicit coordinate dependence then, in each horizontal plane, it would be enough to consider the height-dependent, shearing-periodic boundary conditions given by [EQUATION]', '1406.4864-3-44-3': 'However, the coordinate dependence induced by the terms proportional to [EQUATION] on the right-hand sides of Equations ([REF]) and ([REF]) cannot be eliminated by the same coordinate transformation that removes the [MATH]-dependence in [MATH].', '1406.4864-3-44-4': 'This prevents Equation ([REF]) and ([REF]) from being solved with the shearing-periodic boundary conditions in Equation ([REF]) in a straightforward way.', '1406.4864-3-45-0': 'In what follows we analyze the consequences of proceeding by neglecting the term proportional to the coordinate [MATH] in Equation ([REF]).', '1406.4864-3-45-1': 'We thus approximate this equation as [EQUATION]', '1406.4864-3-45-2': 'With the exception of the background hydrostatic profile which will be dealt with in the next section, this approximation eliminates the explicit coordinate dependence on the right-hand sides of Equations ([REF]) and ([REF]), leading to [EQUATION]', '1406.4864-3-45-3': 'Because the only coordinate dependences arise through the advection operator [MATH], these equations are compatible with the height-dependent, shearing-periodic boundary conditions Equations ([REF]).', '1406.4864-3-45-4': 'By reducing the coordinate dependence down to the advection operator [MATH], and the background equilibrium [MATH] we have brought these equation closer to compatibility with with the height-dependent, shearing-periodic boundary conditions Equations ([REF]).', '1406.4864-3-45-5': 'The background equilibrium will be treated in Section [REF], but first we deal with issues which arise from this step.', '1406.4864-3-45-6': 'The approximation embodied in Equation ([REF]) does, in general, affect the validity of the conservation theorems discussed in Section [REF].', '1406.4864-3-45-7': "We proceed by showing that Equations ([REF]) and ([REF]) do satisfy Kelvin's circulation theorem and Alfven's frozen-in theorem when the underlying global disk model has a barotropic equilibrium or when we consider axisymmetry.", '1406.4864-3-46-0': '### Radially local, Vertically Global Hydrodynamic Disk Models', '1406.4864-3-47-0': 'It can be seen that equation ([REF]) when applied to an inviscid, barotropic, unmagnetized flow, leads to [EQUATION]', '1406.4864-3-47-1': "This means that under the conditions over which Kelvin's circulation theorem is satisfied, the approximation embodied in Equation ([REF]) does not lead to spurious sources of circulation when considering Equation ([REF]) instead of ([REF]).", '1406.4864-3-48-0': 'If the global disk model under consideration has baroclinic equilibrium, the circulation [MATH] is no longer conserved and thus, for physical reasons, [MATH] is no longer expected to vanish.', '1406.4864-3-48-1': 'However, one should also realize that the approximation invoked in Equation ([REF]) leads to a source term that contributes spuriously to the evolution of the circulation [EQUATION]', '1406.4864-3-48-2': 'Here, the dots represent the physical sources of circulation present in fluids which are either viscous or baroclinic.', '1406.4864-3-48-3': 'The spurious source of circulation in Equation ([REF]) vanishes under axisymmetry.', '1406.4864-3-48-4': 'In order to demonstrate this, let us examine the integral involved.', '1406.4864-3-48-5': "Using Stoke's theorem, it follows that [EQUATION]", '1406.4864-3-48-6': 'In axisymmetry, i.e., [MATH], the problem reduces to understanding the dynamics of the fluid in the [MATH] plane.', '1406.4864-3-48-7': 'Under this condition, the line integral over a closed loop vanishes.', '1406.4864-3-49-0': 'Thus, given a global hydrodynamic disk model, corresponding to either barotropic or baroclinic equilibria, Equation ([REF]) can be used to define an associated disk model that is local in horizontal planes but global in height.', '1406.4864-3-49-1': 'This model can be non-axisymmetric for disks with globally barotropic equilibria but should be axisymmetric for disks with globally baroclinic equilibria.', '1406.4864-3-50-0': "The flow of hydrodynamic fluids is constrained by the evolution of the potential vorticity as governed by Ertel's theorem .", '1406.4864-3-50-1': 'In Appendix [REF], we derive the equations associated with the evolution of potential vorticity in the framework of the VGSB and discuss how these relate to the Kelvin circulation theorem alluded to in this section.', '1406.4864-3-50-2': 'The details of this also depend on the final approximations needed to make the model compatible with shearing periodic radial boundaries in Section [REF].', '1406.4864-3-50-3': 'Before that, we discuss the analogous issue which occurs with magnetic fields in the induction equation.', '1406.4864-3-51-0': '### Radially local, Vertically Global MHD Disk Models', '1406.4864-3-52-0': 'Let us now consider the implications for the induction equation.', '1406.4864-3-52-1': 'In general, neglecting the term proportional to the coordinate [MATH] in Equation ([REF]) leads to an approximated induction equation that no longer preserves the solenoidal character of the magnetic field.', '1406.4864-3-52-2': 'More specifically, Equation ([REF]) leads to [EQUATION]', '1406.4864-3-52-3': 'Taken at face value, this implies that Equation ([REF]) induces spurious generation of magnetic monopoles that will break flux freezing even in the absence of dissipation.', '1406.4864-3-52-4': 'Therefore, in the case of a magnetized fluid, while eliminating the explicit [MATH]-dependence that can not be removed by a coordinate transformation allows the induction equation to be solved with shearing-periodic, height-dependent boundary conditions, this approximation, in general, would destroy the solenoidal character of the magnetic field.', '1406.4864-3-53-0': 'However, the spurious evolution of the divergence of the magnetic field that results from the approximation Equation ([REF]) is absent in Equation ([REF]) if the underlying global disk model is has a barotropic equilibrium or axisymmetry, i.e., [MATH], is considered.', '1406.4864-3-54-0': 'Therefore, given a global disk model, it is possible to develop models which preserve the solenoidal character of the magnetic field that are local in horizontal planes and global in height that are amenable to shearing-periodic boundary conditions.', '1406.4864-3-54-1': 'These models can be non-axisymmetric for disks with globally barotropic equilibria but should be axisymmetric for disks with globally baroclinic equilibria.', '1406.4864-3-54-2': 'Under either of these conditions, the approximation invoked in Equation ([REF]), which is necessary to employ the shearing-periodic boundary conditions given by Equation ([REF]), does not lead to spurious source terms that could affect the evolution of the circulation, magnetic flux freezing, or the solenoidal character of the magnetic field, i.e., [EQUATION] and [EQUATION]', '1406.4864-3-55-0': '### Approximation of the Background Equilibrium', '1406.4864-3-56-0': 'The fourth term on the right-hand side of Equation ([REF]) can be dealt with along the lines proposed by [CITATION] for the hydrostatic background quantities [EQUATION] where [MATH] and [MATH] are the leading order terms associated with the mass and internal energy density profiles of the background flow at the fiducial radius.', '1406.4864-3-56-1': 'Within the level of approximation we are working at, we thus have [EQUATION] where the vertical acceleration induced by the background pressure gradient is balanced by gravity at the fiducial radius [EQUATION] with [MATH], the gravitational potential evaluated at the fiducial radius [MATH].', '1406.4864-3-57-0': 'The set of approximations to the background described above can also be considered on more formal grounds by introducing a set of dimensionless parameters that describe the relative scale of the phenomena of interest and the departures from a thin, Keplerian disk.', '1406.4864-3-57-1': 'In Appendix [REF], we provide the details involved in this procedure, emphasizing in particular the handling of the terms related to pressure gradients in the momentum equation.', '1406.4864-3-57-2': 'Our considerations build on, and extend, the analysis carried out in [CITATION].', '1406.4864-3-57-3': 'The analysis suggests the condition [MATH] as a limit on the height of the best modeled part of the domain.', '1406.4864-3-57-4': 'This criterion also applies to the SSB, as we demonstrate in Appendix [REF].', '1406.4864-3-58-0': 'The set of equations that result from considering the approximations described in this section are compatible with the height-dependent, shearing-periodic boundary conditions Equations ([REF]).', '1406.4864-3-58-1': 'This motivates introducing the VGSB.', '1406.4864-3-59-0': '# The Vertically Global Shearing Box (VGSB)', '1406.4864-3-60-0': '## Equations of Motion for the VGSB', '1406.4864-3-61-0': 'Following the steps outlined in the previous section, we arrive at the expressions for the continuity, momentum, induction, and energy equations that define the framework of the vertically global shearing box (VGSB) [EQUATION]', '1406.4864-3-61-1': 'Here, all the operators are defined in a cartesian coordinate system centered at the fiducial radius [MATH].', '1406.4864-3-62-0': 'The only explicit coordinate dependence in the VGSB arises through the advection operator [MATH], which is linear in the coordinate [MATH] for all heights [MATH].', '1406.4864-3-62-1': 'This implies that the equations are suitable for being solved with a vertically varying shear-periodic [MATH]-direction boundary condition.', '1406.4864-3-62-2': 'The radial and azimuthal boundary conditions for mapping a field variable [MATH] in a VGSB of size [MATH] are, respectively, [EQUATION]', '1406.4864-3-62-3': 'In order to completely define the problem, appropriate boundary conditions in the vertical direction must be specified.', '1406.4864-3-63-0': 'The set of Equations ([REF])-([REF]), together with the boundary conditions Equations ([REF])-([REF]) lead to approximate radially local, vertically global disk models if the underlying global disk model has a barotropic equilibrium or if we assume axisymmetry.', '1406.4864-3-64-0': 'It is important to emphasize that the velocity [MATH] is the departure from the local approximation of the bulk flow and thus the total fluid velocity [MATH] in the VGSB is [EQUATION] where [EQUATION] and the local shear rate [MATH] is related to the generalization of the height-independent [MATH]-parameter in the SSB: [EQUATION]', '1406.4864-3-64-1': 'At the midplane, i.e., [MATH], the flow velocity [MATH] is the same as the steady state flow velocity in the SSB.', '1406.4864-3-64-2': 'However, in global disk models where the shear rate of the flow decreases to zero at high altitude, i.e., [MATH] and, therefore, [MATH] for [MATH].', '1406.4864-3-64-3': 'This is just the reflex motion induced by the rotating frame.', '1406.4864-3-64-4': 'This shows that in the limit [MATH], Equations ([REF])-([REF]) reduce to the MHD equations in the absence of a gravitational field.', '1406.4864-3-65-0': '## Connecting the VGSB to the SSB', '1406.4864-3-66-0': 'In order to connect the SSB to the VGSB, with its generalization to baroclinic disk equilibria, it is useful to recall some results that apply to barotropic hydrostatic equilibria.', '1406.4864-3-66-1': 'If the pressure is [MATH] it follows that:', '1406.4864-3-67-0': 'Let us now focus our attention on a global isothermal disk in the gravitational potential of a point source of mass [MATH], i.e., [EQUATION] with [MATH] the gravitational constant.', '1406.4864-3-67-1': 'The pressure is [MATH], with constant sound speed [MATH].', '1406.4864-3-67-2': 'Adopting a power-law density dependence with radius in the midplane, tha hydrostatic balance in the vertical direction implies that the density is [EQUATION] where the constant [MATH] and we defined the Keplerian speed [EQUATION]', '1406.4864-3-67-3': 'In this case, the enthalpy is simply [MATH], and thus the generalized gravito-thermal potential [MATH], is [EQUATION]', '1406.4864-3-67-4': 'The angular frequency can be obtained from Equation ([REF]) as [EQUATION] where the Keplerian frequency is [EQUATION]', '1406.4864-3-67-5': 'The generalized gravito-thermal potential [MATH] and the angular frequency [MATH] are both independent of height, as expected.', '1406.4864-3-68-0': 'Expanding Equations ([REF]) and ([REF]) in radius around [MATH], we obtain expressions for the local values the angular frequency [MATH] and the shear rate [MATH] that can be used, in the framework of the VGSB, to study isothermal disks which are not necessarily thin compared to the local radius [MATH], i.e., [EQUATION] where we have defined [MATH] and [MATH].', '1406.4864-3-69-0': 'These expressions are useful to show that in the limit of a cold, thin disk, i.e., [MATH], the density profile becomes [MATH], with [MATH], and the angular frequency and shear rate become [MATH], [MATH], respectively, and Equations ([REF])-([REF]) reduce to the equations for the SSB.', '1406.4864-3-70-0': '## Examples of VGSB Models for Global Equilibria', '1406.4864-3-71-0': 'We now consider two families of baroclinic, global disk equilibria that are characterized by angular frequencies and shear rates that depend on height, and are thus impossible to study within the standard shearing box approximation.', '1406.4864-3-71-1': 'The VGSB framework, embodied in Equations ([REF])-([REF]), together with their associated boundary conditions, can be used to produce axisymmetric, vertically global, horizontally local models for these astrophysical disks.', '1406.4864-3-71-2': 'After these examples, we summarize the new aspects of the baroclinic VGSB.', '1406.4864-3-72-0': '### VGSB for Global Disk Models with Cylindrical Temperature Structure', '1406.4864-3-73-0': 'There is a family of disk models which are isothermal in the vertical direction at any fiducial radius.', '1406.4864-3-73-1': 'These have been used, for example, by [CITATION].', '1406.4864-3-73-2': 'The corresponding global hydrostatic equilibrium configuration is given by [EQUATION] where [MATH] stands for the sound speed, assuming an ideal gas, [EQUATION]', '1406.4864-3-73-3': 'The local expansions involved in the VGSB for this global baroclinic equilibrium disk model are [EQUATION] with [MATH] and [MATH].', '1406.4864-3-74-0': 'Note that all the explicit [MATH]-dependences in [MATH] and [MATH] arise only when [MATH].', '1406.4864-3-74-1': 'This is in agreement with the isothermal case, corresponding to [MATH], in which the angular frequency is independent of height.', '1406.4864-3-74-2': 'It can also be seen from inspection of these equations that in the thin disk limit, i.e., [MATH], the VGSB reduces to the SSB with [MATH], [MATH], and [MATH], provided that [MATH], even if the temperature depends on radius, i.e., [MATH].', '1406.4864-3-74-3': 'This is what ultimately justifies the use of a collection of SSBs to produce local barotropic equilibrium disk models, with height-independent angular frequency at different radii, even though the underlying global disk model has a baroclinic equilibrium structure.', '1406.4864-3-75-0': '### VGSB for Global Disk Models with Spherical Temperature Structure', '1406.4864-3-76-0': 'A spherical temperature dependence of the form [EQUATION] leads to a global hydrostatic disk configuration given by [EQUATION]', '1406.4864-3-76-1': 'Here, the power-law coefficients [MATH] and [MATH] are related via [EQUATION] and the sound speed is [EQUATION] where [MATH] is the sound speed at the midplane at [MATH].', '1406.4864-3-76-2': 'Such models have been used, for example, by [CITATION].', '1406.4864-3-77-0': 'The expansion of these expressions leads to the VGSB model corresponding to a global disk with a temperature structure varying in spherical shells.', '1406.4864-3-77-1': 'The local angular frequency and shear rate are, respectively, [EQUATION]', '1406.4864-3-77-2': 'In the limiting case in which [MATH]; the angular frequency and the shear rate take the Keplerian values at the midplane, i.e., [MATH] and [MATH], and the parameter [MATH] alone determines the disk thickness.', '1406.4864-3-78-0': '### New Aspects of the VGSB', '1406.4864-3-79-0': 'The approach that we followed in deriving the equations defining the framework of the VGSB is similar in spirit to the one employed in the derivation of the SSB.', '1406.4864-3-79-1': 'This consists of expanding the steady sate bulk flow, retaining only the leading order terms in the ratio [MATH].', '1406.4864-3-79-2': 'The significant difference resides in that we have avoided making any expansion in the vertical direction.', '1406.4864-3-79-3': 'This allows us to retain the full height-dependence of the angular frequency.', '1406.4864-3-79-4': 'This is essential when dealing with disks in which the equilibrium structure is described by a baroclinic equation of state, since their angular frequency is in general a function of height.', '1406.4864-3-80-0': 'In order to illustrate the new aspects that are open to examination by retaining the full height-dependence in Equations ([REF]), ([REF]), and ([REF]), let us consider a baroclinic equilibrium global disk model, in which the temperature is a function of the spherical radius, as in [CITATION].', '1406.4864-3-80-1': 'The four panels in Figure [REF] compare the temperature, density, pressure, and angular frequency corresponding to the global equilibrium of the disk model (labeled "Global"), the local equilibrium defining the VGSB, and local equilibrium that would result in the SSB.', '1406.4864-3-81-0': 'The VGSB provides a local representation of a global baroclinic equilibrium disk model, capturing effects that the SSB would not be able to account for.', '1406.4864-3-81-1': 'Because the lowest order discarded in Equation ([REF]) is [MATH], the radial dependence of the angular frequency is correct, to linear order in radius, for all heights.', '1406.4864-3-81-2': 'Moreover, the VGSB retains the correct global vertical gradients in temperature, density, and pressure, while neglecting the local radial variation of the isopycnic, isothermal, isobaric surfaces.', '1406.4864-3-81-3': 'This is in sharp contrast with the local equilibrium involved in the SSB framework that is unable to capture the variation of the angular frequency with height, preventing its use when modeling disks with a baroclinic equilibrium structure.', '1406.4864-3-81-4': 'Note also that, for the case under consideration, the equilibrium density and pressure profiles involved in the standard isothermal shearing box are less accurate than the ones associated with the VGSB.', '1406.4864-3-82-0': 'In the next two sections, we illustrate how the VGSB framework relates to, and extends, previous treatments of two disk instabilities that are relevant for a wide variety of astrophysical disks; namely the VSI, also known as the Goldreich-Schubert-Fricke (GSF) instability , and the MRI .', '1406.4864-3-82-1': 'These instabilities will be the subject of future, more detailed, work.', '1406.4864-3-83-0': '# VSI in the VGSB', '1406.4864-3-84-0': 'Unmagnetized disks with shear profiles that depend on height have been long suspected to be unstable to various instabilities that feed off this angular frequency gradient.', '1406.4864-3-84-1': 'The pioneering studies of [CITATION] and [CITATION] invoked local approximations in both radius and height, capturing the essence of these instabilities but leaving open questions about their global behavior.', '1406.4864-3-84-2': 'These instabilities have been studied locally, and in conjunction with magnetorotational instability, in the context of accretion disks.', '1406.4864-3-84-3': 'It has been suggested that they can play a role in the low-conductivity regime characterizing protoplanetary disks .', '1406.4864-3-84-4': 'The global, nonlinear evolution of these instabilities has ben recently studied in [CITATION] by performing numerical simulations.', '1406.4864-3-84-5': 'This work includes an extension of the original local analysis, by considering the effects of compressibility and also an approximate, vertically global linear mode analysis.', '1406.4864-3-84-6': 'In this section, we show that the VGSB formalism recovers the local dispersion relation found in [CITATION] and can be used to address their vertically global mode analysis without invoking the approximations related to compressibility considered by the authors.', '1406.4864-3-85-0': 'There are general considerations which are common to both local and global approaches in height.', '1406.4864-3-85-1': 'Let us first write the VGSB equations for an unmagnetized disk for which the local angular frequency and shear rate derive from a global disk model with temperature [MATH] dependent on the cylindrical radius, as described in Section [REF].', '1406.4864-3-85-2': 'Following [CITATION], we make the change of variables [MATH] so that Equations ([REF])-([REF]) become [EQUATION]', '1406.4864-3-85-3': 'In order to derive the equations for the linear mode analysis that lead to the VSI starting from Equations ([REF]) and ([REF]), we proceed as follows: (i) we write the density variable as [MATH], i.e., the sum of its hydrostatic value and a fluctuation over this background, and the pressure as [MATH].', '1406.4864-3-85-4': '(ii) We use the scales of length and time provided by [MATH] and [MATH] to define the dimensionless variables [MATH] and [MATH], so that [MATH] is the dimensionless shear rate, [MATH] is the dimensionless sound speed, etc. (iii) We assume that all the perturbations are small and axisymmetric.', '1406.4864-3-85-5': '(iv) We focus on radial scales that are small compared to the fiducial radius [MATH] and take the Fourier transform of the set of equations ([REF]) and ([REF]), which reduces to making the substitution [MATH], for all flow variables, [MATH], [MATH], [MATH], and [MATH], where the tilde denotes Fourier amplitudes.', '1406.4864-3-85-6': 'This procedure leads to the following set of dimensionless equations [EQUATION]', '1406.4864-3-86-0': '## Vertically Local Analysis of VSI', '1406.4864-3-87-0': 'Let us first show that the equations for the VGSB lead to the dispersion relation derived by [CITATION], which is a generalization of the dispersion relation derived originally by GSF , who considered an incompressible gas.', '1406.4864-3-87-1': 'In order to do this, we restrict the analysis to the vertical location about the height [MATH], focusing on scales that are small in both the radial and vertical direction.', '1406.4864-3-87-2': 'Note that, in this case, the convenient time scale to define dimensionless variables is [MATH].', '1406.4864-3-87-3': 'Taking the Fourier transform of the set of equations ([REF]) and ([REF]) reduces to making the substitution [MATH], so that [EQUATION] where the background quantities are understood to be evaluated at [MATH] and we have used that [MATH], where [MATH] is the local value of the gravitational accelleration.', '1406.4864-3-88-0': 'The characteristic polynomial of the homogeneous system of equations ([REF])-([REF]) yields the dispersion relation in terms of dimensionless variables [EQUATION]', '1406.4864-3-88-1': 'This result is equivalent to the equation one before Equation (32) in [CITATION].', '1406.4864-3-88-2': 'Therefore, the dispersion relation of the VSI present in global disk models with height-dependent angular frequencies is correctly obtained by the VGSB framework.', '1406.4864-3-89-0': '## Vertically Global Analysis of VSI', '1406.4864-3-90-0': 'In this section we perform a vertically global analysis of isothermal, axisymmetric unmagnetized perturbations by solving the coupled set of ordinary differential equations ([REF])-([REF]).', '1406.4864-3-90-1': 'For a fixed radial wavenumber [MATH], the solution to the eigenvalue problem defined by these equations yields a set of eigenfunctions [MATH], [MATH], [MATH], and [MATH] associated with the eigenvalues [MATH].', '1406.4864-3-90-2': 'We are particularly interested in finding the modes with growing amplitude, i.e., where the real part of the eigenvalue [MATH].', '1406.4864-3-91-0': 'We compare the eigenvalues for a given set of parameters to those found through the approximated eigenproblem derived by [CITATION], which invokes additional approximations.', '1406.4864-3-91-1': 'Reasonable agreement is found, as shown in Figure [REF], with the main difference being that the surface modes show slower growth when full compressibility is retained.', '1406.4864-3-92-0': 'We illustrate the eigenmodes present in a disk model with cylindrical temperature structure [MATH], as presented in Section [REF], in Figures [REF] and [REF].', '1406.4864-3-92-1': 'We consider as an example a global disk model with [MATH], [MATH], and the finite domain [MATH], where [MATH] and [MATH].', '1406.4864-3-92-2': 'These parameters match those presented in [CITATION].', '1406.4864-3-92-3': 'We solve the problem corresponding to a single radial wavenumber [MATH], associated to a wavelength of [MATH], as follows.', '1406.4864-3-92-4': 'We discretize the problem in terms of Chebyshev cardinal functions on the Gauss-Lobatto grid, and the boundary condition [MATH] is enforced by the "boundary bordering" method .', '1406.4864-3-92-5': 'We vary the resolution, using a maximum of 300 grid points (yielding a 1200 by 1200 matrix) to obtain the converged eigenvalues shown.', '1406.4864-3-93-0': 'The basic pattern of modes shown in Figure [REF] agrees well with the approximate analysis carried out by [CITATION], who solved a second order differential equation that results from combining and approximate Equations ([REF])-([REF]) .', '1406.4864-3-93-1': 'The important difference between these two approaches is that [CITATION] make an approximation which removes full compressibility from the problem.', '1406.4864-3-93-2': 'We see that the fastest growing modes are relatively damped when full compressibility is retained in our analysis.', '1406.4864-3-93-3': 'In order to facilitate making a connection to their findings, we discuss the modes that we obtained using their terminology.', '1406.4864-3-94-0': 'The eigenvalues have the symmetry in the complex plane [MATH].', '1406.4864-3-94-1': 'The fastest growing modes are a branch of "surface" modes with degenerate eigenvalues.', '1406.4864-3-94-2': 'In Figure [REF] we show eigenfunctions where all components have been normalized by the complex value [EQUATION] where [MATH] is the maximum [MATH] of the domain.', '1406.4864-3-94-3': 'Because the Fourier amplitudes given by solving the eigenproblem as posed in transformed-[MATH] Fourier space for [MATH] and [MATH] are complex conjugates, they correspond to a single real-valued eigenfunction when the [MATH]-direction Fourier transformation is inverted to bring the eigenfunction into real space.', '1406.4864-3-94-4': 'Therefore, we have plotted this real-valued eigenfunction resulting from the [MATH] and [MATH] pair.', '1406.4864-3-94-5': 'The first and second row in Figure [REF] illustrate the fastest growing pair of modes with degenerate eigenvalue.', '1406.4864-3-94-6': 'A branch of eigenvalues proceeding from the origin of the complex plane contains the fundamental body modes, which are associated with "corrugation" and "breathing" modes.', '1406.4864-3-94-7': 'Like in [CITATION], the eigenvalue closest to the origin is associated with the fundamental corrugation mode, shown in the third row of Figure [REF].', '1406.4864-3-94-8': 'Next on this branch is the fundamental breathing mode, shown in the fourth row of Figure [REF].', '1406.4864-3-95-0': 'Repeating the same calculation with a larger finite domain [MATH], and thicker disk [MATH] yields instead the modes shown in Figure [REF].', '1406.4864-3-95-1': 'The remarkable change is the introduction if high frequency oscillations at high altitudes.', '1406.4864-3-95-2': 'The prodigious number of fast growing surface modes which appear in this calculation as the vertical size of the domain is increased suggests that magnetic fields will be a significant consideration in the astrophysical context for vertical shear instabilities.', '1406.4864-3-95-3': 'In the low density regime high above the disk midplane the gas will almost certainly be ionized in many types of astrophysical disks; the presence of magnetic fields ought then to have a significant impact on the dynamics of these instabilities.', '1406.4864-3-96-0': '# MRI in the VGSB', '1406.4864-3-97-0': 'Following, and extending, the general technique of [CITATION], we perform a linear stability analysis of the MRI in the framework of the VGSB.', '1406.4864-3-97-1': 'In order to derive the equations for the linear mode analysis that lead to the MRI, we assume a homogeneous background magnetic field [MATH] and examine the evolution of the perturbations of the form [MATH] for the velocity and magnetic field components [MATH], [MATH], [MATH], [MATH], considering [MATH].', '1406.4864-3-97-2': 'Upon substitution in Equations ([REF])-([REF]) and linearizing in the perturbation amplitudes, the equations for the perturbation are [EQUATION]', '1406.4864-3-97-3': 'In a similar way as we did for the VSI, it is convenient to use the scales of length and time provided by [MATH] and [MATH] in order to define dimensionless variables, i.e., [MATH] and [MATH], etc.', '1406.4864-3-97-4': 'Note that the time unit of the growth rate [MATH] is [MATH], so its actual value will depend on the parameters of the shearing box model employed.', '1406.4864-3-98-0': 'The set of equations ([REF])-([REF]) generalize Equations (12)-(15) in [CITATION] .', '1406.4864-3-98-1': 'to include the effects of height-dependent angular frequency and shear rate that arise in the context of baroclinic equilibrium disk models.', '1406.4864-3-98-2': 'For the sake of simplicity, and in order to make a direct connection with the linear mode analysis in [CITATION], we have neglected the term proportional to [MATH] on the right hand side of Equation ([REF]), which contributes to the secular evolution of the magnetic field.', '1406.4864-3-98-3': 'This is a sensible approximation if the associated modes are localized in height in regions where the vertical shear rate [MATH] is smaller than the growth rate of the unstable modes.', '1406.4864-3-98-4': 'Figure [REF] shows the vertical shear rate [MATH] for two spherical temperature structure VGSB models, and the case which we consider below is roughly compatible with this approximation.', '1406.4864-3-98-5': 'However, though for the thin disk case the instability growth is roughly ten times faster than than the vertical shear rate at [MATH], this separation is such that the possible effects warrant further exploration in the future.', '1406.4864-3-99-0': 'In order to illustrate how the MRI modes are affected, we examine the spectrum of growing modes present in a disk with spherical temperature structure, [MATH], as discussed in Section [REF], taking as an example the [MATH].', '1406.4864-3-99-1': 'These values corresponding to a scale height of [MATH], and magnetic field strength [MATH].', '1406.4864-3-100-0': 'We solve the eigenproblem posed by Equations ([REF])-([REF]) by discretizing in the infinite domain [MATH] in terms of rational Chebyshev cardinal functions on the "roots", or interior grid.', '1406.4864-3-100-1': 'For the set of chosen parameters, 200 grid points were sufficient to obtain converged results.', '1406.4864-3-100-2': 'We adopted the same set of boundary conditions used in [CITATION], i.e., [MATH] and [MATH].', '1406.4864-3-100-3': 'To make the problem numerically tractable, we limit [MATH] to a minimum value of [MATH], which is equivalent to limiting the Alfven speed at high altitudes.', '1406.4864-3-100-4': 'The VGSB model of this particular disk model has one more unstable eigenmode than in the SSB.', '1406.4864-3-100-5': 'Note that our VGSB analysis yields growth rates scaled by [MATH], which is smaller than the Keplerian frequency.', '1406.4864-3-100-6': 'Thus, even though the parameters characterizing the specific VGSB model do not appear explicitly in the linear analysis, the physical values obtained do depend on them.', '1406.4864-3-100-7': 'If we consider [MATH], then in the VGSB model analyzed here [MATH].', '1406.4864-3-100-8': 'This implies that the physical growth rate of the MRI is smaller in the VGSB model of this particular disk than in the SSB.', '1406.4864-3-100-9': 'In addition to solving the eigenproblem for the VGSB, we solve the matching problem for the SSB by using an isothermal disk with the same midplane density and scale height [MATH] as used in the VGSB.', '1406.4864-3-100-10': 'The magnetic field and velocity perturbations of the eigenmodes, shown in Figures [REF] and [REF], are directionally orthogonal in the SSB, whereas in the VGSB this is not the case.', '1406.4864-3-100-11': 'The eigenmodes in the VGSB are generally more complex, and, as it can be seen from these figures, they do not appear to have such a simple ordering in terms of the number of nodes as the SSB eigenmodes do.', '1406.4864-3-101-0': '# Discussion', '1406.4864-3-102-0': 'The framework of the VGSB allows, for the first time, to develop models for astrophysical disks which are local in radius but are global in height, in that no expansion is made vertically.', '1406.4864-3-102-1': 'This is critical to study astrophysical disks where the equilibrium cannot be described by a barotropic equation of state, as these, in general, do not rotate on cylinders and are thus not amenable to the SSB framework (see Figure [REF]).', '1406.4864-3-102-2': 'The VGSB naturally accounts for height-dependent radial and vertical flux of fluid and electromagnetic momentum in the azimuthal direction.', '1406.4864-3-102-3': 'This is relevant for the dynamics of disks with baroclinic equilibrium structure, especially if these cannot be regarded as thin.', '1406.4864-3-102-4': 'Physical domains with a large vertical extent allow for the global magnetic field threading the disk to be efficiently anchored into the surrounding medium.', '1406.4864-3-102-5': 'The coupling between stresses and shear, both radial and vertical, can efficiently transport momentum and energy to/from the disk.', '1406.4864-3-102-6': 'We envision that these effects will have important consequences, for example, in the study of the coupled dynamics between disks and their coronae and winds.', '1406.4864-3-103-0': 'Because of its inherent local character, the framework provided by the SSB is well suited to study physical processes involving scales that are smaller than the characteristic disk scales.', '1406.4864-3-103-1': 'Global disk simulations, which are becoming ever more accessible , are useful for understanding the large-scale disk dynamics but face the challenge of resolving the physical processes at small scales.', '1406.4864-3-103-2': 'In order to understand how local and global processes interact, it is desirable to devise a framework to bridge local and global approaches.', '1406.4864-3-103-3': 'Hence, there have been several initiatives to relax the local character of the SSB.', '1406.4864-3-104-0': '## Previous Works Beyond the Standard Shearing Box', '1406.4864-3-105-0': 'The periodic "shearing disk" annulus introduced in [CITATION] relaxes the condition of azimuthal locality by building a radially periodic annulus and remapping quantities across the radial background according to an imposed power-law radial disk structure.', '1406.4864-3-105-1': 'This approach involves equations which have explicit coordinate dependence in the direction where the computational domain is shear-periodic.', '1406.4864-3-105-2': 'Therefore, the flow properties present a jump at the shearing boundary, which might lead to unphysical effects.', '1406.4864-3-105-3': 'In this context, the formulations presented in [CITATION] and [CITATION] complement each other in the sense that they retain the terms accounting for finite curvature but discard global gradients and vice-versa, but both lead to similar issues.', '1406.4864-3-106-0': 'There have been several attempts to make the SSB global in height.', '1406.4864-3-106-1': 'Foremost, [CITATION] treated the case of a polytropic gas, but only made local expansions in the horizontal direction.', '1406.4864-3-106-2': 'An early example of modifying the vertical gravity in a SSB is given in [CITATION], where the full [MATH]-dependence of the vertical component of the gravitational force is considered with an isothermal gas.', '1406.4864-3-106-3': 'Several other authors have used similar approaches, for both galactic and Keplerian disks .', '1406.4864-3-106-4': 'These works assume that all the global effects in the vertical direction can be accounted for by modifying only the momentum equation in the [MATH]-direction and exclusively through the term [MATH].', '1406.4864-3-106-5': 'When the gas in the global equilibrium configuration is strictly baroptropic, it must rotate on cylinders, and the height-independent shear in the SSB is a consistent approximation (given the caveats in Appendix [REF]).', '1406.4864-3-106-6': 'Importantly, in the case that the gas is assumed to be isothermal, the SSB provides a consistent approach if the global disk configuration, at all radii, is isothermal.', '1406.4864-3-106-7': 'If another equation of state for the hydrostatic structure is used, or if the global configuration of which the shearing box is a small part has a radial temperature gradient, then the VGSB provides a viable framework going beyond these previous approaches, with important modifications beyond the SSB formalism, see Equations ([REF])-([REF]).', '1406.4864-3-107-0': '## Limitations of the VGSB', '1406.4864-3-108-0': 'The generalization of the SSB to account for the full vertical variation of gravity and the presence of vertical shear, inherent to global baroclinic disk models, does posses limitations.', '1406.4864-3-108-1': 'Some of these limitations are also characteristic of the SSB, while others are inherent to the VGSB.', '1406.4864-3-109-0': 'It should be clear that the VGSB, as well as the SSB, framework consists of a set of dynamical equations for the perturbations with respect to a time-independent background representing a smaller section of a steady global disk model.', '1406.4864-3-109-1': 'In this approach, by construction, the fluctuations cannot modify the dynamics of the background.', '1406.4864-3-109-2': 'This could be considered a serious limitation, however, this kind of approach as embodied in the SSB has proven exceptionally useful as a workhorse to understand a wide variety of phenomena in disks.', '1406.4864-3-109-3': 'Another limitation of the VGSB, that is shared by the SSB, is that the local approximation in radius prevents the full consideration curvature terms, which are inherent to the cylindrical geometry usually employed to model disks (see Appendix [REF]).', '1406.4864-3-110-0': 'The set of equations that define the SSB can be obtained by expanding in radius and height either the gravitational potential of a point source or the Keplerian background flow that ensues when radial pressure support can be neglected.', '1406.4864-3-110-1': 'In order to derive the equations defining the VGSB we have approximated the background flow itself.', '1406.4864-3-110-2': 'The body and frame forces which appear in this formulation relate to a fluid element in that flow with the given global thermodynamics, not the forces on a test particle.', '1406.4864-3-110-3': 'Thus, these forces are not the same as an expansion of the tidal potential about a Keplerian orbit.', '1406.4864-3-110-4': 'In the special case of a barotropic global equilibrium, the body forces do correspond to a conservative potential, but in general for baroclinic backgrounds this is not the case.', '1406.4864-3-110-5': 'The advantage of expanding the angular frequency is that its equilibrium profile [MATH] is sensitive to the equilibrium pressure gradient, enabling access to rotation laws beyond strictly Keplerian, in particular those with vertical shear.', '1406.4864-3-111-0': 'The most important limitations of the VGSB stem from the fact that we have opted to explore the consequences of imposing height-dependent shearing-periodic boundary conditions (see Section [REF]).', '1406.4864-3-111-1': 'In order to do this, we had to approximate Equation ([REF]) and use Equation ([REF]) instead.', '1406.4864-3-111-2': "Whereas this seems a reasonable approximation, it had in general undesired consequences for some basic conservation laws such as Kelvin's Circulation Theorem and Alfven's Frozen-in Theorem unless axisymmetry is imposed.", '1406.4864-3-111-3': 'In order to consider height-dependent shearing-periodic boundary conditions we also needed to understand under what circumstances it was acceptable to neglect background pressure gradients.', '1406.4864-3-111-4': 'Even though the VGSB can be considered to be global in height because there is no need to expand the background in the [MATH]-direction, the analysis we carried out of the pressure gradients in Appendix [REF] shows that it is necessary to limit the vertical extent of the domain to [MATH].', '1406.4864-3-111-5': 'This is more restrictive than [MATH], but nevertheless can accommodate for many scale-heights with the added value of being able to account for the full expression of the vertical gravity and vertical shear within the domain.', '1406.4864-3-112-0': 'The derivation of the VGSB formalism has allowed us to shed light into several subtle issues that are usually not addressed in the context of the SSB, see e.g., Appendices [REF] and [REF].', '1406.4864-3-112-1': 'In spite of its limitations, the SSB has proven to be a useful tool to learn about local disk dynamics.', '1406.4864-3-112-2': 'We believe that the VGSB will allow the relaxation of some of the assumptions that have been widely adopted by using the SSB.', '1406.4864-3-112-3': 'In particular, it will enable the investigation of some dynamical aspects of astrophysical disks that cannot be studied with the SSB and for which full global modeling is too demanding.', '1406.4864-3-113-0': '## Applications of the VGSB', '1406.4864-3-114-0': 'We anticipate that the VGSB framework, summarized in Appendix [REF], will benefit the modeling of a wide variety of phenomena in astrophysical disks.', '1406.4864-3-114-1': 'The advances made possible with respect to the SSB will depend on the nature of the physical phenomena under study, e.g.,: Hydrodynamic Disk Instabilities The VGSB framework may be used for the local study of vertical shear instabilities, such as GSF instability and generalizations .', '1406.4864-3-114-2': 'We have demonstrated in Section [REF] that the VGSB captures the correct local linear behavior.', '1406.4864-3-114-3': 'It should also be useful to analyze the propagation of hydrodynamic waves in disks .', '1406.4864-3-115-0': 'Disk Convection Vertical convective instabilities in disks have been studied with the SSB as an angular momentum transport mechanism.', '1406.4864-3-115-1': 'Though early studies gave negative results, [CITATION] have suggested that earlier models are under-resolved, and outward angular momentum transport is possible.', '1406.4864-3-115-2': 'The VGSB framework enables to study the effects that vertical shear can have on the long-term evolution of convective motions in a way which is not accessible with the SSB.', '1406.4864-3-116-0': 'Disk Coronae Previous standard stratified shearing box simulations show the buoyant rise of magnetic field to the upper disk layers , where it is thought to dissipate giving rise to a hot corona .', '1406.4864-3-116-1': "If the disk's equilibrium structure is not barotropic, the fact that the shear, i.e., the source of free energy, decreases with height can have important implications for the turbulent disk dynamics and energetics in the disk corona.", '1406.4864-3-116-2': 'This could affect the vertical disk structure and its corona .', '1406.4864-3-116-3': 'These effects could be important for thick disks, such as advection dominated accretion flows (ADAFs) .', '1406.4864-3-117-0': 'Disk Winds Standard shearing boxes extended in height have been used to study the MRI as a mechanism for launching disk winds though only in the framework provided by modifying the gravitational forces through the term [MATH].', '1406.4864-3-117-1': 'We have shown in Section [REF] that the linear MRI is modified by effects included in the VGSB for disks with baroclinic equilibria.', '1406.4864-3-117-2': 'Because the wind dynamics are particularly sensitive to the forces acting on the fluid far away from the midplane, the VGSB provides a more general framework for local studies of this nature.', '1406.4864-3-118-0': 'Interstellar Medium and Galactic Disks In galactic disks, the VGSB makes it possible to capture critical physics for studying the nonlinear evolution of the magnetorotational, Parker, and magneto-Jeans instabilities .', '1406.4864-3-118-1': 'The framework provided by the VGSB also allows for the inclusion of fundamental height-dependent physical effects that can play an important role in the study of star formation, galactic dynamos, and the structure of the interstellar medium .', '1406.4864-3-119-0': 'We are grateful to Tobias Heinemann, whose valuable insights helped us to better understand the subtleties involved in the derivation of the VGSB and to present it in a transparent way.', '1406.4864-3-119-1': 'We acknowledge thoughtful comments from Scott Tremaine and Eric Blackman, and discussions with Troels Haugbolle, AAke Nordlund, Andrew Jackson, Oliver Gressel, Orkan M. Umurhan, Gopakumar Mohandas, Henrik Latter, Andrea Mignone, Gianluigi Bodo and Sacha Brun.', '1406.4864-3-119-2': 'We are thankful to the anonymous referee, whose tenacious reports led us to better appreciate how the VGSB connects to, and extends, the SSB framework, and to improve the original manuscript significantly.', '1406.4864-3-119-3': "The research leading to these results has received funding from the People Programme (Marie Curie Actions) of the European Union's Seventh Framework Programme (FP7/2007-2013) under REA grant agreement 327995 (CPM), and the European Research Council under the European Union's Seventh Framework Programme (FP/2007-2013) under ERC grant agreement 306614 (MEP).", '1406.4864-3-119-4': 'MEP also acknowledges support from the Young Investigator Programme of the Villum Foundation.', '1406.4864-3-120-0': '# Derivation of the momentum equation for departures from the background bulk flow', '1406.4864-3-121-0': 'Here, we provide the algebraic details involved in obtaining the momentum equation ([REF]) starting from Equation ([REF]).', '1406.4864-3-122-0': 'Let us first expand the left-hand side of Equation ([REF]), LHS for short, after substituting [MATH], we obtain [EQUATION]', '1406.4864-3-122-1': 'We now expand the last three terms, recalling that the differential operators involved are in cylindrical coordinates and using the definition of the bulk flow in terms of the angular frequency [MATH].', '1406.4864-3-122-2': 'This yields [EQUATION]', '1406.4864-3-122-3': 'This completes the expansion of the left-hand side in terms of the new velocity variable [MATH].', '1406.4864-3-123-0': 'We now proceed similarly with the right-hand side of the momentum equation ([REF]), RHS for short, by substituting [MATH] and expanding [EQUATION]', '1406.4864-3-123-1': 'Equating the left-hand side expression (Equation [REF]) and the right-hand side expression (Equation [REF]) and canceling out the term [MATH] which appears on both sides, the momentum Equation ([REF]) becomes, without approximation, [EQUATION] which corresponds to Equation ([REF]).', '1406.4864-3-124-0': 'As stated in Section [REF], a cancellation of terms from the left and right-hand sides of the momentum equation results in the term [MATH] that looks similar to the Coriolis acceleration, i.e., [MATH], but with the angular frequency [MATH] taking the place of the constant angular frequency of the rotating frame [MATH].', '1406.4864-3-125-0': '# Summary of the VGSB Equations', '1406.4864-3-126-0': 'Here we provide a self-contained summary of the equations defining the VGSB.', '1406.4864-3-127-0': 'VGSB Equations The continuity, momentum, induction, and energy equations in the VGSB are, respectively, given by [EQUATION]', '1406.4864-3-127-1': 'Here, [MATH] is the mass density, [MATH] is the velocity with respect to the local bulk flow in the disk, [MATH] is the magnetic field, and [MATH] is the internal energy density.', '1406.4864-3-127-2': 'The pressure [MATH] is determined as a function of [MATH] and [MATH], through an appropriate equation of state.', '1406.4864-3-127-3': 'The current density is [MATH], with [MATH] a constant dependent on the unit system adopted.', '1406.4864-3-127-4': 'All the operators in the VGSB equations are defined in a cartesian coordinate system centered at the fiducial radius [MATH].', '1406.4864-3-128-0': 'For barotropic background equilibrium disc structures, the above equations can be used in fully three dimensions, while for baroclinic background equilibrium disc structures, these equations must by restricted to axisymmetric solutions ([MATH]-invariant).', '1406.4864-3-129-0': 'VGSB Definitions The equations defining the VGSB framework depend on a number of functions of the coordinate [MATH], which result from local radial expansions of the global disk model around the radius [MATH].', '1406.4864-3-129-1': 'The height-dependent advection by the background shear, [MATH], is given by [EQUATION] where [MATH], [MATH], and [MATH] correspond, respectively, to the local values of the background azimuthal velocity, angular frequency, and background shear flow, which are all derived from the angular frequency of the global disk model [MATH], i.e., [EQUATION]', '1406.4864-3-129-2': 'The gravitational potential in the VGSB [MATH] is given by the value of the global gravitational potential [MATH] at [MATH] [EQUATION]', '1406.4864-3-129-3': 'VGSB Boundary Conditions For a domain with size [MATH] the horizontal boundary conditions are [EQUATION] which correspond to a generalization of the shearing-periodic boundary conditions adopted in the SSB.', '1406.4864-3-130-0': '# Neglecting Curvature Terms', '1406.4864-3-131-0': 'The cylindrical coordinate system that we adopt to describe the global disk model leads naturally to the presence of the quadratic terms [MATH], [MATH], [MATH], and [MATH] in the momentum equation ([REF]).', '1406.4864-3-131-1': 'All of these terms are usually neglected in the SSB.', '1406.4864-3-131-2': 'The terms related to the velocity field can be neglected safely when they are small compared to the corresponding components of the acceleration [MATH] and [MATH].', '1406.4864-3-131-3': 'Both of these conditions lead to the inequality [MATH].', '1406.4864-3-131-4': 'The magnetic terms [MATH] and [MATH] in Equation ([REF]) are usually absent in local studies (see for an exception).', '1406.4864-3-131-5': 'If the magnetic field is sufficiently sub-thermal everywhere in the domain, these terms can be neglected .', '1406.4864-3-131-6': 'However, they have been commonly neglected even when this is not the case.', '1406.4864-3-131-7': 'This is perhaps because it is not obvious that retaining them will lead to a physically consistent problem in the framework of the shearing box.', '1406.4864-3-131-8': 'Consider, for example, the case of a shearing box in which a strong net azimuthal field develops as a consequence of the local disk dynamics.', '1406.4864-3-131-9': 'This could lead to the force arising from the term [MATH] to increase with time.', '1406.4864-3-131-10': 'However, this force cannot be balanced by an increase in the centripetal acceleration as the latter is fixed by the choice of the bulk flow .', '1406.4864-3-131-11': 'Note that all of this statements regarding the quadratic terms that reminisce the curvilinear nature of the original cylindrical coordinate system are independent of whether the SSB, or the vertically global version developed in this paper, is considered.', '1406.4864-3-132-0': '# Nondimensionalization and Ordering of the Hydrodynamic Momentum Equation', '1406.4864-3-133-0': 'A more rigorous description of the approximations that lead to the equations for the VGSB can be made by introducing nondimensional variables in order to expose the hierarchy of the various terms involved.', '1406.4864-3-133-1': 'Here, we only deal with the hydrodynamic part of the momentum equation, for the magnetic components we make the assumption that the magnetic field are weak enough to not effect the global equilibrium background configuration.', '1406.4864-3-133-2': 'Our approach follows and generalizes [CITATION].', '1406.4864-3-134-0': 'The hydrodynamic part of the momentum equation ([REF]) is given by [EQUATION]', '1406.4864-3-134-1': 'We define a horizontal length scale [MATH], a vertical length scale [MATH], time scale [MATH], pressure scale [MATH], and density scale [MATH].', '1406.4864-3-134-2': 'Horizontal velocities are nondimensionalized by [MATH] and vertical velocities by [MATH].', '1406.4864-3-134-3': 'We use primes to denote normalized quantities, like [MATH] the radial component of the fluctuation velocity, and [MATH] the radial partial derivative with respect to the radial nondimensional length, with radial length nondimensionalized as [MATH].', '1406.4864-3-134-4': 'The components of the momentum equation are thus [EQUATION] where we have introduced the dimensionless parameters [EQUATION]', '1406.4864-3-134-5': 'Note that [MATH] is most readily identified as the [MATH] appearing at the location of the dynamics in question, not the Keplerian frequency.', '1406.4864-3-134-6': 'Thus [MATH] is not directly the usual nondimensional scale height.', '1406.4864-3-134-7': 'Taylor expanding in small [MATH] about [MATH], keeping only lowest order terms, and transforming to cartesian coordinates as this allows (Section [REF]) yields [EQUATION]', '1406.4864-3-134-8': 'We now decompose the pressure and density into their hydrostatic part and a their associated fluctuation as [EQUATION]', '1406.4864-3-134-9': 'The components of the momentum equation then become [EQUATION]', '1406.4864-3-134-10': 'The fluctuation pressure terms have the same scaling as the background hydrostatic pressure terms, as in [CITATION].', '1406.4864-3-134-11': 'Using the expression for hydrostatic equilibrium given by Equation ([REF]), we obtain the form of the radial and vertical components of the acceleration as [EQUATION]', '1406.4864-3-134-12': 'The vertical acceleration depends only on the gravitational potential, but the radial acceleration depends on the thermodynamics of the hydrostatic equilibrium though the nondimensionalized rotation [MATH].', '1406.4864-3-135-0': 'In order to illustrate the procedure for determining the appropriate expansion of the radial acceleration, we work with the structure of the hydrostatic background provided by the cylindrical temperature profile [MATH], as in Section [REF].', '1406.4864-3-135-1': 'This thermal structure contains a barotropic equilibrium as a special case.', '1406.4864-3-135-2': 'The nondimensional version of the rotation law given by Equation ([REF]) is [EQUATION]', '1406.4864-3-135-3': 'When [MATH] this structure is isothermal, and barotropic, and when [MATH] the disk is thin, with the rotation close to Keplerian.', '1406.4864-3-136-0': '## Case of SSB and VGSB for a Globally Isothermal Background', '1406.4864-3-137-0': 'In order to connect our analysis with [CITATION], who considered the case with [MATH] and [MATH], we examine explicitly here the globally isothermal case with [MATH].', '1406.4864-3-137-1': 'Even though the equations that we presented for the VGSB do not explicitly involve expansions in the vertical direction, at this point in the analysis we do make such an expansion in order to assess the conditions under which it is acceptable to neglect the radial pressure gradients that need to be discarded in order to impose shearing-periodic boundary conditions, as discussed in Section [REF].', '1406.4864-3-137-2': 'Thus, the results of this section apply equally to the VGSB model for a globally isothermal (barotropic) disk, as well as to the SSB.', '1406.4864-3-138-0': 'Expanding the radial and vertical components of the acceleration about [MATH] and [MATH] we obtain, to leading order, [EQUATION]', '1406.4864-3-138-1': 'It is evident, as any finite domain contains arbitrarily small [MATH], that the only rigorous limit in which we can strictly justify discarding every term on the right hand side of equation ([REF]) is the one corresponding to a disk with a constant midplane density, [MATH], and/or infinitely thin, [MATH], in which the characteristic scales of interest in the vertical direction vanish identically, [MATH].', '1406.4864-3-138-2': 'This is certainly not an interesting limit, especially in the case where [MATH].', '1406.4864-3-138-3': 'Perhaps this is the reason for which these terms are usually discarded in the standard shearing box formalism, even though they cannot be strictly neglected in a disk with a radial density gradient and/or non-vanishing scale height when the vertical extent of the domain does not vanish identically.', '1406.4864-3-139-0': 'Having this caveat in mind, let us examine under what conditions we can argue that the terms proportional to [MATH] and [MATH] are sufficiently small that it is acceptable to neglect them.', '1406.4864-3-139-1': 'This requires understanding when each of the dimensionless terms in question is small compared to unity.', '1406.4864-3-139-2': 'In equation ([REF]) for the radial acceleration, the first term can be argued to be sufficiently small for [MATH] provided that [MATH], which implies [MATH].', '1406.4864-3-139-3': 'This condition can be satisfied for power-law indices [MATH] of order unity.', '1406.4864-3-139-4': 'The second term, of order [MATH], is small based on the same condition as [MATH] is at most of order unity.', '1406.4864-3-139-5': 'The third and fourth terms are clearly small for [MATH].', '1406.4864-3-139-6': 'The fifth term in this equation is more difficult to deal with because it scales with the ratio [MATH].', '1406.4864-3-139-7': 'In the case considered by [CITATION], because [MATH], this term is simply proportional to [MATH], and it can be directly neglected in the limit [MATH], along with the higher order terms in the radial acceleration.', '1406.4864-3-139-8': 'When the assumption [MATH] is relaxed, the second term in the radial acceleration is small when [MATH].', '1406.4864-3-139-9': 'Restoring dimensional quantities, this condition becomes [MATH].', '1406.4864-3-139-10': 'This implies that the fifth term in the equation for the radial acceleration is small provided that the vertical extent of the domain is smaller than the geometric mean between its radial extent and the radial location of the box.', '1406.4864-3-139-11': 'Note that this condition is more restrictive that requesting that [MATH], but far less restrictive than imposing that [MATH] vanishes identically.', '1406.4864-3-139-12': 'The higher order terms, such as the sixth term on the right-hand side of equation ([REF]), are negligible provided that [MATH].', '1406.4864-3-139-13': 'The vertical acceleration in equation ([REF]) does not depend on the parameters [MATH] or [MATH].', '1406.4864-3-139-14': 'In this case, it is straightforward to neglect higher order terms proportional to [MATH] and [MATH] with respect to the leading term [MATH].', '1406.4864-3-140-0': '## Case of VGSB for Cylindrical Temperature Profiles', '1406.4864-3-141-0': 'Here, we generalize the above result in order to include a baroclinic background with non-zero radial temperature power law index [MATH].', '1406.4864-3-141-1': 'The generalization of Equation ([REF]) is [EQUATION]', '1406.4864-3-141-2': 'It is clear that there is a one-to-one correspondence between each of the terms present in Equation ([REF]) and the ones appearing in Equation ([REF]).', '1406.4864-3-141-3': 'For any reasonable value of [MATH], the numerical coefficients in both equations are of the same order.', '1406.4864-3-141-4': 'Because of this, the conditions required to neglect each of the terms in Equation ([REF]) and Equation ([REF]) are, within factors of order unity, identical.', '1406.4864-3-141-5': 'The equation for the expansion of the vertical pressure gradient is independent of [MATH] and thus identical to Equation ([REF]).', '1406.4864-3-142-0': 'From the analysis in this appendix, we thus conclude that, within factors of order unity, the requirements to neglect the radial background pressure gradients in the VGSB are expected to be similar to the ones involved in the SSB.', '1406.4864-3-143-0': "# Potential Vorticity and Ertel's Theorem", '1406.4864-3-144-0': "In previous sections of this paper, we have addressed the impact that the approximations embodied in the VGSB framework have on Kelvin's Circulation Theorem and Alfven's Frozen-in Theorem and showed that these are satisfied if the fluid is barotropic or axisymmetric.", '1406.4864-3-144-1': "Another important conservation law is given by Ertel's Theorem, which governs the evolution of the potential vorticity [MATH], where [MATH] is the absolute vorticity and [MATH] is a fluid property advected with the flow, according to [EQUATION]", '1406.4864-3-145-0': "Note that for any conserved scalar field [MATH], Ertel's Theorem leads to a conservation law for the potential vorticity if the flow is barotropic, i.e., [MATH].", '1406.4864-3-145-1': "If [MATH] is taken to be the specific entropy [MATH] in an isentropic flow, i.e. [MATH], Ertel's Theorem provides a conservation law for the potential vorticity even if the flow is baroclinic, because [MATH].", '1406.4864-3-146-0': "In order to shed light into the implications that the approximations embodied in the VGSB have for Ertel's Theorem let us examine the evolution equation for a PV-type quantity defined, in terms of the VGSB background velocity field [MATH] and the velocity fluctuations [MATH], as [EQUATION]", '1406.4864-3-146-1': 'We can assess when such a quantity obeys a conservation law in the form of Equation ([REF]).', '1406.4864-3-146-2': 'Starting from the approximate VGSB momentum equation ([REF]), transforming from the fluctuation velocity [MATH] to the velocity [MATH], and following the usual steps for deriving a potential vorticity evolution equation , we arrive to the following evolution equation [EQUATION]', '1406.4864-3-146-3': 'The first term on the right-hand side can be non-zero, so this is not in general a conservation law of the form Equation ([REF]).', '1406.4864-3-146-4': 'This term has three components within it, the first being a nonconservative tidal force, the second being a coriolis-like force which remains from the velocity transformation, and the final term being due to the approximation made in Equation ([REF]).', '1406.4864-3-146-5': 'A coriolis-like force remains from the velocity transform from [MATH] to [MATH] because in Cartesian coordinates it is just a linear velocity boost in [MATH], and the VGSB momentum equation contains a coriolis-like term which is proportional to the [MATH] velocity.', '1406.4864-3-146-6': 'Interestingly, when a velocity transformation is done in cylindrical coordinates with an analogous change in the [MATH] velocity, the matching change in the coriolis term cancels with components of the cylindrical coordinate advection operator on the left-hand side, as was seen in Appendix [REF].', '1406.4864-3-146-7': 'In the case of a VGSB model with a height-independent [MATH] or an SSB, the first term on the right-hand side of Equation ([REF]) is zero, as the first two components are conservative and curl-free, and the third component is zero.', '1406.4864-3-146-8': 'In those cases, this applies for any choice of [MATH], as [MATH] is constant.', '1406.4864-3-146-9': 'Thus in those cases this PV-type quantity is conserved in the same sense as in Equation ([REF]).', '1406.4864-3-146-10': 'Beyond simply not being in general in the form of a conservation law, Equation ([REF]) also has the property that the Lagrangian derivative term with [MATH] is not zero when the velocity fluctuations [MATH] are zero.', '1406.4864-3-146-11': 'This is because the quantity defined in Equation ([REF]) contains both the background velocity and the fluctuation velocity.', '1406.4864-3-146-12': 'Note that in the absence of fluctuations in an axisymmetric flow, i.e., [MATH], and the first term on the right-hand side of Equation ([REF]) does vanish.', '1406.4864-3-147-0': 'Understanding the implications of the source terms that would appear in general on the right-hand side of Equation ([REF]) is beyond the scope of this work.', '1406.4864-3-147-1': 'In the reminder of this section we have a more modest goal which consists of assessing how the approximations leading to the VGSB impact the dynamics of a PV-type quantity [MATH] defined solely in terms of the velocity fluctuations, [MATH].', '1406.4864-3-148-0': 'In a way similar to that in the SSB, the VGSB framework consists of a set of equations for the fluctuations with respect to a known local equilibrium background (which corresponds to a local approximation of a global equilibrium).', '1406.4864-3-148-1': 'Starting from the momentum equation in terms of the fluid velocity [MATH] in the frame rotating with [MATH], we have derived the exact Equation ([REF]) for the velocity [MATH] characterizing the departures from the background equilibrium [MATH], Equation ([REF]).', '1406.4864-3-148-2': 'In what follows, we derive an evolution equation for the PV-type quantity associated with the velocity fluctuations in an exact way (Equation [REF]), and then repeat the derivation for the VGSB approximation (Equation [REF]) with the goal of comparing both results.', '1406.4864-3-149-0': 'Starting from the exact momentum equation ([REF]), taking the curl and using the continuity equation, we arrive to [EQUATION] where it has been natural to define the quantity [MATH], i.e., the vorticity associated with the fluctuations.', '1406.4864-3-149-1': 'We can obtain the following relation from the continuity equation ([REF]) [EQUATION] and use it to replace the second term on the left-hand side in Equation ([REF]) to obtain [EQUATION]', '1406.4864-3-149-2': 'If we consider a scalar quantity [MATH] which is frozen-in to the flow, i.e., [MATH], we can derive the following identity [EQUATION]', '1406.4864-3-149-3': 'We can combine equations ([REF]) and ([REF]) to obtain an evolution equation for [MATH] as [EQUATION]', '1406.4864-3-149-4': 'This result is exact and it follows directly from Equations ([REF]) and ([REF]) in this paper, provided that [MATH] is frozen-in to the flow.', '1406.4864-3-150-0': 'Following the same procedures in the VGSB context, starting from the approximate VGSB momentum equation ([REF]) and again taking the curl and combining with the continuity equation we arrive at [EQUATION]', '1406.4864-3-150-1': 'Here, it has also been natural to define [MATH], i.e., the voticity associated with the fluctuations with respect to the VGSB background.', '1406.4864-3-150-2': 'This result is the VGSB equivalent of Equation ([REF]).', '1406.4864-3-150-3': 'The notable difference between the VGSB and the exact form is the lack of the [MATH] term.', '1406.4864-3-150-4': 'Considering a scalar quantity frozen-in to the approximate background [MATH], we obtain [EQUATION]', '1406.4864-3-150-5': 'This result is the VGSB version of the exact result in Equation ([REF]).', '1406.4864-3-150-6': 'From this form, we can see that in the shearing box (both SSB and VGSB) the lack of the term [MATH] prevents the thermodynamic driving of vorticity by the background, which would usually lead in a baroclinic disk to phenomena like the Rossby wave instability .', '1406.4864-3-150-7': 'In shearing boxes, this kind of instability can be driven by localized gradients, or for example, by adopting a Boussinesq approximation for the system at the point of Equation ([REF]) and thus retaining the background radial hydrostatic pressure gradient with shear-periodic radial boundary conditions .'}

1604.06985

{'1604.06985-1-0-0': 'This paper extends our previous work on regularization of neural networks using Eigenvalue Decay by employing a soft approximation of the dominant eigenvalue in order to enable the calculation of its derivatives in relation to the synaptic weights, and therefore the application of back-propagation, which is a primary demand for deep learning.', '1604.06985-1-0-1': 'Moreover, we extend our previous theoretical analysis to deep neural networks and multiclass classification problems.', '1604.06985-1-0-2': 'Our method is implemented as an additional regularizer in Keras, a modular neural networks library written in Python, and evaluated in the benchmark data sets Reuters Newswire Topics Classification, IMDB database for binary sentiment classification and MNIST database of handwritten digits.', '1604.06985-1-1-0': '# Introduction', '1604.06985-1-2-0': 'One of the problems in Machine Learning is termed overfitting.', '1604.06985-1-2-1': 'The error on the training data set is driven toward a small value; however, the error is large when new data are presented to the trained algorithm.', '1604.06985-1-2-2': 'This occurs because the algorithm does not learn to generalize when new situations are presented.', '1604.06985-1-2-3': "This phenomenon is related to the models' complexity, in Vapnik sense, and can be minimized by using regularization techniques [CITATION] and [CITATION].", '1604.06985-1-3-0': 'Current deep learning models present good generalization capacity, despite having very high VC dimensions [CITATION].', '1604.06985-1-3-1': 'This is mostly because of recent advances in regularization techniques, which control the size of the hypothesis space [CITATION].', '1604.06985-1-3-2': 'Existing libraries for deep learning allow users to set constraints on network parameters and to apply penalties on parameters [CITATION] or activity of the model layers.', '1604.06985-1-3-3': 'These penalties are usually incorporated into the loss function that the network optimizes on a per-layer basis and can be understood as soft constraints.', '1604.06985-1-4-0': 'In our previous paper [CITATION] we proposed and analyzed a regularization technique named Eigenvalue Decay, aiming at improving the classification margin, which is an effective strategy to decrease the classifier complexity, in Vapnik sense, by taking advantage on geometric properties of the training examples within the feature space.', '1604.06985-1-4-1': 'However, our previous approach requires a highly computational demanding training method based on Genetic Algorithms, which is not suitable for deep learning.', '1604.06985-1-4-2': 'In this paper we utilize a soft approximation of the dominant eigenvalue, in order to enable the calculation of its derivatives in relation to the synaptic weights, aiming at the application of back-propagation.', '1604.06985-1-4-3': 'Moreover, we extend our previous theoretical analysis to deep neural networks and multiclass classification problems.', '1604.06985-1-5-0': 'The paper is organized as follows: Section [REF] briefly reports the state-of-the-art in neural network regularization, while Section [REF] defines the problem of training with Eigenvalue Decay and analyzes the relationship between such regularization method and the classification margin.', '1604.06985-1-5-1': 'In Section [REF] we explain how we implement this method in Keras.', '1604.06985-1-5-2': 'Section [REF] reports the experiments, while Section [REF] summarizes some conclusions.', '1604.06985-1-6-0': '# State-of-the-art', '1604.06985-1-7-0': 'There are many regularization strategies available to the deep learning practitioner, most of them based on regularizing estimators, i.e. trading increased bias for reduced variance [CITATION].', '1604.06985-1-7-1': 'In this section we briefly describe some of the most usual regularization strategies, such as constraining the parameter values of the model, adding extra terms in the objective function to penalize overly high values of the parameters and a recently developed technique, inspired in ensemble methods, which combines multiple hypotheses that explain the training data.', '1604.06985-1-8-0': '[MATH] weight decay is the most usual weight regularizer, and was theoretically analyzed in [CITATION], which concludes that the bounds on the expected risk of a multilayer perceptron (MLP) depends on the magnitude of the parameters rather than the number of parameters.', '1604.06985-1-8-1': 'In the work [CITATION] the author showed that the misclassification probability can be bounded in terms of the empirical risk, the number of training examples, and a scale-sensitive version of the VC-dimension, known as the fat-shattering dimension, which can be upper-bounded in terms of the magnitudes of the network parameters, independently from the number of parameters.', '1604.06985-1-8-2': 'In short, as regards [MATH] weight-decay, the work [CITATION] only shows that such a method can be applied to control the capacity of the classifier space.', '1604.06985-1-8-3': 'However, the best known way to minimize the capacity of the classifier space without damaging the accuracy on the training data is to maximize the classification margin, which is the SVM principle.', '1604.06985-1-8-4': 'Unfortunately, from the best of our knowledge, there is no formal proof that weight decay can maximize the margin.', '1604.06985-1-8-5': 'Therefore, we propose the Eigenvalue Decay, for which it is possible to establish a relationship between the eigenvalue minimization and the classification margin.', '1604.06985-1-9-0': 'Another commonly used weight regularizer is the [MATH] weight decay, which results in a sparse solution, in the sense that some parameters have an optimal value near zero.', '1604.06985-1-9-1': 'The sparsity property induced by [MATH] regularization is useful as a feature selection mechanism, such as in the least absolute shrinkage and selection operator (LASSO) algorithm [CITATION], which integrates an [MATH] penalty with a linear model to lead a subset of the model weights to become zero.', '1604.06985-1-10-0': 'It is also possible to set constraints on network parameters (usually on the norm of the parameters) during optimization, yielding a constrained optimization problem.', '1604.06985-1-10-1': 'If the [MATH] norm is adopted, the weights are constrained to lie in an ball, resulting a smaller hypothesis space.', '1604.06985-1-11-0': 'Early stopping [CITATION] is a commonly employed method to improve the generalization capacity of neural networks (NN).', '1604.06985-1-11-1': 'This method also acts as a regularizer, since it restricts the optimization procedure to a small volume of parameter space within the neighborhood of the initial parameter value [CITATION].', '1604.06985-1-11-2': 'In early stopping, the labeled data are divided into training and validation data sets.', '1604.06985-1-11-3': 'After some number of iterations the NN begins to overfit the data and the error on the validation data set begins to rise.', '1604.06985-1-11-4': 'When the validation error increases during a specified number of iterations, the algorithm stops the training section and applies the weights and biases at the minimum of the validation error to the NN.', '1604.06985-1-12-0': 'The recently proposed DropOut [CITATION] provides a powerful way of regularizing deep models, while maintaining a relatively small computational cost.', '1604.06985-1-12-1': 'DropOut can be understood as a practical technique for constructing bootstrap aggregating (bagging) ensembles [CITATION] of many large NNs, i.e. DropOut trains an ensemble consisting of all sub-NNs that can be formed by removing non-output units from an underlying base NN [CITATION].', '1604.06985-1-12-2': 'However, while in bagging the models are all independent, in DropOut the models share parameters from the parent NN, making it possible to represent an exponential number of models with a tractable amount of memory.', '1604.06985-1-13-0': '# Eigenvalue Decay for deep neural networks', '1604.06985-1-14-0': 'In this section we define the problem of using Eigenvalue Decay in deep learning and show a relationship between this regularizer and the classification margin.', '1604.06985-1-15-0': 'Eigenvalue Decay can be understood as a weight decay regularizer; however, while the usual weight decay regularizers penalize overly high values of weights, Eigenvalue Decay penalizes overly high values of the dominant eigenvalue of [MATH], where [MATH] is the synaptic weight matrix of any arbitrary layer [MATH] of the NN.', '1604.06985-1-15-1': 'Both methods force the NN response to be smoother and less likely to overfit the training data.', '1604.06985-1-16-0': 'We consider a binary or multiclass classification problem where the target output is encoded in one-hot style.', '1604.06985-1-16-1': 'Therefore, assuming [MATH] classes, the target output is a [MATH]-dimensional vector where the position corresponding to the target class has the value [MATH] and all the other [MATH] positions have the value [MATH].', '1604.06985-1-17-0': 'We analyze the classification margin of a deep MLP with [MATH] sigmoidal hidden layer and linear output layer, whose model is given by: [EQUATION] where [MATH] is the output vector of the [MATH] hidden layer, [MATH] is a matrix whose elements are the synaptic weights of layer [MATH], [MATH] is the bias vector of the layer [MATH], [MATH] is the input vector, and [MATH] is a commonly used activation function, such as the sigmoid function.', '1604.06985-1-18-0': 'A single multiclass problem can be reduced into multiple binary classification problems; therefore, a multiclass classifier can be understood as an ensemble of binary classifiers which distinguish between one of the labels and the remainder (i.e. one-versus-all approach).', '1604.06985-1-18-1': 'In our case, a binary classifier for any arbitrary class [MATH] can be built by substituting the matrix [MATH] in ([REF]) by its line [MATH].', '1604.06985-1-18-2': 'Our analysis considers the classification margin per class, i.e. we analyze the relation between the application of Eigenvalue Decay on any hidden layer and the classification margin of the model ([REF]) for any arbitrary class [MATH], i.e. by considering only the line [MATH] of the matrix [MATH] in our analysis.', '1604.06985-1-19-0': 'The MLP training using Eigenvalue Decay is modeled as: [EQUATION] where [EQUATION] where [MATH] can be any commonly used loss function, such as mean squared error (MSE) or the Hinge loss, [MATH] is the dominant eigenvalue of [MATH], [MATH] is a constant that controls the regularization on the layer [MATH] and [MATH], [MATH] are the weight matrices of the MLP model with [MATH] layers, as defined in ([REF]).', '1604.06985-1-20-0': 'As can be seen in ([REF]), our new theoretical analysis assumes the regularization by Eigenvalue Decay for any layer [MATH], since our code enables the application of this regularizer on any hidden layer.', '1604.06985-1-20-1': 'We extended our previous analysis [CITATION] by applying the chain rule to calculate the derivatives of the estimated output of the MLP in relation to the synaptic weights of any hidden layer, thus yielding larger equations and also unavoidable complexity.', '1604.06985-1-20-2': 'So, we recommend the reading of the theoretical derivations in [CITATION] for MLP with only one hidden neuron before the reading of the derivations presented in the present paper.', '1604.06985-1-21-0': 'We start our analysis with the following lemma:', '1604.06985-1-22-0': 'Lemma 1.', '1604.06985-1-22-1': '[CITATION] Let [MATH] denote the field of real numbers, [MATH] a vector space containing all matrices with [MATH] rows and [MATH] columns with entries in [MATH], [MATH] be a symmetric positive-semidefinite matrix and [MATH] be the dominant eigenvalue of [MATH].', '1604.06985-1-22-2': 'Therefore, for any [MATH], the following inequality holds true: [EQUATION]', '1604.06985-1-23-0': 'Our method penalizes the dominant eigenvalue aiming at maximizing the lower bound of the classification margin, as will be shown in Theorem 1.', '1604.06985-1-23-1': 'For the sake of space, we call the value of the output of the [MATH] hidden layer, [MATH], in response to the [MATH] training example as [MATH], and the [MATH] line of [MATH] as [MATH].', '1604.06985-1-24-0': 'We define the classification margin of the output of the [MATH] hidden layer, [MATH], (in response to the training example [MATH]) as the smallest orthogonal distance, [MATH], between [MATH] and the separating hypersurface defined by the MLP layers ahead, i.e. from the layer [MATH] until the output layer, see Figure [REF].', '1604.06985-1-25-0': 'Theorem 1.', '1604.06985-1-25-1': 'Let [MATH] be the classification margin of the hidden output [MATH], in response to the training example [MATH], for any arbitrary class [MATH], and [MATH] be the dominant eigenvalue of [MATH]; then, for [MATH], i.e. an example correctly classified, the following inequality hold true: [EQUATION] where [EQUATION] [EQUATION] [MATH] is the [MATH] projection of the hidden output [MATH] on the separating hypersurface defined by the MLP layers ahead, i.e. from the layer [MATH] until the output layer, as illustrated in Fig. [REF], and [MATH] is the [MATH] target output for the class [MATH], i.e. the position [MATH] of the target vector [MATH].', '1604.06985-1-26-0': 'Proof.', '1604.06985-1-26-1': 'The first step in this proof is the calculation of the gradient of the position [MATH] of the estimated output vector, [MATH], in relation to the output [MATH] of the hidden layer [MATH] (or the MLP input, [MATH], in the case of the first layer), at the projected point [MATH] (see Figure [REF]): [EQUATION]', '1604.06985-1-26-2': 'The vector [EQUATION] is normal to the separating surface, giving the direction from [MATH] to [MATH]; therefore [EQUATION] where [MATH] is the scalar distance between [MATH] and [MATH].', '1604.06985-1-26-3': 'From ([REF]) we have: [EQUATION]', '1604.06985-1-26-4': 'Substituting ([REF]) into ([REF]) and solving for [MATH], yields: [EQUATION]', '1604.06985-1-26-5': 'The sign of [MATH] depends on which side of the decision surface [MATH] is placed.', '1604.06985-1-26-6': 'It means that an example, [MATH], correctly classified whose target value for the class [MATH] is [MATH] corresponds to [MATH].', '1604.06985-1-26-7': 'On the other hand, the classification margin must be positive in cases where examples are correctly classified, and negative in cases of misclassified examples, independently from their target classes.', '1604.06985-1-26-8': 'Therefore, the margin is defined as function of [MATH], where [MATH] is the value of the target output of the [MATH] training example for the class [MATH].', '1604.06985-1-26-9': 'More specifically, the margin, [MATH], is the smallest value of [MATH] in relation to [MATH], that is: [EQUATION]', '1604.06985-1-26-10': 'Substituting ([REF]) in ([REF]) yields: [EQUATION]', '1604.06985-1-26-11': 'For the last hidden layer we have: [EQUATION] as can be derived from ([REF]).', '1604.06985-1-26-12': 'Assuming [MATH] and substituting ([REF]) in ([REF]), yields: [EQUATION]', '1604.06985-1-26-13': 'Note that [MATH] is a symmetric positive-semidefinite matrix, therefore, from Lemma 1, the inequality: [EQUATION]', '1604.06985-1-26-14': 'For the last but one hidden layer we have: [EQUATION]', '1604.06985-1-26-15': 'Substituting ([REF]) in ([REF]), yields: [EQUATION]', '1604.06985-1-26-16': 'Since [MATH] is a symmetric positive-semidefinite matrix: [EQUATION]', '1604.06985-1-26-17': 'From ([REF]) and ([REF]) we can deduce ([REF]) by induction.', '1604.06985-1-27-0': 'Taking into account that [MATH] is the denominator of the bound in ([REF]), the training method based on Eigenvalue Decay decreases [MATH] aiming at increasing the lower bound on the classification margin.', '1604.06985-1-27-1': 'However, Eigenvalue Decay does not assure, by itself, increasing the margin, because [MATH] is function of [MATH].', '1604.06985-1-28-0': '# Using Eigenvalue Decay in Keras', '1604.06985-1-29-0': 'The use of Eigenvalue Decay within a deep learning library, such as Keras [CITATION], requires, not only a lightweight algorithm, but also a formulation that enables the calculation of derivatives of the objective function in relation to the synaptic weights, aiming at the application of back-propagation, which is a main demand for deep learning.', '1604.06985-1-30-0': 'We approximate the dominant eigenvalue by the power method; therefore, assuming [MATH], the eigenvector corresponding to the dominant eigenvalue can be approximated by: [EQUATION] where [MATH] is an initial nonzero approximation of the dominant eigenvector.', '1604.06985-1-30-1': 'We set all the elements of [MATH] equal to one.', '1604.06985-1-30-2': 'Having [MATH], we calculate [MATH] as follows: [EQUATION]', '1604.06985-1-30-3': 'The approximation given by the power method has derivatives in relation to the synaptic weights, i.e. the elements of [MATH], enabling the application of backpropagation in Keras.', '1604.06985-1-30-4': 'We implement Eigenvalue Decay in Keras using Theano functions to model the approximation of [MATH] based on an approximation of [MATH] resulting from eight iterations of the power method.', '1604.06985-1-31-0': 'Beyond the custom regularizer presented in this paper, it is possible to implement a custom objective function in Keras, see our previous work [CITATION], where we implemented a custom version of the Hinge loss, aiming at a SVM-like learning for multiclass classification.', '1604.06985-1-31-1': 'The idea is to make better use of the margin resulting from the use of Eigenvalue Decay, since the Hinge loss penalizes only examples that violate a given margin or are misclassified, i.e. an estimated output smaller than 1 in response to a positive example or an estimated output larger than -1 in response to a negative example (these training examples can be understood as support vectors).', '1604.06985-1-31-2': "The other training examples are ignored during the optimization, i.e. they don't participate in defining the decision surface.", '1604.06985-1-32-0': 'Experiments', '1604.06985-1-33-0': 'In this section our methods are evaluated using the benchmark data sets Reuters Newswire Topics Classification, IMDB database for binary sentiment classification and MNIST database of handwritten digits.', '1604.06985-1-34-0': 'Reuters Newswire Topics Classification (RNTC) is a collection of 11228 newswires from Reuters, labeled over 46 topics.', '1604.06985-1-34-1': 'IMDB Movie Reviews Sentiment Classification Dataset is a collection of 25000 movies reviews from IMDB, labeled by sentiment (positive/negative).', '1604.06985-1-34-2': 'MNIST database of Handwritten Digits Dataset is a collection of 60000 [MATH] grayscale images of the 10 digits, along with a test set of 10000 images.', '1604.06985-1-34-3': 'Table [REF] summarizes the details of the data sets.', '1604.06985-1-35-0': 'For the sake of comparison, we adopt the original models from the examples provided in the Keras repository , which are well adjusted and regularized with DropOut.', '1604.06985-1-35-1': 'Our experiments compare the performance of weight regularizers; therefore, we keep the original DropOut regularization and apply Eigenvalue Decay (ED), [MATH] and [MATH] weight regularizers to compare the accuracy gains over the original models from Keras repository.', '1604.06985-1-36-0': 'Among the models available in the Keras repository, we adopt the MLP with a single hidden layer for RNTC, the deep pipeline composed by an embedding layer, a convolutional neural network [CITATION], a long short term memory (LSTM) network [CITATION] and a dense layer for IMDB, and the deep MLP with two hidden layers for MNIST.', '1604.06985-1-36-1': 'In our experiments we apply the weight regularizers in both layers of the MLP used for RNTC, on the dense and embedding layers of the deep model used for IMBD and on the last two layers of the deep MLP used for MNIST.', '1604.06985-1-37-0': 'To find the optimal values of [MATH] in ([REF]), we exploit a 2D grid using 5-fold cross validation on the training data, keeping the original architectures and loss functions of the Keras models, i.e. categorical cross-entropy for RNTC and MNIST and binary cross-entropy for IMDB.', '1604.06985-1-37-1': 'The accuracy values on the test data and the processing time per training epoch, running in a GPU NVIDIA GeForce GTX 980, are summarized in Table [REF], where [MATH] is the gain over the original model from Keras.', '1604.06985-1-37-2': 'In the case of the IMDB data set the accuracy values were averaged over 10 runs.', '1604.06985-1-38-0': 'As can be seen in Table [REF], the weight regularizers yielded small gains on the accuracy, since the models provided in the Keras repository are well adjusted and regularized with DropOut.', '1604.06985-1-38-1': 'Eigenvalue Decay yielded the largest gains in all the data sets, but it was also the most costly regularizer, which is not a surprise, given the cost associated with the computation of the dominant eigenvalue by the power method.', '1604.06985-1-39-0': '# Conclusion', '1604.06985-1-40-0': 'This work introduces a new option of weight regularizer to the deep learning practitioners.', '1604.06985-1-40-1': 'The analysis presented in this paper indicates that Eigenvalue Decay can increase the classification margin, which can improve the generalization capability of deep models.', '1604.06985-1-41-0': 'In the scope of weight regularizers, the experiments indicate that Eigenvalue Decay can provide better gains on the classification accuracy at the cost of a larger CPU/GPU time.'}

{'1604.06985-2-0-0': 'This paper extends our previous work on regularization of neural networks using Eigenvalue Decay by employing a soft approximation of the dominant eigenvalue in order to enable the calculation of its derivatives in relation to the synaptic weights, and therefore the application of back-propagation, which is a primary demand for deep learning.', '1604.06985-2-0-1': 'Moreover, we extend our previous theoretical analysis to deep neural networks and multiclass classification problems.', '1604.06985-2-0-2': 'Our method is implemented as an additional regularizer in Keras, a modular neural networks library written in Python, and evaluated in the benchmark data sets Reuters Newswire Topics Classification, IMDB database for binary sentiment classification and MNIST database of handwritten digits.', '1604.06985-2-1-0': '# Introduction', '1604.06985-2-2-0': 'One of the problems in Machine Learning is termed overfitting.', '1604.06985-2-2-1': 'The error on the training data set is driven toward a small value; however, the error is large when new data are presented to the trained algorithm.', '1604.06985-2-2-2': 'This occurs because the algorithm does not learn to generalize when new situations are presented.', '1604.06985-2-2-3': "This phenomenon is related to the models' complexity, in Vapnik sense, and can be minimized by using regularization techniques [CITATION] and [CITATION].", '1604.06985-2-3-0': 'Current deep learning models present good generalization capacity, despite having very high VC dimensions [CITATION].', '1604.06985-2-3-1': 'This is mostly because of recent advances in regularization techniques, which control the size of the hypothesis space [CITATION].', '1604.06985-2-3-2': 'Existing libraries for deep learning allow users to set constraints on network parameters and to apply penalties on parameters [CITATION] or activity of the model layers.', '1604.06985-2-3-3': 'These penalties are usually incorporated into the loss function that the network optimizes on a per-layer basis and can be understood as soft constraints.', '1604.06985-2-4-0': 'In our previous paper [CITATION] we proposed and analyzed a regularization technique named Eigenvalue Decay, aiming at improving the classification margin, which is an effective strategy to decrease the classifier complexity, in Vapnik sense, by taking advantage on geometric properties of the training examples within the feature space.', '1604.06985-2-4-1': 'However, our previous approach requires a highly computational demanding training method based on Genetic Algorithms, which is not suitable for deep learning.', '1604.06985-2-4-2': 'In this paper we utilize a soft approximation of the dominant eigenvalue, in order to enable the calculation of its derivatives in relation to the synaptic weights, aiming at the application of back-propagation.', '1604.06985-2-4-3': 'Moreover, we extend our previous theoretical analysis to deep neural networks and multiclass classification problems.', '1604.06985-2-5-0': 'The paper is organized as follows: Section [REF] briefly reports the state-of-the-art in neural network regularization, while Section [REF] defines the problem of training with Eigenvalue Decay and analyzes the relationship between such regularization method and the classification margin.', '1604.06985-2-5-1': 'In Section [REF] we explain how we implement this method in Keras.', '1604.06985-2-5-2': 'Section [REF] reports the experiments, while Section [REF] summarizes some conclusions.', '1604.06985-2-6-0': '# State-of-the-art', '1604.06985-2-7-0': 'There are many regularization strategies available to the deep learning practitioner, most of them based on regularizing estimators, i.e. trading increased bias for reduced variance [CITATION].', '1604.06985-2-7-1': 'In this section we briefly describe some of the most usual regularization strategies, such as constraining the parameter values of the model, adding extra terms in the objective function to penalize overly high values of the parameters and a recently developed technique, inspired in ensemble methods, which combines multiple hypotheses that explain the training data.', '1604.06985-2-8-0': '[MATH] weight decay is the most usual weight regularizer, and was theoretically analyzed in [CITATION], which concludes that the bounds on the expected risk of a multilayer perceptron (MLP) depends on the magnitude of the parameters rather than the number of parameters.', '1604.06985-2-8-1': 'In the work [CITATION] the author showed that the misclassification probability can be bounded in terms of the empirical risk, the number of training examples, and a scale-sensitive version of the VC-dimension, known as the fat-shattering dimension, which can be upper-bounded in terms of the magnitudes of the network parameters, independently from the number of parameters.', '1604.06985-2-8-2': 'In short, as regards [MATH] weight-decay, the work [CITATION] only shows that such a method can be applied to control the capacity of the classifier space.', '1604.06985-2-8-3': 'However, the best known way to minimize the capacity of the classifier space without damaging the accuracy on the training data is to maximize the classification margin, which is the SVM principle.', '1604.06985-2-8-4': 'Unfortunately, from the best of our knowledge, there is no formal proof that weight decay can maximize the margin.', '1604.06985-2-8-5': 'Therefore, we propose the Eigenvalue Decay, for which it is possible to establish a relationship between the eigenvalue minimization and the classification margin.', '1604.06985-2-9-0': 'Another commonly used weight regularizer is the [MATH] weight decay, which results in a sparse solution, in the sense that some parameters have an optimal value near zero.', '1604.06985-2-9-1': 'The sparsity property induced by [MATH] regularization is useful as a feature selection mechanism, such as in the least absolute shrinkage and selection operator (LASSO) algorithm [CITATION], which integrates an [MATH] penalty with a linear model to lead a subset of the model weights to become zero.', '1604.06985-2-10-0': 'It is also possible to set constraints on network parameters (usually on the norm of the parameters) during optimization, yielding a constrained optimization problem.', '1604.06985-2-10-1': 'If the [MATH] norm is adopted, the weights are constrained to lie in an ball, resulting a smaller hypothesis space.', '1604.06985-2-11-0': 'Early stopping [CITATION] is a commonly employed method to improve the generalization capacity of neural networks (NN).', '1604.06985-2-11-1': 'This method also acts as a regularizer, since it restricts the optimization procedure to a small volume of parameter space within the neighborhood of the initial parameter value [CITATION].', '1604.06985-2-11-2': 'In early stopping, the labeled data are divided into training and validation data sets.', '1604.06985-2-11-3': 'After some number of iterations the NN begins to overfit the data and the error on the validation data set begins to rise.', '1604.06985-2-11-4': 'When the validation error increases during a specified number of iterations, the algorithm stops the training section and applies the weights and biases at the minimum of the validation error to the NN.', '1604.06985-2-12-0': 'The recently proposed DropOut [CITATION] provides a powerful way of regularizing deep models, while maintaining a relatively small computational cost.', '1604.06985-2-12-1': 'DropOut can be understood as a practical technique for constructing bootstrap aggregating (bagging) ensembles [CITATION] of many large NNs, i.e. DropOut trains an ensemble consisting of all sub-NNs that can be formed by removing non-output units from an underlying base NN [CITATION].', '1604.06985-2-12-2': 'However, while in bagging the models are all independent, in DropOut the models share parameters from the parent NN, making it possible to represent an exponential number of models with a tractable amount of memory.', '1604.06985-2-13-0': '# Eigenvalue Decay for deep neural networks', '1604.06985-2-14-0': 'In this section we define the problem of using Eigenvalue Decay in deep learning and show a relationship between this regularizer and the classification margin.', '1604.06985-2-15-0': 'Eigenvalue Decay can be understood as a weight decay regularizer; however, while the usual weight decay regularizers penalize overly high values of weights, Eigenvalue Decay penalizes overly high values of the dominant eigenvalue of [MATH], where [MATH] is the synaptic weight matrix of any arbitrary layer [MATH] of the NN.', '1604.06985-2-15-1': 'Both methods force the NN response to be smoother and less likely to overfit the training data.', '1604.06985-2-16-0': 'We consider a binary or multiclass classification problem where the target output is encoded in one-hot style.', '1604.06985-2-16-1': 'Therefore, assuming [MATH] classes, the target output is a [MATH]-dimensional vector where the position corresponding to the target class has the value [MATH] and all the other [MATH] positions have the value [MATH].', '1604.06985-2-17-0': 'We analyze the classification margin of a deep MLP with [MATH] sigmoidal hidden layer and linear output layer, whose model is given by: [EQUATION] where [MATH] is the output vector of the [MATH] hidden layer, [MATH] is a matrix whose elements are the synaptic weights of layer [MATH], [MATH] is the bias vector of the layer [MATH], [MATH] is the input vector, and [MATH] is a commonly used activation function, such as the sigmoid function.', '1604.06985-2-18-0': 'A single multiclass problem can be reduced into multiple binary classification problems; therefore, a multiclass classifier can be understood as an ensemble of binary classifiers which distinguish between one of the labels and the remainder (i.e. one-versus-all approach).', '1604.06985-2-18-1': 'In our case, a binary classifier for any arbitrary class [MATH] can be built by substituting the matrix [MATH] in ([REF]) by its line [MATH].', '1604.06985-2-18-2': 'Our analysis considers the classification margin per class, i.e. we analyze the relation between the application of Eigenvalue Decay on any hidden layer and the classification margin of the model ([REF]) for any arbitrary class [MATH], i.e. by considering only the line [MATH] of the matrix [MATH] in our analysis.', '1604.06985-2-19-0': 'The MLP training using Eigenvalue Decay is modeled as: [EQUATION] where [EQUATION] where [MATH] can be any commonly used loss function, such as mean squared error (MSE) or the Hinge loss, [MATH] is the dominant eigenvalue of [MATH], [MATH] is a constant that controls the regularization on the layer [MATH] and [MATH], [MATH] are the weight matrices of the MLP model with [MATH] layers, as defined in ([REF]).', '1604.06985-2-20-0': 'As can be seen in ([REF]), our new theoretical analysis assumes the regularization by Eigenvalue Decay for any layer [MATH], since our code enables the application of this regularizer on any hidden layer.', '1604.06985-2-20-1': 'We extended our previous analysis [CITATION] by applying the chain rule to calculate the derivatives of the estimated output of the MLP in relation to the synaptic weights of any hidden layer, thus yielding larger equations and also unavoidable complexity.', '1604.06985-2-20-2': 'So, we recommend the reading of the theoretical derivations in [CITATION] for MLP with only one hidden neuron before the reading of the derivations presented in the present paper.', '1604.06985-2-21-0': 'We start our analysis with the following lemma:', '1604.06985-2-22-0': 'Lemma 1.', '1604.06985-2-22-1': '[CITATION] Let [MATH] denote the field of real numbers, [MATH] a vector space containing all matrices with [MATH] rows and [MATH] columns with entries in [MATH], [MATH] be a symmetric positive-semidefinite matrix and [MATH] be the dominant eigenvalue of [MATH].', '1604.06985-2-22-2': 'Therefore, for any [MATH], the following inequality holds true: [EQUATION]', '1604.06985-2-23-0': 'Our method penalizes the dominant eigenvalue aiming at maximizing the lower bound of the classification margin, as will be shown in Theorem 1.', '1604.06985-2-23-1': 'For the sake of space, we call the value of the output of the [MATH] hidden layer, [MATH], in response to the [MATH] training example as [MATH], and the [MATH] line of [MATH] as [MATH].', '1604.06985-2-24-0': 'We define the classification margin of the output of the [MATH] hidden layer, [MATH], (in response to the training example [MATH]) as the smallest orthogonal distance, [MATH], between [MATH] and the separating hypersurface defined by the MLP layers ahead, i.e. from the layer [MATH] until the output layer, see Figure [REF].', '1604.06985-2-25-0': 'Theorem 1.', '1604.06985-2-25-1': 'Let [MATH] be the classification margin of the hidden output [MATH], in response to the training example [MATH], for any arbitrary class [MATH], and [MATH] be the dominant eigenvalue of [MATH]; then, for [MATH], i.e. an example correctly classified, the following inequality hold true: [EQUATION] where [EQUATION] [EQUATION] [MATH] (i.e. the identity matrix), [MATH] is the [MATH] projection of the hidden output [MATH] on the separating hypersurface defined by the MLP layers ahead, i.e. from the layer [MATH] until the output layer, as illustrated in Fig. [REF], [MATH] is the [MATH] target output for the class [MATH], i.e. the position [MATH] of the target vector [MATH] and [MATH] is the simplified notation for [MATH].', '1604.06985-2-26-0': 'Proof.', '1604.06985-2-26-1': 'The first step in this proof is the calculation of the gradient of the position [MATH] of the estimated output vector, [MATH], in relation to the output [MATH] of the hidden layer [MATH] (or the MLP input, [MATH], in the case of the first layer), at the projected point [MATH] (see Figure [REF]): [EQUATION]', '1604.06985-2-26-2': 'The vector [EQUATION] is normal to the separating surface, giving the direction from [MATH] to [MATH]; therefore [EQUATION] where [MATH] is the scalar distance between [MATH] and [MATH].', '1604.06985-2-26-3': 'From ([REF]) we have: [EQUATION]', '1604.06985-2-26-4': 'Substituting ([REF]) into ([REF]) and solving for [MATH], yields: [EQUATION]', '1604.06985-2-26-5': 'The sign of [MATH] depends on which side of the decision surface [MATH] is placed.', '1604.06985-2-26-6': 'It means that an example, [MATH], correctly classified whose target value for the class [MATH] is [MATH] corresponds to [MATH].', '1604.06985-2-26-7': 'On the other hand, the classification margin must be positive in cases where examples are correctly classified, and negative in cases of misclassified examples, independently from their target classes.', '1604.06985-2-26-8': 'Therefore, the margin is defined as function of [MATH], where [MATH] is the value of the target output of the [MATH] training example for the class [MATH].', '1604.06985-2-26-9': 'More specifically, the margin, [MATH], is the smallest value of [MATH] in relation to [MATH], that is: [EQUATION]', '1604.06985-2-26-10': 'Substituting ([REF]) in ([REF]) yields: [EQUATION]', '1604.06985-2-26-11': 'For the last hidden layer we have: [EQUATION] as can be derived from ([REF]).', '1604.06985-2-26-12': 'Assuming [MATH] and substituting ([REF]) in ([REF]), yields: [EQUATION]', '1604.06985-2-26-13': 'Note that [MATH] is a symmetric positive-semidefinite matrix, therefore, from Lemma 1, the inequality: [EQUATION]', '1604.06985-2-26-14': 'For the last but one hidden layer we have: [EQUATION]', '1604.06985-2-26-15': 'Substituting ([REF]) in ([REF]), yields: [EQUATION]', '1604.06985-2-26-16': 'Since [MATH] is a symmetric positive-semidefinite matrix: [EQUATION]', '1604.06985-2-26-17': 'From ([REF]) and ([REF]) we can deduce ([REF]) by induction.', '1604.06985-2-27-0': 'Taking into account that [MATH] is the denominator of the bound in ([REF]), the training method based on Eigenvalue Decay decreases [MATH] aiming at increasing the lower bound on the classification margin.', '1604.06985-2-27-1': 'However, Eigenvalue Decay does not assure, by itself, increasing the margin, because [MATH] is function of [MATH].', '1604.06985-2-28-0': '# Using Eigenvalue Decay in Keras', '1604.06985-2-29-0': 'The use of Eigenvalue Decay within a deep learning library, such as Keras [CITATION], requires, not only a lightweight algorithm, but also a formulation that enables the calculation of derivatives of the objective function in relation to the synaptic weights, aiming at the application of back-propagation, which is a main demand for deep learning.', '1604.06985-2-30-0': 'We approximate the dominant eigenvalue by the power method; therefore, assuming [MATH], the eigenvector corresponding to the dominant eigenvalue can be approximated by: [EQUATION] where [MATH] is an arbitrary positive integer and [MATH] is an initial nonzero approximation of the dominant eigenvector.', '1604.06985-2-30-1': 'We set all the elements of [MATH] equal to one.', '1604.06985-2-30-2': 'Having [MATH], we calculate [MATH] as follows: [EQUATION]', '1604.06985-2-30-3': 'The approximation given by the power method has derivatives in relation to the synaptic weights, i.e. the elements of [MATH], enabling the application of backpropagation in Keras.', '1604.06985-2-30-4': 'We implement Eigenvalue Decay in Keras using Theano functions to model the approximation of [MATH] based on an approximation of [MATH] resulting from nine iterations of the power method.', '1604.06985-2-31-0': 'Beyond the custom regularizer presented in this paper, it is possible to implement a custom objective function in Keras, see our previous work [CITATION], where we implemented a custom version of the Hinge loss, aiming at a SVM-like learning for multiclass classification.', '1604.06985-2-31-1': 'The idea is to make better use of the margin resulting from the use of Eigenvalue Decay, since the Hinge loss penalizes only examples that violate a given margin or are misclassified, i.e. an estimated output smaller than 1 in response to a positive example or an estimated output larger than -1 in response to a negative example (these training examples can be understood as support vectors).', '1604.06985-2-31-2': "The other training examples are ignored during the optimization, i.e. they don't participate in defining the decision surface.", '1604.06985-2-32-0': 'Experiments', '1604.06985-2-33-0': 'In this section our methods are evaluated using the benchmark data sets Reuters Newswire Topics Classification, IMDB database for binary sentiment classification and MNIST database of handwritten digits.', '1604.06985-2-34-0': 'Reuters Newswire Topics Classification (RNTC) is a collection of 11228 newswires from Reuters, labeled over 46 topics.', '1604.06985-2-34-1': 'IMDB Movie Reviews Sentiment Classification Dataset is a collection of 25000 movies reviews from IMDB, labeled by sentiment (positive/negative).', '1604.06985-2-34-2': 'MNIST database of Handwritten Digits Dataset is a collection of 60000 [MATH] grayscale images of the 10 digits, along with a test set of 10000 images.', '1604.06985-2-34-3': 'Table [REF] summarizes the details of the data sets.', '1604.06985-2-35-0': 'For the sake of comparison, we adopt the original models from the examples provided in the Keras repository , which are well adjusted and regularized with DropOut.', '1604.06985-2-35-1': 'Our experiments compare the performance of weight regularizers; therefore, we keep the original DropOut regularization and apply Eigenvalue Decay (ED), [MATH] and [MATH] weight regularizers to compare the accuracy gains over the original models from Keras repository.', '1604.06985-2-36-0': 'Among the models available in the Keras repository, we adopt the MLP with a single hidden layer for RNTC, the deep pipeline composed by an embedding layer, a convolutional neural network [CITATION], a long short term memory (LSTM) network [CITATION] and a dense layer for IMDB, and the deep MLP with two hidden layers for MNIST.', '1604.06985-2-36-1': 'In our experiments we apply the weight regularizers in both layers of the MLP used for RNTC, on the dense and embedding layers of the deep model used for IMBD and on the last two layers of the deep MLP used for MNIST.', '1604.06985-2-37-0': 'To find the optimal values of [MATH] in ([REF]), we exploit a 2D grid using 5-fold cross validation on the training data, keeping the original architectures and loss functions of the Keras models, i.e. categorical cross-entropy for RNTC and MNIST and binary cross-entropy for IMDB.', '1604.06985-2-37-1': 'The accuracy values on the test data and the processing time per training epoch, running in a GPU NVIDIA GeForce GTX 980, are summarized in Table [REF], where [MATH] is the gain over the original model from Keras.', '1604.06985-2-37-2': 'In the case of the IMDB data set the accuracy values were averaged over 10 runs.', '1604.06985-2-38-0': 'As can be seen in Table [REF], the weight regularizers yielded small gains on the accuracy, since the models provided in the Keras repository are well adjusted and regularized with DropOut.', '1604.06985-2-38-1': 'Eigenvalue Decay yielded the largest gains in all the data sets, but it was also the most costly regularizer, which is not a surprise, given the cost associated with the computation of the dominant eigenvalue by the power method.', '1604.06985-2-39-0': '# Conclusion', '1604.06985-2-40-0': 'This work introduces a new option of weight regularizer to the deep learning practitioners.', '1604.06985-2-40-1': 'The analysis presented in this paper indicates that Eigenvalue Decay can increase the classification margin, which can improve the generalization capability of deep models.', '1604.06985-2-41-0': 'In the scope of weight regularizers, the experiments indicate that Eigenvalue Decay can provide better gains on the classification accuracy at the cost of a larger CPU/GPU time.'}

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'1604.06985-3-19-0'], ['1604.06985-2-22-1', '1604.06985-3-22-1'], ['1604.06985-2-22-2', '1604.06985-3-22-2'], ['1604.06985-2-34-0', '1604.06985-3-34-0'], ['1604.06985-2-34-1', '1604.06985-3-34-1'], ['1604.06985-2-34-2', '1604.06985-3-34-2'], ['1604.06985-2-34-3', '1604.06985-3-34-4'], ['1604.06985-2-40-0', '1604.06985-3-40-0'], ['1604.06985-2-40-1', '1604.06985-3-40-1'], ['1604.06985-2-23-0', '1604.06985-3-23-0'], ['1604.06985-2-4-0', '1604.06985-3-4-0'], ['1604.06985-2-4-1', '1604.06985-3-4-1'], ['1604.06985-2-4-2', '1604.06985-3-4-2'], ['1604.06985-2-4-3', '1604.06985-3-4-3'], ['1604.06985-2-26-4', '1604.06985-3-26-4'], ['1604.06985-2-26-5', '1604.06985-3-26-5'], ['1604.06985-2-26-6', '1604.06985-3-26-6'], ['1604.06985-2-26-7', '1604.06985-3-26-7'], ['1604.06985-2-26-8', '1604.06985-3-26-8'], ['1604.06985-2-26-9', '1604.06985-3-26-9'], ['1604.06985-2-26-10', '1604.06985-3-26-10'], ['1604.06985-2-26-13', '1604.06985-3-26-13'], ['1604.06985-2-26-17', '1604.06985-3-26-18'], ['1604.06985-2-29-0', '1604.06985-3-29-0'], ['1604.06985-2-30-0', '1604.06985-3-30-0'], ['1604.06985-2-30-1', '1604.06985-3-30-1'], ['1604.06985-2-30-2', '1604.06985-3-30-2'], ['1604.06985-2-30-3', '1604.06985-3-30-3'], ['1604.06985-2-18-0', '1604.06985-3-18-0'], ['1604.06985-2-18-1', '1604.06985-3-18-1'], ['1604.06985-2-18-2', '1604.06985-3-18-2'], ['1604.06985-2-38-0', '1604.06985-3-38-0'], ['1604.06985-2-38-1', '1604.06985-3-38-1'], ['1604.06985-2-5-0', '1604.06985-3-5-0'], ['1604.06985-2-5-1', '1604.06985-3-5-1'], ['1604.06985-2-5-2', '1604.06985-3-5-2'], ['1604.06985-2-20-0', '1604.06985-3-20-0'], ['1604.06985-2-20-1', '1604.06985-3-20-1'], ['1604.06985-2-20-2', '1604.06985-3-20-2'], ['1604.06985-2-41-0', '1604.06985-3-41-0'], ['1604.06985-2-16-0', '1604.06985-3-16-0'], ['1604.06985-2-16-1', '1604.06985-3-16-1'], ['1604.06985-2-35-0', '1604.06985-3-35-0'], ['1604.06985-2-35-1', '1604.06985-3-35-1'], ['1604.06985-2-7-0', '1604.06985-3-7-0'], ['1604.06985-2-7-1', 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'1604.06985-2-25-1'], ['1604.06985-2-25-1', '1604.06985-3-25-1'], ['1604.06985-2-0-2', '1604.06985-3-0-2'], ['1604.06985-2-26-1', '1604.06985-3-26-1'], ['1604.06985-2-26-2', '1604.06985-3-26-2'], ['1604.06985-2-26-11', '1604.06985-3-26-11'], ['1604.06985-2-26-16', '1604.06985-3-26-14'], ['1604.06985-2-33-0', '1604.06985-3-33-0'], ['1604.06985-2-17-0', '1604.06985-3-17-0'], ['1604.06985-2-37-0', '1604.06985-3-37-0'], ['1604.06985-2-37-2', '1604.06985-3-37-2'], ['1604.06985-2-27-0', '1604.06985-3-27-0'], ['1604.06985-2-36-0', '1604.06985-3-36-0'], ['1604.06985-2-36-1', '1604.06985-3-36-2'], ['1604.06985-2-24-0', '1604.06985-3-24-0'], ['1604.06985-2-23-1', '1604.06985-3-23-1'], ['1604.06985-2-30-4', '1604.06985-3-30-4']]

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['1604.06985-2-29-0', '1604.06985-3-29-0'], ['1604.06985-2-30-0', '1604.06985-3-30-0'], ['1604.06985-2-30-1', '1604.06985-3-30-1'], ['1604.06985-2-30-2', '1604.06985-3-30-2'], ['1604.06985-2-30-3', '1604.06985-3-30-3'], ['1604.06985-2-18-0', '1604.06985-3-18-0'], ['1604.06985-2-18-1', '1604.06985-3-18-1'], ['1604.06985-2-18-2', '1604.06985-3-18-2'], ['1604.06985-2-38-0', '1604.06985-3-38-0'], ['1604.06985-2-38-1', '1604.06985-3-38-1'], ['1604.06985-2-5-0', '1604.06985-3-5-0'], ['1604.06985-2-5-1', '1604.06985-3-5-1'], ['1604.06985-2-5-2', '1604.06985-3-5-2'], ['1604.06985-2-20-0', '1604.06985-3-20-0'], ['1604.06985-2-20-1', '1604.06985-3-20-1'], ['1604.06985-2-20-2', '1604.06985-3-20-2'], ['1604.06985-2-41-0', '1604.06985-3-41-0'], ['1604.06985-2-16-0', '1604.06985-3-16-0'], ['1604.06985-2-16-1', '1604.06985-3-16-1'], ['1604.06985-2-35-0', '1604.06985-3-35-0'], ['1604.06985-2-35-1', '1604.06985-3-35-1'], ['1604.06985-2-7-0', '1604.06985-3-7-0'], ['1604.06985-2-7-1', 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[]

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[]

['1604.06985-1-21-0', '1604.06985-1-22-0', '1604.06985-1-25-0', '1604.06985-1-26-0', '1604.06985-1-26-3', '1604.06985-1-26-15', '1604.06985-1-32-0', '1604.06985-2-21-0', '1604.06985-2-22-0', '1604.06985-2-25-0', '1604.06985-2-26-0', '1604.06985-2-26-3', '1604.06985-2-26-15', '1604.06985-2-32-0', '1604.06985-3-21-0', '1604.06985-3-22-0', '1604.06985-3-25-0', '1604.06985-3-26-0', '1604.06985-3-26-3', '1604.06985-3-26-12', '1604.06985-3-26-16', '1604.06985-3-32-0']

{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '3': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1604.06985

{'1604.06985-3-0-0': 'This paper extends our previous work on regularization of neural networks using Eigenvalue Decay by employing a soft approximation of the dominant eigenvalue in order to enable the calculation of its derivatives in relation to the synaptic weights, and therefore the application of back-propagation, which is a primary demand for deep learning.', '1604.06985-3-0-1': 'Moreover, we extend our previous theoretical analysis to deep neural networks and multiclass classification problems.', '1604.06985-3-0-2': 'Our method is implemented as an additional regularizer in Keras, a modular neural networks library written in Python, and evaluated in the benchmark data sets Reuters Newswire Topics Classification, IMDB database for binary sentiment classification, MNIST database of handwritten digits and CIFAR-10 data set for image classification.', '1604.06985-3-1-0': '# Introduction', '1604.06985-3-2-0': 'One of the problems in Machine Learning is termed overfitting.', '1604.06985-3-2-1': 'The error on the training data set is driven toward a small value; however, the error is large when new data are presented to the trained algorithm.', '1604.06985-3-2-2': 'This occurs because the algorithm does not learn to generalize when new situations are presented.', '1604.06985-3-2-3': "This phenomenon is related to the models' complexity, in Vapnik sense, and can be minimized by using regularization techniques [CITATION] and [CITATION].", '1604.06985-3-3-0': 'Current deep learning models present good generalization capacity, despite having very high VC dimensions [CITATION].', '1604.06985-3-3-1': 'This is mostly because of recent advances in regularization techniques, which control the size of the hypothesis space [CITATION].', '1604.06985-3-3-2': 'Existing libraries for deep learning allow users to set constraints on network parameters and to apply penalties on parameters [CITATION] or activity of the model layers.', '1604.06985-3-3-3': 'These penalties are usually incorporated into the loss function that the network optimizes on a per-layer basis and can be understood as soft constraints.', '1604.06985-3-4-0': 'In our previous paper [CITATION] we proposed and analyzed a regularization technique named Eigenvalue Decay, aiming at improving the classification margin, which is an effective strategy to decrease the classifier complexity, in Vapnik sense, by taking advantage on geometric properties of the training examples within the feature space.', '1604.06985-3-4-1': 'However, our previous approach requires a highly computational demanding training method based on Genetic Algorithms, which is not suitable for deep learning.', '1604.06985-3-4-2': 'In this paper we utilize a soft approximation of the dominant eigenvalue, in order to enable the calculation of its derivatives in relation to the synaptic weights, aiming at the application of back-propagation.', '1604.06985-3-4-3': 'Moreover, we extend our previous theoretical analysis to deep neural networks and multiclass classification problems.', '1604.06985-3-5-0': 'The paper is organized as follows: Section [REF] briefly reports the state-of-the-art in neural network regularization, while Section [REF] defines the problem of training with Eigenvalue Decay and analyzes the relationship between such regularization method and the classification margin.', '1604.06985-3-5-1': 'In Section [REF] we explain how we implement this method in Keras.', '1604.06985-3-5-2': 'Section [REF] reports the experiments, while Section [REF] summarizes some conclusions.', '1604.06985-3-6-0': '# State-of-the-art', '1604.06985-3-7-0': 'There are many regularization strategies available to the deep learning practitioner, most of them based on regularizing estimators, i.e. trading increased bias for reduced variance [CITATION].', '1604.06985-3-7-1': 'In this section we briefly describe some of the most usual regularization strategies, such as constraining the parameter values of the model, adding extra terms in the objective function to penalize overly high values of the parameters and a recently developed technique, inspired in ensemble methods, which combines multiple hypotheses that explain the training data.', '1604.06985-3-8-0': '[MATH] weight decay is the most usual weight regularizer, and was theoretically analyzed in [CITATION], which concludes that the bounds on the expected risk of a multilayer perceptron (MLP) depends on the magnitude of the parameters rather than the number of parameters.', '1604.06985-3-8-1': 'In the work [CITATION] the author showed that the misclassification probability can be bounded in terms of the empirical risk, the number of training examples, and a scale-sensitive version of the VC-dimension, known as the fat-shattering dimension, which can be upper-bounded in terms of the magnitudes of the network parameters, independently from the number of parameters.', '1604.06985-3-8-2': 'In short, as regards [MATH] weight-decay, the work [CITATION] only shows that such a method can be applied to control the capacity of the classifier space.', '1604.06985-3-8-3': 'However, the best known way to minimize the capacity of the classifier space without damaging the accuracy on the training data is to maximize the classification margin, which is the SVM principle.', '1604.06985-3-8-4': 'Unfortunately, from the best of our knowledge, there is no formal proof that weight decay can maximize the margin.', '1604.06985-3-8-5': 'Therefore, we propose the Eigenvalue Decay, for which it is possible to establish a relationship between the eigenvalue minimization and the classification margin.', '1604.06985-3-9-0': 'Another commonly used weight regularizer is the [MATH] weight decay, which results in a sparse solution, in the sense that some parameters have an optimal value near zero.', '1604.06985-3-9-1': 'The sparsity property induced by [MATH] regularization is useful as a feature selection mechanism, such as in the least absolute shrinkage and selection operator (LASSO) algorithm [CITATION], which integrates an [MATH] penalty with a linear model to lead a subset of the model weights to become zero.', '1604.06985-3-10-0': 'It is also possible to set constraints on network parameters (usually on the norm of the parameters) during optimization, yielding a constrained optimization problem.', '1604.06985-3-10-1': 'If the [MATH] norm is adopted, the weights are constrained to lie in an ball, resulting a smaller hypothesis space.', '1604.06985-3-11-0': 'Early stopping [CITATION] is a commonly employed method to improve the generalization capacity of neural networks (NN).', '1604.06985-3-11-1': 'This method also acts as a regularizer, since it restricts the optimization procedure to a small volume of parameter space within the neighborhood of the initial parameter value [CITATION].', '1604.06985-3-11-2': 'In early stopping, the labeled data are divided into training and validation data sets.', '1604.06985-3-11-3': 'After some number of iterations the NN begins to overfit the data and the error on the validation data set begins to rise.', '1604.06985-3-11-4': 'When the validation error increases during a specified number of iterations, the algorithm stops the training section and applies the weights and biases at the minimum of the validation error to the NN.', '1604.06985-3-12-0': 'The recently proposed DropOut [CITATION] provides a powerful way of regularizing deep models, while maintaining a relatively small computational cost.', '1604.06985-3-12-1': 'DropOut can be understood as a practical technique for constructing bootstrap aggregating (bagging) ensembles [CITATION] of many large NNs, i.e. DropOut trains an ensemble consisting of all sub-NNs that can be formed by removing non-output units from an underlying base NN [CITATION].', '1604.06985-3-12-2': 'However, while in bagging the models are all independent, in DropOut the models share parameters from the parent NN, making it possible to represent an exponential number of models with a tractable amount of memory.', '1604.06985-3-13-0': '# Eigenvalue Decay for deep neural networks', '1604.06985-3-14-0': 'In this section we define the problem of using Eigenvalue Decay in deep learning and show a relationship between this regularizer and the classification margin.', '1604.06985-3-15-0': 'Eigenvalue Decay can be understood as a weight decay regularizer; however, while the usual weight decay regularizers penalize overly high values of weights, Eigenvalue Decay penalizes overly high values of the dominant eigenvalue of [MATH], where [MATH] is the synaptic weight matrix of any arbitrary layer [MATH] of the NN.', '1604.06985-3-15-1': 'Both methods force the NN response to be smoother and less likely to overfit the training data.', '1604.06985-3-16-0': 'We consider a binary or multiclass classification problem where the target output is encoded in one-hot style.', '1604.06985-3-16-1': 'Therefore, assuming [MATH] classes, the target output is a [MATH]-dimensional vector where the position corresponding to the target class has the value [MATH] and all the other [MATH] positions have the value [MATH].', '1604.06985-3-17-0': 'We analyze the classification margin of a deep MLP with [MATH] hidden layer and linear output layer, whose model is given by: [EQUATION] where [MATH] is the output vector of the [MATH] hidden layer, [MATH] is a matrix whose elements are the synaptic weights of layer [MATH], [MATH] is the bias vector of the layer [MATH], [MATH] is the input vector, and [MATH] is a commonly used activation function, such as the sigmoid function.', '1604.06985-3-18-0': 'A single multiclass problem can be reduced into multiple binary classification problems; therefore, a multiclass classifier can be understood as an ensemble of binary classifiers which distinguish between one of the labels and the remainder (i.e. one-versus-all approach).', '1604.06985-3-18-1': 'In our case, a binary classifier for any arbitrary class [MATH] can be built by substituting the matrix [MATH] in ([REF]) by its line [MATH].', '1604.06985-3-18-2': 'Our analysis considers the classification margin per class, i.e. we analyze the relation between the application of Eigenvalue Decay on any hidden layer and the classification margin of the model ([REF]) for any arbitrary class [MATH], i.e. by considering only the line [MATH] of the matrix [MATH] in our analysis.', '1604.06985-3-19-0': 'The MLP training using Eigenvalue Decay is modeled as: [EQUATION] where [EQUATION] where [MATH] can be any commonly used loss function, such as mean squared error (MSE) or the Hinge loss, [MATH] is the dominant eigenvalue of [MATH], [MATH] is a constant that controls the regularization on the layer [MATH] and [MATH], [MATH] are the weight matrices of the MLP model with [MATH] layers, as defined in ([REF]).', '1604.06985-3-20-0': 'As can be seen in ([REF]), our new theoretical analysis assumes the regularization by Eigenvalue Decay for any layer [MATH], since our code enables the application of this regularizer on any hidden layer.', '1604.06985-3-20-1': 'We extended our previous analysis [CITATION] by applying the chain rule to calculate the derivatives of the estimated output of the MLP in relation to the synaptic weights of any hidden layer, thus yielding larger equations and also unavoidable complexity.', '1604.06985-3-20-2': 'So, we recommend the reading of the theoretical derivations in [CITATION] for MLP with only one hidden neuron before the reading of the derivations presented in the present paper.', '1604.06985-3-21-0': 'We start our analysis with the following lemma:', '1604.06985-3-22-0': 'Lemma 1.', '1604.06985-3-22-1': '[CITATION] Let [MATH] denote the field of real numbers, [MATH] a vector space containing all matrices with [MATH] rows and [MATH] columns with entries in [MATH], [MATH] be a symmetric positive-semidefinite matrix and [MATH] be the dominant eigenvalue of [MATH].', '1604.06985-3-22-2': 'Therefore, for any [MATH], the following inequality holds true: [EQUATION]', '1604.06985-3-23-0': 'Our method penalizes the dominant eigenvalue aiming at maximizing the lower bound of the classification margin, as will be shown in Theorem 1.', '1604.06985-3-23-1': 'For the sake of space, we call the [MATH] line of [MATH] as [MATH].', '1604.06985-3-24-0': 'We define the classification margin of the input data [MATH] as the smallest orthogonal distance, [MATH], between [MATH] and the separating hypersurface defined by the MLP, see Figure [REF].', '1604.06985-3-25-0': 'Theorem 1.', '1604.06985-3-25-1': 'Let [MATH] be the classification margin of the training example [MATH], for any arbitrary class [MATH], and [MATH] be the dominant eigenvalue of [MATH]; then, for [MATH], i.e. an example correctly classified, and a MLP with [MATH] hidden layers, the following inequality hold true: [EQUATION] where [EQUATION] [EQUATION] [MATH] is the [MATH] projection of [MATH] on the separating hypersurface defined by the MLP, as illustrated in Fig. [REF], [MATH] is the [MATH] target output for the class [MATH], i.e. the position [MATH] of the target vector [MATH] and [MATH] is the dominant eigenvalue of [MATH].', '1604.06985-3-26-0': 'Proof.', '1604.06985-3-26-1': 'The first step in this proof is the calculation of the gradient of the position [MATH] of the estimated output vector, [MATH], in relation to the input [MATH] at the projected point [MATH] (see Figure [REF]): [EQUATION]', '1604.06985-3-26-2': 'The normalized vector [EQUATION] is normal to the separating surface, giving the direction from [MATH] to [MATH]; therefore [EQUATION] where [MATH] is the scalar distance between [MATH] and [MATH].', '1604.06985-3-26-3': 'From ([REF]) we have: [EQUATION]', '1604.06985-3-26-4': 'Substituting ([REF]) into ([REF]) and solving for [MATH], yields: [EQUATION]', '1604.06985-3-26-5': 'The sign of [MATH] depends on which side of the decision surface [MATH] is placed.', '1604.06985-3-26-6': 'It means that an example, [MATH], correctly classified whose target value for the class [MATH] is [MATH] corresponds to [MATH].', '1604.06985-3-26-7': 'On the other hand, the classification margin must be positive in cases where examples are correctly classified, and negative in cases of misclassified examples, independently from their target classes.', '1604.06985-3-26-8': 'Therefore, the margin is defined as function of [MATH], where [MATH] is the value of the target output of the [MATH] training example for the class [MATH].', '1604.06985-3-26-9': 'More specifically, the margin, [MATH], is the smallest value of [MATH] in relation to [MATH], that is: [EQUATION]', '1604.06985-3-26-10': 'Substituting ([REF]) in ([REF]) yields: [EQUATION]', '1604.06985-3-26-11': 'For a MLP with a single hidden layer we have: [EQUATION] as can be derived from ([REF]).', '1604.06985-3-26-12': 'Substituting ([REF]) in ([REF]), yields: [EQUATION]', '1604.06985-3-26-13': 'Note that [MATH] is a symmetric positive-semidefinite matrix, therefore, from Lemma 1, the inequality: [EQUATION]', '1604.06985-3-26-14': 'Since [MATH] is also a symmetric positive-semidefinite matrix: [EQUATION]', '1604.06985-3-26-15': 'For a MLP with two hidden layers we have: [EQUATION]', '1604.06985-3-26-16': 'Substituting ([REF]) in ([REF]), yields: [EQUATION]', '1604.06985-3-26-17': 'Since [MATH], [MATH], [MATH] and [MATH] are symmetric positive-semidefinite matrices: [EQUATION]', '1604.06985-3-26-18': 'From ([REF]) and ([REF]) we can deduce ([REF]) by induction.', '1604.06985-3-27-0': 'Taking into account that [MATH] is in the denominator of the bound in ([REF]), the training method based on Eigenvalue Decay decreases [MATH] aiming at increasing the lower bound on the classification margin.', '1604.06985-3-27-1': 'However, Eigenvalue Decay does not assure, by itself, increasing the margin, because [MATH] is function of [MATH].', '1604.06985-3-28-0': '# Using Eigenvalue Decay in Keras', '1604.06985-3-29-0': 'The use of Eigenvalue Decay within a deep learning library, such as Keras [CITATION], requires, not only a lightweight algorithm, but also a formulation that enables the calculation of derivatives of the objective function in relation to the synaptic weights, aiming at the application of back-propagation, which is a main demand for deep learning.', '1604.06985-3-30-0': 'We approximate the dominant eigenvalue by the power method; therefore, assuming [MATH], the eigenvector corresponding to the dominant eigenvalue can be approximated by: [EQUATION] where [MATH] is an arbitrary positive integer and [MATH] is an initial nonzero approximation of the dominant eigenvector.', '1604.06985-3-30-1': 'We set all the elements of [MATH] equal to one.', '1604.06985-3-30-2': 'Having [MATH], we calculate [MATH] as follows: [EQUATION]', '1604.06985-3-30-3': 'The approximation given by the power method has derivatives in relation to the synaptic weights, i.e. the elements of [MATH], enabling the application of backpropagation in Keras.', '1604.06985-3-30-4': 'We implement Eigenvalue Decay in Keras using Theano functions to model the approximation of [MATH] based on an approximation of [MATH] where [MATH].', '1604.06985-3-30-5': 'Our source code is freely available in Github.', '1604.06985-3-31-0': 'Beyond the custom regularizer presented in this paper, it is possible to implement a custom objective function in Keras, see our previous work [CITATION], where we implemented a custom version of the Hinge loss, aiming at a SVM-like learning for multiclass classification.', '1604.06985-3-31-1': 'The idea is to make better use of the margin resulting from the use of Eigenvalue Decay, since the Hinge loss penalizes only examples that violate a given margin or are misclassified, i.e. an estimated output smaller than 1 in response to a positive example or an estimated output larger than -1 in response to a negative example (these training examples can be understood as support vectors).', '1604.06985-3-31-2': "The other training examples are ignored during the optimization, i.e. they don't participate in defining the decision surface.", '1604.06985-3-32-0': 'Experiments', '1604.06985-3-33-0': 'In this section our methods are evaluated using the benchmark data sets Reuters Newswire Topics Classification, IMDB database for binary sentiment classification, MNIST database of handwritten digits and CIFAR-10 data set for image classification.', '1604.06985-3-34-0': 'Reuters Newswire Topics Classification (RNTC) is a collection of 11228 newswires from Reuters, labeled over 46 topics.', '1604.06985-3-34-1': 'IMDB Movie Reviews Sentiment Classification Dataset is a collection of 25000 movies reviews from IMDB, labeled by sentiment (positive/negative).', '1604.06985-3-34-2': 'MNIST database of Handwritten Digits Dataset is a collection of 60000 [MATH] grayscale images of the 10 digits, along with a test set of 10000 images.', '1604.06985-3-34-3': 'CIFAR-10 data set consists of 60000 [MATH] color images labeled over 10 categories, 50000 training images and 10000 test images.', '1604.06985-3-34-4': 'Table [REF] summarizes the details of the data sets.', '1604.06985-3-35-0': 'For the sake of comparison, we adopt the original models from the examples provided in the Keras repository , which are well adjusted and regularized with DropOut.', '1604.06985-3-35-1': 'Our experiments compare the performance of weight regularizers; therefore, we keep the original DropOut regularization and apply Eigenvalue Decay (ED), [MATH] and [MATH] weight regularizers to compare the accuracy gains over the original models from Keras repository.', '1604.06985-3-36-0': 'Among the models available in the Keras repository, we adopt the MLP with a single hidden layer for RNTC, the deep pipeline composed by an embedding layer, a convolutional neural network (CNN) [CITATION], a long short term memory (LSTM) network [CITATION] and a dense layer for IMDB, the deep MLP with two hidden layers for MNIST and the deep pipeline composed by a MLP stacked on the top of a CNN for CIFAR-10.', '1604.06985-3-36-1': 'The experiment with CIFAR-10 does not use data augmentation.', '1604.06985-3-36-2': 'In our experiments we apply the weight regularizers in both layers of the MLP used for RNTC, on the dense and embedding layers of the deep model used for IMBD, on the last two layers of the deep MLP used for MNIST and on both dense layers of the model used for CIFAR-10.', '1604.06985-3-37-0': 'To find the optimal values of [MATH] in ([REF]), we exploit a 2D grid using 5-fold cross validation on the training data, keeping the original architectures and loss functions of the Keras models, i.e. categorical cross-entropy for RNTC, MNIST and CIFAR-10, and binary cross-entropy for IMDB.', '1604.06985-3-37-1': 'The accuracy values on the test data and the processing time per training epoch, running in a GPU NVIDIA GeForce GTX 980, are summarized in Table [REF], where [MATH] is the gain over the original model from Keras.', '1604.06985-3-37-2': 'In the case of the IMDB and CIFAR-10 data sets the accuracy values were averaged over 10 runs.', '1604.06985-3-38-0': 'As can be seen in Table [REF], the weight regularizers yielded small gains on the accuracy, since the models provided in the Keras repository are well adjusted and regularized with DropOut.', '1604.06985-3-38-1': 'Eigenvalue Decay yielded the largest gains in all the data sets, but it was also the most costly regularizer, which is not a surprise, given the cost associated with the computation of the dominant eigenvalue by the power method.', '1604.06985-3-39-0': '# Conclusion', '1604.06985-3-40-0': 'This work introduces a new option of weight regularizer to the deep learning practitioners.', '1604.06985-3-40-1': 'The analysis presented in this paper indicates that Eigenvalue Decay can increase the classification margin, which can improve the generalization capability of deep models.', '1604.06985-3-41-0': 'In the scope of weight regularizers, the experiments indicate that Eigenvalue Decay can provide better gains on the classification accuracy at the cost of a larger CPU/GPU time.'}

1407.8277

{'1407.8277-1-0-0': 'We study the variance and kurtosis of the net-baryon number in a fluid dynamical model for heavy-ion collisions.', '1407.8277-1-0-1': 'It is based on an effective chiral model with dilatons for the strong coupling regime of QCD.', '1407.8277-1-0-2': 'Taking into account spinodal instabilities, we demonstrate that this model exhibits a diverging quark number susceptibility and kurtosis all along the spinodal lines of the first-order phase transition, with a change of universality class at the critical end point.', '1407.8277-1-0-3': 'During the (3+1) dimensional expansion of a hot and dense fireball, instabilities are created by fluctuations in the explicitly propagated chiral and dilaton field.', '1407.8277-1-0-4': 'We find a clear enhancement of event-by-event fluctuations of the baryon number at the critical point and first-order phase transition in comparison with an evolution through the crossover region.', '1407.8277-1-1-0': 'Keywords: relativistic heavy-ion collisions, relativistic fluid dynamics, dynamic symmetry breaking, fluctuations at first-order phase transitions', '1407.8277-1-2-0': '# Introduction', '1407.8277-1-3-0': 'Disclosing the phase structure of strongly interacting matter is one of the major motivations in the study of finite-temperature quantum chromodynamics (QCD).', '1407.8277-1-3-1': 'The volume independence of susceptibilities from lattice QCD data confirms a crossover chiral and deconfinement transition at small values of the baryochemical potential [MATH] [CITATION].', '1407.8277-1-3-2': 'In the regime of large densities, studies of effective models like the linear sigma or Nambu-Jona-Lasinio (NJL) model suggest a first-order phase transition and a critical end point (CEP) [CITATION].', '1407.8277-1-3-3': 'This can be further supported by investigations within the approach of Dyson-Schwinger equations [CITATION].', '1407.8277-1-3-4': 'Due to various approximations and limitations of all these methods there is, however, no agreement about the location of the CEP and transition line.', '1407.8277-1-3-5': 'During the RHIC beam energy scan, STAR has recently reported measurement of directed flow [CITATION], which might be interpreted as experimental evidence for a first-order phase transition.', '1407.8277-1-4-0': 'In order to determine the transition temperature and chemical potential experimentally, quantities which may signal a chiral phase transition are required.', '1407.8277-1-4-1': 'Of special interest in this context are susceptibilities of conserved charges like the net-baryon number or electric charge, who have been shown to display a peak at a crossover and first-order phase transition and a divergence at a CEP [CITATION].', '1407.8277-1-4-2': 'Such fluctuations have been proposed as experimental observable for the detection of a CEP in a heavy-ion collision [CITATION].', '1407.8277-1-4-3': 'However, measurements in the NA49 experiment could hardly find any non-monotonic behavior [CITATION].', '1407.8277-1-4-4': 'It has later been shown that higher moments or cumulants and their ratios are even more sensitive to a critical structure [CITATION].', '1407.8277-1-4-5': 'Of particular interest here is the kurtosis, a volume-independent quantity which becomes negative on the crossover side of the CEP [CITATION].', '1407.8277-1-4-6': 'The beam-energy scan carried out by the STAR collaboration was able to find deviations of the kurtosis from hadron resonance gas and UrQMD transport model calculations [CITATION].', '1407.8277-1-5-0': 'It is important to note that all predictions have been made under the assumption that the phase transition takes place in equilibrium, resulting both in divergent fluctuations at a CEP and finite ones at the first-order transition.', '1407.8277-1-5-1': 'However, the system produced in a heavy-ion collision is rapidly expanding and cooling which makes it inevitable to consider dynamical effects.', '1407.8277-1-5-2': 'Besides the finite system size, critical slowing down is expected to influence the dynamics near a CEP.', '1407.8277-1-5-3': 'This has been demonstrated phenomenologically in [CITATION] and within a nonequilibrium fluid dynamical model in [CITATION].', '1407.8277-1-5-4': 'At a dynamical first-order phase transition, spinodal instabilities play a crucial role.', '1407.8277-1-5-5': 'Including them in the NJL model, the authors in [CITATION] succeeded to demonstrate how the quark number susceptibility diverges all along the isothermal spinodals.', '1407.8277-1-5-6': 'These divergences result from the convex structure of the pressure and the presence of a mechanically instable region.', '1407.8277-1-5-7': 'Consequently, one would expect large fluctuations not only at a CEP but also, and possibly stronger, at a first-order phase transition.', '1407.8277-1-5-8': 'Here, the fast collective expansion of matter produced after the collision of two nuclei should lead to the formation of a supercooled phase [CITATION].', '1407.8277-1-5-9': 'If nucleation times are large, this phase will spinodally decompose [CITATION], leading to domain formation in the order parameter fields [CITATION] and non-uniform structures like droplets in the baryon density, driven by pressure gradients.', '1407.8277-1-5-10': 'The subsequent hadronization of such droplets would result in non-statistical multiplicity fluctuations and an enhancement of higher flow harmonics [CITATION].', '1407.8277-1-5-11': 'In order to draw final conclusions from the experimental data, models for a dynamical phase transition including critical behavior and also finite size and time effects are required.', '1407.8277-1-5-12': 'This would also allow predictions for future experiments at FAIR [CITATION] and NICA [CITATION] which will cover the region of high densities in the QCD phase diagram.', '1407.8277-1-6-0': 'In this article we present a study of event-by-event fluctuations from a fully dynamical model of heavy-ion collisions.', '1407.8277-1-6-1': 'Starting from a linear sigma model with dilatons [CITATION], we couple a fluid of quarks and gluons to the explicit propagation of the sigma field as the chiral order parameter and the dilaton representing a gluon condensate.', '1407.8277-1-6-2': 'Such an ansatz has been pursued for the first time in [CITATION], where the production and collapse of vacuum bubbles was observed during the expansion of the chiral fluid.', '1407.8277-1-6-3': 'We go beyond this study by augmenting the classical Euler-Lagrange equations for the sigma field with terms for dissipation and noise, considering the proper and full nonequilibrium dynamics from the interaction of the locally thermalized fluid with the out-of-equilibrium evolution of the field.', '1407.8277-1-6-4': 'The corresponding Langevin equation has been derived selfconsistently in [CITATION].', '1407.8277-1-6-5': 'In former dynamical studies the gluons were included on the basis of the Polyakov loop [CITATION], a static quantity defined in Euclidean space-time.', '1407.8277-1-6-6': 'In contrast to this, the dilaton field has two advantages: First, it comes with a kinetic term in the Lagrangian, making its dynamics straightforward to derive.', '1407.8277-1-6-7': 'Second, the problem of negative pressures at a first-order phase transition in the Polyakov loop model [CITATION] can be avoided.', '1407.8277-1-7-0': 'We begin with a description of the model and the equations of motion in Sec. [REF], followed by the calculation of the quark number susceptibility and kurtosis at a nonequilibrium first-order phase transition in the regime of low temperatures in Sec. [REF].', '1407.8277-1-7-1': 'In Sec. [REF], we focus on the impact of a nonequilibrium evolution on fluctuation observables by determining the variance and kurtosis of the net-baryon number distribution in an event-by-event study.', '1407.8277-1-7-2': 'We conclude with a summary and outlook in Sec. [REF].', '1407.8277-1-8-0': '# Nonequilibrium chiral fluid dynamics', '1407.8277-1-9-0': 'We provide a dynamical nonequilibrium model based on a linear sigma model with a dilaton field [CITATION], for which the Lagrangian density reads [EQUATION]', '1407.8277-1-9-1': 'In addition to the usual linear sigma model which describes the melting of the chiral condensate [MATH] at high temperatures or net-baryon densities, it includes a dilaton or glueball field which we may identify with the gluon condensate [MATH].', '1407.8277-1-9-2': 'The term [MATH] in the Lagrangian stands for a constituent gluon field [MATH] which acquires mass from the nonvanishing expectation value of the gluon condensate [MATH].', '1407.8277-1-9-3': 'Its field strength tensor is defined as [MATH].', '1407.8277-1-9-4': 'We consider only the light quarks [MATH] in our present study.', '1407.8277-1-10-0': 'The model captures essential features of QCD in the strong coupling regime, the spontaneous breakdown of chiral symmetry and the trace anomaly.', '1407.8277-1-10-1': 'An alternative approach including the Polyakov loop as the thermal Wilson line over the color-electric field [MATH] has been pursued in [CITATION].', '1407.8277-1-10-2': 'However, this Polyakov-quark-meson (PQM) model [CITATION] yields negative values of the pressure at a first-order phase transition with spinodal instabilities [CITATION].', '1407.8277-1-11-0': 'The masses of both the constituent quarks and gluons are dynamically generated via [MATH] and [MATH].', '1407.8277-1-11-1': 'From this the coupling constants are fixed by reproducing the vacuum nucleon and glueball masses [MATH] MeV and [MATH] GeV.', '1407.8277-1-11-2': 'The term [MATH] with the pion decay constant [MATH] MeV and the pion mass [MATH] MeV explicitly breaks chiral symmetry.', '1407.8277-1-11-3': 'In vacuum, the energy density equals [MATH] which determines the bag constant B.', '1407.8277-1-11-4': 'The dimensionful parameter [MATH] is obtained from setting the vacuum glueball mass [MATH] equal to the second derivative of the potential [MATH].', '1407.8277-1-11-5': 'The self-coupling of the chiral field can be evaluated as [MATH] and thus depends on the vacuum sigma mass.', '1407.8277-1-11-6': 'As shown in [CITATION], a value of [MATH] MeV yields a reasonable behavior of the gluon condensate around the chiral transition in comparison with lattice QCD data.', '1407.8277-1-11-7': 'The corresponding phase diagram has a chiral critical point at [MATH] MeV and [MATH] MeV with an adjacent first-order phase transition line.', '1407.8277-1-11-8': 'Above the temperature of [MATH] MeV, scale symmetry is restored, here the model predicts a strong first-order phase transition which is not seen in lattice QCD.', '1407.8277-1-11-9': 'This defect is nevertheless neglectable for our studies as we are not going to probe the regime of deconfined gluons but focus on the chiral transition only.', '1407.8277-1-12-0': 'Within the mean-field approximation, the effective thermodynamic potential is obtained by a path integration over the quark and gluon fields [EQUATION]', '1407.8277-1-12-1': 'The quark and gluon contributions can be evaluated as [EQUATION] with the quasiparticle energies [MATH] and [MATH], respectively.', '1407.8277-1-12-2': 'A constant term [MATH] is added to ensure zero potential and pressure in vacuum.', '1407.8277-1-13-0': 'Having integrated out the quark and gluon degrees of freedom, we treat them as an ideal fluid to mimic the quark-gluon plasma at high energy densities.', '1407.8277-1-13-1': 'The fluid is described by the energy-momentum tensor [MATH] and the quark number current [MATH].', '1407.8277-1-13-2': 'The pressure is given by [EQUATION] from where energy and quark density are obtained via the standard thermodynamic relations [MATH] and [MATH].', '1407.8277-1-13-3': 'A full nonequilibrium dynamics for the coupled system of the sigma field and quarks has been derived in [CITATION] from the two-particle irreducible effective action.', '1407.8277-1-13-4': 'We adopt the result for the extended model with gluons and dilatons, the Langevin equation of motion for the sigma field reading [EQUATION]', '1407.8277-1-13-5': 'In addition to the classical Euler-Lagrange equation it contains a damping coefficient [MATH] and a stochastic noise field [MATH] describing dissipation and noise in the thermalized heat bath of quarks and gluons.', '1407.8277-1-13-6': 'Physically, the damping occurs due to the decay of one sigma into a quark-antiquark pair.', '1407.8277-1-13-7': 'As the sigma meson becomes light around the phase transition, also [MATH] decreases and eventually vanishes at the critical point.', '1407.8277-1-13-8': 'Its explicit form is given by [EQUATION]', '1407.8277-1-13-9': 'We work in the approximation of Gaussian white noise, with the noise field correlator [EQUATION]', '1407.8277-1-13-10': 'Similar to the sigma in the quark fluid, one might expect the dilaton in the gluonic medium to be damped, too.', '1407.8277-1-13-11': 'The corresponding process would be the emission of two gluons [MATH], according to Eq. ([REF]).', '1407.8277-1-13-12': 'However, as the dilaton mass is of the order of two times the in-medium mass of the gluons, this process is kinematically forbidden.', '1407.8277-1-13-13': 'We therefore apply the classical equation of motion to the dilaton field [EQUATION]', '1407.8277-1-13-14': 'Conservation of the overall energy-momentum as well as the baryon number are ensured by the fluid dynamical equations [EQUATION]', '1407.8277-1-13-15': 'Mainly due to the aforementioned dissipation, the fields lose energy to the fluid which is transferred via the source terms in Eq. ([REF]).', '1407.8277-1-13-16': 'This effect will be especially significant at a first-order phase transition, during the formation of a supercooled phase and its subsequent decay [CITATION].', '1407.8277-1-13-17': 'Since the evolution of the sigma field is stochastic in nature, also the fluid dynamical equations ([REF]), ([REF]) are becoming stochastic via the coupling to the source term.', '1407.8277-1-13-18': 'Fluctuating fluid dynamics has recently attracted attention in the context of heavy-ion collisions [CITATION].', '1407.8277-1-13-19': 'Finally, our set of equations is closed by a nonequilibrium equation of state, where the pressure explicitly depends on the local values of the fields [MATH] and [MATH], cf. Eq. ([REF]).', '1407.8277-1-14-0': '# Susceptibilities in the spinodal region', '1407.8277-1-15-0': 'Of particular interest for the detection and localization of the chiral phase transition are fluctuations of conserved charges like the net-baryon number.', '1407.8277-1-15-1': 'From effective models, one can calculate susceptibilities which describe fluctuations in the quark number density as response to changes in the chemical potential.', '1407.8277-1-15-2': 'They are in general defined as [EQUATION]', '1407.8277-1-15-3': 'Here we focus on two coefficients, namely [MATH], which is proportional to the quark number susceptibility [MATH] and the variance of fluctuations [MATH], and the kurtosis [MATH], given by the ratio of the fourth to the second coefficient.', '1407.8277-1-15-4': 'They are related to fluctuations in the quark number [MATH] as [EQUATION] with [MATH], the deviation from the ensemble average of the quark number distribution.', '1407.8277-1-15-5': 'Note that for [MATH] the volume and temperature dependence cancel to leading order.', '1407.8277-1-15-6': 'In [CITATION] it has been shown that event-by-event fluctuations like [MATH] diverge at a critical point.', '1407.8277-1-15-7': 'The non-monotonic behavior of fluctuations in heavy-ion collisions as a function of beam energy was proposed as an experimental signal.', '1407.8277-1-15-8': 'It was later shown in [CITATION] within the NJL model that susceptibilities also diverge along the spinodal lines of the first-order phase transition.', '1407.8277-1-15-9': 'This requires that nonequilibrium effects, i. e. spinodal instabilities, are taken into account.', '1407.8277-1-15-10': 'The presence of a mechanically instable region with [MATH] then consequently leads to diverging susceptibilities.', '1407.8277-1-15-11': 'For the sigma model with dilatons, we calculate both the quark number susceptibility and the kurtosis as a function of density for fixed temperature [MATH] MeV, where the model exhibits a first-order phase transition, see Fig. [REF].', '1407.8277-1-15-12': 'In the left plot, the susceptibility is shown as a function of the quark density.', '1407.8277-1-15-13': 'Similar to the result from the NJL model, we see divergences at the isothermal spinodal points.', '1407.8277-1-15-14': 'Inside the spinodal region, the susceptibility becomes negative due to instabilities.', '1407.8277-1-15-15': 'On the right hand side we show the kurtosis in the same density range.', '1407.8277-1-15-16': 'Here we also find strong divergences when crossing the spinodal lines.', '1407.8277-1-15-17': 'Interestingly, [MATH] remains positive even inside the coexistence region.', '1407.8277-1-15-18': 'On the other hand, it is known that the kurtosis becomes negative when approaching the critical end point from the crossover side [CITATION].', '1407.8277-1-16-0': 'Further insight into the nature of these fluctuations can be gained by determining the critical exponents which control the strength of the divergences at the CEP or spinodals, respectively.', '1407.8277-1-16-1': 'In the vicinity of the singularity, the behavior of susceptibility and kurtosis may be described by a power law of the form [EQUATION]', '1407.8277-1-16-2': 'We calculate the critical exponents [MATH] and [MATH] both analytically and numerically.', '1407.8277-1-16-3': 'The chiral transition may be described by a Ginzburg-Landau effective theory around the CEP and the spinodal lines [CITATION].', '1407.8277-1-16-4': 'At both points, the first and second derivatives of the effective potential vanish.', '1407.8277-1-16-5': 'Around a zero of the second derivative, it can be expanded in terms of [MATH] [EQUATION]', '1407.8277-1-16-6': 'At [MATH] we have [MATH], so for these coefficients the relation [MATH] and [MATH] holds for small values of [MATH].', '1407.8277-1-16-7': 'From [MATH] we obtain [MATH], and we can assume that [MATH] with [MATH].', '1407.8277-1-16-8': 'At the spinodal point, the leading term reads [MATH], giving [MATH] and consequently [MATH].', '1407.8277-1-16-9': 'For a CEP, we also have [MATH], so from [MATH] we end up with [MATH] and [MATH].', '1407.8277-1-16-10': 'For the fourth generalized susceptibility [MATH], we can immediately state the coefficients to be [MATH] for the CEP and [MATH] for the spinodal.', '1407.8277-1-16-11': 'As the kurtosis is proportional to the ratio of the fourth to the second derivative of the effective potential with respect to [MATH], we get [MATH] for both cases.', '1407.8277-1-16-12': 'The same values for [MATH] and [MATH] have been found within a numerical analysis fitting the forms in Eqs. ([REF]) to the numerically determined susceptibility and kurtosis.', '1407.8277-1-16-13': 'Fig. [REF] shows the analytical and the numerical results versus the reduced chemical potential [MATH] for illustration.', '1407.8277-1-17-0': 'The critical properties for a CEP and an isothermal spinodal point are different due to a change in the universality class, indicating different critical behavior and strength of divergences.', '1407.8277-1-17-1': 'The same result and exponents have been found for a chiral NJL model with finite current quark masses [CITATION].', '1407.8277-1-17-2': 'The critical exponents of the kurtosis are naturally in agreement for both types of transition and are found to be equal to [MATH].', '1407.8277-1-18-0': '# Nonequilibrium enhancement of fluctuation signals', '1407.8277-1-19-0': 'The results from the previous section indicate that in dynamical systems as they are created in heavy-ion collisions, fluctuation signals may not only be enhanced in the vicinity of a critical point, but also in the spinodal region.', '1407.8277-1-19-1': 'This might provide us with a more applicable means to investigate the QCD phase structure as the region around the CEP with enhanced susceptibility is small [CITATION] and subject to finite size and time effects that limit the growth of fluctuations.', '1407.8277-1-19-2': 'We may therefore expect larger fluctuations at a first-order phase transition than at a CEP.', '1407.8277-1-19-3': 'We test this assumption within the nonequilibrium chiral fluid dynamics model introduced in Sec. [REF].', '1407.8277-1-19-4': 'We initialize a spherical droplet of quark-gluon plasma by defining an initial temperature and quark chemical potential, with a Woods-Saxon distribution to ensure a smooth transition to the vacuum at the edges.', '1407.8277-1-19-5': 'Fields and fluid dynamical quantities are initialized with their respective equilibrium values.', '1407.8277-1-19-6': 'By choosing different initial conditions, we are able to observe the expansion and cooling through the crossover, critical and spinodal region.', '1407.8277-1-19-7': 'The total quark number is in each case fixed to [MATH].', '1407.8277-1-19-8': 'In Fig. [REF], we show event-averaged trajectories for the three scenarios in the [MATH]-[MATH]-plane.', '1407.8277-1-19-9': 'The values of the density and temperature in each event are obtained from an averaging over a central volume.', '1407.8277-1-19-10': 'The curves start on the right side proceeding to lower density on the left.', '1407.8277-1-19-11': 'Interestingly, we see that the first-order curve shows a slightly increasing temperature at intermediate densities between [MATH] to [MATH].', '1407.8277-1-19-12': 'This is a result of the reheating effect that occurs after the decomposition of a supercooled phase and is typical for a first-order phase transition.', '1407.8277-1-19-13': 'It has already been found in earlier works of nonequilibrium fluid dynamical models [CITATION].', '1407.8277-1-19-14': 'Note that this is a purely dynamical effect which causes the trajectory to even cross the CEP curve, where the temperature decreases monotonically.', '1407.8277-1-19-15': 'It also implies that for this curve there is some significant deviation from the equilibrium trajectory along the corresponding isentrope.', '1407.8277-1-19-16': 'For the first-order phase transition we also find bubbles created through spinodal decomposition, as has been reported in earlier fluid dynamical studies using a PQM model [CITATION].', '1407.8277-1-19-17': 'However, in the present model these high-density droplets are not stable but start to decay after traversing the spinodal region.', '1407.8277-1-19-18': 'This is a direct consequence of the now strictly positive pressure as has been pointed out already in the introduction.', '1407.8277-1-20-0': 'We show the evolution of event-by-event fluctuations in the net-baryon number [MATH] corresponding to these trajectories as a function of time in Fig. [REF].', '1407.8277-1-20-1': 'The baryon number is extracted directly from the fluid dynamical density.', '1407.8277-1-20-2': 'As a conserved quantity, it will fluctuate only slightly when including a freeze-out and hadronic interactions in the final state [CITATION].', '1407.8277-1-20-3': 'We use two different methods to determine [MATH]: First, within a fixed volume in the center of the collision, with an extension of [MATH] fm in [MATH]-direction and [MATH] fm each in [MATH]- and [MATH]-direction.', '1407.8277-1-20-4': 'Second, by limiting the region of acceptance via rapidity to [MATH] and transverse momentum density to [MATH] as it was performed in recent measurements at STAR [CITATION].', '1407.8277-1-20-5': 'Both methods yield qualitatively similar results.', '1407.8277-1-20-6': 'The variance of fluctuations is enhanced at a CEP in comparison with a crossover transition and even more, by a factor of [MATH] to [MATH], at a first-order phase transition.', '1407.8277-1-20-7': 'As the variance depends on the volume, we observe clear differences in the scales of the two plots.', '1407.8277-1-20-8': 'Furthermore, the volume strongly varies when applying a constant rapidity and momentum cut, therefore a more irregular structure in the time dependence can be found for that case.', '1407.8277-1-21-0': 'For the kurtosis in Fig. [REF], we also find that the crossover transition produces values close to zero, while a clear enhancement can be found at the CEP.', '1407.8277-1-21-1': 'Again, the largest fluctuations occur for a first-order phase transition.', '1407.8277-1-21-2': 'Remarkably, at CEP and first-order transition, both positive and negative values of [MATH] occur during the evolution, confirming the assumption of critical behavior in the spinodal region.', '1407.8277-1-21-3': 'Although the kurtosis is not dependent on the volume, we find its values to be an order of magnitude higher when using a rapidity and momentum cut than in the case of a fixed test volume.', '1407.8277-1-21-4': 'This can be explained considering the overall conservation of quark or baryon number in the system under consideration.', '1407.8277-1-21-5': 'In contrast to that, the baryon number is only on average conserved in a grand canonical ensemble which is used for effective model or lattice QCD calculations.', '1407.8277-1-21-6': 'As shown in [CITATION], this global conservation significantly effects ratios of cumulants, making them dependent on the fraction of measured to total baryons.', '1407.8277-1-22-0': '# Summary and Outlook', '1407.8277-1-23-0': 'We have investigated baryon number fluctuations within a chiral model with dilatons from two different approaches: First, through the calculation of susceptibilities, where we went beyond standard thermodynamics by including spinodal instabilities.', '1407.8277-1-23-1': 'We were able to show that both the quark number susceptibility and kurtosis diverge at the CEP and spinodal lines.', '1407.8277-1-23-2': 'The singularity in the susceptibility at the spinodals becomes suddenly stronger at the CEP, indicated by a larger critical exponent.', '1407.8277-1-23-3': 'The implications of such a behavior for experiment are strong enhancements of event-by-event fluctuations in the net-baryon number, which we investigated in a second dynamical approach.', '1407.8277-1-23-4': 'Propagating the chiral field and the dilaton explicitly on a locally thermalized background of quarks and gluons, we simulated the expansion of the hot and dense plasma created in a heavy-ion collision.', '1407.8277-1-23-5': 'We extracted the variance and kurtosis of the net-baryon number.', '1407.8277-1-23-6': 'Both are stronger at a CEP in comparison with a crossover scenario, and even more enhanced when the system evolves through the spinodal region of the first-order phase transition.', '1407.8277-1-24-0': 'In the future we are going to include hadronic degrees of freedom for a more realistic description of the chirally broken and confined phase.', '1407.8277-1-24-1': 'It is furthermore necessary to study particle distributions from a freeze-out or hadronic afterburner.', '1407.8277-1-24-2': 'This would also allow us to study the momentum anisotropy and the effect of the phase transition on flow.', '1407.8277-1-24-3': 'For the determination of susceptibilities, it would be interesting to include quantum or thermal fluctuations and study their effect on the critical properties near the CEP and first-order phase transition.'}

{'1407.8277-2-0-0': 'We study the variance and kurtosis of the net-baryon number in a fluid dynamical model for heavy-ion collisions.', '1407.8277-2-0-1': 'It is based on an effective chiral model with dilatons for the strong coupling regime of QCD.', '1407.8277-2-0-2': 'Taking into account spinodal instabilities, we demonstrate that this model exhibits a diverging quark number susceptibility and kurtosis all along the spinodal lines of the first-order phase transition, with a change of universality class at the critical end point.', '1407.8277-2-0-3': 'During the (3+1) dimensional expansion of a hot and dense fireball, instabilities are created by fluctuations in the explicitly propagated chiral and dilaton field.', '1407.8277-2-0-4': 'We find a clear enhancement of event-by-event fluctuations of the baryon number at the critical point and first-order phase transition in comparison with an evolution through the crossover region.', '1407.8277-2-1-0': 'Keywords: relativistic heavy-ion collisions, relativistic fluid dynamics, dynamic symmetry breaking, fluctuations at first-order phase transitions', '1407.8277-2-2-0': '# Introduction', '1407.8277-2-3-0': 'Disclosing the phase structure of strongly interacting matter is one of the major motivations in the study of finite-temperature quantum chromodynamics (QCD).', '1407.8277-2-3-1': 'The volume independence of susceptibilities from lattice QCD data confirms a crossover chiral and deconfinement transition at small values of the baryochemical potential [MATH] [CITATION].', '1407.8277-2-3-2': 'In the regime of large densities, studies of effective models like the linear sigma or Nambu-Jona-Lasinio (NJL) model suggest a first-order phase transition and a critical end point (CEP) [CITATION].', '1407.8277-2-3-3': 'This can be further supported by investigations within the approach of Dyson-Schwinger equations [CITATION].', '1407.8277-2-3-4': 'Due to various approximations and limitations of all these methods there is, however, no agreement about the location of the CEP and transition line.', '1407.8277-2-3-5': 'During the RHIC beam energy scan, STAR has recently reported measurement of directed flow [CITATION], which might be interpreted as experimental evidence for a first-order phase transition.', '1407.8277-2-4-0': 'In order to determine the transition temperature and chemical potential experimentally, quantities which may signal a chiral phase transition are required.', '1407.8277-2-4-1': 'Of special interest in this context are susceptibilities of conserved charges like the net-baryon number or electric charge, who have been shown to display a peak at a crossover and first-order phase transition and a divergence at a CEP [CITATION].', '1407.8277-2-4-2': 'Such fluctuations have been proposed as experimental observable for the detection of a CEP in a heavy-ion collision [CITATION].', '1407.8277-2-4-3': 'However, measurements in the NA49 experiment could hardly find any non-monotonic behavior [CITATION].', '1407.8277-2-4-4': 'It has later been shown that higher moments or cumulants and their ratios are even more sensitive to a critical structure [CITATION].', '1407.8277-2-4-5': 'Of particular interest here is the kurtosis, a volume-independent quantity which becomes negative on the crossover side of the CEP [CITATION].', '1407.8277-2-4-6': 'The beam-energy scan carried out by the STAR collaboration was able to find deviations of the kurtosis from hadron resonance gas and UrQMD transport model calculations [CITATION].', '1407.8277-2-4-7': 'As an alternative to the measurement of fluctuations, it has been shown in [CITATION] that the ratio of antiprotons to protons is sensitive to the presence of a CEP due to a focusing of the isentropic trajectories.', '1407.8277-2-5-0': 'It is important to note that all predictions have been made under the assumption that the phase transition takes place in equilibrium, resulting both in divergent fluctuations at a CEP and finite ones at the first-order transition.', '1407.8277-2-5-1': 'However, the system produced in a heavy-ion collision is rapidly expanding and cooling which makes it inevitable to consider dynamical effects.', '1407.8277-2-5-2': 'Besides the finite system size, critical slowing down is expected to influence the dynamics near a CEP.', '1407.8277-2-5-3': 'This has been demonstrated phenomenologically in [CITATION] and within a nonequilibrium fluid dynamical model in [CITATION].', '1407.8277-2-5-4': 'At a dynamical first-order phase transition, spinodal instabilities play a crucial role.', '1407.8277-2-5-5': 'Including them in the NJL model, the authors in [CITATION] succeeded to demonstrate how the quark number susceptibility diverges all along the isothermal spinodals.', '1407.8277-2-5-6': 'These divergences result from the convex structure of the pressure and the presence of a mechanically instable region.', '1407.8277-2-5-7': 'Consequently, one would expect large fluctuations not only at a CEP but also, and possibly stronger, at a first-order phase transition.', '1407.8277-2-5-8': 'Here, the fast collective expansion of matter produced after the collision of two nuclei should lead to the formation of a supercooled phase [CITATION].', '1407.8277-2-5-9': 'If nucleation times are large, this phase will spinodally decompose [CITATION], leading to domain formation in the order parameter fields [CITATION] and non-uniform structures like droplets in the baryon density, driven by pressure gradients.', '1407.8277-2-5-10': 'The subsequent hadronization of such droplets would result in non-statistical multiplicity fluctuations and an enhancement of higher flow harmonics [CITATION].', '1407.8277-2-5-11': 'In order to draw final conclusions from the experimental data, models for a dynamical phase transition including critical behavior and also finite size and time effects are required.', '1407.8277-2-5-12': 'This would also allow predictions for future experiments at FAIR [CITATION] and NICA [CITATION] which will cover the region of high densities in the QCD phase diagram.', '1407.8277-2-6-0': 'In this article we present a study of event-by-event fluctuations from a fully dynamical model of heavy-ion collisions.', '1407.8277-2-6-1': 'Starting from a linear sigma model with dilatons [CITATION], we couple a fluid of quarks and gluons to the explicit propagation of the sigma field as the chiral order parameter and the dilaton representing a gluon condensate.', '1407.8277-2-6-2': 'Such an ansatz has been pursued for the first time in [CITATION], where the production and collapse of vacuum bubbles was observed during the expansion of the chiral fluid.', '1407.8277-2-6-3': 'We go beyond this study by augmenting the classical Euler-Lagrange equations for the sigma field with terms for dissipation and noise, considering the proper and full nonequilibrium dynamics from the interaction of the locally thermalized fluid with the out-of-equilibrium evolution of the field.', '1407.8277-2-6-4': 'The corresponding Langevin equation has been derived selfconsistently in [CITATION].', '1407.8277-2-6-5': 'In former dynamical studies the gluons were included on the basis of the Polyakov loop [CITATION], a static quantity defined in Euclidean space-time.', '1407.8277-2-6-6': 'In contrast to this, the dilaton field has two advantages: First, it comes with a kinetic term in the Lagrangian, making its dynamics straightforward to derive.', '1407.8277-2-6-7': 'Second, the problem of negative pressures at a first-order phase transition in the Polyakov loop model [CITATION] can be avoided.', '1407.8277-2-7-0': 'We begin with a description of the model and the equations of motion in Sec. [REF], followed by the calculation of the quark number susceptibility and kurtosis at a nonequilibrium first-order phase transition in the regime of low temperatures in Sec. [REF].', '1407.8277-2-7-1': 'In Sec. [REF], we focus on the impact of a nonequilibrium evolution on fluctuation observables by determining the variance and kurtosis of the net-baryon number distribution in an event-by-event study.', '1407.8277-2-7-2': 'We conclude with a summary and outlook in Sec. [REF].', '1407.8277-2-8-0': '# Nonequilibrium chiral fluid dynamics', '1407.8277-2-9-0': 'We provide a dynamical nonequilibrium model based on a linear sigma model with a dilaton field [CITATION], for which the Lagrangian density reads [EQUATION]', '1407.8277-2-9-1': 'In addition to the usual linear sigma model which describes the melting of the chiral condensate [MATH] at high temperatures or net-baryon densities, it includes a dilaton or glueball field which we may identify with the gluon condensate [MATH].', '1407.8277-2-9-2': 'The term [MATH] in the Lagrangian stands for a constituent gluon field [MATH] which acquires mass from the nonvanishing expectation value of the gluon condensate [MATH].', '1407.8277-2-9-3': 'Its field strength tensor is defined as [MATH].', '1407.8277-2-9-4': 'We consider only the light quarks [MATH] in our present study.', '1407.8277-2-10-0': 'The model captures essential features of QCD in the strong coupling regime, the spontaneous breakdown of chiral symmetry and the trace anomaly.', '1407.8277-2-10-1': 'An alternative approach including the Polyakov loop as the thermal Wilson line over the color-electric field [MATH] has been pursued in [CITATION].', '1407.8277-2-10-2': 'However, this Polyakov-quark-meson (PQM) model [CITATION] yields negative values of the pressure at a first-order phase transition with spinodal instabilities [CITATION].', '1407.8277-2-11-0': 'The masses of both the constituent quarks and gluons are dynamically generated via [MATH] and [MATH].', '1407.8277-2-11-1': 'From this the coupling constants are fixed by reproducing the vacuum nucleon and glueball masses [MATH] MeV and [MATH] GeV.', '1407.8277-2-11-2': 'The term [MATH] with the pion decay constant [MATH] MeV and the pion mass [MATH] MeV explicitly breaks chiral symmetry.', '1407.8277-2-11-3': 'In vacuum, the energy density equals [MATH] which determines the bag constant B.', '1407.8277-2-11-4': 'The dimensionful parameter [MATH] is obtained from setting the vacuum glueball mass [MATH] equal to the second derivative of the potential [MATH].', '1407.8277-2-11-5': 'The self-coupling of the chiral field can be evaluated as [MATH] and thus depends on the vacuum sigma mass.', '1407.8277-2-11-6': 'As shown in [CITATION], a value of [MATH] MeV yields a reasonable behavior of the gluon condensate around the chiral transition in comparison with lattice QCD data.', '1407.8277-2-11-7': 'At nonzero baryochemical potential, lattice calculations have to rely on sophisticated methods to circumvent the infamous sign problem via reweighting [CITATION] or an imaginary chemical potential [CITATION].', '1407.8277-2-11-8': 'Obtained results are however still inconclusive about both the existence and position of a CEP.', '1407.8277-2-11-9': 'The corresponding phase diagram of our model has a chiral critical point at temperature [MATH] MeV and quark chemical potential [MATH] MeV with an adjacent first-order phase transition line.', '1407.8277-2-11-10': 'The critical chemical potential here is rather large compared with recent results from Dyson-Schwinger equations [CITATION] or a chiral quark-hadron model [CITATION] which predict values of [MATH] below [MATH] MeV.', '1407.8277-2-11-11': 'On the other hand, a similarly large [MATH] and correspondingly small [MATH] are predicted by a PQM model supplemented with fluctuations [CITATION] or the mean-field NJL model [CITATION].', '1407.8277-2-11-12': 'The linear sigma model with dilatons and also the PQM model approach the Stefan-Boltzmann limit at large temperatures which the standard linear sigma model fails to reproduce [CITATION].', '1407.8277-2-11-13': 'At large chemical potentials, however, the equation of state for the PQM model predicts negative values of the pressure in the region with spinodal instabilities.', '1407.8277-2-11-14': 'In contrast to that, the equation of state is well-behaved for the model with dilatons.', '1407.8277-2-11-15': 'Above the temperature of [MATH] MeV, scale symmetry is restored in the dilaton model, here it predicts a strong first-order phase transition which is not seen in lattice QCD.', '1407.8277-2-11-16': 'This defect is nevertheless neglectable for our studies as we are not going to probe the regime of deconfined gluons but focus on the chiral transition only.', '1407.8277-2-12-0': 'Within the mean-field approximation, the effective thermodynamic potential is obtained by a path integration over the quark and gluon fields [EQUATION]', '1407.8277-2-12-1': 'The quark and gluon contributions can be evaluated as [EQUATION] with the quasiparticle energies [MATH] and [MATH], respectively.', '1407.8277-2-12-2': 'Here and in the following, [MATH] denotes the quark chemical potential.', '1407.8277-2-12-3': 'A constant term [MATH] is added to ensure zero potential and pressure in vacuum.', '1407.8277-2-13-0': 'Having integrated out the quark and gluon degrees of freedom, we treat them as an ideal fluid to mimic the quark-gluon plasma at high energy densities.', '1407.8277-2-13-1': 'The fluid is described by the energy-momentum tensor [MATH] and the quark number current [MATH].', '1407.8277-2-13-2': 'The pressure is given by [EQUATION] from where energy and quark density are obtained via the standard thermodynamic relations [MATH] and [MATH].', '1407.8277-2-13-3': 'A full nonequilibrium dynamics for the coupled system of the sigma field and quarks has been derived in [CITATION] from the two-particle irreducible effective action.', '1407.8277-2-13-4': 'We adopt the result for the extended model with gluons and dilatons, the Langevin equation of motion for the sigma field reading [EQUATION]', '1407.8277-2-13-5': 'In addition to the classical Euler-Lagrange equation it contains a damping coefficient [MATH] and a stochastic noise field [MATH] describing dissipation and noise in the thermalized heat bath of quarks and gluons.', '1407.8277-2-13-6': 'Physically, the damping occurs due to the decay of one sigma into a quark-antiquark pair.', '1407.8277-2-13-7': 'As the sigma meson becomes light around the phase transition, also [MATH] decreases and eventually vanishes at the critical point.', '1407.8277-2-13-8': 'Its explicit form is given by [EQUATION]', '1407.8277-2-13-9': 'We work in the approximation of Gaussian white noise, with the noise field correlator [EQUATION]', '1407.8277-2-13-10': 'Similar to the sigma in the quark fluid, one might expect the dilaton in the gluonic medium to be damped, too.', '1407.8277-2-13-11': 'The corresponding process would be the emission of two gluons [MATH], according to Eq. ([REF]).', '1407.8277-2-13-12': 'However, as the dilaton mass is of the order of two times the in-medium mass of the gluons, this process is kinematically forbidden.', '1407.8277-2-13-13': 'We therefore apply the classical equation of motion to the dilaton field [EQUATION]', '1407.8277-2-13-14': 'Conservation of the overall energy-momentum as well as the baryon number are ensured by the fluid dynamical equations [EQUATION]', '1407.8277-2-13-15': 'Mainly due to the aforementioned dissipation, the fields lose energy to the fluid which is transferred via the source terms in Eq. ([REF]).', '1407.8277-2-13-16': 'This effect will be especially significant at a first-order phase transition, during the formation of a supercooled phase and its subsequent decay [CITATION].', '1407.8277-2-13-17': 'Since the evolution of the sigma field is stochastic in nature, also the fluid dynamical equations ([REF]), ([REF]) are becoming stochastic via the coupling to the source term.', '1407.8277-2-13-18': 'Fluctuating fluid dynamics has recently attracted attention in the context of heavy-ion collisions [CITATION].', '1407.8277-2-13-19': 'Finally, our set of equations is closed by a nonequilibrium equation of state, where the pressure explicitly depends on the local values of the fields [MATH] and [MATH], cf. Eq. ([REF]).', '1407.8277-2-14-0': '# Susceptibilities in the spinodal region', '1407.8277-2-15-0': 'Of particular interest for the detection and localization of the chiral phase transition are fluctuations of conserved charges like the net-baryon number.', '1407.8277-2-15-1': 'From effective models, one can calculate susceptibilities which describe fluctuations in the quark number density as response to changes in the chemical potential.', '1407.8277-2-15-2': 'They are in general defined as [EQUATION]', '1407.8277-2-15-3': 'Here we focus on two coefficients, namely [MATH], which is proportional to the quark number susceptibility [MATH] and the variance of fluctuations [MATH], and the kurtosis [MATH], given by the ratio of the fourth to the second coefficient.', '1407.8277-2-15-4': 'They are related to fluctuations in the quark number [MATH] as [EQUATION] with [MATH], the deviation from the ensemble average of the quark number distribution.', '1407.8277-2-15-5': 'Note that for [MATH] the volume and temperature dependence cancel to leading order.', '1407.8277-2-15-6': 'In [CITATION] it has been shown that event-by-event fluctuations like [MATH] diverge at a critical point.', '1407.8277-2-15-7': 'The non-monotonic behavior of fluctuations in heavy-ion collisions as a function of beam energy was proposed as an experimental signal.', '1407.8277-2-15-8': 'It was later shown in [CITATION] within the NJL model that susceptibilities also diverge along the spinodal lines of the first-order phase transition.', '1407.8277-2-15-9': 'This requires that nonequilibrium effects, i. e. spinodal instabilities, are taken into account.', '1407.8277-2-15-10': 'The presence of a mechanically instable region with [MATH] then consequently leads to diverging susceptibilities.', '1407.8277-2-15-11': 'For the sigma model with dilatons, we calculate both the quark number susceptibility and the kurtosis as a function of density for fixed temperature [MATH] MeV, where the model exhibits a first-order phase transition, see Fig. [REF].', '1407.8277-2-15-12': 'In the left plot, the susceptibility is shown as a function of the quark density.', '1407.8277-2-15-13': 'Similar to the result from the NJL model, we see divergences at the isothermal spinodal points.', '1407.8277-2-15-14': 'Inside the spinodal region, the susceptibility becomes negative due to instabilities.', '1407.8277-2-15-15': 'On the right hand side we show the kurtosis in the same density range.', '1407.8277-2-15-16': 'Here we also find strong divergences when crossing the spinodal lines.', '1407.8277-2-15-17': 'Interestingly, [MATH] remains positive even inside the coexistence region.', '1407.8277-2-15-18': 'On the other hand, it is known that the kurtosis becomes negative when approaching the critical end point from the crossover side [CITATION].', '1407.8277-2-16-0': 'Further insight into the nature of these fluctuations can be gained by determining the critical exponents which control the strength of the divergences at the CEP or spinodals, respectively.', '1407.8277-2-16-1': 'In the vicinity of the singularity, the behavior of susceptibility and kurtosis may be described by a power law of the form [EQUATION]', '1407.8277-2-16-2': 'We calculate the critical exponents [MATH] and [MATH] both analytically and numerically.', '1407.8277-2-16-3': 'The chiral transition may be described by a Ginzburg-Landau effective theory around the CEP and the spinodal lines [CITATION].', '1407.8277-2-16-4': 'At both points, the first and second derivatives of the effective potential vanish.', '1407.8277-2-16-5': 'Around a zero of the second derivative, it can be expanded in terms of [MATH] [EQUATION]', '1407.8277-2-16-6': 'At [MATH] we have [MATH], so for these coefficients the relation [MATH] and [MATH] holds for small values of [MATH].', '1407.8277-2-16-7': 'From [MATH] we obtain [MATH], and we can assume that [MATH] with [MATH].', '1407.8277-2-16-8': 'At the spinodal point, the leading term reads [MATH], giving [MATH] and consequently [MATH].', '1407.8277-2-16-9': 'For a CEP, we also have [MATH], so from [MATH] we end up with [MATH] and [MATH].', '1407.8277-2-16-10': 'For the fourth generalized susceptibility [MATH], we can immediately state the coefficients to be [MATH] for the CEP and [MATH] for the spinodal.', '1407.8277-2-16-11': 'As the kurtosis is proportional to the ratio of the fourth to the second derivative of the effective potential with respect to [MATH], we get [MATH] for both cases.', '1407.8277-2-16-12': 'The same values for [MATH] and [MATH] have been found within a numerical analysis fitting the forms in Eqs. ([REF]) to the numerically determined susceptibility and kurtosis.', '1407.8277-2-16-13': 'Fig. [REF] shows the analytical and the numerical results versus the reduced chemical potential [MATH] for illustration.', '1407.8277-2-17-0': 'The critical properties for a CEP and an isothermal spinodal point are different due to a change in the universality class, indicating different critical behavior and strength of divergences.', '1407.8277-2-17-1': 'The same result and exponents have been found for a chiral NJL model with finite current quark masses [CITATION].', '1407.8277-2-17-2': 'The critical exponents of the kurtosis are naturally in agreement for both types of transition and are found to be equal to [MATH].', '1407.8277-2-18-0': '# Nonequilibrium enhancement of fluctuation signals', '1407.8277-2-19-0': 'The results from the previous section indicate that in dynamical systems as they are created in heavy-ion collisions, fluctuation signals may not only be enhanced in the vicinity of a critical point, but also in the spinodal region.', '1407.8277-2-19-1': 'This might provide us with a more applicable means to investigate the QCD phase structure as the region around the CEP with enhanced susceptibility is small [CITATION] and subject to finite size and time effects that limit the growth of fluctuations.', '1407.8277-2-19-2': 'We may therefore expect larger fluctuations at a first-order phase transition than at a CEP.', '1407.8277-2-19-3': 'We test this assumption within the nonequilibrium chiral fluid dynamics model introduced in Sec. [REF].', '1407.8277-2-19-4': 'We initialize a spherical droplet of quark-gluon plasma by defining an initial temperature and quark chemical potential, with a Woods-Saxon distribution to ensure a smooth transition to the vacuum at the edges.', '1407.8277-2-19-5': 'Then the fields are initialized with their respective equilibrium distribution, assuming Gaussian fluctuations around the thermal expectation values [MATH] and [MATH], and finally we calculate the fluid dynamical quantities out of the values for [MATH], [MATH], [MATH] and [MATH].', '1407.8277-2-19-6': 'Fluctuations in the initial conditions have only minor influence on the evolution as these are quickly washed out by the damping and superposed by the stochastic noise.', '1407.8277-2-19-7': 'By choosing appropriate initial values for [MATH] and [MATH], we are able to observe the expansion and cooling through the crossover, critical and spinodal region.', '1407.8277-2-19-8': 'The total quark number is in each case fixed to [MATH].', '1407.8277-2-19-9': 'In Fig. [REF], we show event-averaged trajectories for the three scenarios in the [MATH]-[MATH]-plane.', '1407.8277-2-19-10': 'The values of the density and temperature in a single event are obtained from an averaging over a central volume.', '1407.8277-2-19-11': 'Each single cell follows of course its individual path through the phase diagram, and plotting all of them would yield a blob moving from higher to lower densities.', '1407.8277-2-19-12': 'The volume-averaged trajectories differ from event to event due to different noise configurations.', '1407.8277-2-19-13': 'The curves start on the right side proceeding to lower density on the left.', '1407.8277-2-19-14': 'Interestingly, we see that the first-order curve shows a slightly increasing temperature at intermediate densities between [MATH] to [MATH].', '1407.8277-2-19-15': 'This is a result of the reheating effect that occurs after the decomposition of a supercooled phase and is typical for a first-order phase transition.', '1407.8277-2-19-16': 'It has already been found in earlier works of nonequilibrium fluid dynamical models [CITATION].', '1407.8277-2-19-17': 'Note that this is a purely dynamical effect which causes the trajectory to even cross the CEP curve, where the temperature decreases monotonically.', '1407.8277-2-19-18': 'It also implies that for this curve there is some significant deviation from the equilibrium trajectory along the corresponding isentrope.', '1407.8277-2-19-19': 'This crossing was also observed for the earlier used PQM model [CITATION] if the curves were plotted in the [MATH]-[MATH]-plane and the initial conditions were chosen close enough to each other.', '1407.8277-2-19-20': 'In the present model the initial conditions for a CEP and first-order transiton are inevitably closer as [MATH] is comparatively low.', '1407.8277-2-19-21': 'For the first-order phase transition we also find bubbles created through spinodal decomposition, as has been reported in earlier fluid dynamical studies [CITATION].', '1407.8277-2-19-22': 'However, in the present model these high-density droplets are not stable but start to decay after traversing the spinodal region.', '1407.8277-2-19-23': 'This is a direct consequence of the now strictly positive pressure as has been pointed out already in the introduction.', '1407.8277-2-20-0': 'We show the evolution of event-by-event fluctuations in the net-baryon number [MATH] corresponding to these trajectories as a function of time in Fig. [REF].', '1407.8277-2-20-1': 'The baryon number is extracted directly from the fluid dynamical density.', '1407.8277-2-20-2': 'As a conserved quantity, it will fluctuate only slightly when including a freeze-out and hadronic interactions in the final state [CITATION].', '1407.8277-2-20-3': 'We use two different methods to determine [MATH]: First, within a fixed volume in the center of the collision, with an extension of [MATH] fm in [MATH]-direction and [MATH] fm each in [MATH]- and [MATH]-direction.', '1407.8277-2-20-4': 'Second, by limiting the region of acceptance via rapidity to [MATH] and transverse momentum density to [MATH] as it was performed in recent measurements at STAR [CITATION].', '1407.8277-2-20-5': 'Both methods yield qualitatively similar results.', '1407.8277-2-20-6': 'The variance of fluctuations is enhanced at a CEP in comparison with a crossover transition and even more, by a factor of [MATH] to [MATH], at a first-order phase transition.', '1407.8277-2-20-7': 'As the variance depends on the volume, we observe clear differences in the scales of the two plots.', '1407.8277-2-20-8': 'Furthermore, the volume strongly varies when applying a constant rapidity and momentum cut, therefore a more irregular structure in the time dependence can be found for that case.', '1407.8277-2-21-0': 'For the kurtosis in Fig. [REF], we also find that the crossover transition produces values close to zero, while a clear enhancement can be found at the CEP.', '1407.8277-2-21-1': 'Again, the largest fluctuations occur for a first-order phase transition.', '1407.8277-2-21-2': 'Remarkably, at CEP and first-order transition, both positive and negative values of [MATH] occur during the evolution, confirming the assumption of critical behavior in the spinodal region.', '1407.8277-2-21-3': 'Although the kurtosis is not dependent on the volume, we find its values to be an order of magnitude higher when using a rapidity and momentum cut than in the case of a fixed test volume.', '1407.8277-2-21-4': 'This can be explained considering the overall conservation of quark or baryon number in the system under consideration.', '1407.8277-2-21-5': 'In contrast to that, the baryon number is only on average conserved in a grand canonical ensemble which is used for effective model or lattice QCD calculations.', '1407.8277-2-21-6': 'As shown in [CITATION], this global conservation significantly effects ratios of cumulants, making them dependent on the fraction of measured to total baryons.', '1407.8277-2-22-0': 'At this point we should note that it is nontrivial to draw the connection between the time evolution of the variance and kurtosis and event-by-event fluctuations from experiment, which are supposed to be emitted over a hypersurface of constant energy density or temperature.', '1407.8277-2-22-1': 'The latter can only be measured after freeze-out and therefore a signal from the phase transition can only be extracted if the chemical freeze-out temperature is close to the temperature of hadronization.', '1407.8277-2-22-2': 'Otherwise, the fluctuations may have been washed out after passing the phase transition.', '1407.8277-2-22-3': 'Finally, one needs to relate the baryon number fluctuations to quantities that are actually measured, like fluctuations in the proton number [CITATION], and consider the evolution of these fluctuations in the hadronic phase [CITATION].', '1407.8277-2-23-0': '# Summary and Outlook', '1407.8277-2-24-0': 'We have investigated baryon number fluctuations within a chiral model with dilatons from two different approaches: First, through the calculation of susceptibilities, where we went beyond standard thermodynamics by including spinodal instabilities.', '1407.8277-2-24-1': 'We were able to show that both the quark number susceptibility and kurtosis diverge at the CEP and spinodal lines.', '1407.8277-2-24-2': 'The singularity in the susceptibility at the spinodals becomes suddenly stronger at the CEP, indicated by a larger critical exponent.', '1407.8277-2-24-3': 'The implications of such a behavior for experiment are strong enhancements of event-by-event fluctuations in the net-baryon number, which we investigated in a second dynamical approach.', '1407.8277-2-24-4': 'Propagating the chiral field and the dilaton explicitly on a locally thermalized background of quarks and gluons, we simulated the expansion of the hot and dense plasma created in a heavy-ion collision.', '1407.8277-2-24-5': 'We extracted the variance and kurtosis of the net-baryon number.', '1407.8277-2-24-6': 'Both are stronger at a CEP in comparison with a crossover scenario, and even more enhanced when the system evolves through the spinodal region of the first-order phase transition.', '1407.8277-2-25-0': 'In the future we are going to include hadronic degrees of freedom for a more realistic description of the chirally broken and confined phase.', '1407.8277-2-25-1': 'It is furthermore necessary to study particle distributions from a freeze-out or hadronic afterburner.', '1407.8277-2-25-2': 'This would also allow us to study the momentum anisotropy and the effect of the phase transition on flow.', '1407.8277-2-25-3': 'For the determination of susceptibilities, it would be interesting to include quantum or thermal fluctuations and study their effect on the critical properties near the CEP and first-order phase transition.'}

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['1407.8277-1-3-1', '1407.8277-2-3-1'], ['1407.8277-1-3-2', '1407.8277-2-3-2'], ['1407.8277-1-3-3', '1407.8277-2-3-3'], ['1407.8277-1-3-4', '1407.8277-2-3-4'], ['1407.8277-1-3-5', '1407.8277-2-3-5'], ['1407.8277-1-21-0', '1407.8277-2-21-0'], ['1407.8277-1-21-1', '1407.8277-2-21-1'], ['1407.8277-1-21-2', '1407.8277-2-21-2'], ['1407.8277-1-21-3', '1407.8277-2-21-3'], ['1407.8277-1-21-4', '1407.8277-2-21-4'], ['1407.8277-1-21-5', '1407.8277-2-21-5'], ['1407.8277-1-21-6', '1407.8277-2-21-6'], ['1407.8277-1-4-0', '1407.8277-2-4-0'], ['1407.8277-1-4-1', '1407.8277-2-4-1'], ['1407.8277-1-4-2', '1407.8277-2-4-2'], ['1407.8277-1-4-3', '1407.8277-2-4-3'], ['1407.8277-1-4-4', '1407.8277-2-4-4'], ['1407.8277-1-4-5', '1407.8277-2-4-5'], ['1407.8277-1-4-6', '1407.8277-2-4-6'], ['1407.8277-1-10-0', '1407.8277-2-10-0'], ['1407.8277-1-10-1', '1407.8277-2-10-1'], ['1407.8277-1-10-2', '1407.8277-2-10-2'], ['1407.8277-1-6-0', '1407.8277-2-6-0'], ['1407.8277-1-6-1', '1407.8277-2-6-1'], ['1407.8277-1-6-2', '1407.8277-2-6-2'], ['1407.8277-1-6-3', '1407.8277-2-6-3'], ['1407.8277-1-6-4', '1407.8277-2-6-4'], ['1407.8277-1-6-5', '1407.8277-2-6-5'], ['1407.8277-1-6-6', '1407.8277-2-6-6'], ['1407.8277-1-6-7', '1407.8277-2-6-7'], ['1407.8277-1-11-0', '1407.8277-2-11-0'], ['1407.8277-1-11-1', '1407.8277-2-11-1'], ['1407.8277-1-11-2', '1407.8277-2-11-2'], ['1407.8277-1-11-3', '1407.8277-2-11-3'], ['1407.8277-1-11-4', '1407.8277-2-11-4'], ['1407.8277-1-11-5', '1407.8277-2-11-5'], ['1407.8277-1-11-6', '1407.8277-2-11-6'], ['1407.8277-1-11-9', '1407.8277-2-11-16'], ['1407.8277-1-13-0', '1407.8277-2-13-0'], ['1407.8277-1-13-1', '1407.8277-2-13-1'], ['1407.8277-1-13-2', '1407.8277-2-13-2'], ['1407.8277-1-13-3', '1407.8277-2-13-3'], ['1407.8277-1-13-4', '1407.8277-2-13-4'], ['1407.8277-1-13-5', '1407.8277-2-13-5'], ['1407.8277-1-13-6', '1407.8277-2-13-6'], ['1407.8277-1-13-7', '1407.8277-2-13-7'], ['1407.8277-1-13-8', '1407.8277-2-13-8'], ['1407.8277-1-13-9', '1407.8277-2-13-9'], ['1407.8277-1-13-10', '1407.8277-2-13-10'], ['1407.8277-1-13-11', '1407.8277-2-13-11'], ['1407.8277-1-13-12', '1407.8277-2-13-12'], ['1407.8277-1-13-13', '1407.8277-2-13-13'], ['1407.8277-1-13-14', '1407.8277-2-13-14'], ['1407.8277-1-13-15', '1407.8277-2-13-15'], ['1407.8277-1-13-16', '1407.8277-2-13-16'], ['1407.8277-1-13-17', '1407.8277-2-13-17'], ['1407.8277-1-13-18', '1407.8277-2-13-18'], ['1407.8277-1-13-19', '1407.8277-2-13-19'], ['1407.8277-1-19-0', '1407.8277-2-19-0'], ['1407.8277-1-19-1', '1407.8277-2-19-1'], ['1407.8277-1-19-2', '1407.8277-2-19-2'], ['1407.8277-1-19-3', '1407.8277-2-19-3'], ['1407.8277-1-19-4', '1407.8277-2-19-4'], ['1407.8277-1-19-7', '1407.8277-2-19-8'], ['1407.8277-1-19-8', '1407.8277-2-19-9'], ['1407.8277-1-19-10', '1407.8277-2-19-13'], ['1407.8277-1-19-11', '1407.8277-2-19-14'], ['1407.8277-1-19-12', '1407.8277-2-19-15'], ['1407.8277-1-19-13', '1407.8277-2-19-16'], ['1407.8277-1-19-14', '1407.8277-2-19-17'], ['1407.8277-1-19-15', '1407.8277-2-19-18'], ['1407.8277-1-19-17', '1407.8277-2-19-22'], ['1407.8277-1-19-18', '1407.8277-2-19-23'], ['1407.8277-1-17-0', '1407.8277-2-17-0'], ['1407.8277-1-17-1', '1407.8277-2-17-1'], ['1407.8277-1-17-2', '1407.8277-2-17-2'], ['1407.8277-1-15-0', '1407.8277-2-15-0'], ['1407.8277-1-15-1', '1407.8277-2-15-1'], ['1407.8277-1-15-2', '1407.8277-2-15-2'], ['1407.8277-1-15-3', '1407.8277-2-15-3'], ['1407.8277-1-15-4', '1407.8277-2-15-4'], ['1407.8277-1-15-5', '1407.8277-2-15-5'], ['1407.8277-1-15-6', '1407.8277-2-15-6'], ['1407.8277-1-15-7', '1407.8277-2-15-7'], ['1407.8277-1-15-8', '1407.8277-2-15-8'], ['1407.8277-1-15-9', '1407.8277-2-15-9'], ['1407.8277-1-15-10', '1407.8277-2-15-10'], ['1407.8277-1-15-11', '1407.8277-2-15-11'], ['1407.8277-1-15-12', '1407.8277-2-15-12'], ['1407.8277-1-15-13', '1407.8277-2-15-13'], ['1407.8277-1-15-14', '1407.8277-2-15-14'], ['1407.8277-1-15-15', '1407.8277-2-15-15'], ['1407.8277-1-15-16', '1407.8277-2-15-16'], ['1407.8277-1-15-17', '1407.8277-2-15-17'], ['1407.8277-1-15-18', '1407.8277-2-15-18'], ['1407.8277-1-7-0', '1407.8277-2-7-0'], ['1407.8277-1-7-1', '1407.8277-2-7-1'], ['1407.8277-1-7-2', '1407.8277-2-7-2'], ['1407.8277-1-16-0', '1407.8277-2-16-0'], ['1407.8277-1-16-1', '1407.8277-2-16-1'], ['1407.8277-1-16-2', '1407.8277-2-16-2'], ['1407.8277-1-16-3', '1407.8277-2-16-3'], ['1407.8277-1-16-4', '1407.8277-2-16-4'], ['1407.8277-1-16-5', '1407.8277-2-16-5'], ['1407.8277-1-16-6', '1407.8277-2-16-6'], ['1407.8277-1-16-7', '1407.8277-2-16-7'], ['1407.8277-1-16-8', '1407.8277-2-16-8'], ['1407.8277-1-16-9', '1407.8277-2-16-9'], ['1407.8277-1-16-10', '1407.8277-2-16-10'], ['1407.8277-1-16-11', '1407.8277-2-16-11'], ['1407.8277-1-16-12', '1407.8277-2-16-12'], ['1407.8277-1-16-13', '1407.8277-2-16-13'], ['1407.8277-1-11-7', '1407.8277-2-11-9'], ['1407.8277-1-11-8', '1407.8277-2-11-15'], ['1407.8277-1-19-6', '1407.8277-2-19-7'], ['1407.8277-1-19-9', '1407.8277-2-19-10'], ['1407.8277-1-19-16', '1407.8277-2-19-21']]

[['1407.8277-1-1-0', '1407.8277-2-1-0'], ['1407.8277-1-23-0', '1407.8277-2-24-0'], ['1407.8277-1-23-1', '1407.8277-2-24-1'], ['1407.8277-1-23-2', '1407.8277-2-24-2'], ['1407.8277-1-23-3', '1407.8277-2-24-3'], ['1407.8277-1-23-4', '1407.8277-2-24-4'], ['1407.8277-1-23-5', '1407.8277-2-24-5'], ['1407.8277-1-23-6', '1407.8277-2-24-6'], ['1407.8277-1-9-0', '1407.8277-2-9-0'], ['1407.8277-1-9-1', '1407.8277-2-9-1'], ['1407.8277-1-9-2', '1407.8277-2-9-2'], ['1407.8277-1-9-3', '1407.8277-2-9-3'], ['1407.8277-1-9-4', '1407.8277-2-9-4'], ['1407.8277-1-24-0', '1407.8277-2-25-0'], ['1407.8277-1-24-1', '1407.8277-2-25-1'], ['1407.8277-1-24-2', '1407.8277-2-25-2'], ['1407.8277-1-24-3', '1407.8277-2-25-3'], ['1407.8277-1-5-0', '1407.8277-2-5-0'], ['1407.8277-1-5-1', '1407.8277-2-5-1'], ['1407.8277-1-5-2', '1407.8277-2-5-2'], ['1407.8277-1-5-3', '1407.8277-2-5-3'], ['1407.8277-1-5-4', '1407.8277-2-5-4'], ['1407.8277-1-5-5', '1407.8277-2-5-5'], ['1407.8277-1-5-6', '1407.8277-2-5-6'], ['1407.8277-1-5-7', '1407.8277-2-5-7'], ['1407.8277-1-5-8', '1407.8277-2-5-8'], ['1407.8277-1-5-9', '1407.8277-2-5-9'], ['1407.8277-1-5-10', '1407.8277-2-5-10'], ['1407.8277-1-5-11', '1407.8277-2-5-11'], ['1407.8277-1-5-12', '1407.8277-2-5-12'], ['1407.8277-1-20-0', '1407.8277-2-20-0'], ['1407.8277-1-20-1', '1407.8277-2-20-1'], ['1407.8277-1-20-2', '1407.8277-2-20-2'], ['1407.8277-1-20-3', '1407.8277-2-20-3'], ['1407.8277-1-20-4', '1407.8277-2-20-4'], ['1407.8277-1-20-5', '1407.8277-2-20-5'], ['1407.8277-1-20-6', '1407.8277-2-20-6'], ['1407.8277-1-20-7', '1407.8277-2-20-7'], ['1407.8277-1-20-8', '1407.8277-2-20-8'], ['1407.8277-1-12-0', '1407.8277-2-12-0'], ['1407.8277-1-12-1', '1407.8277-2-12-1'], ['1407.8277-1-12-2', '1407.8277-2-12-3'], ['1407.8277-1-0-0', '1407.8277-2-0-0'], ['1407.8277-1-0-1', '1407.8277-2-0-1'], ['1407.8277-1-0-2', '1407.8277-2-0-2'], ['1407.8277-1-0-3', '1407.8277-2-0-3'], ['1407.8277-1-0-4', '1407.8277-2-0-4'], ['1407.8277-1-3-0', '1407.8277-2-3-0'], ['1407.8277-1-3-1', '1407.8277-2-3-1'], ['1407.8277-1-3-2', '1407.8277-2-3-2'], ['1407.8277-1-3-3', '1407.8277-2-3-3'], ['1407.8277-1-3-4', '1407.8277-2-3-4'], ['1407.8277-1-3-5', '1407.8277-2-3-5'], ['1407.8277-1-21-0', '1407.8277-2-21-0'], ['1407.8277-1-21-1', '1407.8277-2-21-1'], ['1407.8277-1-21-2', '1407.8277-2-21-2'], ['1407.8277-1-21-3', '1407.8277-2-21-3'], ['1407.8277-1-21-4', '1407.8277-2-21-4'], ['1407.8277-1-21-5', '1407.8277-2-21-5'], ['1407.8277-1-21-6', '1407.8277-2-21-6'], ['1407.8277-1-4-0', '1407.8277-2-4-0'], ['1407.8277-1-4-1', '1407.8277-2-4-1'], ['1407.8277-1-4-2', '1407.8277-2-4-2'], ['1407.8277-1-4-3', '1407.8277-2-4-3'], ['1407.8277-1-4-4', '1407.8277-2-4-4'], ['1407.8277-1-4-5', '1407.8277-2-4-5'], ['1407.8277-1-4-6', '1407.8277-2-4-6'], ['1407.8277-1-10-0', '1407.8277-2-10-0'], ['1407.8277-1-10-1', '1407.8277-2-10-1'], ['1407.8277-1-10-2', '1407.8277-2-10-2'], ['1407.8277-1-6-0', '1407.8277-2-6-0'], ['1407.8277-1-6-1', '1407.8277-2-6-1'], ['1407.8277-1-6-2', '1407.8277-2-6-2'], ['1407.8277-1-6-3', '1407.8277-2-6-3'], ['1407.8277-1-6-4', '1407.8277-2-6-4'], ['1407.8277-1-6-5', '1407.8277-2-6-5'], ['1407.8277-1-6-6', '1407.8277-2-6-6'], ['1407.8277-1-6-7', '1407.8277-2-6-7'], ['1407.8277-1-11-0', '1407.8277-2-11-0'], ['1407.8277-1-11-1', '1407.8277-2-11-1'], ['1407.8277-1-11-2', '1407.8277-2-11-2'], ['1407.8277-1-11-3', '1407.8277-2-11-3'], ['1407.8277-1-11-4', '1407.8277-2-11-4'], ['1407.8277-1-11-5', '1407.8277-2-11-5'], ['1407.8277-1-11-6', '1407.8277-2-11-6'], ['1407.8277-1-11-9', '1407.8277-2-11-16'], ['1407.8277-1-13-0', '1407.8277-2-13-0'], ['1407.8277-1-13-1', '1407.8277-2-13-1'], ['1407.8277-1-13-2', '1407.8277-2-13-2'], ['1407.8277-1-13-3', '1407.8277-2-13-3'], ['1407.8277-1-13-4', '1407.8277-2-13-4'], ['1407.8277-1-13-5', '1407.8277-2-13-5'], ['1407.8277-1-13-6', '1407.8277-2-13-6'], ['1407.8277-1-13-7', '1407.8277-2-13-7'], ['1407.8277-1-13-8', '1407.8277-2-13-8'], ['1407.8277-1-13-9', '1407.8277-2-13-9'], ['1407.8277-1-13-10', '1407.8277-2-13-10'], ['1407.8277-1-13-11', '1407.8277-2-13-11'], ['1407.8277-1-13-12', '1407.8277-2-13-12'], ['1407.8277-1-13-13', '1407.8277-2-13-13'], ['1407.8277-1-13-14', '1407.8277-2-13-14'], ['1407.8277-1-13-15', '1407.8277-2-13-15'], ['1407.8277-1-13-16', '1407.8277-2-13-16'], ['1407.8277-1-13-17', '1407.8277-2-13-17'], ['1407.8277-1-13-18', '1407.8277-2-13-18'], ['1407.8277-1-13-19', '1407.8277-2-13-19'], ['1407.8277-1-19-0', '1407.8277-2-19-0'], ['1407.8277-1-19-1', '1407.8277-2-19-1'], ['1407.8277-1-19-2', '1407.8277-2-19-2'], ['1407.8277-1-19-3', '1407.8277-2-19-3'], ['1407.8277-1-19-4', '1407.8277-2-19-4'], ['1407.8277-1-19-7', '1407.8277-2-19-8'], ['1407.8277-1-19-8', '1407.8277-2-19-9'], ['1407.8277-1-19-10', '1407.8277-2-19-13'], ['1407.8277-1-19-11', '1407.8277-2-19-14'], ['1407.8277-1-19-12', '1407.8277-2-19-15'], ['1407.8277-1-19-13', '1407.8277-2-19-16'], ['1407.8277-1-19-14', '1407.8277-2-19-17'], ['1407.8277-1-19-15', '1407.8277-2-19-18'], ['1407.8277-1-19-17', '1407.8277-2-19-22'], ['1407.8277-1-19-18', '1407.8277-2-19-23'], ['1407.8277-1-17-0', '1407.8277-2-17-0'], ['1407.8277-1-17-1', '1407.8277-2-17-1'], ['1407.8277-1-17-2', '1407.8277-2-17-2'], ['1407.8277-1-15-0', '1407.8277-2-15-0'], ['1407.8277-1-15-1', '1407.8277-2-15-1'], ['1407.8277-1-15-2', '1407.8277-2-15-2'], ['1407.8277-1-15-3', '1407.8277-2-15-3'], ['1407.8277-1-15-4', '1407.8277-2-15-4'], ['1407.8277-1-15-5', '1407.8277-2-15-5'], ['1407.8277-1-15-6', '1407.8277-2-15-6'], ['1407.8277-1-15-7', '1407.8277-2-15-7'], ['1407.8277-1-15-8', '1407.8277-2-15-8'], ['1407.8277-1-15-9', '1407.8277-2-15-9'], ['1407.8277-1-15-10', '1407.8277-2-15-10'], ['1407.8277-1-15-11', '1407.8277-2-15-11'], ['1407.8277-1-15-12', '1407.8277-2-15-12'], ['1407.8277-1-15-13', '1407.8277-2-15-13'], ['1407.8277-1-15-14', '1407.8277-2-15-14'], ['1407.8277-1-15-15', '1407.8277-2-15-15'], ['1407.8277-1-15-16', '1407.8277-2-15-16'], ['1407.8277-1-15-17', '1407.8277-2-15-17'], ['1407.8277-1-15-18', '1407.8277-2-15-18'], ['1407.8277-1-7-0', '1407.8277-2-7-0'], ['1407.8277-1-7-1', '1407.8277-2-7-1'], ['1407.8277-1-7-2', '1407.8277-2-7-2'], ['1407.8277-1-16-0', '1407.8277-2-16-0'], ['1407.8277-1-16-1', '1407.8277-2-16-1'], ['1407.8277-1-16-2', '1407.8277-2-16-2'], ['1407.8277-1-16-3', '1407.8277-2-16-3'], ['1407.8277-1-16-4', '1407.8277-2-16-4'], ['1407.8277-1-16-5', '1407.8277-2-16-5'], ['1407.8277-1-16-6', '1407.8277-2-16-6'], ['1407.8277-1-16-7', '1407.8277-2-16-7'], ['1407.8277-1-16-8', '1407.8277-2-16-8'], ['1407.8277-1-16-9', '1407.8277-2-16-9'], ['1407.8277-1-16-10', '1407.8277-2-16-10'], ['1407.8277-1-16-11', '1407.8277-2-16-11'], ['1407.8277-1-16-12', '1407.8277-2-16-12'], ['1407.8277-1-16-13', '1407.8277-2-16-13']]

[['1407.8277-1-11-7', '1407.8277-2-11-9'], ['1407.8277-1-11-8', '1407.8277-2-11-15'], ['1407.8277-1-19-6', '1407.8277-2-19-7'], ['1407.8277-1-19-9', '1407.8277-2-19-10'], ['1407.8277-1-19-16', '1407.8277-2-19-21']]

[]

[]

[]

[]

{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1407.8277

0907.1563

{'0907.1563-1-0-0': '# Introduction', '0907.1563-1-1-0': 'Let [MATH] be the field of complex numbers.', '0907.1563-1-1-1': 'If [MATH] then we write [MATH] for its complex-conjugate and denote by [MATH] the corresponding element of the group [MATH] of automorphisms of [MATH].', '0907.1563-1-1-2': 'We write [MATH] for the algebraic closure of [MATH] in [MATH].', '0907.1563-1-1-3': 'It is well-known that the subfield [MATH] is [MATH]-stable and the natural homomorphism [EQUATION] is surjective.', '0907.1563-1-1-4': 'If [MATH] is a [MATH]-vector space, [MATH]-algebra or [MATH]-Lie algebra then we write [MATH] for the corresponding [MATH]-vector space (respectively, [MATH]-algebra or [MATH]-Lie algebra) [MATH].', '0907.1563-1-2-0': 'Let [MATH] be a polynomial of degree [MATH] without multiple roots.', '0907.1563-1-2-1': 'Suppose that [MATH] is a prime that does not divide [MATH] and a positive integer [MATH] is a power of [MATH].', '0907.1563-1-2-2': 'As usual, [MATH] denotes the Euler function.', '0907.1563-1-2-3': 'Let us fix a primitive [MATH]th root of unity [MATH].', '0907.1563-1-2-4': 'We write [MATH] for the superelliptic curve [MATH] and [MATH] for its jacobian.', '0907.1563-1-2-5': 'Clearly, [MATH] is an abelian variety and [EQUATION].', '0907.1563-1-2-6': 'The periodic automorphism [MATH] of [MATH] induces by Albanese functoriality the periodic automorphism of [MATH] that we denote by [MATH].', '0907.1563-1-2-7': 'It is known [CITATION] that [MATH] gives rise to an embedding of the product [MATH] of cyclotomic fields into the endomorphism algebra [MATH] of [MATH].', '0907.1563-1-2-8': '(If [MATH] then we actually get an embedding [MATH] that sends [MATH] to [MATH].)', '0907.1563-1-2-9': 'More precisely, if [MATH] then the map [MATH] defines the map of curves [MATH], which induces (by Albanese functoriality) the surjective homomorphism [MATH] of complex abelian varieties; we write [MATH] for the identity component of its kernel.', '0907.1563-1-2-10': '(If [MATH] then we put [MATH].)', '0907.1563-1-2-11': 'One may check [CITATION] that [MATH] is isogenous to the product [MATH] and [MATH] gives rise to an embedding [EQUATION].', '0907.1563-1-3-0': 'In a series of papers [CITATION], one of the authors (Y.Z.) was able to prove that [EQUATION] assuming that [MATH] and there exists a subfield [MATH] such that all the coefficients of [MATH] lie in [MATH] and the Galois group of [MATH] over [MATH] is either the full symmetric group [MATH] or the alternating group [MATH].', '0907.1563-1-3-1': 'In particular, [MATH].', '0907.1563-1-3-2': '(The same assertion holds true if [MATH], the prime [MATH] is odd, [MATH] and the Galois group is [MATH].)', '0907.1563-1-4-0': 'Our goal is to study the (reductive [MATH]-algebraic connected) Hodge group [MATH] of [MATH].', '0907.1563-1-4-1': 'Notice that when [MATH] (i.e., in the hyperelliptic case) this group was completely determined in [CITATION] (when [MATH] has "large" Galois group); in particular, in this case the Hodge group is simple and the center of its Lie algebra is [MATH].', '0907.1563-1-4-2': 'So, further we assume that [MATH] and therefore [MATH] is a CM-field.', '0907.1563-1-4-3': 'So, if [MATH] then (see Remark [REF] below) the center [MATH] of the [MATH]-Lie algebra [MATH] of [MATH] lies in [EQUATION] (If [MATH] then [MATH] and [MATH].)', '0907.1563-1-4-4': 'In particular, its dimension does not exceed [MATH]; the equality holds if and only if [MATH] and [MATH] coincides with [MATH].', '0907.1563-1-5-0': 'Let [EQUATION] be the maximal totally real subfield of [MATH].', '0907.1563-1-5-1': 'If [MATH] then [MATH].', '0907.1563-1-5-2': 'We write [MATH] and [MATH] for the algebraic [MATH]-tori obtained by the Weil restricton of scalars of the multiplicative group [MATH] to [MATH] from [MATH] and [MATH] respectively.', '0907.1563-1-5-3': 'The norm map [MATH] induces the natural homomorphism of algebraic [MATH]-tori and and we denote by [MATH] its kernel, i.e., the corresponding norm torus [CITATION].', '0907.1563-1-5-4': 'It is well known that [MATH] is an algebraic [MATH]-torus (in particuluar, it is connected) and [EQUATION].', '0907.1563-1-5-5': 'The embedding [MATH] allows us to identify [MATH] as a certain [MATH]-subalgebra of [MATH] and consider [MATH] and therefore [MATH] as certain [MATH]-algebraic subgroups of the general linear group [MATH] over [MATH].', '0907.1563-1-5-6': 'Then the [MATH]-Lie algebras of [MATH] and [MATH], viewed as [MATH]-Lie subalgebras of [MATH], coincide with [MATH] and [MATH] respectively.', '0907.1563-1-6-0': 'Let [MATH] be the derived group of [MATH].', '0907.1563-1-6-1': 'Let [MATH] be the center of [MATH] and [MATH] the identity component of [MATH].', '0907.1563-1-6-2': 'Since the Hodge group is connected reductive, [MATH] is a semisimple connected algebraic [MATH]-group, [MATH] an algebraic [MATH]-torus and the natural morphism of linear algebraic [MATH]-groups [MATH] is an isogeny.', '0907.1563-1-6-3': 'It follows that the [MATH]-Lie algebra [MATH] of [MATH] coincides with the [MATH]-Lie algebra [MATH] of [MATH] and equals [MATH].', '0907.1563-1-7-0': 'Assume that [MATH] and [MATH] does not divide [MATH].', '0907.1563-1-7-1': 'Let [MATH] be a degree [MATH] polynomial without multiple roots.', '0907.1563-1-7-2': 'If [MATH] then the center [MATH] of the [MATH]-Lie algebra [MATH] of [MATH] has [MATH]-dimension greater or equal than [MATH].', '0907.1563-1-7-3': 'In other words, the center of [MATH] has dimension greater or equal than [MATH].', '0907.1563-1-8-0': 'Assume that [MATH] and [MATH] does not divide [MATH].', '0907.1563-1-8-1': 'Let [MATH] be a degree [MATH] polynomial without multiple roots.', '0907.1563-1-8-2': 'Assume also that [MATH].', '0907.1563-1-9-0': 'Assume that [MATH] and [MATH] does not divide [MATH].', '0907.1563-1-9-1': 'Suppose that there exists a subfield [MATH] of [MATH] that contains all the coefficients of [MATH].', '0907.1563-1-9-2': 'Suppose that [MATH] is irreducible over [MATH] and the Galois group [MATH] of [MATH] over [MATH] is either [MATH] or [MATH].', '0907.1563-1-9-3': 'Assume additionally that either [MATH] or [MATH] and [MATH].', '0907.1563-1-10-0': 'Suppose that [MATH] enjoy the conditions of Theorem [REF].', '0907.1563-1-10-1': 'Assume additionally that [MATH] is odd.', '0907.1563-1-10-2': 'Since [MATH], we conclude that the center [MATH] of the [MATH]-Lie algebra [MATH] of [MATH] has [MATH]-dimension [MATH] and coincides with [MATH].', '0907.1563-1-10-3': 'In addition, the center of [MATH] coincides with [MATH].', '0907.1563-1-11-0': 'In order to describe our results for the whole [MATH] when [MATH], let us put [EQUATION].', '0907.1563-1-12-0': 'Assume that [MATH] and [MATH] does not divide [MATH].', '0907.1563-1-12-1': 'Suppose that there exists a subfield [MATH] of [MATH] that contains all the coefficients of [MATH].', '0907.1563-1-12-2': 'Suppose that [MATH] is irreducible over [MATH] and the Galois group [MATH] of [MATH] over [MATH] is either [MATH] or [MATH].', '0907.1563-1-12-3': 'Assume additionally that either [MATH] or [MATH] and [MATH].', '0907.1563-1-12-4': 'Let us consider the abelian variety [MATH] and its first rational homology group [MATH].', '0907.1563-1-12-5': 'If [MATH] then the center [MATH] of the [MATH]-Lie algebra [MATH] of the Hodge group [MATH] of [MATH] has [MATH]-dimension [MATH] and coincides with [EQUATION].', '0907.1563-1-13-0': 'Let us fix an isogeny [MATH].', '0907.1563-1-13-1': 'Then [MATH] induces an isomorphism of [MATH]-vector spaces [EQUATION].', '0907.1563-1-13-2': 'Clearly, the Hodge group of [MATH] coincides with [MATH].', '0907.1563-1-13-3': 'This implies that if [MATH] then the center of the [MATH]-Lie algebra of [MATH] has [MATH]-dimension [MATH] and coincides with [MATH].', '0907.1563-1-14-0': 'Suppose that [MATH] and [MATH].', '0907.1563-1-14-1': 'Let [MATH] be the [MATH]-algebraic subgroup of [MATH] that is cut out by the polarization and the endomorphisms (compare with [CITATION]).', '0907.1563-1-14-2': 'Theorem [REF] and Remark [REF] imply that (under their assumptions) the center [MATH] of the Lie algebra [MATH] of [MATH] is a proper subspace of [MATH].', '0907.1563-1-14-3': 'One may deduce from that that [MATH] is a proper subgroup of [MATH] and therefore certain self-products of [MATH] admit exotic Hodge classes that could not be presented as linear combinations of products of divisor classes.', '0907.1563-1-15-0': 'The paper is organized as follows.', '0907.1563-1-15-1': 'In Section [REF] we discuss auxiliary results related to CM-fields.', '0907.1563-1-15-2': 'Section [REF] treats complex abelian varieties with multiplication by CM-fields.', '0907.1563-1-15-3': 'Section [REF] contains the proof of main results modulo some arithmetic properties of certain (non-vanishing) Fourier coefficients with respect to the finite commutative group [MATH]; their properties are proved in Section [REF].'}

{'0907.1563-2-0-0': '# Introduction', '0907.1563-2-1-0': 'Let [MATH] be the field of complex numbers.', '0907.1563-2-1-1': 'If [MATH] then we write [MATH] for its complex-conjugate and denote by [MATH] the corresponding element of the group [MATH] of automorphisms of [MATH].', '0907.1563-2-1-2': 'We write [MATH] for the algebraic closure of [MATH] in [MATH].', '0907.1563-2-1-3': 'It is well-known that the subfield [MATH] is [MATH]-stable and the natural homomorphism [EQUATION] is surjective.', '0907.1563-2-1-4': 'If [MATH] is a [MATH]-vector space, [MATH]-algebra or [MATH]-Lie algebra then we write [MATH] for the corresponding [MATH]-vector space (respectively, [MATH]-algebra or [MATH]-Lie algebra) [MATH].', '0907.1563-2-2-0': 'Let [MATH] be a polynomial of degree [MATH] without multiple roots.', '0907.1563-2-2-1': 'Suppose that [MATH] is a prime that does not divide [MATH] and a positive integer [MATH] is a power of [MATH].', '0907.1563-2-2-2': 'As usual, [MATH] denotes the Euler function.', '0907.1563-2-2-3': 'Let us fix a primitive [MATH]th root of unity [MATH].', '0907.1563-2-2-4': 'We write [MATH] for the superelliptic curve [MATH] and [MATH] for its jacobian.', '0907.1563-2-2-5': 'Clearly, [MATH] is an abelian variety and [EQUATION].', '0907.1563-2-2-6': 'The periodic automorphism [MATH] of [MATH] induces by Albanese functoriality the periodic automorphism of [MATH] that we denote by [MATH].', '0907.1563-2-2-7': 'It is known [CITATION] that [MATH] gives rise to an embedding of the product [MATH] of cyclotomic fields into the endomorphism algebra [MATH] of [MATH].', '0907.1563-2-2-8': '(If [MATH] then we actually get an embedding [MATH] that sends [MATH] to [MATH].)', '0907.1563-2-2-9': 'More precisely, if [MATH] then the map [MATH] defines the map of curves [MATH], which induces (by Albanese functoriality) the surjective homomorphism [MATH] of complex abelian varieties; we write [MATH] for the identity component of its kernel.', '0907.1563-2-2-10': '(If [MATH] then we put [MATH].)', '0907.1563-2-2-11': 'One may check [CITATION] that [MATH] is isogenous to the product [MATH] and [MATH] gives rise to an embedding [EQUATION].', '0907.1563-2-3-0': 'In a series of papers [CITATION], one of the authors (Y.Z.) was able to prove that [EQUATION] assuming that [MATH] and there exists a subfield [MATH] such that all the coefficients of [MATH] lie in [MATH] and the Galois group of [MATH] over [MATH] is either the full symmetric group [MATH] or the alternating group [MATH].', '0907.1563-2-3-1': 'In particular, [MATH].', '0907.1563-2-3-2': '(The same assertion holds true if [MATH], the prime [MATH] is odd, [MATH] and the Galois group is [MATH].)', '0907.1563-2-4-0': 'Our goal is to study the (reductive [MATH]-algebraic connected) Hodge group [MATH] of [MATH].', '0907.1563-2-4-1': 'Notice that when [MATH] (i.e., in the hyperelliptic case) this group was completely determined in [CITATION] (when [MATH] has "large" Galois group); in particular, in this case the Hodge group is simple and the center of its Lie algebra is [MATH].', '0907.1563-2-4-2': 'So, further we assume that [MATH] and therefore [MATH] is a CM-field.', '0907.1563-2-4-3': 'So, if [MATH] then (see Remark [REF] below) the center [MATH] of the [MATH]-Lie algebra [MATH] of [MATH] lies in [EQUATION] (If [MATH] then [MATH] and [MATH].)', '0907.1563-2-4-4': 'In particular, its dimension does not exceed [MATH]; the equality holds if and only if [MATH] and [MATH] coincides with [MATH].', '0907.1563-2-5-0': 'Let [EQUATION] be the maximal totally real subfield of [MATH].', '0907.1563-2-5-1': 'If [MATH] then [MATH].', '0907.1563-2-5-2': 'We write [MATH] and [MATH] for the algebraic [MATH]-tori obtained by the Weil restriction of scalars of the multiplicative group [MATH] to [MATH] from [MATH] and [MATH] respectively.', '0907.1563-2-5-3': 'The norm map [MATH] induces the natural homomorphism of algebraic [MATH]-tori and we denote by [MATH] its kernel, i.e., the corresponding norm torus [CITATION].', '0907.1563-2-5-4': 'It is well known that [MATH] is an algebraic [MATH]-torus (in particular, it is connected) and [EQUATION].', '0907.1563-2-5-5': 'The embedding [MATH] allows us to identify [MATH] with a certain [MATH]-subalgebra of [MATH] and consider [MATH] and therefore [MATH] as certain [MATH]-algebraic subgroups of the general linear group [MATH] over [MATH].', '0907.1563-2-5-6': 'Then the [MATH]-Lie algebras of [MATH] and [MATH], viewed as [MATH]-Lie subalgebras of [MATH], coincide with [MATH] and [MATH] respectively.', '0907.1563-2-6-0': 'Recall that [MATH] is an abelian subvariety of the jacobian [MATH] and consider the ([MATH]-invariant) polarization [MATH] on [MATH] induced by the canonical principal polarization on [MATH].', '0907.1563-2-6-1': 'The polarization [MATH] gives rise to a certain nondegenerate alternating [MATH]-bilinear form [EQUATION] such that [MATH].', '0907.1563-2-6-2': '(This form is the imaginary part of the Riemann form of [MATH] [CITATION].)', '0907.1563-2-6-3': 'If [MATH] then we choose a nonzero element [MATH] and a standard construction (see, for instance, [CITATION]) gives us a nondegenerate Hermitian [MATH]-sesquilinear form [EQUATION] such that [MATH].', '0907.1563-2-6-4': 'We write [MATH] for the unitary group of [MATH],', '0907.1563-2-7-0': "viewed as an algebraic (reductive) [MATH]-subgroup of [MATH] (via Weil's restriction of scalars from [MATH] to [MATH] (ibid).", '0907.1563-2-7-1': 'Then the center of [MATH] coincides with [MATH].', '0907.1563-2-8-0': '(The group [MATH] does not depend on a choice of [MATH].)', '0907.1563-2-9-0': 'Since the Hodge group of [MATH] respects the polarization and commutes with endomorphisms of [MATH], [EQUATION].', '0907.1563-2-9-1': 'Recall that the centralizer of [MATH] in [MATH] coincides with [MATH].', '0907.1563-2-9-2': 'This implies that if [MATH] coincides with [MATH] then the center of [MATH] lies in [MATH].', '0907.1563-2-10-0': 'Let [MATH] be the derived subgroup of [MATH].', '0907.1563-2-10-1': 'Let [MATH] be the center of [MATH] and [MATH] the identity component of [MATH].', '0907.1563-2-10-2': 'Since the Hodge group is connected reductive, [MATH] is a semisimple connected algebraic [MATH]-group, [MATH] an algebraic [MATH]-torus and the natural morphism of linear algebraic [MATH]-groups [MATH] is an isogeny.', '0907.1563-2-10-3': 'It follows that the [MATH]-Lie algebra [MATH] of [MATH] coincides with the [MATH]-Lie algebra [MATH] of [MATH] and equals [MATH].', '0907.1563-2-11-0': 'Assume that [MATH] and [MATH] does not divide [MATH].', '0907.1563-2-11-1': 'Let [MATH] be a degree [MATH] polynomial without multiple roots.', '0907.1563-2-11-2': 'If [MATH] then the center [MATH] of the [MATH]-Lie algebra [MATH] of [MATH] has [MATH]-dimension greater or equal than [MATH].', '0907.1563-2-11-3': 'In other words, the center of [MATH] has dimension greater or equal than [MATH].', '0907.1563-2-12-0': 'Suppose that [MATH] enjoy the conditions of Theorem [REF].', '0907.1563-2-12-1': 'Assume additionally that [MATH] is odd.', '0907.1563-2-12-2': 'Since [MATH], we conclude that the center of [MATH] coincides with [MATH].', '0907.1563-2-13-0': 'In Theorem [REF] we prove that the center of the Hodge group of [MATH] is "as large as possible", taking into account that the endomorphism algebra of [MATH] coincides with [MATH].', '0907.1563-2-13-1': 'In fact, our goal was (and still is) to prove that (under the assumptions of Theorem [REF]) the whole Hodge group is "as large as possible", i.e., coincides with [MATH],', '0907.1563-2-14-0': 'which would imply that all Hodge classes on each self-product of [MATH] can be presented as linear combinations of products of divisor classes and, in particular, the validity of the Hodge conjecture for all the self-products [CITATION].', '0907.1563-2-14-1': 'Since the Hodge group is connected reductive, the problem splits naturally in two parts: to prove that the center of [MATH] is "as large as possible" (i.e., coincides with [MATH]) and that the derived subgroup (semisimple part) of [MATH] is "as large as possible" (i.e., coincides with the corresponding special unitary group).', '0907.1563-2-14-2': 'Theorem [REF] settles the first one.', '0907.1563-2-14-3': '(The second problem is solved in [CITATION] under certain additional conditions on [MATH] and [MATH].)', '0907.1563-2-15-0': 'In order to describe our results for the whole [MATH] when [MATH], let us put [EQUATION].', '0907.1563-2-16-0': 'Assume that [MATH] and [MATH] does not divide [MATH].', '0907.1563-2-16-1': 'Suppose that there exists a subfield [MATH] of [MATH] that contains all the coefficients of [MATH].', '0907.1563-2-16-2': 'Suppose that [MATH] is irreducible over [MATH] and the Galois group [MATH] of [MATH] over [MATH] is either [MATH] or [MATH].', '0907.1563-2-16-3': 'Assume additionally that either [MATH] or [MATH] and [MATH].', '0907.1563-2-16-4': 'Let us consider the abelian variety [MATH] and its first rational homology group [MATH].', '0907.1563-2-16-5': 'If [MATH] then the center [MATH] of the [MATH]-Lie algebra [MATH] of the Hodge group [MATH] of [MATH] has [MATH]-dimension [MATH] and coincides with [EQUATION].', '0907.1563-2-17-0': 'Let us fix an isogeny [MATH].', '0907.1563-2-17-1': 'Then [MATH] induces an isomorphism of [MATH]-vector spaces [MATH].', '0907.1563-2-17-2': 'Clearly, the Hodge group of [MATH] coincides with [MATH].', '0907.1563-2-17-3': 'This implies that if [MATH] then the center of the [MATH]-Lie algebra of [MATH] has [MATH]-dimension [MATH] and coincides with [MATH].', '0907.1563-2-18-0': 'We keep the notation and assumptions of Theorem [REF] and Remark [REF].', '0907.1563-2-18-1': 'Let us identify (via [MATH] with [MATH].', '0907.1563-2-19-0': 'Since the Hodge group of [MATH] respects the polarization and commutes with endomorphisms of [MATH], [EQUATION].', '0907.1563-2-19-1': 'Let [MATH] be the reductive [MATH]-algebraic subgroup of [MATH] that is cut out by the polarization and the endomorphisms of [MATH] [CITATION]).', '0907.1563-2-20-0': 'Now assume that [MATH] is odd.', '0907.1563-2-20-1': 'Taking into account that all [MATH] are (mutually nonisomorphic) CM-fields [MATH] and using results from p. 531 of [CITATION], one may easily check that [MATH].', '0907.1563-2-20-2': 'It follows that the center of the [MATH]-Lie algebra of [MATH] coincides with [MATH].', '0907.1563-2-20-3': 'On the other hand, Theorem [REF] and Remark [REF] imply that (under their assumptions) the center [MATH] of the [MATH]-Lie algebra of [MATH] is the proper subspace [MATH] of [MATH].', '0907.1563-2-20-4': 'It follows that [MATH] and therefore [MATH] is a proper subgroup of [MATH].', '0907.1563-2-20-5': 'This implies that a certain self-product of [MATH] admits an exotic Hodge class that could not be presented as a linear combination of products of divisor classes.', '0907.1563-2-20-6': 'The same assertion holds true if [MATH] and [MATH].', '0907.1563-2-21-0': 'Assume that [MATH] and [MATH] does not divide [MATH] Let [MATH] be a degree [MATH] polynomial without multiple roots.', '0907.1563-2-21-1': 'Assume also that [MATH].', '0907.1563-2-22-0': 'Actually, our proof gives a little bit more, namely, that the center [MATH] of [MATH] is a CM-field such that [MATH] is greater or equal than the lower bound given in Theorem [REF].', '0907.1563-2-22-1': '(Notice that [MATH] is a direct summand of the center of [MATH].)', '0907.1563-2-23-0': '[Corollary to Theorem [REF]]', '0907.1563-2-24-0': 'Suppose that [MATH] and a prime [MATH] does not divide [MATH].', '0907.1563-2-24-1': 'Let [MATH] be a degree [MATH] polynomial without multiple roots.', '0907.1563-2-24-2': 'If [MATH] then [MATH] is not isogenous to a product of elliptic curves.', '0907.1563-2-25-0': 'Corollary [REF] asserts that if [MATH] then none of jacobians of [MATH] is totally split in a sense of [CITATION].', '0907.1563-2-26-0': 'Recently D. Ulmer [CITATION], using a construction of L. Berger [CITATION], found out that the rank of the Mordell-Weil group of the jacobian of the curve [MATH] over the function field [MATH] is closely related to the endomorphism algebras of [MATH] (for [MATH]).', '0907.1563-2-26-1': 'One may hope that our results could be useful for the study of the rank of abelian varieties in infinite towers of function fields.', '0907.1563-2-27-0': 'The paper is organized as follows.', '0907.1563-2-27-1': 'In Section [REF] we discuss auxiliary results related to CM-fields.', '0907.1563-2-27-2': 'Section [REF] treats complex abelian varieties with multiplication by CM-fields.', '0907.1563-2-27-3': 'Section [REF] contains the proof of main results modulo some arithmetic properties of certain (non-vanishing) Fourier coefficients with respect to the finite commutative group [MATH]; those properties are proved in Sections [REF] and [REF].', '0907.1563-2-27-4': 'Last section contains an auxiliary result from semilinear algebra.'}

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'0907.1563-3-19-1'], ['0907.1563-2-20-2', '0907.1563-3-19-2'], ['0907.1563-2-20-3', '0907.1563-3-19-3'], ['0907.1563-2-20-4', '0907.1563-3-19-4'], ['0907.1563-2-20-6', '0907.1563-3-19-6'], ['0907.1563-2-22-0', '0907.1563-3-21-0'], ['0907.1563-2-22-1', '0907.1563-3-21-1'], ['0907.1563-2-19-0', '0907.1563-3-18-0'], ['0907.1563-2-19-1', '0907.1563-3-18-1'], ['0907.1563-2-21-1', '0907.1563-3-20-2'], ['0907.1563-2-5-0', '0907.1563-3-5-0'], ['0907.1563-2-5-2', '0907.1563-3-5-2'], ['0907.1563-2-5-3', '0907.1563-3-5-3'], ['0907.1563-2-5-4', '0907.1563-3-5-4'], ['0907.1563-2-5-5', '0907.1563-3-5-5'], ['0907.1563-2-5-6', '0907.1563-3-5-6'], ['0907.1563-2-3-0', '0907.1563-3-3-0'], ['0907.1563-2-3-2', '0907.1563-3-3-2'], ['0907.1563-2-27-0', '0907.1563-3-28-0'], ['0907.1563-2-27-1', '0907.1563-3-28-1'], ['0907.1563-2-27-2', '0907.1563-3-28-2'], ['0907.1563-2-27-3', '0907.1563-3-28-3'], ['0907.1563-2-27-4', '0907.1563-3-28-4'], ['0907.1563-2-4-0', '0907.1563-3-4-0'], ['0907.1563-2-4-1', '0907.1563-3-4-1'], ['0907.1563-2-4-2', '0907.1563-3-4-2'], ['0907.1563-2-4-3', '0907.1563-3-4-3'], ['0907.1563-2-4-4', '0907.1563-3-4-4'], ['0907.1563-2-1-0', '0907.1563-3-1-0'], ['0907.1563-2-1-1', '0907.1563-3-1-1'], ['0907.1563-2-1-2', '0907.1563-3-1-2'], ['0907.1563-2-1-3', '0907.1563-3-1-3'], ['0907.1563-2-1-4', '0907.1563-3-1-4'], ['0907.1563-2-18-0', '0907.1563-3-17-0'], ['0907.1563-2-18-1', '0907.1563-3-17-1'], ['0907.1563-2-7-0', '0907.1563-3-7-0'], ['0907.1563-2-7-1', '0907.1563-3-7-1'], ['0907.1563-2-13-0', '0907.1563-3-12-0'], ['0907.1563-2-14-0', '0907.1563-3-13-0'], ['0907.1563-2-14-1', '0907.1563-3-13-1'], ['0907.1563-2-14-2', '0907.1563-3-13-2'], ['0907.1563-2-14-3', '0907.1563-3-13-3'], ['0907.1563-2-15-0', '0907.1563-3-14-0'], ['0907.1563-2-16-0', '0907.1563-3-15-0'], ['0907.1563-2-16-1', '0907.1563-3-15-1'], ['0907.1563-2-16-2', '0907.1563-3-15-2'], ['0907.1563-2-16-3', '0907.1563-3-15-3'], ['0907.1563-2-16-4', '0907.1563-3-15-4'], ['0907.1563-2-16-5', '0907.1563-3-15-5'], ['0907.1563-2-17-0', '0907.1563-3-16-0'], ['0907.1563-2-17-1', '0907.1563-3-16-1'], ['0907.1563-2-17-2', '0907.1563-3-16-2'], ['0907.1563-2-17-3', '0907.1563-3-16-3'], ['0907.1563-2-6-0', '0907.1563-3-6-0'], ['0907.1563-2-6-3', '0907.1563-3-6-3'], ['0907.1563-2-12-0', '0907.1563-3-11-0'], ['0907.1563-2-12-1', '0907.1563-3-11-1'], ['0907.1563-2-12-2', '0907.1563-3-11-2'], ['0907.1563-2-11-0', '0907.1563-3-10-0'], ['0907.1563-2-11-1', '0907.1563-3-10-1'], ['0907.1563-2-11-2', '0907.1563-3-10-2'], ['0907.1563-2-11-3', '0907.1563-3-10-3'], ['0907.1563-2-2-0', '0907.1563-3-2-0'], ['0907.1563-2-2-1', '0907.1563-3-2-1'], ['0907.1563-2-2-2', '0907.1563-3-2-2'], ['0907.1563-2-2-3', '0907.1563-3-2-3'], ['0907.1563-2-2-4', '0907.1563-3-2-4'], ['0907.1563-2-2-5', '0907.1563-3-2-5'], ['0907.1563-2-2-6', '0907.1563-3-2-6'], ['0907.1563-2-2-7', '0907.1563-3-2-7'], ['0907.1563-2-2-8', '0907.1563-3-2-8'], ['0907.1563-2-2-9', '0907.1563-3-2-9'], ['0907.1563-2-2-11', '0907.1563-3-2-11'], ['0907.1563-2-9-0', '0907.1563-3-8-0'], ['0907.1563-2-9-1', '0907.1563-3-8-1'], ['0907.1563-2-9-2', '0907.1563-3-8-2'], ['0907.1563-2-10-0', '0907.1563-3-9-0'], ['0907.1563-2-10-1', '0907.1563-3-9-1'], ['0907.1563-2-10-2', '0907.1563-3-9-2'], ['0907.1563-2-10-3', '0907.1563-3-9-3'], ['0907.1563-2-26-0', '0907.1563-3-27-0'], ['0907.1563-2-26-1', '0907.1563-3-27-1'], ['0907.1563-3-3-0', '0907.1563-4-3-0'], ['0907.1563-3-3-2', '0907.1563-4-3-2'], ['0907.1563-3-15-0', '0907.1563-4-15-0'], ['0907.1563-3-15-2', '0907.1563-4-15-2'], ['0907.1563-3-15-3', '0907.1563-4-15-3'], ['0907.1563-3-15-4', '0907.1563-4-15-4'], ['0907.1563-3-15-5', '0907.1563-4-15-5'], ['0907.1563-3-25-1', '0907.1563-4-25-1'], ['0907.1563-3-25-2', '0907.1563-4-25-2'], ['0907.1563-3-12-0', '0907.1563-4-12-0'], ['0907.1563-3-18-0', '0907.1563-4-18-0'], ['0907.1563-3-18-1', '0907.1563-4-18-1'], ['0907.1563-3-1-0', '0907.1563-4-1-0'], ['0907.1563-3-1-1', '0907.1563-4-1-1'], ['0907.1563-3-1-2', '0907.1563-4-1-2'], ['0907.1563-3-1-3', '0907.1563-4-1-3'], ['0907.1563-3-1-4', '0907.1563-4-1-4'], ['0907.1563-3-17-0', '0907.1563-4-17-0'], ['0907.1563-3-17-1', '0907.1563-4-17-1'], ['0907.1563-3-7-0', '0907.1563-4-7-0'], ['0907.1563-3-7-1', '0907.1563-4-7-1'], ['0907.1563-3-4-0', '0907.1563-4-4-0'], ['0907.1563-3-4-1', '0907.1563-4-4-1'], ['0907.1563-3-4-2', '0907.1563-4-4-2'], ['0907.1563-3-4-3', '0907.1563-4-4-3'], ['0907.1563-3-4-4', '0907.1563-4-4-4'], ['0907.1563-3-20-0', '0907.1563-4-20-0'], ['0907.1563-3-20-1', '0907.1563-4-20-1'], ['0907.1563-3-20-2', '0907.1563-4-20-2'], ['0907.1563-3-13-0', '0907.1563-4-13-0'], ['0907.1563-3-13-1', '0907.1563-4-13-1'], ['0907.1563-3-13-2', '0907.1563-4-13-2'], ['0907.1563-3-13-3', '0907.1563-4-13-3'], ['0907.1563-3-16-0', '0907.1563-4-16-0'], ['0907.1563-3-16-1', '0907.1563-4-16-1'], ['0907.1563-3-16-2', '0907.1563-4-16-2'], ['0907.1563-3-16-3', '0907.1563-4-16-3'], ['0907.1563-3-26-0', '0907.1563-4-26-0'], ['0907.1563-3-8-0', '0907.1563-4-8-0'], ['0907.1563-3-8-1', '0907.1563-4-8-1'], ['0907.1563-3-8-2', '0907.1563-4-8-2'], ['0907.1563-3-2-0', '0907.1563-4-2-0'], ['0907.1563-3-2-1', '0907.1563-4-2-1'], ['0907.1563-3-2-2', '0907.1563-4-2-2'], ['0907.1563-3-2-3', '0907.1563-4-2-3'], ['0907.1563-3-2-4', '0907.1563-4-2-4'], ['0907.1563-3-2-5', '0907.1563-4-2-5'], ['0907.1563-3-2-6', '0907.1563-4-2-6'], ['0907.1563-3-2-7', '0907.1563-4-2-7'], ['0907.1563-3-2-8', '0907.1563-4-2-8'], ['0907.1563-3-2-9', '0907.1563-4-2-9'], ['0907.1563-3-2-11', '0907.1563-4-2-11'], ['0907.1563-3-14-0', '0907.1563-4-14-0'], ['0907.1563-3-9-0', '0907.1563-4-9-0'], ['0907.1563-3-9-1', '0907.1563-4-9-1'], ['0907.1563-3-9-2', '0907.1563-4-9-2'], ['0907.1563-3-9-3', '0907.1563-4-9-3'], ['0907.1563-3-23-1', '0907.1563-4-23-1'], ['0907.1563-3-23-3', '0907.1563-4-23-3'], ['0907.1563-3-24-0', '0907.1563-4-24-0'], ['0907.1563-3-27-0', '0907.1563-4-27-0'], ['0907.1563-3-27-1', '0907.1563-4-27-1'], ['0907.1563-3-5-0', '0907.1563-4-5-0'], ['0907.1563-3-5-2', '0907.1563-4-5-2'], ['0907.1563-3-5-3', '0907.1563-4-5-3'], ['0907.1563-3-5-4', '0907.1563-4-5-4'], ['0907.1563-3-5-5', '0907.1563-4-5-5'], ['0907.1563-3-5-6', '0907.1563-4-5-6'], ['0907.1563-3-10-0', '0907.1563-4-10-0'], ['0907.1563-3-10-1', '0907.1563-4-10-1'], ['0907.1563-3-10-2', '0907.1563-4-10-2'], ['0907.1563-3-10-3', '0907.1563-4-10-3'], ['0907.1563-3-19-0', '0907.1563-4-19-0'], ['0907.1563-3-19-1', '0907.1563-4-19-1'], ['0907.1563-3-19-2', '0907.1563-4-19-2'], ['0907.1563-3-19-3', '0907.1563-4-19-3'], ['0907.1563-3-19-4', '0907.1563-4-19-4'], ['0907.1563-3-19-5', '0907.1563-4-19-5'], ['0907.1563-3-19-6', '0907.1563-4-19-6'], ['0907.1563-3-21-0', '0907.1563-4-21-0'], ['0907.1563-3-21-1', '0907.1563-4-21-1'], ['0907.1563-3-6-0', '0907.1563-4-6-0'], ['0907.1563-3-6-1', '0907.1563-4-6-1'], ['0907.1563-3-6-2', '0907.1563-4-6-2'], ['0907.1563-3-6-3', '0907.1563-4-6-3'], ['0907.1563-3-11-0', '0907.1563-4-11-0'], ['0907.1563-3-11-1', '0907.1563-4-11-1'], ['0907.1563-3-11-2', '0907.1563-4-11-2'], ['0907.1563-3-28-0', '0907.1563-4-28-0'], ['0907.1563-3-28-1', '0907.1563-4-28-1'], ['0907.1563-3-28-2', '0907.1563-4-28-2'], ['0907.1563-3-28-3', '0907.1563-4-28-3'], ['0907.1563-3-28-4', '0907.1563-4-28-4'], ['0907.1563-2-24-1', '0907.1563-3-23-1']]

[['0907.1563-1-6-0', '0907.1563-2-10-0'], ['0907.1563-1-10-2', '0907.1563-2-12-2'], ['0907.1563-1-15-3', '0907.1563-2-27-3'], ['0907.1563-1-13-1', '0907.1563-2-17-1'], ['0907.1563-1-5-2', '0907.1563-2-5-2'], ['0907.1563-1-5-3', '0907.1563-2-5-3'], ['0907.1563-1-5-4', '0907.1563-2-5-4'], ['0907.1563-1-5-5', '0907.1563-2-5-5'], ['0907.1563-1-14-2', '0907.1563-2-20-3'], ['0907.1563-2-20-5', '0907.1563-3-19-5'], ['0907.1563-2-25-0', '0907.1563-3-26-0'], ['0907.1563-3-25-0', '0907.1563-4-25-0'], ['0907.1563-3-26-1', '0907.1563-4-26-1'], ['0907.1563-3-23-2', '0907.1563-4-23-2'], ['0907.1563-2-24-0', '0907.1563-3-23-0'], ['0907.1563-2-24-2', '0907.1563-3-24-0']]

[]

[['0907.1563-1-14-3', '0907.1563-2-20-5'], ['0907.1563-2-21-0', '0907.1563-3-20-0'], ['0907.1563-2-21-0', '0907.1563-3-20-1'], ['0907.1563-2-6-1', '0907.1563-3-6-1'], ['0907.1563-2-6-2', '0907.1563-3-6-1'], ['0907.1563-3-15-1', '0907.1563-4-15-1'], ['0907.1563-3-23-0', '0907.1563-4-23-0']]

[['0907.1563-1-8-1', '0907.1563-2-24-1'], ['0907.1563-1-8-2', '0907.1563-2-21-1']]

['0907.1563-1-2-10', '0907.1563-1-3-1', '0907.1563-1-5-1', '0907.1563-2-2-10', '0907.1563-2-3-1', '0907.1563-2-5-1', '0907.1563-2-6-4', '0907.1563-2-13-1', '0907.1563-2-23-0', '0907.1563-3-2-10', '0907.1563-3-3-1', '0907.1563-3-5-1', '0907.1563-3-6-4', '0907.1563-3-12-1', '0907.1563-3-22-0', '0907.1563-4-2-10', '0907.1563-4-3-1', '0907.1563-4-5-1', '0907.1563-4-6-4', '0907.1563-4-12-1', '0907.1563-4-22-0']

{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '3': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '4': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/0907.1563

{'0907.1563-3-0-0': '# Introduction', '0907.1563-3-1-0': 'Let [MATH] be the field of complex numbers.', '0907.1563-3-1-1': 'If [MATH] then we write [MATH] for its complex-conjugate and denote by [MATH] the corresponding element of the group [MATH] of automorphisms of [MATH].', '0907.1563-3-1-2': 'We write [MATH] for the algebraic closure of [MATH] in [MATH].', '0907.1563-3-1-3': 'It is well-known that the subfield [MATH] is [MATH]-stable and the natural homomorphism [EQUATION] is surjective.', '0907.1563-3-1-4': 'If [MATH] is a [MATH]-vector space, [MATH]-algebra or [MATH]-Lie algebra then we write [MATH] for the corresponding [MATH]-vector space (respectively, [MATH]-algebra or [MATH]-Lie algebra) [MATH].', '0907.1563-3-2-0': 'Let [MATH] be a polynomial of degree [MATH] without multiple roots.', '0907.1563-3-2-1': 'Suppose that [MATH] is a prime that does not divide [MATH] and a positive integer [MATH] is a power of [MATH].', '0907.1563-3-2-2': 'As usual, [MATH] denotes the Euler function.', '0907.1563-3-2-3': 'Let us fix a primitive [MATH]th root of unity [MATH].', '0907.1563-3-2-4': 'We write [MATH] for the superelliptic curve [MATH] and [MATH] for its jacobian.', '0907.1563-3-2-5': 'Clearly, [MATH] is an abelian variety and [EQUATION].', '0907.1563-3-2-6': 'The periodic automorphism [MATH] of [MATH] induces by Albanese functoriality the periodic automorphism of [MATH] that we denote by [MATH].', '0907.1563-3-2-7': 'It is known [CITATION] that [MATH] gives rise to an embedding of the product [MATH] of cyclotomic fields into the endomorphism algebra [MATH] of [MATH].', '0907.1563-3-2-8': '(If [MATH] then we actually get an embedding [MATH] that sends [MATH] to [MATH].)', '0907.1563-3-2-9': 'More precisely, if [MATH] then the map [MATH] defines the map of curves [MATH], which induces (by Albanese functoriality) the surjective homomorphism [MATH] of complex abelian varieties; we write [MATH] for the identity component of its kernel.', '0907.1563-3-2-10': '(If [MATH] then we put [MATH].)', '0907.1563-3-2-11': 'One may check [CITATION] that [MATH] is isogenous to the product [MATH] and [MATH] gives rise to an embedding [EQUATION].', '0907.1563-3-3-0': 'In a series of papers [CITATION], one of the authors (Y.Z.) was able to prove that [EQUATION] assuming that [MATH] and there exists a subfield [MATH] such that all the coefficients of [MATH] lie in [MATH] and the Galois group of [MATH] over [MATH] is either the full symmetric group [MATH] or the alternating group [MATH].', '0907.1563-3-3-1': 'In particular, [MATH].', '0907.1563-3-3-2': '(The same assertion holds true if [MATH], the prime [MATH] is odd, [MATH] and the Galois group is [MATH].)', '0907.1563-3-4-0': 'Our goal is to study the (reductive [MATH]-algebraic connected) Hodge group [MATH] of [MATH].', '0907.1563-3-4-1': 'Notice that when [MATH] (i.e., in the hyperelliptic case) this group was completely determined in [CITATION] (when [MATH] has "large" Galois group); in particular, in this case the Hodge group is simple and the center of its Lie algebra is [MATH].', '0907.1563-3-4-2': 'So, further we assume that [MATH] and therefore [MATH] is a CM-field.', '0907.1563-3-4-3': 'So, if [MATH] then (see Remark [REF] below) the center [MATH] of the [MATH]-Lie algebra [MATH] of [MATH] lies in [EQUATION] (If [MATH] then [MATH] and [MATH].)', '0907.1563-3-4-4': 'In particular, its dimension does not exceed [MATH]; the equality holds if and only if [MATH] and [MATH] coincides with [MATH].', '0907.1563-3-5-0': 'Let [EQUATION] be the maximal totally real subfield of [MATH].', '0907.1563-3-5-1': 'If [MATH] then [MATH].', '0907.1563-3-5-2': 'We write [MATH] and [MATH] for the algebraic [MATH]-tori obtained by the Weil restriction of scalars of the multiplicative group [MATH] to [MATH] from [MATH] and [MATH] respectively.', '0907.1563-3-5-3': 'The norm map [MATH] induces the natural homomorphism of algebraic [MATH]-tori and we denote by [MATH] its kernel, i.e., the corresponding norm torus [CITATION].', '0907.1563-3-5-4': 'It is well known that [MATH] is an algebraic [MATH]-torus (in particular, it is connected) and [EQUATION].', '0907.1563-3-5-5': 'The embedding [MATH] allows us to identify [MATH] with a certain [MATH]-subalgebra of [MATH] and consider [MATH] and therefore [MATH] as certain [MATH]-algebraic subgroups of the general linear group [MATH] over [MATH].', '0907.1563-3-5-6': 'Then the [MATH]-Lie algebras of [MATH] and [MATH], viewed as [MATH]-Lie subalgebras of [MATH], coincide with [MATH] and [MATH] respectively.', '0907.1563-3-6-0': 'Recall that [MATH] is an abelian subvariety of the jacobian [MATH] and consider the ([MATH]-invariant) polarization [MATH] on [MATH] induced by the canonical principal polarization on [MATH].', '0907.1563-3-6-1': 'The polarization [MATH] gives rise to a certain [MATH]-invariant nondegenerate alternating [MATH]-bilinear form [EQUATION] (This form is the imaginary part of the Riemann form of [MATH] [CITATION].)', '0907.1563-3-6-2': 'The [MATH]-invariance implies that [MATH].', '0907.1563-3-6-3': 'If [MATH] then we choose a nonzero element [MATH] and a standard construction (see, for instance, [CITATION]) gives us a nondegenerate Hermitian [MATH]-sesquilinear form [EQUATION] such that [MATH].', '0907.1563-3-6-4': 'We write [MATH] for the unitary group of [MATH],', '0907.1563-3-7-0': "viewed as an algebraic (reductive) [MATH]-subgroup of [MATH] (via Weil's restriction of scalars from [MATH] to [MATH] (ibid).", '0907.1563-3-7-1': 'Then the center of [MATH] coincides with [MATH].', '0907.1563-3-8-0': 'Since the Hodge group of [MATH] respects the polarization and commutes with endomorphisms of [MATH], [EQUATION].', '0907.1563-3-8-1': 'Recall that the centralizer of [MATH] in [MATH] coincides with [MATH].', '0907.1563-3-8-2': 'This implies that if [MATH] coincides with [MATH] then the center of [MATH] lies in [MATH].', '0907.1563-3-9-0': 'Let [MATH] be the derived subgroup of [MATH].', '0907.1563-3-9-1': 'Let [MATH] be the center of [MATH] and [MATH] the identity component of [MATH].', '0907.1563-3-9-2': 'Since the Hodge group is connected reductive, [MATH] is a semisimple connected algebraic [MATH]-group, [MATH] an algebraic [MATH]-torus and the natural morphism of linear algebraic [MATH]-groups [MATH] is an isogeny.', '0907.1563-3-9-3': 'It follows that the [MATH]-Lie algebra [MATH] of [MATH] coincides with the [MATH]-Lie algebra [MATH] of [MATH] and equals [MATH].', '0907.1563-3-10-0': 'Assume that [MATH] and [MATH] does not divide [MATH].', '0907.1563-3-10-1': 'Let [MATH] be a degree [MATH] polynomial without multiple roots.', '0907.1563-3-10-2': 'If [MATH] then the center [MATH] of the [MATH]-Lie algebra [MATH] of [MATH] has [MATH]-dimension greater or equal than [MATH].', '0907.1563-3-10-3': 'In other words, the center of [MATH] has dimension greater or equal than [MATH].', '0907.1563-3-11-0': 'Suppose that [MATH] enjoy the conditions of Theorem [REF].', '0907.1563-3-11-1': 'Assume additionally that [MATH] is odd.', '0907.1563-3-11-2': 'Since [MATH], we conclude that the center of [MATH] coincides with [MATH].', '0907.1563-3-12-0': 'In Theorem [REF] we prove that the center of the Hodge group of [MATH] is "as large as possible", taking into account that the endomorphism algebra of [MATH] coincides with [MATH].', '0907.1563-3-12-1': 'In fact, our goal was (and still is) to prove that (under the assumptions of Theorem [REF]) the whole Hodge group is "as large as possible", i.e., coincides with [MATH],', '0907.1563-3-13-0': 'which would imply that all Hodge classes on each self-product of [MATH] can be presented as linear combinations of products of divisor classes and, in particular, the validity of the Hodge conjecture for all the self-products [CITATION].', '0907.1563-3-13-1': 'Since the Hodge group is connected reductive, the problem splits naturally in two parts: to prove that the center of [MATH] is "as large as possible" (i.e., coincides with [MATH]) and that the derived subgroup (semisimple part) of [MATH] is "as large as possible" (i.e., coincides with the corresponding special unitary group).', '0907.1563-3-13-2': 'Theorem [REF] settles the first one.', '0907.1563-3-13-3': '(The second problem is solved in [CITATION] under certain additional conditions on [MATH] and [MATH].)', '0907.1563-3-14-0': 'In order to describe our results for the whole [MATH] when [MATH], let us put [EQUATION].', '0907.1563-3-15-0': 'Assume that [MATH] and [MATH] does not divide [MATH].', '0907.1563-3-15-1': 'Suppose that there exists a subfield [MATH] of [MATH] that contains all the coefficients of [MATH].', '0907.1563-3-15-2': 'Suppose that [MATH] is irreducible over [MATH] and the Galois group [MATH] of [MATH] over [MATH] is either [MATH] or [MATH].', '0907.1563-3-15-3': 'Assume additionally that either [MATH] or [MATH] and [MATH].', '0907.1563-3-15-4': 'Let us consider the abelian variety [MATH] and its first rational homology group [MATH].', '0907.1563-3-15-5': 'If [MATH] then the center [MATH] of the [MATH]-Lie algebra [MATH] of the Hodge group [MATH] of [MATH] has [MATH]-dimension [MATH] and coincides with [EQUATION].', '0907.1563-3-16-0': 'Let us fix an isogeny [MATH].', '0907.1563-3-16-1': 'Then [MATH] induces an isomorphism of [MATH]-vector spaces [MATH].', '0907.1563-3-16-2': 'Clearly, the Hodge group of [MATH] coincides with [MATH].', '0907.1563-3-16-3': 'This implies that if [MATH] then the center of the [MATH]-Lie algebra of [MATH] has [MATH]-dimension [MATH] and coincides with [MATH].', '0907.1563-3-17-0': 'We keep the notation and assumptions of Theorem [REF] and Remark [REF].', '0907.1563-3-17-1': 'Let us identify (via [MATH] with [MATH].', '0907.1563-3-18-0': 'Since the Hodge group of [MATH] respects the polarization and commutes with endomorphisms of [MATH], [EQUATION].', '0907.1563-3-18-1': 'Let [MATH] be the reductive [MATH]-algebraic subgroup of [MATH] that is cut out by the polarization and the endomorphisms of [MATH] [CITATION]).', '0907.1563-3-19-0': 'Now assume that [MATH] is odd.', '0907.1563-3-19-1': 'Taking into account that all [MATH] are (mutually nonisomorphic) CM-fields [MATH] and using results from p. 531 of [CITATION], one may easily check that [MATH].', '0907.1563-3-19-2': 'It follows that the center of the [MATH]-Lie algebra of [MATH] coincides with [MATH].', '0907.1563-3-19-3': 'On the other hand, Theorem [REF] and Remark [REF] imply that (under their assumptions) the center [MATH] of the [MATH]-Lie algebra of [MATH] is the proper subspace [MATH] of [MATH].', '0907.1563-3-19-4': 'It follows that [MATH] and therefore [MATH] is a proper subgroup of [MATH].', '0907.1563-3-19-5': 'This implies that a certain self-product of [MATH] admits an exotic Hodge class that could not be presented as a linear combinations of products of divisor classes.', '0907.1563-3-19-6': 'The same assertion holds true if [MATH] and [MATH].', '0907.1563-3-20-0': 'Assume that [MATH] and [MATH] does not divide [MATH].', '0907.1563-3-20-1': 'Let [MATH] be a degree [MATH] polynomial without multiple roots.', '0907.1563-3-20-2': 'Assume also that [MATH].', '0907.1563-3-21-0': 'Actually, our proof gives a little bit more, namely, that the center [MATH] of [MATH] is a CM-field such that [MATH] is greater or equal than the lower bound given in Theorem [REF].', '0907.1563-3-21-1': '(Notice that [MATH] is a direct summand of the center of [MATH].)', '0907.1563-3-22-0': '[Corollary to Theorem [REF]]', '0907.1563-3-23-0': 'Suppose that [MATH] and a positive integer [MATH] does not divide [MATH].', '0907.1563-3-23-1': 'Let [MATH] be a degree [MATH] polynomial without multiple roots.', '0907.1563-3-23-2': 'Assume that [MATH] and [MATH] is neither [MATH] nor [MATH].', '0907.1563-3-23-3': 'Let us consider the superelliptic curve [MATH] and let [MATH] be its jacobian.', '0907.1563-3-24-0': 'Then [MATH] is not isogenous to a product of elliptic curves.', '0907.1563-3-25-0': 'Clearly, [MATH] has a divisor [MATH] such that either [MATH] is a prime [MATH] or [MATH] or [MATH] is a power of [MATH].', '0907.1563-3-25-1': 'The existence of the covering of algebraic curves [EQUATION] implies that [MATH] has a quotient isomorphic to [MATH].', '0907.1563-3-25-2': 'Now the result follows from Theorem [REF] if we take into account that [MATH] is an abelian subvariety of [MATH].', '0907.1563-3-26-0': 'Corollary [REF] implies that if [MATH] then none of jacobians of [MATH] is totally split in a sense of [CITATION].', '0907.1563-3-26-1': 'The same is true for jacobians of [MATH] and [MATH].', '0907.1563-3-27-0': 'Recently D. Ulmer [CITATION], using a construction of L. Berger [CITATION], found out that the rank of the Mordell-Weil group of the jacobian of the curve [MATH] over the function field [MATH] is closely related to the endomorphism algebras of [MATH] (for [MATH]).', '0907.1563-3-27-1': 'One may hope that our results could be useful for the study of the rank of abelian varieties in infinite towers of function fields.', '0907.1563-3-28-0': 'The paper is organized as follows.', '0907.1563-3-28-1': 'In Section [REF] we discuss auxiliary results related to CM-fields.', '0907.1563-3-28-2': 'Section [REF] treats complex abelian varieties with multiplication by CM-fields.', '0907.1563-3-28-3': 'Section [REF] contains the proof of main results modulo some arithmetic properties of certain (non-vanishing) Fourier coefficients with respect to the finite commutative group [MATH]; those properties are proved in Sections [REF] and [REF].', '0907.1563-3-28-4': 'Last section contains an auxiliary result from semilinear algebra.'}

{'0907.1563-4-0-0': '# Introduction', '0907.1563-4-1-0': 'Let [MATH] be the field of complex numbers.', '0907.1563-4-1-1': 'If [MATH] then we write [MATH] for its complex-conjugate and denote by [MATH] the corresponding element of the group [MATH] of automorphisms of [MATH].', '0907.1563-4-1-2': 'We write [MATH] for the algebraic closure of [MATH] in [MATH].', '0907.1563-4-1-3': 'It is well-known that the subfield [MATH] is [MATH]-stable and the natural homomorphism [EQUATION] is surjective.', '0907.1563-4-1-4': 'If [MATH] is a [MATH]-vector space, [MATH]-algebra or [MATH]-Lie algebra then we write [MATH] for the corresponding [MATH]-vector space (respectively, [MATH]-algebra or [MATH]-Lie algebra) [MATH].', '0907.1563-4-2-0': 'Let [MATH] be a polynomial of degree [MATH] without multiple roots.', '0907.1563-4-2-1': 'Suppose that [MATH] is a prime that does not divide [MATH] and a positive integer [MATH] is a power of [MATH].', '0907.1563-4-2-2': 'As usual, [MATH] denotes the Euler function.', '0907.1563-4-2-3': 'Let us fix a primitive [MATH]th root of unity [MATH].', '0907.1563-4-2-4': 'We write [MATH] for the superelliptic curve [MATH] and [MATH] for its jacobian.', '0907.1563-4-2-5': 'Clearly, [MATH] is an abelian variety and [EQUATION].', '0907.1563-4-2-6': 'The periodic automorphism [MATH] of [MATH] induces by Albanese functoriality the periodic automorphism of [MATH] that we denote by [MATH].', '0907.1563-4-2-7': 'It is known [CITATION] that [MATH] gives rise to an embedding of the product [MATH] of cyclotomic fields into the endomorphism algebra [MATH] of [MATH].', '0907.1563-4-2-8': '(If [MATH] then we actually get an embedding [MATH] that sends [MATH] to [MATH].)', '0907.1563-4-2-9': 'More precisely, if [MATH] then the map [MATH] defines the map of curves [MATH], which induces (by Albanese functoriality) the surjective homomorphism [MATH] of complex abelian varieties; we write [MATH] for the identity component of its kernel.', '0907.1563-4-2-10': '(If [MATH] then we put [MATH].)', '0907.1563-4-2-11': 'One may check [CITATION] that [MATH] is isogenous to the product [MATH] and [MATH] gives rise to an embedding [EQUATION].', '0907.1563-4-3-0': 'In a series of papers [CITATION], one of the authors (Y.Z.) was able to prove that [EQUATION] assuming that [MATH] and there exists a subfield [MATH] such that all the coefficients of [MATH] lie in [MATH] and the Galois group of [MATH] over [MATH] is either the full symmetric group [MATH] or the alternating group [MATH].', '0907.1563-4-3-1': 'In particular, [MATH].', '0907.1563-4-3-2': '(The same assertion holds true if [MATH], the prime [MATH] is odd, [MATH] and the Galois group is [MATH].)', '0907.1563-4-4-0': 'Our goal is to study the (reductive [MATH]-algebraic connected) Hodge group [MATH] of [MATH].', '0907.1563-4-4-1': 'Notice that when [MATH] (i.e., in the hyperelliptic case) this group was completely determined in [CITATION] (when [MATH] has "large" Galois group); in particular, in this case the Hodge group is simple and the center of its Lie algebra is [MATH].', '0907.1563-4-4-2': 'So, further we assume that [MATH] and therefore [MATH] is a CM-field.', '0907.1563-4-4-3': 'So, if [MATH] then (see Remark [REF] below) the center [MATH] of the [MATH]-Lie algebra [MATH] of [MATH] lies in [EQUATION] (If [MATH] then [MATH] and [MATH].)', '0907.1563-4-4-4': 'In particular, its dimension does not exceed [MATH]; the equality holds if and only if [MATH] and [MATH] coincides with [MATH].', '0907.1563-4-5-0': 'Let [EQUATION] be the maximal totally real subfield of [MATH].', '0907.1563-4-5-1': 'If [MATH] then [MATH].', '0907.1563-4-5-2': 'We write [MATH] and [MATH] for the algebraic [MATH]-tori obtained by the Weil restriction of scalars of the multiplicative group [MATH] to [MATH] from [MATH] and [MATH] respectively.', '0907.1563-4-5-3': 'The norm map [MATH] induces the natural homomorphism of algebraic [MATH]-tori and we denote by [MATH] its kernel, i.e., the corresponding norm torus [CITATION].', '0907.1563-4-5-4': 'It is well known that [MATH] is an algebraic [MATH]-torus (in particular, it is connected) and [EQUATION].', '0907.1563-4-5-5': 'The embedding [MATH] allows us to identify [MATH] with a certain [MATH]-subalgebra of [MATH] and consider [MATH] and therefore [MATH] as certain [MATH]-algebraic subgroups of the general linear group [MATH] over [MATH].', '0907.1563-4-5-6': 'Then the [MATH]-Lie algebras of [MATH] and [MATH], viewed as [MATH]-Lie subalgebras of [MATH], coincide with [MATH] and [MATH] respectively.', '0907.1563-4-6-0': 'Recall that [MATH] is an abelian subvariety of the jacobian [MATH] and consider the ([MATH]-invariant) polarization [MATH] on [MATH] induced by the canonical principal polarization on [MATH].', '0907.1563-4-6-1': 'The polarization [MATH] gives rise to a certain [MATH]-invariant nondegenerate alternating [MATH]-bilinear form [EQUATION] (This form is the imaginary part of the Riemann form of [MATH] [CITATION].)', '0907.1563-4-6-2': 'The [MATH]-invariance implies that [MATH].', '0907.1563-4-6-3': 'If [MATH] then we choose a nonzero element [MATH] and a standard construction (see, for instance, [CITATION]) gives us a nondegenerate Hermitian [MATH]-sesquilinear form [EQUATION] such that [MATH].', '0907.1563-4-6-4': 'We write [MATH] for the unitary group of [MATH],', '0907.1563-4-7-0': "viewed as an algebraic (reductive) [MATH]-subgroup of [MATH] (via Weil's restriction of scalars from [MATH] to [MATH] (ibid).", '0907.1563-4-7-1': 'Then the center of [MATH] coincides with [MATH].', '0907.1563-4-8-0': 'Since the Hodge group of [MATH] respects the polarization and commutes with endomorphisms of [MATH], [EQUATION].', '0907.1563-4-8-1': 'Recall that the centralizer of [MATH] in [MATH] coincides with [MATH].', '0907.1563-4-8-2': 'This implies that if [MATH] coincides with [MATH] then the center of [MATH] lies in [MATH].', '0907.1563-4-9-0': 'Let [MATH] be the derived subgroup of [MATH].', '0907.1563-4-9-1': 'Let [MATH] be the center of [MATH] and [MATH] the identity component of [MATH].', '0907.1563-4-9-2': 'Since the Hodge group is connected reductive, [MATH] is a semisimple connected algebraic [MATH]-group, [MATH] an algebraic [MATH]-torus and the natural morphism of linear algebraic [MATH]-groups [MATH] is an isogeny.', '0907.1563-4-9-3': 'It follows that the [MATH]-Lie algebra [MATH] of [MATH] coincides with the [MATH]-Lie algebra [MATH] of [MATH] and equals [MATH].', '0907.1563-4-10-0': 'Assume that [MATH] and [MATH] does not divide [MATH].', '0907.1563-4-10-1': 'Let [MATH] be a degree [MATH] polynomial without multiple roots.', '0907.1563-4-10-2': 'If [MATH] then the center [MATH] of the [MATH]-Lie algebra [MATH] of [MATH] has [MATH]-dimension greater or equal than [MATH].', '0907.1563-4-10-3': 'In other words, the center of [MATH] has dimension greater or equal than [MATH].', '0907.1563-4-11-0': 'Suppose that [MATH] enjoy the conditions of Theorem [REF].', '0907.1563-4-11-1': 'Assume additionally that [MATH] is odd.', '0907.1563-4-11-2': 'Since [MATH], we conclude that the center of [MATH] coincides with [MATH].', '0907.1563-4-12-0': 'In Theorem [REF] we prove that the center of the Hodge group of [MATH] is "as large as possible", taking into account that the endomorphism algebra of [MATH] coincides with [MATH].', '0907.1563-4-12-1': 'In fact, our goal was (and still is) to prove that (under the assumptions of Theorem [REF]) the whole Hodge group is "as large as possible", i.e., coincides with [MATH],', '0907.1563-4-13-0': 'which would imply that all Hodge classes on each self-product of [MATH] can be presented as linear combinations of products of divisor classes and, in particular, the validity of the Hodge conjecture for all the self-products [CITATION].', '0907.1563-4-13-1': 'Since the Hodge group is connected reductive, the problem splits naturally in two parts: to prove that the center of [MATH] is "as large as possible" (i.e., coincides with [MATH]) and that the derived subgroup (semisimple part) of [MATH] is "as large as possible" (i.e., coincides with the corresponding special unitary group).', '0907.1563-4-13-2': 'Theorem [REF] settles the first one.', '0907.1563-4-13-3': '(The second problem is solved in [CITATION] under certain additional conditions on [MATH] and [MATH].)', '0907.1563-4-14-0': 'In order to describe our results for the whole [MATH] when [MATH], let us put [EQUATION].', '0907.1563-4-15-0': 'Assume that [MATH] and [MATH] does not divide [MATH].', '0907.1563-4-15-1': 'Let [MATH] be a subfield of [MATH] that contains all the coefficients of [MATH].', '0907.1563-4-15-2': 'Suppose that [MATH] is irreducible over [MATH] and the Galois group [MATH] of [MATH] over [MATH] is either [MATH] or [MATH].', '0907.1563-4-15-3': 'Assume additionally that either [MATH] or [MATH] and [MATH].', '0907.1563-4-15-4': 'Let us consider the abelian variety [MATH] and its first rational homology group [MATH].', '0907.1563-4-15-5': 'If [MATH] then the center [MATH] of the [MATH]-Lie algebra [MATH] of the Hodge group [MATH] of [MATH] has [MATH]-dimension [MATH] and coincides with [EQUATION].', '0907.1563-4-16-0': 'Let us fix an isogeny [MATH].', '0907.1563-4-16-1': 'Then [MATH] induces an isomorphism of [MATH]-vector spaces [MATH].', '0907.1563-4-16-2': 'Clearly, the Hodge group of [MATH] coincides with [MATH].', '0907.1563-4-16-3': 'This implies that if [MATH] then the center of the [MATH]-Lie algebra of [MATH] has [MATH]-dimension [MATH] and coincides with [MATH].', '0907.1563-4-17-0': 'We keep the notation and assumptions of Theorem [REF] and Remark [REF].', '0907.1563-4-17-1': 'Let us identify (via [MATH] with [MATH].', '0907.1563-4-18-0': 'Since the Hodge group of [MATH] respects the polarization and commutes with endomorphisms of [MATH], [EQUATION].', '0907.1563-4-18-1': 'Let [MATH] be the reductive [MATH]-algebraic subgroup of [MATH] that is cut out by the polarization and the endomorphisms of [MATH] [CITATION]).', '0907.1563-4-19-0': 'Now assume that [MATH] is odd.', '0907.1563-4-19-1': 'Taking into account that all [MATH] are (mutually nonisomorphic) CM-fields [MATH] and using results from p. 531 of [CITATION], one may easily check that [MATH].', '0907.1563-4-19-2': 'It follows that the center of the [MATH]-Lie algebra of [MATH] coincides with [MATH].', '0907.1563-4-19-3': 'On the other hand, Theorem [REF] and Remark [REF] imply that (under their assumptions) the center [MATH] of the [MATH]-Lie algebra of [MATH] is the proper subspace [MATH] of [MATH].', '0907.1563-4-19-4': 'It follows that [MATH] and therefore [MATH] is a proper subgroup of [MATH].', '0907.1563-4-19-5': 'This implies that a certain self-product of [MATH] admits an exotic Hodge class that could not be presented as a linear combinations of products of divisor classes.', '0907.1563-4-19-6': 'The same assertion holds true if [MATH] and [MATH].', '0907.1563-4-20-0': 'Assume that [MATH] and [MATH] does not divide [MATH].', '0907.1563-4-20-1': 'Let [MATH] be a degree [MATH] polynomial without multiple roots.', '0907.1563-4-20-2': 'Assume also that [MATH].', '0907.1563-4-21-0': 'Actually, our proof gives a little bit more, namely, that the center [MATH] of [MATH] is a CM-field such that [MATH] is greater or equal than the lower bound given in Theorem [REF].', '0907.1563-4-21-1': '(Notice that [MATH] is a direct summand of the center of [MATH].)', '0907.1563-4-22-0': '[Corollary to Theorem [REF]]', '0907.1563-4-23-0': 'Suppose that [MATH] and [MATH] is a positive integer such that [MATH].', '0907.1563-4-23-1': 'Let [MATH] be a degree [MATH] polynomial without multiple roots.', '0907.1563-4-23-2': 'Assume that [MATH] and [MATH] is neither [MATH] nor [MATH] nor [MATH] nor [MATH].', '0907.1563-4-23-3': 'Let us consider the superelliptic curve [MATH] and let [MATH] be its jacobian.', '0907.1563-4-24-0': 'Then [MATH] is not isogenous to a product of elliptic curves.', '0907.1563-4-25-0': 'Clearly, [MATH] has a divisor [MATH] such that either [MATH] is a prime [MATH] or [MATH] or [MATH].', '0907.1563-4-25-1': 'The existence of the covering of algebraic curves [EQUATION] implies that [MATH] has a quotient isomorphic to [MATH].', '0907.1563-4-25-2': 'Now the result follows from Theorem [REF] if we take into account that [MATH] is an abelian subvariety of [MATH].', '0907.1563-4-26-0': 'Corollary [REF] implies that if [MATH] then none of jacobians of [MATH] is totally split in a sense of [CITATION].', '0907.1563-4-26-1': 'The same is true for the jacobians of [MATH] and [MATH].', '0907.1563-4-27-0': 'Recently D. Ulmer [CITATION], using a construction of L. Berger [CITATION], found out that the rank of the Mordell-Weil group of the jacobian of the curve [MATH] over the function field [MATH] is closely related to the endomorphism algebras of [MATH] (for [MATH]).', '0907.1563-4-27-1': 'One may hope that our results could be useful for the study of the rank of abelian varieties in infinite towers of function fields.', '0907.1563-4-28-0': 'The paper is organized as follows.', '0907.1563-4-28-1': 'In Section [REF] we discuss auxiliary results related to CM-fields.', '0907.1563-4-28-2': 'Section [REF] treats complex abelian varieties with multiplication by CM-fields.', '0907.1563-4-28-3': 'Section [REF] contains the proof of main results modulo some arithmetic properties of certain (non-vanishing) Fourier coefficients with respect to the finite commutative group [MATH]; those properties are proved in Sections [REF] and [REF].', '0907.1563-4-28-4': 'Last section contains an auxiliary result from semilinear algebra.'}

astro-ph-0508234

{'astro-ph-0508234-1-0-0': 'Using observations, we investigate the scaling and structural properties of the ICM entropy in a sample of 10 nearby ([MATH]) relaxed galaxy clusters in the temperature range 2-9 keV.', 'astro-ph-0508234-1-0-1': 'We derive the local entropy-temperature () relation at [MATH] and [MATH].', 'astro-ph-0508234-1-0-2': 'The logarithmic slope of the relation is the same within the [MATH] error at all scaled radii.', 'astro-ph-0508234-1-0-3': 'However, the intrinsic dispersion about the best fitting relation is significantly higher at [MATH].', 'astro-ph-0508234-1-0-4': 'The slope is [MATH] at [MATH], in excellent agreement with previous work.', 'astro-ph-0508234-1-0-5': 'We also investigate the entropy-mass relation at density contrasts [MATH] and 1000.', 'astro-ph-0508234-1-0-6': 'We find a shallower slope than that expected in simple self-similar models, which is in agreement with the observed empirically-determined entropy-temperature and mass-temperature scaling.', 'astro-ph-0508234-1-0-7': 'The dispersion is smaller than for the [MATH]-[MATH] relation.', 'astro-ph-0508234-1-0-8': 'Once scaled appropriately, the entropy profiles appear similar beyond [MATH], with an intrinsic dispersion of [MATH] per cent and a shape consistent with gravitational heating ([MATH]).', 'astro-ph-0508234-1-0-9': 'However, the scatter in scaled entropy profiles increases with smaller scaled radius, to more than [MATH] per cent at [MATH].', 'astro-ph-0508234-1-0-10': 'Our results are in qualitative agreement with models which boost entropy production at the accretion shock.', 'astro-ph-0508234-1-0-11': 'However, localised entropy modification may be needed to explain the dispersion in the inner regions.', 'astro-ph-0508234-1-1-0': 'An update on galaxy cluster entropy scaling', 'astro-ph-0508234-1-2-0': '# Introduction', 'astro-ph-0508234-1-3-0': 'X-ray observations of the hot, gaseous intracluster medium (ICM) have been telling us for well over a decade that physical processes other than gravity are acting to modify the properties of the cluster population.', 'astro-ph-0508234-1-3-1': 'An understanding of the source(s) of this modification is of great importance for our understanding of cluster formation and evolution, and is essential for the use of clusters as precision cosmological probes.', 'astro-ph-0508234-1-4-0': 'Entropy is important because, together with the shape of the potential well, it is the quantity which dictates the observed X-ray properties of the ICM in galaxy clusters .', 'astro-ph-0508234-1-4-1': 'The intracluster entropy is generated in shocks as gas enters the cluster , thus it reflects the accretion history of the ICM.', 'astro-ph-0508234-1-4-2': 'However, the entropy distribution also preserves key information regarding the influence of non-gravitational processes.', 'astro-ph-0508234-1-5-0': 'X-ray measurements of the entropy at [MATH] (hereafter [MATH]) showed that the entropy of the coolest systems is considerably higher than that available from gravitational collapse , and that the entropy-temperature () relation is shallower than expected .', 'astro-ph-0508234-1-5-1': 'More recent spatially resolved entropy profiles indicate that the entropy is higher throughout the ICM, and that, outside the core regions, entropy profiles are structurally similar .', 'astro-ph-0508234-1-5-2': 'At the same time the scatter in [MATH] at a given temperature can be up to a factor of three .', 'astro-ph-0508234-1-6-0': 'Possible entropy modification mechanisms have historically included preheating, where the gas has been heated before being accreted into the potential well, by early supernovae and/or AGN activity , internal heating after accretion , and cooling .', 'astro-ph-0508234-1-6-1': 'The lack of isentropic core entropy profiles in groups and poor clusters has shown that simple preheating is unlikely to be the sole explanation of the observations .', 'astro-ph-0508234-1-6-2': 'Since cooling-only models generally predict a higher stellar mass fraction than observed , attention is now focussing on the interplay between cooling and feedback.', 'astro-ph-0508234-1-6-3': 'Further high quality observations are needed in order to distinguish between these different entropy modification mechanisms.', 'astro-ph-0508234-1-7-0': 'The present paper is the third in a series based on observations of ten clusters in the temperature range from 2 keV to 9 keV.', 'astro-ph-0508234-1-7-1': 'In [CITATION], we investigated the shape and scaling properties of the dark matter distribution, and in [CITATION] we examined the relation between the total mass and the X-ray temperature.', 'astro-ph-0508234-1-7-2': 'In the present paper we extend the work of [CITATION] by exploring the entropy scaling properties over a larger temperature/mass range.', 'astro-ph-0508234-1-8-0': 'All results given below were calculated assuming a [MATH]CDM cosmology with [MATH] and [MATH] and [MATH] km s[MATH] Mpc[MATH].', 'astro-ph-0508234-1-8-1': 'Unless otherwise stated, errors are given at the [MATH] per cent confidence level.', 'astro-ph-0508234-1-9-0': '# Sample and analysis', 'astro-ph-0508234-1-10-0': 'The sample (Table [REF]) comprises 10 systems ranging in temperature from 2 keV to [MATH] keV.', 'astro-ph-0508234-1-10-1': 'In Paper I, spatially resolved temperature and density data were used to calculate gravitating mass profiles.', 'astro-ph-0508234-1-10-2': "These were which then fitted with an NFW model, yielding values for [MATH] and [MATH] ('virial' mass and radius corresponding to a density contrast of 200).", 'astro-ph-0508234-1-10-3': "The mean ('virial') temperature of each system, [MATH], needed to investigate the scaling properties of the profiles, was estimated from fits to a spectrum built from all events in the region [MATH] (see the discussion in Paper II).", 'astro-ph-0508234-1-10-4': 'The data reduction steps are described in detail in Paper I.', 'astro-ph-0508234-1-11-0': 'We calculated the entropy profile of each system using the deprojected, PSF corrected temperature profile and an analytical model for the gas density derived in Paper I. Analytic gas density model parameters are listed in Appendix A.', 'astro-ph-0508234-1-11-1': 'In Fig [REF], the raw entropy profiles are shown in physical units ([MATH] kpc), and in terms of the measured virial radius, [MATH].', 'astro-ph-0508234-1-11-2': 'All profiles increase monotonically with radius and, while the slope of the profile becomes shallower towards the centre in some of the clusters, none shows an isentropic core.', 'astro-ph-0508234-1-12-0': '# Entropy scaling properties', 'astro-ph-0508234-1-13-0': '## Introduction', 'astro-ph-0508234-1-14-0': 'In a self-similar cluster population expected from simple gravitational collapse, the scaled profiles of any physical quantity coincide, and thus measures of these quantities at any scaled radius, should correlate with global quantities such as the mean temperature, or the virial mass.', 'astro-ph-0508234-1-14-1': 'In this standard model, the entropy is expected to scale as [MATH] or [MATH].', 'astro-ph-0508234-1-15-0': 'In purely theoretical terms, the most fundamental characteristic of a cluster is its mass.', 'astro-ph-0508234-1-15-1': 'Previous investigations of the entropy scaling relations were hampered in this respect because they did not have accurate mass information.', 'astro-ph-0508234-1-15-2': 'The present observations, which have excellent mass data (Papers I and II), thus represent an ideal opportunity to examine the entropy-mass relation.', 'astro-ph-0508234-1-15-3': 'In addition, the spatial resolution and radial coverage in the entropy profiles are significantly improved, as compared with previous / studies .', 'astro-ph-0508234-1-15-4': 'In this Section, we investigate both the entropy-temperature ([MATH]-[MATH]) and the entropy-mass ([MATH]-[MATH]) relations.', 'astro-ph-0508234-1-15-5': 'We derive the slope and normalisation of each relation, at various scaled radius/density contrast.', 'astro-ph-0508234-1-15-6': 'We also estimate the raw, statistical and intrinsic scatter about the relations.', 'astro-ph-0508234-1-16-0': '## Method', 'astro-ph-0508234-1-17-0': 'The entropy estimates at a given scaled radius ([MATH]) or density contrast ([MATH]) have to be calculated taking into account the uncertainties on the temperature, the analytical gas density model, and the measurement of the scaled radius/density contrast itself.', 'astro-ph-0508234-1-17-1': 'We thus estimated the entropy at a given [MATH] or [MATH], and the associated errors, using a Monte Carlo method.', 'astro-ph-0508234-1-17-2': 'The physical radius [MATH]/[MATH] values and corresponding uncertainties were estimated from NFW-type modelling of the mass profile (see Papers I and II for details), and subsequently randomised assuming a Gaussian distribution with sigma equal to the [MATH] error.', 'astro-ph-0508234-1-17-3': 'The temperature profile was similarly randomised, then estimated at the relevant (randomised) [MATH]/[MATH] using spline interpolation.', 'astro-ph-0508234-1-17-4': 'The entropy was then calculated in the usual manner, [MATH], including an additional [MATH] per cent systematic error on [MATH] (corresponding to the uncertainties in the surface brightness profile modelling; see Paper I).', 'astro-ph-0508234-1-17-5': 'This procedure was undertaken 1000 times for each cluster; the final entropy value at each [MATH]/ [MATH] was calculated from the mean and standard deviation of the randomised values at that radius.', 'astro-ph-0508234-1-18-0': 'The radius/density contrast of maximum detection of A1983 and MKW9 are [MATH]/1455[MATH] and [MATH], respectively.', 'astro-ph-0508234-1-18-1': 'The temperature profiles of these clusters had thus to be extrapolated to [MATH]/[MATH].', 'astro-ph-0508234-1-18-2': 'To avoid spuriously high/low temperatures, we only allowed extrapolations within [MATH] of the mean temperature of the cluster.', 'astro-ph-0508234-1-19-0': '## The [MATH]-[MATH] relation', 'astro-ph-0508234-1-20-0': 'Our data have the required temperature coverage to investigate the entropy-temperature relation.', 'astro-ph-0508234-1-20-1': 'For easier comparison with previous work, we first investigated the [MATH]-[MATH] relation at different fractions of the virial radius, viz., 0.1, 0.2, 0.3, and [MATH].', 'astro-ph-0508234-1-20-2': 'We fitted a power law model of the form [EQUATION] to the data, [MATH] being defined as in Sect. [REF].', 'astro-ph-0508234-1-20-3': 'We chose a pivot point of 5 keV for consistency with the [MATH]-[MATH] relation of [CITATION].', 'astro-ph-0508234-1-20-4': 'The fit was performed using linear regression in the log-log plane, taking into account the errors on both variables.', 'astro-ph-0508234-1-20-5': 'We have also computed the raw and intrinsic scatter about the best fitting relations in the log-log plane.', 'astro-ph-0508234-1-20-6': 'To estimate the raw scatter, we used the vertical distances to the regression line, weighted by the error.', 'astro-ph-0508234-1-20-7': 'The intrinsic scatter was computed from the quadratic difference between the raw scatter and the scatter expected from the statistical errors.', 'astro-ph-0508234-1-20-8': 'The resulting values are given in Table [REF].', 'astro-ph-0508234-1-21-0': 'In undertaking the fits, we first used the classical weighted least square method, WLS ([MATH] estimator) implemented in the the routine FITEXY from numerical recipes .', 'astro-ph-0508234-1-21-1': 'This regression method is strictly valid only if the intrinsic scatter is negligible as compared to the statistical scatter.', 'astro-ph-0508234-1-21-2': 'This is generally not the case (see Table [REF]).', 'astro-ph-0508234-1-21-3': 'We thus also considered the orthogonal BCES method , with bootstrap resampling.', 'astro-ph-0508234-1-21-4': 'This is the least-biased regression method when both measurement errors and intrinsic scatter are present.', 'astro-ph-0508234-1-21-5': 'However, no error weighting is performed on individual data points.', 'astro-ph-0508234-1-21-6': 'We thus also considered a variation of the WLS method, where a constant term, corresponding to the intrinsic scatter, is added quadratically to the [MATH] error (WLSS method).', 'astro-ph-0508234-1-21-7': 'The best fit is determined such that the reduced [MATH], after minimization over [MATH] and [MATH], is [MATH].', 'astro-ph-0508234-1-22-0': 'The best fitting slopes and intercepts for the different methods are listed in Table [REF].', 'astro-ph-0508234-1-22-1': 'Figure [REF] shows the data and the best-fitting power law relation for each radius under consideration, obtained using the standard WLS and BCES methods.', 'astro-ph-0508234-1-22-2': 'The three methods give best fitting parameters consistent within the [MATH] errors.', 'astro-ph-0508234-1-22-3': 'However, the errors are underestimated with the WLS method, especially at low radii.', 'astro-ph-0508234-1-22-4': 'This is a consequence of the high [MATH] value obtained, reflecting the high intrinsic scatter at small scaled radius (e.g. [MATH] for 8 d.o.f at [MATH]).', 'astro-ph-0508234-1-22-5': 'In this case the standard [MATH] criteria to derive parameter errors is not valid.', 'astro-ph-0508234-1-22-6': 'The BCES and WLSS give results in excellent agreement with each other, both in terms of best-fitting values and associated uncertainties.', 'astro-ph-0508234-1-22-7': 'In the following, for easier comparison with previous work, we will refer to results obtained using the commonly used BCES method.', 'astro-ph-0508234-1-23-0': 'The slope of the entropy-temperature relation is incompatible with the standard self-similar prediction at all radii at which we have measured it, confirming the results of [CITATION].', 'astro-ph-0508234-1-23-1': 'Our best-fitting [MATH]-[MATH] slope ([MATH]) is slightly shallower than that found by [CITATION] from unweighted orthogonal regression on individual data points ([MATH]).', 'astro-ph-0508234-1-23-2': 'A better agreement is observed with the slope derived by these authors from data grouped in temperature bins ([MATH]).', 'astro-ph-0508234-1-23-3': 'However, the slopes are consistent once uncertainties are taken into account.', 'astro-ph-0508234-1-24-0': 'Beyond [MATH], the slope remains remarkably stable.', 'astro-ph-0508234-1-24-1': 'At [MATH], our gas density and temperature measurements are well constrained, and do not suffer potential systematic problems connected to the correction for PSF and projection effects.', 'astro-ph-0508234-1-24-2': 'At this radius, we are well outside the strong cooling cores found in the majority of the objects in our sample.', 'astro-ph-0508234-1-24-3': 'In addition, we note that there are no significant temperature gradients at this radius in the high resolution temperature profiles produced by [CITATION] from Chandra observations.', 'astro-ph-0508234-1-24-4': 'For this reason we consider the measurements at [MATH] to be the most reliable.', 'astro-ph-0508234-1-24-5': 'The slope of the [MATH]-[MATH] relation, obtained using the BCES method, is [MATH], in excellent agreement with that found by [CITATION].', 'astro-ph-0508234-1-25-0': 'Figure [REF] shows that there is noticeable scatter about the [MATH]-[MATH] relation.', 'astro-ph-0508234-1-25-1': 'Table [REF] makes clear that the scatter is reduced at larger scaled radius.', 'astro-ph-0508234-1-25-2': 'The intrinsic scatter remains the dominant contributor to the dispersion in all relations, except at [MATH].', 'astro-ph-0508234-1-26-0': '## The [MATH]-[MATH] relation', 'astro-ph-0508234-1-27-0': 'In investigating the [MATH]-[MATH] relation, it is more logical to work in terms of density contrast [MATH].', 'astro-ph-0508234-1-27-1': 'We used [MATH], 2500, and 1000, which correspond to average fractions of [MATH], [MATH] and [MATH] for the present sample, and fitted a power law of the form [EQUATION] to the data.', 'astro-ph-0508234-1-27-2': 'The pivot point of [MATH] corresponds to a temperature of 5 keV using the best-fitting [MATH]-[MATH] relation of [CITATION].', 'astro-ph-0508234-1-27-3': 'In Fig [REF] we show the data and the best-fitting power law relation for each density contrast under consideration.', 'astro-ph-0508234-1-27-4': 'Since MKW9 appears to be a clear outlier, we also fitted the relation excluding this cluster.', 'astro-ph-0508234-1-27-5': 'The best fitting slopes and intercepts for the relation, with and without MKW9, are listed in Table [REF].', 'astro-ph-0508234-1-28-0': 'The best-fitting [MATH]-[MATH] relation is shallower than the prediction from standard self-similar models.', 'astro-ph-0508234-1-28-1': 'The results for the full sample are entirely consistent with the observed and relations (Sect. [REF] and Paper II).', 'astro-ph-0508234-1-28-2': 'Excluding MKW9, the slope is slightly, although not significantly, steeper.', 'astro-ph-0508234-1-29-0': 'As can clearly be seen in Fig. [REF], and as quantified in Table [REF], the intrinsic dispersion about the relations is less than that for the relations.', 'astro-ph-0508234-1-29-1': 'This is particularly true for the sample excluding MKW9.', 'astro-ph-0508234-1-29-2': 'This is not an artifact of the larger statistical mass errors compared to those on the temperature: the dispersion is similarly smaller for the raw scatter.', 'astro-ph-0508234-1-29-3': 'Table [REF], also indicates that, excluding MKW9, the dispersion is now dominated by the statistical scatter at all density contrasts.', 'astro-ph-0508234-1-30-0': '# Scaled entropy profiles', 'astro-ph-0508234-1-31-0': 'We have confirmed that the [MATH]-[MATH] relation is shallower than expected in the standard self-similar model, and we have shown that this is also true for the [MATH]-[MATH] relation.', 'astro-ph-0508234-1-31-1': 'In this Section, we examine the structural properties of the entropy profiles.', 'astro-ph-0508234-1-31-2': 'We do this by looking at the entropy profiles once scaled by the best fitting relations.', 'astro-ph-0508234-1-32-0': '## Scaled entropy versus [MATH]', 'astro-ph-0508234-1-33-0': 'In Fig. [REF] we show the profiles scaled using the relation [MATH], where [MATH] is the global temperature measured in units of 10 keV.', 'astro-ph-0508234-1-33-1': 'This relation is consistent with our data (Table [REF]), and allows us to compare our results with previous work.', 'astro-ph-0508234-1-33-2': 'As an initial measure of the scatter in scaled entropy profiles, we estimated the dispersion at various radii in the range [MATH]-[MATH].', 'astro-ph-0508234-1-33-3': 'The shaded area in Fig [REF] shows the region enclosed by the mean plus/minus the [MATH] standard deviation.', 'astro-ph-0508234-1-33-4': 'Figure [REF] shows that, outside the core regions, the entropy profiles present a high degree of self-similarity.', 'astro-ph-0508234-1-33-5': 'The relative dispersion in scaled profiles remains approximately constant at [MATH] per cent for [MATH], in excellent agreement with the dispersion found in a smaller subsample by [CITATION].', 'astro-ph-0508234-1-33-6': 'In the core regions, however, the dispersion increases with decreasing radius to reach [MATH] per cent at [MATH].', 'astro-ph-0508234-1-34-0': 'We next fitted the scaled profiles with a power law using the BCES method.', 'astro-ph-0508234-1-34-1': 'The fit was performed in the log-log plane taking into account the errors on both variables.', 'astro-ph-0508234-1-34-2': 'Fitting the data in the radial range [MATH] we find a slope of [MATH], with a dispersion of [MATH] per cent about the best fitting line.', 'astro-ph-0508234-1-34-3': 'The slope is not significantly changed ([MATH]) if the data are fitted in the radial range [MATH], but the intrinsic dispersion is two times smaller ([MATH] per cent).', 'astro-ph-0508234-1-35-0': 'For comparison, [CITATION], using a subsample of the current data (A1983, MKW9, A2717, A1991 and A1413), found a slope of [MATH], while [CITATION], using the same scaling, find a slope of [MATH] for their sample of 13 cool core clusters.', 'astro-ph-0508234-1-35-1': 'In contrast, broken power-law behaviour has been found by [CITATION] for a sample of groups ([MATH] keV).', 'astro-ph-0508234-1-35-2': 'Our slightly higher slope, compared to that of [CITATION], can be explained a slight difference in the temperature profile shape.', 'astro-ph-0508234-1-35-3': 'We note simply that the samples are not equivalent and that a detailed investigation of the slope of the entropy profile will require a carefully controlled, unbiased sample spanning the mass range from groups to massive clusters.', 'astro-ph-0508234-1-35-4': 'Irrespective of the exact slope, that we find [MATH] is rather surprising considering that this value is expected for shock heating in spherical collapse .', 'astro-ph-0508234-1-35-5': 'This is further discussed in Sec. [REF].', 'astro-ph-0508234-1-36-0': '## Scaled entropy versus [MATH]', 'astro-ph-0508234-1-37-0': 'In Figure [REF], we show the entropy profiles scaled using the relation [MATH], where [MATH] is the total mass [MATH] in units of [MATH] M[MATH].', 'astro-ph-0508234-1-37-1': 'Radii are plotted in terms of the density contrast, [MATH].', 'astro-ph-0508234-1-37-2': 'The shaded area in Fig. [REF] shows the region enclosed by the mean plus/minus the [MATH] standard deviation.', 'astro-ph-0508234-1-38-0': 'Again, while we observe similarity in the entropy profiles outside the core regions ([MATH]), there is a significant increase in the dispersion at higher density contrast.', 'astro-ph-0508234-1-38-1': 'It is also clear that MKW9 is a distinct outlier.', 'astro-ph-0508234-1-38-2': 'This may indicate that the mass of this cluster is underestimated, or its temperature is overestimated.', 'astro-ph-0508234-1-38-3': 'Excluding MKW9 and fitting the scaled entropy profiles with a power law using the BCES method, the slope varies from [MATH] when fitting [MATH]), to [MATH], when fitting [MATH]).', 'astro-ph-0508234-1-38-4': 'Thus the entropy does not strictly behave as a power-law in density contrast.', 'astro-ph-0508234-1-38-5': 'The intrinsic dispersion drops from [MATH] per cent to [MATH] per cent depending on the range of density contrast fitted.', 'astro-ph-0508234-1-38-6': 'Outside the core regions, the intrinsic dispersion is smaller than for temperature-scaled profiles, in agreement with the lower dispersion observed for the [MATH]-[MATH] relation compared to that of the [MATH]-[MATH] relation.', 'astro-ph-0508234-1-39-0': '# Discussion', 'astro-ph-0508234-1-40-0': 'Our results put into evidence two main departures from the standard self-similar model of cluster formation.', 'astro-ph-0508234-1-40-1': 'First, beyond the core region ([MATH]) the entropy profiles obey self-similarity, having a shape consistent with expectations but with a modified temperature (or mass) scaling.', 'astro-ph-0508234-1-40-2': 'The scaling relations are shallower than expected.', 'astro-ph-0508234-1-40-3': 'Second, there is a break of similarity in the core region: the dispersion in scaled profiles increases with decreasing radius.', 'astro-ph-0508234-1-40-4': 'In this section, we will discuss in turn both characteristics.', 'astro-ph-0508234-1-41-0': '## Entropy normalisation', 'astro-ph-0508234-1-42-0': 'The modified entropy scaling indicates that there is an excess of entropy, in low mass objects relative to more massive systems, as compared to the expectation from pure shock heating.', 'astro-ph-0508234-1-42-1': 'A comparison with adiabatic numerical simulations allows us to quantify the absolute value of the excess and to examine whether an entropy excess is also present for the most massive systems.', 'astro-ph-0508234-1-43-0': '[CITATION] shows the results of adiabatic numerical simulations of 30 clusters spanning a mass range of more than a factor of ten.', 'astro-ph-0508234-1-43-1': 'Once scaled by the characteristic entropy of the halo, [EQUATION] where [MATH] is the baryon fraction, the simulated profiles are closely self-similar, and can be well fitted in the radial range [MATH] by the power-law relation [MATH].', 'astro-ph-0508234-1-43-2': 'Assuming [MATH] and [MATH]) and typical elemental abundances, we can scale our observed entropy values to [MATH] using the expression [EQUATION]', 'astro-ph-0508234-1-43-3': 'Further scaling the radius by the measured value of [MATH] allows us to compare our results directly with the adiabatic simulations.', 'astro-ph-0508234-1-43-4': 'In the left hand panel of Figure [REF], our observed entropy profiles are compared with the prediction of [CITATION].', 'astro-ph-0508234-1-43-5': 'The right hand panel shows the ratio between the best-fitting adiabatic power-law relation and the observed data.', 'astro-ph-0508234-1-44-0': 'It can be seen that the richer systems all have entropies which are in good agreement (both in slope and normalisation) with the adiabatic prediction, denoted by the solid line in Fig. [REF].', 'astro-ph-0508234-1-44-1': 'On average, their entropy is only slightly higher than predicted (by [MATH] per cent), although the effect is not very significant.', 'astro-ph-0508234-1-44-2': 'We recall that there is also a [MATH] per cent difference in normalisation between the observed [MATH]-[MATH] relation and that predicted by adiabatic simulations .', 'astro-ph-0508234-1-44-3': 'Interestingly, this corresponds to a [MATH] per cent entropy excess at a given mass for [MATH].', 'astro-ph-0508234-1-44-4': 'The (slight) excess of entropy in massive systems is thus consistent with a simple increase of the mean temperature, i.e., of the internal energy of the ICM.', 'astro-ph-0508234-1-44-5': 'However, Fig. [REF] shows explicitly that the poorer systems have a systematically higher entropy normalisation than the richer systems.', 'astro-ph-0508234-1-44-6': 'There is approximately 2.5 times more entropy at [MATH] in the ICM of A1983, the poorest cluster in our sample, than that predicted by gravitational heating.', 'astro-ph-0508234-1-44-7': 'This excess shows that the density of the ICM is also affected at lower mass.', 'astro-ph-0508234-1-45-0': '[CITATION] and [CITATION] independently noted that the ICM entropy is highly sensitive to the density of the incoming gas and suggested that a smoothing of the gas density due to pre-heating in filaments and/or infalling groups would boost the entropy production at the accretion shock.', 'astro-ph-0508234-1-45-1': 'The entropy amplification is more efficient in low mass systems, because they accrete smaller halos more affected by smoothing due to pre-heating.', 'astro-ph-0508234-1-45-2': 'In this interpretation, no substantial isentropic core will develop because the amount of initial preheating needed is substantially less than the characteristic entropy of the final halo.', 'astro-ph-0508234-1-45-3': 'Thus clusters will thus be self-similar down to low mass but with a modified normalisation scaling as discussed above.', 'astro-ph-0508234-1-45-4': 'This is in agreement with our results outside the core region ([MATH]).', 'astro-ph-0508234-1-46-0': 'Recent numerical simulations which mimic preheating by imposing a minimum entropy floor at high [MATH] have confirmed the entropy amplification effect due to smooth accretion .', 'astro-ph-0508234-1-46-1': 'However, the effect seems to be substantially reduced when cooling is also taken into account.', 'astro-ph-0508234-1-46-2': 'Furthermore, the physical origin of the preheating is still unclear.', 'astro-ph-0508234-1-46-3': 'Heating by SNe seems to be too localized to have a significant effect in smoothing the accreting gas .', 'astro-ph-0508234-1-46-4': 'AGN might be better candidates to produce the extra heating.', 'astro-ph-0508234-1-47-0': '## Localised modification', 'astro-ph-0508234-1-48-0': 'While filamentary pre-heating may explain the increased normalisation of poor systems relative to hotter systems, it does not explain the increasing scatter towards the central regions in scaled profiles (Figs. [REF], [REF] and [REF]).', 'astro-ph-0508234-1-48-1': 'The adiabatic numerical simulations of [CITATION] show both a flattening of the slope and an increase in the dispersion of the scaled entropy profiles in the central regions ([MATH]).', 'astro-ph-0508234-1-48-2': 'However, the dispersion in our observed profiles ([MATH] per cent) greatly exceeds that of the simulations ([MATH] per cent, cf Fig. [REF] and Fig. 11 of ).', 'astro-ph-0508234-1-48-3': 'Six clusters out of our total sample of ten (A1991, A2597, A1068, A478, PKS0745 and A2204) have remarkably similar scaled entropy profiles, displaying power-law behaviour down to the smallest radii measured.', 'astro-ph-0508234-1-48-4': 'Fitting these scaled profiles in the radial range [MATH] with power-law using the BCES method, we find a slope of [MATH], with a dispersion of only 13 per cent.', 'astro-ph-0508234-1-48-5': 'This slope is very similar to that found by fitting the [MATH] range.', 'astro-ph-0508234-1-48-6': 'These five clusters all appear to host a bona fide cooling core, each having a central temperature decrement of a factor [MATH] .', 'astro-ph-0508234-1-48-7': 'Similar power-law profiles were found by in a sample of 13 cooling core clusters by [CITATION].', 'astro-ph-0508234-1-48-8': 'Strong radiative cooling thus appears to generate entropy profiles which display power-law behaviour down to very small radii (Fig. [REF]).', 'astro-ph-0508234-1-48-9': 'This is similar to the quasi-steady-state entropy profile in the models of [CITATION], which include radiative cooling.', 'astro-ph-0508234-1-49-0': 'The other four clusters in our sample are characterised by a smaller central temperature decrement, larger cooling times and shallower entropy profiles.', 'astro-ph-0508234-1-49-1': 'Clearly, some mechanism has modified the cooling history of these clusters.', 'astro-ph-0508234-1-49-2': 'Energy input from AGN is regularly invoked as a way of moderating cooling at the centres of galaxy clusters .', 'astro-ph-0508234-1-49-3': 'Our sample contains four clusters which have X-ray evidence for interactions between radio and X-ray plasma (A478, A2204, A2597, and PKS0745), and yet the entropy profiles of all of these clusters increase monotonically outward in the canonical fashion.', 'astro-ph-0508234-1-49-4': 'However, we note that Chandra observations of clusters with moderate to strong radio sources (including A478), show some evidence for a slightly shallower slope ([MATH]) but only in the very inner regions .', 'astro-ph-0508234-1-49-5': 'The effect of AGN activity on the entropy profile will depend on whether the energy input is catastrophic (i.e., occurring in strong bursts) or distributed (more moderated input).', 'astro-ph-0508234-1-49-6': 'It is clear that if the heating is catastrophic in nature, no cluster has yet been found in which this is evident at least from the point of view of the entropy (although see ).', 'astro-ph-0508234-1-49-7': 'The heating is likely more distributed, via e.g., weak shocks , thus preserving the generally increasing form of the entropy profile.', 'astro-ph-0508234-1-50-0': 'Merging events can result in substantial mixing of high and low entropy gas .', 'astro-ph-0508234-1-50-1': 'Relatively little attention has been given to this possibility in the literature (see, however, Fig. 5 of , Fig. 12 of and Fig. 6 of ).', 'astro-ph-0508234-1-50-2': 'Such redistribution of entropy will depend on the scale of the merger, whether the merger has disrupted the structure of the cool core, and the timescale for re-establishment of the cool core if disrupted.', 'astro-ph-0508234-1-50-3': 'In the current sample MKW9, A2717, A1413 and A1983 all have flatter core entropy profiles.', 'astro-ph-0508234-1-50-4': 'We note that the morphological information for the present sample would argue against recent merger activity in these clusters (; Papers I and II).', 'astro-ph-0508234-1-50-5': 'However, this does not rule out entropy modification due to a more ancient merger, particularly if the relaxation time is less than the cooling time .', 'astro-ph-0508234-1-51-0': '# Conclusions', 'astro-ph-0508234-1-52-0': 'The present sample of ten relaxed clusters constitutes the first with precise mass data and wide temperature coverage, allowing the detailed study of the scaling of ICM entropy with both temperature and mass.', 'astro-ph-0508234-1-52-1': 'The entropy profiles are sampled with good spatial resolution up to [MATH], allowing us to examine structural properties avoiding the stacking analysis and extrapolation on which previous ROSAT/ASCA works relied.', 'astro-ph-0508234-1-53-0': 'We have found that the entropy profiles of the present cluster sample are self-similar beyond [MATH], with a shape not significantly different from that expected from shock heating.', 'astro-ph-0508234-1-53-1': 'The entropy scaling relations are shallower than expected from standard self-similar models and adiabatic numerical simulations.', 'astro-ph-0508234-1-53-2': 'The slopes of the entropy scaling relations are independent of radius, reflecting the structural self-similarity.', 'astro-ph-0508234-1-53-3': 'The entropy-mass () relation is consistent with the observed entropy-temperature () and mass-temperature ([MATH]-[MATH]) relations.', 'astro-ph-0508234-1-53-4': 'The dispersion is smaller about the relation, reflecting the fact that the mass of a cluster is its most fundamental characteristic.', 'astro-ph-0508234-1-53-5': 'These results confirm the trends seen in smaller samples , and in stacking analysis of [MATH] data .', 'astro-ph-0508234-1-54-0': 'Comparison of our observed profiles with adiabatic numerical simulations has allowed us to quantify the entropy excess relative to that expected in pure gravitational collapse.', 'astro-ph-0508234-1-54-1': 'The excess is only [MATH] per cent in our highest-mass systems.', 'astro-ph-0508234-1-54-2': 'This is consistent with a simple increase in the mean temperature consistent with the observed difference in normalisation between the observed [MATH]-[MATH] relation and that predicted by adiabatic simulations.', 'astro-ph-0508234-1-54-3': 'However, the excess in low mass systems can be of more than a factor of 2.5, indicating that the density of the ICM is also affected in these systems.', 'astro-ph-0508234-1-54-4': 'These characteristics are in qualitative agreement with a scenario in which entropy production is boosted at the accretion shock.', 'astro-ph-0508234-1-54-5': 'A plausible candidate mechanism is smoothing of the accreted gas due to preheating .', 'astro-ph-0508234-1-55-0': 'We have found new and compelling evidence for an increase in the entropy dispersion in the core regions [MATH] per cent at [MATH].', 'astro-ph-0508234-1-55-1': 'However the five cooling core clusters in our sample have remarkably self-similar power-law profiles with a dispersion of only 13 per cent between 0.01 and [MATH].', 'astro-ph-0508234-1-55-2': 'The observed increase in dispersion towards the central regions argues for localised entropy modification mechanisms.', 'astro-ph-0508234-1-55-3': 'We conclude that AGN activity and/or transient entropy modification due to merging events are good candidates for the modification of cooling in these clusters.', 'astro-ph-0508234-1-56-0': 'These will be interesting questions to address with numerical simulations.', 'astro-ph-0508234-1-56-1': 'On the observational side, a larger, unbiased, sample of clusters will undoubtedly provide greater insights.', 'astro-ph-0508234-1-57-0': 'GWP acknowledges funding from a Marie Curie Intra-European Fellowship under the FP6 programme (Contract No. MEIF-CT-2003-500915).', 'astro-ph-0508234-1-57-1': 'EP aknowledges the financial support of CNES, the French Space Agency, and of the Leverhulme trust (UK).', 'astro-ph-0508234-1-57-2': 'The present work is based on observations obtained with XMM-Newton, an ESA science mission with instruments and contributions directly funded by ESA Member States and the USA (NASA).', 'astro-ph-0508234-1-58-0': '# Surface brightness profile fit results', 'astro-ph-0508234-1-59-0': 'The co-added EPIC azimuthally averaged, vignetting corrected, background subtracted surface brightness profile was computed in the [0.3-3.]', 'astro-ph-0508234-1-59-1': 'keV band for each cluster.', 'astro-ph-0508234-1-59-2': 'The profile was corrected for radial variations in the emissivity due to abundance or temperature gradients as described in [CITATION].', 'astro-ph-0508234-1-59-3': 'The gas density profile was obtained from fitting parametric analytic models, convolved with the EPIC PSF, to the corrected profile.', 'astro-ph-0508234-1-59-4': 'As discussed in Paper I, we tried various models and empirically chose the best fitting model using the [MATH] statistic as a measure of the goodness of fit.', 'astro-ph-0508234-1-59-5': 'The models used were: (i) the double [MATH] (BB) density model ; (ii) the modified double [MATH] (KBB) model, allowing a more concentrated gas density distribution towards the centre ; (iii) the sum of three [MATH] (3B) models for the emission measure, in which a common value of [MATH] is used to ensure smooth behaviour at large radii .', 'astro-ph-0508234-1-59-6': 'The best-fitting models, and model parameters, are listed in Table [REF].'}

{'astro-ph-0508234-2-0-0': 'Using observations, we investigate the scaling and structural properties of the ICM entropy in a sample of 10 nearby ([MATH]) relaxed galaxy clusters in the temperature range 2-9 keV.', 'astro-ph-0508234-2-0-1': 'We derive the local entropy-temperature () relation at [MATH] and [MATH].', 'astro-ph-0508234-2-0-2': 'The logarithmic slope of the relation is the same within the [MATH] error at all scaled radii.', 'astro-ph-0508234-2-0-3': 'However, the intrinsic dispersion about the best fitting relation is significantly higher at [MATH].', 'astro-ph-0508234-2-0-4': 'The slope is [MATH] at [MATH], in excellent agreement with previous work.', 'astro-ph-0508234-2-0-5': 'We also investigate the entropy-mass relation at density contrasts [MATH] and 1000.', 'astro-ph-0508234-2-0-6': 'We find a shallower slope than that expected in simple self-similar models, which is in agreement with the observed empirically-determined entropy-temperature and mass-temperature scaling.', 'astro-ph-0508234-2-0-7': 'The dispersion is smaller than for the [MATH]-[MATH] relation.', 'astro-ph-0508234-2-0-8': 'Once scaled appropriately, the entropy profiles appear similar beyond [MATH], with an intrinsic dispersion of [MATH] per cent and a shape consistent with gravitational heating ([MATH]).', 'astro-ph-0508234-2-0-9': 'However, the scatter in scaled entropy profiles increases with smaller scaled radius, to more than [MATH] per cent at [MATH].', 'astro-ph-0508234-2-0-10': 'Our results are in qualitative agreement with models which boost entropy production at the accretion shock.', 'astro-ph-0508234-2-0-11': 'However, localised entropy modification may be needed to explain the dispersion in the inner regions.', 'astro-ph-0508234-2-1-0': 'An update on galaxy cluster entropy scaling', 'astro-ph-0508234-2-2-0': '# Introduction', 'astro-ph-0508234-2-3-0': 'X-ray observations of the hot, gaseous intracluster medium (ICM) have been telling us for well over a decade that physical processes other than gravity are acting to modify the properties of the cluster population.', 'astro-ph-0508234-2-3-1': 'An understanding of the source(s) of this modification is of great importance for our understanding of cluster formation and evolution, and is essential for the use of clusters as precision cosmological probes.', 'astro-ph-0508234-2-4-0': 'Entropy is important because, together with the shape of the potential well, it is the quantity which dictates the observed X-ray properties of the ICM in galaxy clusters .', 'astro-ph-0508234-2-4-1': 'The intracluster entropy is generated in shocks as gas is drawn into the gravitational potential of the cluster halo , thus it reflects the accretion history of the ICM.', 'astro-ph-0508234-2-4-2': 'However, the entropy distribution also preserves key information regarding the influence of non-gravitational processes.', 'astro-ph-0508234-2-5-0': 'X-ray measurements of the entropy at [MATH] (hereafter [MATH]) showed that the entropy of the coolest systems is considerably higher than that available from gravitational collapse , and that the entropy-temperature () relation is shallower than expected .', 'astro-ph-0508234-2-5-1': 'More recent spatially resolved entropy profiles indicate that the entropy is higher throughout the ICM, and that, outside the core regions, entropy profiles are structurally similar .', 'astro-ph-0508234-2-5-2': 'At the same time the scatter in [MATH] at a given temperature can be up to a factor of three .', 'astro-ph-0508234-2-6-0': 'Possible entropy modification mechanisms have historically included preheating, where the gas has been heated before being accreted into the potential well, by early supernovae and/or AGN activity , internal heating after accretion , and cooling .', 'astro-ph-0508234-2-6-1': 'The lack of isentropic core entropy profiles in groups and poor clusters has shown that simple preheating is unlikely to be the sole explanation of the observations .', 'astro-ph-0508234-2-6-2': 'Since cooling-only models generally predict a higher stellar mass fraction than observed , attention is now focussing on the interplay between cooling and feedback.', 'astro-ph-0508234-2-6-3': 'Further high quality observations are needed in order to distinguish between these different entropy modification mechanisms.', 'astro-ph-0508234-2-7-0': 'The present paper is the third in a series based on observations of ten clusters in the temperature range from 2 keV to 9 keV.', 'astro-ph-0508234-2-7-1': 'In [CITATION], we investigated the shape and scaling properties of the dark matter distribution, and in [CITATION] we examined the relation between the total mass and the X-ray temperature.', 'astro-ph-0508234-2-7-2': 'In the present paper we extend the work of [CITATION] by exploring the entropy scaling properties over a larger temperature/mass range.', 'astro-ph-0508234-2-8-0': 'All results given below were calculated assuming a [MATH]CDM cosmology with [MATH] and [MATH] and [MATH] km s[MATH] Mpc[MATH].', 'astro-ph-0508234-2-8-1': 'Unless otherwise stated, errors are given at the [MATH] per cent confidence level.', 'astro-ph-0508234-2-9-0': '# Sample and analysis', 'astro-ph-0508234-2-10-0': 'The sample (Table [REF]) comprises 10 systems ranging in temperature from 2 keV to [MATH] keV.', 'astro-ph-0508234-2-10-1': 'In Paper I, spatially resolved temperature and density data were used to calculate gravitating mass profiles.', 'astro-ph-0508234-2-10-2': "These were which then fitted with an NFW model, yielding values for [MATH] and [MATH] ('virial' mass and radius corresponding to a density contrast of 200).", 'astro-ph-0508234-2-10-3': 'The mean temperature of each system, [MATH], needed to investigate the scaling properties of the profiles, was estimated from fits to a spectrum built from all events in the region [MATH] (see the discussion in Paper II).', 'astro-ph-0508234-2-10-4': 'The data reduction steps are described in detail in Paper I.', 'astro-ph-0508234-2-11-0': 'We calculated the entropy profile of each system using the deprojected, PSF corrected temperature profile and an analytical model for the gas density derived in Paper I. Analytic gas density model parameters are listed in Appendix A.', 'astro-ph-0508234-2-11-1': 'In Fig [REF], the raw entropy profiles are shown in physical units ([MATH] kpc), and in terms of the measured virial radius, [MATH].', 'astro-ph-0508234-2-11-2': 'All profiles increase monotonically with radius and, while the slope of the profile becomes shallower towards the centre in some of the clusters, none shows an isentropic core.', 'astro-ph-0508234-2-12-0': '# Entropy scaling properties', 'astro-ph-0508234-2-13-0': '## Introduction', 'astro-ph-0508234-2-14-0': 'In a self-similar cluster population expected from simple gravitational collapse, the scaled profiles of any physical quantity coincide, and thus measures of these quantities at any scaled radius should correlate with global quantities such as the mean temperature, or the virial mass.', 'astro-ph-0508234-2-14-1': 'In this standard model, the entropy is expected to scale as [MATH] or [MATH].', 'astro-ph-0508234-2-15-0': 'In purely theoretical terms, the most fundamental characteristic of a cluster is its mass.', 'astro-ph-0508234-2-15-1': 'Previous investigations of the entropy scaling relations were hampered in this respect because they did not have accurate mass information.', 'astro-ph-0508234-2-15-2': 'The present observations, which have excellent mass data (Papers I and II), thus represent an ideal opportunity to examine the entropy-mass relation.', 'astro-ph-0508234-2-15-3': 'In addition, the spatial resolution and radial coverage in the entropy profiles are significantly improved, as compared with previous / studies .', 'astro-ph-0508234-2-15-4': 'In this Section, we investigate both the entropy-temperature ([MATH]-[MATH]) and the entropy-mass ([MATH]-[MATH]) relations.', 'astro-ph-0508234-2-15-5': 'We derive the slope and normalisation of each relation, at various scaled radius/density contrasts.', 'astro-ph-0508234-2-15-6': 'We also estimate the raw, statistical and intrinsic scatter about the relations.', 'astro-ph-0508234-2-16-0': '## Method', 'astro-ph-0508234-2-17-0': 'The entropy estimates at a given scaled radius ([MATH]) or density contrast ([MATH]) have to be calculated taking into account the uncertainties on the temperature, the analytical gas density model, and the measurement of the scaled radius/density contrast itself.', 'astro-ph-0508234-2-17-1': 'We thus estimated the entropy at a given [MATH] or [MATH], and the associated errors, using a Monte Carlo method.', 'astro-ph-0508234-2-17-2': 'The scaled radius [MATH]/[MATH] values and corresponding uncertainties were estimated from NFW-type modelling of the mass profile (see Papers I and II for details), and subsequently randomised assuming a Gaussian distribution with sigma equal to the [MATH] error.', 'astro-ph-0508234-2-17-3': 'The temperature profile was then similarly randomised taking into account the observed uncertainties, and the temperature estimated at the relevant (randomised) [MATH]/[MATH] using spline interpolation.', 'astro-ph-0508234-2-17-4': 'The entropy was then calculated in the usual manner, [MATH], including an additional [MATH] per cent systematic error on [MATH] (corresponding to the uncertainties in the surface brightness profile modelling; see Paper I).', 'astro-ph-0508234-2-17-5': 'This procedure was undertaken 1000 times for each cluster; the final entropy value at each [MATH]/ [MATH] was calculated from the mean and standard deviation of the randomised values at that radius.', 'astro-ph-0508234-2-18-0': 'The temperature profiles of all systems except A1983 and MKW9 are detected up to [MATH] (the scaled radius/density contrast of maximum detection of A1983 and MKW9 are [MATH]/1455[MATH] and [MATH], respectively).', 'astro-ph-0508234-2-18-1': 'The temperature profiles of these clusters had thus to be extrapolated to [MATH]/[MATH].', 'astro-ph-0508234-2-18-2': 'To avoid spuriously high/low temperatures, we only allowed extrapolations within [MATH] times the mean temperature of the cluster (well within the range of temperature gradients observed to date, e.g., ).', 'astro-ph-0508234-2-19-0': '## The [MATH]-[MATH] relation', 'astro-ph-0508234-2-20-0': 'Our data have the required temperature coverage to investigate the entropy-temperature relation.', 'astro-ph-0508234-2-20-1': 'For easier comparison with previous work, we first investigated the [MATH]-[MATH] relation at different fractions of the virial radius, viz., 0.1, 0.2, 0.3, and [MATH].', 'astro-ph-0508234-2-20-2': 'We fitted a power law model of the form [EQUATION] to the data, [MATH] being defined as in Sect. [REF].', 'astro-ph-0508234-2-20-3': 'We chose a pivot point of 5 keV for consistency with the [MATH]-[MATH] relation of [CITATION].', 'astro-ph-0508234-2-20-4': 'The fit was performed using linear regression in the log-log plane, taking into account the errors on both variables.', 'astro-ph-0508234-2-20-5': 'We have also computed the raw and intrinsic scatter about the best fitting relations in the log-log plane.', 'astro-ph-0508234-2-20-6': 'To estimate the raw scatter, we used the vertical distances to the regression line, weighted by the error.', 'astro-ph-0508234-2-20-7': 'The intrinsic scatter was computed from the quadratic difference between the raw scatter and the scatter expected from the statistical errors.', 'astro-ph-0508234-2-20-8': 'The resulting values are given in Table [REF].', 'astro-ph-0508234-2-21-0': 'In undertaking the fits, we first used the classical weighted least square method, WLS ([MATH] estimator) implemented in the the routine FITEXY from numerical recipes .', 'astro-ph-0508234-2-21-1': 'This regression method is strictly valid only if the intrinsic scatter is negligible as compared to the statistical scatter.', 'astro-ph-0508234-2-21-2': 'This is generally not the case (see Table [REF]).', 'astro-ph-0508234-2-21-3': 'We thus also considered the orthogonal BCES method , with bootstrap resampling.', 'astro-ph-0508234-2-21-4': 'This is the least-biased regression method when both measurement errors and intrinsic scatter are present.', 'astro-ph-0508234-2-21-5': 'However, no error weighting is performed on individual data points.', 'astro-ph-0508234-2-21-6': 'We thus also considered a variation of the WLS method, where a constant term, corresponding to the intrinsic scatter, is added quadratically to the [MATH] error (WLSS method).', 'astro-ph-0508234-2-21-7': 'The best fit is determined such that the reduced [MATH], after minimization over [MATH] and [MATH], is [MATH].', 'astro-ph-0508234-2-22-0': 'The best fitting slopes and intercepts for the different methods are listed in Table [REF].', 'astro-ph-0508234-2-22-1': 'Figure [REF] shows the data and the best-fitting power law relation for each radius under consideration, obtained using the standard WLS and BCES methods.', 'astro-ph-0508234-2-22-2': 'The three methods give best fitting parameters consistent within the [MATH] errors.', 'astro-ph-0508234-2-22-3': 'However, the errors are underestimated with the WLS method, especially at low radii.', 'astro-ph-0508234-2-22-4': 'This is a consequence of the high [MATH] value obtained, reflecting the high intrinsic scatter at small scaled radius (e.g. [MATH] for 8 d.o.f at [MATH]).', 'astro-ph-0508234-2-22-5': 'In this case the standard [MATH] criteria to derive parameter errors is not valid.', 'astro-ph-0508234-2-22-6': 'The BCES and WLSS give results in excellent agreement with each other, both in terms of best-fitting values and associated uncertainties.', 'astro-ph-0508234-2-22-7': 'In the following, for easier comparison with previous work, we will refer to results obtained using the commonly used BCES method.', 'astro-ph-0508234-2-23-0': 'The slope of the entropy-temperature relation is incompatible with the standard self-similar prediction at all radii at which we have measured it, confirming the results of [CITATION].', 'astro-ph-0508234-2-23-1': 'Our best-fitting [MATH]-[MATH] slope ([MATH]) is slightly shallower than that found by [CITATION] from unweighted orthogonal regression on individual data points ([MATH]).', 'astro-ph-0508234-2-23-2': 'A better agreement is observed with the slope derived by these authors from data grouped in temperature bins ([MATH]).', 'astro-ph-0508234-2-23-3': 'However, the slopes are consistent once uncertainties are taken into account.', 'astro-ph-0508234-2-24-0': 'Beyond [MATH], the slope remains remarkably stable.', 'astro-ph-0508234-2-24-1': 'At [MATH], our gas density and temperature measurements are well constrained, and do not suffer potential systematic problems connected to the correction for PSF and projection effects.', 'astro-ph-0508234-2-24-2': 'At this radius, we are well outside the strong cooling cores found in the majority of the objects in our sample.', 'astro-ph-0508234-2-24-3': 'In addition, we note that there are no significant temperature gradients at this radius in the high resolution temperature profiles produced by [CITATION] from Chandra observations.', 'astro-ph-0508234-2-24-4': 'For this reason we consider the measurements at [MATH] to be the most reliable.', 'astro-ph-0508234-2-24-5': 'The slope of the [MATH]-[MATH] relation, obtained using the BCES method, is [MATH], in excellent agreement with that found by [CITATION].', 'astro-ph-0508234-2-25-0': 'Figure [REF] shows that there is noticeable scatter about the [MATH]-[MATH] relation.', 'astro-ph-0508234-2-25-1': 'Table [REF] makes clear that the scatter is reduced at larger scaled radius.', 'astro-ph-0508234-2-25-2': 'The intrinsic scatter remains the dominant contributor to the dispersion in all relations, except at [MATH].', 'astro-ph-0508234-2-26-0': '## The [MATH]-[MATH] relation', 'astro-ph-0508234-2-27-0': 'In investigating the [MATH]-[MATH] relation, it is more logical to work in terms of density contrast [MATH].', 'astro-ph-0508234-2-27-1': 'We used [MATH], 2500, and 1000, which correspond to average fractions of [MATH], [MATH] and [MATH] for the present sample, and fitted a power law of the form [EQUATION] to the data.', 'astro-ph-0508234-2-27-2': 'The pivot point of [MATH] corresponds to a temperature of 5 keV using the best-fitting [MATH]-[MATH] relation of [CITATION].', 'astro-ph-0508234-2-27-3': 'In Fig [REF] we show the data and the best-fitting power law relation for each density contrast under consideration.', 'astro-ph-0508234-2-27-4': 'Since MKW9 appears to be a clear outlier, we also fitted the relation excluding this cluster.', 'astro-ph-0508234-2-27-5': 'The best fitting slopes and intercepts for the relation, with and without MKW9, are listed in Table [REF].', 'astro-ph-0508234-2-28-0': 'The best-fitting [MATH]-[MATH] relation is shallower than the prediction from standard self-similar models.', 'astro-ph-0508234-2-28-1': 'The results for the full sample are entirely consistent with the observed and relations (Sect. [REF] and Paper II).', 'astro-ph-0508234-2-28-2': 'Excluding MKW9, the slope is slightly, although not significantly, steeper.', 'astro-ph-0508234-2-29-0': 'As can clearly be seen in Fig. [REF], and as quantified in Table [REF], the intrinsic dispersion about the relations is less than that for the relations.', 'astro-ph-0508234-2-29-1': 'This is particularly true for the sample excluding MKW9.', 'astro-ph-0508234-2-29-2': 'This is not an artifact of the larger statistical mass errors compared to those on the temperature: the dispersion is similarly smaller for the raw scatter.', 'astro-ph-0508234-2-29-3': 'Table [REF] also indicates that, excluding MKW9, the dispersion is now dominated by the statistical scatter at all density contrasts.', 'astro-ph-0508234-2-30-0': '# Scaled entropy profiles', 'astro-ph-0508234-2-31-0': 'We have confirmed that the [MATH]-[MATH] relation is shallower than expected in the standard self-similar model, and we have shown that this is also true for the [MATH]-[MATH] relation.', 'astro-ph-0508234-2-31-1': 'In this Section, we examine the structural properties of the entropy profiles.', 'astro-ph-0508234-2-31-2': 'We do this by looking at the entropy profiles once scaled by the best fitting relations.', 'astro-ph-0508234-2-32-0': '## Scaled entropy versus [MATH]', 'astro-ph-0508234-2-33-0': 'In Fig. [REF] we show the profiles scaled using the relation [MATH], where [MATH] is the global temperature measured in units of 10 keV.', 'astro-ph-0508234-2-33-1': 'This relation is consistent with our data (Table [REF]), and allows us to compare our results with previous work.', 'astro-ph-0508234-2-33-2': 'As an initial measure of the scatter in scaled entropy profiles, we estimated the dispersion at various radii in the range [MATH]-[MATH].', 'astro-ph-0508234-2-33-3': 'The shaded area in Fig [REF] shows the region enclosed by the mean plus/minus the [MATH] standard deviation.', 'astro-ph-0508234-2-33-4': 'Figure [REF] shows that, outside the core regions, the entropy profiles present a high degree of self-similarity.', 'astro-ph-0508234-2-33-5': 'The relative dispersion in scaled profiles remains approximately constant at [MATH] per cent for [MATH], in excellent agreement with the dispersion found in a smaller subsample by [CITATION].', 'astro-ph-0508234-2-33-6': 'In the core regions, however, the dispersion increases with decreasing radius to reach [MATH] per cent at [MATH].', 'astro-ph-0508234-2-34-0': 'We next fitted the scaled profiles with a power law using the BCES method.', 'astro-ph-0508234-2-34-1': 'The fit was performed in the log-log plane taking into account the errors on both variables.', 'astro-ph-0508234-2-34-2': 'Fitting the data in the radial range [MATH] we find a slope of [MATH], with a dispersion of [MATH] per cent about the best fitting line.', 'astro-ph-0508234-2-34-3': 'The slope is not significantly changed ([MATH]) if the data are fitted in the radial range [MATH], but the intrinsic dispersion is two times smaller ([MATH] per cent).', 'astro-ph-0508234-2-35-0': 'For comparison, [CITATION], using a subsample of the current data (A1983, MKW9, A2717, A1991 and A1413), found a slope of [MATH], while [CITATION], using the same scaling, find a slope of [MATH] for their sample of 13 cool core clusters.', 'astro-ph-0508234-2-35-1': 'In contrast, broken power-law behaviour has been found by [CITATION] for a sample of groups ([MATH] keV).', 'astro-ph-0508234-2-35-2': 'Our slightly higher slope, compared to that of [CITATION], can be explained by a slight difference in the temperature profile shape.', 'astro-ph-0508234-2-35-3': 'We note simply that the samples are not equivalent and that a detailed investigation of the slope of the entropy profile will require a carefully controlled, unbiased sample spanning the mass range from groups to massive clusters.', 'astro-ph-0508234-2-35-4': 'Irrespective of the exact slope, that we find [MATH] is interesting considering that this value is expected for shock heating in spherical collapse .', 'astro-ph-0508234-2-35-5': 'This is further discussed in Sec. [REF].', 'astro-ph-0508234-2-36-0': '## Scaled entropy versus [MATH]', 'astro-ph-0508234-2-37-0': 'In Figure [REF], we show the entropy profiles scaled using the relation [MATH], where [MATH] is the total mass [MATH] in units of [MATH] M[MATH].', 'astro-ph-0508234-2-37-1': 'Radii are plotted in terms of the density contrast, [MATH].', 'astro-ph-0508234-2-37-2': 'The shaded area in Fig. [REF] shows the region enclosed by the mean plus/minus the [MATH] standard deviation.', 'astro-ph-0508234-2-38-0': 'Again, while we observe similarity in the entropy profiles outside the core regions ([MATH]), there is a significant increase in the dispersion at higher density contrast.', 'astro-ph-0508234-2-38-1': 'It is also clear that MKW9 is a distinct outlier.', 'astro-ph-0508234-2-38-2': 'This may indicate that the mass of this cluster is underestimated, or its temperature is overestimated.', 'astro-ph-0508234-2-38-3': 'Excluding MKW9 and fitting the scaled entropy profiles with a power law using the BCES method, the slope varies from [MATH] when fitting [MATH]), to [MATH], when fitting [MATH]).', 'astro-ph-0508234-2-38-4': 'Thus the entropy does not strictly behave as a power-law in density contrast.', 'astro-ph-0508234-2-38-5': 'The intrinsic dispersion drops from [MATH] per cent to [MATH] per cent depending on the range of density contrast fitted.', 'astro-ph-0508234-2-38-6': 'Outside the core regions, the intrinsic dispersion is smaller than for temperature-scaled profiles, in agreement with the lower dispersion observed for the [MATH]-[MATH] relation compared to that of the [MATH]-[MATH] relation.', 'astro-ph-0508234-2-39-0': '# Discussion', 'astro-ph-0508234-2-40-0': 'Our results put into evidence two main departures from the standard self-similar model of cluster formation.', 'astro-ph-0508234-2-40-1': 'First, beyond the core region ([MATH]) the entropy profiles obey self-similarity, having a shape consistent with expectations, but with a modified temperature (or mass) scaling.', 'astro-ph-0508234-2-40-2': 'The scaling relations are shallower than expected.', 'astro-ph-0508234-2-40-3': 'Second, there is a break of similarity in the core region: the dispersion in scaled profiles increases with decreasing radius.', 'astro-ph-0508234-2-40-4': 'In this section, we will discuss in turn both characteristics.', 'astro-ph-0508234-2-41-0': '## Entropy normalisation', 'astro-ph-0508234-2-42-0': 'The modified entropy scaling indicates that there is an excess of entropy, in low mass objects relative to more massive systems, as compared to the expectation from pure shock heating.', 'astro-ph-0508234-2-42-1': 'A comparison with adiabatic numerical simulations allows us to quantify the absolute value of the excess and to examine whether an entropy excess is also present for the most massive systems.', 'astro-ph-0508234-2-43-0': '[CITATION] shows the results of adiabatic numerical simulations of 30 clusters spanning a mass range of more than a factor of ten.', 'astro-ph-0508234-2-43-1': 'Once scaled by the characteristic entropy of the halo, [EQUATION] where [MATH] is the baryon fraction, the simulated profiles are closely self-similar, and can be well fitted in the radial range [MATH] by the power-law relation [MATH].', 'astro-ph-0508234-2-43-2': 'Assuming [MATH] and [MATH]) and typical elemental abundances, we can scale our observed entropy values to [MATH] using the expression [EQUATION]', 'astro-ph-0508234-2-43-3': 'Further scaling the radius by the measured value of [MATH] allows us to compare our results directly with the adiabatic simulations.', 'astro-ph-0508234-2-43-4': 'In the left hand panel of Figure [REF], our observed entropy profiles are compared with the prediction of [CITATION].', 'astro-ph-0508234-2-43-5': 'The right hand panel shows the ratio between the best-fitting adiabatic power-law relation and the observed data.', 'astro-ph-0508234-2-44-0': 'It can be seen that the richer systems all have entropies which are in good agreement (both in slope and normalisation) with the adiabatic prediction, denoted by the solid line in Fig. [REF].', 'astro-ph-0508234-2-44-1': 'On average, their entropy is only slightly higher than predicted (by [MATH] per cent), although the effect is not very significant.', 'astro-ph-0508234-2-44-2': 'We recall that there is also a [MATH] per cent difference in normalisation between the observed [MATH]-[MATH] relation and that predicted by adiabatic simulations .', 'astro-ph-0508234-2-44-3': 'Interestingly, this corresponds to a [MATH] per cent entropy excess at a given mass for [MATH].', 'astro-ph-0508234-2-44-4': 'The (slight) excess of entropy in massive systems is thus consistent with a simple increase of the mean temperature, i.e., of the internal energy of the ICM.', 'astro-ph-0508234-2-44-5': 'However, Fig. [REF] shows explicitly that the poorer systems have a systematically higher entropy normalisation than the richer systems.', 'astro-ph-0508234-2-44-6': 'There is approximately 2.5 times more entropy at [MATH] in the ICM of A1983, the poorest cluster in our sample, than that predicted by gravitational heating.', 'astro-ph-0508234-2-44-7': 'This excess shows that the density of the ICM is also affected at lower mass.', 'astro-ph-0508234-2-45-0': '[CITATION] and [CITATION] independently noted that the ICM entropy is highly sensitive to the density of the incoming gas and suggested that a smoothing of the gas density due to pre-heating in filaments and/or infalling groups would boost the entropy production at the accretion shock.', 'astro-ph-0508234-2-45-1': 'The entropy amplification is more efficient in low mass systems, because they accrete smaller halos more affected by smoothing due to pre-heating.', 'astro-ph-0508234-2-45-2': 'In this interpretation, no substantial isentropic core will develop because the amount of initial preheating needed is substantially less than the characteristic entropy of the final halo.', 'astro-ph-0508234-2-45-3': 'Clusters will thus be self-similar down to low mass but with a modified normalisation scaling as discussed above.', 'astro-ph-0508234-2-45-4': 'This is in agreement with our results outside the core region ([MATH]).', 'astro-ph-0508234-2-46-0': 'Recent numerical simulations which mimic preheating by imposing a minimum entropy floor at high [MATH] have confirmed the entropy amplification effect due to smooth accretion .', 'astro-ph-0508234-2-46-1': 'However, the effect seems to be substantially reduced when cooling is also taken into account.', 'astro-ph-0508234-2-46-2': 'Furthermore, the physical origin of the preheating is still unclear.', 'astro-ph-0508234-2-46-3': 'Heating by SNe seems to be too localised to have a significant effect in smoothing the accreting gas .', 'astro-ph-0508234-2-46-4': 'While AGN might be better candidates to produce the extra heating, the observed normalisations require that the AGN affects the entropy distribution at least up to [MATH].', 'astro-ph-0508234-2-46-5': 'Some recent theoretical investigations suggest that this may be possible .', 'astro-ph-0508234-2-47-0': '## Localised modification', 'astro-ph-0508234-2-48-0': 'While filamentary pre-heating may explain the increased normalisation of poor systems relative to hotter systems, it does not explain the increasing scatter towards the central regions in scaled profiles (Figs. [REF], [REF] and [REF]).', 'astro-ph-0508234-2-48-1': 'The adiabatic numerical simulations of [CITATION] show both a flattening of the slope and an increase in the dispersion of the scaled entropy profiles in the central regions ([MATH]).', 'astro-ph-0508234-2-48-2': 'However, the dispersion in our observed profiles ([MATH] per cent) greatly exceeds that of the simulations ([MATH] per cent, cf Fig. [REF] and Fig. 11 of ).', 'astro-ph-0508234-2-48-3': 'Six clusters out of our total sample of ten (A1991, A2597, A1068, A478, PKS0745 and A2204) have remarkably similar scaled entropy profiles, displaying power-law behaviour down to the smallest radii measured.', 'astro-ph-0508234-2-48-4': 'Fitting these scaled profiles in the radial range [MATH] with power-law using the BCES method, we find a slope of [MATH], with a dispersion of only 13 per cent.', 'astro-ph-0508234-2-48-5': 'This slope is very similar to that found by fitting the [MATH] range.', 'astro-ph-0508234-2-48-6': 'These six clusters all appear to host a bona fide cooling core, each having a central temperature decrement of a factor [MATH] .', 'astro-ph-0508234-2-48-7': 'Similar power-law profiles were found by in a sample of 13 cooling core clusters by [CITATION].', 'astro-ph-0508234-2-48-8': 'Strong radiative cooling thus appears to generate entropy profiles which display power-law behaviour down to very small radii (Fig. [REF]).', 'astro-ph-0508234-2-48-9': 'This is similar to the quasi-steady-state entropy profile in the models of [CITATION], which include radiative cooling.', 'astro-ph-0508234-2-49-0': 'The other four clusters in our sample are characterised by a smaller central temperature decrement, larger cooling times and shallower entropy profiles.', 'astro-ph-0508234-2-49-1': 'Clearly, some mechanism has modified the cooling history of these clusters.', 'astro-ph-0508234-2-49-2': 'Energy input from AGN is regularly invoked as a way of moderating cooling at the centres of galaxy clusters .', 'astro-ph-0508234-2-49-3': 'Our sample contains four clusters which have X-ray evidence for interactions between radio and X-ray plasma (A478, A2204, A2597, and PKS0745), and yet the entropy profiles of all of these clusters increase monotonically outward in the canonical fashion.', 'astro-ph-0508234-2-49-4': 'However, we note that Chandra observations of clusters with moderate to strong radio sources (including A478), show some evidence for a slightly shallower slope ([MATH]) but only in the very inner regions .', 'astro-ph-0508234-2-49-5': 'The effect of AGN activity on the entropy profile will depend on whether the energy input is catastrophic (i.e., occurring in strong bursts) or distributed (more moderated input).', 'astro-ph-0508234-2-49-6': 'It is clear that if the heating is catastrophic in nature, no cluster has yet been found in which this is evident at least from the point of view of the entropy (although see ).', 'astro-ph-0508234-2-49-7': 'The heating is likely more distributed, via e.g., weak shocks , thus preserving the generally increasing form of the entropy profile.', 'astro-ph-0508234-2-50-0': 'Merging events can result in substantial mixing of high and low entropy gas .', 'astro-ph-0508234-2-50-1': 'Relatively little attention has been given to this possibility in the literature (see, however, Fig. 5 of , Fig. 12 of and Fig. 6 of ).', 'astro-ph-0508234-2-50-2': 'Such redistribution of entropy will depend on the scale of the merger, whether the merger has disrupted the structure of the cool core, and the timescale for re-establishment of the cool core if disrupted.', 'astro-ph-0508234-2-50-3': 'In the current sample MKW9, A2717, A1413 and A1983 all have flatter core entropy profiles.', 'astro-ph-0508234-2-50-4': 'We note that the morphological information for the present sample would argue against recent merger activity in these clusters (; Papers I and II).', 'astro-ph-0508234-2-50-5': 'However, this does not rule out entropy modification due to a more ancient merger, particularly if the relaxation time is less than the cooling time .', 'astro-ph-0508234-2-51-0': '# Conclusions', 'astro-ph-0508234-2-52-0': 'The present sample of ten relaxed clusters constitutes the first with precise mass data and wide temperature coverage, allowing the detailed study of the scaling of ICM entropy with both temperature and mass.', 'astro-ph-0508234-2-52-1': 'The entropy profiles are sampled with good spatial resolution up to [MATH], allowing us to examine structural properties avoiding the stacking analysis and extrapolation on which previous ROSAT/ASCA works relied.', 'astro-ph-0508234-2-53-0': 'We have found that the entropy profiles of the present cluster sample are self-similar beyond [MATH], with a shape not significantly different from that expected from shock heating.', 'astro-ph-0508234-2-53-1': 'The entropy scaling relations are shallower than expected from standard self-similar models and adiabatic numerical simulations.', 'astro-ph-0508234-2-53-2': 'The slopes of the entropy scaling relations are independent of radius, reflecting the structural self-similarity.', 'astro-ph-0508234-2-53-3': 'The entropy-mass () relation is consistent with the observed entropy-temperature () and mass-temperature ([MATH]-[MATH]) relations.', 'astro-ph-0508234-2-53-4': 'The dispersion is smaller about the relation, reflecting the fact that the mass of a cluster is its most fundamental characteristic.', 'astro-ph-0508234-2-53-5': 'These results confirm the trends seen in smaller samples , and in stacking analysis of [MATH] data .', 'astro-ph-0508234-2-54-0': 'Comparison of our observed profiles with adiabatic numerical simulations has allowed us to quantify the entropy excess relative to that expected in pure gravitational collapse.', 'astro-ph-0508234-2-54-1': 'The excess is only [MATH] per cent in our highest-mass systems.', 'astro-ph-0508234-2-54-2': 'This is in line with a simple increase in the mean temperature consistent with the observed difference in normalisation between the observed [MATH]-[MATH] relation and that predicted by adiabatic simulations.', 'astro-ph-0508234-2-54-3': 'However, the excess in low mass systems can be of more than a factor of 2.5, indicating that the density of the ICM is also affected in these systems.', 'astro-ph-0508234-2-54-4': 'These characteristics are in qualitative agreement with a scenario in which entropy production is boosted at the accretion shock.', 'astro-ph-0508234-2-54-5': 'A plausible candidate mechanism is smoothing of the accreted gas due to preheating .', 'astro-ph-0508234-2-55-0': 'We have found new and compelling evidence for an increase in the entropy dispersion in the core regions [MATH] per cent at [MATH].', 'astro-ph-0508234-2-55-1': 'However the five cooling core clusters in our sample have remarkably self-similar power-law profiles with a dispersion of only 13 per cent between 0.01 and [MATH].', 'astro-ph-0508234-2-55-2': 'The observed increase in dispersion towards the central regions argues for localised entropy modification mechanisms.', 'astro-ph-0508234-2-55-3': 'We conclude that AGN activity and/or transient entropy modification due to merging events are good candidates for the modification of cooling in these clusters.', 'astro-ph-0508234-2-56-0': 'These will be interesting questions to address with numerical simulations.', 'astro-ph-0508234-2-56-1': 'On the observational side, a larger, unbiased, sample of clusters will undoubtedly provide greater insights.', 'astro-ph-0508234-2-57-0': 'GWP thanks E. Belsole and J.P. Henry for useful discussions, and acknowledges funding from a Marie Curie Intra-European Fellowship under the FP6 programme (Contract No. MEIF-CT-2003-500915).', 'astro-ph-0508234-2-57-1': 'EP acknowledges the financial support of CNES, the French Space Agency, and of the Leverhulme trust (UK).', 'astro-ph-0508234-2-57-2': 'The present work is based on observations obtained with XMM-Newton, an ESA science mission with instruments and contributions directly funded by ESA Member States and the USA (NASA).', 'astro-ph-0508234-2-58-0': '# Surface brightness profile fit results', 'astro-ph-0508234-2-59-0': 'The co-added EPIC azimuthally averaged, vignetting corrected, background subtracted surface brightness profile was computed in the [0.3-3.]', 'astro-ph-0508234-2-59-1': 'keV band for each cluster.', 'astro-ph-0508234-2-59-2': 'The profile was corrected for radial variations in the emissivity due to abundance or temperature gradients as described in [CITATION].', 'astro-ph-0508234-2-59-3': 'The gas density profile was obtained from fitting parametric analytic models, convolved with the EPIC PSF, to the corrected profile.', 'astro-ph-0508234-2-59-4': 'As discussed in Paper I, we tried various models and empirically chose the best fitting model using the [MATH] statistic as a measure of the goodness of fit.', 'astro-ph-0508234-2-59-5': 'The models used were: (i) the double [MATH] (BB) density model ; (ii) the modified double [MATH] (KBB) model, allowing a more concentrated gas density distribution towards the centre , and (iii) the sum of three [MATH] (3B) models for the emission measure, in which a common value of [MATH] is used to ensure smooth behaviour at large radii .', 'astro-ph-0508234-2-60-0': 'The BB and KBB models can be written: [EQUATION] where [MATH] for the BB model and [MATH] for the KBB model, and [EQUATION] with [EQUATION]', 'astro-ph-0508234-2-61-0': 'The density profile of the BBB model can be written: [EQUATION] with [EQUATION]', 'astro-ph-0508234-2-61-1': 'The best-fitting models, and model parameters, are listed in Table [REF].'}

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['astro-ph-0508234-1-0-2', 'astro-ph-0508234-2-0-2'], ['astro-ph-0508234-1-0-3', 'astro-ph-0508234-2-0-3'], ['astro-ph-0508234-1-0-4', 'astro-ph-0508234-2-0-4'], ['astro-ph-0508234-1-0-5', 'astro-ph-0508234-2-0-5'], ['astro-ph-0508234-1-0-6', 'astro-ph-0508234-2-0-6'], ['astro-ph-0508234-1-0-7', 'astro-ph-0508234-2-0-7'], ['astro-ph-0508234-1-0-8', 'astro-ph-0508234-2-0-8'], ['astro-ph-0508234-1-0-9', 'astro-ph-0508234-2-0-9'], ['astro-ph-0508234-1-0-10', 'astro-ph-0508234-2-0-10'], ['astro-ph-0508234-1-0-11', 'astro-ph-0508234-2-0-11'], ['astro-ph-0508234-1-29-0', 'astro-ph-0508234-2-29-0'], ['astro-ph-0508234-1-29-1', 'astro-ph-0508234-2-29-1'], ['astro-ph-0508234-1-29-2', 'astro-ph-0508234-2-29-2'], ['astro-ph-0508234-1-27-0', 'astro-ph-0508234-2-27-0'], ['astro-ph-0508234-1-27-1', 'astro-ph-0508234-2-27-1'], ['astro-ph-0508234-1-27-2', 'astro-ph-0508234-2-27-2'], ['astro-ph-0508234-1-27-3', 'astro-ph-0508234-2-27-3'], ['astro-ph-0508234-1-27-4', 'astro-ph-0508234-2-27-4'], ['astro-ph-0508234-1-27-5', 'astro-ph-0508234-2-27-5'], ['astro-ph-0508234-1-11-0', 'astro-ph-0508234-2-11-0'], ['astro-ph-0508234-1-11-1', 'astro-ph-0508234-2-11-1'], ['astro-ph-0508234-1-11-2', 'astro-ph-0508234-2-11-2'], ['astro-ph-0508234-1-14-1', 'astro-ph-0508234-2-14-1'], ['astro-ph-0508234-1-24-0', 'astro-ph-0508234-2-24-0'], ['astro-ph-0508234-1-24-1', 'astro-ph-0508234-2-24-1'], ['astro-ph-0508234-1-24-2', 'astro-ph-0508234-2-24-2'], ['astro-ph-0508234-1-24-3', 'astro-ph-0508234-2-24-3'], ['astro-ph-0508234-1-24-4', 'astro-ph-0508234-2-24-4'], ['astro-ph-0508234-1-24-5', 'astro-ph-0508234-2-24-5'], ['astro-ph-0508234-1-45-0', 'astro-ph-0508234-2-45-0'], ['astro-ph-0508234-1-45-1', 'astro-ph-0508234-2-45-1'], ['astro-ph-0508234-1-45-2', 'astro-ph-0508234-2-45-2'], ['astro-ph-0508234-1-45-4', 'astro-ph-0508234-2-45-4'], ['astro-ph-0508234-1-52-0', 'astro-ph-0508234-2-52-0'], ['astro-ph-0508234-1-52-1', 'astro-ph-0508234-2-52-1'], ['astro-ph-0508234-1-37-0', 'astro-ph-0508234-2-37-0'], ['astro-ph-0508234-1-37-1', 'astro-ph-0508234-2-37-1'], ['astro-ph-0508234-1-37-2', 'astro-ph-0508234-2-37-2'], ['astro-ph-0508234-1-42-0', 'astro-ph-0508234-2-42-0'], ['astro-ph-0508234-1-42-1', 'astro-ph-0508234-2-42-1'], ['astro-ph-0508234-1-17-0', 'astro-ph-0508234-2-17-0'], ['astro-ph-0508234-1-17-1', 'astro-ph-0508234-2-17-1'], ['astro-ph-0508234-1-17-4', 'astro-ph-0508234-2-17-4'], ['astro-ph-0508234-1-17-5', 'astro-ph-0508234-2-17-5'], ['astro-ph-0508234-1-31-0', 'astro-ph-0508234-2-31-0'], ['astro-ph-0508234-1-31-1', 'astro-ph-0508234-2-31-1'], ['astro-ph-0508234-1-31-2', 'astro-ph-0508234-2-31-2'], ['astro-ph-0508234-1-54-0', 'astro-ph-0508234-2-54-0'], ['astro-ph-0508234-1-54-1', 'astro-ph-0508234-2-54-1'], ['astro-ph-0508234-1-54-3', 'astro-ph-0508234-2-54-3'], ['astro-ph-0508234-1-54-4', 'astro-ph-0508234-2-54-4'], ['astro-ph-0508234-1-54-5', 'astro-ph-0508234-2-54-5'], ['astro-ph-0508234-1-20-0', 'astro-ph-0508234-2-20-0'], ['astro-ph-0508234-1-20-1', 'astro-ph-0508234-2-20-1'], ['astro-ph-0508234-1-20-2', 'astro-ph-0508234-2-20-2'], ['astro-ph-0508234-1-20-3', 'astro-ph-0508234-2-20-3'], ['astro-ph-0508234-1-20-4', 'astro-ph-0508234-2-20-4'], ['astro-ph-0508234-1-20-5', 'astro-ph-0508234-2-20-5'], ['astro-ph-0508234-1-20-6', 'astro-ph-0508234-2-20-6'], ['astro-ph-0508234-1-20-7', 'astro-ph-0508234-2-20-7'], ['astro-ph-0508234-1-20-8', 'astro-ph-0508234-2-20-8'], ['astro-ph-0508234-1-38-0', 'astro-ph-0508234-2-38-0'], ['astro-ph-0508234-1-38-1', 'astro-ph-0508234-2-38-1'], ['astro-ph-0508234-1-38-2', 'astro-ph-0508234-2-38-2'], ['astro-ph-0508234-1-38-3', 'astro-ph-0508234-2-38-3'], ['astro-ph-0508234-1-38-4', 'astro-ph-0508234-2-38-4'], ['astro-ph-0508234-1-38-5', 'astro-ph-0508234-2-38-5'], ['astro-ph-0508234-1-38-6', 'astro-ph-0508234-2-38-6'], ['astro-ph-0508234-1-40-0', 'astro-ph-0508234-2-40-0'], ['astro-ph-0508234-1-40-2', 'astro-ph-0508234-2-40-2'], ['astro-ph-0508234-1-40-3', 'astro-ph-0508234-2-40-3'], ['astro-ph-0508234-1-40-4', 'astro-ph-0508234-2-40-4'], ['astro-ph-0508234-1-48-0', 'astro-ph-0508234-2-48-0'], ['astro-ph-0508234-1-48-1', 'astro-ph-0508234-2-48-1'], ['astro-ph-0508234-1-48-2', 'astro-ph-0508234-2-48-2'], ['astro-ph-0508234-1-48-3', 'astro-ph-0508234-2-48-3'], ['astro-ph-0508234-1-48-4', 'astro-ph-0508234-2-48-4'], ['astro-ph-0508234-1-48-5', 'astro-ph-0508234-2-48-5'], ['astro-ph-0508234-1-48-7', 'astro-ph-0508234-2-48-7'], ['astro-ph-0508234-1-48-8', 'astro-ph-0508234-2-48-8'], ['astro-ph-0508234-1-48-9', 'astro-ph-0508234-2-48-9'], ['astro-ph-0508234-1-8-0', 'astro-ph-0508234-2-8-0'], ['astro-ph-0508234-1-8-1', 'astro-ph-0508234-2-8-1'], ['astro-ph-0508234-1-23-0', 'astro-ph-0508234-2-23-0'], ['astro-ph-0508234-1-23-1', 'astro-ph-0508234-2-23-1'], ['astro-ph-0508234-1-23-2', 'astro-ph-0508234-2-23-2'], ['astro-ph-0508234-1-23-3', 'astro-ph-0508234-2-23-3'], ['astro-ph-0508234-1-55-0', 'astro-ph-0508234-2-55-0'], ['astro-ph-0508234-1-55-1', 'astro-ph-0508234-2-55-1'], ['astro-ph-0508234-1-55-2', 'astro-ph-0508234-2-55-2'], ['astro-ph-0508234-1-55-3', 'astro-ph-0508234-2-55-3'], ['astro-ph-0508234-1-43-0', 'astro-ph-0508234-2-43-0'], ['astro-ph-0508234-1-43-1', 'astro-ph-0508234-2-43-1'], ['astro-ph-0508234-1-43-2', 'astro-ph-0508234-2-43-2'], ['astro-ph-0508234-1-43-3', 'astro-ph-0508234-2-43-3'], ['astro-ph-0508234-1-43-4', 'astro-ph-0508234-2-43-4'], ['astro-ph-0508234-1-43-5', 'astro-ph-0508234-2-43-5'], ['astro-ph-0508234-1-44-0', 'astro-ph-0508234-2-44-0'], ['astro-ph-0508234-1-44-1', 'astro-ph-0508234-2-44-1'], ['astro-ph-0508234-1-44-2', 'astro-ph-0508234-2-44-2'], ['astro-ph-0508234-1-44-3', 'astro-ph-0508234-2-44-3'], ['astro-ph-0508234-1-44-4', 'astro-ph-0508234-2-44-4'], ['astro-ph-0508234-1-44-5', 'astro-ph-0508234-2-44-5'], ['astro-ph-0508234-1-44-6', 'astro-ph-0508234-2-44-6'], ['astro-ph-0508234-1-44-7', 'astro-ph-0508234-2-44-7'], ['astro-ph-0508234-1-22-0', 'astro-ph-0508234-2-22-0'], ['astro-ph-0508234-1-22-1', 'astro-ph-0508234-2-22-1'], ['astro-ph-0508234-1-22-2', 'astro-ph-0508234-2-22-2'], ['astro-ph-0508234-1-22-3', 'astro-ph-0508234-2-22-3'], ['astro-ph-0508234-1-22-4', 'astro-ph-0508234-2-22-4'], ['astro-ph-0508234-1-22-5', 'astro-ph-0508234-2-22-5'], ['astro-ph-0508234-1-22-6', 'astro-ph-0508234-2-22-6'], ['astro-ph-0508234-1-22-7', 'astro-ph-0508234-2-22-7'], ['astro-ph-0508234-1-21-0', 'astro-ph-0508234-2-21-0'], ['astro-ph-0508234-1-21-1', 'astro-ph-0508234-2-21-1'], ['astro-ph-0508234-1-21-2', 'astro-ph-0508234-2-21-2'], ['astro-ph-0508234-1-21-3', 'astro-ph-0508234-2-21-3'], ['astro-ph-0508234-1-21-4', 'astro-ph-0508234-2-21-4'], ['astro-ph-0508234-1-21-5', 'astro-ph-0508234-2-21-5'], ['astro-ph-0508234-1-21-6', 'astro-ph-0508234-2-21-6'], ['astro-ph-0508234-1-21-7', 'astro-ph-0508234-2-21-7'], ['astro-ph-0508234-1-4-0', 'astro-ph-0508234-2-4-0'], ['astro-ph-0508234-1-4-2', 'astro-ph-0508234-2-4-2'], ['astro-ph-0508234-1-6-0', 'astro-ph-0508234-2-6-0'], ['astro-ph-0508234-1-6-1', 'astro-ph-0508234-2-6-1'], ['astro-ph-0508234-1-6-2', 'astro-ph-0508234-2-6-2'], ['astro-ph-0508234-1-6-3', 'astro-ph-0508234-2-6-3'], ['astro-ph-0508234-1-15-0', 'astro-ph-0508234-2-15-0'], ['astro-ph-0508234-1-15-1', 'astro-ph-0508234-2-15-1'], ['astro-ph-0508234-1-15-2', 'astro-ph-0508234-2-15-2'], ['astro-ph-0508234-1-15-3', 'astro-ph-0508234-2-15-3'], ['astro-ph-0508234-1-15-4', 'astro-ph-0508234-2-15-4'], ['astro-ph-0508234-1-15-6', 'astro-ph-0508234-2-15-6'], ['astro-ph-0508234-1-25-0', 'astro-ph-0508234-2-25-0'], ['astro-ph-0508234-1-25-1', 'astro-ph-0508234-2-25-1'], ['astro-ph-0508234-1-25-2', 'astro-ph-0508234-2-25-2'], ['astro-ph-0508234-1-50-0', 'astro-ph-0508234-2-50-0'], ['astro-ph-0508234-1-50-1', 'astro-ph-0508234-2-50-1'], ['astro-ph-0508234-1-50-2', 'astro-ph-0508234-2-50-2'], ['astro-ph-0508234-1-50-3', 'astro-ph-0508234-2-50-3'], ['astro-ph-0508234-1-50-4', 'astro-ph-0508234-2-50-4'], ['astro-ph-0508234-1-50-5', 'astro-ph-0508234-2-50-5'], ['astro-ph-0508234-1-18-1', 'astro-ph-0508234-2-18-1'], ['astro-ph-0508234-1-3-0', 'astro-ph-0508234-2-3-0'], ['astro-ph-0508234-1-3-1', 'astro-ph-0508234-2-3-1'], ['astro-ph-0508234-1-7-0', 'astro-ph-0508234-2-7-0'], ['astro-ph-0508234-1-7-1', 'astro-ph-0508234-2-7-1'], ['astro-ph-0508234-1-7-2', 'astro-ph-0508234-2-7-2'], ['astro-ph-0508234-1-35-0', 'astro-ph-0508234-2-35-0'], ['astro-ph-0508234-1-35-1', 'astro-ph-0508234-2-35-1'], ['astro-ph-0508234-1-35-3', 'astro-ph-0508234-2-35-3'], ['astro-ph-0508234-1-35-5', 'astro-ph-0508234-2-35-5'], ['astro-ph-0508234-1-57-2', 'astro-ph-0508234-2-57-2'], ['astro-ph-0508234-1-46-0', 'astro-ph-0508234-2-46-0'], ['astro-ph-0508234-1-46-1', 'astro-ph-0508234-2-46-1'], ['astro-ph-0508234-1-46-2', 'astro-ph-0508234-2-46-2'], ['astro-ph-0508234-1-34-0', 'astro-ph-0508234-2-34-0'], ['astro-ph-0508234-1-34-1', 'astro-ph-0508234-2-34-1'], ['astro-ph-0508234-1-34-2', 'astro-ph-0508234-2-34-2'], 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['astro-ph-0508234-1-1-0', 'astro-ph-0508234-1-57-0', 'astro-ph-0508234-2-1-0', 'astro-ph-0508234-2-57-0']

{'1': 'http://arxiv.org/licenses/assumed-1991-2003/', '2': 'http://arxiv.org/licenses/assumed-1991-2003/'}

https://arxiv.org/abs/astro-ph/0508234

1404.5740

{'1404.5740-1-0-0': 'Inflationary and hence quantum origin of primordial perturbations is on a firmer ground than ever post the BICEP2 observations of primordial gravitational waves.', '1404.5740-1-0-1': 'One crucial ingredient of success of this paradigm rests on explaining the observed classicality of cosmological inhomogeinities despite their quantum origin.', '1404.5740-1-0-2': 'Although decoherence provides a partial understanding of this issue, the question of single outcome motivates the analysis of quantum collapse models in cosmological context which generically modify the dynamics of primordial perturbations and hence can leave their imprints on observables.', '1404.5740-1-0-3': 'We revisit one such recently proposed working model of classicalization by spontaneous collapse [CITATION] in the light of BICEP2 observations to look for possible modifications to tensor power spectra and their implications.', '1404.5740-1-0-4': 'We show that it can potentially change the consistency relation of single-field models and a precise measurement of [MATH] and its running could serve as a test of such dynamics in the early universe.', '1404.5740-1-1-0': '# Introduction', '1404.5740-1-2-0': 'The recent observation of [MATH]mode polarization of Cosmic Microwave Background Radiation (CMBR) by BICEP2 [CITATION] has put the cosmological inflationary paradigm [CITATION] on stronger footing than ever by confirming one of its many observational predictions, namely the generation of primordial tensor modes or primordial gravitational waves.', '1404.5740-1-2-1': "Inflationary dynamics gives rise to both scalar and tensor fluctuations during inflation which then 'redshift' out of the horizon and freeze.", '1404.5740-1-2-2': 'These primordial scalar fluctuations, up on re-entering the horizon at later stages, give rise to [MATH] anisotropy spectrum of the CMBR that has been measured by many observations such as WMAP [CITATION] and PLANCK [CITATION].', '1404.5740-1-2-3': 'On the other hand, the tensor modes give rise to the [MATH] spectrum of CMBR which has recently been detected by BICEP2 [CITATION].', '1404.5740-1-3-0': 'Detection of [MATH]modes of CMBR is of profound importance as according to the simplest model of inflation it sets the scale of inflation in a unique way, which turns out to be surprisingly close to the Grand Unification (GUT) scale ([MATH] GeV).', '1404.5740-1-3-1': 'But this is not all what the detection of [MATH]modes by BICEP2 can provide.', '1404.5740-1-3-2': 'BICEP2 measurement happens to be clearly the first direct evidence of the quantum nature of gravity as these tensor perturbations generated during inflation were of quantum nature.', '1404.5740-1-3-3': 'BICEP2 has measured the power of these tensor modes over the scalar ones on large cosmological scales, called the tensor-to-scalar ratio [MATH] and the measured value of [MATH] by BICEP2 is [CITATION] [EQUATION]', '1404.5740-1-3-4': 'It has been argued in [CITATION] that if these primordial gravitational waves were of classical nature, it would have produced negligible tensor amplitude compared to the scalar one and the value of [MATH] would have at least been suppressed by an extra power of slow-roll parameter [MATH] in comparison to the standard scenario.', '1404.5740-1-3-5': 'That these primordial tensor modes are indeed of quantum nature has also been argued in other literature such as [CITATION].', '1404.5740-1-3-6': 'Primordial scalar perturbations are also generated quantum mechanically and this scenario is well supported by observations.', '1404.5740-1-4-0': 'From this discussion it is evident that the primordial modes, both scalar and tensor, are being generated quantum mechanically at very early times.', '1404.5740-1-4-1': 'But these quantum fluctuations are the ones which give rise to CMBR temperature fluctuations and its [MATH]mode polarization on large angular scales which are classical in nature.', '1404.5740-1-4-2': 'This leads us to the problem of quantum-to-classical transition in the cosmological context, which is a more serious form of the so-called "quantum measurement problem" which also prevails in laboratory systems.', '1404.5740-1-5-0': 'Heuristically, the classical nature of primordial perturbations is argued by the large occupation number of superhorizon modes and approximate vanishing of the commutator of the field variable and its conjugate momentum on superhorizon scales.', '1404.5740-1-5-1': 'However, the perturbations generically evolve into highly squeezed states [CITATION] on superhorizon scales, which are highly non-classical states.', '1404.5740-1-5-2': 'But the quantum nature is not directly evident in observations because it can be shown that quantum expectations of highly squeezed states are indistinguishable from average of a classical stochastic field.', '1404.5740-1-5-3': "This is so-called 'decoherence without decoherence' [CITATION].", '1404.5740-1-5-4': 'Furthermore decoherence, despite the ambiguity in system and environment spilt for a cosmological scenario, selects the field amplitude basis as the pointer basis of the system and justifies the standard calculation.', '1404.5740-1-5-5': 'At this point, what still remains unresolved is the issue of single outcome.', '1404.5740-1-5-6': 'This problem is also present in laboratory systems and only gets more intriguing in cosmological context [CITATION].', '1404.5740-1-5-7': 'One possible way out is to appeal to the many-worlds interpretation.', '1404.5740-1-5-8': 'The other alternative is the so-called collapse models which we explore in this paper.', '1404.5740-1-5-9': 'Just as in laboratory systems, it is important to investigate whether and how collapse models can be distinguished from the standard quantum mechanical setup in a cosmological context.', '1404.5740-1-6-0': 'In a generic collapse model Schrodinger equation is modified by adding stochastic and non-linear terms.', '1404.5740-1-6-1': 'The stochastic nature of the equation helps explaining the probabilistic outcome of quantum measurements without allowing for superluminal communication and the presence of non-linear terms breaks down the underlying superposition principle of quantum mechanics.', '1404.5740-1-6-2': 'For a detailed review on collapse models refer to [CITATION].', '1404.5740-1-6-3': 'Though a proper field theoretic treatment of collapse dynamics is not yet known, a few attempts have been made to apply such collapse models, especially Continuous Spontaneous Localization (CSL) model [CITATION], into inflationary dynamics to resolve the problem of quantum to classical transition in cosmological context, such as [CITATION], [CITATION], [CITATION].', '1404.5740-1-6-4': 'As these collapse models explicitly modify the standard dynamics, we can anticipate that its observational implications would diverge from the standard dynamics and the aim of this brief paper is to determine how and where these collapse mechanisms differ from the standard scenario observationally within the context of an illustrative example in the light of recent BICEP observation.', '1404.5740-1-7-0': '# Generic Single-field slow-roll inflation and its observational implications', '1404.5740-1-8-0': 'The most economic model of inflation is the so-called slow-roll single field model.', '1404.5740-1-8-1': 'The power spectra for scalar and tensor perturbations in such a model can be given as [CITATION] [EQUATION] respectively, where [MATH] is called the pivot scale and for PLANCK it is chosen to be [MATH] Mpc[MATH].', '1404.5740-1-8-2': 'It is to be noted that as the comoving curvature perturbations, denoted by [MATH], and the tensor modes, denoted by [MATH], freeze on superhorizon scales, it is customary to derive the power spectra given above at horizon crossing of each mode [MATH] during inflation.', '1404.5740-1-8-3': 'In the above equations [MATH] is the Hubble parameter during inflation, [MATH] GeV is the reduced Planck mass, [MATH] is the first Hubble slow-roll parameter defined as [EQUATION] [MATH] and [MATH] are the scalar spectral index and the tensor spectral index respectively which are the measures of the scale dependence of the respective spectrum and for this generic slow-roll single-field model turn out to be [EQUATION] where [MATH] is the second Hubble slow-roll parameter defined as [EQUATION] where [MATH] is the inflaton field.', '1404.5740-1-9-0': 'Each of the tensor modes, which are the traceless and transverse part of the metric fluctuations, is associated with two helicity states, often denoted as + and [MATH] polarization.', '1404.5740-1-9-1': 'As has been first pointed out by Grishchuk in [CITATION], the evolution of the Fourier modes of each of the helicity states of these tensor modes is identical to that of a massless scalar in de Sitter space with the correspondence [EQUATION] where [MATH] is the helicity states and [MATH] is the re-defined tensor modes.', '1404.5740-1-9-2': 'Hence the evolution of quantum tensor modes can be reduced to that of two decoupled massless scalar modes.', '1404.5740-1-9-3': 'We will make use of this fact while dealing with CSL-modified inflationary dynamics.', '1404.5740-1-10-0': 'Now, let us consider the observational implications of this simplest model of inflation.', '1404.5740-1-10-1': 'PLANCK measures the amplitude [MATH] and the scalar spectral index [MATH] as [CITATION] [EQUATION] from the [MATH] anisotropy spectrum of the CMBR.', '1404.5740-1-10-2': 'BICEP2, by detecting the [MATH]polarization of CMBR measured tensor-to-scalar ratio [MATH], defined as [EQUATION] to be [MATH].', '1404.5740-1-10-3': 'According to the simplest single field model the tensor-to-scalar ratio turns out to be of the order of slow-roll parameter: [EQUATION] which then yields the consistency relation of this single-field model given as [EQUATION]', '1404.5740-1-10-4': 'Thus independent measurements of [MATH] and [MATH] can unambiguously determine whether the inflationary dynamics was indeed that simple or not.', '1404.5740-1-11-0': 'The recent observation of [MATH] by BICEP2 and the measurement of scalar amplitude [MATH] by PLANCK indirectly provides the amplitude of the tensor perturbations.', '1404.5740-1-11-1': 'But one can see from Eq. ([REF]) that the amplitude of the tensor power solely depends on the Hubble parameter during inflation and considering the central values of observed [MATH] and [MATH] one gets the Hubble parameter during inflation as [EQUATION]', '1404.5740-1-11-2': 'Now, during inflationary era the universe is dominated by the potential energy of the inflaton field and thus the Friedmann equation during inflation is written as [EQUATION]', '1404.5740-1-11-3': 'Hence the BICEP2 measurement of [MATH] also sets the scale of inflation as [EQUATION] which is very close to the GUT scale.', '1404.5740-1-12-0': 'Another implication of the BICEP2 result comes from the Lyth bound [CITATION].', '1404.5740-1-12-1': 'Writing the first slow-roll parameters [MATH] as [EQUATION] one can determine the field excursion during the inflation as [EQUATION] which shows that the inflaton field excursion is super-Planckian during inflation if [MATH], as has been observed by BICEP2.', '1404.5740-1-12-2': 'This leads to some tension with effective field theory description of inflation as the field excursion becomes of the same order of the natural cutoff scale.', '1404.5740-1-13-0': '# CSL-modified single-field dynamics and its observational implications', '1404.5740-1-14-0': '## Scalar perturbations', '1404.5740-1-15-0': 'For this analysis we would keep our focus on the derivations done in [CITATION].', '1404.5740-1-15-1': 'The collapse models modify the dynamics of particles in the Schrodinger picture.', '1404.5740-1-15-2': 'Hence while applying the CSL modifications to the inflationary dynamics one requires to analyze the field dynamics in the Schrodinger picture.', '1404.5740-1-15-3': 'An elaborate analysis of Schrodinger picture dynamics of scalar perturbations during inflation is given in [CITATION].', '1404.5740-1-15-4': 'We would study the primordial scalar perturbations during inflation in terms of the so-called Mukhanov-Sakasi variable, a gauge-invariant quantity denoted by [MATH] , which is related to the comoving curvature perturbation as [EQUATION]', '1404.5740-1-15-5': 'In Schrodinger picture, the standard scalar perturbation is analyzed in terms of its wave-functional [CITATION] defined as [EQUATION] where we have [EQUATION]', '1404.5740-1-15-6': 'These wave functionals satisfy the functional Schrodinger equation as [EQUATION] where the Hamiltonian [MATH] and the ground state solution of the functional Schrodinger equation is written as [EQUATION]', '1404.5740-1-15-7': 'Here [MATH] is related to the mode functions [MATH] in the Heisenberg picture as [EQUATION]', '1404.5740-1-15-8': 'Thus, in Schrodinger picture the power spectrum of [MATH] turns out to be [EQUATION] and thus the power spectrum of the comoving curvature perturbations can be obtained as [EQUATION]', '1404.5740-1-15-9': 'It has been proposed in [CITATION] that CSL-like modifications can be applied to the inflationary perturbations directly in the Fourier space by adding CSL-like non-linear and stochastic terms to the functional Schrodinger equation of [MATH] which then looks like [EQUATION] where the stochastic behavior due to CSL mechanism is encoded in the Wiener process [MATH] and [MATH] is called the collapse parameter.', '1404.5740-1-15-10': 'The most general stochastic wave-functional which satisfies this stochastic functional Schrodinger equation can be written as [EQUATION] where [MATH], [MATH] and [MATH] are real numbers.', '1404.5740-1-16-0': 'In [CITATION], the collapse parameter [MATH] was taken to be constant and it was inferred that such a case is not capable of explaining the quantum-to-classical transition of primordial modes.', '1404.5740-1-16-1': 'Then it was argued in [CITATION] that taking [MATH] to be constant the scenario loses one of the crucial features of CSL dynamics, known as amplification mechanism.', '1404.5740-1-16-2': 'In CSL-modified quantum mechanics, the collapse parameter is taken to be directly proportional to the mass of the system and its number density reflecting the fact that heavier objects become classical faster than the lighter ones.', '1404.5740-1-16-3': 'Similarly, in tune with the expectation that superhorizon modes behave classically one can demand in the cosmological context that the collapse parameter should become stronger as a generic mode starts to cross the horizon facilitating its collapse to one of its field eigenstates.', '1404.5740-1-16-4': 'Hence a phenomenological form of the collapse parameter was proposed in [CITATION] as [EQUATION] where [MATH].', '1404.5740-1-16-5': 'It was also shown in [CITATION] that with [MATH] the CSL-modified scalar dynamics successfully explains the quantum-to-classical transition of primordial scalar modes without destroying the phase coherence of the superhorizon modes essential to explain the peaks and troughs of the CMBR anisotropy spectrum.', '1404.5740-1-17-0': 'Now, let us calculate the power spectrum in this scenario.', '1404.5740-1-17-1': 'In such a case it has been calculated in [CITATION] that [EQUATION] which indicates that the power spectrum would not be time-invariant on superhorizon scale unlike the standard scenario.', '1404.5740-1-17-2': 'This feature has been observed in both [CITATION] and as a cure to it the power was calculated at the end of inflation and not at the horizon-crossing of each mode by using [EQUATION] where [MATH] is the comoving wavenumber of the mode which is at the horizon today [MATH] and [MATH] is the number of efolds the mode has spent outside the horizon after its exit and thus for observationally relevant modes [MATH].', '1404.5740-1-17-3': 'We would henceforth consider [MATH].', '1404.5740-1-17-4': 'Also to make the power spectrum nearly scale-invariant we chose the scale-dependence of [MATH] as [EQUATION]', '1404.5740-1-17-5': 'Then the power spectrum of comoving curvature perturbations turns out to be [EQUATION]', '1404.5740-1-17-6': 'Incorporating the correction to the tilt in the spectrum due to quasi-de Sitter evolution of the background, the power spectrum gets modified to [EQUATION] where the power is now calculated at the end of inflation.', '1404.5740-1-18-0': '## Tensor perturbations', '1404.5740-1-19-0': 'The tensor perturbations would now be straightforward to calculate once the scalar analysis is done as each of the helicity components of the Fourier tensor mode behaves like massless scalar perturbations, as discussed before.', '1404.5740-1-19-1': 'It should be noted that tensor perturbations are gauge-invariant by construction and the redefined tensor perturbations [MATH] of Eq. ([REF]) can be identified as the Mukhanov-Sasaki variable of scalar perturbations defined in the previous section.', '1404.5740-1-19-2': 'Hence in the Schrodinger picture each helicity component [MATH] of the tensor modes can be expressed as functionals given in Eq. ([REF]) following similar functional Schrodinger equation given in Eq. ([REF]) in the standard scenario.', '1404.5740-1-19-3': 'The ground state solutions of the functional Schrodinger equation would also be Gaussian as given in Eq. ([REF]).', '1404.5740-1-19-4': 'Similarly the power spectrum of [MATH] would be same as of [MATH] given in Eq. ([REF]) and can be written as [EQUATION] and following Eq. ([REF]) one can write down the power spectrum for the tensor modes as [EQUATION]', '1404.5740-1-19-5': 'At this point, we assume CSL collapse mechanism affects each helicity mode of the gravitons the same way as it affects the inflatons.', '1404.5740-1-19-6': 'This is the simplest scenario to imagine and is in conformity with the philosophy that the collapse mechanism should be universal in nature.', '1404.5740-1-19-7': 'Since in our way of implementing the collapse mechanism we essentially use the fact that each mode is an independent harmonic oscillator to modify the equation of motion, there is no reason to expect that this modification should be sensitive to details of the underlying nature of the field.', '1404.5740-1-19-8': 'Hence the CSL-modified dynamics of each helicity mode of the gravitons would be same as that of the massless inflatons or the gauge-invariant Mukhanov-Sasaki variable which has been considered in the previous section.', '1404.5740-1-19-9': 'For redefined tensor modes [MATH], [MATH] would have the same form as given in Eq. ([REF]), following which the power spectrum of the tensor modes can be determined as [EQUATION] and considering the tilt in the power due to quasi-de Sitter background evolution one gets [EQUATION]', '1404.5740-1-20-0': '## Observables', '1404.5740-1-21-0': 'Let us now illustrate how the CSL-modified primordial dynamics differ from the standard one observationally.', '1404.5740-1-21-1': 'The first thing to note from Eq. ([REF]) and Eq. ([REF]) is that the tensor-to-scalar ratio remain the same in the modified dynamics: [EQUATION] which indicates from the Lyth bound that the field excursion during inflation would be super-Planckian even in this case.', '1404.5740-1-21-2': 'The scalar spectral index and the tensor spectral index would now become (from Eq. ([REF]) and Eq. ([REF])) [EQUATION] where we have defined [MATH].', '1404.5740-1-21-3': 'We note here that, although the model has three free parameters [MATH], [MATH] and [MATH] to begin with, the modification to spectral indices can be captured in one effective free parameter [MATH].', '1404.5740-1-21-4': 'The observation of the scalar spectral index by PLANCK indicates that the quantity [MATH] can at best be of the order of slow-roll parameters so that the comoving curvature power spectrum remains to be nearly scale-invariant.', '1404.5740-1-22-0': 'We also note from Eq. ([REF]) that the tensor amplitude in such a case does not remain to be a sole function of Hubble parameter, the prime feature which is used to determine the scale of inflation using the BICEP2 observations.', '1404.5740-1-22-1': 'Even though, if we consider that inflation has indeed taken place at that high scale, then that would help estimating the collapse parameter as [EQUATION] which turns out to be extremely small.', '1404.5740-1-22-2': 'A stronger collapse mechanism can then bring down the scale of inflation even though the field excursions would remain to be super-Planckian.', '1404.5740-1-23-0': 'Most interestingly what this modified dynamics does is to change the consistency relation of the single-field models.', '1404.5740-1-23-1': 'In such a scenario the consistency relation turns out to be [EQUATION]', '1404.5740-1-23-2': 'Hence independent accurate measurements of [MATH] and [MATH] would give us a direct handle on [MATH] in this model.', '1404.5740-1-24-0': '# Discussion and Conclusion', '1404.5740-1-25-0': 'Belief in quantum nature of primordial perturbations makes it essential to understand the apparent classicality of inhomogeneity it gives rise to.', '1404.5740-1-25-1': 'The squeezing of the superhorizon modes and decoherence partially explain this conundrum.', '1404.5740-1-25-2': 'But the single outcome problem cannot be addressed without appealing to either many-worlds interpretation or collapse mechanisms.', '1404.5740-1-25-3': 'Collapse mechanisms generically modify the dynamics of the primordial perturbations and so are expected to leave their imprints on cosmological observables.', '1404.5740-1-25-4': 'In [CITATION] some toy models of CSL applied to inflationary dynamics were explored.', '1404.5740-1-25-5': 'In [CITATION], within the context of an illustrative example, it was shown that classicality of perturbations can be achieved while still preserving scale-invariance and phase coherence for a certain parameter range.', '1404.5740-1-25-6': 'In the light of BICEP results which report high enough tensor-to-scalar ratio, inconsistent with purely classical dynamics of the primordial tensor modes [CITATION], we revisit this model to investigate its observational consequences.', '1404.5740-1-26-0': 'The first point to note is that in this illustrative example the tensor-to-scalar ratio remains unchanged indicating super-Planckian field excursions during inflation as demanded by Lyth bound [CITATION].', '1404.5740-1-26-1': 'Secondly, the tensor amplitude no longer directly yields the scale of inflation as the collapse parameter also enters in this conversion.', '1404.5740-1-26-2': 'A stronger collapse parameter would bring down the scale of inflation.', '1404.5740-1-26-3': 'Unfortunately, so far there is no other theoretical or observational guide to estimate the scale of collapse parameter.', '1404.5740-1-26-4': 'Furthermore, the spectral tilts get modified by the one and the same combination of the free parameters of the model which eventually also change the consistency relation of the standard single-field scenario by capturing the deviation from this in one effective free parameter [MATH].', '1404.5740-1-26-5': "From PLANCK's observation of the scalar spectral index, the free parameter [MATH] can at best be of the order of slow-roll parameters.", '1404.5740-1-26-6': 'It is also in principle possible that [MATH] be identically zero in which case the collapse mechanism achieves the required classicality without leaving any imprint on observations.', '1404.5740-1-26-7': 'All the same, it is more reasonable to expect [MATH] to be non-zero which then would make this model testable by observations.', '1404.5740-1-26-8': 'One important feature which distinguishes this kind of modification from other scenarios which also modify the consistency relation like curvaton [CITATION] or multifield [CITATION] models is that generically such extensions of the minimal model modify the scalar sector while leaving the tensor sector untouched, however the collapse models modify both and hence are potentially distinguishable by precision measurement of [MATH].', '1404.5740-1-26-9': 'However modification to initial conditions of tensor modes, i.e. deviation from Bunch-Davies vacuum would also modify the tensor spectral tilt [CITATION] reflecting the scale-dependence of Bogoliubov coefficients.', '1404.5740-1-26-10': 'It is possible to arrange this scale-dependene in such a way as to shift the tensor spectral index by a constant, thus mimicking the effect of the collapse mechanism considered here.', '1404.5740-1-26-11': 'But in principle, the generic scenario of non-Bunch Davies initial condition should be distingushible from the collapse scenario.', '1404.5740-1-26-12': 'This would require precision measurement of running of tensor spectral index which has been argued to be possible in the near future [CITATION].', '1404.5740-1-27-0': 'The work of TPS is supported by a John Templeton Foundation grant [39530].', '1404.5740-1-27-1': 'SD and SS would like to thank Kinjalk Lochan for useful discussions.'}

{'1404.5740-2-0-0': 'Inflationary and hence quantum origin of primordial perturbations is on a firmer ground than ever post the BICEP2 observations of primordial gravitational waves.', '1404.5740-2-0-1': 'One crucial ingredient of success of this paradigm rests on explaining the observed classicality of cosmological inhomogeinities despite their quantum origin.', '1404.5740-2-0-2': 'Although decoherence provides a partial understanding of this issue, the question of single outcome motivates the analysis of quantum collapse models in cosmological context which generically modify the dynamics of primordial perturbations and hence can leave their imprints on observables.', '1404.5740-2-0-3': 'We revisit one such recently proposed working model of classicalization by spontaneous collapse [CITATION] in the light of BICEP2 observations to look for possible modifications to tensor power spectra and their implications.', '1404.5740-2-0-4': 'We show that it can potentially change the consistency relation of single-field models and a precise measurement of [MATH] and its running could serve as a test of such dynamics in the early universe.', '1404.5740-2-1-0': '# Introduction', '1404.5740-2-2-0': 'The recent observation of [MATH]mode polarization of Cosmic Microwave Background Radiation (CMBR) by BICEP2 [CITATION] has put the cosmological inflationary paradigm [CITATION] on stronger footing than ever by confirming one of its many observational predictions, namely the generation of primordial tensor modes or primordial gravitational waves.', '1404.5740-2-2-1': "Inflationary dynamics gives rise to both scalar and tensor fluctuations during inflation which then 'redshift' out of the horizon and freeze.", '1404.5740-2-2-2': 'These primordial scalar fluctuations, up on re-entering the horizon at later stages, give rise to [MATH] anisotropy spectrum of the CMBR that has been measured by many observations such as WMAP [CITATION] and PLANCK [CITATION].', '1404.5740-2-2-3': 'On the other hand, the tensor modes give rise to the [MATH] spectrum of CMBR which has recently been detected by BICEP2 [CITATION].', '1404.5740-2-3-0': 'Detection of [MATH]modes of CMBR is of profound importance as according to the simplest model of inflation it sets the scale of inflation in a unique way, which turns out to be surprisingly close to the Grand Unification (GUT) scale ([MATH] GeV).', '1404.5740-2-3-1': 'But this is not all what the detection of [MATH]modes by BICEP2 can provide.', '1404.5740-2-3-2': 'BICEP2 measurement happens to be clearly the first direct evidence of the quantum nature of gravity as these tensor perturbations generated during inflation were of quantum nature.', '1404.5740-2-3-3': 'BICEP2 has measured the power of these tensor modes over the scalar ones on large cosmological scales, called the tensor-to-scalar ratio [MATH] and the measured value of [MATH] by BICEP2 is [CITATION] [EQUATION]', '1404.5740-2-3-4': 'It has been argued in [CITATION] that if these primordial gravitational waves were of classical nature, it would have produced negligible tensor amplitude compared to the scalar one and the value of [MATH] would have at least been suppressed by an extra power of slow-roll parameter [MATH] in comparison to the standard scenario.', '1404.5740-2-3-5': 'That these primordial tensor modes are indeed of quantum nature has also been argued in other literature such as [CITATION].', '1404.5740-2-3-6': 'Primordial scalar perturbations are also generated quantum mechanically and this scenario is well supported by observations.', '1404.5740-2-4-0': 'From this discussion it is evident that the primordial modes, both scalar and tensor, are being generated quantum mechanically at very early times.', '1404.5740-2-4-1': 'But these quantum fluctuations are the ones which give rise to CMBR temperature fluctuations and its [MATH]mode polarization on large angular scales which are classical in nature.', '1404.5740-2-4-2': 'This leads us to the problem of quantum-to-classical transition in the cosmological context, which is a more serious form of the so-called "quantum measurement problem" which also prevails in laboratory systems.', '1404.5740-2-5-0': 'Heuristically, the classical nature of primordial perturbations is argued by the large occupation number of superhorizon modes and effective irrelevance of the commutator of the field variable and its conjugate momentum on superhorizon scales.', '1404.5740-2-5-1': 'However, the perturbations generically evolve into highly squeezed states [CITATION] on superhorizon scales, which are highly non-classical states.', '1404.5740-2-5-2': 'But the quantum nature is not directly evident in observations because it can be shown that quantum expectations of highly squeezed states are indistinguishable from average of a classical stochastic field.', '1404.5740-2-5-3': "This is so-called 'decoherence without decoherence' [CITATION].", '1404.5740-2-5-4': 'Furthermore decoherence, despite the ambiguity in system and environment spilt for a cosmological scenario, selects the field amplitude basis as the pointer basis of the system and justifies the standard calculation.', '1404.5740-2-5-5': 'At this point, what still remains unresolved is the issue of single outcome.', '1404.5740-2-5-6': 'This problem is also present in laboratory systems and only gets more intriguing in cosmological context [CITATION].', '1404.5740-2-5-7': 'One possible way out is to appeal to the many-worlds interpretation.', '1404.5740-2-5-8': 'The other alternative is the so-called collapse models which we explore in this paper.', '1404.5740-2-5-9': 'Just as in laboratory systems, it is important to investigate whether and how collapse models can be distinguished from the standard quantum mechanical setup in a cosmological context.', '1404.5740-2-6-0': 'In a generic collapse model Schrodinger equation is modified by adding stochastic and non-linear terms.', '1404.5740-2-6-1': 'The stochastic nature of the equation helps explaining the probabilistic outcome of quantum measurements without allowing for superluminal communication and the presence of non-linear terms breaks down the underlying superposition principle of quantum mechanics.', '1404.5740-2-6-2': 'For a detailed review on collapse models refer to [CITATION].', '1404.5740-2-6-3': 'Though a proper field theoretic treatment of collapse dynamics is not yet known, a few attempts have been made to apply such collapse models, especially Continuous Spontaneous Localization (CSL) model [CITATION], into inflationary dynamics to resolve the problem of quantum to classical transition in cosmological context, such as [CITATION], [CITATION], [CITATION].', '1404.5740-2-6-4': 'As these collapse models explicitly modify the standard dynamics, we can anticipate that its observational implications would diverge from the standard dynamics and the aim of this brief paper is to determine how and where these collapse mechanisms differ from the standard scenario observationally within the context of an illustrative example in the light of recent BICEP observation.', '1404.5740-2-7-0': '# Generic Single-field slow-roll inflation and its observational implications', '1404.5740-2-8-0': 'The most economic model of inflation is the so-called slow-roll single field model.', '1404.5740-2-8-1': 'The power spectra for scalar and tensor perturbations in such a model can be given as [CITATION] [EQUATION] respectively, where [MATH] is called the pivot scale and for PLANCK it is chosen to be [MATH] Mpc[MATH].', '1404.5740-2-8-2': 'It is to be noted that as the comoving curvature perturbations, denoted by [MATH], and the tensor modes, denoted by [MATH], freeze on superhorizon scales, it is customary to derive the power spectra given above at horizon crossing of each mode [MATH] during inflation.', '1404.5740-2-8-3': 'In the above equations [MATH] is the Hubble parameter during inflation, [MATH] GeV is the reduced Planck mass, [MATH] is the first Hubble slow-roll parameter defined as [EQUATION] [MATH] and [MATH] are the scalar spectral index and the tensor spectral index respectively which are the measures of the scale dependence of the respective spectrum and for this generic slow-roll single-field model turn out to be [EQUATION] where [MATH] is the second Hubble slow-roll parameter defined as [EQUATION] where [MATH] is the inflaton field.', '1404.5740-2-9-0': 'Each of the tensor modes, which are the traceless and transverse part of the metric fluctuations, is associated with two helicity states, often denoted as + and [MATH] polarization.', '1404.5740-2-9-1': 'As has been first pointed out by Grishchuk in [CITATION], the evolution of the Fourier modes of each of the helicity states of these tensor modes is identical to that of a massless scalar in de Sitter space with the correspondence [EQUATION] where [MATH] is the helicity states and [MATH] is the re-defined tensor modes.', '1404.5740-2-9-2': 'Hence the evolution of quantum tensor modes can be reduced to that of two decoupled massless scalar modes.', '1404.5740-2-9-3': 'We will make use of this fact while dealing with CSL-modified inflationary dynamics.', '1404.5740-2-10-0': 'Now, let us consider the observational implications of this simplest model of inflation.', '1404.5740-2-10-1': 'PLANCK measures the amplitude [MATH] and the scalar spectral index [MATH] as [CITATION] [EQUATION] from the [MATH] anisotropy spectrum of the CMBR.', '1404.5740-2-10-2': 'BICEP2, by detecting the [MATH]polarization of CMBR measured tensor-to-scalar ratio [MATH], defined as [EQUATION] to be [MATH].', '1404.5740-2-10-3': 'According to the simplest single field model the tensor-to-scalar ratio turns out to be of the order of slow-roll parameter: [EQUATION] which then yields the consistency relation of this single-field model given as [EQUATION]', '1404.5740-2-10-4': 'Thus independent measurements of [MATH] and [MATH] can unambiguously determine whether the inflationary dynamics was indeed that simple or not.', '1404.5740-2-11-0': 'The recent observation of [MATH] by BICEP2 and the measurement of scalar amplitude [MATH] by PLANCK indirectly provides the amplitude of the tensor perturbations.', '1404.5740-2-11-1': 'But one can see from Eq. ([REF]) that the amplitude of the tensor power solely depends on the Hubble parameter during inflation and considering the central values of observed [MATH] and [MATH] one gets the Hubble parameter during inflation as [EQUATION]', '1404.5740-2-11-2': 'Now, during inflationary era the universe is dominated by the potential energy of the inflaton field and thus the Friedmann equation during inflation is written as [EQUATION]', '1404.5740-2-11-3': 'Hence the BICEP2 measurement of [MATH] also sets the scale of inflation as [EQUATION] which is very close to the GUT scale.', '1404.5740-2-11-4': 'However, inferring the scale of inflation from tensor amplitude becomes more involved if one invokes large extra dimensions [CITATION] or classical sources of primordial gravitational waves in a multi-field scenario [CITATION].', '1404.5740-2-12-0': 'Another implication of the BICEP2 result comes from the Lyth bound [CITATION].', '1404.5740-2-12-1': 'Writing the first slow-roll parameters [MATH] as [EQUATION] one can determine the field excursion during the inflation as [EQUATION] which shows that the inflaton field excursion is super-Planckian during inflation if [MATH], as has been observed by BICEP2.', '1404.5740-2-12-2': 'This leads to some tension with effective field theory description of inflation as the field excursion becomes of the same order of the natural cutoff scale .', '1404.5740-2-13-0': '# CSL-modified single-field dynamics and its observational implications', '1404.5740-2-14-0': '## Scalar perturbations', '1404.5740-2-15-0': 'For this analysis we would keep our focus on the derivations done in [CITATION].', '1404.5740-2-15-1': 'The collapse models modify the dynamics of particles in the Schrodinger picture.', '1404.5740-2-15-2': 'Hence while applying the CSL modifications to the inflationary dynamics one requires to analyze the field dynamics in the Schrodinger picture.', '1404.5740-2-15-3': 'An elaborate analysis of Schrodinger picture dynamics of scalar perturbations during inflation is given in [CITATION].', '1404.5740-2-15-4': 'We would study the primordial scalar perturbations during inflation in terms of the so-called Mukhanov-Sakasi variable, a gauge-invariant quantity denoted by [MATH] , which is related to the comoving curvature perturbation as [EQUATION]', '1404.5740-2-15-5': 'In Schrodinger picture, the standard scalar perturbation is analyzed in terms of its wave-functional [CITATION] defined as [EQUATION] where we have [EQUATION]', '1404.5740-2-15-6': 'These wave functionals satisfy the functional Schrodinger equation as [EQUATION] where the Hamiltonian [MATH] and the ground state solution of the functional Schrodinger equation is written as [EQUATION]', '1404.5740-2-15-7': 'Here [MATH] is related to the mode functions [MATH] in the Heisenberg picture as [EQUATION]', '1404.5740-2-15-8': 'Thus, in Schrodinger picture the power spectrum of [MATH] turns out to be [EQUATION] and thus the power spectrum of the comoving curvature perturbations can be obtained as [EQUATION]', '1404.5740-2-15-9': 'It has been proposed in [CITATION] that CSL-like modifications can be applied to the inflationary perturbations directly in the Fourier space by adding CSL-like non-linear and stochastic terms to the functional Schrodinger equation of [MATH] which then looks like [EQUATION] where the stochastic behavior due to CSL mechanism is encoded in the Wiener process [MATH] and [MATH] is called the collapse parameter.', '1404.5740-2-15-10': 'The most general stochastic wave-functional which satisfies this stochastic functional Schrodinger equation can be written as [EQUATION] where [MATH], [MATH] and [MATH] are real numbers.', '1404.5740-2-16-0': 'In [CITATION], the collapse parameter [MATH] was taken to be constant and it was inferred that such a case is not capable of explaining the quantum-to-classical transition of primordial modes.', '1404.5740-2-16-1': 'Then it was argued in [CITATION] that taking [MATH] to be constant the scenario loses one of the crucial features of CSL dynamics, known as amplification mechanism.', '1404.5740-2-16-2': 'In CSL-modified quantum mechanics, the collapse parameter is taken to be directly proportional to the mass of the system and its number density reflecting the fact that heavier objects become classical faster than the lighter ones.', '1404.5740-2-16-3': 'Similarly, in tune with the expectation that superhorizon modes behave classically one can demand in the cosmological context that the collapse parameter should become stronger as a generic mode starts to cross the horizon facilitating its collapse to one of its field eigenstates.', '1404.5740-2-16-4': 'Hence a phenomenological form of the collapse parameter was proposed in [CITATION] as [EQUATION] where [MATH].', '1404.5740-2-16-5': 'It was also shown in [CITATION] that with [MATH] the CSL-modified scalar dynamics successfully explains the quantum-to-classical transition of primordial scalar modes without destroying the phase coherence of the superhorizon modes essential to explain the peaks and troughs of the CMBR anisotropy spectrum.', '1404.5740-2-17-0': 'Now, let us calculate the power spectrum in this scenario.', '1404.5740-2-17-1': 'In such a case it has been calculated in [CITATION] that [EQUATION] which indicates that the power spectrum would not be time-invariant on superhorizon scale unlike the standard scenario.', '1404.5740-2-17-2': 'This feature has been observed in both [CITATION] and as a cure to it the power was calculated at the end of inflation and not at the horizon-crossing of each mode by using [EQUATION] where [MATH] is the comoving wavenumber of the mode which is at the horizon today [MATH] and [MATH] is the number of efolds the mode has spent outside the horizon after its exit and thus for observationally relevant modes [MATH].', '1404.5740-2-17-3': 'We would henceforth consider [MATH].', '1404.5740-2-17-4': 'Also to make the power spectrum nearly scale-invariant we chose the scale-dependence of [MATH] as [EQUATION]', '1404.5740-2-17-5': 'Then the power spectrum of comoving curvature perturbations turns out to be [EQUATION]', '1404.5740-2-17-6': 'Incorporating the correction to the tilt in the spectrum due to quasi-de Sitter evolution of the background, the power spectrum gets modified to [EQUATION] where the power is now calculated at the end of inflation.', '1404.5740-2-18-0': '## Tensor perturbations', '1404.5740-2-19-0': 'The tensor perturbations would now be straightforward to calculate once the scalar analysis is done as each of the helicity components of the Fourier tensor mode behaves like massless scalar perturbations, as discussed before.', '1404.5740-2-19-1': 'It should be noted that tensor perturbations are gauge-invariant by construction and the redefined tensor perturbations [MATH] of Eq. ([REF]) can be identified as the Mukhanov-Sasaki variable of scalar perturbations defined in the previous section.', '1404.5740-2-19-2': 'Hence in the Schrodinger picture each helicity component [MATH] of the tensor modes can be expressed as functionals given in Eq. ([REF]) following similar functional Schrodinger equation given in Eq. ([REF]) in the standard scenario.', '1404.5740-2-19-3': 'The ground state solutions of the functional Schrodinger equation would also be Gaussian as given in Eq. ([REF]).', '1404.5740-2-19-4': 'Similarly the power spectrum of [MATH] would be same as of [MATH] given in Eq. ([REF]) and can be written as [EQUATION] and following Eq. ([REF]) one can write down the power spectrum for the tensor modes as [EQUATION]', '1404.5740-2-19-5': 'At this point, we assume CSL collapse mechanism affects each helicity mode of the gravitons the same way as it affects the inflatons.', '1404.5740-2-19-6': 'This is the simplest scenario to imagine and is in conformity with the philosophy that the collapse mechanism should be universal in nature.', '1404.5740-2-19-7': 'Since in our way of implementing the collapse mechanism we essentially use the fact that each mode is an independent harmonic oscillator to modify the equation of motion, there is no reason to expect that this modification should be sensitive to details of the underlying nature of the field.', '1404.5740-2-19-8': 'Hence the CSL-modified dynamics of each helicity mode of the gravitons would be same as that of the massless inflatons or the gauge-invariant Mukhanov-Sasaki variable which has been considered in the previous section.', '1404.5740-2-19-9': 'For redefined tensor modes [MATH], [MATH] would have the same form as given in Eq. ([REF]), following which the power spectrum of the tensor modes can be determined as [EQUATION] and considering the tilt in the power due to quasi-de Sitter background evolution one gets [EQUATION]', '1404.5740-2-20-0': '## Observables', '1404.5740-2-21-0': 'Let us now illustrate how the CSL-modified primordial dynamics differ from the standard one observationally.', '1404.5740-2-21-1': 'The first thing to note from Eq. ([REF]) and Eq. ([REF]) is that the tensor-to-scalar ratio remain the same in the modified dynamics: [EQUATION] which indicates from the Lyth bound that the field excursion during inflation would be super-Planckian even in this case.', '1404.5740-2-21-2': 'The scalar spectral index and the tensor spectral index would now become (from Eq. ([REF]) and Eq. ([REF])) [EQUATION] where we have defined [MATH].', '1404.5740-2-21-3': 'We note here that, although the model has three free parameters [MATH], [MATH] and [MATH] to begin with, the modification to spectral indices can be captured in one effective free parameter [MATH].', '1404.5740-2-21-4': 'The observation of the scalar spectral index by PLANCK indicates that the quantity [MATH] can at best be of the order of slow-roll parameters so that the comoving curvature power spectrum remains to be nearly scale-invariant.', '1404.5740-2-22-0': 'We also note from Eq. ([REF]) that the tensor amplitude in such a case does not remain to be a sole function of Hubble parameter, the prime feature which is used to determine the scale of inflation using the BICEP2 observations.', '1404.5740-2-22-1': 'Even though, if we consider that inflation has indeed taken place at that high scale, then that would help estimating the collapse parameter as [EQUATION] which turns out to be extremely small.', '1404.5740-2-22-2': 'A stronger collapse mechanism can then bring down the scale of inflation even though the field excursions would remain to be super-Planckian.', '1404.5740-2-23-0': 'Most interestingly what this modified dynamics does is to change the consistency relation of the single-field models.', '1404.5740-2-23-1': 'In such a scenario the consistency relation turns out to be [EQUATION]', '1404.5740-2-23-2': 'Hence independent accurate measurements of [MATH] and [MATH] would give us a direct handle on [MATH] in this model.', '1404.5740-2-24-0': '# Discussion and Conclusion', '1404.5740-2-25-0': 'Belief in quantum nature of primordial perturbations makes it essential to understand the apparent classicality of inhomogeneity it gives rise to.', '1404.5740-2-25-1': 'The squeezing of the superhorizon modes and decoherence partially explain this conundrum.', '1404.5740-2-25-2': 'But the single outcome problem cannot be addressed without appealing to either many-worlds interpretation or collapse mechanisms.', '1404.5740-2-25-3': 'Collapse mechanisms generically modify the dynamics of the primordial perturbations and so are expected to leave their imprints on cosmological observables.', '1404.5740-2-25-4': 'In [CITATION] some toy models of CSL applied to inflationary dynamics were explored.', '1404.5740-2-25-5': 'In [CITATION], within the context of an illustrative example, it was shown that classicality of perturbations can be achieved while still preserving scale-invariance and phase coherence for a certain parameter range.', '1404.5740-2-25-6': 'In the light of BICEP results which report high enough tensor-to-scalar ratio, inconsistent with purely classical dynamics of the primordial tensor modes [CITATION], we revisit this model to investigate its observational consequences.', '1404.5740-2-26-0': 'The first point to note is that in this illustrative example the tensor-to-scalar ratio remains unchanged indicating super-Planckian field excursions during inflation as demanded by Lyth bound [CITATION].', '1404.5740-2-26-1': 'Secondly, the tensor amplitude no longer directly yields the scale of inflation as the collapse parameter also enters in this conversion.', '1404.5740-2-26-2': 'A stronger collapse parameter would bring down the scale of inflation.', '1404.5740-2-26-3': 'Unfortunately, so far there is no other theoretical or observational guide to estimate the scale of collapse parameter.', '1404.5740-2-26-4': 'Furthermore, the spectral tilts get modified by the one and the same combination of the free parameters of the model which eventually also change the consistency relation of the standard single-field scenario by capturing the deviation from this in one effective free parameter [MATH].', '1404.5740-2-26-5': "From PLANCK's observation of the scalar spectral index, the free parameter [MATH] can at best be of the order of slow-roll parameters.", '1404.5740-2-26-6': 'It is also in principle possible that [MATH] be identically zero in which case the collapse mechanism achieves the required classicality without leaving any imprint on observations.', '1404.5740-2-26-7': 'All the same, it is more reasonable to expect [MATH] to be non-zero which then would make this model testable by observations.', '1404.5740-2-26-8': 'One important feature which distinguishes this kind of modification from other scenarios which also modify the consistency relation like curvaton [CITATION] or multifield [CITATION] models is that generically such extensions of the minimal model modify the scalar sector while leaving the tensor sector untouched, however the collapse models modify both and hence are potentially distinguishable by precision measurement of [MATH].', '1404.5740-2-26-9': 'However modification to initial conditions of tensor modes, i.e. deviation from Bunch-Davies vacuum would also modify the tensor spectral tilt [CITATION] reflecting the scale-dependence of Bogoliubov coefficients.', '1404.5740-2-26-10': 'It is possible to arrange this scale-dependence in such a way as to shift the tensor spectral index by a constant, thus mimicking the effect of the collapse mechanism considered here.', '1404.5740-2-26-11': 'But in principle, the generic scenario of non-Bunch Davies initial condition should be distinguishable from the collapse scenario.', '1404.5740-2-26-12': 'This would require precision measurement of running of tensor spectral index which has been argued to be possible in the near future [CITATION].', '1404.5740-2-27-0': 'The work of TPS is supported by a John Templeton Foundation grant [39530].', '1404.5740-2-27-1': 'SD and SS would like to thank Kinjalk Lochan for useful discussions.'}

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[]

[['1404.5740-2-3-2', '1404.5740-3-3-2']]

[]

[]

{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '3': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1404.5740

{'1404.5740-3-0-0': 'Inflationary and hence quantum origin of primordial perturbations is on a firmer ground than ever post the BICEP2 observations of primordial gravitational waves.', '1404.5740-3-0-1': 'One crucial ingredient of success of this paradigm rests on explaining the observed classicality of cosmological inhomogeinities despite their quantum origin.', '1404.5740-3-0-2': 'Although decoherence provides a partial understanding of this issue, the question of single outcome motivates the analysis of quantum collapse models in cosmological context which generically modify the dynamics of primordial perturbations and hence can leave their imprints on observables.', '1404.5740-3-0-3': 'We revisit one such recently proposed working model of classicalization by spontaneous collapse [CITATION] in the light of BICEP2 observations to look for possible modifications to tensor power spectra and their implications.', '1404.5740-3-0-4': 'We show that it can potentially change the consistency relation of single-field models and a precise measurement of [MATH] and its running could serve as a test of such dynamics in the early universe.', '1404.5740-3-1-0': '# Introduction', '1404.5740-3-2-0': 'The recent observation of [MATH]mode polarization of Cosmic Microwave Background Radiation (CMBR) by BICEP2 [CITATION] has put the cosmological inflationary paradigm [CITATION] on stronger footing than ever by confirming one of its many observational predictions, namely the generation of primordial tensor modes or primordial gravitational waves.', '1404.5740-3-2-1': "Inflationary dynamics gives rise to both scalar and tensor fluctuations during inflation which then 'redshift' out of the horizon and freeze.", '1404.5740-3-2-2': 'These primordial scalar fluctuations, up on re-entering the horizon at later stages, give rise to [MATH] anisotropy spectrum of the CMBR that has been measured by many observations such as WMAP [CITATION] and PLANCK [CITATION].', '1404.5740-3-2-3': 'On the other hand, the tensor modes give rise to the [MATH] spectrum of CMBR which has recently been detected by BICEP2 [CITATION].', '1404.5740-3-3-0': 'Detection of [MATH]modes of CMBR is of profound importance as according to the simplest model of inflation it sets the scale of inflation in a unique way, which turns out to be surprisingly close to the Grand Unification (GUT) scale ([MATH] GeV).', '1404.5740-3-3-1': 'But this is not all what the detection of [MATH]modes by BICEP2 can provide.', '1404.5740-3-3-2': 'BICEP2 measurement has further reinforced the quantum nature of gravity as it indicates that the tensor perturbations generated during inflation are of quantum nature.', '1404.5740-3-3-3': 'BICEP2 has measured the power of these tensor modes over the scalar ones on large cosmological scales, called the tensor-to-scalar ratio [MATH] and the measured value of [MATH] by BICEP2 is [CITATION] [EQUATION]', '1404.5740-3-3-4': 'It has been argued in [CITATION] that if these primordial gravitational waves were of classical nature, it would have produced negligible tensor amplitude compared to the scalar one and the value of [MATH] would have at least been suppressed by an extra power of slow-roll parameter [MATH] in comparison to the standard scenario.', '1404.5740-3-3-5': 'That these primordial tensor modes are indeed of quantum nature has also been argued in other literature such as [CITATION].', '1404.5740-3-3-6': 'Primordial scalar perturbations are also generated quantum mechanically and this scenario is well supported by observations.', '1404.5740-3-4-0': 'From this discussion it is evident that the primordial modes, both scalar and tensor, are being generated quantum mechanically at very early times.', '1404.5740-3-4-1': 'But these quantum fluctuations are the ones which give rise to CMBR temperature fluctuations and its [MATH]mode polarization on large angular scales which are classical in nature.', '1404.5740-3-4-2': 'This leads us to the problem of quantum-to-classical transition in the cosmological context, which is a more serious form of the so-called "quantum measurement problem" which also prevails in laboratory systems.', '1404.5740-3-5-0': 'Heuristically, the classical nature of primordial perturbations is argued by the large occupation number of superhorizon modes and effective irrelevance of the commutator of the field variable and its conjugate momentum on superhorizon scales.', '1404.5740-3-5-1': 'However, the perturbations generically evolve into highly squeezed states [CITATION] on superhorizon scales, which are highly non-classical states.', '1404.5740-3-5-2': 'But the quantum nature is not directly evident in observations because it can be shown that quantum expectations of highly squeezed states are indistinguishable from average of a classical stochastic field.', '1404.5740-3-5-3': "This is so-called 'decoherence without decoherence' [CITATION].", '1404.5740-3-5-4': 'Furthermore decoherence, despite the ambiguity in system and environment spilt for a cosmological scenario, selects the field amplitude basis as the pointer basis of the system and justifies the standard calculation.', '1404.5740-3-5-5': 'At this point, what still remains unresolved is the issue of single outcome.', '1404.5740-3-5-6': 'This problem is also present in laboratory systems and only gets more intriguing in cosmological context [CITATION].', '1404.5740-3-5-7': 'One possible way out is to appeal to the many-worlds interpretation.', '1404.5740-3-5-8': 'The other alternative is the so-called collapse models which we explore in this paper.', '1404.5740-3-5-9': 'Just as in laboratory systems, it is important to investigate whether and how collapse models can be distinguished from the standard quantum mechanical setup in a cosmological context.', '1404.5740-3-6-0': 'In a generic collapse model Schrodinger equation is modified by adding stochastic and non-linear terms.', '1404.5740-3-6-1': 'The stochastic nature of the equation helps explaining the probabilistic outcome of quantum measurements without allowing for superluminal communication and the presence of non-linear terms breaks down the underlying superposition principle of quantum mechanics.', '1404.5740-3-6-2': 'For a detailed review on collapse models refer to [CITATION].', '1404.5740-3-6-3': 'Though a proper field theoretic treatment of collapse dynamics is not yet known, a few attempts have been made to apply such collapse models, especially Continuous Spontaneous Localization (CSL) model [CITATION], into inflationary dynamics to resolve the problem of quantum to classical transition in cosmological context, such as [CITATION], [CITATION], [CITATION].', '1404.5740-3-6-4': 'As these collapse models explicitly modify the standard dynamics, we can anticipate that its observational implications would diverge from the standard dynamics and the aim of this brief paper is to determine how and where these collapse mechanisms differ from the standard scenario observationally within the context of an illustrative example in the light of recent BICEP observation.', '1404.5740-3-7-0': '# Generic Single-field slow-roll inflation and its observational implications', '1404.5740-3-8-0': 'The most economic model of inflation is the so-called slow-roll single field model.', '1404.5740-3-8-1': 'The power spectra for scalar and tensor perturbations in such a model can be given as [CITATION] [EQUATION] respectively, where [MATH] is called the pivot scale and for PLANCK it is chosen to be [MATH] Mpc[MATH].', '1404.5740-3-8-2': 'It is to be noted that as the comoving curvature perturbations, denoted by [MATH], and the tensor modes, denoted by [MATH], freeze on superhorizon scales, it is customary to derive the power spectra given above at horizon crossing of each mode [MATH] during inflation.', '1404.5740-3-8-3': 'In the above equations [MATH] is the Hubble parameter during inflation, [MATH] GeV is the reduced Planck mass, [MATH] is the first Hubble slow-roll parameter defined as [EQUATION] [MATH] and [MATH] are the scalar spectral index and the tensor spectral index respectively which are the measures of the scale dependence of the respective spectrum and for this generic slow-roll single-field model turn out to be [EQUATION] where [MATH] is the second Hubble slow-roll parameter defined as [EQUATION] where [MATH] is the inflaton field.', '1404.5740-3-9-0': 'Each of the tensor modes, which are the traceless and transverse part of the metric fluctuations, is associated with two helicity states, often denoted as + and [MATH] polarization.', '1404.5740-3-9-1': 'As has been first pointed out by Grishchuk in [CITATION], the evolution of the Fourier modes of each of the helicity states of these tensor modes is identical to that of a massless scalar in de Sitter space with the correspondence [EQUATION] where [MATH] is the helicity states and [MATH] is the re-defined tensor modes.', '1404.5740-3-9-2': 'Hence the evolution of quantum tensor modes can be reduced to that of two decoupled massless scalar modes.', '1404.5740-3-9-3': 'We will make use of this fact while dealing with CSL-modified inflationary dynamics.', '1404.5740-3-10-0': 'Now, let us consider the observational implications of this simplest model of inflation.', '1404.5740-3-10-1': 'PLANCK measures the amplitude [MATH] and the scalar spectral index [MATH] as [CITATION] [EQUATION] from the [MATH] anisotropy spectrum of the CMBR.', '1404.5740-3-10-2': 'BICEP2, by detecting the [MATH]polarization of CMBR measured tensor-to-scalar ratio [MATH], defined as [EQUATION] to be [MATH].', '1404.5740-3-10-3': 'According to the simplest single field model the tensor-to-scalar ratio turns out to be of the order of slow-roll parameter: [EQUATION] which then yields the consistency relation of this single-field model given as [EQUATION]', '1404.5740-3-10-4': 'Thus independent measurements of [MATH] and [MATH] can unambiguously determine whether the inflationary dynamics was indeed that simple or not.', '1404.5740-3-11-0': 'The recent observation of [MATH] by BICEP2 and the measurement of scalar amplitude [MATH] by PLANCK indirectly provides the amplitude of the tensor perturbations.', '1404.5740-3-11-1': 'But one can see from Eq. ([REF]) that the amplitude of the tensor power solely depends on the Hubble parameter during inflation and considering the central values of observed [MATH] and [MATH] one gets the Hubble parameter during inflation as [EQUATION]', '1404.5740-3-11-2': 'Now, during inflationary era the universe is dominated by the potential energy of the inflaton field and thus the Friedmann equation during inflation is written as [EQUATION]', '1404.5740-3-11-3': 'Hence the BICEP2 measurement of [MATH] also sets the scale of inflation as [EQUATION] which is very close to the GUT scale.', '1404.5740-3-11-4': 'However, inferring the scale of inflation from tensor amplitude becomes more involved if one invokes large extra dimensions [CITATION] or classical sources of primordial gravitational waves in a multi-field scenario [CITATION].', '1404.5740-3-12-0': 'Another implication of the BICEP2 result comes from the Lyth bound [CITATION].', '1404.5740-3-12-1': 'Writing the first slow-roll parameters [MATH] as [EQUATION] one can determine the field excursion during the inflation as [EQUATION] which shows that the inflaton field excursion is super-Planckian during inflation if [MATH], as has been observed by BICEP2.', '1404.5740-3-12-2': 'This leads to some tension with effective field theory description of inflation as the field excursion becomes of the same order of the natural cutoff scale .', '1404.5740-3-13-0': '# CSL-modified single-field dynamics and its observational implications', '1404.5740-3-14-0': '## Scalar perturbations', '1404.5740-3-15-0': 'For this analysis we would keep our focus on the derivations done in [CITATION].', '1404.5740-3-15-1': 'The collapse models modify the dynamics of particles in the Schrodinger picture.', '1404.5740-3-15-2': 'Hence while applying the CSL modifications to the inflationary dynamics one requires to analyze the field dynamics in the Schrodinger picture.', '1404.5740-3-15-3': 'An elaborate analysis of Schrodinger picture dynamics of scalar perturbations during inflation is given in [CITATION] (see also [CITATION]).', '1404.5740-3-15-4': 'We would study the primordial scalar perturbations during inflation in terms of the so-called Mukhanov-Sakasi variable, a gauge-invariant quantity denoted by [MATH] , which is related to the comoving curvature perturbation as [EQUATION]', '1404.5740-3-15-5': 'In Schrodinger picture, the standard scalar perturbation is analyzed in terms of its wave-functional [CITATION] defined as [EQUATION] where we have [EQUATION]', '1404.5740-3-15-6': 'These wave functionals satisfy the functional Schrodinger equation as [EQUATION] where the Hamiltonian [MATH] and the ground state solution of the functional Schrodinger equation is written as [EQUATION]', '1404.5740-3-15-7': 'Here [MATH] is related to the mode functions [MATH] in the Heisenberg picture as [EQUATION]', '1404.5740-3-15-8': 'Thus, in Schrodinger picture the power spectrum of [MATH] turns out to be [EQUATION] and thus the power spectrum of the comoving curvature perturbations can be obtained as [EQUATION]', '1404.5740-3-15-9': 'It has been proposed in [CITATION] that CSL-like modifications can be applied to the inflationary perturbations directly in the Fourier space by adding CSL-like non-linear and stochastic terms to the functional Schrodinger equation of [MATH] which then looks like [EQUATION] where the stochastic behavior due to CSL mechanism is encoded in the Wiener process [MATH] and [MATH] is called the collapse parameter.', '1404.5740-3-15-10': 'The most general stochastic wave-functional which satisfies this stochastic functional Schrodinger equation can be written as [EQUATION] where [MATH], [MATH] and [MATH] are real numbers.', '1404.5740-3-16-0': 'In [CITATION], the collapse parameter [MATH] was taken to be constant and it was inferred that such a case is not capable of explaining the quantum-to-classical transition of primordial modes.', '1404.5740-3-16-1': 'Then it was argued in [CITATION] that taking [MATH] to be constant the scenario loses one of the crucial features of CSL dynamics, known as amplification mechanism.', '1404.5740-3-16-2': 'In CSL-modified quantum mechanics, the collapse parameter is taken to be directly proportional to the mass of the system and its number density reflecting the fact that heavier objects become classical faster than the lighter ones.', '1404.5740-3-16-3': 'Similarly, in tune with the expectation that superhorizon modes behave classically one could assume in the cosmological context that the collapse parameter would become stronger as a generic mode starts to cross the horizon facilitating its collapse to one of its field eigenstates.', '1404.5740-3-16-4': 'Hence a phenomenological form of the collapse parameter was proposed in [CITATION] as [EQUATION] where [MATH].', '1404.5740-3-16-5': 'It was also shown in [CITATION] that with [MATH] the CSL-modified scalar dynamics successfully explains the quantum-to-classical transition of primordial scalar modes without destroying the phase coherence of the superhorizon modes essential to explain the peaks and troughs of the CMBR anisotropy spectrum.', '1404.5740-3-17-0': 'Now, let us calculate the power spectrum in this scenario.', '1404.5740-3-17-1': 'In such a case it has been calculated in [CITATION] that [EQUATION] which indicates that the power spectrum would not be time-invariant on superhorizon scale unlike the standard scenario.', '1404.5740-3-17-2': 'This feature has been observed in both [CITATION] and as a cure to it the power was calculated at the end of inflation and not at the horizon-crossing of each mode by using [EQUATION] where [MATH] is the comoving wavenumber of the mode which is at the horizon today [MATH] and [MATH] is the number of efolds the mode has spent outside the horizon after its exit and thus for observationally relevant modes [MATH].', '1404.5740-3-17-3': 'We would henceforth consider [MATH].', '1404.5740-3-17-4': 'Also to make the power spectrum nearly scale-invariant we chose the scale-dependence of [MATH] as [EQUATION]', '1404.5740-3-17-5': 'Then the power spectrum of comoving curvature perturbations turns out to be [EQUATION]', '1404.5740-3-17-6': 'Incorporating the correction to the tilt in the spectrum due to quasi-de Sitter evolution of the background, the power spectrum gets modified to [EQUATION] where the power is now calculated at the end of inflation.', '1404.5740-3-18-0': '## Tensor perturbations', '1404.5740-3-19-0': 'The tensor perturbations would now be straightforward to calculate once the scalar analysis is done as each of the helicity components of the Fourier tensor mode behaves like massless scalar perturbations, as discussed before.', '1404.5740-3-19-1': 'It should be noted that tensor perturbations are gauge-invariant by construction and the redefined tensor perturbations [MATH] of Eq. ([REF]) can be identified as the Mukhanov-Sasaki variable of scalar perturbations defined in the previous section.', '1404.5740-3-19-2': 'Hence in the Schrodinger picture each helicity component [MATH] of the tensor modes can be expressed as functionals given in Eq. ([REF]) following similar functional Schrodinger equation given in Eq. ([REF]) in the standard scenario.', '1404.5740-3-19-3': 'The ground state solutions of the functional Schrodinger equation would also be Gaussian as given in Eq. ([REF]).', '1404.5740-3-19-4': 'Similarly the power spectrum of [MATH] would be same as of [MATH] given in Eq. ([REF]) and can be written as [EQUATION] and following Eq. ([REF]) one can write down the power spectrum for the tensor modes as [EQUATION]', '1404.5740-3-19-5': 'At this point, we assume CSL collapse mechanism affects each helicity mode of the gravitons the same way as it affects the inflatons.', '1404.5740-3-19-6': 'This is the simplest scenario to imagine and is in conformity with the philosophy that the collapse mechanism should be universal in nature.', '1404.5740-3-19-7': 'Since in our way of implementing the collapse mechanism we essentially use the fact that each mode is an independent harmonic oscillator to modify the equation of motion, there is no reason to expect that this modification should be sensitive to details of the underlying nature of the field.', '1404.5740-3-19-8': 'Hence the CSL-modified dynamics of each helicity mode of the gravitons would be same as that of the massless inflatons or the gauge-invariant Mukhanov-Sasaki variable which has been considered in the previous section.', '1404.5740-3-19-9': 'For redefined tensor modes [MATH], [MATH] would have the same form as given in Eq. ([REF]), following which the power spectrum of the tensor modes can be determined as [EQUATION] and considering the tilt in the power due to quasi-de Sitter background evolution one gets [EQUATION]', '1404.5740-3-20-0': '## Observables', '1404.5740-3-21-0': 'Let us now illustrate how the CSL-modified primordial dynamics differ from the standard one observationally.', '1404.5740-3-21-1': 'The first thing to note from Eq. ([REF]) and Eq. ([REF]) is that the tensor-to-scalar ratio remain the same in the modified dynamics: [EQUATION] which indicates from the Lyth bound that the field excursion during inflation would be super-Planckian even in this case.', '1404.5740-3-21-2': 'The scalar spectral index and the tensor spectral index would now become (from Eq. ([REF]) and Eq. ([REF])) [EQUATION] where we have defined [MATH].', '1404.5740-3-21-3': 'We note here that, although the model has three free parameters [MATH], [MATH] and [MATH] to begin with, the modification to spectral indices can be captured in one effective free parameter [MATH].', '1404.5740-3-21-4': 'The observation of the scalar spectral index by PLANCK indicates that the quantity [MATH] can at best be of the order of slow-roll parameters so that the comoving curvature power spectrum remains to be nearly scale-invariant.', '1404.5740-3-22-0': 'We also note from Eq. ([REF]) that the tensor amplitude in such a case does not remain to be a sole function of Hubble parameter, the prime feature which is used to determine the scale of inflation using the BICEP2 observations.', '1404.5740-3-22-1': 'Even though, if we consider that inflation has indeed taken place at that high scale, then that would help estimating the collapse parameter as [EQUATION] which turns out to be extremely small.', '1404.5740-3-22-2': 'A stronger collapse mechanism can then bring down the scale of inflation even though the field excursions would remain to be super-Planckian.', '1404.5740-3-23-0': 'Most interestingly what this modified dynamics does is to change the consistency relation of the single-field models.', '1404.5740-3-23-1': 'In such a scenario the consistency relation turns out to be [EQUATION]', '1404.5740-3-23-2': 'Hence independent accurate measurements of [MATH] and [MATH] would give us a direct handle on [MATH] in this model.', '1404.5740-3-24-0': '# Discussion and Conclusion', '1404.5740-3-25-0': 'Belief in quantum nature of primordial perturbations makes it essential to understand the apparent classicality of inhomogeneity it gives rise to.', '1404.5740-3-25-1': 'The squeezing of the superhorizon modes and decoherence partially explain this conundrum.', '1404.5740-3-25-2': 'But the single outcome problem cannot be addressed without appealing to either many-worlds interpretation or collapse mechanisms.', '1404.5740-3-25-3': 'Collapse mechanisms generically modify the dynamics of the primordial perturbations and so are expected to leave their imprints on cosmological observables.', '1404.5740-3-25-4': 'In [CITATION] some toy models of CSL applied to inflationary dynamics were explored.', '1404.5740-3-25-5': 'In [CITATION], within the context of an illustrative example, it was shown that classicality of perturbations can be achieved while still preserving scale-invariance and phase coherence for a certain parameter range.', '1404.5740-3-25-6': 'In the light of BICEP results which report high enough tensor-to-scalar ratio, inconsistent with purely classical dynamics of the primordial tensor modes [CITATION], we revisit this model to investigate its observational consequences.', '1404.5740-3-26-0': 'The first point to note is that in this illustrative example the tensor-to-scalar ratio remains unchanged indicating super-Planckian field excursions during inflation as demanded by Lyth bound [CITATION].', '1404.5740-3-26-1': 'Secondly, the tensor amplitude no longer directly yields the scale of inflation as the collapse parameter also enters in this conversion.', '1404.5740-3-26-2': 'A stronger collapse parameter would bring down the scale of inflation.', '1404.5740-3-26-3': 'Unfortunately, so far there is no other theoretical or observational guide to estimate the scale of collapse parameter.', '1404.5740-3-26-4': 'Furthermore, the spectral tilts get modified by the one and the same combination of the free parameters of the model which eventually also change the consistency relation of the standard single-field scenario by capturing the deviation from this in one effective free parameter [MATH].', '1404.5740-3-26-5': "From PLANCK's observation of the scalar spectral index, the free parameter [MATH] can at best be of the order of slow-roll parameters.", '1404.5740-3-26-6': 'It is also in principle possible that [MATH] be identically zero in which case the collapse mechanism achieves the required classicality without leaving any imprint on observations.', '1404.5740-3-26-7': 'All the same, it is more reasonable to expect [MATH] to be non-zero which then would make this model testable by observations.', '1404.5740-3-26-8': 'One important feature which distinguishes this kind of modification from other scenarios which also modify the consistency relation like curvaton [CITATION] or multifield [CITATION] models is that generically such extensions of the minimal model modify the scalar sector while leaving the tensor sector untouched, however the collapse models modify both and hence are potentially distinguishable by precision measurement of [MATH].', '1404.5740-3-26-9': 'However modification to initial conditions of tensor modes, i.e. deviation from Bunch-Davies vacuum would also modify the tensor spectral tilt [CITATION] reflecting the scale-dependence of Bogoliubov coefficients.', '1404.5740-3-26-10': 'It is possible to arrange this scale-dependence in such a way as to shift the tensor spectral index by a constant, thus mimicking the effect of the collapse mechanism considered here.', '1404.5740-3-26-11': 'But in principle, the generic scenario of non-Bunch Davies initial condition should be distinguishable from the collapse scenario.', '1404.5740-3-26-12': 'This would require precision measurement of running of tensor spectral index which has been argued to be possible in the near future [CITATION].', '1404.5740-3-27-0': 'The work of TPS is supported by a John Templeton Foundation grant [39530].', '1404.5740-3-27-1': 'SD and SS would like to thank Kinjalk Lochan for useful discussions.'}

hep-ph-0408300

{'hep-ph-0408300-1-0-0': 'We study possible contributions to the [MATH] and [MATH] decay amplitudes.', 'hep-ph-0408300-1-0-1': 'The [MATH] decay amplitude vanishes when the naive factorization is used, while the [MATH] decay amplitude arises due to the annihilation contribution.', 'hep-ph-0408300-1-0-2': 'We find that amplitudes for both decays might be a result of the internal [MATH], [MATH] exchange.', 'hep-ph-0408300-1-0-3': 'The [MATH] amplitude might obtain additional contribution from [MATH] re-scattering.', 'hep-ph-0408300-1-0-4': 'The low experimental bound on the [MATH] rate can be understood as a result of combination of the [MATH] pole dominated annihilation contribution and the [MATH] internal exchanges.', 'hep-ph-0408300-1-0-5': 'The calculated branching fractions for [MATH] and [MATH] are in agreement with the current experimental results.', 'hep-ph-0408300-1-1-0': 'The weak nonleptonic decays of charm mesons were usually approached within factorization ansatz [CITATION].', 'hep-ph-0408300-1-1-1': 'A decade ago it was realized [CITATION] that one has to include the effects of final state interactions (FSI), with the simplest approach being to treat the FSI by assuming the dominance of nearby resonances.', 'hep-ph-0408300-1-1-2': 'This leads to rather good overall agreement with the experimental data [CITATION], however, there are a few cases where none of the existing approaches work.', 'hep-ph-0408300-1-1-3': 'Two such examples are the channels (quoting the PDG experimental values [CITATION]) [EQUATION]', 'hep-ph-0408300-1-1-4': 'The current theoretical approaches usually give that the [MATH] branching fraction is equal [CITATION] or even larger than the branching fraction for the [MATH] decay [CITATION] in contradiction with the present data [REF].', 'hep-ph-0408300-1-2-0': 'On the other hand, the observation of the [MATH] decay ([REF]) has been motivated as a clean signature of the annihilation decay of [MATH] [CITATION].', 'hep-ph-0408300-1-2-1': 'The sizes of annihilation contributions are very important for phenomenological studies, but are also very hard to obtain from theoretical considerations.', 'hep-ph-0408300-1-2-2': 'Understanding the origin of [MATH] transition is thus of great theoretical interest.', 'hep-ph-0408300-1-3-0': 'Let us first discuss the two modes [REF] using factorization approximation for the weak vertex.', 'hep-ph-0408300-1-3-1': 'In this approximation the [MATH] amplitude is zero due to G - parity conservation, which gives a vanishing [MATH] matrix element [CITATION].', 'hep-ph-0408300-1-3-2': 'The [MATH] decay amplitude, on the other hand, already in the factorization limit receives the contribution through the annihilation graph, Fig. 1, [EQUATION] leading to a simple [MATH] pole dominance in the [MATH] matrix element.', 'hep-ph-0408300-1-3-3': 'The analysis of [CITATION] indicates that [MATH] states dominate this annihilation graph, while the contribution of the lowest lying [MATH] is negligible.', 'hep-ph-0408300-1-3-4': 'In [CITATION] we have estimated the size of the annihilation contribution coming from the [MATH] intermediate state.', 'hep-ph-0408300-1-3-5': 'We found [MATH] MeV [CITATION].', 'hep-ph-0408300-1-3-6': 'In the factorization approximation for the weak vertex we then get [EQUATION] where we have used [MATH] MeV, together with the conservative assumptions of [MATH] and [MATH] equal to its upper experimental bound of [MATH] MeV.', 'hep-ph-0408300-1-3-7': 'The interference with other annihilation contributions from intermediate [MATH] and [MATH] states can somewhat change the above estimate (using PCAC, the contribution from [MATH] was found in [CITATION] to be negligible, while the contribution of [MATH] is difficult to estimate due to the lack of experimental data).', 'hep-ph-0408300-1-3-8': 'In addition, also the FSI contributions (to be considered shortly) fall in exactly the same range [CITATION].', 'hep-ph-0408300-1-3-9': 'Therefore, unless there are large cancellations, the value of [MATH] is expected to be near to its present experimental upper bound [REF].', 'hep-ph-0408300-1-4-0': 'In the case of the [MATH] final state there is no such resonance annihilation contribution and one has to explain a relatively large experimental value for [MATH] [REF] in a different way.', 'hep-ph-0408300-1-4-1': 'An important observation is that there are multi-body intermediate states that do have correct values of [MATH] and [MATH], for instance the two-body [MATH] states.', 'hep-ph-0408300-1-4-2': 'As we will show in the rest of the talk, it is possible to explain the experimental value for [MATH] by considering the contributions due to the rescattering of these intermediate states.', 'hep-ph-0408300-1-5-0': 'In estimating the contributions from hidden strangeness intermediate states (that can arise from spectator quark diagrams), we use the following assumptions', 'hep-ph-0408300-1-6-0': 'In addition we have checked that the use of factorization for the [MATH], [MATH] and [MATH] decays gives reasonable estimates of the measured rates (note that we do not need [MATH] in further considerations) [CITATION].', 'hep-ph-0408300-1-6-1': 'In these results the annihilation contributions have been neglected since they are an order of magnitude smaller.', 'hep-ph-0408300-1-7-0': 'The situation in the case of [MATH] intermediate states is not so favorable.', 'hep-ph-0408300-1-7-1': 'To treat the [MATH] mixing we use the approach of ref. [CITATION] with the value of the mixing angle transforming between [MATH] and [MATH] states taken to be [MATH].', 'hep-ph-0408300-1-7-2': 'The factorization approach then gives a reasonable description of [MATH] decay, while it does not reproduce satisfactorily the experimental result for [MATH] [CITATION].', 'hep-ph-0408300-1-7-3': 'This is a known problem as [MATH] rate is very difficult to reproduce by any of present approaches [CITATION].', 'hep-ph-0408300-1-7-4': 'This inevitably introduces some further uncertainty into our approach, yet the resulting uncertainty is not expected to affect significantly our main conclusions.', 'hep-ph-0408300-1-8-0': 'For the weak current matrix elements between [MATH] and vector or pseudoscalar final states we use a common form factors decomposition [CITATION] with the form factors [MATH], [MATH], [MATH]and [MATH].', 'hep-ph-0408300-1-8-1': 'For the [MATH] dependence of the form factors we use results of [CITATION], based on a quark model calculation combined with a fit to lattice and experimental data.', 'hep-ph-0408300-1-8-2': 'Ref. [CITATION] provides a simple fit to their numerical results with the form factors [MATH], [MATH] and [MATH] described by double pole [MATH] dependence [EQUATION] while single pole parameterization [EQUATION] can be used for [MATH], as the contributing resonances have masses farther away from the physical region (note that this parameterization applies also to [MATH] form factor, which however does not contribute in the processes we discuss in this paper).', 'hep-ph-0408300-1-8-3': 'The values of [MATH] and [MATH] are listed in Table [REF] and are taken from [CITATION].', 'hep-ph-0408300-1-8-4': 'We use [MATH] in the expression for [MATH], and [MATH] for all the other form factors [CITATION].', 'hep-ph-0408300-1-8-5': 'Incidentally, the parameterizations of form factors [REF], [REF] make all the loop diagrams in Figs. [REF] and [REF] finite.', 'hep-ph-0408300-1-9-0': 'For the decay constants, defined through [MATH] and [MATH], we use [MATH] and [MATH] as obtained on the lattice [CITATION] and for the rest [MATH], [MATH], [MATH] and [MATH] coming from the experimental measurements [CITATION].', 'hep-ph-0408300-1-10-0': 'The amplitudes for the [MATH] and [MATH] decays can be written as: [EQUATION] with [MATH] the helicity zero polarization vector of the [MATH] or [MATH] vector mesons, while [MATH] is the pion momentum.', 'hep-ph-0408300-1-10-1': 'The reduced amplitudes [MATH] and [MATH] correspond to the diagrams on Figs. [REF] and [REF] respectively.', 'hep-ph-0408300-1-10-2': 'The explicit expressions can be found in Appendix of [CITATION].', 'hep-ph-0408300-1-10-3': 'The numerical values for [MATH] and [MATH] are given in Table [REF].', 'hep-ph-0408300-1-11-0': 'Combining the above results we arrive at the prediction [EQUATION]', 'hep-ph-0408300-1-11-1': 'Note that in this calculation we have used the factorization approximation for the diagram of Fig. [REF], which as stated above, does not work well for [MATH] transition.', 'hep-ph-0408300-1-11-2': 'By including hidden strangeness FSI to the [MATH] decay mode gives an order of magnitude smaller contribution.', 'hep-ph-0408300-1-11-3': 'On the other hand one can use the experimental input to rescale the corresponding amplitudes.', 'hep-ph-0408300-1-11-4': 'This results in the prediction [MATH].', 'hep-ph-0408300-1-11-5': 'We point out that the loop contributions are finite due to double pole parametrization of the form factors.', 'hep-ph-0408300-1-11-6': 'If single pole parametrization is used, one has to regularize the amplitudes.', 'hep-ph-0408300-1-11-7': 'We found that the numerical results do not change significantly in this case when cut-off scale is above but close enough to [MATH] meson mass.', 'hep-ph-0408300-1-11-8': 'We can draw the conclusion that the experimental result for [MATH] can be understood as a result of combined effect of a spectator transition and FSI.', 'hep-ph-0408300-1-11-9': 'Therefore, it makes the attempt to understand the [MATH] amplitude as a result of annihilation contributions unsuccessful.', 'hep-ph-0408300-1-12-0': 'In the case of the [MATH] transition, the FSI contributions alone result in [EQUATION]', 'hep-ph-0408300-1-12-1': 'This is almost exactly the same as our estimate of the upper bound on the annihilation contribution [REF].', 'hep-ph-0408300-1-12-2': 'Both contributions are equal or very close to the present 90% CL upper bound.', 'hep-ph-0408300-1-12-3': 'If there is no destructive interference between these two contributions and the contributions of FSI through higher resonances that we did not take into account, the branching fraction for this decay will be hopefully determined in the near future.', 'hep-ph-0408300-1-12-4': 'Our prediction is in the agreement with the results of other theoretical studies which give the rate for [MATH] to be equal [CITATION] or even larger than the rate for [MATH] decay [CITATION].', 'hep-ph-0408300-1-13-0': 'However, one should consider possible cancellation that might occur.', 'hep-ph-0408300-1-13-1': 'Adding the FSI contribution and the maximal annihilation contributions [REF] with alternating signs gives a fairly large interval [EQUATION]', 'hep-ph-0408300-1-13-2': 'We note that the experimental uncertainties translating in the input parameters can change the values for [MATH] and [MATH] by about 20%.', 'hep-ph-0408300-1-14-0': 'Finally, we mention that the kind of FSI contributions we were considering in this paper is not the leading contribution in the [MATH] transition, which can proceed through spectator quark transition directly.', 'hep-ph-0408300-1-14-1': 'Use of factorization approximation for the weak vertex leads to a prediction [MATH], which is already in excellent agreement with the experimental result of [MATH].', 'hep-ph-0408300-1-14-2': 'We found that inclusion of FSI reduces the theoretical prediction from 4% to [MATH].', 'hep-ph-0408300-1-14-3': 'The size of the shift also indicates that FSI of the type described in the present paper are in the case of [MATH] transition a second order effect.', 'hep-ph-0408300-1-14-4': 'Note as well, that the size of the FSI correction is in agreement with the predictions for [MATH] and [MATH], which are an order of magnitude smaller than [MATH].', 'hep-ph-0408300-1-15-0': 'We summarize that the hidden strangeness final state interactions are very important in understanding the [MATH] and [MATH] decay mechanism.', 'hep-ph-0408300-1-15-1': 'The [MATH] amplitude can be explained fully by this mechanism.', 'hep-ph-0408300-1-15-2': 'In the case of the [MATH] decay rate we obtain a fairly large range due to possible cancellation between FSI and single pole contributions.', 'hep-ph-0408300-1-16-0': 'The measurement of the [MATH] decay rate will considerably improve our understanding of the [MATH] decay mechanism.', 'hep-ph-0408300-1-16-1': 'The hidden strangeness FSI might fully explain the observed decay rate for [MATH].', 'hep-ph-0408300-1-16-2': 'Finally, this kind of FSI give only subdominant contributions in the case of [MATH] decays, which are well described by the factorization approximation.'}

{'hep-ph-0408300-2-0-0': 'We study possible contributions to the [MATH] and [MATH] decay amplitudes.', 'hep-ph-0408300-2-0-1': 'The [MATH] decay amplitude vanishes when naive factorization is used, while the [MATH] decay amplitude arises due to the annihilation contribution.', 'hep-ph-0408300-2-0-2': 'We find that amplitudes for both decays might be a result of the internal [MATH], [MATH] exchange.', 'hep-ph-0408300-2-0-3': 'The [MATH] amplitude might obtain additional contributions from [MATH] re-scattering.', 'hep-ph-0408300-2-0-4': 'The low experimental bound on the [MATH] rate can be understood as a result of combination of the [MATH] pole dominated annihilation contribution and the [MATH] internal exchanges.', 'hep-ph-0408300-2-0-5': 'The calculated branching fractions for [MATH] and [MATH] are in agreement with the current experimental results.', 'hep-ph-0408300-2-1-0': 'The weak nonleptonic decays of charm mesons were usually approached within the factorization ansatz [CITATION].', 'hep-ph-0408300-2-1-1': 'A decade ago it was realized [CITATION] that one has to include the effects of final state interactions (FSI), with the simplest approach being to treat the FSI by assuming the dominance of nearby resonances.', 'hep-ph-0408300-2-1-2': 'This leads to rather good overall agreement with the experimental data [CITATION]; however, there are a few cases where none of the existing approaches work.', 'hep-ph-0408300-2-1-3': 'Two such examples are the channels (quoting the PDG experimental values [CITATION]) [EQUATION]', 'hep-ph-0408300-2-1-4': 'The current theoretical approaches usually predict that the [MATH] branching fraction is equal [CITATION] or even larger than the branching fraction for the [MATH] decay [CITATION] in contradiction with the present data [REF].', 'hep-ph-0408300-2-2-0': 'On the other hand, the observation of the [MATH] decay ([REF]) has been motivated as a clean signature of the annihilation decay of [MATH] [CITATION].', 'hep-ph-0408300-2-2-1': 'The sizes of annihilation contributions are very important for phenomenological studies, but are also very hard to obtain from theoretical considerations (see e.g., [CITATION]).', 'hep-ph-0408300-2-2-2': 'Understanding the origin of the [MATH] transition is thus of great theoretical interest.', 'hep-ph-0408300-2-3-0': 'Let us first discuss the two modes [REF] using factorization approximation for the weak vertex.', 'hep-ph-0408300-2-3-1': 'In this approximation the [MATH] amplitude is zero due to G - parity conservation, which gives a vanishing [MATH] matrix element [CITATION].', 'hep-ph-0408300-2-3-2': 'The [MATH] decay amplitude, on the other hand, already in the factorization limit receives a contribution through the annihilation graph, Fig. 1, [EQUATION] leading to simple [MATH] pole dominance in the [MATH] matrix element.', 'hep-ph-0408300-2-3-3': 'The analysis of [CITATION] indicates that [MATH] states dominate this annihilation graph, while the contribution of the lowest lying [MATH] is negligible.', 'hep-ph-0408300-2-3-4': 'In [CITATION] we have estimated the size of the annihilation contribution coming from the [MATH] intermediate state.', 'hep-ph-0408300-2-3-5': 'We found [MATH] MeV [CITATION].', 'hep-ph-0408300-2-3-6': 'In the factorization approximation for the weak vertex we then get [EQUATION] where we have used [MATH] MeV, together with the conservative assumptions of [MATH] and [MATH] equal to its upper experimental bound of [MATH] MeV.', 'hep-ph-0408300-2-3-7': 'The interference with other annihilation contributions from intermediate [MATH] and [MATH] states can somewhat change the above estimate (using PCAC, the contribution from [MATH] was found in [CITATION] to be negligible, while the contribution of [MATH] is difficult to estimate due to the lack of experimental data).', 'hep-ph-0408300-2-3-8': 'In addition, also the FSI contributions (to be considered shortly) fall in exactly the same range [CITATION].', 'hep-ph-0408300-2-3-9': 'Therefore, unless there are large cancellations, the value of [MATH] is expected to be near to its present experimental upper bound [REF].', 'hep-ph-0408300-2-4-0': 'In the case of the [MATH] final state there is no such resonance annihilation contribution and one has to explain a relatively large experimental value for [MATH] [REF] in a different way.', 'hep-ph-0408300-2-4-1': 'An important observation is that there are multi-body intermediate states that do have the correct values of [MATH] and [MATH], for instance the two-body [MATH] states.', 'hep-ph-0408300-2-4-2': 'As we will show in the rest of the talk, it is possible to explain the experimental value for [MATH] by considering the contributions due to the rescattering of these intermediate states.', 'hep-ph-0408300-2-5-0': 'In estimating the contributions from hidden strangeness intermediate states (that can arise from spectator quark diagrams), we use the following assumptions', 'hep-ph-0408300-2-6-0': 'In addition we have checked that the use of factorization for the [MATH], [MATH] and [MATH] decays gives reasonable estimates of the measured rates (note that we do not need [MATH] in further considerations) [CITATION].', 'hep-ph-0408300-2-6-1': 'In these results the annihilation contributions have been neglected since they are an order of magnitude smaller.', 'hep-ph-0408300-2-7-0': 'The situation in the case of [MATH] intermediate states is not so favorable.', 'hep-ph-0408300-2-7-1': 'To treat the [MATH] mixing we use the approach of Ref. [CITATION] with the value of the mixing angle transforming between [MATH] and [MATH] states taken to be [MATH].', 'hep-ph-0408300-2-7-2': 'The factorization approach then gives a reasonable description of [MATH] decay, while it does not reproduce satisfactorily the experimental result for [MATH] [CITATION].', 'hep-ph-0408300-2-7-3': 'This is a known problem as the [MATH] rate is very difficult to reproduce in any of the present approaches [CITATION].', 'hep-ph-0408300-2-7-4': 'This inevitably introduces some further uncertainty into our approach, yet the resulting uncertainty is not expected to affect significantly our main conclusions.', 'hep-ph-0408300-2-8-0': 'For the weak current matrix elements between [MATH] and vector or pseudoscalar final states we use a common form factor decomposition [CITATION] with the form factors [MATH], [MATH], [MATH]and [MATH].', 'hep-ph-0408300-2-8-1': 'For the [MATH] dependence of the form factors we use results of [CITATION], based on a quark model calculation combined with a fit to lattice and experimental data.', 'hep-ph-0408300-2-8-2': 'Ref. [CITATION] provides a simple fit to their numerical results with the form factors [MATH], [MATH] and [MATH] described by double pole [MATH] dependence [EQUATION] while single pole parameterization [EQUATION] can be used for [MATH], as the contributing resonances have masses farther away from the physical region (note that this parameterization applies also to [MATH] form factor, which however does not contribute in the processes we discuss in this paper).', 'hep-ph-0408300-2-8-3': 'The values of [MATH] and [MATH] are listed in Table [REF] and are taken from [CITATION].', 'hep-ph-0408300-2-8-4': 'We use [MATH] in the expression for [MATH], and [MATH] for all the other form factors [CITATION].', 'hep-ph-0408300-2-8-5': 'Incidentally, the parameterizations of the form factors [REF] and [REF] make all the loop diagrams in Figs. [REF] and [REF] finite.', 'hep-ph-0408300-2-9-0': 'For the decay constants, defined through [MATH] and [MATH], we use [MATH] and [MATH] as obtained on the lattice [CITATION] and for the rest [MATH], [MATH], [MATH] and [MATH] coming from the experimental measurements [CITATION].', 'hep-ph-0408300-2-10-0': 'The amplitudes for the [MATH] and [MATH] decays can be written as: [EQUATION] with [MATH] the helicity zero polarization vector of the [MATH] or [MATH] vector mesons, while [MATH] is the pion momentum.', 'hep-ph-0408300-2-10-1': 'The reduced amplitudes [MATH] and [MATH] correspond to the diagrams in Figs. [REF] and [REF] respectively.', 'hep-ph-0408300-2-10-2': 'The explicit expressions can be found in Appendix of [CITATION].', 'hep-ph-0408300-2-10-3': 'The numerical values for [MATH] and [MATH] are given in Table [REF].', 'hep-ph-0408300-2-11-0': 'Combining the above results we arrive at the prediction [EQUATION]', 'hep-ph-0408300-2-11-1': 'Note that in this calculation we have used the factorization approximation for the diagram of Fig. [REF], which as stated above, does not work well for [MATH] transition.', 'hep-ph-0408300-2-11-2': 'Including hidden strangeness FSI to the [MATH] decay mode gives an order of magnitude smaller contribution.', 'hep-ph-0408300-2-11-3': 'On the other hand one can use the experimental input to rescale the corresponding amplitudes.', 'hep-ph-0408300-2-11-4': 'This results in the prediction [MATH].', 'hep-ph-0408300-2-11-5': 'We point out that the loop contributions are finite due to double pole parametrization of the form factors.', 'hep-ph-0408300-2-11-6': 'If a single pole parametrization is used, one has to regularize the amplitudes.', 'hep-ph-0408300-2-11-7': 'We found that the numerical results do not change significantly in this case when the cut-off scale is above but close enough to the [MATH] meson mass.', 'hep-ph-0408300-2-11-8': 'We can draw the conclusion that the experimental result for [MATH] can be understood as a result of the combined effect of a spectator transition and FSI.', 'hep-ph-0408300-2-11-9': 'Therefore, it makes the attempt to understand the [MATH] amplitude as a result of annihilation contributions unsuccessful.', 'hep-ph-0408300-2-12-0': 'In the case of the [MATH] transition, the FSI contributions alone result in [EQUATION]', 'hep-ph-0408300-2-12-1': 'This is almost exactly the same as our estimate of the upper bound on the annihilation contribution [REF].', 'hep-ph-0408300-2-12-2': 'Both contributions are equal or very close to the present 90% CL upper bound.', 'hep-ph-0408300-2-12-3': 'If there is no destructive interference between these two contributions and the contributions of FSI through higher resonances that we did not take into account, one hopes that the branching fraction for this decay will be determined in the near future.', 'hep-ph-0408300-2-12-4': 'Our prediction is in agreement with the results of other theoretical studies which give the rate for [MATH] to be equal [CITATION] or even larger than the rate for [MATH] decay [CITATION].', 'hep-ph-0408300-2-13-0': 'However, one should consider possible cancellation that might occur.', 'hep-ph-0408300-2-13-1': 'Adding the FSI contribution and the maximal annihilation contributions [REF] with alternating signs gives a fairly large interval [EQUATION]', 'hep-ph-0408300-2-13-2': 'We note that the experimental uncertainties reflected in the input parameters can change the values for [MATH] and [MATH] by about 20%.', 'hep-ph-0408300-2-14-0': 'Finally, we mention that the kind of FSI contributions we were considering in this paper is not the leading contribution in the [MATH] transition, which can proceed through spectator quark transition directly.', 'hep-ph-0408300-2-14-1': 'Use of the factorization approximation for the weak vertex leads to a prediction [MATH], which is already in excellent agreement with the experimental result of [MATH].', 'hep-ph-0408300-2-14-2': 'We found that inclusion of FSI reduces the theoretical prediction from 4% to [MATH].', 'hep-ph-0408300-2-14-3': 'The size of the shift also indicates that FSI of the type described in the present paper are in the case of [MATH] transition a second order effect.', 'hep-ph-0408300-2-14-4': 'Note as well, that the size of the FSI correction is in agreement with the predictions for [MATH] and [MATH], which are an order of magnitude smaller than [MATH].', 'hep-ph-0408300-2-15-0': 'We summarize that the hidden strangeness final state interactions are very important in understanding the [MATH] and [MATH] decay mechanism.', 'hep-ph-0408300-2-15-1': 'The [MATH] amplitude can be explained fully by this mechanism.', 'hep-ph-0408300-2-15-2': 'In the case of the [MATH] decay rate we obtain a fairly large range due to possible cancellation between FSI and single pole contributions.', 'hep-ph-0408300-2-16-0': 'The measurement of the [MATH] decay rate will considerably improve our understanding of the [MATH] decay mechanism.', 'hep-ph-0408300-2-16-1': 'The hidden strangeness FSI might fully explain the observed decay rate for [MATH].', 'hep-ph-0408300-2-16-2': 'Finally, this kind of FSI gives only subdominant contributions in the case of [MATH] decays, which are well described by the factorization approximation.'}

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[]

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[]

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{'1': 'http://arxiv.org/licenses/assumed-1991-2003/', '2': 'http://arxiv.org/licenses/assumed-1991-2003/'}

https://arxiv.org/abs/hep-ph/0408300

0907.5490

{'0907.5490-1-0-0': 'We have developed a contact-free 2-dimensional laser sensor with which the position of wires can be measured in 3 dimensions with an accuracy of better than [MATH]m and with which the tension of the wires can be determined with an accuracy of 0.04 N.', '0907.5490-1-0-1': 'These measurements can be made from a distance of 15 cm.', '0907.5490-1-0-2': 'The sensor consists of commercially available laser pointers, lenses, color filters and photodiodes.', '0907.5490-1-0-3': 'In our application we have used this laser sensor together with an automated 3 dimensional coordinate table.', '0907.5490-1-0-4': 'For a single position measurement, the laser sensor is moved by the 3-dimensional coordinate table in a plane and determines the coordinates at which the wires intersect with this plane.', '0907.5490-1-0-5': 'The position of the plane itself (the third coordinate) is given by the third axis of the measurement table which is perpendicular to this plane.', '0907.5490-1-0-6': 'The control and readout of the table and the readout of the laser sensor were realized with LabVIEW.', '0907.5490-1-0-7': 'The precision of the position measurement in the plane was determined with wires of 0.2 mm and 0.3 mm diameter.', '0907.5490-1-0-8': 'We use the sensor for the quality assurance of the wire electrode modules for the KATRIN neutrino mass experiment.', '0907.5490-1-0-9': 'We expect that the precision is at least comparable or better if the wires are thinner.', '0907.5490-1-0-10': 'Such a device could be well suited for the measurement of wire chamber geometries even with more than one wire layer.', '0907.5490-1-1-0': '# Introduction', '0907.5490-1-2-0': 'Conventional wire sensors measure the position in 1 dimensions along a certain axis and the tension of a wire from a short distance of a few millimeters [CITATION].', '0907.5490-1-2-1': 'The principle of these sensors can be extended to a laser-based sensor, which is sensitive to all three dimensions and which is able to measure from a much larger distance.', '0907.5490-1-2-2': 'Other authors describe wire position measurements with a laser diode with an accuracy of better than [MATH]m for relative wire separations of thin wires ([MATH]m diameter) in a wire chamber but with a principle, which is restricted to 1 dimension [CITATION].', '0907.5490-1-2-3': 'There is an example of a measurement with a single CCD and a precision of [MATH]m for wires of a wire chamber [CITATION] in 1 dimension.', '0907.5490-1-2-4': 'There is also an example [CITATION] of a measurement with two CCDs in 2 dimensions with an accuracy of [MATH]m but from a smaller distance of [MATH] mm.', '0907.5490-1-2-5': 'The last method is the most precise method in 2 dimensions, which we know, but it is not able to measure wire tensions.', '0907.5490-1-3-0': 'We have developed a contact-free laser sensor with which the position of wires can be measured in 2 dimensions (3 dimensions when operated with a 3-dimensional coordinate table) from a rather large distance of 15 cm.', '0907.5490-1-3-1': 'In this paper, we show that the accuracy of position measurements is better than [MATH]m for wires with a diameter of 0.2 mm - 0.3 mm.', '0907.5490-1-3-2': 'In addition the wire tension can be determined by measuring the oscillation frequency of the wire via the reflected laser light.', '0907.5490-1-3-3': 'In other sensors, such as described in [CITATION], an electrical contact between the wire and a sensor has to be established to measure the wire tension.', '0907.5490-1-3-4': 'As a mechanical contact with the wires has to be established, a damage or change of the wire tension cannot be excluded if the wires are thin.', '0907.5490-1-3-5': 'In another well-known method for measuring the wire tension, an external magnetic field has to be applied and mechanical oscillations of the wire are induced by passing an A.C. current through it.', '0907.5490-1-3-6': 'As long as no special techniques are employed one has to scan a frequency range to find the resonant frequency of the wire which gives the tension.', '0907.5490-1-3-7': 'This procedure can take several minutes [CITATION].', '0907.5490-1-3-8': 'Our sensor determines wire tensions in a few seconds.', '0907.5490-1-4-0': 'Our device could be well suited for the measurement of a complex detector which consists of many wires e.g. a large wire chamber.', '0907.5490-1-4-1': 'We are using it to control the quality of the wire modules of the inner electrode of the KATRIN neutrino mass experiment [CITATION] where a precision of 0.2 mm per module has to be ensured [CITATION][CITATION].', '0907.5490-1-4-2': 'Our article is organized in the following way: The principle of the position measurement with our novel sensor is presented in section [REF].', '0907.5490-1-4-3': 'An estimate of the accuracy of the position measurement is given in section [REF], section [REF] presents the tension measurements with this device and in section [REF] we estimate the accuracy of these measurements.', '0907.5490-1-4-4': 'Section [REF] explains the set-up of the actual sensor and in section [REF], we give a conclusion.', '0907.5490-1-5-0': '# Principle of the Position Measurement', '0907.5490-1-6-0': 'In this section we explain the principle of a position measurement of a wire with our laser sensor.', '0907.5490-1-6-1': 'The coordinates x and y of the wire will be determined by the sensor (fig. [REF]) the z coordinate is defined by the position of the plane in which the sensor moves during the measurement.', '0907.5490-1-6-2': 'In the following, we have to work with two different coordinate systems: A coordinate system with the index [MATH] which has its origin in the intersection of the red and green laser beams of the laser sensor (fig. [REF]) and a system with the index [MATH] for an absolute coordinate.', '0907.5490-1-6-3': 'We will also use values with the index [MATH], these are readings of the encoders for the sensor positions on a 3-dimensional coordinate table.', '0907.5490-1-6-4': 'The heart of the sensor are a red and a green laser (fig. [REF]) and two detectors directly next to the lasers.', '0907.5490-1-6-5': 'The detectors measure laser light which is reflected from the wires.', '0907.5490-1-6-6': 'The detectors are equipped with filters such that they detect only the red or green light from the red or green laser which is reflected by the wire.', '0907.5490-1-7-0': 'For a position measurement of the x and y coordinates of a wire, the sensor is being moved across the wire with the help of the 3-dimensional coordinate table along a straight line until both laser beams have hit the wire once and laser light was reflected and detected by the two detectors.', '0907.5490-1-7-1': 'During the motion, the position of the sensor and the intensity of the reflected light are being measured and digitized.', '0907.5490-1-7-2': 'The direction of motion is indicated by the vector [MATH] (fig. [REF]).', '0907.5490-1-7-3': 'This motion is parallel to the x-axis of the coordinate system of the 3-dimensional coordinate table, i.e. [MATH] changes continously.', '0907.5490-1-8-0': 'The wire has to be in the vicinity ([MATH] mm) of the intersection of the laser beams.', '0907.5490-1-8-1': 'This requirement will be explained in section [REF].', '0907.5490-1-8-2': 'For the following, we assume that the wire is between the laser sensor and the intersection of the two laser beams.', '0907.5490-1-8-3': 'The red reflection occurs, when the sensor is at the position [MATH] and the green reflection occurs, when the sensor is at [MATH].', '0907.5490-1-8-4': 'The positions [MATH] and [MATH] are defined as the maxima of the gaussians with which the reflected light distribution can be described (fig. [REF] and [REF]).', '0907.5490-1-9-0': 'Figure [REF] shows, that the y-coordinate [MATH] of the wire relative to the intersection point of the two laser beams is linearly related to the distance between the x-positions of the laser sensor [MATH] and [MATH] at which the reflexes occurred: [EQUATION]', '0907.5490-1-9-1': 'The slope m is defined by the two angles of the laser beams [MATH] and [MATH]: [EQUATION]', '0907.5490-1-9-2': 'We cannot measure the angles of laser beams [MATH] and [MATH] very accurately in a direct way, but we can determine precisely the positions of the sensor defined by the 3-dimensional coordinate table, when the reflections occur.', '0907.5490-1-9-3': 'Therefore, we determine the slope m of eq. ([REF]) as follows: We move the sensor across the wire at an unknown distance y and we record ([MATH]- [MATH]).', '0907.5490-1-9-4': 'Then, we move the sensor to a new position [MATH] which is [MATH] closer to the wire (e.g. [MATH] mm) and repeat the measurement at several distances [MATH] which differ by [MATH].', '0907.5490-1-9-5': 'After several measurements at different positions [MATH], we plot ([MATH] -[MATH]) against the different distances [MATH] from the wire, which differ by [MATH] and to fit the slope [MATH] to these data.', '0907.5490-1-9-6': 'As the distances [MATH] are not necessarily equal to the distance between the intersection of the laser beams and the wires, we do not use eq. ([REF]) as fit function but include an offset [MATH]', '0907.5490-1-10-0': '[EQUATION] in the fit function.', '0907.5490-1-10-1': 'Note that in eq. ([REF]), we write [MATH] and not [MATH].', '0907.5490-1-10-2': 'The offset [MATH] is necessary, because in general, the encoder coordinate [MATH], of the arm which moves the sensor and which was recorded during the measurement does not coincide with the coordinate [MATH].', '0907.5490-1-11-0': 'When a y-position is later determined with the laser sensor, we want to determine [MATH] from [MATH]- [MATH], thus we use formula ([REF]) without the offset [MATH] and the slope [MATH] which we determined with the fit of formula ([REF]) as this gives us the desired value [MATH].', '0907.5490-1-11-1': 'Thus, to calibrate the y-position of the sensor system means to determine the slope [MATH] of equations ([REF]) - ([REF]).', '0907.5490-1-12-0': 'Now we also have to calibrate the x-coordinate in the sensor system with respect to the intersection point of the two laser beams (see fig. 3).', '0907.5490-1-12-1': 'We define [MATH] by the reflection of the green laser.', '0907.5490-1-12-2': 'Therefore the angle [MATH] needs to be determined.', '0907.5490-1-12-3': 'From fig. [REF] we can read: [EQUATION]', '0907.5490-1-12-4': 'From the same calibration data which we used to obtain the slope [MATH] of eq. ([REF]) we can now determine [MATH] in a similar way: Again, the slope [MATH] does not depend on absolute coordinates but on the changes of the coordinates.', '0907.5490-1-12-5': 'Therefore we can plot the point of reflections of the green laser [MATH] versus the various positions [MATH] differing by [MATH] and fit the slope [MATH] to the data.', '0907.5490-1-13-0': 'Now, we have to combine the positions in the coordinate system of the laser sensor (indices [MATH]) and the readings of the encoders [MATH] and [MATH] to obtain absolute positions.', '0907.5490-1-13-1': 'In the previous discussion, we have implicitly assumed, that [MATH] is a positive number i.e. the distance between [MATH] and [MATH] becomes larger, if [MATH] increases.', '0907.5490-1-13-2': 'Thus, in fig. [REF] [MATH] increases from top to bottom.', '0907.5490-1-13-3': 'This implies, that [MATH] has to be subtracted from [MATH] to get an absolute position.', '0907.5490-1-13-4': 'As [MATH] and [MATH] have opposite orientations, they have to be subtracted as well.', '0907.5490-1-13-5': '[EQUATION]', '0907.5490-1-14-0': '# Accuracy of the Position Measurement', '0907.5490-1-15-0': 'To determine the accuracy of the position measurement with the laser sensor, we have mounted a test wire on a set-up which could be moved in x y direction with micrometer screws.', '0907.5490-1-15-1': 'The angle between the two lasers in our sensor is [MATH].', '0907.5490-1-16-0': 'Table [REF] shows the result of this test run.', '0907.5490-1-16-1': 'Positions with the index [MATH] were set with the micrometer screw.', '0907.5490-1-16-2': 'Positions with the index [MATH] were determined with the laser sensor.', '0907.5490-1-16-3': 'The resolution of the 3-dimensional coordinate table which moves the sensor is 0.01 mm.', '0907.5490-1-16-4': 'To estimate the precision of the laser sensor, we assume that the relation between the wire positions and the measured positions can be described by the equations [EQUATION] and [EQUATION] i.e. that there are only arbitrary offsets [MATH] between the laser sensor data and the actual positions.', '0907.5490-1-16-5': 'They result from the initial position of the micrometer screw which moved the test wire.', '0907.5490-1-16-6': 'Equations ([REF]) and ([REF]) were fitted to the data (table [REF]).', '0907.5490-1-16-7': 'The reduced [MATH] of these fits become 1 if we assume an uncertainty of 0.005 mm for the x-measurement and an uncertainty of 0.01 mm for the y-measurement.', '0907.5490-1-16-8': 'The uncertainty of the measurement of the x-coordinate is smaller, than the resolution of the 3-dimensional coordinate table.', '0907.5490-1-16-9': 'This is reasonable as the statistically distributed difference [MATH] is never larger, than the resolution of 0.01 mm itself.', '0907.5490-1-16-10': 'In this case, the intrinsic accuracy of the laser sensor is smaller than the resolution of the 3-dimensional coordinate table.', '0907.5490-1-17-0': '# Principle of the Tension Measurement', '0907.5490-1-18-0': 'If a wire with length [MATH], density [MATH] and cross-section [MATH] is held with a force [MATH] and corresponding tension [MATH] is deflected from its eqilibrium position, it will oscillate.', '0907.5490-1-18-1': 'The fundamental mode has a wavelength [MATH].', '0907.5490-1-18-2': 'Both ends of the wire are fixed.', '0907.5490-1-18-3': 'The speed of sound of a wave on such a wire is given by [EQUATION]', '0907.5490-1-18-4': 'Together with the relation [MATH], one arrives at the formula for the oscillation frequency of the wire: [EQUATION]', '0907.5490-1-18-5': 'This formula is a good approximation because for small oscillation amplitudes, no corrections for the elasticity of the material need to be applied.', '0907.5490-1-19-0': 'We use eq. ([REF]) to determine the tension of the wires in our electrode modules with our laser sensor.', '0907.5490-1-19-1': 'The design values for our wires are e.g. 10 N for wires with 0.3 mm diameter.', '0907.5490-1-19-2': 'The material of the wires is stainless steel type 1.4404 ([MATH]kg/m[MATH]) thus the oscillation frequency is approx. 37 Hz.', '0907.5490-1-19-3': 'To measure this oscillation, we measure the time-dependent reflection of one laser beam, e.g. the red one.', '0907.5490-1-19-4': 'Note, that the wire will reflect light twice per oscillation period, thus the reflected light pattern has a frequency [MATH].', '0907.5490-1-20-0': 'The wire has to be close to the intersection of the two laser beams because the diameter of the laser beam needs to be small.', '0907.5490-1-20-1': 'At this point, the laser beams are focussed to a minimum diameter by lenses (sec. [REF]).', '0907.5490-1-20-2': 'The wire starts to oscillate after it was shot at with short pulse of clean gas (argon with purity 4.6).', '0907.5490-1-20-3': "The nozzle, which is being used to generate this pulse is marked with 'C' in fig. [REF].", '0907.5490-1-20-4': 'The nozzle consists of a metal capillary with an inner diameter of 2 mm.', '0907.5490-1-20-5': 'An electronically controlled valve opens the connection between the nozzle and a reservoir of argon at 8 bar for 10 ms. This results in a pulse which has a diameter of approx. 10 mm at the location of the wire.', '0907.5490-1-20-6': 'The distance which the argon has to travel between the valve and the exit of the nozzle is approximately 130 mm (fig. [REF] and fig. [REF]).', '0907.5490-1-21-0': 'The reflected intensity versus time as recorded by our data acquisition system is plotted in fig. [REF], upper panel.', '0907.5490-1-21-1': 'We let a peak finding algorithm determine the position of the fundamental mode.', '0907.5490-1-21-2': 'The measurement stops if the peak finding algorithm has found a frequency in a certain range.', '0907.5490-1-21-3': 'In our example, this range could be 40 Hz to 90 Hz.', '0907.5490-1-21-4': 'The exact values for the upper and lower limits of this range, which one would choose depend on the variation of the actual wire tension.', '0907.5490-1-21-5': 'Our data acquisition system samples the signal from the detector with [MATH] samples per second, which corresponds to a Nyquist frequency of 4000 Hz.', '0907.5490-1-21-6': 'Since the oscillation of the reflected laser light becomes typically stationary after 1 s and dies away within less then ten seconds, we sample the signal for some seconds and calculate the Fourier transform every 0.5 s i.e. every 4000 samples (fig. 4 bottom).', '0907.5490-1-22-0': 'In order to suppress high frequency noise, we filter the signal with a 3rd order Butterworth lowpass with a cutoff frequency of 100 Hz in LabVIEW (version 8.0) before the power spectrum of the signal is being computed.', '0907.5490-1-23-0': "We use the LabVIEW function 'Power Spectrum' to compute the power spectrum of the signal.", '0907.5490-1-23-1': "The fundamental mode of the signal is being determined with the LabVIEW function 'Harmonic Distortion Analyzer'.", '0907.5490-1-23-2': 'Our 4000 samples at a sampling frequency of 8000 Hz result in a frequency spacing of 2 Hz (see fig. [REF] bottom).', '0907.5490-1-23-3': "The peak algorithm 'Harmonic Distortion Analyzer' can interpolate the oscillating frequency with a higher precision than the frequency spacing.", '0907.5490-1-24-0': '# Accuracy of the tension measurement', '0907.5490-1-25-0': 'In order to determine the precision of the tension measurement, we varied the tension of a 176 cm long test wire with 0.3 mm diameter and measured the frequency with our laser sensor (table [REF]).', '0907.5490-1-25-1': 'The tension was measured independently with a calibrated load cell (WRC-01 USB, model DBBP-20 by Weiss Robotics).', '0907.5490-1-26-0': 'Following eq. [REF], we fit the resulting data with a parabola [EQUATION] where [MATH] was the only free parameter.', '0907.5490-1-26-1': 'Note, that the measured frequency [MATH] is twice the oscillation frequency of the wire [MATH], as mentioned before.', '0907.5490-1-26-2': 'The reduced [MATH] of this fit becomes 1 if we assume an uncertainty of 0.04 N for the tension.', '0907.5490-1-26-3': 'The best fit parameter [MATH] is [MATH].', '0907.5490-1-26-4': 'Figure [REF] shows the parabolic fit of the data.', '0907.5490-1-26-5': 'The plot of the deviations (fig. [REF]) confirms the precision of the method.The parameter [MATH] can also be computed with eq. ([MATH]).', '0907.5490-1-26-6': 'With our parameters ([MATH] m, wire diameter = 0.3 mm and [MATH]), it should be [MATH], which is slightly larger than the experimental value.', '0907.5490-1-26-7': 'The two parameters [MATH] and [MATH] are known with high precision, but an uncertainty of the wire diameter could explain the difference between the theoretical and experimental value of [MATH].', '0907.5490-1-26-8': 'Our experimental value is compatible with theory if one assumes, that the real wire diameter is 0.285 mm instead of 0.3 mm, which is not unrealistic.', '0907.5490-1-27-0': '# Set-Up of the Laser Sensor', '0907.5490-1-28-0': 'Fig. [REF] displays a cross-sectional view of our laser sensor.', '0907.5490-1-28-1': 'Fig. [REF] shows the actual sensor mounted on the measurement head of our 3-dimensional coordinate table.', '0907.5490-1-28-2': 'The weight of this sensor is 250 g.', '0907.5490-1-28-3': 'It is vital that the laser beams have cross sections which are small compared to the diameters of the wires.', '0907.5490-1-28-4': 'To meet this requirement we focus the laser beams with lenses having a focal length (f=150 mm, diam.', '0907.5490-1-28-5': '22 mm) identical to the desired measurement distance between the laser and the wire (fig. [REF]).', '0907.5490-1-29-0': 'The position and frequency measurements are made in the vicinity ([MATH] mm) of the intersection of the laser beams.', '0907.5490-1-29-1': 'This point has a distance of 150 mm to these lenses.', '0907.5490-1-29-2': 'This guarantees, that the laser beams are narrow where the measurement is made.', '0907.5490-1-29-3': 'The laser beams, which are being generated by the laser pointers have diameters of approx. 2 mm.', '0907.5490-1-29-4': 'The lasers are ordinary laser pointers with 1 mW power.', '0907.5490-1-29-5': 'The wavelength of the green laser is 532 nm and the wavelength of the red laser pointer lies between 640 nm and 660 nm.', '0907.5490-1-29-6': 'The battery compartments of the laser pointers were removed (fig. [REF]) and the batteries were replaced with a laboratory power supply and connected with cables (fig. [REF]).', '0907.5490-1-30-0': 'The removal of the batteries helps to keep the weight down and avoids de-adjustments of the lasers due to the torque created by the weight of the batteries.', '0907.5490-1-30-1': 'The reflected light is detected with pin-diodes.', '0907.5490-1-30-2': 'The signal of the pin-diodes is being amplified by a custom-made operational amplifier circuit.', '0907.5490-1-30-3': 'The filters, which sit in front of the diodes are a green and red bandpass filter by Baader Planetarium, Germany (item number 2458304 and 2458307).', '0907.5490-1-30-4': 'As it should be clear from section [REF] the diodes need to be equipped with filters such that they detect only red or green light.', '0907.5490-1-30-5': 'The filters make sure that the device can be operated in an illuminated room.', '0907.5490-1-30-6': 'The detectors are both equipped with special optics which focus the light on the diodes.', '0907.5490-1-30-7': 'The first lens (f=150 mm, diam.', '0907.5490-1-30-8': '22 mm) parallelizes the reflected light and the second lens (f=20 mm) focusses it on the diode.', '0907.5490-1-31-0': 'Our laser sensor is being used with a very robust 3-dimensional coordinate table (weight 2500 kg, System C by Stiefelmayer).', '0907.5490-1-31-1': 'We equipped this coordinate table with three motors (type A-Max by Maxon Motor) for the three axes.', '0907.5490-1-31-2': 'Due to the use of a very robust 3 dimensional coordinate table and speed-regulated direct current motors we could avoid vibrations of the laser sensor which would spoil the measurements.', '0907.5490-1-31-3': 'The motors are controlled with LabVIEW via three RS-232 interfaces of our measurement computer.', '0907.5490-1-31-4': 'This computer also digitizes the signals from the laser sensor with a measurement card (NI PCI-6014 by National Instruments).', '0907.5490-1-31-5': 'The 3-dimensional position of the measurement arm, which holds the laser sensor is measured with three magneto-mechanical encoders.', '0907.5490-1-31-6': 'The resolution of these encoders is 0.01 mm.', '0907.5490-1-31-7': 'The position information of these three encoders is digitized by electronics built in our electronic workshop.', '0907.5490-1-31-8': 'The position information is read in from this electronics into our measurement computer with LabVIEW via the above mentioned measurement card.', '0907.5490-1-32-0': '# Conclusion', '0907.5490-1-33-0': 'We developed a new 2-dimensional laser sensor to determine wire positions as well as wire tensions from a distance of 15 cm.', '0907.5490-1-33-1': 'The laser sensor is moved by a precise 3-dimensional coordinate table and the intensity of the reflected light is recorded resulting in position resolution of better than [MATH]m. Wire oscillations can be excited by blowing a short pulse of clean gas on it.', '0907.5490-1-33-2': 'The Fast Fourier Transform of the reflected light yields the wire tension with a precision of 0.04 N.', '0907.5490-1-33-3': 'An important property for our application is the lightweight construction of the laser sensor ([MATH] g) to avoid a bending of the arm of the 3-dimensional coordinate table.', '0907.5490-1-33-4': 'The key features of our novel sensor are the ability to measure precise 2-dimensional positions from a distance of 15 cm which is much larger than in other sensors.', '0907.5490-1-33-5': 'It also combines position and tension measurements.', '0907.5490-1-33-6': 'The tension measurements can be done contactless within a few seconds.', '0907.5490-1-34-0': 'We apply this wire measurement system for the quality assurance of the wire electrode system of the KATRIN neutrino mass experiment [CITATION].', '0907.5490-1-34-1': 'In a second version of the sensor we have enlarged the detector lenses to a diameter of 40 mm in order to increase the angle of acceptance to [MATH] to measure wires non-perpendicular to the laser plane.'}

{'0907.5490-2-0-0': 'We have developed a contact-less two-dimensional laser sensor which combines position and tension measurements of wires with a diameter of order 0.2 mm.', '0907.5490-2-0-1': 'The sensor consists of commercially available laser pointers, lenses, color filters and photodiodes.', '0907.5490-2-0-2': 'In our application we have used this laser sensor in conjunction with an automated three-dimensional coordinate measuring machine (CMM).', '0907.5490-2-0-3': 'The device allows for a position measurement of wires in three dimensions with an accuracy of about [MATH].', '0907.5490-2-0-4': 'At the same time the wire tension can be determined with an accuracy of 0.04 N.', '0907.5490-2-0-5': 'The device is operated at a distance of 150 mm from the wire.', '0907.5490-2-1-0': 'For each position measurement, the laser sensor is moved by the automated CMM into a plane, where the coordinates at which the wires intersect with this plane are determined.', '0907.5490-2-1-1': 'The position of the plane itself (the third coordinate) is given by the third axis of the CMM which is perpendicular to this plane.', '0907.5490-2-1-2': 'The precision of the device was determined using stainless steel wires with a diameter of 0.2 mm and a tension of 5 N.', '0907.5490-2-1-3': 'We use the sensor for quality assurance of the wire electrode modules for the KATRIN neutrino mass experiment.', '0907.5490-2-1-4': 'These modules are comprised of two layers of wires, which are 70 mm apart.', '0907.5490-2-1-5': 'In general, the device presented here is well suited for the measurement of any complex wire chamber geometry.', '0907.5490-2-2-0': 'section', '0907.5490-2-3-0': '# Introduction', '0907.5490-2-4-0': 'The main spectrometer of the KATRIN neutrino mass experiment [CITATION] will be equipped with 248 wire electrodes (Fig. [REF]).', '0907.5490-2-4-1': 'These wire electrodes consist of two layers of stainless steel wires, which are approximately 70 mm apart.', '0907.5490-2-4-2': 'The diameters of the wires are 0.2 mm and 0.3 mm, respectively for the two layers.', '0907.5490-2-4-3': 'We have developed a special laser sensor, which provides the measurement of the wire positions and their respective tensions in both layers.', '0907.5490-2-4-4': 'The instrument had to meet the following specifications: 1) Measurement from a distance of 150 mm to avoid the necessity to move through a wire layer during a measurement, 2) position measurement with an accuracy of at least 0.1 mm for wires with a diameter of 0.2 mm and 0.3 mm, 3) tension measurement with an accuracy of better than 0.1 N to assure that wire tension is less than 1 N above a critical value, 4) total weight of less than 250 g, and, 5) operation under clean room conditions.', '0907.5490-2-5-0': 'With the ascent of wire chambers in nuclear and particle physics many methods have been developed to precisely measure the position and tension of thin wires ([CITATION], [CITATION], [CITATION], [CITATION], [CITATION], [CITATION]).', '0907.5490-2-5-1': 'Based on this experience our developments focused on the above requirements.', '0907.5490-2-6-0': 'The basic idea of our method is the use of two laser beams of different wavelength which allows for a very fast kind of triangulation of the position of the wire in two dimensions.', '0907.5490-2-6-1': 'The lasers and the corresponding light detectors are precisely moved with a three-dimensional coordinate measurement machine (CMM) to allow an overall three-dimensional position determination.', '0907.5490-2-6-2': 'When the wire is excited to oscillate by a puff of clean gas, the Fourier transform of the reflected light signal is used to determine the wire tension.', '0907.5490-2-6-3': 'The combination of these methods is new and fulfills all the mentioned requirements.', '0907.5490-2-6-4': 'Our device could be well suited for the measurement of a complex detector which consists of many wires e.g. a large wire chamber.', '0907.5490-2-7-0': 'Our article is organized in the following way: in Section II, we describe the setup of our laser sensor.', '0907.5490-2-7-1': 'Section III covers the principle and accuracy of the position measurement with our device.', '0907.5490-2-7-2': 'In Section IV the tension measurement is described.', '0907.5490-2-7-3': 'In Section V we discuss sources of systematic errors.', '0907.5490-2-7-4': 'Finally, in Section VI and VII we give a comparison to other methodes and our conclusions.', '0907.5490-2-8-0': '# Set-Up of the Laser Sensor', '0907.5490-2-9-0': 'Figure [REF] shows our contact-less two-dimensional laser sensor.', '0907.5490-2-9-1': 'The laser beams, which are generated by ordinary laser pointers with 1 mW power have diameters of approx. 2 mm at the exit of the laser pointers.', '0907.5490-2-9-2': 'The wavelength of the green laser is 532 nm.', '0907.5490-2-9-3': 'The wavelength of the red laser pointer is in the range of 640 nm to 660 nm.', '0907.5490-2-9-4': 'The battery compartments of the laser pointers were removed (Fig. [REF]) and the batteries were replaced by cables connected to a laboratory power supply (Fig. [REF]).', '0907.5490-2-9-5': 'The removal of the batteries also led to a reduction in deformation of the laser mount due to the weight of the batteries.', '0907.5490-2-9-6': 'In order to make precise position measurements with our detector, the laser beams need to have diameters which are small compared to the diameter of the wire.', '0907.5490-2-9-7': 'To meet this requirement we focus the laser beams with lenses (f=150 mm, diam.', '0907.5490-2-9-8': '22 mm), where the focal length corresponds to the desired distance between the laser and the wire.', '0907.5490-2-9-9': 'The intersection point of both laser beams coincides with the focal points of both lasers.', '0907.5490-2-10-0': 'The light reflected from the wire is detected with PIN diodes.', '0907.5490-2-10-1': 'The special optics of the light detectors (Fig. [REF]) has the purpose to focus as much light as possible onto the PIN diode.', '0907.5490-2-10-2': 'The first lens (f = 150 mm) parallelizes the light, which is reflected from the wire.', '0907.5490-2-10-3': 'The second lens (f = 20 mm) focuses the light onto the PIN diode.', '0907.5490-2-10-4': 'Green and red bandpass filters [CITATION] sit in front of the PIN diodes.', '0907.5490-2-10-5': 'The light detectors need to be equipped with filters such that they detect only red or green light.', '0907.5490-2-10-6': 'The filters ensure that the device can be operated under standard light conditions and that crosstalk between the two different reflection signals can be excluded.', '0907.5490-2-10-7': 'The signal of the PIN diodes is amplified by a custom-made operational amplifier circuit and recorded by a digitizer card from National Instruments [CITATION].', '0907.5490-2-11-0': 'For the position measurement the laser sensor is moved by a three-dimensional coordinate measurement machine (CMM) across the wire at a distance of [MATH]150 mm and to detect the laser light, which is is reflected from a wire (Fig. [REF]).', '0907.5490-2-11-1': 'This distance needs to be within 5 mm of the focal points due the use of the focusing optics.', '0907.5490-2-11-2': 'Figures [REF] and [REF] show the sensor mounted on the measurement head of our CMM.', '0907.5490-2-11-3': 'Our CMM is a 20 years old and very robust machine (weight 2500 kg, System C by Stiefelmayer, Fig. [REF]).', '0907.5490-2-11-4': 'It has been equipped with three motors (type A-Max by Maxon Motor [CITATION]) for the three axes and modified to make it compliant with the operation in a class 10000 clean room [CITATION].', '0907.5490-2-12-0': 'Due to the use of a very robust CMM and speed-regulated direct current motors we could avoid vibrations of the laser sensor which would otherwise affect the precision of the measurements.', '0907.5490-2-12-1': 'The motors are controlled with LabVIEW via three RS-232 interfaces of a PC, which also digitizes the signals from the laser sensors.', '0907.5490-2-12-2': 'The complete software was written with LabVIEW, version 8.0 [CITATION].', '0907.5490-2-12-3': 'The three-dimensional position of the measurement arm, which holds the laser sensors is measured with three magneto-mechanical encoders with a resolution of 0.01 mm.', '0907.5490-2-12-4': 'The position information of these three encoders is recorded with LabVIEW via the measurement card.', '0907.5490-2-12-5': 'The requirement for the laser sensor to be light-weight results from the maximum load specification of 250 g for a sensor on the CMM, which does not compromise the specified precision.', '0907.5490-2-13-0': '# Position Measurement', '0907.5490-2-14-0': 'In this section we explain the principle of position measurement of a wire with our laser sensor.', '0907.5490-2-14-1': 'The coordinates x and y of the wire will be determined by the sensor (Fig. [REF]) by moving it across the wire in x- respectively y-direction and recording the x(y)-positions [MATH] and [MATH], when the reflections of the green and red lasers from the wire hit the detectors.', '0907.5490-2-14-2': 'The z-coordinate is defined by the CMM-position of the laser sensor.', '0907.5490-2-15-0': 'The basic [MATH] coordinate system is defined by the CMM encoders.', '0907.5490-2-15-1': 'For a given [MATH], a fixed point [MATH] can be set by setting the sensor such, that a reference wire on the device to be scanned is at the intersection of the two laser beams.', '0907.5490-2-15-2': 'The angles of the two laser beams have to be calibrated beforehand.', '0907.5490-2-15-3': 'For this, the sensor has to be moved across a wire at different y-distances (e.g. 5 mm, 10 mm, 15 mm, 20 mm).', '0907.5490-2-15-4': 'The angles of the laser beams can then be calculated by fitting the distance versus the position of the maximum of the reflection in CMM coordinates [MATH]).', '0907.5490-2-16-0': 'The position [MATH] of any other wire can be calculated using this calibration point and the inclinations of the laser beams [MATH] and [MATH]), where [MATH] and [MATH] (cf. Fig. [REF]), as follows: [EQUATION]', '0907.5490-2-17-0': 'In order to determine the accuracy of the position measurement with the laser sensor, we have mounted a test wire with a diameter of 0.2 mm in a set-up which could be moved in both x- and y-direction by means of micrometer screws.', '0907.5490-2-17-1': 'Table [REF] shows the results of these tests.', '0907.5490-2-17-2': 'Positions with index [MATH] were set using the micrometer screws.', '0907.5490-2-17-3': 'Positions with index [MATH] were determined with the laser sensor.', '0907.5490-2-17-4': 'The resolution of the CMM which moves the sensor is 0.01 mm.', '0907.5490-2-17-5': 'To estimate the precision of the laser sensor, we assume that the relation between the wire positions and the measured positions can be described by the equations: [EQUATION]', '0907.5490-2-17-6': 'There can be arbitrary offsets [MATH] and [MATH] between the laser sensor data and the actual positions.', '0907.5490-2-17-7': 'The offsets result from the initial position of the micrometer screw which moved the test wire.', '0907.5490-2-17-8': 'The slopes [MATH] and [MATH] indicate the size of the systematic error.', '0907.5490-2-18-0': 'Equations ([REF]) were fitted to the data (Table [REF]) by a least-squares fit under the assumption of a constant uncertainty [MATH]) for the x- and y-determination by the laser sensor.', '0907.5490-2-18-1': 'This resulted in uncertainties of [MATH] mm and [MATH] mm, respectively.', '0907.5490-2-19-0': 'The uncertainty of the measurement of both coordinates is smaller than the smallest encoder step [MATH] mm of the CMM.', '0907.5490-2-19-1': 'This is reasonable as the statistical limit [MATH] for the statistically distributed difference [MATH] is [EQUATION]', '0907.5490-2-19-2': 'The results in Table [REF] show the precision of the position measurements at a distance of less than 1 mm from the intersection of the two laser beams.', '0907.5490-2-19-3': 'Especially the slope of [MATH] mm/mm could be due to scale errors of the micrometer screw.', '0907.5490-2-19-4': 'However, it is also possible that the deviation of [MATH] from unity points towards a systematic uncertainty in the determination of the position of the wire far away from the intersection of the laser beams.', '0907.5490-2-19-5': 'This shows that the distance between the wire and the intersection point has to be minimized in order to guarantee a precision of about [MATH].', '0907.5490-2-20-0': '# Tension Measurement', '0907.5490-2-21-0': 'When a wire of length [MATH], density [MATH] and cross-sectional area [MATH] is stretched by a force [MATH] between two fixed ends, is excited in transverse direction, it will oscillate.', '0907.5490-2-21-1': 'The fundamental mode has a wavelength of [MATH].', '0907.5490-2-21-2': 'The speed of sound of a wave on such a wire is given by [EQUATION]', '0907.5490-2-21-3': 'With [MATH], the oscillation frequency [MATH] of the fundamental mode of the wire is given by: [EQUATION]', '0907.5490-2-21-4': 'For small oscillation amplitudes this formula is correct to good approximation, since corrections for the elasticity of the material can be neglected.', '0907.5490-2-22-0': 'We use Eq. ([REF]) to determine the tension of the wires in our electrode modules with our laser sensor.', '0907.5490-2-22-1': 'The design value is 10 N for wires with 0.3 mm diameter and 1.8 m length.', '0907.5490-2-22-2': 'The material of the wires is stainless steel type 1.4404 ([MATH]kg/m[MATH]).', '0907.5490-2-22-3': 'Thus the oscillation frequency is expected to be around 37 Hz.', '0907.5490-2-22-4': 'The frequency is determined through measurement of the time-dependent reflection amplitude of one of the laser beams, e.g. the red one.', '0907.5490-2-22-5': 'Note that the wire will reflect light twice per oscillation period, thus the reflected light pattern has a frequency [MATH].', '0907.5490-2-22-6': 'We always observed the lowest frequency at this frequency.', '0907.5490-2-22-7': 'Higher harmonics always appear at multiples of [MATH].', '0907.5490-2-22-8': 'We observed that these always exist in our measurements.', '0907.5490-2-22-9': 'The first harmonic is often dominant directly after the excitation and dies out quickly.', '0907.5490-2-22-10': 'The wire has to be close to the intersection of the two laser beams, because the diameter of the laser beam needs to be small.', '0907.5490-2-22-11': 'The wire starts to oscillate after it was excited by a short puff of clean gas (Ar 4.6).', '0907.5490-2-22-12': 'A single pulse is sufficient since even a small displacement of the wire results in a clearly detectable signal.', '0907.5490-2-22-13': "The nozzle, which is used to direct that puff at the wire is labeled with 'C' in Fig. [REF].", '0907.5490-2-22-14': 'The nozzle consists of a metal capillary with an inner diameter of 2 mm.', '0907.5490-2-22-15': 'An electronically controlled valve opens the connection between the nozzle and a reservoir of argon with a pressure of 8 bar for a duration of 10 ms. This results in a puff which has a diameter of approx. 10 mm at the location of the wire.', '0907.5490-2-22-16': 'The distance which the argon has to travel between the valve and the exit of the nozzle is approximately 130 mm (cf. Fig. [REF] and Fig. [REF]).', '0907.5490-2-22-17': 'One can also excite wire oscillations with a mechanical device, e.g. a small hammer, without damaging the wires.', '0907.5490-2-22-18': 'However, our method is entirely contact-less.', '0907.5490-2-22-19': 'The versatility of the CMM avoids the necessity to adapt any mechanics for different module geometries.', '0907.5490-2-23-0': 'The reflected intensity versus time as recorded by our data acquisition system is plotted in the upper plot of Fig. [REF].', '0907.5490-2-23-1': "Also shown, in the lower plot, is the corresponding Fourier transformation derived with the LabVIEW function 'Power Spectrum'.", '0907.5490-2-23-2': "We use a peak finding algorithm (LabVIEW function 'Harmonic Distortion Analyzer') to determine the position of the fundamental mode, which is able to interpolate in steps of 2 Hz resulting from the sampling frequency of 8000 Hz and the sampling length of 0.5 s.", '0907.5490-2-23-3': 'The measurement stops, when the peak finding algorithm has detected a frequency within a certain range.', '0907.5490-2-23-4': 'In our case, this range was chosen from 40 Hz to 90 Hz to find frequencies around [MATH] Hz.', '0907.5490-2-23-5': 'The exact values to be chosen as upper and lower bounds depend on the variations incurred in the actual wire tensions.', '0907.5490-2-23-6': 'Our data acquisition system samples the signal from the detector with [MATH] samples per second corresponding to a Nyquist frequency of 4000 Hz.', '0907.5490-2-23-7': 'Typically, the oscillation of the reflected laser light is stationary after 1 s and dies out within less then ten seconds.', '0907.5490-2-23-8': 'Therefore, we have chosen to sample the signal for a few seconds and calculate the Fourier transform every 0.5 s i.e. every 4000 samples (Fig. [REF] bottom).', '0907.5490-2-24-0': 'In order to suppress high frequency noise, we filter the signal with a third order Butterworth low-pass with a cutoff frequency of 100 Hz in LabVIEW before the power spectrum of the signal is being computed.', '0907.5490-2-24-1': 'The cutoff frequency is also chosen such that harmonics above the signal frequency of the fundamental mode are suppressed.', '0907.5490-2-25-0': 'The choice of this particular filter has no special reason, but it satisfies our requirements nicely.', '0907.5490-2-25-1': "Using a software filter from LabVIEW's library instead of a hardware filter maintained our flexibility for different edge frequencies, since we commissioned the laser sensor while the first electrode modules for KATRIN were built and the design of the last modules was still undefined.", '0907.5490-2-26-0': 'In order to determine the precision of the tension measurement, we varied the tension of a 176 cm long test wire with 0.3 mm diameter and measured the frequency with our laser sensor (Fig. [REF]).', '0907.5490-2-26-1': 'The tension was measured independently with a calibrated load cell (WRC-01 USB, model DBBP-20 by Weiss Robotics).', '0907.5490-2-26-2': 'Following Eq. [REF], we fit the resulting data with a parabola [EQUATION] where [MATH] is a free parameter.', '0907.5490-2-26-3': 'As mentioned above the measured frequency [MATH] corresponds to twice the oscillation frequency of the wire [MATH].', '0907.5490-2-26-4': 'Assuming a constant uncertainty of the force [MATH] the least-squares fit results in [MATH] N for the uncertainty of the tension determination (Fig. [REF]).', '0907.5490-2-26-5': 'The measured deviations from the fit (Fig. [REF] bottom) confirm the precision of the method.', '0907.5490-2-27-0': 'The fit result [MATH] can be compared to the expectation from Eq. ([REF]).', '0907.5490-2-27-1': 'One has to keep in mind that there are uncertainties of the parameters [MATH], [MATH] and [MATH].', '0907.5490-2-27-2': 'The reflection of the standing wave at the ends of the wire is not necessarily ideal.', '0907.5490-2-27-3': 'We do not know the uncertainty of [MATH] but could measure [MATH] and the wire diameter [MATH] directly.', '0907.5490-2-27-4': 'The measured parameters ([MATH] m and [MATH] mm) imply a theoretical value [MATH].', '0907.5490-2-27-5': 'I.e. the theoretical prediction is unsignificantly larger than the observation.', '0907.5490-2-28-0': '# Error Sources and Limitations', '0907.5490-2-29-0': 'For the stainless steel wires under scrutiny (diam.', '0907.5490-2-29-1': '0.2 mm - 0.3 mm, [MATH] m, [MATH]), we observe that the tension can no longer be determined when it is less than [MATH] 1 N. Around that value, the intensity variation of the reflected laser light is no longer periodical.', '0907.5490-2-29-2': 'An important specification for our sensor is a maximum weight of 250 g.', '0907.5490-2-29-3': 'The weight of the sensor causes the arm of the CMM to sag by 30 [MATH]m when fully extended (1 m).', '0907.5490-2-29-4': 'This sag is tolerable for us, and can be compensated for.', '0907.5490-2-29-5': 'The precision of the position measurements depends on the correct alignment of the two laser beams.', '0907.5490-2-29-6': 'When the laser beams are not perfectly aligned with the measurement plane, the beams hit the wire a slightly too early or too late, when the sensor moves across a wire.', '0907.5490-2-29-7': 'This leads to an error in the determination of [MATH] and [MATH] leading to a systematic error of the wire position.', '0907.5490-2-29-8': 'Suppose that the red laser is correctly aligned, but the green laser is displaced by 0.5 mm with respect to the measurement plane.', '0907.5490-2-29-9': 'Let the measured wire lie in the (x,z)-plane and both angles [MATH] and [MATH] are [MATH].', '0907.5490-2-29-10': 'Let us further assume, that the measured wire has an inclination of [MATH] with respect to the z-axis.', '0907.5490-2-29-11': 'The resulting errors due to this misalignment will be: [EQUATION]', '0907.5490-2-30-0': '# Comparison with other Methods', '0907.5490-2-31-0': 'There are numerous methods which can be used to measure wire tensions and positions from a distance.', '0907.5490-2-31-1': 'In a well-known method for measuring the wire tension, an external magnetic field is applied and mechanical oscillations of the wire are induced by passing an alternating current through it.', '0907.5490-2-31-2': 'As long as no special techniques are employed one has to scan a frequency range in order to find the resonance frequency of the wire, which provides the tension measurement.', '0907.5490-2-31-3': 'This process can take several minutes [CITATION].', '0907.5490-2-31-4': 'This time can be drastically shortened using a sudden excitation of the wire in the magnetic field with a current pulse [CITATION] or an electrostatic excitation through capacitive coupling [CITATION].', '0907.5490-2-31-5': 'In any case, one still needs to connect the wires to an external circuit, which is still time-consuming.', '0907.5490-2-31-6': 'Moreover, it could lead to a damage of the wires.', '0907.5490-2-32-0': 'It has also been reported that one can excite wire oscillations with a small hammer, even if the wire thickness is only 30 [MATH]m, without causing any damage [CITATION].', '0907.5490-2-32-1': 'It could be difficult to adapt the corresponding hardware to changing geometries.', '0907.5490-2-32-2': 'Thus the real advantage of our excitation method is not that we avoid damage, but that we avoid the necessity to adapt mechanics to changing module geometries.', '0907.5490-2-32-3': 'Other authors describe wire position measurements with a laser diode with an accuracy of better than [MATH]m for relative wire separations of thin wires ([MATH]m diameter) in a wire chamber but with a principle which is restricted to one dimension [CITATION].', '0907.5490-2-32-4': 'There is also an example [CITATION] of a measurement with two CCDs in 2 dimensions with an accuracy of [MATH]m but from a smaller distance of [MATH] mm.', '0907.5490-2-32-5': 'The last method is the most precise method in two dimensions of which we are aware, however, it cannot be used to measure wire tensions.', '0907.5490-2-33-0': '# Conclusion', '0907.5490-2-34-0': 'We developed a new two-dimensional laser sensor for the measurement of parameters of thin wires.', '0907.5490-2-34-1': 'The key feature of our novel sensor is the combination of a two-dimensional position measurement with a tension measurement at a distance of 150 mm with respect to the wires.', '0907.5490-2-34-2': 'Both the position and the tension measurements can be performed within a few seconds without touching the wire.', '0907.5490-2-34-3': 'The laser sensor is moved by a precise three-dimensional coordinate table and the intensity of the reflected light is recorded.', '0907.5490-2-34-4': 'With this a position resolution of better than [MATH]m has been achieved.', '0907.5490-2-34-5': 'For the measurement of the wire tension, oscillations are excited by a puff of gas blown at the wire, and the intensity variation of the reflected light is recorded.', '0907.5490-2-34-6': 'The fundamental frequency of this variation is found by means of a fast Fourier transform and yields the wire tension with a precision of 0.04 N.', '0907.5490-2-34-7': 'An important property for our application is the light-weight construction of the laser sensor ([MATH] g).', '0907.5490-2-34-8': 'We apply this wire measurement system for the quality assurance of the wire electrode system of the KATRIN neutrino mass experiment [CITATION].', '0907.5490-2-34-9': 'In a second version of the sensor we have now enlarged the detector lenses to a diameter of 40 mm in order to increase the angle of acceptance to [MATH] to measure wires non-perpendicular to the laser plane.', '0907.5490-2-34-10': 'In practice, our sensor needs [MATH] minutes to measure the tensions of all 120 wires of a module and [MATH] minutes to scan 120 wire position in one plane.', '0907.5490-2-34-11': 'A large fraction of the time, our CMM has to move the sensor from one wire to another.', '0907.5490-2-34-12': 'As soon as the CMM has positioned the sensor on a wire, it takes between two and four seconds to obtain a result for a wire tension.', '0907.5490-2-34-13': 'A position measurement, i.e., a scan across a wire, takes [MATH] 10 s.', '0907.5490-2-34-14': 'If it is required to speed up the measurement process, a faster CMM machine could be used, up to the limit of vibration damping of the sensor.'}

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[]

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['0907.5490-1-13-5', '0907.5490-2-2-0']

{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/0907.5490

1811.00021

{'1811.00021-1-0-0': 'Pyrochlore systems (A[MATH]B[MATH]O[MATH]) with A-site rare-earth local moments and B-site 5d conduction electrons offer excellent material platforms for the discovery of exotic quantum many-body ground states.', '1811.00021-1-0-1': 'Notable examples include U(1) quantum spin liquid (QSL) of the local moments and semi-metallic non-Fermi liquid of the conduction electrons.', '1811.00021-1-0-2': 'Here we investigate emergent quantum phases and their transitions driven by the Kondo-lattice coupling between such highly-entangled quantum ground states.', '1811.00021-1-0-3': 'Using the renormalization group method, it is shown that weak Kondo-lattice coupling is irrelevant, leading to a fractionalized semimetal phase with decoupled local moments and conduction electrons.', '1811.00021-1-0-4': 'Upon increasing the Kondo-lattice coupling, this phase is unstable to the formation of broken symmetry states.', '1811.00021-1-0-5': 'Particularly important is the opposing influence of the Kondo-lattice coupling and long-range Coulomb interaction.', '1811.00021-1-0-6': 'The former prefers to break the particle-hole symmetry while the latter tends to restore it.', '1811.00021-1-0-7': 'The characteristic competition leads to possibly multiple phase transitions, first from a fractionalized semimetal phase to a fractionalized Fermi surface state with particle-hole pockets, followed by the second transition to a fractionalized ferromagnetic state.', '1811.00021-1-0-8': 'Multi-scale quantum critical behaviors appear at non-zero temperatures and with external magnetic field near such quantum phase transitions.', '1811.00021-1-0-9': 'We discuss the implication of these results to the experiments on Pr[MATH]Ir[MATH]O[MATH].', '1811.00021-1-1-0': 'Introduction : Recent advances in correlated electron systems reveal emergent phenomena beyond the Landau paradigm.', '1811.00021-1-1-1': 'Localized magnetic moments may host quantum spin liquid phases characterized by fluctuating gauge fields and fractionalized particles [CITATION].', '1811.00021-1-1-2': 'Itinerant electron systems may show non-Fermi liquid behavior without quasi-particles [CITATION].', '1811.00021-1-1-3': 'Such phenomena and associated quantum phase transitions demand development of new concepts and novel understandings in strongly interacting quantum many body systems [CITATION].', '1811.00021-1-2-0': 'In this work, we study the intertwined model of two emergent phases beyond the Landau paradigm.', '1811.00021-1-2-1': 'We consider the interaction between the local moment system supporting a U(1) quantum spin liquid and a non-Fermi liquid semimetallic state of conduction electrons.', '1811.00021-1-2-2': 'This model is partly motivated by physics of the pyrochlore materials, A[MATH]B[MATH]O[MATH], where the A- and B-site pyrochlore lattices are occupied by rare-earth local moments and 5d conduction electrons, respectively.', '1811.00021-1-2-3': 'The A-site local moments may form a quantum spin liquid with emergent photons, a.k.a quantum spin ice, as suggested in Yb[MATH]Ti[MATH]O[MATH], Pr[MATH]Hf[MATH]O[MATH], and Pr[MATH]Zr[MATH]O[MATH] [CITATION].', '1811.00021-1-2-4': 'When the B-site is occupied by conduction electrons in 5d orbitals, such as J[MATH]=1/2 Kramers doublet of Ir ions, the system supports a non-Fermi liquid semimetal, so-called Luttinger-Abrikosov-Beneslaevski (LAB) state, which is derived from the quadratic band-touching with long-range Coulomb interaction [CITATION].', '1811.00021-1-2-5': 'The quadratic band-touching of Ir conduction electrons is confirmed in the ARPES experiment in the finite-temperature paramagnetic state of Pr[MATH]Ir[MATH]O[MATH] and Nd[MATH]Ir[MATH]O[MATH] [CITATION].', '1811.00021-1-2-6': 'It is believed that the interplay between the two emergent phases mentioned above may play a crucial role in low temperature physics of Pr[MATH]Ir[MATH]O[MATH] [CITATION] .', '1811.00021-1-3-0': 'Considering the Kondo-lattice coupling between the localized moments and conduction electrons, we first construct a low energy effective field theory for the coupled system of the U(1) QSL and the non-Fermi liquid semimetal state.', '1811.00021-1-3-1': 'We use the renormalization group method to investigate emergent phases and phase transitions and find that the Kondo-lattice coupling and the long-range Coulomb interaction shows intriguing interplay physics.', '1811.00021-1-3-2': 'Namely, the former shows the tendency of breaking the particle-hole symmetry but the latter plays the opposite role.', '1811.00021-1-3-3': 'For small Kondo-lattice coupling, the Coulomb interaction prevails, and the two underlying phases remain weakly coupled.', '1811.00021-1-3-4': 'This is the Luttinger semimetal coexisting with fractionalized excitations and emergent photons from the local moments.', '1811.00021-1-3-5': 'For sufficiently strong interaction, either time-reversal or inversion symmetry may be broken and our perturbative renormalization group analysis shows that time-reversal symmetry breaking is the most relevant channel.', '1811.00021-1-3-6': 'Our analysis suggests that the particle-hole symmetry breaking occurs first as the Kondo-lattice coupling becomes dominant over the long-range Coulomb interaction, leading to a fractionalized Fermi surface state with emergent particle-hole pockets.', '1811.00021-1-3-7': 'This is followed by the time-reversal symmetry breaking transition to a ferromagnetically-ordered fractionalized semimetal phase.', '1811.00021-1-3-8': 'We discuss the resulting multi-scaling critical behavior in light of some key experiments in the low temperature phase of Pr[MATH]Ir[MATH]O[MATH].', '1811.00021-1-4-0': 'Model : We start with a generic model Hamiltonian for the pyrochlore system, A[MATH]B[MATH]O[MATH], [EQUATION]', '1811.00021-1-4-1': 'The Hamiltonian for A sites ([MATH]) describes localized magnetic moments ([MATH]), and the Hamiltonian for B sites ([MATH]) describes conduction electrons with annihilation and creation operators ([MATH]).', '1811.00021-1-4-2': 'Greek indices ([MATH]) are for spin quantum numbers, and ([MATH]) and ([MATH]) are indices for A and B sites, respectively.', '1811.00021-1-4-3': 'The Kondo-lattice coupling is described by [MATH].', '1811.00021-1-4-4': 'A generic hopping term ([MATH]), generic exchange interaction ([MATH]), and interaction function ([MATH]) are introduced, which are constrained by lattice symmetry.', '1811.00021-1-4-5': 'The long-range Coulomb interaction with electric charge [MATH] is also introduced.', '1811.00021-1-4-6': 'The local spin operators may be represented in terms of either global or local axes, [MATH].', '1811.00021-1-4-7': 'The basis vectors of the local axes ([MATH]) are commonly used in spin-ice literatures [CITATION], and it is straightforward to find the relations with the basis vectors of the global axes ([MATH]).', '1811.00021-1-5-0': 'We focus on the system where spins at A sites host a U(1) QSL and electrons at B sites form the Luttinger semi-metal, motivated by the ARPES experiments [CITATION].', '1811.00021-1-5-1': 'The low energy effective Hamiltonian of [MATH] may be written as the quantum spin ice Hamiltonian, [MATH] [CITATION].', '1811.00021-1-5-2': 'The "star" index, [MATH], represents the dual diamond lattice sites of the underlying A-site pyrochlore lattice.', '1811.00021-1-5-3': 'The emergent magnetic field [MATH] is proportional to the average of the local spin projections to easy-axis ([111] or equivalent) directions, and the emergent electric field ([MATH]) describes spin fluctuations out of local easy-axis directions.', '1811.00021-1-5-4': 'We stress that the quantum spin ice manifold is defined by the divergence-free condition, [MATH], and thus the [MATH] fluctuations are all transversal.', '1811.00021-1-6-0': 'The Luttinger semi-metal Hamiltonian can be approximated as [MATH] with a four component spinor [MATH], [EQUATION]', '1811.00021-1-6-1': 'At low energy, the functions [MATH] may be written as [EQUATION]', '1811.00021-1-6-2': 'The term with [MATH] breaks the particle-hole symmetry (PHS), and the terms with [MATH] are associated with the [MATH] and [MATH] representations, respectively, of the cubic symmetry.', '1811.00021-1-6-3': 'We emphasize that the charge-neutrality of the system would demand the particle-hole band condition [MATH] if there are no other electron-hole pockets far away from the zone center[CITATION].', '1811.00021-1-6-4': 'We also set [MATH] for simplicity unless otherwise stated.', '1811.00021-1-6-5': 'Note that the presence of the long-range Coulomb interaction makes the particle-hole and SO(3) rotational symmetries emergent in the LAB phase ([MATH] and [MATH])[CITATION].', '1811.00021-1-7-0': 'The two sectors (A,B) host the excitations with fundamentally different dynamics.', '1811.00021-1-7-1': 'Namely, low energy excitations of the A-sites are emergent photons whose dispersion relation is [MATH] with two polarizations [MATH] and their velocities [MATH].', '1811.00021-1-7-2': 'On the other hand, the B-sites have electronic excitations with the long-range Coulomb interaction [MATH] with [MATH].', '1811.00021-1-8-0': 'We first construct a low energy effective coupling term of the Kondo-lattice coupling.', '1811.00021-1-8-1': 'Employing the lattice symmetries and gauge invariance, the lowest order coupling terms may be written as [EQUATION]', '1811.00021-1-8-2': 'The two coupling constants ([MATH] and [MATH]) characterize the Kondo-lattice coupling.', '1811.00021-1-8-3': 'The specific form of the coupling matrix [MATH] is completely determined by the cubic and time-reversal symmetries.', '1811.00021-1-8-4': 'The 4[MATH]4 matrix ([MATH]) of a spin operator [MATH] is used with the explicit form being introduced in SI.', '1811.00021-1-8-5': 'We mainly focus on the coupling to [MATH] because [MATH] couples to conduction electrons through a polarization type coupling [MATH] or a Rashba type coupling [MATH], which is less relevant than the coupling to [MATH] (see SI).', '1811.00021-1-9-0': 'The effective low energy action of the total Hamiltonian in the Euclidean spacetime is [EQUATION] with the density operator, [MATH].', '1811.00021-1-9-1': 'Though the form of the Yukawa coupling with [MATH] may look similar to the ones in previous literatures [CITATION], we emphasize that the U(1) gauge structure in the spin-ice manifold plays a crucially different role here.', '1811.00021-1-10-0': 'With a non-zero small coupling ([MATH]), the dynamics of the [MATH] fields is modified as [EQUATION]', '1811.00021-1-10-1': 'Hereafter, we use the four-vector notation for momentum and frequency, [MATH].', '1811.00021-1-10-2': "The boson self-energy at one-loop order is [EQUATION] with the Fermion Green's function, [MATH].", '1811.00021-1-10-3': 'Introducing an ultraviolet (UV) cut-off [MATH], we find [MATH] for the most symmetric condition ([MATH], [MATH], and [MATH]), which corresponds to [MATH].', '1811.00021-1-10-4': 'The fermion self-energy from the gauge fluctuations is [EQUATION]', '1811.00021-1-10-5': 'Unless [MATH], we find the fermion self-energy, [EQUATION] with non-zero values of [MATH].', '1811.00021-1-10-6': 'For the most symmetric condition, we find [MATH], [MATH].', '1811.00021-1-10-7': 'The two self-energies are UV divergent under the Kondo-lattice coupling.', '1811.00021-1-10-8': 'The absence of logarithmic dependence on the UV cutoff means the bosonic and fermionic excitations remain weakly coupled, and the decoupled ground state ([MATH]) is stable as far as the Kondo-lattice coupling is small.', '1811.00021-1-10-9': 'We call this state the fractionalized Luttinger semi-metal (fLSM).', '1811.00021-1-11-0': 'We stress that the Kondo-lattice coupling generates the PHS breaking term as manifested by [MATH] even with the most symmetric bare Hamiltonian ([MATH]).', '1811.00021-1-11-1': 'This contribution arises from the transversal gauge fluctuations, and the absence of the longitudinal gauge fluctuations is essential.', '1811.00021-1-11-2': 'The competition between the Kondo-lattice coupling and the long-range Coulomb interaction plays an important role in the PHS channel of fLSM.', '1811.00021-1-11-3': 'In the perturbative regime [MATH], the long-range Coulomb interaction is more relevant than the Kondo-lattice coupling [CITATION], and the PHS is realized.', '1811.00021-1-12-0': 'Adjacent phases of fLSM may be obtained by using the lattice symmetries.', '1811.00021-1-12-1': 'Considering time-reversal symmetry and parity as well as the rotational symmetries, a ground state with [MATH] breaks time-reversal symmetry and rotations but not parity.', '1811.00021-1-12-2': 'The resulting time-reversal broken phase hosts nodal conduction electrons in the form of Weyl semi-metals or metals.', '1811.00021-1-12-3': 'We call such a semi-metal phase with broken time-reversal symmetry a fractionalized ferromagnetic semi-metal (fFM-SM).', '1811.00021-1-12-4': 'The presence of both gapless electronic and gauge excitations is one of the main characteristics of fFM-SM, different from other states such as the Coulombic ferromagnetic state [CITATION].', '1811.00021-1-12-5': 'Similarly, a state with [MATH] and [MATH] is naturally dubbed a fractionalized ferroelectric phase, and one with [MATH] may be called a fractionalized multiferroic phase.', '1811.00021-1-12-6': 'As shown below, however, there may exist another transition before the system reaches the fFM-SM.', '1811.00021-1-13-0': "Quantum Phase Transitions : To investigate transitions to symmetry-broken phases, we first extend and apply Landau's mean field analysis, which amounts to ignoring spatial and temporal fluctuations of the emergent fields via [MATH].", '1811.00021-1-13-1': 'Integrating out the fermion excitations, we obtain the effective action of [MATH], and a continuous quantum phase transition between fLSM and fFM-SM is obtained for [MATH] (similar to the one of [CITATION], and also see SI).', '1811.00021-1-13-2': 'At the one-loop level, we find [MATH] for the most symmetric condition.', '1811.00021-1-13-3': 'The continuous transition obtained in the mean-field calculation respects the PHS and [MATH] rotational symmetry because fLSM enjoys those symmetries.', '1811.00021-1-14-0': 'We, however, show that the gauge fluctuations destabilize the continuous transition of the mean-field calculation.', '1811.00021-1-14-1': 'Defining [MATH], we find that the boson self-energy has the form, [EQUATION]', '1811.00021-1-14-2': 'The condition [MATH] enforces that the fluctuations are transversal, and the dimensionless functions ([MATH]) are positive in a wide range of the parameters which are illustrated in SI.', '1811.00021-1-14-3': 'Their positiveness indicates that the gauge fluctuations are stable with the renormalized propagator [MATH].', '1811.00021-1-15-0': 'Let us assume that there is a stable continuous transition between fLSM and fFM-SM.', '1811.00021-1-15-1': 'At the critical point ([MATH]), the dominant boson propagator may be written as [EQUATION] omitting higher order terms.', '1811.00021-1-15-2': 'To control calculations better, one may introduce the flavor number of fermions ([MATH]) and perform [MATH] calculations (see SI).', '1811.00021-1-15-3': 'The PHS breaking term can be obtained by evaluating [EQUATION]', '1811.00021-1-15-4': 'The integral is logarithmically divergent, [EQUATION]', '1811.00021-1-15-5': 'Including both the gauge-fluctuations and the long-range Coulomb interaction, we find [MATH] for the most symmetric condition), which can be interpreted as a divergent [MATH].', '1811.00021-1-15-6': 'The logarithmic divergence demonstrates the PHS cannot be realized at the critical point, which indicates the Kondo-lattice coupling dominates the long-range Coulomb interaction near the critical point.', '1811.00021-1-15-7': 'Thus, there is no continuous quantum phase transition between fLSM and fFM-SM.', '1811.00021-1-16-0': 'The divergence of the PHS breaking term destabilizes not only the validity of the mean-field calculation but also the particle-hole band condition ([MATH]).', '1811.00021-1-16-1': 'We find that the corrections of [MATH] are smaller than the one of [MATH], and thus the particle-hole symmetry condition may break down at long wave-length and low energy.', '1811.00021-1-16-2': 'The charge neutrality condition then enforces the formation of electron and hole pockets near the Brillouin zone center.', '1811.00021-1-17-0': 'We propose, based on the above calculations, that the PHS is broken before the onset of [MATH], whose validity is self-consistently checked a posteriori.', '1811.00021-1-17-1': 'The transition between fLSM and fFM-SM is intervened by an intermediate phase with the electron and hole pockets dubbed the fractionalized Luttinger metal (fLM).', '1811.00021-1-17-2': 'There must be more than one continuous transition between fLSM and fFM-SM as illustrated in Fig. 1.', '1811.00021-1-17-3': 'The transition between fLSM and fLM is likely to be described by the Lifshitz transition.', '1811.00021-1-17-4': 'Once the pockets appear, the scale ([MATH]) associated with the size of the Fermi pockets is emergent.', '1811.00021-1-17-5': 'The long-range Coulomb interaction is screened by the Thomas-Fermi screening, and the Yukawa coupling induces the Landau damping term similar to the one of the Hertz-Millis theory.', '1811.00021-1-17-6': 'Thus, in spite of the presence of the gauge structure, the critical theory becomes [EQUATION]', '1811.00021-1-17-7': 'The term with [MATH] is for the Landau-damping, and the coefficient of the term with [MATH] is normalized to be one.', '1811.00021-1-17-8': 'We omit the term with [MATH] because it is irrelevant at the critical point (say, [MATH]).', '1811.00021-1-17-9': 'Namely, the dynamics of the gauge fluctuations are determined by the damping term, and the critical modes have the dynamical critical exponent [MATH].', '1811.00021-1-17-10': 'Since [MATH], the term [MATH] with [MATH] is irrelevant, and the gauge fluctuations are weakly correlated with [MATH].', '1811.00021-1-17-11': 'The operator scaling dimensions are [MATH], [MATH], and [MATH].', '1811.00021-1-18-0': 'Multi-scale Quantum Criticality : The interplay between the Kondo-lattice coupling and the long-range Coulomb interaction naturally brings about multi-scale quantum criticality around the onset of [MATH].', '1811.00021-1-18-1': 'To see this, let us estimate the energy scale for breaking the particle-hole band condition by using Eqn. [REF].', '1811.00021-1-18-2': 'Setting [MATH], the renormalization group equation is [MATH], and the assumption [MATH] becomes invalid at [MATH].', '1811.00021-1-18-3': 'The associated energy scale is [MATH] with the UV cutoff scale, [MATH], below which the assumption of small [MATH] breaks down.', '1811.00021-1-18-4': 'The energy scale [MATH] is much smaller than the band-width of the conduction electron, which is of the order [MATH].', '1811.00021-1-18-5': 'It is natural to expect that the emergent particle-hole pocket-size scale ([MATH]) in the intermediate phase between fLSM and fFM-SM is of similar order of magnitude, namely [MATH].', '1811.00021-1-19-0': 'Because of the hierarchy of energy scales, three sets of critical exponents would naturally appear in physical quantities near the onset of [MATH].', '1811.00021-1-19-1': 'For example, the emergent photons have [MATH], the non-Fermi liquid excitations have [MATH], and the Hertz-Millis fluctuations have [MATH].', '1811.00021-1-19-2': 'The scaling dimensions of the external magnetic field are easily obtained by considering the coupling to the magnetic field.', '1811.00021-1-19-3': 'The emergent magnetic field couples to the Zeeman external magnetic field via [MATH], and the scaling dimension of the external field is [MATH].', '1811.00021-1-19-4': 'The conduction electron couples to the external field as [MATH], which gives [MATH].', '1811.00021-1-19-5': 'We also show that the Hertz-Millis type fluctuation gives [MATH].', '1811.00021-1-20-0': 'Three different scaling behaviors can naturally arise in all physical quantities.', '1811.00021-1-20-1': 'For example, the magnetic Gruneissen parameter, [MATH] with magnetization ([MATH]) and specific heat ([MATH]) at constant external magnetic field [MATH], has the scaling form, [EQUATION]', '1811.00021-1-20-2': 'The dimensionless function, [MATH], manifests the multi-scale quantum criticality.', '1811.00021-1-20-3': 'For example, when [MATH], one can find [MATH] with three coefficients [MATH] for [MATH].', '1811.00021-1-21-0': 'Possible exponents are [MATH] for the emergent photons and [MATH] for conduction electrons in fLSM.', '1811.00021-1-21-1': 'Most importantly, near the quantum phase transition to the fractionalized ferromagnetic semi-metal state, the Hertz-Millis QCP gives the scaling exponent, [MATH].', '1811.00021-1-21-2': 'It is interesting to note that similar multi-scaling critical behavior in magnetic Gruneisen parameter is seen in Pr[MATH]Ir[MATH]O[MATH] [CITATION].', '1811.00021-1-21-3': 'Our theory naturally explains the appearance of Fermi-pockets at low temperatures with multi-scaling behaviors even though the calculated critical exponents are not exactly the same as the experimentally-determined value.', '1811.00021-1-22-0': 'We also remark that our theory allows two channels, semi-metallic conduction electrons and collective modes of the U(1) QSL, to contribute to magnetic susceptibility and other thermodynamic quantities.', '1811.00021-1-22-1': 'An interesting question is whether the contributions of such unusual excitations to thermodynamic and transport properties can explain various non-Fermi liquid behaviors seen in the experiment on Pr[MATH]Ir[MATH]O[MATH].', '1811.00021-1-22-2': 'We leave this intriguing problem for a future work.', '1811.00021-1-23-0': 'In conclusion, we investigate emergent quantum phenomena arising from the Kondo-lattice coupling between the quantum spin liquid of local moments and non-Fermi liquid conduction electrons in pyrochlore systems A[MATH]B[MATH]O[MATH].', '1811.00021-1-23-1': 'Intertwined actions between the Kondo-lattice coupling and the long-range Coulomb interaction are uncovered.', '1811.00021-1-23-2': 'As an important result, quantum criticality near the onset of ferromagnetic ordering naturally displays multi-scaling behaviors.', '1811.00021-1-23-3': 'Further works on more quantitative analysis and comparison with experiments are highly desired.'}

{'1811.00021-2-0-0': 'Pyrochlore systems ([MATH]) with [MATH]-site rare-earth local moments and [MATH]-site [MATH] conduction electrons offer excellent material platforms for the discovery of exotic quantum many-body ground states.', '1811.00021-2-0-1': 'Notable examples include U(1) quantum spin liquid of the local moments and semimetallic non-Fermi liquid of the conduction electrons.', '1811.00021-2-0-2': 'Here we investigate emergent quantum phases and their transitions driven by the Kondo lattice coupling between such highly entangled quantum ground states.', '1811.00021-2-0-3': 'Using the renormalization group method, it is shown that weak Kondo lattice coupling is irrelevant, leading to a fractionalized semimetal phase with decoupled local moments and conduction electrons.', '1811.00021-2-0-4': 'Upon increasing the Kondo lattice coupling, this phase is unstable to the formation of broken symmetry states.', '1811.00021-2-0-5': 'Particularly important is the opposing influence of the Kondo lattice coupling and long-range Coulomb interaction.', '1811.00021-2-0-6': 'The former prefers to break the particle-hole symmetry while the latter tends to restore it.', '1811.00021-2-0-7': 'The characteristic competition leads to possibly multiple phase transitions, first from a fractionalized semimetal phase to a fractionalized Fermi surface state with particle-hole pockets, followed by the second transition to a fractionalized ferromagnetic state.', '1811.00021-2-0-8': 'Multiscale quantum critical behaviors appear at nonzero temperatures and with external magnetic field near such quantum phase transitions.', '1811.00021-2-0-9': 'We discuss the implication of these results to the experiments on Pr[MATH]Ir[MATH]O[MATH].', '1811.00021-2-1-0': 'Introduction : Recent advances in correlated electron systems reveal emergent phenomena beyond the Landau paradigm.', '1811.00021-2-1-1': 'Localized magnetic moments may host quantum spin liquid (QSL) phases characterized by fluctuating gauge fields and fractionalized particles [CITATION].', '1811.00021-2-1-2': 'Itinerant electron systems may show non-Fermi liquid behavior without quasiparticles [CITATION].', '1811.00021-2-1-3': 'Such phenomena and associated quantum phase transitions demand development of new concepts and novel understandings in strongly interacting quantum many-body systems [CITATION].', '1811.00021-2-2-0': 'In this work, we study the intertwined model of two emergent phases beyond the Landau paradigm.', '1811.00021-2-2-1': 'We consider the interaction between the local moment system supporting a U(1) quantum spin liquid and a non-Fermi liquid semimetallic state of conduction electrons.', '1811.00021-2-2-2': 'This model is partly motivated by physics of the pyrochlore materials, [MATH], where the [MATH]- and [MATH]-site pyrochlore lattices are occupied by rare-earth local moments and [MATH] conduction electrons, respectively.', '1811.00021-2-2-3': 'The [MATH]-site local moments may form a quantum spin liquid with emergent photons, also known as quantum spin-ice, as suggested in Yb[MATH]Ti[MATH]O[MATH], Pr[MATH]Hf[MATH]O[MATH], and Pr[MATH]Zr[MATH]O[MATH] [CITATION].', '1811.00021-2-2-4': 'When the [MATH]-site is occupied by conduction electrons in [MATH] orbitals, such as J[MATH]=1/2 Kramers doublet of Ir ions, the system supports a non-Fermi liquid semimetal, the so-called Luttinger-Abrikosov-Beneslaevski (LAB) state, which is derived from the quadratic band touching with long-range Coulomb interaction [CITATION].', '1811.00021-2-2-5': 'The quadratic band touching of Ir conduction electrons is confirmed in the angle-resolved photoemission spectroscopy (ARPES) experiment in the finite-temperature paramagnetic state of Pr[MATH]Ir[MATH]O[MATH] and Nd[MATH]Ir[MATH]O[MATH] [CITATION].', '1811.00021-2-2-6': 'It is believed that the interplay between the two emergent phases mentioned above may play a crucial role in low-temperature physics of Pr[MATH]Ir[MATH]O[MATH] [CITATION] .', '1811.00021-2-3-0': 'Considering the Kondo lattice coupling between the localized moments and conduction electrons, we first construct a low energy effective field theory for the coupled system of the U(1) QSL and the non-Fermi liquid semimetal state.', '1811.00021-2-3-1': 'We use the renormalization group method to investigate emergent phases and phase transitions and find that the Kondo lattice coupling and the long-range Coulomb interaction show intriguing interplay physics.', '1811.00021-2-3-2': 'Namely, the former shows the tendency of breaking the particle-hole symmetry but the latter plays the opposite role.', '1811.00021-2-3-3': 'For small Kondo lattice coupling, the Coulomb interaction prevails, and the two underlying phases remain weakly coupled.', '1811.00021-2-3-4': 'This is the Luttinger semimetal coexisting with fractionalized excitations and emergent photons from the local moments.', '1811.00021-2-3-5': 'For sufficiently strong interaction, either time reversal or inversion symmetry may be broken and our perturbative renormalization group analysis shows that time reversal symmetry breaking is the most relevant channel.', '1811.00021-2-3-6': 'Our analysis suggests that the particle-hole symmetry breaking occurs first as the Kondo lattice coupling becomes dominant over the long-range Coulomb interaction, leading to a fractionalized Fermi surface state with emergent particle-hole pockets.', '1811.00021-2-3-7': 'This is followed by the time reversal symmetry breaking transition to a ferromagnetically ordered fractionalized semimetal phase.', '1811.00021-2-3-8': 'We discuss the resulting multiscaling critical behavior in light of some key experiments in the low-temperature phase of Pr[MATH]Ir[MATH]O[MATH].', '1811.00021-2-4-0': 'Model : We start with a generic model Hamiltonian for the pyrochlore system, [MATH], [EQUATION]', '1811.00021-2-4-1': 'The Hamiltonian for [MATH] sites ([MATH]) describes localized magnetic moments ([MATH]), and the Hamiltonian for [MATH] sites ([MATH]) describes conduction electrons with annihilation and creation operators ([MATH]).', '1811.00021-2-4-2': 'Greek indices ([MATH]) are for spin quantum numbers, and ([MATH]) and ([MATH]) are indices for [MATH] and [MATH] sites, respectively.', '1811.00021-2-4-3': 'The Kondo lattice coupling is described by [MATH].', '1811.00021-2-4-4': 'A generic hopping term ([MATH]), generic exchange interaction [MATH], and interaction function [MATH] are introduced, which are constrained by lattice symmetry.', '1811.00021-2-4-5': 'The long-range Coulomb interaction with electric charge [MATH] is also introduced.', '1811.00021-2-4-6': 'The local spin operators may be represented in terms of either global or local axes, [MATH].', '1811.00021-2-4-7': 'The basis vectors of the local axes ([MATH]) are commonly used in spin-ice literature [CITATION], and it is straightforward to find the relations with the basis vectors of the global axes ([MATH]).', '1811.00021-2-5-0': 'We focus on the system where spins at [MATH] sites host a U(1) QSL and electrons at [MATH] sites form the Luttinger semi-metal, motivated by the ARPES experiments [CITATION].', '1811.00021-2-5-1': 'The low energy effective Hamiltonian of [MATH] may be written as the quantum spin-ice Hamiltonian, [MATH] [CITATION].', '1811.00021-2-5-2': 'The "star" index, [MATH], represents the dual diamond lattice sites of the underlying [MATH]-site pyrochlore lattice.', '1811.00021-2-5-3': 'The emergent magnetic field [MATH] is proportional to the average of the local spin projections to easy-axis ([111] or equivalent) directions, and the emergent electric field ([MATH]) describes spin fluctuations out of local easy-axis directions.', '1811.00021-2-5-4': 'We stress that the quantum spin-ice manifold is defined by the divergence-free condition, [MATH], and thus the [MATH] fluctuations are all transversal.', '1811.00021-2-6-0': 'The Luttinger semi-metal Hamiltonian can be approximated as [MATH] with a four component spinor [MATH], [EQUATION]', '1811.00021-2-6-1': 'At low energy, the functions [MATH] may be written as [EQUATION]', '1811.00021-2-6-2': 'The term with [MATH] breaks the particle-hole symmetry (PHS), and the terms with [MATH] are associated with the [MATH] and [MATH] representations, respectively, of the cubic symmetry.', '1811.00021-2-6-3': 'We emphasize that the charge neutrality of the system would demand the particle-hole band condition [MATH] if there are no other electron-hole pockets far away from the zone center[CITATION].', '1811.00021-2-6-4': 'We also set [MATH] for simplicity unless otherwise stated.', '1811.00021-2-6-5': 'Note that the presence of the long-range Coulomb interaction makes the particle-hole and SO(3) rotational symmetries emergent in the LAB phase ([MATH] and [MATH])[CITATION].', '1811.00021-2-7-0': 'The two sectors ([MATH]) host the excitations with fundamentally different dynamics.', '1811.00021-2-7-1': 'Namely, low energy excitations of the [MATH]-sites are emergent photons whose dispersion relation is [MATH] with two polarizations [MATH] and their velocities [MATH].', '1811.00021-2-7-2': 'On the other hand, the [MATH]-sites have electronic excitations with the long-range Coulomb interaction [MATH] with [MATH].', '1811.00021-2-8-0': 'We first construct a low energy effective coupling term of the Kondo lattice coupling.', '1811.00021-2-8-1': 'Employing the lattice symmetries and gauge invariance, the lowest order coupling terms may be written as [EQUATION]', '1811.00021-2-8-2': 'The two coupling constants ([MATH] and [MATH]) characterize the Kondo lattice coupling.', '1811.00021-2-8-3': 'The specific form of the coupling matrix [MATH] is completely determined by the cubic and time reversal symmetries.', '1811.00021-2-8-4': 'The 4[MATH]4 matrix ([MATH]) of a spin operator [MATH] is used with the explicit form being introduced in Supplemental Material(SM).', '1811.00021-2-8-5': 'We mainly focus on the coupling to [MATH] because [MATH] couples to conduction electrons through a polarization-type coupling [MATH] or a Rashba-type coupling [MATH], which is less relevant than the coupling to [MATH] (see SM).', '1811.00021-2-9-0': 'The effective low energy action of the total Hamiltonian in the Euclidean spacetime is [EQUATION] with the density operator, [MATH].', '1811.00021-2-9-1': 'Though the form of the Yukawa coupling with [MATH] may look similar to the ones in previous literatures [CITATION], we emphasize that the U(1) gauge structure in the spin-ice manifold plays a crucially different role here.', '1811.00021-2-10-0': 'With a nonzero small coupling ([MATH]), the dynamics of the [MATH] fields is modified as [EQUATION]', '1811.00021-2-10-1': 'Hereafter, we use the four-vector notation for momentum and frequency, [MATH].', '1811.00021-2-10-2': "The boson self-energy at one-loop order is [EQUATION] with the Fermion Green's function [MATH].", '1811.00021-2-10-3': 'Introducing an ultraviolet (UV) cutoff [MATH], we find [MATH] for the most symmetric condition ([MATH], [MATH], and [MATH]), which corresponds to [MATH].', '1811.00021-2-10-4': 'The fermion self-energy from the gauge fluctuations is [EQUATION]', '1811.00021-2-10-5': 'Unless [MATH], we find the fermion self-energy, [EQUATION] with nonzero values of [MATH].', '1811.00021-2-10-6': 'For the most symmetric condition, we find [MATH], [MATH].', '1811.00021-2-10-7': 'The two self-energies are UV divergent under the Kondo lattice coupling.', '1811.00021-2-10-8': 'The absence of logarithmic dependence on the UV cutoff means the bosonic and fermionic excitations remain weakly coupled, and the decoupled ground state ([MATH]) is stable as far as the Kondo lattice coupling is small.', '1811.00021-2-10-9': 'We call this state the fractionalized Luttinger semimetal (F-LSM).', '1811.00021-2-11-0': 'We stress that the Kondo lattice coupling generates the PHS breaking term as manifested by [MATH] even with the most symmetric bare Hamiltonian ([MATH]).', '1811.00021-2-11-1': 'This contribution arises from the transversal gauge fluctuations, and the absence of the longitudinal gauge fluctuations is essential.', '1811.00021-2-11-2': 'The competition between the Kondo lattice coupling and the long-range Coulomb interaction plays an important role in the PHS channel of F-LSM.', '1811.00021-2-11-3': 'In the perturbative regime [MATH], the long-range Coulomb interaction is more relevant than the Kondo lattice coupling [CITATION], and the PHS is realized.', '1811.00021-2-12-0': 'Adjacent phases of F-LSM may be obtained by using the lattice symmetries.', '1811.00021-2-12-1': 'Considering time reversal symmetry and parity as well as the rotational symmetries, a ground state with [MATH] breaks time reversal symmetry and rotations but not parity.', '1811.00021-2-12-2': 'The resulting time reversal broken phase hosts nodal conduction electrons in the form of Weyl semi-metals or metals.', '1811.00021-2-12-3': 'We call such a semi-metal phase with broken time reversal symmetry a fractionalized ferromagnetic semi-metal (F-FMSM).', '1811.00021-2-12-4': 'The presence of both gapless electronic and gauge excitations is one of the main characteristics of F-FMSM, different from other states such as the Coulombic ferromagnetic state [CITATION].', '1811.00021-2-12-5': 'Similarly, a state with [MATH] and [MATH] is naturally dubbed a fractionalized ferroelectric phase, and one with [MATH] may be called a fractionalized multiferroic phase.', '1811.00021-2-12-6': 'As shown below, however, there may exist another transition before the system reaches the F-FMSM.', '1811.00021-2-13-0': "Quantum phase transitions : To investigate transitions to symmetry-broken phases, we first extend and apply Landau's mean- field analysis, which amounts to ignoring spatial and temporal fluctuations of the emergent fields via [MATH].", '1811.00021-2-13-1': 'Integrating out the fermion excitations, we obtain the effective action of [MATH], and a continuous quantum phase transition between F-LSM and F-FMSM is obtained for [MATH] (similar to the one of Ref. [CITATION], and also see SM).', '1811.00021-2-13-2': 'At the one-loop level, we find [MATH] for the most symmetric condition.', '1811.00021-2-13-3': 'The continuous transition obtained in the mean-field calculation respects the PHS and SO(3) rotational symmetry because F-LSM enjoys those symmetries.', '1811.00021-2-14-0': 'We, however, show that the gauge fluctuations destabilize the continuous transition of the mean-field calculation.', '1811.00021-2-14-1': 'Defining [MATH], we find that the boson self-energy has the form, [EQUATION]', '1811.00021-2-14-2': 'The condition [MATH] enforces that the fluctuations are transversal, and the dimensionless functions ([MATH]) are positive in a wide range of the parameters which are illustrated in SM.', '1811.00021-2-14-3': 'Their positiveness indicates that the gauge fluctuations are stable with the renormalized propagator [MATH].', '1811.00021-2-15-0': 'Let us assume that there is a stable continuous transition between F-LSM and F-FMSM.', '1811.00021-2-15-1': 'At the critical point ([MATH]), the dominant boson propagator may be written as [EQUATION] omitting higher order terms.', '1811.00021-2-15-2': 'To control calculations better, one may introduce the flavor number of fermions ([MATH]) and perform [MATH] calculations (see SM).', '1811.00021-2-15-3': 'The PHS breaking term can be obtained by evaluating [EQUATION]', '1811.00021-2-15-4': 'The integral is logarithmically divergent, [EQUATION]', '1811.00021-2-15-5': 'Including both the gauge fluctuations and the long-range Coulomb interaction, we find [MATH] for the most symmetric condition), which can be interpreted as a divergent [MATH].', '1811.00021-2-15-6': 'The logarithmic divergence demonstrates the PHS cannot be realized at the critical point, which indicates that the Kondo lattice coupling dominates the long-range Coulomb interaction near the critical point.', '1811.00021-2-15-7': 'Thus, there is no continuous quantum phase transition between F-LSM and F-FMSM.', '1811.00021-2-16-0': 'The divergence of the PHS breaking term destabilizes not only the validity of the mean-field calculation but also the particle-hole band condition ([MATH]).', '1811.00021-2-16-1': 'We find that the corrections of [MATH] are smaller than the one of [MATH], and thus the particle-hole symmetry condition may break down at long wavelength and low energy.', '1811.00021-2-16-2': 'The charge neutrality condition then enforces the formation of electron and hole pockets near the Brillouin zone center.', '1811.00021-2-17-0': 'We propose, based on the above calculations, that the PHS is broken before the onset of [MATH], whose validity is self-consistently checked a posteriori.', '1811.00021-2-17-1': 'The transition between F-LSM and F-FMSM is intervened by an intermediate phase with the electron and hole pockets dubbed the fractionalized Luttinger metal (F-LM).', '1811.00021-2-17-2': 'There must be more than one continuous transition between F-LSM and F-FMSM as illustrated in Fig. 1.', '1811.00021-2-17-3': 'The transition between F-LSM and F-LM is likely to be described by the Lifshitz transition.', '1811.00021-2-17-4': 'Once the pockets appear, the scale ([MATH]) associated with the size of the Fermi pockets is emergent.', '1811.00021-2-17-5': 'The long-range Coulomb interaction is screened by the Thomas-Fermi screening, and the Yukawa coupling induces the Landau damping term similar to the one of the Hertz-Millis theory.', '1811.00021-2-17-6': 'Thus, in spite of the presence of the gauge structure, the critical theory becomes [EQUATION]', '1811.00021-2-17-7': 'The term with [MATH] is for the Landau damping, and the coefficient of the term with [MATH] is normalized to be one.', '1811.00021-2-17-8': 'We omit the term with [MATH] because it is irrelevant at the critical point (say, [MATH]).', '1811.00021-2-17-9': 'Namely, the dynamics of the gauge fluctuations are determined by the damping term, and the critical modes have the dynamical critical exponent [MATH].', '1811.00021-2-17-10': 'Since [MATH], the term [MATH] with [MATH] is irrelevant, and the gauge fluctuations are weakly correlated with [MATH].', '1811.00021-2-17-11': 'The operator scaling dimensions are [MATH], [MATH], and [MATH].', '1811.00021-2-18-0': 'Multiscale quantum criticality : The interplay between the Kondo lattice coupling and the long-range Coulomb interaction naturally brings about Multiscale quantum criticality around the onset of [MATH].', '1811.00021-2-18-1': 'To see this, let us estimate the energy scale for breaking the particle-hole band condition by using Eq. [REF].', '1811.00021-2-18-2': 'Setting [MATH], the renormalization group equation is [MATH], and the assumption [MATH] becomes invalid at [MATH].', '1811.00021-2-18-3': 'The associated energy scale is [MATH] with the UV cutoff scale, [MATH], below which the assumption of small [MATH] breaks down.', '1811.00021-2-18-4': 'The energy scale [MATH] is much smaller than the band width of the conduction electron, which is of the order [MATH].', '1811.00021-2-18-5': 'It is natural to expect that the emergent particle-hole pocket-size scale ([MATH]) in the intermediate phase between F-LSM and F-FMSM is of similar order of magnitude, namely [MATH].', '1811.00021-2-19-0': 'Because of the hierarchy of energy scales, three sets of critical exponents would naturally appear in physical quantities near the onset of [MATH].', '1811.00021-2-19-1': 'For example, the emergent photons have [MATH], the non-Fermi liquid excitations have [MATH], and the Hertz-Millis fluctuations have [MATH].', '1811.00021-2-19-2': 'The scaling dimensions of the external magnetic field are easily obtained by considering the coupling to the magnetic field.', '1811.00021-2-19-3': 'The emergent magnetic field couples to the Zeeman external magnetic field via [MATH], and the scaling dimension of the external field is [MATH].', '1811.00021-2-19-4': 'The conduction electron couples to the external field as [MATH], which gives [MATH].', '1811.00021-2-19-5': 'We also show that the Hertz-Millis type fluctuation gives [MATH].', '1811.00021-2-20-0': 'Three different scaling behaviors can naturally arise in all physical quantities.', '1811.00021-2-20-1': 'For example, the magnetic Gruneissen parameter, [MATH] with magnetization ([MATH]) and specific heat ([MATH]) at constant external magnetic field [MATH], has the scaling form, [EQUATION]', '1811.00021-2-20-2': 'The dimensionless function, [MATH], manifests the Multiscale quantum criticality.', '1811.00021-2-20-3': 'For example, when [MATH], one can find [MATH] with three coefficients [MATH] for [MATH].', '1811.00021-2-21-0': 'Possible exponents are [MATH] for the emergent photons and [MATH] for conduction electrons in F-LSM.', '1811.00021-2-21-1': 'Most importantly, near the quantum phase transition to the fractionalized ferromagnetic semi-metal state, the Hertz-Millis quantum critical point gives the scaling exponent, [MATH].', '1811.00021-2-21-2': 'It is interesting to note that similar multi scaling critical behavior in magnetic Gruneisen parameter is seen in Pr[MATH]Ir[MATH]O[MATH] [CITATION].', '1811.00021-2-21-3': 'Our theory naturally explains the appearance of Fermi pockets at low temperatures with multi scaling behaviors even though the calculated critical exponents are not exactly the same as the experimentally determined value.', '1811.00021-2-22-0': 'We also remark that our theory allows two channels, semimetallic conduction electrons and collective modes of the U(1) QSL, to contribute to magnetic susceptibility and other thermodynamic quantities.', '1811.00021-2-22-1': 'An interesting question is whether the contributions of such unusual excitations to thermodynamic and transport properties can explain various non-Fermi liquid behaviors seen in the experiment on Pr[MATH]Ir[MATH]O[MATH].', '1811.00021-2-22-2': 'We leave this intriguing problem for a future work.', '1811.00021-2-23-0': 'In conclusion, we investigate emergent quantum phenomena arising from the Kondo lattice coupling between the quantum spin liquid of local moments and non-Fermi liquid conduction electrons in pyrochlore systems [MATH].', '1811.00021-2-23-1': 'Intertwined actions between the Kondo lattice coupling and the long-range Coulomb interaction are uncovered.', '1811.00021-2-23-2': 'As an important result, quantum criticality near the onset of ferromagnetic ordering naturally displays multi scaling behaviors.', '1811.00021-2-23-3': 'Further works on more quantitative analysis and comparison with experiments are highly desired.'}

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[]

[['1811.00021-1-15-0', '1811.00021-2-15-0'], ['1811.00021-1-15-0', '1811.00021-2-15-7'], ['1811.00021-1-15-7', '1811.00021-2-15-7'], ['1811.00021-1-20-2', '1811.00021-2-20-2'], ['1811.00021-1-17-1', '1811.00021-2-17-1'], ['1811.00021-1-17-3', '1811.00021-2-17-3'], ['1811.00021-1-10-9', '1811.00021-2-10-9']]

[]

[]

{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1811.00021

1006.4388

{'1006.4388-1-0-0': 'We examine a model of classical deterministic computing in which the ground state of the classical system is a spatial history of the computation.', '1006.4388-1-0-1': 'This model is relevant to quantum dot cellular automata as well as to recent universal adiabatic quantum computing constructions.', '1006.4388-1-0-2': 'In its most primitive form, systems constructed in this model cannot compute in an error free manner when working at non-zero temperature.', '1006.4388-1-0-3': 'However, by exploiting a mapping between the partition function for this model and probabilistic classical circuits we are able to show that it is possible to make this model effectively error free.', '1006.4388-1-0-4': 'We achieve this by using techniques in fault-tolerant classical computing and the result is that the system can compute effectively error free if the temperature is below a critical temperature.', '1006.4388-1-0-5': 'We further link this model to computational complexity and show that a certain problem concerning finite temperature classical spin systems is complete for the complexity class Merlin-Arthur.', '1006.4388-1-0-6': 'This provides an interesting connection between the physical behavior of certain many-body spin systems and computational complexity.', '1006.4388-1-1-0': 'The creation of massive digital and deterministic computing systems represents one of the greatest triumphs of the last century.', '1006.4388-1-1-1': 'Increasingly, however, as the components in our information processing devices shrink to atomic scales [CITATION], the two defining characteristics of these systems-that they are digital as opposed to analog, and that they are deterministic as opposed to probabilistic-are beginning to be revealed as approximations that arise from considering macroscopic or mesoscopic physical systems.', '1006.4388-1-1-2': 'As computing elements are made smaller they are increasingly subject to noise and an inability to be perfectly controlled [CITATION].', '1006.4388-1-1-3': 'Even further down this route lie quantum computers [CITATION] where, instead of probabilistic time evolution, the computer is made of elements which function according to the laws of quantum information.', '1006.4388-1-1-4': 'These trends point toward the necessity of studying computing elements in settings where digital and deterministic functionality are not a priori guaranteed [CITATION].', '1006.4388-1-1-5': 'Here we consider a physical method for classical computing wherein a computation is encoded into the ground state of a classical many-body system of spins.', '1006.4388-1-1-6': 'At zero temperature in this model, computation is deterministic and thus we label this model classical ground state spin computation.', '1006.4388-1-1-7': 'The model has many predecessors, include quantum-dot cellular automata [CITATION], the broadcast model on trees [CITATION], quantum ground state computing [CITATION], and universal adiabatic quantum computing [CITATION].', '1006.4388-1-1-8': 'We focus our work here on this model when the system has thermalized at a non-zero temperature.', '1006.4388-1-1-9': 'At non-zero temperature, if we do nothing to compensate for the effects of thermal equilibrium, ground state spin computation fails.', '1006.4388-1-1-10': 'Here we show that it is possible to use the ideas of fault-tolerant classical computing to design ground state spin computers which function deterministically with high probability at non-zero temperatures below a critical temperature.', '1006.4388-1-1-11': 'We achieve this by mapping the thermal ensembles of our computational spin systems onto the distribution of ensembles produced during classical probabilistic computations.', '1006.4388-1-1-12': 'This is the main result of the paper: some classical spin systems in thermal equilibrium can be thought of as enacting classical probabilistic computations spatially across the system.', '1006.4388-1-2-0': 'Finally, our mapping leads naturally to problems which are complete for the complexity class promise Merlin-Arthur [CITATION], a result which can be regarded as a finite temperature version of the Cook-Levin theorem [CITATION] from the theory of computational complexity.', '1006.4388-1-2-1': 'The new computational model we consider thus serves as a bridge between statistical mechanics and the classical computational complexity of probabilistic computation.', '1006.4388-1-2-2': 'We also obtain a computational complexity result concerning the difficulty of identifying when a system admits a mapping to a probabilistic circuit by showing that a restricted version of this problem is This implies that the problem of identifying when a physical system can be seen to be performing a computation is itself computationally intractable.', '1006.4388-1-3-0': 'The outline of our paper is as follows.', '1006.4388-1-3-1': 'In Section [REF] we introduce the model of computing with spins in the ground state using a particular energy function for this system.', '1006.4388-1-3-2': 'In Section [REF] we show that at non-zero temperature the model from the previous section fails to properly compute.', '1006.4388-1-3-3': 'We do this for an extremely simple and previously explored model, but present the result using two different methods, one related to reinterpreting the transfer matrix and the other related to classical circuits applied to simple thermal ensembles.', '1006.4388-1-3-4': 'Generalizing these methods allows us to discuss the energy functions arising in Section [REF] as probabilistic circuits.', '1006.4388-1-3-5': 'This more general formulation we discuss using two different methods in Section [REF].', '1006.4388-1-3-6': 'Motivated by the mappings discovered in Section [REF], we then return to our original model and define the broader class of energy functions consistent with these models in Section [REF].', '1006.4388-1-3-7': 'At this point we also point out how our models differ from classical models of quantum-dot cellular automata.', '1006.4388-1-3-8': 'The more general setting leads us to ask questions about how algorithmically hard it is to decide if a given physical system supports computation in these models.', '1006.4388-1-3-9': 'We show that this problem, even in a very weak form, is and thus likely to be intractable.', '1006.4388-1-3-10': 'In Section [REF] we discuss how to design ground state spin models that, unlike the generic case, do compute fault-tolerantly at non-zero temperature.', '1006.4388-1-3-11': "Our construction is intimately related to the original fault-tolerance construction of von Neumann [CITATION] and we show that von Neumann's threshold for fault-tolerance is related to a critical temperature in our system.", '1006.4388-1-3-12': 'Finally in Section [REF] we discuss how the model we consider is related to the Cook-Levin theorem from computational complexity and show how this leads to certain natural problems about our model being Promise-', '1006.4388-1-4-0': '# Classical Ground State Spin Computing', '1006.4388-1-5-0': 'Here we introduce the model of classical ground state spin computing.', '1006.4388-1-5-1': 'This model, in particular restricted cases, is relevant to a variety of different models considered elsewhere.', '1006.4388-1-5-2': 'For instance, quantum dot cellular automata can be partially modeled by classical ground state spin computing [CITATION].', '1006.4388-1-5-3': 'However the model we consider is more general than these specific instances.', '1006.4388-1-5-4': 'This larger generality comes along with certain unphysical assumptions: our models contain, for example, three-body interactions which are not necessarily easily achievable in a physical device.', '1006.4388-1-5-5': 'Physically we imagine that models such as the one we consider might emerge in certain limits where effective many-body interactions can emerge.', '1006.4388-1-5-6': 'However, irrespective of the physical implementation, the model provides a set of classical many-body interacting systems which can be mapped onto probabilistic classical circuits.', '1006.4388-1-5-7': 'This connects statistical mechanics with classical probabilistic circuits thus bridging computer science and physics in a new and interesting manner.', '1006.4388-1-6-0': 'We further note that a quantum version similar to this model has been studied by Mizel et al [CITATION].', '1006.4388-1-6-1': 'This later model contains significant difference owing to the quantum nature of the computation: in particular the ground state does not contain the computation laid out spatially, but instead exists in a superposition of the computation being carried out spatially.', '1006.4388-1-6-2': 'These quantum models are also connected to universal adiabatic quantum computing schemes [CITATION] and piecewise variations on these schemes [CITATION].', '1006.4388-1-6-3': 'By studying the classical analogy of these schemes, we hope to shed light on how these later quantum methods can be made fault-tolerant.', '1006.4388-1-7-0': '## The Model', '1006.4388-1-8-0': 'A combinatorial circuit, [MATH], is a directed acyclic graph, [MATH] with vertices [MATH] and edge set [MATH], where the internal vertices, [MATH], of the graph are logical gates, [MATH] is the set of all logical gates), and the external vertices, [MATH], are the inputs and outputs to the circuit.', '1006.4388-1-8-1': 'External vertices that have no edges leading to them are inputs and external vertices that have no edges that lead away from them our outputs to the circuit.', '1006.4388-1-8-2': 'If a circuit has [MATH] input vertices and [MATH] output vertices, then to such a circuit we can assign a function [MATH] which results from propagating the input through the logic gates to the output vertices.', '1006.4388-1-8-3': 'Note that at this point we require that fan-outs in our circuit are represented by gates and do not allow circuits which have fan-in gates.', '1006.4388-1-8-4': 'A [MATH]-fan-out gate is the boolean function [MATH] given by [MATH].', '1006.4388-1-9-0': 'Consider the following physical system.', '1006.4388-1-9-1': 'For each edge of a combinatorial circuit associate a single subsystem with only two possible states, [MATH] and [MATH], i.e. associate a bit to every edge.', '1006.4388-1-9-2': 'Consider next a logic gate which has incoming edges labeled by bits [MATH] and outgoing edges labeled by [MATH].', '1006.4388-1-9-3': 'For such a logic gate, [MATH], define the following energy function [EQUATION] where [MATH] is the function the logic gate is computing.', '1006.4388-1-9-4': 'Note that this energy contributes [MATH] when the inputs and outputs follow the rules of the logic gate, but is [MATH] otherwise.', '1006.4388-1-9-5': 'Now define the energy for a particular configuration of the bits associated with edges as the sum over all logic gates of these energy terms: [EQUATION] where each [MATH] acts on the appropriate input and output bits and [MATH] is the collection of bits for the entire system.', '1006.4388-1-9-6': 'This energy is a function of the labels on all of the edges.', '1006.4388-1-9-7': 'Its zero energy configurations are ground states which consist of configurations which correctly compute the logical function corresponding to the circuit [MATH].', '1006.4388-1-9-8': 'Note that at this point the ground state is degenerate: every valid input (and corresponding output) defines a valid zero energy configuration.', '1006.4388-1-9-9': 'We call an energy function constructed in this fashion from a circuit a circuit energy function.', '1006.4388-1-10-0': 'Suppose further that, in addition to imposing energy constraints to enforce logic gates, we also impose energy constraints to fix a particular input to the circuit.', '1006.4388-1-10-1': 'Suppose that the input vertices are [MATH] and we wish to force the input [MATH], where [MATH].', '1006.4388-1-10-2': 'Then to each edge whose bit is labeled [MATH] lead away from vertex [MATH] we can associate an energy term [MATH] if [MATH] and [MATH] otherwise.', '1006.4388-1-10-3': 'Taking a sum over all of these terms for the input vertices (and corresponding edges) we can thus add a term such that the ground state configuration consists now only of the input [MATH] propagated to the output of the circuit, [MATH].', '1006.4388-1-10-4': 'In particular for input [MATH] define the [EQUATION] where [EQUATION]', '1006.4388-1-10-5': 'We call an energy function made up of a circuit energy function plus an input forcing energy function a computed circuit energy function.', '1006.4388-1-11-0': 'The above description of how to take a combinatorial circuit, [MATH], plus its input, [MATH], and converting it into a many-spin energy function is what we term the classical ground state spin computing (CGSSC) model.', '1006.4388-1-11-1': 'Just as in quantum dot cellular automata [CITATION], the computation occurs when the system is in its ground state and is spatially spread out across the device.', '1006.4388-1-11-2': 'Unlike in quantum dot cellular automata, however, this model can implement logic gates by using interactions beyond just pairwise interacting spins (or pseudo-spins in the case of the quantum dot configurations.)', '1006.4388-1-11-3': 'For example, constructing a quantum wire in the CGSSC model will correspond directly to an identical model in the semi-classical limit of quantum dot cellular automata models.', '1006.4388-1-11-4': 'Gates in quantum dot cellular automata, however, are implemented in a way which is not directly analogous to the CGSSC model.', '1006.4388-1-11-5': 'In section [REF] we return to this issue and define a more general set of energy functions wherein our results still hold and compare this with quantum cellular automata models.', '1006.4388-1-12-0': '## Example', '1006.4388-1-13-0': "To be explicit, let's consider an example circuit involving four inputs, a few logical gates, and one output as diagramed in Fig. [REF].", '1006.4388-1-13-1': 'The energy function for the AND gate is then, for example, [EQUATION]', '1006.4388-1-13-2': 'The full circuit energy function is given by [EQUATION]', '1006.4388-1-13-3': 'Suppose that we wish to force the input to be [MATH] and [MATH].', '1006.4388-1-13-4': 'Then we would add the term [EQUATION]', '1006.4388-1-14-0': '# Unprotected Ground State Spin Computing Fails at Finite Temperature', '1006.4388-1-15-0': 'Above we have defined an energy function whose ground state deterministically carries out a circuit.', '1006.4388-1-15-1': 'Having defined this energy model we can now consider the physically important question of what happens to this model when the physical system described by this energy function is in thermal equilibrium at a finite temperature.', '1006.4388-1-15-2': 'Thus we are led to consider the Boltzmann distribution corresponding to a non-zero temperature version of a circuit energy function.', '1006.4388-1-15-3': 'It then makes sense to consider whether the conditional probability of output [MATH] given the forced input [MATH] for this circuit is large enough to distinguish this output from a completely random outcome.', '1006.4388-1-15-4': 'It is easy to see that for at least some circuits [MATH] and inputs [MATH], ground state computation will fail to correctly evaluate the function corresponding to the circuit as the size of the circuit being implemented grows.', '1006.4388-1-15-5': 'This follows directly from examining one of the most basic models in statistical physics, the one-dimensional Ising model [CITATION].', '1006.4388-1-15-6': 'This was pointed out in the context of quantum-dot cellular automata in [CITATION] (see also [CITATION]).', '1006.4388-1-15-7': 'Here we reproduce this argument presenting it in two different forms.', '1006.4388-1-15-8': 'We do this not just to be pedagogical, but also because these two methods will generalize from the one dimensional case to more general circuit computing energy functions.', '1006.4388-1-16-0': 'Consider the circuit computing energy function corresponding to inputing the bit [MATH] into a series of [MATH] identity gates: [EQUATION] where [MATH] represents the negation of [MATH]: [MATH] and [MATH].', '1006.4388-1-16-1': "It is convenient here, and in the sequel, to work with [MATH] valued variables instead of the [MATH] valued [MATH]'s.", '1006.4388-1-16-2': "Thus we will define uppercase spin variables to be [MATH] valued versions of the [MATH]'s via [MATH].", '1006.4388-1-16-3': 'Then the above energy function can be written as [EQUATION] which we recognize as the one dimensional Ising model with ferromagnetic couplings and a boundary term which is a local field.', '1006.4388-1-16-4': 'The ground state of this is simply all [MATH]), i.e. the initial [MATH] has been copied by identity gates down the line.', '1006.4388-1-17-0': 'The thermal ensemble arising from this energy function is given by [EQUATION] where [MATH] is the partition function, [EQUATION] and [MATH] is the inverse temperature.', '1006.4388-1-17-1': 'It is well known that the one dimensional Ising model does not order at finite temperature in the thermodynamic limit [CITATION].', '1006.4388-1-17-2': 'From our perspective, we observe that the system will fail to correctly transmit the [MATH] down the line at finite temperature except in a window of size [MATH] which goes to zero as the system size goes to infinity.', '1006.4388-1-18-0': '## Transfer Matrix Reinterpreted as a Probabilistic Circuit', '1006.4388-1-19-0': 'The standard method for solving this model is the transfer matrix method which we review and then reinterpret.', '1006.4388-1-19-1': 'To the energy function add a term dependent on a variable [MATH] for the last spin: [EQUATION]', '1006.4388-1-19-2': 'If we calculate [MATH] for this energy function, we can then use this to calculate the probability that [MATH] is [MATH] via [MATH] where [EQUATION]', '1006.4388-1-19-3': 'Define the two by two matrices [EQUATION] where [MATH], as well as the column vector [EQUATION] and the row vector, [EQUATION]', '1006.4388-1-19-4': 'Explicitly the partition function [MATH] is [EQUATION]', '1006.4388-1-19-5': 'This can then be written as [EQUATION] where the last equation is a row vector, a matrix product, and a column vector thus giving a scalar.', '1006.4388-1-19-6': 'This is the transfer matrix form of the solution: the partition function is written as a product of "transfer matrices" and, in this case, sandwiched between an "initial" and "final" vector.', '1006.4388-1-20-0': 'Define the following matrices and vectors: [EQUATION]', '1006.4388-1-20-1': 'We can rewrite [MATH] as [EQUATION]', '1006.4388-1-20-2': 'It is at this point that we begin to see our reinterpretation of the transfer matrix method.', '1006.4388-1-20-3': 'In particular we note that [MATH] is a vector of probabilities (that is it sums to unity) and [MATH] is a stochastic matrix (its columns sum to unity).', '1006.4388-1-20-4': 'In other words encoded into [MATH] is a classical preparation of a probabilistic bit, followed by an evolution according to probabilistic gates.', '1006.4388-1-20-5': 'Suppose that we let [MATH] denote the output of this probabilistic circuit.', '1006.4388-1-20-6': 'Then [EQUATION] where the sample is taken from the output of the probabilistic circuit.', '1006.4388-1-20-7': 'Notice that at [MATH], [MATH] is a vector made up of all components equaling [MATH].', '1006.4388-1-20-8': 'In this case, [MATH] since then this term is a sum over all the outputs to the probabilistic circuit.', '1006.4388-1-20-9': 'Further [EQUATION] where [MATH].', '1006.4388-1-20-10': 'Thus [EQUATION]', '1006.4388-1-20-11': 'In other words the expectation value of the last spin in the chain is equal to the expectation value for the output of the circuit starting with [MATH] and then applying the probabilistic gates [MATH].', '1006.4388-1-20-12': 'Note that the combinatorial factors [MATH] have canceled out.', '1006.4388-1-21-0': 'The above description tells us that we can think about the one dimensional Ising spin chain with a forced boundary term as a probabilistic circuit related to our ground state spin computation.', '1006.4388-1-21-1': 'In particular the probabilistic circuit starts with the possibility of an incorrectly initialized boundary.', '1006.4388-1-21-2': 'This is formulated in the preparation probability vector [MATH].', '1006.4388-1-21-3': 'Then with probability [MATH] the ground state spin computation of an identity gate is replaced by a bit flip gate.', '1006.4388-1-21-4': 'The probability of the final spin of the chain being in a particular state is then directly given by the probability that the probabilistic circuit outputs a particular value.', '1006.4388-1-22-0': 'In this model the probability of the ground state properly computing the desired identity circuit is such that the information is washed out at finite temperature.', '1006.4388-1-22-1': 'To see this we must actually calculate [MATH].', '1006.4388-1-22-2': 'This can be done most easily by diagonalizing the gates.', '1006.4388-1-22-3': 'In particular if we define the Hadmard matrix, [EQUATION] then [EQUATION] where [EQUATION]', '1006.4388-1-22-4': 'Thus [EQUATION] which can be reduced to [EQUATION]', '1006.4388-1-22-5': 'From this expression we see that the probability that the final spin is aligned with the input quickly drops to [MATH] as a function of [MATH] (unless [MATH], i.e. in a small window of temperature below [MATH].)', '1006.4388-1-22-6': 'Thus the circuit fails with high probability as we make the chain longer and longer.', '1006.4388-1-22-7': 'This is exactly the result of [CITATION].', '1006.4388-1-22-8': 'In this respect, ground state spin computing in this model is not tolerant to errors arising from working at finite temperature.', '1006.4388-1-22-9': 'Note that we do not consider scaling of the temperature to stay within the small window where the computation is correctly performed to be a fault-tolerant method as it requires unreasonable physical resources as the system size gets larger.', '1006.4388-1-23-0': '## A Second Approach Using Controlled-Not Gates', '1006.4388-1-24-0': 'Having shown in the previous subsection how one can reinterpret the transfer matrix method as a probabilistic circuit for the one dimensional Ising model with forced boundary, we next present a second way to derive this observation.', '1006.4388-1-24-1': 'This method is a direct variation on the method used in [CITATION] to study Ising models on Cayley trees.', '1006.4388-1-25-0': 'On the [MATH] spins, define the controlled-not on positions [MATH] and [MATH] as the function, [MATH], which maps [MATH] to [MATH] via [EQUATION]', '1006.4388-1-25-1': 'We call this a controlled-not because it functions as a deterministic gate which, controlled on the spin [MATH] either does nothing to the spin in position [MATH] (if [MATH]), or flips the spin in position [MATH] (if [MATH].)', '1006.4388-1-25-2': "Further define the function which is the composition of controlled-not's starting at the first spin and working forward (note that [MATH]): [EQUATION]", '1006.4388-1-25-3': 'Notice that if we start with spin [MATH] in the state [MATH] and all of the other spins [MATH] in [MATH], then the action of [MATH] is to copy [MATH] down the line: [MATH].', '1006.4388-1-25-4': 'In this sense [MATH] is the operation of applying identity gates as information is propagated down the spin chain.', '1006.4388-1-25-5': 'Note that the [MATH] are bijections as is [MATH] and that [MATH] is its own inverse.', '1006.4388-1-25-6': 'Thus [EQUATION]', '1006.4388-1-25-7': 'Explicitly, [EQUATION] and [EQUATION]', '1006.4388-1-25-8': 'Consider the energy function on [MATH] spins: [EQUATION]', '1006.4388-1-25-9': 'Then clearly a system in the thermal state described by this energy function has each spin in [MATH] with probability [MATH] and [MATH] with probability [MATH].', '1006.4388-1-25-10': 'Suppose that we apply the [MATH] function to this system.', '1006.4388-1-25-11': 'That is consider the above system in its thermal ensemble and consider physically applying the controlled-not gates.', '1006.4388-1-25-12': 'Then certainly this can be thought of as a system in which the first bit is prepared in [MATH] with probability [MATH], with probability [MATH] the controled-nots successfully copy the bit down the line, and with probability [MATH] the output of the controlled-not is flipped.', '1006.4388-1-25-13': 'This is exactly the probabilistic circuit as described in the previous section.', '1006.4388-1-26-0': 'How is this related to our original [MATH]?', '1006.4388-1-26-1': 'Let the unprimed variables be the spins after [MATH] has been applied to the system.', '1006.4388-1-26-2': 'Then [EQUATION] where [EQUATION]', '1006.4388-1-26-3': 'Therefore [EQUATION]', '1006.4388-1-26-4': 'This implies that the probability distribution produced by taking the thermal ensemble for [MATH] and applying [MATH] to the system is equivalent to a thermal ensemble with a new energy function [EQUATION]', '1006.4388-1-26-5': 'A quick calculation using Eq. ([REF]) finds that [EQUATION] which we see is equal to our original energy function [MATH].', '1006.4388-1-26-6': 'Reversing the above argument we thus see that the energy function [MATH] can equally well be thought of as the ensemble corresponding to [MATH] followed by the application of the mapping [MATH].', '1006.4388-1-26-7': 'Further this later ensemble and computation has a clear interpretation in terms of a probabilistic initialization followed by probabilistic gates: when we apply controlled-nots with error target values then this causes a bit flip error in copying the information down the line.', '1006.4388-1-26-8': 'Thus we see that we can derive the same probabilistic gate expression for the one-dimensional Ising model as that which arose by reinterpreting the transfer matrix as a probabilistic circuit.', '1006.4388-1-27-0': '# The Partition Function as a Probabilistic Circuit', '1006.4388-1-28-0': 'We now turn to the more general setting of a circuit energy function for a generic circuit [MATH].', '1006.4388-1-28-1': 'We begin by focusing on circuit energy functions without a forced input.', '1006.4388-1-29-0': 'Let [MATH] be the energy function for the circuit [MATH].', '1006.4388-1-29-1': 'For each logic gate, [MATH], with inputs [MATH] and output [MATH], define the following tensor, [EQUATION] where [MATH] as before.', '1006.4388-1-29-2': 'Given this definition, we can now write the partition function [MATH] as the contraction of a tensor network [CITATION] plus a sum over all inputs and outputs.', '1006.4388-1-29-3': 'What is a tensor network?', '1006.4388-1-29-4': 'Take a graph and map a tensor to every vertex in such a manner that the tensor has an index for every edge of the relevant vertex.', '1006.4388-1-29-5': 'One can then perform a sum over two tensor indices connected via edges in the graph.', '1006.4388-1-29-6': 'A tensor network is thus a graph where vertices are tensors, and edges are tensor indices.', '1006.4388-1-29-7': 'Free edges are indices which have not been summed over and connected edges are ones for which a sum has been performed.', '1006.4388-1-29-8': 'The entire network (graph) then represents itself a tensor with a number of unsummed indices equal to the number of free edges.', '1006.4388-1-30-0': 'Given the tensors [MATH] and the graph given by the circuit (which will have free edges for the inputs and outputs), we can build a tensor network out of these [MATH]s.', '1006.4388-1-30-1': 'This tensor network will itself be a tensor network having indices for the inputs and outputs.', '1006.4388-1-30-2': 'Call this tensor [MATH] for inputs [MATH] and output [MATH].', '1006.4388-1-30-3': 'Then it is easy to see that the partition function is equal to [EQUATION]', '1006.4388-1-30-4': 'In other words, the partition function is given by the sum over all inputs, and all outputs of the value of the tensor network.', '1006.4388-1-31-0': 'What does this have to do with probabilistic circuits?', '1006.4388-1-31-1': 'Well now instead of using the tensors [MATH], instead use [EQUATION] or in other words [EQUATION]', '1006.4388-1-31-2': 'It is easy to check that this defines a stochastic matrix, i.e. a probabilistic gate.', '1006.4388-1-31-3': 'Indeed it is a probabilistic gate which performs the correct function of the gate with probability [MATH] and randomly flips the output to one of the other outputs with equal probability [MATH].', '1006.4388-1-32-0': 'Notice, importantly, that the difference between [MATH] and [MATH] is a constant which depends on only on the number of output bits.', '1006.4388-1-32-1': 'So suppose we consider the tensor network for the circuit made now with [MATH]s and not [MATH]s.', '1006.4388-1-32-2': 'Then we can write the partition function as [EQUATION] where [MATH] is the simple combinatorial factor [EQUATION] and [MATH] is the number of outputs of the [MATH]th gate.', '1006.4388-1-32-3': 'Now examine [MATH].', '1006.4388-1-32-4': 'This is nothing more than the probability that we get output [MATH] given that we had input [MATH] to the probabilistic circuit with probabilistic gates [MATH].', '1006.4388-1-32-5': 'In other words, we can express the partition function as [EQUATION]', '1006.4388-1-32-6': 'Thus the partition function is really hiding a probabilistic computation.', '1006.4388-1-32-7': 'Notice further that we can explicitly calculate that partition function because [MATH], [EQUATION] where [MATH] is the number of input bits.', '1006.4388-1-33-0': 'Now we turn to the case where we force the input to the circuit.', '1006.4388-1-33-1': 'Let [MATH] be the inputs to the circuit [MATH] and [MATH] be the energy function for the forced computation.', '1006.4388-1-33-2': 'Define the modified energy function [EQUATION] where [EQUATION] with [EQUATION] and the sum is over the output bits, [MATH].', '1006.4388-1-33-3': 'Call the partition function associated with this setup [MATH].', '1006.4388-1-34-0': 'Note that [MATH] is equal to the partition function for energy function with a forced input [MATH], [MATH].', '1006.4388-1-34-1': 'Define [EQUATION] where [EQUATION]', '1006.4388-1-34-2': 'Then it is easy to see that [EQUATION] where [EQUATION]', '1006.4388-1-34-3': 'Thus we see that we can interpret [MATH] as a sum over a probabilistic circuit, but now with a probabilistic input corresponding to each bit being flipped with a probability [MATH].', '1006.4388-1-34-4': 'Again, because we are summing over all outputs, we can explicitly perform this sum [EQUATION]', '1006.4388-1-34-5': 'We are interested in computing the probability that the thermal ensemble for the forced input circuit has output given by [MATH].', '1006.4388-1-34-6': 'This can be calculated by [EQUATION]', '1006.4388-1-34-7': 'Note that [EQUATION] so that the partial derivatives may be evaluated only over the first term.', '1006.4388-1-34-8': 'There we find that [EQUATION]', '1006.4388-1-34-9': 'Thus we find that [EQUATION]', '1006.4388-1-34-10': 'This is the main result of this section: the probability of the ground state spin computing model output bits is equal to the probability of running the probabilistic circuit described by [MATH] on inputs described by input probability distribution [MATH].', '1006.4388-1-34-11': 'The gates in [MATH] are simply noisy versions of the deterministic gates which fail with a fixed probability related to the temperature.', '1006.4388-1-35-0': 'This result is interesting in two manners.', '1006.4388-1-35-1': 'First it allows one to use techniques designed for probabilistically failing gates to be ported over to our model.', '1006.4388-1-35-2': 'For instance this will allow us to use methods for constructing fault-tolerant circuits.', '1006.4388-1-35-3': 'It is also interesting in that it shows that certain many-body quantum systems can be efficiently simulated.', '1006.4388-1-35-4': 'Computed circuit energy functions can be simulated by directly implementing the probabilistic gate that these systems correspond to.', '1006.4388-1-35-5': 'Here we have shown that output bits for the circuit energy function are related to the output bits for the probabilistic computation.', '1006.4388-1-35-6': 'A straightforward generalization shows that this is also true of other bits in the system.', '1006.4388-1-35-7': 'We can efficiently simulate systems with computed circuit energy functions by executing the probabilistic computation these energy functions represent.', '1006.4388-1-36-0': '# The Gate Model Derivation', '1006.4388-1-37-0': 'Next let us turn to the same analysis as the prior subsection, but now using the gate trick as we did for the one-dimensional model in Section [REF].', '1006.4388-1-37-1': 'To do this we must first define the equivalent of the controlled-not gate.', '1006.4388-1-37-2': 'Consider a logical gate [MATH] with input [MATH] and output [MATH] which computes the function [MATH].', '1006.4388-1-37-3': 'Define the function [MATH] from [MATH] to [MATH] as [EQUATION]', '1006.4388-1-37-4': 'This function thus computes the function on input [MATH], uncomputes the function on input [MATH], and does nothing otherwise.', '1006.4388-1-37-5': 'Note that [MATH] is a bijection and is self-inverse, [MATH].', '1006.4388-1-37-6': 'Defining an order to gates in our circuit such that the circuit computes properly, [MATH] first, [MATH] second, etc.', '1006.4388-1-37-7': 'Then we can define the function on all our bits of [EQUATION]', '1006.4388-1-37-8': 'Suppose we initially start with input bits initialized to an input [MATH] and all other bits initialized to [MATH].', '1006.4388-1-37-9': 'Then if we apply [MATH] to these bits we will place the result of the computation in their appropriate locations across the circuit.', '1006.4388-1-38-0': 'Next define an energy function on all of our system.', '1006.4388-1-38-1': 'This is most easily defined in terms of the inputs to the full circuit and the outputs to the gates [MATH], [EQUATION] and [EQUATION]', '1006.4388-1-38-2': 'This energy function thus assigns for non-global inputs an energy penalty for output to gates being anything different that [MATH].', '1006.4388-1-38-3': 'For global inputs it adds a penalty for every input which is not the correct input from [MATH].', '1006.4388-1-38-4': 'The thermal ensemble resulting from this system is then trivially described: the input qubits are in the correct input [MATH] with probability [MATH] and the internal output bits of a gate are grouped together and have a probability [MATH] being all zeros and probability [MATH] to be anything else.', '1006.4388-1-39-0': 'Imagine applying [MATH] to the state described by the ensemble for [MATH].', '1006.4388-1-39-1': 'Clearly the ground state of [MATH] is such that the circuit [MATH] will be correctly computed.', '1006.4388-1-39-2': 'Further, because the ensemble related to [MATH] has erred inputs, the resulting probability distribution will have erred inputs.', '1006.4388-1-39-3': 'Finally the gates will function properly only when the output was properly initialized to all [MATH]s.', '1006.4388-1-39-4': 'This occurs with probability [MATH] for logic gate [MATH] and otherwise the gate fails by an equally probable error state with probability [MATH].', '1006.4388-1-39-5': 'In other words the ensemble is exactly the one describing the probabilistic circuit in Section [REF].', '1006.4388-1-39-6': 'Let us now show that this corresponds to the thermal ensemble of our original [MATH] construction.', '1006.4388-1-40-0': 'As in Section [REF] the trick here is to express the probability distribution for the ensemble after applying [MATH] as [EQUATION] where [EQUATION] such that [EQUATION]', '1006.4388-1-40-1': 'Thus the probability distribution resulting from after the application of [MATH] to the [MATH] thermal ensemble is equal to the probability distribution arising from [MATH].', '1006.4388-1-41-0': 'We will now show the [MATH] is exactly the energy function we would define for a ground state spin computing construction for our circuit [MATH].', '1006.4388-1-41-1': 'Recall that [EQUATION]', '1006.4388-1-41-2': 'Note that in evaluating [MATH], which is a sum of terms, we can evaluate the action action of [MATH] on each of these terms separately and then re-sum.', '1006.4388-1-41-3': 'Thus we first examine [EQUATION]', '1006.4388-1-41-4': 'This is because the inputs to the circuit commute through the individual gate elements [MATH].', '1006.4388-1-41-5': 'Indeed this observation along with the order of [MATH] allows one to derive the term arising from each [MATH] term independently.', '1006.4388-1-41-6': 'In particular the only term which is not [MATH] for this energy function is [EQUATION]', '1006.4388-1-41-7': 'But this simply means that [EQUATION] only when [MATH] and is [MATH] otherwise.', '1006.4388-1-41-8': 'This is just our normal definition of the circuit energy function.', '1006.4388-1-42-0': 'Thus we see that using the [MATH] construction one can construct a probabilistic circuit from the thermal ensemble for [MATH] which is the probabilistic circuit described in Section [REF].', '1006.4388-1-42-1': 'Further we have shown that this can be thought of as arising from the original energy function [MATH].', '1006.4388-1-42-2': 'This provides an alternative derivation of our main result.', '1006.4388-1-43-0': '# Generalized Classical Ground State Spin Computation Models', '1006.4388-1-44-0': 'Having shown that classical ground state spin computing at non-zero temperature can be mapped to probabilistic circuits it is interesting to return to our original model and consider what generalizations of the energy function allow for this mapping to occur.', '1006.4388-1-44-1': 'The crucial assumption during our derivation of the probabilistic re-interpretation of the partition function was that we could convert the tensors [MATH] of Eq. ([REF]) to tensors [MATH] of Eq. ([REF]) in such a way that [MATH] could be interpreted as a probablistic gate.', '1006.4388-1-44-2': 'This requires (1) the division of tensor indices into input and output indices and (2) for each input to the circuit the normalization to make the outputs probabilities (positive and sum to unity) is the same.', '1006.4388-1-45-0': 'Let us begin by showing that not all tensors and divisions into input and output vertices can result in a probabilistic interpretation.', '1006.4388-1-45-1': 'This can be demonstrated with a simple example on two bits: [EQUATION]', '1006.4388-1-45-2': 'If we chose [MATH] as an input and [MATH] as an output, this gives rise to the tensor [EQUATION]', '1006.4388-1-45-3': 'In order that the first column of this matrix sum to unity we must divide by [MATH].', '1006.4388-1-45-4': 'However when we do this the second column will not sum to [MATH].', '1006.4388-1-45-5': 'A similar argument occurs if we had chosen [MATH] as the output and [MATH] as the input.', '1006.4388-1-45-6': 'Thus there is no way to scale [MATH] in such a way that it can be interpreted as a probabilistic gate.', '1006.4388-1-46-0': 'A more relevant example of a failure for a model to not be amenable to our scaling arising in quantum-dot cellular automata models.', '1006.4388-1-46-1': 'In quantum dot cellular automata models bistable quantum dots are engineered in such a way that the ground state enacts a classical computation as in our model.', '1006.4388-1-46-2': 'Most of the treatments of these models deal with the quantum mechanics of these devices [CITATION].', '1006.4388-1-46-3': 'One can, however, build a classical model of these systems as was done, for example, by Wang in [CITATION].', '1006.4388-1-46-4': 'In this approximation one derives a classical statistical mechanical many spin system as in our model.', '1006.4388-1-46-5': 'When one does this for a quantum-dot cellular automata wire one ends up with exactly the model we can consider above in Section [REF].', '1006.4388-1-46-6': 'However when one considers this for the majority gate constructions (see [CITATION]) one obtains a model for which our technique cannot be applied.', '1006.4388-1-46-7': 'In particular the majority gate that they propose has a classical energy function which is [EQUATION]', '1006.4388-1-46-8': 'Here [MATH] are the input spins and [MATH] is the output spin.', '1006.4388-1-46-9': '(Note that we have used one less spin than in the traditional presentation, but this does not affect our conclusions as one can think of the extra spin as being connected to the central spin by an identity gate.)', '1006.4388-1-46-10': 'Note that the lowest energy configuration for a given input correctly computes the majority function.', '1006.4388-1-46-11': 'That is if we fix [MATH], [MATH], and [MATH], then the lowest energy value of [MATH] is given by the majority of the inputs, [MATH].', '1006.4388-1-46-12': 'Note, however, that if we sum over all outputs for input [MATH] [EQUATION] but if one uses input [MATH] and [MATH] one obtains [EQUATION]', '1006.4388-1-46-13': 'Thus we cannot properly globally normalize the entire tensor in such a way as to to obtain a probabilistic gate.', '1006.4388-1-47-0': 'However, the same two-spin energy function can be used to describe a fanout gate, in contrast to the many-spin interaction used in Eq. ([REF]), [EQUATION]', '1006.4388-1-47-1': 'Here [MATH] is the input and the other spins are the output.', '1006.4388-1-47-2': 'By changing the direction of the circuit, we can show that this element can be represented as a probabilistic gate.', '1006.4388-1-47-3': 'I.e. the majority gate used in quantum dot cellular automata can not be mapped to a probabilistic circuit, but reversing the role of the inputs and outputs, this gate can be used to construct a fan-out gate.', '1006.4388-1-47-4': 'To see this note that, as for our previous fanout model, the ground state of this system, where the computation is performed, is degenerate.', '1006.4388-1-47-5': 'However now, as opposed to a single excited state there are multiple excited states.', '1006.4388-1-47-6': 'For example if we consider a fan-out to two spins we obtain an [MATH] as given in Table [REF].', '1006.4388-1-48-0': 'From this table we see that while the outputs that are in error have different Boltzman factors, we see that the sum over the output spins is indeed independent of the output.', '1006.4388-1-48-1': 'Indeed when we interpret this as a probabilistic gate by dividing out by [MATH], then we see that this is a probabilistic gate which, instead of having a global failure where all possible output failures have equal probability, the fan-out spins have independent error probabilities.', '1006.4388-1-48-2': 'In other words it is as if the fan-out occurred and then with probability [MATH] each of the outputs is independently flipped.', '1006.4388-1-48-3': 'We will use this fan-out gate in the next section when we discuss fault-tolerance.', '1006.4388-1-49-0': 'What is a necessary and sufficient condition for an energy function to allow for a proper normalization to a probabilistic gate?', '1006.4388-1-49-1': 'If we insist that this work for all temperatures, it is easy to give this answer.', '1006.4388-1-49-2': 'Let [MATH] denote the energy function for a logic gate with inputs [MATH] and output [MATH].', '1006.4388-1-49-3': 'Let [MATH] denote the set of energies for a given input.', '1006.4388-1-49-4': "Then the necessary and sufficient condition is that the [MATH]'s must be permutations of each other.", '1006.4388-1-49-5': 'The sufficiency of this condition is obvious.', '1006.4388-1-49-6': 'Necessity follows from noting that we are requiring this to hold for all temperatures and thus the sum over outputs of the [MATH] must be a rearrangement of this sum and therefore a permutation of the output energies.', '1006.4388-1-49-7': 'Note that this condition implies that the ground states of a logic gate energy function must be degenerate for all possible inputs (but note that this alone is not sufficient for the condition to hold.)', '1006.4388-1-50-0': '## Computational Complexity of Identifying Probabilistic Circuits in Thermal Systems', '1006.4388-1-51-0': 'Having established a necessary and sufficient condition for the possibility of scaling the [MATH] tensors as probabilistic gates, a natural follow up question is how one can determine whether it is possible to take an energy function [MATH] and determine whether or not it can be interpreted as a probabilistic circuit.', '1006.4388-1-51-1': 'Here we will show that this problem is computationally intractable, even when we are given information about the breakdown of [MATH] into terms which we wish to interpret as gates.', '1006.4388-1-52-0': 'Suppose that we are told the [MATH] is made up of a sum over logic gates and preparations [EQUATION] i.e. we are given a specification of the [MATH] in terms of a constant number of bits and we are told that each of these [MATH] will correspond to a logic gate or a preparation of initial inputs.', '1006.4388-1-52-1': 'We can then ask: can we interpret this as a probabilistic circuit with or without prepared inputs?', '1006.4388-1-52-2': 'Every [MATH] involves some inputs and output bits, but in general we will not be told which of these are inputs and which are outputs.', '1006.4388-1-52-3': 'In general an [MATH] may thus have many different tuples of input bits and output bits such that we may interpret [MATH] as a probabilistic gate by proper global normalization.', '1006.4388-1-52-4': 'For example gates which have doubly stochastic [MATH] tensors may be run either backwards or forwards (a doubly stochastic matrix is a stochastic matrix whose rows and columns sum to [MATH].)', '1006.4388-1-52-5': 'Another example is given by a fan-out gate in our original energy function model [EQUATION]', '1006.4388-1-52-6': 'In this case we may interpret the [MATH] arising from this energy function as having input [MATH] and outputs [MATH] and [MATH], input [MATH] and outputs [MATH] and [MATH], or input [MATH] and outputs [MATH] and [MATH].', '1006.4388-1-52-7': 'Contrary to this, suppose that the energy function is that of the one-to-two fanout described in Eq. ([REF]), [EQUATION]', '1006.4388-1-52-8': 'The [MATH] for this energy function has only one possible orientation between inputs and output: [MATH] is an input and [MATH] and [MATH] are outputs.', '1006.4388-1-52-9': 'This in general will lead to the following question: given the possible input and outputs tuples for the [MATH]s is it possible to assign these in such a way as to interpret the resulting total tensor as a probabilistic circuit.', '1006.4388-1-52-10': 'We will now show that this problem is [MATH]-complete and is therefore (unless [MATH]), in general, intractable.', '1006.4388-1-53-0': 'To see that the problem is [MATH]-complete proceed as follows.', '1006.4388-1-53-1': 'Assign to each bit of our physical system a boolean variable, [MATH].', '1006.4388-1-53-2': 'Assume that a given logical energy function [MATH] is involved in only a constant number of bits.', '1006.4388-1-53-3': 'Since [MATH] is involved in a constant number of bits we can explicitly calculate all of the sets of inputs and output bits for which [MATH] can be globally normalized so as to make it a probabilistic gate.', '1006.4388-1-53-4': 'The variables [MATH]s are going to represent whether the corresponding bit is associated is an input or and output.', '1006.4388-1-53-5': 'In particular, TRUE, will represent that the bit is an input and FALSE will represent that the bit is an output.', '1006.4388-1-53-6': 'Thus from the sets of input and output bits for which one can consistently label inputs to [MATH] we can construct a boolean function on the relevant bits such that this function is true if and only if a valid assignment of boolean variables respects a possible input and outputs.', '1006.4388-1-53-7': 'Thus, for example, for [MATH] above in Eq. ([REF]), the boolean function would be [EQUATION]', '1006.4388-1-53-8': 'Constructing one such boolean function for each logic gate we can then construct a boolean function on all bits [MATH] which is the conjunction (logical and) of all of the gate boolean functions.', '1006.4388-1-53-9': 'Thus we see that by calculating for each [MATH] the possible combinations of inputs and outputs that have an allowed interpretation in terms of a probabilistic circuit, we have a one-to-one mapping of the problem to the problem of deciding whether there is a satisfying assignment for the boolean function made up of the conjunction of logical terms with only a constant number of involved bits.', '1006.4388-1-53-10': 'In order to demonstrate that the problem is we need now only show that the boolean function we can construct in this manner are general enough so as to yield a version of the satisfaction problem which is In particular it is clear that this problem is in [MATH], since we can evaluate in polynomial time whether a given assignment of inputs and outputs can be interpreted as a probabilistic circuit.', '1006.4388-1-53-11': 'To demonstrate [MATH]-completeness we need to show that it is at least as hard as another [MATH]-complete problem.', '1006.4388-1-54-0': 'We will show that the problem is [MATH]-complete by showing that some instance of the problem can be mapped to the one-in-three satisfiability problem of Schaefer [CITATION] (now sometimes called the monotone one-in-three satisfiability problem.)', '1006.4388-1-54-1': 'Define [MATH] as the boolean function which is true iff exactly one of [MATH] is true.', '1006.4388-1-54-2': 'Notice that this is exactly the boolean formula of Eq. ([REF]) (with [MATH], [MATH], and [MATH].)', '1006.4388-1-54-3': 'In the one-in-three satisfiability problem one is given a boolean function which is the conjunction (logical and) of clauses each of which are of the form [MATH] for a choice of variables [MATH] in the problem.', '1006.4388-1-54-4': 'Schaefer proved that the one-in-three satisfiability problem is Actually there is a slight subtlety here as the problem Schaefer considers must allow repeated variables.', '1006.4388-1-54-5': 'We will now show that it is possible to take a one-in-three satisfiability problem and convert it into an instance of the problem of spotting whether a given energy function can support interpretation as a classical probabilistic circuit.', '1006.4388-1-55-0': 'As mentioned above the one-in-three satisfiability problem has as input a conjunction of boolean functions which are all of the form [MATH], and where some boolean variables can be repeated, [EQUATION] where [MATH] and none of the [MATH], [MATH], and [MATH] are the same variable for fixed [MATH].', '1006.4388-1-55-1': 'Let [MATH] denote the set of [MATH] variables that are [MATH].', '1006.4388-1-55-2': 'We will define an energy function on [MATH] spins which, when converted into the problem of trying to assign consistent inputs and outputs will yield exactly the boolean function [MATH] of Eq. ([REF]).', '1006.4388-1-55-3': 'Call the first [MATH] spins [MATH] and the last [MATH] spins [MATH].', '1006.4388-1-55-4': 'First define the energy function [EQUATION] where [MATH] is the fan-out energy function defined in Eq. ([REF]).', '1006.4388-1-55-5': 'When one converts this into a logical expression for whether the spins are inputs or outputs, one gets exactly the one-in-three logical clause of Eq. ([REF]).', '1006.4388-1-56-0': 'However, at this point, the variables that will arise in these clauses are all different.', '1006.4388-1-56-1': 'This is what the other [MATH] spins are for.', '1006.4388-1-56-2': 'Define the following energy function for the "identity" gate: [EQUATION]', '1006.4388-1-56-3': 'This energy function corresponds to a logical reversible gate which acts as identity from the [MATH] spins to the [MATH] spins.', '1006.4388-1-56-4': 'This energy function has the property that the [MATH] variables are all either inputs or all outputs (and the [MATH] variables are also all either inputs or outputs, opposite to the [MATH] variable assignment.)', '1006.4388-1-56-5': 'Thus we can use gates of this form to effectively make the spins of the [MATH] gates constructed above the same variable.', '1006.4388-1-56-6': 'In particular define the following energy function [EQUATION] where [MATH] and [MATH] is the [MATH]th element of [MATH] translated over to a [MATH] variable, i.e. [MATH] corresponds to [MATH], [MATH] corresponds to [MATH] and [MATH] corresponds to [MATH].', '1006.4388-1-57-0': 'Now consider the energy function [EQUATION] and stipulate that every term in the sum of [MATH] will correspond to one gate and every term in [MATH] will correspond to one gate.', '1006.4388-1-57-1': 'Then if we convert this problem over into a boolean satisfaction problem as describe above, where each spin is a boolean variable representing whether the spin is an input or an output, we will obtain a boolean expression that is equivalent to the problem described in Eq. ([REF]).', '1006.4388-1-57-2': 'Thus we have shown that we can, for a given one-in-three satisfaction problem, Eq. ([REF]) construct an energy function, along with labeling of the terms in this function corresponding to different gates, such that determining whether this energy function can be reinterpreted as a probabilistic computation is equivalent to the original one-in-three satisfaction problem.', '1006.4388-1-57-3': 'This implies that the problem of determining whether such probabilistic computations exist is at least as hard as this [MATH]-complete problem.', '1006.4388-1-57-4': 'Combined with the fact the problem is in [MATH] this means that the problem of deciding whether a given energy function can be thought of as enacting a probabilistic circuit is [MATH]-complete.', '1006.4388-1-58-0': 'Thus we have shown that the problem of determining a many-spin statistical mechanics system enacts a ground state spin computation, even given the decomposition of this system into logical gates, is We note in passing that this result is of perhaps some philosophical significance: determining whether a system can be thought of as a computer is shown to be computationally intractable (assuming [MATH]).', '1006.4388-1-59-0': '## Models that have a Probabilistic Gate Interpretation', '1006.4388-1-60-0': 'So far we have considered spin models in a very general setting unrelated to physical theories.', '1006.4388-1-60-1': 'We point out, here, however, that there are a variety of models that have arisen in physics which admit such an interpretation.', '1006.4388-1-60-2': 'We have already seen that that one dimensional Ising model admits an interpretation as a series of identity gates which err with some probability.', '1006.4388-1-60-3': 'Here we point out some other models which admit such interpretations.', '1006.4388-1-61-0': '### Ising Models on Trees', '1006.4388-1-62-0': 'A tree is a connected graph, [MATH], with vertex set [MATH], edge set [MATH], and no cycles.', '1006.4388-1-62-1': 'Assign to each spin a vertex, [MATH] for [MATH].', '1006.4388-1-62-2': 'To each edge [MATH] assign an non-zero energy for the Ising coupling, [MATH].', '1006.4388-1-62-3': 'Then the Ising model on the tree has an energy function given by [EQUATION]', '1006.4388-1-62-4': 'Fix a root vertex [MATH].', '1006.4388-1-62-5': 'First consider the model at zero temperature.', '1006.4388-1-62-6': 'At [MATH], following our discussion of the fan-out in Section [REF], it is easy to see that the ground state of this model can be thought of as a series of fan-out gates originating from the root vertex [MATH] with possible bit flips applied to the fan-out depending on the sign of [MATH].', '1006.4388-1-62-7': 'At [MATH] we see that, as in the discussion of the fan-out gate, each of the individual fan-out wires (with possible bit flips), will fail independently with a probability of [EQUATION]', '1006.4388-1-62-8': 'Note that a failure for an antiferromagnetic coupling (bit flip) corresponds to an identity gate.', '1006.4388-1-62-9': 'Ising models on trees have been studied in a variety of settings [CITATION] and in fact the interpretation of this model in terms of a probabilistic circuit has been used extensively in the computer science literature, where the model goes under the name of broadcasting [CITATION].', '1006.4388-1-63-0': '### [MATH] Lattice Pure Gauge Theories', '1006.4388-1-64-0': 'Perhaps slightly more interesting that Ising models on trees is the fact that [MATH] lattice gauge theories [CITATION] can be cast as a probabilistic gate circuits.', '1006.4388-1-64-1': 'For definiteness, first consider a (pure) [MATH] lattice gauge theory on a two dimensional square lattice with closed boundaries.', '1006.4388-1-64-2': 'In this model one places spins on the edges of the lattice and defines the energy function [EQUATION] where [MATH] is the set of all plaquettes of the square lattice, and [MATH] are the spins surrounding plaquette [MATH].', '1006.4388-1-64-3': 'Consider an individual term in this sum, and assume for simplicity that for all plaquettes, [MATH] that [MATH].', '1006.4388-1-64-4': 'We can view the individual term [MATH] as a gate element with either [MATH], [MATH], [MATH], or [MATH] inputs and [MATH], [MATH], [MATH], and [MATH] outputs respectively.', '1006.4388-1-64-5': 'For example if we view it as a gate with [MATH] inputs and [MATH] outputs, then, in its ground state this corresponds to a probabilistic gate which, with equal probability, changes the input to a set of spins with the same parity of [MATH]s as the input.', '1006.4388-1-64-6': 'In other words the logical evolution is [EQUATION]', '1006.4388-1-64-7': 'The gate with [MATH] inputs and [MATH] outputs is deterministic and is given by [MATH].', '1006.4388-1-64-8': 'The gate with [MATH] inputs corresponds to a preparation of [MATH] spins which have an even number of [MATH] spins and this state is prepared with equal probability.', '1006.4388-1-64-9': 'The gate with [MATH] input and [MATH] outputs is a probabilistic gate which outputs an even number of [MATH]s if the input is [MATH] all such possibilities occurring with equal probability.', '1006.4388-1-64-10': 'Similarly it outputs an odd number of [MATH]s if the input is [MATH] all such possibilities occurring with equal probability.', '1006.4388-1-64-11': 'By picking boundary terms to serve as input and output bits, it is clear that it is possible to order the energy terms in the above sum in such a way that we can interpret the ground state as a probabilistic cellular automata like model using the above gates.', '1006.4388-1-64-12': 'At finite gate these probabilistic gates all fail with some probability, where failure results in an equally likely failed output for the gate.', '1006.4388-1-64-13': 'While we have define this for the two dimensional square lattice, it is clear that the above construction can result in a probabilistic circuit for a far greater number of lattices, the main criteria being that there exists a way to label inputs and outputs in such a way that a proper circuit is constructed.', '1006.4388-1-65-0': '# Making Classical Ground State Spin Computation Robust at Finite Temperature', '1006.4388-1-66-0': 'Having shown how the computed circuit energy functions give rise to thermal ensembles which can be interpreted as probabilistic circuits with probabilistic inputs, we now discuss how it is possible to make the model of classical ground state spin computation fault-tolerant at finite temperature.', '1006.4388-1-66-1': 'Indeed this is now rather simple given that we have a mapping between the behavior of these systems at finite temperature and probabilistic circuits which have gates and preparations failing with a fixed probability.', '1006.4388-1-66-2': 'We will need, however, to use the more general class of energy functions considered in Section [REF].', '1006.4388-1-67-0': 'The general theory of computing in the presence of gates which fail goes back at least as far as the work of von Neumann [CITATION].', '1006.4388-1-67-1': 'Von Neumann considered a model in which each logic gate failed with exactly the probability [MATH].', '1006.4388-1-67-2': 'In order to get this model to compute reliably, von Neumann used two major techniques.', '1006.4388-1-67-3': 'The first techniques is that one needs to suitably encode information: in order to do this one encodes a single bit across a bundle of wires in a logic circuit and interprets this as [MATH] (or [MATH]) when a majority of the wires are in the [MATH] (or [MATH].)', '1006.4388-1-67-4': 'Given such an encoding it is then easy to compute in a parallel manner such that the effect of gates failing is simply to decrease the proportion of [MATH]s (or [MATH]s) in an encoded [MATH] (or [MATH].)', '1006.4388-1-67-5': 'However, the fact that the gates fail does cause the ratio of correct bits in an encoded bundle of bits to decrease.', '1006.4388-1-67-6': 'In order to deal with this von Neumann used a second technique whereby faulty gates were used to perform error correction.', '1006.4388-1-67-7': "Actually this portion of von Neumann's analysis is not quite satisfactory, as it requires the use of a random permutation.", '1006.4388-1-67-8': 'However Pippenger [CITATION] has shown that one can de-randomize this construction (one can obtain reasonable parameters for this method by using the expanders defined in [CITATION]).', '1006.4388-1-67-9': 'The end result of this is that one can show that if one wishes to compute a logical circuit of size [MATH] to an accuracy [MATH] (that is, have the circuit fail with probability [MATH]) with gates that fail at [MATH] for some fixed threshold [MATH], one can do this using a circuit with the unreliable gates which is only a [MATH] larger, where [MATH] is a fixed constant.', '1006.4388-1-67-10': 'This effectively means that one can create for all effective purposes reliable logic gates out unreliable logic gates with an overhead which is logarithmic in the inverse of the error desired.', '1006.4388-1-68-0': "Having described von Neumann's construction one can now see how it is possible to make classical ground state spin computing fault-tolerant.", '1006.4388-1-68-1': 'In particular we have shown above how the thermal ensemble for such systems gives rise to probabilistic gates.', '1006.4388-1-68-2': 'For gates constructed using energy functions like Eq. ([REF]) we have shown that the thermal ensemble arising for these functions is equivalent to a circuit in which gates fail with probability [MATH] if the gate has [MATH] output bits.', '1006.4388-1-68-3': 'For [MATH] this is [MATH].', '1006.4388-1-68-4': "We would like to use this for von Neumann's [MATH] failure probability.", '1006.4388-1-68-5': "One complication arises, however, which is that in von Neumann's model the fan-out gates do not fail.", '1006.4388-1-68-6': 'These gates are needed during the error correcting stage of his procedure.', '1006.4388-1-68-7': 'We get around this by using fan-out gates constructed from energy functions like those in Section [REF].', '1006.4388-1-68-8': 'In particular define the fan-out gate via Eq. ([REF]).', '1006.4388-1-68-9': 'As shown in Section [REF] this gives rise in our probabilistic gate setting in which each fan-out resulting wire fails with probability [MATH].', '1006.4388-1-68-10': "These independent failures can then be reinterpreted as errors for the gates that follow them in von Neumann's error correcting construction.", '1006.4388-1-68-11': "One should note that there is a subtlety here in that if we desire to recover von Neuman's error model where the gates fail with exactly probability [MATH] then a complicated calculation is needed in order to figure out [MATH] in this model.", '1006.4388-1-68-12': 'However, for large enough [MATH] the effect will mostly be that the error probabilities add such that [MATH] will be the sum of the fan-out error and the gate error.', '1006.4388-1-69-0': 'Putting this together we can conclude that there is a critical temperature below which one can use fault-tolerant constructions to design ground state spin energy functions which compute correctly with probability [MATH] by using [MATH] more interactions in the energy functions.', '1006.4388-1-69-1': 'This means that while the general model of ground state spin computing is not tolerant to errors, for a cleverly designed setup one can overcome this difficulty and build robust devices, assuming that the temperature is low enough to keep the error probability below the error threshold.', '1006.4388-1-70-0': '# Relationship to Computational Complexity Theory', '1006.4388-1-71-0': 'Now we turn to a relationship between our model and computational complexity.', '1006.4388-1-71-1': 'This connection is provided because the ground state spin computing model at zero temperature is intimately related to the Cook-Levin theorem from computational complexity.', '1006.4388-1-72-0': '## The Cook-Levin Theorem', '1006.4388-1-73-0': 'The Cook-Levin theorem [CITATION] is a fundamental result in the theory of computational complexity.', '1006.4388-1-73-1': 'Here we briefly review this result.', '1006.4388-1-73-2': 'For more details one is referred, for example, to the classic textbook [CITATION].', '1006.4388-1-74-0': 'Traditionally the theory of computational complexity is founded upon the computational model of a Turing machine [CITATION].', '1006.4388-1-74-1': 'The Church-Turing thesis roughly states that the Turing machine model adequately captures the notion of what can be performed by any physical instantiation of a computer and decades of experience have yet to challenge this assumption.', '1006.4388-1-74-2': 'A deterministic Turing machine consists of the machine and a linear tape onto which elements from a finite alphabet, [MATH], can be written into different cells on the tape.', '1006.4388-1-74-3': 'The machine itself has an finite set of internal states, [MATH], a method for reading a symbol written in a cell on the tape, a method to write a symbol in a cell on the tape, and the ability to move either left or right one cell.', '1006.4388-1-74-4': 'Formally the tape is assumed to be infinite or at least infinitely extendable.', '1006.4388-1-74-5': 'Initially an input, consisting of a string of symbols from a finite alphabet [MATH] is placed on the tape and the physical machine is placed on the first symbol of this string.', '1006.4388-1-74-6': 'The Turing machine is assumed to start itself with its internal states in a special state called the start state, [MATH].', '1006.4388-1-74-7': 'A deterministic Turing machine then proceeds to "compute" as follows.', '1006.4388-1-74-8': 'At each time step the deterministic Turing machine examines the symbol of the cell it currently occupies and its internal state.', '1006.4388-1-74-9': 'Then conditional on these two variables the Turing machine writes a symbol onto cell (possibly from a larger, but still finite, alphabet than the alphabet of the initial strings) it occupies, moves either left or right, and modifies its internal state.', '1006.4388-1-74-10': 'These rules can be specified by a function, the transition function, [MATH], where [MATH] and [MATH] denote moving left or right.', '1006.4388-1-74-11': 'The deterministic Turing machine does this indefinitely until it reaches one of a set of its internal states known as halt states.', '1006.4388-1-74-12': 'These halt states are labeled as either "reject" or "accept."', '1006.4388-1-74-13': 'If we initially input the string [MATH] and the deterministic Turing machine halts in accept state, then we say the deterministic Turing machine accepts [MATH].', '1006.4388-1-74-14': 'Similarly if on input [MATH] the machine halts in a reject state, then we say that the machine rejects [MATH].', '1006.4388-1-74-15': 'Note that on some strings the Turing machine may neither accept or reject a string, but continue indefinitely.', '1006.4388-1-74-16': 'The language of a Turing machine is the set of all strings taken from the input alphabet for which the Turing machine accepts for that string being input.', '1006.4388-1-75-0': 'Deterministic Turing machines capture many important features of real modern computers.', '1006.4388-1-75-1': 'For example, the running time of most algorithms on a variety of hardware are closely related to the similar problems cast in terms of deterministic Turing machines.', '1006.4388-1-75-2': 'An important class of languages are those for which a deterministic Turing machine accepts a string from the language in a time bounded by a constant times a polynomial in the size of the string.', '1006.4388-1-75-3': 'Such languages are said to be in the complexity class [MATH].', '1006.4388-1-76-0': 'In contrast to a deterministic Turing machine, non-deterministic Turing machines are a model of computing which is not thought to properly encapsulate what can be efficiently computed on a modern computer.', '1006.4388-1-76-1': 'A non-determinisitic Turing machine works in a manner similar to a deterministic Turing machine: it has an internal state, it can read and write symbols onto a tape, etc.', '1006.4388-1-76-2': "The difference is that at each step in the computation, a non-deterministic Turing machine doesn't have to chose one new state, new symbol and direction to move, but may branch to multiple such triples.", '1006.4388-1-76-3': 'The transition function is now not [MATH] but instead is of the form [MATH] where [MATH] denotes the power set.', '1006.4388-1-76-4': 'Non-deterministic Turing machines thus explore multiple pathways at once.', '1006.4388-1-76-5': 'Such a Turing machine halts when one of the states of any of the multiple branches it is exploring enters a halt state.', '1006.4388-1-76-6': 'Again we can define whether the initial input is accepted or rejected depending on whether the state where the computation halts is a reject or accept state.', '1006.4388-1-76-7': 'Given this we can define the language of the non-determinstic Turing machine as the set of strings in which the machine accepts the given input.', '1006.4388-1-76-8': 'The time complexity of a non-deterministic Turing machine is the number of steps of executed by the Turing machine irrespective of the number of branches explored by the machine.', '1006.4388-1-77-0': 'Because a non-deterministic Turing machine is allowed to explore multiple pathways in its computation it is not clear that such machines can be reasonably mapped (in terms of say, time complexity of problems) to modern computers.', '1006.4388-1-77-1': 'This question can be put on a more formal footing by defining the complexity class [MATH].', '1006.4388-1-77-2': '[MATH] is the class of languages which can be recognized by a non-deterministic Turing machine in a time bounded by a constant times a polynomial in the size of the string being decided.', '1006.4388-1-77-3': 'The statement of whether [MATH] is strictly larger than [MATH] is the famous [MATH] problem, one of the most important open problems in the theory of computational complexity.', '1006.4388-1-77-4': 'That being said, most researchers do believe that [MATH] and that the model of a non-deterministic Turing machine is more powerful than computers we can physically construct.', '1006.4388-1-78-0': 'Given the above brief overview of the basic models of deterministic and non-deterministic Turing machines we can now introduce the Cook-Levin theorem.', '1006.4388-1-78-1': 'To define the Cook-Levin problem we introduce the [MATH] problem.', '1006.4388-1-79-0': 'Boolean Satisfaction Problem (): Given as input a description of a Boolean function from [MATH] boolean inputs to one output, [MATH], decide whether there exists a set of inputs [MATH], with each [MATH], such that [MATH].', '1006.4388-1-80-0': 'Cast in terms of languages, the [MATH] language is the set of strings describing a Boolean function such that the Boolean function has some input which evaluates to [MATH].', '1006.4388-1-80-1': 'The Cook-Levin theorem then states that the [MATH] is complete for [MATH].', '1006.4388-1-80-2': 'This means that every language in [MATH] can be converted by a deterministic Turing machine in polynomial time to a [MATH] language and every [MATH] language can be converted into a language in [MATH].', '1006.4388-1-80-3': 'Thus, in a real sense, [MATH] captures what it means for a lanuage to be in [MATH].', '1006.4388-1-80-4': 'For example, if we could efficiently solve a [MATH] problem in polynomial time we would be able to solve every problem in [MATH] in polynomial time and vice versa.', '1006.4388-1-80-5': 'This is especially important considering that numerous problems beside [MATH] have been identified as [MATH]-complete.', '1006.4388-1-80-6': 'This class of problems are essentially "the most difficult" in [MATH] and are correspondingly the problems for which no known polynomial time algorithms for all instances of the problems in the class are thought to exist.', '1006.4388-1-81-0': 'We are most interested in the proof of the Cook-Levin theorem.', '1006.4388-1-81-1': 'Before reviewing this it is useful to provide another equivalent characterization of [MATH] which uses, instead of a non-deterministic Turing machine, a prover and a verifier.', '1006.4388-1-81-2': 'The prover is assumed to have unlimited computational power while the verifier is limited to running a polynomial time algorithm on a deterministic Turing machine.', '1006.4388-1-81-3': 'A language, [MATH], is in [MATH] if the prover can supply to a verifier a polynomial sized proof [MATH] that a instance [MATH] is in the language.', '1006.4388-1-81-4': 'This proof must always be correct, i.e. the verifier must always be convinced, by using a Turing machine from [MATH], that the instance [MATH] is the in the language.', '1006.4388-1-81-5': 'Similarly the prover must never be able to convince the verifier incorrectly.', '1006.4388-1-81-6': 'In other words if [MATH] is not in [MATH], then the verifier must always reject.', '1006.4388-1-81-7': 'For example [MATH] is in [MATH] by this definition since the prover can simply supply a satisfying input to the boolean function if the function can be satisfied, and the verifier can compute the boolean function on this input in polynomial time.', '1006.4388-1-81-8': 'Similarly if the function does not admit a satisfying assignment the verifier can never be convinced that the instance is not in the language because no string the prover can give will lead to a satisfied boolean function.', '1006.4388-1-81-9': 'Note that in the prover-verifier setting we do not require that the prover try to convince the verifier that an instance [MATH] is not in [MATH], only that if it is true that [MATH] is not in [MATH] that the prover cannot convince the verifier that [MATH] is in [MATH].', '1006.4388-1-82-0': 'Given the prover/verifier definition of [MATH] we can now discuss the Cook-Levin theorem.', '1006.4388-1-82-1': 'The Cook-Levin theorem states that [MATH] is [MATH]-complete.', '1006.4388-1-82-2': 'A problem, [MATH], is [MATH]-complete if and only if every problem in [MATH] can be transformed into [MATH] by a deterministic Turing machine in polynomial time.', '1006.4388-1-82-3': 'Thus [MATH], since it is [MATH]-complete, captures much of what is difficult about problems in all of [MATH]: if there were a polynomial time algorithm for [MATH] then there would be a polynomial time algorithm for every problem in [MATH].', '1006.4388-1-83-0': 'The proof that [MATH] is [MATH]-complete has two directions.', '1006.4388-1-83-1': 'We have already stated the first direction, that [MATH] is a [MATH] problem.', '1006.4388-1-83-2': 'The meat of the proof is in the other direction, that every problem in [MATH] can be converted into a [MATH] problem.', '1006.4388-1-83-3': 'Here we do this in a slightly non-standard manner, using the prover-verifier definition of [MATH].', '1006.4388-1-83-4': 'In particular for any problem, [MATH], in [MATH] we know that there exists a verifier who takes as input the problem instance, [MATH] and a certificate [MATH], runs these on a deterministic Turing machine and verifies whether the instance is in the language.', '1006.4388-1-83-5': 'Imagine now the time history of such a verification by a Turing machine.', '1006.4388-1-83-6': 'Initially the Turing machine is at the beginning of the string, in its initial state, and the input string is on the tape.', '1006.4388-1-83-7': 'At the next time step, the machine can change its state, move left, right or stay in place, and possibly change the symbol on the tape where the machine is located.', '1006.4388-1-83-8': 'Notice, however that the contents of a cell on the tape, whether the machine is at that location, and the internal state of the machine if it is on that cell at the next time step is a function only of the contents of the cell and its neighbors at the prior step, whether the machine was on one of those cells, and its internal state if it was.', '1006.4388-1-83-9': 'In other words the computation can be described locally in space and time.', '1006.4388-1-83-10': 'Given this we can now describe how to convert a verifier, which is just a deterministic Turing machine and the strings representing the instance [MATH] and the proof certificate [MATH] into a problem in [MATH].', '1006.4388-1-84-0': 'Suppose that the verifiers deterministic Turing machine runs in time [MATH] where [MATH] is the size of the instance and the proof and [MATH] is a polynomial function.', '1006.4388-1-84-1': 'Define the following boolean variables, [MATH], [MATH], [MATH] to represent the history of the deterministic Turing machine in verifying the proof.', '1006.4388-1-85-0': 'We think about the first two variables as being arranged in a large [MATH] tableau onto which the computational history of the deterministic Turing machine is carried out.', '1006.4388-1-85-1': 'We can now define a [MATH] problem over these variables which is satisfiable iff the deterministic Turing machine accepts the input string [MATH].', '1006.4388-1-85-2': 'This [MATH] formula is conjunction (logical and) of a clauses which enforce (a) that the initial contents of the tape are [MATH], (b) that the machine is in the correct initial state, (c) the initial location of the head of the tape, (d) that there is only one symbol per tape cell, one state per time, one tape position per time, (e) that the tape cell is unchanged unless it is the one which can be written upon, (f) the machine properly updates according to the Turing machine transition function, and (g) the machine finishes in an accepting state.', '1006.4388-1-85-3': 'For example condition (b) consists of clauses with a single variable [MATH] where [MATH] is the initial state of the Turing machine.', '1006.4388-1-85-4': 'The most complicated clause in this construction is the one that encodes the local update rule for the machine, (f).', '1006.4388-1-85-5': 'This can be done by adding the clauses [EQUATION] for valid transitions of the Turing machine (that is for [MATH] representing a valid change of state, symbol, and direction of movement for the read/write head of the Turing machine.)', '1006.4388-1-85-6': 'Putting this together one thus sees that once can take the deterministic verifier along with the instance and [MATH] and construct a polynomial sized boolean formula [MATH] which is satisfied iff the verifier accepts the proof on the instance.', '1006.4388-1-85-7': 'Note that we have not forced a constraint on the proof input to the circuit, but that the definition of [MATH] in a prover-verifier setting leads to the satisfiability of the formula [MATH] iff the instance [MATH] is in the language.', '1006.4388-1-86-0': 'We can now see that connection between the Cook-Levin theorem and the classical ground state spin computing model.', '1006.4388-1-86-1': 'In particular a central part of the proof of this theorem is that one can construct a satisfiability formula directly from the history of a computation.', '1006.4388-1-86-2': 'In a similar manner classical ground state spin computing constructs an energy function whose ground state is the history of a computation.', '1006.4388-1-86-3': 'The conversion of a logic circuit to such an energy function is essentially the arithmetization technique of computational complexity [CITATION].', '1006.4388-1-86-4': 'Note however that the important role here of the input and outputs to the constructed formula: in the case where the output is forced, we are led to [MATH]-complete problems, but when the input is forced then we are led to efficient constructions.', '1006.4388-1-86-5': 'Motivated by this it is interesting to consider how the above constructions works but now at finite temperature.', '1006.4388-1-87-0': '## A Promise-MA Problem From Classical Ground State Spin Computing', '1006.4388-1-88-0': 'We will now show that it is possible to use classical ground state spin computing to define, in analogy with [MATH] a Promise-[MATH]-complete problem.', '1006.4388-1-88-1': '[MATH] stands for Merlin-Arthur and is essentially a probabilistic generalization of the complexity class [MATH] [CITATION].', '1006.4388-1-88-2': 'It is defined as the follows:', '1006.4388-1-89-0': 'Promise-MA: A language [MATH], [MATH], is a promise problem in Merlin-Arthur (Promise-) if there is a probabilistic polynomial-time Turing machine [MATH] and a polynomial [MATH] such that for all strings [MATH],', '1006.4388-1-90-0': 'Here [MATH] is the alphabet over which the language is defined.', '1006.4388-1-90-1': 'This version of the definition of [MATH] is a promise problem, since [MATH] does not necessarily cover all possible elements of [MATH].', '1006.4388-1-90-2': 'The reason that this class is called Merlin-Arthur is because it can be defined in terms of an interactive proof system between a computationally unbounded prover, Merlin, and a more limited verifier, Arthur, who can perform feasible probabilistic computations (polynomial time probabilistic computations with bounded error, [MATH].)', '1006.4388-1-90-3': 'Merlin is trying to convince Arthur that a certain instance [MATH] is in a language [MATH].', '1006.4388-1-90-4': 'He does this by communicating a polynomial size proof [MATH] to Arthur.', '1006.4388-1-90-5': 'Arthur can then take this proof and run a polynomial time probabilistic computation on this proof the result of which is: if [MATH], then Arthur is, with high probability, convinced the [MATH] is in [MATH], while if [MATH] no matter what proof Merlin supplies Arthur cannot be convinced that [MATH] except with a small probability.', '1006.4388-1-91-0': 'We define the following problem:', '1006.4388-1-92-0': 'Boundary Expectation Value (BEV): Let [MATH] be a positive integer and [MATH] a [MATH] digit number.', '1006.4388-1-92-1': 'Suppose we are given an energy function on [MATH] bits, [MATH] which can be written as a sum of energy functions [EQUATION] in such a way that this energy function can be interpreted as a circuit energy function (see Eq. ([REF])) with [MATH] input bits and [MATH] output bit, and each term in [MATH] is specified with [MATH] bits of precision.', '1006.4388-1-92-2': 'Consider the thermal ensemble generated by this energy function at inverse temperature [MATH].', '1006.4388-1-92-3': 'We promise that either there exists input bits [MATH] and output bit [MATH] such that for the reduced probability distribution on these bits [EQUATION] or for any input bits [MATH], [EQUATION]', '1006.4388-1-92-4': 'The problem BEV is to determine whether the first of these conditions, Eq. ([REF]), holds.', '1006.4388-1-92-5': 'In other words yes instances of this problem satisfy Eq. ([REF]) for some inputs [MATH] and no instances satisfy Eq. ([REF]) for all [MATH].', '1006.4388-1-93-0': 'We claim that BEV is Promise-The proof of this is fairly straightforward given our prior discussion of the Cook-Levin theorem.', '1006.4388-1-94-0': 'First we claim that [MATH] is in Promise-.', '1006.4388-1-94-1': 'To show this we must show that there exists an interactive proof of the form described above for the problem BEV.', '1006.4388-1-94-2': 'To see this note that a computationally unbounded Merlin can sample from the thermal ensemble for an instance of this problem and can check whether there exists an input [MATH] such that Eq. ([REF]) holds or whether for all inputs [MATH], Eq. ([REF]) holds.', '1006.4388-1-94-3': 'This will then serve as the proof, [MATH], in the definition of .', '1006.4388-1-94-4': 'Given the description of [MATH] and the circuit energy function, Arthur can construct a probabilistic Turing machine which implements the circuit related to this energy function using our construction of a probabilistic circuit from Section [REF].', '1006.4388-1-94-5': 'Then Arthur can run this probabilistic Turing machine on the proof that Merlin supplies.', '1006.4388-1-94-6': 'It is clear that if [MATH] is a yes instance of BEV, then the proof [MATH] Merlin supplies will convince Arthur that he has a yes instance of this interactive protocol.', '1006.4388-1-94-7': 'Further if [MATH] is a no instance of BEV it is also clear that no proof that Merlin supplies will convince Arthur that [MATH] is a yes instance.', '1006.4388-1-94-8': 'Thus it is clear that [MATH] is in Promise-.', '1006.4388-1-95-0': 'The other direction of the proof requires that we show that every problem in Promise- can be converted into a BEV problem.', '1006.4388-1-95-1': 'To do this we follow the idea of the Cook-Levin theorem in that, for a problem [MATH], we convert the probabilistic Turing machine [MATH] that verifies the proof [MATH] of the problem into a thermal circuit for which the associated BEV problem has [MATH] as the satisfying input.', '1006.4388-1-95-2': 'Similarly, for an problem [MATH], the thermal circuit corresponding to [MATH] must correspond to a no instance of BEV, since otherwise there would exist inputs [MATH] which would cause [MATH] to accept with probability greater than [MATH].', '1006.4388-1-95-3': 'Clearly to do this we will make use of our probabilistic circuit representation result from Section [REF].', '1006.4388-1-95-4': 'The main tool that we need in such a construction is that we need to be able to construct circuit energy functions which perform probabilistic gates of a form we desire, for a fixed inverse temperature [MATH].', '1006.4388-1-95-5': 'To do this we can proceed as follows.', '1006.4388-1-95-6': 'We can use our fault-tolerant constructions to produce circuits which operate with near deterministic behavior.', '1006.4388-1-95-7': 'Then in order to augment a deterministic computation we need a source of randomness, indeed in most standard definitions of a probabilistic Turing machine one requires randomness which is close to uniformly random across the random bits being used (or in other words one requires bits which are nearly equal probability of being in [MATH] or [MATH].)', '1006.4388-1-95-8': 'To obtain such randomness in our constructions, notice that if we do not force our inputs to be [MATH] or [MATH] then there is, in a thermal ensemble, an equal probability to be [MATH] and [MATH].', '1006.4388-1-95-9': 'This allows one to create random bits in an circuit energy function.', '1006.4388-1-95-10': 'However in order to get this to work with the deterministic part of our Turing machine, we must find a method to get this randomness into encoded bits in our circuits.', '1006.4388-1-95-11': 'This can be accomplished by simply taking the equal probability [MATH] and [MATH] bits and running them through the circuit which performs error correction on the bundle corresponding to these bits.', '1006.4388-1-95-12': 'Thus we see that we can construct at finite [MATH] and circuit energy function which is near deterministic and which can also have input bits which are nearly uniformly random.', '1006.4388-1-95-13': 'Thus we can proceed just as in the Cook-Levin theorem.', '1006.4388-1-95-14': 'For a tableau describing the history of the probabilistic Turing machine, [MATH], (when this probablistic Turing machine is simply a deterministic Turing machine aided by bits of randomness, we can construct an energy circuit function which implements the probabilistic Turing machine spatially across this tableau.', '1006.4388-1-96-0': 'Thus we have shown that BEV is Promise-While the proof of this result was rather straightforward and followed quickly from understanding the Cook-Levin theorem it is interesting to note that very few problems are known.', '1006.4388-1-96-1': 'In fact the only other non-natural problem which is Promise-to our knowledge is the stoquastic 6-problem [CITATION].', '1006.4388-1-96-2': 'The nearest comparable result is the classic result for the complexity of finding the ground state and computing the partition function for the Ising model of Barahona [CITATION].', '1006.4388-1-97-0': '# Quantum Models', '1006.4388-1-98-0': 'Finally we would like to briefly mention the quantum models which show some similarity with our classical model.', '1006.4388-1-98-1': 'In particular these models were the inspiration for investigating this problem and thus this serves as a good location to introduce the open problems concerning these problem.', '1006.4388-1-99-0': 'One of the quantum models relevant to this discussion are universal adiabatic quantum computing schemes.', '1006.4388-1-99-1': 'In these models one shows how to adiabatically drag a many-body quantum system from one easily preparable ground state to the ground state of another many-body quantum system whose ground state is a superposition over the history of a quantum circuit.', '1006.4388-1-99-2': 'Thus, similar to our models, the end result is a ground state which encodes a computation: but in this case the computation is a superposition over the history of the computation.', '1006.4388-1-99-3': 'These models show that adiabatic quantum computing, which previously had been only used for optimization problems [CITATION] can also be used for universal quantum computation [CITATION].', '1006.4388-1-99-4': 'However the final state of these models are systems with small energy gaps and thus the effects of working at non-zero temperature will have a profound effect on these models.', '1006.4388-1-99-5': 'Indeed it is for exactly this reason that such universal adiabatic quantum computing schemes are not known to be fault-tolerant (see, however, [CITATION].)', '1006.4388-1-99-6': 'Similar reasoning follows for an earlier model of ground state quantum computing due to Mizel and co-workers [CITATION].', '1006.4388-1-100-0': 'The problem of having a system whose ground state correctly computes but whose excited states do not is exactly the problem we have addressed in this paper for the classical ground state spin computing.', '1006.4388-1-100-1': 'An interesting question then, when considering the quantum models, is whether there is a similar reinterpretation quantum thermal ensembles as spatially enacted quantum computations.', '1006.4388-1-100-2': 'This is an important open problem which might conceivably lead to fault-tolerant methods for adiabatic quantum computing.', '1006.4388-1-101-0': '# Conclusion', '1006.4388-1-102-0': 'We have introduce a set of energy functions on many bits (spins) which have the property that their ground state can be thought of as a spatially distributed deterministic computation.', '1006.4388-1-102-1': 'At temperature greater than zero we have shown that the thermal ensemble arising from these models can be reinterpreted as a spatially distributed probabilistic computation.', '1006.4388-1-102-2': 'Further, above zero temperature we see that the gates of a ground state spin computer can become unreliable and fail to execute the desired computation with high-fidelity.', '1006.4388-1-102-3': 'However with the mapping of the thermal ensemble to probabilistic circuits we have shown how it is possible to make versions of the desired deterministic circuits which are fault-tolerant.', '1006.4388-1-102-4': 'We have shown that determining whether a given classical energy function can be thought of as enacting a ground state computation is Finally we have shown that a problem concerning the thermal ensembles arising in our model give rise to a rare complete problem for the complexity class Promise-.', '1006.4388-1-102-5': 'The models we have considered here are mostly devoid of connections to actual physical systems of interest, besides connections to quantum-dot cellular automata.', '1006.4388-1-102-6': 'An interesting an important open question about these models is whether they arise in naturally occurring physical systems, or a suitably engineered system.', '1006.4388-1-102-7': 'Another interesting question is the rate at which the ground state spin model thermalizes: a proof that the system reaches thermal equilibrium in a time polynomial in the size of the circuit being implement would be another step towards making this model more physically relevant.', '1006.4388-1-103-0': 'D.B. and I.C. are supported by NSF grants 0621621, 0803478, 0829937, and 091640 and DARPA grant FA9550-09-1-0044.', '1006.4388-1-103-1': 'K.R.B is supported by Georgia Tech.', '1006.4388-1-103-2': 'We acknowledge useful conversations with David Meyer and Maxwell Pierce.'}

{'1006.4388-2-0-0': 'We examine a model of classical deterministic computing in which the ground state of the classical system is a spatial history of the computation.', '1006.4388-2-0-1': 'This model is relevant to quantum dot cellular automata as well as to recent universal adiabatic quantum computing constructions.', '1006.4388-2-0-2': 'In its most primitive form, systems constructed in this model cannot compute in an error free manner when working at non-zero temperature.', '1006.4388-2-0-3': 'However, by exploiting a mapping between the partition function for this model and probabilistic classical circuits we are able to show that it is possible to make this model effectively error free.', '1006.4388-2-0-4': 'We achieve this by using techniques in fault-tolerant classical computing and the result is that the system can compute effectively error free if the temperature is below a critical temperature.', '1006.4388-2-0-5': 'We further link this model to computational complexity and show that a certain problem concerning finite temperature classical spin systems is complete for the complexity class Merlin-Arthur.', '1006.4388-2-0-6': 'This provides an interesting connection between the physical behavior of certain many-body spin systems and computational complexity.', '1006.4388-2-1-0': 'The creation of massive digital and deterministic computing systems represents one of the greatest triumphs of the last century.', '1006.4388-2-1-1': 'Increasingly, however, as the components in our information processing devices shrink to atomic scales [CITATION], the two defining characteristics of these systems-that they are digital as opposed to analog, and that they are deterministic as opposed to probabilistic-are beginning to be revealed as approximations that arise from considering macroscopic or mesoscopic physical systems.', '1006.4388-2-1-2': 'As computing elements are made smaller they are increasingly subject to noise and an inability to be perfectly controlled [CITATION].', '1006.4388-2-1-3': 'Even further down this route lie quantum computers [CITATION] where, instead of probabilistic time evolution, the computer is made of elements which function according to the laws of quantum information.', '1006.4388-2-1-4': 'These trends point toward the necessity of studying computing elements in settings where digital and deterministic functionality are not a priori guaranteed [CITATION].', '1006.4388-2-1-5': 'Here we consider a physical method for classical computing wherein a computation is encoded into the ground state of a classical many-body system of spins.', '1006.4388-2-1-6': 'At zero temperature in this model, computation is deterministic and thus we label this model classical ground state spin computation.', '1006.4388-2-1-7': 'The model has many predecessors, include quantum-dot cellular automata [CITATION], the broadcast model on trees [CITATION], quantum ground state computing [CITATION], and universal adiabatic quantum computing [CITATION].', '1006.4388-2-1-8': 'We focus our work here on this model when the system has thermalized at a non-zero temperature.', '1006.4388-2-1-9': 'At non-zero temperature, if we do nothing to compensate for the effects of thermal equilibrium, ground state spin computation fails.', '1006.4388-2-1-10': 'Here we show that it is possible to use the ideas of fault-tolerant classical computing to design ground state spin computers which function deterministically with high probability at non-zero temperatures below a critical temperature.', '1006.4388-2-1-11': 'We achieve this by mapping the thermal ensembles of our computational spin systems onto the distribution of ensembles produced during classical probabilistic computations.', '1006.4388-2-1-12': 'This is the main result of the paper: some classical spin systems in thermal equilibrium can be thought of as enacting classical probabilistic computations spatially across the system.', '1006.4388-2-2-0': 'Finally, our mapping leads naturally to problems which are complete for the complexity class promise Merlin-Arthur [CITATION], a result which can be regarded as a finite temperature version of the Cook-Levin theorem [CITATION] from the theory of computational complexity.', '1006.4388-2-2-1': 'The new computational model we consider thus serves as a bridge between statistical mechanics and the classical computational complexity of probabilistic computation.', '1006.4388-2-2-2': 'We also obtain a computational complexity result concerning the difficulty of identifying when a system admits a mapping to a probabilistic circuit by showing that a restricted version of this problem is This implies that the problem of identifying when a physical system can be seen to be performing a computation is itself computationally intractable.', '1006.4388-2-3-0': 'The outline of our paper is as follows.', '1006.4388-2-3-1': 'In Section [REF] we introduce the model of computing with spins in the ground state using a particular energy function for this system.', '1006.4388-2-3-2': 'In Section [REF] we show that at non-zero temperature the model from the previous section fails to properly compute.', '1006.4388-2-3-3': 'We do this for an extremely simple and previously explored model, but present the result using two different methods, one related to reinterpreting the transfer matrix and the other related to classical circuits applied to simple thermal ensembles.', '1006.4388-2-3-4': 'Generalizing these methods allows us to discuss the energy functions arising in Section [REF] as probabilistic circuits.', '1006.4388-2-3-5': 'This more general formulation we discuss using two different methods in Section [REF].', '1006.4388-2-3-6': 'Motivated by the mappings discovered in Section [REF], we then return to our original model and define the broader class of energy functions consistent with these models in Section [REF].', '1006.4388-2-3-7': 'At this point we also point out how our models differ from classical models of quantum-dot cellular automata.', '1006.4388-2-3-8': 'The more general setting leads us to ask questions about how algorithmically hard it is to decide if a given physical system supports computation in these models.', '1006.4388-2-3-9': 'We show that this problem, even in a very weak form, is and thus likely to be intractable.', '1006.4388-2-3-10': 'In Section [REF] we discuss how to design ground state spin models that, unlike the generic case, do compute fault-tolerantly at non-zero temperature.', '1006.4388-2-3-11': "Our construction is intimately related to the original fault-tolerance construction of von Neumann [CITATION] and we show that von Neumann's threshold for fault-tolerance is related to a critical temperature in our system.", '1006.4388-2-3-12': 'Finally in Section [REF] we discuss how the model we consider is related to the Cook-Levin theorem from computational complexity and show how this leads to certain natural problems about our model being Promise-', '1006.4388-2-4-0': '# Classical Ground State Spin Computing', '1006.4388-2-5-0': 'Here we introduce the model of classical ground state spin computing.', '1006.4388-2-5-1': 'This model, in particular restricted cases, is relevant to a variety of different models considered elsewhere.', '1006.4388-2-5-2': 'For instance, quantum dot cellular automata can be partially modeled by classical ground state spin computing [CITATION].', '1006.4388-2-5-3': 'However the model we consider is more general than these specific instances.', '1006.4388-2-5-4': 'This larger generality comes along with certain unphysical assumptions: our models contain, for example, three-body interactions which are not necessarily easily achievable in a physical device.', '1006.4388-2-5-5': 'Physically we imagine that models such as the one we consider might emerge in certain limits where effective many-body interactions can emerge.', '1006.4388-2-5-6': 'However, irrespective of the physical implementation, the model provides a set of classical many-body interacting systems which can be mapped onto probabilistic classical circuits.', '1006.4388-2-5-7': 'This connects statistical mechanics with classical probabilistic circuits thus bridging computer science and physics in a new and interesting manner.', '1006.4388-2-6-0': 'We further note that a quantum version similar to this model has been studied by Mizel et al [CITATION].', '1006.4388-2-6-1': 'This later model contains significant difference owing to the quantum nature of the computation: in particular the ground state does not contain the computation laid out spatially, but instead exists in a superposition of the computation being carried out spatially.', '1006.4388-2-6-2': 'These quantum models are also connected to universal adiabatic quantum computing schemes [CITATION] and piecewise variations on these schemes [CITATION].', '1006.4388-2-6-3': 'By studying the classical analogy of these schemes, we hope to shed light on how these later quantum methods can be made fault-tolerant.', '1006.4388-2-7-0': '## The Model', '1006.4388-2-8-0': 'A combinatorial circuit, [MATH], is a directed acyclic graph, [MATH] with vertices [MATH] and edge set [MATH], where the internal vertices, [MATH], of the graph are logical gates, [MATH] is the set of all logical gates), and the external vertices, [MATH], are the inputs and outputs to the circuit.', '1006.4388-2-8-1': 'External vertices that have no edges leading to them are inputs and external vertices that have no edges that lead away from them our outputs to the circuit.', '1006.4388-2-8-2': 'If a circuit has [MATH] input vertices and [MATH] output vertices, then to such a circuit we can assign a function [MATH] which results from propagating the input through the logic gates to the output vertices.', '1006.4388-2-8-3': 'Note that at this point we require that fan-outs in our circuit are represented by gates and do not allow circuits which have fan-in gates.', '1006.4388-2-8-4': 'A [MATH]-fan-out gate is the boolean function [MATH] given by [MATH].', '1006.4388-2-9-0': 'Consider the following physical system.', '1006.4388-2-9-1': 'For each edge of a combinatorial circuit associate a single subsystem with only two possible states, [MATH] and [MATH], i.e. associate a bit to every edge.', '1006.4388-2-9-2': 'Consider next a logic gate which has incoming edges labeled by bits [MATH] and outgoing edges labeled by [MATH].', '1006.4388-2-9-3': 'For such a logic gate, [MATH], define the following energy function [EQUATION] where [MATH] is the function the logic gate is computing.', '1006.4388-2-9-4': 'Note that this energy contributes [MATH] when the inputs and outputs follow the rules of the logic gate, but is [MATH] otherwise.', '1006.4388-2-9-5': 'Now define the energy for a particular configuration of the bits associated with edges as the sum over all logic gates of these energy terms: [EQUATION] where each [MATH] acts on the appropriate input and output bits and [MATH] is the collection of bits for the entire system.', '1006.4388-2-9-6': 'This energy is a function of the labels on all of the edges.', '1006.4388-2-9-7': 'Its zero energy configurations are ground states which consist of configurations which correctly compute the logical function corresponding to the circuit [MATH].', '1006.4388-2-9-8': 'Note that at this point the ground state is degenerate: every valid input (and corresponding output) defines a valid zero energy configuration.', '1006.4388-2-9-9': 'We call an energy function constructed in this fashion from a circuit a circuit energy function.', '1006.4388-2-10-0': 'Suppose further that, in addition to imposing energy constraints to enforce logic gates, we also impose energy constraints to fix a particular input to the circuit.', '1006.4388-2-10-1': 'Suppose that the input vertices are [MATH] and we wish to force the input [MATH], where [MATH].', '1006.4388-2-10-2': 'Then to each edge whose bit is labeled [MATH] lead away from vertex [MATH] we can associate an energy term [MATH] if [MATH] and [MATH] otherwise.', '1006.4388-2-10-3': 'Taking a sum over all of these terms for the input vertices (and corresponding edges) we can thus add a term such that the ground state configuration consists now only of the input [MATH] propagated to the output of the circuit, [MATH].', '1006.4388-2-10-4': 'In particular for input [MATH] define the [EQUATION] where [EQUATION]', '1006.4388-2-10-5': 'We call an energy function made up of a circuit energy function plus an input forcing energy function a computed circuit energy function.', '1006.4388-2-11-0': 'The above description of how to take a combinatorial circuit, [MATH], plus its input, [MATH], and converting it into a many-spin energy function is what we term the classical ground state spin computing (CGSSC) model.', '1006.4388-2-11-1': 'Just as in quantum dot cellular automata [CITATION], the computation occurs when the system is in its ground state and is spatially spread out across the device.', '1006.4388-2-11-2': 'Unlike in quantum dot cellular automata, however, this model can implement logic gates by using interactions beyond just pairwise interacting spins (or pseudo-spins in the case of the quantum dot configurations.)', '1006.4388-2-11-3': 'For example, constructing a quantum wire in the CGSSC model will correspond directly to an identical model in the semi-classical limit of quantum dot cellular automata models.', '1006.4388-2-11-4': 'Gates in quantum dot cellular automata, however, are implemented in a way which is not directly analogous to the CGSSC model.', '1006.4388-2-11-5': 'In section [REF] we return to this issue and define a more general set of energy functions wherein our results still hold and compare this with quantum cellular automata models.', '1006.4388-2-12-0': '## Example', '1006.4388-2-13-0': "To be explicit, let's consider an example circuit involving four inputs, a few logical gates, and one output as diagramed in Fig. [REF].", '1006.4388-2-13-1': 'The energy function for the AND gate is then, for example, [EQUATION]', '1006.4388-2-13-2': 'The full circuit energy function is given by [EQUATION]', '1006.4388-2-13-3': 'Suppose that we wish to force the input to be [MATH] and [MATH].', '1006.4388-2-13-4': 'Then we would add the term [EQUATION]', '1006.4388-2-14-0': '# Unprotected Ground State Spin Computing Fails at Finite Temperature', '1006.4388-2-15-0': 'Above we have defined an energy function whose ground state deterministically carries out a circuit.', '1006.4388-2-15-1': 'Having defined this energy model we can now consider the physically important question of what happens to this model when the physical system described by this energy function is in thermal equilibrium at a finite temperature.', '1006.4388-2-15-2': 'Thus we are led to consider the Boltzmann distribution corresponding to a non-zero temperature version of a circuit energy function.', '1006.4388-2-15-3': 'It then makes sense to consider whether the conditional probability of output [MATH] given the forced input [MATH] for this circuit is large enough to distinguish this output from a completely random outcome.', '1006.4388-2-15-4': 'It is easy to see that for at least some circuits [MATH] and inputs [MATH], ground state computation will fail to correctly evaluate the function corresponding to the circuit as the size of the circuit being implemented grows.', '1006.4388-2-15-5': 'This follows directly from examining one of the most basic models in statistical physics, the one-dimensional Ising model [CITATION].', '1006.4388-2-15-6': 'This was pointed out in the context of quantum-dot cellular automata in [CITATION] (see also [CITATION]).', '1006.4388-2-15-7': 'Here we reproduce this argument presenting it in two different forms.', '1006.4388-2-15-8': 'We do this not just to be pedagogical, but also because these two methods will generalize from the one dimensional case to more general circuit computing energy functions.', '1006.4388-2-16-0': 'Consider the circuit computing energy function corresponding to inputing the bit [MATH] into a series of [MATH] identity gates: [EQUATION] where [MATH] represents the negation of [MATH]: [MATH] and [MATH].', '1006.4388-2-16-1': "It is convenient here, and in the sequel, to work with [MATH] valued variables instead of the [MATH] valued [MATH]'s.", '1006.4388-2-16-2': "Thus we will define uppercase spin variables to be [MATH] valued versions of the [MATH]'s via [MATH].", '1006.4388-2-16-3': 'Then the above energy function can be written as [EQUATION] which we recognize as the one dimensional Ising model with ferromagnetic couplings and a boundary term which is a local field.', '1006.4388-2-16-4': 'The ground state of this is simply all [MATH]), i.e. the initial [MATH] has been copied by identity gates down the line.', '1006.4388-2-17-0': 'The thermal ensemble arising from this energy function is given by [EQUATION] where [MATH] is the partition function, [EQUATION] and [MATH] is the inverse temperature.', '1006.4388-2-17-1': 'It is well known that the one dimensional Ising model does not order at finite temperature in the thermodynamic limit [CITATION].', '1006.4388-2-17-2': 'From our perspective, we observe that the system will fail to correctly transmit the [MATH] down the line at finite temperature except in a window of size [MATH] which goes to zero as the system size goes to infinity.', '1006.4388-2-18-0': '## Transfer Matrix Reinterpreted as a Probabilistic Circuit', '1006.4388-2-19-0': 'The standard method for solving this model is the transfer matrix method which we review and then reinterpret.', '1006.4388-2-19-1': 'To the energy function add a term dependent on a variable [MATH] for the last spin: [EQUATION]', '1006.4388-2-19-2': 'If we calculate [MATH] for this energy function, we can then use this to calculate the probability that [MATH] is [MATH] via [MATH] where [EQUATION]', '1006.4388-2-19-3': 'Define the two by two matrices [EQUATION] where [MATH], as well as the column vector [EQUATION] and the row vector, [EQUATION]', '1006.4388-2-19-4': 'Explicitly the partition function [MATH] is [EQUATION]', '1006.4388-2-19-5': 'This can then be written as [EQUATION] where the last equation is a row vector, a matrix product, and a column vector thus giving a scalar.', '1006.4388-2-19-6': 'This is the transfer matrix form of the solution: the partition function is written as a product of "transfer matrices" and, in this case, sandwiched between an "initial" and "final" vector.', '1006.4388-2-20-0': 'Define the following matrices and vectors: [EQUATION]', '1006.4388-2-20-1': 'We can rewrite [MATH] as [EQUATION]', '1006.4388-2-20-2': 'It is at this point that we begin to see our reinterpretation of the transfer matrix method.', '1006.4388-2-20-3': 'In particular we note that [MATH] is a vector of probabilities (that is it sums to unity) and [MATH] is a stochastic matrix (its columns sum to unity).', '1006.4388-2-20-4': 'In other words encoded into [MATH] is a classical preparation of a probabilistic bit, followed by an evolution according to probabilistic gates.', '1006.4388-2-20-5': 'Suppose that we let [MATH] denote the output of this probabilistic circuit.', '1006.4388-2-20-6': 'Then [EQUATION] where the sample is taken from the output of the probabilistic circuit.', '1006.4388-2-20-7': 'Notice that at [MATH], [MATH] is a vector made up of all components equaling [MATH].', '1006.4388-2-20-8': 'In this case, [MATH] since then this term is a sum over all the outputs to the probabilistic circuit.', '1006.4388-2-20-9': 'Further [EQUATION] where [MATH].', '1006.4388-2-20-10': 'Thus [EQUATION]', '1006.4388-2-20-11': 'In other words the expectation value of the last spin in the chain is equal to the expectation value for the output of the circuit starting with [MATH] and then applying the probabilistic gates [MATH].', '1006.4388-2-20-12': 'Note that the combinatorial factors [MATH] have canceled out.', '1006.4388-2-21-0': 'The above description tells us that we can think about the one dimensional Ising spin chain with a forced boundary term as a probabilistic circuit related to our ground state spin computation.', '1006.4388-2-21-1': 'In particular the probabilistic circuit starts with the possibility of an incorrectly initialized boundary.', '1006.4388-2-21-2': 'This is formulated in the preparation probability vector [MATH].', '1006.4388-2-21-3': 'Then with probability [MATH] the ground state spin computation of an identity gate is replaced by a bit flip gate.', '1006.4388-2-21-4': 'The probability of the final spin of the chain being in a particular state is then directly given by the probability that the probabilistic circuit outputs a particular value.', '1006.4388-2-22-0': 'In this model the probability of the ground state properly computing the desired identity circuit is such that the information is washed out at finite temperature.', '1006.4388-2-22-1': 'To see this we must actually calculate [MATH].', '1006.4388-2-22-2': 'This can be done most easily by diagonalizing the gates.', '1006.4388-2-22-3': 'In particular if we define the Hadmard matrix, [EQUATION] then [EQUATION] where [EQUATION]', '1006.4388-2-22-4': 'Thus [EQUATION] which can be reduced to [EQUATION]', '1006.4388-2-22-5': 'From this expression we see that the probability that the final spin is aligned with the input quickly drops to [MATH] as a function of [MATH] (unless [MATH], i.e. in a small window of temperature below [MATH].)', '1006.4388-2-22-6': 'Thus the circuit fails with high probability as we make the chain longer and longer.', '1006.4388-2-22-7': 'This is exactly the result of [CITATION].', '1006.4388-2-22-8': 'In this respect, ground state spin computing in this model is not tolerant to errors arising from working at finite temperature.', '1006.4388-2-22-9': 'Note that we do not consider scaling of the temperature to stay within the small window where the computation is correctly performed to be a fault-tolerant method as it requires unreasonable physical resources as the system size gets larger.', '1006.4388-2-23-0': '## A Second Approach Using Controlled-Not Gates', '1006.4388-2-24-0': 'Having shown in the previous subsection how one can reinterpret the transfer matrix method as a probabilistic circuit for the one dimensional Ising model with forced boundary, we next present a second way to derive this observation.', '1006.4388-2-24-1': 'This method is a direct variation on the method used in [CITATION] to study Ising models on Cayley trees.', '1006.4388-2-25-0': 'On the [MATH] spins, define the controlled-not on positions [MATH] and [MATH] as the function, [MATH], which maps [MATH] to [MATH] via [EQUATION]', '1006.4388-2-25-1': 'We call this a controlled-not because it functions as a deterministic gate which, controlled on the spin [MATH] either does nothing to the spin in position [MATH] (if [MATH]), or flips the spin in position [MATH] (if [MATH].)', '1006.4388-2-25-2': "Further define the function which is the composition of controlled-not's starting at the first spin and working forward (note that [MATH]): [EQUATION]", '1006.4388-2-25-3': 'Notice that if we start with spin [MATH] in the state [MATH] and all of the other spins [MATH] in [MATH], then the action of [MATH] is to copy [MATH] down the line: [MATH].', '1006.4388-2-25-4': 'In this sense [MATH] is the operation of applying identity gates as information is propagated down the spin chain.', '1006.4388-2-25-5': 'Note that the [MATH] are bijections as is [MATH] and that [MATH] is its own inverse.', '1006.4388-2-25-6': 'Thus [EQUATION]', '1006.4388-2-25-7': 'Explicitly, [EQUATION] and [EQUATION]', '1006.4388-2-25-8': 'Consider the energy function on [MATH] spins: [EQUATION]', '1006.4388-2-25-9': 'Then clearly a system in the thermal state described by this energy function has each spin in [MATH] with probability [MATH] and [MATH] with probability [MATH].', '1006.4388-2-25-10': 'Suppose that we apply the [MATH] function to this system.', '1006.4388-2-25-11': 'That is consider the above system in its thermal ensemble and consider physically applying the controlled-not gates.', '1006.4388-2-25-12': 'Then certainly this can be thought of as a system in which the first bit is prepared in [MATH] with probability [MATH], with probability [MATH] the controled-nots successfully copy the bit down the line, and with probability [MATH] the output of the controlled-not is flipped.', '1006.4388-2-25-13': 'This is exactly the probabilistic circuit as described in the previous section.', '1006.4388-2-26-0': 'How is this related to our original [MATH]?', '1006.4388-2-26-1': 'Let the unprimed variables be the spins after [MATH] has been applied to the system.', '1006.4388-2-26-2': 'Then [EQUATION] where [EQUATION]', '1006.4388-2-26-3': 'Therefore [EQUATION]', '1006.4388-2-26-4': 'This implies that the probability distribution produced by taking the thermal ensemble for [MATH] and applying [MATH] to the system is equivalent to a thermal ensemble with a new energy function [EQUATION]', '1006.4388-2-26-5': 'A quick calculation using Eq. ([REF]) finds that [EQUATION] which we see is equal to our original energy function [MATH].', '1006.4388-2-26-6': 'Reversing the above argument we thus see that the energy function [MATH] can equally well be thought of as the ensemble corresponding to [MATH] followed by the application of the mapping [MATH].', '1006.4388-2-26-7': 'Further this later ensemble and computation has a clear interpretation in terms of a probabilistic initialization followed by probabilistic gates: when we apply controlled-nots with error target values then this causes a bit flip error in copying the information down the line.', '1006.4388-2-26-8': 'Thus we see that we can derive the same probabilistic gate expression for the one-dimensional Ising model as that which arose by reinterpreting the transfer matrix as a probabilistic circuit.', '1006.4388-2-27-0': '# The Partition Function as a Probabilistic Circuit', '1006.4388-2-28-0': 'We now turn to the more general setting of a circuit energy function for a generic circuit [MATH].', '1006.4388-2-28-1': 'We begin by focusing on circuit energy functions without a forced input.', '1006.4388-2-29-0': 'Let [MATH] be the energy function for the circuit [MATH].', '1006.4388-2-29-1': 'For each logic gate, [MATH], with inputs [MATH] and output [MATH], define the following tensor, [EQUATION] where [MATH] as before.', '1006.4388-2-29-2': 'Given this definition, we can now write the partition function [MATH] as the contraction of a tensor network [CITATION] plus a sum over all inputs and outputs.', '1006.4388-2-29-3': 'What is a tensor network?', '1006.4388-2-29-4': 'Take a graph and map a tensor to every vertex in such a manner that the tensor has an index for every edge of the relevant vertex.', '1006.4388-2-29-5': 'One can then perform a sum over two tensor indices connected via edges in the graph.', '1006.4388-2-29-6': 'A tensor network is thus a graph where vertices are tensors, and edges are tensor indices.', '1006.4388-2-29-7': 'Free edges are indices which have not been summed over and connected edges are ones for which a sum has been performed.', '1006.4388-2-29-8': 'The entire network (graph) then represents itself a tensor with a number of unsummed indices equal to the number of free edges.', '1006.4388-2-30-0': 'Given the tensors [MATH] and the graph given by the circuit (which will have free edges for the inputs and outputs), we can build a tensor network out of these [MATH]s.', '1006.4388-2-30-1': 'This tensor network will itself be a tensor network having indices for the inputs and outputs.', '1006.4388-2-30-2': 'Call this tensor [MATH] for inputs [MATH] and output [MATH].', '1006.4388-2-30-3': 'Then it is easy to see that the partition function is equal to [EQUATION]', '1006.4388-2-30-4': 'In other words, the partition function is given by the sum over all inputs, and all outputs of the value of the tensor network.', '1006.4388-2-31-0': 'What does this have to do with probabilistic circuits?', '1006.4388-2-31-1': 'Well now instead of using the tensors [MATH], instead use [EQUATION] or in other words [EQUATION]', '1006.4388-2-31-2': 'It is easy to check that this defines a stochastic matrix, i.e. a probabilistic gate.', '1006.4388-2-31-3': 'Indeed it is a probabilistic gate which performs the correct function of the gate with probability [MATH] and randomly flips the output to one of the other outputs with equal probability [MATH].', '1006.4388-2-32-0': 'Notice, importantly, that the difference between [MATH] and [MATH] is a constant which depends on only on the number of output bits.', '1006.4388-2-32-1': 'So suppose we consider the tensor network for the circuit made now with [MATH]s and not [MATH]s.', '1006.4388-2-32-2': 'Then we can write the partition function as [EQUATION] where [MATH] is the simple combinatorial factor [EQUATION] and [MATH] is the number of outputs of the [MATH]th gate.', '1006.4388-2-32-3': 'Now examine [MATH].', '1006.4388-2-32-4': 'This is nothing more than the probability that we get output [MATH] given that we had input [MATH] to the probabilistic circuit with probabilistic gates [MATH].', '1006.4388-2-32-5': 'In other words, we can express the partition function as [EQUATION]', '1006.4388-2-32-6': 'Thus the partition function is really hiding a probabilistic computation.', '1006.4388-2-32-7': 'Notice further that we can explicitly calculate that partition function because [MATH], [EQUATION] where [MATH] is the number of input bits.', '1006.4388-2-33-0': 'Now we turn to the case where we force the input to the circuit.', '1006.4388-2-33-1': 'Let [MATH] be the inputs to the circuit [MATH] and [MATH] be the energy function for the forced computation.', '1006.4388-2-33-2': 'Define the modified energy function [EQUATION] where [EQUATION] with [EQUATION] and the sum is over the output bits, [MATH].', '1006.4388-2-33-3': 'Call the partition function associated with this setup [MATH].', '1006.4388-2-34-0': 'Note that [MATH] is equal to the partition function for energy function with a forced input [MATH], [MATH].', '1006.4388-2-34-1': 'Define [EQUATION] where [EQUATION]', '1006.4388-2-34-2': 'Then it is easy to see that [EQUATION] where [EQUATION]', '1006.4388-2-34-3': 'Thus we see that we can interpret [MATH] as a sum over a probabilistic circuit, but now with a probabilistic input corresponding to each bit being flipped with a probability [MATH].', '1006.4388-2-34-4': 'Again, because we are summing over all outputs, we can explicitly perform this sum [EQUATION]', '1006.4388-2-34-5': 'We are interested in computing the probability that the thermal ensemble for the forced input circuit has output given by [MATH].', '1006.4388-2-34-6': 'This can be calculated by [EQUATION]', '1006.4388-2-34-7': 'Note that [EQUATION] so that the partial derivatives may be evaluated only over the first term.', '1006.4388-2-34-8': 'There we find that [EQUATION]', '1006.4388-2-34-9': 'Thus we find that [EQUATION]', '1006.4388-2-34-10': 'This is the main result of this section: the probability of the ground state spin computing model output bits is equal to the probability of running the probabilistic circuit described by [MATH] on inputs described by input probability distribution [MATH].', '1006.4388-2-34-11': 'The gates in [MATH] are simply noisy versions of the deterministic gates which fail with a fixed probability related to the temperature.', '1006.4388-2-35-0': 'This result is interesting in two manners.', '1006.4388-2-35-1': 'First it allows one to use techniques designed for probabilistically failing gates to be ported over to our model.', '1006.4388-2-35-2': 'For instance this will allow us to use methods for constructing fault-tolerant circuits.', '1006.4388-2-35-3': 'It is also interesting in that it shows that certain many-body quantum systems can be efficiently simulated.', '1006.4388-2-35-4': 'Computed circuit energy functions can be simulated by directly implementing the probabilistic gate that these systems correspond to.', '1006.4388-2-35-5': 'Here we have shown that output bits for the circuit energy function are related to the output bits for the probabilistic computation.', '1006.4388-2-35-6': 'A straightforward generalization shows that this is also true of other bits in the system.', '1006.4388-2-35-7': 'We can efficiently simulate systems with computed circuit energy functions by executing the probabilistic computation these energy functions represent.', '1006.4388-2-36-0': '# The Gate Model Derivation', '1006.4388-2-37-0': 'Next let us turn to the same analysis as the prior subsection, but now using the gate trick as we did for the one-dimensional model in Section [REF].', '1006.4388-2-37-1': 'To do this we must first define the equivalent of the controlled-not gate.', '1006.4388-2-37-2': 'Consider a logical gate [MATH] with input [MATH] and output [MATH] which computes the function [MATH].', '1006.4388-2-37-3': 'Define the function [MATH] from [MATH] to [MATH] as [EQUATION]', '1006.4388-2-37-4': 'This function thus computes the function on input [MATH], uncomputes the function on input [MATH], and does nothing otherwise.', '1006.4388-2-37-5': 'Note that [MATH] is a bijection and is self-inverse, [MATH].', '1006.4388-2-37-6': 'Defining an order to gates in our circuit such that the circuit computes properly, [MATH] first, [MATH] second, etc.', '1006.4388-2-37-7': 'Then we can define the function on all our bits of [EQUATION]', '1006.4388-2-37-8': 'Suppose we initially start with input bits initialized to an input [MATH] and all other bits initialized to [MATH].', '1006.4388-2-37-9': 'Then if we apply [MATH] to these bits we will place the result of the computation in their appropriate locations across the circuit.', '1006.4388-2-38-0': 'Next define an energy function on all of our system.', '1006.4388-2-38-1': 'This is most easily defined in terms of the inputs to the full circuit and the outputs to the gates [MATH], [EQUATION] and [EQUATION]', '1006.4388-2-38-2': 'This energy function thus assigns for non-global inputs an energy penalty for output to gates being anything different that [MATH].', '1006.4388-2-38-3': 'For global inputs it adds a penalty for every input which is not the correct input from [MATH].', '1006.4388-2-38-4': 'The thermal ensemble resulting from this system is then trivially described: the input qubits are in the correct input [MATH] with probability [MATH] and the internal output bits of a gate are grouped together and have a probability [MATH] being all zeros and probability [MATH] to be anything else.', '1006.4388-2-39-0': 'Imagine applying [MATH] to the state described by the ensemble for [MATH].', '1006.4388-2-39-1': 'Clearly the ground state of [MATH] is such that the circuit [MATH] will be correctly computed.', '1006.4388-2-39-2': 'Further, because the ensemble related to [MATH] has erred inputs, the resulting probability distribution will have erred inputs.', '1006.4388-2-39-3': 'Finally the gates will function properly only when the output was properly initialized to all [MATH]s.', '1006.4388-2-39-4': 'This occurs with probability [MATH] for logic gate [MATH] and otherwise the gate fails by an equally probable error state with probability [MATH].', '1006.4388-2-39-5': 'In other words the ensemble is exactly the one describing the probabilistic circuit in Section [REF].', '1006.4388-2-39-6': 'Let us now show that this corresponds to the thermal ensemble of our original [MATH] construction.', '1006.4388-2-40-0': 'As in Section [REF] the trick here is to express the probability distribution for the ensemble after applying [MATH] as [EQUATION] where [EQUATION] such that [EQUATION]', '1006.4388-2-40-1': 'Thus the probability distribution resulting from after the application of [MATH] to the [MATH] thermal ensemble is equal to the probability distribution arising from [MATH].', '1006.4388-2-41-0': 'We will now show the [MATH] is exactly the energy function we would define for a ground state spin computing construction for our circuit [MATH].', '1006.4388-2-41-1': 'Recall that [EQUATION]', '1006.4388-2-41-2': 'Note that in evaluating [MATH], which is a sum of terms, we can evaluate the action action of [MATH] on each of these terms separately and then re-sum.', '1006.4388-2-41-3': 'Thus we first examine [EQUATION]', '1006.4388-2-41-4': 'This is because the inputs to the circuit commute through the individual gate elements [MATH].', '1006.4388-2-41-5': 'Indeed this observation along with the order of [MATH] allows one to derive the term arising from each [MATH] term independently.', '1006.4388-2-41-6': 'In particular the only term which is not [MATH] for this energy function is [EQUATION]', '1006.4388-2-41-7': 'But this simply means that [EQUATION] only when [MATH] and is [MATH] otherwise.', '1006.4388-2-41-8': 'This is just our normal definition of the circuit energy function.', '1006.4388-2-42-0': 'Thus we see that using the [MATH] construction one can construct a probabilistic circuit from the thermal ensemble for [MATH] which is the probabilistic circuit described in Section [REF].', '1006.4388-2-42-1': 'Further we have shown that this can be thought of as arising from the original energy function [MATH].', '1006.4388-2-42-2': 'This provides an alternative derivation of our main result.', '1006.4388-2-43-0': '# Generalized Classical Ground State Spin Computation Models', '1006.4388-2-44-0': 'Having shown that classical ground state spin computing at non-zero temperature can be mapped to probabilistic circuits it is interesting to return to our original model and consider what generalizations of the energy function allow for this mapping to occur.', '1006.4388-2-44-1': 'The crucial assumption during our derivation of the probabilistic re-interpretation of the partition function was that we could convert the tensors [MATH] of Eq. ([REF]) to tensors [MATH] of Eq. ([REF]) in such a way that [MATH] could be interpreted as a probablistic gate.', '1006.4388-2-44-2': 'This requires (1) the division of tensor indices into input and output indices and (2) for each input to the circuit the normalization to make the outputs probabilities (positive and sum to unity) is the same.', '1006.4388-2-45-0': 'Let us begin by showing that not all tensors and divisions into input and output vertices can result in a probabilistic interpretation.', '1006.4388-2-45-1': 'This can be demonstrated with a simple example on two bits: [EQUATION]', '1006.4388-2-45-2': 'If we chose [MATH] as an input and [MATH] as an output, this gives rise to the tensor [EQUATION]', '1006.4388-2-45-3': 'In order that the first column of this matrix sum to unity we must divide by [MATH].', '1006.4388-2-45-4': 'However when we do this the second column will not sum to [MATH].', '1006.4388-2-45-5': 'A similar argument occurs if we had chosen [MATH] as the output and [MATH] as the input.', '1006.4388-2-45-6': 'Thus there is no way to scale [MATH] in such a way that it can be interpreted as a probabilistic gate.', '1006.4388-2-46-0': 'A more relevant example of a failure for a model to not be amenable to our scaling arising in quantum-dot cellular automata models.', '1006.4388-2-46-1': 'In quantum dot cellular automata models bistable quantum dots are engineered in such a way that the ground state enacts a classical computation as in our model.', '1006.4388-2-46-2': 'Most of the treatments of these models deal with the quantum mechanics of these devices [CITATION].', '1006.4388-2-46-3': 'One can, however, build a classical model of these systems as was done, for example, by Wang in [CITATION].', '1006.4388-2-46-4': 'In this approximation one derives a classical statistical mechanical many spin system as in our model.', '1006.4388-2-46-5': 'When one does this for a quantum-dot cellular automata wire one ends up with exactly the model we can consider above in Section [REF].', '1006.4388-2-46-6': 'However when one considers this for the majority gate constructions (see [CITATION]) one obtains a model for which our technique cannot be applied.', '1006.4388-2-46-7': 'In particular the majority gate that they propose has a classical energy function which is [EQUATION]', '1006.4388-2-46-8': 'Here [MATH] are the input spins and [MATH] is the output spin.', '1006.4388-2-46-9': '(Note that we have used one less spin than in the traditional presentation, but this does not affect our conclusions as one can think of the extra spin as being connected to the central spin by an identity gate.)', '1006.4388-2-46-10': 'Note that the lowest energy configuration for a given input correctly computes the majority function.', '1006.4388-2-46-11': 'That is if we fix [MATH], [MATH], and [MATH], then the lowest energy value of [MATH] is given by the majority of the inputs, [MATH].', '1006.4388-2-46-12': 'Note, however, that if we sum over all outputs for input [MATH] [EQUATION] but if one uses input [MATH] and [MATH] one obtains [EQUATION]', '1006.4388-2-46-13': 'Thus we cannot properly globally normalize the entire tensor in such a way as to to obtain a probabilistic gate.', '1006.4388-2-47-0': 'However, the same two-spin energy function can be used to describe a fanout gate, in contrast to the many-spin interaction used in Eq. ([REF]), [EQUATION]', '1006.4388-2-47-1': 'Here [MATH] is the input and the other spins are the output.', '1006.4388-2-47-2': 'By changing the direction of the circuit, we can show that this element can be represented as a probabilistic gate.', '1006.4388-2-47-3': 'I.e. the majority gate used in quantum dot cellular automata can not be mapped to a probabilistic circuit, but reversing the role of the inputs and outputs, this gate can be used to construct a fan-out gate.', '1006.4388-2-47-4': 'To see this note that, as for our previous fanout model, the ground state of this system, where the computation is performed, is degenerate.', '1006.4388-2-47-5': 'However now, as opposed to a single excited state there are multiple excited states.', '1006.4388-2-47-6': 'For example if we consider a fan-out to two spins we obtain an [MATH] as given in Table [REF].', '1006.4388-2-48-0': 'From this table we see that while the outputs that are in error have different Boltzman factors, we see that the sum over the output spins is indeed independent of the output.', '1006.4388-2-48-1': 'Indeed when we interpret this as a probabilistic gate by dividing out by [MATH], then we see that this is a probabilistic gate which, instead of having a global failure where all possible output failures have equal probability, the fan-out spins have independent error probabilities.', '1006.4388-2-48-2': 'In other words it is as if the fan-out occurred and then with probability [MATH] each of the outputs is independently flipped.', '1006.4388-2-48-3': 'We will use this fan-out gate in the next section when we discuss fault-tolerance.', '1006.4388-2-49-0': 'What is a necessary and sufficient condition for an energy function to allow for a proper normalization to a probabilistic gate?', '1006.4388-2-49-1': 'If we insist that this work for all temperatures, it is easy to give this answer.', '1006.4388-2-49-2': 'Let [MATH] denote the energy function for a logic gate with inputs [MATH] and output [MATH].', '1006.4388-2-49-3': 'Let [MATH] denote the set of energies for a given input.', '1006.4388-2-49-4': "Then the necessary and sufficient condition is that the [MATH]'s must be permutations of each other.", '1006.4388-2-49-5': 'The sufficiency of this condition is obvious.', '1006.4388-2-49-6': 'Necessity follows from noting that we are requiring this to hold for all temperatures and thus the sum over outputs of the [MATH] must be a rearrangement of this sum and therefore a permutation of the output energies.', '1006.4388-2-49-7': 'Note that this condition implies that the ground states of a logic gate energy function must be degenerate for all possible inputs (but note that this alone is not sufficient for the condition to hold.)', '1006.4388-2-50-0': '## Computational Complexity of Identifying Probabilistic Circuits in Thermal Systems', '1006.4388-2-51-0': 'Having established a necessary and sufficient condition for the possibility of scaling the [MATH] tensors as probabilistic gates, a natural follow up question is how one can determine whether it is possible to take an energy function [MATH] and determine whether or not it can be interpreted as a probabilistic circuit.', '1006.4388-2-51-1': 'Here we will show that this problem is computationally intractable, even when we are given information about the breakdown of [MATH] into terms which we wish to interpret as gates.', '1006.4388-2-52-0': 'Suppose that we are told the [MATH] is made up of a sum over logic gates and preparations [EQUATION] i.e. we are given a specification of the [MATH] in terms of a constant number of bits and we are told that each of these [MATH] will correspond to a logic gate or a preparation of initial inputs.', '1006.4388-2-52-1': 'We can then ask: can we interpret this as a probabilistic circuit with or without prepared inputs?', '1006.4388-2-52-2': 'Every [MATH] involves some inputs and output bits, but in general we will not be told which of these are inputs and which are outputs.', '1006.4388-2-52-3': 'In general an [MATH] may thus have many different tuples of input bits and output bits such that we may interpret [MATH] as a probabilistic gate by proper global normalization.', '1006.4388-2-52-4': 'For example gates which have doubly stochastic [MATH] tensors may be run either backwards or forwards (a doubly stochastic matrix is a stochastic matrix whose rows and columns sum to [MATH].)', '1006.4388-2-52-5': 'Another example is given by a fan-out gate in our original energy function model [EQUATION]', '1006.4388-2-52-6': 'In this case we may interpret the [MATH] arising from this energy function as having input [MATH] and outputs [MATH] and [MATH], input [MATH] and outputs [MATH] and [MATH], or input [MATH] and outputs [MATH] and [MATH].', '1006.4388-2-52-7': 'Contrary to this, suppose that the energy function is that of the one-to-two fanout described in Eq. ([REF]), [EQUATION]', '1006.4388-2-52-8': 'The [MATH] for this energy function has only one possible orientation between inputs and output: [MATH] is an input and [MATH] and [MATH] are outputs.', '1006.4388-2-52-9': 'This in general will lead to the following question: given the possible input and outputs tuples for the [MATH]s is it possible to assign these in such a way as to interpret the resulting total tensor as a probabilistic circuit.', '1006.4388-2-52-10': 'We will now show that this problem is [MATH]-complete and is therefore (unless [MATH]), in general, intractable.', '1006.4388-2-53-0': 'To see that the problem is [MATH]-complete proceed as follows.', '1006.4388-2-53-1': 'Assign to each bit of our physical system a boolean variable, [MATH].', '1006.4388-2-53-2': 'Assume that a given logical energy function [MATH] is involved in only a constant number of bits.', '1006.4388-2-53-3': 'Since [MATH] is involved in a constant number of bits we can explicitly calculate all of the sets of inputs and output bits for which [MATH] can be globally normalized so as to make it a probabilistic gate.', '1006.4388-2-53-4': 'The variables [MATH]s are going to represent whether the corresponding bit is associated is an input or and output.', '1006.4388-2-53-5': 'In particular, TRUE, will represent that the bit is an input and FALSE will represent that the bit is an output.', '1006.4388-2-53-6': 'Thus from the sets of input and output bits for which one can consistently label inputs to [MATH] we can construct a boolean function on the relevant bits such that this function is true if and only if a valid assignment of boolean variables respects a possible input and outputs.', '1006.4388-2-53-7': 'Thus, for example, for [MATH] above in Eq. ([REF]), the boolean function would be [EQUATION]', '1006.4388-2-53-8': 'Constructing one such boolean function for each logic gate we can then construct a boolean function on all bits [MATH] which is the conjunction (logical and) of all of the gate boolean functions.', '1006.4388-2-53-9': 'Thus we see that by calculating for each [MATH] the possible combinations of inputs and outputs that have an allowed interpretation in terms of a probabilistic circuit, we have a one-to-one mapping of the problem to the problem of deciding whether there is a satisfying assignment for the boolean function made up of the conjunction of logical terms with only a constant number of involved bits.', '1006.4388-2-53-10': 'In order to demonstrate that the problem is we need now only show that the boolean function we can construct in this manner are general enough so as to yield a version of the satisfaction problem which is In particular it is clear that this problem is in [MATH], since we can evaluate in polynomial time whether a given assignment of inputs and outputs can be interpreted as a probabilistic circuit.', '1006.4388-2-53-11': 'To demonstrate [MATH]-completeness we need to show that it is at least as hard as another [MATH]-complete problem.', '1006.4388-2-54-0': 'We will show that the problem is [MATH]-complete by showing that some instance of the problem can be mapped to the one-in-three satisfiability problem of Schaefer [CITATION] (now sometimes called the monotone one-in-three satisfiability problem.)', '1006.4388-2-54-1': 'Define [MATH] as the boolean function which is true iff exactly one of [MATH] is true.', '1006.4388-2-54-2': 'Notice that this is exactly the boolean formula of Eq. ([REF]) (with [MATH], [MATH], and [MATH].)', '1006.4388-2-54-3': 'In the one-in-three satisfiability problem one is given a boolean function which is the conjunction (logical and) of clauses each of which are of the form [MATH] for a choice of variables [MATH] in the problem.', '1006.4388-2-54-4': 'Schaefer proved that the one-in-three satisfiability problem is Actually there is a slight subtlety here as the problem Schaefer considers must allow repeated variables.', '1006.4388-2-54-5': 'We will now show that it is possible to take a one-in-three satisfiability problem and convert it into an instance of the problem of spotting whether a given energy function can support interpretation as a classical probabilistic circuit.', '1006.4388-2-55-0': 'As mentioned above the one-in-three satisfiability problem has as input a conjunction of boolean functions which are all of the form [MATH], and where some boolean variables can be repeated, [EQUATION] where [MATH] and none of the [MATH], [MATH], and [MATH] are the same variable for fixed [MATH].', '1006.4388-2-55-1': 'Let [MATH] denote the set of [MATH] variables that are [MATH].', '1006.4388-2-55-2': 'We will define an energy function on [MATH] spins which, when converted into the problem of trying to assign consistent inputs and outputs will yield exactly the boolean function [MATH] of Eq. ([REF]).', '1006.4388-2-55-3': 'Call the first [MATH] spins [MATH] and the last [MATH] spins [MATH].', '1006.4388-2-55-4': 'First define the energy function [EQUATION] where [MATH] is the fan-out energy function defined in Eq. ([REF]).', '1006.4388-2-55-5': 'When one converts this into a logical expression for whether the spins are inputs or outputs, one gets exactly the one-in-three logical clause of Eq. ([REF]).', '1006.4388-2-56-0': 'However, at this point, the variables that will arise in these clauses are all different.', '1006.4388-2-56-1': 'This is what the other [MATH] spins are for.', '1006.4388-2-56-2': 'Define the following energy function for the "identity" gate: [EQUATION]', '1006.4388-2-56-3': 'This energy function corresponds to a logical reversible gate which acts as identity from the [MATH] spins to the [MATH] spins.', '1006.4388-2-56-4': 'This energy function has the property that the [MATH] variables are all either inputs or all outputs (and the [MATH] variables are also all either inputs or outputs, opposite to the [MATH] variable assignment.)', '1006.4388-2-56-5': 'Thus we can use gates of this form to effectively make the spins of the [MATH] gates constructed above the same variable.', '1006.4388-2-56-6': 'In particular define the following energy function [EQUATION] where [MATH] and [MATH] is the [MATH]th element of [MATH] translated over to a [MATH] variable, i.e. [MATH] corresponds to [MATH], [MATH] corresponds to [MATH] and [MATH] corresponds to [MATH].', '1006.4388-2-57-0': 'Now consider the energy function [EQUATION] and stipulate that every term in the sum of [MATH] will correspond to one gate and every term in [MATH] will correspond to one gate.', '1006.4388-2-57-1': 'Then if we convert this problem over into a boolean satisfaction problem as describe above, where each spin is a boolean variable representing whether the spin is an input or an output, we will obtain a boolean expression that is equivalent to the problem described in Eq. ([REF]).', '1006.4388-2-57-2': 'Thus we have shown that we can, for a given one-in-three satisfaction problem, Eq. ([REF]) construct an energy function, along with labeling of the terms in this function corresponding to different gates, such that determining whether this energy function can be reinterpreted as a probabilistic computation is equivalent to the original one-in-three satisfaction problem.', '1006.4388-2-57-3': 'This implies that the problem of determining whether such probabilistic computations exist is at least as hard as this [MATH]-complete problem.', '1006.4388-2-57-4': 'Combined with the fact the problem is in [MATH] this means that the problem of deciding whether a given energy function can be thought of as enacting a probabilistic circuit is [MATH]-complete.', '1006.4388-2-58-0': 'Thus we have shown that the problem of determining a many-spin statistical mechanics system enacts a ground state spin computation, even given the decomposition of this system into logical gates, is We note in passing that this result is of perhaps some philosophical significance: determining whether a system can be thought of as a computer is shown to be computationally intractable (assuming [MATH]).', '1006.4388-2-59-0': '## Models that have a Probabilistic Gate Interpretation', '1006.4388-2-60-0': 'So far we have considered spin models in a very general setting unrelated to physical theories.', '1006.4388-2-60-1': 'We point out, here, however, that there are a variety of models that have arisen in physics which admit such an interpretation.', '1006.4388-2-60-2': 'We have already seen that that one dimensional Ising model admits an interpretation as a series of identity gates which err with some probability.', '1006.4388-2-60-3': 'Here we point out some other models which admit such interpretations.', '1006.4388-2-61-0': '### Ising Models on Trees', '1006.4388-2-62-0': 'A tree is a connected graph, [MATH], with vertex set [MATH], edge set [MATH], and no cycles.', '1006.4388-2-62-1': 'Assign to each spin a vertex, [MATH] for [MATH].', '1006.4388-2-62-2': 'To each edge [MATH] assign an non-zero energy for the Ising coupling, [MATH].', '1006.4388-2-62-3': 'Then the Ising model on the tree has an energy function given by [EQUATION]', '1006.4388-2-62-4': 'Fix a root vertex [MATH].', '1006.4388-2-62-5': 'First consider the model at zero temperature.', '1006.4388-2-62-6': 'At [MATH], following our discussion of the fan-out in Section [REF], it is easy to see that the ground state of this model can be thought of as a series of fan-out gates originating from the root vertex [MATH] with possible bit flips applied to the fan-out depending on the sign of [MATH].', '1006.4388-2-62-7': 'At [MATH] we see that, as in the discussion of the fan-out gate, each of the individual fan-out wires (with possible bit flips), will fail independently with a probability of [EQUATION]', '1006.4388-2-62-8': 'Note that a failure for an antiferromagnetic coupling (bit flip) corresponds to an identity gate.', '1006.4388-2-62-9': 'Ising models on trees have been studied in a variety of settings [CITATION] and in fact the interpretation of this model in terms of a probabilistic circuit has been used extensively in the computer science literature, where the model goes under the name of broadcasting [CITATION].', '1006.4388-2-63-0': '### [MATH] Lattice Pure Gauge Theories', '1006.4388-2-64-0': 'Perhaps slightly more interesting that Ising models on trees is the fact that [MATH] lattice gauge theories [CITATION] can be cast as a probabilistic gate circuits.', '1006.4388-2-64-1': 'For definiteness, first consider a (pure) [MATH] lattice gauge theory on a two dimensional square lattice with closed boundaries.', '1006.4388-2-64-2': 'In this model one places spins on the edges of the lattice and defines the energy function [EQUATION] where [MATH] is the set of all plaquettes of the square lattice, and [MATH] are the spins surrounding plaquette [MATH].', '1006.4388-2-64-3': 'Consider an individual term in this sum, and assume for simplicity that for all plaquettes, [MATH] that [MATH].', '1006.4388-2-64-4': 'We can view the individual term [MATH] as a gate element with either [MATH], [MATH], [MATH], or [MATH] inputs and [MATH], [MATH], [MATH], and [MATH] outputs respectively.', '1006.4388-2-64-5': 'For example if we view it as a gate with [MATH] inputs and [MATH] outputs, then, in its ground state this corresponds to a probabilistic gate which, with equal probability, changes the input to a set of spins with the same parity of [MATH]s as the input.', '1006.4388-2-64-6': 'In other words the logical evolution is [EQUATION]', '1006.4388-2-64-7': 'The gate with [MATH] inputs and [MATH] outputs is deterministic and is given by [MATH].', '1006.4388-2-64-8': 'The gate with [MATH] inputs corresponds to a preparation of [MATH] spins which have an even number of [MATH] spins and this state is prepared with equal probability.', '1006.4388-2-64-9': 'The gate with [MATH] input and [MATH] outputs is a probabilistic gate which outputs an even number of [MATH]s if the input is [MATH] all such possibilities occurring with equal probability.', '1006.4388-2-64-10': 'Similarly it outputs an odd number of [MATH]s if the input is [MATH] all such possibilities occurring with equal probability.', '1006.4388-2-64-11': 'By picking boundary terms to serve as input and output bits, it is clear that it is possible to order the energy terms in the above sum in such a way that we can interpret the ground state as a probabilistic cellular automata like model using the above gates.', '1006.4388-2-64-12': 'At finite gate these probabilistic gates all fail with some probability, where failure results in an equally likely failed output for the gate.', '1006.4388-2-64-13': 'While we have define this for the two dimensional square lattice, it is clear that the above construction can result in a probabilistic circuit for a far greater number of lattices, the main criteria being that there exists a way to label inputs and outputs in such a way that a proper circuit is constructed.', '1006.4388-2-65-0': '# Making Classical Ground State Spin Computation Robust at Finite Temperature', '1006.4388-2-66-0': 'Having shown how the computed circuit energy functions give rise to thermal ensembles which can be interpreted as probabilistic circuits with probabilistic inputs, we now discuss how it is possible to make the model of classical ground state spin computation fault-tolerant at finite temperature.', '1006.4388-2-66-1': 'Indeed this is now rather simple given that we have a mapping between the behavior of these systems at finite temperature and probabilistic circuits which have gates and preparations failing with a fixed probability.', '1006.4388-2-66-2': 'We will need, however, to use the more general class of energy functions considered in Section [REF].', '1006.4388-2-67-0': 'The general theory of computing in the presence of gates which fail goes back at least as far as the work of von Neumann [CITATION].', '1006.4388-2-67-1': 'Von Neumann considered a model in which each logic gate failed with exactly the probability [MATH].', '1006.4388-2-67-2': 'In order to get this model to compute reliably, von Neumann used two major techniques.', '1006.4388-2-67-3': 'The first techniques is that one needs to suitably encode information: in order to do this one encodes a single bit across a bundle of wires in a logic circuit and interprets this as [MATH] (or [MATH]) when a majority of the wires are in the [MATH] (or [MATH].)', '1006.4388-2-67-4': 'Given such an encoding it is then easy to compute in a parallel manner such that the effect of gates failing is simply to decrease the proportion of [MATH]s (or [MATH]s) in an encoded [MATH] (or [MATH].)', '1006.4388-2-67-5': 'However, the fact that the gates fail does cause the ratio of correct bits in an encoded bundle of bits to decrease.', '1006.4388-2-67-6': 'In order to deal with this von Neumann used a second technique whereby faulty gates were used to perform error correction.', '1006.4388-2-67-7': "Actually this portion of von Neumann's analysis is not quite satisfactory, as it requires the use of a random permutation.", '1006.4388-2-67-8': 'However Pippenger [CITATION] has shown that one can de-randomize this construction (one can obtain reasonable parameters for this method by using the expanders defined in [CITATION]).', '1006.4388-2-67-9': 'The end result of this is that one can show that if one wishes to compute a logical circuit of size [MATH] to an accuracy [MATH] (that is, have the circuit fail with probability [MATH]) with gates that fail at [MATH] for some fixed threshold [MATH], one can do this using a circuit with the unreliable gates which is only a [MATH] larger, where [MATH] is a fixed constant.', '1006.4388-2-67-10': 'This effectively means that one can create for all effective purposes reliable logic gates out unreliable logic gates with an overhead which is logarithmic in the inverse of the error desired.', '1006.4388-2-68-0': "Having described von Neumann's construction one can now see how it is possible to make classical ground state spin computing fault-tolerant.", '1006.4388-2-68-1': 'In particular we have shown above how the thermal ensemble for such systems gives rise to probabilistic gates.', '1006.4388-2-68-2': 'For gates constructed using energy functions like Eq. ([REF]) we have shown that the thermal ensemble arising for these functions is equivalent to a circuit in which gates fail with probability [MATH] if the gate has [MATH] output bits.', '1006.4388-2-68-3': 'For [MATH] this is [MATH].', '1006.4388-2-68-4': "We would like to use this for von Neumann's [MATH] failure probability.", '1006.4388-2-68-5': "One complication arises, however, which is that in von Neumann's model the fan-out gates do not fail.", '1006.4388-2-68-6': 'These gates are needed during the error correcting stage of his procedure.', '1006.4388-2-68-7': 'We get around this by using fan-out gates constructed from energy functions like those in Section [REF].', '1006.4388-2-68-8': 'In particular define the fan-out gate via Eq. ([REF]).', '1006.4388-2-68-9': 'As shown in Section [REF] this gives rise in our probabilistic gate setting in which each fan-out resulting wire fails with probability [MATH].', '1006.4388-2-68-10': "These independent failures can then be reinterpreted as errors for the gates that follow them in von Neumann's error correcting construction.", '1006.4388-2-68-11': "One should note that there is a subtlety here in that if we desire to recover von Neuman's error model where the gates fail with exactly probability [MATH] then a complicated calculation is needed in order to figure out [MATH] in this model.", '1006.4388-2-68-12': 'However, for large enough [MATH] the effect will mostly be that the error probabilities add such that [MATH] will be the sum of the fan-out error and the gate error.', '1006.4388-2-69-0': 'Putting this together we can conclude that there is a critical temperature below which one can use fault-tolerant constructions to design ground state spin energy functions which compute correctly with probability [MATH] by using [MATH] more interactions in the energy functions.', '1006.4388-2-69-1': 'This means that while the general model of ground state spin computing is not tolerant to errors, for a cleverly designed setup one can overcome this difficulty and build robust devices, assuming that the temperature is low enough to keep the error probability below the error threshold.', '1006.4388-2-70-0': '# Relationship to Computational Complexity Theory', '1006.4388-2-71-0': 'Now we turn to a relationship between our model and computational complexity.', '1006.4388-2-71-1': 'This connection is provided because the ground state spin computing model at zero temperature is intimately related to the Cook-Levin theorem from computational complexity.', '1006.4388-2-72-0': '## The Cook-Levin Theorem', '1006.4388-2-73-0': 'The Cook-Levin theorem [CITATION] is a fundamental result in the theory of computational complexity.', '1006.4388-2-73-1': 'Here we briefly review this result.', '1006.4388-2-73-2': 'For more details one is referred, for example, to the classic textbook [CITATION].', '1006.4388-2-74-0': 'Traditionally the theory of computational complexity is founded upon the computational model of a Turing machine [CITATION].', '1006.4388-2-74-1': 'The Church-Turing thesis roughly states that the Turing machine model adequately captures the notion of what can be performed by any physical instantiation of a computer and decades of experience have yet to challenge this assumption.', '1006.4388-2-74-2': 'A deterministic Turing machine consists of the machine and a linear tape onto which elements from a finite alphabet, [MATH], can be written into different cells on the tape.', '1006.4388-2-74-3': 'The machine itself has an finite set of internal states, [MATH], a method for reading a symbol written in a cell on the tape, a method to write a symbol in a cell on the tape, and the ability to move either left or right one cell.', '1006.4388-2-74-4': 'Formally the tape is assumed to be infinite or at least infinitely extendable.', '1006.4388-2-74-5': 'Initially an input, consisting of a string of symbols from a finite alphabet [MATH] is placed on the tape and the physical machine is placed on the first symbol of this string.', '1006.4388-2-74-6': 'The Turing machine is assumed to start itself with its internal states in a special state called the start state, [MATH].', '1006.4388-2-74-7': 'A deterministic Turing machine then proceeds to "compute" as follows.', '1006.4388-2-74-8': 'At each time step the deterministic Turing machine examines the symbol of the cell it currently occupies and its internal state.', '1006.4388-2-74-9': 'Then conditional on these two variables the Turing machine writes a symbol onto cell (possibly from a larger, but still finite, alphabet than the alphabet of the initial strings) it occupies, moves either left or right, and modifies its internal state.', '1006.4388-2-74-10': 'These rules can be specified by a function, the transition function, [MATH], where [MATH] and [MATH] denote moving left or right.', '1006.4388-2-74-11': 'The deterministic Turing machine does this indefinitely until it reaches one of a set of its internal states known as halt states.', '1006.4388-2-74-12': 'These halt states are labeled as either "reject" or "accept."', '1006.4388-2-74-13': 'If we initially input the string [MATH] and the deterministic Turing machine halts in accept state, then we say the deterministic Turing machine accepts [MATH].', '1006.4388-2-74-14': 'Similarly if on input [MATH] the machine halts in a reject state, then we say that the machine rejects [MATH].', '1006.4388-2-74-15': 'Note that on some strings the Turing machine may neither accept or reject a string, but continue indefinitely.', '1006.4388-2-74-16': 'The language of a Turing machine is the set of all strings taken from the input alphabet for which the Turing machine accepts for that string being input.', '1006.4388-2-75-0': 'Deterministic Turing machines capture many important features of real modern computers.', '1006.4388-2-75-1': 'For example, the running time of most algorithms on a variety of hardware are closely related to the similar problems cast in terms of deterministic Turing machines.', '1006.4388-2-75-2': 'An important class of languages are those for which a deterministic Turing machine accepts a string from the language in a time bounded by a constant times a polynomial in the size of the string.', '1006.4388-2-75-3': 'Such languages are said to be in the complexity class [MATH].', '1006.4388-2-76-0': 'In contrast to a deterministic Turing machine, non-deterministic Turing machines are a model of computing which is not thought to properly encapsulate what can be efficiently computed on a modern computer.', '1006.4388-2-76-1': 'A non-determinisitic Turing machine works in a manner similar to a deterministic Turing machine: it has an internal state, it can read and write symbols onto a tape, etc.', '1006.4388-2-76-2': "The difference is that at each step in the computation, a non-deterministic Turing machine doesn't have to chose one new state, new symbol and direction to move, but may branch to multiple such triples.", '1006.4388-2-76-3': 'The transition function is now not [MATH] but instead is of the form [MATH] where [MATH] denotes the power set.', '1006.4388-2-76-4': 'Non-deterministic Turing machines thus explore multiple pathways at once.', '1006.4388-2-76-5': 'Such a Turing machine halts when one of the states of any of the multiple branches it is exploring enters a halt state.', '1006.4388-2-76-6': 'Again we can define whether the initial input is accepted or rejected depending on whether the state where the computation halts is a reject or accept state.', '1006.4388-2-76-7': 'Given this we can define the language of the non-determinstic Turing machine as the set of strings in which the machine accepts the given input.', '1006.4388-2-76-8': 'The time complexity of a non-deterministic Turing machine is the number of steps of executed by the Turing machine irrespective of the number of branches explored by the machine.', '1006.4388-2-77-0': 'Because a non-deterministic Turing machine is allowed to explore multiple pathways in its computation it is not clear that such machines can be reasonably mapped (in terms of say, time complexity of problems) to modern computers.', '1006.4388-2-77-1': 'This question can be put on a more formal footing by defining the complexity class [MATH].', '1006.4388-2-77-2': '[MATH] is the class of languages which can be recognized by a non-deterministic Turing machine in a time bounded by a constant times a polynomial in the size of the string being decided.', '1006.4388-2-77-3': 'The statement of whether [MATH] is strictly larger than [MATH] is the famous [MATH] problem, one of the most important open problems in the theory of computational complexity.', '1006.4388-2-77-4': 'That being said, most researchers do believe that [MATH] and that the model of a non-deterministic Turing machine is more powerful than computers we can physically construct.', '1006.4388-2-78-0': 'Given the above brief overview of the basic models of deterministic and non-deterministic Turing machines we can now introduce the Cook-Levin theorem.', '1006.4388-2-78-1': 'To define the Cook-Levin problem we introduce the [MATH] problem.', '1006.4388-2-79-0': 'Boolean Satisfaction Problem (): Given as input a description of a Boolean function from [MATH] boolean inputs to one output, [MATH], decide whether there exists a set of inputs [MATH], with each [MATH], such that [MATH].', '1006.4388-2-80-0': 'Cast in terms of languages, the [MATH] language is the set of strings describing a Boolean function such that the Boolean function has some input which evaluates to [MATH].', '1006.4388-2-80-1': 'The Cook-Levin theorem then states that the [MATH] is complete for [MATH].', '1006.4388-2-80-2': 'This means that every language in [MATH] can be converted by a deterministic Turing machine in polynomial time to a [MATH] language and every [MATH] language can be converted into a language in [MATH].', '1006.4388-2-80-3': 'Thus, in a real sense, [MATH] captures what it means for a lanuage to be in [MATH].', '1006.4388-2-80-4': 'For example, if we could efficiently solve a [MATH] problem in polynomial time we would be able to solve every problem in [MATH] in polynomial time and vice versa.', '1006.4388-2-80-5': 'This is especially important considering that numerous problems beside [MATH] have been identified as [MATH]-complete.', '1006.4388-2-80-6': 'This class of problems are essentially "the most difficult" in [MATH] and are correspondingly the problems for which no known polynomial time algorithms for all instances of the problems in the class are thought to exist.', '1006.4388-2-81-0': 'We are most interested in the proof of the Cook-Levin theorem.', '1006.4388-2-81-1': 'Before reviewing this it is useful to provide another equivalent characterization of [MATH] which uses, instead of a non-deterministic Turing machine, a prover and a verifier.', '1006.4388-2-81-2': 'The prover is assumed to have unlimited computational power while the verifier is limited to running a polynomial time algorithm on a deterministic Turing machine.', '1006.4388-2-81-3': 'A language, [MATH], is in [MATH] if the prover can supply to a verifier a polynomial sized proof [MATH] that a instance [MATH] is in the language.', '1006.4388-2-81-4': 'This proof must always be correct, i.e. the verifier must always be convinced, by using a Turing machine from [MATH], that the instance [MATH] is the in the language.', '1006.4388-2-81-5': 'Similarly the prover must never be able to convince the verifier incorrectly.', '1006.4388-2-81-6': 'In other words if [MATH] is not in [MATH], then the verifier must always reject.', '1006.4388-2-81-7': 'For example [MATH] is in [MATH] by this definition since the prover can simply supply a satisfying input to the boolean function if the function can be satisfied, and the verifier can compute the boolean function on this input in polynomial time.', '1006.4388-2-81-8': 'Similarly if the function does not admit a satisfying assignment the verifier can never be convinced that the instance is not in the language because no string the prover can give will lead to a satisfied boolean function.', '1006.4388-2-81-9': 'Note that in the prover-verifier setting we do not require that the prover try to convince the verifier that an instance [MATH] is not in [MATH], only that if it is true that [MATH] is not in [MATH] that the prover cannot convince the verifier that [MATH] is in [MATH].', '1006.4388-2-82-0': 'Given the prover/verifier definition of [MATH] we can now discuss the Cook-Levin theorem.', '1006.4388-2-82-1': 'The Cook-Levin theorem states that [MATH] is [MATH]-complete.', '1006.4388-2-82-2': 'A problem, [MATH], is [MATH]-complete if and only if every problem in [MATH] can be transformed into [MATH] by a deterministic Turing machine in polynomial time.', '1006.4388-2-82-3': 'Thus [MATH], since it is [MATH]-complete, captures much of what is difficult about problems in all of [MATH]: if there were a polynomial time algorithm for [MATH] then there would be a polynomial time algorithm for every problem in [MATH].', '1006.4388-2-83-0': 'The proof that [MATH] is [MATH]-complete has two directions.', '1006.4388-2-83-1': 'We have already stated the first direction, that [MATH] is a [MATH] problem.', '1006.4388-2-83-2': 'The meat of the proof is in the other direction, that every problem in [MATH] can be converted into a [MATH] problem.', '1006.4388-2-83-3': 'Here we do this in a slightly non-standard manner, using the prover-verifier definition of [MATH].', '1006.4388-2-83-4': 'In particular for any problem, [MATH], in [MATH] we know that there exists a verifier who takes as input the problem instance, [MATH] and a certificate [MATH], runs these on a deterministic Turing machine and verifies whether the instance is in the language.', '1006.4388-2-83-5': 'Imagine now the time history of such a verification by a Turing machine.', '1006.4388-2-83-6': 'Initially the Turing machine is at the beginning of the string, in its initial state, and the input string is on the tape.', '1006.4388-2-83-7': 'At the next time step, the machine can change its state, move left, right or stay in place, and possibly change the symbol on the tape where the machine is located.', '1006.4388-2-83-8': 'Notice, however that the contents of a cell on the tape, whether the machine is at that location, and the internal state of the machine if it is on that cell at the next time step is a function only of the contents of the cell and its neighbors at the prior step, whether the machine was on one of those cells, and its internal state if it was.', '1006.4388-2-83-9': 'In other words the computation can be described locally in space and time.', '1006.4388-2-83-10': 'Given this we can now describe how to convert a verifier, which is just a deterministic Turing machine and the strings representing the instance [MATH] and the proof certificate [MATH] into a problem in [MATH].', '1006.4388-2-84-0': 'Suppose that the verifiers deterministic Turing machine runs in time [MATH] where [MATH] is the size of the instance and the proof and [MATH] is a polynomial function.', '1006.4388-2-84-1': 'Define the following boolean variables, [MATH], [MATH], [MATH] to represent the history of the deterministic Turing machine in verifying the proof.', '1006.4388-2-85-0': 'We think about the first two variables as being arranged in a large [MATH] tableau onto which the computational history of the deterministic Turing machine is carried out.', '1006.4388-2-85-1': 'We can now define a [MATH] problem over these variables which is satisfiable iff the deterministic Turing machine accepts the input string [MATH].', '1006.4388-2-85-2': 'This [MATH] formula is conjunction (logical and) of a clauses which enforce (a) that the initial contents of the tape are [MATH], (b) that the machine is in the correct initial state, (c) the initial location of the head of the tape, (d) that there is only one symbol per tape cell, one state per time, one tape position per time, (e) that the tape cell is unchanged unless it is the one which can be written upon, (f) the machine properly updates according to the Turing machine transition function, and (g) the machine finishes in an accepting state.', '1006.4388-2-85-3': 'For example condition (b) consists of clauses with a single variable [MATH] where [MATH] is the initial state of the Turing machine.', '1006.4388-2-85-4': 'The most complicated clause in this construction is the one that encodes the local update rule for the machine, (f).', '1006.4388-2-85-5': 'This can be done by adding the clauses [EQUATION] for valid transitions of the Turing machine (that is for [MATH] representing a valid change of state, symbol, and direction of movement for the read/write head of the Turing machine.)', '1006.4388-2-85-6': 'Putting this together one thus sees that once can take the deterministic verifier along with the instance and [MATH] and construct a polynomial sized boolean formula [MATH] which is satisfied iff the verifier accepts the proof on the instance.', '1006.4388-2-85-7': 'Note that we have not forced a constraint on the proof input to the circuit, but that the definition of [MATH] in a prover-verifier setting leads to the satisfiability of the formula [MATH] iff the instance [MATH] is in the language.', '1006.4388-2-86-0': 'We can now see that connection between the Cook-Levin theorem and the classical ground state spin computing model.', '1006.4388-2-86-1': 'In particular a central part of the proof of this theorem is that one can construct a satisfiability formula directly from the history of a computation.', '1006.4388-2-86-2': 'In a similar manner classical ground state spin computing constructs an energy function whose ground state is the history of a computation.', '1006.4388-2-86-3': 'The conversion of a logic circuit to such an energy function is essentially the arithmetization technique of computational complexity [CITATION].', '1006.4388-2-86-4': 'Note however that the important role here of the input and outputs to the constructed formula: in the case where the output is forced, we are led to [MATH]-complete problems, but when the input is forced then we are led to efficient constructions.', '1006.4388-2-86-5': 'Motivated by this it is interesting to consider how the above constructions works but now at finite temperature.', '1006.4388-2-87-0': '## A Promise-MA Problem From Classical Ground State Spin Computing', '1006.4388-2-88-0': 'We will now show that it is possible to use classical ground state spin computing to define, in analogy with [MATH] a Promise-[MATH]-complete problem.', '1006.4388-2-88-1': '[MATH] stands for Merlin-Arthur and is essentially a probabilistic generalization of the complexity class [MATH] [CITATION].', '1006.4388-2-88-2': 'It is defined as the follows:', '1006.4388-2-89-0': 'Promise-MA: A language [MATH], [MATH], is a promise problem in Merlin-Arthur (Promise-) if there is a probabilistic polynomial-time Turing machine [MATH] and a polynomial [MATH] such that for all strings [MATH],', '1006.4388-2-90-0': 'Here [MATH] is the alphabet over which the language is defined.', '1006.4388-2-90-1': 'This version of the definition of [MATH] is a promise problem, since [MATH] does not necessarily cover all possible elements of [MATH].', '1006.4388-2-90-2': 'The reason that this class is called Merlin-Arthur is because it can be defined in terms of an interactive proof system between a computationally unbounded prover, Merlin, and a more limited verifier, Arthur, who can perform feasible probabilistic computations (polynomial time probabilistic computations with bounded error, [MATH].)', '1006.4388-2-90-3': 'Merlin is trying to convince Arthur that a certain instance [MATH] is in a language [MATH].', '1006.4388-2-90-4': 'He does this by communicating a polynomial size proof [MATH] to Arthur.', '1006.4388-2-90-5': 'Arthur can then take this proof and run a polynomial time probabilistic computation on this proof the result of which is: if [MATH], then Arthur is, with high probability, convinced the [MATH] is in [MATH], while if [MATH] no matter what proof Merlin supplies Arthur cannot be convinced that [MATH] except with a small probability.', '1006.4388-2-91-0': 'We define the following problem:', '1006.4388-2-92-0': 'Boundary Expectation Value (BEV): Let [MATH] be a positive integer and [MATH] a [MATH] digit number.', '1006.4388-2-92-1': 'Suppose we are given an energy function on [MATH] bits, [MATH] which can be written as a sum of energy functions [EQUATION] in such a way that this energy function can be interpreted as a circuit energy function (see Eq. ([REF])) with [MATH] input bits and [MATH] output bit, and each term in [MATH] is specified with [MATH] bits of precision.', '1006.4388-2-92-2': 'Consider the thermal ensemble generated by this energy function at inverse temperature [MATH].', '1006.4388-2-92-3': 'We promise that either there exists input bits [MATH] and output bit [MATH] such that for the reduced probability distribution on these bits [EQUATION] or for any input bits [MATH], [EQUATION]', '1006.4388-2-92-4': 'The problem BEV is to determine whether the first of these conditions, Eq. ([REF]), holds.', '1006.4388-2-92-5': 'In other words yes instances of this problem satisfy Eq. ([REF]) for some inputs [MATH] and no instances satisfy Eq. ([REF]) for all [MATH].', '1006.4388-2-93-0': 'We claim that BEV is Promise-The proof of this is fairly straightforward given our prior discussion of the Cook-Levin theorem.', '1006.4388-2-94-0': 'First we claim that [MATH] is in Promise-.', '1006.4388-2-94-1': 'To show this we must show that there exists an interactive proof of the form described above for the problem BEV.', '1006.4388-2-94-2': 'To see this note that a computationally unbounded Merlin can sample from the thermal ensemble for an instance of this problem and can check whether there exists an input [MATH] such that Eq. ([REF]) holds or whether for all inputs [MATH], Eq. ([REF]) holds.', '1006.4388-2-94-3': 'This will then serve as the proof, [MATH], in the definition of .', '1006.4388-2-94-4': 'Given the description of [MATH] and the circuit energy function, Arthur can construct a probabilistic Turing machine which implements the circuit related to this energy function using our construction of a probabilistic circuit from Section [REF].', '1006.4388-2-94-5': 'Then Arthur can run this probabilistic Turing machine on the proof that Merlin supplies.', '1006.4388-2-94-6': 'It is clear that if [MATH] is a yes instance of BEV, then the proof [MATH] Merlin supplies will convince Arthur that he has a yes instance of this interactive protocol.', '1006.4388-2-94-7': 'Further if [MATH] is a no instance of BEV it is also clear that no proof that Merlin supplies will convince Arthur that [MATH] is a yes instance.', '1006.4388-2-94-8': 'Thus it is clear that [MATH] is in Promise-.', '1006.4388-2-95-0': 'The other direction of the proof requires that we show that every problem in Promise- can be converted into a BEV problem.', '1006.4388-2-95-1': 'To do this we follow the idea of the Cook-Levin theorem in that, for a problem [MATH], we convert the probabilistic Turing machine [MATH] that verifies the proof [MATH] of the problem into a thermal circuit for which the associated BEV problem has [MATH] as the satisfying input.', '1006.4388-2-95-2': 'Similarly, for an problem [MATH], the thermal circuit corresponding to [MATH] must correspond to a no instance of BEV, since otherwise there would exist inputs [MATH] which would cause [MATH] to accept with probability greater than [MATH].', '1006.4388-2-95-3': 'Clearly to do this we will make use of our probabilistic circuit representation result from Section [REF].', '1006.4388-2-95-4': 'The main tool that we need in such a construction is that we need to be able to construct circuit energy functions which perform probabilistic gates of a form we desire, for a fixed inverse temperature [MATH].', '1006.4388-2-95-5': 'To do this we can proceed as follows.', '1006.4388-2-95-6': 'We can use our fault-tolerant constructions to produce circuits which operate with near deterministic behavior.', '1006.4388-2-95-7': 'Then in order to augment a deterministic computation we need a source of randomness, indeed in most standard definitions of a probabilistic Turing machine one requires randomness which is close to uniformly random across the random bits being used (or in other words one requires bits which are nearly equal probability of being in [MATH] or [MATH].)', '1006.4388-2-95-8': 'To obtain such randomness in our constructions, notice that if we do not force our inputs to be [MATH] or [MATH] then there is, in a thermal ensemble, an equal probability to be [MATH] and [MATH].', '1006.4388-2-95-9': 'This allows one to create random bits in an circuit energy function.', '1006.4388-2-95-10': 'However in order to get this to work with the deterministic part of our Turing machine, we must find a method to get this randomness into encoded bits in our circuits.', '1006.4388-2-95-11': 'This can be accomplished by simply taking the equal probability [MATH] and [MATH] bits and running them through the circuit which performs error correction on the bundle corresponding to these bits.', '1006.4388-2-95-12': 'Thus we see that we can construct at finite [MATH] and circuit energy function which is near deterministic and which can also have input bits which are nearly uniformly random.', '1006.4388-2-95-13': 'Thus we can proceed just as in the Cook-Levin theorem.', '1006.4388-2-95-14': 'For a tableau describing the history of the probabilistic Turing machine, [MATH], (when this probablistic Turing machine is simply a deterministic Turing machine aided by bits of randomness, we can construct an energy circuit function which implements the probabilistic Turing machine spatially across this tableau.', '1006.4388-2-96-0': 'Thus we have shown that BEV is Promise-While the proof of this result was rather straightforward and followed quickly from understanding the Cook-Levin theorem it is interesting to note that very few problems are known.', '1006.4388-2-96-1': 'In fact the only other non-natural problem which is Promise-to our knowledge is the stoquastic 6-problem [CITATION].', '1006.4388-2-96-2': 'The nearest comparable result is the classic result for the complexity of finding the ground state and computing the partition function for the Ising model of Barahona [CITATION].', '1006.4388-2-97-0': '# Quantum Models', '1006.4388-2-98-0': 'Finally we would like to briefly mention the quantum models which show some similarity with our classical model.', '1006.4388-2-98-1': 'In particular these models were the inspiration for investigating this problem and thus this serves as a good location to introduce the open problems concerning these problem.', '1006.4388-2-99-0': 'One of the quantum models relevant to this discussion are universal adiabatic quantum computing schemes.', '1006.4388-2-99-1': 'In these models one shows how to adiabatically drag a many-body quantum system from one easily preparable ground state to the ground state of another many-body quantum system whose ground state is a superposition over the history of a quantum circuit.', '1006.4388-2-99-2': 'Thus, similar to our models, the end result is a ground state which encodes a computation: but in this case the computation is a superposition over the history of the computation.', '1006.4388-2-99-3': 'These models show that adiabatic quantum computing, which previously had been only used for optimization problems [CITATION] can also be used for universal quantum computation [CITATION].', '1006.4388-2-99-4': 'However the final state of these models are systems with small energy gaps and thus the effects of working at non-zero temperature will have a profound effect on these models.', '1006.4388-2-99-5': 'Indeed it is for exactly this reason that such universal adiabatic quantum computing schemes are not known to be fault-tolerant (see, however, [CITATION].)', '1006.4388-2-99-6': 'Similar reasoning follows for an earlier model of ground state quantum computing due to Mizel and co-workers [CITATION].', '1006.4388-2-100-0': 'The problem of having a system whose ground state correctly computes but whose excited states do not is exactly the problem we have addressed in this paper for the classical ground state spin computing.', '1006.4388-2-100-1': 'An interesting question then, when considering the quantum models, is whether there is a similar reinterpretation quantum thermal ensembles as spatially enacted quantum computations.', '1006.4388-2-100-2': 'This is an important open problem which might conceivably lead to fault-tolerant methods for adiabatic quantum computing.', '1006.4388-2-101-0': '# Conclusion', '1006.4388-2-102-0': 'We have introduce a set of energy functions on many bits (spins) which have the property that their ground state can be thought of as a spatially distributed deterministic computation.', '1006.4388-2-102-1': 'At temperature greater than zero we have shown that the thermal ensemble arising from these models can be reinterpreted as a spatially distributed probabilistic computation.', '1006.4388-2-102-2': 'Further, above zero temperature we see that the gates of a ground state spin computer can become unreliable and fail to execute the desired computation with high-fidelity.', '1006.4388-2-102-3': 'However with the mapping of the thermal ensemble to probabilistic circuits we have shown how it is possible to make versions of the desired deterministic circuits which are fault-tolerant.', '1006.4388-2-102-4': 'We have shown that determining whether a given classical energy function can be thought of as enacting a ground state computation is Finally we have shown that a problem concerning the thermal ensembles arising in our model give rise to a rare complete problem for the complexity class Promise-.', '1006.4388-2-102-5': 'The models we have considered here are mostly devoid of connections to actual physical systems of interest, besides connections to quantum-dot cellular automata.', '1006.4388-2-102-6': 'An interesting an important open question about these models is whether they arise in naturally occurring physical systems, or a suitably engineered system.', '1006.4388-2-102-7': 'Another interesting question is the rate at which the ground state spin model thermalizes: a proof that the system reaches thermal equilibrium in a time polynomial in the size of the circuit being implement would be another step towards making this model more physically relevant.', '1006.4388-2-103-0': 'D.B. and E.C. are supported by NSF grants 0621621, 0803478, 0829937, and 091640 and DARPA grant FA9550-09-1-0044.', '1006.4388-2-103-1': 'K.R.B is supported by Georgia Tech.', '1006.4388-2-103-2': 'We acknowledge useful conversations with David Meyer and Maxwell Pierce.'}

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['1006.4388-1-2-1', '1006.4388-2-2-1'], ['1006.4388-1-2-2', '1006.4388-2-2-2']]

[]

[]

[]

[]

['1006.4388-1-20-10', '1006.4388-1-25-6', '1006.4388-1-26-3', '1006.4388-1-32-3', '1006.4388-1-41-1', '1006.4388-1-88-2', '1006.4388-1-89-0', '1006.4388-1-91-0', '1006.4388-1-103-0', '1006.4388-2-20-10', '1006.4388-2-25-6', '1006.4388-2-26-3', '1006.4388-2-32-3', '1006.4388-2-41-1', '1006.4388-2-88-2', '1006.4388-2-89-0', '1006.4388-2-91-0', '1006.4388-2-103-0']

{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1006.4388

1903.01385

{'1903.01385-1-0-0': 'One of the well-known formulations of human perception is a hierarchical inference model based on the interaction between conceptual knowledge and sensory stimuli from the partially observable environment.', '1903.01385-1-0-1': 'This model helps human to learn inductive biases and guides their behaviors by minimizing their surprise of observations.', '1903.01385-1-0-2': 'However, most model-based reinforcement learning still lacks the support of object-based physical reasoning.', '1903.01385-1-0-3': 'In this paper, we propose Object-based Perception Control (OPC).', '1903.01385-1-0-4': 'It combines the learning of perceiving objects from the scene and that of control of the objects in the perceived environments by the free-energy principle.', '1903.01385-1-0-5': 'Extensive experiments on high-dimensional pixel environments show that OPC outperforms several strong baselines in accumulated rewards and the quality of perceptual grouping.', '1903.01385-1-1-0': '# Introduction', '1903.01385-1-2-0': '# Preliminary on Bayesian Brain Hypothesis', '1903.01385-1-3-0': 'Many of the classical hierarchical perception models on vision sensory inputs include the Restricted Boltzmann Machines (RBMs) [CITATION], the predictive coding [CITATION], and the Helmholtz Machine [CITATION].', '1903.01385-1-3-1': 'They share a general principle of Bayesian brain hypothesis, which formulates human perception as a hierarchical inference process based on the interaction between sensory stimuli (a bottom-up recognition model [MATH]) and conceptual knowledge (a top-down generative model [MATH]) [CITATION].', '1903.01385-1-3-2': 'The perception then comes as the recognition model formalizing beliefs about the cause of observations, i.e., to infer and maximize the posterior [MATH] (the probability of different hidden states given the observation) by inverting the likelihood model [MATH] (the probability of observation given their causes).', '1903.01385-1-3-3': 'The inversion requires maximizing the model evidence [MATH] (the prediction ability of the generative model regarding the observation [MATH]), which is equivalent to minimizing the surprise [EQUATION] where [MATH] is the bottom-up recognition model.', '1903.01385-1-3-4': 'Eq. ([REF]) is called as the free energy [CITATION], which is an upper bound of the surprise of observations as the KL term in Eq. ([REF]) is non-negative.', '1903.01385-1-3-5': 'The bound becomes tight when the KL term equals to zero, i.e., the difference between the recognition model and the posterior probability is minimized.', '1903.01385-1-4-0': '# Object-based Perception Control', '1903.01385-1-5-0': '## Environment Setting', '1903.01385-1-6-0': 'We define the environment as a partially observable Markov Decision Process (POMDP) represented by tuple [MATH], where [MATH] are the state space, the action space, and the observation space, respectively.', '1903.01385-1-6-1': 'Given two sets [MATH] and [MATH], we use [MATH] to denote the Cartesian product of [MATH] and [MATH], i.e., [MATH].', '1903.01385-1-7-0': 'For an agent performing in this environment, we consider its received observation [MATH] at time step [MATH] as a visual image (a matrix of pixels) composited of [MATH] objects, where each pixel [MATH] is determined by exactly one object.', '1903.01385-1-7-1': 'At each time step [MATH], we denote as [MATH] the latent state which encodes the unknown true pixel assignments, such that [MATH] iff pixel [MATH] was generated by component [MATH], and [MATH].', '1903.01385-1-7-2': 'The observation [MATH] is generated by the environment following the conditional observation distribution [MATH].', '1903.01385-1-7-3': 'Concretely, each pixel [MATH] is rendered by one of the object representations [MATH], which are transformed by a differentiable non-linear function [MATH] into variables [MATH] for separate pixel-wise distributions [MATH] .', '1903.01385-1-7-4': "Whether an observation is probable (or not surprised) given an agent's current state and action is quantified by its observed reward [MATH] provided by the environment according to the reward function [MATH].", '1903.01385-1-7-5': 'When the environment receives an action [MATH], it moves to a new state [MATH] following the transition function [MATH].', '1903.01385-1-7-6': 'In the following sections, we also use [MATH] to represent the state-action tuple [MATH] at time step [MATH].', '1903.01385-1-8-0': '## Learning Object-based Perception Control', '1903.01385-1-9-0': 'To formalize the belief about the way the observation [MATH] is caused, the agent maintains a perception model [MATH] to approximate the probability of different latent states [MATH].', '1903.01385-1-9-1': "Using the encoded sufficient statistic of the history, agent's inferred belief about the latent state could serve as the input to its decision-making module (the policy) [MATH].", '1903.01385-1-9-2': 'The prediction ability of the perception model at time step [MATH] is given by the model evidence as [EQUATION]', '1903.01385-1-9-3': 'See details of derivation in the supplementary material.', '1903.01385-1-9-4': "The goal of the agent is to learn a perception model with high prediction ability by minimizing the average surprise [EQUATION] over trajectories [MATH] induced by agent's policy [MATH], We denote the distribution over initial state as [MATH].", '1903.01385-1-9-5': '[EQUATION]', '1903.01385-1-9-6': 'See derivation in the supplementary material.', '1903.01385-1-9-7': 'The first term in Eq. ([REF]) is identical to the expected free energy [CITATION] along trajectory [MATH] with respect to a set of object representations [MATH], and is also minimized by improving the perception [MATH] about the true posterior [MATH].', '1903.01385-1-9-8': 'Meanwhile, the second term in Eq. ([REF]) is the negative of the total reward along trajectory [MATH], and is minimized by taking actions on the environment to change the sensory input.', '1903.01385-1-9-9': 'The optimization of these terms are discussed in Sect. [REF] and Sect. [REF] respectively.', '1903.01385-1-10-0': '## Perception Model Update', '1903.01385-1-11-0': 'Assume that the estimate [MATH] is given when the expected free energy is already minimized with respect to [MATH], which happens when the KL term in Eq. ([REF]) equals to 0, i.e., [MATH], resulting in a soft-assignment of each pixel to one of the [MATH] objects [EQUATION]', '1903.01385-1-11-1': 'We then find [MATH] to minimize the expected free energy as [EQUATION]', '1903.01385-1-11-2': 'See derivation in the supplementary material.', '1903.01385-1-11-3': 'Note that Eq. ([REF]) returns a set of points that maximize [MATH], and we choose [MATH] to be any value within this set.', '1903.01385-1-12-0': 'To update the perception model by minimizing the expected free energy with respect to [MATH] and [MATH], we compute Eq. [REF] and Eq. [REF] iteratively.', '1903.01385-1-12-1': 'However, an analytical solution to Eq. [REF] is not available because we use a differentiable non-linear function [MATH] to map from object representations [MATH] into [MATH].', '1903.01385-1-12-2': 'Instead, we get [MATH] by [EQUATION] where [MATH] is the learning rate (see details of derivation in the supplementary material).', '1903.01385-1-12-3': 'We regard the iterative process as [MATH]-step rollout of [MATH] copies of a recurrent neural network with hidden states [MATH] receiving [MATH] as input (see the inner loop of Algorithm [REF]).', '1903.01385-1-12-4': 'Each copy generates a new [MATH], which is then used to re-estimate the soft-assignments [MATH].', '1903.01385-1-12-5': 'We parameterize the Jacobian [MATH] and the differentiable non-linear function [MATH] using a convolutional encoder-decoder architecture with a recurrent neural network as bottleneck, which linearly combine the output of the encoder with [MATH] from the previous timestep.', '1903.01385-1-12-6': 'The training process is organized in an unsupervised manner by minimizing [MATH].', '1903.01385-1-13-0': '## Convergence of the Perception Model Update', '1903.01385-1-14-0': 'Under main assumptions and lemmas as introduced below, we demonstrate the convergence of a sequence of average surprise values [MATH] generated by the perception update.', '1903.01385-1-14-1': 'The proof is presented by showing that the learning process follows the Global Convergence Theorem [CITATION].', '1903.01385-1-15-0': '[MATH] is compact for any [MATH].', '1903.01385-1-16-0': '[MATH] is continuous in [MATH] and differentiable in the interior of [MATH].', '1903.01385-1-17-0': 'The above assumptions lead to the fact that [MATH] is bounded for any [MATH].', '1903.01385-1-18-0': 'Let [MATH] be the set of stationary points in the interior of [MATH], then the mapping [MATH] from Eq. [REF] is closed over [MATH] (the complement of [MATH]).', '1903.01385-1-19-0': 'See [CITATION].', '1903.01385-1-19-1': 'A sufficient condition is that [MATH] is continuous in both [MATH] and [MATH].', '1903.01385-1-20-0': 'Let [MATH] be the set of stationary points in the interior of [MATH], then (i.) [MATH] and (ii.)', '1903.01385-1-20-1': '[MATH].', '1903.01385-1-21-0': 'Note that (i.) holds true given the condition.', '1903.01385-1-21-1': 'To prove (ii.)', '1903.01385-1-21-2': ', consider any [MATH], we have [EQUATION]', '1903.01385-1-21-3': 'Hence [MATH] is not maximized at [MATH].', '1903.01385-1-21-4': 'Given the perception update described by Eq. ([REF]), we therefore have [MATH] which implies [MATH].', '1903.01385-1-22-0': 'Let [MATH] be a sequence generated by the mapping from Eq. [REF], [MATH] be the set of stationary points in the interior of [MATH].', '1903.01385-1-22-1': 'If Assumptions [REF] [REF], Lemma [REF], and Proposition [REF] are met, then all the limit points of [MATH] are stationary points (local minima) and [MATH] converges monotonically to [MATH] for some stationary point [MATH].', '1903.01385-1-23-0': 'Suppose that [MATH] is a limit point of the sequence [MATH].', '1903.01385-1-23-1': 'Given Assumptions [REF] [REF] and Proposition [REF].', '1903.01385-1-23-2': 'i), we have that the sequence [MATH] are contained in a compact set [MATH].', '1903.01385-1-23-3': 'Thus, there is a subsequence [MATH] of [MATH] such that [MATH] as [MATH] and [MATH].', '1903.01385-1-24-0': 'We first show that [MATH] as [MATH].', '1903.01385-1-24-1': 'Given [MATH] is continuous in [MATH] (Assumption [REF]), we have [MATH] as [MATH] and [MATH], which means [EQUATION]', '1903.01385-1-24-2': 'Given Proposition [REF] and Eq. ([REF]), [MATH] is therefore monotonically decreasing on the sequence [MATH], which gives [EQUATION]', '1903.01385-1-24-3': 'Given Eq. ([REF]), for any [MATH], we have [EQUATION]', '1903.01385-1-24-4': 'Given Eq. ([REF]) and Eq. ([REF]), we therefore have [MATH] as [MATH].', '1903.01385-1-24-5': 'We then prove that the limit point [MATH] is a stationary point.', '1903.01385-1-24-6': 'Suppose [MATH] is not a stationary point, i.e., [MATH], we consider the sub-sequence [MATH], which are also contained in the compact set [MATH].', '1903.01385-1-24-7': 'Thus, there is a subsequence [MATH] of [MATH] such that [MATH] as [MATH] and [MATH], yielding [MATH] as [MATH] and [MATH], which gives [EQUATION]', '1903.01385-1-24-8': 'On the other hand, since the mapping from Eq. ([REF]) is closed over [MATH] (Lemma [REF]), and [MATH], we therefore have [MATH], yielding [MATH] (Proposition [REF].', '1903.01385-1-24-9': 'ii), which contradicts Eq. ([REF]).', '1903.01385-1-25-0': '[t] Learning the Object-based Perception Control', '1903.01385-1-26-0': 'Initialize [MATH] Create [MATH] environments that will execute in parallel training not finished Initialize the history [MATH] with environment rollouts for [MATH] time-steps under current policy [MATH] Get [MATH] from [MATH] Get [MATH] from [MATH] respectively Feed [MATH] into each of the [MATH] RNN copy to get [MATH] and forward-output [MATH] Compute [MATH] by Eq. [REF] [MATH] by Eq. ([REF]) Perform [MATH] according to policy [MATH] Receive reward [MATH] and new observation [MATH] Store [MATH] in [MATH] respectively Feed [MATH] into each of the [MATH] RNN copy to get [MATH] Perform synchronous update of [MATH] using [MATH], of [MATH] using [MATH], and of [MATH] using [MATH]', '1903.01385-1-27-0': '## Decision-making Module Update', '1903.01385-1-28-0': 'The second term in Eq. ([REF]) is the negative of the total reward along trajectory [MATH], which is straightforward to understand as an agent with a better perception model would receive a higher total reward by conditioning its policy on its inferred belief about the latent state.', '1903.01385-1-28-1': 'Intuitively, the additional knowledge provided by the decomposition of high-dimensional observations with respect to disentangled and structured object abstractions could ease the burden of decision-making.', '1903.01385-1-29-0': "To maximize the total reward along trajectory [MATH], we follow the conventional Temporal-Difference learning approach [CITATION] by passing the object abstractions to a small multilayer perceptron (MLP) [CITATION] to produce an [MATH]-dimensional vector, which is split into a [MATH]-dimensional vector of [MATH]'s (the ‘actor’) logits, and a baseline scalar [MATH] (the ‘critic’).", '1903.01385-1-29-1': 'The [MATH] logits are normalized using a softmax function, and used as multinomial distribution from which an action is sampled.', '1903.01385-1-29-2': 'The [MATH] is an estimate of the state-value function at the current state, which is given by the last hidden state [MATH] of the [MATH]-step RNN rollout.', '1903.01385-1-29-3': 'On training the decision-making module, the [MATH] is used to compute the temporal-difference error given by [EQUATION] where [MATH] is a discount factor.', '1903.01385-1-29-4': '[MATH] is used both to optimize [MATH] to generate actions with larger total rewards than [MATH] predicts by updating [MATH] with respect to the policy gradient [EQUATION] and to optimize [MATH] to more accurately estimate state values by updating [MATH].', '1903.01385-1-29-5': 'Also, differentiating [MATH] with respect to [MATH] enables the gradient-based optimizers to update the perception model.', '1903.01385-1-29-6': 'We provide the pseudo-code for one-step TD-learning of the proposed model in Algorithm [REF].', '1903.01385-1-29-7': 'By grouping objects with respect to the reward, our model could support distinguishing objects with high visual similarities but different meanings, thus helping to accelerate the learning procedure and better understand the environment.', '1903.01385-1-30-0': '# Related Work', '1903.01385-1-31-0': '# Experiments', '1903.01385-1-32-0': '# Conclusions', '1903.01385-1-33-0': 'In this paper, we propose Object-based Perception Control (OPC), demonstrating the mutual facilitation of hierarchical object-based perception and reinforcement learning under the Bayesian brain hypothesis derived from the free-energy principle.', '1903.01385-1-33-1': "Extensive experiments on high-dimensional pixel environments show that OPC outperforms several strong baselines in terms of accumulated rewards within agent's life-cycle, and the quality and consistency of the perceptual grouping.", '1903.01385-1-33-2': 'In future work, we intend to investigate OPC with more types of inductive biases and test the model performance in a wider variety of environments.', '1903.01385-1-34-0': '0.5', '1903.01385-1-35-0': '# Detailed Review on Bayesian Brain Hypothesis', '1903.01385-1-36-0': 'Many of the classical hierarchical perception models on vision sensory inputs include the Helmholtz Machineappendixdayan1995helmholtz, the predictive codingappendixmcclelland1981interactive,rao1999nature,clark2013whatever, the Restricted Boltzmann Machines (RBMs)appendixHinton2012, and the Variational Auto-Encoders (VAEs)appendixkingma2013auto.', '1903.01385-1-36-1': 'Although they vary from the architecture design, the training procedure, or the optimization goals, they share a general principle that the brain is constantly generating and amending hypotheses of sensory input at varying levels of abstraction.', '1903.01385-1-36-2': 'This principle comes from the Bayesian brain hypothesis, which derives from the idea of analysis-by-synthesisappendixneisser2014cognitive and uses Bayesian probability theory to formulate human perception as a hierarchical inference process based on the interaction between sensory stimuli (a bottom-up recognition model [MATH]) and conceptual knowledge (a top-down generative model [MATH]) bruner1947value,bruner1949perception,mackay:automatastudies1956,neal1992connectionist.', '1903.01385-1-36-3': 'The perception model actively predicts and explains its sensations while being updated using sensory inputsappendixvon2005treatise, e.g., raw pixels of high-dimensional environment observations.', '1903.01385-1-36-4': 'The generative model is decomposed into a likelihood [MATH] (the probability of observation given their causes) and the prior belief [MATH] (the a priori probability of those causes).', '1903.01385-1-36-5': 'The perception then comes as the recognition model formalizing beliefs about the way observations are caused, i.e., to infer the posterior [MATH] (the probability of different hidden states given the observation) by inverting the likelihood model.', '1903.01385-1-36-6': 'This inversion is the same as minimizing the difference between the recognition model and the posterior probability to optimize free energy, as we shall show below.', '1903.01385-1-37-0': 'The inference of [MATH] via Bayesian theory requires us to compute and maximize the model evidence [MATH], which quantifies the prediction ability of the generative model regarding the observation [MATH]: the better the model, the higher the probability of the observation [MATH].', '1903.01385-1-37-1': 'Maximizing model evidence is equivalent to minimizing the surprise [MATH], which follows the primary goal of any intelligent agent operating in a changing environment.', '1903.01385-1-37-2': 'However, an intelligent agent cannot directly decide whether an observation is probable, since the evaluation of surprise by direct marginalization [MATH] is intractable due to the integration over the hidden variable [MATH].', '1903.01385-1-37-3': 'Instead, the agent keeps a recognition model [MATH] and therefore manages to minimize [EQUATION]', '1903.01385-1-37-4': "The inequality in Eq. ([REF]) stands on the convex property of the logarithm and the Jensen's inequality, i.e., the convex transformation of a mean is less than or equal to the mean applied after convex transformationappendixjensen1906fonctions.", '1903.01385-1-37-5': 'Eq. ([REF]) is called as the free energyappendixNeal:1999:VEA:308574.308679,Hinton:1993:KNN:168304.168306, which is an upper bound of the surprise of observations as the KL term in Eq. ([REF]) is non-negative.', '1903.01385-1-37-6': 'The bound becomes tight when the KL term equals to zero, i.e., the difference between the recognition model and the posterior probability is minimized.', '1903.01385-1-37-7': 'The minimization of free energy is thus tractable by two distinct ways: (1) change perceptual representations (predictions) by updating the perception model [MATH], so that they approximate the conditional density on the causes of particular observations [MATH]FristonNature2010; and (2) change sensory input (observations) by taking actions on the environment to change its states, so that they conform to expectationsappendixadams2013predictions,friston2017active.', '1903.01385-1-38-0': '# Additional Related Work', '1903.01385-1-39-0': 'Human-like computing, which aims at endowing machines with human-like perceptual, reasoning and learning abilities, has drawn considerate attention from the recent progress in artificial intelligence (AI) lake2014towards,Lake1332,GravesHybrid2016,BakerRational17.', '1903.01385-1-39-1': 'Reinforcement learning (RL), which learns how to map observations to actions to maximize a numerical reward signal, is considered to be the closest form of learning that humans and other animals do rescorla1972theory,BartoSuttonAnderson83,barto1989learning,SuttonBarto90.', '1903.01385-1-39-2': 'Being capable of reasoning about the motions of inanimate objects even at early ages in infancy baillargeon1985object,SPELKE199029, humans gradually develop their cortical systems to learn physical conceptions with the interactive environment feedback spelke1992origins, leading to a unified hierarchical and behavioral-correlated perception model to perceive events and objects from the environment mumford1992computational,lee2003hierarchical.', '1903.01385-1-39-3': 'Different from model-free methods silver2017mastering, which treats learning as making good predictions by discovering patterns correlated to large rewards directly from the environments, model-based deep reinforcement learning algorithms have been shown to be more effective in certain tasks levine2014learning,watter2015embed,levine2016end,Gu:2016:CDQ:3045390.3045688,finn2017deep,pmlr-v80-igl18a.', '1903.01385-1-40-0': 'The object-based approach, which recognizes decomposed objects from the environment observations, has attracted considerable attention in the field of reinforcement learning as well asai2017classical,NIPS2018_8187,NIPS2018_7811.', '1903.01385-1-40-1': 'LatPlan asai2017classical uses a variational auto-encoder (VAE) to generate a problem representation from raw pixel observations in an unsupervised manner, but requires certain constraint on the environment as objects can only appear in a limited number of discrete locations, e.g., each piece of an image-based instance of the 8-puzzle.', '1903.01385-1-40-2': 'OODP NIPS2018_8187 is very close related to our work by proposing a complicated compositional architecture combining background extraction, static object detection, and dynamic object detection.', '1903.01385-1-40-3': 'One of the limitations of OODP is that it requires the background to be fixed through time.', '1903.01385-1-40-4': 'MOREL NIPS2018_7811 applies optical flow in video sequences and feeds the learned features to model-free RL frameworks as the position and velocity information of moving objects.', '1903.01385-1-40-5': 'Although MOREL ignores the relational reasoning among objects, it could be served as a supplementary module to our proposed method to better benefit from existing advances in computer vision.', '1903.01385-1-40-6': 'On the other hand, OPC follows an unsupervised manner to extract object abstractions by minimizing the free energy of unsupervised perceptual grouping van2018relational.', '1903.01385-1-40-7': 'By biasing the discovered object representations through the temporal-difference error, he decision-making process can also benefit from the inductive bias by representing states, actions and policies using the relational language, which facilitates the generalization over goals, states, and actions dvzeroski2001relational.', '1903.01385-1-41-0': 'In addition to the exchange between the agent and the environment, we also consider the physical interactions between object-pairs from the raw sensory input by equipping the perception model with the relational mechanism, which has been extensively discussed in both physical reasoning chang2016compositional,battaglia2016interaction,santoro2017simple and reinforcement learning literature dvzeroski2001relational,driessens2004integrating,Zambaldi2018RelationalDR.', '1903.01385-1-42-0': '# Experiment Details', '1903.01385-1-43-0': 'OPC In all experiments we trained the perception model using ADAM kingma2014adam with default parameters and a batch size of [MATH].', '1903.01385-1-43-1': 'Each input consists of a sequence of binary [MATH] images containing two poison objects (two circles) and one food object (a rectangle) that start in random positions and float within the image for [MATH] steps.', '1903.01385-1-43-2': 'These frames were thresholded at [MATH] to obtain binary images and added with bit-flip noise ([MATH]).', '1903.01385-1-43-3': 'We used a convolutional encoder-decoder architecture inspired by recent GANs Chen2016InfoGAN with a recurrent neural network as bottleneck, where the encoder used the same network architecture from mnih2013playing as', '1903.01385-1-44-0': '[MATH] conv.', '1903.01385-1-44-1': '16 ELU.', '1903.01385-1-44-2': 'stride 4.', '1903.01385-1-44-3': 'layer norm [MATH] conv.', '1903.01385-1-44-4': '32 ELU.', '1903.01385-1-44-5': 'stride 2.', '1903.01385-1-44-6': 'layer norm fully connected.', '1903.01385-1-44-7': '256 ELU.', '1903.01385-1-44-8': 'layer norm recurrent.', '1903.01385-1-44-9': '250 Sigmoid.', '1903.01385-1-44-10': 'layer norm on the output fully connected.', '1903.01385-1-44-11': '256 RELU.', '1903.01385-1-44-12': 'layer norm fully connected.', '1903.01385-1-44-13': '[MATH] RELU.', '1903.01385-1-44-14': 'layer norm [MATH] reshape 2 nearest-neighbour, conv.', '1903.01385-1-44-15': '16 RELU.', '1903.01385-1-44-16': 'layer norm [MATH] reshape 4 nearest-neighbour, conv.', '1903.01385-1-44-17': '1 Sigmoid', '1903.01385-1-45-0': 'We used the Advantage Actor-Critic (A2C) pmlr-v48-mniha16 with an MLP policy as the decision making module of OPC.', '1903.01385-1-45-1': 'The MLP policy added a 512-unit fully connected layer with rectifier nonlinearity after layer 4 of the perception model.', '1903.01385-1-45-2': 'The decision making module had two set of outputs – a softmax output with one entry per action representing the probability of selecting the action, and a single linear output representing the value function.', '1903.01385-1-45-3': 'The decision making module was trained using RMSProp tieleman2012lecture with a learning rate of [MATH], a reward discount factor [MATH], an RMSProp decay factor of [MATH], and performed updates after every [MATH] actions.', '1903.01385-1-46-0': 'A2C We used the same convolutional architecture as the encoder of the perception model of OPC (layer 1 to 3), followed by a fully connected layer with 512 hidden units followed by a rectifier nonlinearity.', '1903.01385-1-46-1': 'The A2C was trained using the same setting as the decision making module of OPC.', '1903.01385-1-47-0': 'WM-A2C We use the same setting as ha2018worldmodels to separately train the V model and the M model.', '1903.01385-1-47-1': 'The experience is generated off-line by a random policy operating in the Pixel Waterworld environment.', '1903.01385-1-47-2': 'We concatenated the output of the V model and the M model as the A2C input, and trained A2C using the same setting as introduced above.', '1903.01385-1-48-0': '# Details of Derivation', '1903.01385-1-49-0': '## Derivation of Eq. ([REF])', '1903.01385-1-50-0': '[EQUATION]', '1903.01385-1-51-0': '## Derivation of Eq. ([REF])', '1903.01385-1-52-0': '[EQUATION]', '1903.01385-1-53-0': '## Derivation of Eq. ([REF])', '1903.01385-1-54-0': '[EQUATION]', '1903.01385-1-55-0': '## Derivation of Eq. ([REF])', '1903.01385-1-56-0': '[EQUATION]'}

{'1903.01385-2-0-0': 'This paper is concerned with object-based perception control (OPC), which allows for joint optimization of hierarchical object-based perception and decision making.', '1903.01385-2-0-1': 'We define the OPC framework by extending the Bayesian brain hypothesis to support object-based latent representations and propose an unsupervised end-to-end solution method.', '1903.01385-2-0-2': 'We develop a practical algorithm and analyze the convergence of the perception model update.', '1903.01385-2-0-3': 'Experiments on a high-dimensional pixel environment justify the learning effectiveness of our object-based perception control approach.', '1903.01385-2-1-0': '# Introduction', '1903.01385-2-2-0': 'Reinforcement learning (RL), which learns how to map environment observations to actions to maximize total-expected returns, is considered to be the closest form of learning that humans and other animals do [CITATION].', '1903.01385-2-2-1': 'However, many of the current advances in RL lie in the model-free paradigm [CITATION], which usually requires large amounts of experience to explore until learning a good policy, thus suffering from the sample-efficiency problem [CITATION].', '1903.01385-2-2-2': 'On the other hand, humans usually learn and behave by minimizing their surprise of the observation outcome, e.g., trying to keep homeostasis [CITATION].', '1903.01385-2-2-3': 'To do this, people maintain and update a good environment model to help to learn new concepts from a few examples efficiently [CITATION].', '1903.01385-2-2-4': 'The idea has since innovated many model-based reinforcement learning (MBRL) approaches to learn a concise model of the world [CITATION].', '1903.01385-2-2-5': 'MBRL methods help to generalize features across observations and actions in high-dimensional environments, thus reducing the amount of training data [CITATION].', '1903.01385-2-3-0': "However, most existing MBRL methods often fail to facilitate common-sense physical reasoning [CITATION], or learning the inductive biases, which is the prior knowledge of observed objects' visual properties, such as color, material, locations, and shapes.", '1903.01385-2-3-1': 'On the other hand, humans learn inductive biases with the interactive environment feedback throughout their life-cycles [CITATION], leading to a unified hierarchical and behavioral-correlated perception model to perceive events and objects from the environment [CITATION].', '1903.01385-2-3-2': 'By learning the structured properties of the environment, inductive biases help human make inferences that go beyond the observation [CITATION].', '1903.01385-2-3-3': 'For example, humans exploit the shape bias in early lexical learning, i.e., we usually assume that objects with the same name tend to have the same shape.', '1903.01385-2-3-4': 'Children, therefore, attend to shape more often than other properties when generalizing a novel name to new examples [CITATION].', '1903.01385-2-3-5': 'We argue that learning inductive biases for MBRL is essential to create AI capable of simulating human-like learning perceptual, reasoning, and learning abilities [CITATION].', '1903.01385-2-3-6': 'Mainly, we shall build a model to create knowledge through execution-time optimization, rather than one merely generating static products from offline training [CITATION].', '1903.01385-2-4-0': 'In this paper, we propose Object-based Perception Control (OPC), a perception model in the context of RL to infer inductive biases from raw pixel observations.', '1903.01385-2-4-1': 'We develop the object-based inference model by: 1) minimizing the surprise for unsupervised perceptual grouping [CITATION], and 2) imposing the trial-and-error on the perceptual grouping process to emphasize the behavioral correlates of perception updating.', '1903.01385-2-4-2': 'The coherent framework of perception and control becomes beneficial as the perception model helps decision-making with inductive biases, while the decision-making module provides the temporal-difference error [CITATION] from interactions with the environment to bias the perception learning towards semantically relevant representations.', '1903.01385-2-4-3': 'Experiments on the Pixel Waterworld environment show that OPC outperforms several strong baselines in terms of accumulated rewards, and the quality and consistency of the perceptual grouping.', '1903.01385-2-5-0': '# Preliminary on Bayesian Brain Hypothesis', '1903.01385-2-6-0': 'Many of the classical hierarchical perception models on vision sensory inputs include the Helmholtz Machine [CITATION], the predictive coding [CITATION], and the Restricted Boltzmann Machines (RBMs) [CITATION].', '1903.01385-2-6-1': 'They share a general principle of Bayesian brain hypothesis [CITATION], which formulates human perception as a hierarchical inference process based on the interaction between sensory stimuli (a bottom-up recognition model [MATH]) and conceptual knowledge (a top-down generative model [MATH]) [CITATION].', '1903.01385-2-6-2': 'The perception then comes as the recognition model formalizing beliefs about the cause of observations, i.e., to infer and maximize the posterior [MATH] (the probability of different hidden states given the observation) by inverting the likelihood model [MATH] (the probability of observation given their causes).', '1903.01385-2-6-3': 'The inversion requires maximizing the model evidence [MATH] (the prediction ability of the generative model regarding the observation [MATH]), which is equivalent to minimizing the surprise [EQUATION] where [MATH] is the bottom-up recognition model.', '1903.01385-2-6-4': 'Eq. ([REF]) is usually referred to the free energy [CITATION], which is an upper bound of the surprise of observations.', '1903.01385-2-6-5': 'The bound becomes tight when the KL term equals to zero, i.e., the difference between the recognition model and the posterior probability is minimized.', '1903.01385-2-7-0': '# Object-based Perception Control', '1903.01385-2-8-0': '## Environment Setting', '1903.01385-2-9-0': 'We define the environment as a partially observable Markov Decision Process (POMDP) represented by the tuple [MATH], where [MATH] are the state space, the action space, and the observation space, respectively.', '1903.01385-2-9-1': 'Given two sets [MATH] and [MATH], we use [MATH] to denote the Cartesian product of [MATH] and [MATH], i.e., [MATH].', '1903.01385-2-10-0': 'For an agent performing in this environment, we consider its received observation [MATH] at time step [MATH] as a visual image (a matrix of pixels) composited of [MATH] objects, where each pixel [MATH] is determined by exactly one object.', '1903.01385-2-10-1': 'At each time step [MATH], we denote as [MATH] the latent state which encodes the unknown true pixel assignments, such that [MATH] iff pixel [MATH] was generated by component [MATH].', '1903.01385-2-10-2': 'The observation [MATH] is generated by the environment following the conditional observation distribution [MATH].', '1903.01385-2-10-3': 'Concretely, each pixel [MATH] is rendered by one of the object representations [MATH], which are transformed by a differentiable non-linear function [MATH] into variables [MATH] for separate pixel-wise distributions [MATH] .', '1903.01385-2-10-4': "Whether an observation is probable (or not surprised) given an agent's current state and action, i.e., [MATH], is quantified by its observed reward [MATH] provided by the environment according to the reward function [MATH].", '1903.01385-2-10-5': 'When the environment receives an action [MATH], it moves to a new state [MATH] following the transition function [MATH].', '1903.01385-2-11-0': '## Learning Object-based Perception Control', '1903.01385-2-12-0': 'To formalize the belief about the cause of the observation [MATH], the agent maintains a perception model [MATH] to approximate the probability of different latent states [MATH].', '1903.01385-2-12-1': "Using the encoded sufficient statistic of the history, agent's inferred belief about the latent state could serve as the input to its decision-making module (the policy) [MATH].", '1903.01385-2-12-2': 'The prediction ability of the perception model at time step [MATH] is given by the model evidence as [EQUATION]', '1903.01385-2-12-3': 'See details of derivation in the supplementary material.', '1903.01385-2-12-4': "The goal of the agent is to learn a perception model with high prediction ability by minimizing the average surprise [EQUATION] over trajectories [MATH] induced by agent's policy [MATH], We denote the distribution over initial state as [MATH].", '1903.01385-2-13-0': 'Learning the OPC via minimizing the average surprise from Eq. ([REF]) is typically achieved by marginalization of the joint distribution as', '1903.01385-2-14-0': '[EQUATION]', '1903.01385-2-15-0': 'See derivation in the supplementary material.', '1903.01385-2-15-1': 'The first term in Eq. ([REF]) is identical to the expected free energy [CITATION] along trajectory [MATH] with respect to a set of object representations [MATH], and is minimized by improving the perception [MATH] about the true posterior [MATH].', '1903.01385-2-15-2': 'Meanwhile, the second term in Eq. ([REF]) is the negative of the total reward along trajectory [MATH], and is minimized by taking actions on the environment to change the sensory input.', '1903.01385-2-15-3': 'The optimization of these terms are discussed in Sect. [REF] and Sect. [REF] respectively.', '1903.01385-2-16-0': '## Perception Model Update', '1903.01385-2-17-0': 'We formulate the perception model update as an iterative process.', '1903.01385-2-17-1': 'Given that the expected free energy is minimized with respect to [MATH] at time step [MATH], i.e., [MATH], we can generate a soft-assignment of each pixel to one of the [MATH] objects as [EQUATION]', '1903.01385-2-17-2': 'We then find [MATH] to minimize the expected free energy as [EQUATION]', '1903.01385-2-17-3': 'See derivation in the supplementary material.', '1903.01385-2-17-4': 'Note that Eq. ([REF]) returns a set of points that maximize [MATH], and we choose [MATH] to be any value within this set.', '1903.01385-2-18-0': 'To update the perception model by minimizing the expected free energy with respect to [MATH] and [MATH], we compute Eq. ([REF]) and Eq. ([REF]) iteratively.', '1903.01385-2-18-1': 'However, an analytical solution to Eq. ([REF]) is not available because we use a differentiable non-linear function [MATH] to map from object representations [MATH] into [MATH].', '1903.01385-2-18-2': 'Threfore, we get [MATH] by [EQUATION] where [MATH] is the learning rate (see details of derivation in the supplementary material).', '1903.01385-2-18-3': 'We regard the iterative process as [MATH]-step rollout of [MATH] copies of a recurrent neural network with hidden states [MATH] receiving [MATH] as input (see the inner loop of Algorithm [REF]).', '1903.01385-2-18-4': 'Each copy generates a new [MATH], which is then used to re-estimate the soft-assignments [MATH].', '1903.01385-2-18-5': 'We parameterize the Jacobian [MATH] and the differentiable non-linear function [MATH] using a convolutional encoder-decoder architecture with a recurrent neural network as bottleneck, which linearly combine the output of the encoder with [MATH] from the previous timestep.', '1903.01385-2-18-6': 'The training process is organized in an unsupervised manner by minimizing [MATH].', '1903.01385-2-19-0': '## Convergence of the Perception Model Update', '1903.01385-2-20-0': 'Under main assumptions and lemmas as introduced below, we demonstrate the convergence of a sequence of average surprise values [MATH] generated by the perception update.', '1903.01385-2-20-1': 'The proof is presented by showing that the learning process follows the Global Convergence Theorem [CITATION].', '1903.01385-2-21-0': '[MATH] is compact for any [MATH].', '1903.01385-2-22-0': '[MATH] is continuous in [MATH] and differentiable in the interior of [MATH].', '1903.01385-2-23-0': 'The above assumptions lead to the fact that [MATH] is bounded for any [MATH].', '1903.01385-2-24-0': 'Let [MATH] be the set of stationary points in the interior of [MATH], then the mapping [MATH] from Eq. [REF] is closed over [MATH] (the complement of [MATH]).', '1903.01385-2-25-0': 'See [CITATION].', '1903.01385-2-25-1': 'A sufficient condition is that [MATH] is continuous in both [MATH] and [MATH].', '1903.01385-2-26-0': 'Let [MATH] be the set of stationary points in the interior of [MATH], then (i.) [MATH] and (ii.)', '1903.01385-2-26-1': '[MATH].', '1903.01385-2-27-0': 'Note that (i.) holds true given the condition.', '1903.01385-2-27-1': 'To prove (ii.)', '1903.01385-2-27-2': ', consider any [MATH], we have [EQUATION].', '1903.01385-2-27-3': 'Hence [MATH] is not maximized at [MATH].', '1903.01385-2-27-4': 'Given the perception update described by Eq. ([REF]), we therefore have [MATH] which implies [MATH].', '1903.01385-2-28-0': 'Let [MATH] be a sequence generated by the mapping from Eq. [REF], [MATH] be the set of stationary points in the interior of [MATH].', '1903.01385-2-28-1': 'If Assumptions [REF] [REF], Lemma [REF], and Proposition [REF] are met, then all the limit points of [MATH] are stationary points (local minima) and [MATH] converges monotonically to [MATH] for some stationary point [MATH].', '1903.01385-2-29-0': 'Suppose that [MATH] is a limit point of the sequence [MATH].', '1903.01385-2-29-1': 'Given Assumptions [REF] [REF] and Proposition [REF].', '1903.01385-2-29-2': 'i), we have that the sequence [MATH] are contained in a compact set [MATH].', '1903.01385-2-29-3': 'Thus, there is a subsequence [MATH] of [MATH] such that [MATH] as [MATH] and [MATH].', '1903.01385-2-30-0': 'We first show that [MATH] as [MATH].', '1903.01385-2-30-1': 'Given [MATH] is continuous in [MATH] (Assumption [REF]), we have [MATH] as [MATH] and [MATH], which means [EQUATION]', '1903.01385-2-30-2': 'Given Proposition [REF] and Eq. ([REF]), [MATH] is therefore monotonically decreasing on the sequence [MATH], which gives [EQUATION]', '1903.01385-2-30-3': 'Given Eq. ([REF]), for any [MATH], we have [EQUATION]', '1903.01385-2-30-4': 'Given Eq. ([REF]) and Eq. ([REF]), we therefore have [MATH] as [MATH].', '1903.01385-2-30-5': 'We then prove that the limit point [MATH] is a stationary point.', '1903.01385-2-30-6': 'Suppose [MATH] is not a stationary point, i.e., [MATH], we consider the sub-sequence [MATH], which are also contained in the compact set [MATH].', '1903.01385-2-30-7': 'Thus, there is a subsequence [MATH] of [MATH] such that [MATH] as [MATH] and [MATH], yielding [MATH] as [MATH] and [MATH], which gives [EQUATION]', '1903.01385-2-30-8': 'On the other hand, since the mapping from Eq. ([REF]) is closed over [MATH] (Lemma [REF]), and [MATH], we therefore have [MATH], yielding [MATH] (Proposition [REF].', '1903.01385-2-30-9': 'ii), which contradicts Eq. ([REF]).', '1903.01385-2-31-0': '[t] Learning the Object-based Perception Control', '1903.01385-2-32-0': 'Initialize [MATH], [MATH] Create [MATH] environments that will execute in parallel training not finished Initialize the history [MATH] with environment rollouts for [MATH] time-steps under current policy [MATH] Get [MATH] from [MATH] Get [MATH] from [MATH] respectively Feed [MATH] into each of the [MATH] RNN copy to get [MATH] and forward-output [MATH] Compute [MATH] by Eq. [REF] [MATH] by Eq. ([REF]) Perform [MATH] according to policy [MATH] Receive reward [MATH] and new observation [MATH] Store [MATH] in [MATH] respectively Feed [MATH] into each of the [MATH] RNN copy to get [MATH] Perform synchronous update of [MATH] using [MATH], of [MATH] using [MATH], and of [MATH] using [MATH]', '1903.01385-2-33-0': '## Decision-making Module Update', '1903.01385-2-34-0': 'The second term in Eq. ([REF]) is the negative of the total reward along trajectory [MATH], which is straightforward to understand as an agent with a better perception model would receive a higher total reward by conditioning its policy on its inferred belief about the latent state.', '1903.01385-2-34-1': 'Intuitively, the additional knowledge provided by the decomposition of high-dimensional observations for disentangled and structured object abstractions could ease the burden of decision-making.', '1903.01385-2-35-0': "To maximize the total reward along trajectory [MATH], we follow the conventional temporal-difference (TD) learning approach [CITATION] by feeding the object abstractions to a small multilayer perceptron (MLP) [CITATION] to produce an [MATH]-dimensional vector, which is split into a [MATH]-dimensional vector of [MATH]'s (the 'actor') logits, and a baseline scalar [MATH] (the 'critic').", '1903.01385-2-35-1': 'The [MATH] logits are normalized using a softmax function, and used as the multinomial distribution from which an action is sampled.', '1903.01385-2-35-2': 'The [MATH] is an estimate of the state-value function at the current state, which is given by the last hidden state [MATH] of the [MATH]-step RNN rollout.', '1903.01385-2-35-3': 'On training the decision-making module, the [MATH] is used to compute the temporal-difference error given by [EQUATION] where [MATH] is a discount factor.', '1903.01385-2-35-4': '[MATH] is used both to optimize [MATH] to generate actions with larger total rewards than [MATH] predicts by updating [MATH] with respect to the policy gradient [EQUATION] and to optimize [MATH] to more accurately estimate state values by updating [MATH].', '1903.01385-2-35-5': 'Also, differentiating [MATH] with respect to [MATH] enables the gradient-based optimizers to update the perception model.', '1903.01385-2-35-6': 'We provide the pseudo-code for one-step TD-learning of the proposed model in Algorithm [REF].', '1903.01385-2-35-7': 'By grouping objects concerning the reward, our model distinguishes objects with high visual similarities but different semantics, thus helping the agent to better understand the environment.', '1903.01385-2-36-0': '# Related Work', '1903.01385-2-37-0': 'Model-based deep reinforcement learning algorithms have been shown to be more effective than model-free alternatives in certain tasks [CITATION].', '1903.01385-2-37-1': 'However, these models typically produce entangled latent representations for pixel observations, making them unable to facilitate physical reasoning and learn inductive biases.', '1903.01385-2-37-2': 'Although [CITATION] have used the relational mechanism to discover and reason about relevant entities, their model needs additional supervision to label entities with location information.', '1903.01385-2-37-3': 'The method most closely related to us is the World Model [CITATION], which consists of separately trained models for visual, memorizing, and control purposes, thus preventing the formerly trained components from the guidance provided by latter components.', '1903.01385-2-37-4': 'On the contrary, OPC provides the decision-making process with object-based abstractions of high-dimensional observations, which naturally contain object-based position information and contribute to faster learning.', '1903.01385-2-38-0': 'The object-based approach, which recognizes decomposed objects from the environment observations, has attracted considerable attention in RL as well [CITATION].', '1903.01385-2-38-1': 'However, most models usually assume prior knowledge of object-based representations rather than extracting from high-dimensional observation space [CITATION].', '1903.01385-2-38-2': 'When objects are extracted through learning methods, these models usually require supervised modeling of the object definition, by either comparing the activation spectrum generated from neural network filters with existing types [CITATION] or leveraging the bounding boxes generated by standard object detection algorithms in computer vision [CITATION].', '1903.01385-2-38-3': 'MOREL [CITATION] applies optical flow in video sequences and feeds the learned features to model-free RL frameworks as the position and velocity information of moving objects.', '1903.01385-2-38-4': 'On the other hand, OPC follows an unsupervised manner to extract object abstractions by minimizing the surprise of observations.', '1903.01385-2-39-0': 'Unsupervised object segmentation and representation learning have seen several recent breakthroughs, such as IODINE [CITATION] and MONet [CITATION].', '1903.01385-2-39-1': 'Although OPC is built upon previous unsupervised object segmentation back-end [CITATION], we explore one step forward by combining perceptual grouping with decision-making, which helps to break down raw pixels into structured objects in a manner appropriate to the particular task faced by the agent.', '1903.01385-2-39-2': 'Our framework also adheres to the Bayesian brain hypothesis by maintaining and updating a compact perception model towards the cause of particular observations [CITATION].', '1903.01385-2-40-0': 'Several recent works have investigated the unsupervised object extraction for reinforcement learning.', '1903.01385-2-40-1': 'OODP [CITATION] is closely related to our work by proposing a multi-step architecture combining background extraction, static object detection, and dynamic object detection.', '1903.01385-2-40-2': 'However, the background has to be fixed through time.', '1903.01385-2-40-3': 'LatPlan [CITATION] uses a variational auto-encoder (VAE) to generate a problem representation from raw pixel observations in an unsupervised manner, but requires objects to only appear in a limited number of discrete locations, e.g., each piece of an image-based instance of the 8-puzzle.', '1903.01385-2-40-4': 'A recent work by [CITATION] proposes to learn keypoints from object movements between video frames, thus do not apply to environments with static objects.', '1903.01385-2-40-5': 'COBRA [CITATION] also proposes an object-oriented MBRL framework but was based upon MONet to learn a static scene representation.', '1903.01385-2-40-6': 'Different from the above approaches, OPC learns to adjust the discovered object representations through the temporal-difference error, thus benefiting the decision-making process with the inductive bias and better generalization over goals, states, and actions [CITATION].', '1903.01385-2-41-0': '# Experiments', '1903.01385-2-42-0': '## Pixel Waterworld', '1903.01385-2-43-0': 'We demonstrate the mutual facilitation of object-based perception and reinforcement learning by applying OPC on an environment similar to the one used in COBRA [CITATION], a modified Waterworld environment [CITATION], where the observations are [MATH] grayscale raw pixel images composited of an agent and two types of bouncing targets: the poison and the food, as illustrated in Fig. ([REF]).', '1903.01385-2-43-1': 'The agent can control its velocity by choosing from four available actions: to apply thrusters to the left, right, up and down.', '1903.01385-2-43-2': 'The environment dynamics then integrate the velocity of the agent to change its position.', '1903.01385-2-43-3': 'A negative reward is awarded to the agent if it touches any poison target, while a positive reward for making contact with any food target.', '1903.01385-2-43-4': 'The optimal strategy depends on the number, the moving speed, and the size of objects, thus requiring the agent to infer the underlying dynamics of the environment within a given amount of observations.', '1903.01385-2-44-0': "The intuition of this environment is to test whether the agent can quickly learn the dynamics of a new environment without any prior knowledge, i.e., the execution-time optimization throughout the agent's life-cycle to imitate the inductive biases learning process of humans.", '1903.01385-2-44-1': 'We choose Advantage Actor-Critic (A2C) [CITATION] as the decision-making module of OPC without loss of generality, although other online learning methods are also applicable.', '1903.01385-2-44-2': 'For OPC, we use [MATH] and [MATH] for all experiments except Sect. [REF], where we analyze the effectiveness of the hyper-parameter setting.', '1903.01385-2-45-0': '## Facilitate RL by Inductive Biases: Accumulated Reward Comparisons', '1903.01385-2-46-0': 'To verify inductive biases facilitating reinforcement learning, we compare OPC against a set of baseline algorithms, including: 1) the standard A2C, which uses convolutional layers to transform the raw pixel observations to low dimensional vectors as input for the same MLP described in Sect. [REF], 2) the World Model [CITATION] (WM), a state-of-the-art model-based approach, which learns separate models for visual, memorizing, and control purposes respectively, and 3) the random policy.', '1903.01385-2-46-1': 'For both the baseline A2C and the decision-making module of OPC, we follow the convention of [CITATION] by running the algorithm in the forward view and using the same mix of [MATH]-step returns to update both the [MATH] and the [MATH].', '1903.01385-2-46-2': 'We build the environment with two high-speed poison objects and one food object, and set the size of the agent [MATH] times smaller than the target.', '1903.01385-2-46-3': 'Results are reported with separate runs of three random seeds.', '1903.01385-2-46-4': 'Note that the training procedure of WM includes independent off-line training for each component, thus requiring many more observation samples than OPC.', '1903.01385-2-46-5': 'Following its original early-stopping criteria, we report that WM requires 300 times more observation samples to give the results presented in Fig. ([REF]).', '1903.01385-2-47-0': 'Fig. ([REF]) shows the result of accumulated rewards after each agent has experienced the same amount of observations in its life-cycle, where the agent with OPC achieves the best performance as having the highest accumulated reward than agents with any other models performing in the environment.', '1903.01385-2-47-1': "We believe this advantage owes to the help of inductive biases learned by the perception model (compared to entangled representations extracted by the CNN used in standard A2C), and the joint learning of perception and control of OPC instead of the separate learning of WM. To illustrate agent's learning process through time, we also present the period reward, which is the accumulated reward during a given period of environment interactions ([MATH] of experienced observations) along the agent's life-cycle.", '1903.01385-2-47-2': 'As illustrated in Fig. ([REF]), OPC significantly improves the sample efficiency of A2C, making the agent performing in the environment find an optimal strategy more quickly than agents with baseline models.', '1903.01385-2-47-3': 'We also find that the standard version of A2C with four parallel running threads gives roughly the same result as the single-thread version of A2C (the same as the decision-making module of OPC), eliminating the potential drawback of single-thread learning.', '1903.01385-2-48-0': '## Facilitate Perception Update by the TD Signal: Perceptual Grouping Results', '1903.01385-2-49-0': 'To demonstrate the facilitation of perception update by reinforcement learning, we provide an example rollout at the early stage of training in Fig. ([REF]).', '1903.01385-2-49-1': 'All objects are shown to be grouped semantically: the agent in blue, the food in green, and both poisons in red.', '1903.01385-2-49-2': 'During joint-training of the perception model and the decision-making module, the soft-assignment gradually constitutes a semantic segmentation as the temporal-difference signal guides the perception model to recognize the agent.', '1903.01385-2-49-3': 'The TD signal then improves the recognition when grouping pixels into different types based on the interaction with the environment, thus specializing the learned inductive biases semantically.', '1903.01385-2-49-4': 'Consequently, the learned object representations [MATH] is a semantic interpretation of the raw pixels grouped into semantic perceptual objects.', '1903.01385-2-50-0': 'To gain insights into knowledge acquired by the inductive biases learning, we further compare OPC against OP, a perception model with the same architecture as OPC but no guidance of the TD signal from RL, and show the soft-assignment [MATH] produced by both perception models in Fig. ([REF]).', '1903.01385-2-50-1': 'Results are produced by models running in environments with the same random seed.', '1903.01385-2-51-0': 'As illustrated in Fig. ([REF]), OPC outperforms OP in terms of the accumulated rewards through time with a given amount of observations.', '1903.01385-2-51-1': 'This performance difference owes to the fact shown in Fig. ([REF]), here the [MATH] of OP are overlaid, and the shapes are not delineated in bold colors but are a mixture, showing that OP has not learned to segment the objects clearly.', '1903.01385-2-51-2': 'On the other hand, the quality and consistency of the soft-assignment generated by OPC are improved with the TD signal, suggesting that RL facilitates the learning of the perception model.', '1903.01385-2-51-3': 'Furthermore, the interaction between objects shown in Fig. ([REF]) demonstrates the agent is moving away from the poison (even when the food is nearby).', '1903.01385-2-51-4': 'Because of the environment setting of the high moving speed and more poison objects, we believe that agents are given strong indications to stay away from all stimuli.', '1903.01385-2-51-5': 'Thus, the result shows an understanding of visual reasoning that the agent can separate itself from the rest of the objects.', '1903.01385-2-52-0': '## Effectiveness of the Hyper-parameter Setting', '1903.01385-2-53-0': 'We further investigate the influence of hyper-parameters towards the learning ability of OPC, by changing: 1) the number of recurrent copies [MATH], 2) the rollout steps for recurrent iteration [MATH], and 3) the use of relational mechanism across object representations [MATH] described in [CITATION].', '1903.01385-2-53-1': 'We present results of the period reward across different hyper-parameter settings in Fig. ([REF]) and Fig. ([REF]).', '1903.01385-2-54-0': "As illustrated in Fig. ([REF]), the number of recurrent copies affects the stability of OPC learning, as OPC with [MATH] has experienced larger variance during the agent's life-cycle.", '1903.01385-2-54-1': 'We believe the difference comes from the environment dynamics as we have visually four objects in the environment.', '1903.01385-2-54-2': 'During the earlier stage of interacting with the environment, OPC tries to group each object into a distinct class; thus, a different number of [MATH] against the number of objects in the environment confuse the perception model and lead to unstable learning.', '1903.01385-2-54-3': 'Although different [MATH] settings might affect the learning stability and slow down the convergence, OPC can still find an optimal strategy within a given amount of observations, thus being superior to other baseline models.', '1903.01385-2-54-4': 'Meanwhile, the use of relational mechanism has limited impact on OPC, possibly because OPC does not rely on the prediction ability of the perception model, but instead benefits from the learned inductive biases.', '1903.01385-2-55-0': 'In Fig. ([REF]), we compare OPC with different steps of recurrent rollout.', '1903.01385-2-55-1': 'A shorter rollout means fewer rounds of perception update, thus suffering from slower convergence in terms of the number of experienced observations.', '1903.01385-2-55-2': 'We believe that the choice of [MATH] depends on the difficulty of the environment.', '1903.01385-2-55-3': 'For simpler environments, a smaller [MATH] can help to find the optimal strategy more quickly.', '1903.01385-2-56-0': '# Conclusions', '1903.01385-2-57-0': 'In this paper, we propose Object-based Perception Control (OPC), demonstrating the mutual facilitation of hierarchical object-based perception and reinforcement learning under the Bayesian brain hypothesis.', '1903.01385-2-57-1': 'We provide the convergence proof of OPC perception model update, and demonstrate the execution-time optimization ability of OPC in a high-dimensional pixel environment.', '1903.01385-2-57-2': "Notably, we show that OPC outperforms several strong baselines in terms of accumulated rewards within the agent's life-cycle, and the quality and consistency of the perceptual grouping.", '1903.01385-2-57-3': 'OPC agent can quickly learn the dynamics of a new environment without any prior knowledge, imitating the inductive biases learning process of humans.', '1903.01385-2-57-4': 'In future work, we would like to investigate OPC with more types of inductive biases and test the model performance in a wider variety of environments.', '1903.01385-2-58-0': '# Experiment Details', '1903.01385-2-59-0': 'OPC In all experiments we trained the perception model using ADAM kingma2014adam with default parameters and a batch size of [MATH].', '1903.01385-2-59-1': 'Each input consists of a sequence of binary [MATH] images containing two poison objects (two circles) and one food object (a rectangle) that start in random positions and move within the image for [MATH] steps.', '1903.01385-2-59-2': 'These frames were thresholded at [MATH] to obtain binary images and added with bit-flip noise ([MATH]).', '1903.01385-2-59-3': 'We used a convolutional encoder-decoder architecture inspired by recent GANs Chen2016InfoGAN with a recurrent neural network as bottleneck, where the encoder used the same network architecture from mnih2013playing as', '1903.01385-2-60-0': '[MATH] conv.', '1903.01385-2-60-1': '16 ELU.', '1903.01385-2-60-2': 'stride 4.', '1903.01385-2-60-3': 'layer norm [MATH] conv.', '1903.01385-2-60-4': '32 ELU.', '1903.01385-2-60-5': 'stride 2.', '1903.01385-2-60-6': 'layer norm fully connected.', '1903.01385-2-60-7': '256 ELU.', '1903.01385-2-60-8': 'layer norm recurrent.', '1903.01385-2-60-9': '250 Sigmoid.', '1903.01385-2-60-10': 'layer norm on the output fully connected.', '1903.01385-2-60-11': '256 RELU.', '1903.01385-2-60-12': 'layer norm fully connected.', '1903.01385-2-60-13': '[MATH] RELU.', '1903.01385-2-60-14': 'layer norm [MATH] reshape 2 nearest-neighbour, conv.', '1903.01385-2-60-15': '16 RELU.', '1903.01385-2-60-16': 'layer norm [MATH] reshape 4 nearest-neighbour, conv.', '1903.01385-2-60-17': '1 Sigmoid', '1903.01385-2-61-0': 'We used the Advantage Actor-Critic (A2C) pmlr-v48-mniha16 with an MLP policy as the decision making module of OPC.', '1903.01385-2-61-1': 'The MLP policy added a 512-unit fully connected layer with rectifier nonlinearity after layer 4 of the perception model.', '1903.01385-2-61-2': 'The decision making module had two set of outputs: 1) a softmax output with one entry per action representing the probability of selecting the action, and 2) a single linear output representing the value function.', '1903.01385-2-61-3': 'The decision making module was trained using RMSProp tieleman2012lecture with a learning rate of [MATH], a reward discount factor [MATH], an RMSProp decay factor of [MATH], and performed updates after every [MATH] actions.', '1903.01385-2-62-0': 'A2C We used the same convolutional architecture as the encoder of the perception model of OPC (layer 1 to 3), followed by a fully connected layer with 512 hidden units followed by a rectifier nonlinearity.', '1903.01385-2-62-1': 'The A2C was trained using the same setting as the decision making module of OPC.', '1903.01385-2-63-0': 'WM-A2C We used the same setting as ha2018worldmodels to separately train the V model and the M model.', '1903.01385-2-63-1': 'The experience was generated off-line by a random policy operating in the Pixel Waterworld environment.', '1903.01385-2-63-2': 'We concatenated the output of the V model and the M model as the A2C input, and trained A2C using the same setting as introduced above.', '1903.01385-2-64-0': '# Details of Derivation', '1903.01385-2-65-0': '## Derivation of Eq. ([REF])', '1903.01385-2-66-0': '[EQUATION]', '1903.01385-2-67-0': '## Derivation of Eq. ([REF])', '1903.01385-2-68-0': '[EQUATION]', '1903.01385-2-69-0': '## Derivation of Eq. ([REF])', '1903.01385-2-70-0': '[EQUATION]', '1903.01385-2-71-0': '## Derivation of Eq. ([REF])', '1903.01385-2-72-0': '[EQUATION]'}

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[]

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{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1903.01385

quant-ph-0411202

{'quant-ph-0411202-1-0-0': 'We suggest the application of nitronylnitroxide substituted with methyl group and 2,2,6,6-tetramethylpiperidin organic radicals as [MATH]-spin qubits for self-assembled monolayer quantum devices.', 'quant-ph-0411202-1-0-1': 'We show that the oscillating cantilever driven adiabatic reversals technique can provide the read-out of the spin states.', 'quant-ph-0411202-1-0-2': 'We compute components of the [MATH]-tensor and dipole-dipole interaction tensor for these radicals.', 'quant-ph-0411202-1-0-3': 'We show that the delocalization of the spin in the radical may significantly influence the dipole-dipole interaction between the spins.', 'quant-ph-0411202-1-1-0': '# 1.', 'quant-ph-0411202-1-1-1': 'Introduction', 'quant-ph-0411202-1-2-0': 'The self assembled monolayer (SAM) molecular systems with no or small amount of defects are the promising candidates for many electronic devices [CITATION].', 'quant-ph-0411202-1-2-1': 'SAM systems containing radicals with unpaired spin [MATH] can be used in quantum logic devices [CITATION].', 'quant-ph-0411202-1-2-2': 'It was suggested using micro-wires, which produce a magnetic field gradient, and rf pulses, which induce Rabi transitions, to manipulate with the radical spins and create the quantum entanglement between them.', 'quant-ph-0411202-1-2-3': 'Later the conditions for the molecules appropriate for the quantum logic devices have been formulated, and 1,3-diketone radicals have been suggested and theoretically analyzed from the point of view of their application to the quantum logical devices [CITATION].', 'quant-ph-0411202-1-3-0': 'This paper pursues two objectives.', 'quant-ph-0411202-1-3-1': 'First, we suggest two stable organic radicals, nitronylnitroxide substituted with methyl group (NITRO) and 2,2,6,6-tetramethylpiperidin (TEMPO), for SAM quantum devices.', 'quant-ph-0411202-1-3-2': 'In order to stimulate the experiments we compute the [MATH]-tensor and zz-component of the dipole-dipole interaction tensor.', 'quant-ph-0411202-1-3-3': 'Second, we suggest the novel oscillating cantilever driven adiabatic reversals (OSCAR) technique as the tool for read-out of the spin states.', 'quant-ph-0411202-1-3-4': 'The OSCAR technique has been recently successfully implemented for the single-spin detection [CITATION].', 'quant-ph-0411202-1-3-5': 'In section II we describe the effective spin Hamiltonian for the two interacting radicals, in section III we consider the method of creation and detection entanglement between the two spins, in section III we describe the methods of computation of the [MATH]-tensor and the dipole-dipole interaction tensors, and in section IV discuss the results of our quantum chemical computations.', 'quant-ph-0411202-1-4-0': '# 2.', 'quant-ph-0411202-1-4-1': 'Spin Hamiltonian for two qubits', 'quant-ph-0411202-1-5-0': 'Let us discuss the simplest possible quantum computer element - the two qubit system.', 'quant-ph-0411202-1-5-1': 'In SAM the two organic radicals are embedded in a monolayer.', 'quant-ph-0411202-1-5-2': 'Electron spins [MATH] localized on the radicals represent the two qubit system.', 'quant-ph-0411202-1-5-3': 'We use geometrical arrangement the same as in the Ref. [[CITATION]], where external magnetic field [MATH] is oriented along the [MATH]-axis and radicals are separated by the distance [MATH] along the [MATH]-axis (see Fig. 1).', 'quant-ph-0411202-1-5-4': 'In addition to constant external magnetic field [MATH] a gradient of external magnetic field [MATH] along the [MATH]-axis is applied on the SAM.', 'quant-ph-0411202-1-5-5': 'If radicals in SAM are oriented in such a way that the principle [MATH]-axis of the electronic g-tensor corresponds to the orientation of the external magnetic field [MATH], then the effective Hamiltonian for the spin group is [EQUATION] where [MATH], [MATH] are respectively radical one and radical two spin projection operators, [MATH] - the Bohr magneton, [MATH] - the component of the electronic g-tensor of a radical (we assume that the radicals are the same in the spin group) and [MATH] is the dipole-dipole interaction tensor component.', 'quant-ph-0411202-1-5-6': 'The Hamiltonian ([REF]) can be treated in electron spin functions basis [MATH], [MATH] = 0 or 1, where 0 stands for the spin ground state and 1 for the spin excited state).', 'quant-ph-0411202-1-5-7': 'As in our system the magnetic field gradient along the [MATH]-axis exists, both radicals in the spin group (first and second) have distinguished spin transition frequencies, which depend on the local magnetic field strength at the radical position and additionally shifted by the dipole-dipole interaction between the radicals.', 'quant-ph-0411202-1-6-0': 'Under these conditions, energy levels of the system are defined by the following expressions: [EQUATION] ([MATH] and [MATH] are negative), the external magnetic field on the second spin is [MATH] and on the first spin is [MATH]).', 'quant-ph-0411202-1-6-1': 'The energy levels diagram of two qubit system with four possible transitions is given in Fig. 2.', 'quant-ph-0411202-1-6-2': 'The transition frequencies are determined by the following expressions ([MATH]) : [EQUATION].', 'quant-ph-0411202-1-6-3': 'Here [MATH] means the transition frequency for the spin "i" ([MATH]) if the neighboring spin is in the state [MATH]).', 'quant-ph-0411202-1-6-4': 'Given formulas for radical spin energy levels involve two molecular parameters: electronic g-tensor component [MATH] (property of radical itself) and dipole-dipole coupling parameter [MATH], which is the property of the two radical system.', 'quant-ph-0411202-1-6-5': 'Both these properties can be efficiently evaluated using quantum chemistry methods and influence of various parameters (radicals geometries, distance between radicals in spin group, radical relative orientation to each other, etc.) on these properties can be investigated.', 'quant-ph-0411202-1-7-0': '# 3.', 'quant-ph-0411202-1-7-1': 'Entangled spin states', 'quant-ph-0411202-1-8-0': 'In this section, we consider the creation and detection of the entangled state [EQUATION] where [MATH] is a nonsignificant phase factor.', 'quant-ph-0411202-1-9-0': 'First, one applies a [MATH]-pulse with the frequency [MATH] (we assume that initially both spins are in their ground state [MATH], as it is shown in Fig. 1).', 'quant-ph-0411202-1-9-1': 'This pulse drives the spin system into the state [EQUATION].', 'quant-ph-0411202-1-9-2': 'Now one applies a [MATH]-pulse with the frequency [MATH].', 'quant-ph-0411202-1-9-3': 'This pulse drives the second spin if the first spin is in the state [MATH].', 'quant-ph-0411202-1-9-4': 'Thus, the new state of the system is [EQUATION].', 'quant-ph-0411202-1-9-5': 'This is a desired entangled state of the two-qubit system.', 'quant-ph-0411202-1-10-0': 'To detect this state one uses the OSCAR technique.', 'quant-ph-0411202-1-10-1': 'In the OSCAR technique the cantilever tip with the ferromagnetic particle oscillates along the line which is parallel to the sample surface.', 'quant-ph-0411202-1-10-2': 'The distance between the ferromagnetic particle and the selected radical spin changes periodically.', 'quant-ph-0411202-1-10-3': 'When the cantilever tip is in its end point one starts to apply the rf field.', 'quant-ph-0411202-1-10-4': 'The magnetic field on the spin changes its magnitude with the cantilever period.', 'quant-ph-0411202-1-10-5': 'Suppose that the equilibrium position of the cantilever corresponds to the resonant condition for the spin.', 'quant-ph-0411202-1-10-6': 'Then, in the rotating system of coordinates the effective field on the spin changes its direction from +z to -z in the x-z plane (we assume that the rotating rf field points in the positive x-direction of the rotating system).', 'quant-ph-0411202-1-10-7': 'The condition of the adiabatic reversals is [EQUATION] where [MATH] is the effective magnetic field in the rotating frame, [MATH] is the rf field amplitude, and [MATH] is the free electron g-factor.', 'quant-ph-0411202-1-10-8': 'If the condition of the adiabatic reversals is satisfied the spin follows the effective field.', 'quant-ph-0411202-1-11-0': 'The back action of the spin on the cantilever tip causes the frequency shift of the cantilever vibrations, which can be measured with high accuracy.', 'quant-ph-0411202-1-11-1': 'The cantilever frequency shift can take two values [MATH] depending on the spin direction relatively the effective field.', 'quant-ph-0411202-1-11-2': 'Measuring the sign of the frequency shift one can find the initial direction of the spin relatively to the effective magnetic field.', 'quant-ph-0411202-1-11-3': 'We assume that initially (when we start application of the rf field) the effective magnetic field has the same direction as the external magnetic field.', 'quant-ph-0411202-1-11-4': 'In this case the ground state of the resonant spin corresponds to the negative frequency shift of the cantilever vibrations [CITATION].', 'quant-ph-0411202-1-12-0': 'How to verify the entangled state of the two-spin system?', 'quant-ph-0411202-1-12-1': 'The entangled state manifests itself in the two outcomes of the measurement: [MATH] or [MATH].', 'quant-ph-0411202-1-12-2': 'Let, for example, we apply the rf field with the frequency [MATH].', 'quant-ph-0411202-1-12-3': 'In general, we have three possible outcomes for the measurement of the cantilever frequency shift: 1) The cantilever frequency shift is negative.', 'quant-ph-0411202-1-12-4': 'It means that the second (right) spin is in the ground state, and the first (left) spin is also in the ground state.', 'quant-ph-0411202-1-12-5': 'Thus, our system collapsed to the state [MATH] (in the system of coordinates connected to the effective field).', 'quant-ph-0411202-1-12-6': '2) The frequency shift is zero.', 'quant-ph-0411202-1-12-7': 'It means that our system initially collapsed to the states [MATH] or [MATH] (or their superposition), which are insensitive to the frequency [MATH].', 'quant-ph-0411202-1-12-8': 'In this case, after the integer number of the cantilever cycles we may change the rf field frequency from [MATH] to, for example, [MATH].', 'quant-ph-0411202-1-12-9': 'Now we have two possible outcomes: 2a) For the state [MATH] the left resonant spin will provide the positive cantilever frequency shift; 2b) For the state [MATH] there is no resonant spin, and the frequency shift is zero.', 'quant-ph-0411202-1-12-10': '3) The frequency shift is positive.', 'quant-ph-0411202-1-12-11': 'It means the second resonant spin is in the excited state [MATH], and the first spin is in the ground state [MATH], i.e. our system has collapsed to [MATH].', 'quant-ph-0411202-1-12-12': 'Thus, the repeated outcomes 1) or 2b) (with equal probability) correspond to the initial entangled state, which collapses to the states [MATH] or [MATH].', 'quant-ph-0411202-1-12-13': 'The outcomes 2a) or 3) will show that the system is not in the expected entangled state.', 'quant-ph-0411202-1-12-14': 'One can see that the OSCAR technique provides the simple verification of the quantum entanglement.', 'quant-ph-0411202-1-12-15': 'Note that the verification experiment must be conducted for the time interval smaller than the characteristic time between the spin quantum jumps.', 'quant-ph-0411202-1-13-0': '# 4.', 'quant-ph-0411202-1-13-1': 'Computational methods', 'quant-ph-0411202-1-14-0': 'A. Electronic [MATH]-tensor', 'quant-ph-0411202-1-15-0': 'According to Spin Hamiltonian (see Eq. [REF]) electronic [MATH]-tensor of a radical is defined as the second derivative of the molecular electronic energy [MATH] [EQUATION]', 'quant-ph-0411202-1-15-1': 'For molecules described by a Breit-Pauli Hamiltonian, where the spin and the magnetic field as well as all other relativistic corrections are treated in perturbation theory framework, the molecular electronic [MATH]-tensor can be evaluated using expression (up to second order in the fine structure constant [MATH]) [CITATION]: [EQUATION]', 'quant-ph-0411202-1-15-2': 'In this equation, the first terms is the free electron g-factor, which comes from the electronic Zeeman operator in the Breit-Pauli Hamiltonian, with the radiative corrections from quantum electrodynamics introduced into the Hamiltonian empirically.', 'quant-ph-0411202-1-15-3': 'The next three terms originate from first order perturbation theory applied to the Breit-Pauli Hamiltonian and are the mass-velocity and the one- and two electron corrections to the electronic Zeeman effect.', 'quant-ph-0411202-1-15-4': 'The last two terms are, respectively, the one- and two-electron corrections contributing to the electronic g-tensor to second order in perturbation theory as cross terms between the spin-orbit operators and the orbital Zeeman operator.', 'quant-ph-0411202-1-15-5': 'All terms except the free electron g-factor value, [MATH], in Eq. ([REF]) contribute to the electronic g-tensor shift [MATH], which accounts for the influence of the local electronic environment in the molecule on the unpaired electrons.', 'quant-ph-0411202-1-15-6': 'From the contributions to the g-tensor shift given in Eq. ([REF]), the first three terms are evaluated straightforwardly according to Eq. ([REF]) from expectation values of the corresponding Breit-Pauli Hamiltonian operators.', 'quant-ph-0411202-1-15-7': 'The last two terms, namely spin-orbit (SO) corrections, are considerably more involving computationally, as they are defined in terms of second-order perturbation theory, i.e. their calculation formally require a knowledge of all relevant excited states energies and wave functions.', 'quant-ph-0411202-1-15-8': 'The evaluation of these terms using density functional theory (DFT) methods are most often done using various kinds of approximate sum-over-states approaches or response theory methods [CITATION].', 'quant-ph-0411202-1-16-0': 'In the following part of this section we briefly describe electronic [MATH]-tensor evaluation approach implemented in quantum chemistry code DALTON [CITATION], which is based on a restricted DFT linear response theory [CITATION].', 'quant-ph-0411202-1-16-1': 'In this approach from the first-order perturbation theory contributions to the electronic g-tensor shift, we only include those terms that involve one-electron operators as DFT in principle can not handle two-electron operators [CITATION].', 'quant-ph-0411202-1-16-2': 'The Cartesian [MATH] components of these contributions to [MATH] tensor are [EQUATION] where [MATH] is the canonical linear momentum of electron [MATH], [MATH] the z-component of the spin operator of electron [MATH], [MATH] and [MATH] are the position vectors of electron [MATH] relative to nucleus [MATH] and the magnetic gauge origin [MATH], respectively.', 'quant-ph-0411202-1-16-3': 'In the above equations [MATH] is chosen to be the ground-state wave function with maximum spin projection [MATH].', 'quant-ph-0411202-1-16-4': 'Neglecting of the two-electron gauge correction, [MATH], the contribution to the [MATH]-tensor shift in DFT calculations does not influence the accuracy of the g-tensor evaluation as this term only gives a correction from a two-electron screening of the [MATH] and considering the smallness of the one-electron gauge correction term itself is justified [CITATION].', 'quant-ph-0411202-1-16-5': 'The major contributions to [MATH] tensor arising from second-order perturbation theory, so-called one- and two-electron SO corrections, are evaluated as linear response functions [EQUATION] where the spectral representation of the linear response function at zero frequency for two arbitrary operators [MATH] and [MATH] is given by [EQUATION]', 'quant-ph-0411202-1-16-6': 'In Eq. (8) [MATH] is the cartesian [MATH] component of the angular momentum operator of electron [MATH], and [MATH] and [MATH] are the Cartesian [MATH] components of the one- and two-electron SO operators.', 'quant-ph-0411202-1-16-7': 'The Cartesian component [MATH] of the one-electron SO operator is defined as [EQUATION]', 'quant-ph-0411202-1-16-8': 'As mentioned above in DFT two-electron operators can not be evaluated properly and one need to introduce one or another approximation of the two-electron operators in order to perform calculations within limits of formalism.', 'quant-ph-0411202-1-16-9': 'For g-tensor calculations performed in this work we selected to approximate two-electron SO operators by an Atomic Mean Field (AMFI) SO operator [CITATION] as previous experience with the AMFI SO approximation in ab intio and DFT works devoted to electronic g-tensors calculations have been very encouraging and no significant problems with the accuracy of the AMFI SO approximation has been reported [CITATION].', 'quant-ph-0411202-1-17-0': 'B. Dipole-dipole coupling', 'quant-ph-0411202-1-18-0': 'Formally, in the case of two radicals with the spins [MATH], the interaction operator and corresponding [MATH] tensor may be written as [EQUATION] where [MATH], [MATH] are respectively radical one and radical two effective spin operators, [MATH] - the [MATH] position vector with respect to [MATH].', 'quant-ph-0411202-1-18-1': 'In above definition of the dipole-dipole interaction tensor we assumed that each radical electronic g-tensor is isotropic and equal to the free electron g-factor.', 'quant-ph-0411202-1-18-2': 'In this work we selected two methods for calculation of the dipole-dipole interaction tensor required for determination of transition frequencies in the two qubit system (see Eqs. (3)): classical point dipole approach and "point dipole-spin density" approach.', 'quant-ph-0411202-1-18-3': 'In case of classical point-dipole approximation delocalization of unpaired electrons is neglected and D tensor components are evaluated (in our two spin system geometry, r=(0,a,0)) as [EQUATION].', 'quant-ph-0411202-1-18-4': 'One can expect this approach to be accurate for large separation of radicals, then the unpaired electrons delocalization region are very small comparing to the distance between the radicals.', 'quant-ph-0411202-1-18-5': 'However, in case of moderate separation between radicals (0.5-1.5 nm) unpaired electrons delocalization in radicals can not be neglected and should be explicitly taken into account evaluating spin-dipole interaction tensor D.', 'quant-ph-0411202-1-18-6': 'The simplest approach, which partially accounts for unpaired electrons distribution in radicals, is "point dipole-spin density" approach in which one radical unpaired electron treated as delocalized and another radical electron magnetic moment represented as a point dipole.', 'quant-ph-0411202-1-18-7': 'In this approximation D tensor components are evaluated as [EQUATION] where [MATH] is the electron spin density distribution in radical and [MATH] is the electron position with respect to the magnetic dipole position [MATH], [MATH].', 'quant-ph-0411202-1-18-8': 'The described approach only partially takes into account delocalization of the unpaired electrons in [MATH] tensor calculations, but in order to obtain qualitative picture of the electron delocalization influence on the dipole-dipole interaction of the two radicals is sufficient.', 'quant-ph-0411202-1-18-9': 'In this work we use this approach to determine limits of the classical point dipole approach for evaluation of the dipole-dipole interaction tensor [MATH] in two radicals system.', 'quant-ph-0411202-1-19-0': '# 5.', 'quant-ph-0411202-1-19-1': 'Computational details', 'quant-ph-0411202-1-20-0': 'As the promising candidates for qubits in SAM of organic radicals we selected two well-known stable organic radicals, namely methyl group substituted nitronyl nitroxide (NITRO) and 2,2,6,6-tetramethylpiperidin (TEMPO) (see Fig. 3), which form stable organic crystals and already have applications in material sciences as well as biochemistry.', 'quant-ph-0411202-1-20-1': 'Both chosen radicals possess one unpaired electron and therefore each radical can be recognized as a single qubit.', 'quant-ph-0411202-1-20-2': 'The geometry of NITRO and TEMPO radicals used in calculations of electronic g-tensors and of dipole-dipole interaction tensor D have been obtained by performing geometry optimization of single radical in 6-311G(d,p) basis set [CITATION] at the B3LYP [CITATION] level .', 'quant-ph-0411202-1-20-3': 'All geometry optimizations have been carried out in Gaussian-98 program [CITATION].', 'quant-ph-0411202-1-20-4': 'Apart from optimizing geometries of NITRO and TEMPO radicals we performed geometry optimization of the NITRO and TEMPO radicals derivatives with [MATH] tails (see Fig. 4).', 'quant-ph-0411202-1-20-5': 'The [MATH] groups were added to radicals in order to simulate conventional structure of the compounds used in formation of SAM, which usually feature long [MATH] groups or similar tails.', 'quant-ph-0411202-1-20-6': 'Obtained structures of NITRO and TEMPO radicals with [MATH] tails have been used to investigate influence of the tails on the properties of the NITRO and TEMPO derivatives compared to free NITRO and TEMPO radicals and allowed us to estimate feasibility of the NITRO and TEMPO radicals usage as basic building blocks of the spin arrays in SAM.', 'quant-ph-0411202-1-21-0': 'Calculations of electronic g-tensors for NITRO and TEMPO radicals as well as their derivatives with [MATH] tails have been carried out using BP86 exchange-correlation functional [CITATION], which gives accurate results for organic radicals.', 'quant-ph-0411202-1-21-1': 'In all calculations of g-tensors we employed IGLO-II basis set [CITATION] especially designed for evaluation of magnetic properties.', 'quant-ph-0411202-1-21-2': 'In order to estimate influence of the [MATH] (n=1,2) tails, which usually added to radicals in order to enable formation of SAM, on the electronic g-tensors of the radicals we carried out electronic g-tensors calculations for single NITRO and TEMPO radicals (see Fig. 3) as their derivatives with [MATH] tails (see Fig. 4).', 'quant-ph-0411202-1-22-0': 'Calculations of the dipole-dipole interaction tensor between two NITRO or two TEMPO radicals have been performed using previously described point dipole and "point dipole-spin density" approaches varying the distance between radicals, [MATH], from 1 nm to 2 nm.', 'quant-ph-0411202-1-22-1': 'Single radicals optimized geometries have been employed in calculations using "point dipole-spin density" approach.', 'quant-ph-0411202-1-22-2': "All calculations have been performed at the B3LYP level using Duning's double zeta basis set [CITATION], which allow adequate description of the electron density distribution in investigated radicals.", 'quant-ph-0411202-1-22-3': 'In calculations of the dipole-dipole interaction tensor we limited ourselves only by investigation of the [MATH] component dependence on the distance between radicals [MATH] and contrary to investigation of the electronic g-tensors do not carried out investigation of the [MATH] tails influence on dipole-dipole interaction tensor [MATH], as our g-tensor calculations indicated negligible influence of the [MATH] tails on unpaired electron density distribution in both NITRO and TEMPO radicals.', 'quant-ph-0411202-1-23-0': '# 6.', 'quant-ph-0411202-1-23-1': 'Results and discussion', 'quant-ph-0411202-1-24-0': 'A. Electronic g-tensor', 'quant-ph-0411202-1-25-0': 'Electronic g-tensor calculations results for single NITRO and TEMPO molecules as well as their derivatives with [MATH] n=1,2 tails are tabulated in Table 1.', 'quant-ph-0411202-1-25-1': '[REF] The results of electronic g-tensor calculations for NITRO compound with [MATH] tail separated for two different conformations of this compound, which differ only by orientation of the tail with one [MATH] unit.', 'quant-ph-0411202-1-25-2': 'However, already for tail consisting of the two [MATH] units, there is no difference between NITRO A and NITRO B conformations due to the increased flexibility of tail, which leads geometry optimization procedure converges to same structure independently on the starting geometry.', 'quant-ph-0411202-1-25-3': 'Here, we note even thought our calculations predict small radical tail rotation around C-C bonds it does not correspond to the "real" behavior of the [MATH] tail in SAM, as the motion of the tail is constrained by surrounding molecule tails in SAM.', 'quant-ph-0411202-1-25-4': 'Now let us turn discussion from radicals geometrical structure features to their electronic g-tensors, which are one of the key quantities in our two qubit system Spin Hamiltonian.', 'quant-ph-0411202-1-26-0': 'All electronic g-tensor components, presented in Table 1, [REF] are only slightly altered by addition of [MATH] tail chain units, suggesting small distortion of unpaired electron density in radical by such chemical modification.', 'quant-ph-0411202-1-26-1': 'The electronic g-tensor shifts almost converge, and extension of the chain further from two to three [MATH] units changes g-shift components in range 1-5 ppm.', 'quant-ph-0411202-1-26-2': 'Therefore, we conclude that electronic g-tensors of the NITRO and TEMPO radicals are only slightly affected by chemical addition of the [MATH] tail, which is required for growth of the SAM, i.e. both radicals preserve their properties in SAM.', 'quant-ph-0411202-1-26-3': 'Selected radicals have highly anisotropic electronic g-tensor (see Table 1 [REF] ) with large [MATH] and [MATH] components (for orientation of the electronic g-tensor axes see Fig. 1).', 'quant-ph-0411202-1-26-4': 'Only [MATH] is small, and therefore along this axis we have molecular g-tensor component close to the free electron g-factor.', 'quant-ph-0411202-1-26-5': 'The large anisotropy of the electronic g-tensor poses significant difficulties for the quantum computation prospective as in order to have well defined frequencies of the transitions it is essential to have fixed orientation of the electronic g-tensor principal axes with respect to the external magnetic field, i.e. radicals in SAM should be fixed in their position and rotations of the radicals with respect to their g-tensor principle axes must be restrained.', 'quant-ph-0411202-1-27-0': 'B. Dipole-dipole interaction', 'quant-ph-0411202-1-28-0': 'Dipole-dipole interaction tensor between two radicals is one of the key parameters in the spin Hamiltonian, which describes the two qubit system.', 'quant-ph-0411202-1-28-1': 'We employed different approaches, namely classical point dipole approach and "point dipole-spin density" approach, for evaluation of the [MATH].', 'quant-ph-0411202-1-28-2': 'The second approach as discussed previously partially accounts electron distribution in molecule and therefore results are directly dependent on the unpaired electron density localization in the molecule.', 'quant-ph-0411202-1-28-3': 'Contrary, the first approach does not account for unpaired electron delocalization in radical and therefore gives the same results for all radicals.', 'quant-ph-0411202-1-28-4': 'The dipole-dipole interaction dependence on the distance between radicals [MATH] is plotted in Fig. 5, where results for both classical point dipole approach (denoted classical) and "point dipole-spin density" approach (denoted Nitro and Tempo for NITRO and TEMPO radicals, respectively).', 'quant-ph-0411202-1-28-5': 'Quick inspection of these plots indicates a substantial influence of the unpaired electron delocalization on the [MATH] value especially at the small [MATH] region.', 'quant-ph-0411202-1-28-6': 'Therefore, unpaired electron delocalization can not be neglected in evaluation of the [MATH] tensor and previous data, which have been used in modeling of the qubits system, obtained with classical point dipole approach should be carefully reexamined for [MATH] range of the 10-15 AA.', 'quant-ph-0411202-1-28-7': 'Another implication of the non-negligible contribution from unpaired electron delocalization to the dipole-dipole interaction is that orientation of the radical can influence [MATH] tensor components values.', 'quant-ph-0411202-1-28-8': 'Therefore, similarly to electronic g-tensor calculations the dipole-dipole interaction modeling results suggest that the rigid fixation of the radical in SAM with specific orientation is one of the key requirements in production of SAM in order to make them useful in quantum computing.', 'quant-ph-0411202-1-29-0': '# 7.', 'quant-ph-0411202-1-29-1': 'Conclusion', 'quant-ph-0411202-1-30-0': 'We have suggested using NITRO and TEMPO radicals as spin qubits for a quantum logical device based on SAM systems.', 'quant-ph-0411202-1-30-1': 'In order to stimulate experimental implementation of our idea we have computed the components of g-tensor and dipole-dipole interaction tensor for these radicals.', 'quant-ph-0411202-1-30-2': 'We have shown that adding [MATH] tail chain does not influence significantly the radical electron g-factor.', 'quant-ph-0411202-1-30-3': 'Delocalization of the electron spin in radicals influences the dipole-dipole interaction between the radicals.', 'quant-ph-0411202-1-30-4': 'We suggested a scheme for detection of the entanglement between the two radicals based on the novel OSCAR technique, which has been recently used for the single-spin detection.', 'quant-ph-0411202-1-31-0': '# ACKNOWLEDGEMENTS', 'quant-ph-0411202-1-32-0': 'We are grateful to G. D. Doolen for useful discussions.', 'quant-ph-0411202-1-32-1': 'This work was supported by the Department of Energy (DOE) under Contract No. W-7405-ENG-36, by the National Security Agency (NSA) and Advanced Research and Development Activity (ARDA) under Army Research Office (ARO) contract 707003.'}

{'quant-ph-0411202-2-0-0': 'We suggest the application of nitronylnitroxide substituted with methyl group and 2,2,6,6-tetramethylpiperidin organic radicals as [MATH]-spin qubits for self-assembled monolayer quantum devices.', 'quant-ph-0411202-2-0-1': 'We show that the oscillating cantilever driven adiabatic reversals technique can provide the read-out of the spin states.', 'quant-ph-0411202-2-0-2': 'We compute components of the [MATH]-tensor and dipole-dipole interaction tensor for these radicals.', 'quant-ph-0411202-2-0-3': 'We show that the delocalization of the spin in the radical may significantly influence the dipole-dipole interaction between the spins.', 'quant-ph-0411202-2-1-0': '# 1.', 'quant-ph-0411202-2-1-1': 'Introduction', 'quant-ph-0411202-2-2-0': 'The self assembled monolayer (SAM) molecular systems with no or small amount of defects are the promising candidates for many electronic devices [CITATION].', 'quant-ph-0411202-2-2-1': 'SAM systems containing radicals with unpaired spin [MATH] can be used in quantum logic devices [CITATION].', 'quant-ph-0411202-2-2-2': 'It was suggested using micro-wires, which produce a magnetic field gradient, and rf pulses, which induce Rabi transitions, to manipulate with the radical spins and create the quantum entanglement between them.', 'quant-ph-0411202-2-2-3': 'Later the conditions for the molecules appropriate for the quantum logic devices have been formulated, and 1,3-diketone radicals have been suggested and theoretically analyzed from the point of view of their application to the quantum logical devices [CITATION].', 'quant-ph-0411202-2-3-0': 'This paper pursues two objectives.', 'quant-ph-0411202-2-3-1': 'First, we suggest two stable organic radicals, nitronylnitroxide substituted with methyl group (NITRO) and 2,2,6,6-tetramethylpiperidin (TEMPO), for SAM quantum devices.', 'quant-ph-0411202-2-3-2': 'In order to stimulate the experiments we compute the [MATH]-tensor and zz-component of the dipole-dipole interaction tensor.', 'quant-ph-0411202-2-3-3': 'Second, we suggest the novel oscillating cantilever driven adiabatic reversals (OSCAR) technique as the tool for read-out of the spin states.', 'quant-ph-0411202-2-3-4': 'The OSCAR technique has been recently successfully implemented for the single-spin detection [CITATION].', 'quant-ph-0411202-2-3-5': 'In section II we describe the effective spin Hamiltonian for the two interacting radicals, in section III we consider the method of creation and detection entanglement between the two spins, in section III we describe the methods of computation of the [MATH]-tensor and the dipole-dipole interaction tensors, and in section IV discuss the results of our quantum chemical computations.', 'quant-ph-0411202-2-4-0': '# 2.', 'quant-ph-0411202-2-4-1': 'Spin Hamiltonian for two qubits', 'quant-ph-0411202-2-5-0': 'Let us discuss the simplest possible quantum computer element - the two qubit system.', 'quant-ph-0411202-2-5-1': 'In SAM the two organic radicals are embedded in a monolayer.', 'quant-ph-0411202-2-5-2': 'Electron spins [MATH] localized on the radicals represent the two qubit system.', 'quant-ph-0411202-2-5-3': 'We use geometrical arrangement the same as in the Ref. [[CITATION]], where external magnetic field [MATH] is oriented along the [MATH]-axis and radicals are separated by the distance [MATH] along the [MATH]-axis (see Fig. 1).', 'quant-ph-0411202-2-5-4': 'In addition to constant external magnetic field [MATH] a gradient of external magnetic field [MATH] along the [MATH]-axis is applied on the SAM.', 'quant-ph-0411202-2-5-5': 'If radicals in SAM are oriented in such a way that the principle [MATH]-axis of the electronic g-tensor corresponds to the orientation of the external magnetic field [MATH], then the effective Hamiltonian for the spin group is [EQUATION] where [MATH], [MATH] are respectively radical one and radical two spin projection operators, [MATH] - the Bohr magneton, [MATH] - the component of the electronic g-tensor of a radical (we assume that the radicals are the same in the spin group) and [MATH] is the dipole-dipole interaction tensor component.', 'quant-ph-0411202-2-5-6': 'The Hamiltonian ([REF]) can be treated in electron spin functions basis [MATH], [MATH] = 0 or 1, where 0 stands for the spin ground state and 1 for the spin excited state).', 'quant-ph-0411202-2-5-7': 'As in our system the magnetic field gradient along the [MATH]-axis exists, both radicals in the spin group (first and second) have distinguished spin transition frequencies, which depend on the local magnetic field strength at the radical position and additionally shifted by the dipole-dipole interaction between the radicals.', 'quant-ph-0411202-2-6-0': 'Under these conditions, energy levels of the system are defined by the following expressions: [EQUATION] ([MATH] and [MATH] are negative), the external magnetic field on the second spin is [MATH] and on the first spin is [MATH]).', 'quant-ph-0411202-2-6-1': 'The energy levels diagram of two qubit system with four possible transitions is given in Fig. 2.', 'quant-ph-0411202-2-6-2': 'The transition frequencies are determined by the following expressions ([MATH]) : [EQUATION].', 'quant-ph-0411202-2-6-3': 'Here [MATH] means the transition frequency for the spin "i" ([MATH]) if the neighboring spin is in the state [MATH]).', 'quant-ph-0411202-2-6-4': 'Given formulas for radical spin energy levels involve two molecular parameters: electronic g-tensor component [MATH] (property of radical itself) and dipole-dipole coupling parameter [MATH], which is the property of the two radical system.', 'quant-ph-0411202-2-6-5': 'Both these properties can be efficiently evaluated using quantum chemistry methods and influence of various parameters (radicals geometries, distance between radicals in spin group, radical relative orientation to each other, etc.) on these properties can be investigated.', 'quant-ph-0411202-2-7-0': '# 3.', 'quant-ph-0411202-2-7-1': 'Entangled spin states', 'quant-ph-0411202-2-8-0': 'In this section, we consider the creation and detection of the entangled state [EQUATION] where [MATH] is a nonsignificant phase factor.', 'quant-ph-0411202-2-9-0': 'First, one applies a [MATH]-pulse with the frequency [MATH] (we assume that initially both spins are in their ground state [MATH], as it is shown in Fig. 1).', 'quant-ph-0411202-2-9-1': 'This pulse drives the spin system into the state [EQUATION].', 'quant-ph-0411202-2-9-2': 'Now one applies a [MATH]-pulse with the frequency [MATH].', 'quant-ph-0411202-2-9-3': 'This pulse drives the second spin if the first spin is in the state [MATH].', 'quant-ph-0411202-2-9-4': 'Thus, the new state of the system is [EQUATION].', 'quant-ph-0411202-2-9-5': 'This is a desired entangled state of the two-qubit system.', 'quant-ph-0411202-2-10-0': 'To detect this state one uses the OSCAR technique.', 'quant-ph-0411202-2-10-1': 'In the OSCAR technique the cantilever tip with the ferromagnetic particle oscillates along the line which is parallel to the sample surface.', 'quant-ph-0411202-2-10-2': 'The distance between the ferromagnetic particle and the selected radical spin changes periodically.', 'quant-ph-0411202-2-10-3': 'When the cantilever tip is in its end point one starts to apply the rf field.', 'quant-ph-0411202-2-10-4': 'The magnetic field on the spin changes its magnitude with the cantilever period.', 'quant-ph-0411202-2-10-5': 'Suppose that the equilibrium position of the cantilever corresponds to the resonant condition for the spin.', 'quant-ph-0411202-2-10-6': 'Then, in the rotating system of coordinates the effective field on the spin changes its direction from +z to -z in the x-z plane (we assume that the rotating rf field points in the positive x-direction of the rotating system).', 'quant-ph-0411202-2-10-7': 'The condition of the adiabatic reversals is [EQUATION] where [MATH] is the effective magnetic field in the rotating frame, [MATH] is the rf field amplitude, and [MATH] is the free electron g-factor.', 'quant-ph-0411202-2-10-8': 'If the condition of the adiabatic reversals is satisfied the spin follows the effective field.', 'quant-ph-0411202-2-11-0': 'The back action of the spin on the cantilever tip causes the frequency shift of the cantilever vibrations, which can be measured with high accuracy.', 'quant-ph-0411202-2-11-1': 'The cantilever frequency shift can take two values [MATH] depending on the spin direction relatively the effective field.', 'quant-ph-0411202-2-11-2': 'Measuring the sign of the frequency shift one can find the initial direction of the spin relatively to the effective magnetic field.', 'quant-ph-0411202-2-11-3': 'We assume that initially (when we start application of the rf field) the effective magnetic field has the same direction as the external magnetic field.', 'quant-ph-0411202-2-11-4': 'In this case the ground state of the resonant spin corresponds to the negative frequency shift of the cantilever vibrations [CITATION].', 'quant-ph-0411202-2-12-0': 'How to verify the entangled state of the two-spin system?', 'quant-ph-0411202-2-12-1': 'The entangled state manifests itself in the two outcomes of the measurement: [MATH] or [MATH].', 'quant-ph-0411202-2-12-2': 'Let, for example, we apply the rf field with the frequency [MATH].', 'quant-ph-0411202-2-12-3': 'In general, we have three possible outcomes for the measurement of the cantilever frequency shift: 1) The cantilever frequency shift is negative.', 'quant-ph-0411202-2-12-4': 'It means that the second (right) spin is in the ground state, and the first (left) spin is also in the ground state.', 'quant-ph-0411202-2-12-5': 'Thus, our system collapsed to the state [MATH] (in the system of coordinates connected to the effective field).', 'quant-ph-0411202-2-12-6': '2) The frequency shift is zero.', 'quant-ph-0411202-2-12-7': 'It means that our system initially collapsed to the states [MATH] or [MATH] (or their superposition), which are insensitive to the frequency [MATH].', 'quant-ph-0411202-2-12-8': 'In this case, after the integer number of the cantilever cycles we may change the rf field frequency from [MATH] to, for example, [MATH].', 'quant-ph-0411202-2-12-9': 'Now we have two possible outcomes: 2a) For the state [MATH] the left resonant spin will provide the positive cantilever frequency shift; 2b) For the state [MATH] there is no resonant spin, and the frequency shift is zero.', 'quant-ph-0411202-2-12-10': '3) The frequency shift is positive.', 'quant-ph-0411202-2-12-11': 'It means the second resonant spin is in the excited state [MATH], and the first spin is in the ground state [MATH], i.e. our system has collapsed to [MATH].', 'quant-ph-0411202-2-12-12': 'Thus, the repeated outcomes 1) or 2b) (with equal probability) correspond to the initial entangled state, which collapses to the states [MATH] or [MATH].', 'quant-ph-0411202-2-12-13': 'The outcomes 2a) or 3) will show that the system is not in the expected entangled state.', 'quant-ph-0411202-2-12-14': 'One can see that the OSCAR technique provides the simple verification of the quantum entanglement.', 'quant-ph-0411202-2-12-15': 'Note that the verification experiment must be conducted for the time interval smaller than the characteristic time between the spin quantum jumps.', 'quant-ph-0411202-2-13-0': '# 4.', 'quant-ph-0411202-2-13-1': 'Computational methods', 'quant-ph-0411202-2-14-0': 'A. Electronic [MATH]-tensor', 'quant-ph-0411202-2-15-0': 'According to Spin Hamiltonian (see Eq. [REF]) electronic [MATH]-tensor of a radical is defined as the second derivative of the molecular electronic energy [MATH] [EQUATION]', 'quant-ph-0411202-2-15-1': 'For molecules described by a Breit-Pauli Hamiltonian, where the spin and the magnetic field as well as all other relativistic corrections are treated in perturbation theory framework, the molecular electronic [MATH]-tensor can be evaluated using expression (up to second order in the fine structure constant [MATH]) [CITATION]: [EQUATION]', 'quant-ph-0411202-2-15-2': 'In this equation, the first terms is the free electron g-factor, which comes from the electronic Zeeman operator in the Breit-Pauli Hamiltonian, with the radiative corrections from quantum electrodynamics introduced into the Hamiltonian empirically.', 'quant-ph-0411202-2-15-3': 'The next three terms originate from first order perturbation theory applied to the Breit-Pauli Hamiltonian and are the mass-velocity and the one- and two electron corrections to the electronic Zeeman effect.', 'quant-ph-0411202-2-15-4': 'The last two terms are, respectively, the one- and two-electron corrections contributing to the electronic g-tensor to second order in perturbation theory as cross terms between the spin-orbit operators and the orbital Zeeman operator.', 'quant-ph-0411202-2-15-5': 'All terms except the free electron g-factor value, [MATH], in Eq. ([REF]) contribute to the electronic g-tensor shift [MATH], which accounts for the influence of the local electronic environment in the molecule on the unpaired electrons.', 'quant-ph-0411202-2-15-6': 'From the contributions to the g-tensor shift given in Eq. ([REF]), the first three terms are evaluated straightforwardly according to Eq. ([REF]) from expectation values of the corresponding Breit-Pauli Hamiltonian operators.', 'quant-ph-0411202-2-15-7': 'The last two terms, namely spin-orbit (SO) corrections, are considerably more involving computationally, as they are defined in terms of second-order perturbation theory, i.e. their calculation formally require a knowledge of all relevant excited states energies and wave functions.', 'quant-ph-0411202-2-15-8': 'The evaluation of these terms using density functional theory (DFT) methods are most often done using various kinds of approximate sum-over-states approaches or response theory methods [CITATION].', 'quant-ph-0411202-2-16-0': 'In the following part of this section we briefly describe electronic [MATH]-tensor evaluation approach implemented in quantum chemistry code DALTON [CITATION], which is based on a restricted DFT linear response theory [CITATION].', 'quant-ph-0411202-2-16-1': 'In this approach from the first-order perturbation theory contributions to the electronic g-tensor shift, we only include those terms that involve one-electron operators as DFT in principle can not handle two-electron operators [CITATION].', 'quant-ph-0411202-2-16-2': 'The Cartesian [MATH] components of these contributions to [MATH] tensor are [EQUATION] where [MATH] is the canonical linear momentum of electron [MATH], [MATH] the z-component of the spin operator of electron [MATH], [MATH] and [MATH] are the position vectors of electron [MATH] relative to nucleus [MATH] and the magnetic gauge origin [MATH], respectively.', 'quant-ph-0411202-2-16-3': 'In the above equations [MATH] is chosen to be the ground-state wave function with maximum spin projection [MATH].', 'quant-ph-0411202-2-16-4': 'Neglecting of the two-electron gauge correction, [MATH], the contribution to the [MATH]-tensor shift in DFT calculations does not influence the accuracy of the g-tensor evaluation as this term only gives a correction from a two-electron screening of the [MATH] and considering the smallness of the one-electron gauge correction term itself is justified [CITATION].', 'quant-ph-0411202-2-16-5': 'The major contributions to [MATH] tensor arising from second-order perturbation theory, so-called one- and two-electron SO corrections, are evaluated as linear response functions [EQUATION] where the spectral representation of the linear response function at zero frequency for two arbitrary operators [MATH] and [MATH] is given by [EQUATION]', 'quant-ph-0411202-2-16-6': 'In Eq. (8) [MATH] is the cartesian [MATH] component of the angular momentum operator of electron [MATH], and [MATH] and [MATH] are the Cartesian [MATH] components of the one- and two-electron SO operators.', 'quant-ph-0411202-2-16-7': 'The Cartesian component [MATH] of the one-electron SO operator is defined as [EQUATION]', 'quant-ph-0411202-2-16-8': 'As mentioned above in DFT two-electron operators can not be evaluated properly and one need to introduce one or another approximation of the two-electron operators in order to perform calculations within limits of formalism.', 'quant-ph-0411202-2-16-9': 'For g-tensor calculations performed in this work we selected to approximate two-electron SO operators by an Atomic Mean Field (AMFI) SO operator [CITATION] as previous experience with the AMFI SO approximation in ab intio and DFT works devoted to electronic g-tensors calculations have been very encouraging and no significant problems with the accuracy of the AMFI SO approximation has been reported [CITATION].', 'quant-ph-0411202-2-17-0': 'B. Dipole-dipole coupling', 'quant-ph-0411202-2-18-0': 'Formally, in the case of two radicals with the spins [MATH], the interaction operator and corresponding [MATH] tensor may be written as [EQUATION] where [MATH], [MATH] are respectively radical one and radical two effective spin operators, [MATH] - the [MATH] position vector with respect to [MATH].', 'quant-ph-0411202-2-18-1': 'In above definition of the dipole-dipole interaction tensor we assumed that each radical electronic g-tensor is isotropic and equal to the free electron g-factor.', 'quant-ph-0411202-2-18-2': 'In this work we selected two methods for calculation of the dipole-dipole interaction tensor required for determination of transition frequencies in the two qubit system (see Eqs. (3)): classical point dipole approach and "point dipole-spin density" approach.', 'quant-ph-0411202-2-18-3': 'In case of classical point-dipole approximation delocalization of unpaired electrons is neglected and D tensor components are evaluated (in our two spin system geometry, r=(0,a,0)) as [EQUATION].', 'quant-ph-0411202-2-18-4': 'One can expect this approach to be accurate for large separation of radicals, then the unpaired electrons delocalization region are very small comparing to the distance between the radicals.', 'quant-ph-0411202-2-18-5': 'However, in case of moderate separation between radicals (0.5-1.5 nm) unpaired electrons delocalization in radicals can not be neglected and should be explicitly taken into account evaluating spin-dipole interaction tensor D.', 'quant-ph-0411202-2-18-6': 'The simplest approach, which partially accounts for unpaired electrons distribution in radicals, is "point dipole-spin density" approach in which one radical unpaired electron treated as delocalized and another radical electron magnetic moment represented as a point dipole.', 'quant-ph-0411202-2-18-7': 'In this approximation D tensor components are evaluated as [EQUATION] where [MATH] is the electron spin density distribution in radical and [MATH] is the electron position with respect to the magnetic dipole position [MATH], [MATH].', 'quant-ph-0411202-2-18-8': 'The described approach only partially takes into account delocalization of the unpaired electrons in [MATH] tensor calculations, but in order to obtain qualitative picture of the electron delocalization influence on the dipole-dipole interaction of the two radicals is sufficient.', 'quant-ph-0411202-2-18-9': 'In this work we use this approach to determine limits of the classical point dipole approach for evaluation of the dipole-dipole interaction tensor [MATH] in two radicals system.', 'quant-ph-0411202-2-19-0': '# 5.', 'quant-ph-0411202-2-19-1': 'Computational details', 'quant-ph-0411202-2-20-0': 'As the promising candidates for qubits in SAM of organic radicals we selected two well-known stable organic radicals, namely methyl group substituted nitronyl nitroxide (NITRO) and 2,2,6,6-tetramethylpiperidin (TEMPO) (see Fig. 3), which form stable organic crystals and already have applications in material sciences as well as biochemistry.', 'quant-ph-0411202-2-20-1': 'Both chosen radicals possess one unpaired electron and therefore each radical can be recognized as a single qubit.', 'quant-ph-0411202-2-20-2': 'The geometry of NITRO and TEMPO radicals used in calculations of electronic g-tensors and of dipole-dipole interaction tensor D have been obtained by performing geometry optimization of single radical in 6-311G(d,p) basis set [CITATION] at the B3LYP [CITATION] level .', 'quant-ph-0411202-2-20-3': 'All geometry optimizations have been carried out in Gaussian-98 program [CITATION].', 'quant-ph-0411202-2-20-4': 'Apart from optimizing geometries of NITRO and TEMPO radicals we performed geometry optimization of the NITRO and TEMPO radicals derivatives with [MATH] tails (see Fig. 4).', 'quant-ph-0411202-2-20-5': 'The [MATH] groups were added to radicals in order to simulate conventional structure of the compounds used in formation of SAM, which usually feature long [MATH] groups or similar tails.', 'quant-ph-0411202-2-20-6': 'Obtained structures of NITRO and TEMPO radicals with [MATH] tails have been used to investigate influence of the tails on the properties of the NITRO and TEMPO derivatives compared to free NITRO and TEMPO radicals and allowed us to estimate feasibility of the NITRO and TEMPO radicals usage as basic building blocks of the spin arrays in SAM.', 'quant-ph-0411202-2-21-0': 'Calculations of electronic g-tensors for NITRO and TEMPO radicals as well as their derivatives with [MATH] tails have been carried out using BP86 exchange-correlation functional [CITATION], which gives accurate results for organic radicals.', 'quant-ph-0411202-2-21-1': 'In all calculations of g-tensors we employed IGLO-II basis set [CITATION] especially designed for evaluation of magnetic properties.', 'quant-ph-0411202-2-21-2': 'In order to estimate influence of the [MATH] (n=1,2) tails, which usually added to radicals in order to enable formation of SAM, on the electronic g-tensors of the radicals we carried out electronic g-tensors calculations for single NITRO and TEMPO radicals (see Fig. 3) as their derivatives with [MATH] tails (see Fig. 4).', 'quant-ph-0411202-2-22-0': 'Calculations of the dipole-dipole interaction tensor between two NITRO or two TEMPO radicals have been performed using previously described point dipole and "point dipole-spin density" approaches varying the distance between radicals, [MATH], from 1 nm to 2 nm.', 'quant-ph-0411202-2-22-1': 'Single radicals optimized geometries have been employed in calculations using "point dipole-spin density" approach.', 'quant-ph-0411202-2-22-2': "All calculations have been performed at the B3LYP level using Duning's double zeta basis set [CITATION], which allow adequate description of the electron density distribution in investigated radicals.", 'quant-ph-0411202-2-22-3': 'In calculations of the dipole-dipole interaction tensor we limited ourselves only by investigation of the [MATH] component dependence on the distance between radicals [MATH] and contrary to investigation of the electronic g-tensors do not carried out investigation of the [MATH] tails influence on dipole-dipole interaction tensor [MATH], as our g-tensor calculations indicated negligible influence of the [MATH] tails on unpaired electron density distribution in both NITRO and TEMPO radicals.', 'quant-ph-0411202-2-23-0': '# 6.', 'quant-ph-0411202-2-23-1': 'Results and discussion', 'quant-ph-0411202-2-24-0': 'A. Electronic g-tensor', 'quant-ph-0411202-2-25-0': 'Electronic g-tensor calculations results for single NITRO and TEMPO molecules as well as their derivatives with [MATH] n=1,2 tails are tabulated in Table 1.', 'quant-ph-0411202-2-25-1': '[REF] The results of electronic g-tensor calculations for NITRO compound with [MATH] tail separated for two different conformations of this compound, which differ only by orientation of the tail with one [MATH] unit.', 'quant-ph-0411202-2-25-2': 'However, already for tail consisting of the two [MATH] units, there is no difference between NITRO A and NITRO B conformations due to the increased flexibility of tail, which leads geometry optimization procedure converges to same structure independently on the starting geometry.', 'quant-ph-0411202-2-25-3': 'Here, we note even thought our calculations predict small radical tail rotation around C-C bonds it does not correspond to the "real" behavior of the [MATH] tail in SAM, as the motion of the tail is constrained by surrounding molecule tails in SAM.', 'quant-ph-0411202-2-25-4': 'Now let us turn discussion from radicals geometrical structure features to their electronic g-tensors, which are one of the key quantities in our two qubit system Spin Hamiltonian.', 'quant-ph-0411202-2-26-0': 'All electronic g-tensor components, presented in Table 1, [REF] are only slightly altered by addition of [MATH] tail chain units, suggesting small distortion of unpaired electron density in radical by such chemical modification.', 'quant-ph-0411202-2-26-1': 'The electronic g-tensor shifts almost converge, and extension of the chain further from two to three [MATH] units changes g-shift components in range 1-5 ppm.', 'quant-ph-0411202-2-26-2': 'Therefore, we conclude that electronic g-tensors of the NITRO and TEMPO radicals are only slightly affected by chemical addition of the [MATH] tail, which is required for growth of the SAM, i.e. both radicals preserve their properties in SAM.', 'quant-ph-0411202-2-26-3': 'Selected radicals have highly anisotropic electronic g-tensor (see Table 1 [REF] ) with large [MATH] and [MATH] components (for orientation of the electronic g-tensor axes see Fig. 1).', 'quant-ph-0411202-2-26-4': 'Only [MATH] is small, and therefore along this axis we have molecular g-tensor component close to the free electron g-factor.', 'quant-ph-0411202-2-26-5': 'The large anisotropy of the electronic g-tensor poses significant difficulties for the quantum computation prospective as in order to have well defined frequencies of the transitions it is essential to have fixed orientation of the electronic g-tensor principal axes with respect to the external magnetic field, i.e. radicals in SAM should be fixed in their position and rotations of the radicals with respect to their g-tensor principle axes must be restrained.', 'quant-ph-0411202-2-27-0': 'B. Dipole-dipole interaction', 'quant-ph-0411202-2-28-0': 'Dipole-dipole interaction tensor between two radicals is one of the key parameters in the spin Hamiltonian, which describes the two qubit system.', 'quant-ph-0411202-2-28-1': 'We employed different approaches, namely classical point dipole approach and "point dipole-spin density" approach, for evaluation of the [MATH].', 'quant-ph-0411202-2-28-2': 'The second approach as discussed previously partially accounts electron distribution in molecule and therefore results are directly dependent on the unpaired electron density localization in the molecule.', 'quant-ph-0411202-2-28-3': 'Contrary, the first approach does not account for unpaired electron delocalization in radical and therefore gives the same results for all radicals.', 'quant-ph-0411202-2-28-4': 'The dipole-dipole interaction dependence on the distance between radicals [MATH] is plotted in Fig. 5, where results for both classical point dipole approach (denoted classical) and "point dipole-spin density" approach (denoted Nitro and Tempo for NITRO and TEMPO radicals, respectively).', 'quant-ph-0411202-2-28-5': 'Quick inspection of these plots indicates a substantial influence of the unpaired electron delocalization on the [MATH] value especially at the small [MATH] region.', 'quant-ph-0411202-2-28-6': 'Therefore, unpaired electron delocalization can not be neglected in evaluation of the [MATH] tensor and previous data, which have been used in modeling of the qubits system, obtained with classical point dipole approach should be carefully reexamined for [MATH] range of the 10-15 AA.', 'quant-ph-0411202-2-28-7': 'Another implication of the non-negligible contribution from unpaired electron delocalization to the dipole-dipole interaction is that orientation of the radical can influence [MATH] tensor components values.', 'quant-ph-0411202-2-28-8': 'Therefore, similarly to electronic g-tensor calculations the dipole-dipole interaction modeling results suggest that the rigid fixation of the radical in SAM with specific orientation is one of the key requirements in production of SAM in order to make them useful in quantum computing.', 'quant-ph-0411202-2-29-0': '# 7.', 'quant-ph-0411202-2-29-1': 'Conclusion', 'quant-ph-0411202-2-30-0': 'We have suggested using NITRO and TEMPO radicals as spin qubits for a quantum logical device based on SAM systems.', 'quant-ph-0411202-2-30-1': 'In order to stimulate experimental implementation of our idea we have computed the components of g-tensor and dipole-dipole interaction tensor for these radicals.', 'quant-ph-0411202-2-30-2': 'We have shown that adding [MATH] tail chain does not influence significantly the radical electron g-factor.', 'quant-ph-0411202-2-30-3': 'Delocalization of the electron spin in radicals influences the dipole-dipole interaction between the radicals.', 'quant-ph-0411202-2-30-4': 'We suggested a scheme for detection of the entanglement between the two radicals based on the novel OSCAR technique, which has been recently used for the single-spin detection.', 'quant-ph-0411202-2-31-0': '# ACKNOWLEDGEMENTS', 'quant-ph-0411202-2-32-0': 'We are grateful to G. D. Doolen for useful discussions.', 'quant-ph-0411202-2-32-1': 'This work was supported by the Department of Energy (DOE) under Contract No. W-7405-ENG-36, by the National Security Agency (NSA) and Advanced Research and Development Activity (ARDA) under Army Research Office (ARO) contract 707003.'}

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['quant-ph-0411202-1-1-1', 'quant-ph-0411202-1-4-0', 'quant-ph-0411202-1-4-1', 'quant-ph-0411202-1-7-1', 'quant-ph-0411202-1-13-1', 'quant-ph-0411202-1-14-0', 'quant-ph-0411202-1-17-0', 'quant-ph-0411202-1-19-1', 'quant-ph-0411202-1-23-1', 'quant-ph-0411202-1-24-0', 'quant-ph-0411202-1-27-0', 'quant-ph-0411202-1-29-1', 'quant-ph-0411202-1-32-1', 'quant-ph-0411202-2-1-1', 'quant-ph-0411202-2-4-0', 'quant-ph-0411202-2-4-1', 'quant-ph-0411202-2-7-1', 'quant-ph-0411202-2-13-1', 'quant-ph-0411202-2-14-0', 'quant-ph-0411202-2-17-0', 'quant-ph-0411202-2-19-1', 'quant-ph-0411202-2-23-1', 'quant-ph-0411202-2-24-0', 'quant-ph-0411202-2-27-0', 'quant-ph-0411202-2-29-1', 'quant-ph-0411202-2-32-1']

{'1': 'http://arxiv.org/licenses/assumed-1991-2003/', '2': 'http://arxiv.org/licenses/assumed-1991-2003/'}

https://arxiv.org/abs/quant-ph/0411202

1708.03791

{'1708.03791-1-0-0': 'Recent events highlight the widespread risks malicious hacking poses to a modern internet-connected society.', '1708.03791-1-0-1': 'As automotive vehicles increase their level of computerized sophistication, they have become, and will continue to be, targets of opportunity.', '1708.03791-1-0-2': 'Here, we study how the beneficial emergent collective motion of autonomous vehicles can be weaponized and offer strategies to preemptively inoculate against this attack.', '1708.03791-1-0-3': 'Our approach combines empirical measurements with simulations of an active matter model to demonstrate that autonomous vehicles enhance traffic throughput, and that a relatively unsophisticated attack can have disproportionately negative consequences.', '1708.03791-1-1-0': 'Autonomous self-driving vehicles are slated to bring substantial transformation to the automotive industry with foreseeable ripple effects in commercial and private sectors[CITATION] [see SM].', '1708.03791-1-1-1': 'These disruptions stem from the ability of autonomous vehicles to sense and rapidly respond to their environment.', '1708.03791-1-1-2': 'Inevitably, however, the software enabling these functions has a greater "surface area" for weakness to be exploited.', '1708.03791-1-1-3': 'As the recent public disclosure of stockpiled NSA hacking tools[CITATION] and the subsequent global-scale ransomware attacks show[CITATION], software exploits can be weaponized by malicious actors.', '1708.03791-1-1-4': 'Already, proof-of-concept hacks in 2015 and 2016 have demonstrated the ability to remotely control the brakes, acceleration, steering, transmission, locks, stereo, and windshield wipers of a targeted vehicle by exploiting weaknesses including the internet-connected entertainment console[CITATION].', '1708.03791-1-1-5': 'While attacks narrowly aimed at individual vehicles will continue to occur, another class of hacks that weaponizes emergent collective motion appears just as inevitable [see SM].', '1708.03791-1-1-6': 'Fortunately, the physics of active matter[CITATION] offers a framework for understanding how collective motion can be weaponized without knowing the details of a specific software exploit a priori.', '1708.03791-1-2-0': 'Here, we study the emergent collective motion of human and autonomously-driven vehicles.', '1708.03791-1-2-1': 'We first focus on the benefits of autonomous driving where fast reaction times and uniform driving speeds enable smooth traffic flow, even at high densities.', '1708.03791-1-2-2': 'Just as "white hat" hackers work to proactively identify software vulnerabilities, we shift our attention to study how immobilized hacked vehicles disrupt traffic.', '1708.03791-1-2-3': 'While the collective motion of autonomous vehicles that comes from their uniformity is beneficial, a malicious attack can lead to the undesirable emergence of obstructed flow.', '1708.03791-1-3-0': 'From the various approaches used to study vehicular collective motion[CITATION], we chose a force-based model[CITATION] that represents each vehicle by their position [MATH] at time [MATH] on a straight road of length [MATH].', '1708.03791-1-3-1': 'Numerical simulations were performed using velocity Verlet integration for [MATH] vehicles seeded with uniformly distributed random initial positions, zero initial velocity, zero initial acceleration, and periodic boundary conditions.', '1708.03791-1-3-2': 'The position of each vehicle was evolved using a self-propulsion force [MATH], and a repulsive collision-avoidance force [MATH] according to [EQUATION] where [MATH] and [MATH] are the velocity and acceleration of a given vehicle [see SM].', '1708.03791-1-3-3': 'Each vehicle has a preferred speed [MATH] and a characteristic response time [MATH] that it takes to equilibrate to this speed; in the absence of other vehicles, [MATH] and direct integration shows [MATH].', '1708.03791-1-3-4': 'When the number of vehicles [MATH], the repulsive force acts to slow a given vehicle as it approaches another vehicle from behind.', '1708.03791-1-3-5': 'The strength of this force increases from zero as the vehicle-vehicle separation [MATH] becomes smaller than the interaction threshold distance [MATH].', '1708.03791-1-3-6': 'This threshold reflects the 2-second rule, which states that a minimum safe distance between two vehicles is the distance traveled during 2 seconds.', '1708.03791-1-3-7': 'The functional form of the repulsion force follows from the compression of two elastic spheres[CITATION], and only applies to the trailing vehicle, as drivers respond to traffic ahead of them more strongly than behind.', '1708.03791-1-3-8': 'Given [MATH] is finite for all [MATH], this term is numerically stable in multi-vehicle simulations.', '1708.03791-1-3-9': 'However, the finite magnitude requires we fix the coefficient [MATH] in terms of other variables so that the maximum propulsion force does not lead to unphysical effects such as one vehicle passing through another.', '1708.03791-1-3-10': 'Balancing propulsion and repulsion forces when two vehicles are bumper-to-bumper, we set [MATH] m as the typical size of a vehicle and find [MATH].', '1708.03791-1-4-0': 'We focus on the emergent collective motion of two types of vehicles: human-driven and fully autonomous.', '1708.03791-1-4-1': 'In Eq. ([REF]), this simplification amounts to distinct values for the variables with subscript [MATH], where we use [MATH] for human-driven vehicles and [MATH] for autonomous vehicles.', '1708.03791-1-4-2': 'For each of the [MATH] human drivers, we draw their preferred velocity [MATH] from a Gaussian distribution with mean 29 m/s (65 mi/h) and standard deviation 6.7 m/s (15 mi/h), approximating typical highway conditions.', '1708.03791-1-4-3': 'The only remaining free parameter in the equations of motion for human-driven vehicles is the response time [MATH], which generally varies from 0.5 s to several seconds, depending on attentiveness, age, driving conditions, visibility, and level of distractions.', '1708.03791-1-4-4': 'To empirically measure [MATH], we analyzed three sources of driving data including (i) a highway web cam in the San Francisco/Oakland Bay, California area (northbound 101 at N 1[MATH] street) [Fig. [REF](a)][CITATION], (ii) digital footage we filmed in Atlanta, Georgia (southbound I85 exit 249C) [Fig. [REF](b)], and (iii) previously published Department of Transportation (DOT) traffic data from Los Angeles, California (southbound US 101; NGSIM June 15, 2005, 7:50 to 8:05 am)[CITATION].', '1708.03791-1-4-5': 'While the DOT data provided vehicle coordinates [MATH], we used quantitative image analysis on the first two video sources to obtain this information [see SM].', '1708.03791-1-4-6': 'Differentiating and smoothing with a Savitzky-Golay filter allowed us to identify vehicles that stopped and accelerated over a time [MATH] s due to stop-and-go traffic.', '1708.03791-1-4-7': 'Assuming the repulsion force is negligible during the "go" phase, we fit this segment of the velocity data from each vehicle using non-linear least squares to the functional form [MATH] resulting in measurements of [MATH] [Fig. [REF](c)].', '1708.03791-1-4-8': 'In high-density traffic, we observed drivers tend to accelerate over longer periods of time, whereas in low-density traffic, drivers tend to react faster.', '1708.03791-1-4-9': 'Because Eq. ([REF]) simplifies driver response time to be density-independent, we perform a weighted average of the empirical measurements to set [MATH] s [see SM], which is [MATH] the typical human reaction time to visual stimulus, and reasonably models attentive drivers.', '1708.03791-1-5-0': 'In simulations of an [MATH] lane road, [MATH] km long, and with [MATH] human-driven vehicles, we find a brief transient [MATH] integration time steps followed by steady-state dynamics.', '1708.03791-1-5-1': 'Consistent with our empirical observations [see SM], we subsequently observed backwards-propagating density waves superimposed on the 1D line of traffic.', '1708.03791-1-5-2': 'These density waves cause stop-and-go motion of individual vehicles, and are otherwise known as "phantom traffic jams."', '1708.03791-1-5-3': 'This emergent collective phenomenon has been extensively studied[CITATION], and is generally understood as arising from a finite driver response time.', '1708.03791-1-6-0': 'To perform multi-lane simulations, we require a model for lane-switching behavior.', '1708.03791-1-6-1': 'Currently, our only source of stochasticity in the steady-state dynamics are the [MATH] random velocities [MATH].', '1708.03791-1-6-2': 'To preserve this feature of the model, we use a deterministic force-based approach where each vehicle is in a lane [MATH] that takes discrete values (e.g. [MATH]).', '1708.03791-1-6-3': 'The evolution of [MATH] for each vehicle is calculated by [MATH], where [MATH] is the equation for an overdamped oscillator driven by a force with magnitude [MATH] [see SM].', '1708.03791-1-6-4': 'This ad hoc model for lane switching leads to stable lane preferences with vehicles traveling together at similar speeds, and self-sorting so that faster drivers switch lanes to bypass slower drivers.', '1708.03791-1-6-5': 'We fixed all parameters governing the dynamics of [MATH] at values that reproduce familiar phenomenology and validated our approach by comparing simulated and empirical density-dependent speed distributions [see SM].', '1708.03791-1-7-0': 'Beyond qualitative observations of phantom traffic jams, the density-dependent flux [MATH] is a useful quantity to study emergent collective properties of traffic flow [Fig. [REF]].', '1708.03791-1-7-1': 'In empirical measurements and simulations with [MATH] lanes, we use a portion of the road [MATH] m to calculate [MATH] and [MATH], where the index [MATH] runs over all vehicles in [MATH] sampled at statistically independent temporal intervals.', '1708.03791-1-7-2': 'When the density is low, [MATH] [Fig. [REF], [MATH] cars/km/lane].', '1708.03791-1-7-3': 'As the density increases to a critical value [MATH] cars/km/lane, the flux peaks, and subsequently declines for increasing [MATH].', '1708.03791-1-7-4': 'Both in empirical measurements and simulations, we find [MATH] corresponds to an average of one car per 25 m of road per lane.', '1708.03791-1-7-5': 'This distance is [MATH] s of travel time between vehicles, which places them well-within the repulsive-force interaction distance [MATH].', '1708.03791-1-7-6': 'As such, when [MATH], the system effectively becomes a 1D continuum with density waves (phantom traffic jams), and hence [MATH] declines.', '1708.03791-1-8-0': 'Turning to self-driving vehicles, we model each of the [MATH] autonomous drivers by fixing their preferred velocity [MATH] m/s to be constant and equal to the typical speed limit (65 mi/hr).', '1708.03791-1-8-1': 'This reflects the notion that autonomous vehicles will be engineered to travel at the legally allowed maximum speed to optimize traffic flow.', '1708.03791-1-8-2': 'We also expect autonomous vehicles to utilize a combination of local sensor information, wirelessly shared non-local traffic conditions, and cloud-connected AI to respond to driving conditions more rapidly than human drivers.', '1708.03791-1-8-3': 'As an order of magnitude estimate, we set the autonomous vehicle response time [MATH].', '1708.03791-1-8-4': 'Thus, the only difference between human-driven and autonomous vehicles in this active matter model are (i) response time [MATH] and (ii) preferred speed [MATH].', '1708.03791-1-8-5': 'Simulations with all autonomous vehicles immediately show important differences in the flux, highlighting a key advantage of autonomous driving [Fig. [REF]].', '1708.03791-1-8-6': 'Specifically, [MATH] still peaks near the same critical density [MATH], however, the subsequent decrease in flux associated with phantom traffic jams essentially vanishes and instead declines a modest [MATH] with a 4-fold increase in density [Fig. [REF], upper dark line].', '1708.03791-1-8-7': 'This trend is a significant improvement compared to human-driven vehicles where empirical observations and simulations show [MATH] loss in flux over the same range in [MATH] [Fig. [REF]].', '1708.03791-1-8-8': 'Interestingly, [MATH] in all-autonomous vehicle simulations can be predicted from Eq. ([REF]).', '1708.03791-1-8-9': 'Given the uniform preferred speed and rapid response time, [MATH].', '1708.03791-1-8-10': 'Under steady conditions, propulsion and repulsion forces balance so that [MATH] and [MATH], which leads to [MATH].', '1708.03791-1-8-11': 'The flux is therefore the real part of [MATH], in agreement with simulations [Fig. [REF], dashed line].', '1708.03791-1-8-12': 'Substituting [MATH] shows [MATH] is independent of [MATH] in the limit of uniformly distributed autonomous vehicles.', '1708.03791-1-9-0': 'Given that autonomous vehicles can be hacked, it seems self-evident they will be hacked.', '1708.03791-1-9-1': 'Our goal is to look beyond the specifics of a given exploit, and to anticipate how autonomous vehicles can be weaponized.', '1708.03791-1-9-2': 'One possibility is that hacked vehicles are critically disabled causing them to become immobile obstacles on the road.', '1708.03791-1-9-3': 'This type of "bricking" attack would cause significant social harm: emergency vehicles would be unable to respond to calls for help, food shipments to grocery stores would be delayed, and long-distance commuters would be unable to get to work.', '1708.03791-1-9-4': 'Urban cores and suburban towns, which are key centers of economic activity, would cease to function normally.', '1708.03791-1-9-5': 'Relative to its unsophisticated nature, a bricking attack would have disproportionate effects.', '1708.03791-1-9-6': 'Within the context of our active matter model, bricking amounts to setting [MATH], and the subsequent collective properties can be understood through measurements of [MATH].', '1708.03791-1-9-7': 'In the case of an all autonomous fleet, the net flux simply goes to zero for all [MATH] [Fig. [REF], lower dark line].', '1708.03791-1-9-8': 'However, even with equal numbers of human and autonomous vehicles, the flux is dramatically suppressed [Fig. [REF], lower light line].', '1708.03791-1-9-9': 'Essentially, the malicious hack causes the [MATH] autonomous vehicles to become irregularly spaced obstacles clogging[CITATION] the road, and obstructing flow of the remaining [MATH] human-driven vehicles.', '1708.03791-1-10-0': 'For a more complete view of how the flux is affected by malicious hacks, we treat [MATH] as an order parameter and generate the phase diagram composed of vehicle density [MATH] and the fraction of autonomously driven vehicles [MATH] [Fig. [REF](a)].', '1708.03791-1-10-1': 'As expected from the simulations of all human, all autonomous, and 50/50 human/autonomous drivers, the phase diagram pre-malicious hack has three regions corresponding to linear steady flow ([MATH]), a saturated steady flow where [MATH] is maximized and weakly dependent on [MATH], and an unsteady flow regime where density waves decrease [MATH].', '1708.03791-1-10-2': 'Post-malicious hack, the phase diagram collapses with unsteady and near-zero flow dominating while saturated steady flow appears at very low densities of autonomously-driven vehicles [Fig. [REF](b)].', '1708.03791-1-10-3': 'These differences between the pre- and post-hack phase diagrams show collective motion, or rather, the lack of collective motion, has significant disruptive potential.', '1708.03791-1-11-0': 'A remarkable feature of the post-malicious hack phase diagram is the relatively low fraction of autonomous vehicles required for traffic to be appreciably disrupted [Fig. [REF](b), boundary between steady and unsteady flow].', '1708.03791-1-11-1': 'In fact, this feature is surprisingly general and can be understood from combinatorial counting arguments [see SM].', '1708.03791-1-11-2': 'Approximating the continuous road of length [MATH] as a discrete grid with dimensions [MATH], we randomly select [MATH] grid sites to be occupied by bricked autonomous vehicles.', '1708.03791-1-11-3': 'Traffic is stopped if [MATH] vehicles are randomly assigned side-by-side positions, causing a blockage across all [MATH] lanes.', '1708.03791-1-11-4': 'For [MATH], the probability of at least one blockage occurring when [MATH] is [MATH], which is one minus the probability of no blockages occurring, and [MATH].', '1708.03791-1-11-5': 'Using the same [MATH] km road from simulations, the probability of traffic being completely stopped by a malicious bricking hack is 50 for [MATH] cars/km/lane, and 99 when [MATH] cars/km/lane.', '1708.03791-1-11-6': 'An analogous calculation for [MATH] shows the probabilities are 50 and 99 for [MATH] and [MATH] cars/km/lane, respectively.', '1708.03791-1-11-7': 'As such, vehicle densities well-within the range of empirical observations [Fig. [REF], green data] have a substantial chance of completely obstructed flow post-malicious hack.', '1708.03791-1-11-8': 'These counting arguments help ground our understanding of the phase diagram [Fig. [REF](b)], and ultimately suggest the physics of clogging[CITATION], random pinning in glasses[CITATION], and percolation[CITATION] may offer useful insights.', '1708.03791-1-12-0': 'Anticipating how emergent collective motion can be weaponized ultimately suggests how to inoculate against this type of attack.', '1708.03791-1-12-1': 'One solution is to guarantee the existence of an unobstructed route for human drivers by limiting the [MATH] autonomous vehicles to a subset of lanes.', '1708.03791-1-12-2': 'Running simulations where human-driven vehicles were allowed on three lanes, but autonomously-driven vehicles were allowed on only two [Fig. [REF](c-d)] or one [Fig. [REF](e-f)] shows that benefits from autonomously driven vehicles can be retained while limiting the effects of a bricking attack, albeit with reduced flow rates [see SM].', '1708.03791-1-13-0': 'By removing human variances in driving behavior, the collective motion of autonomous vehicles produces smooth, steady flowing traffic.', '1708.03791-1-13-1': 'This emergent property is highly desired for its environmental benefits, shortened commute times, and lower accident rates.', '1708.03791-1-13-2': 'Nevertheless, new technological achievements bring new potential risks.', '1708.03791-1-13-3': 'By disabling autonomous vehicles, malicious hackers can suppress this desirable collective motion leaving behind the highly disrupted collective motion of human drivers navigating in and around obstacles.', '1708.03791-1-13-4': 'As a bi-disperse active matter model, this situation leads to connections in the physics of clogging, random pinning, and percolation.', '1708.03791-1-13-5': 'Additionally, the absence of phantom traffic jams in the flow of autonomous vehicles suggests measurements of the dispersion relation and density of states may be useful probes to study the cross-over from human-driven to autonomously-driven vehicles.', '1708.03791-1-13-6': 'Working on these questions through the lens of the anticipate-and-inoculate mindset used here has the potential to identify how other emergent collective phenomena can be weaponized and preemptive disarmed.', '1708.03791-1-13-7': 'Ultimately, this approach should lead to safer roads and a greater good.'}

{'1708.03791-2-0-0': 'While much effort has been invested in studies of traffic flow as a physics problem, two emerging trends in technology have broadened the subject for new investigations.', '1708.03791-2-0-1': 'The first trend is the development of self-driving vehicles.', '1708.03791-2-0-2': 'This highly-anticipated shift from human- to autonomous-drivers is expected to offer substantial benefits for traffic throughput by streamlining large-scale collective behavior.', '1708.03791-2-0-3': 'The second trend is the widespread hacking of Internet-connected devices, which as of 2015, includes vehicles.', '1708.03791-2-0-4': 'While the first proof-of-concept automobile hack was done at the single-vehicle scale, undesirable collective effects can easily arise if this activity becomes more common.', '1708.03791-2-0-5': 'Motivated by these two trends, we explore the phenomena that arise in an active matter model with lanes and lane-changing behavior.', '1708.03791-2-0-6': 'Our model incorporates a simplified minimal description of essential differences between human- and autonomous-drivers.', '1708.03791-2-0-7': 'We study the emergent collective behavior as the population of vehicles shifts from all-human to all-autonomous.', '1708.03791-2-0-8': 'Within the context of our model, we explore a worst-case scenario where Internet-connected autonomous vehicles are disabled simultaneously and en masse.', '1708.03791-2-0-9': 'Our approach reveals a model-independent role for percolation in interpreting the results.', '1708.03791-2-0-10': 'A broad lesson our work highlights is that seemingly minor malicious activity can ultimately have major impacts when magnified through the action of collective behavior.', '1708.03791-2-1-0': 'Autonomous vehicles are highly sought for their potential to revolutionize transportation.', '1708.03791-2-1-1': 'This convergence between automotive and IT industries is expected to ripple beyond the immediate in-vehicle experience, and yield a broader array of benefits for society with changes in both commercial and private sectors[CITATION] [see SM].', '1708.03791-2-1-2': 'However, this focus on the value of technology-driven disruption leaves potential blind spots with respect to emergent risks brought by automotive computerization.', '1708.03791-2-1-3': 'For example, proof-of-concept hacks in 2015 and 2016 demonstrated the ability to remotely control an Internet-connected vehicle, including the brakes, acceleration, and steering [CITATION].', '1708.03791-2-1-4': 'Given that the level of computerization will only increase with self-driving vehicles, there is a deep need to understand the consequences of hackers leveraging remote automation to achieve undesirable ends.', '1708.03791-2-1-5': 'We therefore consider a worst-case scenario where various numbers of connected vehicles are simultaneously and en masse disabled during transit.', '1708.03791-2-1-6': 'While the details of how such a hack would be executed are beyond the scope of our work here, similar concerns have been raised by cybersecurity experts[CITATION].', '1708.03791-2-1-7': 'To explore this critical blind spot, we model human and autonomously driven vehicles as an active matter system[CITATION].', '1708.03791-2-1-8': 'Unlike conventional flowing systems subject to the kinetic phenomenon of clogging[CITATION], the incorporation of lanes and lane changing behavior motivates the geometric phenomenon of percolation as the relevant framework for understanding collective behavior that emerges in the presence of disabled vehicles.', '1708.03791-2-2-0': 'We begin this study by defining a microscopic model for human and autonomously driven vehicles.', '1708.03791-2-2-1': 'Of the different approaches to modeling vehicular collective motion[CITATION], we chose a force-based approach[CITATION] that represents each vehicle by their position [MATH] at time [MATH] on a straight road of length [MATH].', '1708.03791-2-2-2': 'The main appeal of this type of model is the rich complexity of interesting collective behavior that can be generated by its simple set of equations and correspondingly small set of parameters.', '1708.03791-2-2-3': 'Numerical simulations were performed using velocity Verlet integration for [MATH] vehicles seeded with uniformly distributed random initial positions, zero initial velocity, zero initial acceleration, and periodic boundary conditions.', '1708.03791-2-2-4': 'The position of each vehicle was evolved using a self-propulsion force [MATH], and a repulsive collision-avoidance force [MATH] according to [EQUATION] where [MATH] and [MATH] are the velocity and acceleration of a given vehicle [see SM].', '1708.03791-2-2-5': 'Each vehicle has a preferred speed [MATH] and a characteristic response time [MATH] that it takes to equilibrate to this speed.', '1708.03791-2-2-6': 'In the limit of a single vehicle without interactions, [MATH] and direct integration shows [MATH].', '1708.03791-2-2-7': 'As the number of vehicles increases and they begin to interact, the repulsive force slows a given vehicle down as it approaches another vehicle from behind.', '1708.03791-2-2-8': 'The strength of this force increases smoothly from zero as the vehicle-vehicle separation [MATH] becomes smaller than the interaction threshold distance [MATH].', '1708.03791-2-2-9': 'This threshold models the 2-second rule on highway-like conditions, which states a minimum safe distance between two vehicles is the distance traveled during 2 seconds at typical speeds of 29 m/s (65 mi/h).', '1708.03791-2-2-10': 'The functional form of the repulsion force in Eqs. ([REF]) follows from the compression of two elastic spheres[CITATION], and only applies to the trailing vehicle, as drivers respond to traffic ahead of them more strongly than behind.', '1708.03791-2-2-11': 'Under these conditions, we balance propulsion and repulsion forces when two vehicles are bumper-to-bumper, and set [MATH] m as the typical size of a vehicle to find [MATH].', '1708.03791-2-2-12': 'Thus, the distance between vehicles [MATH] is always larger than the size of a vehicle [MATH], but the interaction between vehicles is only experienced when [MATH].', '1708.03791-2-3-0': 'For simplicity, we focus on the emergent collective motion of two types of vehicles: human-driven and fully autonomous.', '1708.03791-2-3-1': 'In Eqs. ([REF]), this simplification amounts to distinct values for the variables with subscript [MATH], where we use [MATH] for human-driven vehicles and [MATH] for autonomous vehicles.', '1708.03791-2-3-2': 'For example, [MATH] denotes the number of human drivers and [MATH] denotes the number of autonomous drivers.', '1708.03791-2-3-3': 'Each of the [MATH] human drivers is given a randomly chosen preferred velocity [MATH] chosen from a Gaussian distribution with mean 29 m/s and standard deviation 6.7 m/s (15 mi/h).', '1708.03791-2-3-4': 'For the [MATH] autonomous drivers, we fix their preferred velocity [MATH] m/s to be constant and equal to the typical speed limit.', '1708.03791-2-3-5': 'This homogeneity reflects the notion that autonomous vehicles will be engineered to travel at the legally allowed maximum speed to optimize traffic flow and safety.', '1708.03791-2-4-0': 'The only remaining free parameter in the equations of motion for human-driven vehicles is the response time [MATH], which generally varies from 0.5 s to several seconds, depending on attentiveness, driving conditions, visibility, and level of distractions.', '1708.03791-2-4-1': 'To empirically measure [MATH], we analyzed three sources of driving data including a highway webcam [Fig. [REF](a)][CITATION], digital footage we recorded [Fig. [REF](b)], and previously published Department of Transportation (DOT) traffic data from Los Angeles, California (southbound US 101; NGSIM June 15, 2005, 7:50 to 8:05 am)[CITATION].', '1708.03791-2-4-2': 'By measuring the velocity in "stop-and-go" traffic and assuming the repulsion force is negligible during the initial "go," we could fit this data to [MATH] resulting in measurements of [MATH] [Fig. [REF](c)].', '1708.03791-2-4-3': 'In higher-density traffic, we observed drivers tend to accelerate over longer periods of time, whereas in lower-density traffic, drivers tend to react faster.', '1708.03791-2-4-4': 'Because Eqs. ([REF]) simplifies driver response time to be density-independent, we set [MATH] s, which comes from the average of empirical measurements that were found to be [MATH] s.', '1708.03791-2-5-0': 'In simulations with [MATH] human-driven vehicles on an [MATH] lane road, [MATH] km long, we find a brief transient regime that lasts for [MATH] integration time steps, followed by steady-state dynamics.', '1708.03791-2-5-1': 'Superimposed on this 1D line of traffic, we observe backwards-propagating density waves known as "phantom traffic jams" that cause stop-and-go motion of individual vehicles [Fig. [REF]a].', '1708.03791-2-5-2': 'This emergent collective phenomenon has been extensively studied in other models of vehicle traffic[CITATION], and is generally known to arise from the non-zero driver response time [MATH].', '1708.03791-2-5-3': 'Here, we see these density waves as an important validation of our active matter model, as their spontaneous emergence indicates broad consistency with the known phenomenology of vehicle traffic.', '1708.03791-2-6-0': 'Going beyond one-lane simulations and reproducing essential features of multi-lane traffic where [MATH] requires a model for lane changing behavior.', '1708.03791-2-6-1': 'Otherwise, quantitative metrics of traffic flow would be dominated by the slowest moving vehicle.', '1708.03791-2-6-2': 'One method for realistically capturing these dynamics is the Minimizing Overall Breaking Induced By Lane changes (MOBIL) framework[CITATION], which offers a simple force-based rule to determine when a vehicle should change lanes.', '1708.03791-2-6-3': 'Specifically, MOBIL considers whether a vehicle and its neighbors would better match their preferred speeds if a given vehicle changes lanes [see SM].', '1708.03791-2-6-4': 'In our implementation, this algorithm effectively minimizes [MATH] over all vehicles involved, and enables quantitative predictions of traffic flow beyond the qualitative observations of phantom traffic jams [Fig. [REF](a)].', '1708.03791-2-6-5': 'For example, we use the density-dependent flux [MATH] as a measure to study the emergent collective properties of multi-lane traffic flow [Fig. [REF](b)].', '1708.03791-2-6-6': 'In empirical measurements and simulations with [MATH] lanes, we use a portion of the road [MATH] m to calculate [MATH] and [MATH], where the index [MATH] runs over all vehicles in [MATH], sampled at statistically independent temporal intervals.', '1708.03791-2-6-7': 'Comparing observational data [Fig. [REF](b), green dots] to simulation predictions with MOBIL lane changing [Fig. [REF](b), black line and blue-shaded band] shows a broad consistency with no fitting parameters.', '1708.03791-2-6-8': 'Both in empirical measurements and simulations, we find that when the density is low, [MATH] [Fig. [REF](b), [MATH] cars/km/lane].', '1708.03791-2-6-9': 'As the density increases to a critical value [MATH] cars/km/lane, the flux peaks, and subsequently declines for increasing [MATH].', '1708.03791-2-6-10': 'In simulations, we find [MATH] corresponds to an average of one vehicle per 33 m of road per lane.', '1708.03791-2-6-11': 'This distance is [MATH] s of travel time between vehicles, placing them within the repulsive-force interaction distance [MATH].', '1708.03791-2-6-12': 'As such, when [MATH], the system is effectively a 1D continuum with density waves [Fig. [REF](a), horizontal bands] that produce the decline of [MATH] at larger [MATH].', '1708.03791-2-7-0': 'Thus far, our simulations and analysis have focused on human-driven vehicles.', '1708.03791-2-7-1': 'Incorporating autonomous vehicles requires that we fix the final parameter [MATH], which is the autonomous driver response time.', '1708.03791-2-7-2': 'Given these highly computerized vehicles are expected to utilize a combination of local sensor information, wirelessly shared non-local traffic conditions, and cloud-connected AI, we can reasonably assume they will respond to driving conditions more rapidly than human drivers.', '1708.03791-2-7-3': 'As an order of magnitude estimate, we set the autonomous vehicle response time [MATH] s, which is a fraction of the human response time [MATH], but still larger than the simulation time step.', '1708.03791-2-7-4': 'Simulations with all autonomous vehicles immediately show important differences in the flux [MATH], highlighting a key advantage of autonomous vehicles over human drivers that underlies their anticipated impact on automotive transport.', '1708.03791-2-7-5': 'Specifically, [MATH] still peaks near the same critical density [MATH], however, with all autonomous vehicles [MATH] is nearly 20% larger [Fig. [REF](a), upper light-bronze data].', '1708.03791-2-7-6': 'Even with a 50/50 mix of autonomous and human vehicles there are substantial advantages at the highest densities where the flow is steadier relative to results with all human drivers [Fig. [REF](a), upper light-red data, [MATH] cars/km/lane].', '1708.03791-2-7-7': 'The combination of higher throughput and steadier flow translate to reduced congestion, reduced fuel consumption, and less time spent traveling between destinations.', '1708.03791-2-8-0': 'Interestingly, [MATH] in all-autonomous vehicle simulations can be predicted from Eqs. ([REF]).', '1708.03791-2-8-1': 'Given the uniform preferred speed and rapid response time, [MATH].', '1708.03791-2-8-2': 'Under steady conditions, propulsion and repulsion forces balance so that [MATH] and [MATH], which leads to [MATH].', '1708.03791-2-8-3': 'The flux is therefore the real part of [MATH], where we have used the expression for [MATH] derived from force balance to find the final equation.', '1708.03791-2-8-4': 'Comparing this prediction to simulations shows strong agreement [Fig. [REF](a), solid black line].', '1708.03791-2-9-0': 'From the perspective of the flux-density relationship, there are measurable benefits to having an increased number of autonomous vehicles on the road due to the generally increased throughput of traffic.', '1708.03791-2-9-1': 'However, as indicated by recent proof-of-concept hacks, Internet-connected vehicles are endowed with enhanced abilities at the cost of new risks.', '1708.03791-2-9-2': 'The worst-case scenario we consider is one in which hacked vehicles are disabled during transit, which in the context of our model, amounts to setting [MATH] after the simulation has reached steady-state.', '1708.03791-2-9-3': 'This type of event has actually been proposed as an active self-defense mechanism for vehicles that detect their systems have been compromised, though it was offered in the context where only a single vehicle was being considered[CITATION].', '1708.03791-2-9-4': 'If we assume this strategy was broadly deployed, a wide-spread hack that compromises many vehicles simultaneously would most certainly give rise to new collective behavior that affects any functional vehicles still on the road.', '1708.03791-2-9-5': 'For example, a 50/50 mixture of human and autonomous drivers in this circumstance has a substantial loss of flow with a nearly 60% reduction in the peak flux and a total loss of flow at higher densities [Fig. [REF](a), lower dark-red data [MATH] cars/km/lane].', '1708.03791-2-10-0': 'Generating heat maps for [MATH] with varying fractions of autonomously driven vehicles [MATH] offers a more complete view of the benefits of autonomous vehicles [Fig. [REF](b)] as well as the effects on flow when these vehicles are disabled [Fig. [REF](c)].', '1708.03791-2-10-1': 'Comparing the pre-hack to the post-hack results show the flow largely collapses when vehicles are disabled, leaving behind unsteady and near-zero flux, even at relatively low fractions of autonomous vehicles [Fig. [REF](b) and (c), increased size of red region].', '1708.03791-2-10-2': 'In the post-hack scenario we observe the flux substantially drops when the fraction of disabled autonomous vehicles exceeds 10 to 20% [Fig. [REF](c), boundary of lower green band].', '1708.03791-2-10-3': 'At the same time, this number of self-driving vehicles corresponds to a rather small benefit in the pre-hack scenario relative to bounds at the [MATH] and [MATH] limits [Fig. [REF](b)].', '1708.03791-2-10-4': 'Evidently, the risks introduced by connected-car technologies have an earlier onset than the benefits, as measured by vehicle throughput.', '1708.03791-2-11-0': 'To better understand the emergent collective behavior of the scenario being considered, we note the resulting flux contours of the post-hack heat map appear to follow lines of constant autonomous vehicle density [MATH] [Fig. [REF](c), solid and dashed lines].', '1708.03791-2-11-1': 'This observation suggests [MATH] is a quantity of significant explanatory power for how disabled vehicles affect traffic flow.', '1708.03791-2-11-2': 'We therefore use the data that generated the heat map to calculate the probability that [MATH] as a function of [MATH], revealing a well-defined sigmoidal-like relationship [Fig. [REF](a), solid red squares].', '1708.03791-2-11-3': 'Furthermore, when we plot the actual velocity of vehicles [MATH] against the mean-field velocity [MATH], we see the data separates onto two distinct curves: the [MATH] curve where simulation and prediction agree up-to fluctuations [Fig. [REF](a) left inset, red data along black diagonal line, 11% of simulation data], and the [MATH] no-flow curve [Fig. [REF](a) left inset, red data along black horizontal line, 89% of the simulation data].', '1708.03791-2-11-4': 'Taken together, the combination of the bimodal separation of velocities and the smooth sigmoidal relationship between flow and no-flow outcomes strongly suggests the presence of a percolation phase transition.', '1708.03791-2-12-0': 'A simple description of percolation is the study of flow through porous materials[CITATION].', '1708.03791-2-12-1': "When the material's pores are connected such that a continuous path exists between two surfaces, fluid is able to move through the material.", '1708.03791-2-12-2': 'Conversely, when no continuous path between surfaces can be found, fluid cannot flow.', '1708.03791-2-12-3': "Complex phenomenology is then introduced through the non-trivial network of connections among the material's pores.", '1708.03791-2-12-4': 'Often, random lattice models are used to recapitulate the statistical properties of pore connectivity, and the prominent features studied are the emergence of a flow to no-flow phase transition where universality and scaling phenomena are readily found.', '1708.03791-2-13-0': 'In our model, there appears to be a percolation phase transition where the flow of traffic is analogous to the flow of fluid, the [MATH]-lane road is the material, and disabled vehicles create a network of obstacles reminiscent of a series of connected pores.', '1708.03791-2-13-1': 'Freely-moving vehicles approaching a disabled vehicle from behind continue to flow, as would a conventional fluid, although the underlying mechanism relies on lane switching behavior [Fig. [REF](b)].', '1708.03791-2-13-2': 'When disabled vehicles align across all [MATH] lanes, they form an immobile barrier obstructing flow [Fig. [REF](c)].', '1708.03791-2-13-3': 'In this sense, the percolation of obstacles appears to be the critical factor determining traffic flow properties, especially when asking whether a given simulation produces either a flow or no-flow condition [Fig. [REF](a), left inset, diagonal and horizontal branches].', '1708.03791-2-13-4': 'Obstacle percolation should be differentiated from the formation of clogs[CITATION], which are dynamic in nature, depend on observation time, and have statistical properties distinct from percolation that do not match our results.', '1708.03791-2-13-5': 'Nevertheless, if percolation is the correct framework, then stochastically placing obstacles at density [MATH] on an [MATH]-laned road and computing the probability that these obstacles percolate should reproduce the measured probability distribution for [MATH].', '1708.03791-2-13-6': 'In this calculation, we set the length for steric interactions of disabled autonomous vehicles and moving human-driven vehicles equal to the balance of repulsion and propulsion forces, described by Eqs. ([REF]).', '1708.03791-2-13-7': 'On average, these values are 8.5 m ([MATH]) for human-driven vehicles and 4.5 m ([MATH]) for autonomous vehicles, where the larger-than-expected average value for human-driven vehicles arises from heterogeneity in the preferred velocity [MATH] and the non-linear repulsion force.', '1708.03791-2-13-8': 'Remarkably, this minimal calculation, which only depends on the geometry of percolation, is able to accurately reproduce the probability distribution for [MATH] [Fig. [REF](a), solid red line].', '1708.03791-2-14-0': 'In light of the different effective sizes for human and autonomous vehicles, we can use geometric obstacle percolation to predict the probability that [MATH] when the road contains all autonomous vehicles, but only a fraction are disabled [Fig. [REF](a), solid blue line].', '1708.03791-2-14-1': 'Comparing this new scenario to a direct microscopic simulation again shows a bimodal separation of flow and no-flow outcomes [Fig. [REF](a), two branches of right inset].', '1708.03791-2-14-2': 'Moreover, the probability distribution for [MATH] computed separately by microscopic simulation and geometric obstacle percolation accurately agree with one another [Fig. [REF](a) solid blue circle match solid blue line].', '1708.03791-2-14-3': 'We can further show the emergence of percolation is a model-independent result when explaining the effects of disabled vehicles.', '1708.03791-2-14-4': 'We implemented the Nagel-Schrekenberg cellular automata model of traffic flow on an [MATH]-lane road with steric interaction of 4.5 and 7 m, disabled a number vehicles at density [MATH] by setting their speed to 0, and computed the probability that [MATH].', '1708.03791-2-14-5': 'Even in this completely different model for traffic flow, the microscopic numerical simulations [Fig. [REF](a), open red squares and circles] continue to match the prediction of obstacle percolation [Fig. [REF](a), dashed red and blue lines].', '1708.03791-2-14-6': 'Thus, as as suggested by the contours of constant [MATH] in the post-hack heat map for [MATH] [Fig. [REF](c)], the density of autonomous vehicles [MATH] explains the emergent flow phenomenology.', '1708.03791-2-15-0': 'We motivated this work by exploring the unintentional collective effects that can arise when Internet-connected autonomous vehicles are disabled suddenly and en masse.', '1708.03791-2-15-1': 'Our approach utilized a lane-based active matter model and uncovered a model-independent role for percolation in interpreting the results.', '1708.03791-2-15-2': 'Looking more broadly at the immediate consequences of obstacle percolation, we see a striking set of potential outcomes: emergency vehicles would be unable to respond to calls for help, food shipments to grocery stores would be delayed, and long-distance commuters would be unable to get to work.', '1708.03791-2-15-3': 'Urban cores and suburban towns, which are key centers of economic activity, would cease to function normally.', '1708.03791-2-15-4': 'Given the magnitude of these potential harms, there is a clear need to proactively develop strategies that mitigate the risks posed by obstacle percolation.', '1708.03791-2-16-0': 'Fortunately, our results on the statistics of obstacle percolation indirectly suggest a way to achieve this risk-mitigation.', '1708.03791-2-16-1': 'The probability of no-flow is small when the densities of disabled vehicles are correspondingly low [Fig. [REF](a)], so the goal should be to minimize the number of affected vehicles, as one would naturally guess.', '1708.03791-2-16-2': 'The probability of geometric percolation allows us to assess how different strategies result in different effective risk levels.', '1708.03791-2-16-3': 'One approach could be to distribute vehicle communications across multiple independent networks in such a fashion that a hack affecting one network would have no effect on the others.', '1708.03791-2-16-4': 'These multiple non-overlapping networks would appear inefficient from the perspective of communication standards, but it offers a greater degree of robustness with respect to the cyber-physical issues motivating this study.', '1708.03791-2-16-5': 'While such strategies are yet to be developed and deployed, only by anticipating these worst-case scenarios can we inoculate for their consequences.', '1708.03791-2-16-6': 'For example, if there were 10 separate networks, each connected to 1/10 of all vehicles, then a network-wide hack would have the effect of disabling only 10% of the vehicles on the road.', '1708.03791-2-16-7': 'In light of our simulation results [Fig. [REF](c), [MATH]], this relatively modest number of disabled vehicles would still allow traffic to generally flow.', '1708.03791-2-16-8': 'To achieve higher impact, a hacker would have to breach multiple separate networks, each with their own security and communications protocols, increasing the level of difficulty and sophistication required to realize a worst-case scenario.', '1708.03791-2-16-9': 'While these particular simulations were computed with specific ratios of human and autonomous drivers, the physics of percolation [Fig. [REF](a)] ensures similar outcomes.', '1708.03791-2-16-10': 'Indeed, this generality is the strength of the obstacle percolation interpretation uncovered here.'}

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[]

{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1708.03791

1705.05841

{'1705.05841-1-0-0': 'The idea of statistical transmutation plays a crucial role in descriptions of the fractional quantum Hall effect.', '1705.05841-1-0-1': 'However, a recently conjectured duality between a critical boson and a massless 2-component Dirac fermion extends this notion to gapless systems.', '1705.05841-1-0-2': 'This duality may shed light on highly non-trivial problems such as the half-filled Landau level, the superconductor-insulator transition, and surface states of strongly coupled topological insulators.', '1705.05841-1-0-3': 'Although this boson-fermion duality has undergone many consistency checks, it has remained unproven.', '1705.05841-1-0-4': 'We describe the duality in a non-perturbative fashion using an exact UV mapping of partition functions on a 3D Euclidean lattice.', '1705.05841-1-0-5': 'Our approach is purely analytic and has the advantage of being simple and exact.', '1705.05841-1-1-0': '# Introduction', '1705.05841-1-2-0': 'It is well-known that in [MATH] dimensions, there can be exotic quantum effects, such as fractional statistics and statistical transmutation.', '1705.05841-1-2-1': 'In a system with a gap, these features can be simply understood as Berry phase effects arising in adiabatic transport [CITATION].', '1705.05841-1-2-2': 'Statistical transmutation in a long-wavelength description is implemented via coupling matter to Chern-Simons (CS) gauge fields [CITATION].', '1705.05841-1-2-3': 'These effective field theories motivate mean-field treatments of the fractional quantum Hall effect based on "flux attachment", and have enjoyed much success [CITATION].', '1705.05841-1-3-0': 'A more non-trivial question is whether similar ideas survive in a gapless context, say near a critical point between two quantum Hall phases.', '1705.05841-1-3-1': 'Since such systems possess gapless excitations and are in general strongly coupled, the demonstration of statistical transmutation in this context is a much more challenging problem.', '1705.05841-1-3-2': 'Recently, significant progress towards such a description has been made in the form of a conjectured duality between critical bosons coupled to a level-1 Chern-Simons gauge field and a massless two component Dirac fermion [CITATION].', '1705.05841-1-3-3': 'More explicitly, in 3D Euclidean spacetime, the duality is between two theories described by the Lagrangian densities [EQUATION]', '1705.05841-1-3-4': 'Here [MATH] is a complex boson, [MATH] a dynamical [MATH] gauge field, [MATH] a 2-component Dirac fermion with [MATH] the Pauli matrices, and [MATH] a background [MATH] gauge field (more precisely, [MATH] is a [MATH] gauge field, but this difference will not alter our discussions below).', '1705.05841-1-3-5': 'This boson-fermion duality holds in the gapped case [CITATION] with [MATH], and, as conjectured, can be extended to the gapless case [MATH].', '1705.05841-1-3-6': 'In the gapless case, [MATH] is at the Wilson-Fisher fixed point.', '1705.05841-1-4-0': 'There is another duality that has been familiar for decades: the boson-vortex duality between a charged scalar field (XY model) and a gauged [MATH] Wilson-Fisher theory (Abelian Higgs model) [CITATION].', '1705.05841-1-4-1': 'The recent boson-fermion duality combined with the familiar boson-vortex duality generate an entire web of dualities [CITATION] and have unified our understanding of many different branches of theoretical physics.', '1705.05841-1-4-2': 'The boson-vortex duality was originally presented as an essentially exact mapping of lattice gauge theories.', '1705.05841-1-4-3': 'On the other hand, while the boson-fermion duality has undergone many non-trivial consistency checks and is almost certainly correct, it has not been proven at the same level as the boson-vortex duality.', '1705.05841-1-5-0': 'In this letter, we demonstrate an exact mapping between two 3D Euclidean lattice gauge theories.', '1705.05841-1-5-1': 'The first theory is an XY model minimally coupled to a [MATH] Chern-Simons gauge field.', '1705.05841-1-5-2': "We implement the Chern-Simons theory on the lattice using gapped Wilson's lattice fermion, thereby manifestly preserving gauge invariance and the compactness of the gauge group.", '1705.05841-1-5-3': "The second theory is a free, massless Dirac fermion, also realized by Wilson's lattice fermion.", '1705.05841-1-5-4': 'We show that there is an exact mapping relating these two theories which holds even at criticality.', '1705.05841-1-5-5': 'Remarkably, the mapping does not even require a Villain approximation for the XY model.', '1705.05841-1-5-6': 'Our method can be generalized to other members of the web of dualities (most notably the fermion-fermion duality [CITATION] that has been proposed to describe the half-filled Landau level [CITATION] and the surface states of strongly coupled topological insulators [CITATION]), can be used to discover new non-relativistic dualities, can be used to derive mappings between theories with non-abelian gauge groups [CITATION], and many more [CITATION].', '1705.05841-1-5-7': 'We will develop these ideas elsewhere.', '1705.05841-1-6-0': '# Basic Ingredients', '1705.05841-1-7-0': 'We consider bosons and fermions on a 3D Euclidean spacetime lattice, taken to be a cubic lattice for simplicity.', '1705.05841-1-7-1': 'There are two basic ingredients in the model.', '1705.05841-1-7-2': 'The first is a 3D XY model, which has a partition function [EQUATION] where [MATH] labels lattice sites, [MATH] are elementary displacement vectors, and [MATH] labels the link between the sites [MATH] and [MATH].', '1705.05841-1-7-3': 'The XY model consists of angles [MATH] on each site [MATH], and is coupled to a [MATH] gauge field [MATH] on each link [MATH].', '1705.05841-1-7-4': 'Note that although the lattice XY "spins" [MATH] have unit length, upon coarse graining, the amplitude will no longer be constrained.', '1705.05841-1-7-5': 'Thus, at distances large compared to the lattice spacing, the XY model describes a complex self-interacting scalar close to its Wilson-Fisher fixed point [CITATION].', '1705.05841-1-8-0': 'The second ingredient will be two sets of Grassmann fields [MATH] describing 2-component Dirac fermions on the lattice sites.', '1705.05841-1-8-1': "We will use Wilson's lattice fermions, which have the following partition function: [EQUATION] [MATH] is the bare theory of massive Dirac fermions on a Euclidean lattice, with [MATH] a constant to be discussed below.", '1705.05841-1-8-2': 'The first two terms of the second line of Eq. [REF] are associated with the lattice links whereas the third term, the Dirac mass term, is associated with lattice sites.', '1705.05841-1-8-3': '[MATH] is some lattice scale interaction of strength [MATH] that will be chosen later (not necessarily physically motivated) for the purpose of constructing a simple exact lattice duality.', '1705.05841-1-9-0': 'Suppose we turn off [MATH] for now.', '1705.05841-1-9-1': 'Consistent with the doubling theorem [CITATION], there are [MATH] 2-component Dirac fermion modes in the IR (see appendix).', '1705.05841-1-9-2': 'The parameter [MATH] was introduced by Wilson as a way to assign different Dirac masses to these Dirac fermion modes [CITATION].', '1705.05841-1-9-3': 'It represents a momentum-dependent correction to fermion masses and is otherwise invisible in the continuum limit.', '1705.05841-1-9-4': 'In 3D, even though [MATH] gaps out the doubled fermions, these modes never decouple.', '1705.05841-1-9-5': 'Instead, they produce Chern-Simons terms.', '1705.05841-1-9-6': 'As is well-known, a massive 2-component Dirac fermion with mass [MATH] produces a CS term of coefficient [MATH].', '1705.05841-1-9-7': 'Thus, when all 8 fermion modes are gapped, integrating them out results in the following CS term in the long wavelength limit: [EQUATION]', '1705.05841-1-9-8': 'The coefficient [MATH] depends on [MATH] as follows [CITATION] (see appendix): [EQUATION]', '1705.05841-1-9-9': 'We will address the effects of [MATH] later.', '1705.05841-1-10-0': 'The partition function for a boson coupled to a CS term can now be written: [EQUATION] where [MATH] is a fluctuating [MATH] gauge field that lives on the links.', '1705.05841-1-10-1': 'We have not imposed a gauge fixing condition for [MATH], since the gauge redundancy is just an overall factor.', '1705.05841-1-10-2': 'Provided that [MATH] satisfies certain UV boundary conditions to be described below, integrating out the fermions above will lead to a level-1 CS term for the difference [MATH].', '1705.05841-1-10-3': 'Since the XY boson couples to [MATH], we expect this will impart statistical transmutation and convert the boson into a fermion.', '1705.05841-1-11-0': '# Exact UV Mapping', '1705.05841-1-12-0': 'We next present an exact mapping from the partition function [MATH] above to that of a Wilson fermion with a different mass [MATH].', '1705.05841-1-12-1': 'To do this, we work in the representation of boson currents [MATH] defined on the links, obtained by Fourier series [CITATION]: [EQUATION] where [MATH] is the [MATH]th modified Bessel function.', '1705.05841-1-12-2': 'It represents a sum over all possible boson tunneling terms, each term resulting in moving an integer [MATH] number of bosons across the link, with a tunneling amplitude [MATH].', '1705.05841-1-12-3': 'Integrating out [MATH] leads to a "Gauss\'s Law" constraint on each site: [EQUATION]', '1705.05841-1-12-4': 'The divergence-free condition is of course just a restatement of the [MATH] conservation of the original XY model.', '1705.05841-1-13-0': 'We then consider the Wilson fermions.', '1705.05841-1-13-1': 'Following Wilson, we set [MATH].', '1705.05841-1-13-2': 'The choice of the magnitude [MATH] does not affect the IR physics, but it simplifies the lattice model as [MATH] project out one linear combination of the 2-component fermion.', '1705.05841-1-13-3': 'For reasons to be made apparent below, for [MATH] we choose [MATH] to be a hopping-hopping interaction: [EQUATION]', '1705.05841-1-13-4': 'Due to the Grassmann algebra of the fermion fields, we may expand the contribution to [MATH] on each link exactly (see Fig. [REF]).', '1705.05841-1-13-5': 'Each link [MATH] contributes a factor [MATH] which is [EQUATION]', '1705.05841-1-13-6': 'The remarkable feature of the expression above is that the integration over the dynamical gauge field [MATH] can be performed exactly.', '1705.05841-1-13-7': 'Doing so, we find the boson tunneling current [MATH] must be equal to the fermion tunneling current, which can only take values [MATH] as seen in the Grassmann expansion above.', '1705.05841-1-13-8': 'This shows the bosons and the fermions always form bound states.', '1705.05841-1-13-9': "Note that the Gauss's Law constraint [MATH] for bosons is automatically satisfied, because under the Grassmann integral, fermion tunneling currents must form closed loops to have non-vanishing amplitude, as shown in Fig. [REF].", '1705.05841-1-14-0': 'Thus, after integrating out [MATH], [EQUATION] which describes fermions with renormalized tunneling amplitudes and self-interactions on each link.', '1705.05841-1-14-1': 'It is now clear why we have chosen [MATH] to be the hopping-hopping interaction.', '1705.05841-1-14-2': 'Redefining the fermion fields [MATH] (and likewise for [MATH]), we can see a hopping-hopping interaction is generated from the coupling to bosons, so we want to include a bare [MATH] that has the same form as the generated interaction, to act as a "counterterm" and ultimately to produce a free fermion in the IR (see next section).', '1705.05841-1-14-3': 'More explicitly, the total partition function of the system in terms of [MATH] reads [EQUATION] which describes a Wilson fermion with modified [MATH] and [MATH]; the function [MATH] increases from [MATH] to [MATH] as [MATH] increases from [MATH] to [MATH].', '1705.05841-1-14-4': 'This is the exact lattice duality we claimed.', '1705.05841-1-15-0': 'By similar steps, it is also easy to verify that the correlation functions satisfy [EQUATION] under any background [MATH] configuration.', '1705.05841-1-15-1': 'This again manifests the point that the fermion [MATH] is the bound state formed by the boson [MATH] and the fermion [MATH] due to the mediation of the gauge field [MATH].', '1705.05841-1-15-2': 'Note that [MATH] is invariant under the [MATH] gauge transformation of [MATH], as [MATH] should be.', '1705.05841-1-16-0': '# UV Boundary Conditions', '1705.05841-1-17-0': 'In the previous section, we have shown that the 3D XY model coupled to a massive interacting Wilson fermion [MATH] can be exactly mapped in the UV to another (generically massive and interacting) Wilson fermion [MATH].', '1705.05841-1-17-1': 'In order to realize the conjectured IR duality, we will need to adjust the parameters of the UV theory, so that [MATH] is massive and can be integrated out to implement a level-1 CS, while [MATH] has a free massless Dirac fermion mode in the IR.', '1705.05841-1-17-2': 'For concreteness, to realize the IR behavior of [MATH], we can set [MATH], so that the Dirac mode near lattice momentum [MATH] becomes massless, while the other 7 Dirac modes contribute a level-1/2 CS for the background field [MATH] (see appendix).', '1705.05841-1-17-3': 'From Eq. [REF], we see that [MATH] are realized when the UV parameters [MATH] of [MATH] satisfy [EQUATION] where [MATH] is the value of the "temperature" [MATH] of the XY model required to hit the critical point.', '1705.05841-1-17-4': 'Such values of [MATH] and [MATH] lie on a section of a parabola shown by the gray dashed curve in Fig. [REF].', '1705.05841-1-17-5': 'However, now [MATH] must be non-zero.', '1705.05841-1-17-6': 'We know that when [MATH], a Wilson fermion with [MATH] can be integrated out to generate a CS term of level [MATH].', '1705.05841-1-17-7': 'Now we need to show [MATH] continues to hold when [MATH] lie on the said section of the parabola, at least in the vicinity [MATH].', '1705.05841-1-17-8': 'This way, the heavy Wilson fermion [MATH] continues to implement the level-1 CS term for [MATH] in the IR.', '1705.05841-1-18-0': 'Consider the phase diagram in [MATH] and [MATH].', '1705.05841-1-18-1': 'Massive fermions with [MATH] and [MATH] are in different topological phases, so there must be a gapless phase boundary separating the two phase regions, and we know this phase boundary goes through the point [MATH].', '1705.05841-1-18-2': 'We claim the phase boundary looks like the red line in Fig. [REF] in the vicinity of the point [MATH].', '1705.05841-1-18-3': 'To justify this claim, it is sufficient to consider [MATH] and determine the slope [MATH] such that the Dirac mode near [MATH] remains massless.', '1705.05841-1-18-4': 'That is, we equate [MATH], where [MATH] is the self-energy evaluated at [MATH] to first order in [MATH].', '1705.05841-1-18-5': 'The slope thus computed is [MATH], as drawn in Fig. [REF].', '1705.05841-1-18-6': 'Therefore, at least in the vicinity [MATH], points satisfying Eq. [REF] lie in the [MATH] phase as desired.', '1705.05841-1-18-7': 'This completes the verification that the constructed UV duality reproduces the conjectured IR duality.', '1705.05841-1-18-8': 'One can also check that when [MATH] deviates from [MATH], the IR mass of [MATH] has the expected sign.', '1705.05841-1-19-0': 'The reader may naturally wonder whether our analysis of the UV boundary condition amounts to fine-tuning beyond that which is required to reach a critical point.', '1705.05841-1-19-1': 'However, since [MATH] is irrelevant near the free IR fixed point, its only effect is to renormalize the IR mass.', '1705.05841-1-19-2': 'Put differently, [MATH] are not independent parameters that need to be tuned (although we tuned to [MATH] for demonstration).', '1705.05841-1-19-3': 'The irrelevancy of the interaction in the IR also guarantees that first-order transitions are unlikely for small deviations from the point [MATH].', '1705.05841-1-19-4': 'For large deviations from this point, a perturbative analysis is not controlled and other possibilities may occur.', '1705.05841-1-20-0': 'On the other hand, the behavior of the phase boundary(ies) between the [MATH] and [MATH] phases in the vicinity of [MATH] remains unclear.', '1705.05841-1-21-0': '# Conclusion', '1705.05841-1-22-0': 'We have constructed an exact UV duality on a 3D Euclidean lattice, and shown it reproduces the conjectured boson-fermion duality in the IR.', '1705.05841-1-22-1': 'Our idea of realizing [MATH] as the bound state [MATH] is not new - it resembles the so-called "parton construction" widely used in theories of fractional quantum Hall systems [CITATION].', '1705.05841-1-22-2': 'However, the usual parton construction in the continuum is not mathematically well-defined, and the partons are at best heuristic constructions whose justification can only be made a posteriori.', '1705.05841-1-22-3': 'On the other hand, in our lattice construction, those desired behaviors - that the boson [MATH] and the fermion [MATH] form a bound state [MATH], that the fermion [MATH] is massive and can be integrated out to produce level-1 Chern-Simons, and that the fermionic bound state [MATH] becomes massless - are all derived exactly.', '1705.05841-1-22-4': 'In the IR duality, the fermion [MATH] is viewed as the bound state [MATH] where [MATH] is the so-called "monopole operator".', '1705.05841-1-22-5': 'Clearly [MATH] has been realized as the coarse-grained version of [MATH].', '1705.05841-1-22-6': 'Moreover, [MATH] can indeed be realized by the heavy fermion [MATH], because a heavy fermion excitation moving in a Dirac sea in the [MATH] phase attains a [MATH] flux on itself.', '1705.05841-1-23-0': 'The exact mapping of lattice gauge theories presented here may be extended to other members of the web of dualities.', '1705.05841-1-23-1': 'The non-perturbative lattice regularizations of these theories enables the study of non-relativistic dualities, often applicable in condensed matter contexts.', '1705.05841-1-23-2': 'Our methods can also be used to study dualities involving matter fields with non-abelian gauge fields.', '1705.05841-1-23-3': 'We will report on progress in these directions in forthcoming publications.'}

{'1705.05841-2-0-0': 'The idea of statistical transmutation plays a crucial role in descriptions of the fractional quantum Hall effect.', '1705.05841-2-0-1': 'However, a recently conjectured duality between a critical boson and a massless 2-component Dirac fermion extends this notion to gapless systems.', '1705.05841-2-0-2': 'This duality sheds light on highly non-trivial problems such as the half-filled Landau level, the superconductor-insulator transition, and surface states of strongly coupled topological insulators.', '1705.05841-2-0-3': 'Although this boson-fermion duality has undergone many consistency checks, it has remained unproven.', '1705.05841-2-0-4': 'We describe the duality in a non-perturbative fashion using an exact UV mapping of partition functions on a 3D Euclidean lattice.', '1705.05841-2-1-0': '# Introduction', '1705.05841-2-2-0': 'The idea of duality underlies some of the most fascinating aspects of quantum statistical mechanics.', '1705.05841-2-2-1': 'A system that appears strongly coupled and nearly intractable may be dual to a weakly interacting system.', '1705.05841-2-2-2': 'Duality thus enables us to solve some highly non-trivial problems in theoretical physics.', '1705.05841-2-2-3': 'In [MATH] dimensions, a class of duality mappings known as particle-vortex duality follow from the fact that conserved currents can be expressed in terms of dual electric and magnetic fluxes.', '1705.05841-2-2-4': 'The simplest such example is the boson-vortex duality that maps a boson such as a Cooper pair to a dual vortex degree of freedom [CITATION], and plays a key role in our understanding of superfluid-insulator quantum phase transitions.', '1705.05841-2-2-5': 'In this letter, we study another particle-vortex duality which transmutes bosonic and fermionic statistics.', '1705.05841-2-2-6': 'We demonstrate an exact mapping of partition functions of a strongly coupled boson, and its free fermion vortex.', '1705.05841-2-3-0': 'It is well-known that in [MATH] dimensions, there can be exotic quantum effects such as fractional statistics and statistical transmutation.', '1705.05841-2-3-1': 'In a system with a gap, these features can be simply understood as Berry phase effects arising in adiabatic transport [CITATION].', '1705.05841-2-3-2': 'Statistical transmutation in a long-wavelength description is implemented via coupling matter to Chern-Simons (CS) gauge fields [CITATION].', '1705.05841-2-3-3': 'These effective field theories motivate mean-field treatments of the fractional quantum Hall effect based on "flux attachment", and have enjoyed much success [CITATION].', '1705.05841-2-4-0': 'A more non-trivial question is whether similar ideas survive in a gapless context, say near a critical point between two quantum Hall phases (see e.g. [CITATION] for related discussions).', '1705.05841-2-4-1': 'Since such systems possess gapless excitations and are in general strongly coupled, the demonstration of statistical transmutation becomes a much more challenging problem.', '1705.05841-2-4-2': 'Recently, significant progress towards such a description has been made in the form of a conjectured duality between critical bosons coupled to a level-1 Chern-Simons gauge field and a massless two component Dirac fermion [CITATION] (see details below).', '1705.05841-2-4-3': 'More interestingly, this conjectured boson-fermion duality combined with the familiar boson-vortex duality generate an entire web of dualities [CITATION].', '1705.05841-2-4-4': 'Members of this web have important potential applications.', '1705.05841-2-4-5': 'Most notably, the fermion-fermion duality [CITATION] from this web has been proposed to describe the half-filled Landau level [CITATION] and the surface states of strongly coupled topological insulators [CITATION].', '1705.05841-2-4-6': 'On the other hand, from a more formal perspective, along this line people have been exploring non-abelian dualities [CITATION] and many more extensions [CITATION].', '1705.05841-2-4-7': 'It is seen the recent progress on boson-fermion duality has unified many different branches of theoretical physics.', '1705.05841-2-5-0': 'Explicitly, in 3D Euclidean spacetime, the boson-fermion duality maps between the Lagrangian densities [EQUATION]', '1705.05841-2-5-1': 'Here [MATH] is a complex boson, [MATH] a dynamical [MATH] gauge field, [MATH] a 2-component Dirac fermion with [MATH] the Pauli matrices, and [MATH] a background [MATH] gauge field (more precisely, [MATH] is a [MATH] gauge field, but this difference does not alter our discussion below).', '1705.05841-2-5-2': 'This boson-fermion duality holds in the gapped case [CITATION] with [MATH], and, as conjectured, can be extended to the gapless case [MATH].', '1705.05841-2-5-3': 'In the gapless case, [MATH] is at the Wilson-Fisher fixed point.', '1705.05841-2-6-0': 'While this boson-fermion duality is extremely useful and has undergone many non-trivial consistency checks, it has remained unproven (however, see the demonstrations [CITATION] and [CITATION]).', '1705.05841-2-6-1': 'Recall the familiar boson-vortex duality was originally presented as an essentially exact mapping between the lattice gauge theories of an XY model and an abelian Higgs model [CITATION].', '1705.05841-2-6-2': 'A proof of the boson-fermion duality to such rigor is in need.', '1705.05841-2-7-0': 'In this letter, we demonstrate an exact mapping between two 3D Euclidean lattice gauge theories.', '1705.05841-2-7-1': 'The first theory is an XY model minimally coupled to a [MATH] Chern-Simons gauge field.', '1705.05841-2-7-2': "We implement the Chern-Simons theory on the lattice using gapped Wilson's lattice fermion, thereby manifestly preserving gauge invariance and the compactness of the gauge group.", '1705.05841-2-7-3': "The second theory is a free, massless Dirac fermion, also realized by Wilson's lattice fermion.", '1705.05841-2-7-4': 'We show there is an exact mapping relating these two theories which holds even at criticality.', '1705.05841-2-7-5': 'Our method is so simple that it can be generalized to other members of the web of dualities [CITATION], and beyond [CITATION].', '1705.05841-2-7-6': 'We will develop these ideas elsewhere.', '1705.05841-2-8-0': '# Basic Ingredients', '1705.05841-2-9-0': 'We consider bosons and fermions on a 3D Euclidean spacetime lattice, taken to be a cubic lattice for simplicity.', '1705.05841-2-9-1': 'There are two basic ingredients in the model.', '1705.05841-2-9-2': 'The first is a 3D XY model, which has a partition function [EQUATION] where [MATH] labels lattice sites, [MATH] are elementary displacement vectors, and [MATH] labels the link between the sites [MATH] and [MATH].', '1705.05841-2-9-3': 'The XY model consists of angles [MATH] on each site [MATH], and is coupled to a [MATH] gauge field [MATH] on each link [MATH].', '1705.05841-2-9-4': 'Note that although the lattice XY "spins" [MATH] have unit length, upon coarse graining, the amplitude will no longer be constrained.', '1705.05841-2-9-5': 'Thus, at distances large compared to the lattice spacing, the XY model describes a complex self-interacting scalar close to its Wilson-Fisher fixed point [CITATION].', '1705.05841-2-10-0': 'The second ingredient will be two sets of Grassmann fields [MATH] describing 2-component Dirac fermions on the lattice sites.', '1705.05841-2-10-1': "We will use Wilson's lattice fermions, which have the following partition function: [EQUATION] [MATH] is the bare theory of massive Dirac fermions on a Euclidean lattice, with [MATH] a constant to be discussed below.", '1705.05841-2-10-2': 'The first two terms of the second line of Eq. [REF] are associated with the lattice links whereas the third term, the Dirac mass term, is associated with lattice sites.', '1705.05841-2-10-3': '[MATH] is some lattice scale interaction of strength [MATH] that will be chosen later (not necessarily physically motivated) for the purpose of constructing a simple exact lattice duality.', '1705.05841-2-11-0': 'Suppose we turn off [MATH] for now.', '1705.05841-2-11-1': 'Consistent with the doubling theorem [CITATION], there are [MATH] 2-component Dirac fermion modes in the IR (see appendix).', '1705.05841-2-11-2': 'The parameter [MATH] was introduced by Wilson as a way to assign different Dirac masses to these Dirac fermion modes [CITATION].', '1705.05841-2-11-3': 'It represents a momentum-dependent correction to fermion masses and is otherwise invisible in the continuum limit.', '1705.05841-2-11-4': 'In 3D, even though [MATH] gaps out the doubled fermions, these modes never decouple.', '1705.05841-2-11-5': 'Instead, they produce Chern-Simons terms.', '1705.05841-2-11-6': 'As is well-known, a massive 2-component Dirac fermion with mass [MATH] produces a CS term of coefficient [MATH].', '1705.05841-2-11-7': 'Thus, when all 8 fermion modes are gapped, integrating them out results in the following CS term in the long wavelength limit: [EQUATION]', '1705.05841-2-11-8': 'The coefficient [MATH] depends on [MATH] as follows [CITATION] (see appendix): [EQUATION]', '1705.05841-2-11-9': 'We will address the effects of [MATH] later.', '1705.05841-2-12-0': 'The partition function for a boson coupled to a CS term can now be written: [EQUATION] where [MATH] is a fluctuating [MATH] gauge field that lives on the links.', '1705.05841-2-12-1': 'We have not imposed a gauge fixing condition for [MATH], since the gauge redundancy is just an overall factor.', '1705.05841-2-12-2': 'Provided that [MATH] satisfies certain UV boundary conditions to be described below, integrating out the fermions above will lead to a level-1 CS term for the difference [MATH].', '1705.05841-2-12-3': 'Since the XY boson couples to [MATH], we expect this will impart statistical transmutation and convert the boson into a fermion.', '1705.05841-2-13-0': '# Exact UV Mapping', '1705.05841-2-14-0': 'We next present an exact mapping from the partition function [MATH] above to that of a Wilson fermion with a different mass [MATH].', '1705.05841-2-14-1': 'To do this, we work in the representation of boson currents [MATH] defined on the links, obtained by Fourier series [CITATION]: [EQUATION] where [MATH] is the [MATH]th modified Bessel function.', '1705.05841-2-14-2': 'It represents a sum over all possible boson tunneling terms, each term resulting in moving an integer [MATH] number of bosons across the link, with a tunneling amplitude [MATH].', '1705.05841-2-14-3': 'Integrating out [MATH] leads to a "Gauss\'s Law" constraint on each site: [EQUATION]', '1705.05841-2-14-4': 'The divergence-free condition is of course just a restatement of the [MATH] conservation of the original XY model.', '1705.05841-2-15-0': 'We then consider the Wilson fermions.', '1705.05841-2-15-1': 'Following Wilson, we set [MATH].', '1705.05841-2-15-2': 'The choice of the magnitude [MATH] does not affect the IR physics, but it simplifies the lattice model as [MATH] project out one linear combination of the 2-component fermion.', '1705.05841-2-15-3': 'For reasons to be made apparent below, for [MATH] we choose [MATH] to be a hopping-hopping interaction: [EQUATION]', '1705.05841-2-15-4': 'Due to the Grassmann algebra of the fermion fields, we may expand the contribution to [MATH] on each link exactly (see Fig. [REF]).', '1705.05841-2-15-5': 'Each link [MATH] contributes a factor [MATH] which is [EQUATION]', '1705.05841-2-15-6': 'The remarkable feature of the expression above is that the integration over the dynamical gauge field [MATH] can be performed exactly.', '1705.05841-2-15-7': 'Doing so, we find the boson tunneling current [MATH] must be equal to the fermion tunneling current, which can only take values [MATH] as seen in the Grassmann expansion above.', '1705.05841-2-15-8': 'This shows the bosons and the fermions always form composite particles.', '1705.05841-2-15-9': "Note that the Gauss's Law constraint [MATH] for bosons is automatically satisfied, because under the Grassmann integral, fermion tunneling currents must form closed loops to have non-vanishing amplitude, as shown in Fig. [REF].", '1705.05841-2-16-0': 'Thus, after integrating out [MATH], [EQUATION] which describes fermions with renormalized tunneling amplitudes and self-interactions on each link.', '1705.05841-2-16-1': 'It is now clear why we have chosen [MATH] to be the hopping-hopping interaction.', '1705.05841-2-16-2': 'Redefining the fermion fields [MATH] (and likewise for [MATH]), we can see a hopping-hopping interaction is generated from the coupling to bosons, so we want to include a bare [MATH] that has the same form as the generated interaction, to act as a "counterterm" and ultimately to produce a free fermion in the IR (see next section).', '1705.05841-2-16-3': 'More explicitly, the total partition function of the system in terms of [MATH] reads [EQUATION] which describes a Wilson fermion with modified [MATH] and [MATH]; the function [MATH] increases from [MATH] to [MATH] as [MATH] increases from [MATH] to [MATH].', '1705.05841-2-16-4': 'This is the exact lattice duality we claimed.', '1705.05841-2-17-0': 'By similar steps, it is also easy to verify that the correlation functions satisfy [EQUATION] under any background [MATH] configuration.', '1705.05841-2-17-1': 'This again manifests the point that the fermion [MATH] is the composite particle formed by the boson [MATH] and the fermion [MATH] due to the mediation of the gauge field [MATH].', '1705.05841-2-17-2': 'Note that [MATH] is invariant under the [MATH] gauge transformation of [MATH], as [MATH] should be.', '1705.05841-2-18-0': '# UV Boundary Conditions', '1705.05841-2-19-0': 'In the previous section, we have shown that the 3D XY model coupled to a massive interacting Wilson fermion [MATH] can be exactly mapped in the UV to another (generically massive and interacting) Wilson fermion [MATH].', '1705.05841-2-19-1': 'In order to realize the conjectured IR duality, we will need to adjust the parameters of the UV theory, so that [MATH] is massive and can be integrated out before the gauge field [MATH] to implement a level-1 CS, while [MATH] has a free massless Dirac fermion mode in the IR.', '1705.05841-2-19-2': 'For concreteness, to realize the IR behavior of [MATH], we can set [MATH], so that the Dirac mode near lattice momentum [MATH] becomes massless, while the other 7 Dirac modes contribute a level-1/2 CS for the background field [MATH] (see appendix).', '1705.05841-2-19-3': 'From Eq. [REF], we see that [MATH] are realized when the UV parameters [MATH] of [MATH] satisfy [EQUATION] where [MATH] is the value of the "temperature" [MATH] of the XY model required to hit the critical point.', '1705.05841-2-19-4': 'Such values of [MATH] and [MATH] lie on a section of a parabola shown by the gray dashed curve in Fig. [REF].', '1705.05841-2-19-5': 'However, now [MATH] must be non-zero.', '1705.05841-2-19-6': 'We know that when [MATH], a Wilson fermion with [MATH] can be integrated out to generate a CS term of level [MATH].', '1705.05841-2-19-7': 'Now we need to show [MATH] continues to hold when [MATH] lie on the said section of the parabola, at least in the vicinity [MATH].', '1705.05841-2-19-8': 'This way, the heavy Wilson fermion [MATH] continues to implement the level-1 CS term for [MATH] in the IR.', '1705.05841-2-20-0': 'Consider the phase diagram in [MATH] and [MATH].', '1705.05841-2-20-1': 'Massive fermions with [MATH] and [MATH] are in different topological phases, so there must be a gapless phase boundary separating the two phase regions, and we know this phase boundary goes through the point [MATH].', '1705.05841-2-20-2': 'We claim the phase boundary looks like the red line in Fig. [REF] in the vicinity of the point [MATH].', '1705.05841-2-20-3': 'To justify this claim, it is sufficient to consider [MATH] and determine the slope [MATH] such that the Dirac mode near [MATH] remains massless.', '1705.05841-2-20-4': 'That is, we equate [MATH], where [MATH] is the self-energy evaluated at [MATH] to first order in [MATH].', '1705.05841-2-20-5': 'The slope thus computed is [MATH], as drawn in Fig. [REF].', '1705.05841-2-20-6': 'Therefore, at least in the vicinity [MATH], points satisfying Eq. [REF] lie in the [MATH] phase as desired.', '1705.05841-2-20-7': 'This completes the verification that the constructed UV duality reproduces the conjectured IR duality.', '1705.05841-2-20-8': 'One can also check that when [MATH] deviates from [MATH], the IR mass of [MATH] has the expected sign.', '1705.05841-2-21-0': 'The reader may naturally wonder whether our analysis of the UV boundary condition amounts to fine-tuning beyond that which is required to reach a critical point.', '1705.05841-2-21-1': 'However, since [MATH] is irrelevant near the free IR fixed point, its only effect is to renormalize the IR mass.', '1705.05841-2-21-2': 'Put differently, [MATH] are not independent parameters that need to be tuned (although we tuned to [MATH] for demonstration).', '1705.05841-2-21-3': 'The irrelevancy of the interaction in the IR also guarantees that first-order transitions are unlikely for small deviations from the point [MATH].', '1705.05841-2-21-4': 'For large deviations from this point, a perturbative analysis is not controlled and other possibilities may occur.', '1705.05841-2-22-0': 'On the other hand, the behavior of the phase boundary(ies) between the [MATH] and [MATH] phases in the vicinity of [MATH] remains unclear.', '1705.05841-2-23-0': '# Conclusion', '1705.05841-2-24-0': 'In this letter, we solved a strongly interacting 3D lattice gauge theory exactly and showed that the solution reproduces a conjectured boson-fermion duality in the long distance limit.', '1705.05841-2-25-0': 'Our idea of realizing [MATH] as the composite particle [MATH] is not new - it resembles the so-called "parton construction" widely used in theories of fractional quantum Hall systems [CITATION].', '1705.05841-2-25-1': 'In our lattice construction, the following facts are all derived exactly: 1) the boson [MATH] and the fermion [MATH] form a composite particle [MATH], 2) the fermion [MATH] is massive and can be integrated out to produce level-1 Chern-Simons, and 3) the fermionic composite particle [MATH] becomes massless.', '1705.05841-2-25-2': 'In the IR duality, the fermion [MATH] is viewed as the composite particle [MATH] where [MATH] is the so-called "monopole operator".', '1705.05841-2-25-3': 'Clearly [MATH] has been realized as the coarse-grained version of [MATH].', '1705.05841-2-25-4': 'Moreover, [MATH] can indeed be realized by the heavy fermion [MATH], because a heavy fermion excitation moving in a Dirac sea in the [MATH] phase attains a [MATH] flux on itself.', '1705.05841-2-26-0': 'One may ask to what extent our "XY + heavy fermion" lattice construction represents the "Wilson-Fisher + CS" side of continuum duality.', '1705.05841-2-26-1': 'In a sense, our construction can be viewed as giving an operational definition to the strongly coupled continuum Lagrangian by providing a non-perturbative regularization.', '1705.05841-2-26-2': 'Then our claim is that, after integrating out the heavy fermion [MATH] and the fast modes of bosonic fields [MATH] and [MATH], the slow modes of the bosonic fields "look sufficiently like" the continuum Lagrangian of Wilson-Fisher + CS.', '1705.05841-2-26-3': 'Claims of such kind, generally not justifiable by analytic means, are implicitly understood in common applications of lattice gauge theory such as lattice QCD [CITATION] and the original presentation of the boson-vortex duality [CITATION].', '1705.05841-2-26-4': 'Our procedure is a derivation of the duality within such criterion.'}

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[]

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[]

[]

{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1705.05841

1603.03384

{'1603.03384-1-0-0': 'Trapped ions are a promising tool for building a large-scale quantum computer.', '1603.03384-1-0-1': 'The number of required radiation fields for the realisation of quantum gates in any proposed ion-based architecture scales with the number of ions inside the quantum computer, posing a major challenge when imagining a device with millions of qubits.', '1603.03384-1-0-2': 'Here we present a fundamentally different approach, where this scaling entirely vanishes.', '1603.03384-1-0-3': 'The method is based on individually controlled voltages applied to each logic gate location to facilitate the actual gate operation analogous to a traditional transistor architecture within a classical computer processor.', '1603.03384-1-0-4': 'We demonstrate the key principle of this approach by implementing a versatile quantum gate method based on long-wavelength radiation where we generate a maximally entangled two-qubit state with fidelity 0.985(12).', '1603.03384-1-0-5': 'This quantum gate also constitutes a simple-to-implement tool for quantum metrology, sensing and simulation.', '1603.03384-1-1-0': 'The control of the internal and external degrees of freedom of trapped ions using laser light has allowed unprecedented advances in the creation of multi-particle entangled states [CITATION], quantum simulation [CITATION], frequency standards [CITATION] and scalable quantum logic [CITATION].', '1603.03384-1-1-1': 'A major goal is now to construct a large-scale quantum computer by scaling current systems up to a significantly larger number of ions.', '1603.03384-1-1-2': 'The circuit-model approach for quantum information processing requires the realization of arbitrary single qubit gates and a two-qubit entanglement operation [CITATION].', '1603.03384-1-1-3': 'The use of laser light for the implementation of these quantum logic operations has been extremely successful, with gate fidelities in the fault-tolerant regime having been achieved for single [CITATION] as well as two-qubit gates [CITATION].', '1603.03384-1-2-0': 'Despite these successes, developing alternative approaches to using laser light is desirable due to the significant technological challenges which arise with an increasing number of qubits.', '1603.03384-1-2-1': 'This becomes particularly obvious when considering that a useful quantum computer may require millions of ions which would need thousands of laser beams to be carefully aligned to micrometer precision.', '1603.03384-1-2-2': 'Long-wavelength radiation in the microwave and radio-frequency (RF) domain is comparably simple to generate, highly stable and is naturally broadcast over a large area using a small number of standard off-the-shelf emitters, essentially removing these alignment requirements.', '1603.03384-1-2-3': 'Microwave fields have already been used to implement single qubit gates with errors of only [MATH], far surpassing fault-tolerant thresholds [CITATION].', '1603.03384-1-2-4': "Free-running long-wavelength radiation on its own is however not sufficient for the implementation of multi-qubit gates, as it only weakly drives the ions' motion due to the vanishingly small Lamb-Dicke parameter.", '1603.03384-1-2-5': 'This drawback was first addressed in the seminal work by Mintert and Wunderlich in 2001 who showed that combining a static magnetic field gradient with externally applied long-wavelength radiation creates a sizable effective Lamb-Dicke parameter [CITATION].', '1603.03384-1-2-6': 'More recently, Ospelkaus et al. proposed using the oscillating magnetic field gradients experienced by an ion trapped in the near-field of a microwave waveguide to perform multi-qubit gates [CITATION].', '1603.03384-1-2-7': 'This scheme was subsequently used to perform the first microwave-based two-qubit gate by Ospelkaus et al. [CITATION].', '1603.03384-1-2-8': 'This scheme requires ions to be trapped close to a surface incorporating the microwave waveguide and therefore the effects of motional heating must be considered.', '1603.03384-1-2-9': 'When scaling this approach, especially considering complicated electrode geometries such as X-junctions, relevant individual microwave impedance matching for each entanglement zone across the whole architecture must be assured.', '1603.03384-1-2-10': 'Addressing of individual ions would typically require the use of destructive interference incorporating all microwave fields applied within the range of the ion or other sophisticated methods [CITATION].', '1603.03384-1-3-0': 'The approach using a static magnetic field gradient in conjunction with externally applied global long-wavelength radiation is not subject to the above constraints and has also been used to implement a two-qubit gate between nearest as well as non-nearest neighbour ions.', '1603.03384-1-3-1': "In stark contrast to the work presented in this manuscript, the first demonstration of this method was implemented making use of an 'undriven' magnetic gradient induced coupling [CITATION].", '1603.03384-1-3-2': 'However, in this scheme, the dominant source of noise is ambient magnetic field fluctuations as naturally occurring states with different magnetic moments must be used, ruling out the use of a so-called clock qubit [CITATION].', '1603.03384-1-3-3': 'This made it appear that fault-tolerant multi-qubit gate operations and the subsequent use in a long-wavelength-radiation-based ion trap quantum computer may be more challenging.', '1603.03384-1-4-0': "In this work we remove this remaining challenge by quantum engineering an effective clock qubit making use of 'dressed states' [CITATION].", '1603.03384-1-4-1': 'We follow the seminal proposal by Timoney et al. [CITATION] to use such states to implement a Molmer-Sorensen two-qubit gate [CITATION] utilizing long-wavelength radiation and a static magnetic field gradient.', '1603.03384-1-4-2': 'By demonstrating entanglement using quantum-engineered clock qubits we negate the requirement of using naturally occurring highly noise-sensitive qubit states which so far had to be used, thereby opening up the possibility for fault-tolerant quantum operations to be implemented.', '1603.03384-1-4-3': 'We then outline a remarkably simple approach which utilizes this gate technology to scale to an, in principle, arbitrarily large number of fault-tolerant two-qubit gate operations in parallel where the number of required fields is no longer correlated with the number of pairs being entangled.', '1603.03384-1-4-4': 'This provides a simple and powerful architecture which forms the core element within a wider industrial blueprint to build a large-scale microwave ion trap quantum computer [CITATION].', '1603.03384-1-5-0': 'The two-qubit gate is performed on two [MATH]Yb[MATH] ions in a Paul trap with an ion-electrode distance of 310 [MATH]m [CITATION].', '1603.03384-1-5-1': 'We place permanent magnets close to the ion trap with an ion-to-nearest-magnet distance of approximately 6 mm as shown in Fig. 1.', '1603.03384-1-5-2': 'This provides a static magnetic field gradient of 23.6(3) T/m which is approximately constant across the ion string [CITATION].', '1603.03384-1-5-3': 'We slightly displace the ions from the magnetic field nil, which lifts the degeneracy of the [MATH]S[MATH] F=1 manifolds by 12.0 MHz and 14.8 MHz for ions 1 and 2 respectively, and defines the internal-state quantisation axis to lie along the trap axis.', '1603.03384-1-5-4': 'Laser light near-resonant with the [MATH]S[MATH]P[MATH] transition is used for Doppler laser cooling and for initial state preparation as well as state detection.', '1603.03384-1-5-5': 'State-dependent fluorescence is collected on a photo-multiplier tube, and the fluorescence measurements are normalised to remove preparation and detection errors (supplementary text).', '1603.03384-1-6-0': 'To broadcast the required long-wavelength radiation we only require a standard off-the-shelf microwave horn and a 3-turn rf-emitting copper coil placed outside the ultra-high vacuum environment, thereby providing an extremely straightforward approach to scaling this scheme to many more ions.', '1603.03384-1-6-1': 'We note that in a large-scale architecture our approach utilizes submerged static currents to be applied to microfabricated chip traps to give rise to the required static magnetic field gradients.', '1603.03384-1-6-2': 'The ion-surface distance requirement in this case is not very stringent.', '1603.03384-1-6-3': 'Simulations show magnetic field gradients in excess of 150 T/m with an ion-electrode distance of approximately 150 [MATH] can be achieved, using realistic values of applied current that have already been applied to an ion trapping chip of this type [CITATION].', '1603.03384-1-6-4': 'Such a relatively large ion-electrode distance minimizes decoherence due to charge fluctuations from the electrode surface.', '1603.03384-1-7-0': "Instead of using a naturally occurring magnetic field sensitive qubit we quantum engineer a highly noise-resilient 'clock-like' qubit by first addressing each ion with a pair of microwave fields coupling the [MATH]S[MATH] with the [MATH]S[MATH] and [MATH]S[MATH] (Fig. 1).", '1603.03384-1-7-1': 'In the appropriate interaction picture this results in three dressed-states, including the well-protected state [MATH] [CITATION].', '1603.03384-1-7-2': 'We combine this state with the intrinsically well-protected state [MATH] to obtain a quantum-engineered clock qubit [MATH] (supplementary text).', '1603.03384-1-7-3': 'Unlike a standard clock transition, which has a fixed energy splitting, the qubit transition frequency is tunable by using a magnetic-field, enabling individual qubit addressing with global radiation fields.', '1603.03384-1-7-4': 'We prepare and detect the engineered clock qubit using the method developed by Randall et al. [CITATION].', '1603.03384-1-7-5': 'Arbitrary single qubit gates between [MATH] and [MATH] are implemented using an RF field resonant with the [MATH] transition [CITATION].', '1603.03384-1-7-6': 'The degeneracy in frequency between this and the [MATH] transition is lifted by the second-order Zeeman shift.', '1603.03384-1-7-7': 'Using a Ramsey type experiment we measure the coherence time of this qubit to be 650 ms, significantly longer than the [MATH] ms coherence time of the bare state qubits that have so far been used for two-qubit gates with a static magnetic field gradient.', '1603.03384-1-8-0': 'We create a maximally entangled state using a Molmer-Sorensen type gate.', '1603.03384-1-8-1': 'The application of this gate to our qubit has been investigated in detail theoretically [CITATION] and forms the basis of our experimental implementation.', '1603.03384-1-8-2': 'We implement the gate on the axial stretch mode with a frequency of [MATH] kHz, where [MATH] is the axial centre-of-mass mode frequency, giving an effective Lamb-Dicke parameter [CITATION] [MATH], where [MATH].', '1603.03384-1-8-3': 'This mode is sideband cooled to [MATH] using a variant of the scheme described in ref. [CITATION] (supplementary text) before the internal states are prepared in the state [MATH].', '1603.03384-1-8-4': 'A pair of RF fields is then applied to each ion with frequencies close to the red and blue sidebands (carrier Rabi frequency [MATH] kHz).', '1603.03384-1-8-5': 'The frequencies are set to be symmetric about the carrier frequency, corresponding to detunings [MATH].', '1603.03384-1-8-6': 'The gate detuning [MATH] is set to [MATH] Hz in order that at time [MATH] ms, the ions are ideally prepared in a maximally entangled spin state [MATH] (supplementary text).', '1603.03384-1-8-7': 'Fig. 2 (A) shows the evolution of the spin state populations as a function of time.', '1603.03384-1-8-8': 'To measure the coherence of the entangled state, a carrier [MATH] pulse is applied to each ion after the gate pulse.', '1603.03384-1-8-9': 'Fig. 2 (B) shows the parity [MATH] as a function of the phase [MATH] of the [MATH] pulse (supplementary text).', '1603.03384-1-8-10': 'The amplitude of the parity oscillation (A) along with the populations at [MATH] allows the fidelity of the obtained density matrix [MATH] with respect to the ideal outcome [MATH] to be calculated using [MATH] [CITATION].', '1603.03384-1-8-11': 'We measure the populations at [MATH] to be [MATH] and a fit to the parity scan shown in Fig. 2 (B) gives an amplitude of [MATH].', '1603.03384-1-8-12': 'From this we extract a Bell state fidelity of [MATH].', '1603.03384-1-9-0': 'The most significant contributions to the infidelity stem from heating of the vibrational mode of motion ([MATH]) used during the gate operation and depolarisation of the qubit ([MATH]).', '1603.03384-1-9-1': 'Both sources of error can be significantly reduced by increasing the gate speed using a larger static magnetic field gradient and by increasing [MATH].', '1603.03384-1-9-2': 'The depolarisation error can be further reduced by improving our microwave setup as a result of which a coherence time of seconds should be achievable as already demonstrated by Baumgart et al. [CITATION].', '1603.03384-1-9-3': 'Additional small sources of infidelity are discussed in the supplementary text.', '1603.03384-1-10-0': 'Achieving gate fidelities that would enable fault-tolerant operation using long-wavelength radiation can be realized either by the use of ion trap microchips or by a slight modification of our setup.', '1603.03384-1-10-1': 'By reducing the ion-to-nearest-magnet distance in a modified trap design to 2.4 mm, a magnetic field gradient of 150 T/m would result.', '1603.03384-1-10-2': 'This gives a large increase of the motional coupling strength, enabling a significant reduction of the error terms.', '1603.03384-1-10-3': 'Following a full numerical simulation of the system, a fidelity far in the fault-tolerant regime would result using already demonstrated parameters (supplementary text).', '1603.03384-1-11-0': 'Scaling current multi-qubit gates to a very large number of ions for quantum computing while maintaining fault-tolerant fidelities is a major challenge.', '1603.03384-1-11-1': 'The gate presented here provides a significant reduction in experimental complexity, being based on well-developed microwave and RF technology.', '1603.03384-1-11-2': 'Furthermore, in previously envisioned ion trap quantum computing architectures the number of radiation fields required for quantum gate implementation is strongly correlated with the number of ions used.', '1603.03384-1-11-3': 'This constitutes a substantial challenge in the construction of a large-scale quantum computer, which may require manipulation of millions or billions of ions.', '1603.03384-1-11-4': 'We have developed an approach that completely removes this undesirable correlation which we now summarise.', '1603.03384-1-11-5': 'In our approach ions will be located in individual gate zones that are contained within an array of X-junctions as part of a microfabricated ion trap architecture [CITATION].', '1603.03384-1-11-6': 'Currents applied locally to each gate zone create magnetic field gradients of 150 T/m, to be used for entanglement generation.', '1603.03384-1-11-7': 'In order to select any arbitrary set of gate zones for single- or two-qubit gate execution, one simply shifts the position of the ion within these zones axially with respect to the magnetic field gradient by an appropriate amount using local DC electrodes already used for ion shuttling.', '1603.03384-1-11-8': 'In a magnetic field gradient, such shifts in the ion positions result in a variation of the local offset magnetic field.', '1603.03384-1-11-9': 'The energy splitting of the engineered clock-qubit used in this work can be changed using such offset magnetic fields while remaining protected against magnetic field noise making it resonant or off-resonant to globally applied long-wavelength radiation fields.', '1603.03384-1-11-10': 'Therefore, ions in any arbitrary zone can be tuned into resonance with a set of globally applied microwave and RF fields, providing parallel execution of gates in relevant zones while all other zones on the chip remain off-resonant.', '1603.03384-1-11-11': 'Alternatively, instead of using displacement of the ions to change the offset magnetic field, an offset magnetic field could also be realized using additional local magnetic field coils located under each gate zone.', '1603.03384-1-11-12': 'Microwave horns and antennas located outside the vacuum system broadcast this set of microwave and RF fields over the entire microchip or quantum computer architecture.', '1603.03384-1-11-13': 'Quantum operations are then applied in parallel to an arbitrarily large collection of ions with negligible crosstalk (supplementary text) using a small number of offset magnetic fields and associated sets of global microwave and RF fields, as shown in Fig. 3.', '1603.03384-1-11-14': 'This approach, particularly when viewed within the context of a blueprint providing necessary technical details of a realistic device [CITATION] may provide some foundation for the assertion that the construction of a practical ion trap quantum computer is now within reach of current technology.'}

{'1603.03384-2-0-0': 'Trapped ions are a promising tool for building a large-scale quantum computer.', '1603.03384-2-0-1': 'However, the number of required radiation fields for the realisation of quantum gates in any proposed ion-based architecture scales with the number of ions within the quantum computer, posing a major obstacle when imagining a device with millions of ions.', '1603.03384-2-0-2': 'Here we present a fundamentally different concept for trapped-ion quantum computing where this detrimental scaling entirely vanishes, replacing millions of radiation fields with only a handful of fields.', '1603.03384-2-0-3': 'The method is based on individually controlled voltages applied to each logic gate location to facilitate the actual gate operation analogous to a traditional transistor architecture within a classical computer processor.', '1603.03384-2-0-4': 'To demonstrate the key principle of this approach we implement a versatile quantum gate method based on long-wavelength radiation and use this method to generate a maximally entangled state of two quantum engineered clock-qubits with fidelity 0.985(12).', '1603.03384-2-0-5': 'This quantum gate also constitutes a simple-to-implement tool for quantum metrology, sensing and simulation.', '1603.03384-2-1-0': 'The control of the internal and external degrees of freedom of trapped ions using laser light has allowed unprecedented advances in the creation of multi-particle entangled states [CITATION], quantum simulation [CITATION], frequency standards [CITATION], quantum sensing [CITATION] and quantum logic [CITATION].', '1603.03384-2-1-1': 'A major goal is now to construct a large-scale quantum computer by scaling current systems up to a significantly larger number of ions [CITATION].', '1603.03384-2-1-2': 'The circuit-model approach for quantum information processing requires the realization of single qubit gates and a two-qubit entanglement operation [CITATION].', '1603.03384-2-1-3': 'The use of laser light for the implementation of these quantum logic operations has been extremely successful, with gate fidelities in the fault-tolerant regime having been achieved for single [CITATION] as well as two-qubit gates [CITATION].', '1603.03384-2-2-0': 'Instead of using laser light it is also possible to use long-wavelength radiation in the microwave and RF regime to implement quantum logic.', '1603.03384-2-2-1': 'Such fields are comparably simple to generate and highly stable and have already been used to implement single-qubit gates with errors of only [MATH], far surpassing fault-tolerant thresholds [CITATION].', '1603.03384-2-2-2': "Free-running long-wavelength radiation on its own is however not sufficient for the implementation of multi-qubit gates, as it only weakly drives the ions' motion due to the vanishingly small Lamb-Dicke parameter.", '1603.03384-2-2-3': 'This drawback was first addressed in the seminal work by Mintert and Wunderlich in 2001 who showed that combining a static magnetic field gradient with externally applied long-wavelength radiation creates a sizable effective Lamb-Dicke parameter [CITATION].', '1603.03384-2-2-4': 'More recently, Ospelkaus et al. proposed using the oscillating magnetic field gradients experienced by an ion trapped in the near-field of a microwave waveguide to perform multi-qubit gates [CITATION].', '1603.03384-2-2-5': 'This scheme was subsequently used to perform the first microwave-based two-qubit gate by Ospelkaus et al. [CITATION].', '1603.03384-2-2-6': 'The scheme requires ions to be trapped close to a surface incorporating the microwave waveguide and therefore the effects of motional heating must be more carefully considered.', '1603.03384-2-2-7': 'When scaling this approach, especially considering complicated electrode geometries such as X-junctions, relevant individual microwave impedance matching for each gate zone across the whole architecture must be assured.', '1603.03384-2-2-8': 'Addressing of individual ions would typically require the use of destructive interference incorporating all microwave fields applied within the range of the ion or other sophisticated methods [CITATION].', '1603.03384-2-3-0': 'The approach of using a static magnetic field gradient in conjunction with externally applied long-wavelength radiation is not subject to the above constraints (of course the effects of motional heating should still be considered) and has also been used to implement a two-qubit gate between nearest as well as non-nearest neighbour ions [CITATION].', '1603.03384-2-3-1': "In stark contrast to the work presented in this manuscript, the first demonstration of using a static magnetic field gradient to implement a two-qubit gate made use of an 'undriven' magnetic gradient induced coupling.", '1603.03384-2-3-2': 'However, in this scheme the dominant source of noise is ambient magnetic field fluctuations as naturally occurring states with different magnetic moments must be used, ruling out the use of a so-called clock qubit.', '1603.03384-2-3-3': "A promising approach to circumvent this drawback is to use 'dressed states' [CITATION] where one can quantum engineer an effective clock qubit which is highly protected from magnetic field fluctuations while maintaining a strong sensitivity to a static magnetic field gradient.", '1603.03384-2-3-4': 'They have already been used in single qubit operations [CITATION] and to cool an ion close to its ground state of motion [CITATION] and their use to implement a two qubit gate would constitute a significant breakthrough for quantum computing with long-wavelength radiation.', '1603.03384-2-4-0': 'Despite these successes, scaling these laser or long-wavelength radiation based operations to a much larger number of ions constitutes a tremendous challenge.', '1603.03384-2-4-1': 'This becomes particularly obvious when considering that a large-scale universal quantum computer, say of the size large enough to break RSA encoding, would require millions or even billions of qubits [CITATION].', '1603.03384-2-4-2': 'Gate operations need to be carried out in parallel for the quantum computer to work.', '1603.03384-2-4-3': "The implication of this is that a large-scale quantum computer may require millions of spatially separated 'gate zones' where quantum gates are executed.", '1603.03384-2-4-4': 'This results in the requirement of utilizing millions of laser or long-wavelength radiation fields for the implementation of quantum gates when considering all previous proposals to build a large-scale trapped-ion quantum computer [CITATION].', '1603.03384-2-4-5': 'This detrimental scaling between the number of ions and the required number of radiation fields constitutes a significant obstacle to scaling to the desired large system sizes.', '1603.03384-2-5-0': 'In this work we remove this obstacle.', '1603.03384-2-5-1': 'We present a new concept for trapped-ion quantum computing where parallel quantum gate operations in arbitrarily many selected gate zones can be executed using individually controlled voltages applied to each gate zone.', '1603.03384-2-5-2': 'Instead of millions of laser or long-wavelength radiation fields this remarkably simple approach only requires a handful of global radiation fields where the number of radiation fields only depends on the number of different types of quantum gates to be executed in parallel.', '1603.03384-2-5-3': 'This then provides a simple and powerful concept for quantum computing which forms the core element within a wider engineering blueprint to build a large-scale microwave-based trapped-ion quantum computer [CITATION].', '1603.03384-2-5-4': 'A key element of our approach is the use of qubits which feature a widely tunable transition frequency while maintaining its protected nature with respect to ambient magnetic field fluctuations.', '1603.03384-2-5-5': 'Quantum engineered clock qubits meet this requirement and therefore constitute an ideal system for this purpose.', '1603.03384-2-5-6': 'We demonstrate the key element of our approach by generating entanglement between microwave-based quantum-engineered clock qubits in a Molmer-Sorensen-type interaction utilizing long-wavelength radiation and a static magnetic field gradient [CITATION].', '1603.03384-2-6-0': 'The two-qubit gate is performed on two [MATH]Yb[MATH] ions in a Paul trap with an ion-electrode distance of 310 [MATH]m [CITATION].', '1603.03384-2-6-1': 'We place permanent magnets close to the ion trap with an ion-to-nearest-magnet distance of approximately 6 mm as shown in Fig. 1.', '1603.03384-2-6-2': 'This provides a static magnetic field gradient of 23.6(3) T/m which is approximately constant across the ion string [CITATION].', '1603.03384-2-6-3': 'We slightly displace the ions from the magnetic field nil, which lifts the degeneracy of the [MATH]S[MATH] F=1 manifolds by 12.0 MHz and 14.8 MHz for ions 1 and 2 respectively, and defines the internal-state quantisation axis to lie along the trap axis.', '1603.03384-2-6-4': 'Laser light near-resonant with the [MATH]S[MATH]P[MATH] transition is used for Doppler laser cooling and for initial state preparation as well as state detection.', '1603.03384-2-6-5': 'State-dependent fluorescence is collected on a photo-multiplier tube, and the fluorescence measurements are normalised to remove preparation and detection errors (Supplementary Methods).', '1603.03384-2-7-0': 'To globally broadcast the required long-wavelength radiation we only require a standard off-the-shelf microwave horn and a 3-turn rf-emitting copper coil placed outside the ultra-high vacuum environment.', '1603.03384-2-7-1': 'We note that in a large-scale architecture our approach utilizes submerged static currents incorporated into the microfabricated chip traps to give rise to the required static magnetic field gradients.', '1603.03384-2-7-2': 'The ion-surface distance requirement in this case is not very stringent.', '1603.03384-2-7-3': 'Simulations show magnetic field gradients in excess of 150 T/m with an ion-electrode distance of approximately 150 [MATH] can be achieved, using realistic values of applied current that have already been applied to an ion trapping chip of this type [CITATION].', '1603.03384-2-7-4': 'Such a relatively large ion-electrode distance minimizes motional decoherence due to charge fluctuations from the electrode surface.', '1603.03384-2-8-0': "Instead of using a naturally occurring magnetic field sensitive qubit we quantum engineer a tunable highly noise-resilient 'clock-like' qubit by first addressing each ion with a pair of microwave fields coupling the [MATH]S[MATH] with the [MATH]S[MATH] and [MATH]S[MATH] (Fig. 1).", '1603.03384-2-8-1': 'In the appropriate interaction picture this results in three dressed-states, including the well-protected state [MATH] [CITATION].', '1603.03384-2-8-2': 'We combine this state with the intrinsically well-protected state [MATH] to obtain a quantum-engineered clock qubit [MATH] (Supplementary Methods).', '1603.03384-2-8-3': 'Unlike a standard clock transition, which has a fixed transition frequency, the qubit transition frequency is tunable using a magnetic-field, enabling individual qubit addressing with global radiation fields.', '1603.03384-2-8-4': 'This is a critical feature when viewed within the context of the concept for a trapped-ion quantum computer outlined below.', '1603.03384-2-8-5': 'We prepare and detect the engineered clock qubit using the method developed by Randall et al. [CITATION].', '1603.03384-2-8-6': 'Arbitrary single qubit gates between [MATH] and [MATH] are implemented using an RF field resonant with the [MATH] transition [CITATION].', '1603.03384-2-8-7': 'The degeneracy in frequency between this and the [MATH] transition is lifted by the second-order Zeeman shift.', '1603.03384-2-8-8': 'Using a Ramsey type experiment we measure the coherence time of this qubit to be 650 ms, significantly longer than the [MATH] ms coherence time of the bare state qubits that have so far been used for two-qubit gates with a static magnetic field gradient.', '1603.03384-2-9-0': 'We create a maximally entangled state using a Molmer-Sorensen type gate.', '1603.03384-2-9-1': 'The application of this gate to our qubit has been investigated in detail theoretically [CITATION] and forms the basis of our experimental implementation.', '1603.03384-2-9-2': 'We implement the gate on the axial stretch mode with a frequency of [MATH] kHz, where [MATH] is the axial centre-of-mass mode frequency, giving an effective Lamb-Dicke parameter [CITATION] [MATH], where [MATH].', '1603.03384-2-9-3': 'This mode is sideband cooled to [MATH] using a variant of the scheme described in ref. [CITATION] (Supplementary Methods) before the internal states are prepared in the state [MATH].', '1603.03384-2-9-4': 'A pair of RF fields is then applied to each ion with frequencies close to the red and blue sidebands (carrier Rabi frequency [MATH] kHz).', '1603.03384-2-9-5': 'The frequencies are set to be symmetric about the carrier frequency, corresponding to detunings [MATH].', '1603.03384-2-9-6': 'The gate detuning [MATH] is set to [MATH] Hz in order that at time [MATH] ms, the ions are ideally prepared in a maximally entangled spin state [MATH] (Supplementary Methods).', '1603.03384-2-9-7': 'Fig. 2 (a) shows the evolution of the spin state populations as a function of time.', '1603.03384-2-9-8': 'To measure the coherence of the entangled state, a carrier [MATH] pulse is applied to each ion after the gate pulse.', '1603.03384-2-9-9': 'Fig. 2 (b) shows the parity [MATH] as a function of the phase [MATH] of the [MATH] pulse (Supplementary Methods).', '1603.03384-2-9-10': 'The amplitude of the parity oscillation (a) along with the populations at [MATH] allows the fidelity of the obtained density matrix [MATH] with respect to the ideal outcome [MATH] to be calculated using [MATH] [CITATION].', '1603.03384-2-9-11': 'We measure the populations at [MATH] to be [MATH] and a fit to the parity scan shown in Fig. 2 (b) gives an amplitude of [MATH].', '1603.03384-2-9-12': 'From this we extract a Bell state fidelity of [MATH].', '1603.03384-2-10-0': 'The most significant contributions to the infidelity stem from heating of the vibrational mode of motion ([MATH]) used during the gate operation and depolarisation of the qubit ([MATH]).', '1603.03384-2-10-1': 'Both sources of error can be significantly reduced by increasing the gate speed using a larger static magnetic field gradient and by increasing [MATH].', '1603.03384-2-10-2': 'The depolarisation error can be further reduced by improving our microwave setup as a result of which a coherence time of seconds should be achievable as already demonstrated by Baumgart et al. [CITATION].', '1603.03384-2-10-3': 'Additional small sources of infidelity are discussed in the Supplementary Methods.', '1603.03384-2-11-0': 'Achieving gate fidelities that would enable fault-tolerant operation using long-wavelength radiation can be realized either by the use of ion trap microchips or by a slight modification of our setup.', '1603.03384-2-11-1': 'By reducing the ion-to-nearest-magnet distance in a modified trap design to 2.4 mm, a magnetic field gradient of 150 T/m would result.', '1603.03384-2-11-2': 'This gives a large increase of the motional coupling strength, enabling a significant reduction of the error terms.', '1603.03384-2-11-3': 'Following a full numerical simulation of the system, a fidelity far in the fault-tolerant regime would result using already demonstrated parameters (Supplementary Methods).', '1603.03384-2-12-0': 'We now describe how the gate method explained above gives rise to a highly efficient approach to quantum computing with trapped ions.', '1603.03384-2-12-1': 'In previously envisioned trapped-ion quantum computing architectures the number of radiation fields required for quantum gate implementation is strongly correlated with the number of ions used [CITATION].', '1603.03384-2-12-2': 'This constitutes a substantial challenge in the construction of a large-scale quantum computer, which may require the manipulation of millions or billions of ions.', '1603.03384-2-12-3': 'We will now outline an approach that completely removes this undesirable correlation where millions of laser or long-wavelength radiation fields are replaced with only a handful of long-wavelength radiation fields.', '1603.03384-2-13-0': 'Ions are located in individual gate zones that are contained within an array of X-junctions as part of a microfabricated ion trap architecture (see Fig. [REF]).', '1603.03384-2-13-1': 'Currents applied locally to each gate zone create magnetic field gradients of 150 T/m, to be used for entanglement generation.', '1603.03384-2-13-2': 'In order to select any arbitrary set of gate zones for single- or two-qubit gate execution, one simply shifts the position of the ion(s) within these zones axially with respect to the magnetic field gradient by an appropriate amount using local DC electrodes already used for ion transport within the ion trap array.', '1603.03384-2-13-3': 'In a magnetic field gradient, such shifts in the ion positions result in a variation of the local offset magnetic field.', '1603.03384-2-13-4': 'The transition frequency of the quantum engineered clock-qubit used in this work can be changed using such offset magnetic fields.', '1603.03384-2-13-5': 'This provides the ability to tune the quantum engineered clock qubit into and out of resonance with globally applied long-wavelength radiation fields.', '1603.03384-2-13-6': 'Therefore, ions in any arbitrary zone can be tuned into resonance with a set of globally applied microwave and RF fields (of the sort used to implement the two-qubit gate presented in this manuscript), providing parallel execution of gates in relevant zones while all other zones on the architecture remain off-resonant.', '1603.03384-2-13-7': 'Alternatively, instead of using the displacement of the ions to change the offset magnetic field, an offset magnetic field could be generated using additional local magnetic field coils located under each gate zone.', '1603.03384-2-13-8': 'Microwave horns and antennas located outside the vacuum system broadcast the required set of microwave and RF fields over the entire microchip or quantum computer architecture.', '1603.03384-2-13-9': 'Quantum operations are then applied in parallel to arbitrarily many sets of qubits with negligible crosstalk (Supplementary Methods) using a small number of offset magnetic fields and associated sets of global microwave and RF fields, as shown in Fig. 3.', '1603.03384-2-13-10': 'This approach, particularly when viewed within the context of a blueprint providing necessary technical details of a realistic device [CITATION] may provide some foundation for the assertion that the construction of a practical trapped-ion quantum computer is now within reach of current technology.'}

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[]

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[]

[]

{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1603.03384

quant-ph-0701239

{'quant-ph-0701239-1-0-0': 'We propose a quantum repeater protocol which builds on the well-known DLCZ protocol [L.M. Duan, M.D. Lukin, J.I. Cirac, and P. Zoller, Nature 414, 413 (2001)], but which uses photon pair sources in combination with memories that allow to store a large number of temporal modes.', 'quant-ph-0701239-1-0-1': 'We suggest to realize such multi-mode memories based on the principle of photon echo, using solids doped with rare-earth ions.', 'quant-ph-0701239-1-0-2': 'The use of multi-mode memories promises a speedup in entanglement generation by several orders of magnitude and a significant reduction in stability requirements compared to the DLCZ protocol.', 'quant-ph-0701239-1-1-0': 'The distribution of entanglement over long distances is an important challenge in quantum information.', 'quant-ph-0701239-1-1-1': "It would extend the range for tests of Bell's inequalities, quantum key distribution and quantum networks.", 'quant-ph-0701239-1-1-2': 'The direct distribution of entangled states is limited by transmission losses.', 'quant-ph-0701239-1-1-3': 'For example, 1000 km of standard telecommunications optical fiber have a transmission of order [MATH].', 'quant-ph-0701239-1-1-4': 'To distribute entanglement over such distances, quantum repeaters [CITATION] are likely to be required.', 'quant-ph-0701239-1-1-5': 'Implementations of quantum repeaters have been proposed in various systems, including atomic ensembles [CITATION], single atoms [CITATION], NV centers [CITATION] and quantum dots [CITATION].', 'quant-ph-0701239-1-1-6': 'A basic element of all such protocols is the creation of entanglement between neighboring nodes [MATH] and [MATH], which is typically conditional on the outcome of a measurement, e.g. the detection of one or more photons at a central station between two nodes.', 'quant-ph-0701239-1-1-7': 'In order to profit from a nested repeater protocol [CITATION], the entanglement connection operations creating entanglement between non-neighboring nodes can only be performed once one knows the relevant measurement outcomes.', 'quant-ph-0701239-1-1-8': 'This requires a communication time of order [MATH], where [MATH] is the distance between [MATH] and [MATH].', 'quant-ph-0701239-1-1-9': 'Conventional repeater protocols are limited to a single entanglement generation attempt per elementary link per time interval [MATH].', 'quant-ph-0701239-1-1-10': 'Here we propose to overcome this limitation using a scheme that combines photon pair sources and memories that can store a large number of distinguishable temporal modes.', 'quant-ph-0701239-1-1-11': 'We also show that such memories could be realized based on the principle of photon echo, using solids doped with rare-earth ions.', 'quant-ph-0701239-1-2-0': 'Our scheme is inspired by the DLCZ protocol [CITATION], which uses Raman transitions in atomic ensembles that lead to non-classical correlations between atomic excitations and emitted photons [CITATION].', 'quant-ph-0701239-1-2-1': 'The basic procedure for entanglement creation between two remote locations [MATH] and [MATH] in our protocol requires one memory and one source of photon pairs at each location, denoted [MATH] and [MATH] respectively.', 'quant-ph-0701239-1-2-2': 'The two sources are coherently excited such that each has a small probability [MATH] of creating a pair, corresponding to a state [EQUATION]', 'quant-ph-0701239-1-2-3': 'Here [MATH] and [MATH] and [MATH]) are the two modes corresponding to [MATH]), and [MATH] is the vacuum state.', 'quant-ph-0701239-1-2-4': 'The [MATH] term introduces errors in the protocol, leading to the requirement that [MATH] has to be kept small, cf. below.', 'quant-ph-0701239-1-2-5': 'The photons in modes [MATH] and [MATH] are stored in the local memories [MATH] and [MATH].', 'quant-ph-0701239-1-2-6': 'The modes [MATH] and [MATH] are coupled into optical fibers and combined on a beam splitter at a central station between [MATH] and [MATH].', 'quant-ph-0701239-1-2-7': 'The modes after the beam splitter are [MATH], where [MATH] are the phases acquired by the photons on their way to the central station.', 'quant-ph-0701239-1-2-8': 'Detection of a single photon in [MATH], for example, creates a state [MATH] (neglecting [MATH] corrections), where [MATH] and [MATH] are now stored in the memories.', 'quant-ph-0701239-1-2-9': 'This can be rewritten as an entangled state of the two memories, [EQUATION] where [MATH] denotes the empty state of [MATH], [MATH] denotes the state storing a single photon, and [MATH].', 'quant-ph-0701239-1-3-0': 'This entanglement can be extended via entanglement swapping as in Ref. [CITATION].', 'quant-ph-0701239-1-3-1': 'Starting from entangled states [MATH] between memories [MATH] and [MATH], and [MATH] between [MATH] and [MATH], one can create an entangled state between [MATH] and [MATH] by converting the memory excitations of [MATH] and [MATH] back into propagating photonic modes and combining these modes on a beam splitter.', 'quant-ph-0701239-1-3-2': 'Detection of a single photon after the beam splitter, e.g. in the mode [MATH], will create an entangled state of the same type between [MATH] and [MATH], namely [MATH].', 'quant-ph-0701239-1-3-3': 'In this way it is possible to establish entanglement between more distant memories, which can be used for quantum communication as follows [CITATION].', 'quant-ph-0701239-1-4-0': 'Suppose that location [MATH]) contains a pair of memories [MATH] and [MATH] and [MATH]), and that entanglement has been established between [MATH] and [MATH], and between [MATH] and [MATH], i.e. that we have a state [MATH].', 'quant-ph-0701239-1-4-1': 'By post-selecting the cases where there is one memory excitation on each side, this corresponds to an effective state of the form [EQUATION] which is analogous to conventional polarization or time-bin entangled states.', 'quant-ph-0701239-1-4-2': 'Measurements in arbitrary bases can be performed by converting the memory modes back into photonic modes and combining modes [MATH] and [MATH] (and also [MATH] and [MATH]) on beam splitters with appropriate transmission coefficients and phases.', 'quant-ph-0701239-1-5-0': 'The repeater scheme described above is attractive because it requires only pair sources, photon memories and linear optical components.', 'quant-ph-0701239-1-5-1': 'The reliance on a single detection for the elementary entanglement creation makes it less sensitive to fiber losses than schemes based on coincident two-photon detections [CITATION].', 'quant-ph-0701239-1-5-2': 'The price to pay is the requirement of phase stability, cf. below.', 'quant-ph-0701239-1-5-3': 'Note that the modes [MATH] and [MATH], e.g., do not need to have the same wavelength.', 'quant-ph-0701239-1-5-4': 'One can choose the wavelength of [MATH] such that it has minimum transmission loss in the fiber, whereas the wavelength of [MATH] can be adapted to achieve optimal storage efficiency, since the memory can be located very close to the source such that transmission losses play no role for this mode.', 'quant-ph-0701239-1-6-0': 'The time required for a successful creation of an entangled state of the form Eq. ([REF]) is given by [EQUATION]', 'quant-ph-0701239-1-6-1': 'Here [MATH], where [MATH] is the total distance and [MATH] is the nesting level of the repeater.', 'quant-ph-0701239-1-6-2': 'The basic clock interval is [MATH], the time required for the photons to propagate from the sources to the central station and for the information about the result to propagate back to the memories.', 'quant-ph-0701239-1-6-3': 'The success probability for entanglement creation for a single elementary link is denoted by [MATH]; [MATH] is the success probability for entanglement swapping at the [MATH]-th level, and [MATH] is the probability for a successful post-selection of the state Eq. ([REF]).', 'quant-ph-0701239-1-6-4': 'The probabilities [MATH] and [MATH], which are calculated in Ref. [CITATION], depend on the detection and memory efficiencies.', 'quant-ph-0701239-1-6-5': 'The factors of [MATH] arise because entanglement has to be generated for two links before every entanglement connection.', 'quant-ph-0701239-1-6-6': 'If the average waiting time for entanglement generation for one link is [MATH], there will be a success for one of the two after [MATH]; then one has to wait a time [MATH] on average for the second one, giving a total of [MATH].', 'quant-ph-0701239-1-7-0': 'For a single creation of the state Eq. ([REF]) per interval [MATH] the success probability [MATH] would be equal to [MATH], where [MATH] is the photon detection efficiency and [MATH], with [MATH] the fiber attenuation length.', 'quant-ph-0701239-1-7-1': 'This probability is typically very small.', 'quant-ph-0701239-1-7-2': 'However, photon pair sources can have repetition rates much higher than [MATH], which is of order 1 kHz for [MATH] of order 200 km.', 'quant-ph-0701239-1-7-3': 'This leads one to ask whether it is possible to make several entanglement creation attempts per interval [MATH].', 'quant-ph-0701239-1-7-4': 'The source [MATH] then produces pairs of photons in correlated pairs of temporal modes ("time-bins") [MATH], with [MATH].', 'quant-ph-0701239-1-7-5': 'All the modes [MATH] are stored in the memory [MATH], and analogously for [MATH] and [MATH].', 'quant-ph-0701239-1-7-6': 'If there is a detection behind the central beam splitter for the [MATH]-th time-bin, for example in the mode [MATH], then we know that a state [MATH] is stored in the memories [MATH] and [MATH], cf. Fig. 1(a).', 'quant-ph-0701239-1-7-7': 'Running the same protocol for another pair of sources [MATH] and memories [MATH], there may be a detection in the [MATH]-th time-bin, leading to a state [MATH] being stored in the memories [MATH] and [MATH].', 'quant-ph-0701239-1-7-8': 'One can then perform entanglement swapping by re-converting the memory modes [MATH] and [MATH] into photonic modes and combining them on a beam splitter, cf. Fig. 1(b).', 'quant-ph-0701239-1-7-9': 'This leads to an entangled state [MATH] between the [MATH]-th mode stored in [MATH] and the [MATH]-th mode stored in [MATH].', 'quant-ph-0701239-1-7-10': 'Entanglement of the type Eq. ([REF]) can be created as before, again by combining the appropriate time-bins, cf. Fig. 1(c).', 'quant-ph-0701239-1-8-0': 'In order for the described protocol to be possible, one requires memories that allow to store and retrieve the various temporal modes [MATH] etc., preserving their distinguishability.', 'quant-ph-0701239-1-8-1': 'We refer to such memories as multi-mode memories (MMMs).', 'quant-ph-0701239-1-8-2': 'We describe below how MMMs can be realized based on the principle of photon echo, which ensures that photons absorbed at different times are emitted at different times.', 'quant-ph-0701239-1-8-3': 'With MMMs, working with [MATH] attempts per interval increases the success probability from [MATH] to [MATH], which is approximately equal to [MATH] for [MATH].', 'quant-ph-0701239-1-8-4': 'This means that the overall success rate of the repeater is increased by a factor of [MATH].', 'quant-ph-0701239-1-8-5': 'Note that our approach based on pair sources and MMMs can be used to speed up other protocols by the same factor, in particular schemes based on coincident two-photon detection [CITATION].', 'quant-ph-0701239-1-8-6': 'This is because the speedup occurs at the most basic level, that of elementary entanglement generation.', 'quant-ph-0701239-1-8-7': 'To our knowledge, there is no simple equivalent to the use of MMMs as described above within the Raman-transition based approach of Ref. [CITATION], since in that approach all stored modes would be retrieved at the same time (when the relevant control beam is turned on), making them indistinguishable.', 'quant-ph-0701239-1-8-8': 'Other forms of multiplexing (spatial, frequency) can be applied in a similar way both to our protocol and the DLCZ protocol [CITATION].', 'quant-ph-0701239-1-9-0': 'We now discuss how to realize the elements of our proposal in practice.', 'quant-ph-0701239-1-9-1': 'Photon pair sources with the required properties (sufficiently high [MATH], appropriate bandwidth, cf. below) can be realized both with parametric down-conversion [CITATION] and with atomic ensembles [CITATION].', 'quant-ph-0701239-1-9-2': 'Several approaches to the realization of photon memories have been proposed and experimentally investigated, including EIT [CITATION], off-resonant interactions [CITATION] and photon echo [CITATION].', 'quant-ph-0701239-1-9-3': 'The echo approach lends itself in a natural way to the storage of a large number of temporal modes.', 'quant-ph-0701239-1-9-4': 'Storage and retrieval of up to 1760-pulse sequences has been demonstrated [CITATION].', 'quant-ph-0701239-1-9-5': 'The temporal information is stored in the relative phases of atomic excitations at different frequencies.', 'quant-ph-0701239-1-9-6': 'Photon echoes based on controlled reversible inhomogeneous broadening (CRIB) [CITATION] allow in principle perfect reconstruction of the stored light.', 'quant-ph-0701239-1-9-7': 'The method is well adapted to atomic ensembles in the solid state, e.g. crystals doped with rare-earth ions.', 'quant-ph-0701239-1-10-0': 'To implement such a memory, one has to prepare a narrow absorption line inside a wide spectral hole, using optical pumping techniques [CITATION].', 'quant-ph-0701239-1-10-1': 'The line is artificially inhomogeneously broadened, e.g. by applying an electric field gradient.', 'quant-ph-0701239-1-10-2': 'Then the light can be absorbed, e.g. a train of pulses as described above.', 'quant-ph-0701239-1-10-3': 'After the absorption the electric field is turned off, and atoms in the excited state are transferred by a [MATH]-pulse to a second low-lying state, such as a different hyperfine state.', 'quant-ph-0701239-1-10-4': 'For recall, the population is transferred back to the excited state by a counter-propagating [MATH]-pulse, and the electric field is turned back on with the opposite sign, thereby inverting the inhomogeneous broadening.', 'quant-ph-0701239-1-10-5': 'This procedure leads to a time reversal of the absorption process.', 'quant-ph-0701239-1-10-6': 'The pulse train is re-emitted in inverted order, with a retrieval efficiency that is not limited by re-absorption [CITATION].', 'quant-ph-0701239-1-10-7': 'Photons absorbed in different memories at different times can be re-emitted simultaneously (as required in Fig. 1) by choosing appropriate times for the sign-flip of the applied electric field.', 'quant-ph-0701239-1-11-0': 'The achievable memory efficiency is [CITATION] [EQUATION] where for simplicity we consider the case of square spectral atomic distributions, both for the initial narrow line and the artificially broadened line.', 'quant-ph-0701239-1-11-1': 'Here [MATH] is the optical depth of the medium before the artificial broadening, [MATH] is the initial spectral distribution width, [MATH] is the width after broadening, and [MATH] is the time before transfer to the hyperfine state (we neglect hyperfine decoherence).', 'quant-ph-0701239-1-11-2': 'The above formula is exact for all pulse shapes whose spectral support is completely inside the square atomic distribution.', 'quant-ph-0701239-1-11-3': 'Otherwise there are additional losses due to spectral truncation of the pulse.', 'quant-ph-0701239-1-11-4': 'The width [MATH] has to be large enough to allow for pulse durations significantly shorter than the interval between pulses [MATH], in order to avoid errors due to pulse overlap.', 'quant-ph-0701239-1-11-5': 'For truncated Gaussian pulses choosing [MATH] is sufficient for such errors to be negligible.', 'quant-ph-0701239-1-11-6': 'On the other hand, [MATH] is required to be smaller than the separation between the hyperfine states used in the memory protocol and whatever state is used for shelving unwanted atoms in the preparation of the initial spectral hole (typically another hyperfine state).', 'quant-ph-0701239-1-11-7': 'The initial width [MATH] has to fulfill [MATH], where [MATH] is the homogeneous linewidth of the relevant transition [CITATION]; [MATH] should be chosen such as to optimize [MATH], the average of Eq. ([REF]) over all time-bins, i.e. for [MATH] between [MATH] and [MATH].', 'quant-ph-0701239-1-11-8': 'One can show that [MATH] can be expressed as a function of the two variables [MATH] and [MATH].', 'quant-ph-0701239-1-11-9': 'By adjusting [MATH] one can choose, for a given value of [MATH], the value of [MATH] that maximizes [MATH].', 'quant-ph-0701239-1-11-10': 'Then [MATH] becomes a function of [MATH] only, which is plotted in Fig. 2.', 'quant-ph-0701239-1-12-0': 'We now discuss how to achieve high values for [MATH] and [MATH] experimentally.', 'quant-ph-0701239-1-12-1': 'Pr:Y[MATH]SiO[MATH] is a very promising material for initial experiments, since hyperfine coherence times as long as 30 s [CITATION] and memory efficiencies of order 13 % [CITATION] for macroscopic light pulses have already been demonstrated.', 'quant-ph-0701239-1-12-2': 'The main drawback of Pr is the small hyperfine separation (of order a few MHz), which limits the possible pulse bandwidth and thus [MATH], since by definition [MATH] .', 'quant-ph-0701239-1-12-3': 'Neodymium and Erbium have hyperfine separations of hundreds of MHz [CITATION].', 'quant-ph-0701239-1-12-4': 'Nd also has strong absorption, e.g. Nd:YVO[MATH] with a Nd content of 10 ppm has an absorption coefficient [MATH]/cm [CITATION] at 879 nm.', 'quant-ph-0701239-1-12-5': 'Choosing [MATH] MHz (which gives [MATH] ns) and [MATH] kHz, which is well compatible with [MATH] kHz as measured for Nd in [CITATION], our above calculations show that e.g. [MATH] and [MATH] would be possible with [MATH], which could be achieved with a multi-pass configuration or cavity.', 'quant-ph-0701239-1-12-6': 'While Erbium-doped materials typically have lower absorption, they can combine good optical coherence and large inhomogeneous linewidth, e.g. [MATH] kHz and [MATH] GHz for Er:LiNbO[MATH] [CITATION].', 'quant-ph-0701239-1-12-7': 'This makes Er a natural candidate for the implementation of frequency multiplexing in addition to temporal multiplexing as described here.', 'quant-ph-0701239-1-12-8': 'This might allow one to gain an order of magnitude or more for [MATH] compared to our Nd example.', 'quant-ph-0701239-1-13-0': 'We now give an example for the potential performance of our scheme.', 'quant-ph-0701239-1-13-1': 'Consider a distance [MATH] km, and a fiber attenuation of 0.2 dB/km, corresponding to telecom wavelength photons.', 'quant-ph-0701239-1-13-2': 'Assume [MATH]=0.9 and photon-number-resolving detectors with a detection efficiency [MATH].', 'quant-ph-0701239-1-13-3': 'Highly efficient number-resolving detectors are being developed based on avalanche multiplication in semiconductors [CITATION] and on the superconducting-to-normal transition [CITATION].', 'quant-ph-0701239-1-13-4': 'One can show that the optimal nesting level for the repeater protocol for these values is [MATH], corresponding to [MATH] elementary links, which gives [MATH] km.', 'quant-ph-0701239-1-13-5': 'Using Eq. ([REF]) and Ref. [CITATION] one can show that [EQUATION] where [MATH] and [MATH], with [MATH] km, and [MATH] m/s in the fiber.', 'quant-ph-0701239-1-13-6': 'One can show by explicit calculation of the errors due to double emissions [CITATION] that the fidelity [MATH] of the final entangled state compared to the ideal maximally entangled state for a repeater with [MATH] levels is approximately [MATH].', 'quant-ph-0701239-1-13-7': 'If one wants, for example, [MATH] one therefore has to choose [MATH], which finally gives [MATH] s.', 'quant-ph-0701239-1-13-8': 'If one can achieve [MATH] one finds [MATH] s. High-efficiency MMMs as discussed above could thus reduce [MATH] for 1000 km to a few seconds (or less than a second with frequency multiplexing).', 'quant-ph-0701239-1-14-0': 'Finally we show that MMMs can also help to significantly alleviate the stability requirements for the repeater protocol.', 'quant-ph-0701239-1-14-1': 'We have seen that the entanglement in Eq. ([REF]) depends on the phase difference [MATH], which can be rewritten as [MATH].', 'quant-ph-0701239-1-14-2': 'Here [MATH]) is the time when the first (second) entangled state of the type of Eq. ([REF]) is created.', 'quant-ph-0701239-1-14-3': 'For simplicity, let us just consider an elementary link from our above example, with [MATH] km.', 'quant-ph-0701239-1-14-4': '(In reality, the final phase has contributions from all elementary links.)', 'quant-ph-0701239-1-14-5': 'It is possible to use the same fibers to connect [MATH] and [MATH] and respectively [MATH] and [MATH] to the central station.', 'quant-ph-0701239-1-14-6': 'The phase difference is therefore related to the fluctuations in the length of the two fibers in the interval [MATH].', 'quant-ph-0701239-1-14-7': 'In the case without MMMs the mean value of [MATH] is [MATH], which is of order 10 s for our above example, with [MATH] of order 1 ms and [MATH] of order [MATH].', 'quant-ph-0701239-1-14-8': 'Over such long timescales, the fiber lengths are expected to fluctuate significantly.', 'quant-ph-0701239-1-14-9': 'For example, let us assume that the temperature of one fiber changes by 1 K relative to the other one over a time of 1 h. Taking a typical thermal expansion coefficient of [MATH]K, this corresponds to a drift in the path length difference of 350 [MATH]m/s, i.e. by many wavelengths.', 'quant-ph-0701239-1-14-10': 'Active stabilization would thus definitely be required.', 'quant-ph-0701239-1-14-11': 'With MMMs, [MATH] can be made of order [MATH] (for [MATH] of order [MATH]).', 'quant-ph-0701239-1-14-12': 'It then becomes realistic to work only with states Eq. ([REF]) where the initial entanglement between [MATH] and [MATH] and between [MATH] and [MATH] was created in the same interval [MATH].', 'quant-ph-0701239-1-14-13': 'This increases the time [MATH] by a factor of order [MATH] (i.e. of order 10), but it reduces the mean value of [MATH] to of order [MATH], which is of order 20 [MATH]s for our example.', 'quant-ph-0701239-1-14-14': 'In 20 [MATH]s, the fiber length will fluctuate by only 7 nm for our example, which has a negligible effect on the final entanglement fidelity.', 'quant-ph-0701239-1-14-15': 'Active stabilization to interferometric precision might thus not be required with MMMs.', 'quant-ph-0701239-1-15-0': 'In conclusion the combination of photon pair sources and multi-mode memories should allow the realization of a quantum repeater protocol that is much faster and more robust than the protocol of Ref. [CITATION] while retaining its attractive features, in particular the use of linear optical elements and of single photon detections for entanglement generation and swapping.'}

{'quant-ph-0701239-2-0-0': 'We propose a quantum repeater protocol which builds on the well-known DLCZ protocol [L.M. Duan, M.D. Lukin, J.I. Cirac, and P. Zoller, Nature 414, 413 (2001)], but which uses photon pair sources in combination with memories that allow to store a large number of temporal modes.', 'quant-ph-0701239-2-0-1': 'We suggest to realize such multi-mode memories based on the principle of photon echo, using solids doped with rare-earth ions.', 'quant-ph-0701239-2-0-2': 'The use of multi-mode memories promises a speedup in entanglement generation by several orders of magnitude and a significant reduction in stability requirements compared to the DLCZ protocol.', 'quant-ph-0701239-2-1-0': 'The distribution of entanglement over long distances is an important challenge in quantum information.', 'quant-ph-0701239-2-1-1': "It would extend the range for tests of Bell's inequalities, quantum key distribution and quantum networks.", 'quant-ph-0701239-2-1-2': 'The direct distribution of entangled states is limited by transmission losses.', 'quant-ph-0701239-2-1-3': 'For example, 1000 km of standard telecommunications optical fiber have a transmission of order [MATH].', 'quant-ph-0701239-2-1-4': 'To distribute entanglement over such distances, quantum repeaters [CITATION] are likely to be required.', 'quant-ph-0701239-2-1-5': 'Implementations of quantum repeaters have been proposed in various systems, including atomic ensembles [CITATION], single atoms [CITATION], NV centers [CITATION] and quantum dots [CITATION].', 'quant-ph-0701239-2-1-6': 'A basic element of all protocols is the creation of entanglement between neighboring nodes [MATH] and [MATH], typically conditional on the outcome of a measurement, e.g. the detection of one or more photons at a station between two nodes.', 'quant-ph-0701239-2-1-7': 'In order to profit from a nested repeater protocol [CITATION], the entanglement connection operations creating entanglement between non-neighboring nodes can only be performed once one knows the relevant measurement outcomes.', 'quant-ph-0701239-2-1-8': 'This requires a communication time of order [MATH], where [MATH] is the distance between [MATH] and [MATH].', 'quant-ph-0701239-2-1-9': 'Conventional repeater protocols are limited to a single entanglement generation attempt per elementary link per time interval [MATH].', 'quant-ph-0701239-2-1-10': 'Here we propose to overcome this limitation using a scheme that combines photon pair sources and memories that can store a large number of distinguishable temporal modes.', 'quant-ph-0701239-2-1-11': 'We also show that such memories could be realized based on the principle of photon echo, using solids doped with rare-earth ions.', 'quant-ph-0701239-2-2-0': 'Our scheme is inspired by the DLCZ protocol [CITATION], which uses Raman transitions in atomic ensembles that lead to non-classical correlations between atomic excitations and emitted photons [CITATION].', 'quant-ph-0701239-2-2-1': 'The basic procedure for entanglement creation between two remote locations [MATH] and [MATH] in our protocol requires one memory and one source of photon pairs at each location, denoted [MATH] and [MATH] respectively.', 'quant-ph-0701239-2-2-2': 'The two sources are coherently excited such that each has a small probability [MATH] of creating a pair, corresponding to a state [EQUATION]', 'quant-ph-0701239-2-2-3': 'Here [MATH] and [MATH] and [MATH]) are the two modes corresponding to [MATH]), [MATH]) is the phase of the pump laser at location [MATH]), and [MATH] is the vacuum state.', 'quant-ph-0701239-2-2-4': 'The [MATH] term introduces errors in the protocol, leading to the requirement that [MATH] has to be kept small, cf. below.', 'quant-ph-0701239-2-2-5': 'The photons in modes [MATH] and [MATH] are stored in the local memories [MATH] and [MATH].', 'quant-ph-0701239-2-2-6': 'The modes [MATH] and [MATH] are coupled into fibers and combined on a beam splitter at a station between [MATH] and [MATH].', 'quant-ph-0701239-2-2-7': 'The modes after the beam splitter are [MATH], where [MATH] are the phases acquired by the photons on their way to the central station.', 'quant-ph-0701239-2-2-8': 'Detection of a single photon in [MATH], for example, creates a state [MATH] (neglecting [MATH] corrections), with [MATH], where [MATH] and [MATH] are now stored in the memories.', 'quant-ph-0701239-2-2-9': 'This can be rewritten as an entangled state of the two memories, [EQUATION] where [MATH] denotes the empty state of [MATH], [MATH] denotes the state storing a single photon, and [MATH].', 'quant-ph-0701239-2-3-0': 'This entanglement can be extended via entanglement swapping as in Ref. [CITATION].', 'quant-ph-0701239-2-3-1': 'Starting from entangled states [MATH] between memories [MATH] and [MATH], and [MATH] between [MATH] and [MATH], one can create an entangled state between [MATH] and [MATH] by converting the memory excitations of [MATH] and [MATH] back into propagating photonic modes and combining these modes on a beam splitter.', 'quant-ph-0701239-2-3-2': 'Detection of a single photon after the beam splitter, e.g. in the mode [MATH], will create an entangled state of the same type between [MATH] and [MATH], namely [MATH].', 'quant-ph-0701239-2-3-3': 'In this way it is possible to establish entanglement between more distant memories, which can be used for quantum communication as follows [CITATION].', 'quant-ph-0701239-2-4-0': 'Suppose that location [MATH]) contains a pair of memories [MATH] and [MATH] and [MATH]), and that entanglement has been established between [MATH] and [MATH], and between [MATH] and [MATH], i.e. that we have a state [MATH].', 'quant-ph-0701239-2-4-1': 'The projection of this state onto the subspace with one memory excitation in each location is [EQUATION] which is analogous to conventional polarization or time-bin entangled states.', 'quant-ph-0701239-2-4-2': 'The required projection can be performed post-selectively by converting the memory excitations back into photons and counting the number of photons in each location.', 'quant-ph-0701239-2-4-3': 'Measurements in arbitrary bases are possible by combining modes [MATH] and [MATH] (and also [MATH] and [MATH]) on beam splitters with appropriate transmission coefficients and phases.', 'quant-ph-0701239-2-5-0': 'The repeater scheme described above is attractive because it requires only pair sources, photon memories and linear optical components.', 'quant-ph-0701239-2-5-1': 'The reliance on a single detection for the elementary entanglement creation makes it less sensitive to fiber losses than schemes based on coincident two-photon detections [CITATION].', 'quant-ph-0701239-2-5-2': 'The price to pay is the requirement of phase stability, cf. below.', 'quant-ph-0701239-2-6-0': 'The time required for a successful creation of an entangled state of the form Eq. ([REF]) is given by [EQUATION]', 'quant-ph-0701239-2-6-1': 'Here [MATH], where [MATH] is the total distance and [MATH] is the nesting level of the repeater.', 'quant-ph-0701239-2-6-2': 'The basic clock interval is [MATH], the time required for the photons to propagate from the sources to the central station and for the information about the result to propagate back to the memories.', 'quant-ph-0701239-2-6-3': 'The success probability for entanglement creation for a single elementary link is denoted by [MATH]; [MATH] is the success probability for entanglement swapping at the [MATH]-th level, and [MATH] is the probability for a successful projection onto the state Eq. ([REF]).', 'quant-ph-0701239-2-6-4': 'The probabilities [MATH] and [MATH], which are calculated in Ref. [CITATION], depend on the detection and memory efficiencies.', 'quant-ph-0701239-2-6-5': 'The factors of [MATH] arise because entanglement has to be generated for two links before every entanglement connection.', 'quant-ph-0701239-2-6-6': 'If the average waiting time for entanglement generation for one link is [MATH], there will be a success for one of the two after [MATH]; then one has to wait a time [MATH] on average for the second one, giving a total of [MATH].', 'quant-ph-0701239-2-7-0': 'For a single creation of the state Eq. ([REF]) per interval [MATH] the success probability [MATH] is equal to [MATH], where [MATH] is the photon detection efficiency and [MATH], with [MATH] the fiber attenuation length; [MATH] is typically very small.', 'quant-ph-0701239-2-7-1': 'However, photon pair sources can have repetition rates much higher than [MATH], which is of order 1 kHz for [MATH] of order 200 km.', 'quant-ph-0701239-2-7-2': 'This leads one to ask whether it is possible to make several entanglement creation attempts per interval [MATH].', 'quant-ph-0701239-2-7-3': 'The source [MATH] then produces pairs of photons in correlated pairs of temporal modes ("time bins") [MATH], with [MATH].', 'quant-ph-0701239-2-7-4': 'All the modes [MATH] are stored in the memory [MATH], and analogously for [MATH] and [MATH].', 'quant-ph-0701239-2-7-5': 'If there is a detection behind the central beam splitter for the [MATH]-th time bin, for example in the mode [MATH], then we know that a state [MATH] is stored in the memories [MATH] and [MATH], cf. Fig. 1(a).', 'quant-ph-0701239-2-7-6': 'Running the same protocol for another pair of sources [MATH] and memories [MATH], there may be a detection in the [MATH]-th time bin, leading to a state [MATH] being stored in the memories [MATH] and [MATH].', 'quant-ph-0701239-2-7-7': 'One can then perform entanglement swapping by re-converting the memory modes [MATH] and [MATH] into photonic modes and combining them on a beam splitter, cf. Fig. 1(b).', 'quant-ph-0701239-2-7-8': 'This leads to an entangled state [MATH] between the [MATH]-th mode stored in [MATH] and the [MATH]-th mode stored in [MATH].', 'quant-ph-0701239-2-7-9': 'Entanglement of the type Eq. ([REF]) can be created as before, again by combining the appropriate time bins, cf. Fig. 1(c).', 'quant-ph-0701239-2-8-0': 'The described protocol requires memories that allow to store and retrieve the various temporal modes [MATH] etc., preserving their distinguishability.', 'quant-ph-0701239-2-8-1': 'We refer to such memories as multi-mode memories (MMMs).', 'quant-ph-0701239-2-8-2': 'We describe below how MMMs can be realized based on the photon echo principle, which ensures that photons absorbed at different times are emitted at different times.', 'quant-ph-0701239-2-8-3': 'With MMMs, working with [MATH] attempts per interval increases the success probability from [MATH] to [MATH], which is approximately equal to [MATH] for [MATH], increasing the overall success rate of the repeater by a factor of [MATH].', 'quant-ph-0701239-2-8-4': 'Our approach based on pair sources and MMMs can be used to speed up other protocols by the same factor, in particular schemes based on coincident two-photon detection [CITATION], because the speedup occurs at the most basic level, that of elementary entanglement generation.', 'quant-ph-0701239-2-8-5': 'There is no obvious equivalent to the use of MMMs as described above within the Raman-transition based approach of Ref. [CITATION], since all stored modes would be retrieved at the same time, when the relevant control beam is turned on.', 'quant-ph-0701239-2-8-6': 'Other forms of multiplexing (spatial, frequency) can be applied in a similar way both to our protocol and the DLCZ protocol [CITATION].', 'quant-ph-0701239-2-9-0': 'We now discuss how to realize the elements of our proposal in practice.', 'quant-ph-0701239-2-9-1': 'Photon pair sources with the required properties (sufficiently high [MATH], appropriate bandwidth, cf. below) can be realized both with parametric down-conversion [CITATION] and with atomic ensembles [CITATION].', 'quant-ph-0701239-2-9-2': 'Several approaches to the realization of photon memories have been proposed and studied experimentally, including EIT [CITATION], off-resonant interactions [CITATION] and photon echo [CITATION].', 'quant-ph-0701239-2-9-3': 'The echo approach lends itself naturally to the storage of many temporal modes.', 'quant-ph-0701239-2-9-4': 'Storage and retrieval of up to 1760-pulse sequences has been demonstrated [CITATION].', 'quant-ph-0701239-2-9-5': 'The temporal information is stored in the relative phases of atomic excitations at different frequencies.', 'quant-ph-0701239-2-9-6': 'Photon echoes based on controlled reversible inhomogeneous broadening (CRIB) [CITATION] allow in principle perfect reconstruction of the stored light.', 'quant-ph-0701239-2-9-7': 'The method is well adapted to atomic ensembles in solids, e.g. crystals doped with rare-earth ions.', 'quant-ph-0701239-2-9-8': 'To implement such a memory, one has to prepare a narrow absorption line inside a wide spectral hole, using optical pumping techniques [CITATION].', 'quant-ph-0701239-2-9-9': 'The line is artificially inhomogeneously broadened, e.g. by applying an electric field gradient.', 'quant-ph-0701239-2-9-10': 'Then the light can be absorbed, e.g. a train of pulses as described above.', 'quant-ph-0701239-2-9-11': 'After the absorption the electric field is turned off, and atoms in the excited state are transferred by a [MATH]-pulse to a second low-lying state, e.g. a different hyperfine state.', 'quant-ph-0701239-2-9-12': 'For recall, the population is transferred back to the excited state by a counter-propagating [MATH]-pulse, and the electric field is turned back on with the opposite sign, thereby inverting the inhomogeneous broadening.', 'quant-ph-0701239-2-9-13': 'This leads to a time reversal of the absorption.', 'quant-ph-0701239-2-9-14': 'The pulse train is re-emitted in inverted order, with a retrieval efficiency that is not limited by re-absorption [CITATION].', 'quant-ph-0701239-2-9-15': 'Photons absorbed in different memories at different times can be re-emitted simultaneously (as in Fig. 1) by choosing appropriate times for the sign-flip of the applied electric field.', 'quant-ph-0701239-2-10-0': 'The achievable memory efficiency is [CITATION] [EQUATION] where for simplicity we consider square spectral atomic distributions, both for the initial narrow line and the artificially broadened line.', 'quant-ph-0701239-2-10-1': 'Here [MATH] is the optical depth of the medium before the artificial broadening, [MATH] is the initial spectral distribution width, [MATH] is the width after broadening, and [MATH] is the time before transfer to the hyperfine state (neglecting hyperfine decoherence).', 'quant-ph-0701239-2-10-2': 'The above formula is exact for all pulse shapes whose spectral support is completely inside the square atomic distribution.', 'quant-ph-0701239-2-10-3': 'Otherwise there are additional losses due to spectral truncation of the pulse.', 'quant-ph-0701239-2-10-4': 'The width [MATH] has to be large enough to allow for pulse durations significantly shorter than the interval between pulses [MATH], in order to avoid errors due to pulse overlap.', 'quant-ph-0701239-2-10-5': 'For truncated Gaussian pulses choosing [MATH] is sufficient for such errors to be negligible.', 'quant-ph-0701239-2-10-6': 'On the other hand, [MATH] is required to be smaller than the separation between the hyperfine states used in the memory protocol and whatever state is used for shelving unwanted atoms in the preparation of the initial spectral hole (e.g. another hyperfine state).', 'quant-ph-0701239-2-10-7': 'The initial width [MATH] has to fulfill [MATH], where [MATH] is the homogeneous linewidth of the relevant transition [CITATION]; [MATH] should be chosen such as to optimize [MATH], the average of Eq. ([REF]) over all time-bins, i.e. for [MATH] between [MATH] and [MATH].', 'quant-ph-0701239-2-10-8': 'One can show that [MATH] can be expressed as a function of the two variables [MATH] and [MATH].', 'quant-ph-0701239-2-10-9': 'By adjusting [MATH] one can choose, for a given value of [MATH], the value of [MATH] that maximizes [MATH].', 'quant-ph-0701239-2-10-10': 'Then [MATH] becomes a function of [MATH] only, which is plotted in Fig. 2.', 'quant-ph-0701239-2-11-0': 'The storage has to be phase-preserving, so as to conserve the entanglement for the states of Eqs. (2-3).', 'quant-ph-0701239-2-11-1': 'Decoherence can affect the excited states during the absorption of the pulse train, and the hyperfine ground states during the long-term storage.', 'quant-ph-0701239-2-11-2': 'In rare-earth ions, excited state coherence times ranging from tens of [MATH]s to 6 ms [CITATION] and hyperfine coherence times as long as 30 s [CITATION] have been demonstrated.', 'quant-ph-0701239-2-11-3': 'For a recent experimental investigation of phase coherence in photon echo with rare-earth ions see Ref. [CITATION].', 'quant-ph-0701239-2-12-0': 'We now discuss how to achieve high values for [MATH] and [MATH] experimentally.', 'quant-ph-0701239-2-12-1': 'Pr:Y[MATH]SiO[MATH] is a very promising material for initial experiments, since excellent hyperfine coherence [CITATION] and memory efficiencies of order 13 % [CITATION] for macroscopic light pulses have already been demonstrated.', 'quant-ph-0701239-2-12-2': 'The main drawback of Pr is the small hyperfine separation (of order a few MHz), which limits the possible pulse bandwidth and thus [MATH], since by definition [MATH] .', 'quant-ph-0701239-2-12-3': 'Neodymium and Erbium have hyperfine separations of hundreds of MHz [CITATION].', 'quant-ph-0701239-2-12-4': 'Nd also has strong absorption, e.g. Nd:YVO[MATH] with a Nd content of 10 ppm has an absorption coefficient [MATH]/cm [CITATION] at 879 nm.', 'quant-ph-0701239-2-12-5': 'Choosing [MATH] MHz (which gives [MATH] ns) and [MATH] kHz, which is well compatible with [MATH] kHz as measured for Nd in [CITATION], our above calculations show that e.g. [MATH] and [MATH] would be possible with [MATH], which could be achieved with a multi-pass configuration.', 'quant-ph-0701239-2-12-6': 'Erbium-doped materials can combine good optical coherence and large inhomogeneous linewidth, e.g. [MATH] kHz and [MATH] GHz for Er:LiNbO[MATH] [CITATION], which makes Er a natural candidate for the implementation of frequency multiplexing in addition to temporal multiplexing.', 'quant-ph-0701239-2-12-7': 'The protocol could be run in parallel for a large number of frequency channels.', 'quant-ph-0701239-2-12-8': 'Even taking into account the lower absorption for Er, this might allow one to gain an order of magnitude or more for the overall value of [MATH] compared to our Nd example.', 'quant-ph-0701239-2-13-0': 'To assess the potential performance of our scheme, consider a distance [MATH] km, and a fiber attenuation of 0.2 dB/km, corresponding to telecom wavelength photons.', 'quant-ph-0701239-2-13-1': 'Note that the wavelengths of the photon propagating in the fiber and of the photon stored in the memory can be different in our scheme.', 'quant-ph-0701239-2-13-2': 'Assume [MATH]=0.9 and photon-number-resolving detectors with efficiency [MATH].', 'quant-ph-0701239-2-13-3': 'Highly efficient number-resolving detectors are being developed [CITATION].', 'quant-ph-0701239-2-13-4': 'One can show that the optimal nesting level for the repeater protocol for these values is [MATH], corresponding to [MATH] elementary links, which gives [MATH] km.', 'quant-ph-0701239-2-13-5': 'Using Eq. ([REF]) and Ref. [CITATION] one can show that the total time for creating a state of the form Eq. (3) using the scheme of Fig. 1 is [EQUATION] where [MATH] and [MATH], with [MATH] km, and [MATH] m/s in the fiber.', 'quant-ph-0701239-2-13-6': 'One can show by explicit calculation of the errors due to double emissions [CITATION] that the fidelity [MATH] of the final entangled state compared to the ideal maximally entangled state for a repeater with [MATH] levels is approximately [MATH].', 'quant-ph-0701239-2-13-7': 'If one wants, for example, [MATH] one therefore has to choose [MATH], which finally gives [MATH] s.', 'quant-ph-0701239-2-13-8': 'If one can achieve [MATH] one finds [MATH] s. High-efficiency MMMs as discussed above could thus reduce [MATH] for 1000 km to a few seconds (or less with frequency multiplexing).', 'quant-ph-0701239-2-14-0': 'MMMs can also help to significantly alleviate the stability requirements for the repeater protocol.', 'quant-ph-0701239-2-14-1': 'For simplicity, let us just consider an elementary link (between locations [MATH] and [MATH]) from our above example, with [MATH] km.', 'quant-ph-0701239-2-14-2': 'The entanglement in Eq. ([REF]) depends on the phase difference [MATH], which can be rewritten as [MATH].', 'quant-ph-0701239-2-14-3': 'Here [MATH]) is the time when the first (second) entangled state of the type of Eq. ([REF]) is created.', 'quant-ph-0701239-2-14-4': 'The phases thus have to remain very stable on the timescale given by the typical value of [MATH] [CITATION].', 'quant-ph-0701239-2-14-5': 'In the case without MMMs the mean value of [MATH] is [MATH], which is of order 10 s for our above example ([MATH] of order 1 ms and [MATH] of order [MATH]).', 'quant-ph-0701239-2-14-6': 'Over such long timescales, both the phases of the pump lasers and the fiber lengths are expected to fluctuate significantly.', 'quant-ph-0701239-2-14-7': 'Active stabilization would thus definitely be required.', 'quant-ph-0701239-2-14-8': 'With MMMs, for large values of [MATH], [MATH] can be made sufficiently large that it becomes realistic to work only with states Eq. ([REF]) where the initial entanglement between [MATH] and [MATH] and between [MATH] and [MATH] was created in the same interval [MATH] (the probability for such a double success is [MATH]).', 'quant-ph-0701239-2-14-9': 'For our above example, [MATH] can be made of order [MATH] for [MATH] of order [MATH].', 'quant-ph-0701239-2-14-10': 'Working only with entangled states from the same interval increases the time [MATH] by a factor of order [MATH], but it reduces the mean value of [MATH] to of order [MATH], which is of order 20 [MATH]s for our example.', 'quant-ph-0701239-2-14-11': 'For such short time scales, active stabilization of the laser and fiber phases may not be required.', 'quant-ph-0701239-2-15-0': 'In conclusion the combination of photon pair sources and multi-mode memories should allow the realization of a quantum repeater protocol that is much faster and more robust than the protocol of Ref. [CITATION] while retaining its attractive features, in particular the use of linear optical elements and of single photon detections for entanglement generation and swapping.'}

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[]

[['quant-ph-0701239-1-14-8', 'quant-ph-0701239-2-14-6'], ['quant-ph-0701239-1-14-12', 'quant-ph-0701239-2-14-8'], ['quant-ph-0701239-1-14-13', 'quant-ph-0701239-2-14-10'], ['quant-ph-0701239-1-4-1', 'quant-ph-0701239-2-4-1'], ['quant-ph-0701239-1-4-2', 'quant-ph-0701239-2-4-2'], ['quant-ph-0701239-1-4-2', 'quant-ph-0701239-2-4-3'], ['quant-ph-0701239-1-8-5', 'quant-ph-0701239-2-8-4'], ['quant-ph-0701239-1-8-6', 'quant-ph-0701239-2-8-4'], ['quant-ph-0701239-1-8-7', 'quant-ph-0701239-2-8-5'], ['quant-ph-0701239-1-2-3', 'quant-ph-0701239-2-2-3'], ['quant-ph-0701239-1-12-6', 'quant-ph-0701239-2-12-6'], ['quant-ph-0701239-1-12-7', 'quant-ph-0701239-2-12-6'], ['quant-ph-0701239-1-12-8', 'quant-ph-0701239-2-12-8'], ['quant-ph-0701239-1-13-3', 'quant-ph-0701239-2-13-3'], ['quant-ph-0701239-1-13-5', 'quant-ph-0701239-2-13-5'], ['quant-ph-0701239-1-9-3', 'quant-ph-0701239-2-9-3'], ['quant-ph-0701239-1-10-5', 'quant-ph-0701239-2-9-13']]

[]

[]

{'1': 'http://arxiv.org/licenses/assumed-1991-2003/', '2': 'http://arxiv.org/licenses/assumed-1991-2003/'}

https://arxiv.org/abs/quant-ph/0701239

1609.06536

{'1609.06536-1-0-0': "We present a deep learning technique for facial performance capture, i.e., the transfer of video footage into a motion sequence of a 3D mesh representing an actor's face.", '1609.06536-1-0-1': 'Specifically, we build on a conventional capture pipeline based on computer vision and multi-view video, and use its results to train a deep neural network to produce similar output from a monocular video sequence.', '1609.06536-1-0-2': 'Once trained, our network produces high-quality results for unseen inputs with greatly reduced effort compared to the conventional system.', '1609.06536-1-1-0': 'In practice, we have found that approximately 10 minutes worth of high-quality data is sufficient for training a network that can then automatically process as much footage from video to 3D as needed.', '1609.06536-1-1-1': 'This yields major savings in the development of modern narrative-driven video games involving digital doubles of actors and potentially hours of animated dialogue per character.', '1609.06536-1-2-0': '# Introduction', '1609.06536-1-3-0': "Using digital doubles of human actors is a key component in modern video games' strive for realism.", '1609.06536-1-3-1': 'Transferring the essence of a character into digital domain has many challenging technical problems, but the accurate capture of facial movement remains especially tricky.', '1609.06536-1-3-2': "Due to humans' innate sensitivity to the slightest facial cues, it is difficult to surpass the uncanny valley, where an otherwise believable rendering of a character appears lifeless or otherwise unnatural.", '1609.06536-1-4-0': 'Various tools are available for building facial capture pipelines that take video footage into 3D in one form or another, but their accuracy leaves room for improvement.', '1609.06536-1-4-1': 'In practice, high-quality results are generally achievable only with significant amount of manual polishing of output data.', '1609.06536-1-4-2': 'This can be a major cost in a large video game production.', '1609.06536-1-4-3': 'Furthermore, the animators doing the fixing need to be particularly skilled, or otherwise this editing may introduce distracting, unnatural motion.', '1609.06536-1-5-0': 'In this paper, we introduce a neural network based solution to facial performance capture.', '1609.06536-1-5-1': 'Our goal is not to remove the need for manual work entirely, but to dramatically reduce the extent to which it is required.', '1609.06536-1-5-2': 'In our approach, a conventional capture pipeline needs to be applied only to a small subset of input footage, in order to generate enough data for training a neural network.', '1609.06536-1-5-3': 'The bulk of the footage can then be processed using the trained network, skipping the conventional labor-intensive capture pipeline entirely.', '1609.06536-1-5-4': 'Our approach is outlined in Figure [REF].', '1609.06536-1-6-0': '## Problem statement', '1609.06536-1-7-0': "We assume that the input for the capture pipeline is one or more video streams of the actor's head captured under controlled conditions.", '1609.06536-1-7-1': 'The positions of the cameras remain fixed, the lighting and background are standardized, and the actor is to remain at approximately the same position relative to the cameras throughout all shots.', '1609.06536-1-7-2': 'Naturally, some amount of movement needs to be allowed, and we achieve this through input data augmentation in the training phase (Section [REF]).', '1609.06536-1-8-0': 'In our case, the outputs of the capture pipeline are the per-frame positions of the control vertices of a facial mesh, as illustrated in Figure [REF].', '1609.06536-1-8-1': 'There are various other ways to encode the facial expression, including rig parameters or blend shape weights.', '1609.06536-1-8-2': 'In the system where our network was developed in, those kind of encodings are introduced in later stages, mainly for compression and rendering purposes, but the primary capture output consists of the positions of approximately 5000 animated vertices on a fixed-topology facial mesh.', '1609.06536-1-9-0': '## Existing capture pipeline at Remedy', '1609.06536-1-10-0': "Target data necessary for training the neural network was generated using Remedy Entertainment's existing capture pipeline based on the commercial DI4D PRO system [CITATION] that employs nine video cameras.", '1609.06536-1-10-1': 'The benefit of this system is that it captures the nuanced interactions of the skull, muscles, fascia and skin of an actor so as to bypass complex and expensive facial rigging and tissue simulations for digital doubles.', '1609.06536-1-11-0': 'First an unstructured mesh with texture and optical flow data is created from the images for each frame of a facial performance.', '1609.06536-1-11-1': 'A fixed-topology template mesh, created prior to the capture work using a separate photogrammetry pipeline, is then projected on to the unstructured mesh and associated with the optical flow.', '1609.06536-1-11-2': 'The template mesh is tracked across the performance and any issues are fixed semi-automatically in the DI4DTrack software by a tracking artist.', '1609.06536-1-11-3': 'The position and orientation of the head are then stabilized using a few key vertices of the tracking mesh.', '1609.06536-1-11-4': 'Finally a point cache of the facial performance is exported for the fixed-topology template mesh.', '1609.06536-1-11-5': 'The point cache file contains the positions of each vertex in the mesh for each frame of animation in the shot.', '1609.06536-1-12-0': 'Additional automated deformations are later applied to the point cache to fix the remaining issues.', '1609.06536-1-12-1': 'These deformations were not applied to the point caches in the training set.', '1609.06536-1-12-2': 'For example, the eyelids are deformed to meet the eyeball exactly and to slide slightly with motion of the eyes.', '1609.06536-1-12-3': 'Also, opposite vertices of the lips are smoothly brought together to improve lip contacts when needed.', '1609.06536-1-12-4': "After animating the eye directions the results are compressed for runtime use in Remedy's Northlight engine using 416 facial joints.", '1609.06536-1-12-5': 'Pose space deformation is used to augment the facial animation with detailed wrinkle normal map blending.', '1609.06536-1-13-0': '# Previous Work', '1609.06536-1-14-0': '# Network Architecture', '1609.06536-1-15-0': 'Our input footage is divided into a number of shots, with each shot typically consisting of 100-2000 frames at 30 FPS.', '1609.06536-1-15-1': 'Data for each input frame consists of a 1200[MATH]1600 pixel image from each of the nine cameras.', '1609.06536-1-15-2': 'As explained above, the output is the per-frame vertex positions for each of the [MATH]5000 facial mesh vertices, i.e., [MATH]15000 scalars ([MATH]) in total.', '1609.06536-1-16-0': 'As the input for the network, we take the 1200[MATH]1600 video frame from the central camera, crop it with a fixed rectangle so that the face remains in the picture, and scale the remaining portion to 240[MATH]320 resolution.', '1609.06536-1-16-1': 'Furthermore, we convert the image to grayscale, resulting in a total of 76800 scalars to be fed to the network.', '1609.06536-1-16-2': 'The resolution may seem low, but numerous tests confirmed that increasing it did not improve the results.', '1609.06536-1-17-0': 'During the course of the project, we experimented with two neural network architectures.', '1609.06536-1-17-1': 'Our initial efforts concentrated on a fully connected network, but it was soon discovered that a convolutional network architecture was better suited for the task.', '1609.06536-1-17-2': 'For the sake of completeness, we detail the fully connected network in Appendix [REF], where we also briefly characterize its strengths and weaknesses compared to the convolutional network.', '1609.06536-1-18-0': '## Convolutional network', '1609.06536-1-19-0': 'Our convolutional network is based on the all-convolutional architecture [CITATION] extended with two fully connected layers to produce the full-resolution vertex data at output.', '1609.06536-1-19-1': 'The input is a whitened version of the 240[MATH]320 grayscale image.', '1609.06536-1-19-2': 'For whitening, we calculate the mean and variance over all pixels in the training images, and bias and scale the input so that these are normalized to zero and one, respectively.', '1609.06536-1-20-0': 'Note that same whitening coefficients, fixed at training time, are used for all input images during training, validation, and production use.', '1609.06536-1-20-1': 'If the whitening were done on a per-image or per-shot basis, we would lose part of the benefits of the standardized lighting environment.', '1609.06536-1-20-2': "For example, variation in the color of the actor's shirt between shots would end up affecting the brightness of the face.", '1609.06536-1-20-3': 'The layers of the network are listed in the table below.', '1609.06536-1-21-0': 'The output layer is initialized by precomputing a PCA basis for the output meshes based on the target meshes from the training data.', '1609.06536-1-21-1': 'Allowing 160 basis vectors explains approximately 99.9% of the variance seen in the meshes, which was considered to be sufficient.', '1609.06536-1-21-2': 'If we made the weights of the output layer fixed, i.e., made it a non-trainable layer, that would effectively train the remainder of the network to output the 160 PCA coefficients.', '1609.06536-1-21-3': 'However, we found that allowing the last layer to be trainable as well improved the results.', '1609.06536-1-22-0': 'Note that if we merged the last two layers together, we would have a single [MATH] fully connected layer, which would have almost 40 times the number of trainable weights compared to the combination of two layers.', '1609.06536-1-22-1': 'Because 160 PCA basis vectors are sufficient for accurately covering the space of target meshes, these degrees of freedom would be unnecessary and only make the training more difficult and prone to overfitting.', '1609.06536-1-23-0': "The convolutional network, when trained with proper input augmentation (Section [REF]), is not sensitive to the position and orientation of the actor's head in the input images.", '1609.06536-1-23-1': 'Hence no image stabilization is required as a pre-process.', '1609.06536-1-24-0': 'It should be noted that the quality of the results is not overly sensitive to the exact composition of the network.', '1609.06536-1-24-1': 'Changing the geometric progression of the number of feature maps, removing some or all of the [MATH] stride convolution layers, or adding more such layers, did not substantially change the results.', '1609.06536-1-24-2': 'The architecture listed above was found to perform consistently well, and was possible to train in a reasonable time, so it was chosen for use in production.', '1609.06536-1-24-3': 'All results in this paper were computed using this architecture.', '1609.06536-1-25-0': '# Training', '1609.06536-1-26-0': 'For each actor, the training set consists of four parts, totaling approximately 10 minutes of footage.', '1609.06536-1-26-1': 'The composition of the training set is as follows.', '1609.06536-1-27-0': 'Range of motion.', '1609.06536-1-27-1': 'In order to capture the maximal extents of the facial motion, a single range-of-motion shot is taken where the actor goes through a pre-defined set of exteme expressions.', '1609.06536-1-27-2': 'These include, e.g., opening the mouth as wide as possible, moving the jaw sideways and front as far as possible, pursing the lips, opening the eyes wide and forcing them shut, etc.', '1609.06536-1-28-0': 'Expressions.', '1609.06536-1-28-1': 'Unlike the range-of-motion shot that contains exaggerated expressions, this set contains normal expressions such as squinting of the eyes, expression of disgust, etc.', '1609.06536-1-28-2': 'These kind of expressions must be included in the training set, as otherwise the network would not be able to replicate them in production use.', '1609.06536-1-29-0': 'Pangrams.', '1609.06536-1-29-1': 'This set attempts to cover the set of possible facial motions during normal speech for a given target language, in our case English.', '1609.06536-1-29-2': 'The actor speaks one to three pangrams, i.e., sentences that are designed to contain as many different phonemes as possible, in several different emotional tones.', '1609.06536-1-29-3': 'A pangram fitting the emotion would be optimal but in practice this is not always feasible.', '1609.06536-1-30-0': 'In-character material.', '1609.06536-1-30-1': "This set leverages the fact that an actor's performance of a character is often heavily biased in terms of emotional and expressive range for various dramatic and narrative reasons.", '1609.06536-1-30-2': 'This material is composed of the preliminary version of the script, or it may be otherwise prepared for the training.', '1609.06536-1-30-3': "Only the shots that are deemed to support the different aspects of the character are selected so as to ensure that the trained network produces output that stays in character even if the inference isn't perfect, or completely novel or out of character acting is encountered.", '1609.06536-1-31-0': 'The composition of the training is typically roughly as follows: one minute of range-of-motion and expression shots, three minutes of pangrams across emotional states and up to six minutes of in-character performances of varying intensity and scenario.', '1609.06536-1-31-1': 'Natural delivery is preferred so as to ensure that the characteristics of the acting and the shoot as a session are reflected in the training set.', '1609.06536-1-32-0': '## Data augmentation', '1609.06536-1-33-0': 'We perform several transformations to the input images during training in order to make the network resistant to variations in input data.', '1609.06536-1-33-1': 'These transformations are executed on CPU concurrently with network evaluation and training that occurs on the GPU.', '1609.06536-1-33-2': 'Augmentation is not used when evaluating the validation loss or when processing unseen input data in production use.', '1609.06536-1-33-3': 'Examples of augmented input images are shown in Figure [REF].', '1609.06536-1-34-0': "The main transformations are translation, rotation and zoom, which account for the motion of the actor's head during capture.", '1609.06536-1-34-1': 'The magnitudes of these augmentations are set so that they cover at least all of the variation expected in the input data.', '1609.06536-1-35-0': 'In addition to geometric transformations, we vary the brightness and contrast of the input images during training, in order to account for variations in lighting over the capture process.', '1609.06536-1-35-1': 'Our cameras pick a slight periodic flicker from the 50 Hz LED lights in the capture room, and it is possible that some of the bulbs degrade during the capture period that may take place over several days or weeks.', '1609.06536-1-36-0': 'We also tried adding noise to the images, but this was found to be detrimental to learning, as was applying a variable gamma correction factor to approximate varying skin glossiness.', '1609.06536-1-36-1': 'Similarly, small 2D perspective transformations-an attempt to crudely mimic the non-linear effects of rotations of the head-were not found to be beneficial.', '1609.06536-1-37-0': '## Training parameters', '1609.06536-1-38-0': 'We train the network for 200 epochs using the ADAM [CITATION] optimization algorithm with parameters set to values recommended in the paper.', '1609.06536-1-38-1': 'The learning rate is ramped up using a geometric progression during the first training epoch, and then decreased according to [MATH] schedule.', '1609.06536-1-38-2': 'During the last 30 epochs we ramp the learning rate down to zero using a smooth curve, and simultaneously ramp ADAM [MATH] parameter from 0.9 to 0.5.', '1609.06536-1-38-3': 'The ramp-up removes an occasional glitch where the network does not start learning at all, and the ramp-down ensures that the network converges to a local minimum.', '1609.06536-1-38-4': 'Minibatch size is set to 50, and each epoch processes all training frames in randomized order.', '1609.06536-1-38-5': 'Weights are initialized using the initialization scheme of He et al. [CITATION], except for the last output layer which is initialized using a PCA transformation matrix as explained in Section [REF].', '1609.06536-1-39-0': 'The strength of all augmentation transformations is ramped up linearly during the first five training epochs, starting from zero.', '1609.06536-1-39-1': 'This prevented a rare but annoying effect where the network fails to start learning properly, and gets stuck at clearly sub-optimal local minimum.', '1609.06536-1-39-2': 'The augmentation ramp-up process can be seen as a form of curriculum learning [CITATION].', '1609.06536-1-40-0': 'Our loss function is simply the mean square error between the predicted vertex positions produced by the network and the target vertex positions in the training data.', '1609.06536-1-40-1': 'We experimented with a more complicated loss function (Appendix [REF]), but did not find it useful in the end.', '1609.06536-1-41-0': 'Our implementation is written in Python using Theano [CITATION] and Lasagne [CITATION].', '1609.06536-1-41-1': 'On a computer with a modern CPU and a NVIDIA GTX Titan X GPU, the training of one network with a typical training set containing 15000-18000 training frames ([MATH]8-10 minutes at 30Hz) takes approximately 8-10 hours.', '1609.06536-1-42-0': '# Results and Discussion', '1609.06536-1-43-0': 'Given the inevitable variability in the manual work involved in using the conventional capture pipeline, we could not hope that our network would reach a numerically exact match with manually prepared validation data.', '1609.06536-1-43-1': 'The goal of performance capture is to generate believable facial motion, and therefore the perceptual quality of the results-as judged by professional artists-is ultimately what decides whether a capture system is useful or not.', '1609.06536-1-44-0': 'We shall first examine the numerical results, and then turn to perceptual results and production-related aspects that were learned during the project.', '1609.06536-1-45-0': '## Numerical results', '1609.06536-1-46-0': 'Figures [REF] and [REF] illustrate the accuracy of our trained network on a selection of interesting frames in two validation shots that were not used in training.', '1609.06536-1-46-1': 'Per-frame RMSE plot for the same validation shots is shown in Figure [REF].', '1609.06536-1-46-2': 'We found that the neural network was very efficient in producing consistent output even when input data had variability due to inaccuracies of the conventional capture pipeline.', '1609.06536-1-46-3': 'As highlighted in Figure [REF], the data obtained from the conventional capture pipeline sometimes contained systematic errors that the capture artists did not notice and thus correct.', '1609.06536-1-46-4': 'As long as overfitting is avoided, a neural network only learns the consistent features of the input-output mapping, and its output therefore does not fluctuate in the same way.', '1609.06536-1-46-5': 'We believe this explains some of the discrepancies between our output and the validation data.', '1609.06536-1-47-0': 'Figure [REF] demonstrates a problem with our decision to use only one input image as the input to the network: The single frontal image apparently does not contain enough information for the network to detect when the jaw is pushed to the front.', '1609.06536-1-47-1': 'We did not observe this issue until late in the project, and thus these errors need to be manually corrected when using our current system.', '1609.06536-1-47-2': 'We however expect that adding a secondary camera to the side of the actor and using both images as inputs to the network should correct this issue.', '1609.06536-1-48-0': 'Figure [REF] shows the convergence of the network that was used for creating the outputs in Figures [REF], [REF] and [REF].', '1609.06536-1-48-1': 'As previously explained, our loss function is the MSE between network outputs and target positions from the training/validation set.', '1609.06536-1-48-2': 'The vertex coordinates are measured in centimeters in our data, so the final validation loss of 0.0028 corresponds to RMSE of 0.92 millimeters.', '1609.06536-1-48-3': 'With longer training the training loss could be pushed arbitrarily close to zero, but this did not improve the validation loss or the subjective quality of the results.', '1609.06536-1-49-0': 'The most unexpected result was that the network output was temporally perfectly stable, despite the fact that we do not employ recurrent networks or smooth the generated vertex positions temporally in any way.', '1609.06536-1-50-0': '## Perceptual quality and production aspects', '1609.06536-1-51-0': 'The original goal of the project was to cover the bulk of the facial tracking work specifically in facial performance heavy productions involving digital doubles of real actors, by inferring facial motion for non-cinematic parts of the game.', '1609.06536-1-51-1': 'The system is however currently being evaluated at Remedy Entertainment also for high-quality cinematic use, given the encouraging results.', '1609.06536-1-52-0': 'The convolutional neural network was tested in the wild using footage from a later session.', '1609.06536-1-52-1': 'The lighting conditions and facial features exhibited in the training set were carefully preserved.', '1609.06536-1-52-2': 'The inference was evaluated numerically and perceptually in relation to a manually tracked ground truth.', '1609.06536-1-52-3': 'For extreme close-ups manual tracking still results in slightly better quality despite being almost imperceptible at times in a blind experiment.', '1609.06536-1-52-4': 'The results exhibit extremely plausible motion and are temporally stable.', '1609.06536-1-52-5': 'Subtle lip contact and stickiness is an issue but this affects manual tracking as well, hence the aforementioned additional deformations are applied after inference.', '1609.06536-1-52-6': 'The network produces perfectly stabilized vertex positions whereas human operators inevitably vary in their work between sequences.', '1609.06536-1-52-7': 'The results can also be improved by adding data to the training set if deemed necessary.', '1609.06536-1-53-0': 'The 10 minute dataset requirement for high quality output means that the actor needs a big enough role in the game to justify the cost.', '1609.06536-1-53-1': 'Building the dataset for the neural network however enables tracking work to start prior to having a locked screenplay for the project.', '1609.06536-1-53-2': 'This means that late stage pick-up shoots and screenplay changes are much faster to deliver at a high quality.', '1609.06536-1-53-3': 'The dataset shoot allows the actor to familiarize himself with the script and the capture system and to work with the director to discover who the character is instead of practicing expressions just to cover facial muscle activations often with mixed results.', '1609.06536-1-53-4': 'Future work at Remedy will focus on capturing datasets using helmet-mounted cameras for true performance capture of the face and body simultaneously.', '1609.06536-1-54-0': '## Perceptual loss function', '1609.06536-1-55-0': 'In validation shots for which numerical quality results could be computed, the perceptual quality of network output did not always follow the value of the loss function.', '1609.06536-1-55-1': 'For example, a shot with low loss value might exhibit unnatural movement of lips or eyes, whereas a shot with a higher loss value may look more believable.', '1609.06536-1-55-2': 'In the latter case, the higher loss could often be explained by large skin areas being slightly offset tangentially, which is extremely difficult to spot as an artifact by a human observer.', '1609.06536-1-55-3': 'It was not always clear whether these errors were in the validation data or if the network output was amiss.', '1609.06536-1-56-0': 'This suggests that it should be beneficial to design a more perceptually oriented loss function for facial expressions, similar in spirit to how structural similarity metrics have been developed for comparing images [CITATION].', '1609.06536-1-56-1': 'It seems clear that a better loss function would result in a more capable network, as it would learn to focus more on areas that require the highest precision.', '1609.06536-1-56-2': 'In our output, the vertex density is the highest around eyes and lips, which may already partially focus the attention of the network in this way, but nonetheless, a properly motivated perceptual loss function would be an interesting avenue for future work.', '1609.06536-1-57-0': '# Conclusion', '1609.06536-1-58-0': 'We have presented a neural network based facial capture method that has been found accurate enough to be applied in an upcoming game production based on thorough pre-production testing.', '1609.06536-1-59-0': 'Even though the convolutional network may seem like an opaque building block, our approach retains all of the artistic freedom because we output simple 3D point clouds that can be further edited using standard tools, and compressed into standard character rigs.', '1609.06536-1-59-1': 'We feel that many other aspects of the production pipelines of modern games could benefit from similar, selective use of deep learning for bypassing or accelerating manual processing steps that have known but tedious or expensive solutions.', '1609.06536-1-60-0': 'Our initial training data was extracted from an older, massive facial motion database created using the conventional tracking pipeline.', '1609.06536-1-60-1': "Despite the tracking artists' efforts to stabilize the facial mesh as part of the capture process, millimeter-scale inconsistencies remained between shots.", '1609.06536-1-60-2': 'This was problematic for training and validation, because the positional errors produced by the network were of similar magnitude.', '1609.06536-1-60-3': 'As a result, training could not progress beyond the point where the errors present in the training data started dominating the loss function.', '1609.06536-1-60-4': 'Experimenting with different network architectures, etc., was difficult, as most networks could reach this plateau and hence the numerical results always ended up looking similar.', '1609.06536-1-61-0': 'In order to make progress, we devised two techniques for coping with the stabilization errors in the vertex positions in the training data.', '1609.06536-1-61-1': 'The first one was using affine transform invariant loss function for training and validation, and the second was performing additional per-shot mesh stabilization as a pre-process.', '1609.06536-1-62-0': 'As the project progressed to the point of evaluating the feasibility of producing a single, well-curated training set for each actor, it became possible to improve the quality of input by having the tracking artists pay especially close attention to the problematic areas, including head stabilization.', '1609.06536-1-63-0': '## Affine-invariant loss function', '1609.06536-1-64-0': 'A natural choice for loss function is the mean square distance between the predicted vertex positions and the target positions in the training data.', '1609.06536-1-64-1': 'However, as explained above, this leads to problems because the target positions do not necessarily represent a ground truth result due to stabilization errors.', '1609.06536-1-65-0': 'As an attempt to remove the effect of stabilization errors on the loss, we experimented with a loss function that is invariant to affine transformations between the predicted positions and the target positions.', '1609.06536-1-65-1': 'To achieve this, the loss function first finds the [MATH]-optimal affine transformation between the predicted positions and the target positions, and then takes the mean square error between the predicted positions taken through this transform and the target positions.', '1609.06536-1-66-0': 'The predicted positions, transformed with the [MATH]-optimal affine transform [CITATION], can be calculated as:', '1609.06536-1-67-0': '[EQUATION] where [MATH] is a [MATH] matrix whose rows contain the coordinates of each predicted position padded with a [MATH], i.e., [MATH], [MATH] is a similar matrix containing the target positions, and [MATH] receives the transformed predicted positions.', '1609.06536-1-67-1': 'Note that [MATH] is always a [MATH] matrix, so the required matrix inversion is easy to compute analytically.', '1609.06536-1-67-2': 'The final loss is then [MATH], where the squaring is done per-element, and the sum is taken over all elements.', '1609.06536-1-67-3': 'As the rightmost column of [MATH] is always ones (up to arithmetic precision), it does not contribute to the loss, and calculating it can be skipped.', '1609.06536-1-68-0': 'Note that it is crucial that we determine the affine transformation from predicted positions to target positions, instead of vice versa.', '1609.06536-1-68-1': 'If we allowed an affine transformation to be applied to the target positions before evaluating the MSE, the network would recognize that simply outputting a smaller mesh produces smaller post-transformation error.', '1609.06536-1-68-2': 'In the limit, placing all output positions in the same position would result in zero loss.', '1609.06536-1-69-0': 'Problems.', '1609.06536-1-69-1': 'The affine-invariant loss function removes the effects of stabilization errors on the loss, but it was found to have severe negative consequences, as it removes any encouragement for the network to produce the output in a fixed position, size, or orientation.', '1609.06536-1-69-2': 'If employed in production, this would necessitate post-process stabilization of the inferred results, which is undesirable.', '1609.06536-1-70-0': 'If we make the output layer weights fixed-corresponding to the network outputting PCA coefficients, as explained in Section [REF]-the affine-invariant loss function works fine, as the fixed PCA basis at the output prevents spurious movement of the output mesh.', '1609.06536-1-70-1': 'This, however, prevents training the output layer weights, which is not optimal.', '1609.06536-1-70-2': 'Moreover, the output PCA acts as such a strong prior for the network that the results are practically identical to using the standard loss function.', '1609.06536-1-71-0': '## Pre-process mesh stabilization', '1609.06536-1-72-0': 'Our second attempt to deal with stabilization errors in the target positions was to run an additional stabilization pass on the training data.', '1609.06536-1-72-1': 'Our ad-hoc shot stabilization method works as follows.', '1609.06536-1-72-2': 'First we compute, for each frame of the shot, the mean squared distance between the target point set and the rest pose, and sort the frames according to this metric.', '1609.06536-1-72-3': 'Then we select 50% of the frames that had the smallest difference to the rest pose, with the goal of avoiding frames where the face differs significantly from the rest pose.', '1609.06536-1-72-4': 'After this, we find a single affine transformation that minimizes the mean Euclidean distance between the target point sets in the chosen frames and the rest pose.', '1609.06536-1-72-5': 'This [MATH]-optimal transformation is computed using iterative gradient descent.', '1609.06536-1-73-0': 'This approach allows keeping the output layer weights trainable, but it cannot account for drift or non-affine shape errors in the shot.', '1609.06536-1-73-1': 'However, it allowed us to progress forward with the development of the network architectures and training methods, and was consistently used until higher-quality training datasets became available.', '1609.06536-1-74-0': '## Discussion', '1609.06536-1-75-0': 'We currently employ neither of these techniques during training, validation, or inference, because the training data made for production use is sufficiently well stabilized.', '1609.06536-1-75-1': 'As mentioned above, affine-invariant loss function forbids training of the output layer, and performing additional stabilization to already stable vertex positions can only make the situation worse.', '1609.06536-1-75-2': 'Hence not touching the data at all is the best we can do.', '1609.06536-1-75-3': 'However, when experimenting with less than perfectly stabilized vertex data, we found the techniques described above useful, and therefore chose to include them in this appendix for the benefit of those in a similar situation.', '1609.06536-1-76-0': 'When experimenting with fully connected networks, we achieved the best results by transforming the input images into 3000 PCA coefficients.', '1609.06536-1-76-1': 'The PCA basis is pre-calculated based on the input frames from the training set, and the chosen number of basis images captures approximately 99.9% of the variance in the data.', '1609.06536-1-76-2': 'The layers of the network are listed in the table below.', '1609.06536-1-77-0': 'The position and the orientation of the head in the input images varies, which in practice necessitates stabilizing the input images prior to taking the PCA transformation.', '1609.06536-1-77-1': 'For this we used the facial landmark detector of Kazemi and Sullivan [CITATION], an implementation of which can be found in the dlib machine learning package.', '1609.06536-1-77-2': 'Rotation angle and median line of the face were estimated from the landmark points surrounding the eyes.', '1609.06536-1-77-3': 'Because these were found to shift vertically during blinking of the eyes, the vertical position of the face was determined from the landmark points on the nose.', '1609.06536-1-77-4': 'The image was then rotated to a fixed orientation, and translated so that the point midway between the eyes remained at a fixed position.', '1609.06536-1-78-0': 'Even though the network may seem overly simplistic, we did not find a way to improve the results by adding more layers or changing the widths of the existing layers.', '1609.06536-1-78-1': 'We experimented with different regularization schemes, but simply adding two dropout layers was found to yield the best results.', '1609.06536-1-78-2': 'The output layer is initialized using a PCA basis for the output meshes computed as in the convolutional network (Section [REF]).', '1609.06536-1-79-0': '## Comparison', '1609.06536-1-80-0': 'Ultimately, the only aspect in which the fully connected network remained superior to the convolutional network was training time.', '1609.06536-1-80-1': 'Whereas the convolutional network takes 8-10 hours to train in a typical case, the fully conneted network would converge in as little as one hour.', '1609.06536-1-80-2': 'Even though it was initially speculated that fast training times could be beneficial in production use, it ended up not mattering much as long as training could be completed overnight.', '1609.06536-1-81-0': 'Stabilizing the input images for the fully connected network turned out to be problematic because of residual motion that remained due to inaccuracies in the facial landmark detection.', '1609.06536-1-81-1': 'This residual jitter of input images sometimes caused spurious and highly distracting motion of output vertices.', '1609.06536-1-81-2': 'We tried hardening the fully connected network to this effect by applying a similar jitter to inputs during training, i.e., presenting the same stabilized input image to the network in slightly different positions, but this did not help.', '1609.06536-1-82-0': 'We suspect that it may be too difficult for the fully connected network to understand that slightly offset images should produce the same result, perhaps partially due to the input PCA transform.', '1609.06536-1-82-1': 'Nonetheless, the results with input PCA transform were better than using the raw image as the input.', '1609.06536-1-82-2': 'The convolutional network does not suffer from these problems, as input image stabilization is not required.', '1609.06536-1-83-0': 'The fully connected network often produced numerically better results than the convolutional network (Figure [REF]), but perceptually the results were significantly worse, as the fully connected network appeared to generally attenuate the facial motion.', '1609.06536-1-83-1': 'Even in individual shots where the fully connected network produced a numerically clearly superior result, the facial motion was judged to lack expressivity and was not as temporally stable compared to the results produced by the convolutional network.'}

{'1609.06536-2-0-0': "We present a real-time deep learning framework for video-based facial performance capture-the dense 3D tracking of an actor's face given a monocular video.", '1609.06536-2-0-1': 'Our pipeline begins with accurately capturing a subject using a high-end production facial capture pipeline based on multi-view stereo tracking and artist-enhanced animations.', '1609.06536-2-0-2': 'With 5-10 minutes of captured footage, we train a convolutional neural network to produce high-quality output, including self-occluded regions, from a monocular video sequence of that subject.', '1609.06536-2-0-3': 'Since this 3D facial performance capture is fully automated, our system can drastically reduce the amount of labor involved in the development of modern narrative-driven video games or films involving realistic digital doubles of actors and potentially hours of animated dialogue per character.', '1609.06536-2-0-4': 'We compare our results with several state-of-the-art monocular real-time facial capture techniques and demonstrate compelling animation inference in challenging areas such as eyes and lips.', '1609.06536-2-1-0': '<ccs2012> <concept> <concept_id>10010147.10010371.10010352</concept_id> <concept_desc>Computing methodologies Animation</concept_desc> <concept_significance>500</concept_significance> </concept> <concept> <concept_id>10010147.10010257.10010293.10010294</concept_id> <concept_desc>Computing methodologies Neural networks</concept_desc> <concept_significance>500</concept_significance> </concept> <concept> <concept_id>10010147.10010257.10010258.10010259.10010264</concept_id> <concept_desc>Computing methodologies Supervised learning by regression</concept_desc> <concept_significance>300</concept_significance> </concept> </ccs2012>', '1609.06536-2-2-0': '[500]Computing methodologies Animation [500]Computing methodologies Neural networks [300]Computing methodologies Supervised learning by regression', '1609.06536-2-3-0': '# Introduction', '1609.06536-2-4-0': 'The use of visually compelling digital doubles of human actors is a key component for increasing realism in any modern narrative-driven video game.', '1609.06536-2-4-1': "Facial performance capture poses many challenges in computer animation and due to a human's innate sensitivity to the slightest facial cues, it is difficult to surpass the uncanny valley, where otherwise believable renderings of a character appear lifeless or unnatural.", '1609.06536-2-5-0': 'Despite dramatic advancements in automated facial performance capture systems and their wide deployment for scalable production, it is still not possible to obtain a perfect tracking for highly complex expressions, especially in challenging but critical areas such as lips and eye regions.', '1609.06536-2-5-1': 'In most cases, manual clean-up and corrections by skilled artists are necessary to ensure high-quality output that is free from artifacts and noise.', '1609.06536-2-5-2': 'Conventional facial animation pipelines can easily result in drastic costs, especially in settings such as video game production where hours of footage may need to be processed.', '1609.06536-2-6-0': 'In this paper, we introduce a deep learning framework for real-time and production-quality facial performance capture.', '1609.06536-2-6-1': 'Our goal is not to fully eliminate the need for manual work, but to significantly reduce the extent to which it is required.', '1609.06536-2-6-2': 'We apply an offline, multi-view stereo capture pipeline with manual clean-up to a small subset of the input video footage, and use it to generate enough data to train a deep neural network.', '1609.06536-2-6-3': 'The trained network can then be used to automatically process the remaining video footage at rates as fast as 870 fps, skipping the conventional labor-intensive capture pipeline entirely.', '1609.06536-2-7-0': 'Furthermore, we only require a single view as input during runtime which makes our solution attractive for head cam-based facial capture.', '1609.06536-2-7-1': 'Our approach is real-time and does not even need sequential processing, so every frame can be processed independently.', '1609.06536-2-7-2': 'Furthermore, we demonstrate qualitatively superior results compared to state-of-the-art monocular real-time facial capture solutions.', '1609.06536-2-7-3': 'Our pipeline is outlined in Figure [REF].', '1609.06536-2-8-0': '## Problem Statement', '1609.06536-2-9-0': "We assume that the input for the capture pipeline is multiple-view videos of the actor's head captured under controlled conditions to generate training data for the neural network.", '1609.06536-2-9-1': 'The input to the neural network at runtime is video from a single view.', '1609.06536-2-9-2': 'The positions of the cameras remain fixed, the lighting and background are standardized, and the actor is to remain at approximately the same position relative to the cameras throughout the recording.', '1609.06536-2-9-3': 'Naturally, some amount of movement needs to be allowed, and we achieve this through data augmentation during training (Section [REF]).', '1609.06536-2-10-0': 'The output of the capture pipeline is the set of per-frame positions of facial mesh vertices, as illustrated in Figure [REF].', '1609.06536-2-10-1': 'Other face encodings such as blendshape weights or joint positions are introduced in later stages of our pipeline, mainly for compression and rendering purposes, but the primary capture output consists of the positions of approximately 5000 animated vertices on a fixed-topology facial mesh.', '1609.06536-2-11-0': '## Offline Capture Pipeline', '1609.06536-2-12-0': "The training data used for the deep neural network was generated using Remedy Entertainment's in-house capture pipeline based on a cutting edge commercial DI4D PRO system [CITATION] that employs nine video cameras.", '1609.06536-2-13-0': 'First, an unstructured mesh with texture and optical flow data is created from the images for each frame of an input video.', '1609.06536-2-13-1': 'A fixed-topology template mesh is created prior to the capture work by applying Agisoft [CITATION], a standard multi-view stereo reconstruction software, on data from 26 DSLR cameras and two cross polarized flashes.', '1609.06536-2-13-2': 'The mesh is then warped onto the unstructured scan of the first frame.', '1609.06536-2-13-3': 'The template mesh is tracked using optical flow through the entire sequence.', '1609.06536-2-13-4': 'Possible artifacts are manually fixed using the DI4DTrack software by a clean-up artist.', '1609.06536-2-13-5': 'The position and orientation of the head are then stabilized using a few key vertices of the tracking mesh.', '1609.06536-2-13-6': 'The system then outputs the positions of each of the vertices on the fixed-topology template mesh.', '1609.06536-2-14-0': 'Additional automated deformations are later applied to the vertices to fix remaining issues.', '1609.06536-2-14-1': 'For instance, the eyelids are deformed to meet the eyeballs exactly and to slide slightly with motion of the eyes.', '1609.06536-2-14-2': 'Also, opposite vertices of the lips are smoothly brought together to improve lip contacts when needed.', '1609.06536-2-14-3': "After animating the eye directions the results are compressed for runtime use in Remedy's Northlight engine using 416 facial joints.", '1609.06536-2-14-4': 'Pose space deformation is used to augment the facial animation with detailed wrinkle normal map blending.', '1609.06536-2-14-5': 'These ad-hoc deformations were not applied in the training set.', '1609.06536-2-15-0': '# Related Work', '1609.06536-2-16-0': '# Network Architecture', '1609.06536-2-17-0': 'Our input footage is divided into a number of shots, with each shot typically consisting of 100-2000 frames at 30 FPS.', '1609.06536-2-17-1': 'Data for each input frame consists of a 1200[MATH]1600 pixel image from each of the nine cameras.', '1609.06536-2-17-2': 'As explained above, the output is the per-frame vertex position for each of the [MATH]5000 facial mesh vertices.', '1609.06536-2-18-0': 'As input for the network, we take the 1200[MATH]1600 video frame from the central camera, crop it with a fixed rectangle so that the face remains in the picture, and scale the remaining portion to 240[MATH]320 resolution.', '1609.06536-2-18-1': 'Furthermore, we convert the image to grayscale, resulting in a total of 76800 scalars to be fed to the network.', '1609.06536-2-18-2': 'The resolution may seem low, but numerous tests confirmed that increasing it did not improve the results.', '1609.06536-2-19-0': '## Convolutional Network', '1609.06536-2-20-0': 'Our convolutional network is based on the all-convolutional architecture [CITATION] extended with two fully connected layers to produce the full-resolution vertex data at output.', '1609.06536-2-20-1': 'The input is a whitened version of the 240[MATH]320 grayscale image.', '1609.06536-2-20-2': 'For whitening, we calculate the mean and variance over all pixels in the training images, and bias and scale the input so that these are normalized to zero and one, respectively.', '1609.06536-2-21-0': 'Note that the same whitening coefficients, fixed at training time, are used for all input images during training, validation, and production use.', '1609.06536-2-21-1': 'If the whitening were done on a per-image or per-shot basis, we would lose part of the benefits of the standardized lighting environment.', '1609.06536-2-21-2': "For example, variation in the color of the actor's shirt between shots would end up affecting the brightness of the face.", '1609.06536-2-21-3': 'The layers of the network are listed in the table below.', '1609.06536-2-22-0': 'The output layer is initialized by precomputing a PCA basis for the output meshes based on the target meshes from the training data.', '1609.06536-2-22-1': 'Allowing 160 basis vectors explains approximately 99.9% of the variance seen in the meshes, which was considered to be sufficient.', '1609.06536-2-22-2': 'If we fixed the weights of the output layer and did not train them, that would effectively train the remainder of the network to output the 160 PCA coefficients.', '1609.06536-2-22-3': 'However, we found that allowing the last layer to be trainable as well improved the results.', '1609.06536-2-22-4': 'This would seem to suggest that the optimization is able to find a slightly better intermediate basis than the initial PCA basis.', '1609.06536-2-23-0': '# Training', '1609.06536-2-24-0': 'For each actor, the training set consists of four parts, totaling approximately 5-10 minutes of footage.', '1609.06536-2-24-1': 'The composition of the training set is as follows.', '1609.06536-2-25-0': 'Extreme Expressions.', '1609.06536-2-25-1': 'In order to capture the maximal extents of the facial motion, a single range-of-motion shot is taken where the actor goes through a pre-defined set of extreme expressions.', '1609.06536-2-25-2': 'These include but are not limited to opening the mouth as wide as possible, moving the jaw sideways and front as far as possible, pursing the lips, and opening the eyes wide and forcing them shut.', '1609.06536-2-26-0': 'FACS-Like Expressions.', '1609.06536-2-26-1': 'Unlike the range-of-motion shot that contains exaggerated expressions, this set contains regular FACS-like expressions such as squinting of the eyes or an expression of disgust.', '1609.06536-2-26-2': 'These kind of expressions must be included in the training set as otherwise the network would not be able to replicate them in production use.', '1609.06536-2-27-0': 'Pangrams.', '1609.06536-2-27-1': 'This set attempts to cover the set of possible facial motions during normal speech for a given target language, in our case English.', '1609.06536-2-27-2': 'The actor speaks one to three pangrams, which are sentences that are designed to contain as many different phonemes as possible, in several different emotional tones.', '1609.06536-2-27-3': 'A pangram fitting the emotion would be optimal but in practice this is not always feasible.', '1609.06536-2-28-0': 'In-Character Material.', '1609.06536-2-28-1': "This set leverages the fact that an actor's performance of a character is often heavily biased in terms of emotional and expressive range for various dramatic and narrative reasons.", '1609.06536-2-28-2': 'This material is composed of the preliminary version of the script, or it may be otherwise prepared for the training.', '1609.06536-2-28-3': "Only the shots that are deemed to support the different aspects of the character are selected so as to ensure that the trained network produces output that stays in character even if the inference isn't perfect or if completely novel or out of character acting is encountered.", '1609.06536-2-29-0': 'The training set is typically comprised of roughly 10% of range-of-motion and expression shots, 30% of pangrams across emotional states, and 60% of in-character performances of varying intensity and scenario.', '1609.06536-2-30-0': '## Data Augmentation', '1609.06536-2-31-0': 'We perform several transformations to the input images during training in order to make the network resistant to variations in input data.', '1609.06536-2-31-1': 'These transformations are executed on CPU concurrently with network evaluation and training that occurs on the GPU.', '1609.06536-2-31-2': 'Augmentation is not used when evaluating the validation loss or when processing unseen input data in production use.', '1609.06536-2-31-3': 'Examples of augmented input images are shown in Figure [REF].', '1609.06536-2-32-0': "The main transformations are translation, rotation and zoom, which account for the motion of the actor's head during capture.", '1609.06536-2-32-1': 'The magnitudes of these augmentations are set so that they cover at least all of the variation expected in the input data.', '1609.06536-2-32-2': 'This kind of image-based augmentation does not cover large-scale changes in head pose, and thus our method does not tolerate that unless such effects are present in the training data.', '1609.06536-2-33-0': 'In addition to geometric transformations, we vary the brightness and contrast of the input images during training, in order to account for variations in lighting over the capture process.', '1609.06536-2-33-1': 'Our cameras pick a slight periodic flicker from the 50 Hz LED lights in the capture room, and it is possible that some of the bulbs degrade during the capture period that may take place over several days or weeks.', '1609.06536-2-34-0': '## Training Parameters', '1609.06536-2-35-0': 'We train the network for 200 epochs using the Adam [CITATION] optimization algorithm with parameters set to values recommended in the paper.', '1609.06536-2-35-1': 'The learning rate is ramped up using a geometric progression during the first training epoch, and then decreased according to [MATH] schedule.', '1609.06536-2-35-2': 'During the last 30 epochs we ramp the learning rate down to zero using a smooth curve, and simultaneously ramp Adam [MATH] parameter from 0.9 to 0.5.', '1609.06536-2-35-3': 'The ramp-up removes an occasional glitch where the network does not start learning at all, and the ramp-down ensures that the network converges to a local minimum.', '1609.06536-2-35-4': 'Minibatch size is set to 50, and each epoch processes all training frames in randomized order.', '1609.06536-2-35-5': 'Weights are initialized using the initialization scheme of He et al. [CITATION], except for the last output layer which is initialized using a PCA transformation matrix as explained in Section [REF].', '1609.06536-2-36-0': 'The strength of all augmentation transformations is ramped up linearly during the first five training epochs, starting from zero.', '1609.06536-2-36-1': 'This prevented a rare but annoying effect where the network fails to start learning properly, and gets stuck at clearly sub-optimal local minimum.', '1609.06536-2-36-2': 'The augmentation ramp-up process can be seen as a form of curriculum learning [CITATION].', '1609.06536-2-37-0': 'Our loss function is simply the mean square error between the predicted vertex positions produced by the network and the target vertex positions in the training data.', '1609.06536-2-38-0': 'Our implementation is written in Python using Theano [CITATION] and Lasagne [CITATION].', '1609.06536-2-38-1': 'On a computer with a modern CPU and a NVIDIA Titan X GPU, the training of one network with a typical training set containing 10000-18000 training frames ([MATH]5-10 minutes at 30Hz) takes approximately 5-10 hours.', '1609.06536-2-39-0': '# Results', '1609.06536-2-40-0': 'We tested the trained network using footage from a later session.', '1609.06536-2-40-1': 'The lighting conditions and facial features exhibited in the training set were carefully preserved.', '1609.06536-2-40-2': 'The inference was evaluated numerically and perceptually in relation to a manually tracked ground truth.', '1609.06536-2-41-0': 'We will first evaluate our choices in the design and training of our neural network, followed by examination of the numerical results.', '1609.06536-2-41-1': 'We then turn to visual comparisons with recent monocular real-time facial performance capture methods.', '1609.06536-2-41-2': 'Finally, we explore the limitations of our pipeline.', '1609.06536-2-42-0': 'The quality of the results can be best assessed from the accompanying video.', '1609.06536-2-42-1': 'In the video an interesting observation is that our results are not only accurate but also perfectly stable temporally despite the fact that we do not employ recurrent networks or smooth the generated vertex positions temporally in any way.', '1609.06536-2-42-2': 'It is very difficult for human operators to achieve similar temporal stability as they inevitably vary in their work between sequences.', '1609.06536-2-43-0': '## Network Architecture Evaluation', '1609.06536-2-44-0': 'All results in this paper were computed using the architecture described in Section [REF].', '1609.06536-2-44-1': 'It should be noted that the quality of the results is not overly sensitive to the exact composition of the network.', '1609.06536-2-44-2': 'Changing the dimensions of the convolutional layers or removing or adding the [MATH] stride convolution layers only changed performance by a slight margin.', '1609.06536-2-44-3': 'The architecture described in Section [REF] was found to perform slightly better compared to other all-convolutional architectures that could be trained in a reasonable amount of time, so it was chosen for use in production.', '1609.06536-2-45-0': 'In addition to using an all-convolutional neural network, we also experimented with fully connected networks.', '1609.06536-2-45-1': 'When experimenting with fully connected networks, we achieved the best results by transforming the input images into 3000 PCA coefficients.', '1609.06536-2-45-2': 'The PCA basis is pre-calculated based on the input frames from the training set, and the chosen number of basis images captures approximately 99.9% of the variance in the data.', '1609.06536-2-45-3': 'The layers of the network are listed in the table below.', '1609.06536-2-46-0': 'The position and the orientation of the head in the input images varies, which in practice necessitates stabilizing the input images prior to taking the PCA transformation.', '1609.06536-2-46-1': 'For this we used the facial landmark detector of Kazemi and Sullivan [CITATION].', '1609.06536-2-46-2': 'Rotation angle and median line of the face were estimated from the landmark points surrounding the eyes.', '1609.06536-2-46-3': 'Because these were found to shift vertically during blinking of the eyes, the vertical position of the face was determined from the landmark points on the nose.', '1609.06536-2-46-4': 'The image was then rotated to a fixed orientation, and translated so that the point midway between the eyes remained at a fixed position.', '1609.06536-2-47-0': 'Even though the network may seem overly simplistic, similarly to the all-convolutional architecture, we did not find a way to improve the results by adding more layers or changing the widths of the existing layers.', '1609.06536-2-47-1': 'We experimented with different regularization schemes, but simply adding two dropout layers was found to yield the best results.', '1609.06536-2-47-2': 'The output layer is initialized using a PCA basis for the output meshes computed as in the convolutional network.', '1609.06536-2-48-0': 'Ultimately, the only aspect in which the fully connected network remained superior to the convolutional network was training time.', '1609.06536-2-48-1': 'Whereas the convolutional network takes 8-10 hours to train in a typical case, the fully connected network would converge in as little as one hour.', '1609.06536-2-48-2': 'Even though it was initially speculated that fast training times could be beneficial in production use, it ended up not mattering much as long as training could be completed overnight.', '1609.06536-2-49-0': 'One disadvantage of using a fully connected network is that stabilizing the input images for the fully connected network turned out to be problematic because of residual motion that remained due to inaccuracies in the facial landmark detection.', '1609.06536-2-49-1': 'This residual jitter of input images sometimes caused spurious and highly distracting motion of output vertices.', '1609.06536-2-49-2': 'We tried hardening the fully connected network to this effect by applying a similar jitter to inputs during training in order to present the same stabilized input image to the network in slightly different positions, but this did not help.', '1609.06536-2-50-0': 'We suspect that it may be too difficult for the fully connected network to understand that slightly offset images should produce the same result, perhaps partially due to the input PCA transform.', '1609.06536-2-50-1': 'Nonetheless, the results with input PCA transform were better than using the raw image as the input.', '1609.06536-2-51-0': "On the other hand, the convolutional network, when trained with proper input augmentation (Section [REF]), is not sensitive to the position and orientation of the actor's head in the input images.", '1609.06536-2-51-1': 'Hence the convolutional network carries an advantage in that no image stabilization is required as a pre-process.', '1609.06536-2-52-0': 'We see in Figures [REF] and [REF] that the fully connected network often produced numerically better results than the convolutional network, but visually the results were significantly worse as the fully connected network appeared to generally attenuate the facial motion.', '1609.06536-2-52-1': 'Even in individual shots where the fully connected network produced a numerically clearly superior result, the facial motion was judged to lack expressiveness and was not as temporally stable compared to the results produced by the convolutional network.', '1609.06536-2-52-2': 'We further discuss this general discrepancy between numerical and visual quality below.', '1609.06536-2-53-0': '## Training Process Evaluation', '1609.06536-2-54-0': 'In Section [REF] we described several data augmentations we performed that made the network more resistant to variations in input data and eliminated the need for stabilization as a pre-process for our all-convolutional network.', '1609.06536-2-54-1': 'Additionally, we also tried augmenting the data by adding noise to the images and applying a variable gamma correction factor to approximate varying skin glossiness.', '1609.06536-2-54-2': 'However, both of these augmentations were found to be detrimental to learning.', '1609.06536-2-54-3': 'Similarly, small 2D perspective transformations-an attempt to crudely mimic the non-linear effects of head rotations-were not found to be beneficial.', '1609.06536-2-55-0': '## Numerical Results', '1609.06536-2-56-0': 'Figure [REF] shows the convergence of the network for Character 1, trained using 15173 input frames.', '1609.06536-2-56-1': 'The training set for Character 2 contained 10078 frames.', '1609.06536-2-56-2': 'As previously explained, our loss function is the MSE between the network output and target positions from the training/validation set.', '1609.06536-2-56-3': 'The vertex coordinates are measured in centimeters in our data, so the final validation loss of 0.0028 corresponds to RMSE of 0.92 millimeters.', '1609.06536-2-56-4': 'With longer training the training loss could be pushed arbitrarily close to zero, but this did not improve the validation loss or the subjective quality of the results.', '1609.06536-2-57-0': 'Figure [REF] illustrates the numerical accuracy of our trained network on a selection of interesting frames in validation shots that were not used in training.', '1609.06536-2-57-1': 'Note that the RMSE of the frames shown in the figure are higher than average since the validation data mostly consist of more neutral material than the frames shown in the figure.', '1609.06536-2-57-2': 'Per-frame RMSE plot for the validation shots for Character 1 is shown in Figure [REF].', '1609.06536-2-58-0': 'We found that the neural network was very efficient in producing consistent output even when there were variations in the input data because of inaccuracies in the conventional capture pipeline.', '1609.06536-2-58-1': 'In the first four rows of Figure [REF], we can see that, especially in the regions around the hairline and above the eyebrows, the target data obtained from the conventional capture pipeline sometimes contained systematic errors that the capture artists did not notice and thus did not correct.', '1609.06536-2-58-2': 'Because a neural network only learns the consistent features of the input-output mapping as long as overfitting is avoided, our network output does not fluctuate in the same way as the manual target positions do.', '1609.06536-2-58-3': 'In fact, visually it is often not clear whether the manually tracked target positions or the inferred positions are closer to the ground truth.', '1609.06536-2-58-4': 'We believe this explains some of the numerical discrepancies between our output and the validation data.', '1609.06536-2-59-0': 'Given the inevitable variability in the manual work involved in using the conventional capture pipeline, we could not hope that our network would reach a numerically exact match with manually prepared validation data.', '1609.06536-2-59-1': 'The goal of performance capture is to generate believable facial motion, and therefore the perceptual quality of the results-as judged by professional artists-is ultimately what determines whether a capture system is useful or not in our production environment.', '1609.06536-2-60-0': '## Comparison', '1609.06536-2-61-0': 'We visually compare our method in Figure [REF] to Thies et al. Thies2016 and Cao et al. Cao2014, two state-of-the-art monocular real-time facial performance capture methods that do not require a rig.', '1609.06536-2-61-1': 'Since the comparison methods generalize to any identity and our method assumes the identity of one user, in order to make the comparison more fair, we use the method of Li et al. li2010example to fix the identity mesh for the comparison methods and only track the expressions.', '1609.06536-2-61-2': 'Visually our method appears to be more accurate than [CITATION] and [CITATION], but we note that they bear significant advantages in that they do not require per-user calibration and allow for less constrained head movements.', '1609.06536-2-61-3': 'We also note that if we allowed the comparison methods to retrain for new identities and restricted head movement in all their inputs, their accuracy could be improved to more closely match our levels.', '1609.06536-2-61-4': 'An advantage our method poses over the comparison methods is that it is capable of inferring plausible animations for self-occluded or difficult to track regions such as details surrounding the mouth and eyes.', '1609.06536-2-61-5': 'In a production setting where we have resources to constrain the head movement and perform per-user training and would like to capture the user as accurately and plausibly as possible across all regions of the head, our method is advantageous over other existing methods.', '1609.06536-2-62-0': '## Performance', '1609.06536-2-63-0': 'Our system runs comfortably in real-time.', '1609.06536-2-63-1': 'As seen in Table [REF], we achieve 287 frames per second when used online and up to 870 frames per second if batch processing is used offline with a batch size of 200 frames.', '1609.06536-2-63-2': 'Meanwhile, other real-time methods are only able to achieve 28 frames per second.', '1609.06536-2-63-3': 'This allows our system to process a large amount of footage in a short amount of time.', '1609.06536-2-64-0': '## Limitations', '1609.06536-2-65-0': 'We have proposed a system that can achieve high levels of accuracy for facial performance capture while drastically reducing the amount of manual work involved in a production setting.', '1609.06536-2-65-1': 'However, we have observed several limitations in our system and suggest future work we can explore.', '1609.06536-2-66-0': 'Non-Optimal Loss Function.', '1609.06536-2-66-1': 'In validation shots for which numerical quality results could be computed, the visual quality of the network output did not always follow the value of the loss function.', '1609.06536-2-66-2': 'For example, a shot with a low loss value might exhibit unnatural movement of lips or eyes, whereas a shot with a higher loss value may look more believable.', '1609.06536-2-66-3': 'This suggests that our current loss function does not get the optimal results possible from a deep neural network, and it should be beneficial to design a more perceptually oriented loss function for facial expressions similar in spirit of how structural similarity metrics have been developed for comparing images [CITATION].', '1609.06536-2-66-4': 'It seems clear that a better loss function would result in a more capable network, as it would learn to focus more on areas that require the highest precision.', '1609.06536-2-67-0': 'Per-User Calibration.', '1609.06536-2-67-1': 'Despite our ability to capture details more accurately than other methods, one strong limitation of our method compared to other state-of-the-art monocular facial performance capture methods is that we require performing a per-user calibration of retraining the network for each new identity.', '1609.06536-2-67-2': 'In the future, we would like to further reduce the amount of manual labor involved in our pipeline and create a system that can achieve the same level of accuracy while also generalizing to all users.', '1609.06536-2-68-0': '# Conclusion', '1609.06536-2-69-0': 'We have presented a neural network based facial capture method that has proven accurate enough to be applied in an upcoming game production based on thorough pre-production testing while also requiring much less labor than other current facial performance capture pipelines used in game production.', '1609.06536-2-69-1': 'Another advantage our method holds in the production setting is that building the dataset for the network enables tracking work to start any time, so pick-up shoots and screenplay changes are much faster to deliver with high quality.', '1609.06536-2-70-0': 'We have evaluated our network architecture and training pipeline against other network and pipeline variations, and we determined the proposed architecture and augmentation methods to yield a very good balance between optimal visual results and reasonable training time for production purposes.', '1609.06536-2-71-0': 'We have also shown our method to surpass other state-of-the-art monocular real-time facial performance capture methods in our ability to infer a plausible mesh around regions that are invisible or difficult to track such as the area surrounding the eye and mouth.', '1609.06536-2-71-1': 'However, our system has a significant drawback as we require per-user calibration.', '1609.06536-2-71-2': 'The 5-10 minute dataset required for each new identity for high-quality output typically means that the actor needs an important enough role in the game to justify the cost.', '1609.06536-2-72-0': 'Even though the convolutional network may seem like an opaque building block, our approach retains all of the artistic freedom because we output simple 3D point clouds that can be further edited using standard tools, and compressed into standard character rigs.', '1609.06536-2-72-1': 'We feel that many other aspects of the production pipelines of modern games could benefit from similar, selective use of deep learning for bypassing or accelerating manual processing steps that have known but tedious or expensive solutions.', '1609.06536-2-73-0': 'Future Work.', '1609.06536-2-73-1': 'Future work may include addressing the limitations of our system mentioned earlier and developing a more accurate pipeline that does not require per-user calibration.', '1609.06536-2-73-2': 'Additional work will also focus on capturing datasets using helmet-mounted cameras for true performance capture of the face and body simultaneously.', '1609.06536-2-73-3': 'Nevertheless, we have presented a system that has drastically reduced the amount of manual work in high quality facial performance capture, and our system represents an important step in the direction of fully automated, high quality facial and body capture.'}

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[]

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{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1609.06536

1401.3206

{'1401.3206-1-0-0': 'We discuss on some families of skew product maps on a square.', '1401.3206-1-0-1': 'For a kind of skew product maps with coupled-expanding property, we estimate Hausdorff dimension of its attractor.', '1401.3206-1-0-2': 'And we prove that there exists an ergodic measure with full Hausdorff dimension for this kind of skew product maps.', '1401.3206-1-1-0': 'Keywords skew product map, Hausdorff dimension, coupled-expanding property, chaotic dynamical system', '1401.3206-1-2-0': '# Introduction', '1401.3206-1-3-0': 'In this paper we study dimensional theoretical properties of a kind of skew product maps with coupled-expanding property.', '1401.3206-1-3-1': 'By dimensional theoretical properties we mean Hausdorff dimension of invariant sets of dynamical systems and ergodic measures on these sets.', '1401.3206-1-4-0': 'In the modern theory of dynamical systems geometrical invariants like Hausdorff dimension of invariant sets and measures seem to have their place beside classical invariants like entropy and Lyapnov exponents [CITATION].', '1401.3206-1-4-1': 'In the dimension theoretical study on dynamical systems, following two problems are regarded as important.', '1401.3206-1-4-2': 'The first one is to estimate Hausdorff dimension of invariant sets.', '1401.3206-1-4-3': 'The second one is the problem about existence of ergodic measure with full Hausdorff dimension in these invariant sets.', '1401.3206-1-4-4': 'If such an ergodic measure does not exist, it is problem whether at least the variational principle for Hausdorff dimension holds or not.', '1401.3206-1-4-5': 'The variational principle means that there exists a sequence of ergodic measures such that the Hausdorff dimension of the invariant set can be approximated by their Hausdorff dimensions.', '1401.3206-1-4-6': 'It seems to be well accepted by experts that these questions are of great important in developing the dimension theory of dynamical systems [CITATION].', '1401.3206-1-5-0': 'The problem of estimating the Hausdorff dimension of invariant subsets has been studied in several papers for different classes of dynamical system (a good overview can be found in [CITATION]).', '1401.3206-1-5-1': 'In [CITATION], for some noninjective skew product maps defined on a square without finite Markov partition, the author estimated Hausdorff dimension of attractor under some conditions.', '1401.3206-1-5-2': 'In generally the skew product map [MATH] with [MATH] is defined by [MATH], where [MATH] is piecewise monotonic and the map [MATH] is a contraction on [MATH] for all [MATH].', '1401.3206-1-5-3': 'In [CITATION], it was proved that for the skew product map [MATH], Hausdorff dimension of the attractor [MATH] of [MATH] is [MATH] under some conditions, where [MATH] and [MATH] is the topological entropy of [MATH].', '1401.3206-1-6-0': 'On the other hand Baker\'s map which can be considered as a special form of the skew product map is well known as a simple example of a "chaotic" dynamical system.', '1401.3206-1-6-1': "When it comes to dimensional theoretical study for this map, in [CITATION] it was considered that, for parameters in some domain, Baker's transformation has ergodic measures with full Hausdorff dimension in its invariant sets but, for parameters in some other domain, variational principle for Hausdorff dimension doesn't hold, that is, there is not a sequence of ergodic measures of which Hausdorff dimension approximates the Hausdorff dimension of the invariant set.", '1401.3206-1-6-2': "In [CITATION] more improved results than one in [CITATION] on the dimensional theoretical properties of the generalized Baker's map were obtained.", '1401.3206-1-7-0': "According to our search on prior works, it seems that there is no result considered for the existence problem of ergodic measure with full Hausdorff dimension or for the problem on variational principle for Hausdorff dimension for the skew product map yet, whereas such work has done for the Baker's map.", '1401.3206-1-7-1': 'And in [CITATION] rather strong condition including continuously differentiability for the map [MATH] and [MATH] was supposed.', '1401.3206-1-7-2': 'When it comes to coupled-expanding property, strictly coupled-expanding map implies "chaos" in several senses (see [CITATION]).', '1401.3206-1-8-0': 'From the above mentioned results, in this paper we study on dimensional theoretical properties for a kind of skew product maps [MATH] where [MATH] is a coupled-expanding map.', '1401.3206-1-8-1': 'We estimate Hausdorff dimension of the attractor of the skew product map [MATH] (in section 3) and prove that there exists ergodic measure with full Hausdorff dimension in the attractor for the map [MATH] (in section 4).', '1401.3206-1-9-0': '# Basic notation', '1401.3206-1-10-0': 'Let [MATH] be a metric space.', '1401.3206-1-10-1': 'Recall that Hausdorff dimension of [MATH] is given by [EQUATION] where [MATH] denotes the [MATH]-dimensional Hausdorff measure, i.e. [EQUATION] and [MATH] denotes the diameter of a covering element [MATH] (see [CITATION]).', '1401.3206-1-11-0': 'Let [MATH] be a positive integer and [MATH] be a contraction similitude with contraction ratio [MATH], i.e. [MATH] for any [MATH].', '1401.3206-1-11-1': 'A set [MATH] is said to be invariant set with respect to [MATH] if [MATH].', '1401.3206-1-11-2': 'If there exists a finite set [MATH] of contraction similitudes with contraction ratio [MATH] such that [MATH] is a invariant set with respect to [MATH], and if [MATH] for [MATH] such that [MATH], then [MATH] is said to be self-similar set, where [MATH] is [MATH]-dimensional Hausdorff measure and [MATH] (see [CITATION]).', '1401.3206-1-11-3': 'Let [MATH] and [MATH].', '1401.3206-1-11-4': 'If there exists [MATH] subsets [MATH] of [MATH] with [MATH] and [MATH] for all [MATH], such that [EQUATION] then [MATH] is said to be coupled-expanding map in [MATH] (see [CITATION]).', '1401.3206-1-12-0': '# Hausdorff dimension of attractor of a skew product map with coupled expanding property', '1401.3206-1-13-0': 'Let [MATH] be an interval.', '1401.3206-1-13-1': 'Suppose that [MATH] is monotonic on [MATH] and satisfies [MATH] and [MATH].', '1401.3206-1-13-2': 'Assume that [MATH] and that [MATH] is strictly monotonic and satisfies [MATH].', '1401.3206-1-13-3': 'Then the attractor [MATH] of the map [MATH] defined by [MATH], has Hausdorff dimension [EQUATION]', '1401.3206-1-14-0': '(Remark 1) From the conditions of Theorem 3.1, the map [MATH] is strictly expanding map (see [CITATION]) and therefore the entropy of [MATH] is [MATH] ([CITATION]).', '1401.3206-1-14-1': 'So we can see that above expression (1) is analogue to the one of [CITATION].', '1401.3206-1-15-0': 'Proof of Theorem 3.1.', '1401.3206-1-15-1': 'For convenience of expressing, define follows; [EQUATION]', '1401.3206-1-15-2': 'And set [EQUATION]', '1401.3206-1-15-3': 'Then, for any [MATH], it follows that [MATH] since [EQUATION]', '1401.3206-1-15-4': 'On the other hand, we have [EQUATION] from [EQUATION]', '1401.3206-1-15-5': 'From the definition of the map [MATH], it follows that [EQUATION]', '1401.3206-1-15-6': 'Since [MATH], the set [MATH] forms a band with bandwidth [MATH] across the lines [MATH] and [MATH].', '1401.3206-1-15-7': 'Therefore [MATH] is a piece of the curve across the lines [MATH] and [MATH].', '1401.3206-1-15-8': 'Then the set [MATH] consists of uncountable pieces of curve, that is, [EQUATION]', '1401.3206-1-15-9': 'Next, let’s consider the skew product map [MATH] defined by [EQUATION] where [MATH] is defined by [MATH] and [MATH].', '1401.3206-1-16-0': 'With the same way as we have considered for [MATH] above, we can consider the construction of the attractor of [MATH] by [MATH].', '1401.3206-1-16-1': 'For [MATH] let [MATH] be the set for [MATH] corresponding to [MATH].', '1401.3206-1-16-2': 'Then we can see that the set [MATH] is a segment with length [MATH].', '1401.3206-1-17-0': "Now let's consider the map [MATH] defined by [MATH].", '1401.3206-1-17-1': 'The map [MATH] maps the point at which [MATH] intersects the line [MATH], to the point at which [MATH] intersects the line [MATH], where [MATH].', '1401.3206-1-17-2': 'Then [MATH] is Lipschitz towards [MATH]-ax direction, that is, for any fixing [MATH] we have [EQUATION]', '1401.3206-1-17-3': 'Therefore [EQUATION] by using the result of [CITATION] (see Fig.1).', '1401.3206-1-18-0': 'Fig. 1.', '1401.3206-1-18-1': 'Imagine of the point [MATH] by [MATH]', '1401.3206-1-19-0': 'Let [MATH] be the intersection of [MATH] and line [MATH].', '1401.3206-1-19-1': 'Then [MATH] is self-similar set with respect to the contraction similitudes with contraction ratio [MATH] and we have [MATH] (see [CITATION]).', '1401.3206-1-19-2': 'Since [MATH], we have [EQUATION]', '1401.3206-1-19-3': 'This implies that [EQUATION]', '1401.3206-1-19-4': "To prove the opposite inequality, let's consider the map defined by [EQUATION] where [MATH] and [MATH].", '1401.3206-1-20-0': 'By the same procedure as we just have done for [MATH] and [MATH], we can consider the construction of the attractor [MATH] for [MATH].', '1401.3206-1-20-1': 'Let [MATH] be the set corresponding to [MATH], then [MATH] is also a segment with length [MATH].', '1401.3206-1-20-2': 'Let [MATH] be a map which maps intersecting point of the line [MATH] and [MATH] to intersecting point of the line [MATH] and [MATH].', '1401.3206-1-20-3': 'Then [MATH] is also Lipshitz towards [MATH]-ax direction like as [MATH] (see Fig.2).', '1401.3206-1-21-0': 'Fig. 2.', '1401.3206-1-21-1': 'Imagine of the point [MATH] by [MATH]', '1401.3206-1-22-0': 'Therefore [EQUATION]', '1401.3206-1-22-1': 'Let [MATH] be the intersection of the line [MATH] and [MATH].', '1401.3206-1-22-2': 'Then [MATH] is also self-similar set with respect to the contraction similitudes with contraction ratio [MATH].', '1401.3206-1-22-3': 'This means that [MATH].', '1401.3206-1-22-4': 'Since [MATH], we have [EQUATION]', '1401.3206-1-22-5': 'From (1) and (2) it follows that [EQUATION]', '1401.3206-1-22-6': 'The proof is thus complete.', '1401.3206-1-23-0': '# The existence of ergodic measure with full Hausdorff dimension on attractor', '1401.3206-1-24-0': 'Let [MATH] be the space of all [MATH]-invariant and ergodic measures on [MATH].', '1401.3206-1-25-0': 'Proof.', '1401.3206-1-25-1': 'Assume that [MATH].', '1401.3206-1-25-2': 'Then [MATH] and [MATH] because [MATH].', '1401.3206-1-25-3': 'Since [MATH] is [MATH]-ergodic, [MATH] or [MATH].', '1401.3206-1-25-4': 'Therefore [MATH] is [MATH]-ergodic measure on [MATH].', '1401.3206-1-25-5': 'And let [MATH] be a measurable set.', '1401.3206-1-25-6': 'Since [MATH] is [MATH]-invariant, we have [EQUATION]', '1401.3206-1-25-7': 'Therefore the proof is complete.', '1401.3206-1-26-0': 'Suppose that the map [EQUATION] satisfies the assumptions of Theorem 3.1 and that [MATH] is linear on intervals [MATH] and [MATH].', '1401.3206-1-26-1': 'Then there exists a [MATH]-invariant ergodic measure with full dimension on the attractor [MATH].', '1401.3206-1-27-0': 'Proof.', '1401.3206-1-27-1': 'Let [MATH] be a normalized Lebesgue measure on interval [MATH].', '1401.3206-1-27-2': 'Since [MATH] is self similar set with respect to two self similar contractions with contraction ratio [MATH], it is homeomorphism to [MATH].', '1401.3206-1-28-0': 'Let [MATH] be a Bernoulli measure on [MATH] and [MATH] be a measure corresponding to [MATH] on [MATH].', '1401.3206-1-28-1': 'Define [MATH] as follows; [EQUATION]', '1401.3206-1-28-2': 'Then [MATH] is [MATH] ergodic measure defined on [MATH] and it is obvious that [EQUATION] by composing of [MATH].', '1401.3206-1-28-3': 'By using proposition 4.1, it follows that [MATH] is [MATH]-ergodic measure.', '1401.3206-1-28-4': 'Let [MATH].', '1401.3206-1-28-5': 'From Theorem 3.1 it follows that [EQUATION]', '1401.3206-1-28-6': 'Since [MATH], from the definition of Hausdorff dimension we have [EQUATION]', '1401.3206-1-28-7': 'Remember [MATH] is Lipshitz towards [MATH]-ax direction, then we have [EQUATION]', '1401.3206-1-28-8': 'Thus it follows that [EQUATION]', '1401.3206-1-28-9': 'Considering [EQUATION] we have [EQUATION]', '1401.3206-1-28-10': 'The proof is thus complete.', '1401.3206-1-29-0': '# Conclusion and further study', '1401.3206-1-30-0': 'In this paper, for the skew product maps with coupled-expanding property, we estimated Hausdorff dimension of its attractor and proved that there exists an ergodic measure with full Hausdorff dimension for the attractor.', '1401.3206-1-30-1': 'In further study we are going to be weaker the limit for the skew product map [MATH] which is actually assumed to avoid overlapping of its image here.'}

{'1401.3206-2-0-0': 'We discuss on some families of skew product maps on a square.', '1401.3206-2-0-1': 'For a kind of skew product maps with coupled-expanding property, we estimate Hausdorff dimension of its attractor.', '1401.3206-2-0-2': 'And we prove that there exists an ergodic measure with full Hausdorff dimension for this kind of skew product maps.', '1401.3206-2-1-0': 'Keywords skew product map, Hausdorff dimension, coupled-expanding property, chaotic dynamical system', '1401.3206-2-2-0': '# Introduction', '1401.3206-2-3-0': 'In this paper we study dimensional theoretical properties of a kind of skew product maps with coupled-expanding property.', '1401.3206-2-3-1': 'By dimensional theoretical properties we mean Hausdorff dimension of invariant sets of dynamical systems and ergodic measures on these sets.', '1401.3206-2-4-0': 'In the modern theory of dynamical systems geometrical invariants like Hausdorff dimension of invariant sets and measures seem to have their place beside classical invariants like entropy and Lyapnov exponents [CITATION].', '1401.3206-2-4-1': 'In the dimension theoretical study on dynamical systems, following two problems are regarded as important.', '1401.3206-2-4-2': 'The first one is to estimate Hausdorff dimension of invariant sets.', '1401.3206-2-4-3': 'The second one is the problem about existence of ergodic measure with full Hausdorff dimension in these invariant sets.', '1401.3206-2-4-4': 'If such an ergodic measure does not exist, it is problem whether at least the variational principle for Hausdorff dimension holds or not.', '1401.3206-2-4-5': 'The variational principle means that there exists a sequence of ergodic measures such that the Hausdorff dimension of the invariant set can be approximated by their Hausdorff dimensions.', '1401.3206-2-4-6': 'It seems to be well accepted by experts that these questions are of great important in developing the dimension theory of dynamical systems [CITATION].', '1401.3206-2-5-0': 'The problem of estimating the Hausdorff dimension of invariant subsets has been studied in several papers for different classes of dynamical system (a good overview can be found in [CITATION]).', '1401.3206-2-5-1': 'In [CITATION], for some noninjective skew product maps defined on a square without finite Markov partition, the author estimated Hausdorff dimension of attractor under some conditions.', '1401.3206-2-5-2': 'In generally the skew product map [MATH] with [MATH] is defined by [MATH], where [MATH] is piecewise monotonic and the map [MATH] is a contraction on [MATH] for all [MATH].', '1401.3206-2-5-3': 'In [CITATION], it was proved that for the skew product map [MATH], Hausdorff dimension of the attractor [MATH] of [MATH] is [MATH] under some conditions, where [MATH] and [MATH] is the topological entropy of [MATH].', '1401.3206-2-6-0': 'On the other hand Baker\'s map which can be considered as a special form of the skew product map is well known as a simple example of a "chaotic" dynamical system.', '1401.3206-2-6-1': "When it comes to dimensional theoretical study for this map, in [CITATION] it was considered that, for parameters in some domain, Baker's transformation has ergodic measures with full Hausdorff dimension in its invariant sets but, for parameters in some other domain, variational principle for Hausdorff dimension doesn't hold, that is, there is not a sequence of ergodic measures of which Hausdorff dimension approximates the Hausdorff dimension of the invariant set.", '1401.3206-2-6-2': "In [CITATION] more improved results than one in [CITATION] on the dimensional theoretical properties of the generalized Baker's map were obtained.", '1401.3206-2-7-0': "According to our search on prior works, it seems that there is no result considered for the existence problem of ergodic measure with full Hausdorff dimension or for the problem on variational principle for Hausdorff dimension for the skew product map yet, whereas such work has done for the Baker's map.", '1401.3206-2-7-1': 'And in [CITATION] rather strong condition including continuously differentiability for the map [MATH] and [MATH] was supposed.', '1401.3206-2-7-2': 'When it comes to coupled-expanding property, strictly coupled-expanding map implies "chaos" in several senses (see [CITATION]).', '1401.3206-2-8-0': 'From the above mentioned results, in this paper we study on dimensional theoretical properties for a kind of skew product maps [MATH] where [MATH] is a coupled-expanding map.', '1401.3206-2-8-1': 'We estimate Hausdorff dimension of the attractor of the skew product map [MATH] (in section 3) and prove that there exists ergodic measure with full Hausdorff dimension in the attractor for the map [MATH] (in section 4).', '1401.3206-2-9-0': '# Basic notation', '1401.3206-2-10-0': 'Let [MATH] be a metric space.', '1401.3206-2-10-1': 'Recall that Hausdorff dimension of [MATH] is given by [EQUATION] where [MATH] denotes the [MATH]-dimensional Hausdorff measure, i.e. [EQUATION] and [MATH] denotes the diameter of a covering element [MATH] (see [CITATION]).', '1401.3206-2-11-0': 'Let [MATH] be a positive integer and [MATH] be a contraction similitude with contraction ratio [MATH], i.e. [MATH] for any [MATH].', '1401.3206-2-11-1': 'A set [MATH] is said to be invariant set with respect to [MATH] if [MATH].', '1401.3206-2-11-2': 'If there exists a finite set [MATH] of contraction similitudes with contraction ratio [MATH] such that [MATH] is a invariant set with respect to [MATH], and if [MATH] for [MATH] such that [MATH], then [MATH] is said to be self-similar set, where [MATH] is [MATH]-dimensional Hausdorff measure and [MATH] (see [CITATION]).', '1401.3206-2-11-3': 'Let [MATH] and [MATH].', '1401.3206-2-11-4': 'If there exists [MATH] subsets [MATH] of [MATH] with [MATH] and [MATH] for all [MATH], such that [EQUATION] then [MATH] is said to be coupled-expanding map in [MATH] (see [CITATION]).', '1401.3206-2-12-0': '# Hausdorff dimension of attractor of a skew product map with coupled expanding property', '1401.3206-2-13-0': 'Let [MATH] be an interval.', '1401.3206-2-13-1': 'Suppose that [MATH] is monotonic on [MATH] and satisfies [MATH] and [MATH].', '1401.3206-2-13-2': 'Assume that [MATH] and that [MATH] is strictly monotonic and satisfies [MATH].', '1401.3206-2-13-3': 'Then the attractor [MATH] of the map [MATH] defined by [MATH], has Hausdorff dimension [EQUATION]', '1401.3206-2-14-0': '(Remark 1) From the conditions of Theorem 3.1, the map [MATH] is strictly expanding map (see [CITATION]) and therefore the entropy of [MATH] is [MATH] ([CITATION]).', '1401.3206-2-14-1': 'So we can see that above expression (1) is analogue to the one of [CITATION].', '1401.3206-2-15-0': 'Proof of Theorem 3.1.', '1401.3206-2-15-1': 'For convenience of expressing, define follows; [EQUATION]', '1401.3206-2-15-2': 'And set [EQUATION]', '1401.3206-2-15-3': 'Then, for any [MATH], it follows that [MATH] since [EQUATION]', '1401.3206-2-15-4': 'On the other hand, we have [EQUATION] from [EQUATION]', '1401.3206-2-15-5': 'From the definition of the map [MATH], it follows that [EQUATION]', '1401.3206-2-15-6': 'Since [MATH], the set [MATH] forms a band with bandwidth [MATH] across the lines [MATH] and [MATH].', '1401.3206-2-15-7': 'Therefore [MATH] is a piece of the curve across the lines [MATH] and [MATH].', '1401.3206-2-15-8': 'Then the set [MATH] consists of uncountable pieces of curve, that is, [EQUATION]', '1401.3206-2-15-9': 'Next, let’s consider the skew product map [MATH] defined by [EQUATION] where [MATH] is defined by [MATH] and [MATH].', '1401.3206-2-16-0': 'With the same way as we have considered for [MATH] above, we can consider the construction of the attractor of [MATH] by [MATH].', '1401.3206-2-16-1': 'For [MATH] let [MATH] be the set for [MATH] corresponding to [MATH].', '1401.3206-2-16-2': 'Then we can see that the set [MATH] is a segment with length [MATH].', '1401.3206-2-17-0': "Now let's consider the map [MATH] defined by [MATH].", '1401.3206-2-17-1': 'The map [MATH] maps the point at which [MATH] intersects the line [MATH], to the point at which [MATH] intersects the line [MATH], where [MATH].', '1401.3206-2-17-2': 'Then [MATH] is Lipschitz towards [MATH]-ax direction, that is, for any fixing [MATH] we have [EQUATION]', '1401.3206-2-17-3': 'Therefore [EQUATION] by using the result of [CITATION] (see Fig.1).', '1401.3206-2-18-0': 'Fig. 1.', '1401.3206-2-18-1': 'Imagine of the point [MATH] by [MATH]', '1401.3206-2-19-0': 'Let [MATH] be the intersection of [MATH] and line [MATH].', '1401.3206-2-19-1': 'Then [MATH] is self-similar set with respect to the contraction similitudes with contraction ratio [MATH] and we have [MATH] (see [CITATION]).', '1401.3206-2-19-2': 'Since [MATH], we have [EQUATION]', '1401.3206-2-19-3': 'This implies that [EQUATION]', '1401.3206-2-19-4': "To prove the opposite inequality, let's consider the map defined by [EQUATION] where [MATH] and [MATH].", '1401.3206-2-20-0': 'By the same procedure as we just have done for [MATH] and [MATH], we can consider the construction of the attractor [MATH] for [MATH].', '1401.3206-2-20-1': 'Let [MATH] be the set corresponding to [MATH], then [MATH] is also a segment with length [MATH].', '1401.3206-2-20-2': 'Let [MATH] be a map which maps intersecting point of the line [MATH] and [MATH] to intersecting point of the line [MATH] and [MATH].', '1401.3206-2-20-3': 'Then [MATH] is also Lipshitz towards [MATH]-ax direction like as [MATH] (see Fig.2).', '1401.3206-2-21-0': 'Fig. 2.', '1401.3206-2-21-1': 'Imagine of the point [MATH] by [MATH]', '1401.3206-2-22-0': 'Therefore [EQUATION]', '1401.3206-2-22-1': 'Let [MATH] be the intersection of the line [MATH] and [MATH].', '1401.3206-2-22-2': 'Then [MATH] is also self-similar set with respect to the contraction similitudes with contraction ratio [MATH].', '1401.3206-2-22-3': 'This means that [MATH].', '1401.3206-2-22-4': 'Since [MATH], we have [EQUATION]', '1401.3206-2-22-5': 'From (1) and (2) it follows that [EQUATION]', '1401.3206-2-22-6': 'The proof is thus complete.', '1401.3206-2-23-0': '# The existence of ergodic measure with full Hausdorff dimension on attractor', '1401.3206-2-24-0': 'Let [MATH] be the space of all [MATH]-invariant and ergodic measures on [MATH].', '1401.3206-2-25-0': 'Proof.', '1401.3206-2-25-1': 'Assume that [MATH].', '1401.3206-2-25-2': 'Then [MATH] and [MATH] because [MATH].', '1401.3206-2-25-3': 'Since [MATH] is [MATH]-ergodic, [MATH] or [MATH].', '1401.3206-2-25-4': 'Therefore [MATH] is [MATH]-ergodic measure on [MATH].', '1401.3206-2-25-5': 'And let [MATH] be a measurable set.', '1401.3206-2-25-6': 'Since [MATH] is [MATH]-invariant, we have [EQUATION]', '1401.3206-2-25-7': 'Therefore the proof is complete.', '1401.3206-2-26-0': 'Suppose that the map [EQUATION] satisfies the assumptions of Theorem 3.1 and that [MATH] is linear on intervals [MATH] and [MATH].', '1401.3206-2-26-1': 'Then there exists a [MATH]-invariant ergodic measure with full dimension on the attractor [MATH].', '1401.3206-2-27-0': 'Proof.', '1401.3206-2-27-1': 'Let [MATH] be a normalized Lebesgue measure on interval [MATH].', '1401.3206-2-27-2': 'Since [MATH] is self similar set with respect to two self similar contractions with contraction ratio [MATH], it is homeomorphism to [MATH].', '1401.3206-2-28-0': 'Let [MATH] be a Bernoulli measure on [MATH] and [MATH] be a measure corresponding to [MATH] on [MATH].', '1401.3206-2-28-1': 'Define [MATH] as follows; [EQUATION]', '1401.3206-2-28-2': 'Then [MATH] is [MATH] ergodic measure defined on [MATH] and it is obvious that [EQUATION] by composing of [MATH].', '1401.3206-2-28-3': 'By using proposition 4.1, it follows that [MATH] is [MATH]-ergodic measure.', '1401.3206-2-28-4': 'Let [MATH].', '1401.3206-2-28-5': 'From Theorem 3.1 it follows that [EQUATION]', '1401.3206-2-28-6': 'Since [MATH], from the definition of Hausdorff dimension we have [EQUATION]', '1401.3206-2-28-7': 'Remember [MATH] is Lipshitz towards [MATH]-ax direction, then we have [EQUATION]', '1401.3206-2-28-8': 'Thus it follows that [EQUATION]', '1401.3206-2-28-9': 'Considering [EQUATION] we have [EQUATION]', '1401.3206-2-28-10': 'The proof is thus complete.', '1401.3206-2-29-0': '# Conclusion and further study', '1401.3206-2-30-0': 'In this paper, for the skew product maps with coupled-expanding property, we estimated Hausdorff dimension of its attractor and proved that there exists an ergodic measure with full Hausdorff dimension for the attractor.', '1401.3206-2-30-1': 'In further study we are going to weaken the limit for the skew product map [MATH] which is actually assumed to avoid overlapping of its image here.'}

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[['1401.3206-1-30-1', '1401.3206-2-30-1']]

[]

[]

[]

['1401.3206-1-11-3', '1401.3206-1-15-0', '1401.3206-1-15-2', '1401.3206-1-18-0', '1401.3206-1-18-1', '1401.3206-1-21-0', '1401.3206-1-21-1', '1401.3206-1-22-0', '1401.3206-1-25-0', '1401.3206-1-25-1', '1401.3206-1-25-2', '1401.3206-1-25-3', '1401.3206-1-25-4', '1401.3206-1-25-5', '1401.3206-1-25-6', '1401.3206-1-25-7', '1401.3206-1-27-0', '1401.3206-1-28-4', '1401.3206-2-11-3', '1401.3206-2-15-0', '1401.3206-2-15-2', '1401.3206-2-18-0', '1401.3206-2-18-1', '1401.3206-2-21-0', '1401.3206-2-21-1', '1401.3206-2-22-0', '1401.3206-2-25-0', '1401.3206-2-25-1', '1401.3206-2-25-2', '1401.3206-2-25-3', '1401.3206-2-25-4', '1401.3206-2-25-5', '1401.3206-2-25-6', '1401.3206-2-25-7', '1401.3206-2-27-0', '1401.3206-2-28-4']

{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1401.3206

1403.7333

{'1403.7333-1-0-0': 'A framework for macroscopic correlations where the notion of temporal ordering is not considered as fundamental is constructed by taking the macroscopic (classical) limit of a recent framework for quantum correlations by Oreshkov, Costa, and Brukner [Nat.', '1403.7333-1-0-1': 'Commun.', '1403.7333-1-0-2': '3, 1092 (2012)].', '1403.7333-1-0-3': 'Because the same holds for two parties, it was conjectured that also in the multi-party setting, a predefined causal order emerges in that limit.', '1403.7333-1-0-4': 'We show that counter to this belief, the framework for macroscopic correlations does allow for deterministic signaling between three or more parties incompatible with any predefined causal order.', '1403.7333-1-1-0': '# Introduction', '1403.7333-1-2-0': 'Non-local correlations, as predicted by quantum theory and observed in nature [CITATION], refute local realism [CITATION] if we assume the common world view that spatial and temporal separation are fundamental, and that spatially separated settings can be chosen independently.', '1403.7333-1-2-1': 'With this refutation of local realism we lose some explanatory power of quantum theory beyond its mathematical formalism.', '1403.7333-1-2-2': 'Already Einstein, Podolsky and Rosen asked that a theory should give a local description of physical reality and concluded that quantum theory, under this condition, is incomplete [CITATION].', '1403.7333-1-2-3': 'A promising idea in finding an explanatory description of the predicted correlations is to consider time and space not as fundamental, but rather as emerging from fundamental principles.', '1403.7333-1-2-4': 'A step into this direction was initiated by Hardy in his program of merging the theory of general relativity with quantum theory [CITATION], by proposing to extend quantum theory to dynamical causal structures.', '1403.7333-1-2-5': 'Oreshkov, Costa, and Brukner developed a framework for quantum correlations [CITATION] based on this idea, where they drop the assumption of a global background time, while keeping the assumptions that locally, nature is described by quantum theory, and no logical paradoxes arise.', '1403.7333-1-2-6': "Interestingly, in a scenario with two parties, if we consider the macroscopic limit of the quantum systems, i.e., enforce both parties' physics to be described by classical probability theory instead of quantum theory, a predefined causal order emerges, suggesting that also for any number of parties a fixed temporal ordering might result in the macroscopic limit.", '1403.7333-1-3-0': 'This work studies the framework for correlations with no causal order in the macroscopic limit where locally, nature is described by classical probability theory.', '1403.7333-1-3-1': 'We show that for three parties or more, deterministic classical correlations incompatible with any predefined causal order arise, and thus, a predefined causal order does not emerge in that limit.', '1403.7333-1-4-0': 'For that purpose we continue by defining causality, followed by a description of a causal game, with an upper bound on the game winning probability in a world with predefined causal order.', '1403.7333-1-4-1': 'Thereafter we develop the macroscopic framework for correlations with no causal order, and show how it can be used to perfectly win the causal game for any number [MATH] of parties, where [MATH] is larger than two.', '1403.7333-1-5-0': '# Causal order', '1403.7333-1-6-0': 'When observing a physical process we can ultimately speak only of the measurements we chose to make, and of their outcomes.', '1403.7333-1-6-1': 'Therefore, physical systems can be thought of as black boxes that we probe.', '1403.7333-1-6-2': 'Taking this perspective, we describe all physical quantities, in the following called quantities, as functions of variables.', '1403.7333-1-6-3': 'Each party [MATH] is described by a set of variables [MATH] that are chosen freely by the party (we will not give a definition of free choice, rather we define each variable as being intrinsically free) and by a set of quantities [MATH] the party can access.', '1403.7333-1-6-4': 'Quantities are therein deterministic functions of variables.', '1403.7333-1-6-5': 'Probabilistic quantities are derandomized deterministic functions, where the random essence is simply another variable.', '1403.7333-1-6-6': 'We say a quantity [MATH] as a function of a variable [MATH], causally depends on [MATH], and is in the causal future of [MATH], denoted by [MATH] or [MATH].', '1403.7333-1-6-7': 'Conversely, the variable [MATH] is in the causal past of [MATH].', '1403.7333-1-6-8': 'The negation of these relations are denoted by [MATH] and [MATH].', '1403.7333-1-6-9': 'This definition does not induce a causal order between quantities, between variables, nor between any quantity and the variables it does not depend on.', '1403.7333-1-7-0': 'Let us introduce a second party [MATH] described by the set of variables [MATH], with access to the quantities [MATH].', '1403.7333-1-7-1': 'Now, quantities could depend on variables from both parties.', '1403.7333-1-7-2': 'If, for instance, party [MATH] has a quantity that depends on a variable of [MATH], then we say [MATH] can signal to [MATH] (see Figure [REF]).', '1403.7333-1-7-3': 'In the remaining part of this work we will assume unidirectional signaling, e.g., if [MATH] can signal to [MATH], then [MATH] cannot signal to [MATH].', '1403.7333-1-7-4': 'This allows us to causally order parties.', '1403.7333-1-7-5': 'If at least one quantity of [MATH] depends on a variable of [MATH], but no quantity of [MATH] depends on any variable of [MATH] (which is the condition for unidirectional signaling), then [MATH] is in the causal past of [MATH].', '1403.7333-1-7-6': 'Formally, if [MATH], such that [MATH], and [MATH], [MATH], then [MATH].', '1403.7333-1-8-0': 'Consider a two-party scenario with parties [MATH], each having a respective single variable [MATH] and a respective single quantity [MATH].', '1403.7333-1-8-1': 'We call a theory causal realistic if and only if all achievable probability distributions [MATH], which we call processes, result from a convex mixture of possible causal orders, i.e., [EQUATION] where [MATH] is the event [MATH] and [MATH] is a shared quantity depending on a random variable neither chosen by [MATH] nor by [MATH].', '1403.7333-1-8-2': 'The realism attribute of a causal realistic theory is that the causal order is predefined.', '1403.7333-1-8-3': 'For more than two parties, the definition of causal realism becomes more subtle: Suppose we have three parties [MATH], [MATH], and [MATH], and [MATH] is in the causal past of [MATH] and [MATH].', '1403.7333-1-8-4': 'Then [MATH] can specify the causal order between [MATH] and [MATH], while the causal order remains predefined [CITATION].', '1403.7333-1-8-5': 'Hence, the definition of an [MATH]-party causal realistic theory with parties [MATH], respective variables [MATH], and respective quantities [MATH], reads [EQUATION] where [MATH] is the event that party [MATH] precedes the others, i.e., [MATH], and [MATH] is the process compatible with the causal structure where [MATH] is first.', '1403.7333-1-8-6': 'In order for [MATH] to sum up to unity, we assign all causal structures where multiple parties [MATH] are first to the event [MATH].', '1403.7333-1-9-0': '# Causal game', '1403.7333-1-10-0': 'The following multi-party game cannot be won in a causal realistic scenario.', '1403.7333-1-10-1': 'Denote by [MATH] the [MATH] parties participating in the game.', '1403.7333-1-10-2': 'Each party [MATH] has a uniformly distributed binary variable [MATH] and access to the binary quantity [MATH], and to the shared quantity [MATH], which is a function of a variable [MATH] uniformly distributed on the range [MATH] that belongs to a dummy party (we just need it as a resource of shared randomness).', '1403.7333-1-10-3': "One can think of the party's variable [MATH] as its input, and the party's quantity [MATH] as its output.", '1403.7333-1-10-4': 'For a given [MATH], the game is won whenever party [MATH] outputs [MATH] as the parity of the inputs to all other parties, i.e., [MATH].', '1403.7333-1-10-5': 'Therefore, the success probability of winning the game is [EQUATION]', '1403.7333-1-10-6': 'In a causal realistic setup, this success probability is upper bounded by [MATH].', '1403.7333-1-10-7': 'To see this, note that if [MATH] is first, then it will remain first.', '1403.7333-1-10-8': 'However, for [MATH], the last party can be specified by [MATH].', '1403.7333-1-10-9': "Thus, all but one summands (where the first party itself has to guess the parity of the others' inputs) of the success probability expression are 1, the remaining summand is [MATH], as the output [MATH] of [MATH] will match the desired bit with a chance of [MATH].", '1403.7333-1-10-10': 'By repeating the experiment [MATH] times, where [MATH] grows asymptotically faster than [MATH], e.g., [MATH], one can bring the winning probability arbitrarily close to zero.', '1403.7333-1-11-0': '# Framework for macroscopic correlations with no causal order', '1403.7333-1-12-0': 'Instead of assuming that locally, nature is described by quantum theory [CITATION], we take the macroscopic limit of the systems, and thus assume that locally, nature is described by classical probability theory.', '1403.7333-1-12-1': 'Next to this assumption, we ask the probabilities of the outcomes to be non-negative and to sum up to unity, which forbids causal loops [CITATION].', '1403.7333-1-12-2': 'Suppose each party has a closed laboratory, meaning they do not interact, and each laboratory is opened once, during which the only interaction happens.', '1403.7333-1-12-3': 'When a laboratory is opened, it receives a state, manipulates it, and outputs a state.', '1403.7333-1-12-4': 'Thus, in the setting with local classical probability theory, such a laboratory is described by a conditional probability distribution [MATH], where [MATH] is the input space to the laboratory, and [MATH] is the output space of the laboratory.', '1403.7333-1-13-0': 'Let us consider the parties as described in the causal game.', '1403.7333-1-13-1': 'We denote the input space of party [MATH] with [MATH], and its output space with [MATH].', '1403.7333-1-13-2': 'Therefore, the [MATH]th local laboratory is described by the distribution [MATH], where [MATH] is the space of its quantity [MATH], and [MATH] is the space of its variable [MATH].', '1403.7333-1-13-3': 'As we do not have other global assumptions than that the overall picture should describe a probability distribution, we can describe everything outside the laboratories by the distribution [MATH], which we call [MATH] (see Figure [REF]), with the restriction that for any choice of [MATH], i.e., for any [MATH], the global probability distribution [MATH] sums up to unity, and in particular, the distribution [MATH] sums up to unity.', '1403.7333-1-14-0': 'To tackle this restriction formally, we represent a probability distribution [MATH] as a real, positive, diagonal matrix, having unit trace, with the diagonal entries [MATH].', '1403.7333-1-14-1': 'We use the symbol [MATH] to denote such a matrix.', '1403.7333-1-14-2': 'A conditional probability distribution [MATH] is a collection of non-conditional probability distributions for each value of [MATH].', '1403.7333-1-14-3': 'Thus, we represent [MATH] equivalently but with trace [MATH], and use the symbol [MATH].', '1403.7333-1-14-4': 'The condition that for a fixed [MATH] the probability [MATH] sums up to unity is reflected in the condition that, if we trace out [MATH] from the matrix [MATH], we are left with the identity matrix.', '1403.7333-1-14-5': 'The product of two distributions [MATH] and [MATH] is defined by all combinations of [MATH] and [MATH].', '1403.7333-1-14-6': 'Thus, the product in the matrix representation corresponds to the tensor product denoted by [MATH].', '1403.7333-1-14-7': 'To obtain the marginal distribution from a joint distribution we use the partial trace.', '1403.7333-1-14-8': 'From this follows that the output state of a laboratory [MATH] given the input state [MATH] is [MATH], where [MATH] is the identity matrix.', '1403.7333-1-14-9': 'This ultimately allows us to use the framework of Oreshkov, Costa, and Brukner [CITATION], restricting ourselves to diagonal matrices, i.e., all objects ([MATH] and local operations) are simultaneously diagonalizable in the computational basis, and thus can be expressed using the identity matrix [MATH] and the Pauli [MATH] matrix [MATH].', '1403.7333-1-14-10': 'In order to avoid causal loops, we know from their framework [CITATION] that, if we express [MATH] as a matrix [MATH], where [MATH] is a constant depending on the dimension of [MATH] and [MATH], where the superscripts denote the respective spaces, then [MATH] must either be the identity matrix, or have at least one party with identity on the output space and a traceless part on the input space, i.e., [MATH], either [MATH] or, [MATH], such that [MATH] and [MATH].', '1403.7333-1-15-0': '# Winning the causal game perfectly', '1403.7333-1-16-0': 'In order to win the causal game using this framework, we need to provide the distribution [MATH] and all local distributions describing the laboratories.', '1403.7333-1-16-1': 'For that purpose we use that if a set [MATH] of matrices with [MATH] eigenvalues together with the matrix multiplication form an Abelian group, then [MATH] is a positive semi-definite matrix.', '1403.7333-1-16-2': 'An Abelian group is closed, i.e., [MATH], [MATH], such that [MATH], is associative, has an identity element [MATH], i.e., the identity matrix, each element has an inverse, and all elements mutually commute.', '1403.7333-1-16-3': 'To prove positivity, take the eigenvector [MATH] that has the smallest eigenvalue [MATH], i.e., [MATH], where [MATH] is the eigenvalue of [MATH] with respect to the eigenvector [MATH].', '1403.7333-1-16-4': 'Because all elements mutually commute, they share a common set of eigenvalues, and thus, the eigenvalues of the sum of the elements are the sum of the eigenvalues of the single elements.', '1403.7333-1-16-5': 'Let [MATH] be an element contributing negatively to [MATH], i.e., [MATH].', '1403.7333-1-16-6': 'As the set forms a closed group, for every [MATH], there exists a [MATH], such that [MATH].', '1403.7333-1-16-7': 'From this follows [MATH], implying [MATH].', '1403.7333-1-17-0': 'We construct the distribution [MATH] for [MATH] odd and larger than two.', '1403.7333-1-17-1': "Let [MATH] be the set of all matrices that can be written as [MATH], where [MATH], with an even number of [MATH]'s for each [MATH].", '1403.7333-1-17-2': "From the equality [MATH] follows that for any [MATH], the product [MATH] again has an even number of [MATH]'s, and thus is an element in the set, forming a closed group.", '1403.7333-1-17-3': 'Furthermore, all elements mutually commute, have [MATH] eigenvalues, and each element is its own inverse.', '1403.7333-1-17-4': 'Thus, their sum is a positive semi-definite matrix.', '1403.7333-1-17-5': 'The distribution [MATH] as a matrix [MATH] is built by taking the sum over all group elements, where the matrix [MATH] of the group element [MATH] is associated to the input space of the [MATH]th party, i.e., [MATH], and to the output space of the [MATH]th party, i.e., [MATH], [EQUATION]', '1403.7333-1-17-6': 'By construction, [MATH] is positive semi-definite, reflecting that all probabilities are positive.', '1403.7333-1-17-7': 'Each group element [MATH] for [MATH] contains at least one [MATH] matrix preceded by a [MATH] matrix.', '1403.7333-1-17-8': 'Let such a [MATH] matrix of [MATH] be at position [MATH], i.e., [MATH], then the [MATH]th party has identity on the output and a traceless part on the input space, which is a requirement for the summand to be valid.', '1403.7333-1-17-9': 'This furthermore implies that tracing out the spaces [MATH] from [MATH] yields the identity matrix, reflecting that for each condition (input to [MATH]) the probabilities sum up to unity.', '1403.7333-1-18-0': 'This construction, however, works only if [MATH] is odd and larger than two, because, for [MATH] even, the group would contain the element [MATH], leading to a causal loop.', '1403.7333-1-18-1': 'For this last case, we enlarge one input and one output space, and construct the distribution as follows.', '1403.7333-1-18-2': 'Let [MATH] be the group used to construct [MATH] for [MATH] parties.', '1403.7333-1-18-3': 'The set for [MATH] parties is [MATH], where [MATH] are the first two [MATH] matrices of [MATH], i.e., [MATH], and [MATH] are the [MATH] elements with [MATH] and [MATH] exchanged, i.e., [MATH].', '1403.7333-1-18-4': 'This set forms an Abelian group, as all required properties follow immediately from [MATH].', '1403.7333-1-19-0': 'The distribution [MATH] as a matrix [MATH] is then constructed as before, with the exception that [MATH] is considered as single submatrix, i.e., [MATH] is associated to the input space of the first party, and to the output space of the [MATH]th party, [EQUATION]', '1403.7333-1-19-1': 'The first party has a two-bit input and the [MATH]th party has a two-bit output.', '1403.7333-1-19-2': 'Again, by construction, [MATH] is positive semi-definite.', '1403.7333-1-19-3': 'Each summand different from identity contains at least one identity submatrix preceded by a traceless submatrix, fulfilling the requirements that only valid summands appear, and that under each condition, the probabilities sum up to unity.', '1403.7333-1-20-0': 'In the [MATH] odd scenario, the local distribution for the [MATH]th party, given [MATH], to win the game is [EQUATION] where [MATH] for [MATH], and [MATH] otherwise.', '1403.7333-1-20-1': 'The strategies for [MATH] even are equivalent, except that [MATH], has a two-bit input space, respectively output, space, and depending on [MATH], uses the first, second, or both (with the parity as the relevant bit) bit-spaces to receive, send the desired bit.', '1403.7333-1-20-2': 'All local operations are classical since they are diagonal, i.e., consist only of measuring and preparing states in the [MATH] basis.', '1403.7333-1-21-0': 'Let [MATH] be the operation of the [MATH]th party.', '1403.7333-1-21-1': 'The distribution [MATH] is [EQUATION]', '1403.7333-1-21-2': 'This is because, when summing over all elements of [MATH], each term except [MATH] and [MATH] is either zero or depends on a variable [MATH], which, in the process of marginalization over [MATH], cancels.', '1403.7333-1-21-3': 'For each [MATH], the winning probability is [MATH].', '1403.7333-1-21-4': 'Therefore, the game is won with certainty.', '1403.7333-1-22-0': '# Conclusion', '1403.7333-1-23-0': 'In an attempt to find a framework that combines aspects of general relativity and quantum theory, Oreshkov, Costa, and Brukner developed a formalism for quantum correlations with no causal order [CITATION].', '1403.7333-1-23-1': 'They proved that some achievable correlations are incompatible with any a priori causal order and, therefore, are not causal realistic.', '1403.7333-1-23-2': 'We consider the macroscopic limit of this framework and show that, in sharp contrast to the two-party scenario, macroscopic multi-party correlations surprisingly can be incompatible with any a priori causal order.', '1403.7333-1-23-3': 'To show this, we first propose a game that cannot be won in a causal realistic scenario, but is won with certainty when no causal order is fixed.', '1403.7333-1-23-4': 'It is left open whether a different game exists among more than two parties which reopens the gap between the classical and the quantum success probability, and what additional assumptions to the quantum framework render its macroscopic limit causal realistic.', '1403.7333-1-24-0': 'Recently, the idea of indefinite causal order was applied to quantum computation [CITATION].', '1403.7333-1-24-1': 'Furthermore, Aaronson and Watrous showed that closed timelike curves render classical and quantum computing equivalent [CITATION], a result similar to ours, in the sense that the winning probability of the causal game is the same for the quantum and for the classical framework.', '1403.7333-1-24-2': 'Closed timelike curves are an interpretation of the framework for correlations with no causal order, as the [MATH] object in Figure [REF] could be thought of as a channel back in time, and are consistent with the theory of general relativity [CITATION].', '1403.7333-1-24-3': "However, Aaronson and Watrous take Deutsch's approach [CITATION] to closed timelike curves, which, as opposed to the framework studied here, is a non-linear extension of quantum theory.", '1403.7333-1-25-0': 'Given the common world view that time and space are fundamental, quantum theory and all so far known interpretations (see Bell [CITATION] for a review, or the parallel lives interpretation [CITATION] as an example) fail in giving an explanatory description of nature, e.g., non-local correlations, leaving us with a mere mathematical formalism.', '1403.7333-1-25-1': 'Thus, it would be interesting to study theories that take space and time not as fundamental, but as emerging from other principles [CITATION] (this idea dates back around 2500 years [CITATION]), as we believe they might help in understanding such correlations.', '1403.7333-1-26-0': 'We thank Caslav Brukner, Fabio Costa, Christina Giarmatzi, Issam Ibnouhsein, Ognyan Oreshkov, and Jibran Rashid for helpful discussions.', '1403.7333-1-26-1': 'We thank Ognyan Oreshkov for pointing out that whenever the matrix [MATH] is diagonal in the computational basis, all local operations can be reduced to objects diagonal in the same basis.', '1403.7333-1-26-2': 'This work was supported by the Swiss National Science Foundation (SNF), the NCCR QSIT, and the COST action on Fundamental Problems in Quantum Physics.'}

{'1403.7333-2-0-0': 'Quantum theory in a global space-time leads to effects, such as non-local correlations, still asking for an explicit causal explanation.', '1403.7333-2-0-1': 'Oreshkov, Costa, and Brukner (2012) proposed a framework in which quantum theory is valid locally but where, at the same time, no global causal structure is assumed (except for the absence of logical paradoxes).', '1403.7333-2-0-2': 'It was shown for the two-party case, however, that a global causal structure always emerges in the macroscopic limit.', '1403.7333-2-0-3': 'Quite naturally, it has been conjectured that the same also holds in the multi-party setting.', '1403.7333-2-0-4': 'We show that counter to this belief, macroscopic correlations locally compatible with classical probability theory exist that allow for deterministic signaling between three or more parties incompatible with any predefined causal order.', '1403.7333-2-1-0': 'Introduction.-Non-local correlations, as predicted by quantum theory and observed in nature [CITATION], are incompatible with local realism [CITATION] if we assume that global space-time is fundamental, i.e., given a priori, and that spatially separated settings can be chosen (at least partially [CITATION]) freely and independently.', '1403.7333-2-1-1': 'The correlations of physical properties as measured on spatially separated parts of a system may be stronger than what can be explained by shared classical information.', '1403.7333-2-1-2': 'In other words, the classical simulation of quantum correlations requires communication.', '1403.7333-2-1-3': 'Strangely enough, this means that there are not predefined yet correlated physical quantities emerging in a space-like separated way (hence, without communication between the measurement events).', '1403.7333-2-1-4': "This is inconsistent with Reichenbach's principle [CITATION], stating that any correlation must be due to a common cause (in our case: locally predefined values of the physical quantities - also called local realism -; ruled out here by Bell's theorem [CITATION]) or direct causation (in our case: communication not faster than at the speed of light; ruled out here by special relativity).", '1403.7333-2-1-5': 'In other words, the principle requires correlations to have an explicit physical origin (beyond the mere mathematical formalism, i.e., the wave function); we call explanations of correlations referring to such causes descriptive.', '1403.7333-2-1-6': 'It was in the same spirit that Einstein, Podolsky, and Rosen had asked for a physical theory to give a local description of physical reality - before concluding that quantum theory was, in this sense, incomplete [CITATION].', '1403.7333-2-1-7': 'Although their program as such may have failed, it continues to seem natural to ask for a descriptive explanation of non-local correlations.', '1403.7333-2-1-8': 'One approach is to refrain from considering space-time as fundamental (after all, a quantum state describing a physical system exhibiting non-local correlations cannot be mathematically separated into the spatially isolated parts of the physical system), but rather as emerging (possibly along with other macroscopic quantities) from a deeper fundament - comparably to temperature.', '1403.7333-2-1-9': 'A step in this direction was taken by Hardy with his program of merging general relativity with quantum theory [CITATION], in which he proposes to extend the latter to dynamical causal structures, a feature of general relativity (see [CITATION] for a recent review on quantum theory and causality).', '1403.7333-2-1-10': "By dropping the assumption of a global background time while keeping the assumptions that locally, nature is described by quantum theory and that no logical paradoxes arise, Oreshkov, Costa, and Brukner developed a framework for quantum correlations [CITATION] based on Hardy's idea.", '1403.7333-2-1-11': 'The causal structures emerging from this framework can be indefinite, qualitatively [CITATION] and quantitatively [CITATION] - just like other physical quantities which are not predefined [CITATION].', '1403.7333-2-1-12': "If, in the two-party case, we consider the macroscopic limit of the quantum systems, i.e., enforce both parties' physics to be described by classical probability theory (instead of quantum theory), then a predefined causal order always emerges.", '1403.7333-2-1-13': 'This is in accordance with our experience and, hence, natural and unsurprising; it strongly indicates that the same may hold in the multi-party case [CITATION].', '1403.7333-2-1-14': 'This, however, fails to be true, as we show in the present work.', '1403.7333-2-2-0': "We start by defining logical causality, followed by a description of a causal game, and prove an upper bound on the game's winning probability in a world with predefined causal order.", '1403.7333-2-2-1': 'We develop a framework for macroscopic correlations with no causal order and show how it can be used to derive a strategy for perfectly winning the causal game for any number [MATH] of parties.', '1403.7333-2-3-0': 'Causal realism.-By definition, measurement settings and outcomes are classical, i.e., perfectly distinguishable.', '1403.7333-2-3-1': 'Therefore, we think of physical systems as black boxes which we probe with classical inputs and that respond with classical outputs.', '1403.7333-2-3-2': 'When taking this perspective, we describe all physical quantities, in the following simply called quantities, as functions of variables.', '1403.7333-2-3-3': 'The party [MATH] is described by a set of variables [MATH] chosen freely by [MATH], and by a set of quantities [MATH] the party can access.', '1403.7333-2-3-4': 'Instantiations of the variables and quantities are denoted by the same letters but in lowercase.', '1403.7333-2-3-5': 'We will not give a definition of free choice, but rather postulate each variable as being intrinsically free.', '1403.7333-2-3-6': 'Quantities are functions of variables.', '1403.7333-2-3-7': 'Based on this, we define logical - as opposed to space-time - causality and use the former to make statements about the latter.', '1403.7333-2-3-8': 'If a quantity [MATH] is a function of [MATH], we say that [MATH] causally depends on [MATH] and is in the causal future of [MATH] or, equivalently, that [MATH] is in the causal past of [MATH], denoted by [MATH] or [MATH].', '1403.7333-2-3-9': 'The negations of these relations are denoted by [MATH] and [MATH].', '1403.7333-2-3-10': 'This definition does neither induce a causal order between quantities nor between variables nor between any quantity and the variables it does not depend on.', '1403.7333-2-4-0': 'Let us introduce a second party [MATH] described by the set of variables [MATH] and with access to the quantities [MATH].', '1403.7333-2-4-1': 'Quantities can depend on variables of both parties.', '1403.7333-2-4-2': 'If party [MATH] has a quantity that depends on a variable of [MATH], then we say [MATH] can signal to [MATH] (see Figure [REF]).', '1403.7333-2-4-3': 'In the following we will assume unidirectional signaling: If [MATH] can signal to [MATH], then [MATH] cannot signal to [MATH].', '1403.7333-2-4-4': 'This enables us to causally order parties.', '1403.7333-2-4-5': 'If at least one quantity of [MATH] depends on a variable of [MATH], but no quantity of [MATH] depends on any variable of [MATH] (which is the condition for unidirectional signaling), then [MATH] is in the causal past of [MATH].', '1403.7333-2-4-6': 'Formally, if there exist [MATH] and [MATH] fulfilling [MATH] and if for all [MATH] and for all [MATH], the relation [MATH] holds, then we have [MATH].', '1403.7333-2-5-0': 'Consider a two-party scenario with parties [MATH], each having a single variable [MATH], a single quantity [MATH], respectively, and a shared quantity [MATH].', '1403.7333-2-5-1': 'We call a theory causal realistic if all achievable probability distributions [MATH] result from a convex mixture of possible causal orders, i.e., [EQUATION] where [MATH] is the event [MATH], and [MATH] is an instantiation of a [MATH] that depends on a variable not in either of the sets [MATH] or [MATH].', '1403.7333-2-5-2': 'The realism attribute of a causal-realistic theory asks for the causal order and the physical quantities to be predefined.', '1403.7333-2-5-3': 'For more than two parties, the definition of causal realism becomes more subtle.', '1403.7333-2-5-4': 'Suppose we have three parties [MATH], [MATH], and [MATH], where [MATH] is in the causal past of both [MATH] and [MATH].', '1403.7333-2-5-5': 'We call a causal order predefined even when [MATH] is free to choose the causal order between [MATH] and [MATH] [CITATION].', '1403.7333-2-5-6': 'In general, in a predefined causal order, a party is allowed to determine the causal order between all parties in her causal future.', '1403.7333-2-5-7': 'Hence, a theory with the parties [MATH], variables [MATH] (shorthand [MATH]), and quantities [MATH]), respectively, is causal realistic if all achievable probability distributions [MATH] can be written as [EQUATION] where [MATH] is the event that each party [MATH] either is in the causal past of [MATH]) or has no causal relation with [MATH] and [MATH]).', '1403.7333-2-5-8': 'The term [MATH] is a convex mixture of distributions compatible with the causal structures in which [MATH] is first.', '1403.7333-2-6-0': 'Causal game.-The following multi-party game cannot be won in a causal-realistic scenario.', '1403.7333-2-6-1': 'Denote by [MATH] the parties that participate in the game.', '1403.7333-2-6-2': 'Each party [MATH] has a uniformly distributed binary variable [MATH] as well as access to the binary quantity [MATH] and to the shared quantity [MATH], where the variable [MATH] is uniformly distributed in the range [MATH] and belongs to a dummy party (we need it as a source of shared randomness).', '1403.7333-2-6-3': "One can think of the party [MATH]'s variable [MATH] as her input and the quantity [MATH] as her output.", '1403.7333-2-6-4': "For given [MATH], the game is won whenever [MATH]'s output [MATH] is the parity of the inputs to all other parties, i.e., [MATH].", '1403.7333-2-6-5': 'Therefore, the success probability for winning the game is [EQUATION]', '1403.7333-2-6-6': 'In a causal-realistic setup, this success probability is upper bounded by [MATH].', '1403.7333-2-6-7': 'To see this, note that if, without loss of generality, [MATH] is first, she will remain first.', '1403.7333-2-6-8': 'For [MATH], the last party can be specified by [MATH].', '1403.7333-2-6-9': "Thus, all summands of the sum in expression [REF] are [MATH] except for the summand, reflecting the fact that [MATH] herself has to guess the parity of the other's inputs, which is [MATH], since the output [MATH] of [MATH] will match the desired bit with a chance of [MATH].", '1403.7333-2-6-10': 'By repeating the experiment [MATH] times, one can bring the winning probability arbitrarily closely to [MATH].', '1403.7333-2-7-0': 'Framework for macroscopic correlations with no causal order.-Instead of assuming that locally, nature is described by quantum theory [CITATION], we take the macroscopic (i.e., classical) limit of the systems, and thus assume that locally, nature is described by classical probability theory.', '1403.7333-2-7-1': 'In addition to this assumption, we require the probabilities of the outcomes to be non-negative and to sum up to [MATH]; this excludes causal loops [CITATION].', '1403.7333-2-7-2': 'Suppose that each party has a closed laboratory that can be opened once, which is when the only interaction with the environment happens.', '1403.7333-2-7-3': 'When a laboratory is opened, the party receives, manipulates, and outputs a state.', '1403.7333-2-7-4': 'Thus, in the setting of local validity of classical probability theory, such a laboratory is described by a conditional probability distribution [MATH], where [MATH] is the input to and [MATH] is the output from the laboratory.', '1403.7333-2-8-0': 'Let us consider the parties as described in the causal game.', '1403.7333-2-8-1': 'We denote the input of [MATH] by [MATH] and her output by [MATH].', '1403.7333-2-8-2': 'Therefore, the [MATH]th local laboratory is described by the distribution [MATH].', '1403.7333-2-8-3': 'As we do not make global assumptions other than that the overall picture should describe a probability distribution, we can describe everything outside the laboratories by the distribution [MATH] (see Figure [REF]) satisfying the condition that for any choice of [MATH], i.e., for any probability distribution [MATH], the values of the product with [MATH], i.e., the values of [MATH], and in general of [MATH], sum up to [MATH].', '1403.7333-2-8-4': 'For tackling this condition formally, we represent a probability distribution [MATH] as a real positive diagonal matrix [MATH] with trace [MATH] and diagonal entries [MATH].', '1403.7333-2-8-5': 'A conditional probability distribution [MATH] is a collection of (unconditional) probability distributions [MATH] for each value of [MATH].', '1403.7333-2-8-6': 'Thus, we represent [MATH] equivalently, yet with trace [MATH], where [MATH] is the set of values [MATH] can take, and we use the symbol [MATH].', '1403.7333-2-8-7': 'The condition that the probabilities [MATH] sum up to [MATH] for fixed [MATH] is reflected by the condition that if we trace out [MATH] from the matrix [MATH], we are left with the identity.', '1403.7333-2-8-8': 'The product of two distributions [MATH] and [MATH] in the matrix representation corresponds to the tensor product denoted by [MATH].', '1403.7333-2-8-9': 'To obtain the marginal distribution from a joint distribution, we use the partial trace.', '1403.7333-2-8-10': 'This implies that the output state of a laboratory [MATH], given the input state [MATH], is [MATH].', '1403.7333-2-8-11': 'This allows us to use the framework of Oreshkov, Costa, and Brukner [CITATION], where we restrict ourselves to diagonal matrices, i.e., all objects ([MATH] and local operations) are simultaneously diagonalizable in the computational basis and can, hence, be expressed using the identity [MATH] and the Pauli matrix [MATH].', '1403.7333-2-8-12': 'We know from their framework [CITATION] that if we express [MATH] as a matrix [MATH], where [MATH] is a constant which depends on the dimension of [MATH] and [MATH], and where the superscripts label the respective inputs and outputs, then, in order to avoid causal loops, [MATH] must either be the identity or have at least one party with identity as output and a traceless part as input, i.e., for all [MATH] we have either [MATH] or there exists [MATH] such that [MATH] and [MATH].', '1403.7333-2-9-0': 'Winning the causal game perfectly.-To win the causal game using this framework, we need to provide the distribution [MATH] and all distributions describing the laboratories.', '1403.7333-2-9-1': 'For that purpose, we use the fact that if a set [MATH] of matrices with all eigenvalues in [MATH] forms an Abelian group with respect to matrix multiplication, then [MATH] is a positive semi-definite matrix.', '1403.7333-2-9-2': 'To prove this, take the eigenvector [MATH] which has the smallest eigenvalue [MATH], i.e., [MATH], where [MATH] is the eigenvalue of [MATH] with respect to the eigenvector [MATH].', '1403.7333-2-9-3': 'Let [MATH] be an element contributing negatively to [MATH], i.e., [MATH].', '1403.7333-2-9-4': 'As the set forms a group, there exists for every [MATH] a [MATH] such that [MATH].', '1403.7333-2-9-5': 'This implies [MATH] and [MATH].', '1403.7333-2-10-0': 'We construct the distribution [MATH] for odd [MATH].', '1403.7333-2-10-1': "Let [MATH] be the set of matrices that can be written as [MATH], with the objects [MATH], and with an even number of [MATH]'s for each [MATH].", '1403.7333-2-10-2': "The fact [MATH] implies that the product [MATH], for every [MATH], is a tensor product of [MATH] and [MATH] with an even number of [MATH]'s, and is thus an element of [MATH].", '1403.7333-2-10-3': 'Furthermore, all elements mutually commute, have all eigenvalues in [MATH] and, hence, each element is an involution.', '1403.7333-2-10-4': 'Therefore, their sum is a positive semi-definite matrix.', '1403.7333-2-10-5': 'The distribution [MATH] as a matrix [MATH] is built by taking the sum over all group elements, where the matrix [MATH] of the group element [MATH] is the input [MATH] of the [MATH]th party, and the output [MATH] of the [MATH]th party, [EQUATION]', '1403.7333-2-10-6': 'By construction, [MATH] is positive semi-definite, i.e., all probabilities are positive.', '1403.7333-2-10-7': 'Because [MATH] is odd, there exists for each group element [MATH]) at least one position [MATH] such that [MATH].', '1403.7333-2-10-8': "Then the [MATH]th party's input [MATH] is a traceless object, and her output [MATH] is [MATH] (which is a requirement for the summand to be valid).", '1403.7333-2-10-9': 'This implies that tracing out [MATH] from [MATH] yields the identity, which is the requirement that for each input to [MATH], the probabilities of the outputs sum up to [MATH].', '1403.7333-2-11-0': 'The above construction works for odd [MATH].', '1403.7333-2-11-1': 'For even [MATH], the group contains the element [MATH], which leads to a causal loop since all inputs are correlated to all outputs [CITATION].', '1403.7333-2-11-2': 'In this case, we double the dimensions of the input of the first party and of the output of the last party, and construct the distribution based on the group of matrices for the case of [MATH].', '1403.7333-2-11-3': 'Let [MATH] be the group used to construct [MATH].', '1403.7333-2-11-4': 'The set for [MATH] parties is the Abelian subgroup [MATH], where [MATH] and [MATH].', '1403.7333-2-12-0': 'The distribution [MATH] as a matrix [MATH] is constructed as before, with the exception that [MATH] is considered a single submatrix, [EQUATION]', '1403.7333-2-12-1': 'Again, by construction, [MATH] fulfills all requirements.', '1403.7333-2-13-0': 'For odd [MATH], the strategy [MATH] for the [MATH]th party to win the game is [EQUATION] where [MATH] for [MATH], and [MATH] otherwise.', '1403.7333-2-13-1': 'The strategies for even [MATH] are equivalent to the strategies for odd [MATH], except that [MATH] has a two-bit input and [MATH] has a two-bit output.', '1403.7333-2-13-2': 'Depending on [MATH], they use the first, second, or both (with the parity as the relevant bit) bit(s) to receive or send the desired bit.', '1403.7333-2-13-3': 'All local operations are classical since they are diagonal, i.e., consist only of measuring and preparing states in the [MATH] basis.', '1403.7333-2-14-0': 'The distribution [MATH] is [EQUATION]', '1403.7333-2-14-1': 'This is because when one sums over all elements of [MATH], then each term except [MATH] and [MATH] is either zero or depends on a variable [MATH] which, in the process of marginalization over [MATH], cancels out.', '1403.7333-2-14-2': 'For each [MATH], the winning probability is [EQUATION]', '1403.7333-2-14-3': 'Therefore, the game is won with certainty.', '1403.7333-2-15-0': 'Conclusion.-In an attempt to construct a theory that combines aspects of general relativity and quantum theory, Oreshkov, Costa, and Brukner proposed a framework for quantum correlations with no causal order [CITATION].', '1403.7333-2-15-1': 'They proved that some correlations are incompatible with any a priori causal order and, therefore, are not causal realistic although they satisfy quantum theory locally.', '1403.7333-2-15-2': 'We consider the macroscopic limit of this framework and show that in sharp contrast to the two-party scenario [CITATION], macroscopic multi-party correlations can be incompatible with any predefined causal order.', '1403.7333-2-15-3': 'To show this, we propose a game that cannot be won in a causal-realistic scenario, but is won with certainty when no causal order is fixed.', '1403.7333-2-16-0': 'Recently, the ideas of indefinite causal order and of superpositions of causal orders were applied to quantum computation [CITATION].', '1403.7333-2-16-1': 'Furthermore, Aaronson and Watrous showed that closed timelike curves render classical and quantum computing equivalent [CITATION].', '1403.7333-2-16-2': 'Our result is similar in the sense that the winning probability of the causal game is the same for the quantum and for the classical framework.', '1403.7333-2-16-3': 'Since the [MATH]-object in Figure [REF] can be thought of as a channel back in time, closed timelike curves can be interpreted as being part of the framework.', '1403.7333-2-16-4': 'Closed timelike curves per se are consistent with general relativity [CITATION].', '1403.7333-2-16-5': "However, Aaronson and Watrous take Deutsch's approach [CITATION] to closed timelike curves which, as opposed to the framework studied here, is a non-linear extension of quantum theory - such extensions are known to allow for communication faster than at the speed of light [CITATION].", '1403.7333-2-17-0': 'If space-time is fundamental, i.e., given a priori, quantum theory and all its traditional interpretations (see [CITATION] for a review or [CITATION] for a recent example) fail in giving a descriptive explanation of non-local correlations, leaving us with a mere mathematical formalism.', '1403.7333-2-17-1': 'It would be interesting to study theories that refrain from taking space-time as fundamental but that view it as emerging from a different fundament [CITATION] - an idea being at least 2500 years old [CITATION].', '1403.7333-2-17-2': 'It is conceivable that non-local correlations can be better understood in a setting where space-time and classical information arise together.', '1403.7333-2-18-0': 'We thank Caslav Brukner, Fabio Costa, Christina Giarmatzi, Issam Ibnouhsein, Ognyan Oreshkov, and Jibran Rashid for helpful discussions.', '1403.7333-2-18-1': 'We thank Ognyan Oreshkov for pointing out that whenever the matrix [MATH] is diagonal in the computational basis, all local operations can be reduced to objects diagonal in the same basis.', '1403.7333-2-18-2': 'The present work was supported by the Swiss National Science Foundation (SNF), the National Centre of Competence in Research "Quantum Science and Technology" (QSIT), the COST action on Fundamental Problems in Quantum Physics, the European Commission Project RAQUEL, the John Templeton Foundation, FQXi, and the Austrian Science Fund (FWF) through CoQuS, SFB FoQuS, and the Individual Project 24621.'}

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['1403.7333-1-7-3', '1403.7333-2-4-3'], ['1403.7333-1-2-0', '1403.7333-2-1-0'], ['1403.7333-1-2-2', '1403.7333-2-1-6'], ['1403.7333-1-2-4', '1403.7333-2-1-9'], ['1403.7333-1-2-6', '1403.7333-2-1-12'], ['1403.7333-1-12-1', '1403.7333-2-7-1'], ['1403.7333-1-12-2', '1403.7333-2-7-2'], ['1403.7333-1-12-3', '1403.7333-2-7-3'], ['1403.7333-1-18-3', '1403.7333-2-11-4'], ['1403.7333-1-6-1', '1403.7333-2-3-1'], ['1403.7333-1-6-3', '1403.7333-2-3-3'], ['1403.7333-1-6-3', '1403.7333-2-3-5'], ['1403.7333-1-25-0', '1403.7333-2-17-0'], ['1403.7333-1-25-1', '1403.7333-2-17-1'], ['1403.7333-1-16-1', '1403.7333-2-9-1'], ['1403.7333-1-26-2', '1403.7333-2-18-2'], ['1403.7333-1-23-1', '1403.7333-2-15-1'], ['1403.7333-1-10-3', '1403.7333-2-6-3'], ['1403.7333-1-10-7', '1403.7333-2-6-7'], ['1403.7333-1-10-9', '1403.7333-2-6-9'], ['1403.7333-1-10-10', '1403.7333-2-6-10'], ['1403.7333-1-8-1', '1403.7333-2-5-1'], ['1403.7333-1-8-2', '1403.7333-2-5-2'], ['1403.7333-1-8-3', '1403.7333-2-5-3'], ['1403.7333-1-8-3', 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[]

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[]

['1403.7333-1-0-1', '1403.7333-1-0-2', '1403.7333-1-7-6']

{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '3': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1403.7333

{'1403.7333-3-0-0': 'Quantum theory in a global space-time gives rise to non-local correlations, which cannot be explained causally in a satisfactory way; this motivates the study of theories with reduced global assumptions.', '1403.7333-3-0-1': 'Oreshkov, Costa, and Brukner (2012) proposed a framework in which quantum theory is valid locally but where, at the same time, no global space-time, i.e., predefined causal order, is assumed beyond the absence of logical paradoxes.', '1403.7333-3-0-2': 'It was shown for the two-party case, however, that a global causal order always emerges in the classical limit.', '1403.7333-3-0-3': 'Quite naturally, it has been conjectured that the same also holds in the multi-party setting.', '1403.7333-3-0-4': 'We show that counter to this belief, classical correlations locally compatible with classical probability theory exist that allow for deterministic signaling between three or more parties incompatible with any predefined causal order.', '1403.7333-3-1-0': '# Motivation and main result', '1403.7333-3-2-0': 'According to Bell [CITATION], correlations cry out for explanation.', '1403.7333-3-2-1': 'In such a spirit, already Einstein, Podolsky, and Rosen (EPR) [CITATION] had asked for an extension of quantum theory that incorporates a causal explanation [CITATION] of the correlations arising when two parts of an entangled quantum state are measured.', '1403.7333-3-2-2': 'Such an explanation can describe the emergence of the correlations either through pre-shared information or through influences.', '1403.7333-3-2-3': 'Because of relativity, EPR argued further, the latter cannot be the cause of such correlations.', '1403.7333-3-2-4': 'Later, (finite-speed) influences were ruled out by theory [CITATION] and experiments [CITATION].', '1403.7333-3-2-5': "Therefore, still according to EPR's reasoning, physical quantities need to be predefined.", '1403.7333-3-2-6': 'This, however, had been rejected by Bell [CITATION] under the assumption that spatially separated settings can be chosen (at least partially [CITATION]) freely and independently; such correlations are called non-local.', '1403.7333-3-2-7': 'Remarkably, this means that there are not predefined yet correlated physical quantities emerging in a space-like separated way.', '1403.7333-3-2-8': 'However, although the EPR program as such may have failed, it seems natural to continue to ask for a causal explanation of non-local correlations.', '1403.7333-3-2-9': 'A possible approach is to refrain from considering space-time as fundamental, treating it as emerging (potentially along with other macroscopic quantities) from a deeper fundament [CITATION] - comparably to temperature.', '1403.7333-3-2-10': 'A step in this direction was taken by Hardy [CITATION] with his program of merging general relativity with quantum theory, in which he proposes to extend the latter to dynamical causal orders, a feature of relativity (see [CITATION] for a recent review on quantum theory and causality).', '1403.7333-3-2-11': 'Chiribella, D\'Ariano, and Perinotti [CITATION] studied quantum supermaps called "quantum combs" that allow for superpositions of causal orders.', '1403.7333-3-2-12': "Based on Hardy's idea, Oreshkov, Costa, and Brukner [CITATION] developed a framework for quantum correlations without predefined causal order by dropping the assumption of a global background time while keeping the assumptions that locally, nature is described by quantum theory and that no logical paradoxes arise.", '1403.7333-3-2-13': 'Some causal structures emerging from this framework cannot be predefined [CITATION] - just like physical quantities exhibiting non-local correlations [CITATION].', '1403.7333-3-2-14': "If, in the two-party case, we consider the classical limit of the quantum systems, i.e., enforce both parties' physics to be described by classical probability theory (instead of quantum theory), then a predefined causal order always emerges [CITATION].", '1403.7333-3-2-15': 'This is in accordance with our experience and, hence, natural and unsurprising; it strongly indicates that the same may hold in the multi-party case [CITATION].', '1403.7333-3-2-16': 'This, however, fails to be true, as we show in the present work.', '1403.7333-3-3-0': '# Input-output systems, and causal order', '1403.7333-3-4-0': 'By definition, measurement settings and outcomes are classical, i.e., perfectly distinguishable.', '1403.7333-3-4-1': 'Therefore, we think of physical systems as black boxes which we probe with classical inputs and that respond with classical outputs.', '1403.7333-3-4-2': 'When taking this perspective, we describe all physical quantities, i.e., outputs, as functions of inputs.', '1403.7333-3-4-3': 'The party [MATH] is described by a set of inputs [MATH] chosen freely by [MATH], and by a set of outputs [MATH] the party can access (instantiations of the inputs and outputs are denoted by the same letters but in lowercase).', '1403.7333-3-4-4': 'Since we refrain from assuming global space-time as given a priori, we cannot define free randomness based on such a causal structure, as done elsewhere [CITATION].', '1403.7333-3-4-5': 'Instead, we take the concept of free randomness as fundamental - in accordance with a recent trend to derive properties of quantum theory from information-theoretic principles [CITATION] - and postulate inputs as being free.', '1403.7333-3-5-0': 'Outputs are functions of inputs.', '1403.7333-3-5-1': 'Based on this relationship, we define causal order.', '1403.7333-3-5-2': 'If an output [MATH] is a function of [MATH], we say that [MATH] causally depends on [MATH] and is in the causal future of [MATH] or, equivalently, that [MATH] is in the causal past of [MATH], denoted by [MATH] or [MATH].', '1403.7333-3-5-3': 'The negations of these relations are denoted by [MATH] and [MATH].', '1403.7333-3-5-4': 'This definition does neither induce a causal order between outputs nor between inputs nor between any output and the input it does not depend on.', '1403.7333-3-6-0': 'Let us introduce a second party [MATH] described by the set of inputs [MATH] and with access to the outputs [MATH].', '1403.7333-3-6-1': 'Outputs can depend on inputs of both parties.', '1403.7333-3-6-2': 'If party [MATH] has an output that depends on an inputs of [MATH], then we say that [MATH] can signal to [MATH] (see Figure [REF]).', '1403.7333-3-6-3': 'In the following, we will assume unidirectional signaling: If [MATH] can signal to [MATH], then [MATH] cannot signal to [MATH].', '1403.7333-3-6-4': 'This enables us to causally order parties.', '1403.7333-3-6-5': 'If at least one output of [MATH] depends on an input of [MATH], but no output of [MATH] depends on any input of [MATH] (which is the condition for unidirectional signaling), then [MATH] is in the causal future of [MATH].', '1403.7333-3-6-6': 'Formally, if there exist [MATH] and [MATH] fulfilling [MATH] and if for all [MATH] and for all [MATH], the relation [MATH] holds, then we have [MATH].', '1403.7333-3-7-0': 'Consider a two-party scenario with parties [MATH], each having a single input [MATH], a single output [MATH], respectively, and a shared random variable [MATH].', '1403.7333-3-7-1': 'We call a theory compatible with predefined causal order if all achievable probability distributions [MATH] can be written as a convex mixture of possible causal orders, i.e., [EQUATION] where [MATH] is the event [MATH], and [MATH] is an instantiation of [MATH] that depends on a input not in either of the sets [MATH] or [MATH].', '1403.7333-3-7-2': 'For more than two parties, the definition of predefined causal order becomes more subtle.', '1403.7333-3-7-3': 'Suppose we have three parties [MATH], [MATH], and [MATH], where [MATH] is in the causal past of both [MATH] and [MATH].', '1403.7333-3-7-4': 'We call a causal order predefined even if [MATH] is free to choose the causal order between [MATH] and [MATH] [CITATION].', '1403.7333-3-7-5': 'In general, in a predefined causal order, a party is allowed to determine the causal order between all parties in her causal future.', '1403.7333-3-7-6': 'Hence, a theory with the parties [MATH], inputs [MATH] (shorthand [MATH]), and outputs [MATH]), respectively, is compatible with predefined causal order if all achievable probability distributions [MATH] can be written as [EQUATION] where [MATH] is the event that each party [MATH] either is in the causal future of [MATH]) or has no causal relation with [MATH] and [MATH]).', '1403.7333-3-7-7': 'The term [MATH] is a convex mixture of distributions compatible with the causal structures in which [MATH] is first and chooses the causal order between the remaining parties.', '1403.7333-3-8-0': '# Game', '1403.7333-3-9-0': 'The following multi-party game cannot be won in a scenario with predefined causal order.', '1403.7333-3-9-1': 'Denote by [MATH] the parties that participate in the game.', '1403.7333-3-9-2': 'Each party [MATH] has a uniformly distributed binary input [MATH] as well as a binary output [MATH] and access to the shared random variable [MATH] uniformly distributed in the range [MATH].', '1403.7333-3-9-3': 'The random variable [MATH] belongs to a dummy party (we need her as a source of shared randomness).', '1403.7333-3-9-4': "For given [MATH], the game is won whenever [MATH]'s output [MATH] is the parity of the inputs to all other parties, i.e., [MATH].", '1403.7333-3-9-5': 'Therefore, the success probability for winning the game is [EQUATION]', '1403.7333-3-9-6': 'In a setup with predefined causal order, this success probability is upper bounded by [MATH].', '1403.7333-3-9-7': 'To see this, note that if, without loss of generality, [MATH] is first, she will remain first.', '1403.7333-3-9-8': 'For [MATH], the last party can be specified by [MATH].', '1403.7333-3-9-9': "Thus, all the terms of the sum in expression [REF] are [MATH] except for the first summand, reflecting the fact that [MATH] herself has to guess the parity of the other's inputs, which is [MATH].", '1403.7333-3-9-10': 'By repeating the experiment [MATH] times, one can bring the winning probability arbitrarily closely to [MATH].', '1403.7333-3-10-0': '# Framework for classical correlations without causal order', '1403.7333-3-11-0': 'Instead of assuming that locally, nature is described by quantum theory [CITATION], we take the classical limit of the systems and thus assume that locally, nature is described by classical probability theory.', '1403.7333-3-11-1': 'In addition to this, we require the probabilities of the outcomes to be non-negative and to sum up to [MATH]; this excludes logical paradoxes [CITATION].', '1403.7333-3-11-2': 'We suppose that each party has a closed laboratory that can be opened once - which is when the only interaction with the environment happens.', '1403.7333-3-11-3': 'When a laboratory is opened, the party receives, manipulates, and outputs a state.', '1403.7333-3-11-4': 'Thus, in the setting of local validity of classical probability theory, such a laboratory is described by a conditional probability distribution [MATH], where [MATH] is the input to and [MATH] is the output from the laboratory.', '1403.7333-3-12-0': 'Let us consider the parties as described in the game.', '1403.7333-3-12-1': 'We denote the input to [MATH] by [MATH] and the output from [MATH] by [MATH].', '1403.7333-3-12-2': 'Therefore, the [MATH]th local laboratory is described by the distribution [MATH].', '1403.7333-3-12-3': 'As we do not make global assumptions other than that the overall picture should describe a probability distribution, we describe everything outside the laboratories by the distribution [MATH] (see Figure [REF]) satisfying the condition that for any choice of [MATH], i.e., for any probability distribution [MATH], the values of the product with [MATH], i.e., the values of [EQUATION] and in general of [EQUATION] are non-negative and sum up to [MATH].', '1403.7333-3-12-4': 'For tackling this condition formally, we represent a probability distribution [MATH] as a real positive diagonal matrix [MATH] with trace [MATH] and diagonal entries [MATH].', '1403.7333-3-12-5': 'A conditional probability distribution [MATH] is a collection of (unconditional) probability distributions [MATH] for each value of [MATH].', '1403.7333-3-12-6': 'Thus, we represent [MATH] similarly, yet with trace [MATH], where [MATH] is the set of values [MATH] can take, and we use the symbol [MATH].', '1403.7333-3-12-7': 'The values [MATH] are ordered with respect to the ordering of the subscripts of [MATH], e.g., for binary [MATH] and [MATH] the matrix [MATH] is [EQUATION]', '1403.7333-3-12-8': 'The condition that the probabilities [MATH] sum up to [MATH] for fixed [MATH] is reflected by the condition that if we trace out [MATH] from the matrix [MATH] (denoted by [MATH]), we are left with the identity.', '1403.7333-3-12-9': 'The product of two distributions [MATH] and [MATH] in the matrix representation corresponds to the tensor product denoted by [MATH].', '1403.7333-3-12-10': 'To obtain the marginal distribution from a joint distribution, we use the partial trace.', '1403.7333-3-12-11': 'This implies that the output state of a laboratory [MATH], given the input state [MATH], is [MATH], where [MATH] is the identity matrix with the same dimension as [MATH].', '1403.7333-3-12-12': 'This allows us to use the framework of Oreshkov, Costa, and Brukner [CITATION], where we restrict ourselves to diagonal matrices, i.e., all objects ([MATH] and local operations) are simultaneously diagonalizable in the computational basis and can, hence, be expressed using the identity [MATH] and the Pauli matrix [MATH].', '1403.7333-3-12-13': 'We know from their framework [CITATION] that if we express [MATH] as a matrix [MATH], where [MATH] is a normalization constant and [MATH].', '1403.7333-3-12-14': 'For every [MATH], the summand [MATH] represents a channel from all [MATH] with [MATH] to all [MATH] with [MATH].', '1403.7333-3-12-15': 'In order to avoid logical paradoxes, [MATH] must describe a channel where at least one party is a recipient without being a sender [CITATION].', '1403.7333-3-12-16': 'In other words, [MATH] must either be the identity or there exists [MATH] such that [MATH] and [MATH].', '1403.7333-3-13-0': '# Winning the game perfectly', '1403.7333-3-14-0': 'To win the game using this framework, we need to provide the distribution [MATH] and all distributions describing the laboratories.', '1403.7333-3-14-1': 'For that purpose, we use the fact that if a set [MATH] of matrices with all eigenvalues in [MATH] forms an Abelian group with respect to matrix multiplication, then [MATH] is a positive semi-definite matrix.', '1403.7333-3-14-2': 'To prove this, take the eigenvector [MATH] which has the smallest eigenvalue [MATH], i.e., [EQUATION] where [MATH] is the eigenvalue of [MATH] with respect to the eigenvector [MATH].', '1403.7333-3-14-3': 'Let [MATH] be an element contributing negatively to [MATH], i.e., [MATH].', '1403.7333-3-14-4': 'As the set forms a group, for every [MATH] there exists a [MATH] such that [MATH].', '1403.7333-3-14-5': 'This implies [MATH] and [MATH].', '1403.7333-3-15-0': '## Construction of [MATH] for odd [MATH]', '1403.7333-3-16-0': 'We construct the distribution [MATH] for odd [MATH].', '1403.7333-3-16-1': "Let [MATH] be the set of matrices that can be written as [MATH], with the objects [MATH], and with an even number of [MATH]'s for each [MATH].", '1403.7333-3-16-2': 'We use the notation [MATH] to denote the matrix [MATH].', '1403.7333-3-16-3': "The fact [MATH] implies that the product [MATH], for every [MATH], is a tensor product of [MATH] and [MATH] with an even number of [MATH]'s, and is thus an element of [MATH].", '1403.7333-3-16-4': 'Furthermore, all elements mutually commute, have all eigenvalues in [MATH] and, hence, each element is an involution.', '1403.7333-3-16-5': 'Therefore, their sum is a positive semi-definite matrix.', '1403.7333-3-16-6': 'The distribution [MATH] as a matrix [MATH] is built by taking the sum over all group elements, where the matrix [MATH] of the group element [MATH] contributes to the input [MATH] of party [MATH], and to the output [MATH] of the party labeled by [MATH], [EQUATION]', '1403.7333-3-16-7': 'By construction, [MATH] is positive semi-definite, i.e., all probabilities are positive.', '1403.7333-3-16-8': 'Because [MATH] is odd, there exists for each group element [MATH]) at least one position [MATH] such that [MATH], which excludes logical paradoxes.', '1403.7333-3-16-9': 'Furthermore, for every [MATH] the object [MATH] contains the channel from all parties [MATH] to [MATH] - a condition to perfectly win the game.', '1403.7333-3-17-0': '## Example: [MATH]', '1403.7333-3-18-0': 'For illustration, we construct [MATH].', '1403.7333-3-18-1': 'The group from which [MATH] is constructed is [MATH] with the group elements [EQUATION]', '1403.7333-3-18-2': 'The matrix [MATH] is thus [EQUATION]', '1403.7333-3-18-3': 'The second summand of [MATH] represents a channel from [MATH] to [MATH], the third summand represents a channel from [MATH] to [MATH], and finally, the last summand represents a channel from [MATH] to [MATH].', '1403.7333-3-19-0': 'It can easily be verified that if the three parties [MATH], [MATH], and [MATH] input [MATH] into [MATH], then [MATH] outputs the distribution [EQUATION] where [MATH].', '1403.7333-3-19-1': 'Therefore, [MATH] implements a uniform mixture of the loops where the input of party [MATH] is sent to party [MATH], and where the input of party [MATH] is flipped and sent to [MATH] (see Figure [REF]) [CITATION].', '1403.7333-3-19-2': 'It is evident from Figure [REF] that logical paradoxes are not possible.', '1403.7333-3-19-3': 'If all intermediate parties forward what they receive (by applying any reversible transformation), both loops (see Figure [REF]) cancel each other out, i.e., the correlations interfere destructively.', '1403.7333-3-19-4': 'Then again, if one intermediate party does not forward what she receives, the loop is broken.', '1403.7333-3-19-5': 'Conversely, any party can signal to her predecessor in the loop, because then an even number of bit-flips are applied, and thus the correlations interfere constructively.', '1403.7333-3-19-6': 'The same reasoning holds for any [MATH] for odd [MATH].', '1403.7333-3-20-0': '## Construction of [MATH] for even [MATH]', '1403.7333-3-21-0': 'The above construction works for odd [MATH].', '1403.7333-3-21-1': 'For even [MATH], the group contains the element [MATH], which leads to a logical paradox since all inputs are correlated to all outputs [CITATION].', '1403.7333-3-21-2': 'This can also be seen in Figure [REF], where for even [MATH], the sum of both channels leads to a logical paradox.', '1403.7333-3-21-3': 'In this case ([MATH] even), we double the dimensions of the output of the second-to-last party and of the input of the last party, and construct the distribution based on the group of matrices for the case of [MATH].', '1403.7333-3-21-4': 'Let [MATH] be the group used to construct [MATH].', '1403.7333-3-21-5': 'The set for [MATH] parties is the Abelian subgroup [MATH], where [MATH] and [MATH].', '1403.7333-3-22-0': 'The distribution [MATH] as a matrix [MATH] is constructed as before, with the exception that [MATH] is considered a single submatrix, [EQUATION]', '1403.7333-3-22-1': 'Again, by construction, [MATH] fulfills all requirements and contains all channels required to perfectly win the game.', '1403.7333-3-23-0': '## Example: [MATH]', '1403.7333-3-24-0': 'As an example, we construct the matrix [MATH].', '1403.7333-3-24-1': 'The group [MATH] for [MATH] is constructed from the group [MATH] and has the elements [EQUATION]', '1403.7333-3-24-2': 'The matrix [MATH] is thus [EQUATION]', '1403.7333-3-24-3': 'The second to the fifth summands represent the channels that are used to perfectly win the game.', '1403.7333-3-25-0': 'The conditional probability distribution [MATH] responds to input [MATH] with the following output [EQUATION] where [MATH] are both bits of the random variable [MATH], [MATH] are both bits of the random variable [MATH], and where [MATH].', '1403.7333-3-25-1': 'Therefore, [MATH] implements a uniform distribution of four circular channels (see Figure [REF]).', '1403.7333-3-26-0': 'By construction of [MATH], no logical paradox arises.', '1403.7333-3-26-1': "More intuitively, in any strategy that does not break any of the four circular channels of Figure [REF] (i.e., every party's output depends on its input), parties [MATH] and [MATH] use the first bit, the second, or both bits to communicate.", '1403.7333-3-26-2': 'If they use the first bit, then the correlations arising from the first two loops and the last two loops of Figure [REF] interfere constructively.', '1403.7333-3-26-3': 'Both pairs together, however, break the loop.', '1403.7333-3-26-4': 'If they use the second bit, then the first and the third loop, and the second with the last loop yield the same output in every cycle.', '1403.7333-3-26-5': 'In total, all loops cancel each other out.', '1403.7333-3-26-6': 'For the last case as well, where both bits are used for communication, the correlations from the first and the last loop interfere constructively, and so do the second and the third.', '1403.7333-3-26-7': 'Ultimately, again, all loops cancel each other out, and no logical paradox can be created.', '1403.7333-3-27-0': 'For larger even [MATH], the conditional probability distribution [MATH] as well is constructed out of four loops, as in Figure [REF], that cancel each other out when one tries to build a logical paradox.', '1403.7333-3-27-1': 'For [MATH], the same construction does not work, because the two-bit channel cannot be used to signal from its source to its destination - it can only be used when combined with other channels.', '1403.7333-3-27-2': 'In a two-party scenario, however, in order to win the game, each party needs to signal to the other.', '1403.7333-3-28-0': '## Winning strategy', '1403.7333-3-29-0': 'For odd [MATH], the strategy [MATH] for party [MATH] to win the game is [EQUATION] where [MATH] for [MATH], and [MATH] otherwise.', '1403.7333-3-29-1': 'The strategies for even [MATH] are equivalent to the strategies for odd [MATH], except that [MATH] has a two-bit output and [MATH] has a two-bit input.', '1403.7333-3-29-2': 'Depending on [MATH], they use the first, second, or both bit(s) to receive or send the desired bit.', '1403.7333-3-29-3': 'All local operations are classical since they are diagonal, i.e., consist only of measuring and preparing states in the [MATH] basis.', '1403.7333-3-30-0': 'The distribution [MATH] is [EQUATION] where we rearranged the submatrices of [MATH] in the trace expression such that the ordering of the conditional probabilities in [MATH] match.', '1403.7333-3-30-1': 'This result is obtained because after taking the trace, each term except [MATH] and [MATH] is either zero or depends on a variable [MATH] which, in the process of marginalization over [MATH], cancels out.', '1403.7333-3-30-2': 'For each [MATH], the winning probability is [EQUATION]', '1403.7333-3-30-3': 'Therefore, the game is won with certainty.', '1403.7333-3-31-0': '## Example: [MATH]', '1403.7333-3-32-0': 'The probability of obtaining [MATH] in the case [MATH] is [EQUATION]', '1403.7333-3-32-1': 'Therefore, the probability of the event [MATH] is [MATH].', '1403.7333-3-32-2': 'The distribution of [MATH] in the case [MATH] is [EQUATION] and, finally, the distribution of [MATH] in the case [MATH] is [EQUATION]', '1403.7333-3-32-3': 'The probabilities of the events [MATH] and [MATH] are both [MATH].', '1403.7333-3-32-4': 'Therefore, the game is won with certainty.', '1403.7333-3-33-0': 'Intuitively, in the case [MATH], party [MATH] sends [MATH] on both circular channels of Figure [REF].', '1403.7333-3-33-1': 'Thus, party [MATH] receives the uniform mixture of [MATH] (left channel of Figure [REF]) and [MATH] (right channel of Figure [REF]).', '1403.7333-3-33-2': 'Party [MATH] thereafter sends [MATH], i.e., the uniform mixture of [MATH] and [MATH], on both circular channels, yielding the deterministic input [MATH] to party [MATH].', '1403.7333-3-34-0': '## Example: [MATH]', '1403.7333-3-35-0': 'In the example [MATH], we explicitly write the local operations for the third and fourth parties, as the third party has a two-bit output, and the fourth party has a two-bit input.', '1403.7333-3-35-1': 'The local operations for the third party ([MATH]) are [EQUATION] with [EQUATION]', '1403.7333-3-35-2': 'Party [MATH]) uses [EQUATION] where we use shorthand [MATH] for [EQUATION]', '1403.7333-3-35-3': 'The distributions of [MATH], [MATH], [MATH], [MATH], under the condition [MATH], [MATH], [MATH], [MATH], respectively, are [EQUATION]', '1403.7333-3-35-4': 'Therefore, the event [MATH], given [MATH], the event [MATH], in the case [MATH], the event [MATH], if [MATH], and the event [MATH] in the case [MATH] have probability [MATH].', '1403.7333-3-35-5': 'Which implies that the game is won with certainty.', '1403.7333-3-36-0': 'By consulting Figure [REF], we can describe the strategy in the following way.', '1403.7333-3-36-1': 'If [MATH], then party [MATH] sends [MATH] to the next party by using all four channels of Figure [REF].', '1403.7333-3-36-2': 'Each of the next two parties in clockwise orientation, i.e., party [MATH] and party [MATH], sends the parity of what she receives from the previous party and her input [MATH], [MATH], respectively).', '1403.7333-3-36-3': 'Depending on [MATH], parties [MATH] and [MATH] use the first, the second, or both single-bit channels.', '1403.7333-3-36-4': 'In particular, if [MATH], then [MATH] uses the first channel to communicate to [MATH] - the second channel is ignored.', '1403.7333-3-36-5': 'For [MATH] they use both channels, i.e., the parity of the inputs to both channels is equal to the bit [MATH] sends.', '1403.7333-3-36-6': 'For [MATH], the two-bit channel between [MATH] and [MATH] is ignored.', '1403.7333-3-36-7': 'Finally, for [MATH] they use the second channel.', '1403.7333-3-36-8': 'By doing so, [MATH] obtains [MATH], as the introduced bit-flips from the four channels (see Figure [REF]) cancel each other out.', '1403.7333-3-37-0': '# Conclusion', '1403.7333-3-38-0': 'In an attempt to construct a theory that combines aspects of general relativity and quantum theory, Oreshkov, Costa, and Brukner [CITATION] proposed a framework for quantum correlations without causal order.', '1403.7333-3-38-1': 'They proved that some correlations are incompatible with any a priori causal order and, therefore, are not compatible with predefined causal order although they satisfy quantum theory locally.', '1403.7333-3-38-2': 'We consider the classical limit of this framework and show that in sharp contrast to the two-party scenario [CITATION], classical and logically consistent multi-party correlations can be incompatible with any predefined causal order.', '1403.7333-3-38-3': 'To show this, we propose a game that cannot be won in a scenario with predefined causal order, but is won with certainty when no causal order is fixed.', '1403.7333-3-39-0': 'Recently, the ideas of indefinite causal order and of superpositions of causal orders were applied to quantum computation [CITATION].', '1403.7333-3-39-1': 'Furthermore, Aaronson and Watrous [CITATION] showed that closed timelike curves render classical and quantum computing equivalent.', '1403.7333-3-39-2': 'Our result is similar in the sense that the winning probability of the game is the same for the quantum and for the classical framework.', '1403.7333-3-39-3': 'Since the [MATH] object in Figure [REF] can be thought of as a channel back in time, closed timelike curves can be interpreted as being part of the framework.', '1403.7333-3-39-4': 'Closed timelike curves per se are consistent with general relativity [CITATION].', '1403.7333-3-39-5': "However, Aaronson and Watrous take Deutsch's approach [CITATION] to closed timelike curves which, as opposed to the framework studied here, is a non-linear extension of quantum theory - such extensions are known to allow for communication faster than at the speed of light [CITATION].", '1403.7333-3-40-0': 'We thank Caslav Brukner, Fabio Costa, Christina Giarmatzi, Issam Ibnouhsein, Ognyan Oreshkov, and Jibran Rashid for helpful discussions.', '1403.7333-3-40-1': 'We thank Caslav Brukner for the interpretation of [MATH] as two loops.', '1403.7333-3-40-2': 'We thank Ognyan Oreshkov for pointing out that whenever the matrix [MATH] is diagonal in the computational basis, all local operations can be reduced to objects diagonal in the same basis.', '1403.7333-3-40-3': 'We thank three anonymous reviewers for helpful comments on the presentation of the results - especially the last reviewer for the detailed comments.', '1403.7333-3-40-4': 'The present work was supported by the Swiss National Science Foundation (SNF), the National Centre of Competence in Research "Quantum Science and Technology" (QSIT), the COST action on Fundamental Problems in Quantum Physics, the European Commission Project RAQUEL, the John Templeton Foundation, FQXi, and the Austrian Science Fund (FWF) through CoQuS, SFB FoQuS, and the Individual Project 24621.'}

1802.09525

{'1802.09525-1-0-0': 'The discovery of a repeating fast radio burst (FRB) has led to the first precise localization, an association with a dwarf galaxy, and the identification of a coincident persistent radio source.', '1802.09525-1-0-1': 'However, further localizations are required to determine the nature of FRBs, the sources powering them, and the possibility of multiple populations.', '1802.09525-1-0-2': 'Here we investigate the use of associated persistent radio sources to establish FRB counterparts, taking into account the localization area and the persistent source flux density.', '1802.09525-1-0-3': 'Due to the lower areal number density of radio sources compared to faint optical sources, robust associations can be achieved for less precise localizations as compared to direct optical host galaxy associations.', '1802.09525-1-0-4': 'For generally larger localizations which preclude robust associations, the number of candidate hosts can be reduced based on the ratio of radio-to-optical brightness.', '1802.09525-1-0-5': 'We find that confident associations with sources having a flux density of [MATH]1 mJy, comparable to the luminosity of the persistent source associated with FRB121102 over the redshift range [MATH], require FRB localizations of [MATH].', '1802.09525-1-0-6': 'We demonstrate that even in the absence of a robust association, constraints can be placed on the luminosity of an associated radio source as a function of localization and DM.', '1802.09525-1-0-7': 'We find that for DM [MATH], an upper limit comparable to the luminosity of the FRB121102 persistent source can be placed if the localization is [MATH].', '1802.09525-1-0-8': 'We apply our analysis to the case of the ASKAP FRB170107, using optical and radio observations of the localization region.', '1802.09525-1-0-9': 'We identify two candidate hosts based on a ratio of radio-to-optical brightness of [MATH].', '1802.09525-1-0-10': 'We find that if one of these is indeed associated with FRB170107, the resulting radio luminosity ([MATH], as constrained from the DM value) is comparable to the luminosity of the FRB121102 persistent source ([MATH]).', '1802.09525-1-1-0': '# Introduction', '1802.09525-1-2-0': 'Fast radio bursts (FRBs) are extremely bright, millisecond-duration pulses of coherent radio emission, with large dispersion measures (DMs) that exceed typical Galactic values, hence pointing to an extragalactic origin.', '1802.09525-1-2-1': 'Since the discovery of the first FRB in archival data , roughly 30 additional FRBs have been detected, both in archival and real-time searches (; ; ; ; ; ).', '1802.09525-1-2-2': "Despite a growing number of FRB detections, the lack of precise localizations (typically [MATH]) has led to a wide range of suggested progenitor systems: giant pulses and magnetar flares (; ; ' ), mergers of compact objects (; ), the collapse of supramassive neutron stars to black holes (), and emission from rapidly spinning magnetars (; ).", '1802.09525-1-3-0': 'The discovery of the repeating FRB121102 (; ) led to the first precise localization and the identification of a low metallicity dwarf star forming host galaxy at [MATH] , with properties remarkably similar to the host galaxies of both long-duration gamma-ray bursts (LGRBs) and hydrogen-poor superluminous supernovae (SLSN) .', '1802.09525-1-3-1': 'Radio observations also revealed the presence of a persistent radio source, co-located with the bursts to [MATH] mas .', '1802.09525-1-3-2': 'The large angular offset between the radio source and the optical center of the galaxy (170[MATH]300 mas) argue against an active galactic nuclei (AGN) origin .', '1802.09525-1-3-3': 'Similarly, the resulting ratio of radio-to-optical brightness for the persistent source and host galaxy ([MATH]) is much larger than expected for a star formation origin.', '1802.09525-1-3-4': 'This has led to the suggestion that the radio source is a nebula associated with FRB121102 (; ).', '1802.09525-1-3-5': 'More recently, the discovery of a large rotation measure with [MATH] variations on half-year timescales suggests the presence of a dynamic and highly magnetized environment surrounding the source of the FRB .', '1802.09525-1-4-0': 'The properties of the host galaxy, the repeating nature of the bursts, and the persistent radio source are consistent with a model in which FRBs are powered by young, millisecond magnetars (; see also ; ), the same engines that have been argued to power SLSN and perhaps LGRBs (; ).', '1802.09525-1-4-1': 'In this model, individual bursts are emitted via the dissipation of rotational or magnetic energy, while the quiescent radio emission is due to the magnetar wind nebula or the shock interaction between the supernova ejecta and the surrounding circumstellar medium (; ).', '1802.09525-1-5-0': 'In this broad scenario, we expect that FRBs should be located preferentially in dwarf galaxies and be coincident with quiescent radio sources similar to the one observed in FRB121102 .', '1802.09525-1-5-1': '[CITATION] demonstrate that [MATH] FRB localizations will be sufficient to test various formation channels.', '1802.09525-1-5-2': 'Given the high event rate for FRBs ([MATH] above 1 Jy; ) and the improved sensitivities of upcoming telescopes and surveys, the number of detected bursts is expected to increase drastically over the next few years as new radio facilities come on-line.', '1802.09525-1-5-3': 'However, due to a wide range of localization capabilities, precise localizations will continue to be a challenge.', '1802.09525-1-6-0': 'In [CITATION] ([CITATION]; hereafter, Paper I), we explored how robustly FRBs with different localization regions can be associated with host galaxies based on the optical brightness of the galaxy.', '1802.09525-1-6-1': 'Due to the large areal number density of faint optical sources, we showed that sub-arcsecond localizations are required for confident associations with dwarf galaxies at [MATH], whereas localizations of up to [MATH] may suffice if the hosts are instead generally [MATH] galaxies.', '1802.09525-1-7-0': 'Here we explore the likelihood of constraining similar associations with radio sources, motivated by the persistent radio source associated with FRB121102 and assuming that all FRBs are coincident with such sources, potentially with a resulting high radio-to-optical flux ratio.', '1802.09525-1-7-1': 'We also explore what limits can be placed on the presence and luminosity of a persistent radio source (as a function of localization area) even in the absence of a robust association.', '1802.09525-1-7-2': 'We assess our results in the context of existing and upcoming radio facilities.', '1802.09525-1-7-3': 'As an example of our method, we present radio (VLA) and optical (Magellan) observations of the localization region of the ASKAP FRB170107 .', '1802.09525-1-7-4': 'We search for radio counterparts in the region using the radio-to-optical flux ratio to identify potential candidates.', '1802.09525-1-8-0': 'The paper is structured as follows.', '1802.09525-1-8-1': 'We begin by using radio source number counts to determine the probability of chance coincidence for an FRB and a persistent radio source as a function of localization region and persistent source flux density ([REF]).', '1802.09525-1-8-2': 'Next, in [REF] we use these results to constrain the upper limit on the radio luminosity of a source in the absence of a robust association.', '1802.09525-1-8-3': 'In [REF], we discuss various techniques which may further aid in the identification of associated counterparts.', '1802.09525-1-8-4': 'We present an example of our approach in [REF], using radio and optical observations of the localization region of FRB170107.', '1802.09525-1-8-5': 'Finally, we discuss our results in the context of existing and planned FRB search facilities in [REF].', '1802.09525-1-9-0': '# Assessing Probability of Chance Coincidence with Radio Source Counts', '1802.09525-1-10-0': 'We assess the likelihood that an FRB is associated with a persistent radio source by calculating the probability of chance coincidence for a source of a given brightness within a localization region.', '1802.09525-1-10-1': 'We use the 1.4 GHz Euclidean-normalized differential source counts presented in [CITATION] to calculate the number density of radio sources above some limiting flux density, [MATH]; see top left panel of Figure [REF].', '1802.09525-1-10-2': 'These source counts are measured over the range [MATH]Jy [MATH] 100 Jy and extrapolated to lower flux densities imposing an evolutionary model for the spectral luminosity function of extragalactic radio sources (; ).', '1802.09525-1-10-3': 'The source counts are comprised of both AGN and star forming galaxies, with the latter dominating the counts below [MATH] mJy , although radio-quiet AGN have been shown to comprise a sizable fraction of the sub-mJy population (; ; ; ).', '1802.09525-1-10-4': 'We assume a Poisson distribution of radio sources across the sky and calculate the chance coincidence probability as: [EQUATION]', '1802.09525-1-10-5': 'The localization region, [MATH], is parameterized by [MATH], where [MATH] is the [MATH] localization radius of the FRB.', '1802.09525-1-11-0': 'Using the integrated source counts, we plot in the top right panel of Figure [REF] the number of radio sources above some limiting flux density [i.e., [MATH])] as a function of [MATH].', '1802.09525-1-11-1': 'We plot the results over the range [MATH]Jy to 1 Jy, in order of magnitude increments.', '1802.09525-1-11-2': 'We find an expectation value of about one source at [MATH] mJy within the typical localization regions of single-dish telescopes ([MATH]).', '1802.09525-1-11-3': 'Conversely, at the faint end ([MATH] mJy), we expect one source within [MATH].', '1802.09525-1-12-0': 'We plot the resulting chance coincidence probability contours as a function of flux density and [MATH] in the lower panel of Figure [REF].', '1802.09525-1-12-1': 'We denote the contours corresponding to [MATH] and [MATH].', '1802.09525-1-12-2': 'We also show the flux density of the persistent radio source associated with FRB121102 over the redshift range [MATH], as well as the nominal sensitivities for a number of existing and future radio observatories.', '1802.09525-1-12-3': 'We note that the FRB 121102 persistent source falls well below [MATH].', '1802.09525-1-12-4': 'While most existing facilities are sensitive to FRB121102-like persistent sources across a range of redshifts ([MATH]; see Table 1), detecting such a source at [MATH] (with an expected flux density of a few [MATH]Jy) is challenging even with the VLA.', '1802.09525-1-12-5': 'On the other hand, the advent of the ngVLA and SKA will push the achievable sensitivity to sub-[MATH]Jy levels, and hence the detection of such persistent sources to [MATH].', '1802.09525-1-13-0': 'At [MATH] mJy (i.e., FRB121102-like persistent sources at [MATH]), confident associations [MATH] require localizations of [MATH].', '1802.09525-1-13-1': 'At higher redshifts ([MATH]), localizations of [MATH] are required.', '1802.09525-1-13-2': 'While this level of localization is not feasible for most FRB search telescopes, it can be achieved using the VLA in an extended configuration, or with very long baseline interferometry (VLBI), as in the case of FRB121102, which was localized to [MATH] .', '1802.09525-1-13-3': 'However, a number of facilities, including the VLA, MeerKAT, ASKAP, and DSA-10, will be able to provide the [MATH] localizations required for robust associations with sub-mJy sources at moderate redshifts ([MATH]).', '1802.09525-1-13-4': 'Conversely, at [MATH], confident associations with sub-mJy sources are not feasible.', '1802.09525-1-13-5': 'This corresponds to the localization regime for CHIME, UTMOST-2D and Apertif (as well as single-dish telescopes).', '1802.09525-1-13-6': 'These facilities can provide robust associations only if the counterparts generally have flux densities of [MATH] mJy, but this would require some of the persistent radio counterparts to be much more luminous than the source associated with FRB121102.', '1802.09525-1-13-7': 'Finally, the poor localizations from single dish telescopes are not sufficient for robust associations with all but the brightest ([MATH]) sources.', '1802.09525-1-13-8': 'Although these telescopes may reveal additional bursts from FRBs in the case of repetitions, they will not be able to directly provide localizations that will lead to associations with persistent radio sources at any reasonable confidence level.', '1802.09525-1-14-0': 'In Table 1, we list a number of radio facilities designed to detect FRBs.', '1802.09525-1-14-1': 'We sort these by anticipated or known localization capability.', '1802.09525-1-14-2': 'We also list the flux density of a radio source in the respective localization region that would have [MATH] and 0.1, as well as the maximum redshift [MATH] out to which an association with an FRB121102-like persistent source can be made.', '1802.09525-1-14-3': 'These values are extracted directly from the probability contours in Figure [REF].', '1802.09525-1-14-4': 'We find that while the VLA, ASKAP, DSA-10, and MeerKAT are capable of probing these sources out to [MATH] (and [MATH] for VLBI), the large flux densities required for [MATH] for facilities with localizations of [MATH] preclude associations with these sources at [MATH].', '1802.09525-1-14-5': 'Given the expectation from radio source number counts and the sensitivity of current instruments, localizations below [MATH] will not improve the association confidence markedly, as existing facilities cannot achieve the [MATH]Jy levels required for [MATH].', '1802.09525-1-15-0': 'We note that for the same localization requirement of [MATH], optically-based host galaxy associations are impractical, requiring host luminosities several times brighter than [MATH] at redshifts below [MATH] (Paper I).', '1802.09525-1-15-1': 'Assuming that FRB121102 is representative of the FRB population as a whole and that we can expect continuum radio sources coincident with FRBs, the lower areal number density of radio sources on the sky enables more robust associations at a given localization precision; we demonstrate this further using optical and radio observations of the localization region of FRB170107 in [REF].', '1802.09525-1-16-0': '# Placing an Upper Limit on the Radio Luminosity', '1802.09525-1-17-0': 'We use the results of [REF] to investigate upper limits on the radio luminosity of an associated persistent source in the absence of a robust association.', '1802.09525-1-17-1': 'For a given localization radius, we determine the typical brightest source expected within the region (using the top-right panel of Figure [REF]).', '1802.09525-1-17-2': 'For a given intergalactic DM value (excluding the host and Milky Way contribution), we then estimate [MATH], using the DM-redshift relation of [CITATION] (see also ) and incorporating the uncertainty due to IGM inhomogeneities as parameterized by [CITATION].', '1802.09525-1-17-3': 'We follow the procedure described in detail in Paper I.', '1802.09525-1-17-4': 'In Figure [REF], we plot the maximum luminosity of a radio source as a function of [MATH] for a range of DM values.', '1802.09525-1-17-5': 'As in Figure [REF], we overlay the localization capabilities of the various FRB search facilities for reference.', '1802.09525-1-17-6': 'We also denote lines corresponding to the luminosity of star-forming galaxies with star formation rates of 1 and 100 [MATH], and we indicate the dividing line above which radio sources are almost exclusively AGN ([MATH]).', '1802.09525-1-17-7': 'Finally, we mark the achievable luminosity limits as a function of redshift for several facilities (ATCA, VLA, and the SKA/ngVLA).', '1802.09525-1-18-0': 'lcccc 5 0pt Radio facilities and their localization capabilities.', '1802.09525-1-19-0': 'Telescope & [MATH] & [MATH] & [MATH] & [MATH]', '1802.09525-1-20-0': '& [arcsec] & [mJy] & [mJy] & ([MATH])', '1802.09525-1-21-0': 'VLBA / EVN & [MATH] & [MATH] & [MATH] & [MATH]', '1802.09525-1-22-0': 'VLA & [MATH] & 0.01 & [MATH] & 0.6', '1802.09525-1-23-0': 'ASKAP & [MATH] & 0.03 & 0.002 & 0.4', '1802.09525-1-24-0': 'DSA-10 & [MATH] & 0.04 & 0.005 & 0.4', '1802.09525-1-25-0': 'MeerKAT & [MATH] & 0.2 & 0.03 & 0.2', '1802.09525-1-26-0': 'UTMOST-2D & [MATH] & 2 & 0.2 & 0.1', '1802.09525-1-27-0': 'Apertif[MATH] & [MATH] & 20 & 0.8 & [MATH]', '1802.09525-1-28-0': 'CHIME & [MATH] & 200 & 30 & [MATH]', '1802.09525-1-29-0': 'UTMOST & [MATH] & 400 & 60 & [MATH]', '1802.09525-1-30-0': 'Arecibo & [MATH] & 500 & 100 & [MATH]', '1802.09525-1-31-0': 'Parkes & [MATH] & 1000 & 400 & [MATH]', '1802.09525-1-32-0': 'GBT & [MATH] & 1000 & 400 & [MATH]', '1802.09525-1-33-0': 'Radio facilities capable of detecting FRBs, ordered by approximate localization capability.', '1802.09525-1-33-1': '[MATH] Limits are used to denote flux values which extend below the radio source number counts.', '1802.09525-1-33-2': '[MATH] In conjunction with LOFAR, the Apertif LOFAR Exploration of the Radio Transient Sky (ALERT) survey can provide more accurate (arcsecond) localizations (; see also http://alert.eu/).', '1802.09525-1-34-0': 'The results suggest that for DM [MATH], an upper limit comparable to the luminosity of the FRB121102 persistent radio source ([MATH]; ) can be placed if [MATH].', '1802.09525-1-34-1': 'These limits would also rule out star formation at the level of [MATH].', '1802.09525-1-34-2': 'A similar localization in the optical would only constrain the host galaxy luminosity to [MATH] (see Paper I, Figure 3).', '1802.09525-1-34-3': 'However, these limits are below the nominal sensitivities for existing radio telescopes.', '1802.09525-1-34-4': 'For example, although the VLA can provide the required localization precision, the [MATH] limiting flux density at 6 GHz corresponds to a luminosity upper limit of [MATH].', '1802.09525-1-34-5': 'For lower DMs, similar constraints on the luminosity of a quiescent radio source can be placed for larger localizations, i.e., [MATH] for DM [MATH].', '1802.09525-1-35-0': 'Although the upcoming SKA and ngVLA will provide the sensitivities required for meaningful upper limits, similar limits can be placed with existing facilities only for lower DM values (DM [MATH]).', '1802.09525-1-36-0': '# Rejecting Spurious Radio Associations', '1802.09525-1-37-0': 'While the probability of chance coincidence analysis in [REF] provides a direct measure of the confidence level of associating an FRB with a persistent radio source, we are also interested in exploring additional ways of rejecting spurious associations with unrelated radio emission due to AGN or star forming galaxies (i.e., the sources that make up the extragalactic radio source counts).', '1802.09525-1-37-1': 'First, we discuss a number of methods that can be used to identify radio emission due to star formation and separate these sources from FRB host candidates.', '1802.09525-1-37-2': 'Next, we discuss the distinction between AGN and putative FRB hosts, which is generally more complicated due to the wide range of AGN radio properties.', '1802.09525-1-38-0': '## Rejecting Star Forming Galaxies', '1802.09525-1-39-0': 'Motivated by the properties of the FRB121102 persistent radio counterpart, the ratio of radio-to-optical flux can be used to rule out radio emission due to star formation.', '1802.09525-1-39-1': 'This ratio, commonly defined as [MATH], where [MATH] is the [MATH]-band flux density, has previously been used as a discriminant between radio-loud AGN and star forming galaxies, where starburst galaxies have [MATH] (; ; ; ;; ).', '1802.09525-1-39-2': 'In Figure [REF], we plot the distribution of radio-to-optical flux ratios for radio sources from the literature (; ; ) and for the FRB121102 persistent source .', '1802.09525-1-39-3': 'In the case of FRB121102, the presence of bright radio emission, coupled with a faint optical host, leads to [MATH] and implies that the radio emission does not arise from star formation.', '1802.09525-1-39-4': 'Assuming that this is generally the case for FRB counterparts, we can set a threshold of [MATH] which effectively reduces the source counts by a factor of about two.', '1802.09525-1-40-0': 'The angular extent of the radio emission can also be used to distinguish star forming galaxies from compact FRB radio counterparts .', '1802.09525-1-40-1': 'The radio emission from the latter will have a scale of [MATH] pc, as in the case of FRB121102 ([MATH] pc; ), appearing as unresolved sources.', '1802.09525-1-40-2': 'Star forming regions at this scale would point to extremely high star formation rates per unit area ([MATH]; ).', '1802.09525-1-40-3': 'This level of star formation activity is expected in only the most extreme star forming regions .', '1802.09525-1-40-4': 'Typically, brightness temperatures of [MATH] K are expected for star formation .', '1802.09525-1-40-5': 'Follow-up VLBI observations can be used to determine the angular scale and brightness temperature, where resolved sources with [MATH] K will point to a star formation origin.', '1802.09525-1-41-0': '## Rejecting Active Galactic Nuclei', '1802.09525-1-42-0': 'The wide range of radio emission properties in AGN makes their rejection more challenging.', '1802.09525-1-42-1': 'For example, the radio-to-optical flux ratios for AGN exhibit a range of values.', '1802.09525-1-42-2': '[CITATION] claimed the first precise localization of an FRB based on the detection of a contemporaneous radio counterpart within the Parkes Telescope localization region.', '1802.09525-1-42-3': 'The counterpart was subsequently shown to be a variable AGN undergoing strong refractive scintillation .', '1802.09525-1-42-4': 'A search for counterparts in the FRB131104 localization region similarly revealed a coincident variable AGN .', '1802.09525-1-42-5': 'The approach we advocate here is to use various multi-wavelength data to argue whether a source is consistent with an AGN, and hence not securely associated with an FRB.', '1802.09525-1-43-0': 'Optical emission lines can be used to identify AGN using the Baldwin, Phillips [MATH] Terlevich (BPT) diagram .', '1802.09525-1-43-1': 'However, if we impose a high radio-to-optical flux ratio, then obtaining optical spectra of the associated host galaxies at [MATH] will be beyond the reach of most ground-based facilities.', '1802.09525-1-44-0': 'Sources that are precisely coincident with the nuclei of their host galaxies are more likely to have an AGN origin.', '1802.09525-1-44-1': 'However, this is less clear in the case of dwarf galaxies in which the radio source may be offset from the optical center of the galaxy and the optical center may not be well defined.', '1802.09525-1-44-2': 'The fraction of dwarf galaxies which host AGN is thought to be quite small, however, with an occupation fraction of 0.5[MATH]3% (; ).', '1802.09525-1-45-0': 'llccc 4 0pt Radio sources in the FRB170107 localization region.', '1802.09525-1-46-0': 'RA & DEC & Flux Density & [MATH] & [MATH]', '1802.09525-1-47-0': '& & [[MATH]]', '1802.09525-1-48-0': '11:23:14.739 & [MATH]04:58:20.47 & 86[MATH]15 & 20.4 & 0.5', '1802.09525-1-49-0': '11:23:24.431 & [MATH]04:59:33.21 & 112[MATH]15& 21.4 & 1.0', '1802.09525-1-50-0': '11:23:20.400 & [MATH]05:00:38.59 & 76[MATH]14 & 22.0 & 1.1', '1802.09525-1-51-0': '11:23:11.227 & [MATH]04:57:14.38 & 129[MATH]23 & [MATH] & [MATH]', '1802.09525-1-52-0': '11:23:26.385 & [MATH]04:57:41.65 & 158[MATH]16 & [MATH] & [MATH]', '1802.09525-1-53-0': 'Optical non-detections are listed as the [MATH] limiting magnitude.', '1802.09525-1-54-0': 'Finally, X-ray observations can be used to discern AGN activity, with X-ray luminosities of [MATH] indicative of AGN, although low-luminosity AGN may exhibit lower values.', '1802.09525-1-54-1': 'In the case of FRB121102, X-ray emission is not detected, corresponding to a [MATH] upper limit of [MATH] at 0.5[MATH]10 keV .', '1802.09525-1-54-2': 'The possibility of a low-luminosity, radio-loud AGN cannot be precluded from this result, however.', '1802.09525-1-54-3': 'In general, the absence of detectable X-ray emission may not be constraining, while a direct detection can be used to argue for an AGN origin.', '1802.09525-1-55-0': '# A Practical Example: FRB170107', '1802.09525-1-56-0': 'FRB170107 was the first FRB detected by the Australian Square Kilometer Array Pathfinder (ASKAP; ; ) using the phased array feed system.', '1802.09525-1-56-1': 'This led to the detection of the pulse in three separate beams, allowing for a localization region smaller than the size of an individual beam: the 90% confidence ellipse has semi-major and semi-minor axes of [MATH] with a position angle of [MATH].', '1802.09525-1-56-2': 'Motivated by the smaller size of the localization region compared to previous FRBs and the analysis presented above, we obtained radio and optical follow-up observations shortly after publication of the event to search for a host galaxy via a radio source with a large radio-to-optical flux ratio.', '1802.09525-1-57-0': '## Radio Observations', '1802.09525-1-58-0': 'We obtained radio observations with the Karl G. Jansky Very Large Array (VLA) on 2017 Jun 2, 146 days after the FRB.', '1802.09525-1-58-1': 'The observation lasted 30 minutes and the pointing center was [MATH], [MATH], matching the FRB position centroid reported by [CITATION].', '1802.09525-1-58-2': 'The correlator was configured in the standard "C band" wideband continuum mode using the 3-bit samplers, covering the full bandwidth of 4-8 GHz.', '1802.09525-1-58-3': 'Standard calibration techniques were used.', '1802.09525-1-58-4': 'The bandpass and flux density calibrator was 3C286 and the complex gain calibrator was the blazar PKS J[MATH].', '1802.09525-1-59-0': 'After this observation was conducted, we were provided with an updated localization probability map with a new centroid that differed from the published position (R. Shannon and K. Bannister, 2017, priv.', '1802.09525-1-59-1': 'comm.)', '1802.09525-1-59-2': 'On 2017 Jun 22 we conducted a second observation using the improved pointing center of [MATH], [MATH].', '1802.09525-1-59-3': 'The two positions are separated by almost exactly the half-width at half power (HWHP) of the VLA at 6 GHz.', '1802.09525-1-59-4': 'The other characteristics of this observation were identical to those of the first.', '1802.09525-1-60-0': 'We calibrated and imaged both observations within the CASA software environment using standard techniques.', '1802.09525-1-60-1': 'We imaged the fields separately, creating images of [MATH] pixels at a scale of 1[MATH] per pixel using multi-frequency synthesis and [MATH]-projection with 128 planes .', '1802.09525-1-60-2': 'The images of the first and second pointing centers achieved rms values of 8 and 10 [MATH] at the pointing center, respectively.', '1802.09525-1-61-0': 'We manually identified radio sources and measured their flux densities following a primary beam correction.', '1802.09525-1-61-1': 'We impose a [MATH] threshold and find five radio sources above this limit.', '1802.09525-1-61-2': 'The radio image from our second observation (2017 Jun 22) is shown in Figure [REF] and the identified sources are listed in Table 2.', '1802.09525-1-62-0': '## Optical Observations', '1802.09525-1-63-0': 'We obtained [MATH]-band observations covering the FRB localization region with IMACS on the 6.5 m Magellan Baade Telescope.', '1802.09525-1-63-1': 'We processed the images using standard procedures in IRAF and calibrated the resulting magnitude measurements using field stars in common with the Pan-STARRS1 [MATH] survey.', '1802.09525-1-63-2': 'We identify optical counterparts to three of the five radio sources (see Figure [REF]).', '1802.09525-1-63-3': 'We list the apparent [MATH]-band magnitudes (corrected for Galactic extinction; ) for each source in Table 2.', '1802.09525-1-63-4': 'The two optical non-detections correspond to a 3[MATH] limiting magnitude of 24.3 mag.', '1802.09525-1-64-0': '## Candidate Sources', '1802.09525-1-65-0': 'We manually identify five sources above our [MATH]Jy) confidence threshold.', '1802.09525-1-65-1': 'Based on the flux densities of the detected sources ([MATH]Jy), we estimate the expected number of sources above 75 [MATH]Jy from the radio source number counts.', '1802.09525-1-65-2': 'To compare our sources at 6 GHz to the expectation from the 1.4 GHz source counts, we assume a spectral index of [MATH] , corresponding to a limiting flux density of 210 [MATH]Jy at 1.4 GHz.', '1802.09525-1-65-3': 'From our analysis in [REF], we therefore expect to find [MATH] radio sources in the localization region, consistent with our results.', '1802.09525-1-65-4': 'The flux densities of the detected sources preclude a robust association, however.', '1802.09525-1-65-5': 'Namely, even for the brightest detected source, [MATH]; [MATH] requires a [MATH] Jy source.', '1802.09525-1-66-0': 'Motivated by the persistent radio source associated with FRB121102, we determine the ratio of radio-to-optical flux, [MATH], for each source.', '1802.09525-1-66-1': 'Of the five identified radio sources, two lack optical counterparts, corresponding to large radio-to-optical flux ratios ([MATH] and [MATH]), comparable to the value observed for the FRB121102 counterpart ([MATH]) and well in excess of expected values for star forming galaxies (see Figure [REF]).', '1802.09525-1-66-2': 'Conversely, the three sources with optical counterparts have [MATH], within the regime of star forming galaxies.', '1802.09525-1-66-3': 'We therefore consider the two radio sources with optical non-detections as potential hosts of FRB170107, given that we can rule out star formation as the source of the radio emission.', '1802.09525-1-66-4': 'We refer to the 158 and 129 [MATH]Jy sources as G1 and G2, respectively.', '1802.09525-1-67-0': 'Assuming a nominal host DM contribution of [MATH] (as in ), we estimate an upper bound on the redshift using the same approach as in [REF].', '1802.09525-1-67-1': 'We find that the DM-inferred redshift can be characterized as a normal distribution with [MATH] and a 68% confidence range of [MATH].', '1802.09525-1-67-2': 'The range of possible luminosities is therefore [MATH], comparable to the luminosity of the FRB121102 persistent source ([MATH]; ).', '1802.09525-1-67-3': 'The mean luminosities (assuming [MATH]) are [MATH] and [MATH] for G1 and G2, respectively.', '1802.09525-1-68-0': 'The [MATH] limiting magnitudes in the optical ([MATH]) correspond to [MATH] at [MATH] (or [MATH] at [MATH] and [MATH] at [MATH]).', '1802.09525-1-68-1': 'These are broadly consistent with the host galaxy of FRB121102 which has a luminosity of [MATH] .', '1802.09525-1-69-0': 'As both the radio and optical properties of our candidate sources are consistent with the properties of the FRB121102 host galaxy and radio counterpart, we argue that these sources are viable counterparts to FRB170107.', '1802.09525-1-69-1': 'The large localization precludes a robust association with either source, however, given that the expectation value for such a source is [MATH] ([REF]).', '1802.09525-1-69-2': 'Targeted follow-up observations with radio interferometers may reveal additional bursts from these sources, leading to an unambiguous host identification.', '1802.09525-1-70-0': '# Conclusions', '1802.09525-1-71-0': 'Motivated by the persistent radio source associated with the repeating FRB121102, we explored the likelihood of identifying persistent radio sources associated with FRBs as a function of their flux density and the FRB localization radius, assuming that all FRBs are associated with such sources.', '1802.09525-1-71-1': 'We also quantified the limits that can be placed on a radio counterpart in the absence of a robust association.', '1802.09525-1-71-2': 'Finally, we applied our analysis to the case of FRB170107.', '1802.09525-1-71-3': 'Our main results can be summarized as follows:', '1802.09525-1-72-0': 'Combined radio and optical follow-up observations of FRB localization regions can provide a powerful tool for identifying host associations.', '1802.09525-1-72-1': 'If FRB121102 is representative of the FRB population, candidate sources can be identified on the basis of their radio-to-optical flux ratio, and targeted follow-up observations can be used to search for repetitions directly from these sources.', '1802.09525-1-72-2': 'Although continued monitoring with ASKAP or single dish telescopes may reveal additional bursts from FRBs, these will not provide unambiguous host identifications.', '1802.09525-1-72-3': 'Detections of additional bursts with targeted phased-array observations with the VLA would not only demonstrate repetitions, confirming the existence of a class of repeating FRBs, but would also immediately lead to a host identification.', '1802.09525-1-72-4': 'This technique may therefore provide a novel framework for identifying counterparts and localizing future FRBs.', '1802.09525-1-72-5': 'However, in the absence of repetitions or very precise localizations, definitive associations remain challenging, particularly due to the broad region of parameter space occupied by AGN.'}

{'1802.09525-2-0-0': 'The discovery of a repeating fast radio burst (FRB) has led to the first precise localization, an association with a dwarf galaxy, and the identification of a coincident persistent radio source.', '1802.09525-2-0-1': 'However, further localizations are required to determine the nature of FRBs, the sources powering them, and the possibility of multiple populations.', '1802.09525-2-0-2': 'Here we investigate the use of associated persistent radio sources to establish FRB counterparts, taking into account the localization area and the persistent source flux density.', '1802.09525-2-0-3': 'Due to the lower areal number density of radio sources compared to faint optical sources, robust associations can be achieved for less precise localizations as compared to direct optical host galaxy associations.', '1802.09525-2-0-4': 'For generally larger localizations which preclude robust associations, the number of candidate hosts can be reduced based on the ratio of radio-to-optical brightness.', '1802.09525-2-0-5': 'We find that confident associations with sources having a flux density of [MATH]1 mJy, comparable to the luminosity of the persistent source associated with FRB121102 over the redshift range [MATH], require FRB localizations of [MATH].', '1802.09525-2-0-6': 'We demonstrate that even in the absence of a robust association, constraints can be placed on the luminosity of an associated radio source as a function of localization and DM.', '1802.09525-2-0-7': 'We find that for DM [MATH], an upper limit comparable to the luminosity of the FRB121102 persistent source can be placed if the localization is [MATH].', '1802.09525-2-0-8': 'We apply our analysis to the case of the ASKAP FRB170107, using optical and radio observations of the localization region.', '1802.09525-2-0-9': 'We identify two candidate hosts based on a ratio of radio-to-optical brightness of [MATH].', '1802.09525-2-0-10': 'We find that if one of these is indeed associated with FRB170107, the resulting radio luminosity ([MATH], as constrained from the DM value) is comparable to the luminosity of the FRB121102 persistent source ([MATH]).', '1802.09525-2-1-0': '# Introduction', '1802.09525-2-2-0': 'Fast radio bursts (FRBs) are extremely bright, millisecond-duration pulses of coherent radio emission, with large dispersion measures (DMs) that exceed typical Galactic values, hence pointing to an extragalactic origin.', '1802.09525-2-2-1': 'Since the discovery of the first FRB in archival data , roughly 30 additional FRBs have been detected, both in archival and real-time searches (; ; ; ; ; ).', '1802.09525-2-2-2': "Despite a growing number of FRB detections, the lack of precise localizations (typically [MATH]) has led to a wide range of suggested progenitor systems: giant pulses and magnetar flares (; ; ' ), mergers of compact objects (; ), the collapse of supramassive neutron stars to black holes (), and emission from rapidly spinning magnetars (; ).", '1802.09525-2-3-0': 'The discovery of the repeating FRB121102 (; ) led to the first precise localization and the identification of a low metallicity dwarf star forming host galaxy at [MATH] , with properties remarkably similar to the host galaxies of both long-duration gamma-ray bursts (LGRBs) and hydrogen-poor superluminous supernovae (SLSN) .', '1802.09525-2-3-1': 'Radio observations with the European VLBI Network (EVN) revealed the presence of a compact, persistent radio source with a projected linear size of [MATH] pc, co-located with the bursts to [MATH] mas .', '1802.09525-2-3-2': 'The large angular offset between the radio source and the optical center of the galaxy (170[MATH]300 mas; ) generally argue against an active galactic nuclei (AGN) origin, although this premise is less clear in the case of dwarf galaxies.', '1802.09525-2-3-3': 'Similarly, the resulting ratio of radio-to-optical brightness for the persistent source and host galaxy ([MATH]) is much larger than expected for a star formation origin.', '1802.09525-2-3-4': 'This has led to the suggestion that the radio source may be a nebula associated with FRB121102 .', '1802.09525-2-3-5': 'More recently, the discovery of a large rotation measure with [MATH] variations on half-year timescales suggests the presence of a dynamic and highly magnetized environment surrounding the source of the FRB .', '1802.09525-2-4-0': 'The properties of the host galaxy, the repeating nature of the bursts, and the persistent radio source are consistent with a model in which FRBs are powered by young, millisecond magnetars (; see also ; ), the same engines that have been argued to power SLSN and perhaps LGRBs (; ).', '1802.09525-2-4-1': 'In this model, individual bursts are emitted via the dissipation of rotational or magnetic energy, while the quiescent radio emission is due to the magnetar wind nebula or the shock interaction between the supernova ejecta and the surrounding circumstellar medium (; ).', '1802.09525-2-5-0': 'In this broad scenario, we expect that FRBs should be located preferentially in dwarf galaxies and be coincident with quiescent radio sources similar to the one observed in FRB121102 .', '1802.09525-2-5-1': '[CITATION] demonstrate that [MATH] FRB localizations will be sufficient to test various formation channels.', '1802.09525-2-5-2': 'Given the high event rate for FRBs ([MATH] above 1 Jy; ) and the improved sensitivities of upcoming telescopes and surveys, the number of detected bursts is expected to increase drastically over the next few years as new radio facilities come on-line.', '1802.09525-2-5-3': 'However, due to a wide range of localization capabilities, precise localizations will continue to be a challenge.', '1802.09525-2-6-0': 'In [CITATION] ([CITATION]; hereafter, Paper I), we explored how robustly FRBs with different localization regions can be associated with host galaxies based on the optical brightness of the galaxy.', '1802.09525-2-6-1': 'Due to the large areal number density of faint optical sources, we showed that sub-arcsecond localizations are required for confident associations with dwarf galaxies at [MATH], whereas localizations of up to [MATH] may suffice if the hosts are instead generally [MATH] galaxies, where [MATH] denotes the characteristic luminosity of a bright galaxy.', '1802.09525-2-7-0': 'Here we explore the likelihood of constraining similar associations with radio sources, motivated by the persistent radio source associated with FRB121102 and assuming that all FRBs are coincident with such sources, potentially with a resulting high radio-to-optical flux ratio.', '1802.09525-2-7-1': 'We also explore what limits can be placed on the presence and luminosity of a persistent radio source (as a function of localization area) even in the absence of a robust association.', '1802.09525-2-7-2': 'We assess our results in the context of existing and upcoming radio facilities.', '1802.09525-2-7-3': 'As an example of our method, we present radio (VLA) and optical (Magellan) observations of the localization region of the ASKAP FRB170107 .', '1802.09525-2-7-4': 'We search for radio counterparts in the region using the radio-to-optical flux ratio to identify potential candidates.', '1802.09525-2-8-0': 'The paper is structured as follows.', '1802.09525-2-8-1': 'We begin by using radio source number counts to determine the probability of chance coincidence for an FRB and a persistent radio source as a function of localization region and persistent source flux density ([REF]).', '1802.09525-2-8-2': 'Next, in [REF] we use these results to constrain the upper limit on the radio luminosity of a source in the absence of a robust association.', '1802.09525-2-8-3': 'In [REF], we discuss various techniques which may further aid in the identification of associated counterparts.', '1802.09525-2-8-4': 'We present an example of our approach in [REF], using radio and optical observations of the localization region of FRB170107.', '1802.09525-2-8-5': 'Finally, we discuss our results in the context of existing and planned FRB search facilities in [REF].', '1802.09525-2-9-0': '# Assessing Probability of Chance Coincidence with Radio Source Counts', '1802.09525-2-10-0': 'We assess the likelihood that an FRB is associated with a persistent radio source by calculating the probability of chance coincidence for a source of a given brightness within a localization region.', '1802.09525-2-10-1': 'We use the 1.4 GHz Euclidean-normalized differential source counts presented in [CITATION] to calculate the number density of radio sources above some limiting flux density, [MATH]; see top left panel of Figure [REF].', '1802.09525-2-10-2': 'These source counts are measured over the range [MATH]Jy [MATH] 100 Jy and extrapolated to lower flux densities imposing an evolutionary model for the spectral luminosity function of extragalactic radio sources (; ).', '1802.09525-2-10-3': 'The source counts are comprised of both AGN and star forming galaxies, with the latter dominating the counts below [MATH] mJy , although radio-quiet AGN have been shown to comprise a sizable fraction of the sub-mJy population (; ; ; ).', '1802.09525-2-10-4': 'We assume a Poisson distribution of radio sources across the sky and calculate the chance coincidence probability as: [EQUATION]', '1802.09525-2-10-5': 'The localization region, [MATH], is parameterized by [MATH], where [MATH] is the [MATH] localization radius of the FRB.', '1802.09525-2-11-0': 'Using the integrated source counts, we plot in the top right panel of Figure [REF] the number of radio sources above some limiting flux density [i.e., [MATH])] as a function of [MATH].', '1802.09525-2-11-1': 'We plot the results over the range [MATH]Jy to 1 Jy, in order of magnitude increments.', '1802.09525-2-11-2': 'We find an expectation value of about one source at [MATH] mJy within the typical localization regions of single-dish telescopes ([MATH]).', '1802.09525-2-11-3': 'Conversely, at the faint end ([MATH] mJy), we expect one source within [MATH].', '1802.09525-2-12-0': 'We plot the resulting chance coincidence probability contours as a function of flux density and [MATH] in the lower panel of Figure [REF].', '1802.09525-2-12-1': 'We denote the contours corresponding to [MATH] and [MATH].', '1802.09525-2-12-2': 'We also show the flux density of the persistent radio source associated with FRB121102 over the redshift range [MATH], as well as the nominal sensitivities for a number of existing and future radio observatories.', '1802.09525-2-12-3': 'We note that the FRB 121102 persistent source falls well below [MATH].', '1802.09525-2-12-4': 'While most existing facilities are sensitive to FRB121102-like persistent sources across a range of redshifts ([MATH]; see Table 1), detecting such a source at [MATH] (with an expected flux density of a few [MATH]Jy) is challenging even with the VLA.', '1802.09525-2-12-5': 'On the other hand, the advent of the ngVLA and SKA will push the achievable sensitivity to sub-[MATH]Jy levels, and hence the detection of such persistent sources to [MATH].', '1802.09525-2-13-0': 'At [MATH] mJy (i.e., FRB121102-like persistent sources at [MATH]), confident associations [MATH] require localizations of [MATH].', '1802.09525-2-13-1': 'At higher redshifts ([MATH]), localizations of [MATH] are required.', '1802.09525-2-13-2': 'While this level of localization is not feasible for most FRB search telescopes, it can be achieved using the VLA in an extended configuration, or with very long baseline interferometry (VLBI), as in the case of FRB121102, which was localized to [MATH] mas .', '1802.09525-2-13-3': 'However, a number of facilities, including the VLA, MeerKAT, ASKAP, and DSA-10, will be able to provide the [MATH] localizations required for robust associations with sub-mJy sources at moderate redshifts ([MATH]).', '1802.09525-2-13-4': 'Conversely, at [MATH], confident associations with sub-mJy sources are not feasible.', '1802.09525-2-13-5': 'This corresponds to the localization regime for CHIME, UTMOST-2D and Apertif (as well as single-dish telescopes).', '1802.09525-2-13-6': 'These facilities can provide robust associations only if the counterparts generally have flux densities of [MATH] mJy, but this would require some of the persistent radio counterparts to be much more luminous than the source associated with FRB121102.', '1802.09525-2-13-7': 'Finally, the poor localizations from single dish telescopes are not sufficient for robust associations with all but the brightest ([MATH]) sources.', '1802.09525-2-13-8': 'Although these telescopes may reveal additional bursts from FRBs in the case of repetitions, they will not be able to directly provide localizations that will lead to associations with persistent radio sources at any reasonable confidence level.', '1802.09525-2-14-0': 'In Table 1, we list a number of radio facilities designed to detect FRBs.', '1802.09525-2-14-1': 'We sort these by anticipated or known localization capability.', '1802.09525-2-14-2': 'We also list the flux density of a radio source in the respective localization region that would have [MATH] and 0.1, as well as the maximum redshift [MATH] out to which an association with an FRB121102-like persistent source can be made.', '1802.09525-2-14-3': 'These values are extracted directly from the probability contours in Figure [REF].', '1802.09525-2-14-4': 'We find that while the VLA, ASKAP, DSA-10, and MeerKAT are capable of probing these sources out to [MATH] (and [MATH] for VLBI), the large flux densities required for [MATH] for facilities with localizations of [MATH] preclude associations with these sources at [MATH].', '1802.09525-2-14-5': 'Given the expectation from radio source number counts and the sensitivity of current instruments, localizations below [MATH] will not improve the association confidence markedly, as existing facilities cannot achieve the [MATH]Jy levels required for [MATH].', '1802.09525-2-15-0': 'We note that for the same localization requirement of [MATH], optically-based host galaxy associations are impractical, requiring host luminosities several times brighter than [MATH] at redshifts below [MATH] (Paper I).', '1802.09525-2-15-1': 'Assuming that FRB121102 is representative of the FRB population as a whole and that we can expect continuum radio sources coincident with FRBs, the lower areal number density of radio sources on the sky enables more robust associations at a given localization precision; we demonstrate this further using optical and radio observations of the localization region of FRB170107 in [REF].', '1802.09525-2-16-0': '# Placing an Upper Limit on the Radio Luminosity', '1802.09525-2-17-0': 'We use the results of [REF] to investigate upper limits on the radio luminosity of an associated persistent source in the absence of a robust association.', '1802.09525-2-17-1': 'For a given localization radius, we determine the typical brightest source expected within the region (using the top-right panel of Figure [REF]).', '1802.09525-2-17-2': 'For a given intergalactic DM value (excluding the host and Milky Way contribution), we then estimate [MATH], using the DM-redshift relation of [CITATION] (see also ) and incorporating the uncertainty due to IGM inhomogeneities as parameterized by [CITATION].', '1802.09525-2-17-3': 'We follow the procedure described in detail in Paper I.', '1802.09525-2-17-4': 'In Figure [REF], we plot the maximum luminosity of a radio source as a function of [MATH] for a range of DM values.', '1802.09525-2-17-5': 'As in Figure [REF], we overlay the localization capabilities of the various FRB search facilities for reference.', '1802.09525-2-17-6': 'We also denote lines corresponding to the luminosity of star-forming galaxies with star formation rates of 1 and 100 [MATH], and we indicate the dividing line above which radio sources are almost exclusively AGN ([MATH]).', '1802.09525-2-17-7': 'Finally, we mark the achievable luminosity limits as a function of redshift for several facilities (ATCA, VLA, and the SKA/ngVLA).', '1802.09525-2-18-0': 'lcccc 5 0pt Radio facilities and their localization capabilities.', '1802.09525-2-19-0': 'Telescope & [MATH] & [MATH] & [MATH] & [MATH]', '1802.09525-2-20-0': '& [arcsec] & [mJy] & [mJy] & ([MATH])', '1802.09525-2-21-0': 'VLBA / EVN & [MATH] & [MATH] & [MATH] & [MATH]', '1802.09525-2-22-0': 'VLA & [MATH] & 0.01 & [MATH] & 0.6', '1802.09525-2-23-0': 'ASKAP & [MATH] & 0.03 & 0.002 & 0.4', '1802.09525-2-24-0': 'DSA-10 & [MATH] & 0.04 & 0.005 & 0.4', '1802.09525-2-25-0': 'MeerKAT & [MATH] & 0.2 & 0.03 & 0.2', '1802.09525-2-26-0': 'UTMOST-2D & [MATH] & 2 & 0.2 & 0.1', '1802.09525-2-27-0': 'Apertif[MATH] & [MATH] & 20 & 0.8 & [MATH]', '1802.09525-2-28-0': 'CHIME & [MATH] & 200 & 30 & [MATH]', '1802.09525-2-29-0': 'UTMOST & [MATH] & 400 & 60 & [MATH]', '1802.09525-2-30-0': 'Arecibo & [MATH] & 500 & 100 & [MATH]', '1802.09525-2-31-0': 'Parkes & [MATH] & 1000 & 400 & [MATH]', '1802.09525-2-32-0': 'GBT & [MATH] & 1000 & 400 & [MATH]', '1802.09525-2-33-0': 'Radio facilities capable of detecting FRBs, ordered by approximate localization capability.', '1802.09525-2-33-1': '[MATH] Limits are used to denote flux values which extend below the radio source number counts.', '1802.09525-2-33-2': '[MATH] In conjunction with LOFAR, the Apertif LOFAR Exploration of the Radio Transient Sky (ALERT) survey can provide more accurate (arcsecond) localizations (; see also http://alert.eu/).', '1802.09525-2-34-0': 'The results suggest that for DM [MATH], an upper limit comparable to the luminosity of the FRB121102 persistent radio source ([MATH]; ) can be placed if [MATH].', '1802.09525-2-34-1': 'These limits would also rule out star formation at the level of [MATH].', '1802.09525-2-34-2': 'A similar localization in the optical would only constrain the host galaxy luminosity to [MATH] (see Paper I, Figure 3).', '1802.09525-2-34-3': 'However, these limits are below the nominal sensitivities for existing radio telescopes.', '1802.09525-2-34-4': 'For example, although the VLA can provide the required localization precision, the [MATH] limiting flux density at 6 GHz corresponds to a luminosity upper limit of [MATH].', '1802.09525-2-34-5': 'For lower DMs, similar constraints on the luminosity of a quiescent radio source can be placed for larger localizations, i.e., [MATH] for DM [MATH].', '1802.09525-2-35-0': 'Although the upcoming SKA and ngVLA will provide the sensitivities required for meaningful upper limits, similar limits can be placed with existing facilities only for lower DM values (DM [MATH]).', '1802.09525-2-36-0': '# Rejecting Spurious Radio Associations', '1802.09525-2-37-0': 'While the probability of chance coincidence analysis in [REF] provides a direct measure of the confidence level of associating an FRB with a persistent radio source, we are also interested in exploring additional ways of rejecting spurious associations with unrelated radio emission due to AGN or star forming galaxies (i.e., the sources that make up the extragalactic radio source counts).', '1802.09525-2-37-1': 'First, we discuss a number of methods that can be used to identify radio emission due to star formation and separate these sources from FRB host candidates.', '1802.09525-2-37-2': 'Next, we discuss the distinction between AGN and putative FRB hosts, which is generally more complicated due to the wide range of AGN radio properties.', '1802.09525-2-38-0': '## Rejecting Star Forming Galaxies', '1802.09525-2-39-0': 'Motivated by the properties of the FRB121102 persistent radio counterpart, the ratio of radio-to-optical flux can be used to rule out radio emission due to star formation.', '1802.09525-2-39-1': 'This ratio, commonly defined as [MATH], where [MATH] is the [MATH]-band flux density, has previously been used as a discriminant between radio-loud AGN and star forming galaxies, where starburst galaxies have [MATH] (; ; ; ;; ).', '1802.09525-2-39-2': 'In Figure [REF], we plot the distribution of radio-to-optical flux ratios for radio sources from the literature (; ; ) and for the FRB121102 persistent source .', '1802.09525-2-39-3': 'In the case of FRB121102, the presence of bright radio emission, coupled with a faint optical host, leads to [MATH] and implies that the radio emission does not arise from star formation.', '1802.09525-2-39-4': 'Assuming that this is generally the case for FRB counterparts, we can set a threshold of [MATH] which effectively reduces the source counts by a factor of about two.', '1802.09525-2-40-0': 'The angular extent of the radio emission can also be used to distinguish star forming galaxies from compact FRB radio counterparts .', '1802.09525-2-40-1': 'The radio emission from the latter will have a scale of [MATH] pc, as in the case of FRB121102 ([MATH] pc; ), appearing as unresolved sources.', '1802.09525-2-40-2': 'Star forming regions at this scale would point to extremely high star formation rates per unit area ([MATH]; ).', '1802.09525-2-40-3': 'This level of star formation activity is expected in only the most extreme star forming regions .', '1802.09525-2-40-4': 'This argument is borne out by VLBI observations, which show that about [MATH] of radio sources brighter than 1 mJy are resolved, thereby implicating extended regions of star formation or FRII AGN .', '1802.09525-2-40-5': 'Thus, VLBI follow-up of any candidate FRB counterparts can be used to reject extended radio sources.', '1802.09525-2-41-0': '## Rejecting Active Galactic Nuclei', '1802.09525-2-42-0': 'The wide range of radio emission properties in AGN makes their rejection more challenging.', '1802.09525-2-42-1': 'For example, the radio-to-optical flux ratios for AGN exhibit a range of values.', '1802.09525-2-42-2': '[CITATION] claimed the first precise localization of an FRB based on the detection of a contemporaneous radio counterpart within the Parkes Telescope localization region.', '1802.09525-2-42-3': 'The counterpart was subsequently shown to be a variable AGN undergoing strong refractive scintillation .', '1802.09525-2-42-4': 'A search for counterparts in the FRB131104 localization region similarly revealed a coincident variable AGN .', '1802.09525-2-42-5': 'The approach we advocate here is to use various multi-wavelength data to argue whether a source is consistent with an AGN, and hence not securely associated with an FRB.', '1802.09525-2-43-0': 'Optical emission lines can be used to identify AGN using the Baldwin, Phillips [MATH] Terlevich (BPT) diagram .', '1802.09525-2-43-1': 'However, if we impose a high radio-to-optical flux ratio, then obtaining optical spectra of the associated host galaxies at [MATH] will be beyond the reach of most ground-based facilities.', '1802.09525-2-44-0': 'Sources that are precisely coincident with the nuclei of their host galaxies are more likely to have an AGN origin.', '1802.09525-2-44-1': 'However, this is less clear in the case of dwarf galaxies in which the radio source may be offset from the optical center of the galaxy and the optical center may not be well defined.', '1802.09525-2-44-2': 'The fraction of dwarf galaxies which host AGN is thought to be quite small, however, with an occupation fraction of 0.5[MATH]3% (; ).', '1802.09525-2-45-0': 'llccc 4 0pt Radio sources in the FRB170107 localization region.', '1802.09525-2-46-0': 'RA & DEC & Flux Density & [MATH] & [MATH]', '1802.09525-2-47-0': '& & [[MATH]]', '1802.09525-2-48-0': '11:23:14.739 & [MATH]04:58:20.47 & 86[MATH]15 & 20.4 & 0.5', '1802.09525-2-49-0': '11:23:24.431 & [MATH]04:59:33.21 & 112[MATH]15& 21.4 & 1.0', '1802.09525-2-50-0': '11:23:20.400 & [MATH]05:00:38.59 & 76[MATH]14 & 22.0 & 1.1', '1802.09525-2-51-0': '11:23:11.227 & [MATH]04:57:14.38 & 129[MATH]23 & [MATH] & [MATH]', '1802.09525-2-52-0': '11:23:26.385 & [MATH]04:57:41.65 & 158[MATH]16 & [MATH] & [MATH]', '1802.09525-2-53-0': 'Optical non-detections are listed as the [MATH] limiting magnitude.', '1802.09525-2-54-0': 'Finally, X-ray observations can be used to discern AGN activity, with X-ray luminosities of [MATH] indicative of AGN, although low-luminosity AGN may exhibit lower values.', '1802.09525-2-54-1': 'In the case of FRB121102, X-ray emission is not detected, corresponding to a [MATH] upper limit of [MATH] at 0.5[MATH]10 keV .', '1802.09525-2-54-2': 'The possibility of a low-luminosity, radio-loud AGN cannot be precluded from this result, however.', '1802.09525-2-54-3': 'In general, the absence of detectable X-ray emission may not be constraining, while a direct detection can be used to argue for an AGN origin.', '1802.09525-2-55-0': '# A Practical Example: FRB170107', '1802.09525-2-56-0': 'FRB170107 was the first FRB detected by the Australian Square Kilometer Array Pathfinder (ASKAP; ; ) using the phased array feed system.', '1802.09525-2-56-1': 'This led to the detection of the pulse in three separate beams, allowing for a localization region smaller than the size of an individual beam: the 90% confidence ellipse has semi-major and semi-minor axes of [MATH] with a position angle of [MATH].', '1802.09525-2-56-2': 'Motivated by the smaller size of the localization region compared to previous FRBs and the analysis presented above, we obtained radio and optical follow-up observations shortly after publication of the event to search for a host galaxy via a radio source with a large radio-to-optical flux ratio.', '1802.09525-2-57-0': '## Radio Observations', '1802.09525-2-58-0': 'We obtained radio observations with the Karl G. Jansky Very Large Array (VLA) on 2017 Jun 2, 146 days after the FRB.', '1802.09525-2-58-1': 'The observation lasted 30 minutes and the pointing center was [MATH], [MATH], matching the FRB position centroid reported by [CITATION].', '1802.09525-2-58-2': 'The correlator was configured in the standard "C band" wideband continuum mode using the 3-bit samplers, covering the full bandwidth of 4-8 GHz.', '1802.09525-2-58-3': 'Standard calibration techniques were used.', '1802.09525-2-58-4': 'The bandpass and flux density calibrator was 3C286 and the complex gain calibrator was the blazar PKS J[MATH].', '1802.09525-2-59-0': 'After this observation was conducted, we were provided with an updated localization probability map with a new centroid that differed from the published position (R. Shannon and K. Bannister, 2017, priv.', '1802.09525-2-59-1': 'comm.)', '1802.09525-2-59-2': 'On 2017 Jun 22 we conducted a second observation using the improved pointing center of [MATH], [MATH].', '1802.09525-2-59-3': 'The two positions are separated by almost exactly the half-width at half power (HWHP) of the VLA at 6 GHz.', '1802.09525-2-59-4': 'The other characteristics of this observation were identical to those of the first.', '1802.09525-2-60-0': 'We calibrated and imaged both observations within the CASA software environment using standard techniques.', '1802.09525-2-60-1': 'We imaged the fields separately, creating images of [MATH] pixels at a scale of 1[MATH] per pixel using multi-frequency synthesis and [MATH]-projection with 128 planes .', '1802.09525-2-60-2': 'The images of the first and second pointing centers achieved rms values of 8 and 10 [MATH] at the pointing center, respectively.', '1802.09525-2-61-0': 'We visually identified radio sources and measured their flux densities following a primary beam correction.', '1802.09525-2-61-1': 'Due to the decrease in sensitivity away from the pointing center, we impose a 75 [MATH]Jy threshold and find five radio sources above this limit.', '1802.09525-2-61-2': 'The radio image from our second observation (2017 Jun 22) is shown in Figure [REF] and the identified sources are listed in Table 2.', '1802.09525-2-62-0': '## Optical Observations', '1802.09525-2-63-0': 'We obtained [MATH]-band observations covering the FRB localization region with IMACS on the 6.5 m Magellan Baade Telescope.', '1802.09525-2-63-1': 'We processed the images using standard procedures in IRAF and calibrated the resulting magnitude measurements using field stars in common with the Pan-STARRS1 [MATH] survey.', '1802.09525-2-63-2': 'We identify optical counterparts to three of the five radio sources (see Figure [REF]).', '1802.09525-2-63-3': 'We list the apparent [MATH]-band magnitudes (corrected for Galactic extinction; ) for each source in Table 2.', '1802.09525-2-63-4': 'The two optical non-detections correspond to a 3[MATH] limiting magnitude of 24.3 mag.', '1802.09525-2-64-0': '## Candidate Sources', '1802.09525-2-65-0': 'Based on the flux densities of the detected sources ([MATH]Jy), we estimate the expected number of sources above 75 [MATH]Jy from the radio source number counts.', '1802.09525-2-65-1': 'To compare our sources at 6 GHz to the expectation from the 1.4 GHz source counts, we assume a spectral index of [MATH] , corresponding to an effective limiting flux density of 210 [MATH]Jy at 1.4 GHz.', '1802.09525-2-65-2': 'From our analysis in [REF], we therefore expect to find [MATH] radio sources in the localization region, consistent with our results.', '1802.09525-2-65-3': 'The flux densities of the detected sources preclude a robust association, however.', '1802.09525-2-65-4': 'Namely, even for the brightest detected source, [MATH]; [MATH] requires a [MATH] Jy source.', '1802.09525-2-66-0': 'Motivated by the persistent radio source associated with FRB121102, we determine the ratio of radio-to-optical flux, [MATH], for each source.', '1802.09525-2-66-1': 'Of the five identified radio sources, two lack optical counterparts, corresponding to large radio-to-optical flux ratios ([MATH] and [MATH]), comparable to the value observed for the FRB121102 counterpart ([MATH]) and well in excess of expected values for star forming galaxies (see Figure [REF]).', '1802.09525-2-66-2': 'Conversely, the three sources with optical counterparts have [MATH], within the regime of star forming galaxies.', '1802.09525-2-66-3': 'We therefore consider the two radio sources with optical non-detections as potential hosts of FRB170107, given that we can rule out star formation as the source of the radio emission.', '1802.09525-2-66-4': 'We refer to the 158 and 129 [MATH]Jy sources as G1 and G2, respectively.', '1802.09525-2-67-0': 'Assuming a nominal host DM contribution of [MATH] (as in ), we estimate an upper bound on the redshift using the same approach as in [REF].', '1802.09525-2-67-1': 'We find that the DM-inferred redshift can be characterized as a normal distribution with [MATH] and a 68% confidence range of [MATH].', '1802.09525-2-67-2': 'The range of possible luminosities is therefore [MATH], comparable to the luminosity of the FRB121102 persistent source ([MATH]; ).', '1802.09525-2-67-3': 'The mean luminosities (assuming [MATH]) are [MATH] and [MATH] for G1 and G2, respectively.', '1802.09525-2-68-0': 'The [MATH] limiting magnitudes in the optical ([MATH]) correspond to [MATH] at [MATH] (or [MATH] at [MATH] and [MATH] at [MATH]).', '1802.09525-2-68-1': 'These are broadly consistent with the host galaxy of FRB121102 which has a luminosity of [MATH] .', '1802.09525-2-69-0': 'As both the radio and optical properties of our candidate sources are consistent with the properties of the FRB121102 host galaxy and radio counterpart, we argue that these sources are viable counterparts to FRB170107.', '1802.09525-2-69-1': 'The large localization precludes a robust association with either source, however, given that the expectation value for such a source is [MATH] ([REF]).', '1802.09525-2-69-2': 'Targeted follow-up observations with radio interferometers may reveal additional bursts from these sources, leading to an unambiguous host identification.', '1802.09525-2-70-0': '# Conclusions', '1802.09525-2-71-0': 'Motivated by the persistent radio source associated with the repeating FRB121102, we explored the likelihood of identifying persistent radio sources associated with FRBs as a function of their flux density and the FRB localization radius, assuming that all FRBs are associated with such sources.', '1802.09525-2-71-1': 'We also quantified the limits that can be placed on a radio counterpart in the absence of a robust association.', '1802.09525-2-71-2': 'Finally, we applied our analysis to the case of FRB170107.', '1802.09525-2-71-3': 'Our main results can be summarized as follows:', '1802.09525-2-72-0': 'Combined radio and optical follow-up observations of FRB localization regions can provide a powerful tool for identifying host associations.', '1802.09525-2-72-1': 'If FRB121102 is representative of the FRB population, candidate sources can be identified on the basis of their radio-to-optical flux ratio, and targeted follow-up observations can be used to search for repetitions directly from these sources.', '1802.09525-2-72-2': 'Although continued monitoring with ASKAP or single dish telescopes may reveal additional bursts from FRBs, these will not provide unambiguous host identifications.', '1802.09525-2-72-3': 'Detections of additional bursts with targeted phased-array observations with the VLA would not only demonstrate repetitions, confirming the existence of a class of repeating FRBs, but would also immediately lead to a host identification.', '1802.09525-2-72-4': 'This technique may therefore provide a novel framework for identifying counterparts and localizing future FRBs.', '1802.09525-2-72-5': 'However, in the absence of repetitions or very precise localizations, definitive associations remain challenging, particularly due to the broad region of parameter space occupied by AGN.'}

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['1802.09525-1-18-0', '1802.09525-1-19-0', '1802.09525-1-20-0', '1802.09525-1-21-0', '1802.09525-1-22-0', '1802.09525-1-23-0', '1802.09525-1-24-0', '1802.09525-1-25-0', '1802.09525-1-26-0', '1802.09525-1-27-0', '1802.09525-1-28-0', '1802.09525-1-29-0', '1802.09525-1-30-0', '1802.09525-1-31-0', '1802.09525-1-32-0', '1802.09525-1-45-0', '1802.09525-1-46-0', '1802.09525-1-47-0', '1802.09525-1-48-0', '1802.09525-1-49-0', '1802.09525-1-50-0', '1802.09525-1-51-0', '1802.09525-1-52-0', '1802.09525-1-53-0', '1802.09525-1-59-1', '1802.09525-1-71-3', '1802.09525-2-18-0', '1802.09525-2-19-0', '1802.09525-2-20-0', '1802.09525-2-21-0', '1802.09525-2-22-0', '1802.09525-2-23-0', '1802.09525-2-24-0', '1802.09525-2-25-0', '1802.09525-2-26-0', '1802.09525-2-27-0', '1802.09525-2-28-0', '1802.09525-2-29-0', '1802.09525-2-30-0', '1802.09525-2-31-0', '1802.09525-2-32-0', '1802.09525-2-45-0', '1802.09525-2-46-0', '1802.09525-2-47-0', '1802.09525-2-48-0', '1802.09525-2-49-0', '1802.09525-2-50-0', '1802.09525-2-51-0', '1802.09525-2-52-0', '1802.09525-2-53-0', '1802.09525-2-59-1', '1802.09525-2-71-3']

{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1802.09525

1707.08619

{'1707.08619-1-0-0': "Elections seem simple-aren't they just counting?", '1707.08619-1-0-1': 'But they have a unique, challenging combination of security and privacy requirements.', '1707.08619-1-0-2': 'The stakes are high; the context is adversarial; the electorate needs to be convinced that the results are correct; and the secrecy of the ballot must be ensured.', '1707.08619-1-0-3': 'And they have practical constraints: time is of the essence, and voting systems need to be affordable and maintainable, and usable by voters, election officials, and pollworkers.', '1707.08619-1-0-4': 'It is thus not surprising that voting is a rich research area spanning theory, applied cryptography, practical systems analysis, usable security, and statistics.', '1707.08619-1-0-5': 'Election integrity involves two key concepts: convincing evidence that outcomes are correct and privacy, which amounts to convincing assurance that there is no evidence about how any given person voted.', '1707.08619-1-0-6': 'These are obviously in tension.', '1707.08619-1-0-7': 'We examine how current systems walk this tightrope.', '1707.08619-1-1-0': '# Introduction: What is the evidence?', '1707.08619-1-2-0': 'The Russians did three things [MATH] The third is that they tried, and they were not successful, but they still tried, to get access to voting machines and vote counting software, to play with the resultsFormer CIA Acting Director Michael Morell, Mar. 15, 2017', '1707.08619-1-3-0': 'These are baseless allegations substantiated with nothing, done on a rather amateurish, emotional levelKremlin spokesman Dmitry Peskov, Jan. 9, 2017', '1707.08619-1-4-0': 'It would take an army to hack into our voting system.', '1707.08619-1-4-1': 'Tom Hicks, EAC Commissioner, Oct. 6, 2016', '1707.08619-1-5-0': 'It is not enough for an election to produce the correct outcome.', '1707.08619-1-5-1': 'The electorate must also be convinced that the announced result reflects the will of the people.', '1707.08619-1-5-2': 'And for a rational person to be convinced requires evidence.', '1707.08619-1-6-0': 'Modern technology-computer and communications systems-is fragile and vulnerable to programming errors and undetectable manipulation.', '1707.08619-1-6-1': 'No current system that relies on electronic technology alone to capture and tally votes can provide convincing evidence that election results are accurate without endangering or sacrificing the anonymity of votes.', '1707.08619-1-7-0': 'Paper ballots, on the other hand, have some very helpful security properties: they are readable (and countable, and re-countable) by humans; they are relatively durable; and they are tamper-evident.', '1707.08619-1-7-1': 'Votes cast on paper can be counted using electronic technology; then the accuracy of the count can be checked manually to ensure that the technology functioned adequately well.', '1707.08619-1-7-2': 'Statistical methods allow the accuracy of the count to be assessed by examining only a fraction of the ballots manually, often a very small fraction.', '1707.08619-1-7-3': 'If there is also convincing evidence that the collection of ballots has been conserved (no ballots added, lost, or modified) then this combination-voter-verifiable paper ballots, a mechanized count, and a manual check of the accuracy of that count-can provide convincing evidence that announced electoral outcomes are correct.', '1707.08619-1-8-0': 'Conversely, absent convincing evidence that the paper trail has been conserved, a manual double-check of electronic results against the paper trail will not be convincing.', '1707.08619-1-8-1': 'If the paper trail has been conserved adequately, then a full manual tally of the ballots can correct the electronic count if the electronic count is incorrect.', '1707.08619-1-9-0': 'These considerations have led many election integrity advocates to push for a voter-verifiable paper trail (VVPAT).', '1707.08619-1-10-0': 'In the 2016 presidential election, about three quarters of Americans voted using systems that generated voter-verifiable paper records.', '1707.08619-1-10-1': 'The aftermath of the election proved that even if 100% of voters had used such systems, it would not have sufficed to provide convincing evidence that the reported results are accurate.', '1707.08619-1-11-0': "The bottom line is that the paper trail is not worth the paper it's printed on.", '1707.08619-1-11-1': 'Clearly this must change.', '1707.08619-1-12-0': "Other techniques like software independence and end-to-end verifiability can offer far greater assurance in the accuracy of an election's outcome, but these methods have not been broadly applied.", '1707.08619-1-13-0': '## Why so hard?', '1707.08619-1-14-0': 'Several factors make it difficult to generate convincing evidence that reported results are correct.', '1707.08619-1-14-1': 'The first is the trust model.', '1707.08619-1-15-0': 'No one is trusted In any significant election, voters, election officials, and equipment and software cannot necessarily be trusted by anyone with a stake in the outcome.', '1707.08619-1-15-1': 'Voters, operators, system designers, manufacturers, and external parties are all potential adversaries.', '1707.08619-1-16-0': 'The need for evidence Because officials and equipment may not be trustworthy, elections should be evidence-based.', '1707.08619-1-16-1': 'Any observer should be able to verify the reported results based on trustworthy evidence from the voting system.', '1707.08619-1-16-2': 'Many in-person voting systems fail to provide sufficient evidence; and as we shall see Internet systems scarcely provide any at all.', '1707.08619-1-17-0': 'The secret ballot Perhaps the most distinctive element of elections is the secret ballot, a critical safeguard that defends against vote selling and voter coercion.', '1707.08619-1-17-1': 'In practical terms, voters should not be able to prove how they voted to anyone, even if they wish to do so.', '1707.08619-1-17-2': 'This restricts the types of evidence that can be produced by the voting system.', '1707.08619-1-17-3': 'Encryption alone is not sufficient, since the voters may choose to reveal their selections in response to bribery or coercion.', '1707.08619-1-18-0': 'The challenge of voting is thus to use fragile technology to produce trustworthy, convincing evidence of the correctness of the outcome while protecting voter privacy in a world where no person or machine may be trusted.', '1707.08619-1-18-1': 'The resulting voting system and its security features must also be usable by regular voters.', '1707.08619-1-19-0': 'The aim of this paper is to explain the important requirements of secure elections and the solutions already available from e-voting research, then to identify the most important directions for research.', '1707.08619-1-20-0': 'Prior to delving into our discussion, we need to make a distinction in terminology.', '1707.08619-1-20-1': 'Pollsite voting systems are those in which voters record and cast ballots at predetermined locations, often in public areas with strict monitoring.', '1707.08619-1-20-2': 'Remote voting refers to a system where voters fill out ballots anywhere, and then send them to a central location to cast them, either physically mailing them in the case of vote-by-mail, or sending them over the Internet in the case of Internet voting.', '1707.08619-1-21-0': 'The next section defines the requirements, beginning with notions of election evidence, then considering privacy, and concluding with more general usability and security requirements.', '1707.08619-1-21-1': 'Section [REF] describes the cryptographic, statistical, and engineering tools that have been developed for designing voting systems with verifiably correct election outcomes.', '1707.08619-1-21-2': 'Section [REF] discusses the challenge of satisfying our requirements for security using the tools presented in real-world election systems.', '1707.08619-1-21-3': 'Section [REF] concludes with the promise and problems associated with Internet voting.', '1707.08619-1-22-0': '# Requirements for Secure Voting', '1707.08619-1-23-0': "Trustworthiness before trustOnora O'Neill", '1707.08619-1-24-0': '## Trust, Verifiability, and Evidence', '1707.08619-1-25-0': 'For an election to be accepted as legitimate, the outcome should be convincing to all-and in particular to the losers-leaving no valid grounds to challenge the outcome.', '1707.08619-1-25-1': 'Whether elections are conducted by counting paper ballots by hand or using computer technology, the possibility of error or fraud necessitates assurances of the accuracy of the outcome.', '1707.08619-1-26-0': 'It is clear that a naive introduction of computers into voting introduces the possibility of wholesale and largely undetectable fraud.', '1707.08619-1-26-1': "If we can't detect it, how can we prevent it?", '1707.08619-1-27-0': '### Risk-Limiting Audits', '1707.08619-1-28-0': 'Statistical post-election audits provide assurance that a reported outcome is correct, by examining some or all of an audit trail consisting of durable, tamper-evident, voter-verifiable records.', '1707.08619-1-28-1': 'Typically the audit trail consists of paper ballots.', '1707.08619-1-29-0': 'The outcome of an election is the set of winners.', '1707.08619-1-29-1': 'An outcome is incorrect if it differs from the set of winners output by a perfectly accurate manual tabulation of the audit trail.', '1707.08619-1-30-0': 'An audit of an election contest is a risk-limiting audit (RLA) with risk limit [MATH] if it has the following two properties:', '1707.08619-1-31-0': 'If the reported contest outcome under audit is incorrect, the probability that the audit leads to correcting the outcome is at least [MATH].', '1707.08619-1-31-1': 'The audit never indicates a need to alter a reported outcome that is correct.', '1707.08619-1-32-0': '(In this context, "correct" means "what a full manual tally of the paper trail would show."', '1707.08619-1-32-1': 'If the paper trail is unreliable, a RLA in general cannot detect that.', '1707.08619-1-32-2': 'RLAs should be preceded by "compliance audits" that check whether the audit trail itself is adequately reliable to determine who won.)', '1707.08619-1-32-3': 'Together, these two properties imply that post-RLA, either the reported set of winners is the set that a perfectly accurate hand count of the audit trail would show, or an event with probability no larger than [MATH] has occurred.', '1707.08619-1-32-4': '(That event is that the outcome was incorrect, but the RLA did not lead to correcting the outcome.)', '1707.08619-1-32-5': 'RLAs amount to a limited form of probabilistic error correction: by relying on appropriate random sampling of the audit trail and hypothesis tests, they have a known minimum probability of correcting the outcome.', '1707.08619-1-32-6': 'They are not designed to ensure that the reported numerical tally is correct, only that the outcome is correct.', '1707.08619-1-33-0': 'The following procedure is a trivial RLA: with probability [MATH], perform a full manual tally of the audit trail.', '1707.08619-1-33-1': 'Amend the outcome to match the set of winners the full hand count shows if that set is different.', '1707.08619-1-34-0': 'The art in constructing RLAs consists of maintaining the risk limit while performing less work than a full hand count when the outcome is correct.', '1707.08619-1-34-1': 'Typically, this involves framing the audit as a sequential test of the statistical hypothesis that the outcome is incorrect.', '1707.08619-1-34-2': 'To reject that hypothesis is to conclude that the outcome is correct.', '1707.08619-1-34-3': "RLAs have been developed for majority contests, plurality contests, and vote-for-[MATH] contests and complex social choice functions including D'Hondt and other proportional representation rules-see below.", '1707.08619-1-34-4': 'RLAs have also been devised to check more than one election contest simultaneously [CITATION].', '1707.08619-1-35-0': '### Software Independence', '1707.08619-1-36-0': 'Rivest and Wack introduced a definition targeted specifically at detecting misbehavior in computer-based elections:', '1707.08619-1-37-0': '[CITATION] A voting system is software independent if an undetected change or error in its software cannot cause an undetectable change or error in an election outcome.', '1707.08619-1-38-0': 'Software independence clearly expresses that it should not be necessary to trust software to determine election outcomes, but it does not say what procedures or types of evidence should be trusted instead.', '1707.08619-1-38-1': 'A system that is not software independent cannot produce a convincing evidence trail, but neither can a paper-based system that does not ensure that the paper trail is complete and intact, a cryptographic voting system that relies on an invalid cryptographic assumption, or a system that relies on audit procedures but lacks a means of assuring that those procedures are properly followed.', '1707.08619-1-38-2': 'We could likewise demand independence of many other kinds of trust assumptions: hardware, paper chain-of-custody, cryptographic setup, computational hardness, procedures, good randomness generation etc.', '1707.08619-1-39-0': 'Rivest and Wack also define a stronger form of the property that includes error recovery:', '1707.08619-1-40-0': '[CITATION] A voting system is strongly software independent if it is software independent and a detected change or error in an election outcome (due to the software) can be corrected without rerunning the election.', '1707.08619-1-41-0': 'A strongly software-independent system can recover from software errors or bugs, but that recovery in turn is generally based on some other trail of evidence.', '1707.08619-1-42-0': 'A software independent system can be viewed as a form of tamper-evident system: a material software problem leaves a detectable trace.', '1707.08619-1-42-1': 'Strongly software independent systems are resilient: not only do material software problems leave a trace, the overall election system can recover from a detected problem.', '1707.08619-1-43-0': "One mechanism to provide software independence is to record votes on a paper record that provides physical evidence of voter's intent, can be inspected by the voter prior to casting the vote, and-if preserved intact-can later be manually audited to check the election outcome.", '1707.08619-1-43-1': 'Risk-limiting audits (see Section [REF]) can then achieve a pre-specified level of assurance that results are correct; machine assisted risk-limiting audits [CITATION], can help minimize the amount of labor required for legacy systems that do not provide a cast-vote record for every ballot, linked to the corresponding ballot.', '1707.08619-1-44-0': '0.97Open problems:', '1707.08619-1-45-0': '### End-to-end verifiability', '1707.08619-1-46-0': 'The concern regarding fraud and desire for transparency has motivated the security and crypto communities to develop another approach to voting system assurance: end-to-end verifiability (E2E-V).', '1707.08619-1-46-1': 'An election that is end-to-end verifiable achieves software independence together with the analagous notion of hardware independence as well as independence from actions of election personnel and vendors.', '1707.08619-1-46-2': 'Rather than attempting to verify thousands of lines of code or closely monitor all of the many processes in an election, E2E-V focuses on providing a means to detect errors or fraud in the process of voting and counting.', '1707.08619-1-46-3': 'The idea behind E2E-V is to enable voters themselves to monitor the integrity of the election; democracy for the people by the people, as it were.', '1707.08619-1-46-4': 'This is challenging because total transparency is not possible without undermining the secret ballot, hence the mechanisms to generate such evidence have to be carefully designed.', '1707.08619-1-47-0': '(adapted from [CITATION]) A voting system is end-to-end verifiable if it has the following three kinds of verifiability:', '1707.08619-1-48-0': 'If verification relies on trusting entities, software, or hardware, the voter and/or auditor should be able to choose them freely.', '1707.08619-1-48-1': 'Trusted procedures, if there are any, must be open to meaningful observation by every voter.', '1707.08619-1-49-0': 'Note that the above definition allows each voter to check that her vote is correctly collected, thus ensuring that attempts to change or delete cast votes are detected.', '1707.08619-1-49-1': 'In addition, it should also be possible to check the list of voters who cast ballots, to ensure that votes are not added to the collection (i.e., to prevent ballot-box stuffing).', '1707.08619-1-49-2': 'This is called eligibility verifiability [CITATION].', '1707.08619-1-50-0': '### Collection Accountability', '1707.08619-1-51-0': 'In an E2E-V election protocol, voters can check whether their votes have been properly counted, but if they discover a problem, there may not be adequate evidence to correct it.', '1707.08619-1-51-1': 'An election system that is collection-accountable provides voters with evidence of any failure to collect their votes.', '1707.08619-1-52-0': 'An election system is collection accountable if any voter who detects that her vote has not been collected has, as part of the vote-casting protocol, convincing evidence that can be presented to an independent party to demonstrate that the vote has not been collected.', '1707.08619-1-53-0': 'Another form of evidence involves providing each voter with a code representing her votes, such that knowledge of a correct code is evidence of casting a particular vote [CITATION].', '1707.08619-1-53-1': 'Yet another mechanism is a suitable paper receipt.', '1707.08619-1-53-2': 'Forensic analysis may provide evidence that this receipt was not forged by a voter [CITATION].', '1707.08619-1-54-0': '0.97Open problems:', '1707.08619-1-55-0': '### Dispute Resolution', '1707.08619-1-56-0': 'While accountability helps secure the election process, it is not very useful if there is no way to handle disputes.', '1707.08619-1-56-1': 'If a voter claims, on the basis of accountability checks provided by a system, that something has gone wrong, there needs to be a mechanism to address this.', '1707.08619-1-56-2': 'This is known as dispute resolution:', '1707.08619-1-57-0': '[CITATION] A voting system is said to have dispute resolution if, when there is a dispute between two participants regarding honest participation, a third party can correctly resolve the dispute.', '1707.08619-1-58-0': 'An alternative to dispute resolution is dispute-freeness:', '1707.08619-1-59-0': '[CITATION] A dispute-free voting system has built-in prevention mechanisms that eliminate disputes among the active participants; any third party can check whether an active participant has cheated.', '1707.08619-1-60-0': '### From Verifiable to Verified', '1707.08619-1-61-0': 'Constructing a voting system that creates sufficient evidence to reveal problems is not enough on its own.', '1707.08619-1-61-1': 'That evidence must actually be used-and used appropriately-to ensure the accuracy of election outcomes.', '1707.08619-1-62-0': 'An election result may not be verified, even if it is generated by an end-to-end verifiable voting system.', '1707.08619-1-62-1': 'For verification of the result, we need several further conditions to be satisfied:', '1707.08619-1-63-0': "These issues involve complex human factors, including voters' incentives to participate in verification.", '1707.08619-1-63-1': 'Little work has been done on this aspect of the problem.', '1707.08619-1-64-0': 'An E2E-V system might give an individual voter assurance that her vote has not been tampered with if that voter performs certain checks.', '1707.08619-1-64-1': 'However, sufficiently many voters must do this in order to provide evidence that the election outcome as a whole is correct.', '1707.08619-1-64-2': 'Combining risk-limiting audits with E2E-V systems can provide a valuable layer of protection in the case that an insufficient number of voters participate in verification.', '1707.08619-1-65-0': 'Finally, another critical verification problem that has received little attention to date is how to make schemes that are recoverable in the face of errors.', '1707.08619-1-65-1': 'We do not want to have to abort and rerun an election every time a check a fails.', '1707.08619-1-65-2': 'Certain levels of detected errors can be shown to be highly unlikely if the outcome is incorrect, and hence can be tolerated.', '1707.08619-1-65-3': 'Other types and patterns of error cast doubt on the outcome and may require either full inspection or retabulation of the paper trail or, if the paper trail cannot be relied upon, a new election.', '1707.08619-1-66-0': 'Both Kusters et al. [CITATION] and Kiayias et al. [CITATION] model voter-initiated auditing [CITATION] and its implications for detection of an incorrect election result.', '1707.08619-1-66-1': 'Both definitions turn uncertainty about voter initiated auditing into a bound on the probability of detecting deviations of the announced election result from the truth.', '1707.08619-1-67-0': '0.97Open problems:', '1707.08619-1-68-0': '## Voter Authentication', '1707.08619-1-69-0': 'A significant challenge for election systems is the credentialing of voters to ensure that all eligible voters, and no one else, can cast votes.', '1707.08619-1-69-1': 'This presents numerous questions: what kinds of credentials should be used?', '1707.08619-1-69-2': 'How should they be issued?', '1707.08619-1-69-3': 'Can they be revoked or de-activated?', '1707.08619-1-69-4': 'Are credentials good for a single election or for an extended period?', '1707.08619-1-69-5': 'How difficult are they to share, transfer, steal, or forge?', '1707.08619-1-69-6': 'Can the ability to create genuine-looking forgeries help prevent coercion?', '1707.08619-1-69-7': 'These questions must be answered carefully, and until they are satisfied for remote voting, pollsite voting is the only robust way to address these questions-and even then, in-person credentialing is subject to forgery, distribution, and revocation concerns (for instance, the Dominican Republic recently held a pollsite election where voters openly sold their credentials [CITATION]).', '1707.08619-1-69-8': 'In the U.S., there is concern that requiring in-person credentialing, in the form of voter ID, disenfranchises legitimate voters.', '1707.08619-1-70-0': '0.97Open problems:', '1707.08619-1-71-0': '## Privacy, Receipt Freeness, and Coercion Resistance', '1707.08619-1-72-0': 'In most security applications, privacy and confidentiality are synonymous.', '1707.08619-1-72-1': 'In elections, however, privacy has numerous components that go well beyond typical confidentiality.', '1707.08619-1-72-2': 'Individual privacy can be compromised by "normal" election processes such as a unanimous result.', '1707.08619-1-72-3': 'Voters may be coerced if they can produce a proof of how they voted, even if they have to work to do so.', '1707.08619-1-73-0': "Privacy for votes is a means to an end: if voters don't express their true preferences then the election may not produce the right outcome.", '1707.08619-1-73-1': 'This section gives an overview of increasingly strong definitions of what it means for voters to be free of coercion.', '1707.08619-1-74-0': '### Basic Confidentiality', '1707.08619-1-75-0': 'We will take ballot privacy to mean that the election does not leak any information about how any voter voted beyond what can be deduced from the announced results.', '1707.08619-1-75-1': 'Confidentiality is not the only privacy requirement in elections, but even simple confidentiality poses significant challenges.', '1707.08619-1-75-2': 'It is remarkable how many deployed e-voting systems have been shown to lack even the most basic confidentiality properties (e.g., [CITATION]).', '1707.08619-1-76-0': 'Perhaps more discouraging to basic privacy is the fact that remote voting systems (both paper and electronic) inherently allow voters to eschew confidentiality.', '1707.08619-1-76-1': "Because remote systems enable voters to fill out their ballots outside a controlled environment, anyone can watch over the voter's shoulder while she fills out her ballot.", '1707.08619-1-77-0': 'In an election-unlike, say, in a financial transaction-even the candidate receiving an encrypted vote should not be able to decrypt it.', '1707.08619-1-77-1': 'Instead, an encrypted (or otherwise shrouded) vote must remain confidential to keep votes from being directly visible to election authorities.', '1707.08619-1-78-0': 'Some systems, such as code voting [CITATION] and the Norwegian and Swiss Internet voting schemes, defend privacy against an attacker who controls the computer used for voting; however, this relies on assumptions about the privacy and integrity of the code sheet.', '1707.08619-1-78-1': 'Some schemes, such as JCJ/Civitas [CITATION], obscure who has voted while providing a proof that only eligible votes were included in the tally.', '1707.08619-1-79-0': 'Several works [CITATION] [CITATION], following Benaloh [CITATION] formalize the notion of privacy as preventing an attacker from noticing when two parties swap their votes.', '1707.08619-1-80-0': '0.97Open problems:', '1707.08619-1-81-0': '### Everlasting Privacy', '1707.08619-1-82-0': "Moran and Naor expressed concern over what might happen to encrypted votes that can still be linked to their voter's name some decades into the future, and hence decrypted by superior technology.", '1707.08619-1-82-1': 'They define a requirement to prevent this:', '1707.08619-1-83-0': '[CITATION] A voting scheme has everlasting privacy if its privacy does not depend on assumptions of cryptographic hardness.', '1707.08619-1-84-0': 'Their solution uses perfectly hiding commitments to the votes, which are aggregated homomorphically.', '1707.08619-1-84-1': 'Instead of privacy depending upon a cryptographic hardness assumption, it is the integrity of an election that depends upon a hardness assumption; and only a real-time compromise of the assumption can have an impact.', '1707.08619-1-85-0': '### Systemic Privacy Loss', '1707.08619-1-86-0': 'We generally accept that without further information, a voter is more likely to have voted for a candidate who has received more votes, but additional data is commonly released which can further erode voter privacy.', '1707.08619-1-86-1': 'Even if we exclude privacy compromises, there are other privacy risks which must be managed.', '1707.08619-1-86-2': 'If voters achieve privacy by encrypting their selections, the holders of decryption keys can view their votes.', '1707.08619-1-86-3': 'If voters make their selections on devices out of their immediate control (e.g. official election equipment), then it is difficult to assure them that these devices are not retaining information that could later compromise their privacy.', '1707.08619-1-86-4': 'If voters make their selections on their own devices, then there is an even greater risk that these devices could be infected with malware that records (and perhaps even alters) their selections (see, for instance, the Estonian system [CITATION]).', '1707.08619-1-87-0': '0.97Open problems:', '1707.08619-1-88-0': '### Receipt-freeness', '1707.08619-1-89-0': 'Preventing coercion and vote-selling was considered solved with the introduction of the Australian ballot.', '1707.08619-1-89-1': 'The process of voting privately within a public environment where privacy can be monitored and enforced prevents improper influence.', '1707.08619-1-89-2': 'Recent systems have complicated this notion, however.', '1707.08619-1-89-3': 'If a voting protocol provides a receipt but is not carefully designed, the receipt can be a channel for information to the coercive adversary.', '1707.08619-1-90-0': 'Benaloh and Tuinstra [CITATION] pointed out that passive privacy is insufficient for resisting coercion in elections:', '1707.08619-1-91-0': 'A voting system is receipt free if a voter is unable to prove how she voted even if she actively colludes with a coercer and deviates from the protocol in order to try to produce a proof.', '1707.08619-1-92-0': 'Traditional elections may fail receipt-freeness too.', '1707.08619-1-92-1': 'In general, if a vote consists of a long list of choices, the number of possible votes may be much larger than the number of likely voters.', '1707.08619-1-92-2': 'This is sometimes called (a failure of) the short ballot assumption [CITATION].', '1707.08619-1-92-3': 'Prior to each election, coercers assign a particular voting pattern to each voter.', '1707.08619-1-92-4': 'When the individual votes are made public, any voter who did not cast their pattern can then be found out.', '1707.08619-1-92-5': 'This is sometimes called the Italian attack, after a once prevalent practice in Sicily.', '1707.08619-1-92-6': 'It can be easily mitigated when a vote can be broken up, but is difficult to mitigate in systems like IRV in which the vote is complex but must be kept together.', '1707.08619-1-92-7': 'Mitigations are discussed in Sections [REF] and [REF].', '1707.08619-1-93-0': 'Incoercibility has been defined and examined in the universally composable framework in the context of general multiparty computation [CITATION].', '1707.08619-1-93-1': 'These definitions sidestep the question of whether the voting function itself allows coercion (by publishing individual complex ballots, or by revealing a unanimous result for example)-they examine whether the protocol introduces additional opportunities for coercion.', '1707.08619-1-93-2': 'With some exceptions (such as [CITATION]), they usually focus on a passive notion of receipt-freeness, which is not strong enough for voting.', '1707.08619-1-94-0': '### Coercion Resistance', '1707.08619-1-95-0': 'Schemes can be receipt-free, but not entirely resistant to coercion.', '1707.08619-1-95-1': 'Schemes like Pret a Voter [CITATION] that rely on randomization for receipt-freeness can be susceptible to forced randomization, where a coercer forces a voter to always choose the first choice on the ballot.', '1707.08619-1-95-2': 'Due to randomized candidate order, the resulting vote will be randomly distributed.', '1707.08619-1-95-3': 'If a specific group of voters are coerced in this way, it can have a disproportionate impact on the election outcome.', '1707.08619-1-96-0': 'If voting rolls are public and voting is not mandatory, this has an effect equivalent to prevent forced abstention, wherein a coercer refuses to let a voter vote.', '1707.08619-1-96-1': 'Schemes that rely on credentialing are also susceptible to coercion by forced surrender of credentials.', '1707.08619-1-97-0': 'One way to fully resist forced abstention is to obscure who voted.', '1707.08619-1-97-1': 'However, this is difficult to reconcile with the opportunity to verify that only eligible voters have voted (eligibility verifiability), though some schemes achieve both [CITATION].', '1707.08619-1-98-0': 'Moran and Naor [CITATION] provide a strong definition of receipt freeness in which a voter may deviate actively from the protocol in order to convince a coercer that she obeyed.', '1707.08619-1-98-1': 'Their model accommodates forced randomization.', '1707.08619-1-98-2': 'A scheme is resistant to coercion if the voter can always pretend to have obeyed while actually voting as she likes.', '1707.08619-1-99-0': "A voting scheme is coercion resistant if there exists a way for a coerced voter to cast her vote such that her coercer cannot distinguish whether or not she followed the coercer's instructions.", '1707.08619-1-100-0': 'Coercion resistance is defined in [CITATION] to include receipt freeness and defence against forced-randomization, forced abstention and the forced surrender of credentials.', '1707.08619-1-100-1': "More general definitions include [CITATION], which incorporates all these attacks along with Moran and Naor's notion of a coercion resistance strategy.", '1707.08619-1-101-0': 'Note that if the coercer can monitor the voter throughout the vote casting period, then resistance is futile.', '1707.08619-1-101-1': 'For in-person voting, we assume that the voter is isolated from any coercer while she is in the booth (although this is questionable in the era of mobile phones).', '1707.08619-1-101-2': 'For remote voting, we need to assume that voters will have some time when they can interact with the voting system (or the credential-granting system) unobserved.', '1707.08619-1-102-0': '### More Coercion Considerations', '1707.08619-1-103-0': 'Some authors have tried to provide some protection against coercion without achieving full coercion resistance.', '1707.08619-1-103-1': 'Caveat coercitor [CITATION] proposes the notion of coercion evidence and allows voters to cast multiple votes using the same credential.', '1707.08619-1-104-0': '0.97Open problem:', '1707.08619-1-105-0': '## Usability', '1707.08619-1-106-0': 'A voting system must be usable by voters, poll-workers, election officials, observers, and so on.', '1707.08619-1-106-1': 'Voters who may not be computer literate-and sometimes not literate at all-should be able to vote with very low error rates.', '1707.08619-1-106-2': 'Although some error is regarded as inevitable, it is also critical that the interface not drive errors in a particular direction.', '1707.08619-1-106-3': 'For instance, a list of candidates that crosses a page boundary could cause the candidates on the second page to be missed.', '1707.08619-1-106-4': 'Whatever security mechanisms we add to the voting process should operate without degrading usability, otherwise the resulting system will likely be unacceptable.', '1707.08619-1-106-5': 'A full treatment of usability in voting is beyond the scope of this paper.', '1707.08619-1-106-6': 'However, we note that E2E-V systems (and I-voting systems, even when not E2E-V) add additional processes for voters and poll workers to follow.', '1707.08619-1-106-7': "If verification processes can't be used properly by real voters, the outcome will not be properly verified.", '1707.08619-1-106-8': 'One great advantage of statistical audits is to shift complexity from voters to auditors.', '1707.08619-1-107-0': '0.97Open problems:', '1707.08619-1-108-0': '## Local Regulatory Requirements', '1707.08619-1-109-0': 'A variety of other mechanical requirements are often imposed by legal requirements that vary among jurisdictions.', '1707.08619-1-109-1': 'For example:', '1707.08619-1-110-0': 'Newer electronic and I-voting systems raise important policy challenges for real-world adoption.', '1707.08619-1-110-1': 'For example, in STAR-Vote [CITATION], there will be multiple copies of every vote record: mostly electronic records, but also paper records.', '1707.08619-1-110-2': 'There may be instances where one is damaged or destroyed and the other is all that remains.', '1707.08619-1-110-3': 'When laws speak to retention of "the ballot", that term is no longer well-defined.', '1707.08619-1-110-4': 'Such requirements may need to be adapted to newer voting systems.', '1707.08619-1-111-0': '### Complex Election Methods', '1707.08619-1-112-0': 'Many countries allow voters to select, score, or rank candidates or parties.', '1707.08619-1-112-1': 'Votes can then be tallied in a variety of complex ways [CITATION].', '1707.08619-1-112-2': 'None of the requirements for privacy, coercion-resistance, or the provision of verifiable evidence change.', '1707.08619-1-112-3': 'However, many tools that achieve these properties for traditional "first-past-the-post" elections need to be redesigned.', '1707.08619-1-113-0': 'An election method might be complex at the voting or the tallying end.', '1707.08619-1-113-1': "For example, party-list methods such as D'Hondt and Sainte-Lague have simple voting, in which voters select their candidate or party, but complex proportional seat allocation.", '1707.08619-1-113-2': 'Borda, Range Voting, and Approval Voting allow votes to be quite expressive but are simple to tally by addition.', '1707.08619-1-113-3': "Condorcet's method and related functions [CITATION] can be arbitrarily complex, as they can combine with any social choice function.", '1707.08619-1-113-4': 'Instant Runoff Voting (IRV) and the Single Transferable Vote (STV) are both expressive and complicated to tally.', '1707.08619-1-113-5': 'This makes for several challenges.', '1707.08619-1-114-0': '# How can we secure voting?', '1707.08619-1-115-0': 'These truths are self-evident but not self-enforcing Barack Obama', '1707.08619-1-116-0': 'The goal of this section and the next is to provide a state-of-the-art picture of current solutions to voting problems and ongoing voting research, to motivate further work on open problems, and to define clear directions both in research and election policy.', '1707.08619-1-117-0': '## The Role of Paper and Ceremonies', '1707.08619-1-118-0': 'Following security problems with direct-recording electronic voting systems (DREs) [CITATION], many parts of the USA returned to the use of paper ballots.', '1707.08619-1-118-1': 'If secure custody of the paper ballots is assumed, paper provides durable evidence required to determine the correctness of the election outcome.', '1707.08619-1-118-2': 'For this reason, when humans vote from untrusted computers, cryptographic voting system specifications often use paper for security, included in the notions of dispute-freeness, dispute resolution, collection accountability and accountability [CITATION] (all as defined in Section [REF]).', '1707.08619-1-119-0': 'Note that the standard approach to dispute resolution, based on non-repudiation, cannot be applied to the voting problem in the standard fashion, because the human voter does not have the ability to check digital signatures or digitally sign the vote (or other messages that may be part of the protocol) unassisted.', '1707.08619-1-120-0': 'Dispute-freeness or accountability are often achieved in a polling place through the use of cast paper ballots, and the evidence of their chain of custody (e.g., wet-ink signatures).', '1707.08619-1-120-1': 'Paper provides an interface for data entry for the voter-not simply to enter the vote, but also to enter other messages that the protocol might require-and data on unforgeable paper serves many of the purposes of digitally signed data.', '1707.08619-1-120-2': 'Thus, for example, when a voter marks a Pret a Voter [CITATION] or Scantegrity [CITATION] ballot, she is providing an instruction that the voting system cannot pretend was something else.', '1707.08619-1-120-3': 'The resulting vote encryption has been physically committed to by the voting system-by the mere act of printing the ballot-before the voter "casts" her vote.', '1707.08619-1-121-0': 'Physical ceremony, such as can be witnessed while the election is ongoing, also supports verifiable cryptographic election protocols (see Section [REF]).', '1707.08619-1-121-1': 'Such ceremonies include the verification of voter credentials, any generation of randomness if required for the choice between cast and audit, any vote-encryption-verification performed by election officials, etc.', '1707.08619-1-122-0': 'The key aspect of these ceremonies is the chance for observers to see that they are properly conducted.', '1707.08619-1-123-0': '0.97Open problem:', '1707.08619-1-124-0': '## Statistics and Auditing', '1707.08619-1-125-0': '## Cryptographic Tools and Designs', '1707.08619-1-126-0': '### Major Approaches to Voting Cryptography', '1707.08619-1-127-0': 'Typically E2E-V involves providing each voter with a protected receipt-an encrypted or encoded version of their vote-at the time the vote is cast.', '1707.08619-1-127-1': 'The voter can later use her receipt to check whether her vote is included correctly in the tabulation process.', '1707.08619-1-127-2': 'Furthermore, given the set of encrypted votes (as well as other relevant information, like the public keys), the tabulation is universally verifiable: anyone can check whether it is correct.', '1707.08619-1-127-3': 'To achieve this, most E2E-V systems rely on a public bulletin board, where the set of encrypted ballots is published in an append-only fashion.', '1707.08619-1-128-0': 'The votes can then be turned into a tally in one of two main ways.', '1707.08619-1-128-1': 'Homomorphic encryption schemes [CITATION] allow the tally to be produced on encrypted votes.', '1707.08619-1-128-2': 'Verifiable shuffling transforms a list of encrypted votes into a shuffled list that can be decrypted without the input votes being linked to the (decrypted) output.', '1707.08619-1-128-3': 'There are efficient ways to prove that the input list exactly matches the output [CITATION].', '1707.08619-1-129-0': '### Techniques for Cast-as-Intended Verification', '1707.08619-1-130-0': 'How can a voter verify that her cast vote is the one she wanted?', '1707.08619-1-130-1': 'Code Voting, first introduced by Chaum [CITATION], gives each voter a sheet of codes for each candidate.', '1707.08619-1-130-2': 'Assuming the code sheet is valid, the voter can cast a vote on an untrusted machine by entering the code corresponding to her chosen candidate and waiting to receive the correct confirmation code.', '1707.08619-1-130-3': 'Modern interpretations of code voting include [CITATION].', '1707.08619-1-131-0': 'Code voting only provides assurance that the correct voting code reached the server, it does not of itself provide any guarantees that the code will subsequently be correctly counted.', '1707.08619-1-131-1': 'A scheme that improves on this is Pretty Good Democracy [CITATION], where knowledge of the codes is threshold shared in such a way that receipt of the correct confirmation code provides assurance that the voting code has been registered on the bulletin board by a threshold set of trustees, and hence subsequently counted.', '1707.08619-1-132-0': 'The alternative is to ask the machine to encrypt a vote directly, but verify that it does so correctly.', '1707.08619-1-132-1': 'Benaloh [CITATION] developed a simple protocol to enable vote encryption on an untrusted voting machine.', '1707.08619-1-132-2': 'A voter uses a voting machine to encrypt any number of votes, and casts only one of these encrypted votes.', '1707.08619-1-132-3': 'All the other votes may be "audited" by the voter.', '1707.08619-1-132-4': 'If the encryption is audited, the voting system provides a proof that it encrypted the vote correctly, and the proof is public.', '1707.08619-1-132-5': 'The corresponding ballot cannot be cast as the correspondence between the encryption and the ballot is now public, and the vote is no longer secret.', '1707.08619-1-132-6': 'Voters take home receipts corresponding to the encryptions of their cast ballots as well as any ballots that are to be audited.', '1707.08619-1-132-7': 'They may check the presence of these on a bulletin board, and the correctness proofs of the audited encryptions using software obtained from any of several sources.', '1707.08619-1-132-8': 'However, even the most dilligent voters need only check that their receipts match the public record and that any ballots selected for audit display correct candidate selections.', '1707.08619-1-132-9': 'The correctness proofs are part of the public record that can be verified by any individual or observer that is verifying correct tallying.', '1707.08619-1-133-0': '### Formal models and security analyses of cast-as-intended verification protocols', '1707.08619-1-134-0': 'In addition to the work of Adida on assisted-human interactive proofs (AHIPs, see [CITATION]), there has been some work on a rigorous understanding of one or more properties of single protocols, including the work of Moran and Naor [CITATION] and Kusters et al. [CITATION].', '1707.08619-1-135-0': 'There have also been formalizations of voting protocols with human participants, such as by Moran and Naor [CITATION] (for a polling protocol using tamper-evident seals on envelopes) and Kiayias et al. [CITATION].', '1707.08619-1-135-1': 'However, there is no one model that is sufficient for the rigorous understanding of the prominent protocols used/proposed for use in real elections.', '1707.08619-1-135-2': 'The absence of proofs has led to the overlooking of vulnerabilities in the protocols in the past, see [CITATION].', '1707.08619-1-136-0': "Many systems use a combination of paper, cryptography, and auditing to achieve E2E-V in the polling place, including Markpledge [CITATION], Wombat [CITATION], Demos [CITATION], Pret a Voter [CITATION], STAR-Vote [CITATION], and Moran and Naor's scheme [CITATION].", '1707.08619-1-136-1': 'Their properties are summarised more thoroughly in the following section.', '1707.08619-1-137-0': 'The cryptographic literature has numerous constructions of end-to-end verifiable election schemes (e.g., [CITATION]).', '1707.08619-1-137-1': 'There are also detailed descriptions of what it means to verify the correctness of the output of E2E-V systems (e.g., [CITATION]).', '1707.08619-1-137-2': 'Others have attempted to define alternative forms of the E2E-V properties [CITATION].', '1707.08619-1-137-3': 'There are also less technical explanations of E2E-V intended for voters and election officials [CITATION].', '1707.08619-1-138-0': '0.97Open problem:', '1707.08619-1-139-0': '### Techniques for Coercion Resistance', '1707.08619-1-140-0': 'Some simple approaches to coercion resistance have been suggested in the literature.', '1707.08619-1-140-1': 'These include allowing multiple votes with only the last counting and allowing in-person voting to override remotely cast votes (both used in Estonian, Norwegian, and Utah elections [CITATION]).', '1707.08619-1-140-2': 'It is not clear that this mitigates coercion at all.', '1707.08619-1-140-3': 'Alarm codes can also be provided to voters: seemingly real but actually fake election credentials, along with the ability for voters to create their own fake credentials.', '1707.08619-1-140-4': 'Any such approach can be considered a partial solution at best, particularly given the usability challenges.', '1707.08619-1-141-0': 'One voting system, Civitas [CITATION], based on a protocol by Juels, Catalano and Jakobsson [CITATION], allows voters to vote with fake credentials to lead the coercive adversary into believing the desired vote was cast.', '1707.08619-1-141-1': 'Note that the protocol must enable universal verification of the tally from a list of votes cast with both genuine and fake credentials, proving to the verifier that only the ones with genuine credentials were tallied, without identifying which ones they were.', '1707.08619-1-142-0': '0.97Open problem:', '1707.08619-1-143-0': '### Cryptographic Solutions in Complex Elections', '1707.08619-1-144-0': 'Cast-as-intended verification based on creating and then challenging a vote works regardless of the scheme (e.g. Benaloh challenges).', '1707.08619-1-144-1': 'Cut-and-choose based schemes such as Pret a Voter and Scantegrity II need to be modified to work.', '1707.08619-1-145-0': 'Both uses of end-to-end verifiable voting schemes in government elections, the Takoma Park run of Scantegrity II and the Victorian run of Pret a Voter, used IRV (and one used STV).', '1707.08619-1-145-1': "Verifiable IRV/STV counting that doesn't expose individual votes to the Italian attack has been considered [CITATION], but may not be efficient enough for use in large elections in practice, and was not employed in either practical implementation.", '1707.08619-1-146-0': '0.97Open problems:', '1707.08619-1-147-0': '### Blockchains as a Cryptographic Solution', '1707.08619-1-148-0': 'Blockchains provide an unexpectedly effective answer to a long-standing problem in computer science-how to form a consistent public ledger in a dynamic and fully distributed environment in which there is no leader and participants may join and leave at any time [CITATION].', '1707.08619-1-148-1': 'In fact, the blockchain process effectively selects a "random" leader at each step to move things forward, so this seems at first to be a natural fit for elections-citizens post their preferences onto a blockchain and everyone can see and agree upon the outcome of the election.', '1707.08619-1-149-0': 'However, blockchains and elections differ in significant ways.', '1707.08619-1-149-1': 'Elections typically already have central authorities to play the leadership role, an entity that administrates the election: what will be voted on, when, who is allowed to vote, etc.).', '1707.08619-1-149-2': 'This authority can also be tasked with publishing a public ledger of events.', '1707.08619-1-149-3': 'Note that (as with blockchains) there need be no special trust in a central authority as these tasks are all publicly observable.', '1707.08619-1-149-4': 'So to begin with, by simply posting something on a (digitally signed) web page, an election office can do in a single step what blockchains do with a cumbersome protocol involving huge amounts of computation.', '1707.08619-1-150-0': 'Blockchains are inherently unaccountable.', '1707.08619-1-150-1': 'Blockchain miners are individually free to include or reject any transactions they desire-this is considered a feature.', '1707.08619-1-150-2': "To function properly in elections, a blockchain needs a mechanism to ensure all legitimate votes are included in the ledger, which leads to another problem: there's also no certainty in traditional blockchain schemes.", '1707.08619-1-150-3': 'Disputes are typically resolved with a "longest chain wins" rule.', '1707.08619-1-150-4': 'Miners may have inconsistent views of the contents of blockchains, but the incentives are structured so that the less widely held views eventually fade away-usually.', '1707.08619-1-150-5': 'This lack of certainty is not a desirable property in elections.', '1707.08619-1-151-0': 'In addition to lacking certainty and accountability, blockchains also lack anonymity.', '1707.08619-1-151-1': 'While modifications can be made to blockchain protocols to add anonymity, certainty, and accountability, balancing these modifications on top of the additional constraints of voting is difficult, and simpler solutions already exist as we discuss.', '1707.08619-1-152-0': 'In short, blockchains do not address any of the fundamental problems in elections, and their use actually makes things worse.', '1707.08619-1-153-0': '# Current Solutions', '1707.08619-1-154-0': 'I am committed to helping Ohio deliver its electoral votes to the president next year.', '1707.08619-1-154-1': "Walden O'Dell, Diebold CEO, 2003", '1707.08619-1-155-0': 'Below we provide a brief analysis of several real-world voting systems developed by the scientific community.', '1707.08619-1-155-1': 'These systems use the properties discussed in Sections [REF] and [REF].', '1707.08619-1-155-2': 'We include both pollsite and remote systems.', '1707.08619-1-155-3': 'This collection is by no means exhaustive, but hopefully the abundance of verifiable, evidence-based voting systems will convince the reader that there are significant technological improvements that can greatly improve election security.', '1707.08619-1-155-4': 'Our analysis is graphically represented in Table 1.', '1707.08619-1-156-0': '## Pollsite Systems', '1707.08619-1-157-0': 'The systems below were developed specifically with the requirements from Section [REF] in mind.', '1707.08619-1-157-1': 'As such, all satisfy the end-to-end verifiability criteria from Section [REF], and to a varying degree provide collection accountability, receipt-freeness, and coercion resistance.', '1707.08619-1-158-0': '### Pret a Voter', '1707.08619-1-159-0': '### Scantegrity', '1707.08619-1-160-0': 'The Scantegrity [CITATION] voter marks ballots that are very similar to optical scan ballots, with a single important difference.', '1707.08619-1-160-1': 'Each oval has printed on it, in invisible ink, a confirmation code-the encryption corresponding to this vote choice.', '1707.08619-1-160-2': 'When voters fill the oval with a special pen, the confirmation number becomes visible.', '1707.08619-1-160-3': 'The same functionality can be achieved through the use of scratch-off surfaces.', '1707.08619-1-161-0': 'Scantegrity II was used by the City of Takoma Park for its municipal elections in 2009 and 2011 [CITATION], the first secret-ballot election for public office known to use an E2E voting system within the U.S.', '1707.08619-1-162-0': '### VeriScan', '1707.08619-1-163-0': 'VeriScan [CITATION], like Scantegrity, uses optical scan ballots.', '1707.08619-1-163-1': 'But the ballots are ordinary - using regular ink - and are filled by voters using ordinary pens.', '1707.08619-1-163-2': 'Optical scanners used by VeriScan are augmented to hold the ballot deposited by a voter and to print a receipt consisting of an encryption of the selections made by the voter (or a hash thereof).', '1707.08619-1-164-0': 'Once the receipt has been given to the voter by the scanner, the voter can instruct the scanner to either retain the ballot or to return the ballot to the voter.', '1707.08619-1-164-1': 'A returned ballot should be automatically marked as no longer suitable for casting and effectively becomes a challenge ballot as in STAR-Vote (below).', '1707.08619-1-165-0': 'All encrypted ballots - whether cast or retained by a voter - are posted to a public web page where they can be checked against voter receipts.', '1707.08619-1-165-1': 'The cast ballots are listed only in encrypted form, but the retained ballots are listed in both encrypted and decrypted form so that voters can check the decryptions against their own copies of the ballots.', '1707.08619-1-166-0': '### STAR-Vote', '1707.08619-1-167-0': '### PPAT', '1707.08619-1-168-0': 'While many of the above schemes provide most of the required properties laid out in Section [REF], most do not account for everlasting privacy.', '1707.08619-1-168-1': 'However, by integrating the Perfectly Private Audit Trail (PPAT) [CITATION], many of the previously discussed systems can attain everlasting privacy.', '1707.08619-1-168-2': 'Notably, PPAT can be implemented both with mixnet schemes like Scantegrity [CITATION] and Helios [CITATION] as well as with homomorphic schemes like that used in STAR-Vote [CITATION].', '1707.08619-1-169-0': '## Remote Systems', '1707.08619-1-170-0': '### Remotegrity', '1707.08619-1-171-0': 'The Remotegrity [CITATION] voting system specification provides a layer over local coded voting systems specifications to enable their use in a remote setting.', '1707.08619-1-171-1': 'It is the only known specification that enables the voter to detect and prove attempts by adversaries to change the remote vote.', '1707.08619-1-172-0': 'Voters are mailed a package containing a coded-vote ballot and a credential sheet.', '1707.08619-1-172-1': 'The sheet contains authorization codes and lock-in codes under scratch-offs, and a return code.', '1707.08619-1-172-2': 'To vote, voters scratch-off an authorization code at random and use it as a credential to enter the candidate code.', '1707.08619-1-172-3': 'The election website displays the entered information and the return code, which indicates to the voter that the vote was received.', '1707.08619-1-172-4': 'If the website displays the correct information, the voter locks it in with a random lock-in code.', '1707.08619-1-172-5': 'If not, the voter uses another computer to vote, scratching-off another authorization code.', '1707.08619-1-172-6': 'For voter-verifiability, voters may receive multiple ballots, one of which is voted on, and the others audited.', '1707.08619-1-173-0': 'The credential authority (an insider adversary) can use the credentials to vote instead of the voter.', '1707.08619-1-173-1': 'If this happens, the voter can show the unscratched-off surface to prove the existence of a problem.', '1707.08619-1-173-2': 'Remotegrity thus achieves E2E-V, collection accountability, and software independence.', '1707.08619-1-173-3': 'Since there is no secret ballot guarantee, there is no coercion resistance.', '1707.08619-1-174-0': 'Remotegrity was made available to absentee voters in the 2011 election of the City of Takoma Park, alongside in-person voting provided by Scantegrity.', '1707.08619-1-175-0': '### Helios', '1707.08619-1-176-0': '### Selene', '1707.08619-1-177-0': '0.97Open problems:', '1707.08619-1-178-0': '# Internet Voting', '1707.08619-1-179-0': '"People of Dulsford," began Boris, "I want to assure you that as your newly elected mayor I will not just represent the people who voted for me ..."', '1707.08619-1-180-0': '"That\'s good," said Derrick, "because no-one voted for him."', '1707.08619-1-181-0': '"But the people who didn\'t vote for me as well," said Boris.', '1707.08619-1-182-0': 'There was a smattering of half-hearted clapping from the crowd.', '1707.08619-1-182-1': 'R. A. Spratt, Nanny Piggins and the Race to Power', '1707.08619-1-183-0': 'In this section we present the challenges of secure Internet voting through a set of (possibly contradictory) requirements.', '1707.08619-1-183-1': 'No system has addressed the challenges sufficiently so far, and whether it is possible to do so remains an open problem.', '1707.08619-1-183-2': 'We begin by introducing prominent contemporary instances of I-voting as case studies.', '1707.08619-1-183-3': 'Then we examine the Internet voting threat model, along the way showing how these Internet systems have failed to adequately defend themselves.', '1707.08619-1-183-4': "We look at voter authentication, verification of the correctness of a voting system's output, voter privacy and coercion resistance, protections against denial-of-service, and finally the usability and regulatory constraints faced by voting systems.", '1707.08619-1-184-0': 'One major roadblock faced exclusively by I-voting is the underlying infrastructure of the Internet.', '1707.08619-1-184-1': 'The primary security mechanism for Internet communication is Transport Layer Security (TLS), which is constantly evolving in response to vulnerabilities.', '1707.08619-1-184-2': 'For instance, the website used in the iVote system was vulnerable to the TLS FREAK [CITATION] and LogJam [CITATION] vulnerabilities.', '1707.08619-1-184-3': 'Researchers discovered this during the election period and demonstrated that they could exploit it to steal votes [CITATION].', '1707.08619-1-184-4': 'At the time, LogJam had not been publicly disclosed, highlighting the risk to I-voting from zero-day vulnerabilities.', '1707.08619-1-184-5': 'Internet voting systems must find ways to rely on properties like software independence and E2E-V before they can be considered trusted.', '1707.08619-1-185-0': 'In 2015, the U.S. Vote Foundation issued an export report on the viability of using E2E-verifiability for Internet voting [CITATION].', '1707.08619-1-185-1': 'The first two conclusions of the report were as follows.', '1707.08619-1-186-0': 'Any public elections conducted over the Internet must be end-to-end verifiable.', '1707.08619-1-186-1': 'No Internet voting system of any kind should be used for public elections before end-to-end verifiable in-person voting systems have been widely deployed and experience has been gained from their use.', '1707.08619-1-187-0': 'Many of the possible attacks on I-voting systems could be performed on postal voting systems too.', '1707.08619-1-187-1': 'The main difference is the likelihood that a very small number of people could automate the manipulation of a very large number of votes, or a carefully chosen few important votes, without detection.', '1707.08619-1-188-0': '## E2E-V I-voting in Government Elections', '1707.08619-1-189-0': 'Internet voting presents numerous challenges that have not been adequately addressed.', '1707.08619-1-189-1': 'First among these is the coercion problem which is shared with other remote voting systems in widespread use today (such as vote-by-mail).', '1707.08619-1-189-2': 'However, I-voting exacerbates the problem by making coercion and vote-selling a simple matter of a voter providing credentials to another individual.', '1707.08619-1-190-0': 'Client malware poses another significant obstacle.', '1707.08619-1-190-1': 'While E2E-verifiability mitigates the malware risks by providing voters with alternate means to ensure that their votes have been properly recorded and counted, many voters will not avail themselves of these capabilities.', '1707.08619-1-190-2': 'We could therefore have a situation were a large-scale fraud is observed by a relatively small number of voters.', '1707.08619-1-190-3': 'While the detection of a small number of instances of malfeasance can bring a halt to an election which provides collection accountability, the required evidence can be far more fleeting and difficult to validate in an Internet setting.', '1707.08619-1-190-4': 'An election should not be overturned by a small number of complaints if there is no substantive evidence to support these complaints.', '1707.08619-1-191-0': 'Targeted denial-of-service is another serious unresolved threat to I-voting.', '1707.08619-1-191-1': 'Ordinary denial-of-service (DoS) is a common threat on the Internet, and means have been deployed to mitigate - although not eliminate - these threats.', '1707.08619-1-191-2': 'The unique aspect in elections is that while ordinary DoS can slow commerce or block access to a web site for a period, the effects of a targeted DoS attack on an election can be far more severe.', '1707.08619-1-191-3': 'Since voting paterns are far from homogeneous, an attacker can launch a targeted DoS attack against populations and regions which are likely to favor a particular candidate or position.', '1707.08619-1-191-4': 'By merely making it more difficult for people in targeted populations to vote, the result of an election can be altered.', '1707.08619-1-191-5': 'As yet, we have no effective mitigations for such attacks.', '1707.08619-1-192-0': "Finally, as was observed in the U.S. Vote Foundation study [CITATION], we simply don't yet have much experience with large-scale deployments of E2E-verifiable election systems in the simpler and more manageable setting of in-person voting.", '1707.08619-1-192-1': 'It would be angerous to jump directly to the far more challenging setting of Internet voting with a heavy dependence on a technology that has not previously been deployed at scale.', '1707.08619-1-193-0': '## Alternatives to Internet Voting', '1707.08619-1-194-0': 'There are numerous alternatives to Internet voting that can help enfranchise voters who can not easily access a poll site on the day of an election.', '1707.08619-1-195-0': 'Early voting is in widespread use throughout the U.S. By extending the voting window from a single day to as much as three weeks, voters who may be away or busy on the date of an election can be afforded an opportunity to vote in person, at their convenience, at a poll site with traditional safeguards.', '1707.08619-1-195-1': 'Early voting also mitigates many of the risks of traditional systems since, for example, an equipment failure ten days prior to the close of an election is far less serious than one that takes place during a single day of voting.', '1707.08619-1-196-0': 'Some U.S. jurisdictions have adopted a vote center system in which voters may vote in person outside of their home precincts.', '1707.08619-1-196-1': 'This option has been facilitated by the use of electronic poll books, and it allows voters to, for instance, vote during a lunch break from work if they will be away from their homes during voting hours.', '1707.08619-1-196-2': 'The vote center model could potentially be extended from the current model of voters away from their home precincts but still within their home counties by allowing voters to use any poll site in the state or country.', '1707.08619-1-196-3': 'It would even be possible to establish remote voting kiosks overseas in embassies, conslates, or other official sites, and roming voting kiosks could be established with as little as two poll workers and a laptop computer.', '1707.08619-1-196-4': 'Security and accountability in all of these non-local voting scenarios can be greatly enhanced by the use of E2E-verifiability.', '1707.08619-1-197-0': 'Blank-ballot electronic delivery is another option which has gained in popularity.', '1707.08619-1-197-1': 'While there are numerous risks in using the Internet for casting of ballots, the risks a far less in simply providing blank ballots to voters.', '1707.08619-1-197-2': 'Electronic delivery of blank-ballots can save half of the round-trip time that is typical in absentee voting, and traditional methods of ballot return can be used which are less susceptible to the large-scale attacks that are possible with full Internet voting.', '1707.08619-1-198-0': '# A Look Ahead', '1707.08619-1-199-0': 'There is no remedy now to a process that was so opaque that it could have been manipulated at any stageMichael Meyer-Resende and Mirjam Kunkler, on the Iranian 2009 Presidential election', '1707.08619-1-200-0': 'Voting has always used available technology, whether pebbles dropped in an urn or marked paper put in a ballot box; it now uses computers, networks, and cryptography.', '1707.08619-1-200-1': "The core requirement, to provide public evidence of the right result from secret ballots, hasn't changed in 2500 years.", '1707.08619-1-201-0': 'Computers can improve convenience and accessibility over plain paper and manual counting.', '1707.08619-1-201-1': 'In the polling place there are good solutions, including Risk Limiting Audits and end-to-end verifiable systems.', '1707.08619-1-201-2': 'These must be more widely deployed and their options for verifying the election result must actually be used.', '1707.08619-1-202-0': 'Many of the open problems described in this paper-usable and accessible voting systems, dispute resolution, incoercibility-come together in the challenge of a remote voting system that is verifiable and usable without supervision.', '1707.08619-1-202-1': 'The open problem of a system specification that (a) does not use any paper at all and (b) is based on a simple procedure for voters and poll workers, will motivate researchers for a long time.', '1707.08619-1-202-2': 'Perhaps a better goal is a hybrid system combining paper evidence with some auditing or cryptographic verification.', '1707.08619-1-203-0': 'Research in voting brings together knowledge in many fields-cryptography, systems security, statistics, usability and accessibility, software verification, elections, law and policy to name a few-to address a critical real-world problem.', '1707.08619-1-204-0': 'The peaceful transfer of power depends on confidence in the electoral process.', '1707.08619-1-204-1': 'That confidence should not automatically be given to any outcome that seems plausible-it must be earned by producing evidence that the election result is what the people chose.', '1707.08619-1-204-2': 'Insisting on evidence reduces the opportunities for fraud, hence bringing greater security to citizens the world over.'}

{'1707.08619-2-0-0': "Elections seem simple-aren't they just counting?", '1707.08619-2-0-1': 'But they have a unique, challenging combination of security and privacy requirements.', '1707.08619-2-0-2': 'The stakes are high; the context is adversarial; the electorate needs to be convinced that the results are correct; and the secrecy of the ballot must be ensured.', '1707.08619-2-0-3': 'And they have practical constraints: time is of the essence, and voting systems need to be affordable and maintainable, and usable by voters, election officials, and pollworkers.', '1707.08619-2-0-4': 'It is thus not surprising that voting is a rich research area spanning theory, applied cryptography, practical systems analysis, usable security, and statistics.', '1707.08619-2-0-5': 'Election integrity involves two key concepts: convincing evidence that outcomes are correct and privacy, which amounts to convincing assurance that there is no evidence about how any given person voted.', '1707.08619-2-0-6': 'These are obviously in tension.', '1707.08619-2-0-7': 'We examine how current systems walk this tightrope.', '1707.08619-2-1-0': '# Introduction: What is the evidence?', '1707.08619-2-2-0': 'The Russians did three things [MATH] The third is that they tried, and they were not successful, but they still tried, to get access to voting machines and vote counting software, to play with the resultsFormer CIA Acting Director Michael Morell, Mar. 15, 2017', '1707.08619-2-3-0': 'These are baseless allegations substantiated with nothing, done on a rather amateurish, emotional levelKremlin spokesman Dmitry Peskov, Jan. 9, 2017', '1707.08619-2-4-0': 'It would take an army to hack into our voting system.', '1707.08619-2-4-1': 'Tom Hicks, EAC Commissioner, Oct. 6, 2016', '1707.08619-2-5-0': 'It is not enough for an election to produce the correct outcome.', '1707.08619-2-5-1': 'The electorate must also be convinced that the announced result reflects the will of the people.', '1707.08619-2-5-2': 'And for a rational person to be convinced requires evidence.', '1707.08619-2-6-0': 'Modern technology-computer and communications systems-is fragile and vulnerable to programming errors and undetectable manipulation.', '1707.08619-2-6-1': 'No current system that relies on electronic technology alone to capture and tally votes can provide convincing evidence that election results are accurate without endangering or sacrificing the anonymity of votes.', '1707.08619-2-7-0': 'Paper ballots, on the other hand, have some very helpful security properties: they are readable (and countable, and re-countable) by humans; they are relatively durable; and they are tamper-evident.', '1707.08619-2-7-1': 'Votes cast on paper can be counted using electronic technology; then the accuracy of the count can be checked manually to ensure that the technology functioned adequately well.', '1707.08619-2-7-2': 'Statistical methods allow the accuracy of the count to be assessed by examining only a fraction of the ballots manually, often a very small fraction.', '1707.08619-2-7-3': 'If there is also convincing evidence that the collection of ballots has been conserved (no ballots added, lost, or modified) then this combination-voter-verifiable paper ballots, a mechanized count, and a manual check of the accuracy of that count-can provide convincing evidence that announced electoral outcomes are correct.', '1707.08619-2-8-0': 'Conversely, absent convincing evidence that the paper trail has been conserved, a manual double-check of electronic results against the paper trail will not be convincing.', '1707.08619-2-8-1': 'If the paper trail has been conserved adequately, then a full manual tally of the ballots can correct the electronic count if the electronic count is incorrect.', '1707.08619-2-9-0': 'These considerations have led many election integrity advocates to push for a voter-verifiable paper trail (VVPAT).', '1707.08619-2-10-0': 'In the 2016 presidential election, about three quarters of Americans voted using systems that generated voter-verifiable paper records.', '1707.08619-2-10-1': 'The aftermath of the election proved that even if 100% of voters had used such systems, it would not have sufficed to provide convincing evidence that the reported results are accurate.', '1707.08619-2-11-0': "The bottom line is that the paper trail is not worth the paper it's printed on.", '1707.08619-2-11-1': 'Clearly this must change.', '1707.08619-2-12-0': "Other techniques like software independence and end-to-end verifiability can offer far greater assurance in the accuracy of an election's outcome, but these methods have not been broadly applied.", '1707.08619-2-13-0': '## Why so hard?', '1707.08619-2-14-0': 'Several factors make it difficult to generate convincing evidence that reported results are correct.', '1707.08619-2-14-1': 'The first is the trust model.', '1707.08619-2-15-0': 'No one is trusted In any significant election, voters, election officials, and equipment and software cannot necessarily be trusted by anyone with a stake in the outcome.', '1707.08619-2-15-1': 'Voters, operators, system designers, manufacturers, and external parties are all potential adversaries.', '1707.08619-2-16-0': 'The need for evidence Because officials and equipment may not be trustworthy, elections should be evidence-based.', '1707.08619-2-16-1': 'Any observer should be able to verify the reported results based on trustworthy evidence from the voting system.', '1707.08619-2-16-2': 'Many in-person voting systems fail to provide sufficient evidence; and as we shall see Internet systems scarcely provide any at all.', '1707.08619-2-17-0': 'The secret ballot Perhaps the most distinctive element of elections is the secret ballot, a critical safeguard that defends against vote selling and voter coercion.', '1707.08619-2-17-1': 'In practical terms, voters should not be able to prove how they voted to anyone, even if they wish to do so.', '1707.08619-2-17-2': 'This restricts the types of evidence that can be produced by the voting system.', '1707.08619-2-17-3': 'Encryption alone is not sufficient, since the voters may choose to reveal their selections in response to bribery or coercion.', '1707.08619-2-18-0': 'The challenge of voting is thus to use fragile technology to produce trustworthy, convincing evidence of the correctness of the outcome while protecting voter privacy in a world where no person or machine may be trusted.', '1707.08619-2-18-1': 'The resulting voting system and its security features must also be usable by regular voters.', '1707.08619-2-19-0': 'The aim of this paper is to explain the important requirements of secure elections and the solutions already available from e-voting research, then to identify the most important directions for research.', '1707.08619-2-20-0': 'Prior to delving into our discussion, we need to make a distinction in terminology.', '1707.08619-2-20-1': 'Pollsite voting systems are those in which voters record and cast ballots at predetermined locations, often in public areas with strict monitoring.', '1707.08619-2-20-2': 'Remote voting refers to a system where voters fill out ballots anywhere, and then send them to a central location to cast them, either physically mailing them in the case of vote-by-mail, or sending them over the Internet in the case of Internet voting.', '1707.08619-2-21-0': 'The next section defines the requirements, beginning with notions of election evidence, then considering privacy, and concluding with more general usability and security requirements.', '1707.08619-2-21-1': 'Section [REF] describes the cryptographic, statistical, and engineering tools that have been developed for designing voting systems with verifiably correct election outcomes.', '1707.08619-2-21-2': 'Section [REF] discusses the challenge of satisfying our requirements for security using the tools presented in real-world election systems.', '1707.08619-2-21-3': 'Section [REF] concludes with the promise and problems associated with Internet voting.', '1707.08619-2-22-0': '# Requirements for Secure Voting', '1707.08619-2-23-0': "Trustworthiness before trustOnora O'Neill", '1707.08619-2-24-0': '## Trust, Verifiability, and Evidence', '1707.08619-2-25-0': 'For an election to be accepted as legitimate, the outcome should be convincing to all-and in particular to the losers-leaving no valid grounds to challenge the outcome.', '1707.08619-2-25-1': 'Whether elections are conducted by counting paper ballots by hand or using computer technology, the possibility of error or fraud necessitates assurances of the accuracy of the outcome.', '1707.08619-2-26-0': 'It is clear that a naive introduction of computers into voting introduces the possibility of wholesale and largely undetectable fraud.', '1707.08619-2-26-1': "If we can't detect it, how can we prevent it?", '1707.08619-2-27-0': '### Risk-Limiting Audits', '1707.08619-2-28-0': 'Statistical post-election audits provide assurance that a reported outcome is correct, by examining some or all of an audit trail consisting of durable, tamper-evident, voter-verifiable records.', '1707.08619-2-28-1': 'Typically the audit trail consists of paper ballots.', '1707.08619-2-29-0': 'The outcome of an election is the set of winners.', '1707.08619-2-29-1': 'An outcome is incorrect if it differs from the set of winners output by a perfectly accurate manual tabulation of the audit trail.', '1707.08619-2-30-0': 'An audit of an election contest is a risk-limiting audit (RLA) with risk limit [MATH] if it has the following two properties:', '1707.08619-2-31-0': 'If the reported contest outcome under audit is incorrect, the probability that the audit leads to correcting the outcome is at least [MATH].', '1707.08619-2-31-1': 'The audit never indicates a need to alter a reported outcome that is correct.', '1707.08619-2-32-0': '(In this context, "correct" means "what a full manual tally of the paper trail would show."', '1707.08619-2-32-1': 'If the paper trail is unreliable, a RLA in general cannot detect that.', '1707.08619-2-32-2': 'RLAs should be preceded by "compliance audits" that check whether the audit trail itself is adequately reliable to determine who won.)', '1707.08619-2-32-3': 'Together, these two properties imply that post-RLA, either the reported set of winners is the set that a perfectly accurate hand count of the audit trail would show, or an event with probability no larger than [MATH] has occurred.', '1707.08619-2-32-4': '(That event is that the outcome was incorrect, but the RLA did not lead to correcting the outcome.)', '1707.08619-2-32-5': 'RLAs amount to a limited form of probabilistic error correction: by relying on appropriate random sampling of the audit trail and hypothesis tests, they have a known minimum probability of correcting the outcome.', '1707.08619-2-32-6': 'They are not designed to ensure that the reported numerical tally is correct, only that the outcome is correct.', '1707.08619-2-33-0': 'The following procedure is a trivial RLA: with probability [MATH], perform a full manual tally of the audit trail.', '1707.08619-2-33-1': 'Amend the outcome to match the set of winners the full hand count shows if that set is different.', '1707.08619-2-34-0': 'The art in constructing RLAs consists of maintaining the risk limit while performing less work than a full hand count when the outcome is correct.', '1707.08619-2-34-1': 'Typically, this involves framing the audit as a sequential test of the statistical hypothesis that the outcome is incorrect.', '1707.08619-2-34-2': 'To reject that hypothesis is to conclude that the outcome is correct.', '1707.08619-2-34-3': "RLAs have been developed for majority contests, plurality contests, and vote-for-[MATH] contests and complex social choice functions including D'Hondt and other proportional representation rules-see below.", '1707.08619-2-34-4': 'RLAs have also been devised to check more than one election contest simultaneously [CITATION].', '1707.08619-2-35-0': '### Software Independence', '1707.08619-2-36-0': 'Rivest and Wack introduced a definition targeted specifically at detecting misbehavior in computer-based elections:', '1707.08619-2-37-0': '[CITATION] A voting system is software independent if an undetected change or error in its software cannot cause an undetectable change or error in an election outcome.', '1707.08619-2-38-0': 'Software independence clearly expresses that it should not be necessary to trust software to determine election outcomes, but it does not say what procedures or types of evidence should be trusted instead.', '1707.08619-2-38-1': 'A system that is not software independent cannot produce a convincing evidence trail, but neither can a paper-based system that does not ensure that the paper trail is complete and intact, a cryptographic voting system that relies on an invalid cryptographic assumption, or a system that relies on audit procedures but lacks a means of assuring that those procedures are properly followed.', '1707.08619-2-38-2': 'We could likewise demand independence of many other kinds of trust assumptions: hardware, paper chain-of-custody, cryptographic setup, computational hardness, procedures, good randomness generation etc.', '1707.08619-2-39-0': 'Rivest and Wack also define a stronger form of the property that includes error recovery:', '1707.08619-2-40-0': '[CITATION] A voting system is strongly software independent if it is software independent and a detected change or error in an election outcome (due to the software) can be corrected without rerunning the election.', '1707.08619-2-41-0': 'A strongly software-independent system can recover from software errors or bugs, but that recovery in turn is generally based on some other trail of evidence.', '1707.08619-2-42-0': 'A software independent system can be viewed as a form of tamper-evident system: a material software problem leaves a detectable trace.', '1707.08619-2-42-1': 'Strongly software independent systems are resilient: not only do material software problems leave a trace, the overall election system can recover from a detected problem.', '1707.08619-2-43-0': "One mechanism to provide software independence is to record votes on a paper record that provides physical evidence of voter's intent, can be inspected by the voter prior to casting the vote, and-if preserved intact-can later be manually audited to check the election outcome.", '1707.08619-2-43-1': 'Risk-limiting audits (see Section [REF]) can then achieve a pre-specified level of assurance that results are correct; machine assisted risk-limiting audits [CITATION], can help minimize the amount of labor required for legacy systems that do not provide a cast-vote record for every ballot, linked to the corresponding ballot.', '1707.08619-2-44-0': '0.97Open problems:', '1707.08619-2-45-0': '### End-to-end verifiability', '1707.08619-2-46-0': 'The concern regarding fraud and desire for transparency has motivated the security and crypto communities to develop another approach to voting system assurance: end-to-end verifiability (E2E-V).', '1707.08619-2-46-1': 'An election that is end-to-end verifiable achieves software independence together with the analagous notion of hardware independence as well as independence from actions of election personnel and vendors.', '1707.08619-2-46-2': 'Rather than attempting to verify thousands of lines of code or closely monitor all of the many processes in an election, E2E-V focuses on providing a means to detect errors or fraud in the process of voting and counting.', '1707.08619-2-46-3': 'The idea behind E2E-V is to enable voters themselves to monitor the integrity of the election; democracy for the people by the people, as it were.', '1707.08619-2-46-4': 'This is challenging because total transparency is not possible without undermining the secret ballot, hence the mechanisms to generate such evidence have to be carefully designed.', '1707.08619-2-47-0': '(adapted from [CITATION]) A voting system is end-to-end verifiable if it has the following three kinds of verifiability:', '1707.08619-2-48-0': 'If verification relies on trusting entities, software, or hardware, the voter and/or auditor should be able to choose them freely.', '1707.08619-2-48-1': 'Trusted procedures, if there are any, must be open to meaningful observation by every voter.', '1707.08619-2-49-0': 'Note that the above definition allows each voter to check that her vote is correctly collected, thus ensuring that attempts to change or delete cast votes are detected.', '1707.08619-2-49-1': 'In addition, it should also be possible to check the list of voters who cast ballots, to ensure that votes are not added to the collection (i.e., to prevent ballot-box stuffing).', '1707.08619-2-49-2': 'This is called eligibility verifiability [CITATION].', '1707.08619-2-50-0': '### Collection Accountability', '1707.08619-2-51-0': 'In an E2E-V election protocol, voters can check whether their votes have been properly counted, but if they discover a problem, there may not be adequate evidence to correct it.', '1707.08619-2-51-1': 'An election system that is collection-accountable provides voters with evidence of any failure to collect their votes.', '1707.08619-2-52-0': 'An election system is collection accountable if any voter who detects that her vote has not been collected has, as part of the vote-casting protocol, convincing evidence that can be presented to an independent party to demonstrate that the vote has not been collected.', '1707.08619-2-53-0': 'Another form of evidence involves providing each voter with a code representing her votes, such that knowledge of a correct code is evidence of casting a particular vote [CITATION].', '1707.08619-2-53-1': 'Yet another mechanism is a suitable paper receipt.', '1707.08619-2-53-2': 'Forensic analysis may provide evidence that this receipt was not forged by a voter [CITATION].', '1707.08619-2-54-0': '0.97Open problems:', '1707.08619-2-55-0': '### Dispute Resolution', '1707.08619-2-56-0': 'While accountability helps secure the election process, it is not very useful if there is no way to handle disputes.', '1707.08619-2-56-1': 'If a voter claims, on the basis of accountability checks provided by a system, that something has gone wrong, there needs to be a mechanism to address this.', '1707.08619-2-56-2': 'This is known as dispute resolution:', '1707.08619-2-57-0': '[CITATION] A voting system is said to have dispute resolution if, when there is a dispute between two participants regarding honest participation, a third party can correctly resolve the dispute.', '1707.08619-2-58-0': 'An alternative to dispute resolution is dispute-freeness:', '1707.08619-2-59-0': '[CITATION] A dispute-free voting system has built-in prevention mechanisms that eliminate disputes among the active participants; any third party can check whether an active participant has cheated.', '1707.08619-2-60-0': '### From Verifiable to Verified', '1707.08619-2-61-0': 'Constructing a voting system that creates sufficient evidence to reveal problems is not enough on its own.', '1707.08619-2-61-1': 'That evidence must actually be used-and used appropriately-to ensure the accuracy of election outcomes.', '1707.08619-2-62-0': 'An election result may not be verified, even if it is generated by an end-to-end verifiable voting system.', '1707.08619-2-62-1': 'For verification of the result, we need several further conditions to be satisfied:', '1707.08619-2-63-0': "These issues involve complex human factors, including voters' incentives to participate in verification.", '1707.08619-2-63-1': 'Little work has been done on this aspect of the problem.', '1707.08619-2-64-0': 'An E2E-V system might give an individual voter assurance that her vote has not been tampered with if that voter performs certain checks.', '1707.08619-2-64-1': 'However, sufficiently many voters must do this in order to provide evidence that the election outcome as a whole is correct.', '1707.08619-2-64-2': 'Combining risk-limiting audits with E2E-V systems can provide a valuable layer of protection in the case that an insufficient number of voters participate in verification.', '1707.08619-2-65-0': 'Finally, another critical verification problem that has received little attention to date is how to make schemes that are recoverable in the face of errors.', '1707.08619-2-65-1': 'We do not want to have to abort and rerun an election every time a check a fails.', '1707.08619-2-65-2': 'Certain levels of detected errors can be shown to be highly unlikely if the outcome is incorrect, and hence can be tolerated.', '1707.08619-2-65-3': 'Other types and patterns of error cast doubt on the outcome and may require either full inspection or retabulation of the paper trail or, if the paper trail cannot be relied upon, a new election.', '1707.08619-2-66-0': 'Both Kusters et al. [CITATION] and Kiayias et al. [CITATION] model voter-initiated auditing [CITATION] and its implications for detection of an incorrect election result.', '1707.08619-2-66-1': 'Both definitions turn uncertainty about voter initiated auditing into a bound on the probability of detecting deviations of the announced election result from the truth.', '1707.08619-2-67-0': '0.97Open problems:', '1707.08619-2-68-0': '## Voter Authentication', '1707.08619-2-69-0': 'A significant challenge for election systems is the credentialing of voters to ensure that all eligible voters, and no one else, can cast votes.', '1707.08619-2-69-1': 'This presents numerous questions: what kinds of credentials should be used?', '1707.08619-2-69-2': 'How should they be issued?', '1707.08619-2-69-3': 'Can they be revoked or de-activated?', '1707.08619-2-69-4': 'Are credentials good for a single election or for an extended period?', '1707.08619-2-69-5': 'How difficult are they to share, transfer, steal, or forge?', '1707.08619-2-69-6': 'Can the ability to create genuine-looking forgeries help prevent coercion?', '1707.08619-2-69-7': 'These questions must be answered carefully, and until they are satisfied for remote voting, pollsite voting is the only robust way to address these questions-and even then, in-person credentialing is subject to forgery, distribution, and revocation concerns (for instance, the Dominican Republic recently held a pollsite election where voters openly sold their credentials [CITATION]).', '1707.08619-2-69-8': 'In the U.S., there is concern that requiring in-person credentialing, in the form of voter ID, disenfranchises legitimate voters.', '1707.08619-2-70-0': '0.97Open problems:', '1707.08619-2-71-0': '## Privacy, Receipt Freeness, and Coercion Resistance', '1707.08619-2-72-0': 'In most security applications, privacy and confidentiality are synonymous.', '1707.08619-2-72-1': 'In elections, however, privacy has numerous components that go well beyond typical confidentiality.', '1707.08619-2-72-2': 'Individual privacy can be compromised by "normal" election processes such as a unanimous result.', '1707.08619-2-72-3': 'Voters may be coerced if they can produce a proof of how they voted, even if they have to work to do so.', '1707.08619-2-73-0': "Privacy for votes is a means to an end: if voters don't express their true preferences then the election may not produce the right outcome.", '1707.08619-2-73-1': 'This section gives an overview of increasingly strong definitions of what it means for voters to be free of coercion.', '1707.08619-2-74-0': '### Basic Confidentiality', '1707.08619-2-75-0': 'We will take ballot privacy to mean that the election does not leak any information about how any voter voted beyond what can be deduced from the announced results.', '1707.08619-2-75-1': 'Confidentiality is not the only privacy requirement in elections, but even simple confidentiality poses significant challenges.', '1707.08619-2-75-2': 'It is remarkable how many deployed e-voting systems have been shown to lack even the most basic confidentiality properties (e.g., [CITATION]).', '1707.08619-2-76-0': 'Perhaps more discouraging to basic privacy is the fact that remote voting systems (both paper and electronic) inherently allow voters to eschew confidentiality.', '1707.08619-2-76-1': "Because remote systems enable voters to fill out their ballots outside a controlled environment, anyone can watch over the voter's shoulder while she fills out her ballot.", '1707.08619-2-77-0': 'In an election-unlike, say, in a financial transaction-even the candidate receiving an encrypted vote should not be able to decrypt it.', '1707.08619-2-77-1': 'Instead, an encrypted (or otherwise shrouded) vote must remain confidential to keep votes from being directly visible to election authorities.', '1707.08619-2-78-0': 'Some systems, such as code voting [CITATION] and the Norwegian and Swiss Internet voting schemes, defend privacy against an attacker who controls the computer used for voting; however, this relies on assumptions about the privacy and integrity of the code sheet.', '1707.08619-2-78-1': 'Some schemes, such as JCJ/Civitas [CITATION], obscure who has voted while providing a proof that only eligible votes were included in the tally.', '1707.08619-2-79-0': 'Several works [CITATION] [CITATION], following Benaloh [CITATION] formalize the notion of privacy as preventing an attacker from noticing when two parties swap their votes.', '1707.08619-2-80-0': '0.97Open problems:', '1707.08619-2-81-0': '### Everlasting Privacy', '1707.08619-2-82-0': "Moran and Naor expressed concern over what might happen to encrypted votes that can still be linked to their voter's name some decades into the future, and hence decrypted by superior technology.", '1707.08619-2-82-1': 'They define a requirement to prevent this:', '1707.08619-2-83-0': '[CITATION] A voting scheme has everlasting privacy if its privacy does not depend on assumptions of cryptographic hardness.', '1707.08619-2-84-0': 'Their solution uses perfectly hiding commitments to the votes, which are aggregated homomorphically.', '1707.08619-2-84-1': 'Instead of privacy depending upon a cryptographic hardness assumption, it is the integrity of an election that depends upon a hardness assumption; and only a real-time compromise of the assumption can have an impact.', '1707.08619-2-85-0': '### Systemic Privacy Loss', '1707.08619-2-86-0': 'We generally accept that without further information, a voter is more likely to have voted for a candidate who has received more votes, but additional data is commonly released which can further erode voter privacy.', '1707.08619-2-86-1': 'Even if we exclude privacy compromises, there are other privacy risks which must be managed.', '1707.08619-2-86-2': 'If voters achieve privacy by encrypting their selections, the holders of decryption keys can view their votes.', '1707.08619-2-86-3': 'If voters make their selections on devices out of their immediate control (e.g. official election equipment), then it is difficult to assure them that these devices are not retaining information that could later compromise their privacy.', '1707.08619-2-86-4': 'If voters make their selections on their own devices, then there is an even greater risk that these devices could be infected with malware that records (and perhaps even alters) their selections (see, for instance, the Estonian system [CITATION]).', '1707.08619-2-87-0': '0.97Open problems:', '1707.08619-2-88-0': '### Receipt-freeness', '1707.08619-2-89-0': 'Preventing coercion and vote-selling was considered solved with the introduction of the Australian ballot.', '1707.08619-2-89-1': 'The process of voting privately within a public environment where privacy can be monitored and enforced prevents improper influence.', '1707.08619-2-89-2': 'Recent systems have complicated this notion, however.', '1707.08619-2-89-3': 'If a voting protocol provides a receipt but is not carefully designed, the receipt can be a channel for information to the coercive adversary.', '1707.08619-2-90-0': 'Benaloh and Tuinstra [CITATION] pointed out that passive privacy is insufficient for resisting coercion in elections:', '1707.08619-2-91-0': 'A voting system is receipt free if a voter is unable to prove how she voted even if she actively colludes with a coercer and deviates from the protocol in order to try to produce a proof.', '1707.08619-2-92-0': 'Traditional elections may fail receipt-freeness too.', '1707.08619-2-92-1': 'In general, if a vote consists of a long list of choices, the number of possible votes may be much larger than the number of likely voters.', '1707.08619-2-92-2': 'This is sometimes called (a failure of) the short ballot assumption [CITATION].', '1707.08619-2-92-3': 'Prior to each election, coercers assign a particular voting pattern to each voter.', '1707.08619-2-92-4': 'When the individual votes are made public, any voter who did not cast their pattern can then be found out.', '1707.08619-2-92-5': 'This is sometimes called the Italian attack, after a once prevalent practice in Sicily.', '1707.08619-2-92-6': 'It can be easily mitigated when a vote can be broken up, but is difficult to mitigate in systems like IRV in which the vote is complex but must be kept together.', '1707.08619-2-92-7': 'Mitigations are discussed in Sections [REF] and [REF].', '1707.08619-2-93-0': 'Incoercibility has been defined and examined in the universally composable framework in the context of general multiparty computation [CITATION].', '1707.08619-2-93-1': 'These definitions sidestep the question of whether the voting function itself allows coercion (by publishing individual complex ballots, or by revealing a unanimous result for example)-they examine whether the protocol introduces additional opportunities for coercion.', '1707.08619-2-93-2': 'With some exceptions (such as [CITATION]), they usually focus on a passive notion of receipt-freeness, which is not strong enough for voting.', '1707.08619-2-94-0': '### Coercion Resistance', '1707.08619-2-95-0': 'Schemes can be receipt-free, but not entirely resistant to coercion.', '1707.08619-2-95-1': 'Schemes like Pret a Voter [CITATION] that rely on randomization for receipt-freeness can be susceptible to forced randomization, where a coercer forces a voter to always choose the first choice on the ballot.', '1707.08619-2-95-2': 'Due to randomized candidate order, the resulting vote will be randomly distributed.', '1707.08619-2-95-3': 'If a specific group of voters are coerced in this way, it can have a disproportionate impact on the election outcome.', '1707.08619-2-96-0': 'If voting rolls are public and voting is not mandatory, this has an effect equivalent to prevent forced abstention, wherein a coercer refuses to let a voter vote.', '1707.08619-2-96-1': 'Schemes that rely on credentialing are also susceptible to coercion by forced surrender of credentials.', '1707.08619-2-97-0': 'One way to fully resist forced abstention is to obscure who voted.', '1707.08619-2-97-1': 'However, this is difficult to reconcile with the opportunity to verify that only eligible voters have voted (eligibility verifiability), though some schemes achieve both [CITATION].', '1707.08619-2-98-0': 'Moran and Naor [CITATION] provide a strong definition of receipt freeness in which a voter may deviate actively from the protocol in order to convince a coercer that she obeyed.', '1707.08619-2-98-1': 'Their model accommodates forced randomization.', '1707.08619-2-98-2': 'A scheme is resistant to coercion if the voter can always pretend to have obeyed while actually voting as she likes.', '1707.08619-2-99-0': "A voting scheme is coercion resistant if there exists a way for a coerced voter to cast her vote such that her coercer cannot distinguish whether or not she followed the coercer's instructions.", '1707.08619-2-100-0': 'Coercion resistance is defined in [CITATION] to include receipt freeness and defence against forced-randomization, forced abstention and the forced surrender of credentials.', '1707.08619-2-100-1': "More general definitions include [CITATION], which incorporates all these attacks along with Moran and Naor's notion of a coercion resistance strategy.", '1707.08619-2-101-0': 'Note that if the coercer can monitor the voter throughout the vote casting period, then resistance is futile.', '1707.08619-2-101-1': 'For in-person voting, we assume that the voter is isolated from any coercer while she is in the booth (although this is questionable in the era of mobile phones).', '1707.08619-2-101-2': 'For remote voting, we need to assume that voters will have some time when they can interact with the voting system (or the credential-granting system) unobserved.', '1707.08619-2-102-0': '### More Coercion Considerations', '1707.08619-2-103-0': 'Some authors have tried to provide some protection against coercion without achieving full coercion resistance.', '1707.08619-2-103-1': 'Caveat coercitor [CITATION] proposes the notion of coercion evidence and allows voters to cast multiple votes using the same credential.', '1707.08619-2-104-0': '0.97Open problem:', '1707.08619-2-105-0': '## Usability', '1707.08619-2-106-0': 'A voting system must be usable by voters, poll-workers, election officials, observers, and so on.', '1707.08619-2-106-1': 'Voters who may not be computer literate-and sometimes not literate at all-should be able to vote with very low error rates.', '1707.08619-2-106-2': 'Although some error is regarded as inevitable, it is also critical that the interface not drive errors in a particular direction.', '1707.08619-2-106-3': 'For instance, a list of candidates that crosses a page boundary could cause the candidates on the second page to be missed.', '1707.08619-2-106-4': 'Whatever security mechanisms we add to the voting process should operate without degrading usability, otherwise the resulting system will likely be unacceptable.', '1707.08619-2-106-5': 'A full treatment of usability in voting is beyond the scope of this paper.', '1707.08619-2-106-6': 'However, we note that E2E-V systems (and I-voting systems, even when not E2E-V) add additional processes for voters and poll workers to follow.', '1707.08619-2-106-7': "If verification processes can't be used properly by real voters, the outcome will not be properly verified.", '1707.08619-2-106-8': 'One great advantage of statistical audits is to shift complexity from voters to auditors.', '1707.08619-2-107-0': '0.97Open problems:', '1707.08619-2-108-0': '## Local Regulatory Requirements', '1707.08619-2-109-0': 'A variety of other mechanical requirements are often imposed by legal requirements that vary among jurisdictions.', '1707.08619-2-109-1': 'For example:', '1707.08619-2-110-0': 'Newer electronic and I-voting systems raise important policy challenges for real-world adoption.', '1707.08619-2-110-1': 'For example, in STAR-Vote [CITATION], there will be multiple copies of every vote record: mostly electronic records, but also paper records.', '1707.08619-2-110-2': 'There may be instances where one is damaged or destroyed and the other is all that remains.', '1707.08619-2-110-3': 'When laws speak to retention of "the ballot", that term is no longer well-defined.', '1707.08619-2-110-4': 'Such requirements may need to be adapted to newer voting systems.', '1707.08619-2-111-0': '### Complex Election Methods', '1707.08619-2-112-0': 'Many countries allow voters to select, score, or rank candidates or parties.', '1707.08619-2-112-1': 'Votes can then be tallied in a variety of complex ways [CITATION].', '1707.08619-2-112-2': 'None of the requirements for privacy, coercion-resistance, or the provision of verifiable evidence change.', '1707.08619-2-112-3': 'However, many tools that achieve these properties for traditional "first-past-the-post" elections need to be redesigned.', '1707.08619-2-113-0': 'An election method might be complex at the voting or the tallying end.', '1707.08619-2-113-1': "For example, party-list methods such as D'Hondt and Sainte-Lague have simple voting, in which voters select their candidate or party, but complex proportional seat allocation.", '1707.08619-2-113-2': 'Borda, Range Voting, and Approval Voting allow votes to be quite expressive but are simple to tally by addition.', '1707.08619-2-113-3': "Condorcet's method and related functions [CITATION] can be arbitrarily complex, as they can combine with any social choice function.", '1707.08619-2-113-4': 'Instant Runoff Voting (IRV) and the Single Transferable Vote (STV) are both expressive and complicated to tally.', '1707.08619-2-113-5': 'This makes for several challenges.', '1707.08619-2-114-0': '# How can we secure voting?', '1707.08619-2-115-0': 'These truths are self-evident but not self-enforcing Barack Obama', '1707.08619-2-116-0': 'The goal of this section and the next is to provide a state-of-the-art picture of current solutions to voting problems and ongoing voting research, to motivate further work on open problems, and to define clear directions both in research and election policy.', '1707.08619-2-117-0': '## The Role of Paper and Ceremonies', '1707.08619-2-118-0': 'Following security problems with direct-recording electronic voting systems (DREs) [CITATION], many parts of the USA returned to the use of paper ballots.', '1707.08619-2-118-1': 'If secure custody of the paper ballots is assumed, paper provides durable evidence required to determine the correctness of the election outcome.', '1707.08619-2-118-2': 'For this reason, when humans vote from untrusted computers, cryptographic voting system specifications often use paper for security, included in the notions of dispute-freeness, dispute resolution, collection accountability and accountability [CITATION] (all as defined in Section [REF]).', '1707.08619-2-119-0': 'Note that the standard approach to dispute resolution, based on non-repudiation, cannot be applied to the voting problem in the standard fashion, because the human voter does not have the ability to check digital signatures or digitally sign the vote (or other messages that may be part of the protocol) unassisted.', '1707.08619-2-120-0': 'Dispute-freeness or accountability are often achieved in a polling place through the use of cast paper ballots, and the evidence of their chain of custody (e.g., wet-ink signatures).', '1707.08619-2-120-1': 'Paper provides an interface for data entry for the voter-not simply to enter the vote, but also to enter other messages that the protocol might require-and data on unforgeable paper serves many of the purposes of digitally signed data.', '1707.08619-2-120-2': 'Thus, for example, when a voter marks a Pret a Voter [CITATION] or Scantegrity [CITATION] ballot, she is providing an instruction that the voting system cannot pretend was something else.', '1707.08619-2-120-3': 'The resulting vote encryption has been physically committed to by the voting system-by the mere act of printing the ballot-before the voter "casts" her vote.', '1707.08619-2-121-0': 'Physical ceremony, such as can be witnessed while the election is ongoing, also supports verifiable cryptographic election protocols (see Section [REF]).', '1707.08619-2-121-1': 'Such ceremonies include the verification of voter credentials, any generation of randomness if required for the choice between cast and audit, any vote-encryption-verification performed by election officials, etc.', '1707.08619-2-122-0': 'The key aspect of these ceremonies is the chance for observers to see that they are properly conducted.', '1707.08619-2-123-0': '0.97Open problem:', '1707.08619-2-124-0': '## Statistics and Auditing', '1707.08619-2-125-0': '## Cryptographic Tools and Designs', '1707.08619-2-126-0': '### Major Approaches to Voting Cryptography', '1707.08619-2-127-0': 'Typically E2E-V involves providing each voter with a protected receipt-an encrypted or encoded version of their vote-at the time the vote is cast.', '1707.08619-2-127-1': 'The voter can later use her receipt to check whether her vote is included correctly in the tabulation process.', '1707.08619-2-127-2': 'Furthermore, given the set of encrypted votes (as well as other relevant information, like the public keys), the tabulation is universally verifiable: anyone can check whether it is correct.', '1707.08619-2-127-3': 'To achieve this, most E2E-V systems rely on a public bulletin board, where the set of encrypted ballots is published in an append-only fashion.', '1707.08619-2-128-0': 'The votes can then be turned into a tally in one of two main ways.', '1707.08619-2-128-1': 'Homomorphic encryption schemes [CITATION] allow the tally to be produced on encrypted votes.', '1707.08619-2-128-2': 'Verifiable shuffling transforms a list of encrypted votes into a shuffled list that can be decrypted without the input votes being linked to the (decrypted) output.', '1707.08619-2-128-3': 'There are efficient ways to prove that the input list exactly matches the output [CITATION].', '1707.08619-2-129-0': '### Techniques for Cast-as-Intended Verification', '1707.08619-2-130-0': 'How can a voter verify that her cast vote is the one she wanted?', '1707.08619-2-130-1': 'Code Voting, first introduced by Chaum [CITATION], gives each voter a sheet of codes for each candidate.', '1707.08619-2-130-2': 'Assuming the code sheet is valid, the voter can cast a vote on an untrusted machine by entering the code corresponding to her chosen candidate and waiting to receive the correct confirmation code.', '1707.08619-2-130-3': 'Modern interpretations of code voting include [CITATION].', '1707.08619-2-131-0': 'Code voting only provides assurance that the correct voting code reached the server, it does not of itself provide any guarantees that the code will subsequently be correctly counted.', '1707.08619-2-131-1': 'A scheme that improves on this is Pretty Good Democracy [CITATION], where knowledge of the codes is threshold shared in such a way that receipt of the correct confirmation code provides assurance that the voting code has been registered on the bulletin board by a threshold set of trustees, and hence subsequently counted.', '1707.08619-2-132-0': 'The alternative is to ask the machine to encrypt a vote directly, but verify that it does so correctly.', '1707.08619-2-132-1': 'Benaloh [CITATION] developed a simple protocol to enable vote encryption on an untrusted voting machine.', '1707.08619-2-132-2': 'A voter uses a voting machine to encrypt any number of votes, and casts only one of these encrypted votes.', '1707.08619-2-132-3': 'All the other votes may be "audited" by the voter.', '1707.08619-2-132-4': 'If the encryption is audited, the voting system provides a proof that it encrypted the vote correctly, and the proof is public.', '1707.08619-2-132-5': 'The corresponding ballot cannot be cast as the correspondence between the encryption and the ballot is now public, and the vote is no longer secret.', '1707.08619-2-132-6': 'Voters take home receipts corresponding to the encryptions of their cast ballots as well as any ballots that are to be audited.', '1707.08619-2-132-7': 'They may check the presence of these on a bulletin board, and the correctness proofs of the audited encryptions using software obtained from any of several sources.', '1707.08619-2-132-8': 'However, even the most dilligent voters need only check that their receipts match the public record and that any ballots selected for audit display correct candidate selections.', '1707.08619-2-132-9': 'The correctness proofs are part of the public record that can be verified by any individual or observer that is verifying correct tallying.', '1707.08619-2-133-0': '### Formal models and security analyses of cast-as-intended verification protocols', '1707.08619-2-134-0': 'In addition to the work of Adida on assisted-human interactive proofs (AHIPs, see [CITATION]), there has been some work on a rigorous understanding of one or more properties of single protocols, including the work of Moran and Naor [CITATION] and Kusters et al. [CITATION].', '1707.08619-2-135-0': 'There have also been formalizations of voting protocols with human participants, such as by Moran and Naor [CITATION] (for a polling protocol using tamper-evident seals on envelopes) and Kiayias et al. [CITATION].', '1707.08619-2-135-1': 'However, there is no one model that is sufficient for the rigorous understanding of the prominent protocols used/proposed for use in real elections.', '1707.08619-2-135-2': 'The absence of proofs has led to the overlooking of vulnerabilities in the protocols in the past, see [CITATION].', '1707.08619-2-136-0': "Many systems use a combination of paper, cryptography, and auditing to achieve E2E-V in the polling place, including Markpledge [CITATION], Wombat [CITATION], Demos [CITATION], Pret a Voter [CITATION], STAR-Vote [CITATION], and Moran and Naor's scheme [CITATION].", '1707.08619-2-136-1': 'Their properties are summarised more thoroughly in the following section.', '1707.08619-2-137-0': 'The cryptographic literature has numerous constructions of end-to-end verifiable election schemes (e.g., [CITATION]).', '1707.08619-2-137-1': 'There are also detailed descriptions of what it means to verify the correctness of the output of E2E-V systems (e.g., [CITATION]).', '1707.08619-2-137-2': 'Others have attempted to define alternative forms of the E2E-V properties [CITATION].', '1707.08619-2-137-3': 'There are also less technical explanations of E2E-V intended for voters and election officials [CITATION].', '1707.08619-2-138-0': '0.97Open problem:', '1707.08619-2-139-0': '### Techniques for Coercion Resistance', '1707.08619-2-140-0': 'Some simple approaches to coercion resistance have been suggested in the literature.', '1707.08619-2-140-1': 'These include allowing multiple votes with only the last counting and allowing in-person voting to override remotely cast votes (both used in Estonian, Norwegian, and Utah elections [CITATION]).', '1707.08619-2-140-2': 'It is not clear that this mitigates coercion at all.', '1707.08619-2-140-3': 'Alarm codes can also be provided to voters: seemingly real but actually fake election credentials, along with the ability for voters to create their own fake credentials.', '1707.08619-2-140-4': 'Any such approach can be considered a partial solution at best, particularly given the usability challenges.', '1707.08619-2-141-0': 'One voting system, Civitas [CITATION], based on a protocol by Juels, Catalano and Jakobsson [CITATION], allows voters to vote with fake credentials to lead the coercive adversary into believing the desired vote was cast.', '1707.08619-2-141-1': 'Note that the protocol must enable universal verification of the tally from a list of votes cast with both genuine and fake credentials, proving to the verifier that only the ones with genuine credentials were tallied, without identifying which ones they were.', '1707.08619-2-142-0': '0.97Open problem:', '1707.08619-2-143-0': '### Cryptographic Solutions in Complex Elections', '1707.08619-2-144-0': 'Cast-as-intended verification based on creating and then challenging a vote works regardless of the scheme (e.g. Benaloh challenges).', '1707.08619-2-144-1': 'Cut-and-choose based schemes such as Pret a Voter and Scantegrity II need to be modified to work.', '1707.08619-2-145-0': 'Both uses of end-to-end verifiable voting schemes in government elections, the Takoma Park run of Scantegrity II and the Victorian run of Pret a Voter, used IRV (and one used STV).', '1707.08619-2-145-1': "Verifiable IRV/STV counting that doesn't expose individual votes to the Italian attack has been considered [CITATION], but may not be efficient enough for use in large elections in practice, and was not employed in either practical implementation.", '1707.08619-2-146-0': '0.97Open problems:', '1707.08619-2-147-0': '### Blockchains as a Cryptographic Solution', '1707.08619-2-148-0': 'Blockchains provide an unexpectedly effective answer to a long-standing problem in computer science-how to form a consistent public ledger in a dynamic and fully distributed environment in which there is no leader and participants may join and leave at any time [CITATION].', '1707.08619-2-148-1': 'In fact, the blockchain process effectively selects a "random" leader at each step to move things forward, so this seems at first to be a natural fit for elections-citizens post their preferences onto a blockchain and everyone can see and agree upon the outcome of the election.', '1707.08619-2-149-0': 'However, blockchains and elections differ in significant ways.', '1707.08619-2-149-1': 'Elections typically already have central authorities to play the leadership role, an entity that administrates the election: what will be voted on, when, who is allowed to vote, etc.).', '1707.08619-2-149-2': 'This authority can also be tasked with publishing a public ledger of events.', '1707.08619-2-149-3': 'Note that (as with blockchains) there need be no special trust in a central authority as these tasks are all publicly observable.', '1707.08619-2-149-4': 'So to begin with, by simply posting something on a (digitally signed) web page, an election office can do in a single step what blockchains do with a cumbersome protocol involving huge amounts of computation.', '1707.08619-2-150-0': 'Blockchains are inherently unaccountable.', '1707.08619-2-150-1': 'Blockchain miners are individually free to include or reject any transactions they desire-this is considered a feature.', '1707.08619-2-150-2': "To function properly in elections, a blockchain needs a mechanism to ensure all legitimate votes are included in the ledger, which leads to another problem: there's also no certainty in traditional blockchain schemes.", '1707.08619-2-150-3': 'Disputes are typically resolved with a "longest chain wins" rule.', '1707.08619-2-150-4': 'Miners may have inconsistent views of the contents of blockchains, but the incentives are structured so that the less widely held views eventually fade away-usually.', '1707.08619-2-150-5': 'This lack of certainty is not a desirable property in elections.', '1707.08619-2-151-0': 'In addition to lacking certainty and accountability, blockchains also lack anonymity.', '1707.08619-2-151-1': 'While modifications can be made to blockchain protocols to add anonymity, certainty, and accountability, balancing these modifications on top of the additional constraints of voting is difficult, and simpler solutions already exist as we discuss.', '1707.08619-2-152-0': 'In short, blockchains do not address any of the fundamental problems in elections, and their use actually makes things worse.', '1707.08619-2-153-0': '# Current Solutions', '1707.08619-2-154-0': 'I am committed to helping Ohio deliver its electoral votes to the president next year.', '1707.08619-2-154-1': "Walden O'Dell, Diebold CEO, 2003", '1707.08619-2-155-0': 'Below we provide a brief analysis of several real-world voting systems developed by the scientific community.', '1707.08619-2-155-1': 'These systems use the properties discussed in Sections [REF] and [REF].', '1707.08619-2-155-2': 'We include both pollsite and remote systems.', '1707.08619-2-155-3': 'This collection is by no means exhaustive, but hopefully the abundance of verifiable, evidence-based voting systems will convince the reader that there are significant technological improvements that can greatly improve election security.', '1707.08619-2-155-4': 'Our analysis is graphically represented in Table 1.', '1707.08619-2-156-0': '## Pollsite Systems', '1707.08619-2-157-0': 'The systems below were developed specifically with the requirements from Section [REF] in mind.', '1707.08619-2-157-1': 'As such, all satisfy the end-to-end verifiability criteria from Section [REF], and to a varying degree provide collection accountability, receipt-freeness, and coercion resistance.', '1707.08619-2-158-0': '### Pret a Voter', '1707.08619-2-159-0': '### Scantegrity', '1707.08619-2-160-0': 'The Scantegrity [CITATION] voter marks ballots that are very similar to optical scan ballots, with a single important difference.', '1707.08619-2-160-1': 'Each oval has printed on it, in invisible ink, a confirmation code-the encryption corresponding to this vote choice.', '1707.08619-2-160-2': 'When voters fill the oval with a special pen, the confirmation number becomes visible.', '1707.08619-2-160-3': 'The same functionality can be achieved through the use of scratch-off surfaces.', '1707.08619-2-161-0': 'Scantegrity II was used by the City of Takoma Park for its municipal elections in 2009 and 2011 [CITATION], the first secret-ballot election for public office known to use an E2E voting system within the U.S.', '1707.08619-2-162-0': '### VeriScan', '1707.08619-2-163-0': 'VeriScan [CITATION], like Scantegrity, uses optical scan ballots.', '1707.08619-2-163-1': 'But the ballots are ordinary - using regular ink - and are filled by voters using ordinary pens.', '1707.08619-2-163-2': 'Optical scanners used by VeriScan are augmented to hold the ballot deposited by a voter and to print a receipt consisting of an encryption of the selections made by the voter (or a hash thereof).', '1707.08619-2-164-0': 'Once the receipt has been given to the voter by the scanner, the voter can instruct the scanner to either retain the ballot or to return the ballot to the voter.', '1707.08619-2-164-1': 'A returned ballot should be automatically marked as no longer suitable for casting and effectively becomes a challenge ballot as in STAR-Vote (below).', '1707.08619-2-165-0': 'All encrypted ballots - whether cast or retained by a voter - are posted to a public web page where they can be checked against voter receipts.', '1707.08619-2-165-1': 'The cast ballots are listed only in encrypted form, but the retained ballots are listed in both encrypted and decrypted form so that voters can check the decryptions against their own copies of the ballots.', '1707.08619-2-166-0': '### STAR-Vote', '1707.08619-2-167-0': '### PPAT', '1707.08619-2-168-0': 'While many of the above schemes provide most of the required properties laid out in Section [REF], most do not account for everlasting privacy.', '1707.08619-2-168-1': 'However, by integrating the Perfectly Private Audit Trail (PPAT) [CITATION], many of the previously discussed systems can attain everlasting privacy.', '1707.08619-2-168-2': 'Notably, PPAT can be implemented both with mixnet schemes like Scantegrity [CITATION] and Helios [CITATION] as well as with homomorphic schemes like that used in STAR-Vote [CITATION].', '1707.08619-2-169-0': '## Remote Systems', '1707.08619-2-170-0': '### Remotegrity', '1707.08619-2-171-0': 'The Remotegrity [CITATION] voting system specification provides a layer over local coded voting systems specifications to enable their use in a remote setting.', '1707.08619-2-171-1': 'It is the only known specification that enables the voter to detect and prove attempts by adversaries to change the remote vote.', '1707.08619-2-172-0': 'Voters are mailed a package containing a coded-vote ballot and a credential sheet.', '1707.08619-2-172-1': 'The sheet contains authorization codes and lock-in codes under scratch-offs, and a return code.', '1707.08619-2-172-2': 'To vote, voters scratch-off an authorization code at random and use it as a credential to enter the candidate code.', '1707.08619-2-172-3': 'The election website displays the entered information and the return code, which indicates to the voter that the vote was received.', '1707.08619-2-172-4': 'If the website displays the correct information, the voter locks it in with a random lock-in code.', '1707.08619-2-172-5': 'If not, the voter uses another computer to vote, scratching-off another authorization code.', '1707.08619-2-172-6': 'For voter-verifiability, voters may receive multiple ballots, one of which is voted on, and the others audited.', '1707.08619-2-173-0': 'The credential authority (an insider adversary) can use the credentials to vote instead of the voter.', '1707.08619-2-173-1': 'If this happens, the voter can show the unscratched-off surface to prove the existence of a problem.', '1707.08619-2-173-2': 'Remotegrity thus achieves E2E-V, collection accountability, and software independence.', '1707.08619-2-173-3': 'Since there is no secret ballot guarantee, there is no coercion resistance.', '1707.08619-2-174-0': 'Remotegrity was made available to absentee voters in the 2011 election of the City of Takoma Park, alongside in-person voting provided by Scantegrity.', '1707.08619-2-175-0': '### Helios', '1707.08619-2-176-0': '### Selene', '1707.08619-2-177-0': '0.97Open problems:', '1707.08619-2-178-0': '# Internet Voting', '1707.08619-2-179-0': '"People of Dulsford," began Boris, "I want to assure you that as your newly elected mayor I will not just represent the people who voted for me ..."', '1707.08619-2-180-0': '"That\'s good," said Derrick, "because no-one voted for him."', '1707.08619-2-181-0': '"But the people who didn\'t vote for me as well," said Boris.', '1707.08619-2-182-0': 'There was a smattering of half-hearted clapping from the crowd.', '1707.08619-2-182-1': 'R. A. Spratt, Nanny Piggins and the Race to Power', '1707.08619-2-183-0': 'In this section we present the challenges of secure Internet voting through a set of (possibly contradictory) requirements.', '1707.08619-2-183-1': 'No system has addressed the challenges sufficiently so far, and whether it is possible to do so remains an open problem.', '1707.08619-2-183-2': 'We begin by introducing prominent contemporary instances of I-voting as case studies.', '1707.08619-2-183-3': 'Then we examine the Internet voting threat model, along the way showing how these Internet systems have failed to adequately defend themselves.', '1707.08619-2-183-4': "We look at voter authentication, verification of the correctness of a voting system's output, voter privacy and coercion resistance, protections against denial-of-service, and finally the usability and regulatory constraints faced by voting systems.", '1707.08619-2-184-0': 'One major roadblock faced exclusively by I-voting is the underlying infrastructure of the Internet.', '1707.08619-2-184-1': 'The primary security mechanism for Internet communication is Transport Layer Security (TLS), which is constantly evolving in response to vulnerabilities.', '1707.08619-2-184-2': 'For instance, the website used in the iVote system was vulnerable to the TLS FREAK [CITATION] and LogJam [CITATION] vulnerabilities.', '1707.08619-2-184-3': 'Researchers discovered this during the election period and demonstrated that they could exploit it to steal votes [CITATION].', '1707.08619-2-184-4': 'At the time, LogJam had not been publicly disclosed, highlighting the risk to I-voting from zero-day vulnerabilities.', '1707.08619-2-184-5': 'Internet voting systems must find ways to rely on properties like software independence and E2E-V before they can be considered trusted.', '1707.08619-2-185-0': 'In 2015, the U.S. Vote Foundation issued an export report on the viability of using E2E-verifiability for Internet voting [CITATION].', '1707.08619-2-185-1': 'The first two conclusions of the report were as follows.', '1707.08619-2-186-0': 'Any public elections conducted over the Internet must be end-to-end verifiable.', '1707.08619-2-186-1': 'No Internet voting system of any kind should be used for public elections before end-to-end verifiable in-person voting systems have been widely deployed and experience has been gained from their use.', '1707.08619-2-187-0': 'Many of the possible attacks on I-voting systems could be performed on postal voting systems too.', '1707.08619-2-187-1': 'The main difference is the likelihood that a very small number of people could automate the manipulation of a very large number of votes, or a carefully chosen few important votes, without detection.', '1707.08619-2-188-0': '## E2E-V I-voting in Government Elections', '1707.08619-2-189-0': 'Internet voting presents numerous challenges that have not been adequately addressed.', '1707.08619-2-189-1': 'First among these is the coercion problem which is shared with other remote voting systems in widespread use today (such as vote-by-mail).', '1707.08619-2-189-2': 'However, I-voting exacerbates the problem by making coercion and vote-selling a simple matter of a voter providing credentials to another individual.', '1707.08619-2-190-0': 'Client malware poses another significant obstacle.', '1707.08619-2-190-1': 'While E2E-verifiability mitigates the malware risks by providing voters with alternate means to ensure that their votes have been properly recorded and counted, many voters will not avail themselves of these capabilities.', '1707.08619-2-190-2': 'We could therefore have a situation were a large-scale fraud is observed by a relatively small number of voters.', '1707.08619-2-190-3': 'While the detection of a small number of instances of malfeasance can bring a halt to an election which provides collection accountability, the required evidence can be far more fleeting and difficult to validate in an Internet setting.', '1707.08619-2-190-4': 'An election should not be overturned by a small number of complaints if there is no substantive evidence to support these complaints.', '1707.08619-2-191-0': 'Targeted denial-of-service is another serious unresolved threat to I-voting.', '1707.08619-2-191-1': 'Ordinary denial-of-service (DoS) is a common threat on the Internet, and means have been deployed to mitigate - although not eliminate - these threats.', '1707.08619-2-191-2': 'The unique aspect in elections is that while ordinary DoS can slow commerce or block access to a web site for a period, the effects of a targeted DoS attack on an election can be far more severe.', '1707.08619-2-191-3': 'Since voting paterns are far from homogeneous, an attacker can launch a targeted DoS attack against populations and regions which are likely to favor a particular candidate or position.', '1707.08619-2-191-4': 'By merely making it more difficult for people in targeted populations to vote, the result of an election can be altered.', '1707.08619-2-191-5': 'As yet, we have no effective mitigations for such attacks.', '1707.08619-2-192-0': "Finally, as was observed in the U.S. Vote Foundation study [CITATION], we simply don't yet have much experience with large-scale deployments of E2E-verifiable election systems in the simpler and more manageable setting of in-person voting.", '1707.08619-2-192-1': 'It would be angerous to jump directly to the far more challenging setting of Internet voting with a heavy dependence on a technology that has not previously been deployed at scale.', '1707.08619-2-193-0': '## Alternatives to Internet Voting', '1707.08619-2-194-0': 'There are numerous alternatives to Internet voting that can help enfranchise voters who can not easily access a poll site on the day of an election.', '1707.08619-2-195-0': 'Early voting is in widespread use throughout the U.S. By extending the voting window from a single day to as much as three weeks, voters who may be away or busy on the date of an election can be afforded an opportunity to vote in person, at their convenience, at a poll site with traditional safeguards.', '1707.08619-2-195-1': 'Early voting also mitigates many of the risks of traditional systems since, for example, an equipment failure ten days prior to the close of an election is far less serious than one that takes place during a single day of voting.', '1707.08619-2-196-0': 'Some U.S. jurisdictions have adopted a vote center system in which voters may vote in person outside of their home precincts.', '1707.08619-2-196-1': 'This option has been facilitated by the use of electronic poll books, and it allows voters to, for instance, vote during a lunch break from work if they will be away from their homes during voting hours.', '1707.08619-2-196-2': 'The vote center model could potentially be extended from the current model of voters away from their home precincts but still within their home counties by allowing voters to use any poll site in the state or country.', '1707.08619-2-196-3': 'It would even be possible to establish remote voting kiosks overseas in embassies, conslates, or other official sites, and roming voting kiosks could be established with as little as two poll workers and a laptop computer.', '1707.08619-2-196-4': 'Security and accountability in all of these non-local voting scenarios can be greatly enhanced by the use of E2E-verifiability.', '1707.08619-2-197-0': 'Blank-ballot electronic delivery is another option which has gained in popularity.', '1707.08619-2-197-1': 'While there are numerous risks in using the Internet for casting of ballots, the risks a far less in simply providing blank ballots to voters.', '1707.08619-2-197-2': 'Electronic delivery of blank-ballots can save half of the round-trip time that is typical in absentee voting, and traditional methods of ballot return can be used which are less susceptible to the large-scale attacks that are possible with full Internet voting.', '1707.08619-2-198-0': '# A Look Ahead', '1707.08619-2-199-0': 'There is no remedy now to a process that was so opaque that it could have been manipulated at any stageMichael Meyer-Resende and Mirjam Kunkler, on the Iranian 2009 Presidential election', '1707.08619-2-200-0': 'Voting has always used available technology, whether pebbles dropped in an urn or marked paper put in a ballot box; it now uses computers, networks, and cryptography.', '1707.08619-2-200-1': "The core requirement, to provide public evidence of the right result from secret ballots, hasn't changed in 2500 years.", '1707.08619-2-201-0': 'Computers can improve convenience and accessibility over plain paper and manual counting.', '1707.08619-2-201-1': 'In the polling place there are good solutions, including Risk Limiting Audits and end-to-end verifiable systems.', '1707.08619-2-201-2': 'These must be more widely deployed and their options for verifying the election result must actually be used.', '1707.08619-2-202-0': 'Many of the open problems described in this paper-usable and accessible voting systems, dispute resolution, incoercibility-come together in the challenge of a remote voting system that is verifiable and usable without supervision.', '1707.08619-2-202-1': 'The open problem of a system specification that (a) does not use any paper at all and (b) is based on a simple procedure for voters and poll workers, will motivate researchers for a long time.', '1707.08619-2-202-2': 'Perhaps a better goal is a hybrid system combining paper evidence with some auditing or cryptographic verification.', '1707.08619-2-203-0': 'Research in voting brings together knowledge in many fields-cryptography, systems security, statistics, usability and accessibility, software verification, elections, law and policy to name a few-to address a critical real-world problem.', '1707.08619-2-204-0': 'The peaceful transfer of power depends on confidence in the electoral process.', '1707.08619-2-204-1': 'That confidence should not automatically be given to any outcome that seems plausible-it must be earned by producing evidence that the election result is what the people chose.', '1707.08619-2-204-2': 'Insisting on evidence reduces the opportunities for fraud, hence bringing greater security to citizens the world over.'}

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'1707.08619-2-62-0'], ['1707.08619-1-99-0', '1707.08619-2-99-0'], ['1707.08619-1-195-0', '1707.08619-2-195-0'], ['1707.08619-1-195-1', '1707.08619-2-195-1'], ['1707.08619-1-2-0', '1707.08619-2-2-0'], ['1707.08619-1-204-0', '1707.08619-2-204-0'], ['1707.08619-1-204-1', '1707.08619-2-204-1'], ['1707.08619-1-204-2', '1707.08619-2-204-2'], ['1707.08619-1-134-0', '1707.08619-2-134-0'], ['1707.08619-1-12-0', '1707.08619-2-12-0']]

[]

[]

[]

[]

['1707.08619-1-23-0', '1707.08619-1-30-0', '1707.08619-1-36-0', '1707.08619-1-39-0', '1707.08619-1-44-0', '1707.08619-1-47-0', '1707.08619-1-54-0', '1707.08619-1-56-2', '1707.08619-1-58-0', '1707.08619-1-62-1', '1707.08619-1-67-0', '1707.08619-1-70-0', '1707.08619-1-80-0', '1707.08619-1-82-1', '1707.08619-1-87-0', '1707.08619-1-90-0', '1707.08619-1-104-0', '1707.08619-1-107-0', '1707.08619-1-109-1', '1707.08619-1-115-0', '1707.08619-1-123-0', '1707.08619-1-138-0', '1707.08619-1-142-0', '1707.08619-1-146-0', '1707.08619-1-177-0', '1707.08619-1-180-0', '1707.08619-2-23-0', '1707.08619-2-30-0', '1707.08619-2-36-0', '1707.08619-2-39-0', '1707.08619-2-44-0', '1707.08619-2-47-0', '1707.08619-2-54-0', '1707.08619-2-56-2', '1707.08619-2-58-0', '1707.08619-2-62-1', '1707.08619-2-67-0', '1707.08619-2-70-0', '1707.08619-2-80-0', '1707.08619-2-82-1', '1707.08619-2-87-0', '1707.08619-2-90-0', '1707.08619-2-104-0', '1707.08619-2-107-0', '1707.08619-2-109-1', '1707.08619-2-115-0', '1707.08619-2-123-0', '1707.08619-2-138-0', '1707.08619-2-142-0', '1707.08619-2-146-0', '1707.08619-2-177-0', '1707.08619-2-180-0']

{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1707.08619

cond-mat-9701119

{'cond-mat-9701119-1-0-0': 'We performed numerical calculations of the local density of states (LDOS) at disorder induced localization-delocalization transitions.', 'cond-mat-9701119-1-0-1': 'The LDOS defines a spatial measure for fixed energy and a spectral measure for fixed position.', 'cond-mat-9701119-1-0-2': 'At the mobility edge both measures are multifractal and their generalized dimensions [MATH] and [MATH] are found to be proportional: [MATH], where [MATH] is the dimension of the system.', 'cond-mat-9701119-1-0-3': 'This observation is consistent with the identification of the frequency-dependent length scale [MATH] as an effective system size.', 'cond-mat-9701119-1-0-4': 'The calculations are performed for two- and three-dimensional dynamical network models with local time evolution operators.', 'cond-mat-9701119-1-0-5': 'The energy dependence of the LDOS is obtained from the time evolution of the local wavefunction amplitude of a wave packet, providing a numerically efficient way to obtain information about the multifractal exponents of the system.', 'cond-mat-9701119-1-1-0': ']', 'cond-mat-9701119-1-2-0': 'In the presence of disorder electronic states in phase-coherent systems can get localized.', 'cond-mat-9701119-1-2-1': 'For dimensions [MATH] such a disordered system can exhibit a transition from localized to extended eigenstates as a function of energy or strength of the disorder, similar to continuous thermodynamic phase transitions [CITATION].', 'cond-mat-9701119-1-2-2': 'This transition is accompanied by a diverging length scale [MATH] that can be identified for localized states with the localization length.', 'cond-mat-9701119-1-2-3': 'Here [MATH] is the critical energy of the transition.', 'cond-mat-9701119-1-2-4': 'On length scales shorter than [MATH] the eigenstates exhibit strong fluctuations [CITATION].', 'cond-mat-9701119-1-2-5': 'Sufficiently close to the transition the localization length exceeds the system size so that the wave functions fluctuate on all length scales up to the system size [MATH].', 'cond-mat-9701119-1-2-6': 'The squared modulus of these critical eigenstates forms a multifractal measure [MATH] [CITATION].', 'cond-mat-9701119-1-2-7': 'The scaling behavior of multifractal measures is determined by an infinite set of exponents [MATH], called generalized dimensions.', 'cond-mat-9701119-1-2-8': 'The multifractal fluctuations lead to anomalous behavior of the diffusion coefficient [CITATION].', 'cond-mat-9701119-1-3-0': 'The localization properties of the wavefunction are also reflected in the spectral properties of the system.', 'cond-mat-9701119-1-3-1': 'Localized states correspond to a pure point like spectrum, extended states to an absolutely continuous spectrum, and critical states to a singular continuous spectrum.', 'cond-mat-9701119-1-3-2': 'The latter kind of spectrum was observed in quasiperiodic systems [CITATION], systems on incommensurate structures [CITATION], in crystals in a magnetic field [CITATION], and also in random one-dimensional systems [CITATION].', 'cond-mat-9701119-1-3-3': 'For these systems the spectral measure excited by a wave packet is multifractal.', 'cond-mat-9701119-1-3-4': 'Of particular interest has been the implications of the local spectra for dynamical properties [CITATION].', 'cond-mat-9701119-1-3-5': 'Ketzmerick, Petschel, and Geisel [CITATION] found that the return probability of a wave packet decays in time [MATH] asymptotically as [MATH], where [MATH] is the correlation dimension of the associated spectral measure, the local density of states (LDOS).', 'cond-mat-9701119-1-4-0': 'The two aspects of spatial and spectral multifractality were brought together by Huckestein and Schweitzer who showed that enhanced return probability of wavepackets at the mobility edge of quantum Hall systems could be interpreted both from the spectral [CITATION] as well as the spatial properties of the local density of states [CITATION].', 'cond-mat-9701119-1-4-1': 'They showed that the generalized dimensions [MATH] and [MATH] characterizing the second moments of the spatial and spectral measures, respectively, are related by the spatial dimension [MATH] of the system, [MATH].', 'cond-mat-9701119-1-4-2': 'This follows from dynamical scaling by which the two-particle spectral function [MATH] becomes a function of [MATH] at criticality [CITATION].', 'cond-mat-9701119-1-4-3': 'Here, [MATH] can be interpreted as the system size with mean level spacing [MATH] and [MATH] is the density of states at the mobility edge [CITATION].', 'cond-mat-9701119-1-4-4': 'The length scale [MATH] introduced by the finite frequency [MATH] cuts off correlations and acts as an effective system size [CITATION].', 'cond-mat-9701119-1-4-5': 'Recently, the same relation was shown to hold in three dimensional systems at the Anderson transition[CITATION].', 'cond-mat-9701119-1-5-0': 'In this letter, we show that at the mobility edge the spatial and spectral structures of the LDOS are intimately related: their respective generalized dimensions [MATH] and [MATH] are proportional, [EQUATION] where [MATH] is the space dimension of the system.', 'cond-mat-9701119-1-5-1': 'This relation, generalizing the result of ref. [CITATION] to arbitrary [MATH], is a consequence of the length scale [MATH] acting quite generally as an effective system size, not only for correlation functions but also for moments of the LDOS.', 'cond-mat-9701119-1-5-2': 'We test the relation numerically for two- and three-dimensional network models [CITATION].', 'cond-mat-9701119-1-5-3': 'Introducing a time evolution into these models [CITATION] allows us to calculate the LDOS by studying the time evolution of wave packets.', 'cond-mat-9701119-1-5-4': 'This provides a new efficient method to obtain the multifractal exponents of the LDOS without the need to diagonalize large matrices.', 'cond-mat-9701119-1-6-0': 'Before motivating the relation ([REF]) let us define our multifractal analysis.', 'cond-mat-9701119-1-6-1': 'We study a normalized multifractal density [MATH] on a [MATH]-dimensional domain [MATH].', 'cond-mat-9701119-1-6-2': 'Here, [MATH] is the two- or three-dimensional space or the one-dimensional energy axis, depending on whether we study the spatial or spectral aspect of the LDOS.', 'cond-mat-9701119-1-6-3': 'We study the scaling of the box probabilities [EQUATION] with respect to the box sizes [MATH].', 'cond-mat-9701119-1-6-4': '[MATH] is the volume of the [MATH]th of [MATH] non-overlapping hypercubes of linear size [MATH] covering the whole domain [MATH] of linear size [MATH].', 'cond-mat-9701119-1-6-5': 'For a multifractal density the averaged [MATH]th moments, [MATH] show power law dependence on [MATH] for all real [MATH], [EQUATION]', 'cond-mat-9701119-1-6-6': 'The distinguishing feature of a multifractal is that the exponents [MATH] are a non-linear function of [MATH].', 'cond-mat-9701119-1-6-7': 'A multifractal is thus described by a non-countable set of exponents [MATH].', 'cond-mat-9701119-1-6-8': 'If the multifractal measure is taken from a statistical ensemble, the system average over the box probabilities can be replaced by an ensemble average for a single box.', 'cond-mat-9701119-1-7-0': 'The LDOS is given by [MATH], where the sum runs over the quantum numbers [MATH] with corresponding eigenstates [MATH].', 'cond-mat-9701119-1-7-1': 'At energies [MATH] close to the mobility edge [MATH] the LDOS exhibits multifractal properties on length scales between microscopic lengths, like the lattice constant [MATH] or the elastic mean free path, and the system size [MATH] or the correlation length [MATH].', 'cond-mat-9701119-1-7-2': 'The corresponding lower and upper energy scales are the mean level spacing [MATH] and [MATH], respectively.', 'cond-mat-9701119-1-7-3': 'For energies [MATH] less than [MATH] the correlation length [MATH] exceeds the system size [MATH].', 'cond-mat-9701119-1-8-0': 'We now argue that the existence of a single frequency-dependent length scale [MATH] implies the relation ([REF]).', 'cond-mat-9701119-1-8-1': 'Consider box probabilities of the LDOS with respect to both space and energy [EQUATION] where we consider an energy interval in the critical region described above.', 'cond-mat-9701119-1-8-2': 'For energies [MATH] less than the mean level spacing [MATH] these box probabilities scale like single eigenfunctions, [MATH].', 'cond-mat-9701119-1-8-3': 'For energies larger than the mean level spacing, we expect the disorder average of the moments of these box probabilities to scale with the same exponents but with the system size [MATH] replaced by a frequency-dependent effective system size [MATH], [EQUATION]', 'cond-mat-9701119-1-8-4': 'In principle, the frequency-dependent length scale introduced in eq. ([REF]) could depend on [MATH].', 'cond-mat-9701119-1-8-5': 'However, if we assume that there exists only one such length scale, then we can identify [MATH] with [MATH] by considering the first moment [EQUATION]', 'cond-mat-9701119-1-8-6': 'The scaling of the spectral measure defined by the LDOS follows from eq. ([REF]) by choosing the spatial box size [MATH] of the order of the microscopic length scale [MATH], below which the wavefunctions become smooth functions of coordinate.', 'cond-mat-9701119-1-8-7': 'The scaling of the spectral measure with respect to energy defines the exponents [MATH], [EQUATION] leading to eq. ([REF]).', 'cond-mat-9701119-1-9-0': 'For our computations, we consider two normalized measures, the spatial measure [MATH] and the spectral measure [MATH].', 'cond-mat-9701119-1-9-1': 'The spatial measure is normalized since the eigenfunctions are normalized, and the spectral measure is explicitly normalized on an energy interval [MATH], satisfying [MATH].', 'cond-mat-9701119-1-9-2': 'To support the validity of eq. ([REF]) we now present results of numerical calculations.', 'cond-mat-9701119-1-9-3': 'Fig. ([REF]) shows the functions [MATH] and [MATH] calculated for a two-dimensional network model at the quantum Hall critical point [CITATION].', 'cond-mat-9701119-1-9-4': 'The functions agree within the error bars.', 'cond-mat-9701119-1-9-5': 'In order to check the dependence on the spatial dimension [MATH] of the system we study a three-dimensional network introduced recently [CITATION].', 'cond-mat-9701119-1-9-6': 'In contrast to the two-dimensional network here a band of extended states appears.', 'cond-mat-9701119-1-9-7': 'At the mobility edge of this system we again find good agreement with eq. ([REF]) as seen in Fig. ([REF]).', 'cond-mat-9701119-1-10-0': 'We now describe our new numerical method of obtaining the LDOS.', 'cond-mat-9701119-1-10-1': 'We use two- and three-dimensional network models, describing the integer quantum Hall effect [CITATION] and the so called quantum Hall-insulator [CITATION], respectively.', 'cond-mat-9701119-1-10-2': 'Both models are extended to dynamical network models by providing them with a unitary time evolution operator [MATH] for discrete microscopic time steps [CITATION].', 'cond-mat-9701119-1-10-3': 'Within these models the evolution-involving full quantum interference-of arbitrary initial states can be obtained easily by iterative application of [MATH].', 'cond-mat-9701119-1-10-4': "In particular, the time evolution of a state [MATH] initially sharply peaked at coordinate [MATH] yields the temporal Green's function [MATH].", 'cond-mat-9701119-1-10-5': 'The LDOS is then the Fourier transform of this quantity.', 'cond-mat-9701119-1-10-6': 'This provides an efficient method of calculating the LDOS, without the need to diagonalize the operator [MATH] or the associated Hamiltonian.', 'cond-mat-9701119-1-11-0': 'Starting point of our calculations is the 2D-network model introduced by Chalker and Coddington [CITATION] to describe non-interacting spinless electrons in the integer Quantum Hall regime.', 'cond-mat-9701119-1-11-1': 'It exhibits a localization-delocalization-transition characteristic for the QHE: except for the critical energy [MATH] all states are localized, with a localization length [MATH], [MATH].', 'cond-mat-9701119-1-11-2': 'Origin of the model and exact definitions can be found in refs. [CITATION] and [CITATION], here we give only a brief description restricted to our purposes.', 'cond-mat-9701119-1-12-0': 'The network consists of [MATH] scattering matrices as nodes which are arranged on a square lattice and connected by one-dimensional unidirectional channels, called links.', 'cond-mat-9701119-1-12-1': 'The scattering matrices contain coefficients [MATH] describing transitions from electron states on incoming links [MATH] to outgoing link states [MATH].', 'cond-mat-9701119-1-12-2': 'At the critical point the transmission amplitudes are of constant value [MATH], while the disorder is given by randomly distributed arguments of the coefficients [MATH].', 'cond-mat-9701119-1-12-3': 'States (or wave functions) [MATH] on the network are [MATH]-dimensional complex vectors [MATH], where the [MATH]-th component denotes the complex amplitude on link [MATH].', 'cond-mat-9701119-1-12-4': 'The network operator [MATH] is defined by its action on single link states [MATH], [EQUATION] where [MATH] and [MATH] are the transmission coefficients from an incoming link [MATH] into two outgoing links [MATH] and [MATH] at a node[CITATION].', 'cond-mat-9701119-1-13-0': 'Here we use [MATH] as the time evolution operator in discrete time steps for states on the network.', 'cond-mat-9701119-1-13-1': 'To motivate this, consider a particle at critical energy [MATH] in the incoming link [MATH] of node [MATH], that is described by the state [MATH].', 'cond-mat-9701119-1-13-2': 'After a characteristic time [MATH] (we choose [MATH] in the following) the incident wavepacket has passed the scatterer and thereby split into two outgoing packets in the channels [MATH] and [MATH] (see fig. 1 in ref. [CITATION]).', 'cond-mat-9701119-1-13-3': 'This process corresponds just to the acting of [MATH] on [MATH], [EQUATION]', 'cond-mat-9701119-1-13-4': 'Since this happens in the same way at all scatterers we generalize relation ([REF]) to the time evolution of an arbitrary state and define [EQUATION] for integer [MATH].', 'cond-mat-9701119-1-13-5': 'With the latter definition the original static network becomes a quantum-dynamical model of a disordered system.', 'cond-mat-9701119-1-13-6': 'We can relate the eigenvectors [MATH] with eigenvalues [MATH] of [MATH] to eigenfunctions and energy spectrum of a Hamiltonian [MATH] by identifying [CITATION] [EQUATION]', 'cond-mat-9701119-1-13-7': "To obtain the LDOS [MATH] (the sum runs over all eigenvalues [MATH] of [MATH]) at link [MATH], we calculate numerically the temporal Green's function [MATH].", 'cond-mat-9701119-1-13-8': "In terms of the eigenstates of [MATH] the Green's function is given by [EQUATION]", 'cond-mat-9701119-1-13-9': 'Hence the LDOS [MATH] can be obtained by an inverse Fourier transform of [MATH], which on the other hand can be generated iteratively according to ([REF]), [EQUATION]', 'cond-mat-9701119-1-13-10': 'Fig. ([REF]) shows as an example the energy dependence of the LDOS obtained in that way.', 'cond-mat-9701119-1-14-0': 'The three-dimensional network investigated here is very similar to the one studied by Chalker and Dohmen [CITATION].', 'cond-mat-9701119-1-14-1': 'It is built out of layered two-dimensional networks with additional inter-layer couplings, that we choose slightly different from Chalker and Dohmen[CITATION].', 'cond-mat-9701119-1-15-0': 'In conclusion, we studied numerically the spatial and spectral multifractal measures defined by the local density of states (LDOS) at the mobility edge of two- and three-dimensional disordered electron systems.', 'cond-mat-9701119-1-15-1': 'We have presented evidence that both of these measures are equivalent, as the ratio of their respective generalized dimensions [MATH] and [MATH] is simply given by the dimension of the system.', 'cond-mat-9701119-1-15-2': 'This result is interpreted as a consequence of the occurrence of a single energy-dependent length scale [MATH] acting as the effective system size.', 'cond-mat-9701119-1-15-3': 'The numerical calculations were performed for two- and three-dimensional network models.', 'cond-mat-9701119-1-15-4': 'These model, endowed with a discrete time evolution, turned out to be especially suitable for determining the LDOS as a function of energy.', 'cond-mat-9701119-1-16-0': 'We gratefully acknowledge illuminating discussions with J. Hajdu, M. Janssen, M. Metzler, and L. Schweitzer.', 'cond-mat-9701119-1-16-1': 'This work was performed within the research program of the Sonderforschungsbereich 341 of the Deutsche Forschungsgemeinschaft.'}

{'cond-mat-9701119-2-0-0': 'We performed numerical calculations of the local density of states (LDOS) at disorder induced localization-delocalization transitions.', 'cond-mat-9701119-2-0-1': 'The LDOS defines a spatial measure for fixed energy and a spectral measure for fixed position.', 'cond-mat-9701119-2-0-2': 'At the mobility edge both measures are multifractal and their generalized dimensions [MATH] and [MATH] are found to be proportional: [MATH], where [MATH] is the dimension of the system.', 'cond-mat-9701119-2-0-3': 'This observation is consistent with the identification of the frequency-dependent length scale [MATH] as an effective system size.', 'cond-mat-9701119-2-0-4': 'The calculations are performed for two- and three-dimensional dynamical network models with local time evolution operators.', 'cond-mat-9701119-2-0-5': 'The energy dependence of the LDOS is obtained from the time evolution of the local wavefunction amplitude of a wave packet, providing a numerically efficient way to obtain information about the multifractal exponents of the system.', 'cond-mat-9701119-2-1-0': ']', 'cond-mat-9701119-2-2-0': 'In the presence of disorder electronic states in phase-coherent systems can get localized.', 'cond-mat-9701119-2-2-1': 'For dimensions [MATH] such a disordered system can exhibit a transition from localized to extended eigenstates as a function of energy or strength of the disorder, similar to continuous thermodynamic phase transitions [CITATION].', 'cond-mat-9701119-2-2-2': 'This transition is accompanied by a diverging length scale [MATH] that can be identified for localized states with the localization length.', 'cond-mat-9701119-2-2-3': 'Here [MATH] is the critical energy of the transition.', 'cond-mat-9701119-2-2-4': 'On length scales shorter than [MATH] the eigenstates exhibit strong fluctuations [CITATION].', 'cond-mat-9701119-2-2-5': 'Sufficiently close to the transition the localization length exceeds the system size so that the wave functions fluctuate on all length scales up to the system size [MATH].', 'cond-mat-9701119-2-2-6': 'The squared modulus of these critical eigenstates forms a multifractal measure [MATH] [CITATION].', 'cond-mat-9701119-2-2-7': 'The scaling behavior of multifractal measures is determined by an infinite set of exponents [MATH], called generalized dimensions.', 'cond-mat-9701119-2-2-8': 'The multifractal fluctuations lead to anomalous behavior of the diffusion coefficient [CITATION].', 'cond-mat-9701119-2-3-0': 'The localization properties of the wavefunction are also reflected in the spectral properties of the system.', 'cond-mat-9701119-2-3-1': 'Localized states correspond to a pure point like spectrum, extended states to an absolutely continuous spectrum, and critical states to a singular continuous spectrum.', 'cond-mat-9701119-2-3-2': 'The latter kind of spectrum was observed in quasiperiodic systems [CITATION], systems on incommensurate structures [CITATION], in crystals in a magnetic field [CITATION], and also in random one-dimensional systems [CITATION].', 'cond-mat-9701119-2-3-3': 'For these systems the spectral measure excited by a wave packet is multifractal.', 'cond-mat-9701119-2-3-4': 'Of particular interest has been the implications of the local spectra for dynamical properties [CITATION].', 'cond-mat-9701119-2-3-5': 'Ketzmerick, Petschel, and Geisel [CITATION] found that the return probability of a wave packet decays in time [MATH] asymptotically as [MATH], where [MATH] is the correlation dimension of the associated spectral measure, the local density of states (LDOS).', 'cond-mat-9701119-2-4-0': 'The two aspects of spatial and spectral multifractality were brought together by Huckestein and Schweitzer who showed that enhanced return probability of wavepackets at the mobility edge of quantum Hall systems could be interpreted both from the spectral [CITATION] as well as the spatial properties of the local density of states [CITATION].', 'cond-mat-9701119-2-4-1': 'They showed that the generalized dimensions [MATH] and [MATH] characterizing the second moments of the spatial and spectral measures, respectively, are related by the spatial dimension [MATH] of the system, [MATH].', 'cond-mat-9701119-2-4-2': 'This follows from dynamical scaling by which the two-particle spectral function [MATH] becomes a function of [MATH] at criticality [CITATION].', 'cond-mat-9701119-2-4-3': 'Here, [MATH] can be interpreted as the system size with mean level spacing [MATH] and [MATH] is the density of states at the mobility edge [CITATION].', 'cond-mat-9701119-2-4-4': 'The length scale [MATH] introduced by the finite frequency [MATH] cuts off correlations and acts as an effective system size [CITATION].', 'cond-mat-9701119-2-4-5': 'Recently, the same relation was shown to hold in three dimensional systems at the Anderson transition[CITATION].', 'cond-mat-9701119-2-5-0': 'In this letter, we show that at the mobility edge the spatial and spectral structures of the LDOS are intimately related: their respective generalized dimensions [MATH] and [MATH] are proportional, [EQUATION] where [MATH] is the space dimension of the system.', 'cond-mat-9701119-2-5-1': 'This relation, generalizing the result of ref. [CITATION] to arbitrary [MATH], is a consequence of the length scale [MATH] acting quite generally as an effective system size, not only for correlation functions but also for moments of the LDOS.', 'cond-mat-9701119-2-5-2': 'We test the relation numerically for two- and three-dimensional network models [CITATION].', 'cond-mat-9701119-2-5-3': 'Introducing a time evolution into these models [CITATION] allows us to calculate the LDOS by studying the time evolution of wave packets.', 'cond-mat-9701119-2-5-4': 'This provides a new efficient method to obtain the multifractal exponents of the LDOS without the need to diagonalize large matrices.', 'cond-mat-9701119-2-6-0': 'Before motivating the relation ([REF]) let us define our multifractal analysis.', 'cond-mat-9701119-2-6-1': 'We study a normalized multifractal density [MATH] on a [MATH]-dimensional domain [MATH].', 'cond-mat-9701119-2-6-2': 'Here, [MATH] is the two- or three-dimensional space or the one-dimensional energy axis, depending on whether we study the spatial or spectral aspect of the LDOS.', 'cond-mat-9701119-2-6-3': 'We study the scaling of the box probabilities [EQUATION] with respect to the box sizes [MATH].', 'cond-mat-9701119-2-6-4': '[MATH] is the volume of the [MATH]th of [MATH] non-overlapping hypercubes of linear size [MATH] covering the whole domain [MATH] of linear size [MATH].', 'cond-mat-9701119-2-6-5': 'For a multifractal density the averaged [MATH]th moments, [MATH] show power law dependence on [MATH] for all real [MATH], [EQUATION]', 'cond-mat-9701119-2-6-6': 'The distinguishing feature of a multifractal is that the exponents [MATH] are a non-linear function of [MATH].', 'cond-mat-9701119-2-6-7': 'A multifractal is thus described by a non-countable set of exponents [MATH].', 'cond-mat-9701119-2-6-8': 'If the multifractal measure is taken from a statistical ensemble, the system average over the box probabilities can be replaced by an ensemble average for a single box.', 'cond-mat-9701119-2-7-0': 'The LDOS is given by [MATH], where the sum runs over the quantum numbers [MATH] with corresponding eigenstates [MATH].', 'cond-mat-9701119-2-7-1': 'At energies [MATH] close to the mobility edge [MATH] the LDOS exhibits multifractal properties on length scales between microscopic lengths, like the lattice constant [MATH] or the elastic mean free path, and the system size [MATH] or the correlation length [MATH].', 'cond-mat-9701119-2-7-2': 'The corresponding lower and upper energy scales are the mean level spacing [MATH] and [MATH], respectively.', 'cond-mat-9701119-2-7-3': 'For energies [MATH] less than [MATH] the correlation length [MATH] exceeds the system size [MATH].', 'cond-mat-9701119-2-8-0': 'We now argue that the existence of a single frequency-dependent length scale [MATH] implies the relation ([REF]).', 'cond-mat-9701119-2-8-1': 'Consider box probabilities of the LDOS with respect to both space and energy [EQUATION] where we consider an energy interval in the critical region described above.', 'cond-mat-9701119-2-8-2': 'For energies [MATH] less than the mean level spacing [MATH] these box probabilities scale like single eigenfunctions, [MATH].', 'cond-mat-9701119-2-8-3': 'For energies larger than the mean level spacing, we expect the disorder average of the moments of these box probabilities to scale with the same exponents but with the system size [MATH] replaced by a frequency-dependent effective system size [MATH], [EQUATION]', 'cond-mat-9701119-2-8-4': 'In principle, the frequency-dependent length scale introduced in eq. ([REF]) could depend on [MATH].', 'cond-mat-9701119-2-8-5': 'However, if we assume that there exists only one such length scale, then we can identify [MATH] with [MATH] by considering the first moment [EQUATION]', 'cond-mat-9701119-2-8-6': 'The scaling of the spectral measure defined by the LDOS follows from eq. ([REF]) by choosing the spatial box size [MATH] of the order of the microscopic length scale [MATH], below which the wavefunctions become smooth functions of coordinate.', 'cond-mat-9701119-2-8-7': 'The scaling of the spectral measure with respect to energy defines the exponents [MATH], [EQUATION] leading to eq. ([REF]).', 'cond-mat-9701119-2-9-0': 'For our computations, we consider two normalized measures, the spatial measure [MATH] and the spectral measure [MATH].', 'cond-mat-9701119-2-9-1': 'The spatial measure is normalized since the eigenfunctions are normalized, and the spectral measure is explicitly normalized on an energy interval [MATH], satisfying [MATH].', 'cond-mat-9701119-2-9-2': 'To support the validity of eq. ([REF]) we now present results of numerical calculations.', 'cond-mat-9701119-2-9-3': 'Fig. ([REF]) shows the functions [MATH] and [MATH] calculated for a two-dimensional network model at the quantum Hall critical point [CITATION].', 'cond-mat-9701119-2-9-4': 'The functions agree within the error bars.', 'cond-mat-9701119-2-9-5': 'In order to check the dependence on the spatial dimension [MATH] of the system we study a three-dimensional network introduced recently [CITATION].', 'cond-mat-9701119-2-9-6': 'In contrast to the two-dimensional network here a band of extended states appears.', 'cond-mat-9701119-2-9-7': 'At the mobility edge of this system we again find good agreement with eq. ([REF]) as seen in Fig. ([REF]).', 'cond-mat-9701119-2-10-0': 'We now describe our new numerical method of obtaining the LDOS.', 'cond-mat-9701119-2-10-1': 'We use two- and three-dimensional network models, describing the integer quantum Hall effect [CITATION] and the so called quantum Hall-insulator [CITATION], respectively.', 'cond-mat-9701119-2-10-2': 'Both models are extended to dynamical network models by providing them with a unitary time evolution operator [MATH] for discrete microscopic time steps [CITATION].', 'cond-mat-9701119-2-10-3': 'Within these models the evolution-involving full quantum interference-of arbitrary initial states can be obtained easily by iterative application of [MATH].', 'cond-mat-9701119-2-10-4': "In particular, the time evolution of a state [MATH] initially sharply peaked at coordinate [MATH] yields the temporal Green's function [MATH].", 'cond-mat-9701119-2-10-5': 'The LDOS is then the Fourier transform of this quantity.', 'cond-mat-9701119-2-10-6': 'This provides an efficient method of calculating the LDOS, without the need to diagonalize the operator [MATH] or the associated Hamiltonian.', 'cond-mat-9701119-2-11-0': 'Starting point of our calculations is the 2D-network model introduced by Chalker and Coddington [CITATION] to describe non-interacting spinless electrons in the integer Quantum Hall regime.', 'cond-mat-9701119-2-11-1': 'It exhibits a localization-delocalization-transition characteristic for the QHE: except for the critical energy [MATH] all states are localized, with a localization length [MATH], [MATH].', 'cond-mat-9701119-2-11-2': 'Origin of the model and exact definitions can be found in refs. [CITATION] and [CITATION], here we give only a brief description restricted to our purposes.', 'cond-mat-9701119-2-12-0': 'The network consists of [MATH] scattering matrices as nodes which are arranged on a square lattice and connected by one-dimensional unidirectional channels, called links.', 'cond-mat-9701119-2-12-1': 'The scattering matrices contain coefficients [MATH] describing transitions from electron states on incoming links [MATH] to outgoing link states [MATH].', 'cond-mat-9701119-2-12-2': 'At the critical point the transmission amplitudes are of constant value [MATH], while the disorder is given by randomly distributed arguments of the coefficients [MATH].', 'cond-mat-9701119-2-12-3': 'States (or wave functions) [MATH] on the network are [MATH]-dimensional complex vectors [MATH], where the [MATH]-th component denotes the complex amplitude on link [MATH].', 'cond-mat-9701119-2-12-4': 'The network operator [MATH] is defined by its action on single link states [MATH], [EQUATION] where [MATH] and [MATH] are the transmission coefficients from an incoming link [MATH] into two outgoing links [MATH] and [MATH] at a node[CITATION].', 'cond-mat-9701119-2-13-0': 'Here we use [MATH] as the time evolution operator in discrete time steps for states on the network.', 'cond-mat-9701119-2-13-1': 'To motivate this, consider a particle at critical energy [MATH] in the incoming link [MATH] of node [MATH], that is described by the state [MATH].', 'cond-mat-9701119-2-13-2': 'After a characteristic time [MATH] (we choose [MATH] in the following) the incident wavepacket has passed the scatterer and thereby split into two outgoing packets in the channels [MATH] and [MATH] (see fig. 1 in ref. [CITATION]).', 'cond-mat-9701119-2-13-3': 'This process corresponds just to the acting of [MATH] on [MATH], [EQUATION]', 'cond-mat-9701119-2-13-4': 'Since this happens in the same way at all scatterers we generalize relation ([REF]) to the time evolution of an arbitrary state and define [EQUATION] for integer [MATH].', 'cond-mat-9701119-2-13-5': 'With the latter definition the original static network becomes a quantum-dynamical model of a disordered system.', 'cond-mat-9701119-2-13-6': 'We can relate the eigenvectors [MATH] with eigenvalues [MATH] of [MATH] to eigenfunctions and energy spectrum of a Hamiltonian [MATH] by identifying [CITATION] [EQUATION]', 'cond-mat-9701119-2-13-7': "To obtain the LDOS [MATH] (the sum runs over all eigenvalues [MATH] of [MATH]) at link [MATH], we calculate numerically the temporal Green's function [MATH].", 'cond-mat-9701119-2-13-8': "In terms of the eigenstates of [MATH] the Green's function is given by [EQUATION]", 'cond-mat-9701119-2-13-9': 'Hence the LDOS [MATH] can be obtained by an inverse Fourier transform of [MATH], which on the other hand can be generated iteratively according to ([REF]), [EQUATION]', 'cond-mat-9701119-2-13-10': 'Fig. ([REF]) shows as an example the energy dependence of the LDOS obtained in that way.', 'cond-mat-9701119-2-14-0': 'The three-dimensional network investigated here is very similar to the one studied by Chalker and Dohmen [CITATION].', 'cond-mat-9701119-2-14-1': 'It is built out of layered two-dimensional networks with additional inter-layer couplings, that we choose slightly different from Chalker and Dohmen[CITATION].', 'cond-mat-9701119-2-15-0': 'In conclusion, we studied numerically the spatial and spectral multifractal measures defined by the local density of states (LDOS) at the mobility edge of two- and three-dimensional disordered electron systems.', 'cond-mat-9701119-2-15-1': 'We have presented evidence that both of these measures are equivalent, as the ratio of their respective generalized dimensions [MATH] and [MATH] is simply given by the dimension of the system.', 'cond-mat-9701119-2-15-2': 'This result is interpreted as a consequence of the occurrence of a single energy-dependent length scale [MATH] acting as the effective system size.', 'cond-mat-9701119-2-15-3': 'The numerical calculations were performed for two- and three-dimensional network models.', 'cond-mat-9701119-2-15-4': 'These model, endowed with a discrete time evolution, turned out to be especially suitable for determining the LDOS as a function of energy.', 'cond-mat-9701119-2-16-0': 'We gratefully acknowledge illuminating discussions with J. Hajdu, M. Janssen, M. Metzler, and L. Schweitzer.', 'cond-mat-9701119-2-16-1': 'This work was performed within the research program of the Sonderforschungsbereich 341 of the Deutsche Forschungsgemeinschaft.'}

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['cond-mat-9701119-1-15-1', 'cond-mat-9701119-2-15-1'], ['cond-mat-9701119-1-15-2', 'cond-mat-9701119-2-15-2'], ['cond-mat-9701119-1-15-3', 'cond-mat-9701119-2-15-3'], ['cond-mat-9701119-1-15-4', 'cond-mat-9701119-2-15-4'], ['cond-mat-9701119-1-2-0', 'cond-mat-9701119-2-2-0'], ['cond-mat-9701119-1-2-1', 'cond-mat-9701119-2-2-1'], ['cond-mat-9701119-1-2-2', 'cond-mat-9701119-2-2-2'], ['cond-mat-9701119-1-2-3', 'cond-mat-9701119-2-2-3'], ['cond-mat-9701119-1-2-4', 'cond-mat-9701119-2-2-4'], ['cond-mat-9701119-1-2-5', 'cond-mat-9701119-2-2-5'], ['cond-mat-9701119-1-2-6', 'cond-mat-9701119-2-2-6'], ['cond-mat-9701119-1-2-7', 'cond-mat-9701119-2-2-7'], ['cond-mat-9701119-1-2-8', 'cond-mat-9701119-2-2-8'], ['cond-mat-9701119-1-12-0', 'cond-mat-9701119-2-12-0'], ['cond-mat-9701119-1-12-1', 'cond-mat-9701119-2-12-1'], ['cond-mat-9701119-1-12-2', 'cond-mat-9701119-2-12-2'], ['cond-mat-9701119-1-12-3', 'cond-mat-9701119-2-12-3'], ['cond-mat-9701119-1-12-4', 'cond-mat-9701119-2-12-4'], ['cond-mat-9701119-1-3-0', 'cond-mat-9701119-2-3-0'], ['cond-mat-9701119-1-3-1', 'cond-mat-9701119-2-3-1'], ['cond-mat-9701119-1-3-2', 'cond-mat-9701119-2-3-2'], ['cond-mat-9701119-1-3-3', 'cond-mat-9701119-2-3-3'], ['cond-mat-9701119-1-3-4', 'cond-mat-9701119-2-3-4'], ['cond-mat-9701119-1-3-5', 'cond-mat-9701119-2-3-5'], ['cond-mat-9701119-1-6-0', 'cond-mat-9701119-2-6-0'], ['cond-mat-9701119-1-6-1', 'cond-mat-9701119-2-6-1'], ['cond-mat-9701119-1-6-2', 'cond-mat-9701119-2-6-2'], ['cond-mat-9701119-1-6-3', 'cond-mat-9701119-2-6-3'], ['cond-mat-9701119-1-6-4', 'cond-mat-9701119-2-6-4'], ['cond-mat-9701119-1-6-5', 'cond-mat-9701119-2-6-5'], ['cond-mat-9701119-1-6-6', 'cond-mat-9701119-2-6-6'], ['cond-mat-9701119-1-6-7', 'cond-mat-9701119-2-6-7'], ['cond-mat-9701119-1-6-8', 'cond-mat-9701119-2-6-8'], ['cond-mat-9701119-1-16-0', 'cond-mat-9701119-2-16-0'], ['cond-mat-9701119-1-16-1', 'cond-mat-9701119-2-16-1'], ['cond-mat-9701119-1-5-0', 'cond-mat-9701119-2-5-0'], ['cond-mat-9701119-1-5-1', 'cond-mat-9701119-2-5-1'], ['cond-mat-9701119-1-5-2', 'cond-mat-9701119-2-5-2'], ['cond-mat-9701119-1-5-3', 'cond-mat-9701119-2-5-3'], ['cond-mat-9701119-1-5-4', 'cond-mat-9701119-2-5-4'], ['cond-mat-9701119-1-0-0', 'cond-mat-9701119-2-0-0'], ['cond-mat-9701119-1-0-1', 'cond-mat-9701119-2-0-1'], ['cond-mat-9701119-1-0-2', 'cond-mat-9701119-2-0-2'], ['cond-mat-9701119-1-0-3', 'cond-mat-9701119-2-0-3'], ['cond-mat-9701119-1-0-4', 'cond-mat-9701119-2-0-4'], ['cond-mat-9701119-1-0-5', 'cond-mat-9701119-2-0-5'], ['cond-mat-9701119-1-8-0', 'cond-mat-9701119-2-8-0'], ['cond-mat-9701119-1-8-1', 'cond-mat-9701119-2-8-1'], ['cond-mat-9701119-1-8-2', 'cond-mat-9701119-2-8-2'], ['cond-mat-9701119-1-8-3', 'cond-mat-9701119-2-8-3'], ['cond-mat-9701119-1-8-4', 'cond-mat-9701119-2-8-4'], ['cond-mat-9701119-1-8-5', 'cond-mat-9701119-2-8-5'], ['cond-mat-9701119-1-8-6', 'cond-mat-9701119-2-8-6'], ['cond-mat-9701119-1-8-7', 'cond-mat-9701119-2-8-7'], ['cond-mat-9701119-1-4-0', 'cond-mat-9701119-2-4-0'], ['cond-mat-9701119-1-4-1', 'cond-mat-9701119-2-4-1'], ['cond-mat-9701119-1-4-2', 'cond-mat-9701119-2-4-2'], ['cond-mat-9701119-1-4-3', 'cond-mat-9701119-2-4-3'], ['cond-mat-9701119-1-4-4', 'cond-mat-9701119-2-4-4'], ['cond-mat-9701119-1-4-5', 'cond-mat-9701119-2-4-5']]

[]

[]

[]

[]

['cond-mat-9701119-1-1-0', 'cond-mat-9701119-2-1-0']

{'1': 'http://arxiv.org/licenses/assumed-1991-2003/', '2': 'http://arxiv.org/licenses/assumed-1991-2003/'}

https://arxiv.org/abs/cond-mat/9701119

1112.4452

{'1112.4452-1-0-0': 'We establish an interaction Morawetz estimate for the magnetic Schrodinger equation for [MATH] under certain smallness conditions on the gauge potentials.', '1112.4452-1-0-1': 'As an application, we prove global wellposedness and scattering in [MATH] for the cubic defocusing magnetic Schrodinger equation for [MATH].', '1112.4452-1-1-0': '# Introduction', '1112.4452-1-2-0': 'The purpose of this article is to initiate the study of the interaction Morawetz estimates for the magnetic Schrodinger equation.', '1112.4452-1-2-1': 'Morawetz type estimates have their origins in [CITATION] and [CITATION].', '1112.4452-1-2-2': 'The first interaction Morawetz estimate was established for the cubic defocusing NLS [CITATION], and it reads as follows [EQUATION]', '1112.4452-1-2-3': 'In particular, it allowed for a simpler proof of scattering obtained previously in [CITATION].', '1112.4452-1-2-4': 'The estimate was extended to [MATH] in [CITATION] giving [EQUATION]', '1112.4452-1-2-5': 'Then building on an idea of Hassell and other advances, a new proof was obtained in [CITATION] that applies to all dimensions [MATH].', '1112.4452-1-2-6': 'An independent proof was also achieved in [CITATION].', '1112.4452-1-2-7': 'For a more detailed background on Morawetz type estimates we refer to [CITATION].', '1112.4452-1-2-8': 'Now let [MATH] and consider the magnetic nonlinear Schrodinger equation [EQUATION] where [EQUATION]', '1112.4452-1-2-9': 'We use the standard notation, that the greek indices range from [MATH] to [MATH], and Roman indices range from [MATH] to [MATH].', '1112.4452-1-2-10': 'We also sum over repeated indices.', '1112.4452-1-2-11': 'The case of [MATH] is called defocusing and [MATH] is called focusing.', '1112.4452-1-2-12': 'We suppose we are in the Coulomb gauge, [MATH].', '1112.4452-1-2-13': 'The main result is Let [MATH] and let [MATH] solve the defocusing mNLS.', '1112.4452-1-2-14': 'Then the following estimate holds [EQUATION]', '1112.4452-1-2-15': 'The conditions on the gauge potentials [MATH] will soon be discussed in more detail in Section [REF].', '1112.4452-1-2-16': 'As an application we show Let [MATH] satisfy [REF]-[REF], [REF]-[REF] and [REF]-[REF].', '1112.4452-1-2-17': 'Then for given initial data in [MATH], mNLS with a defocusing cubic nonlinearity is globally wellposed and scatters to the linear magnetic Schrodinger equation.', '1112.4452-1-3-0': 'While the theory of existence and uniqueness has been considered before for the nonlinear mNLS (see [CITATION]) this is the first result (to our knowledge) on scattering for the nonlinear equation.', '1112.4452-1-3-1': 'Scattering for the one particle mNLS without the nonlinearity has been considered by many authors.', '1112.4452-1-3-2': 'We refer the reader to [CITATION] and references therein.', '1112.4452-1-3-3': 'The proof of Theorem [REF] relies on the commutator vector operators method developed in [CITATION].', '1112.4452-1-3-4': 'The main ingredient comes from the local conservation laws.', '1112.4452-1-3-5': 'In the case of the classical NLS the momentum is conserved.', '1112.4452-1-3-6': 'In the case of mNLS, we obtain only a balance law (see [REF] and [REF]-[REF] for precise definitions) [EQUATION] which eventually results in a need to control a term of the form [EQUATION].', '1112.4452-1-3-7': 'If [MATH] were positive, we could just ignore it (see for example the proof of [REF]).', '1112.4452-1-3-8': 'However as shown in the appendix, this cannot be expected in general.', '1112.4452-1-3-9': 'Another way to handle this term follows the path used by Fanelli and Vega [CITATION] for the linear magnetic Schrodinger equation.', '1112.4452-1-3-10': "Moreover, applying the results of [CITATION], D'Ancona, Fanelli, Vega and Visciglia established a family of Strichartz estimates [CITATION].", '1112.4452-1-3-11': 'Their work motivates us to assume similar conditions on our gauge potentials.', '1112.4452-1-3-12': 'As a result, we can control the term [MATH] leading to interaction Morawetz estimates for mNLS.', '1112.4452-1-3-13': 'To show Theorem [REF] we need an inhomogeneous Strichartz estimate.', '1112.4452-1-3-14': 'This is a simple consequence of the Christ-Kiselev Lemma and Strichartz estimates from [CITATION] (stated in Theorem [REF] and Theorem [REF]), and we record it here for completeness.', '1112.4452-1-4-0': 'Consider an inhomogeneous linear magnetic Schrodinger equation with zero initial data on [MATH].', '1112.4452-1-4-1': '[EQUATION] and suppose [MATH] is a solution of ([REF]) and that [MATH] satisfy [REF]-[REF], [REF]-[REF].', '1112.4452-1-4-2': 'Then [EQUATION] for Schrodinger admissible Strichartz pairs [MATH] such that both admissible pairs satisfy [EQUATION] and where [MATH] denotes the Holder dual exponent of [MATH].', '1112.4452-1-5-0': 'The dispersive properties of the magnetic Schrodinger equations have been studied also by [CITATION].', '1112.4452-1-5-1': 'We would like to investigate in the future if the interaction Morawetz estimates could be recaptured in the setting of these works.', '1112.4452-1-5-2': 'Also see [CITATION].', '1112.4452-1-6-0': 'The organization of the paper is as follows.', '1112.4452-1-6-1': 'In Section [REF] we gather some identities and known estimates.', '1112.4452-1-6-2': 'In Section [REF] we derive conservation laws and the generalized magnetic virial identity, which are then applied in Section [REF] to show Theorem [REF].', '1112.4452-1-6-3': 'In Section [REF] we prove the inhomogeneous Strichartz estimate.', '1112.4452-1-6-4': 'Section [REF] is devoted to the proof of Theorem [REF].', '1112.4452-1-7-0': '# Preliminaries', '1112.4452-1-8-0': 'We start by stating the following identities, which are easily verified by a direct computation [EQUATION]', '1112.4452-1-8-1': 'We recall the standard Schrodinger estimates [CITATION] .', '1112.4452-1-8-2': 'If [MATH] is Schrodinger admissible, i.e., [EQUATION] then [EQUATION] where [MATH] are Holder dual exponents of a Schrodinger admissible pair [MATH].', '1112.4452-1-9-0': 'We also use the following local smoothing estimate (see [CITATION] for historical remarks) [CITATION] If [MATH] is Schrodinger admissible, then [EQUATION] where [MATH].', '1112.4452-1-10-0': '## Conditions on the gauge potentials', '1112.4452-1-11-0': 'The curvature, [MATH], of the gauge potential [MATH] is a two-form given by [EQUATION] where [MATH] is given in [REF].', '1112.4452-1-11-1': 'From [MATH] we can extract the magnetic field [MATH] by only considering the spatial coordinates of [MATH] in [REF].', '1112.4452-1-11-2': 'Similarly we can extract the electric field from the temporal-spatial components.', '1112.4452-1-11-3': 'In [MATH] dimensions the magnetic field is often identified with a vector field [MATH].', '1112.4452-1-11-4': 'It was observed in [CITATION] that the trapping component, [MATH], of the magnetic field given by [EQUATION] was an obstruction to the dispersion.', '1112.4452-1-11-5': 'This can be thought of as the tangential component of the magnetic field with respect to the unit sphere.', '1112.4452-1-11-6': 'In higher dimensions the trapping component can be rephrased as [EQUATION] where [MATH] is a matrix with the [MATH] entry given by [MATH].', '1112.4452-1-11-7': "Thus the [MATH]'th entry of the vector [MATH] is [MATH].", '1112.4452-1-11-8': 'Next, if we take the radial derivative of [MATH] and decompose it into positive and negative parts [EQUATION] then the positive part can also affect dispersion [CITATION].', '1112.4452-1-11-9': 'The conditions that were used in [CITATION] are [EQUATION] for some [MATH] (see [CITATION]), and where [EQUATION].', '1112.4452-1-11-10': 'With those assumptions Fanelli and Vega were able to show some weak dispersion properties of the solutions of the linear mNLS [CITATION].', '1112.4452-1-11-11': 'The following is a part of Theorems 1.9 and 1.10 [CITATION].', '1112.4452-1-11-12': '[CITATION] Let [MATH], [MATH] satisfy [REF]-[REF], and let [MATH] denote the projection of [MATH] on the tangent space to the unit sphere [MATH], [MATH], then [EQUATION]', '1112.4452-1-11-13': 'Following the proof of Theorem [REF] we can establish analogs of these estimates for the nonlinear, defocusing mNLS.', '1112.4452-1-12-0': 'With the same assumptions as in Theorem [REF] we have [EQUATION] for any [MATH], where [MATH] solves mNLS with a defocusing nonlinearity [MATH], [MATH] and satisfies [MATH] and [MATH] is as in [REF].', '1112.4452-1-13-0': 'This follows immediately from Theorems 1.9 and 1.10 in [CITATION] once we observe that the proofs of these theorems rely on the generalized virial identity.', '1112.4452-1-13-1': 'The virial identity [CITATION] is for the homogeneous equation, but the addition of the defocusing nonlinearity leads to an addition of a term (see Lemma [REF] and Corollary [REF]) that is positive with [MATH] as in [CITATION] and results in an identical proof as before.', '1112.4452-1-14-0': '### Interaction Morawetz: curvature conditions', '1112.4452-1-15-0': 'In order to establish Theorem [REF] in addition to conditions [REF]-[REF] we impose the following (compare with [REF] and [REF] below).', '1112.4452-1-15-1': 'Let [EQUATION] and we assume there is [MATH] satisfying the following: [EQUATION]', '1112.4452-1-15-2': 'For [MATH], [EQUATION] and for [MATH] [EQUATION]', '1112.4452-1-15-3': 'Note that in comparison to [REF], the assumptions are made on the whole curvature and not just the projected components.', '1112.4452-1-15-4': 'On the other hand, we do not require the curvature to be small in these norms as in [REF], but merely to be bounded.', '1112.4452-1-15-5': 'In addition, the norms for the temporal component [MATH] are the same as [REF] whereas the magnetic field [MATH] is using now a slightly stronger norm.', '1112.4452-1-15-6': 'Finally, we observe that for example [MATH] satisfies the conditions with [MATH].', '1112.4452-1-16-0': '### Inhomogeneous Strichartz estimate: gauge potential conditions', '1112.4452-1-17-0': 'Now, to establish the inhomogeneous Strichartz estimate, besides [REF]-[REF] we need additional conditions found in [CITATION].', '1112.4452-1-17-1': '(We do not require here [REF]-[REF].)', '1112.4452-1-17-2': 'They are [EQUATION] where [MATH] is the Kato norm defined by [EQUATION] and where [MATH] is as in [REF].', '1112.4452-1-18-0': '## Magnetic Schrodinger Strichartz and other estimates used.', '1112.4452-1-19-0': '[EQUATION]', '1112.4452-1-19-1': 'As one consequence we have a boundedness of [MATH] on [MATH] as follows.', '1112.4452-1-19-2': 'First apply [REF] for an operator with [MATH], and then [REF] to get [EQUATION] and hence [EQUATION]', '1112.4452-1-19-3': 'For future reference, we remark [MATH].', '1112.4452-1-19-4': 'Next, from the proof of Theorem [REF] we have With the same assumptions as in Theorem [REF] we have [EQUATION]', '1112.4452-1-19-5': 'For [REF] interpolate (2.5) and (2.7) in [CITATION].', '1112.4452-1-19-6': '[REF] is (2.12) in [CITATION].', '1112.4452-1-20-0': 'The homogenous Strichartz estimate was established in [CITATION] [magnetic Schrodinger Strichartz, [CITATION]] Let [MATH].', '1112.4452-1-20-1': 'If [MATH] satisfy [REF]-[REF], [REF]-[REF], then for any Schrodinger admissible pair [MATH], the following Strichartz estimates hold: [EQUATION] and if [MATH], then at the endpoint we have [EQUATION]', '1112.4452-1-20-2': 'In the proof of the inhomogeneous Strichartz estimate we rely on the Christ-Kiselev Lemma.', '1112.4452-1-20-3': '[Christ-Kiselev Lemma [CITATION] and see [CITATION] ] Let [MATH] be Banach spaces and suppose [EQUATION] where [MATH] is an operator given by [EQUATION] for some operator-valued kernel [MATH] from [MATH] to [MATH], and let [MATH] satisfy [EQUATION] where [MATH] and [MATH] is independent of [MATH].', '1112.4452-1-21-0': '# Local Conservation Laws and Virial Identity', '1112.4452-1-22-0': 'Recall [EQUATION] where [MATH] and [MATH] is a real valued [MATH] function such that [MATH].', '1112.4452-1-22-1': 'For the convenience of the computations we write down an equivalent form of mNLS as [EQUATION]', '1112.4452-1-22-2': 'The virial identity for the linear magnetic Schrodinger equations was already established in [CITATION] with a potential [MATH] (which is [MATH] in the above equation) satisfying [EQUATION].', '1112.4452-1-22-3': 'We discuss local conservation laws.', '1112.4452-1-23-0': '## Local conservation laws', '1112.4452-1-24-0': 'Let [MATH] be a real valued function such that [EQUATION]', '1112.4452-1-24-1': 'Define pseudo-stress energy tensors as [EQUATION] for [MATH].', '1112.4452-1-24-2': 'We have the first local conservation law [EQUATION] which can be checked easily as follows.', '1112.4452-1-24-3': '[EQUATION]', '1112.4452-1-24-4': 'Now we compute [EQUATION]', '1112.4452-1-24-5': 'Hence [MATH] as needed.', '1112.4452-1-25-0': 'Next, we show we have [EQUATION]', '1112.4452-1-25-1': 'To establish [REF] we compute [EQUATION]', '1112.4452-1-25-2': 'Since by [REF] [EQUATION] we have [EQUATION]', '1112.4452-1-25-3': 'Next observe [EQUATION]', '1112.4452-1-25-4': 'Hence [EQUATION]', '1112.4452-1-25-5': 'Now [EQUATION]', '1112.4452-1-25-6': 'It follows [EQUATION]', '1112.4452-1-25-7': 'Combining and using [REF] we have [EQUATION]', '1112.4452-1-25-8': 'Since [MATH] from [REF], [EQUATION] and [EQUATION] as needed.', '1112.4452-1-25-9': 'We are now ready to proceed to the virial identity.', '1112.4452-1-26-0': '## Virial identity for mNLS', '1112.4452-1-27-0': 'Let [MATH].', '1112.4452-1-27-1': 'Define (gauged) Morawetz action by [EQUATION]', '1112.4452-1-27-2': "Note from Holder's inequality and the definition of [MATH], we immediately have [EQUATION]", '1112.4452-1-27-3': 'This can be refined just like it was in the classical case in [CITATION].', '1112.4452-1-27-4': 'Using [CITATION] we have (we note the statement of the lemma gives [MATH], but the following can be deduced from the proof) [EQUATION] if we assume [MATH] to be bounded, which they always are in our case.', '1112.4452-1-27-5': 'Next, following [CITATION] we obtain the following lemma.', '1112.4452-1-27-6': '[Generalized virial identity] Let [MATH], and [MATH] be a solution of (mNLS).', '1112.4452-1-27-7': 'Then [EQUATION]', '1112.4452-1-27-8': 'By [REF], [REF] and integration by parts, [EQUATION] [REF] now follows by the fundamental theorem of calculus.', '1112.4452-1-28-0': 'If [MATH] is convex and [MATH] we can further conclude [EQUATION]', '1112.4452-1-28-1': 'This is easy to see since if [MATH] is convex we can first show that [EQUATION]', '1112.4452-1-28-2': 'Indeed, we know if a function [MATH] is convex then for [MATH], [EQUATION]', '1112.4452-1-28-3': 'We apply this twice to conclude [REF].', '1112.4452-1-28-4': 'Define vectors [MATH], [MATH] by [EQUATION]', '1112.4452-1-28-5': 'Next since for general [MATH], [EQUATION] we have [EQUATION] by [REF].', '1112.4452-1-28-6': 'Finally since [MATH] is convex and the Hessian, [MATH] is positive-semidefinite, the trace, [MATH], which implies [EQUATION] and the result follows.', '1112.4452-1-29-0': 'We end this section by a brief discussion of the conservation of mass and energy for the mNLS.', '1112.4452-1-29-1': 'From [CITATION] we have [EQUATION] where [MATH].', '1112.4452-1-29-2': 'This in particular implies conservation of mass and energy for the linear magnetic Schrodinger equations.', '1112.4452-1-29-3': 'In case of mNLS we have [Conservation of mass and energy] Let [MATH] be self-adjoint and positive on [MATH], [MATH] and let [MATH] solve mNLS.', '1112.4452-1-29-4': 'Then for every [MATH] [EQUATION] [REF] follows by integrating in space [MATH], and [REF] by a direct computation using the equation.', '1112.4452-1-30-0': '# Interaction Morawetz Estimates', '1112.4452-1-31-0': 'As in [CITATION] we use the following notation [EQUATION] and [EQUATION] where [MATH].', '1112.4452-1-31-1': 'Then we rewrite the local conservation laws as [EQUATION]', '1112.4452-1-32-0': '## Proof of Theorem [REF] using the commutator vector operators', '1112.4452-1-33-0': 'The Morawetz action [REF] for a tensor product of two solutions [MATH] with [MATH] can be rewritten as [EQUATION]', '1112.4452-1-33-1': 'Following [CITATION] we use operators [MATH] and [MATH] defined by [EQUATION] so [EQUATION]', '1112.4452-1-33-2': 'Further, a computation shows [EQUATION] where [EQUATION] and [EQUATION] where [MATH].', '1112.4452-1-33-3': 'The crucial observation made in [CITATION] was that the derivatives of [MATH] are positive definite.', '1112.4452-1-33-4': 'Using the above operators we write [EQUATION].', '1112.4452-1-33-5': 'Then [EQUATION].', '1112.4452-1-33-6': 'By [REF], and [REF] [EQUATION]', '1112.4452-1-33-7': 'And by [REF] and [REF] [EQUATION]', '1112.4452-1-33-8': 'It follows [EQUATION] where [EQUATION]', '1112.4452-1-33-9': 'We discuss positivity of each term.', '1112.4452-1-33-10': 'This analysis is also the same as in [CITATION], but the difference is that the momentum vector [MATH] is now covariant, and we also have to address [MATH].', '1112.4452-1-33-11': 'We sketch the details for [MATH] through [MATH] for completeness.', '1112.4452-1-33-12': 'Since [MATH] is positive definite, [MATH].', '1112.4452-1-33-13': 'For [MATH] define the two point momentum vector [EQUATION].', '1112.4452-1-33-14': 'Then (see [CITATION] for details) [EQUATION] since again [MATH] is positive definite.', '1112.4452-1-33-15': 'For [MATH] using [MATH] , [EQUATION] and [EQUATION] as long as [MATH].', '1112.4452-1-33-16': 'Integrating in time we have, [EQUATION] so the estimate follows by [REF] if we can handle the last term [MATH].', '1112.4452-1-34-0': 'We cannot expect [MATH] to be positive (see the appendix).', '1112.4452-1-34-1': 'Examples when [MATH] were given in [CITATION] (note this still leaves the term involving [MATH]).', '1112.4452-1-34-2': 'In general, as shown below, we can control [MATH] by imposing the conditions [REF]-[REF] as they allow us to take advantage of the smoothing estimates proved in [CITATION].', '1112.4452-1-34-3': 'In addition, we also require [REF]-[REF].', '1112.4452-1-35-0': '## [MATH] Replacement of positivity condition by bounds on [MATH]', '1112.4452-1-36-0': 'Suppose [REF]-[REF] hold.', '1112.4452-1-36-1': 'Then [EQUATION]', '1112.4452-1-37-0': '### Estimates for [MATH].', '1112.4452-1-38-0': 'Choose [MATH] and let [MATH].', '1112.4452-1-38-1': 'Impose [REF]- [REF].', '1112.4452-1-38-2': 'Since [MATH] we get [EQUATION]', '1112.4452-1-38-3': 'Next [EQUATION] by [REF] and [REF].', '1112.4452-1-38-4': '[EQUATION] by [REF] and [REF].', '1112.4452-1-38-5': 'To estimate [MATH] note that [MATH] is independent in time and [MATH].', '1112.4452-1-38-6': 'Then [EQUATION] by [REF] and [REF].', '1112.4452-1-38-7': 'The estimates for [MATH] are analogous.', '1112.4452-1-39-0': '### Estimates for [MATH].', '1112.4452-1-40-0': 'Just as before, we write [EQUATION] where [MATH] is estimated using [REF] and [REF].', '1112.4452-1-40-1': 'For [MATH] we have [EQUATION] by [REF] and [REF].', '1112.4452-1-40-2': 'Next, [EQUATION] by [REF] and [REF].', '1112.4452-1-41-0': '# Proof of the Inhomogeneous Strichartz Estimate, Theorem [REF]', '1112.4452-1-42-0': 'Let [MATH], [MATH] be a space-time function which is sufficiently regular and [MATH] be the solution of [REF].', '1112.4452-1-42-1': "Note that [EQUATION] by Duhamel's principle and define [MATH].", '1112.4452-1-42-2': 'By the Christ-Kiselev lemma, [EQUATION]', '1112.4452-1-42-3': 'So it is enough to show [EQUATION] for any [MATH], [MATH].', '1112.4452-1-42-4': 'From the definition of [MATH], Strichartz estimate and self-adjointness of [MATH], [EQUATION] and proof is completed.', '1112.4452-1-43-0': '# Application to Magnetic Nonlinear Schro dinger equations', '1112.4452-1-44-0': 'In this section, we show applications of previous estimates to global existence and scattering.', '1112.4452-1-44-1': 'For simplicity, we consider magnetic NLS with defocusing cubic nonlinearity in [MATH].', '1112.4452-1-44-2': '[EQUATION]', '1112.4452-1-44-3': 'We note that by virtue of [REF] and [REF] we have [EQUATION]', '1112.4452-1-44-4': 'To establish Theorem [REF] we begin with a local theory (see [CITATION] for related works).', '1112.4452-1-45-0': '## Local existence', '1112.4452-1-46-0': 'Let [MATH].', '1112.4452-1-46-1': 'Note that by [REF], Theorem [REF], and [REF] [EQUATION] where [MATH].', '1112.4452-1-46-2': 'We show a solution of [REF] uniquely exists locally in time in the space [EQUATION] where [MATH], and [MATH] is the maximum of the constant [MATH], and the constants [MATH] that appear in the estimates below.', '1112.4452-1-46-3': 'Define the sequence of Picard iterates by [EQUATION] where [EQUATION]', '1112.4452-1-46-4': 'By [REF] and Theorem [REF] [EQUATION] and again by [REF], Theorem [REF] and [REF], [EQUATION].', '1112.4452-1-46-5': 'Now suppose that for [MATH].', '1112.4452-1-46-6': 'Then by Theorem [REF], Theorem [REF] and Sobolev embedding, [EQUATION]', '1112.4452-1-46-7': 'Hence, if [MATH] is small enough, i.e., [EQUATION] and using conservation of mass [REF], we have [EQUATION].', '1112.4452-1-46-8': 'Finally, [EQUATION]', '1112.4452-1-46-9': 'If we require [EQUATION] then [EQUATION] which shows the sequence [MATH] belongs to [MATH].', '1112.4452-1-46-10': 'To show the sequence converges, we need to consider the differences.', '1112.4452-1-46-11': 'The estimates are similar, and we only show some of the details.', '1112.4452-1-47-0': 'Let [EQUATION] then we can write [EQUATION].', '1112.4452-1-47-1': 'Now consider [EQUATION] by [REF].', '1112.4452-1-47-2': 'We obtain the same bounds for term [MATH], and for the other norms in [MATH], which show the sequence of the iterates is Cauchy and hence it converges as needed.', '1112.4452-1-48-0': '## Global existence', '1112.4452-1-49-0': 'Let [MATH] be the solution of [REF] obtained from local existence.', '1112.4452-1-49-1': 'First note that by [REF], [REF] and [REF], the [MATH] norm of [MATH] is bounded.', '1112.4452-1-49-2': 'Let [MATH] be the supremum of [MATH] norm of [MATH].', '1112.4452-1-49-3': 'Now, let [MATH] be the maximal time interval that solution exists and suppose [MATH].', '1112.4452-1-49-4': 'Observe, if the global existence fails, then we have an increasing sequence [MATH] such that [EQUATION] where [MATH] is defined as in the local existence proof.', '1112.4452-1-49-5': 'We shall prove that this gives a contradiction.', '1112.4452-1-49-6': 'From the conserved quantities of the equation, [MATH] and [MATH] are uniformly bounded on [MATH].', '1112.4452-1-49-7': 'We consider an interval [MATH] so that [MATH] for some [MATH] which will be chosen later.', '1112.4452-1-49-8': 'Assume [MATH].', '1112.4452-1-49-9': 'Then by the Strichartz estimate [EQUATION]', '1112.4452-1-49-10': 'Since [MATH] is uniformly bounded, if [MATH] is chosen to be sufficiently small, a standard argument of the method of continuity gives [MATH].', '1112.4452-1-49-11': 'Now suppose [MATH] is a sequence satisfying ([REF]) and ([REF]).', '1112.4452-1-49-12': 'Then by the triangle inequality, Strichartz estimate and Sobolev embedding [EQUATION]', '1112.4452-1-49-13': 'The right-hand side is less than [MATH] for sufficiently large [MATH], which is a contradiction to the assumption of [MATH].', '1112.4452-1-50-0': '## Scattering', '1112.4452-1-51-0': 'In this section we consider the question of scattering (asymptotic completeness).', '1112.4452-1-51-1': 'We take a point of view analogous to the classical NLS.', '1112.4452-1-51-2': 'Hence we set out to show that given a solution of the nonlinear mNLS, [MATH], there exists a solution of the linear mNLS, [MATH] such that the [MATH] norm of the difference of the two solutions goes to [MATH] as [MATH] (note, due to [REF] this also gives convergence of [MATH]).', '1112.4452-1-51-3': 'Now, following the classical NLS setup for scattering, let [MATH] be the solution to the cubic defocusing mNLS with initial data [MATH].', '1112.4452-1-51-4': 'We define [EQUATION].', '1112.4452-1-51-5': 'The convergence in [MATH] of the difference of [MATH] and [MATH] is then immediate if we can show [EQUATION] converges in [MATH].', '1112.4452-1-51-6': 'Therefore, equivalently, we need to show [EQUATION]', '1112.4452-1-51-7': 'We prove ([REF]) for [MATH] and [MATH] separately.', '1112.4452-1-51-8': 'For [MATH], we need to show [EQUATION] as [MATH].', '1112.4452-1-51-9': 'Note that [EQUATION]', '1112.4452-1-51-10': 'We used interpolation inequality [MATH].', '1112.4452-1-51-11': 'The last quantity converges to [MATH] as [MATH] since [MATH] is finite.', '1112.4452-1-51-12': 'For [MATH], we need the following lemma.', '1112.4452-1-52-0': 'For a solution [MATH] of the given equation, [MATH] is finite.', '1112.4452-1-53-0': 'By Sobolev embedding, [EQUATION]', '1112.4452-1-53-1': 'We now subdivide [MATH] into finitely many disjoint intervals [MATH] so that [EQUATION] for some [MATH] which will be chosen later.', '1112.4452-1-53-2': 'On each interval [MATH], we have [EQUATION]', '1112.4452-1-53-3': 'We take small enough [MATH] to apply the continuity method.', '1112.4452-1-53-4': 'Note that [MATH] only depends on the implicit constant of the Strichartz estimate and the size of the initial data.', '1112.4452-1-53-5': 'By the method of continuity, we conclude [MATH] is finite on each interval [MATH].', '1112.4452-1-53-6': 'Since we have only finitely many intervals, [MATH] is finite on [MATH], and the result follows by [REF].', '1112.4452-1-54-0': 'Now it is easy to show scattering as follows.', '1112.4452-1-54-1': '[EQUATION] and [EQUATION]', '1112.4452-1-54-2': 'We used interpolation inequality [MATH].', '1112.4452-1-54-3': 'Since [MATH] is finite, the last quantity vanishes as [MATH] which completes the proof of scattering.', '1112.4452-1-55-0': '# Failure of pointwise nonnegativity of [MATH]', '1112.4452-1-56-0': 'Let [MATH] and [MATH] be time independent, divergence-free.', '1112.4452-1-56-1': 'Then [EQUATION].', '1112.4452-1-56-2': 'And since [MATH] the above formula is [EQUATION]', '1112.4452-1-56-3': 'Since [MATH] is parallel to [MATH] as long as [MATH], for any given [MATH], we can find [MATH] so that [MATH].', '1112.4452-1-56-4': 'Similarly, we can find [MATH] so that [MATH] and [MATH] form an angle less than [MATH].', '1112.4452-1-56-5': 'Alternatively, we can view [MATH] as [EQUATION] and again as long as [MATH], then this is a dot product of the momentum vector with a component of [MATH] tangent to the unit sphere centered at [MATH] and the second term is the radial component of [MATH] with respect to the sphere centered at [MATH] (compare to the trapping component in [CITATION]).', '1112.4452-1-56-6': 'Therefore, as we move [MATH] around, pointwise nonnegativity is not possible.'}

{'1112.4452-2-0-0': 'We establish an interaction Morawetz estimate for the magnetic Schrodinger equation for [MATH] under certain smallness conditions on the gauge potentials, but with almost optimal decay.', '1112.4452-2-0-1': 'As an application, we prove global wellposedness and scattering in [MATH] for the cubic defocusing magnetic Schrodinger equation for [MATH].', '1112.4452-2-1-0': '# Introduction', '1112.4452-2-2-0': 'The purpose of this article is to study the interaction Morawetz estimates for the magnetic Schrodinger equation.', '1112.4452-2-2-1': 'Morawetz type estimates have their origins in [CITATION] and [CITATION].', '1112.4452-2-2-2': 'The first interaction Morawetz estimate was established for the cubic defocusing NLS [CITATION], and it reads as follows [EQUATION]', '1112.4452-2-2-3': 'In particular, it allowed for a simpler proof of scattering obtained previously in [CITATION].', '1112.4452-2-2-4': 'The estimate was extended to [MATH] in [CITATION] giving [EQUATION]', '1112.4452-2-2-5': 'Then building on an idea of Hassell and other advances, a new proof was obtained in [CITATION] that applies to all dimensions [MATH].', '1112.4452-2-2-6': 'An independent proof was also achieved in [CITATION].', '1112.4452-2-2-7': 'For a more detailed background on Morawetz type estimates we refer to [CITATION].', '1112.4452-2-2-8': 'Now let [MATH] and consider the magnetic nonlinear Schrodinger equation [EQUATION] where [EQUATION]', '1112.4452-2-2-9': 'We use the standard notation, that the greek indices range from [MATH] to [MATH], and Roman indices range from [MATH] to [MATH].', '1112.4452-2-2-10': 'We also sum over repeated indices.', '1112.4452-2-2-11': 'The case of [MATH] is usually called defocusing and [MATH] is called focusing.', '1112.4452-2-2-12': 'We suppose we are in the Coulomb gauge, [MATH].', '1112.4452-2-2-13': 'The main result is Let [MATH] and let [MATH] solve the defocusing mNLS.', '1112.4452-2-2-14': 'Then the following estimate holds [EQUATION]', '1112.4452-2-2-15': 'The conditions on the gauge potentials [MATH] will soon be discussed in more detail in Section [REF].', '1112.4452-2-2-16': 'As an application we show Let [MATH] satisfy [REF]-[REF], [REF]-[REF] and [REF]-[REF].', '1112.4452-2-2-17': 'Then for given initial data in [MATH], mNLS with a defocusing cubic nonlinearity is globally wellposed and scatters to the linear magnetic Schrodinger equation.', '1112.4452-2-3-0': 'While the theory of existence and uniqueness has been considered before for the nonlinear mNLS (see [CITATION]) this is the first result (to our knowledge) on scattering for the nonlinear equation.', '1112.4452-2-3-1': 'Scattering for the one particle mNLS without the nonlinearity has been considered by many authors.', '1112.4452-2-3-2': 'We refer the reader to [CITATION] and references therein.', '1112.4452-2-4-0': 'We also would like to note that in the proof of local well-posedness we do not reproduce the same contraction argument usually done for the cubic NLS (see for example [CITATION]).', '1112.4452-2-4-1': 'The reason for this is that even though we have Strichartz estimates for the mNLS, to fully benefit from them we would either need to extend estimate [REF] below to other [MATH] besides [MATH] or establish a variant of a product rule in [MATH].', '1112.4452-2-4-2': 'Doing that does not seem easier than using alternate Strichartz exponents.', '1112.4452-2-4-3': 'The current approach has a flavor of what is usually done for the critical equations, and hence it is more involved than if we were going to just use the standard subcritical methods.', '1112.4452-2-5-0': 'The proof of Theorem [REF] relies on the commutator vector operators method developed in [CITATION], where it was used to obtain the interaction Morawetz estimate for the classical NLS.', '1112.4452-2-5-1': 'We show in this article that the method is robust and can be extended to the magnetic case.', '1112.4452-2-6-0': 'The main ingredient comes from the local conservation laws.', '1112.4452-2-6-1': 'In the case of the classical NLS the momentum is conserved.', '1112.4452-2-6-2': 'In the case of mNLS, we obtain only a balance law (see [REF] and [REF]-[REF] for precise definitions) [EQUATION] which eventually results in a need to control a term of the form [EQUATION].', '1112.4452-2-6-3': 'If [MATH] were positive, we could just ignore it (see for example the proof of [REF]).', '1112.4452-2-6-4': 'However as shown in the appendix, this cannot be expected in general.', '1112.4452-2-6-5': 'Another way to handle this term follows the path used by Fanelli and Vega [CITATION] for the linear magnetic Schrodinger equation.', '1112.4452-2-6-6': "Moreover, applying the results of [CITATION], D'Ancona, Fanelli, Vega and Visciglia established a family of Strichartz estimates [CITATION].", '1112.4452-2-6-7': 'Their work motivates us to assume similar conditions on the gauge potentials.', '1112.4452-2-6-8': 'As a result, we can control the term [MATH], and hence obtain the interaction Morawetz estimates for mNLS.', '1112.4452-2-6-9': 'To show Theorem [REF] we need an inhomogeneous Strichartz estimate.', '1112.4452-2-6-10': 'This is a simple consequence of the Christ-Kiselev Lemma and Strichartz estimates from [CITATION] (stated in Theorem [REF] and Theorem [REF]), and we record it here only for completeness.', '1112.4452-2-7-0': 'Consider an inhomogeneous linear magnetic Schrodinger equation with zero initial data on [MATH].', '1112.4452-2-7-1': '[EQUATION] and suppose [MATH] is a solution of ([REF]) and that [MATH] satisfy [REF]-[REF], [REF]-[REF].', '1112.4452-2-7-2': 'Then [EQUATION] for Schrodinger admissible Strichartz pairs [MATH] such that both admissible pairs satisfy [EQUATION] and where [MATH] denotes the Holder dual exponent of [MATH].', '1112.4452-2-8-0': 'The dispersive properties of the magnetic Schrodinger equations have been studied also by [CITATION].', '1112.4452-2-8-1': 'We would like to investigate in the future if the interaction Morawetz estimates could be recaptured in the setting of these works.', '1112.4452-2-8-2': 'Also see [CITATION].', '1112.4452-2-9-0': 'The organization of the paper is as follows.', '1112.4452-2-9-1': 'In Section [REF] we gather some identities and known estimates.', '1112.4452-2-9-2': 'In Section [REF] we derive conservation laws and the generalized magnetic virial identity, which are then applied in Section [REF] to show Theorem [REF].', '1112.4452-2-9-3': 'In Section [REF] we prove the inhomogeneous Strichartz estimate.', '1112.4452-2-9-4': 'Section [REF] is devoted to the proof of Theorem [REF].', '1112.4452-2-10-0': 'The authors would like to thank the referees for the helpful comments.', '1112.4452-2-10-1': 'The first author was supported in part by NSERC grant RGP250233-07.', '1112.4452-2-10-2': 'The second author was partially supported by a grant from the Simons Foundation 246255.', '1112.4452-2-11-0': '# Preliminaries', '1112.4452-2-12-0': 'We start by stating the following identities, which are easily verified by a direct computation [EQUATION]', '1112.4452-2-12-1': 'We recall the standard Strichartz estimates [CITATION].', '1112.4452-2-12-2': 'If [MATH] is Schrodinger admissible, i.e., [EQUATION] then [EQUATION] where [MATH] are Holder dual exponents of a Schrodinger admissible pair [MATH].', '1112.4452-2-13-0': 'We also use the following local smoothing estimate (see [CITATION] for historical remarks) [CITATION] If [MATH] is Schrodinger admissible, then [EQUATION] where [MATH].', '1112.4452-2-14-0': '## Conditions on the gauge potentials', '1112.4452-2-15-0': 'The curvature, [MATH], of the gauge potential [MATH] is a two-form given by [EQUATION] where [MATH] is given in [REF].', '1112.4452-2-15-1': 'From [MATH] we can extract the magnetic field [MATH] by only considering the spatial coordinates of [MATH] in [REF].', '1112.4452-2-15-2': 'Similarly we can extract the electric field from the temporal-spatial components.', '1112.4452-2-15-3': 'In [MATH] dimensions the magnetic field is often identified with a vector field [MATH].', '1112.4452-2-15-4': 'It was observed in [CITATION] that the trapping component, [MATH], of the magnetic field given by [EQUATION] was an obstruction to the dispersion.', '1112.4452-2-15-5': 'This can be thought of as the tangential component of the magnetic field with respect to the unit sphere.', '1112.4452-2-15-6': 'In higher dimensions the trapping component can be rephrased as [EQUATION] where [MATH] is a matrix with the [MATH] entry given by [MATH].', '1112.4452-2-15-7': "Thus the [MATH]'th entry of the vector [MATH] is [MATH].", '1112.4452-2-15-8': 'Next, if we take the radial derivative of [MATH] and decompose it into positive and negative parts [EQUATION] then the positive part can also affect dispersion [CITATION].', '1112.4452-2-15-9': 'The conditions that were used in [CITATION] are [EQUATION] for some [MATH] (see [CITATION]), and where [EQUATION].', '1112.4452-2-15-10': 'With those assumptions Fanelli and Vega were able to show some weak dispersion properties of the solutions of the linear mNLS [CITATION].', '1112.4452-2-15-11': 'The following is a part of Theorems 1.9 and 1.10 [CITATION].', '1112.4452-2-15-12': 'Note with [MATH], linear mNLS, can be written as [EQUATION] so [MATH] below refers to the solution with initial data [MATH].', '1112.4452-2-16-0': '[CITATION] Let [MATH], [MATH] satisfy [REF]-[REF], and let [MATH] denote the projection of [MATH] on the tangent space to the unit sphere [MATH], [MATH], then [EQUATION]', '1112.4452-2-16-1': 'Following the proof of Theorem [REF] one can establish analogs of these estimates for the nonlinear, defocusing mNLS.', '1112.4452-2-17-0': 'With the same assumptions as in Theorem [REF] we have [EQUATION] for any [MATH], where [MATH] solves mNLS with a defocusing nonlinearity [MATH], [MATH] and satisfies [MATH] and [MATH] is as in [REF].', '1112.4452-2-18-0': 'This follows immediately from Theorems 1.9 and 1.10 in [CITATION] once we observe that the proofs of these theorems rely on the generalized virial identity.', '1112.4452-2-18-1': 'The virial identity [CITATION] is for the homogeneous equation, but the addition of the defocusing nonlinearity leads to an addition of a term (see Lemma [REF] and Corollary [REF]) that is positive with [MATH] as in [CITATION] and results in an identical proof as before.', '1112.4452-2-19-0': '### Interaction Morawetz: curvature conditions', '1112.4452-2-20-0': 'In order to establish Theorem [REF] in addition to conditions [REF]-[REF] we impose the following (compare with [REF] and [REF] below).', '1112.4452-2-20-1': 'Let [EQUATION] and we assume there is [MATH] satisfying the following: [EQUATION]', '1112.4452-2-20-2': 'For [MATH], [EQUATION] and for [MATH] [EQUATION]', '1112.4452-2-20-3': 'Note that in comparison to [REF], the assumptions are made on the whole curvature and not just the projected components.', '1112.4452-2-20-4': 'On the other hand, we do not require the curvature to be small in these norms as in [REF], but merely to be bounded.', '1112.4452-2-20-5': 'In addition, the norms for the temporal component [MATH] are the same as [REF] whereas the magnetic field [MATH] is using now a slightly stronger norm.', '1112.4452-2-20-6': 'Finally, observe that the magnetic field [MATH] satisfies the conditions with [MATH].', '1112.4452-2-20-7': 'Such magnetic field corresponds to [MATH] decaying like [MATH].', '1112.4452-2-20-8': 'Similarly, [MATH] satisfies the needed conditions.', '1112.4452-2-20-9': 'This type of decay for [MATH] is almost optimal [CITATION].', '1112.4452-2-20-10': 'Hence Theorem [REF] implies interaction Morawetz estimates for potentials with almost optimal decay.', '1112.4452-2-21-0': '### Inhomogeneous Strichartz estimate: gauge potential conditions', '1112.4452-2-22-0': 'Now, to establish the inhomogeneous Strichartz estimate, besides [REF]-[REF] we need additional conditions found in [CITATION].', '1112.4452-2-22-1': '(We do not require here [REF]-[REF].)', '1112.4452-2-22-2': 'They are [EQUATION] where [MATH] is the Kato norm defined by [EQUATION] and where [MATH] is as in [REF].', '1112.4452-2-23-0': '## Magnetic Schrodinger Strichartz and other estimates used.', '1112.4452-2-24-0': '[EQUATION]', '1112.4452-2-24-1': 'As one consequence we have a boundedness of [MATH] on [MATH] as follows.', '1112.4452-2-24-2': 'First apply [REF] for an operator with [MATH], and then [REF] to get [EQUATION] and hence [EQUATION]', '1112.4452-2-24-3': 'For future reference, we remark [MATH].', '1112.4452-2-24-4': 'Next, from the proof of Theorem [REF] we have With the same assumptions as in Theorem [REF] we have [EQUATION]', '1112.4452-2-24-5': 'For [REF] interpolate (2.5) and (2.7) in [CITATION].', '1112.4452-2-24-6': '[REF] is (2.12) in [CITATION].', '1112.4452-2-25-0': 'The homogenous Strichartz estimate was established in [CITATION] [magnetic Schrodinger Strichartz, [CITATION]] Let [MATH].', '1112.4452-2-25-1': 'If [MATH] satisfy [REF]-[REF], [REF]-[REF], then for any Schrodinger admissible pair [MATH], the following Strichartz estimates hold: [EQUATION] and if [MATH], then at the endpoint we have [EQUATION]', '1112.4452-2-25-2': 'In the proof of the inhomogeneous Strichartz estimate we rely on the Christ-Kiselev Lemma.', '1112.4452-2-25-3': '[Christ-Kiselev Lemma [CITATION] and see [CITATION] ] Let [MATH] be Banach spaces and suppose [EQUATION] where [MATH] is an operator given by [EQUATION] for some operator-valued kernel [MATH] from [MATH] to [MATH], and let [MATH] satisfy [EQUATION] where [MATH] and [MATH] is independent of [MATH].', '1112.4452-2-26-0': '# Local Conservation Laws and Virial Identity', '1112.4452-2-27-0': 'Recall [EQUATION] where [MATH] and [MATH] is a real valued [MATH] function such that [MATH].', '1112.4452-2-27-1': 'For the convenience of the computations we write down an equivalent form of mNLS as [EQUATION]', '1112.4452-2-27-2': 'The virial identity for the linear magnetic Schrodinger equations was already established in [CITATION] with a potential [MATH] (which is [MATH] in the above equation) satisfying [EQUATION].', '1112.4452-2-27-3': 'We discuss local conservation laws.', '1112.4452-2-28-0': '## Local conservation laws', '1112.4452-2-29-0': 'Let [MATH] be a real valued function such that [EQUATION]', '1112.4452-2-29-1': 'Define pseudo-stress energy tensors as [EQUATION] for [MATH].', '1112.4452-2-29-2': 'We have the first local conservation law [EQUATION] which can be checked easily as follows.', '1112.4452-2-29-3': '[EQUATION]', '1112.4452-2-29-4': 'Now we compute [EQUATION]', '1112.4452-2-29-5': 'Hence [MATH] as needed.', '1112.4452-2-30-0': 'Next, we show we have [EQUATION]', '1112.4452-2-30-1': 'To establish [REF] we compute [EQUATION]', '1112.4452-2-30-2': 'Since by [REF] [EQUATION] we have [EQUATION]', '1112.4452-2-30-3': 'Next observe [EQUATION]', '1112.4452-2-30-4': 'Hence [EQUATION]', '1112.4452-2-30-5': 'Now [EQUATION]', '1112.4452-2-30-6': 'It follows [EQUATION]', '1112.4452-2-30-7': 'Combining and using [REF] we have [EQUATION]', '1112.4452-2-30-8': 'Since [MATH] from [REF], [EQUATION] and [EQUATION] as needed.', '1112.4452-2-30-9': 'We are now ready to proceed to the virial identity.', '1112.4452-2-31-0': '## Virial identity for mNLS', '1112.4452-2-32-0': 'Let [MATH].', '1112.4452-2-32-1': 'Define (gauged) Morawetz action by [EQUATION]', '1112.4452-2-32-2': "Note from Holder's inequality and the definition of [MATH], we immediately have [EQUATION]", '1112.4452-2-32-3': 'This can be refined just like it was in the classical case in [CITATION].', '1112.4452-2-32-4': 'Using [CITATION] we have (we note the statement of the lemma gives [MATH], but the following can be deduced from the proof) [EQUATION] if we assume [MATH] to be bounded, which they always are in our case.', '1112.4452-2-32-5': 'Next, following [CITATION] we obtain the following lemma.', '1112.4452-2-32-6': '[Generalized virial identity] Let [MATH], and [MATH] be a solution of (mNLS).', '1112.4452-2-32-7': 'Then [EQUATION]', '1112.4452-2-32-8': 'By [REF], [REF] and integration by parts, [EQUATION] [REF] now follows by the fundamental theorem of calculus.', '1112.4452-2-33-0': 'If [MATH] is convex and [MATH] we can further conclude [EQUATION]', '1112.4452-2-33-1': 'This is easy to see since if [MATH] is convex, we can first show that [EQUATION]', '1112.4452-2-33-2': 'Indeed, we know if a function [MATH] is convex then for [MATH], [EQUATION]', '1112.4452-2-33-3': 'We apply this twice to conclude [REF].', '1112.4452-2-33-4': 'Define vectors [MATH], [MATH] by [EQUATION]', '1112.4452-2-33-5': 'Next since for general [MATH], [EQUATION] we have [EQUATION] by [REF].', '1112.4452-2-33-6': 'Finally since [MATH] is convex and the Hessian, [MATH] is positive-semidefinite, the trace, [MATH], which implies [EQUATION] and the result follows.', '1112.4452-2-34-0': 'We end this section by a brief discussion of the conservation of mass and energy for the mNLS.', '1112.4452-2-34-1': 'From [CITATION] we have [EQUATION] where [MATH].', '1112.4452-2-34-2': 'This in particular implies conservation of mass and energy for the linear magnetic Schrodinger equations.', '1112.4452-2-34-3': 'In case of mNLS we have [Conservation of mass and energy] Let [MATH] be self-adjoint and positive on [MATH], [MATH] and let [MATH] solve mNLS.', '1112.4452-2-34-4': 'Then for every [MATH] [EQUATION] [REF] follows by integrating in space [MATH], and [REF] by a direct computation using the equation.', '1112.4452-2-35-0': '# Interaction Morawetz Estimates', '1112.4452-2-36-0': 'As in [CITATION] we use the following notation [EQUATION] and [EQUATION] where [MATH].', '1112.4452-2-36-1': 'Then we rewrite the local conservation laws as [EQUATION]', '1112.4452-2-37-0': '## Proof of Theorem [REF] using the commutator vector operators', '1112.4452-2-38-0': 'The Morawetz action [REF] for a tensor product of two solutions [MATH] with [MATH] can be rewritten as [EQUATION]', '1112.4452-2-38-1': 'Following [CITATION] we use operators [MATH] and [MATH] defined by [EQUATION] so [EQUATION]', '1112.4452-2-38-2': 'Further, a computation shows [EQUATION] where [EQUATION] and [EQUATION] where [MATH].', '1112.4452-2-38-3': 'The crucial observation made in [CITATION] was that the derivatives of [MATH] are positive definite.', '1112.4452-2-38-4': 'Using the above operators we write [EQUATION].', '1112.4452-2-38-5': 'Then [EQUATION].', '1112.4452-2-38-6': 'By [REF], and [REF] [EQUATION]', '1112.4452-2-38-7': 'And by [REF] [EQUATION]', '1112.4452-2-38-8': 'It follows [EQUATION] where [EQUATION]', '1112.4452-2-38-9': 'We discuss the positivity of each term.', '1112.4452-2-38-10': 'This analysis is also the same as in [CITATION], but the difference is that the momentum vector [MATH] is now covariant, and we also have to address [MATH].', '1112.4452-2-38-11': 'We briefly sketch the main ideas for [MATH] through [MATH] for completeness (for details see [CITATION]).', '1112.4452-2-39-0': 'Since [MATH] is positive definite, [MATH].', '1112.4452-2-39-1': 'For [MATH] define the two point momentum vector [EQUATION].', '1112.4452-2-39-2': 'Then (see [CITATION] for details) [EQUATION] since again [MATH] is positive definite.', '1112.4452-2-39-3': 'For [MATH] using [MATH] , [EQUATION] and [EQUATION] as long as [MATH].', '1112.4452-2-40-0': 'Now, integrating in time we have [EQUATION] so the estimate follows by [REF] if we can handle the last term [MATH].', '1112.4452-2-41-0': 'We cannot expect [MATH] to be positive (see the appendix).', '1112.4452-2-41-1': 'Examples when [MATH] were given in [CITATION] (note this still leaves the term involving [MATH]).', '1112.4452-2-41-2': 'In general, as shown below, we can control [MATH] by imposing the conditions [REF]-[REF] as they allow us to take advantage of the smoothing estimates proved in [CITATION].', '1112.4452-2-41-3': 'In addition, we also require [REF]-[REF].', '1112.4452-2-42-0': '## [MATH] Replacement of positivity condition by bounds on [MATH]', '1112.4452-2-43-0': 'Suppose [REF]-[REF] hold.', '1112.4452-2-43-1': 'Then [EQUATION]', '1112.4452-2-44-0': '### Estimates for [MATH].', '1112.4452-2-45-0': 'Choose [MATH].', '1112.4452-2-45-1': 'Impose [REF]- [REF].', '1112.4452-2-45-2': 'Since [MATH] we get [EQUATION]', '1112.4452-2-45-3': 'Next [EQUATION] by [REF] and [REF].', '1112.4452-2-45-4': '[EQUATION] by [REF] and [REF].', '1112.4452-2-45-5': 'To estimate [MATH] note that [MATH] is independent in time and [MATH].', '1112.4452-2-45-6': 'Then [EQUATION] by [REF] and [REF].', '1112.4452-2-45-7': 'The estimates for [MATH] are analogous.', '1112.4452-2-46-0': '### Estimates for [MATH].', '1112.4452-2-47-0': 'Just as before, we write [EQUATION] where [MATH] is estimated using [REF] and [REF].', '1112.4452-2-47-1': 'For [MATH] we have [EQUATION] by [REF] and [REF].', '1112.4452-2-47-2': 'Next, [EQUATION] by [REF] and [REF].', '1112.4452-2-48-0': '# Proof of the Inhomogeneous Strichartz Estimate, Theorem [REF]', '1112.4452-2-49-0': 'Let [MATH], [MATH] be a space-time function which is sufficiently regular and [MATH] be the solution of [REF].', '1112.4452-2-49-1': "Note that [EQUATION] by Duhamel's principle and define [MATH].", '1112.4452-2-49-2': 'By the Christ-Kiselev lemma, [EQUATION]', '1112.4452-2-49-3': 'So it is enough to show [EQUATION] for any [MATH], [MATH].', '1112.4452-2-49-4': 'From the definition of [MATH], Strichartz estimate and self-adjointness of [MATH], [EQUATION] and proof is completed.', '1112.4452-2-50-0': '# Application to Magnetic Nonlinear Schro dinger equations', '1112.4452-2-51-0': 'In this section, we show applications of previous estimates to global existence and scattering.', '1112.4452-2-51-1': 'For simplicity, we consider magnetic NLS with defocusing cubic nonlinearity in [MATH].', '1112.4452-2-51-2': '[EQUATION]', '1112.4452-2-51-3': 'We note that by virtue of [REF] and [REF] we have [EQUATION]', '1112.4452-2-51-4': 'To establish Theorem [REF] we begin with a local theory (see [CITATION] for related works).', '1112.4452-2-51-5': 'As mentioned in the introduction, the arguments below resemble what is usually done for the critical NLS (see for example [CITATION]).', '1112.4452-2-52-0': '## Local existence', '1112.4452-2-53-0': 'Let [MATH].', '1112.4452-2-53-1': 'Let [MATH] and suppose [EQUATION]', '1112.4452-2-53-2': 'We show that if [MATH] is small enough we obtain local existence.', '1112.4452-2-53-3': 'We note that we do not require small data.', '1112.4452-2-53-4': 'Also, for any [MATH], we can always assume [REF] if the time interval is small enough.', '1112.4452-2-53-5': 'To see that, by [REF], Theorem [REF], and [REF], we have [EQUATION] so [MATH] is finite.', '1112.4452-2-53-6': 'Hence the time interval can be shrunk enough to make [REF] hold.', '1112.4452-2-54-0': 'We construct a unique solution in the space [EQUATION] where [MATH], and [MATH] is the maximum of the constant [MATH], and the constants [MATH] that appear in the estimates below.', '1112.4452-2-54-1': 'Define the sequence of Picard iterates by [EQUATION] where [EQUATION]', '1112.4452-2-54-2': 'By [REF] and Theorem [REF] [EQUATION] and by [REF], Theorem [REF] and [REF], [EQUATION].', '1112.4452-2-54-3': 'Now suppose that for [MATH].', '1112.4452-2-54-4': 'Then by Theorem [REF], Corollary [REF], Theorem [REF] and Sobolev embedding, [EQUATION]', '1112.4452-2-54-5': 'Hence, if [MATH] is small enough, i.e., [EQUATION].', '1112.4452-2-54-6': 'Similarly [EQUATION]', '1112.4452-2-54-7': 'Let [EQUATION]', '1112.4452-2-54-8': 'Then [EQUATION].', '1112.4452-2-54-9': 'Next, [EQUATION]', '1112.4452-2-54-10': 'If we require [EQUATION] then [EQUATION] which shows the sequence [MATH] belongs to [MATH].', '1112.4452-2-54-11': 'To show the sequence converges, we need to consider the differences.', '1112.4452-2-54-12': 'The estimates are similar, and we only show some of the details.', '1112.4452-2-55-0': 'Let [EQUATION] then we can write [EQUATION].', '1112.4452-2-55-1': 'Now consider [EQUATION] by [REF].', '1112.4452-2-55-2': 'We obtain the same bounds for term [MATH], and for the other norms in [MATH], which show the sequence of the iterates is Cauchy and hence it converges as needed.', '1112.4452-2-56-0': '## Global existence', '1112.4452-2-57-0': 'Let [MATH] be the solution of [REF] obtained from local existence on time interval [MATH].', '1112.4452-2-57-1': 'Suppose [MATH].', '1112.4452-2-57-2': 'We show this leads to a contradiction by showing we can extend the solution.', '1112.4452-2-58-0': 'First, by [REF], [REF] and [REF], the [MATH] norm of [MATH] is uniformly bounded.', '1112.4452-2-58-1': 'So by the local well-posedness argument to extend the solution, it is enough to show the existence of [MATH] and [MATH] such that [EQUATION] where [MATH] is specified by [REF] (and [MATH]) and, [EQUATION] due to [REF] and [REF] respectively.', '1112.4452-2-58-2': 'But similarly as in [REF], we have [EQUATION] for all [MATH].', '1112.4452-2-58-3': 'So we can find [MATH] small enough such that [EQUATION].', '1112.4452-2-58-4': 'Then by [REF] and the continuity of the integral, we can find [MATH] such that [REF] holds together with [REF] as needed (by taking [MATH] smaller if necessary).', '1112.4452-2-59-0': '## Scattering', '1112.4452-2-60-0': 'In this section we consider the question of scattering (asymptotic completeness).', '1112.4452-2-60-1': 'We take a point of view analogous to the classical NLS.', '1112.4452-2-60-2': 'Hence we set out to show that given a solution of the nonlinear mNLS, [MATH], there exists a solution of the linear mNLS, [MATH], such that the [MATH] norm of the difference of the two solutions goes to [MATH] as [MATH] (note, due to [REF] this also gives convergence of [MATH]).', '1112.4452-2-60-3': 'Now, following the classical NLS setup for scattering, let [MATH] be the solution to the cubic defocusing mNLS with initial data [MATH].', '1112.4452-2-60-4': 'We define [EQUATION].', '1112.4452-2-60-5': 'The convergence in [MATH] of the difference of [MATH] and [MATH] is then immediate if we can show [EQUATION] converges in [MATH].', '1112.4452-2-60-6': 'Therefore, equivalently, we need to show [EQUATION]', '1112.4452-2-60-7': 'We prove ([REF]) for [MATH] and [MATH] separately.', '1112.4452-2-60-8': 'For [MATH], we need to show [EQUATION] as [MATH].', '1112.4452-2-60-9': 'Note that [EQUATION]', '1112.4452-2-60-10': 'We used interpolation inequality [MATH].', '1112.4452-2-60-11': 'The last quantity converges to [MATH] as [MATH] since [MATH] is finite.', '1112.4452-2-60-12': 'For [MATH], we need the following lemma.', '1112.4452-2-61-0': 'For a solution [MATH] of the given equation, [MATH] is finite.', '1112.4452-2-62-0': 'By Sobolev embedding, [EQUATION]', '1112.4452-2-62-1': 'We now subdivide [MATH] into finitely many disjoint intervals [MATH] so that [EQUATION] for some [MATH] which will be chosen later.', '1112.4452-2-62-2': 'On each interval [MATH], we have [EQUATION]', '1112.4452-2-62-3': 'We take small enough [MATH] to apply the continuity method.', '1112.4452-2-62-4': 'Note that [MATH] only depends on the implicit constant of the Strichartz estimate and the size of the initial data.', '1112.4452-2-62-5': 'By the method of continuity, we conclude [MATH] is finite on each interval [MATH].', '1112.4452-2-62-6': 'Since we have only finitely many intervals, [MATH] is finite on [MATH], and the result follows by [REF].', '1112.4452-2-63-0': 'Now for [MATH] we have [EQUATION] and [EQUATION]', '1112.4452-2-63-1': 'We used interpolation inequality [MATH].', '1112.4452-2-63-2': 'Since [MATH] is finite by Lemma [REF], the last quantity vanishes as [MATH] which completes the proof of scattering.', '1112.4452-2-64-0': '# Failure of pointwise nonnegativity of [MATH]', '1112.4452-2-65-0': 'Let [MATH] and [MATH] be time independent, divergence-free.', '1112.4452-2-65-1': 'Then the terms that appear in the integral in [MATH] are [EQUATION].', '1112.4452-2-65-2': 'Since [MATH] where [MATH] is the Hodge star operator on forms, the above formula is [EQUATION]', '1112.4452-2-65-3': 'Note [MATH] is parallel to [MATH] as long as [MATH], so for any given [MATH], we can find [MATH] so that [MATH].', '1112.4452-2-65-4': 'Similarly, we can find [MATH] so that [MATH] and [MATH] form an angle less than [MATH].', '1112.4452-2-65-5': 'Alternatively, we can write [REF] as [EQUATION] and again as long as [MATH], then this is a dot product of the momentum vector with a component of [MATH] tangent to the unit sphere centered at [MATH] and the second term is the radial component of [MATH] with respect to the sphere centered at [MATH] (compare to the trapping component in [CITATION]).', '1112.4452-2-65-6': 'Therefore, as we move [MATH] around, pointwise nonnegativity in general is not possible.'}

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'1112.4452-2-49-1'], ['1112.4452-1-42-2', '1112.4452-2-49-2'], ['1112.4452-1-42-3', '1112.4452-2-49-3'], ['1112.4452-1-42-4', '1112.4452-2-49-4'], ['1112.4452-1-38-2', '1112.4452-2-45-2'], ['1112.4452-1-38-5', '1112.4452-2-45-5'], ['1112.4452-1-38-7', '1112.4452-2-45-7'], ['1112.4452-1-47-0', '1112.4452-2-55-0'], ['1112.4452-1-47-1', '1112.4452-2-55-1'], ['1112.4452-1-47-2', '1112.4452-2-55-2'], ['1112.4452-1-13-0', '1112.4452-2-18-0'], ['1112.4452-1-13-1', '1112.4452-2-18-1'], ['1112.4452-1-53-0', '1112.4452-2-62-0'], ['1112.4452-1-53-1', '1112.4452-2-62-1'], ['1112.4452-1-53-2', '1112.4452-2-62-2'], ['1112.4452-1-53-3', '1112.4452-2-62-3'], ['1112.4452-1-53-4', '1112.4452-2-62-4'], ['1112.4452-1-53-5', '1112.4452-2-62-5'], ['1112.4452-1-53-6', '1112.4452-2-62-6'], ['1112.4452-1-31-0', '1112.4452-2-36-0'], ['1112.4452-1-31-1', '1112.4452-2-36-1'], ['1112.4452-1-20-0', '1112.4452-2-25-0'], ['1112.4452-1-20-1', '1112.4452-2-25-1'], ['1112.4452-1-20-2', '1112.4452-2-25-2'], 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'1112.4452-2-49-1'], ['1112.4452-1-42-2', '1112.4452-2-49-2'], ['1112.4452-1-42-3', '1112.4452-2-49-3'], ['1112.4452-1-42-4', '1112.4452-2-49-4'], ['1112.4452-1-38-2', '1112.4452-2-45-2'], ['1112.4452-1-38-5', '1112.4452-2-45-5'], ['1112.4452-1-38-7', '1112.4452-2-45-7'], ['1112.4452-1-47-0', '1112.4452-2-55-0'], ['1112.4452-1-47-1', '1112.4452-2-55-1'], ['1112.4452-1-47-2', '1112.4452-2-55-2'], ['1112.4452-1-13-0', '1112.4452-2-18-0'], ['1112.4452-1-13-1', '1112.4452-2-18-1'], ['1112.4452-1-53-0', '1112.4452-2-62-0'], ['1112.4452-1-53-1', '1112.4452-2-62-1'], ['1112.4452-1-53-2', '1112.4452-2-62-2'], ['1112.4452-1-53-3', '1112.4452-2-62-3'], ['1112.4452-1-53-4', '1112.4452-2-62-4'], ['1112.4452-1-53-5', '1112.4452-2-62-5'], ['1112.4452-1-53-6', '1112.4452-2-62-6'], ['1112.4452-1-31-0', '1112.4452-2-36-0'], ['1112.4452-1-31-1', '1112.4452-2-36-1'], ['1112.4452-1-20-0', '1112.4452-2-25-0'], ['1112.4452-1-20-1', '1112.4452-2-25-1'], ['1112.4452-1-20-2', '1112.4452-2-25-2'], 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'1112.4452-2-34-0'], ['1112.4452-1-29-1', '1112.4452-2-34-1'], ['1112.4452-1-29-2', '1112.4452-2-34-2'], ['1112.4452-1-29-3', '1112.4452-2-34-3'], ['1112.4452-1-29-4', '1112.4452-2-34-4'], ['1112.4452-1-28-0', '1112.4452-2-33-0'], ['1112.4452-1-28-2', '1112.4452-2-33-2'], ['1112.4452-1-28-3', '1112.4452-2-33-3'], ['1112.4452-1-28-4', '1112.4452-2-33-4'], ['1112.4452-1-28-5', '1112.4452-2-33-5'], ['1112.4452-1-28-6', '1112.4452-2-33-6'], ['1112.4452-1-33-0', '1112.4452-2-38-0'], ['1112.4452-1-33-1', '1112.4452-2-38-1'], ['1112.4452-1-33-2', '1112.4452-2-38-2'], ['1112.4452-1-33-3', '1112.4452-2-38-3'], ['1112.4452-1-33-4', '1112.4452-2-38-4'], ['1112.4452-1-33-8', '1112.4452-2-38-8'], ['1112.4452-1-33-10', '1112.4452-2-38-10'], ['1112.4452-1-11-0', '1112.4452-2-15-0'], ['1112.4452-1-11-1', '1112.4452-2-15-1'], ['1112.4452-1-11-2', '1112.4452-2-15-2'], ['1112.4452-1-11-3', '1112.4452-2-15-3'], ['1112.4452-1-11-4', '1112.4452-2-15-4'], ['1112.4452-1-11-5', '1112.4452-2-15-5'], ['1112.4452-1-11-6', '1112.4452-2-15-6'], ['1112.4452-1-11-7', '1112.4452-2-15-7'], ['1112.4452-1-11-8', '1112.4452-2-15-8'], ['1112.4452-1-11-9', '1112.4452-2-15-9'], ['1112.4452-1-11-10', '1112.4452-2-15-10'], ['1112.4452-1-11-11', '1112.4452-2-15-11'], ['1112.4452-1-11-12', '1112.4452-2-16-0'], ['1112.4452-1-3-0', '1112.4452-2-3-0'], ['1112.4452-1-3-1', '1112.4452-2-3-1'], ['1112.4452-1-3-2', '1112.4452-2-3-2'], ['1112.4452-1-3-4', '1112.4452-2-6-0'], ['1112.4452-1-3-5', '1112.4452-2-6-1'], ['1112.4452-1-3-6', '1112.4452-2-6-2'], ['1112.4452-1-3-7', '1112.4452-2-6-3'], ['1112.4452-1-3-8', '1112.4452-2-6-4'], ['1112.4452-1-3-9', '1112.4452-2-6-5'], ['1112.4452-1-3-10', '1112.4452-2-6-6'], ['1112.4452-1-3-13', '1112.4452-2-6-9']]

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[]

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[]

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{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1112.4452

1502.03942

{'1502.03942-1-0-0': 'We present scalable parallel algorithms with sublinear communication volume and low latency for several fundamental problems related to finding the most relevant elements in a set: the classical selection problem with unsorted input, its variant with locally sorted input, bulk parallel priority queues, multicriteria selection using threshold algorithms, and finding the most frequent objects.', '1502.03942-1-0-1': 'All of these algorithms push the owner-computes rule to extremes.', '1502.03942-1-0-2': 'The output of these algorithms might unavoidably be unevenly distributed over the processors.', '1502.03942-1-0-3': 'We therefore also explain how to redistribute this data with minimal communication.', '1502.03942-1-1-0': 'Keywords: branch-and-bound, data redistribution, frequent elements, priority queue, sampling, selection, threshold algorithm', '1502.03942-1-2-0': '# Introduction', '1502.03942-1-3-0': 'Overheads due to communication latency and bandwidth limitations of the communication networks are one of the main limiting factors for parallel computing.', '1502.03942-1-3-1': 'Parallel algorithm theory has considered the latency part of this issue since its beginning.', '1502.03942-1-3-2': 'In particular, execution times polylogarithmic in the number [MATH] of processing elements (PEs) were a main focus.', '1502.03942-1-3-3': 'Borkar [CITATION] argues that an exascale computer could only be cost effective if the communication capabilities (bisection width) scale highly sublinearly with the amount of computation done in the connected subsystems.', '1502.03942-1-3-4': 'In a previous paper [CITATION] we therefore proposed to look more intensively for algorithms with sublinear communication volume.', '1502.03942-1-3-5': 'In this paper, we look at the combination of both aspects.', '1502.03942-1-3-6': 'We consider some fundamental algorithmic problems that have a large input (size [MATH]) and a relatively small output.', '1502.03942-1-3-7': 'Since potentially many similar problems have to be solved, both latency and communication volume have to be very small - ideally polylogarithmic in [MATH] and the input parameters.', '1502.03942-1-3-8': 'More precisely, we consider problems which ask for the [MATH] most "relevant" results from a large set of possibilities.', '1502.03942-1-4-0': 'In the simplest case, these are totally ordered elements and we ask for the [MATH] smallest of them - the classical selection problem.', '1502.03942-1-4-1': 'Several variants of this problem are studied in Section [REF].', '1502.03942-1-4-2': 'For the classical variant with unsorted inputs, a careful analysis of a known algorithm [CITATION] shows that the previously assumed random allocation of the inputs is actually not necessary and we get running time [MATH] - essentially, with careful implementation, load imbalance during the algorithm execution can be hidden in the [MATH] term.', '1502.03942-1-4-3': 'For locally sorted input, this "trick" does not work and we get latency [MATH].', '1502.03942-1-4-4': 'Nevertheless, we can return to a single logarithm if we are willing to somewhat relax the output size [MATH].', '1502.03942-1-5-0': 'A data structure generalization of the selection problem are priority queues.', '1502.03942-1-5-1': 'Section [REF] shows that the results for selection transfer to this case.', '1502.03942-1-5-2': 'We get an efficient bulk parallel priority queue by simply inserting new elements into a local sequential priority queue and using the distributed selection algorithms from Section [REF] for deleting the [MATH] smallest elements.', '1502.03942-1-5-3': 'The key to making this work is to use an appropriately augmented search tree data structure as the local queue.', '1502.03942-1-5-4': 'In contrast, previous parallel priority queues are not communication efficient in the sense that they are moving around the elements either by sorting [CITATION] or by random allocation [CITATION].', '1502.03942-1-6-0': 'A prominent problem in information retrieval is to extract the [MATH] most relevant objects (e.g. documents), where relevance is determined by a monotonous function that maps several individual scoring functions to the overall relevance.', '1502.03942-1-6-1': 'For each individual score, a list of objects is precomputed that stores the objects in order of decreasing score.', '1502.03942-1-6-2': 'This is a multicriteria generalization of the sorted top-[MATH] selection problem discussed in Section [REF].', '1502.03942-1-6-3': 'In Section [REF] we parallelize established sequential algorithms for this problem.', '1502.03942-1-6-4': 'The single-criterion algorithms are used as subroutines - the sorted one for approximating the list elements scanned by the sequential algorithm and the unsorted one to actually identify the output.', '1502.03942-1-6-5': 'The algorithm has polylogarithmic overhead for coordinating the PEs and manages to contain unavoidable load imbalance in a single phase of local computation.', '1502.03942-1-6-6': 'This is achieved with a fast estimator of the output size generated with a given number of scanned objects.', '1502.03942-1-7-0': 'A fundamental problem in data mining is finding the most frequently occurring objects.', '1502.03942-1-7-1': 'This is challenging in a distributed setting since the globally most frequent elements do not have to be locally frequent on any particular PE.', '1502.03942-1-7-2': 'In Section [REF] we develop very fast sampling based algorithms that find a probably approximately correct answer, i.e., with probability [MATH] the output is correct within [MATH].', '1502.03942-1-7-3': 'The algorithms run in time logarithmic in [MATH], [MATH], and [MATH].', '1502.03942-1-7-4': 'From a simple algorithm with running time factor [MATH] we go to more sophisticated ones with factor around [MATH] that additionally do exact counting of the most frequent elements (possibly for more than [MATH] elements).', '1502.03942-1-7-5': 'If a significant slope exists in the frequency distribution of the elements, a similar technique allows the answer to be exactly correct with probability [MATH].', '1502.03942-1-8-0': 'All the above algorithms have the unavoidable limitation that the output may be unevenly distributed over the PEs for general distributions of the input objects.', '1502.03942-1-8-1': 'This can lead to load imbalance affecting the efficiency of the overall application.', '1502.03942-1-8-2': 'Offsetting that imbalance will require communication so that one could argue that striving for communication efficiency in the selection process is in vain.', '1502.03942-1-8-3': 'However, we have several advantages over non-communication efficient selection algorithms.', '1502.03942-1-8-4': 'First of all, priorities can be ignored for the redistribution since all the selected elements are relevant.', '1502.03942-1-8-5': 'Hence, we can use any data redistribution algorithm we want.', '1502.03942-1-8-6': 'In particular, we can use an adaptive algorithm that only moves the data that is really necessary.', '1502.03942-1-8-7': 'In Section [REF] we give one such algorithm that combines prefix sums and merging to minimize data movement and incurs only logarithmic additional delay.', '1502.03942-1-8-8': 'Delegating data movement to the responsibility of the application also has the advantage that we can exploit properties of the application that a general top-[MATH] selection algorithm cannot exploit.', '1502.03942-1-8-9': 'For example, multicriteria selection algorithms as discussed in Section [REF] are used in search engines processing a stream of many queries.', '1502.03942-1-8-10': 'Therefore, it is enough to do load balancing over a large number of queries and we can resolve persistent hot spots by adaptive data migration or replication.', '1502.03942-1-8-11': 'Another example are branch-and-bound applications [CITATION].', '1502.03942-1-8-12': 'By randomizing necessary data migrations, we can try to steer the system away from situations with bad data distribution (a proper analysis of such an algorithm is an open problem though).', '1502.03942-1-9-0': '# Preliminaries', '1502.03942-1-10-0': 'Our input usually consists of a multiset [MATH] of [MATH] objects, each represented by a single machine word.', '1502.03942-1-10-1': 'When these objects are ordered, we assume that their total ordering is unique.', '1502.03942-1-10-2': 'This is without loss of generality since we can make the value [MATH] of object [MATH] unique by replacing it by the pair [MATH] for tie breaking.', '1502.03942-1-11-0': 'Consider [MATH] processing elements (PEs) connected by a network.', '1502.03942-1-11-1': 'PEs are numbered [MATH] where [MATH] is a shorthand for [MATH] throughout this paper.', '1502.03942-1-11-2': 'Sending a message of size [MATH] machine words takes time [MATH] assuming that a PE can send and receive at most one message at a time (full-duplex, single ported communication).', '1502.03942-1-11-3': 'We will treat [MATH] and [MATH] as variables in asymptotic analysis.', '1502.03942-1-11-4': 'A running time of [MATH] will then allow us to discuss the internal work [MATH], communication volume [MATH] and latency [MATH] separately.', '1502.03942-1-12-0': 'We present some of the algorithms using high-level pseudocode in a single program multiple data (SPMD) style - the same algorithm runs on each PE, which perform work on their local data and communicate predominantly through collective operations.', '1502.03942-1-12-1': 'Variables are local by default.', '1502.03942-1-12-2': 'We can denote data on remote PEs using the notation [MATH] which refers to the value of variable [MATH] on PE [MATH].', '1502.03942-1-12-3': 'For example, [MATH] denotes the global sum of [MATH] over all PEs.', '1502.03942-1-13-0': 'Collective communication.', '1502.03942-1-13-1': 'Broadcasting sends a message to all PEs.', '1502.03942-1-13-2': 'Reduction applies an associative operation (e.g., sum, maximum, or minimum) to a vector of length [MATH].', '1502.03942-1-13-3': 'An all-reduction also broadcasts this result to all PEs.', '1502.03942-1-13-4': 'A prefix-sum (scan) computes [MATH] on PE [MATH] where [MATH] is a vector of length [MATH].', '1502.03942-1-13-5': 'The scatter operation distributes a message of size [MATH] to a set of [MATH] PEs such that each of them gets a piece of size [MATH].', '1502.03942-1-13-6': 'Symmetrically, a gather operations collects [MATH] pieces of size [MATH] on a single PE.All of these operations can be performed in time [MATH] [CITATION].', '1502.03942-1-13-7': 'In an all-to-all personalized communication (all-to-all for short) each PE sends one message of size [MATH] to every other PE.This can be done in time [MATH] using direct point-to-point delivery or in time [MATH] using indirect delivery [CITATION].', '1502.03942-1-13-8': 'The all-to-all broadcast (aka gossiping) starts with a single message of size [MATH] on each PE and ends with all PEs having all these messages.', '1502.03942-1-13-9': 'This operation works in time [MATH].', '1502.03942-1-14-0': 'Fast inefficient Sorting.', '1502.03942-1-14-1': 'Sorting [MATH] objects can be done in time [MATH] using a brute force algorithm performing all pairwise object comparisons in parallel (e.g., [CITATION]).', '1502.03942-1-15-0': 'Search trees.', '1502.03942-1-15-1': 'Search trees can represent a sorted sequence of objects such that a variety of operations can be supported in logarithmic time.', '1502.03942-1-15-2': 'In particular, one can insert and remove objects and search for the next largest object given a key.', '1502.03942-1-15-3': 'The operation [MATH] splits T into two search trees [MATH] such that [MATH] contains the objects of [MATH] with key [MATH] and [MATH] contains the objects of [MATH] with key [MATH].', '1502.03942-1-15-4': 'Similarly, if the keys in a tree [MATH] are smaller than the keys in [MATH] then [MATH] returns a search tree representing the concatenation of the two sequences.', '1502.03942-1-15-5': 'If one additionally stores the size of subtrees, one can also support the select operation [MATH], which returns the [MATH]-th largest object in [MATH], and operation [MATH], which returns the number of objects in [MATH] of value at most [MATH].', '1502.03942-1-15-6': 'For an example of such a data structure see [CITATION].', '1502.03942-1-16-0': 'Chernoff bounds.', '1502.03942-1-16-1': 'We use the following Chernoff bounds [CITATION] to bound the probability that a sum [MATH] of [MATH] independent indicator random variables deviates substantially from its expected value [MATH].', '1502.03942-1-16-2': 'For [MATH], we have [EQUATION]', '1502.03942-1-16-3': 'Bernoulli sampling.', '1502.03942-1-16-4': 'A simple way to obtain a sample of a set [MATH] of objects is to select each object with probability [MATH] independent of the other objects.', '1502.03942-1-16-5': 'For small [MATH], the naive implementation can be significantly accelerated from time [MATH] to [MATH] by generating skip values - a value of [MATH] indicates that the following [MATH] elements are not selected and the [MATH]-th element is selected for the sample.', '1502.03942-1-16-6': 'These skip values have a geometric distribution with parameter [MATH] and can be generated in constant time.', '1502.03942-1-17-0': 'Distributed FR-Selection [CITATION] Floyd and Rivest [CITATION] developed a modification of quickselect using two pivots with the objective to achieve a number of comparisons close to the lower bound.', '1502.03942-1-17-1': 'This algorithm, FR-select, can be adapted to a distributed memory parallel setting [CITATION], see Figure [REF].', '1502.03942-1-17-2': 'FR-select picks the pivots based on sorting a random sample [MATH] of [MATH] objects, which can be sorted in logarithmic time.', '1502.03942-1-17-3': 'Pivots [MATH] and [MATH] are chosen as the sample objects with ranks [MATH] where [MATH] for some small constant [MATH].', '1502.03942-1-17-4': 'For the analysis, one can choose [MATH] which ensures that with high probability the range of possible values shrinks by a factor [MATH] in every level of recursion.', '1502.03942-1-18-0': 'It is shown that a constant number of recursion levels suffice if [MATH] and when the objects are distributed randomly.', '1502.03942-1-18-1': 'Note that the latter assumption limits the communication efficiency of the algorithm since it requires moving all objects to random PEs for general inputs.', '1502.03942-1-19-0': 'Bulk Parallel Priority Queues.', '1502.03942-1-19-1': 'A natural way to parallelize priority queues is to use bulk operations.', '1502.03942-1-19-2': 'In particular, operation deleteMin[MATH] supports deletion of the [MATH] smallest objects of the queue.', '1502.03942-1-19-3': 'Such a data structure can be based on a heap where nodes store sorted sequences rather than objects [CITATION].', '1502.03942-1-19-4': 'However, an even faster and simple randomized way is to use multiple sequential priority queues - one on each PE [CITATION].', '1502.03942-1-19-5': 'This data structure adopts the idea of Karp and Zhang [CITATION] to give every PE a representative approximate view of the global situation by sending inserted objects to random queues.', '1502.03942-1-19-6': 'However, in contrast to Karp and Zhang [CITATION], [CITATION] implements an exact deleteMin[MATH] using parallel selection and a few further tricks.', '1502.03942-1-19-7': 'Note that the random insertions limit communication efficiency in this case.', '1502.03942-1-20-0': '# More Related Work', '1502.03942-1-21-0': 'Plaxton [CITATION] show a superlogarithmic lower bound for selection on a large class of interconnection networks.', '1502.03942-1-21-1': 'This bound does not apply to our model, since we assume a more powerful network.', '1502.03942-1-22-0': 'There has been intensive work on parallel priority queues.', '1502.03942-1-22-1': "The most scalable solutions are our randomized priority queue [CITATION] and Deo and Prasad's parallel heap [CITATION].", '1502.03942-1-22-2': 'Refer to [CITATION] for a recent overview of further approaches, most of which are for shared memory architectures and mostly on centralized data structures with limited scalability.', '1502.03942-1-23-0': 'Considerable work has been done on distributed top-[MATH] computations in wide area networks and sensor networks [CITATION].', '1502.03942-1-23-1': 'However, these papers use a master-worker approach where a centralized master communicates with the individual nodes.', '1502.03942-1-23-2': 'This implies severe scalability limitations compared to our fully distributed algorithms.', '1502.03942-1-23-3': 'In particular, an algorithm working with communication cost [MATH] in the distributed streaming model as used by Yi and Zhang [CITATION] means that the master has to handle communication volume [MATH] in our model.', '1502.03942-1-23-4': 'Furthermore, each such centralized communication step incurs [MATH] startup overheads for the master.', '1502.03942-1-24-0': 'Data redistribution using prefix sums is standard [CITATION].', '1502.03942-1-24-1': 'But that combining it with merging allows adaptive communication volume as described in Section [REF] seems to be new.', '1502.03942-1-24-2': 'There are algorithms for minimizing communication cost of data distribution in arbitrary networks.', '1502.03942-1-24-3': 'However, these are much more expensive.', '1502.03942-1-24-4': 'Meyer auf der Heide et al. [CITATION] use maximum flow computations.', '1502.03942-1-24-5': 'They also give an algorithm for meshes, which needs more steps than the lower bound though.', '1502.03942-1-24-6': 'Solving the diffusion equations minimizes the sum of the squares of the data to be moved in order to achieve perfect load balance [CITATION].', '1502.03942-1-24-7': 'However, solving this equation is expensive and for parallel computing we are more interested in the bottleneck communication volume.', '1502.03942-1-25-0': '# Selection', '1502.03942-1-26-0': 'We consider the problem of identifying the elements of rank [MATH] from a set of [MATH] objects, distributed over [MATH] processors.', '1502.03942-1-26-1': 'First, we analyze the classical problem with unsorted input, before turning to locally sorted input in Sections [REF] and [REF].', '1502.03942-1-27-0': '## Unsorted Input', '1502.03942-1-28-0': 'Our first result is that using some minor adaptations and a more careful analysis, the parallel FR-algorithm from Section [REF] does not actually need randomly distributed data.', '1502.03942-1-29-0': 'Algorithm [REF] can be implemented to run in expected time [EQUATION] (Outline) The algorithm from [CITATION] computes the pivots based on a sample [MATH] of size [MATH].', '1502.03942-1-29-1': 'There, this is easy since the objects are randomly distributed in all levels of recursion.', '1502.03942-1-29-2': 'Here we can make no such assumption and rather assume Bernoulli sampling with probability [MATH].', '1502.03942-1-29-3': 'Although this can be done in time proportional to the local sample size, we have to be careful since (in some level of recursion) the input might be so skewed that most samples come from the same processor.', '1502.03942-1-29-4': 'Since the result on sorting only works when the input data is uniformly distributed over the PEs, we have to account additional time for spreading the samples over the PEs.', '1502.03942-1-30-0': 'Let [MATH] denote the PE which maximizes [MATH] in the current level of recursion in the algorithm from Figure [REF].', '1502.03942-1-30-1': 'The total problem size shrinks by a factor of [MATH] in each level of recursion with high probability.', '1502.03942-1-30-2': 'Hence, after a constant number of recursion levels, [MATH] is [MATH] as well.', '1502.03942-1-30-3': 'Moreover, after [MATH] further levels of recursion, the problem is solved completely with high probability.', '1502.03942-1-30-4': 'Overall, the total number of recursion levels is [MATH].', '1502.03942-1-31-0': 'This already implies the claimed [MATH] bound on internal computation - [MATH] times [MATH] until [MATH] has size [MATH] and a geometrically shrinking amount of work after that.', '1502.03942-1-32-0': 'Of course, the same bound [MATH] also applies to the communication volume.', '1502.03942-1-32-1': 'However, we also know that even if all samples come from a single PE, the communication volume in a single level of recursion is [MATH].', '1502.03942-1-33-0': 'Finally, the number of startups can be limited to [MATH] per level of recursion by using a collective scatter operation on those PEs that have more than one sample object.', '1502.03942-1-33-1': 'This yields [MATH].', '1502.03942-1-34-0': 'If [MATH] and [MATH] are viewed as constants, the bound from Theorem [REF] reduces to [MATH].', '1502.03942-1-35-0': 'The bound from Theorem [REF] immediately reduces to [MATH] for constant [MATH] and [MATH].', '1502.03942-1-35-1': 'We can further simplify this by observing that when the [MATH] term dominates [MATH], then [MATH].', '1502.03942-1-36-0': '## Locally Sorted Input', '1502.03942-1-37-0': 'Selection on locally sorted input is easier than the unsorted problem from Section [REF] since we only have to consider the locally smallest objects and are able to locate keys in logarithmic time.', '1502.03942-1-37-1': 'Indeed, this problem has been studied as the multisequence selection problem [CITATION].', '1502.03942-1-38-0': 'In [CITATION], we propose a particularly simple and intuitive method based on an adaptation of the well-known quickselect algorithm [CITATION].', '1502.03942-1-38-1': 'For self-containedness, we also give that algorithm in Appendix [REF], adapting it to the need in the present paper.', '1502.03942-1-38-2': 'This algorithm needs running time [MATH].', '1502.03942-1-38-3': 'Algorithm [REF] can be viewed as a step backwards compared to our algorithm from Section [REF] as using a single random pivot forces us to do a deeper recursion.', '1502.03942-1-38-4': 'We do that because it makes it easy to tolerate unbalanced input sizes in the deeper recursion levels.', '1502.03942-1-38-5': 'However, it is possible to reduce the recursion depth to some extent by choosing pivots more carefully and by using multiple pivots at once.', '1502.03942-1-38-6': 'We study this below for a variant of the selection problem where we are flexible about the number [MATH] of objects to be selected.', '1502.03942-1-39-0': '## Flexible k, Locally Sorted Input', '1502.03942-1-40-0': 'The [MATH] startups incurred in Section [REF] can be reduced to [MATH] if we are willing to give up some control of the number of objects that are actually returned.', '1502.03942-1-40-1': 'We now give two input parameters [MATH] and [MATH] and the algorithm returns the [MATH] smallest objects so that [MATH].', '1502.03942-1-40-2': 'We begin with a simple algorithm that runs in logarithmic time if [MATH] and then explain how to refine that for the case [MATH].', '1502.03942-1-41-0': 'The basic idea for the simple algorithm is to take a Bernoulli sample of the input with success probability [MATH] for [MATH].', '1502.03942-1-41-1': 'Then, the expected rank of the smallest sample object is [MATH], i.e., we have a truthful estimator for an object with the desired rank.', '1502.03942-1-41-2': 'Moreover, this object can be computed efficiently when working with locally sorted data: The local rank of the smallest local sample is geometrically distributed with parameter [MATH].', '1502.03942-1-41-3': 'Such a number can be generated in constant time.', '1502.03942-1-41-4': 'By computing the global minimum of these locally smallest samples, we can get the globally smallest sample [MATH] in time [MATH].', '1502.03942-1-41-5': 'We can also count the exact number [MATH] of input objects bounded by this estimate in time [MATH] - we locate [MATH] in each local data set in time [MATH] and then sum the found positions.', '1502.03942-1-41-6': 'If [MATH], we are done.', '1502.03942-1-41-7': 'Otherwise, we can use the acquired information as in any variant of quickselect.', '1502.03942-1-42-0': 'At least in the recursive calls, it can happen that [MATH] is close to the total input size [MATH].', '1502.03942-1-42-1': 'Then it is a better strategy to use a dual algorithm based on computing a global maximum of a Bernoulli sample.', '1502.03942-1-42-2': 'Figure [REF] gives pseudocode for a combined algorithm dynamically choosing between these two cases.', '1502.03942-1-42-3': 'It is an interesting problem which value should be chosen for [MATH].', '1502.03942-1-42-4': 'The formula used in Figure [REF] maximizes the probability that [MATH].', '1502.03942-1-42-5': 'This value is close to the arithmetic mean when [MATH] but it is significantly smaller otherwise.', '1502.03942-1-42-6': 'The reason for this asymmetry is that larger sampling rates decrease the variance of the geometric distribution.', '1502.03942-1-43-0': 'If [MATH], then Algorithm [MATH] from Figure [REF] can be implemented to run in expected time [MATH].', '1502.03942-1-44-0': '(Outline) One level of recursion takes time [MATH] for collective communication operations (min, max, or sum reduction) and time [MATH] for locating the pivot [MATH].', '1502.03942-1-44-1': 'It remains to show that the expected recursion depth is constant.', '1502.03942-1-45-0': 'We actually analyze a weaker algorithm that keeps retrying with the same parameters rather than using recursion and that uses probe [MATH].', '1502.03942-1-45-1': 'We show that, nevertheless, there is a constant success probability (i.e., [MATH] with constant probability).', '1502.03942-1-45-2': 'The rank of [MATH] is geometrically distributed with parameter [MATH].', '1502.03942-1-45-3': 'The success probability becomes [EQUATION] which is a positive constant when [MATH].', '1502.03942-1-46-0': 'Multiple Concurrent Trials.', '1502.03942-1-46-1': 'The running time of Algorithm [REF] is dominated by the logarithmic number of startup overheads for the two reduction operations it uses.', '1502.03942-1-46-2': 'We can exploit that reductions can process long vectors using little additional time.', '1502.03942-1-46-3': 'The idea is to take [MATH] Bernoulli samples of the input and to compute [MATH] estimates for an object of rank [MATH].', '1502.03942-1-46-4': 'If any of these estimates turns out to have exact rank between [MATH] and [MATH], the recursion can be stopped.', '1502.03942-1-46-5': 'Otherwise, we solve a recursive instance consisting of those objects enclosed by the largest underestimate and the smallest overestimate found.', '1502.03942-1-47-0': 'If [MATH], an algorithm processing batches of [MATH] Bernoulli samples can be implemented to run in expected time [MATH].', '1502.03942-1-48-0': '(Outline) A single level of recursion runs in time [MATH].', '1502.03942-1-48-1': 'Analogous to the proof of Theorem [REF], it can be shown that the success probability is [MATH] for a single sample.', '1502.03942-1-48-2': 'This implies that the probability that any of the [MATH] independent samples is successful is constant.', '1502.03942-1-49-0': 'For example, setting [MATH], we obtain communication time [MATH] and [MATH] time for internal work.', '1502.03942-1-50-0': '# Bulk Parallel Priority Queues', '1502.03942-1-51-0': 'Our basic idea is to make the queue from [CITATION] communication efficient by abstaining from moving inserted objects to random PEs - insertions simply access the local queue of each PE and thus work in time [MATH] without any communication.', '1502.03942-1-51-1': 'This makes operation deleteMin[MATH] more difficult since PEs no longer have a representative view of the global situation.', '1502.03942-1-51-2': 'Moreover, PEs with many of the smallest objects might introduce bottlenecks.', '1502.03942-1-52-0': 'We therefore replace the ordinary priority queues used in [CITATION] by search tree data structures that support insertion, deletion, selection, ranking, splitting and concatenation of objects in logarithmic time (see also Section [REF]).', '1502.03942-1-52-1': 'To become independent of the actual tree size of up to [MATH], we furthermore augment the trees with two arrays storing the path to the smallest and largest object respectively.', '1502.03942-1-52-2': 'This way, all required operations can be implemented to run in time [MATH] rather than [MATH].', '1502.03942-1-53-0': 'Operation [MATH] now becomes very similar to the multi-sequence selection algorithms from Appendix [REF] and Section [REF].', '1502.03942-1-53-1': 'The only difference is that instead of sorted arrays, we are now working on search trees.', '1502.03942-1-53-2': 'This implies that selecting a local object with specified local rank (during sampling) now takes time [MATH] rather than constant time.', '1502.03942-1-53-3': 'However, asymptotically, this makes no difference since for any such selection step, we also perform a ranking step which takes time [MATH] anyway in both representations.', '1502.03942-1-54-0': 'One way to implement recursion is via splitting.', '1502.03942-1-54-1': 'Since the split-operation is destructive, after returning from the recursion, we have to reassemble the previous state using concatenation.', '1502.03942-1-54-2': 'Another way that might be faster and simpler in practice is to represent a subsequence of [MATH] by [MATH] itself plus cursor information specifying rank and key of the first and last object of the subsequence.', '1502.03942-1-55-0': 'Now, we attain the following results applying Theorems [REF], [REF], and [REF], respectively:', '1502.03942-1-56-0': 'Operation [MATH] with a batch size of [MATH] can be implemented to run in expected time [MATH].', '1502.03942-1-57-0': 'Operation [MATH] with flexible batch size in [MATH] can be implemented to run in expected time [MATH] for [MATH].', '1502.03942-1-58-0': 'Operation [MATH] with flexible batch size in [MATH] can be implemented to run in expected time [MATH] for [MATH].', '1502.03942-1-59-0': 'Note that flexible batch sizes might be adequate for many applications.', '1502.03942-1-59-1': 'For example, the parallel branch-and-bound algorithm from [CITATION] can easily be adapted: In iteration [MATH] of its main loop, it deletes the smallest [MATH] elements (tree nodes) from the queue, expands these nodes in parallel, and inserts newly generated elements (child nodes of the processed nodes).', '1502.03942-1-59-2': 'Let [MATH] denote the total number of nodes expanded by the parallel algorithm.', '1502.03942-1-59-3': 'One can easily generalize the proof from [CITATION] to show that [MATH] where [MATH] is the number of nodes expanded by a sequential best first algorithm and [MATH] is the length of the path from the root to the optimal solution.', '1502.03942-1-59-4': 'Also note that a typical branch-and-bound computation will insert significantly more nodes than it removes the remaining queue is discarded after the optimal solutions are found.', '1502.03942-1-59-5': 'Hence, the local insertions of our communication efficient queue are a big advantage over previous algorithms which move all nodes [CITATION].', '1502.03942-1-60-0': '# Multicriteria Top-k', '1502.03942-1-61-0': 'In the sequential setting, we consider the following problem: Consider [MATH] lists [MATH] of scores that are sorted in decreasing order.', '1502.03942-1-61-1': 'Overall relevance of an object is determined by a scoring function [MATH] that is monotonous in all its parameters.', '1502.03942-1-61-2': 'For example, there could be [MATH] keywords for a disjunctive query to a fulltext search engine, and for each pair of a keyword and an object it is known how relevant this keyword is for this object.', '1502.03942-1-61-3': "Many algorithms used for this setting are based on Fagin's threshold algorithm [CITATION].", '1502.03942-1-61-4': 'The lists are partially scanned and the algorithm maintains lower bounds for the relevance of the scanned objects as well as upper bounds for the unscanned objects.', '1502.03942-1-61-5': 'Bounds for a scanned object [MATH] can be tightened by retrieving a score value for a dimension for which [MATH] has not been scanned yet.', '1502.03942-1-61-6': 'These random accesses are more expensive than scanning, in particular when the lists are stored on disk.', '1502.03942-1-61-7': 'Once the top-[MATH] scanned objects have better lower bounds than the best upper bound of an unscanned object, no more scanning is necessary.', '1502.03942-1-61-8': 'For determining the exact relevance of the top-[MATH] scanned objects, further random accesses may be necessary.', '1502.03942-1-61-9': 'Various strategies for scanning and random access yield a variety of variants of the threshold algorithm [CITATION].', '1502.03942-1-62-0': 'The original threshold algorithm works as follows [CITATION]: In each of [MATH] iterations of the main loop, scan one object from each list and determine its exact score using random accesses.', '1502.03942-1-62-1': 'Let [MATH] denote the smallest score of a scanned object in [MATH].', '1502.03942-1-62-2': 'Once at least [MATH] scanned objects have score at least [MATH], stop, and output them.', '1502.03942-1-63-0': 'We consider a distributed setting where each PE has a subset of the objects and [MATH] sorted lists ranking its locally present objects.', '1502.03942-1-63-1': 'We describe communication efficient distributed algorithms that approximate the original threshold algorithm (TA) [CITATION].', '1502.03942-1-63-2': 'First, we give a simple algorithm assuming random data distribution of the input objects (RDTA) and then describe a more complex algorithm for arbitrary data distribution (DTA).', '1502.03942-1-64-0': 'Random Data Distribution.', '1502.03942-1-64-1': 'Since the data placement is independent of the relevance of the objects, the top-[MATH] objects are randomly distributed over the PEs.', '1502.03942-1-64-2': 'Well known balls-into-bins bounds give us tight high probability bounds on the maximal number [MATH] of top-[MATH] objects per PE [CITATION].', '1502.03942-1-64-3': 'Here, we work with the simple bound [MATH].', '1502.03942-1-64-4': 'The RDTA algorithm simply runs TA locally to retrieve the [MATH] locally most relevant objects on each PE as result candidates.', '1502.03942-1-64-5': 'It then computes a global threshold as the maximum of the local thresholds and verifies whether at least [MATH] candidate objects are above this global threshold.', '1502.03942-1-64-6': 'In the positive case, the [MATH] most relevant candidates are found using the selection algorithm from [CITATION].', '1502.03942-1-64-7': 'Otherwise, [MATH] is increased and the algorithm is restarted.', '1502.03942-1-64-8': 'In these subsequent iterations, PEs whose local threshold is worse than the relevance of the [MATH]-th best element seen so far do not need to continue scanning.', '1502.03942-1-64-9': 'Hence, we can also trade less local work for more rounds of communication by deliberately working with an insufficiently small value of [MATH].', '1502.03942-1-65-0': 'Arbitrary Data Distribution.', '1502.03942-1-65-1': 'Algorithm DTA "guesses" the number [MATH] of list rows scanned by TA by exponential search.', '1502.03942-1-65-2': 'This yields [MATH] rounds of DTA.In each round, the approximate multisequence selection algorithm from Section [REF] is used to approximate the globally [MATH]-th largest score [MATH] in each list.', '1502.03942-1-65-3': 'This takes time [MATH] by Theorem [REF].', '1502.03942-1-65-4': 'Accumulating the searches for all [MATH] lists yields time [MATH].', '1502.03942-1-65-5': 'Call an object with relevance above the threshold [MATH] which is selected by the selection algorithm a hit.', '1502.03942-1-65-6': 'DTA estimates the number of hits using sampling.', '1502.03942-1-65-7': 'For each PE and list separately, [MATH] objects are sampled from the result of the preceding multiselection.', '1502.03942-1-65-8': 'To eliminate bias for objects appearing in multiple lists, DTA only counts an object [MATH] if it is sampled in the first list containing it.', '1502.03942-1-65-9': 'Otherwise, [MATH] is ignored.', '1502.03942-1-65-10': 'Algorithm DTA also counts the number of ignored samples [MATH].', '1502.03942-1-65-11': 'Let [MATH] denote the number of nonrejected hits in the sample.', '1502.03942-1-65-12': 'Then, [MATH] is a truthful estimate for the length of the list with eliminated duplicates and [MATH] is a truthful estimate for the number of hits for the considered list and PE.Summing these [MATH] values using a reduction operation yields a truthful estimate for the overall number of hits.', '1502.03942-1-65-13': 'Algorithm DTA stops once this estimate is large enough such that with high probability the actual number of hits is at least [MATH].', '1502.03942-1-65-14': 'The output of DTA is a prefix of each list such that the union of all these prefixes contains the [MATH] most relevant ones with high probability.', '1502.03942-1-65-15': 'It also outputs the threshold [MATH].', '1502.03942-1-65-16': 'In total, all of this combined takes time [MATH].', '1502.03942-1-66-0': 'Actually computing the [MATH] most frequent objects amounts to scanning the result lists to find all hits and, if desired, running a selection algorithm to identify the [MATH] most relevant among them.', '1502.03942-1-66-1': 'The scanning step may involve some load imbalance because in the worst case, all hits are concentrated on a single PE.', '1502.03942-1-66-2': 'This seems unavoidable unless one requires random distribution of the objects.', '1502.03942-1-66-3': 'However, in practice it may be possible to equalize the imbalance over a large number of concurrent queries.', '1502.03942-1-67-0': 'Refinements.', '1502.03942-1-67-1': 'We can further reduce the latency of DTA by trying several values of [MATH] in each iteration of algorithm DTA.Since this involves access to only few objects, the overhead in internal work will be limited.', '1502.03942-1-67-2': 'In practice, it may also be possible to make good guesses on [MATH] based on previous executions of the algorithm.', '1502.03942-1-68-0': '# Top-[MATH] Most Frequent Objects', '1502.03942-1-69-0': 'We describe two probably approximately correct (PAC) algorithms that compute the top-[MATH] most frequent objects of a multiset [MATH] with [MATH], followed by a probably exactly correct (PEC) algorithm for suitable inputs in Section [REF].', '1502.03942-1-69-1': 'In all three algorithms, sublinear communication is achieved by transmitting only a small random sample of the input.', '1502.03942-1-70-0': 'To achieve readable bounds, we assume that the input is distributed over the [MATH] PEs so that none has more than [MATH] objects.', '1502.03942-1-70-1': "This requirement is not necessary; indeed the algorithms' running times increase linearly with the maximum fraction of the input concentrated at one PE.", '1502.03942-1-71-0': '## Basic PAC Algorithm', '1502.03942-1-72-0': 'First, we take a Bernoulli sample of the input.', '1502.03942-1-72-1': 'Sampling is done locally.', '1502.03942-1-72-2': 'The frequencies of the sampled objects are counted using distributed hashing - a local object count with key [MATH] is sent to PE [MATH] for a hash function [MATH] that we here expect to behave like a random function.', '1502.03942-1-72-3': 'We then select the [MATH] most frequently sampled objects using the selection algorithm from Section [REF].', '1502.03942-1-72-4': 'An example is illustrated in Figure [REF].', '1502.03942-1-73-0': 'For sampling probability [MATH], our algorithm needs expected time [MATH].', '1502.03942-1-74-0': '(Outline) Bernoulli sampling is done in expected time [MATH] by generating skip values with a geometric distribution.', '1502.03942-1-74-1': 'Since the number of elements in a Bernoulli sample is a random variable, so is the running time.', '1502.03942-1-74-2': 'To count the sampled objects, each PE inserts its share of the sample into a distributed hash table [CITATION] whose hash function we assume to behave like a random function, distributing the objects randomly among the PEs.', '1502.03942-1-74-3': 'The elements are communicated using indirect delivery to maintain logarithmic latency.', '1502.03942-1-74-4': 'This requires [MATH] time in expectation.', '1502.03942-1-75-0': 'From this hash table, we select the object with rank [MATH] in expected time [MATH] using Algorithm [REF].', '1502.03942-1-75-1': 'This pivot is broadcast to all PEs, which then determine their subset of at least as frequent sample objects in expected time [MATH].', '1502.03942-1-75-2': 'These elements are returned.', '1502.03942-1-75-3': 'Overall, the claimed time complexity follows using the estimate [MATH] and [MATH] in order to simplify the [MATH]-term.', '1502.03942-1-76-0': "In the following, we will bound the algorithm's error.", '1502.03942-1-76-1': 'We define the absolute error [MATH] as the count of the most frequent object that was missed by the algorithm minus that of the least frequent object that was output, or 0 if the result was exact.', '1502.03942-1-77-0': 'Let [MATH] be the sampling probability.', '1502.03942-1-77-1': "Then, our PAC algorithm's error exceeds a value of [MATH] with probability at most [EQUATION].", '1502.03942-1-78-0': "We bound the error probability as follows: the probability that the error [MATH] exceeds some value [MATH] is at most [MATH] times the probability that a single object's value estimate deviates from its true value by more than [MATH] in either direction.", '1502.03942-1-78-1': 'This probability can be bounded using Chernoff bounds.', '1502.03942-1-78-2': 'We denote the count of element [MATH] in the input by [MATH] and in the sample by [MATH].', '1502.03942-1-78-3': 'Further, let [MATH] be the probability that the error for element [MATH] exceeds [MATH] in either direction.', '1502.03942-1-78-4': 'Let [MATH], and observe that [MATH].', '1502.03942-1-78-5': 'Using Equations [REF] and [REF] with [MATH], [MATH], and [MATH], we obtain: [EQUATION]', '1502.03942-1-78-6': 'This leaves us with the most frequent [MATH] elements, whose counts can be bounded as [MATH].', '1502.03942-1-78-7': 'As overestimating them is not a concern, we obtain [MATH] from the Chernoff bound in Equation [REF].', '1502.03942-1-78-8': 'In sum, all error probabilities add up to the claimed value.', '1502.03942-1-79-0': 'We can now sum these terms over their [MATH] and [MATH] occurrences, respectively, and bound the result by an error probability [MATH].', '1502.03942-1-79-1': 'Lemma [REF] then allows us to calculate the minimum required sample size given [MATH] and [MATH].', '1502.03942-1-79-2': 'Solving the above equation for [MATH] yields [EQUATION] which is dominated by the latter term in most cases and yields expected sample size [MATH].', '1502.03942-1-80-0': 'Our PAC algorithm exceeding relative error [MATH] with probability at most [MATH] can be implemented to run in expected time [EQUATION].', '1502.03942-1-81-0': 'Equation ([REF]) yields [MATH].', '1502.03942-1-81-1': 'The claimed running time follows from Lemma [REF].', '1502.03942-1-82-0': '## Increasing Communication Efficiency', '1502.03942-1-83-0': 'We can reduce the size of the sample if we perform exact counting of the most frequently sampled objects.', '1502.03942-1-83-1': 'This way, we can improve communication efficiency at the cost of increased local computation.', '1502.03942-1-84-0': 'Again, we begin by taking a Bernoulli sample of the input.', '1502.03942-1-84-1': 'Then we find the [MATH] most frequent objects in the sample and count their frequency in the overall input exactly.', '1502.03942-1-84-2': 'This can be done using the selection algorithm from Section [REF].', '1502.03942-1-84-3': 'The identity of these objects is then broadcast to all PEs using an all-gather (gossiping, all-to-all broadcast) collective communication operation.', '1502.03942-1-84-4': 'After local counting, a global reduction sums up the local counts to exact global values.', '1502.03942-1-84-5': 'The [MATH] most frequent of these are then returned.', '1502.03942-1-85-0': 'Let [MATH] be a relative error bound so that for some [MATH], [MATH].', '1502.03942-1-85-1': 'The expected running time of our algorithm with counting is [EQUATION].', '1502.03942-1-86-0': 'Sampling and hashing are done as in Section [REF].', '1502.03942-1-86-1': 'The sampling probability [MATH] is determined in the next paragraph.', '1502.03942-1-86-2': 'Now, we select the object with rank [MATH] as a pivot.', '1502.03942-1-86-3': 'This requires expected time [MATH], again using the algorithm from Section [REF].', '1502.03942-1-86-4': 'The pivot is broadcast to all PEs, which then determine their subset of at least as frequent sample objects in expected time [MATH].', '1502.03942-1-86-5': 'Next, these [MATH] most frequent objects in the sample are distributed to all PEs using an all-gather operation in time [MATH].', '1502.03942-1-86-6': "Now, the PEs count the received objects' occurrences in their share of the input using time [MATH].", '1502.03942-1-86-7': 'These counts are summed up using a vector-valued reduction, again requiring time [MATH].', '1502.03942-1-86-8': 'We then apply the selection algorithm from Section [REF] a second time to determine the [MATH]-th most frequent of these objects.', '1502.03942-1-86-9': 'This is broadcast, and each PE returns those exactly counted elements that were assigned to it by the hash function.', '1502.03942-1-86-10': 'Overall, the claimed time complexity follows by substituting the sampling probability [MATH] computed in the next paragraph.', '1502.03942-1-87-0': 'With the given error bounds, we can derive the required sampling probability [MATH] from Lemma [REF], and obtain a total expected sample size of [MATH].', '1502.03942-1-87-1': 'For [MATH], which is nearly always the case, the latter term dominates the overall sample size greatly.', '1502.03942-1-87-2': 'Substituting [MATH] then yields [MATH].', '1502.03942-1-88-0': 'Note that the above term for the sample size grows with [MATH] instead of [MATH], at the cost of a [MATH] speedup instead of [MATH] in the overall running time.', '1502.03942-1-88-1': 'However, this decreases the total communication volume to [MATH] words.', '1502.03942-1-88-2': 'Of course, we can only work with the above value of [MATH] if it is at least [MATH].', '1502.03942-1-88-3': 'Otherwise, need to choose [MATH].', '1502.03942-1-89-0': 'To continue the example from Figure [REF], we may set [MATH].', '1502.03942-1-89-1': 'Then, the [MATH] most frequently sampled objects [MATH] with [MATH] occurrences, respectively, will be counted exactly.', '1502.03942-1-89-2': 'The result would now be correct.', '1502.03942-1-90-0': '## Probably Exactly Correct Algorithm', '1502.03942-1-91-0': 'If any significant gap exists in the frequency distribution of the objects (see Figure [REF] for an example), we can choose [MATH] to ensure a sufficiently large difference between [MATH] and [MATH], i.e., [MATH].', '1502.03942-1-91-1': 'We will call this difference the gap.', '1502.03942-1-91-2': 'It consists of two parts, the first being the margin of error around element [MATH], the second around element [MATH], as illustrated by the densely dotted line in Figure [REF].', '1502.03942-1-91-3': 'From the proof of Lemma [REF], we obtain that [MATH].', '1502.03942-1-91-4': 'Now, all we need to do is to apply the algorithm from Section [REF], substituting our modified value of [MATH] for [MATH] and using [MATH] as above.', '1502.03942-1-91-5': 'This works, because all objects that could be among the [MATH] most frequent ones are counted exactly with high probability.', '1502.03942-1-92-0': 'Thus, a probably exactly correct (PEC) algorithm to compute the top-[MATH] most frequent objects of a multiset whose frequency distribution is sufficiently sloped (i.e. we can attain a gap) can be formulated as follows: Take a very small sample of size [MATH], whose exact size we will determine later.', '1502.03942-1-92-1': 'From this small sample, we determine [MATH] so that [MATH] with high probability.', '1502.03942-1-92-2': 'Now we apply the algorithm from Section [REF].', '1502.03942-1-92-3': 'If, after exact counting, it becomes apparent that the choice of [MATH] was not sufficient, we determine its new value from the sample and the exactly counted elements, and count these elements exactly as well, until [MATH] is fulfilled.', '1502.03942-1-93-0': 'To determine the size [MATH] of the first sample, we need to formalize the gap requirement.', '1502.03942-1-93-1': 'In particular, we require that the [MATH]-th most frequent object have an absolute count of [MATH] for some variable [MATH].', '1502.03942-1-93-2': 'Then, [MATH] - the expected number of samples for the [MATH]-th most frequent element - needs to be significant, i.e. large enough to allow for useful estimates of [MATH].', '1502.03942-1-93-3': 'To be on the safe side, we can simply choose [MATH] as in Section [REF].', '1502.03942-1-93-4': 'While this is slightly wasteful, the running time involving the first sample is dominated by the time spent on the second sample anyway.', '1502.03942-1-94-0': "### Zipf's Law", '1502.03942-1-95-0': "In its simplest form, Zipf's law states that the frequency of an object from a multiset [MATH] with [MATH] is inversely proportional to its rank among the objects ordered by frequency.", '1502.03942-1-95-1': 'Here, we consider the general case with exponent parameter [MATH], i.e. [MATH], where [MATH] is the [MATH]-th generalized harmonic number.', '1502.03942-1-96-0': "If the frequencies of the objects in the input are distributed according to Zipf's law with an exponent of [MATH], a sample size of [MATH] is sufficient.", '1502.03942-1-96-1': 'We can determine the PEC top-[MATH] most frequent objects in expected time [EQUATION].', '1502.03942-1-97-0': 'For [MATH], the sample size expression reduces to [MATH], where [MATH] is the harmonic number, for which [MATH].', '1502.03942-1-97-1': 'Then, the communication volume comprises [MATH] words.', '1502.03942-1-98-0': 'We choose [MATH].', '1502.03942-1-98-1': 'From this, we can infer [MATH] as [MATH].', '1502.03942-1-98-2': 'Now, we can compute the sample size that is required to obtain the probably exactly correct top-[MATH] most frequent objects like before as [MATH], which simplifies to the claimed value.', '1502.03942-1-99-0': 'Observe that the latency is very low in the above running time term, and is expected to be dominated by the communication volume and the time for exact counting in the input.', '1502.03942-1-100-0': '## Refinements', '1502.03942-1-101-0': 'When implementing such an algorithm, there are a number of considerations to be taken into account to achieve optimal performance.', '1502.03942-1-101-1': 'Perhaps most importantly, one should apply local aggregation when inserting the sample into the distributed hash table to reduce the amount of information that needs to be communicated in practice.', '1502.03942-1-101-2': 'We now discuss other potential improvements.', '1502.03942-1-102-0': 'Choice of [MATH].', '1502.03942-1-102-1': "In practice, the choice of [MATH] in Section [REF] depends on the communication channel's characteristics [MATH], and, to a lesser extent, [MATH], in addition to the problem parameters.", '1502.03942-1-102-2': 'Thus, an optimized implementation should take them into account when determining the number of objects to be counted exactly.', '1502.03942-1-103-0': 'Adaptive Two-Pass Sampling.', '1502.03942-1-103-1': 'The objectives of the basic PAC algorithm and its variant using exact counting could be unified as follows: we sample in two passes.', '1502.03942-1-103-2': 'In the first pass, we use a small sample size [MATH] to determine the nature of the input distribution.', '1502.03942-1-103-3': 'From the insights gained from this first sample, we compute a larger sample size [MATH].', '1502.03942-1-103-4': 'We then determine and return the [MATH] most frequent objects of this second sample.', '1502.03942-1-104-0': 'Additionally, we can further refine this algorithm if we can already tell from the first sample that with high probability, there is no slope.', '1502.03942-1-104-1': 'If the absolute counts of the objects in the sample are large enough to return the [MATH] most frequent objects in the sample with confidence, then taking a second sample would be of little benefit and we can return the [MATH] most frequent objects from the first sample.', '1502.03942-1-105-0': 'Using Distributed Bloom Filters.', '1502.03942-1-105-1': 'Communication efficiency of the algorithm using exact counting could be improved further by counting sample elements with a distributed single-shot bloom filter (dSBF) [CITATION] instead of a distributed hash table.', '1502.03942-1-105-2': 'Instead of collecting the sampled elements in a distributed hash table, we transmit their hash values and locally aggregated counts.', '1502.03942-1-105-3': 'As multiple keys might be assigned the same hash value, we need to determine the element of rank [MATH] for some safety margin [MATH] instead of [MATH].', '1502.03942-1-105-4': 'We request the keys of all elements with higher rank, and replace the (hash, value) pairs with (key, value) pairs, splitting them where hash collisions occurred.', '1502.03942-1-105-5': 'We now determine the element of rank [MATH] on the [MATH] elements.', '1502.03942-1-105-6': 'If an element whose rank is at most [MATH] was part of the original [MATH] elements, we are finished.', '1502.03942-1-105-7': 'Otherwise, we have to increase [MATH] to determine the missing elements.', '1502.03942-1-105-8': 'Observe that if the frequent objects are dominated by hash collisions, this implies that the input distribution is flat and there exist a large number of nearly equally frequent elements.', '1502.03942-1-105-9': 'Thus, we may not need to count additional elements in this case.', '1502.03942-1-106-0': '# Data Redistribution', '1502.03942-1-107-0': 'Let [MATH] denote the number of data objects present at PE [MATH].', '1502.03942-1-107-1': 'Let [MATH].', '1502.03942-1-107-2': 'We want to redistribute the data such that afterwards each PE has at most [MATH] objects and such that PEs with more than [MATH] only send data (at most [MATH] objects) and PEs with at most [MATH] only receive data (at most [MATH] objects).', '1502.03942-1-107-3': 'We split the PEs into separately numbered groups of senders [MATH] and receivers [MATH].', '1502.03942-1-107-4': 'We also compute the deficit [MATH] on receivers and the surplus [MATH] on senders.', '1502.03942-1-107-5': 'Then we compute the prefix sums [MATH] and [MATH] of these sequences (i.e., [MATH] and [MATH]).', '1502.03942-1-107-6': 'Effectively, [MATH] enumerates the empty slots able to receive objects and [MATH] enumerates the elements to be moved.', '1502.03942-1-107-7': 'Now we match receiving slots and elements to be moved by merging the sequences [MATH] and [MATH].', '1502.03942-1-107-8': "This is possible in time [MATH] using Batcher's parallel merging algorithm [CITATION].", '1502.03942-1-107-9': 'A subsequence of the form [MATH] indicates that sending PEs [MATH] move their surplus to receiving PE [MATH] (where sending PE [MATH] only moves its items numbered [MATH]).', '1502.03942-1-107-10': 'This is a gather operations.', '1502.03942-1-107-11': 'Sending PE [MATH] moves its remaining elements to receiving PEs [MATH].', '1502.03942-1-107-12': 'This is a scatter operation.', '1502.03942-1-107-13': 'These segments of PE numbers can be determined using segmented prefix operations [CITATION].', '1502.03942-1-107-14': 'Overall, this can be implemented to run in time [MATH].', '1502.03942-1-107-15': 'This operation cannot remove worst case bottlenecks.', '1502.03942-1-107-16': 'But it can significantly reduce network traffic.', '1502.03942-1-108-0': '# Conclusions', '1502.03942-1-109-0': 'We have demonstrated that a variety of top-[MATH] selection problems can be solved in a communication efficient way, both with respect to communication volume and with respect to latencies.', '1502.03942-1-109-1': 'The basic methods are simple and versatile - the owner-computes rule, collective communication, and sampling.', '1502.03942-1-109-2': 'Considering the significant previous work on some of the considered problems, it is a bit surprising that such simple algorithms give improved results for such fundamental problems.', '1502.03942-1-109-3': 'However, it seems that the combination of communication efficiency and parallel scalability has been neglected for many problems.', '1502.03942-1-109-4': 'It seems that our methods might have particular impact on those applications where previous work has concentrated on methods with a pure master worker scheme.', '1502.03942-1-110-0': 'It is therefore likely that our approach can also be used for further important problems.', '1502.03942-1-110-1': 'For example, distributed streaming algorithms that generalize the centralized model of Yi and Zhang [CITATION] seem very promising.', '1502.03942-1-110-2': 'The same holds for lower bounds which so far have also neglected multiparty communication with point-to-point communication (see also [CITATION]).', '1502.03942-1-111-0': 'Closer to the problems considered here, there are also a number of interesting open questions.', '1502.03942-1-111-1': 'For the sorted selection problem from Section [REF], it would be interesting to see whether there is a scalable parallel algorithm which makes an information theoretically optimal number of comparisons as in the sequential algorithm of Varman et al. [CITATION].', '1502.03942-1-111-2': 'Our analysis of Algorithm [REF] is ignores the case when [MATH].', '1502.03942-1-111-3': 'It can probably be shown to run in expected time [MATH].', '1502.03942-1-112-0': 'For the multicriteria top-[MATH] problem from Section [REF], we could consider parallelization of advanced algorithms that scan less elements and perform less random accesses, such as [CITATION].', '1502.03942-1-113-0': 'In database terminology, looking for the top-[MATH] most frequent objects is called a count-aggregation.', '1502.03942-1-113-1': 'It is likely that the more general sum-aggregation problem, where arbitrary numbers are added, can be addressed with similar sampling-based techniques (at least for non-negative numbers).', '1502.03942-1-113-2': 'However, we have to handle the case that a small number of objects dominate the overall sum.', '1502.03942-1-114-0': '# Multisequence Selection [CITATION]', '1502.03942-1-115-0': 'Figure [REF] gives high level pseudo code.', '1502.03942-1-115-1': 'The base case occurs if there is only a single object (and [MATH]).', '1502.03942-1-115-2': 'We can also restrict the search to the first [MATH] objects of each local sequence.', '1502.03942-1-115-3': 'A random object is selected as a pivot.', '1502.03942-1-115-4': 'This can be done in parallel by choosing the same random number between 1 and [MATH] on all PEs.', '1502.03942-1-115-5': 'Using a prefix sum over the sizes of the sequences, this object can be located easily in time [MATH].', '1502.03942-1-115-6': 'Where ordinary quickselect has to partition the input doing linear work, we can exploit the sortedness of the sequences to obtain the same information in time [MATH] with [MATH] by doing binary search in parallel on each PE.If items are evenly distributed, we have [MATH], and thus only time [MATH] for the search, which partitions all the sequences into two parts.', '1502.03942-1-115-7': 'Deciding whether we have to continue searching in the left or the right parts needs a global reduction operations taking time [MATH].', '1502.03942-1-115-8': 'As in ordinary quickselect, the expected depth of the recursion is [MATH].', '1502.03942-1-115-9': 'We obtain the following result.', '1502.03942-1-116-0': 'Algorithm [REF] can be implemented to run in expected time [EQUATION]'}

{'1502.03942-2-0-0': 'We present scalable parallel algorithms with sublinear per-processor communication volume and low latency for several fundamental problems related to finding the most relevant elements in a set, for various notions of relevance: We begin with the classical selection problem with unsorted input.', '1502.03942-2-0-1': 'We present generalizations with locally sorted inputs, dynamic content (bulk-parallel priority queues), and multiple criteria.', '1502.03942-2-0-2': 'Then we move on to finding frequent objects and top-[MATH] sum aggregation.', '1502.03942-2-0-3': 'Since it is unavoidable that the output of these algorithms might be unevenly distributed over the processors, we also explain how to redistribute this data with minimal communication.', '1502.03942-2-1-0': 'Keywords: selection, frequent elements, sum aggregation, priority queue, sampling, branch-and-bound, data redistribution, threshold algorithm', '1502.03942-2-2-0': '# Introduction', '1502.03942-2-3-0': 'Overheads due to communication latency and bandwidth limitations of the communication networks are one of the main limiting factors for distributed computing.', '1502.03942-2-3-1': 'Parallel algorithm theory has considered the latency part of this issue since its beginning.', '1502.03942-2-3-2': 'In particular, execution times polylogarithmic in the number [MATH] of processing elements (PEs) were a main focus.', '1502.03942-2-3-3': 'Borkar [CITATION] argues that an exascale computer could only be cost-effective if the communication capabilities (bisection width) scale highly sublinearly with the amount of computation done in the connected subsystems.', '1502.03942-2-3-4': 'Google confirms that at data center scale, the network is the most scarce resource and state that they "don\'t know how to build big networks that deliver lots of bandwidth" [CITATION].', '1502.03942-2-3-5': 'In a previous paper [CITATION] we therefore proposed to look more intensively for algorithms that require bottleneck communication volume sublinear in the local input size.', '1502.03942-2-3-6': 'More precisely, consider an input consisting of [MATH] machine words distributed over the PEs such that each PE holds [MATH] words.', '1502.03942-2-3-7': 'Then sublinear communication volume means that no PE sends or receives more than [MATH] machine words of data.', '1502.03942-2-4-0': 'Here, we combine the latency and data volume aspects.', '1502.03942-2-4-1': 'We consider some fundamental algorithmic problems that have a large input (size [MATH]) and a relatively small output (size [MATH]).', '1502.03942-2-4-2': 'Since potentially many similar problems have to be solved, both latency and communication volume have to be very small-ideally polylogarithmic in [MATH] and the input parameters.', '1502.03942-2-4-3': 'More precisely, we consider problems that ask for the [MATH] most "relevant" results from a large set of possibilities, and aim to obtain low bottleneck communication overhead.', '1502.03942-2-5-0': 'In the simplest case, these are totally ordered elements and we ask for the [MATH] smallest of them-the classical selection problem.', '1502.03942-2-5-1': 'Several variants of this problem are studied in Section [REF].', '1502.03942-2-5-2': 'For the classical variant with unsorted inputs, a careful analysis of a known algorithm [CITATION] shows that the previously assumed random allocation of the inputs is actually not necessary and we get running time [MATH].', '1502.03942-2-5-3': 'For locally sorted input we get latency [MATH].', '1502.03942-2-5-4': 'Interestingly, we can return to logarithmic latency if we are willing to relax the output size [MATH] by a constant factor.', '1502.03942-2-5-5': 'This uses a new technique for obtaining a pivot element with a given rank that is much simpler than the previously proposed techniques based on sorting.', '1502.03942-2-6-0': 'A data structure generalization of the selection problem are bulk-parallel priority queues.', '1502.03942-2-6-1': 'Previous parallel priority queues are not communication efficient in the sense that they move the elements around, either by sorting [CITATION] or by random allocation [CITATION].', '1502.03942-2-6-2': 'Section [REF] generalizes the results on selection from Section [REF].', '1502.03942-2-6-3': 'The key to making this work is to use an appropriately augmented search tree data structure to efficiently support insertion, deletion, and all the operations needed by the parallel selection algorithm.', '1502.03942-2-7-0': 'A prominent problem in information retrieval is to extract the [MATH] most relevant objects (e.g. documents), where relevance is determined by a monotonous function that maps several individual scoring functions to the overall relevance.', '1502.03942-2-7-1': 'For each individual score, a list of objects is precomputed that stores the objects in order of decreasing score.', '1502.03942-2-7-2': 'This is a multicriteria generalization of the sorted top-[MATH] selection problem discussed in Section [REF].', '1502.03942-2-7-3': 'In Section [REF] we parallelize established sequential algorithms for this problem.', '1502.03942-2-7-4': 'The single-criterion selection algorithms are used as subroutines- the sorted one for approximating the list elements scanned by the sequential algorithm and the unsorted one to actually identify the output.', '1502.03942-2-7-5': 'The algorithm has polylogarithmic overhead for coordinating the PEs and manages to contain unavoidable load imbalance in a single phase of local computation.', '1502.03942-2-7-6': 'This is achieved with a fast estimator of the output size generated with a given number of scanned objects.', '1502.03942-2-8-0': 'A fundamental problem in data mining is finding the most frequently occurring objects.', '1502.03942-2-8-1': 'This is challenging in a distributed setting since the globally most frequent elements do not have to be locally frequent on any particular PE.', '1502.03942-2-8-2': 'In Section [REF] we develop very fast sampling-based algorithms that find a [MATH]-approximation or probably approximately correct answer, i.e., with probability at least [MATH] the output is correct within [MATH].', '1502.03942-2-8-3': 'The algorithms run in time logarithmic in [MATH], [MATH], and [MATH].', '1502.03942-2-8-4': 'From a simple algorithm with running time factor [MATH] we go to more sophisticated ones with factor [MATH].', '1502.03942-2-8-5': 'We also show how to compute the exact result with probability at least [MATH] if the elements are non-uniformly distributed.', '1502.03942-2-9-0': 'Subsequently, we generalize these results to sum aggregation, where object occurrences are associated with a value.', '1502.03942-2-9-1': 'In Section [REF], we are thus looking for the objects whose values add up to the highest sums.', '1502.03942-2-10-0': 'All of the above algorithms have the unavoidable limitation that the output may be unevenly distributed over the PEs for general distributions of the input objects.', '1502.03942-2-10-1': 'This can lead to load imbalance affecting the efficiency of the overall application.', '1502.03942-2-10-2': 'Offsetting this imbalance will require communication so that one might argue that striving for communication efficiency in the selection process is in vain.', '1502.03942-2-10-3': 'However, our methods have several advantages over non-communication efficient selection algorithms.', '1502.03942-2-10-4': 'First of all, priorities can be ignored during redistribution since all selected elements are relevant.', '1502.03942-2-10-5': 'Hence, we can employ any data redistribution algorithm we want.', '1502.03942-2-10-6': 'In particular, we can use an adaptive algorithm that only moves data if that is really necessary.', '1502.03942-2-10-7': 'In Section [REF], we give one such algorithm that combines prefix sums and merging to minimize data movement and incurs only logarithmic additional delay.', '1502.03942-2-10-8': 'Delegating data movement to the responsibility of the application also has the advantage that we can exploit properties of the application that a general top-[MATH] selection algorithm cannot.', '1502.03942-2-10-9': 'For example, multicriteria selection algorithms as discussed in Section [REF] are used in search engines processing a stream of many queries.', '1502.03942-2-10-10': 'Therefore, it is enough to do load balancing over a large number of queries and we can resolve persistent hot spots by adaptive data migration or replication.', '1502.03942-2-10-11': 'Another example are branch-and-bound applications [CITATION].', '1502.03942-2-10-12': 'By randomizing necessary data migrations, we can try to steer the system away from situations with bad data distribution (a proper analysis of such an algorithm is an open problem though).', '1502.03942-2-11-0': 'Some of our main results are listed in Table [REF].', '1502.03942-2-11-1': 'Refer to Appendix [REF] for proofs and further discussion.', '1502.03942-2-12-0': '# Preliminaries', '1502.03942-2-13-0': 'Our input usually consists of a multiset [MATH] of [MATH] objects, each represented by a single machine word.', '1502.03942-2-13-1': 'If these objects are ordered, we assume that their total ordering is unique.', '1502.03942-2-13-2': 'This is without loss of generality since we can make the value [MATH] of object [MATH] unique by replacing it by the pair [MATH] for tie breaking.', '1502.03942-2-14-0': 'Consider [MATH] processing elements (PEs) connected by a network.', '1502.03942-2-14-1': 'PEs are numbered [MATH], where [MATH] is a shorthand for [MATH] throughout this paper.', '1502.03942-2-14-2': 'Assuming that a PE can send and receive at most one message at a time (full-duplex, single-ported communication), sending a message of size [MATH] machine words takes time [MATH].', '1502.03942-2-14-3': 'We often treat the time to initiate a connection [MATH] and to send a single machine word [MATH] as variables in asymptotic analysis.', '1502.03942-2-14-4': 'A running time of [MATH] then allows us to discuss internal work [MATH], communication volume [MATH], and latency [MATH] separately.', '1502.03942-2-14-5': 'Depending on the situation, all three aspects may be important, and combining them into a single expression for running time allows us to specify them concisely.', '1502.03942-2-15-0': 'We present some of the algorithms using high level pseudocode in a single program multiple data (SPMD) style-the same algorithm runs on each PE, which perform work on their local data and communicate predominantly through collective operations.', '1502.03942-2-15-1': 'Variables are local by default.', '1502.03942-2-15-2': 'We can denote data on remote PEs using the notation [MATH], which refers to the value of variable [MATH] on PE [MATH].', '1502.03942-2-15-3': 'For example, [MATH] denotes the global sum of [MATH] over all PEs, which can be computed using a sum (all-)reduction.', '1502.03942-2-16-0': 'Collective communication.', '1502.03942-2-16-1': 'Broadcasting sends a message to all PEs.', '1502.03942-2-16-2': 'Reduction applies an associative operation (e.g., sum, maximum, or minimum) to a vector of length [MATH].', '1502.03942-2-16-3': 'An all-reduction also broadcasts this result to all PEs.', '1502.03942-2-16-4': 'A prefix-sum (scan) computes [MATH] on PE [MATH] where [MATH] is a vector of length [MATH].', '1502.03942-2-16-5': 'The scatter operation distributes a message of size [MATH] to a set of [MATH] PEs such that each of them gets a piece of size [MATH].', '1502.03942-2-16-6': 'Symmetrically, a gather operations collects [MATH] pieces of size [MATH] on a single PE.All of these operations can be performed in time [MATH] [CITATION].', '1502.03942-2-16-7': 'In an all-to-all personalized communication (all-to-all for short) each PE sends one message of size [MATH] to every other PE.This can be done in time [MATH] using direct point-to-point delivery or in time [MATH] using indirect delivery [CITATION].', '1502.03942-2-16-8': 'The all-to-all broadcast (aka gossiping) starts with a single message of size [MATH] on each PE and ends with all PEs having all these messages.', '1502.03942-2-16-9': 'This operation works in time [MATH].', '1502.03942-2-17-0': 'Fast inefficient Sorting.', '1502.03942-2-17-1': 'Sorting [MATH] objects can be done in time [MATH] using a brute force algorithm performing all pairwise object comparisons in parallel (e.g., [CITATION]).', '1502.03942-2-18-0': 'Search trees.', '1502.03942-2-18-1': 'Search trees can represent a sorted sequence of objects such that a variety of operations can be supported in logarithmic time.', '1502.03942-2-18-2': 'In particular, one can insert and remove objects and search for the next largest object given a key.', '1502.03942-2-18-3': 'The operation [MATH] splits T into two search trees [MATH] such that [MATH] contains the objects of [MATH] with key [MATH] and [MATH] contains the objects of [MATH] with key [MATH].', '1502.03942-2-18-4': 'Similarly, if the keys in a tree [MATH] are smaller than the keys in [MATH] then [MATH] returns a search tree representing the concatenation of the two sequences.', '1502.03942-2-18-5': 'If one additionally stores the size of subtrees, one can also support the select operation [MATH], which returns the [MATH]-th largest object in [MATH], and operation [MATH], which returns the number of objects in [MATH] of value at most [MATH].', '1502.03942-2-18-6': 'For an example of such a data structure see [CITATION].', '1502.03942-2-19-0': 'Chernoff bounds.', '1502.03942-2-19-1': 'We use the following Chernoff bounds [CITATION] to bound the probability that a sum [MATH] of [MATH] independent indicator random variables deviates substantially from its expected value [MATH].', '1502.03942-2-19-2': 'For [MATH], we have [EQUATION]', '1502.03942-2-19-3': 'Bernoulli sampling.', '1502.03942-2-19-4': 'A simple way to obtain a sample of a set [MATH] of objects is to select each object with probability [MATH] independent of the other objects.', '1502.03942-2-19-5': 'For small sampling probability [MATH], the naive implementation can be significantly accelerated from time [MATH] to expected time [MATH] by using skip values-a value of [MATH] indicates that the following [MATH] elements are skipped and the [MATH]-th element is sampled.', '1502.03942-2-19-6': 'These skip values follow a geometric distribution with parameter [MATH] and can be generated in constant time.', '1502.03942-2-20-0': 'Distributed FR-Selection [CITATION] Floyd and Rivest [CITATION] developed a modification of quickselect using two pivots in order to achieve a number of comparisons that is close to the lower bound.', '1502.03942-2-20-1': 'This algorithm, FR-select, can be adapted to a distributed memory parallel setting [CITATION].', '1502.03942-2-20-2': 'FR-select picks the pivots based on sorting a random sample [MATH] of [MATH] objects, which can easily be done in logarithmic time (e.g., the simple algorithm described in [CITATION] performs brute-force comparisons of all possible pairs of sample elements).', '1502.03942-2-20-3': 'Pivots [MATH] and [MATH] are chosen as the sample objects with ranks [MATH] where [MATH] for some small constant [MATH].', '1502.03942-2-20-4': 'For the analysis, one can choose [MATH], which ensures that with high probability the range of possible values shrinks by a factor [MATH] in every level of recursion.', '1502.03942-2-21-0': 'It is shown that a constant number of recursion levels suffice if [MATH] and if the objects are distributed randomly.', '1502.03942-2-21-1': 'Note that the latter assumption limits communication efficiency of the algorithm since it requires moving all objects to random PEs for general inputs.', '1502.03942-2-22-0': 'Bulk Parallel Priority Queues.', '1502.03942-2-22-1': 'A natural way to parallelize priority queues is to use bulk operations.', '1502.03942-2-22-2': 'In particular, operation deleteMin[MATH] supports deletion of the [MATH] smallest objects of the queue.', '1502.03942-2-22-3': 'Such a data structure can be based on a heap where nodes store sorted sequences rather than objects [CITATION].', '1502.03942-2-22-4': 'However, an even faster and simpler randomized way is to use multiple sequential priority queues-one on each PE [CITATION].', '1502.03942-2-22-5': 'This data structure adopts the idea of Karp and Zhang [CITATION] to give every PE a representative approximate view of the global situation by sending inserted objects to random queues.', '1502.03942-2-22-6': 'However, in contrast to Karp and Zhang [CITATION], [CITATION] implements an exact deleteMin[MATH] using parallel selection and a few further tricks.', '1502.03942-2-22-7': 'Note that the random insertions limit communication efficiency in this case.', '1502.03942-2-23-0': '# More Related Work', '1502.03942-2-24-0': 'The fact that the amount of communication is important for parallel computing is a widely studied issue.', '1502.03942-2-24-1': 'However, there are very few results on achieving sublinear communication volume per processor.', '1502.03942-2-24-2': 'In part, this is because models like parallel external memory or resource-oblivious models assume that the input is located in a global memory so that processing it requires loading all of it into the local caches at least once.', '1502.03942-2-24-3': 'Refer to [CITATION] for a more detailed discussion.', '1502.03942-2-25-0': 'A good framework for studying bottleneck communication volume is the BSP model [CITATION] where a communication round with maximum amount of data per PE [MATH] (maximized over sent and received data) takes time [MATH] for some machine parameters [MATH] (latency) and [MATH] (gap).', '1502.03942-2-25-1': 'Summing the values of [MATH] over all rounds yields the bottleneck communication volume we consider.', '1502.03942-2-25-2': 'We do not use the BSP model directly because we make heavy use of collective communication routines which are not directly represented in the BSP model.', '1502.03942-2-25-3': 'Further, the latency parameter [MATH] is typically equal to [MATH] in real-world implementations of the BSP model (see also [CITATION]).', '1502.03942-2-25-4': 'Also, we are not aware of any concrete results in the BSP model on sublinear communication volume in general or for top-[MATH] problems in particular.', '1502.03942-2-25-5': 'A related recent line of research considers algorithms based on MapReduce (e.g., [CITATION]).', '1502.03942-2-25-6': 'Communication volume is an important issue there but it seems difficult to achieve sublinear communication volume since the MapReduce primitive does not model locality well.', '1502.03942-2-26-0': 'Recently, communication avoiding algorithms have become an important research direction for computations in high performance linear algebra and related problems, e.g. [CITATION].', '1502.03942-2-26-1': 'However, these results only apply to nested loops with array accesses following a data independent affine pattern and the proven bounds are most relevant for computations with work superlinear in the input size.', '1502.03942-2-26-2': 'We go into the opposite direction looking at problems with linear or even sublinear work.', '1502.03942-2-27-0': 'Selection.', '1502.03942-2-27-1': 'Plaxton [CITATION] shows a superlogarithmic lower bound for selection on a large class of interconnection networks.', '1502.03942-2-27-2': 'This bound does not apply to our model, since we assume a more powerful network.', '1502.03942-2-28-0': 'Parallel Priority Queues.', '1502.03942-2-28-1': 'There has been intensive work on parallel priority queues.', '1502.03942-2-28-2': "The most scalable solutions are our randomized priority queue [CITATION] and Deo and Prasad's parallel heap [CITATION].", '1502.03942-2-28-3': 'Refer to [CITATION] for a recent overview of further approaches, most of which are for shared memory architectures and mostly operate on centralized data structures with limited scalability.', '1502.03942-2-29-0': 'Multicriteria, Most Frequent Objects.', '1502.03942-2-29-1': 'Considerable work has been done on distributed top-[MATH] computations in wide area networks, sensor networks and for distributed streaming models [CITATION].', '1502.03942-2-29-2': 'However, these papers use a master-worker approach where all communication has to go over a master node.', '1502.03942-2-29-3': 'This implies up to [MATH] times higher communication volume compared to our results.', '1502.03942-2-29-4': 'Only rarely do randomized versions with better running times exist, but nevertheless, communication volume at the master node still increases with [MATH][CITATION].', '1502.03942-2-29-5': 'Moreover, the scalability of TPUT [CITATION] and KLEE [CITATION] is severely limited by the requirement that the number of workers cannot exceed the number of criteria.', '1502.03942-2-29-6': 'Furthermore, the top-[MATH] most frequent objects problem has received little attention in distributed streaming algorithms.', '1502.03942-2-29-7': 'The only work we could find requires [MATH] working memory at the master node, where [MATH] is the number of distinct objects in the union of the streams [CITATION].', '1502.03942-2-29-8': 'The majority of papers instead considers the significantly easier problem of identifying the heavy hitters, i.e. those objects whose occurrences account for more than a fixed proportion of the input, or frequency tracking, which tracks the approximate frequencies of all items, but requires an additional selection step to obtain the most frequent ones [CITATION].', '1502.03942-2-30-0': 'Sum Aggregation Much work has been done on aggregation in parallel and distributed settings, e.g. in the database community [CITATION].', '1502.03942-2-30-1': 'However, these papers all ask for exact results for all objects, not approximations for the [MATH] most important ones.', '1502.03942-2-30-2': 'We do not believe that exact queries are can be answered in a communication-efficient manner, nor could we find any works that consider the same problem, regardless of communication efficiency.', '1502.03942-2-31-0': 'Data redistribution using prefix sums is standard [CITATION].', '1502.03942-2-31-1': 'But combining it with merging allows adaptive communication volume as described in Section [REF], which seems to be a new result.', '1502.03942-2-31-2': 'There are algorithms for minimizing communication cost of data distribution in arbitrary networks.', '1502.03942-2-31-3': 'However, these are much more expensive.', '1502.03942-2-31-4': 'Meyer auf der Heide et al. [CITATION] use maximum flow computations.', '1502.03942-2-31-5': 'They also give an algorithm for meshes, which needs more steps than the lower bound though.', '1502.03942-2-31-6': 'Solving the diffusion equations minimizes the sum of the squares of the data to be moved in order to achieve perfect load balance [CITATION].', '1502.03942-2-31-7': 'However, solving this equation is expensive and for parallel computing we are more interested in the bottleneck communication volume.', '1502.03942-2-32-0': '# Selection', '1502.03942-2-33-0': 'We consider the problem of identifying the elements of rank [MATH] from a set of [MATH] objects, distributed over [MATH] processors.', '1502.03942-2-33-1': 'First, we analyze the classical problem with unsorted input, before turning to locally sorted input in Sections [REF] and [REF].', '1502.03942-2-34-0': '## Unsorted Input', '1502.03942-2-35-0': 'Our first result is that using some minor adaptations and a more careful analysis, the parallel FR-algorithm from Section [REF] does not actually need randomly distributed data.', '1502.03942-2-36-0': 'Consider [MATH] elements distributed over [MATH] PE such that each PE holds [MATH] elements.', '1502.03942-2-36-1': 'The [MATH] globally smallest of these elements can be identified in expected time [EQUATION] (Outline) The algorithm from [CITATION] computes the pivots based on a sample [MATH] of size [MATH].', '1502.03942-2-36-2': 'There, this is easy since the objects are randomly distributed in all levels of recursion.', '1502.03942-2-36-3': 'Here we can make no such assumption and rather assume Bernoulli sampling with probability [MATH].', '1502.03942-2-36-4': 'Although this can be done in time proportional to the local sample size, we have to be careful since (in some level of recursion) the input might be so skewed that most samples come from the same processor.', '1502.03942-2-36-5': 'Since the result on sorting only works when the input data is uniformly distributed over the PEs, we have to account additional time for spreading the samples over the PEs.', '1502.03942-2-37-0': 'Let [MATH] denote the PE which maximizes [MATH] in the current level of recursion in the algorithm from Figure [REF].', '1502.03942-2-37-1': 'The total problem size shrinks by a factor of [MATH] in each level of recursion with high probability.', '1502.03942-2-37-2': 'Hence, after a constant number of recursion levels, [MATH] is [MATH] as well.', '1502.03942-2-37-3': 'Moreover, after [MATH] further levels of recursion, the problem is solved completely with high probability.', '1502.03942-2-37-4': 'Overall, the total number of recursion levels is [MATH].', '1502.03942-2-38-0': 'This already implies the claimed [MATH] bound on internal computation - [MATH] times [MATH] until [MATH] has size [MATH] and a geometrically shrinking amount of work after that.', '1502.03942-2-39-0': 'Of course, the same bound [MATH] also applies to the communication volume.', '1502.03942-2-39-1': 'However, we also know that even if all samples come from a single PE, the communication volume in a single level of recursion is [MATH].', '1502.03942-2-40-0': 'Finally, the number of startups can be limited to [MATH] per level of recursion by using a collective scatter operation on those PEs that have more than one sample object.', '1502.03942-2-40-1': 'This yields [MATH].', '1502.03942-2-41-0': 'If [MATH] and [MATH] are viewed as constants, the bound from Theorem [REF] reduces to [MATH].', '1502.03942-2-42-0': 'The bound from Theorem [REF] immediately reduces to [MATH] for constant [MATH] and [MATH].', '1502.03942-2-42-1': 'We can further simplify this by observing that when the [MATH] term dominates [MATH], then [MATH].', '1502.03942-2-43-0': '## Locally Sorted Input', '1502.03942-2-44-0': 'Selection on locally sorted input is easier than the unsorted problem from Section [REF] since we only have to consider the locally smallest objects and are able to locate keys in logarithmic time.', '1502.03942-2-44-1': 'Indeed, this problem has been studied as the multisequence selection problem [CITATION].', '1502.03942-2-45-0': 'In [CITATION], we propose a particularly simple and intuitive method based on an adaptation of the well-known quickselect algorithm [CITATION].', '1502.03942-2-45-1': 'For self-containedness, we also give that algorithm in Appendix [REF], adapting it to the need in the present paper.', '1502.03942-2-45-2': "This algorithm needs running time [MATH] and has been on the slides of Sanders' lecture on parallel algorithms since 2008 [CITATION].", '1502.03942-2-45-3': 'Algorithm [REF] can be viewed as a step backwards compared to our algorithm from Section [REF] as using a single random pivot forces us to do a deeper recursion.', '1502.03942-2-45-4': 'We do that because it makes it easy to tolerate unbalanced input sizes in the deeper recursion levels.', '1502.03942-2-45-5': 'However, it is possible to reduce the recursion depth to some extent by choosing pivots more carefully and by using multiple pivots at once.', '1502.03942-2-45-6': 'We study this below for a variant of the selection problem where we are flexible about the number [MATH] of objects to be selected.', '1502.03942-2-46-0': '[Flexible k, Locally Sorted Input]Flexible [MATH], Locally Sorted Input', '1502.03942-2-47-0': 'The [MATH] startups incurred in Section [REF] can be reduced to [MATH] if we are willing to give up some control over the number of objects that are actually returned.', '1502.03942-2-47-1': 'We now give two input parameters [MATH] and [MATH] and the algorithm returns the [MATH] smallest objects so that [MATH].', '1502.03942-2-47-2': 'We begin with a simple algorithm that runs in logarithmic time if [MATH] and then explain how to refine that for the case [MATH].', '1502.03942-2-48-0': 'The basic idea for the simple algorithm is to take a Bernoulli sample of the input using a success probability of [MATH], for [MATH].', '1502.03942-2-48-1': 'Then, the expected rank of the smallest sample object is [MATH], i.e., we have a truthful estimator for an object with the desired rank.', '1502.03942-2-48-2': 'Moreover, this object can be computed efficiently if working with locally sorted data: the local rank of the smallest local sample is geometrically distributed with parameter [MATH].', '1502.03942-2-48-3': 'Such a number can be generated in constant time.', '1502.03942-2-48-4': 'By computing the global minimum of these locally smallest samples, we can get the globally smallest sample [MATH] in time [MATH].', '1502.03942-2-48-5': 'We can also count the exact number [MATH] of input objects bounded by this estimate in time [MATH]-we locate [MATH] in each local data set in time [MATH] and then sum the found positions.', '1502.03942-2-48-6': 'If [MATH], we are done.', '1502.03942-2-48-7': 'Otherwise, we can use the acquired information as in any variant of quickselect.', '1502.03942-2-49-0': 'At least in the recursive calls, it can happen that [MATH] is close to the total input size [MATH].', '1502.03942-2-49-1': 'Then it is a better strategy to use a dual algorithm based on computing a global maximum of a Bernoulli sample.', '1502.03942-2-49-2': 'In Algorithm [REF] we give pseudocode for a combined algorithm dynamically choosing between these two cases.', '1502.03942-2-49-3': 'It is an interesting problem which value should be chosen for [MATH].', '1502.03942-2-49-4': 'The formula used in Algorithm [REF] maximizes the probability that [MATH].', '1502.03942-2-49-5': 'This value is close to the arithmetic mean when [MATH] but it is significantly smaller otherwise.', '1502.03942-2-49-6': 'The reason for this asymmetry is that larger sampling rates decrease the variance of the geometric distribution.', '1502.03942-2-50-0': 'If [MATH], then Algorithm amsSelect from Figure [REF] finds the [MATH] smallest elements with [MATH] in expected time [MATH].', '1502.03942-2-51-0': '(Outline) One level of recursion takes time [MATH] for collective communication operations (min, max, or sum reduction) and time [MATH] for locating the pivot [MATH].', '1502.03942-2-51-1': 'It remains to show that the expected recursion depth is constant.', '1502.03942-2-52-0': 'We actually analyze a weaker algorithm that keeps retrying with the same parameters rather than using recursion and that uses probe [MATH].', '1502.03942-2-52-1': 'We show that, nevertheless, there is a constant success probability (i.e., [MATH] with constant probability).', '1502.03942-2-52-2': 'The rank of [MATH] is geometrically distributed with parameter [MATH].', '1502.03942-2-52-3': 'The success probability becomes [EQUATION] which is a positive constant if [MATH].', '1502.03942-2-53-0': 'Multiple Concurrent Trials.', '1502.03942-2-53-1': 'The running time of Algorithm [REF] is dominated by the logarithmic number of startup overheads for the two reduction operations it uses.', '1502.03942-2-53-2': 'We can exploit that reductions can process long vectors using little additional time.', '1502.03942-2-53-3': 'The idea is to take [MATH] Bernoulli samples of the input and to compute [MATH] estimates for an object of rank [MATH].', '1502.03942-2-53-4': 'If any of these estimates turns out to have exact rank between [MATH] and [MATH], the recursion can be stopped.', '1502.03942-2-53-5': 'Otherwise, we solve a recursive instance consisting of those objects enclosed by the largest underestimate and the smallest overestimate found.', '1502.03942-2-54-0': 'If [MATH], an algorithm processing batches of [MATH] Bernoulli samples can be implemented to run in expected time [MATH].', '1502.03942-2-55-0': '(Outline) A single level of recursion runs in time [MATH].', '1502.03942-2-55-1': 'Analogous to the proof of Theorem [REF], it can be shown that the success probability is [MATH] for a single sample.', '1502.03942-2-55-2': 'This implies that the probability that any of the [MATH] independent samples is successful is constant.', '1502.03942-2-56-0': 'For example, setting [MATH], we obtain communication time [MATH] and [MATH] time for internal work.', '1502.03942-2-57-0': '# Bulk Parallel Priority Queues', '1502.03942-2-58-0': 'We build a global bulk-parallel priority queue from local sequential priority queues as in [CITATION], but never actually move elements.', '1502.03942-2-58-1': 'This immediately implies that insertions simply go to the local queue and thus require only [MATH] time without any communication.', '1502.03942-2-58-2': 'Of course, this complicates operation deleteMin[MATH].', '1502.03942-2-58-3': 'The number of elements to be retrieved from the individual local queues can vary arbitrarily and the set of elements stored locally is not at all representative for the global content of the queue.', '1502.03942-2-58-4': 'We can not even afford to individually remove the objects output by deleteMin[MATH] from their local queues.', '1502.03942-2-59-0': 'We therefore replace the ordinary priority queues used in [CITATION] by search tree data structures that support insertion, deletion, selection, ranking, splitting and concatenation of objects in logarithmic time (see also Section [REF]).', '1502.03942-2-59-1': 'To become independent of the actual tree size of up to [MATH], we furthermore augment the trees with two arrays storing the path to the smallest and largest object respectively.', '1502.03942-2-59-2': 'This way, all required operations can be implemented to run in time [MATH] rather than [MATH].', '1502.03942-2-60-0': 'Operation [MATH] now becomes very similar to the multi-sequence selection algorithms from Section [REF] and Appendix [REF].', '1502.03942-2-60-1': 'The only difference is that instead of sorted arrays, we are now working on search trees.', '1502.03942-2-60-2': 'This implies that selecting a local object with specified local rank (during sampling) now takes time [MATH] rather than constant time.', '1502.03942-2-60-3': 'However, asymptotically, this makes no difference since for any such selection step, we also perform a ranking step, which takes time [MATH] anyway in both representations.', '1502.03942-2-61-0': 'One way to implement the recursion in the selection algorithms is via splitting.', '1502.03942-2-61-1': 'Since the split operation is destructive, after returning from the recursion, we have to reassemble the previous state using concatenation.', '1502.03942-2-61-2': 'Another way that might be faster and simpler in practice is to represent a subsequence of [MATH] by [MATH] itself plus cursor information specifying rank and key of the first and last object of the subsequence.', '1502.03942-2-62-0': 'Now, we obtain the following by applying our results on selection from Section [REF].', '1502.03942-2-63-0': 'Operation [MATH] can be implemented to run in the following expected times.', '1502.03942-2-63-1': 'With fixed batch size [MATH], expected time [MATH] suffices.', '1502.03942-2-63-2': 'For flexible batch size in [MATH], where [MATH], we need expected time [MATH].', '1502.03942-2-63-3': 'If [MATH], expected time [MATH] is sufficient.', '1502.03942-2-64-0': 'Note that flexible batch sizes might be adequate for many applications.', '1502.03942-2-64-1': 'For example, the parallel branch-and-bound algorithm from [CITATION] can easily be adapted: In iteration [MATH] of its main loop, it deletes the smallest [MATH] elements (tree nodes) from the queue, expands these nodes in parallel, and inserts newly generated elements (child nodes of the processed nodes).', '1502.03942-2-64-2': 'Let [MATH] denote the total number of nodes expanded by the parallel algorithm.', '1502.03942-2-64-3': 'One can easily generalize the proof from [CITATION] to show that [MATH] where [MATH] is the number of nodes expanded by a sequential best first algorithm and [MATH] is the length of the path from the root to the optimal solution.', '1502.03942-2-64-4': 'Also note that a typical branch-and-bound computation will insert significantly more nodes than it removes-the remaining queue is discarded after the optimal solutions are found.', '1502.03942-2-64-5': 'Hence, the local insertions of our communication efficient queue are a big advantage over previous algorithms, which move all nodes [CITATION].', '1502.03942-2-65-0': '[Multicriteria Top-k]Multicriteria Top-[MATH]', '1502.03942-2-66-0': 'In the sequential setting, we consider the following problem: Consider [MATH] lists [MATH] of scores that are sorted in decreasing order.', '1502.03942-2-66-1': 'Overall relevance of an object is determined by a scoring function [MATH] that is monotonous in all its parameters.', '1502.03942-2-66-2': 'For example, there could be [MATH] keywords for a disjunctive query to a fulltext search engine, and for each pair of a keyword and an object, it is known how relevant this keyword is for this object.', '1502.03942-2-66-3': "Many algorithms used for this setting are based on Fagin's threshold algorithm [CITATION].", '1502.03942-2-66-4': 'The lists are partially scanned and the algorithm maintains lower bounds for the relevance of the scanned objects as well as upper bounds for the unscanned objects.', '1502.03942-2-66-5': 'Bounds for a scanned object [MATH] can be tightened by retrieving a score value for a dimension for which [MATH] has not been scanned yet.', '1502.03942-2-66-6': 'These random accesses are more expensive than scanning, in particular if the lists are stored on disk.', '1502.03942-2-66-7': 'Once the top-[MATH] scanned objects have better lower bounds than the best upper bound of an unscanned object, no more scanning is necessary.', '1502.03942-2-66-8': 'For determining the exact relevance of the top-[MATH] scanned objects, further random accesses may be required.', '1502.03942-2-66-9': 'Various strategies for scanning and random access yield a variety of variants of the threshold algorithm [CITATION].', '1502.03942-2-67-0': 'The original threshold algorithm works as follows [CITATION]: In each of [MATH] iterations of the main loop, scan one object from each list and determine its exact score using random accesses.', '1502.03942-2-67-1': 'Let [MATH] denote the smallest score of a scanned object in [MATH].', '1502.03942-2-67-2': 'Once at least [MATH] scanned objects have score at least [MATH], stop, and output them.', '1502.03942-2-68-0': 'We consider a distributed setting where each PE has a subset of the objects and [MATH] sorted lists ranking its locally present objects.', '1502.03942-2-68-1': 'We describe communication efficient distributed algorithms that approximate the original threshold algorithm (TA) [CITATION].', '1502.03942-2-68-2': 'First, we give a simple algorithm assuming random data distribution of the input objects (RDTA) and then describe a more complex algorithm for arbitrary data distribution (DTA).', '1502.03942-2-69-0': 'Random Data Distribution.', '1502.03942-2-69-1': 'Since the data placement is independent of the relevance of the objects, the top-[MATH] objects are randomly distributed over the PEs.', '1502.03942-2-69-2': 'Well known balls-into-bins bounds give us tight high probability bounds on the maximal number [MATH] of top-[MATH] objects on each PE [CITATION].', '1502.03942-2-69-3': 'Here, we work with the simple bound [MATH].', '1502.03942-2-69-4': 'The RDTA algorithm simply runs TA locally to retrieve the [MATH] locally most relevant objects on each PE as result candidates.', '1502.03942-2-69-5': 'It then computes a global threshold as the maximum of the local thresholds and verifies whether at least [MATH] candidate objects are above this global threshold.', '1502.03942-2-69-6': 'In the positive case, the [MATH] most relevant candidates are found using the selection algorithm from [CITATION].', '1502.03942-2-69-7': 'Otherwise, [MATH] is increased and the algorithm is restarted.', '1502.03942-2-69-8': 'In these subsequent iterations, PEs whose local threshold is worse than the relevance of the [MATH]-th best element seen so far do not need to continue scanning.', '1502.03942-2-69-9': 'Hence, we can also trade less local work for more rounds of communication by deliberately working with an insufficiently small value of [MATH].', '1502.03942-2-70-0': 'Arbitrary Data Distribution.', '1502.03942-2-71-0': 'We give pseudocode for Algorithm DTA in Figure [REF].', '1502.03942-2-72-0': 'Algorithm DTA requires expected time [MATH] to identify a set of at most [MATH] objects that contains the set of [MATH] objects scanned by TA.', '1502.03942-2-73-0': '(Outline) Algorithm DTA "guesses" the number [MATH] of list rows scanned by TA using exponential search.', '1502.03942-2-73-1': 'This yields [MATH] rounds of DTA.In each round, the approximate multisequence selection algorithm from Section [REF] is used to approximate the globally [MATH]-th largest score [MATH] in each list.', '1502.03942-2-73-2': 'Define [MATH] as the prefix of [MATH] containing all objects of score at least [MATH].', '1502.03942-2-73-3': 'This takes time [MATH] in expectation by Theorem [REF].', '1502.03942-2-73-4': 'Accumulating the searches for all [MATH] lists yields [MATH] time.', '1502.03942-2-73-5': 'Call an object selected by the selection algorithm a hit if its relevance is above the threshold [MATH].', '1502.03942-2-73-6': 'DTA estimates the number of hits using sampling.', '1502.03942-2-73-7': 'For each PE and list [MATH] separately, [MATH] objects are sampled from [MATH].', '1502.03942-2-73-8': 'Each sample takes time [MATH] for evaluating [MATH].', '1502.03942-2-73-9': 'Multiplying this with [MATH] lists and [MATH] iterations gives a work bound of [MATH].', '1502.03942-2-73-10': 'Note that this is sublinear in the work done by the sequential algorithm which takes time [MATH] to scan the [MATH] most important objects in every list.', '1502.03942-2-73-11': 'To eliminate bias for objects selected in multiple lists [MATH], DTA only counts an object [MATH] if it is sampled in the first list that contains it.', '1502.03942-2-73-12': 'Otherwise, [MATH] is rejected.', '1502.03942-2-73-13': 'DTA also counts the number of rejected samples [MATH].', '1502.03942-2-73-14': 'Let [MATH] denote the number of nonrejected hits in the sample.', '1502.03942-2-73-15': 'Then, [MATH] is a truthful estimate for the length of the list with eliminated duplicates and [MATH] is a truthful estimate for the number of hits for the considered list and PE.Summing these [MATH] values using a reduction operation yields a truthful estimate for the overall number of hits.', '1502.03942-2-73-16': 'DTA stops once this estimate is large enough such that with high probability the actual number of hits is at least [MATH].', '1502.03942-2-73-17': 'The output of DTA are the prefixes [MATH] of the lists on each PE, the union of which contains the [MATH] most relevant objects with high probability.', '1502.03942-2-73-18': 'It also outputs the threshold [MATH].', '1502.03942-2-73-19': 'In total, all of this combined takes expected time [MATH].', '1502.03942-2-74-0': 'Actually computing the [MATH] most frequent objects amounts to scanning the result lists to find all hits and, if desired, running a selection algorithm to identify the [MATH] most relevant among them.', '1502.03942-2-74-1': 'The scanning step may involve some load imbalance, as in the worst case, all hits are concentrated on a single PE.', '1502.03942-2-74-2': 'This seems unavoidable unless one requires random distribution of the objects.', '1502.03942-2-74-3': 'However, in practice it may be possible to equalize the imbalance over a large number of concurrently executed queries.', '1502.03942-2-75-0': 'Refinements.', '1502.03942-2-75-1': 'We can further reduce the latency of DTA by trying several values of [MATH] in each iteration of algorithm DTA.Since this involves access to only few objects, the overhead in internal work will be limited.', '1502.03942-2-75-2': 'In practice, it may also be possible to make good guesses on [MATH] based on previous executions of the algorithm.', '1502.03942-2-76-0': '[Top-k Most Frequent Objects]Top-[MATH] Most Frequent Objects', '1502.03942-2-77-0': 'We describe two probably approximately correct (PAC) algorithms to compute the top-[MATH] most frequent objects of a multiset [MATH] with [MATH], followed by a probably exactly correct (PEC) algorithm for suitable inputs.', '1502.03942-2-77-1': 'Sublinear communication is achieved by transmitting only a small random sample of the input.', '1502.03942-2-77-2': 'For bound readability, we assume that [MATH] is distributed over the [MATH] PEs so that none has more than [MATH] objects.', '1502.03942-2-77-3': "This is not restricting, as the algorithms' running times scale linearly with the maximum fraction of the input concentrated at one PE.", '1502.03942-2-78-0': "We express the algorithms' error relative to the total input size.", '1502.03942-2-78-1': 'This is reasonable-consider a large input where [MATH] objects occur twice, and all others once.', '1502.03942-2-78-2': "If we expressed the error relative to the objects' frequencies, this input would be infeasible without communicating all elements.", '1502.03942-2-78-3': "Thus, we refer to the algorithms' relative error as [MATH], defined so that the absolute error [MATH] is the count of the most frequent object that was not output minus that of the least frequent object that was output, or 0 if the result was exact.", '1502.03942-2-78-4': 'Let [MATH] limit the probability that the algorithms exceeds bound [MATH], i.e. [MATH].', '1502.03942-2-78-5': 'We refer to the result as an [MATH]-approximation.', '1502.03942-2-79-0': '## Basic Approximation Algorithm', '1502.03942-2-80-0': 'First, we take a Bernoulli sample of the input.', '1502.03942-2-80-1': 'Sampling is done locally.', '1502.03942-2-80-2': 'The frequencies of the sampled objects are counted using distributed hashing-a local object count with key [MATH] is sent to PE [MATH] for a hash function [MATH] that we here expect to behave like a random function.', '1502.03942-2-80-3': 'We then select the [MATH] most frequently sampled objects using the unsorted selection algorithm from Section [REF].', '1502.03942-2-80-4': 'An example is illustrated in Figure [REF].', '1502.03942-2-81-0': 'Algorithm PACcan be implemented to compute an [MATH]-approximation of the top-[MATH] most frequent objects in expected time [MATH].', '1502.03942-2-82-0': 'For sampling probability [MATH], Algorithm PACruns in [MATH] time in expectation.', '1502.03942-2-83-0': '(Outline) Bernoulli sampling is done in expected time [MATH] by generating skip values with a geometric distribution using success probability [MATH].', '1502.03942-2-83-1': 'Since the number of sample elements in a Bernoulli sample is a random variable, so is the running time.', '1502.03942-2-83-2': 'To count the sampled objects, each PE aggregates its local samples in a hash table, counting the occurrence of each sample object during the sampling process.', '1502.03942-2-83-3': 'It inserts its share of the sample into a distributed hash table [CITATION] whose hash function we assume to behave like a random function, thus distributing the objects randomly among the PEs.', '1502.03942-2-83-4': 'The elements are communicated using indirect delivery to maintain logarithmic latency.', '1502.03942-2-83-5': 'This requires [MATH] time in expectation.', '1502.03942-2-83-6': 'To minimize worst-case communication volume, the incoming sample counts are merged with a hash table in each step of the reduction.', '1502.03942-2-83-7': 'Thus, each PE receives at most one message per object assigned to it by the hash function.', '1502.03942-2-84-0': 'From this hash table, we select the object with rank [MATH] using Algorithm [REF] in expected time [MATH].', '1502.03942-2-84-1': 'This pivot is broadcast to all PEs, which then determine their subset of at least as frequent sample objects in expected time [MATH].', '1502.03942-2-84-2': 'These elements are returned.', '1502.03942-2-84-3': 'Overall, the claimed time complexity follows using the estimate [MATH] and [MATH] in order to simplify the [MATH]-term.', '1502.03942-2-85-0': 'Let [MATH] be the sampling probability.', '1502.03942-2-85-1': 'Then, Algorithm PAC error [MATH] exceeds a value of [MATH] with probability [EQUATION].', '1502.03942-2-86-0': "We bound the error probability as follows: the probability that the error [MATH] exceeds some value [MATH] is at most [MATH] times the probability that a single object's value estimate deviates from its true value by more than [MATH] in either direction.", '1502.03942-2-86-1': 'This probability can be bounded using Chernoff bounds.', '1502.03942-2-86-2': 'We denote the count of element [MATH] in the input by [MATH] and in the sample by [MATH].', '1502.03942-2-86-3': 'Further, let [MATH] be the probability that the error for element [MATH] exceeds [MATH] in either direction.', '1502.03942-2-86-4': 'Let [MATH], and observe that [MATH].', '1502.03942-2-86-5': 'Using Equations [REF] and [REF] with [MATH], [MATH], and [MATH], we obtain: [EQUATION]', '1502.03942-2-86-6': 'This leaves us with the most frequent [MATH] elements, whose counts can be bounded as [MATH].', '1502.03942-2-86-7': 'As overestimating them is not a concern, we apply the Chernoff bound in Equation [REF] and obtain [MATH].', '1502.03942-2-86-8': 'In sum, all error probabilities add up to the claimed value.', '1502.03942-2-87-0': 'We bound this result by an error probability [MATH].', '1502.03942-2-87-1': 'This allows us to calculate the minimum required sample size given [MATH] and [MATH].', '1502.03942-2-87-2': 'Solving the above equation for [MATH] yields [EQUATION] which is dominated by the latter term in most cases and yields [MATH] for the expected sample size.', '1502.03942-2-88-0': '(Theorem [REF]) Equation [REF] yields [MATH].', '1502.03942-2-88-1': 'The claimed running time bound then follows from Lemma [REF].', '1502.03942-2-89-0': 'Note that if we can afford to aggregate the local input, we can also use the Sum Aggregation algorithm from Section [REF] and associate a value of 1 with each object.', '1502.03942-2-90-0': '## Increasing Communication Efficiency', '1502.03942-2-91-0': 'Sample sizes proportional to [MATH] quickly become unacceptably large as [MATH] decreases.', '1502.03942-2-91-1': "To remedy this, we iterate over the local input a second time and count the most frequently sampled objects' occurrences exactly.", '1502.03942-2-91-2': 'This allows us to reduce the sample size and improve communication efficiency at the cost of increased local computation.', '1502.03942-2-91-3': 'We call this Algorithm EC for exact counting.', '1502.03942-2-91-4': 'Again, we begin by taking a Bernoulli sample.', '1502.03942-2-91-5': 'Then we find the [MATH] globally most frequent objects in the sample using the unsorted selection algorithm from Section [REF], and count their frequency in the overall input exactly.', '1502.03942-2-91-6': 'The identity of these objects is broadcast to all PEs using an all-gather (gossiping, all-to-all broadcast) collective communication operation.', '1502.03942-2-91-7': 'After local counting, a global reduction sums up the local counts to exact global values.', '1502.03942-2-91-8': 'The [MATH] most frequent of these are then returned.', '1502.03942-2-92-0': "When counting the [MATH] most frequently sampled objects' occurrences exactly, a sample size of [MATH] suffices to ensure that the result of Algorithm ECis an [MATH]-approximation of the top-[MATH] most frequent objects.", '1502.03942-2-93-0': 'With the given error bounds, we can derive the required sampling probability [MATH] similar to Lemma [REF].', '1502.03942-2-93-1': 'However, we need not consider overestimation of the [MATH] most frequent objects, as their counts are known exactly.', '1502.03942-2-93-2': 'We can also allow the full margin of error towards underestimating their frequency ([MATH] instead of [MATH]) and can ignore overestimation.', '1502.03942-2-93-3': 'This way, we obtain a total expected sample size of [MATH].', '1502.03942-2-94-0': 'We can now calculate the value of [MATH] that minimizes the total communication volume and obtain [MATH].', '1502.03942-2-94-1': 'Substituting this into the sample size equation of Lemma [REF] and adapting the running time bound of Lemma [REF] then yields the following theorem.', '1502.03942-2-95-0': 'Algorithm ECcan be implemented to compute an [MATH]-approximation of the top-[MATH] most frequent objects in [MATH] time in expectation.', '1502.03942-2-96-0': 'Sampling and hashing are done as in Algorithm PAC (Section [REF]).', '1502.03942-2-96-1': 'We select the object with rank [MATH] as a pivot.', '1502.03942-2-96-2': 'This requires expected time [MATH] using Algorithm [REF].', '1502.03942-2-96-3': 'The pivot is broadcast to all PEs, which then determine their subset of at least as frequent sample objects using [MATH] time in expectation.', '1502.03942-2-96-4': 'Next, these [MATH] most frequently sampled objects are distributed to all PEs using an all-gather operation in time [MATH].', '1502.03942-2-96-5': "Now, the PEs count the received objects' occurrences in their local input, which takes [MATH] time.", '1502.03942-2-96-6': 'These counts are summed up using a vector-valued reduction, again requiring [MATH] time.', '1502.03942-2-96-7': 'We then apply Algorithm [REF] a second time to determine the [MATH] most frequent of these objects.', '1502.03942-2-96-8': 'Overall, the claimed time complexity follows by substituting [MATH] for [MATH] in the sampling probability from Lemma [REF].', '1502.03942-2-97-0': 'Substituting [MATH] from before then yields a total required sample size of [MATH] for Algorithm EC.', '1502.03942-2-97-1': 'Note that this term grows with [MATH] instead of [MATH], reducing per-PE communication volume to [MATH] words.', '1502.03942-2-98-0': 'To continue the example from Figure [REF], we may set [MATH].', '1502.03942-2-98-1': 'Then, the [MATH] most frequently sampled objects [MATH] with [MATH] occurrences, respectively, will be counted exactly.', '1502.03942-2-98-2': 'The result would now be correct.', '1502.03942-2-99-0': 'Note that Algorithm ECcan be less communication efficient than Algorithm PACif [MATH] is large, i.e. the result is very approximate.', '1502.03942-2-99-1': 'Then, [MATH] can be prohibitively large, and the necessity to communicate the identity of the objects to be counted exactly, requiring time [MATH], can cause a loss in communication efficiency.', '1502.03942-2-100-0': '## Probably Exactly Correct Algorithm', '1502.03942-2-101-0': 'If any significant gap exists in the frequency distribution of the objects (see Figure [REF] for an example), we perform exact counting on all likely relevant objects, determined from their sample count.', '1502.03942-2-101-1': 'Thus, choose [MATH] to ensure that the top-[MATH] most frequent objects in the input are among the top-[MATH] most frequent objects in the sample with probability at least [MATH].', '1502.03942-2-102-0': 'A probably exactly correct (PEC) algorithm to compute the top-[MATH] most frequent objects of a multiset whose frequency distribution is sufficiently sloped can therefore be formulated as follows.', '1502.03942-2-102-1': 'Take a small sample with sampling probability [MATH], the value of which we will consider later.', '1502.03942-2-102-2': 'From this small sample, we deduce the required value of [MATH] to fulfill the above requirements.', '1502.03942-2-102-3': 'Now, we apply Algorithm ECusing this value of [MATH].', '1502.03942-2-102-4': 'Let [MATH] be the object of rank [MATH] in the input, and [MATH] that of rank [MATH] in the small sample.', '1502.03942-2-103-0': 'It suffices to choose [MATH] in such a way that [MATH] to ensure correctness of Algorithm PECresult with probability at least [MATH].', '1502.03942-2-104-0': 'We use the Chernoff bound from Equation [REF] to bound the probability of a top-[MATH] object not being among the top-[MATH] objects of the sample.', '1502.03942-2-104-1': 'Define [MATH] as the number of samples of the object with input rank [MATH] in the first sample, and [MATH] to be the exact number of occurrences in the input of the object with rank [MATH] in the first sample.', '1502.03942-2-104-2': 'Using [MATH], [MATH], and [MATH], we obtain [EQUATION]', '1502.03942-2-104-3': "We bound this value by the algorithm's error probability [MATH] and solve for [MATH], which yields the claimed value.", '1502.03942-2-105-0': 'This only works for sufficiently sloped distributions, as otherwise [MATH] would be necessary.', '1502.03942-2-105-1': 'Furthermore, it is clear that the choice of [MATH] presents a trade-off between the time and communication spent on the first sample and the exactness of [MATH], which we have to estimate more conservatively if the first sample is small.', '1502.03942-2-105-2': 'This is due to less precise estimations of [MATH] if [MATH] is small.', '1502.03942-2-105-3': 'To keep things simple, we can choose a relative error bound [MATH] and use the sample size from our PAC algorithm of Theorem [REF].', '1502.03942-2-105-4': 'The value of [MATH]-and thus [MATH]-that minimizes the communication volume depends on the distribution of the input data.', '1502.03942-2-106-0': 'If the value of [MATH] computed from Lemma [REF] satisfies [MATH], then Algorithm PECrequires time asymptotically equal to the sum of the running times of algorithms PACand ECfrom Theorems [REF] and [REF].', '1502.03942-2-107-0': '(Outline) In the first sampling step, we are free to choose an arbitrary relative error tolerance [MATH].', '1502.03942-2-107-1': 'The running time of this stage is [MATH] by Theorem [REF].', '1502.03942-2-107-2': 'We then estimate [MATH] by substituting the high-probability bound [MATH] (whp) for its expected value in Lemma [REF] (note that as [MATH] increases with growing sample size and thus grows with falling [MATH], the precision of this bound increases).', '1502.03942-2-107-3': 'In the second stage, we can calculate the required value of [MATH] from [MATH] by solving the expression for [MATH] in the proof of Theorem [REF] for [MATH], and obtain [MATH].', '1502.03942-2-107-4': "Since [MATH], the second stage's running time is as in Theorem [REF].", '1502.03942-2-107-5': 'In sum, the algorithm requires the claimed running time.', '1502.03942-2-108-0': "Zipf's Law In its simplest form, Zipf's Law states that the frequency of an object from a multiset [MATH] with [MATH] is inversely proportional to its rank among the objects ordered by frequency.", '1502.03942-2-108-1': 'Here, we consider the general case with exponent parameter [MATH], i.e. [MATH], where [MATH] is the [MATH]-th generalized harmonic number.', '1502.03942-2-109-0': "For inputs distributed according to Zipf's law with exponent [MATH], a sample size of [MATH] is sufficient to compute a probably exactly correct result.", '1502.03942-2-109-1': 'Algorithm PECthen requires expected time [MATH] to compute the [MATH] most frequent objects with probability at least [MATH].', '1502.03942-2-110-0': 'Knowing the value distribution, we do not need to take the first sample.', '1502.03942-2-110-1': 'Instead, we can calculate the expected value of [MATH] directly from the proof of Lemma [REF] and obtain [EQUATION].', '1502.03942-2-110-2': 'This immediately yields [MATH], and we choose [MATH], i.e. twice the required minimum value.', '1502.03942-2-110-3': 'This gives us the claimed sample size.', '1502.03942-2-110-4': 'Now, we obtain [MATH].', '1502.03942-2-110-5': 'In particular, [MATH] is only a constant factor away from [MATH].', '1502.03942-2-110-6': 'Plugging the above sampling probability into the running time formula for our algorithm using exact counting, we obtain the exact top-[MATH] most frequent objects with probability at least [MATH] in the claimed running time.', '1502.03942-2-111-0': 'Note that the number of frequent objects decreases sharply with [MATH], as the [MATH]-th most frequent one has a relative frequency of only [MATH].', '1502.03942-2-111-1': 'The [MATH] term in the communication volume is thus small in practice, and, in fact, unavoidable in a sampling-based algorithm.', '1502.03942-2-111-2': "(One can easily verify this claim by observing that this factor is the reciprocal of the [MATH]-th most frequent object's relative frequency.", '1502.03942-2-111-3': 'It is clear that this object needs at least one occurrence in the sample for the algorithm to be able to find it, and that the sample size must thus scale with [MATH].)', '1502.03942-2-112-0': '## Refinements', '1502.03942-2-113-0': 'When implementing such an algorithm, there are a number of considerations to be taken into account to achieve optimal performance.', '1502.03942-2-113-1': 'Perhaps most importantly, one should apply local aggregation when inserting the sample into the distributed hash table to reduce the amount of information that needs to be communicated in practice.', '1502.03942-2-113-2': 'We now discuss other potential improvements.', '1502.03942-2-114-0': 'Choice of [MATH].', '1502.03942-2-114-1': "In practice, the choice of [MATH] in Section [REF] depends on the communication channel's characteristics [MATH], and, to a lesser extent, [MATH], in addition to the problem parameters.", '1502.03942-2-114-2': 'Thus, an optimized implementation should take them into account when determining the number of objects to be counted exactly.', '1502.03942-2-115-0': 'Adaptive Two-Pass Sampling (Outline).', '1502.03942-2-115-1': 'The objectives of the basic PAC algorithm and its variant using exact counting could be unified as follows: we sample in two passes.', '1502.03942-2-115-2': 'In the first pass, we use a small sample size [MATH] to determine the nature of the input distribution.', '1502.03942-2-115-3': 'From the insights gained from this first sample, we compute a larger sample size [MATH].', '1502.03942-2-115-4': 'We then determine and return the [MATH] most frequent objects of this second sample.', '1502.03942-2-116-0': 'Additionally, we can further refine this algorithm if we can already tell from the first sample that with high probability, there is no slope.', '1502.03942-2-116-1': 'If the absolute counts of the objects in the sample are large enough to return the [MATH] most frequent objects in the sample with confidence, then taking a second sample would be of little benefit and we can return the [MATH] most frequent objects from the first sample.', '1502.03942-2-117-0': 'Using Distributed Bloom Filters.', '1502.03942-2-117-1': 'Communication efficiency of the algorithm using exact counting could be improved further by counting sample elements with a distributed single-shot bloom filter (dSBF) [CITATION] instead of a distributed hash table.', '1502.03942-2-117-2': 'We transmit their hash values and locally aggregated counts.', '1502.03942-2-117-3': 'As multiple keys might be assigned the same hash value, we need to determine the element of rank [MATH] instead of [MATH], for some safety margin [MATH].', '1502.03942-2-117-4': 'We request the keys of all elements with higher rank, and replace the (hash, value) pairs with (key, value) pairs, splitting them where hash collisions occurred.', '1502.03942-2-117-5': 'We now determine the element of rank [MATH] on the [MATH] elements.', '1502.03942-2-117-6': 'If an element whose rank is at most [MATH] was part of the original [MATH] elements, we are finished.', '1502.03942-2-117-7': 'Otherwise, we have to increase [MATH] to determine the missing elements.', '1502.03942-2-117-8': 'Observe that if the frequent objects are dominated by hash collisions, this implies that the input distribution is flat and there exist a large number of nearly equally frequent elements.', '1502.03942-2-117-9': 'Thus, we may not need to count additional elements in this case.', '1502.03942-2-118-0': '[Top-k Sum Aggregation]Top-[MATH] Sum Aggregation', '1502.03942-2-119-0': 'Generalizing from Section [REF], we now consider an input multiset of keys with associated non-negative counts, and ask for the [MATH] keys whose counts have the largest sums.', '1502.03942-2-119-1': 'Again, the input [MATH] is distributed over the [MATH] PEs so that no PE has more than [MATH] objects.', '1502.03942-2-119-2': "Define [MATH] as the sum of all counts, and let no PE's local sum exceed [MATH].", '1502.03942-2-119-3': 'We additionally assume that the local input and a key-aggregation thereof-e.g. a hash table mapping keys to their local sums-fit into RAM at every PE.', '1502.03942-2-120-0': 'It is easy to see that except for the sampling process, the algorithms of Section [REF] carry over directly, but a different approach is required in the analysis.', '1502.03942-2-121-0': '## Sampling', '1502.03942-2-122-0': 'Let [MATH] be the desired sample size, and define [MATH] as the expected count required to yield a sample.', '1502.03942-2-122-1': 'When sampling an object [MATH], its expected sample count is thus [MATH].', '1502.03942-2-122-2': 'To retain constant time per object, we add [MATH] samples directly, and one additional sample with probability [MATH] using a Bernoulli trial.', '1502.03942-2-122-3': 'We can again use a geometric distribution to reduce the number of calls to the random number generator.', '1502.03942-2-123-0': 'To improve accuracy and speed up exact counting, we aggregate the local input in a hash table, and sample the aggregate counts.', '1502.03942-2-123-1': "This allows us to analyze the algorithms' error independent of the input objects' distribution.", '1502.03942-2-123-2': 'A direct consequence is that for each key and PE, the number of samples deviates from its expected value by at most [MATH], and the overall deviation per key [MATH] is at most [MATH].', '1502.03942-2-124-0': '## Probably Approximately Correct Algorithms', '1502.03942-2-125-0': 'We can compute an [MATH]-approximation of the top-[MATH] highest summing items in expected time [MATH].', '1502.03942-2-126-0': 'Sampling is done using local aggregation as described in Section [REF].', '1502.03942-2-126-1': 'From then on, we proceed exactly as in Algorithm PACfrom Section [REF].', '1502.03942-2-127-0': "The part of an element's sample count that is actually determined by sampling is the sum of up to [MATH] Bernoulli trials [MATH] with differing success probabilities.", '1502.03942-2-127-1': "Therefore, its expected value [MATH] is the sum of the probabilities, and we can use Hoeffding's inequality to bound the probability of significant deviations from this value.", '1502.03942-2-127-2': 'Let [MATH].', '1502.03942-2-127-3': 'Then, [EQUATION]', '1502.03942-2-127-4': 'We now use this to bound the likelihood that an object has been very mis-sampled.', '1502.03942-2-127-5': 'Consider an element [MATH] with exact sum [MATH] and sample sum [MATH].', '1502.03942-2-127-6': 'For some threshold [MATH], consider the element mis-sampled if [MATH], i.e. its estimated sum deviates from the true value by more than [MATH] in either direction.', '1502.03942-2-127-7': 'Thus, we substitute [MATH] into Equation ([REF]) and bound the result by [MATH] to account for all elements.', '1502.03942-2-127-8': 'Solving for [MATH], we obtain [MATH].', '1502.03942-2-128-0': "In total, we require time [MATH] for sampling, [MATH] for insertion (as no PE's local sum exceeds [MATH] O(s/p)) samples), and [MATH] for selection.", '1502.03942-2-128-1': 'We then obtain the claimed bound as sum of the components.', '1502.03942-2-129-0': 'As in Section [REF], we can use exact summation to obtain a more precise answer.', '1502.03942-2-129-1': 'We do not go into details here, as the procedure is nearly identical.', '1502.03942-2-129-2': 'The main difference is that a lookup in the local aggregation result now suffices to obtain exact local sums without requiring consultation of the input.', '1502.03942-2-130-0': '## Probably Exactly Correct Algorithm', '1502.03942-2-131-0': 'If there is a gap in the input distribution, we can make the algorithm probably exactly correct using exact counting.', '1502.03942-2-132-0': 'If a [MATH] so that [MATH] exists, we can compute the top-[MATH] highest summing objects with probability at least [MATH] in expected time [MATH]', '1502.03942-2-133-0': 'We use a two-pass sampling approach to avoid requiring prior knowledge of the input distribution.', '1502.03942-2-133-1': 'In a sufficiently sized second sample, we can use the fact that sample counts deviate from their expected values by at most [MATH] to obtain a probably exactly correct result.', '1502.03942-2-133-2': 'We accomplish this by counting all [MATH] objects with at least [MATH] samples in the second sample exactly.', '1502.03942-2-134-0': 'Note that we can also make the algorithm exactly correct by leveraging the maximum sampling error per object.', '1502.03942-2-134-1': 'With a larger second sample, exactly counting all objects with at least [MATH] samples yields an exactly correct result without changing the asymptotic bounds.', '1502.03942-2-135-0': '# Data Redistribution', '1502.03942-2-136-0': 'Let [MATH] denote the number of data objects present at PE [MATH].', '1502.03942-2-136-1': 'Let [MATH].', '1502.03942-2-136-2': 'We want to redistribute the data such that afterwards each PE has at most [MATH] objects and such that PEs with more than [MATH] objects only send data (at most [MATH] objects) and PEs with at most [MATH] objects only receive data (at most [MATH] objects).', '1502.03942-2-136-3': 'We split the PEs into separately numbered groups of senders [MATH] and receivers [MATH].', '1502.03942-2-136-4': 'We also compute the deficit [MATH] on receivers and the surplus [MATH] on senders.', '1502.03942-2-136-5': 'Then we compute the prefix sums [MATH] and [MATH] of these sequences (i.e., [MATH] and [MATH]).', '1502.03942-2-136-6': 'Effectively, [MATH] enumerates the empty slots able to receive objects and [MATH] enumerates the elements to be moved.', '1502.03942-2-136-7': 'Now we match receiving slots and elements to be moved by merging the sequences [MATH] and [MATH].', '1502.03942-2-136-8': "This is possible in time [MATH] using Batcher's parallel merging algorithm [CITATION].", '1502.03942-2-136-9': 'A subsequence of the form [MATH] indicates that sending PEs [MATH] move their surplus to receiving PE [MATH] (where sending PE [MATH] only moves its items numbered [MATH]).', '1502.03942-2-136-10': 'This is a gather operation.', '1502.03942-2-136-11': 'Sending PE [MATH] moves its remaining elements to receiving PEs [MATH].', '1502.03942-2-136-12': 'This is a scatter operation.', '1502.03942-2-136-13': 'These segments of PE numbers can be determined using segmented prefix operations [CITATION].', '1502.03942-2-136-14': 'Overall, this can be implemented to run in time [MATH].', '1502.03942-2-136-15': 'Even though this operation cannot remove worst case bottlenecks, it can significantly reduce network traffic.', '1502.03942-2-137-0': '# Experiments', '1502.03942-2-138-0': 'We now present an experimental evaluation of the unsorted selection from Section [REF] and the top-[MATH] most frequent objects algorithms from Section [REF].', '1502.03942-2-139-0': 'Experimental Setup.', '1502.03942-2-139-1': 'All algorithms were implemented in C++11 using OpenMPI 1.8 and Boost.MPI 1.59 for inter-process communication.', '1502.03942-2-139-2': "Additionally, Intel's Math Kernel Library in version 11.2 was used for random number generation.", '1502.03942-2-139-3': 'All code was compiled with the clang++ compiler in version 3.7 using optimization level -Ofast and instruction set specification -march=sandybridge.', '1502.03942-2-139-4': 'The experiments were conducted on InstitutsCluster II at Karlsruhe Institute of Technology, a distributed system consisting of 480 computation nodes, of which 128 were available to us.', '1502.03942-2-139-5': 'Each node is equipped with two Intel Xeon E5-2670 processors for a total of 16 cores with a nominal frequency of 2.6GHz, and 64GiB of main memory.', '1502.03942-2-139-6': 'In total, 2048 cores were available to us.', '1502.03942-2-139-7': 'An Infiniband 4X QDR interconnect provides networking between the nodes.', '1502.03942-2-140-0': 'Methodology.', '1502.03942-2-140-1': 'We run weak scaling benchmarks, which show how wall-time develops for fixed per-PE problem size [MATH] as [MATH] increases.', '1502.03942-2-140-2': 'We consider [MATH] to [MATH] PEs, doubling [MATH] in each step.', '1502.03942-2-140-3': 'Each PE is mapped to one physical core in the cluster.', '1502.03942-2-141-0': "Zipf's Lawstates that the frequency of an object from a multiset [MATH] with [MATH] is inversely proportional to its rank among the objects ordered by frequency.", '1502.03942-2-141-1': 'Here, we consider the general case with exponent parameter [MATH], i.e. [MATH], where [MATH] is the [MATH]-th generalized harmonic number.', '1502.03942-2-142-0': '## Unsorted Selection', '1502.03942-2-143-0': 'Input Generation.', '1502.03942-2-143-1': 'We select values from the high tail of Zipf distributions.', '1502.03942-2-143-2': 'Per PE, we consider [MATH], [MATH], and [MATH] integer elements.', '1502.03942-2-143-3': "To test with non-uniformly distributed data, the PE's distribution parameters are randomized.", '1502.03942-2-143-4': "The Zipf distributions comprise between [MATH] and [MATH] elements, with each PE's value chosen uniformly at random.", '1502.03942-2-143-5': 'Similarly, the exponent [MATH] is uniformly distributed between [MATH] and [MATH].', '1502.03942-2-143-6': 'This ensures that several PEs contribute to the result, so that the distribution is asymmetric, without the computation becoming a local operation at one PE, which has all of the largest elements.', '1502.03942-2-144-0': 'We used several values of [MATH], namely [MATH], [MATH], and [MATH].', '1502.03942-2-144-1': 'We do not consider smaller values than 1024, as for values this small, it would be more efficient to locally select the [MATH] largest (or smallest) elements, and run a simple distributed selection on those.', '1502.03942-2-145-0': 'Results.', '1502.03942-2-145-1': 'Figure [REF] shows the results for selecting the [MATH]-th largest values from the input, for the above values of [MATH].', '1502.03942-2-145-2': 'We can see that in most cases, the algorithm scales even better than the bounds lead us to expect-running time decreases as more PEs are added.', '1502.03942-2-145-3': 'This is especially prominent when selecting an element of high rank ([MATH] in Figure [REF]).', '1502.03942-2-145-4': 'The majority of the time is spent in partitioning, i.e. local work, dominating the time spent on communication.', '1502.03942-2-145-5': 'This underlines the effect of communication efficiency.', '1502.03942-2-146-0': '[Top-k Most Frequent Objects]Top-[MATH] Most Frequent Objects', '1502.03942-2-147-0': 'As we could not find any competitors to compare our methods against, we use two naive centralized algorithm as baseline.', '1502.03942-2-147-1': 'The first algorithm, Naive, samples the input with the same probability as algorithm PAC, but instead of using a distributed hash table and distributed selection, each PE sends its aggregated local sample to a coordinator.', '1502.03942-2-147-2': 'The coordinator then uses quickselect to determine the elements of rank [MATH] in the global sample, which it returns.', '1502.03942-2-147-3': 'Algorithm Naive Treeproceeds similarly, but uses a tree reduction to send the elements to the coordinator to reduce latency.', '1502.03942-2-147-4': 'Similar to Algorithm PAChash table insertion operation, this reduction aggregates the counts in each step to keep communication volume low.', '1502.03942-2-148-0': 'Input Generation.', '1502.03942-2-148-1': 'We consider [MATH], [MATH] and [MATH] elements per PE, which are generated according to different random distributions.', '1502.03942-2-148-2': "First, we consider elements distributed according to Zipf's Law with [MATH] possible values.", '1502.03942-2-148-3': 'These values are very concentrated and model word frequencies in natural languages, city population sizes, and many other rankings well [CITATION], the most frequent element being [MATH]-times more frequent than that of rank [MATH].', '1502.03942-2-148-4': 'Next, we study a negative binomial distribution with [MATH] and success probability [MATH].', '1502.03942-2-148-5': 'This distribution has a rather wide plateau, resulting in the most frequent objects and their surrounding elements all being of very similar frequency.', '1502.03942-2-148-6': 'For simplicity, each PE generates objects according to the same distribution, as the distributed hash table into which the sample is inserted distributes elements randomly.', '1502.03942-2-148-7': 'Thus, tests with non-uniformly distributed data would not add substantially to the evaluation.', '1502.03942-2-149-0': 'Approximation Quality.', '1502.03942-2-149-1': 'To evaluate different accuracy levels, we consider the [MATH] pairs [MATH] and [MATH].', '1502.03942-2-149-2': 'This allows us to evaluate how running time develops under different accuracy requirements.', '1502.03942-2-150-0': 'We then select the [MATH] most frequent elements from the input according to the above requirements.', '1502.03942-2-150-1': 'We do not vary the parameter [MATH] here, as it has very little impact on overall performance.', '1502.03942-2-150-2': 'Instead, we refer to Section [REF] for experiments on unsorted selection, which is the only subroutine affected by increasing [MATH] and shows no increase up to [MATH].', '1502.03942-2-151-0': 'Results.', '1502.03942-2-151-1': 'Figure [REF] shows the results for [MATH] elements per PE using [MATH] and [MATH].', '1502.03942-2-151-2': 'We can clearly see that Algorithm Naivedoes not scale beyond a single node at all ([MATH]).', '1502.03942-2-151-3': 'In fact, its running time is directly proportional to [MATH], which is consistent with the coordinator receiving [MATH] messages-every other PE sends its key-aggregated sample to the coordinator.', '1502.03942-2-151-4': 'Algorithm Naive Treefares better, and actually improves as more PEs are added.', '1502.03942-2-151-5': 'This is easily explained by the reduced sample size per PE as [MATH] increases, decreasing sampling time.', '1502.03942-2-151-6': 'However, communication time begins to dominate, as the decrease in overall running time is nowhere near as strong as the decrease in local sample size.', '1502.03942-2-151-7': 'This becomes clear when comparing it to Algorithm PAC, which outperforms Naive Treefor any number of PEs.', '1502.03942-2-151-8': 'We can see that it scales nearly perfectly-doubling the number of PEs (and thereby total input size) roughly halves running time.', '1502.03942-2-151-9': 'Since these three algorithms all use the same sampling rate, any differences in running time are completely due to time spent on communication.', '1502.03942-2-152-0': 'Lastly, let us consider Algorithm EC.', '1502.03942-2-152-1': 'In the beginning, it benefits from its much smaller sample size (see Section [REF]), but as [MATH] grows, the local work for exact counting dominates overall running time strongly and Algorithm EC is no longer competitive.', '1502.03942-2-152-2': 'Since local work remains constant with increasing [MATH], we see nearly no change in overall running time.', '1502.03942-2-152-3': 'To see the benefits of Algorithm EC, we need to consider stricter accuracy requirements.', '1502.03942-2-153-0': 'In Figure [REF], we consider [MATH] and [MATH].', '1502.03942-2-153-1': 'For Algorithms PAC, Naive, and Naive Tree, this requires considering the entire input for any number of PEs, as sample size is proportional to [MATH], which the other terms cannot offset here.', '1502.03942-2-153-2': 'Conversely, Algorithm ECsample size depends only linearly on [MATH], resulting in sample sizes orders of magnitude below those of the other algorithms.', '1502.03942-2-154-0': 'Again, we can see that Algorithm Naiveis completely unscalable.', '1502.03942-2-154-1': 'Algorithm Naive Treeperforms much better, with running times remaining roughly constant at around 6.5 seconds as soon as multiple nodes are used.', '1502.03942-2-154-2': 'Algorithm PACsuffers a similar fate, however it is slightly faster at 6.2 seconds.', '1502.03942-2-154-3': 'This difference stems from reduced communication volume.', '1502.03942-2-154-4': 'However, both are dominated by the time spent on aggregating the input.', '1502.03942-2-154-5': 'Lastly, Algorithm ECis consistently fastest, requiring 4.1 seconds, of which 3.7 seconds are spent on exact counting.', '1502.03942-2-154-6': 'This clearly demonstrates that Algorithm ECis superior for small [MATH].', '1502.03942-2-155-0': 'Smaller local input sizes do not yield significant differences, and preliminary experiments with elements distributed according to a negative binomial distribution proved unspectacular and of little informational value, as the aggregated samples have much fewer elements than in a Zipfian distribution-an easy case for selection.', '1502.03942-2-156-0': '# Conclusions', '1502.03942-2-157-0': 'We have demonstrated that a variety of top-[MATH] selection problems can be solved in a communication efficient way, with respect to both communication volume and latencies.', '1502.03942-2-157-1': 'The basic methods are simple and versatile-the owner-computes rule, collective communication, and sampling.', '1502.03942-2-157-2': 'Considering the significant previous work on some of these problems, it is a bit surprising that such simple algorithms give improved results for such fundamental problems.', '1502.03942-2-157-3': 'However, it seems that the combination of communication efficiency and parallel scalability has been neglected for many problems.', '1502.03942-2-157-4': 'Our methods might have particular impact on applications where previous work has concentrated on methods with a pure master-worker scheme.', '1502.03942-2-158-0': 'It is therefore likely that our approach can also be applied to further important problems.', '1502.03942-2-158-1': 'For example, distributed streaming algorithms that generalize the centralized model of Yi and Zhang [CITATION] seem very promising.', '1502.03942-2-158-2': 'The same holds for lower bounds, which so far have also neglected multiparty communication with point-to-point communication (see also [CITATION]).', '1502.03942-2-159-0': 'Closer to the problems considered here, there is also a number of interesting open questions.', '1502.03942-2-159-1': 'For the sorted selection problem from Section [REF], it would be interesting to see whether there is a scalable parallel algorithm that makes an information theoretically optimal number of comparisons as in the sequential algorithm of Varman et al. [CITATION].', '1502.03942-2-159-2': 'Our analysis of approximate multiselection ignores the case where [MATH].', '1502.03942-2-159-3': 'It can probably be shown to run in expected time [MATH].', '1502.03942-2-159-4': 'For the multicriteria top-[MATH] problem from Section [REF], we could consider parallelization of advanced algorithms that scan less elements and perform less random accesses, such as [CITATION].', '1502.03942-2-160-0': 'Regarding the top-[MATH] most frequent objects and sum aggregation, we expect to be able to conduct fully distributed monitoring queries without a substantial increase in communication volume over our one-shot algorithm.', '1502.03942-2-161-0': '[Multisequence Selection]Multisequence Selection [CITATION]', '1502.03942-2-162-0': 'Figure [REF] gives high level pseudo code.', '1502.03942-2-162-1': 'The base case occurs if there is only a single object (and [MATH]).', '1502.03942-2-162-2': 'We can also restrict the search to the first [MATH] objects of each local sequence.', '1502.03942-2-162-3': 'A random object is selected as a pivot.', '1502.03942-2-162-4': 'This can be done in parallel by choosing the same random number between 1 and [MATH] on all PEs.', '1502.03942-2-162-5': 'Using a prefix sum over the sizes of the sequences, this object can be located easily in time [MATH].', '1502.03942-2-162-6': 'Where ordinary quickselect has to partition the input doing linear work, we can exploit the sortedness of the sequences to obtain the same information in time [MATH] with [MATH] by doing binary search in parallel on each PE.If items are evenly distributed, we have [MATH], and thus only time [MATH] for the search, which partitions all the sequences into two parts.', '1502.03942-2-162-7': 'Deciding whether we have to continue searching in the left or the right parts needs a global reduction operations taking time [MATH].', '1502.03942-2-162-8': 'As in ordinary quickselect, the expected depth of the recursion is [MATH].', '1502.03942-2-162-9': 'We obtain the following result.', '1502.03942-2-163-0': 'Algorithm [REF] can be implemented to run in expected time [EQUATION]', '1502.03942-2-164-0': '# Running Times of Existing Algorithms', '1502.03942-2-165-0': 'We now prove the running times given for previous works in Table [REF].', '1502.03942-2-166-0': '### Unsorted Selection', '1502.03942-2-167-0': 'Previous algorithms, see [CITATION], rely on randomly distributed input data or they explicitly redistribute the data.', '1502.03942-2-167-1': 'Thus, they have to move [MATH] elements in the worst case.', '1502.03942-2-167-2': 'The remainder of the bound follows trivially.', '1502.03942-2-168-0': '## Sorted Selection', '1502.03942-2-169-0': 'We could not find any prior results on distributed multiselection from sorted lists and thus list a sequential result by Varman et al. [CITATION].', '1502.03942-2-170-0': '## Bulk Parallel Priority Queue', '1502.03942-2-171-0': 'The result in [CITATION] relies on randomly distributed input data.', '1502.03942-2-171-1': 'Therefore, in operation insert*, each PE needs to send its [MATH] elements to random PEs, costing [MATH] time.', '1502.03942-2-171-2': 'Then, operation deleteMin* is fairly straight-forward and mostly amounts to a selection.', '1502.03942-2-171-3': 'The deterministic parallel heap [CITATION] needs to sort inserted elements and then they travel through [MATH] levels of the data structure, which is allocated to different PEs.', '1502.03942-2-171-4': 'This alone means communication cost [MATH].', '1502.03942-2-172-0': '## Heavy Hitters Monitoring', '1502.03942-2-173-0': 'Huang et al. give a randomized heavy hitters monitoring algorithm [CITATION] that, for constant failure probability [MATH], requires time [MATH] in our model.', '1502.03942-2-174-0': 'All communication is between the controller node and a monitor node, thus the maximum amount of communication is at the controller node.', '1502.03942-2-174-1': 'Each update, which consists of a constant number of words, is transmitted separately.', '1502.03942-2-174-2': 'Thus, the communication term given by the authors transfers directly into our model (except that the number of monitor nodes [MATH] is [MATH] here).', '1502.03942-2-175-0': '## Top-k Frequent Objects Monitoring', '1502.03942-2-176-0': 'Monitoring Query 1 of [CITATION] performs top-[MATH] most frequent object monitoring, for which it incurs a running time of [MATH] for relative error bound [MATH], where [MATH] corresponds to [MATH] in their notation.', '1502.03942-2-176-1': 'This algorithm also has further restrictions: It does not provide approximate counts of the objects in the top-[MATH] set.', '1502.03942-2-176-2': 'It can only handle a small number [MATH] of distinct objects, all of which must be known in advance.', '1502.03942-2-176-3': 'It requires [MATH] memory on the coordinator node, which is prohibitive if [MATH] and [MATH] are large.', '1502.03942-2-176-4': 'It must also be initialized with a top-[MATH] set for the beginning of the streams, using an algorithm such as TA [CITATION].', '1502.03942-2-176-5': 'We now present a family of inputs for which the algorithm uses the claimed amount of communication.', '1502.03942-2-177-0': 'Initialize with [MATH] items [MATH] that all have the same frequency.', '1502.03942-2-177-1': 'Thus, the initial top-k set comprises an arbitrary [MATH] of these.', '1502.03942-2-177-2': 'Choose one of the two objects that are not in the top-[MATH] set, and refer to this object as [MATH], and pick a peer (PE) [MATH].', '1502.03942-2-177-3': 'Now, we send [MATH] to [MATH] repeatedly, and all other items to all other peers in an evenly distributed manner (each peer receives around the same number of occurrences of each object).', '1502.03942-2-177-4': 'After at most [MATH] steps, the top-[MATH] set has become invalid and an expensive full resolution step is required.', '1502.03942-2-177-5': 'As we expect [MATH] to be on the large side, we choose [MATH] and [MATH] for [MATH] as per the instructions in [CITATION].', '1502.03942-2-177-6': 'We can repeat this cycle to obtain an instance of the example family for any input size [MATH].', '1502.03942-2-177-7': 'Note that the number of "cheap" resolution steps during this cycle depends on the choice of the [MATH] values, for which Babcock and Olston give rough guidelines of what might constitute a good choice, but do not present a theoretical analysis of how it influences communication.', '1502.03942-2-177-8': 'Here, we ignore their cost and focus solely on the cost of "expensive" resolutions.', '1502.03942-2-178-0': 'By the above, an "expensive" resolution round is required every (at most) [MATH] items in the input.', '1502.03942-2-178-1': 'Since the resolution set contains [MATH] objects (the top-[MATH] set plus the non-top-[MATH] object with a constraint violation), each "expensive" resolution has communication cost [MATH].', '1502.03942-2-178-2': 'Thus, we obtain a total communication cost for expensive resolutions of [MATH], for a given relative error bound [MATH].', '1502.03942-2-178-3': 'Additionally, each item requires at least constant time in local processing, accounting for the additive [MATH] term.', '1502.03942-2-178-4': 'The actual worst-case communication cost is likely even higher, but this example suffices to show that the approach of [CITATION] is not communication-efficient in our model.', '1502.03942-2-179-0': '## Multicriteria Top-k', '1502.03942-2-180-0': 'Previous algorithms such as TPUT [CITATION] or KLEE [CITATION] are not directly comparable to our approach for a number of reasons.', '1502.03942-2-180-1': 'First, they do not support arbitrary numbers of processing elements, but limit [MATH] to the number of criteria [MATH].', '1502.03942-2-180-2': 'Each PE is assigned one or more complete index lists, whereas our approach splits the objects among the PEs, storing the index lists of the locally present objects on each PE.', '1502.03942-2-180-3': "This severely limits TPUT's and KLEE's scalability.", '1502.03942-2-180-4': 'Secondly, as noted in the introduction, these algorithms use a centralized master-worker approach, where the master (or coordinator) node handles all communication.', '1502.03942-2-180-5': 'This further limits scalability and leads to an inherent increase in communication volume by a factor of up to [MATH].', '1502.03942-2-180-6': 'Thirdly, they are explicitly designed for wide-area networks (WANs), whereas our algorithms are designed with strongly interconnected PEs in mind, as they might be found in a data center.', '1502.03942-2-180-7': 'Since the modeling assumptions are too different to provide a meaningful comparison, we refrain from giving a communication analysis of these algorithms (nor was one provided in the original papers).'}

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{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1502.03942

1304.5054

{'1304.5054-1-0-0': 'Real-time magnetic resonance imaging (MRI) methods generally shorten the measuring time by acquiring less data than needed according to the sampling theorem.', '1304.5054-1-0-1': 'In order to obtain a proper image from such undersampled data, the reconstruction is commonly defined as the solution of an inverse problem, which is regularized by a priori assumptions about the object.', '1304.5054-1-0-2': 'While practical realizations have hitherto been surprisingly successful, strong assumptions about the continuity of image features may affect the temporal fidelity of the estimated images.', '1304.5054-1-0-3': 'Here we propose a novel approach for the reconstruction of serial real-time MRI data which integrates the deformations between nearby frames into the data consistency term.', '1304.5054-1-0-4': 'The method is not required to be affine or rigid and does not need additional measurements.', '1304.5054-1-0-5': 'Moreover, it handles multi-channel MRI data by simultaneously determining the image and its coil sensitivity profiles in a nonlinear formulation which also adapts to non-Cartesian (e.g., radial) sampling schemes.', '1304.5054-1-0-6': 'Experimental results of a motion phantom with controlled speed and in vivo measurements of rapid tongue movements demonstrate image improvements in preserving temporal fidelity and removing residual artifacts.', '1304.5054-1-1-0': 'Keywords: inverse problems, motion estimation, aggregated imaging, nonlinear inversion, real-time MRI, parallel imaging.', '1304.5054-1-2-0': '# Introduction', '1304.5054-1-3-0': 'Imaging speed is crucial in real-time MRI studies of physiologic processes, ranging from cardiovascular imaging to noninvasive monitoring of interventional (surgical) procedures.', '1304.5054-1-3-1': 'Because the physical acceleration of the data acquisition process is limited by physiologic regulations to prevent peripheral nerve stimulation, most strategies ultimately reduce the measuring time by acquiring less data, while attempting to preserve the quality of the reconstructed image.', '1304.5054-1-3-2': 'A first development along this line is the parallel imaging concept which takes advantage of multiple receiver coils to acquire data simultaneously.', '1304.5054-1-3-3': 'Such techniques benefit from encoding part of the spatial information of an object into spatially complementary coil sensitivities, which are generally unknown and also depend on the actual experimental condition.', '1304.5054-1-3-4': 'Therefore, coil sensitivity profiles are either explicitly pre-calibrated in image space, like SENSE [CITATION], or implicitly in [MATH]-space, like SMASH [CITATION] and GRAPPA [CITATION].', '1304.5054-1-3-5': 'Unfortunately, however, such pre-calibration techniques make only suboptimal use of the available data from multiple receiver channels, so that respective errors in the estimated coil profiles may lead to artifacts in the iteratively optimized image - already for moderate acceleration factors of about two to three.', '1304.5054-1-3-6': 'An improved strategy is to compute spin density maps and coil profiles at the same time by means of a nonlinear formulation of the inverse reconstruction problem [CITATION].', '1304.5054-1-3-7': 'In this case, when aiming for high temporal resolution, the use of strongly undersampled data introduces additional ill-posedness to the reconstruction problem.', '1304.5054-1-3-8': 'In order to stabilize the problem and obtain plausible solutions, it is necessary to incorporate a priori information about the unknown object (described by its spin density) and the coil profiles into the reconstruction method.', '1304.5054-1-3-9': 'In [CITATION], a temporal [MATH]-regularization on the object was studied, which is, however, usually too weak to remove residual artifacts.', '1304.5054-1-3-10': 'For example, temporally flickering artifacts are observed for a radial sampling scheme which employs complementary sets of spatial encodings in consecutively acquired datasets.', '1304.5054-1-3-11': 'In practice, a temporal median filter may effectively diminish residual streaking artifacts though at the expense of degrading the true temporal resolution.', '1304.5054-1-3-12': 'Alternatively, the total variance or total generalized variance were used for regularization [CITATION], which also reduce streaking artifacts, but fail to recover small-scale details of the object and therefore sacrifice spatial resolution.', '1304.5054-1-3-13': 'In general, regularization methods alone seem to be unable to provide artifact-free images with high spatial and temporal resolution in real-time MRI scenarios with pronounced undersampling.', '1304.5054-1-4-0': 'An alternative strategy for improved image quality is to integrate information about moving object features into the reconstruction by exploiting multiple measurements at different (neighboring) time points.', '1304.5054-1-4-1': "One of the most effective means for motion compensation in MRI is the 'navigator echo' technique and its variants [CITATION].", '1304.5054-1-4-2': 'In these methods, a navigator signal is repetitively acquired during the scan to extract specific motion information.', '1304.5054-1-4-3': "The need to insert multiple navigator modules into the MRI sequence may be avoided by 'self-navigating' techniques which may determine motions from the actual data.", '1304.5054-1-4-4': 'Early applications [CITATION] used partial or full [MATH]-space data in a block-based or parametric manner, but failed to detect complex motions such as elastic deformations.', '1304.5054-1-4-5': 'Lately, more flexible motion-detection techniques were developed for free-breathing cine MRI studies [CITATION], but they can only compensate for slow (e.g., respiratory) movements that affect a faster (e.g., cardiac) motion of interest.', '1304.5054-1-4-6': 'Motion compensation was also combined with conventional parallel MRI reconstructions [CITATION].', '1304.5054-1-5-0': "To overcome the aforementioned limitations, this work presents a novel reconstruction method for real-time MRI by integrating the idea of a 'self-navigating' motion into a nonlinear formulation of the inverse problem which simultaneously estimates spin density and coil sensitivity profiles.", '1304.5054-1-5-1': 'Based on a non-parametric motion estimation, the new method generates images with high temporal fidelity and reduced residual artifacts.', '1304.5054-1-5-2': 'The validation of the approach in comparison to current algorithms employed a motion phantom with controlled speed as well as real-time MRI studies of movements during human sound production.', '1304.5054-1-6-0': '# Theory', '1304.5054-1-7-0': 'The proposed method is based on a recently developed reconstruction from highly undersampled radial MRI acquisitions with multiple receiver coils [CITATION].', '1304.5054-1-7-1': 'It generalizes the respective data consistency term to incorporate an aggregated reconstruction from multiple frames with non-parametric motion correction (i.e., AME = Aggregated Motion Estimation) and is schematically outlined in Figure [REF].', '1304.5054-1-8-0': '## Real-time Magnetic Resonance Imaging', '1304.5054-1-9-0': 'The real-time MRI data acquisition of the [MATH]-th ([MATH]) frame from multiple receiver coils is given by [EQUATION] where [MATH] denotes the spin density, [MATH] the sensitivity profiles of [MATH] individual receiver coils, [MATH] the noise, [MATH] the sampling operator at the positions [MATH] in [MATH]-space, and [MATH] the Fourier transform.', '1304.5054-1-9-1': 'The goal is to obtain a serial stream of images [MATH] with high spatial and temporal resolution from the measured data [MATH].', '1304.5054-1-9-2': 'The MRI measuring time per frame is mainly determined by the number of samples [MATH] (times the physical repetition time TR needed for radiofrequency excitation and spatial encoding), which therefore is kept as small as possible.', '1304.5054-1-9-3': 'On the other hand, for pronounced undersampling conditions, equation ([REF]) becomes increasingly ill-conditioned.', '1304.5054-1-9-4': 'As a consequence, the inversion of the system leads to an amplification of noise which in turn results in low-resolution images.', '1304.5054-1-9-5': 'Thus, a proper choice of [MATH] should be a sensible trade-off between temporal and spatial resolution.', '1304.5054-1-10-0': '## Aggregated Motion Estimation for Nonlinear Reconstruction', '1304.5054-1-11-0': 'For explanation purpose, in this subsection, we first assume that the motion (or deformation) [MATH] from [MATH] to [MATH], i.e. [MATH], is known for every [MATH], with some [MATH].', '1304.5054-1-11-1': 'For example, [MATH] can be [MATH], which corresponds to 5 successive acquisitions.', '1304.5054-1-11-2': 'In this case, only two future frames are included corresponding to an ignorable waiting time of around 60 ms in our applications.', '1304.5054-1-11-3': 'By variable substitution, the spin density of the [MATH]-th frame satisfies [EQUATION]', '1304.5054-1-11-4': 'Thus, if successive frames rely on complementary data samples in [MATH]-space, the reconstruction takes advantage of [MATH] samples for recovering [MATH], while the temporal resolution remains unchanged, i.e. corresponding to [MATH] times the repetition time TR.', '1304.5054-1-11-5': 'Accordingly, while keeping the temporal resolution, the approach may obtain images with higher spatial resolution from ([REF]).', '1304.5054-1-11-6': 'For sake of clarity, we rewrite ([REF]) by an abstract nonlinear operator equation [EQUATION]', '1304.5054-1-11-7': 'Here, [MATH] and [MATH], for every [MATH].', '1304.5054-1-12-0': 'These equations are solved for the unknown [MATH] by a Newton-type method, whose key idea consists in repeatedly linearizing the operator equation [MATH], [MATH], around some approximate solution [MATH], and solving the linearized problems [EQUATION]', '1304.5054-1-12-1': 'As the real-time MRI problem is highly ill-posed, [MATH] is not bounded or seriously ill-conditioned.', '1304.5054-1-12-2': 'The standard Newton method is not applicable and may not even be well defined for noise-free data, because [MATH] is not guaranteed to lie in the range of [MATH].', '1304.5054-1-12-3': 'Therefore, some regularization method has to be employed for solving the linearized equation ([REF]).', '1304.5054-1-13-0': 'Because only the product of the spin density and coil profiles is determined, the real-time MRI problem is undetermined even in the fully sampled case.', '1304.5054-1-13-1': 'Although the image may contain fine structures, the coil profiles are generally rather smooth.', '1304.5054-1-13-2': 'As in [CITATION], this can be ensured by introducing a term promoting smoothness which may be given by a Sobolev norm [MATH], [MATH].', '1304.5054-1-13-3': 'It penalizes high spatial frequencies by a polynomial of degree [MATH] as a function of the distance to the centre of [MATH]-space.', '1304.5054-1-13-4': 'The object (i.e., its spin density) usually deforms continuously and smoothly from frame to frame, so an efficient regularization penalizes the differences between neighboring frames to ensure temporal continuity.', '1304.5054-1-13-5': 'By combining the standard Newton method and the aforementioned regularization, we obtain the well-known iteratively regularized Gauss-Newton method (IRGNM) [CITATION] for solving ([REF]) [EQUATION] where [MATH], [MATH] and [MATH] the initial guess.', '1304.5054-1-13-6': 'If the initialization is close enough to the true solution, the choice of [MATH], with [MATH], is usually sufficient for convergence (cf. [CITATION]).', '1304.5054-1-13-7': 'Because the IRGNM method reduces to a Gauss-Newton method for [MATH], it uses the more robust descent direction at the beginning of the iterative process (far from the solution) and the faster convergent algorithm at the end (near the solution).', '1304.5054-1-13-8': 'The choice of parameter [MATH] serves as a balance between penalization of the spin density and coil profiles.', '1304.5054-1-13-9': 'The quadratic optimization is equivalent to solve its normal equation, precisely a linear equation with a symmetric coefficient matrix.', '1304.5054-1-13-10': 'Unfortunately, this linear equation is numerically ill-conditioned for large [MATH].', '1304.5054-1-13-11': 'A simple preconditioning by the following variable substitution can significantly reduce its condition number, making it numerically stable.', '1304.5054-1-13-12': 'Let [EQUATION] with [MATH].', '1304.5054-1-13-13': 'If [MATH], then an equivalent form of IRGNM is given by [EQUATION]', '1304.5054-1-13-14': 'Explicitly, the optimality condition for this quadratic optimization is [EQUATION]', '1304.5054-1-13-15': 'Its discretized form can efficiently be solved by the conjugate gradient (CG) algorithm.', '1304.5054-1-13-16': 'Together with the motion estimation described in the next subsection, this strategy represents our novel AME-based nonlinear reconstruction method for high temporal and spatial resolution (see Figure [REF]).', '1304.5054-1-13-17': 'In this manner, multiple acquisition frames are exploited for reconstruction with proper motion correction, implicitly increasing the sampling rate while preserving temporal sharpness.', '1304.5054-1-14-0': '## Motion Estimation', '1304.5054-1-15-0': 'In general, the differential positional displacements between nearby frames are not known.', '1304.5054-1-15-1': 'Therefore, we first precompute each frame by the nonlinear inversion (NLINV) method introduced in [CITATION], which corresponds to [MATH] as defined in the previous subsection.', '1304.5054-1-15-2': 'Subsequently, the motion is estimated on these precomputed images, using the TV-[MATH] optical flow model (cf. [CITATION]).', '1304.5054-1-15-3': 'In detail, the motion [MATH] from [MATH] to [MATH] is estimated by [MATH], with [MATH] given by the solution to [EQUATION]', '1304.5054-1-15-4': 'Because [MATH], the auxiliary variable [MATH] models the varying reconstruction artifacts in different frames.', '1304.5054-1-15-5': 'For radial MRI acquisitions, the residual streaking artifacts have a relatively low total variation in comparison to the object which contains all local structures.', '1304.5054-1-15-6': 'Therefore, it is expected that [MATH] can only capture the true motion of the object instead of the undesirable motion of the artifacts contained in the precomputed images.', '1304.5054-1-15-7': 'In order to avoid the impact of outliers on motion estimation, we only used the [MATH] norm.', '1304.5054-1-15-8': 'This non-smooth minimization can efficiently be solved by the first-order primal-dual algorithm proposed in [CITATION].', '1304.5054-1-16-0': '## Discretization', '1304.5054-1-17-0': 'By denoting [MATH], with [MATH], for [MATH], [MATH], a detailed formula of ([REF]) is [EQUATION]', '1304.5054-1-17-1': 'This equation will be solved by the CG algorithm which requires repeated application of the operations [MATH] and [MATH].', '1304.5054-1-17-2': 'For numerical computation, every function involved needs to be approximated by a discretized form of points on a rectangular grid.', '1304.5054-1-17-3': 'Since density and coil profiles are compactly supported, the Fourier transform [MATH] can be computed by fast Fourier transform (FFT) with proper periodic extension.', '1304.5054-1-18-0': 'If the sampling trajectory represents a non-Cartesian radial scheme as used for real-time MRI in [CITATION], the computation involving the sampling operators [MATH] and [MATH] is not straightforward.', '1304.5054-1-18-1': 'With [MATH] and [MATH], we have [MATH].', '1304.5054-1-18-2': 'Then, [MATH] can be computed by inverse FFT after gridding [CITATION], or nonuniform FFT [CITATION].', '1304.5054-1-18-3': 'With respect to [MATH], we have [EQUATION] where [MATH].', '1304.5054-1-18-4': 'It can be computed by two FFTs and one inverse FFT, with [MATH] given by the gridding algorithm.', '1304.5054-1-18-5': 'In the Cartesian case [MATH] equals ones at measured points and zeros elsewhere.', '1304.5054-1-18-6': 'To sum up, equation ([REF]) can numerically be solved in an efficient way.', '1304.5054-1-19-0': '# Methods', '1304.5054-1-20-0': '## Data Acquisition', '1304.5054-1-21-0': 'The proposed reconstruction technique was evaluated for real-time MRI measurements of a motion phantom as well as for different parts of the human body in vivo.', '1304.5054-1-21-1': 'All studies were conducted on a 3T MRI system (Siemens Magnetom TIM Trio, Erlangen, Germany).', '1304.5054-1-21-2': 'Continuous data acquisition was achieved by using a radiofrequency-spoiled radial FLASH (fast low angle shot) pulse sequence developed for real-time MRI [CITATION].', '1304.5054-1-21-3': 'T1-weighted images were generated by a short repetition time TR (approximately 2 ms) and a low flip angle of the RF excitation pulse (5 to 10 degree).', '1304.5054-1-21-4': 'A highly undersampled radial [MATH]-space encoding scheme was employed with an interleaved arrangement of spokes for five successive datasets (i.e., frames).', '1304.5054-1-21-5': 'Each single turn corresponded to a full image and contained only a small number of spokes that were equally distributed over a full 360circle in order to homogeneously sample [MATH]-space.', '1304.5054-1-21-6': 'To prevent aliasing effects from object structures outside the selected field-of-view, a readout oversampling by a factor of two was used during data acquisition without compromising imaging speed or signal, for details see [CITATION].', '1304.5054-1-21-7': 'For human studies, healthy subjects with no known illness were recruited among the university students and written informed consent was obtained in all cases prior to each examination.', '1304.5054-1-22-0': 'The motion phantom consisted of a polyacetal disc rotating with respect to its geometric center.', '1304.5054-1-22-1': 'Three water-filled tubes with approximately 10 mm diameter were fixed on the disc with a distance to the center of 25 mm, 37.75 mm, and 55 mm, respectively.', '1304.5054-1-22-2': 'The MRI signals were acquired using a 32-channel head coil (Siemens Healthcare, Erlangen, Germany) and the measurements were performed with three different rotational speeds at angular velocities of 0.5 Hz, 1.0 Hz, and 1.5 Hz, respectively.', '1304.5054-1-23-0': 'Real-time MRI of the human body was performed in a supine position for studies of the heart and movements of the tongue during playing a plastic mouthpiece of a brass instrument.', '1304.5054-1-23-1': 'In the latter case subjects were asked to perform rapid tongue movements (staccato) at a rate of about 5 Hz.', '1304.5054-1-23-2': 'A mid-sagittal image was chosen to cover the oropharyngolaryngeal area, while MRI signals were acquired by combining a 4-channel small flexible receiver coil (Siemens Healthcare, Erlangen, Germany) and a bilateral [MATH] array coil (NORAS MRI products, Hoechberg, Germany), using the same setup as previously reported for real-time MRI of speech generation [CITATION].', '1304.5054-1-23-3': 'Cardiac MRI was performed during free breathing and without synchronization to the electrocardiogram [CITATION] using a 32-channel body coil consisting of an anterior and a posterior array with 16 elements each.', '1304.5054-1-23-4': 'Online image control employed conventional NLINV reconstructions with a post-processing temporal median filter (NLINV-MED).', '1304.5054-1-23-5': 'Details of the imaging parameters are summarized in Table [REF].', '1304.5054-1-24-0': '## Image Reconstruction', '1304.5054-1-25-0': 'All reconstructions were done offline using an in-house software package written in Matlab (R2012a, The MathWorks, Natick, MA).', '1304.5054-1-25-1': 'In the first step, data from up to 32 receiver channels were combined into a small set of 10 virtual channels based on a principal component analysis, as previously described in [CITATION].', '1304.5054-1-25-2': 'For the interpolation in [MATH]-space from radial spokes to Cartesian grids, a Kaiser-Bessel window function with [MATH], [MATH] and a 1.5 fold oversampling was used [CITATION].', '1304.5054-1-25-3': 'To speed up the process, the interpolation coefficients were precalculated and stored in a look-up table.', '1304.5054-1-25-4': 'In the next step the interpolated data were normalized such that the [MATH] norm equaled 100.', '1304.5054-1-25-5': 'This allows for choosing the reconstruction parameters independent from the data acquisition parameters, which minimizes the operator interference and also maintains the quality of the results.', '1304.5054-1-26-0': 'For AME reconstruction of experimental datasets, we found empirically that it is sufficient to choose [MATH] to exploit the complementary information from 5 successive acquisitions with interleaved radial encodings.', '1304.5054-1-26-1': 'Numerically, [MATH] gives similar results for both simulated and real data (not shown) but increases computational complexity.', '1304.5054-1-26-2': 'The preliminary images were reconstructed by NLINV, with almost identical parameters for regularization and penalization of coil sensitivity profiles.', '1304.5054-1-26-3': 'The same initialization was used with the spin density set to ones and coil sensitivities to zeros for the first frame.', '1304.5054-1-26-4': 'Later both were replaced by the reconstruction results from the previous frame.', '1304.5054-1-26-5': 'For motion estimation, the spatial deformation of the object was implemented with a bicubic interpolation, and parameters of the model were set to [MATH] and [MATH].', '1304.5054-1-27-0': 'For comparison, the same data was also reconstructed by the standard NLINV method [CITATION] which was implemented in the same software environment.', '1304.5054-1-27-1': 'In addition, a temporal median filter was applied to the images to reduce residual streaking artifacts as proposed in [CITATION].', '1304.5054-1-27-2': 'It was implemented as a post-processing step with a window width covering 5 neighboring images (NLINV-MED).', '1304.5054-1-28-0': '# Results', '1304.5054-1-29-0': '## Motion Estimation', '1304.5054-1-30-0': 'The principle of the proposed motion estimation is demonstrated in Figure [REF] using data for the human heart.', '1304.5054-1-30-1': 'Two frames at end diastole (NLINV1) and end systole (NLINV2) were selected to depict distinct differences due to myocardial contraction in preliminary NLINV reconstructions (arrows).', '1304.5054-1-30-2': 'The calculated deformation of NLINV1 by incorporating the estimated motion is shown as DEFORM.', '1304.5054-1-30-3': 'It clearly identifies the contraction of the myocardium, whereas the streaking artifacts at the top-left corner of the image remain similar as in NLINV1.', '1304.5054-1-30-4': 'The example demonstrates that the information of the moving object has correctly been captured by the motion estimation, while the image artifacts, which may also change from time to time, are appropriately excluded.', '1304.5054-1-30-5': 'This can also be visualized in the difference between DEFORM and NLINV2 (DIFF), where the artifacts are dominant and the structure of the heart is less visible.', '1304.5054-1-30-6': 'In the next subsection, we show how the artifacts will be removed, rather than enhanced, by aggregating the estimated motions in AME (cf. Figure [REF]).', '1304.5054-1-31-0': '## AME in Action', '1304.5054-1-32-0': '### Motion Phantom', '1304.5054-1-33-0': 'Figure [REF] compares reconstructions for a phantom moving at different speeds that were obtained by NLINV, NLINV-MED, and the proposed AME method, respectively.', '1304.5054-1-33-1': 'For the lowest velocity, all three methods produce acceptable results, although the latter two surpass NLINV in reducing streaking artifacts (arrow in top row).', '1304.5054-1-33-2': 'At moderate velocity, NLINV suffers from stronger artifacts due to faster motions, while NLINV-MED even distorts the structure of the fastest moving outermost tube (arrow in middle row).', '1304.5054-1-33-3': 'The stretched shape of the circular tube is a typical effect from the temporal median filter.', '1304.5054-1-33-4': 'In contrast, the AME reconstruction offers a proper image with almost no motion or streaking artifacts.', '1304.5054-1-33-5': 'Finally, for the highest velocity, both NLINV and NLINV-MED result in severely deformed shapes for almost all tubes as well as pronounced streaking artifacts.', '1304.5054-1-33-6': 'Again, the AME method shows best results with only very mild and barely visible artifacts.', '1304.5054-1-33-7': 'Furthermore, the signal-to-noise ratio (SNR) of the AME reconstruction is higher in all cases compared with the two other methods.', '1304.5054-1-34-0': '### Human Tongue Movements', '1304.5054-1-35-0': 'Figure [REF] demonstrates tongue movements during playing the mouthpiece of a brass instrument.', '1304.5054-1-35-1': 'For this particular task the tongue tip of the subject had to rapidly move forward and backward touching the upper teeth ridge.', '1304.5054-1-35-2': 'To better demonstrate the temporal evolution of the motion, a reference line is placed at the tongue tip to derive corresponding 2D spatiotemporal intensity profiles.', '1304.5054-1-35-3': 'Thus, the flickering of the residual artifacts at every 5-th frame is clearly visualized for NLINV.', '1304.5054-1-35-4': 'For NLINV-MED the residual artifacts are effectively removed at the expense of blurring the tongue movements by the temporal median filter.', '1304.5054-1-35-5': 'On the contrary, the proposed AME method preserves the sharp intensity changes associated with the rapid tongue movements even better than in the original NLINV reconstruction, while at the same time successfully minimizing residual streaking artifacts.', '1304.5054-1-36-0': '# Discussion', '1304.5054-1-37-0': 'In comparison to NLINV reconstructions with and without temporal median filter, the proposed AME reconstruction for real-time MRI with pronounced radial undersampling yields serial images with improved temporal acuity and less residual artifacts.', '1304.5054-1-37-1': 'The new approach emerges as an expansion of the previously introduced NLINV reconstruction with an aggregated motion estimation which estimates respective movements from multiple consecutive data sets with complementary spatial encodings.', '1304.5054-1-37-2': 'The additional information is incorporated into the data consistency term of the nonlinear inverse problem for a simultaneous determination of spin density and coil sensitivities.', '1304.5054-1-38-0': 'Extending other approaches for motion estimation in MRI, the present work is not limited to affine or rigid motions.', '1304.5054-1-38-1': 'Moreover, the combination of AME with nonlinear reconstruction permits an arbitrary choice of [MATH] which defines the set of frames used for reconstructing the actual frame.', '1304.5054-1-38-2': 'Extensive experimental studies (not shown) demonstrate that a choice of [MATH] smaller than the number of frames with complementary spatial encodings fails to remove the temporally flickering artifacts.', '1304.5054-1-38-3': 'On the other hand, choosing [MATH] greater than the number of differently encoded frames does not further improve the reconstruction but yields comparable image quality.', '1304.5054-1-38-4': 'Because the computational complexity increases as [MATH] increases, this behavior explains our choice of [MATH] in all experiments.', '1304.5054-1-38-5': "As a stopping criterion of the iterations in NLINV, NLINV-MED and AME, the well-known Morozov's discrepancy principle was initially considered, but it forbids a unique choice of the threshold value for every frame because the energy of the signal slightly changes with time even for normalized [MATH]-space data.", '1304.5054-1-38-6': 'In our applications, we have chosen a fixed number of iterations (i.e., Newton steps) for each method, respectively, which gives satisfactory results.', '1304.5054-1-38-7': 'A data-driven choice certainly appears to be more sensible and might be considered in future research.', '1304.5054-1-39-0': 'A limitation of the AME method, which may deteriorate its performance, stems from errors in the preceding NLINV reconstructions that may lead to an unfaithful motion estimation.', '1304.5054-1-39-1': 'This can be seen from the slightly blurred temporal profile in Figure [REF].', '1304.5054-1-39-2': 'A natural way to overcome this problem would be to run the whole AME procedure at least twice using previous AME (rather than NLINV) reconstructions for more accurate motion estimations, though at the expense of further increasing the computational demand.', '1304.5054-1-39-3': 'In fact, at this time the high computational cost, which is about 20 times that of a comparable NLINV implementation on a laptop with MATLAB, is currently the major obstacle for more extended practical applications.', '1304.5054-1-39-4': 'However, because the computations for each receiver coil and of different frames are independent, AME is highly adaptable to parallel computing.', '1304.5054-1-39-5': 'Apart from interpolation, the involved calculations are simplified to point-wise operations, fast Fourier transform, and scalar products.', '1304.5054-1-39-6': 'As is shown in Section II-D, the interpolation for non-Cartesian data may be separated from the iterative optimization, through a convolution with the point-spread function.', '1304.5054-1-39-7': 'These features further ensure a possible speed-up by an implementation on graphical processing units.', '1304.5054-1-40-0': '# Conclusion', '1304.5054-1-41-0': 'This work introduces a new reconstruction method for real-time MRI that offers improved temporal fidelity for visualizing rapid dynamic changes.', '1304.5054-1-41-1': 'Preliminary results for an experimental phantom and in vivo human data demonstrate the practical performance and improved quality which is based on the incorporation of estimated object motions into the nonlinear inverse reconstruction process.', '1304.5054-1-41-2': 'Future improvements are expected by exploiting new regularization methods and by accelerating the computational speed.'}

{'1304.5054-2-0-0': 'Real-time magnetic resonance imaging (MRI) methods generally shorten the measuring time by acquiring less data than needed according to the sampling theorem.', '1304.5054-2-0-1': 'In order to obtain a proper image from such undersampled data, the reconstruction is commonly defined as the solution of an inverse problem, which is regularized by a priori assumptions about the object.', '1304.5054-2-0-2': 'While practical realizations have hitherto been surprisingly successful, strong assumptions about the continuity of image features may affect the temporal fidelity of the estimated images.', '1304.5054-2-0-3': 'Here we propose a novel approach for the reconstruction of serial real-time MRI data which integrates the deformations between nearby frames into the data consistency term.', '1304.5054-2-0-4': 'The method is not required to be affine or rigid and does not need additional measurements.', '1304.5054-2-0-5': 'Moreover, it handles multi-channel MRI data by simultaneously determining the image and its coil sensitivity profiles in a nonlinear formulation which also adapts to non-Cartesian (e.g., radial) sampling schemes.', '1304.5054-2-0-6': 'Experimental results of a motion phantom with controlled speed and in vivo measurements of rapid tongue movements demonstrate image improvements in preserving temporal fidelity and removing residual artifacts.', '1304.5054-2-1-0': 'Keywords: inverse problems, motion estimation, aggregated imaging, nonlinear inversion, real-time MRI, parallel imaging.', '1304.5054-2-2-0': '# Introduction', '1304.5054-2-3-0': 'Imaging speed is crucial in real-time MRI studies of physiologic processes, ranging from cardiovascular imaging to noninvasive monitoring of interventional (surgical) procedures.', '1304.5054-2-3-1': 'Because the physical acceleration of the data acquisition process is limited by physiologic regulations to prevent peripheral nerve stimulation, most strategies ultimately reduce the measuring time by acquiring less data, while attempting to preserve the quality of the reconstructed image.', '1304.5054-2-3-2': 'A first development along this line is the parallel imaging concept which takes advantage of multiple receiver coils to acquire data simultaneously.', '1304.5054-2-3-3': 'Such techniques benefit from encoding part of the spatial information of an object into spatially complementary coil sensitivities, which are generally unknown and also depend on the actual experimental condition.', '1304.5054-2-3-4': 'Therefore, coil sensitivity profiles are either explicitly pre-calibrated in image space, like SENSE [CITATION], or implicitly in [MATH]-space, like SMASH [CITATION] and GRAPPA [CITATION].', '1304.5054-2-3-5': 'Unfortunately, however, such pre-calibration techniques make only suboptimal use of the available data from multiple receiver channels, so that respective errors in the estimated coil profiles may lead to artifacts in the iteratively optimized image - already for moderate acceleration factors of about two to three.', '1304.5054-2-3-6': 'An improved strategy is to compute spin density maps and coil profiles at the same time by means of a nonlinear formulation of the inverse reconstruction problem [CITATION].', '1304.5054-2-3-7': 'In this case, when aiming for high temporal resolution, the use of strongly undersampled data introduces additional ill-posedness to the reconstruction problem.', '1304.5054-2-3-8': 'In order to stabilize the problem and obtain plausible solutions, it is necessary to incorporate a priori information about the unknown object (described by its spin density) and the coil profiles into the reconstruction method.', '1304.5054-2-3-9': 'In [CITATION], a temporal [MATH]-regularization on the object was studied, which is, however, usually too weak to remove residual artifacts.', '1304.5054-2-3-10': 'For example, temporally flickering artifacts are observed for a radial sampling scheme which employs complementary sets of spatial encodings in consecutively acquired datasets.', '1304.5054-2-3-11': 'In practice, a temporal median filter may effectively diminish residual streaking artifacts though at the expense of degrading the true temporal resolution.', '1304.5054-2-3-12': 'Alternatively, the total variance or total generalized variance were used for regularization [CITATION], which also reduce streaking artifacts, but fail to recover small-scale details of the object and therefore sacrifice spatial resolution.', '1304.5054-2-3-13': 'In general, regularization methods alone seem to be unable to provide artifact-free images with high spatial and temporal resolution in real-time MRI scenarios with pronounced undersampling.', '1304.5054-2-4-0': 'An alternative strategy for improved image quality is to integrate information about moving object features into the reconstruction by exploiting multiple measurements at different (neighboring) time points.', '1304.5054-2-4-1': "One of the most effective means for motion compensation in MRI is the 'navigator echo' technique and its variants [CITATION].", '1304.5054-2-4-2': 'In these methods, a navigator signal is repetitively acquired during the scan to extract specific motion information.', '1304.5054-2-4-3': "The need to insert multiple navigator modules into the MRI sequence may be avoided by 'self-navigating' techniques which may determine motions from the actual data.", '1304.5054-2-4-4': 'Early applications [CITATION] used partial or full [MATH]-space data in a block-based or parametric manner, but failed to detect complex motions such as elastic deformations.', '1304.5054-2-4-5': 'Lately, more flexible motion-detection techniques were developed for free-breathing cine MRI studies [CITATION], but they can only compensate for slow (e.g., respiratory) movements that affect a faster (e.g., cardiac) motion of interest.', '1304.5054-2-4-6': 'Motion compensation was also combined with conventional parallel MRI reconstructions [CITATION].', '1304.5054-2-5-0': "To overcome the aforementioned limitations, this work presents a novel reconstruction method for real-time MRI by integrating the idea of a 'self-navigating' motion into a nonlinear formulation of the inverse problem which simultaneously estimates spin density and coil sensitivity profiles.", '1304.5054-2-5-1': 'Based on a non-parametric motion estimation, the new method generates images with high temporal fidelity and reduced residual artifacts.', '1304.5054-2-5-2': 'The validation of the approach in comparison to current algorithms employed a motion phantom with controlled speed as well as real-time MRI studies of movements during human sound production.', '1304.5054-2-6-0': '# Theory', '1304.5054-2-7-0': 'The proposed method is based on a recently developed reconstruction from highly undersampled radial MRI acquisitions with multiple receiver coils [CITATION].', '1304.5054-2-7-1': 'It generalizes the respective data consistency term to incorporate an aggregated reconstruction from multiple frames with non-parametric motion correction (i.e., AME = Aggregated Motion Estimation) and is schematically outlined in Figure [REF].', '1304.5054-2-8-0': '## Real-time Magnetic Resonance Imaging', '1304.5054-2-9-0': 'The real-time MRI data acquisition of the [MATH]-th ([MATH]) frame from multiple receiver coils is given by [EQUATION] where [MATH] denotes the spin density, [MATH] the sensitivity profiles of [MATH] individual receiver coils, [MATH] the noise, [MATH] the sampling operator at the positions [MATH] in [MATH]-space, and [MATH] the Fourier transform.', '1304.5054-2-9-1': 'The goal is to obtain a serial stream of images [MATH] with high spatial and temporal resolution from the measured data [MATH].', '1304.5054-2-9-2': 'The MRI measuring time per frame is mainly determined by the number of samples [MATH] (times the physical repetition time TR needed for radiofrequency excitation and spatial encoding), which therefore is kept as small as possible.', '1304.5054-2-9-3': 'On the other hand, for pronounced undersampling conditions, equation ([REF]) becomes increasingly ill-conditioned.', '1304.5054-2-9-4': 'As a consequence, the inversion of the system leads to an amplification of noise which in turn results in low-resolution images.', '1304.5054-2-9-5': 'Thus, a proper choice of [MATH] should be a sensible trade-off between temporal and spatial resolution.', '1304.5054-2-10-0': '## Aggregated Motion Estimation for Nonlinear Reconstruction', '1304.5054-2-11-0': 'For explanation purpose, in this subsection, we first assume that the motion (or deformation) [MATH] from [MATH] to [MATH], i.e. [MATH], is known for every [MATH], with some [MATH].', '1304.5054-2-11-1': 'For example, [MATH] can be [MATH], which corresponds to 5 successive acquisitions.', '1304.5054-2-11-2': 'In this case, only two future frames are included corresponding to an ignorable waiting time of around 60 ms in our applications.', '1304.5054-2-11-3': 'By variable substitution, the spin density of the [MATH]-th frame satisfies [EQUATION]', '1304.5054-2-11-4': 'Thus, if successive frames rely on complementary data samples in [MATH]-space, the reconstruction takes advantage of [MATH] samples for recovering [MATH], while the temporal resolution remains unchanged, i.e. corresponding to [MATH] times the repetition time TR.', '1304.5054-2-11-5': 'Accordingly, while keeping the temporal resolution, the approach may obtain images with higher spatial resolution from ([REF]).', '1304.5054-2-11-6': 'For sake of clarity, we rewrite ([REF]) by an abstract nonlinear operator equation [EQUATION]', '1304.5054-2-11-7': 'Here, [MATH] and [MATH], for every [MATH].', '1304.5054-2-12-0': 'These equations are solved for the unknown [MATH] by a Newton-type method, whose key idea consists in repeatedly linearizing the operator equation [MATH], [MATH], around some approximate solution [MATH], and solving the linearized problems [EQUATION]', '1304.5054-2-12-1': 'As the real-time MRI problem is highly ill-posed, [MATH] is not bounded or seriously ill-conditioned.', '1304.5054-2-12-2': 'The standard Newton method is not applicable and may not even be well defined for noise-free data, because [MATH] is not guaranteed to lie in the range of [MATH].', '1304.5054-2-12-3': 'Therefore, some regularization method has to be employed for solving the linearized equation ([REF]).', '1304.5054-2-13-0': 'Because only the product of the spin density and coil profiles is determined, the real-time MRI problem is undetermined even in the fully sampled case.', '1304.5054-2-13-1': 'Although the image may contain fine structures, the coil profiles are generally rather smooth.', '1304.5054-2-13-2': 'As in [CITATION], this can be ensured by introducing a term promoting smoothness which may be given by a Sobolev norm [MATH], [MATH].', '1304.5054-2-13-3': 'It penalizes high spatial frequencies by a polynomial of degree [MATH] as a function of the distance to the centre of [MATH]-space.', '1304.5054-2-13-4': 'The object (i.e., its spin density) usually deforms continuously and smoothly from frame to frame, so an efficient regularization penalizes the differences between neighboring frames to ensure temporal continuity.', '1304.5054-2-13-5': 'By combining the standard Newton method and the aforementioned regularization, we obtain the well-known iteratively regularized Gauss-Newton method (IRGNM) [CITATION] for solving ([REF]) [EQUATION] where [MATH], [MATH] and [MATH] the initial guess.', '1304.5054-2-13-6': 'If the initialization is close enough to the true solution, the choice of [MATH], with [MATH], is usually sufficient for convergence (cf. [CITATION]).', '1304.5054-2-13-7': 'Because the IRGNM method reduces to a Gauss-Newton method for [MATH], it uses the more robust descent direction at the beginning of the iterative process (far from the solution) and the faster convergent algorithm at the end (near the solution).', '1304.5054-2-13-8': 'The choice of parameter [MATH] serves as a balance between penalization of the spin density and coil profiles.', '1304.5054-2-13-9': 'The quadratic optimization is equivalent to solve its normal equation, precisely a linear equation with a symmetric coefficient matrix.', '1304.5054-2-13-10': 'Unfortunately, this linear equation is numerically ill-conditioned for large [MATH].', '1304.5054-2-13-11': 'A simple preconditioning by the following variable substitution can significantly reduce its condition number, making it numerically stable.', '1304.5054-2-13-12': 'Let [EQUATION] with [MATH].', '1304.5054-2-13-13': 'If [MATH], then an equivalent form of IRGNM is given by [EQUATION]', '1304.5054-2-13-14': 'Explicitly, the optimality condition for this quadratic optimization is [EQUATION]', '1304.5054-2-13-15': 'Its discretized form can efficiently be solved by the conjugate gradient (CG) algorithm.', '1304.5054-2-13-16': 'Together with the motion estimation described in the next subsection, this strategy represents our novel AME-based nonlinear reconstruction method for high temporal and spatial resolution (see Figure [REF]).', '1304.5054-2-13-17': 'In this manner, multiple acquisition frames are exploited for reconstruction with proper motion correction, implicitly increasing the sampling rate while preserving temporal sharpness.', '1304.5054-2-14-0': '## Motion Estimation', '1304.5054-2-15-0': 'In general, the differential positional displacements between nearby frames are not known.', '1304.5054-2-15-1': 'Therefore, we first precompute each frame by the nonlinear inversion (NLINV) method introduced in [CITATION], which corresponds to [MATH] as defined in the previous subsection.', '1304.5054-2-15-2': 'Subsequently, the motion is estimated on these precomputed images, using the TV-[MATH] optical flow model (cf. [CITATION]).', '1304.5054-2-15-3': 'In detail, the motion [MATH] from [MATH] to [MATH] is estimated by [MATH], with [MATH] given by the solution to [EQUATION]', '1304.5054-2-15-4': 'Because [MATH], the auxiliary variable [MATH] models the varying reconstruction artifacts in different frames.', '1304.5054-2-15-5': 'For radial MRI acquisitions, the residual streaking artifacts have a relatively low total variation in comparison to the object which contains all local structures.', '1304.5054-2-15-6': 'Therefore, it is expected that [MATH] can only capture the true motion of the object instead of the undesirable motion of the artifacts contained in the precomputed images.', '1304.5054-2-15-7': 'In order to avoid the impact of outliers on motion estimation, we only used the [MATH] norm.', '1304.5054-2-15-8': 'This non-smooth minimization can efficiently be solved by the first-order primal-dual algorithm proposed in [CITATION].', '1304.5054-2-16-0': '## Discretization', '1304.5054-2-17-0': 'By denoting [MATH], with [MATH], for [MATH], [MATH], a detailed formula of ([REF]) is [EQUATION]', '1304.5054-2-17-1': 'This equation will be solved by the CG algorithm which requires repeated application of the operations [MATH] and [MATH].', '1304.5054-2-17-2': 'For numerical computation, every function involved needs to be approximated by a discretized form of points on a rectangular grid.', '1304.5054-2-17-3': 'Since density and coil profiles are compactly supported, the Fourier transform [MATH] can be computed by fast Fourier transform (FFT) with proper periodic extension.', '1304.5054-2-18-0': 'If the sampling trajectory represents a non-Cartesian radial scheme as used for real-time MRI in [CITATION], the computation involving the sampling operators [MATH] and [MATH] is not straightforward.', '1304.5054-2-18-1': 'With [MATH] and [MATH], we have [MATH].', '1304.5054-2-18-2': 'Then, [MATH] can be computed by inverse FFT after gridding [CITATION], or nonuniform FFT [CITATION].', '1304.5054-2-18-3': 'With respect to [MATH], we have [EQUATION] where [MATH].', '1304.5054-2-18-4': 'It can be computed by two FFTs and one inverse FFT, with [MATH] given by the gridding algorithm.', '1304.5054-2-18-5': 'In the Cartesian case [MATH] equals ones at measured points and zeros elsewhere.', '1304.5054-2-18-6': 'To sum up, equation ([REF]) can numerically be solved in an efficient way.', '1304.5054-2-19-0': '# Methods', '1304.5054-2-20-0': '## Data Acquisition', '1304.5054-2-21-0': 'The proposed reconstruction technique was evaluated for real-time MRI measurements of a motion phantom as well as for different parts of the human body in vivo.', '1304.5054-2-21-1': 'All studies were conducted on a 3T MRI system (Siemens Magnetom TIM Trio, Erlangen, Germany).', '1304.5054-2-21-2': 'Continuous data acquisition was achieved by using a radiofrequency-spoiled radial FLASH (fast low angle shot) pulse sequence developed for real-time MRI [CITATION].', '1304.5054-2-21-3': 'T1-weighted images were generated by a short repetition time TR (approximately 2 ms) and a low flip angle of the RF excitation pulse (5 to 10 degree).', '1304.5054-2-21-4': 'A highly undersampled radial [MATH]-space encoding scheme was employed with an interleaved arrangement of spokes for five successive datasets (i.e., frames).', '1304.5054-2-21-5': 'Each single turn corresponded to a full image and contained only a small number of spokes that were equally distributed over a full 360circle in order to homogeneously sample [MATH]-space.', '1304.5054-2-21-6': 'To prevent aliasing effects from object structures outside the selected field-of-view, a readout oversampling by a factor of two was used during data acquisition without compromising imaging speed or signal, for details see [CITATION].', '1304.5054-2-21-7': 'For human studies, healthy subjects with no known illness were recruited among the university students and written informed consent was obtained in all cases prior to each examination.', '1304.5054-2-22-0': 'The motion phantom consisted of a polyacetal disc rotating with respect to its geometric center.', '1304.5054-2-22-1': 'Three water-filled tubes with approximately 10 mm diameter were fixed on the disc with a distance to the center of 25 mm, 37.75 mm, and 55 mm, respectively.', '1304.5054-2-22-2': 'The MRI signals were acquired using a 32-channel head coil (Siemens Healthcare, Erlangen, Germany) and the measurements were performed with three different rotational speeds at angular velocities of 0.5 Hz, 1.0 Hz, and 1.5 Hz, respectively.', '1304.5054-2-23-0': 'Real-time MRI of the human body was performed in a supine position for studies of the heart and movements of the tongue during playing a plastic mouthpiece of a brass instrument.', '1304.5054-2-23-1': 'In the latter case subjects were asked to perform rapid tongue movements (staccato) at a rate of about 5 Hz.', '1304.5054-2-23-2': 'A mid-sagittal image was chosen to cover the oropharyngolaryngeal area, while MRI signals were acquired by combining a 4-channel small flexible receiver coil (Siemens Healthcare, Erlangen, Germany) and a bilateral [MATH] array coil (NORAS MRI products, Hoechberg, Germany), using the same setup as previously reported for real-time MRI of speech generation [CITATION].', '1304.5054-2-23-3': 'Cardiac MRI was performed during free breathing and without synchronization to the electrocardiogram [CITATION] using a 32-channel body coil consisting of an anterior and a posterior array with 16 elements each.', '1304.5054-2-23-4': 'Online image control employed conventional NLINV reconstructions with a post-processing temporal median filter (NLINV-MED).', '1304.5054-2-23-5': 'Details of the imaging parameters are summarized in Table [REF].', '1304.5054-2-24-0': '## Image Reconstruction', '1304.5054-2-25-0': 'All reconstructions were done offline using an in-house software package written in Matlab (R2012a, The MathWorks, Natick, MA).', '1304.5054-2-25-1': 'In the first step, data from up to 32 receiver channels were combined into a small set of 10 virtual channels based on a principal component analysis, as previously described in [CITATION].', '1304.5054-2-25-2': 'For the interpolation in [MATH]-space from radial spokes to Cartesian grids, a Kaiser-Bessel window function with [MATH], [MATH] and a 1.5 fold oversampling was used [CITATION].', '1304.5054-2-25-3': 'To speed up the process, the interpolation coefficients were precalculated and stored in a look-up table.', '1304.5054-2-25-4': 'In the next step the interpolated data were normalized such that the [MATH] norm equaled 100.', '1304.5054-2-25-5': 'This allows for choosing the reconstruction parameters independent from the data acquisition parameters, which minimizes the operator interference and also maintains the quality of the results.', '1304.5054-2-26-0': 'For AME reconstruction of experimental datasets, we found empirically that it is sufficient to choose [MATH] to exploit the complementary information from 5 successive acquisitions with interleaved radial encodings.', '1304.5054-2-26-1': 'Numerically, [MATH] gives similar results for both simulated and real data (not shown) but increases computational complexity.', '1304.5054-2-26-2': 'The preliminary images were reconstructed by NLINV, with almost identical parameters for regularization and penalization of coil sensitivity profiles.', '1304.5054-2-26-3': 'The same initialization was used with the spin density set to ones and coil sensitivities to zeros for the first frame.', '1304.5054-2-26-4': 'Later both were replaced by the reconstruction results from the previous frame.', '1304.5054-2-26-5': 'For motion estimation, the spatial deformation of the object was implemented with a bicubic interpolation, and parameters of the model were set to [MATH] and [MATH].', '1304.5054-2-27-0': 'For comparison, the same data was also reconstructed by the standard NLINV method [CITATION] which was implemented in the same software environment.', '1304.5054-2-27-1': 'In addition, a temporal median filter was applied to the images to reduce residual streaking artifacts as proposed in [CITATION].', '1304.5054-2-27-2': 'It was implemented as a post-processing step with a window width covering 5 neighboring images (NLINV-MED).', '1304.5054-2-28-0': '# Results', '1304.5054-2-29-0': '## Motion Estimation', '1304.5054-2-30-0': 'The principle of the proposed motion estimation is demonstrated in Figure [REF] using data for the human heart.', '1304.5054-2-30-1': 'Two frames at end diastole (NLINV1) and end systole (NLINV2) were selected to depict distinct differences due to myocardial contraction in preliminary NLINV reconstructions (arrows).', '1304.5054-2-30-2': 'The calculated deformation of NLINV1 by incorporating the estimated motion is shown as DEFORM.', '1304.5054-2-30-3': 'It clearly identifies the contraction of the myocardium, whereas the streaking artifacts at the top-left corner of the image remain similar as in NLINV1.', '1304.5054-2-30-4': 'The example demonstrates that the information of the moving object has correctly been captured by the motion estimation, while the image artifacts, which may also change from time to time, are appropriately excluded.', '1304.5054-2-30-5': 'This can also be visualized in the difference between DEFORM and NLINV2 (DIFF), where the artifacts are dominant and the structure of the heart is less visible.', '1304.5054-2-30-6': 'In the next subsection, we show how the artifacts will be removed, rather than enhanced, by aggregating the estimated motions in AME (cf. Figure [REF]).', '1304.5054-2-31-0': '## AME in Action', '1304.5054-2-32-0': '### Motion Phantom', '1304.5054-2-33-0': 'Figure [REF] compares reconstructions for a phantom moving at different speeds that were obtained by NLINV, NLINV-MED, and the proposed AME method, respectively.', '1304.5054-2-33-1': 'For the lowest velocity, all three methods produce acceptable results, although the latter two surpass NLINV in reducing streaking artifacts (arrow in top row).', '1304.5054-2-33-2': 'At moderate velocity, NLINV suffers from stronger artifacts due to faster motions, while NLINV-MED even distorts the structure of the fastest moving outermost tube (arrow in middle row).', '1304.5054-2-33-3': 'The stretched shape of the circular tube is a typical effect from the temporal median filter.', '1304.5054-2-33-4': 'In contrast, the AME reconstruction offers a proper image with almost no motion or streaking artifacts.', '1304.5054-2-33-5': 'Finally, for the highest velocity, both NLINV and NLINV-MED result in severely deformed shapes for almost all tubes as well as pronounced streaking artifacts.', '1304.5054-2-33-6': 'Again, the AME method shows best results with only very mild and barely visible artifacts.', '1304.5054-2-33-7': 'Furthermore, the signal-to-noise ratio (SNR) of the AME reconstruction is higher in all cases compared with the two other methods.', '1304.5054-2-34-0': '### Human Tongue Movements', '1304.5054-2-35-0': 'Figure [REF] demonstrates tongue movements during playing the mouthpiece of a brass instrument.', '1304.5054-2-35-1': 'For this particular task the tongue tip of the subject had to rapidly move forward and backward touching the upper teeth ridge.', '1304.5054-2-35-2': 'To better demonstrate the temporal evolution of the motion, a reference line is placed at the tongue tip to derive corresponding 2D spatiotemporal intensity profiles.', '1304.5054-2-35-3': 'Thus, the flickering of the residual artifacts at every 5-th frame is clearly visualized for NLINV.', '1304.5054-2-35-4': 'For NLINV-MED the residual artifacts are effectively removed at the expense of blurring the tongue movements by the temporal median filter.', '1304.5054-2-35-5': 'On the contrary, the proposed AME method preserves the sharp intensity changes associated with the rapid tongue movements even better than in the original NLINV reconstruction, while at the same time successfully minimizing residual streaking artifacts.', '1304.5054-2-36-0': '# Discussion', '1304.5054-2-37-0': 'In comparison to NLINV reconstructions with and without temporal median filter, the proposed AME reconstruction for real-time MRI with pronounced radial undersampling yields serial images with improved temporal acuity and less residual artifacts.', '1304.5054-2-37-1': 'The new approach emerges as an expansion of the previously introduced NLINV reconstruction with an aggregated motion estimation which estimates respective movements from multiple consecutive data sets with complementary spatial encodings.', '1304.5054-2-37-2': 'The additional information is incorporated into the data consistency term of the nonlinear inverse problem for a simultaneous determination of spin density and coil sensitivities.', '1304.5054-2-38-0': 'Extending other approaches for motion estimation in MRI, the present work is not limited to affine or rigid motions.', '1304.5054-2-38-1': 'Moreover, the combination of AME with nonlinear reconstruction permits an arbitrary choice of [MATH] which defines the set of frames used for reconstructing the actual frame.', '1304.5054-2-38-2': 'Extensive experimental studies (not shown) demonstrate that a choice of [MATH] smaller than the number of frames with complementary spatial encodings fails to remove the temporally flickering artifacts.', '1304.5054-2-38-3': 'On the other hand, choosing [MATH] greater than the number of differently encoded frames does not further improve the reconstruction but yields comparable image quality.', '1304.5054-2-38-4': 'Because the computational complexity increases as [MATH] increases, this behavior explains our choice of [MATH] in all experiments.', '1304.5054-2-38-5': "As a stopping criterion of the iterations in NLINV, NLINV-MED and AME, the well-known Morozov's discrepancy principle was initially considered, but it forbids a unique choice of the threshold value for every frame because the energy of the signal slightly changes with time even for normalized [MATH]-space data.", '1304.5054-2-38-6': 'In our applications, we have chosen a fixed number of iterations (i.e., Newton steps) for each method, respectively, which gives satisfactory results.', '1304.5054-2-38-7': 'A data-driven choice certainly appears to be more sensible and might be considered in future research.', '1304.5054-2-39-0': 'A limitation of the AME method, which may deteriorate its performance, stems from errors in the preceding NLINV reconstructions that may lead to an unfaithful motion estimation.', '1304.5054-2-39-1': 'This can be seen from the slightly blurred temporal profile in Figure [REF].', '1304.5054-2-39-2': 'A natural way to overcome this problem would be to run the whole AME procedure at least twice using previous AME (rather than NLINV) reconstructions for more accurate motion estimations, though at the expense of further increasing the computational demand.', '1304.5054-2-39-3': 'In fact, at this time the high computational cost, which is about 20 times that of a comparable NLINV implementation on a laptop with MATLAB, is currently the major obstacle for more extended practical applications.', '1304.5054-2-39-4': 'However, because the computations for each receiver coil and of different frames are independent, AME is highly adaptable to parallel computing.', '1304.5054-2-39-5': 'Apart from interpolation, the involved calculations are simplified to point-wise operations, fast Fourier transform, and scalar products.', '1304.5054-2-39-6': 'As is shown in Section II-D, the interpolation for non-Cartesian data may be separated from the iterative optimization, through a convolution with the point-spread function.', '1304.5054-2-39-7': 'These features further ensure a possible speed-up by an implementation on graphical processing units.', '1304.5054-2-40-0': '# Conclusion', '1304.5054-2-41-0': 'This work introduces a new reconstruction method for real-time MRI that offers improved temporal fidelity for visualizing rapid dynamic changes.', '1304.5054-2-41-1': 'Preliminary results for an experimental phantom and in vivo human data demonstrate the practical performance and improved quality which is based on the incorporation of estimated object motions into the nonlinear inverse reconstruction process.', '1304.5054-2-41-2': 'Future improvements are expected by exploiting new regularization methods and by accelerating the computational speed.'}

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'1304.5054-3-30-6'], ['1304.5054-2-15-0', '1304.5054-3-15-0'], ['1304.5054-2-15-1', '1304.5054-3-15-1'], ['1304.5054-2-15-2', '1304.5054-3-15-2'], ['1304.5054-2-15-3', '1304.5054-3-15-3'], ['1304.5054-2-15-4', '1304.5054-3-15-4'], ['1304.5054-2-15-5', '1304.5054-3-15-5'], ['1304.5054-2-15-6', '1304.5054-3-15-6'], ['1304.5054-2-15-7', '1304.5054-3-15-7'], ['1304.5054-2-15-8', '1304.5054-3-15-8'], ['1304.5054-2-17-0', '1304.5054-3-17-0'], ['1304.5054-2-17-1', '1304.5054-3-17-1'], ['1304.5054-2-17-2', '1304.5054-3-17-2'], ['1304.5054-2-17-3', '1304.5054-3-17-3'], ['1304.5054-2-3-0', '1304.5054-3-3-0'], ['1304.5054-2-3-1', '1304.5054-3-3-1'], ['1304.5054-2-3-2', '1304.5054-3-3-2'], ['1304.5054-2-3-3', '1304.5054-3-3-3'], ['1304.5054-2-3-4', '1304.5054-3-3-4'], ['1304.5054-2-3-5', '1304.5054-3-3-5'], ['1304.5054-2-3-6', '1304.5054-3-3-6'], ['1304.5054-2-3-7', '1304.5054-3-3-7'], ['1304.5054-2-3-8', '1304.5054-3-3-8'], ['1304.5054-2-3-9', '1304.5054-3-3-9'], ['1304.5054-2-3-10', '1304.5054-3-3-10'], ['1304.5054-2-3-11', '1304.5054-3-3-11'], ['1304.5054-2-3-12', '1304.5054-3-3-12'], ['1304.5054-2-3-13', '1304.5054-3-3-13'], ['1304.5054-2-21-0', '1304.5054-3-21-0'], ['1304.5054-2-21-1', '1304.5054-3-21-1'], ['1304.5054-2-21-2', '1304.5054-3-21-2'], ['1304.5054-2-21-3', '1304.5054-3-21-3'], ['1304.5054-2-21-4', '1304.5054-3-21-4'], ['1304.5054-2-21-5', '1304.5054-3-21-5'], ['1304.5054-2-21-6', '1304.5054-3-21-6'], ['1304.5054-2-21-7', '1304.5054-3-21-7'], ['1304.5054-2-38-0', '1304.5054-3-38-0'], ['1304.5054-2-38-1', '1304.5054-3-38-1'], ['1304.5054-2-38-2', '1304.5054-3-38-2'], ['1304.5054-2-38-3', '1304.5054-3-38-3'], ['1304.5054-2-38-4', '1304.5054-3-38-4'], ['1304.5054-2-38-5', '1304.5054-3-38-5'], ['1304.5054-2-38-6', '1304.5054-3-38-6'], ['1304.5054-2-38-7', '1304.5054-3-38-7'], ['1304.5054-2-22-0', '1304.5054-3-22-0'], ['1304.5054-2-22-1', '1304.5054-3-22-1'], ['1304.5054-2-22-2', '1304.5054-3-22-2'], ['1304.5054-2-25-0', '1304.5054-3-25-0'], ['1304.5054-2-25-1', '1304.5054-3-25-1'], ['1304.5054-2-25-2', '1304.5054-3-25-2'], ['1304.5054-2-25-3', '1304.5054-3-25-3'], ['1304.5054-2-25-4', '1304.5054-3-25-4'], ['1304.5054-2-25-5', '1304.5054-3-25-5'], ['1304.5054-2-11-0', '1304.5054-3-11-0'], ['1304.5054-2-11-1', '1304.5054-3-11-1'], ['1304.5054-2-11-2', '1304.5054-3-11-2'], ['1304.5054-2-11-3', '1304.5054-3-11-3'], ['1304.5054-2-11-4', '1304.5054-3-11-4'], ['1304.5054-2-11-5', '1304.5054-3-11-5'], ['1304.5054-2-11-6', '1304.5054-3-11-6'], ['1304.5054-2-11-7', '1304.5054-3-11-7'], ['1304.5054-2-4-0', '1304.5054-3-4-0'], ['1304.5054-2-4-1', '1304.5054-3-4-1'], ['1304.5054-2-4-2', '1304.5054-3-4-2'], ['1304.5054-2-4-3', '1304.5054-3-4-3'], ['1304.5054-2-4-4', '1304.5054-3-4-4'], ['1304.5054-2-4-5', '1304.5054-3-4-5'], ['1304.5054-2-4-6', '1304.5054-3-4-6']]

[]

[]

[]

[]

['1304.5054-1-13-12', '1304.5054-2-13-12', '1304.5054-3-13-12']

{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '3': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1304.5054

{'1304.5054-3-0-0': 'Real-time magnetic resonance imaging (MRI) methods generally shorten the measuring time by acquiring less data than needed according to the sampling theorem.', '1304.5054-3-0-1': 'In order to obtain a proper image from such undersampled data, the reconstruction is commonly defined as the solution of an inverse problem, which is regularized by a priori assumptions about the object.', '1304.5054-3-0-2': 'While practical realizations have hitherto been surprisingly successful, strong assumptions about the continuity of image features may affect the temporal fidelity of the estimated images.', '1304.5054-3-0-3': 'Here we propose a novel approach for the reconstruction of serial real-time MRI data which integrates the deformations between nearby frames into the data consistency term.', '1304.5054-3-0-4': 'The method is not required to be affine or rigid and does not need additional measurements.', '1304.5054-3-0-5': 'Moreover, it handles multi-channel MRI data by simultaneously determining the image and its coil sensitivity profiles in a nonlinear formulation which also adapts to non-Cartesian (e.g., radial) sampling schemes.', '1304.5054-3-0-6': 'Experimental results of a motion phantom with controlled speed and in vivo measurements of rapid tongue movements demonstrate image improvements in preserving temporal fidelity and removing residual artifacts.', '1304.5054-3-1-0': 'Keywords: inverse problems, motion estimation, aggregated imaging, nonlinear inversion, real-time MRI, parallel imaging.', '1304.5054-3-2-0': '# Introduction', '1304.5054-3-3-0': 'Imaging speed is crucial in real-time MRI studies of physiologic processes, ranging from cardiovascular imaging to noninvasive monitoring of interventional (surgical) procedures.', '1304.5054-3-3-1': 'Because the physical acceleration of the data acquisition process is limited by physiologic regulations to prevent peripheral nerve stimulation, most strategies ultimately reduce the measuring time by acquiring less data, while attempting to preserve the quality of the reconstructed image.', '1304.5054-3-3-2': 'A first development along this line is the parallel imaging concept which takes advantage of multiple receiver coils to acquire data simultaneously.', '1304.5054-3-3-3': 'Such techniques benefit from encoding part of the spatial information of an object into spatially complementary coil sensitivities, which are generally unknown and also depend on the actual experimental condition.', '1304.5054-3-3-4': 'Therefore, coil sensitivity profiles are either explicitly pre-calibrated in image space, like SENSE [CITATION], or implicitly in [MATH]-space, like SMASH [CITATION] and GRAPPA [CITATION].', '1304.5054-3-3-5': 'Unfortunately, however, such pre-calibration techniques make only suboptimal use of the available data from multiple receiver channels, so that respective errors in the estimated coil profiles may lead to artifacts in the iteratively optimized image - already for moderate acceleration factors of about two to three.', '1304.5054-3-3-6': 'An improved strategy is to compute spin density maps and coil profiles at the same time by means of a nonlinear formulation of the inverse reconstruction problem [CITATION].', '1304.5054-3-3-7': 'In this case, when aiming for high temporal resolution, the use of strongly undersampled data introduces additional ill-posedness to the reconstruction problem.', '1304.5054-3-3-8': 'In order to stabilize the problem and obtain plausible solutions, it is necessary to incorporate a priori information about the unknown object (described by its spin density) and the coil profiles into the reconstruction method.', '1304.5054-3-3-9': 'In [CITATION], a temporal [MATH]-regularization on the object was studied, which is, however, usually too weak to remove residual artifacts.', '1304.5054-3-3-10': 'For example, temporally flickering artifacts are observed for a radial sampling scheme which employs complementary sets of spatial encodings in consecutively acquired datasets.', '1304.5054-3-3-11': 'In practice, a temporal median filter may effectively diminish residual streaking artifacts though at the expense of degrading the true temporal resolution.', '1304.5054-3-3-12': 'Alternatively, the total variance or total generalized variance were used for regularization [CITATION], which also reduce streaking artifacts, but fail to recover small-scale details of the object and therefore sacrifice spatial resolution.', '1304.5054-3-3-13': 'In general, regularization methods alone seem to be unable to provide artifact-free images with high spatial and temporal resolution in real-time MRI scenarios with pronounced undersampling.', '1304.5054-3-4-0': 'An alternative strategy for improved image quality is to integrate information about moving object features into the reconstruction by exploiting multiple measurements at different (neighboring) time points.', '1304.5054-3-4-1': "One of the most effective means for motion compensation in MRI is the 'navigator echo' technique and its variants [CITATION].", '1304.5054-3-4-2': 'In these methods, a navigator signal is repetitively acquired during the scan to extract specific motion information.', '1304.5054-3-4-3': "The need to insert multiple navigator modules into the MRI sequence may be avoided by 'self-navigating' techniques which may determine motions from the actual data.", '1304.5054-3-4-4': 'Early applications [CITATION] used partial or full [MATH]-space data in a block-based or parametric manner, but failed to detect complex motions such as elastic deformations.', '1304.5054-3-4-5': 'Lately, more flexible motion-detection techniques were developed for free-breathing cine MRI studies [CITATION], but they can only compensate for slow (e.g., respiratory) movements that affect a faster (e.g., cardiac) motion of interest.', '1304.5054-3-4-6': 'Motion compensation was also combined with conventional parallel MRI reconstructions [CITATION].', '1304.5054-3-5-0': "To overcome the aforementioned limitations, this work presents a novel reconstruction method for real-time MRI by integrating the idea of a 'self-navigating' motion into a nonlinear formulation of the inverse problem which simultaneously estimates spin density and coil sensitivity profiles.", '1304.5054-3-5-1': 'Based on a non-parametric motion estimation, the new method generates images with high temporal fidelity and reduced residual artifacts.', '1304.5054-3-5-2': 'The validation of the approach in comparison to current algorithms employed a motion phantom with controlled speed as well as real-time MRI studies of movements during human sound production.', '1304.5054-3-6-0': '# Theory', '1304.5054-3-7-0': 'The proposed method is based on a recently developed reconstruction from highly undersampled radial MRI acquisitions with multiple receiver coils [CITATION].', '1304.5054-3-7-1': 'It generalizes the respective data consistency term to incorporate an aggregated reconstruction from multiple frames with non-parametric motion correction (i.e., AME = Aggregated Motion Estimation) and is schematically outlined in Figure [REF].', '1304.5054-3-8-0': '## Real-time Magnetic Resonance Imaging', '1304.5054-3-9-0': 'The real-time MRI data acquisition of the [MATH]-th ([MATH]) frame from multiple receiver coils is given by [EQUATION] where [MATH] denotes the spin density, [MATH] the sensitivity profiles of [MATH] individual receiver coils, [MATH] the noise, [MATH] the sampling operator at the positions [MATH] in [MATH]-space, and [MATH] the Fourier transform.', '1304.5054-3-9-1': 'The goal is to obtain a serial stream of images [MATH] with high spatial and temporal resolution from the measured data [MATH].', '1304.5054-3-9-2': 'The MRI measuring time per frame is mainly determined by the number of samples [MATH] (times the physical repetition time TR needed for radiofrequency excitation and spatial encoding), which therefore is kept as small as possible.', '1304.5054-3-9-3': 'On the other hand, for pronounced undersampling conditions, equation ([REF]) becomes increasingly ill-conditioned.', '1304.5054-3-9-4': 'As a consequence, the inversion of the system leads to an amplification of noise which in turn results in low-resolution images.', '1304.5054-3-9-5': 'Thus, a proper choice of [MATH] should be a sensible trade-off between temporal and spatial resolution.', '1304.5054-3-10-0': '## Aggregated Motion Estimation for Nonlinear Reconstruction', '1304.5054-3-11-0': 'For explanation purpose, in this subsection, we first assume that the motion (or deformation) [MATH] from [MATH] to [MATH], i.e. [MATH], is known for every [MATH], with some [MATH].', '1304.5054-3-11-1': 'For example, [MATH] can be [MATH], which corresponds to 5 successive acquisitions.', '1304.5054-3-11-2': 'In this case, only two future frames are included corresponding to an ignorable waiting time of around 60 ms in our applications.', '1304.5054-3-11-3': 'By variable substitution, the spin density of the [MATH]-th frame satisfies [EQUATION]', '1304.5054-3-11-4': 'Thus, if successive frames rely on complementary data samples in [MATH]-space, the reconstruction takes advantage of [MATH] samples for recovering [MATH], while the temporal resolution remains unchanged, i.e. corresponding to [MATH] times the repetition time TR.', '1304.5054-3-11-5': 'Accordingly, while keeping the temporal resolution, the approach may obtain images with higher spatial resolution from ([REF]).', '1304.5054-3-11-6': 'For sake of clarity, we rewrite ([REF]) by an abstract nonlinear operator equation [EQUATION]', '1304.5054-3-11-7': 'Here, [MATH] and [MATH], for every [MATH].', '1304.5054-3-12-0': 'These equations are solved for the unknown [MATH] by a Newton-type method, whose key idea consists in repeatedly linearizing the operator equation [MATH], [MATH], around some approximate solution [MATH], and solving the linearized problems [EQUATION]', '1304.5054-3-12-1': 'As the real-time MRI problem is highly ill-posed, [MATH] is not bounded or seriously ill-conditioned.', '1304.5054-3-12-2': 'The standard Newton method is not applicable and may not even be well defined for noise-free data, because [MATH] is not guaranteed to lie in the range of [MATH].', '1304.5054-3-12-3': 'Therefore, some regularization method has to be employed for solving the linearized equation ([REF]).', '1304.5054-3-13-0': 'Because only the product of the spin density and coil profiles is determined, the real-time MRI problem is undetermined even in the fully sampled case.', '1304.5054-3-13-1': 'Although the image may contain fine structures, the coil profiles are generally rather smooth.', '1304.5054-3-13-2': 'As in [CITATION], this can be ensured by introducing a term promoting smoothness which may be given by a Sobolev norm [MATH], [MATH].', '1304.5054-3-13-3': 'It penalizes high spatial frequencies by a polynomial of degree [MATH] as a function of the distance to the centre of [MATH]-space.', '1304.5054-3-13-4': 'The object (i.e., its spin density) usually deforms continuously and smoothly from frame to frame, so an efficient regularization penalizes the differences between neighboring frames to ensure temporal continuity.', '1304.5054-3-13-5': 'By combining the standard Newton method and the aforementioned regularization, we obtain the well-known iteratively regularized Gauss-Newton method (IRGNM) [CITATION] for solving ([REF]) [EQUATION] where [MATH], [MATH] and [MATH] the initial guess.', '1304.5054-3-13-6': 'If the initialization is close enough to the true solution, the choice of [MATH], with [MATH], is usually sufficient for convergence (cf. [CITATION]).', '1304.5054-3-13-7': 'Because the IRGNM method reduces to a Gauss-Newton method for [MATH], it uses the more robust descent direction at the beginning of the iterative process (far from the solution) and the faster convergent algorithm at the end (near the solution).', '1304.5054-3-13-8': 'The choice of parameter [MATH] serves as a balance between penalization of the spin density and coil profiles.', '1304.5054-3-13-9': 'The quadratic optimization is equivalent to solve its normal equation, precisely a linear equation with a symmetric coefficient matrix.', '1304.5054-3-13-10': 'Unfortunately, this linear equation is numerically ill-conditioned for large [MATH].', '1304.5054-3-13-11': 'A simple preconditioning by the following variable substitution can significantly reduce its condition number, making it numerically stable.', '1304.5054-3-13-12': 'Let [EQUATION] with [MATH].', '1304.5054-3-13-13': 'If [MATH], then an equivalent form of IRGNM is given by [EQUATION]', '1304.5054-3-13-14': 'Explicitly, the optimality condition for this quadratic optimization is [EQUATION]', '1304.5054-3-13-15': 'Its discretized form can efficiently be solved by the conjugate gradient (CG) algorithm.', '1304.5054-3-13-16': 'Together with the motion estimation described in the next subsection, this strategy represents our novel AME-based nonlinear reconstruction method for high temporal and spatial resolution (see Figure [REF]).', '1304.5054-3-13-17': 'In this manner, multiple acquisition frames are exploited for reconstruction with proper motion correction, implicitly increasing the sampling rate while preserving temporal sharpness.', '1304.5054-3-14-0': '## Motion Estimation', '1304.5054-3-15-0': 'In general, the differential positional displacements between nearby frames are not known.', '1304.5054-3-15-1': 'Therefore, we first precompute each frame by the nonlinear inversion (NLINV) method introduced in [CITATION], which corresponds to [MATH] as defined in the previous subsection.', '1304.5054-3-15-2': 'Subsequently, the motion is estimated on these precomputed images, using the TV-[MATH] optical flow model (cf. [CITATION]).', '1304.5054-3-15-3': 'In detail, the motion [MATH] from [MATH] to [MATH] is estimated by [MATH], with [MATH] given by the solution to [EQUATION]', '1304.5054-3-15-4': 'Because [MATH], the auxiliary variable [MATH] models the varying reconstruction artifacts in different frames.', '1304.5054-3-15-5': 'For radial MRI acquisitions, the residual streaking artifacts have a relatively low total variation in comparison to the object which contains all local structures.', '1304.5054-3-15-6': 'Therefore, it is expected that [MATH] can only capture the true motion of the object instead of the undesirable motion of the artifacts contained in the precomputed images.', '1304.5054-3-15-7': 'In order to avoid the impact of outliers on motion estimation, we only used the [MATH] norm.', '1304.5054-3-15-8': 'This non-smooth minimization can efficiently be solved by the first-order primal-dual algorithm proposed in [CITATION].', '1304.5054-3-16-0': '## Discretization', '1304.5054-3-17-0': 'By denoting [MATH], with [MATH], for [MATH], [MATH], a detailed formula of ([REF]) is [EQUATION]', '1304.5054-3-17-1': 'This equation will be solved by the CG algorithm which requires repeated application of the operations [MATH] and [MATH].', '1304.5054-3-17-2': 'For numerical computation, every function involved needs to be approximated by a discretized form of points on a rectangular grid.', '1304.5054-3-17-3': 'Since density and coil profiles are compactly supported, the Fourier transform [MATH] can be computed by fast Fourier transform (FFT) with proper periodic extension.', '1304.5054-3-18-0': 'If the sampling trajectory represents a non-Cartesian radial scheme as used for real-time MRI in [CITATION], the computation involving the sampling operators [MATH] and [MATH] is not straightforward.', '1304.5054-3-18-1': 'With [MATH] and [MATH], we have [MATH].', '1304.5054-3-18-2': 'Then, [MATH] can be computed by inverse FFT after gridding [CITATION], or nonuniform FFT [CITATION].', '1304.5054-3-18-3': 'With respect to [MATH], we have [EQUATION] where [MATH].', '1304.5054-3-18-4': 'It can be computed by two FFTs and one inverse FFT, with [MATH] given by the gridding algorithm.', '1304.5054-3-18-5': 'In the Cartesian case [MATH] equals ones at measured points and zeros elsewhere.', '1304.5054-3-18-6': 'To sum up, equation ([REF]) can numerically be solved in an efficient way.', '1304.5054-3-19-0': '# Methods', '1304.5054-3-20-0': '## Data Acquisition', '1304.5054-3-21-0': 'The proposed reconstruction technique was evaluated for real-time MRI measurements of a motion phantom as well as for different parts of the human body in vivo.', '1304.5054-3-21-1': 'All studies were conducted on a 3T MRI system (Siemens Magnetom TIM Trio, Erlangen, Germany).', '1304.5054-3-21-2': 'Continuous data acquisition was achieved by using a radiofrequency-spoiled radial FLASH (fast low angle shot) pulse sequence developed for real-time MRI [CITATION].', '1304.5054-3-21-3': 'T1-weighted images were generated by a short repetition time TR (approximately 2 ms) and a low flip angle of the RF excitation pulse (5 to 10 degree).', '1304.5054-3-21-4': 'A highly undersampled radial [MATH]-space encoding scheme was employed with an interleaved arrangement of spokes for five successive datasets (i.e., frames).', '1304.5054-3-21-5': 'Each single turn corresponded to a full image and contained only a small number of spokes that were equally distributed over a full 360circle in order to homogeneously sample [MATH]-space.', '1304.5054-3-21-6': 'To prevent aliasing effects from object structures outside the selected field-of-view, a readout oversampling by a factor of two was used during data acquisition without compromising imaging speed or signal, for details see [CITATION].', '1304.5054-3-21-7': 'For human studies, healthy subjects with no known illness were recruited among the university students and written informed consent was obtained in all cases prior to each examination.', '1304.5054-3-22-0': 'The motion phantom consisted of a polyacetal disc rotating with respect to its geometric center.', '1304.5054-3-22-1': 'Three water-filled tubes with approximately 10 mm diameter were fixed on the disc with a distance to the center of 25 mm, 37.75 mm, and 55 mm, respectively.', '1304.5054-3-22-2': 'The MRI signals were acquired using a 32-channel head coil (Siemens Healthcare, Erlangen, Germany) and the measurements were performed with three different rotational speeds at angular velocities of 0.5 Hz, 1.0 Hz, and 1.5 Hz, respectively.', '1304.5054-3-23-0': 'Real-time MRI of the human body was performed in a supine position for studies of the heart and movements of the tongue during playing a plastic mouthpiece of a brass instrument.', '1304.5054-3-23-1': 'In the latter case subjects were asked to perform rapid tongue movements (staccato) at a rate of about 5 Hz.', '1304.5054-3-23-2': 'A mid-sagittal image was chosen to cover the oropharyngolaryngeal area, while MRI signals were acquired by combining a 4-channel small flexible receiver coil (Siemens Healthcare, Erlangen, Germany) and a bilateral [MATH] array coil (NORAS MRI products, Hoechberg, Germany), using the same setup as previously reported for real-time MRI of speech generation [CITATION].', '1304.5054-3-23-3': 'Cardiac MRI was performed during free breathing and without synchronization to the electrocardiogram [CITATION] using a 32-channel body coil consisting of an anterior and a posterior array with 16 elements each.', '1304.5054-3-23-4': 'Online image control employed conventional NLINV reconstructions with a post-processing temporal median filter (NLINV-MED).', '1304.5054-3-23-5': 'Details of the imaging parameters are summarized in Table [REF].', '1304.5054-3-24-0': '## Image Reconstruction', '1304.5054-3-25-0': 'All reconstructions were done offline using an in-house software package written in Matlab (R2012a, The MathWorks, Natick, MA).', '1304.5054-3-25-1': 'In the first step, data from up to 32 receiver channels were combined into a small set of 10 virtual channels based on a principal component analysis, as previously described in [CITATION].', '1304.5054-3-25-2': 'For the interpolation in [MATH]-space from radial spokes to Cartesian grids, a Kaiser-Bessel window function with [MATH], [MATH] and a 1.5 fold oversampling was used [CITATION].', '1304.5054-3-25-3': 'To speed up the process, the interpolation coefficients were precalculated and stored in a look-up table.', '1304.5054-3-25-4': 'In the next step the interpolated data were normalized such that the [MATH] norm equaled 100.', '1304.5054-3-25-5': 'This allows for choosing the reconstruction parameters independent from the data acquisition parameters, which minimizes the operator interference and also maintains the quality of the results.', '1304.5054-3-26-0': 'For AME reconstruction of experimental datasets, we found empirically that it is sufficient to choose [MATH] to exploit the complementary information from 5 successive acquisitions with interleaved radial encodings.', '1304.5054-3-26-1': 'Numerically, [MATH] gives similar results for both simulated and real data (not shown) but increases computational complexity.', '1304.5054-3-26-2': 'The preliminary images were reconstructed by NLINV, with almost identical parameters for regularization and penalization of coil sensitivity profiles.', '1304.5054-3-26-3': 'The same initialization was used with the spin density set to ones and coil sensitivities to zeros for the first frame.', '1304.5054-3-26-4': 'Later both were replaced by the reconstruction results from the previous frame.', '1304.5054-3-26-5': 'For motion estimation, the spatial deformation of the object was implemented with a bicubic interpolation, and parameters of the model were set to [MATH] and [MATH].', '1304.5054-3-27-0': 'For comparison, the same data was also reconstructed by the standard NLINV method [CITATION] which was implemented in the same software environment.', '1304.5054-3-27-1': 'In addition, a temporal median filter was applied to the images to reduce residual streaking artifacts as proposed in [CITATION].', '1304.5054-3-27-2': 'It was implemented as a post-processing step with a window width covering 5 neighboring images (NLINV-MED).', '1304.5054-3-28-0': '# Results', '1304.5054-3-29-0': '## Motion Estimation', '1304.5054-3-30-0': 'The principle of the proposed motion estimation is demonstrated in Figure [REF] using data for the human heart.', '1304.5054-3-30-1': 'Two frames at end diastole (NLINV1) and end systole (NLINV2) were selected to depict distinct differences due to myocardial contraction in preliminary NLINV reconstructions (arrows).', '1304.5054-3-30-2': 'The calculated deformation of NLINV1 by incorporating the estimated motion is shown as DEFORM.', '1304.5054-3-30-3': 'It clearly identifies the contraction of the myocardium, whereas the streaking artifacts at the top-left corner of the image remain similar as in NLINV1.', '1304.5054-3-30-4': 'The example demonstrates that the information of the moving object has correctly been captured by the motion estimation, while the image artifacts, which may also change from time to time, are appropriately excluded.', '1304.5054-3-30-5': 'This can also be visualized in the difference between DEFORM and NLINV2 (DIFF), where the artifacts are dominant and the structure of the heart is less visible.', '1304.5054-3-30-6': 'In the next subsection, we show how the artifacts will be removed, rather than enhanced, by aggregating the estimated motions in AME (cf. Figure [REF]).', '1304.5054-3-31-0': '## AME in Action', '1304.5054-3-32-0': '### Motion Phantom', '1304.5054-3-33-0': 'Figure [REF] compares reconstructions for a phantom moving at different speeds that were obtained by NLINV, NLINV-MED, and the proposed AME method, respectively.', '1304.5054-3-33-1': 'For the lowest velocity, all three methods produce acceptable results, although the latter two surpass NLINV in reducing streaking artifacts (arrow in top row).', '1304.5054-3-33-2': 'At moderate velocity, NLINV suffers from stronger artifacts due to faster motions, while NLINV-MED even distorts the structure of the fastest moving outermost tube (arrow in middle row).', '1304.5054-3-33-3': 'The stretched shape of the circular tube is a typical effect from the temporal median filter.', '1304.5054-3-33-4': 'In contrast, the AME reconstruction offers a proper image with almost no motion or streaking artifacts.', '1304.5054-3-33-5': 'Finally, for the highest velocity, both NLINV and NLINV-MED result in severely deformed shapes for almost all tubes as well as pronounced streaking artifacts.', '1304.5054-3-33-6': 'Again, the AME method shows best results with only very mild and barely visible artifacts.', '1304.5054-3-33-7': 'Furthermore, the signal-to-noise ratio (SNR) of the AME reconstruction is higher in all cases compared with the two other methods.', '1304.5054-3-34-0': '### Human Tongue Movements', '1304.5054-3-35-0': 'Figure [REF] demonstrates tongue movements during playing the mouthpiece of a brass instrument.', '1304.5054-3-35-1': 'For this particular task the tongue tip of the subject had to rapidly move forward and backward touching the upper teeth ridge.', '1304.5054-3-35-2': 'To better demonstrate the temporal evolution of the motion, a reference line is placed at the tongue tip to derive corresponding 2D spatiotemporal intensity profiles.', '1304.5054-3-35-3': 'Thus, the flickering of the residual artifacts at every 5-th frame is clearly visualized for NLINV.', '1304.5054-3-35-4': 'For NLINV-MED the residual artifacts are effectively removed at the expense of blurring the tongue movements by the temporal median filter.', '1304.5054-3-35-5': 'On the contrary, the proposed AME method preserves the sharp intensity changes associated with the rapid tongue movements even better than in the original NLINV reconstruction, while at the same time successfully minimizing residual streaking artifacts.', '1304.5054-3-36-0': '# Discussion', '1304.5054-3-37-0': 'In comparison to NLINV reconstructions with and without temporal median filter, the proposed AME reconstruction for real-time MRI with pronounced radial undersampling yields serial images with improved temporal acuity and less residual artifacts.', '1304.5054-3-37-1': 'The new approach emerges as an expansion of the previously introduced NLINV reconstruction with an aggregated motion estimation which estimates respective movements from multiple consecutive data sets with complementary spatial encodings.', '1304.5054-3-37-2': 'The additional information is incorporated into the data consistency term of the nonlinear inverse problem for a simultaneous determination of spin density and coil sensitivities.', '1304.5054-3-38-0': 'Extending other approaches for motion estimation in MRI, the present work is not limited to affine or rigid motions.', '1304.5054-3-38-1': 'Moreover, the combination of AME with nonlinear reconstruction permits an arbitrary choice of [MATH] which defines the set of frames used for reconstructing the actual frame.', '1304.5054-3-38-2': 'Extensive experimental studies (not shown) demonstrate that a choice of [MATH] smaller than the number of frames with complementary spatial encodings fails to remove the temporally flickering artifacts.', '1304.5054-3-38-3': 'On the other hand, choosing [MATH] greater than the number of differently encoded frames does not further improve the reconstruction but yields comparable image quality.', '1304.5054-3-38-4': 'Because the computational complexity increases as [MATH] increases, this behavior explains our choice of [MATH] in all experiments.', '1304.5054-3-38-5': "As a stopping criterion of the iterations in NLINV, NLINV-MED and AME, the well-known Morozov's discrepancy principle was initially considered, but it forbids a unique choice of the threshold value for every frame because the energy of the signal slightly changes with time even for normalized [MATH]-space data.", '1304.5054-3-38-6': 'In our applications, we have chosen a fixed number of iterations (i.e., Newton steps) for each method, respectively, which gives satisfactory results.', '1304.5054-3-38-7': 'A data-driven choice certainly appears to be more sensible and might be considered in future research.', '1304.5054-3-39-0': 'A limitation of the AME method, which may deteriorate its performance, stems from errors in the preceding NLINV reconstructions that may lead to an unfaithful motion estimation.', '1304.5054-3-39-1': 'This can be seen from the slightly blurred temporal profile in Figure [REF].', '1304.5054-3-39-2': 'A natural way to overcome this problem would be to run the whole AME procedure at least twice using previous AME (rather than NLINV) reconstructions for more accurate motion estimations, though at the expense of further increasing the computational demand.', '1304.5054-3-39-3': 'In fact, at this time the high computational cost, which is about 20 times that of a comparable NLINV implementation on a laptop with MATLAB, is currently the major obstacle for more extended practical applications.', '1304.5054-3-39-4': 'However, because the computations for each receiver coil and of different frames are independent, AME is highly adaptable to parallel computing.', '1304.5054-3-39-5': 'Apart from interpolation, the involved calculations are simplified to point-wise operations, fast Fourier transform, and scalar products.', '1304.5054-3-39-6': 'As is shown in Section II-D, the interpolation for non-Cartesian data may be separated from the iterative optimization, through a convolution with the point-spread function.', '1304.5054-3-39-7': 'These features further ensure a possible speed-up by an implementation on graphical processing units.', '1304.5054-3-40-0': '# Conclusion', '1304.5054-3-41-0': 'This work introduces a new reconstruction method for real-time MRI that offers improved temporal fidelity for visualizing rapid dynamic changes.', '1304.5054-3-41-1': 'Preliminary results for an experimental phantom and in vivo human data demonstrate the practical performance and improved quality which is based on the incorporation of estimated object motions into the nonlinear inverse reconstruction process.', '1304.5054-3-41-2': 'Future improvements are expected by exploiting new regularization methods and by accelerating the computational speed.'}

hep-ph-9903217

{'hep-ph-9903217-1-0-0': 'We study the [MATH] and [MATH] decays into [MATH], [MATH] using a chiral unitary approach to deal with the final state interaction of the [MATH] system.', 'hep-ph-9903217-1-0-1': 'The final state interaction modifies only moderately the large momenta tail of the photon spectrum of the [MATH] decay.', 'hep-ph-9903217-1-0-2': 'In the case of [MATH] decay the contribution to [MATH] and [MATH] decay proceeds via kaonic loops, which are finite, and gives a distribution of [MATH] invariant masses in which the [MATH] resonance shows up with a very distinct peak.', 'hep-ph-9903217-1-0-3': 'The spectrum found for [MATH] decay agrees with the recent experimental results obtained at Novosibirsk.', 'hep-ph-9903217-1-1-0': 'PACS: 13.25.', 'hep-ph-9903217-1-1-1': 'Jx 12.39.', 'hep-ph-9903217-1-1-2': 'Fe 13.40.', 'hep-ph-9903217-1-1-3': 'Hq', 'hep-ph-9903217-1-2-0': 'In this work we investigate the reactions [MATH], [MATH] and [MATH], [MATH] treating the final state interaction of the two mesons with techniques of chiral unitary theory recently developed.', 'hep-ph-9903217-1-2-1': 'The energies of the two pion system are too big in both the [MATH] and [MATH] decay to be treated with standard chiral perturbation theory, [MATH] [CITATION].', 'hep-ph-9903217-1-2-2': 'However, a unitary coupled channels method, which makes use of the standard chiral Lagrangians together with an expansion of Re [MATH] instead of the [MATH] matrix, has proved to be very efficient in decribing the meson meson interactions in all channels up to energies around 1.2 GeV [CITATION].', 'hep-ph-9903217-1-2-3': 'The method is analogous to the effective range expansion in Quantum Mechanics.', 'hep-ph-9903217-1-2-4': 'The work of [CITATION] establishes a direct connection with [MATH] at low energies and gives the same numerical results as the work of [CITATION] where tadpoles and loops in the cross channels are not evaluated but are reabsorbed into the [MATH] coefficients of the second order Lagrangian of [MATH].', 'hep-ph-9903217-1-2-5': 'A technically much simpler approach is done in [CITATION] where, only for [MATH], it is shown that the effect of the second order Lagrangian can be suitably incorporated by means of the Bethe-Salpeter equation using the lowest order Lagrangian as a source of the potential and a suitable cut off, of the order of 1 GeV, to regularize the loops.', 'hep-ph-9903217-1-2-6': 'This latter approach will be the one used here, where the two pions interact in s-wave.', 'hep-ph-9903217-1-3-0': 'The diagrammatic description for the [MATH] decay is shown in Fig. 1', 'hep-ph-9903217-1-4-0': 'In Fig. 1 the intermediate states in the loops attached to the photon, [MATH], can be [MATH] or [MATH].', 'hep-ph-9903217-1-4-1': 'However, the other loops involving only the meson meson interaction can be also [MATH] or [MATH] in the coupled channel approach of [CITATION].', 'hep-ph-9903217-1-5-0': 'For the case of [MATH] decay only the diagrams with at least one loop contribute, [MATH] in Fig. 1.', 'hep-ph-9903217-1-6-0': 'The case of the [MATH] decay is analogous to the [MATH] decay.', 'hep-ph-9903217-1-6-1': 'Indeed, the terms [MATH] of Fig. 1 do not contribute since we do not have direct [MATH] coupling.', 'hep-ph-9903217-1-6-2': 'Furthermore, there is another novelty since only [MATH] contributes to the loop with a photon attached.', 'hep-ph-9903217-1-7-0': 'The procedure followed here in the case of [MATH] production is analogous to the one used in [CITATION].', 'hep-ph-9903217-1-7-1': 'Depending on the gauge and renormalization scheme chosen, other diagrams can appear [CITATION] but the whole set is calculated using gauge invariant arguments, as done here, with the same result.', 'hep-ph-9903217-1-7-2': 'The novelty in the present work is that the strong interaction [MATH] is evaluated using the unitary chiral amplitudes instead of the lowest order used in [CITATION].', 'hep-ph-9903217-1-8-0': 'The basic couplings needed to evaluate the diagrams of Fig. 1 are [EQUATION] with [MATH], [MATH] the [MATH] momenta.', 'hep-ph-9903217-1-8-1': 'The vertices are easily generalized to the case of [MATH].', 'hep-ph-9903217-1-8-2': 'In the first two couplings one has an extra factor [MATH] [CITATION] and the last coupling is the same.', 'hep-ph-9903217-1-8-3': 'The couplings for [MATH] and [MATH] which are needed for the [MATH] decay are like the two first couplings of Eq. ([REF]) substituting [MATH] by [MATH] and [MATH] by [MATH].', 'hep-ph-9903217-1-8-4': 'The values of [MATH], [MATH] extracted from the decay widths of the [MATH] are [EQUATION]', 'hep-ph-9903217-1-8-5': 'The evaluation of the [MATH] width for the first three diagrams [MATH] of Fig. 1 is straightforward and has been done before [CITATION].', 'hep-ph-9903217-1-8-6': 'We rewrite the results in a convenient way for our purposes [EQUATION] where [EQUATION]', 'hep-ph-9903217-1-8-7': 'In Eq. ([REF]), [MATH] is the invariant mass of the two [MATH] system and [MATH] the angle between the [MATH] meson and the photon in the frame where the [MATH] system is at rest.', 'hep-ph-9903217-1-8-8': 'The quantity [MATH] stands for the contribution of the first diagram alone, Fig. 1 [MATH], [MATH] for the second and third [MATH] and [MATH] for the interference between the first diagram and the other two.', 'hep-ph-9903217-1-8-9': 'They are given by [EQUATION] where [MATH] are the momenta of the meson and the photon in the rest frame of the [MATH] system, and [MATH] the meson propagators in the [MATH] Bremsstrahlung diagrams [EQUATION]', 'hep-ph-9903217-1-8-10': 'The technology to introduce the final state interactions is available from the study of [MATH] in [CITATION].', 'hep-ph-9903217-1-8-11': 'There it was shown that the strong [MATH] matrix for the [MATH] transition factorizes with their on shell values in the loops with a photon attached.', 'hep-ph-9903217-1-8-12': 'The same was proved for the loops of the Bethe-Salpeter equation in the meson meson interaction description of [CITATION].', 'hep-ph-9903217-1-8-13': 'On the other hand the sum of the diagrams [MATH] could be done using arguments of gauge invariance which led to a finite contribution for the sum of the loops [CITATION].', 'hep-ph-9903217-1-8-14': 'Following the steps of [CITATION] it is easy to include the effect of the final state interaction of the mesons.', 'hep-ph-9903217-1-8-15': 'The sum of the diagrams [MATH] and further iterated loops of the meson-meson interaction, [MATH] is shown to have the same structure as the contact term of [MATH] in the Coulomb gauge, which one chooses to evaluate the amplitudes.', 'hep-ph-9903217-1-8-16': 'The sum of all terms including loops is readily accomplished by multiplying the contribution of the contact term by the factor [MATH] [EQUATION] where [MATH] are the strong transition matrix elements in s-wave evaluated in [CITATION] and [MATH] is given by [EQUATION] and [MATH] a function given analytically in [CITATION].', 'hep-ph-9903217-1-8-17': 'By using isospin Clebsch Gordan coefficients the amplitudes [MATH] can be written in terms of the isospin amplitudes of [CITATION] as [EQUATION] neglecting the small [MATH] amplitudes.', 'hep-ph-9903217-1-8-18': 'In Eq. ([REF]), one factor [MATH] for each [MATH] state has been introduced, since the isospin amplitudes of [CITATION] used in Eq. ([REF]) are written in a unitary normalization which includes an extra factor [MATH] for each [MATH] state.', 'hep-ph-9903217-1-9-0': 'The invariant mass distribution is now given by Eqs. ([REF], [REF], [REF]) by changing in Eq. ([REF]) [EQUATION]', 'hep-ph-9903217-1-9-1': 'The [MATH] width is readily obtained by omitting the terms [MATH] and also omitting the first term (the unity) in the definition of the [MATH] factor in Eq. ([REF]).', 'hep-ph-9903217-1-10-0': 'The evaluation of the [MATH] decay is straightforward by noting that the tree level contributions, diagrams [MATH] are not present now, and that only kaonic loops attached to photons contribute in this case.', 'hep-ph-9903217-1-10-1': 'Hence, the invariant mass distribution for [MATH] is given in this case by Eq. ([REF]), changing [MATH], with [EQUATION]', 'hep-ph-9903217-1-10-2': "For [MATH] the cross section is the same divided by a factor two to account fot the identity of the two [MATH]'s.", 'hep-ph-9903217-1-11-0': 'In Fig. 2 we show [MATH] for [MATH] decay, with [MATH] the photon momentum in the [MATH] rest frame [MATH], for the case when only the tree diagrams [MATH] are considered and also with the corrections of final state interaction.', 'hep-ph-9903217-1-11-1': 'The process is infrared divergent and we plot the distribution for photons with energy [MATH] MeV, where the experimental measurements exist [CITATION].', 'hep-ph-9903217-1-11-2': 'We have also added the experimental data, given in [CITATION] with arbitrary normalization, normalized to our results.', 'hep-ph-9903217-1-12-0': 'As one can see in Fig. 2, the shape of the distribution of photon momenta is well reproduced.', 'hep-ph-9903217-1-12-1': 'As for the total normalization we obtain a branching ratio to the total width of the [MATH] [EQUATION] which compares favourably with the experimental number [CITATION], [MATH] for [MATH] MeV.', 'hep-ph-9903217-1-13-0': 'The effect of the final state interaction is small and mostly visible at high photon energies, where it increases [MATH] by about 20%.', 'hep-ph-9903217-1-13-1': 'The branching ratio for [MATH] that we obtain is [MATH] which can be interpreted in our case as [MATH] since the [MATH] interaction is dominated by the [MATH] pole in the energy regime where it appears here.', 'hep-ph-9903217-1-13-2': 'This result is very similar to the one obtained in [CITATION].', 'hep-ph-9903217-1-13-3': 'Accordingly, the branching ratio for [MATH] should be twice this number.', 'hep-ph-9903217-1-13-4': 'However, the difference between the ratios for [MATH] with and without final state interaction is [MATH].', 'hep-ph-9903217-1-13-5': 'This difference of about a factor 20 is due to the fact that the important contributions of FSI in this latter case come from interference terms between the diagram [MATH] and those with loops plus diagrams [MATH] and those with loops (mostly from the first case).', 'hep-ph-9903217-1-13-6': 'The contribution from the modulus squared of the loop terms (the only one in [MATH]) is about one order of magnitude smaller than the interference terms.', 'hep-ph-9903217-1-13-7': 'Estimates for [MATH] decay (equivalent to [MATH] in present nomenclature) of the order of [MATH] were made in [CITATION] using vector meson dominance and current algebra arguments.', 'hep-ph-9903217-1-13-8': 'The present availability of good [MATH] and [MATH] amplitudes, together with the chiral techniques used here, allow us to make more precise evaluations of the contribution of the scalar meson formation while at the same time realizing the importance of the interference terms.', 'hep-ph-9903217-1-14-0': 'As for the [MATH] decay, as we pointed above the [MATH] rate is twice the one of the [MATH].', 'hep-ph-9903217-1-14-1': 'We have evaluated the invariant mass distribution for these decay channels and in Fig. 3 we plot the distribution [MATH] for [MATH] which allows us to see the [MATH] contribution since the [MATH] is the important scalar resonance appearing in the [MATH] amplitude [CITATION].', 'hep-ph-9903217-1-14-2': 'We compare our results with the recent ones of the Novosibirsk experiment [CITATION], which appeared published after completion of the present work.', 'hep-ph-9903217-1-14-3': 'We can see that the shape of the spectrum is well reproduced, as well as the strength, considering statistical and systematic errors (the latter ones not shown in the figure).', 'hep-ph-9903217-1-15-0': 'The finite total branching ratio which we find for [MATH] is [MATH] and correspondingly [MATH] for the [MATH].', 'hep-ph-9903217-1-15-1': 'This latter number compares favourably with the result given in [CITATION] of [MATH] where the first error is statistical and the second one systematic.', 'hep-ph-9903217-1-16-0': 'The contribution of [MATH], obtained by integrating [MATH] assuming an approximate Breit-Wigner form to the left of the [MATH] peak, gives us a branching ratio [MATH].', 'hep-ph-9903217-1-16-1': 'As argued above, the branching ratio for [MATH] is one half of [MATH], which should not be compared to the one given in [CITATION] since there the assumption that all the strength of the spectrum is due to the [MATH] excitation is done.', 'hep-ph-9903217-1-16-2': 'As one can see in Fig. 2, we find also an appreciable strength for [MATH].', 'hep-ph-9903217-1-17-0': 'Our result for [MATH] is about twice as large as the one obtained in [CITATION] owed to the use of the unitary [MATH] amplitude instead of the lowest order chiral one.', 'hep-ph-9903217-1-17-1': 'The shape of the distribution found here is also rather different than the one obtained in [CITATION], showing the important contribution of the [MATH] resonance which appears naturally in the unitary chiral approach.', 'hep-ph-9903217-1-18-0': 'The [MATH] decay has been advocated as an important source of information on the nature of the [MATH] resonance and experiments are been conducted at Novosibirsk [CITATION] and are also planned at Frascati [CITATION], trying to magnify the signal for [MATH] production through interference with initial and final state radiation in the [MATH] reaction [CITATION].', 'hep-ph-9903217-1-18-1': 'The completion of the experiment [CITATION] is a significant step forward.', 'hep-ph-9903217-1-19-0': 'Present evaluations for [MATH] are based on models assuming a [MATH] molecule for the [MATH] [CITATION] with a branching ratio 1-[MATH], a [MATH] structure with a value [MATH] [CITATION] and a [MATH] structure with a value [MATH] [CITATION].', 'hep-ph-9903217-1-20-0': 'The model for [MATH] assumed in Fig. 1 is similar to the one of [CITATION] where the production also proceeds via the kaonic loops.', 'hep-ph-9903217-1-20-1': 'There a [MATH] molecule is assumed for the [MATH] resonance while here the realistic [MATH] amplitude of [CITATION] is used.', 'hep-ph-9903217-1-20-2': 'Emphasis is made in the importance of going beyond the zero width approximation for the resonance in [CITATION].', 'hep-ph-9903217-1-20-3': 'Our approach automatically takes into account this since the [MATH] amplitude correctly incorporates the width of the [MATH] resonance [CITATION].', 'hep-ph-9903217-1-21-0': 'We would also like to warn that the peak of the [MATH] seen in Fig. 3 cannot be trivially interpreted as a resonant contribution on top of a background, since there are important interference effects between the [MATH] production and the [MATH] background.', 'hep-ph-9903217-1-21-1': 'The strength of the peak comes in our case in about equal amounts from the real and the imaginary parts of the amplitude for the process.', 'hep-ph-9903217-1-22-0': 'The agreement found between our results for the [MATH] and experiment provides an important endorsement for the chiral unitary approach used here.', 'hep-ph-9903217-1-22-1': 'Improvements in the future, reducing the experimental errors, should put further constraints on avalaible theoretical approaches for this reaction.'}

{'hep-ph-9903217-2-0-0': 'We study the [MATH] and [MATH] decays into [MATH], [MATH] and [MATH] into [MATH] using a chiral unitary approach to deal with the final state interaction of the [MATH] system.', 'hep-ph-9903217-2-0-1': 'The final state interaction modifies only moderately the large momenta tail of the photon spectrum of the [MATH] decay.', 'hep-ph-9903217-2-0-2': 'In the case of [MATH] decay the contribution to [MATH] and [MATH] decay proceeds via kaonic loops and gives a distribution of [MATH] invariant masses in which the [MATH] resonance shows up with a very distinct peak.', 'hep-ph-9903217-2-0-3': 'The spectrum found for [MATH] decay agrees with the recent experimental results obtained at Novosibirsk.', 'hep-ph-9903217-2-0-4': 'The branching ratio for [MATH], dominated by the [MATH], is also in agreement with recent Novosibirsk results.', 'hep-ph-9903217-2-1-0': 'PACS: 13.25.', 'hep-ph-9903217-2-1-1': 'Jx 12.39.', 'hep-ph-9903217-2-1-2': 'Fe 13.40.', 'hep-ph-9903217-2-1-3': 'Hq', 'hep-ph-9903217-2-2-0': 'In this work we investigate the reactions [MATH], [MATH] and [MATH], [MATH], [MATH], treating the final state interaction of the two mesons with techniques of chiral unitary theory recently developed.', 'hep-ph-9903217-2-2-1': 'The energies of the two meson system are too big in both the [MATH] and [MATH] decay to be treated with standard chiral perturbation theory, [MATH] [CITATION].', 'hep-ph-9903217-2-2-2': 'However, a unitary coupled channels method, which makes use of the standard chiral Lagrangians together with an expansion of Re [MATH] instead of the [MATH] matrix, has proved to be very efficient in describing the meson meson interactions in all channels up to energies around 1.2 GeV [CITATION].', 'hep-ph-9903217-2-2-3': 'The method is analogous to the effective range expansion in Quantum Mechanics.', 'hep-ph-9903217-2-2-4': 'The work of [CITATION] establishes a direct connection with [MATH] at low energies and gives the same numerical results as the work of [CITATION] where tadpoles and loops in the crossed channels are not evaluated but are reabsorbed into the [MATH] coefficients of the second order Lagrangian of [MATH].', 'hep-ph-9903217-2-2-5': 'A technically much simpler approach is done in [CITATION] where, only for [MATH], it is shown that the effect of the second order Lagrangian can be suitably incorporated by means of the Bethe-Salpeter equation using the lowest order Lagrangian as a source of the potential and a suitable cut off, of the order of 1 GeV, to regularize the loops.', 'hep-ph-9903217-2-2-6': 'This latter approach will be the one used here, where the two pions interact in s-wave.', 'hep-ph-9903217-2-3-0': 'The diagrammatic description for the [MATH] decay is shown in Fig. 1', 'hep-ph-9903217-2-4-0': 'In Fig. 1 the intermediate states in the loops attached to the photon, [MATH], can be [MATH] or [MATH].', 'hep-ph-9903217-2-4-1': 'However, the other loops involving only the meson meson interaction can be also [MATH] or [MATH] in the coupled channel approach of [CITATION].', 'hep-ph-9903217-2-5-0': 'For the case of [MATH] decay only the diagrams with at least one loop contribute, [MATH] in Fig. 1.', 'hep-ph-9903217-2-6-0': 'The case of the [MATH] decay is analogous to the [MATH] decay.', 'hep-ph-9903217-2-6-1': 'Indeed, the terms [MATH] of Fig. 1 do not contribute since we do not have direct [MATH] coupling.', 'hep-ph-9903217-2-6-2': 'Furthermore, there is another novelty since only [MATH] contributes to the loop with a photon attached.', 'hep-ph-9903217-2-7-0': 'The procedure followed here in the cases of [MATH] and [MATH] production is analogous to the one used in [CITATION].', 'hep-ph-9903217-2-7-1': 'Depending on the renormalization scheme chosen, other diagrams can appear [CITATION] but the whole set is calculated using gauge invariant arguments, as done here, with the same result.', 'hep-ph-9903217-2-7-2': 'The novelty in the present work is that the strong interaction [MATH] is evaluated using the unitary chiral amplitudes instead of the lowest order used in [CITATION].', 'hep-ph-9903217-2-8-0': 'We shall make use of the chiral Lagrangians for vector mesons of [CITATION] and follow the lines of ref. [CITATION] in the treatment of the radiative rho decay.', 'hep-ph-9903217-2-8-1': 'The Lagrangian coupling vector mesons to pseudoscalar mesons and photons is given by [EQUATION] where [MATH] is a [MATH] matrix of antisymmetric tensor fields representing the octet of vector mesons, [MATH], [MATH], [MATH].', 'hep-ph-9903217-2-8-2': 'All magnitudes involved in Eq. ([REF]) are defined in [CITATION].', 'hep-ph-9903217-2-8-3': 'The coupling [MATH] is deduced from the [MATH] decay and the [MATH] coupling from [MATH].', 'hep-ph-9903217-2-8-4': 'We take the values chosen in [CITATION], [MATH] MeV, [MATH] MeV.', 'hep-ph-9903217-2-8-5': 'The [MATH] meson is introduced in the scheme by means of a singlet, [MATH], going from SU(3) to U(3) through the substitution [MATH], with [MATH] the [MATH] diagonal matrix.', 'hep-ph-9903217-2-8-6': 'Then, assuming ideal mixing for the [MATH] and [MATH] mesons [EQUATION] one obtains the Lagrangian of Eq. ([REF]) substituting [MATH] by [MATH], given by [EQUATION]', 'hep-ph-9903217-2-8-7': 'From there one can obtain the couplings corresponding to [MATH] vector and [MATH] pseudoscalar) and [MATH] with the [MATH] term or the [MATH] with the [MATH] term.', 'hep-ph-9903217-2-9-0': 'The basic couplings needed to evaluate the diagrams of Fig. 1 are [EQUATION] with [MATH], [MATH] the [MATH] momenta, [MATH], [MATH] the [MATH] and photon momenta and [MATH] the pion decay constant which we take as [MATH] MeV.', 'hep-ph-9903217-2-10-0': 'The vertices of Eq. ([REF]) are easily generalized to the case of [MATH].', 'hep-ph-9903217-2-10-1': 'Using the Lagrangian of Eq. ([REF]), in the first two couplings one has an extra factor [MATH] and the last coupling is the same.', 'hep-ph-9903217-2-10-2': 'The couplings for [MATH] and [MATH] which are needed for the [MATH] decay are like the two first couplings of Eq. ([REF]) substituting [MATH] by [MATH], [MATH] by [MATH] and multiplying by [MATH].', 'hep-ph-9903217-2-10-3': 'In addition we shall take the values [MATH] MeV and [MATH] MeV which are suited to the [MATH] and [MATH] decay widths respectively.', 'hep-ph-9903217-2-11-0': 'The evaluation of the [MATH] width for the first three diagrams [MATH] of Fig. 1 is straightforward and has been done before [CITATION] and in [CITATION] following the present formalism.', 'hep-ph-9903217-2-11-1': 'We rewrite the results in a convenient way for our purposes [EQUATION] where [EQUATION]', 'hep-ph-9903217-2-11-2': 'In Eq. ([REF]), [MATH] is the invariant mass of the two [MATH] system and [MATH] the angle between the [MATH] meson and the photon in the frame where the [MATH] system is at rest.', 'hep-ph-9903217-2-11-3': 'The quantity [MATH] stands for the contribution of the first diagram alone, Fig. 1 [MATH], [MATH] for the second and third [MATH] and [MATH] for the interference between the first diagram and the other two.', 'hep-ph-9903217-2-11-4': 'They are given by [EQUATION] where [MATH] is the photon momentum in the [MATH] rest frame and [MATH] are the momenta of the meson and the photon in the rest frame of the [MATH] system, and [MATH] the meson propagators in the [MATH] Bremsstrahlung diagrams, conveniently written in terms of [MATH] and [MATH].', 'hep-ph-9903217-2-12-0': 'The first term of the contact term, [MATH], in Eq. ([REF]) is not gauge invariant.', 'hep-ph-9903217-2-12-1': 'It requires the addition of the diagrams [MATH] and [MATH] of Fig. 1 to have a gauge invariant set.', 'hep-ph-9903217-2-12-2': 'On the other hand the second term in the contact term ([MATH] part) is gauge invariant by itself.', 'hep-ph-9903217-2-12-3': 'When considering final state interaction of the mesons this means that the [MATH] part of the contact term, diagram [MATH], must be complemented by diagrams [MATH], [MATH], [MATH] to form the gauge invariant set.', 'hep-ph-9903217-2-12-4': 'On the other hand the [MATH] part of the contact term appears in the [MATH] diagram which is gauge invariant by itself.', 'hep-ph-9903217-2-13-0': 'The technology to introduce the final state interactions is available from the study of [MATH] in [CITATION].', 'hep-ph-9903217-2-13-1': 'There it was shown that the strong [MATH] matrix for the [MATH] transition factorizes with their on shell values in the loops with a photon attached.', 'hep-ph-9903217-2-13-2': 'The same was proved for the loops of the Bethe-Salpeter equation in the meson meson interaction description of [CITATION].', 'hep-ph-9903217-2-13-3': 'On the other hand the sum of the diagrams [MATH], which appears now with the [MATH] part of the contact term (diagram [MATH]), could be done using arguments of gauge invariance which led to a finite contribution for the sum of the loops [CITATION].', 'hep-ph-9903217-2-13-4': 'A sketch of the procedure is given here.', 'hep-ph-9903217-2-13-5': 'The [MATH] amplitude can be written as [MATH] and the structure of the loops in Fig. 1 is such that [EQUATION] where [MATH], [MATH] are the [MATH] meson and photon momenta respectively.', 'hep-ph-9903217-2-13-6': 'Gauge invariance [MATH] forces [MATH] and [MATH].', 'hep-ph-9903217-2-13-7': 'Furthermore, in the Coulomb gauge only the [MATH] term of Eq. ([REF]) contributes and the coefficient [MATH] is calculated from the [MATH] coefficient, to which only the diagrams [MATH], [MATH], of Fig. 1 contribute.', 'hep-ph-9903217-2-13-8': 'For dimensional reasons the loop integral contains two powers less in the internal variables than the pieces contributing to the [MATH] term from these diagrams, since the product [MATH] is factorized out of the integral.', 'hep-ph-9903217-2-13-9': 'This makes the [MATH] coefficient finite.', 'hep-ph-9903217-2-13-10': 'Furthermore, the [MATH] vertices appearing there have the structure [MATH], which can be recast as [MATH].', 'hep-ph-9903217-2-13-11': 'The first two terms in the sum give the on shell contribution and the third one the off shell part.', 'hep-ph-9903217-2-13-12': 'This latter term kills one of the meson propagators in the loops and does not contribute to the [MATH] term in Eq. ([REF]).', 'hep-ph-9903217-2-13-13': 'Hence, the meson meson amplitudes factorize outside the loop integral with their on shell values.', 'hep-ph-9903217-2-13-14': 'A more detailed description, done for a similar problem, can be seen in [CITATION], following the steps from Eqs. (13) to (23).', 'hep-ph-9903217-2-14-0': 'Following these steps, as done in [CITATION], it is easy to include the effect of the final state interaction of the mesons.', 'hep-ph-9903217-2-14-1': 'The sum of the diagrams [MATH] and further iterated loops of the meson-meson interaction, [MATH] is shown to have the same structure as the contact term of [MATH] in the Coulomb gauge, which one chooses to evaluate the amplitudes.', 'hep-ph-9903217-2-14-2': 'The sum of all terms including loops is readily accomplished by multiplying the [MATH] part of the contact term by the factor [MATH] [EQUATION] where [MATH] are the strong transition matrix elements in s-wave evaluated in [CITATION] and [MATH] is given by [EQUATION] with [MATH] a function given analytically in [CITATION].', 'hep-ph-9903217-2-14-3': 'The [MATH] part of the contact term is iterated by means of diagrams [MATH], [MATH] in order to account for final state interaction.', 'hep-ph-9903217-2-14-4': 'Here the loop function is the ordinary two meson propagator function, [MATH], of the Bethe-Salpeter equation, [MATH], for the meson-meson scattering and which is regularized in [CITATION] by means of a cut-off in order to fit the scattering data.', 'hep-ph-9903217-2-14-5': 'The sum of all these diagrams is readily accomplished by multiplying the [MATH] part of the contact term by the factor', 'hep-ph-9903217-2-15-0': '[EQUATION]', 'hep-ph-9903217-2-15-1': 'By using isospin Clebsch Gordan coefficients the amplitudes [MATH] can be written in terms of the isospin amplitudes of [CITATION] as [EQUATION] neglecting the small [MATH] amplitudes.', 'hep-ph-9903217-2-15-2': 'In Eq. ([REF]), one factor [MATH] for each [MATH] state has been introduced, since the isospin amplitudes of [CITATION] used in Eq. ([REF]) are written in a unitary normalization which includes an extra factor [MATH] for each [MATH] state.', 'hep-ph-9903217-2-16-0': 'The invariant mass distribution in the presence of final state interaction is now given by Eqs. ([REF], [REF], [REF]) by changing in Eq. ([REF]) [EQUATION]', 'hep-ph-9903217-2-16-1': 'The [MATH] width is readily obtained by omitting the terms [MATH] and also omitting the first term (the unity) in the definition of the [MATH], [MATH] factors in Eqs. ([REF]) and ([REF]) and dividing by a factor two the width to account for the identity of the particles.', 'hep-ph-9903217-2-17-0': 'The evaluation of the [MATH] decay is straightforward by noting that the tree level contributions, diagrams [MATH] are not present now, and that only kaonic loops attached to photons contribute in this case.', 'hep-ph-9903217-2-17-1': 'Hence, the invariant mass distribution for [MATH] is given in this case by Eq. ([REF]), changing [MATH], with [EQUATION]', 'hep-ph-9903217-2-17-2': "For [MATH] the cross section is the same divided by a factor two to account fot the identity of the two [MATH]'s.", 'hep-ph-9903217-2-18-0': 'For the [MATH] case we have [EQUATION]', 'hep-ph-9903217-2-18-1': 'In Fig. 2 we show [MATH] for [MATH] decay, [MATH].', 'hep-ph-9903217-2-18-2': 'The dashed-dotted line shows the contribution of diagrams [MATH] and taking [MATH].', 'hep-ph-9903217-2-18-3': 'The dashed line shows again the contribution coming from diagrams [MATH] but now considering also the [MATH] contributions.', 'hep-ph-9903217-2-18-4': 'Finally, the solid line includes the full set of diagrams in Fig. 1 to account for final state interaction and with the [MATH] and [MATH] contributions.', 'hep-ph-9903217-2-18-5': 'The process is infrared divergent and we plot the distribution for photons with energy bigger than 50 MeV, where the experimental measurements exist [CITATION].', 'hep-ph-9903217-2-18-6': 'We have also added the experimental data, given in [CITATION] with arbitrary normalization, normalized to our results.', 'hep-ph-9903217-2-19-0': 'As one can see in Fig. 2, the shape of the distribution of photon momenta is well reproduced.', 'hep-ph-9903217-2-19-1': 'For the total contribution we obtain a branching ratio to the total width of the [MATH] [EQUATION] which compares favourably with the experimental number [CITATION], [MATH] for [MATH] MeV.', 'hep-ph-9903217-2-20-0': 'The changes induced by the [MATH] term found here reconfirm the findings of [CITATION].', 'hep-ph-9903217-2-20-1': 'The effect of the final state interaction is small and mostly visible at high photon energies, where it increases [MATH] by about 25%.', 'hep-ph-9903217-2-20-2': 'The branching ratio for [MATH] that we obtain is [MATH] which can be interpreted in our case as [MATH] since the [MATH] interaction is dominated by the [MATH] pole in the energy regime where it appears here.', 'hep-ph-9903217-2-20-3': 'This result is very similar to the one obtained in [CITATION].', 'hep-ph-9903217-2-20-4': 'In the case one considers [MATH], the result obtained is [MATH].', 'hep-ph-9903217-2-20-5': 'The measurement of this quantity may serve as a test for the sign of the [MATH] product.', 'hep-ph-9903217-2-21-0': 'As for the [MATH] decay, as we pointed above the [MATH] rate is twice the one of the [MATH].', 'hep-ph-9903217-2-21-1': 'We have evaluated the invariant mass distribution for these decay channels and in Fig. 3 we plot the distribution [MATH] for [MATH] which allows us to see the [MATH] contribution since the [MATH] is the important scalar resonance appearing in the [MATH] amplitude [CITATION].', 'hep-ph-9903217-2-21-2': 'The solid curve shows our prediction, with [MATH], the sign predicted by vector meson dominance [CITATION].', 'hep-ph-9903217-2-21-3': 'The dashed curve is obtained considering [MATH].', 'hep-ph-9903217-2-21-4': 'We compare our results with the recent ones of the Novosibirsk experiment [CITATION].', 'hep-ph-9903217-2-21-5': 'We can see that the shape of the spectrum is relatively well reproduced considering statistical and systematic errors (the latter ones not shown in the figure).', 'hep-ph-9903217-2-21-6': 'The results considering [MATH] are in complete disagreement with the data.', 'hep-ph-9903217-2-22-0': 'The finite total branching ratio which we find for [MATH] is [MATH] and correspondingly [MATH] for the [MATH].', 'hep-ph-9903217-2-22-1': 'This latter number is slightly smaller than the result given in [CITATION], [MATH], where the first error is statistical and the second one systematic.', 'hep-ph-9903217-2-22-2': 'The result given in [CITATION] is [MATH], compatible with our prediction.', 'hep-ph-9903217-2-22-3': 'The branching ratio measured in [CITATION] for [MATH] is [MATH].', 'hep-ph-9903217-2-23-0': 'The branching ratio obtained for the case [MATH] is [MATH].', 'hep-ph-9903217-2-23-1': 'The results obtained at Novosibirsk are [CITATION] [MATH] and [CITATION] [MATH].', 'hep-ph-9903217-2-23-2': 'The spectrum, not shown, is dominated by the [MATH] contribution.', 'hep-ph-9903217-2-24-0': 'The contribution of [MATH], obtained by integrating [MATH] assuming an approximate Breit-Wigner form to the left of the [MATH] peak, gives us a branching ratio [MATH].', 'hep-ph-9903217-2-24-1': 'As argued above, the branching ratio for [MATH] is one half of [MATH], which should not be compared to the one given in [CITATION] since there the assumption that all the strength of the spectrum is due to the [MATH] excitation is done.', 'hep-ph-9903217-2-24-2': 'As one can see in Fig. 3, we find also an appreciable strength for [MATH].', 'hep-ph-9903217-2-25-0': 'We should also warn not to compare our predicted rate for [MATH] directly with experiment.', 'hep-ph-9903217-2-25-1': 'Indeed, the experiment is done using the reaction [MATH], which interferes with the [MATH] contribution [MATH] at the tail of the [MATH] mass distribution in the [MATH] mass region [CITATION].', 'hep-ph-9903217-2-25-2': 'Also the results in [CITATION] are based on model dependent assumptions.', 'hep-ph-9903217-2-25-3': 'For these reasons, as quoted in [CITATION], the [MATH] mode is more efficient to study the [MATH] mass spectrum.', 'hep-ph-9903217-2-26-0': 'Our result for [MATH] is 50 % larger than the one obtained in [CITATION] owed to the use of the unitary [MATH] amplitude instead of the lowest order chiral one.', 'hep-ph-9903217-2-26-1': 'The shape of the distribution found here is, however, rather different than the one obtained in [CITATION], showing the important contribution of the [MATH] resonance which appears naturally in the unitary chiral approach.', 'hep-ph-9903217-2-27-0': 'The [MATH] decay has been advocated as an important source of information on the nature of the [MATH] resonance and experiments have been conducted at Novosibirsk [CITATION] and are also planned at Frascati [CITATION], trying to magnify the signal for [MATH] production through interference with initial and final state radiation in the [MATH] reaction [CITATION].', 'hep-ph-9903217-2-27-1': 'The completion of the experiments [CITATION] is a significant step forward.', 'hep-ph-9903217-2-28-0': 'Present evaluations for [MATH] are based on models assuming a [MATH] molecule for the [MATH] [CITATION] with a branching ratio 1-[MATH], a [MATH] structure with a value [MATH] [CITATION] and a [MATH] structure with a value [MATH] [CITATION].', 'hep-ph-9903217-2-29-0': 'The model for [MATH] assumed in Fig. 1 is similar to the one of [CITATION] where the production also proceeds via the kaonic loops.', 'hep-ph-9903217-2-29-1': 'There a [MATH] molecule is assumed for the [MATH] resonance while here the realistic [MATH] amplitude of [CITATION] is used.', 'hep-ph-9903217-2-29-2': 'Emphasis is made in the importance of going beyond the zero width approximation for the resonance in [CITATION].', 'hep-ph-9903217-2-29-3': 'Our approach automatically takes this into account since the [MATH] amplitude correctly incorporates the width of the [MATH] resonance [CITATION].', 'hep-ph-9903217-2-30-0': 'We would also like to warn that the peak of the [MATH] seen in Fig. 3 cannot be trivially interpreted as a resonant contribution on top of a background, since there are important interference effects between the [MATH] production and the [MATH] background.', 'hep-ph-9903217-2-30-1': 'The strength of the peak comes in our case in about equal amounts from the real and the imaginary parts of the amplitude for the process.', 'hep-ph-9903217-2-31-0': 'The agreement found between our results for the [MATH], [MATH] and experiment provides an important endorsement for the chiral unitary approach used here.', 'hep-ph-9903217-2-31-1': 'Improvements in the future, reducing the experimental errors, should put further constraints on avalaible theoretical approaches for this reaction.'}

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[]

[['hep-ph-9903217-1-9-1', 'hep-ph-9903217-2-16-1'], ['hep-ph-9903217-1-11-0', 'hep-ph-9903217-2-18-1'], ['hep-ph-9903217-1-14-2', 'hep-ph-9903217-2-21-4'], ['hep-ph-9903217-1-15-1', 'hep-ph-9903217-2-22-1'], ['hep-ph-9903217-1-8-0', 'hep-ph-9903217-2-9-0']]

[]

['hep-ph-9903217-1-1-0', 'hep-ph-9903217-1-1-1', 'hep-ph-9903217-1-1-2', 'hep-ph-9903217-1-1-3', 'hep-ph-9903217-2-1-0', 'hep-ph-9903217-2-1-1', 'hep-ph-9903217-2-1-2', 'hep-ph-9903217-2-1-3', 'hep-ph-9903217-2-15-0']

{'1': 'http://arxiv.org/licenses/assumed-1991-2003/', '2': 'http://arxiv.org/licenses/assumed-1991-2003/'}

https://arxiv.org/abs/hep-ph/9903217

1312.0576

{'1312.0576-1-0-0': 'Based on a variant of frequency function, we improve the vanishing order of solutions for Schr[MATH]dinger equations which describes quantitative behavior of strong uniqueness continuation property.', '1312.0576-1-0-1': 'For the first time, we investigate the quantitative uniqueness of higher order elliptic equations and show the vanishing order of solutions.', '1312.0576-1-0-2': 'Furthermore, strong unique continuation is established for higher order elliptic equations using this variant of frequency function.', '1312.0576-1-1-0': '# Introduction', '1312.0576-1-2-0': 'We say the vanishing order of solution at [MATH] is [MATH], if [MATH] is the largest integer such that [MATH] for all [MATH].', '1312.0576-1-2-1': 'It describes quantitative behavior of strong unique continuation property.', '1312.0576-1-2-2': 'It is well known that all zeros of nontrivial solutions of second order linear equations on smooth compact Riemannian manifold are of finite order [CITATION].', '1312.0576-1-2-3': 'In the papers [CITATION], [CITATION], Donnelly and Fefferman showed that if [MATH] is an eigenfunction on a compact smooth Riemannian manifold [MATH], that is, [EQUATION] for some [MATH], then the maximal vanishing order of [MATH] on [MATH] is less than [MATH], here [MATH] only depends on the manifold [MATH].', '1312.0576-1-2-4': 'Kukavica in [CITATION] considered the vanishing order of solutions of Schr[MATH]dinger equation [EQUATION] where [MATH].', '1312.0576-1-2-5': 'He established that the vanishing order of solution in ([REF]) is everywhere less than [EQUATION] where [MATH] and [MATH] only depends on the underlying domain [MATH].', '1312.0576-1-2-6': 'If [MATH], he showed that the upper bound of vanishing order is less than [MATH], where [MATH].', '1312.0576-1-2-7': "Based on the Donnelly and Fefferman's work in [CITATION], Kukavica conjectured that the rate of of vanishing order of [MATH] is less than [MATH] for the cases of [MATH] and [MATH].", '1312.0576-1-2-8': "While the upper bound he obtained agrees with Donnelly and Fefferman's results in the eigenvalue case [MATH].", '1312.0576-1-2-9': 'Recently Kenig [CITATION] considered a similar problem which is motivated by his work with Bourgain in [CITATION] on Anderson localization for the Bernoulli model.', '1312.0576-1-2-10': 'Kenig investigated the following normalized model.', '1312.0576-1-2-11': 'Let [EQUATION] where [MATH] is a ball centered at origin with radius [MATH] in [MATH].', '1312.0576-1-2-12': 'Assume that [MATH] and [MATH].', '1312.0576-1-2-13': 'Kenig established the following result [EQUATION] where [MATH] depends only on [MATH], [MATH] and [MATH] depends on [MATH].', '1312.0576-1-2-14': 'By exploiting the Carleman estimates, Kenig proved the [MATH].', '1312.0576-1-2-15': "He also pointed out that the exponent [MATH] of [MATH] is sharp for complex valued [MATH] based on Meshkov's example in [CITATION].", '1312.0576-1-2-16': "On the basis of Donnelly and Fefferman's work, Kenig asked if [MATH] can be achieved for real [MATH].", '1312.0576-1-2-17': 'Very recently, Bakri in [CITATION] considered ([REF]) in the case of [MATH].', '1312.0576-1-2-18': 'He obtained that the vanishing order of solutions in ([REF]) is less than [MATH].', '1312.0576-1-2-19': 'His proof is an extension of the Carleman estimates in [CITATION].', '1312.0576-1-2-20': 'It is worthwhile to mention that the vanishing order of solutions is closely related to eigenfunctions on manifold.', '1312.0576-1-2-21': 'We refer to the survey [CITATION] for detailed account.', '1312.0576-1-3-0': 'Our first goal in this paper is to address the above problems.', '1312.0576-1-3-1': 'Relied on a variant of frequency function, we are able to verify that [MATH] is indeed true for the case of [MATH].', '1312.0576-1-3-2': 'In particular, our result also confirms that the vanishing order of solutions in ([REF]) is less than [MATH] if [MATH].', '1312.0576-1-4-0': 'Assume that [MATH].', '1312.0576-1-4-1': 'Under the assumptions in ([REF]) with [MATH], the maximal order of vanishing of [MATH] in ([REF]) is less than [MATH], where [MATH] depends on [MATH] and [MATH].', '1312.0576-1-5-0': 'Generally speaking, the Carleman estimates and frequency function are two principal ways to obtain quantitative uniqueness results and strong unique continuation for solutions of partial differential equations.', '1312.0576-1-5-1': 'Carleman estimates were introduced by Carleman, when he studied the strong unique continuation property.', '1312.0576-1-5-2': 'Carleman estimates are weighted integral inequalities.', '1312.0576-1-5-3': 'See e.g. [CITATION], [CITATION], [CITATION], [CITATION], [CITATION], [CITATION], to just mention a few.', '1312.0576-1-5-4': "In order to obtain the quantitative uniqueness results for solutions, one uses the Carleman estimates with a special choice of weight functions to obtain a type of Hadamard's three-ball theorem, then doubling estimates follow.", '1312.0576-1-5-5': 'The vanishing order will come from the doubling estimates.', '1312.0576-1-5-6': 'The frequency function controls the local growth rate of [MATH] and is a local measure of its "degree" as a polynomial like function in [MATH].', '1312.0576-1-5-7': 'See e.g. [CITATION], [CITATION], [CITATION], [CITATION], [CITATION], [CITATION], etc.', '1312.0576-1-5-8': 'The frequency function was first observed by Almgren for harmonic functions.', '1312.0576-1-5-9': 'Garofalo and Lin in [CITATION], [CITATION] showed its powerful applications in strong unique continuation problem.', '1312.0576-1-5-10': 'The frequency function Garofalo and Lin investigated is given by [EQUATION] where [MATH] and [MATH].', '1312.0576-1-5-11': 'After one proves the certain monotonicity of [MATH], the doubling estimates will follow by a standard argument.', '1312.0576-1-5-12': 'In [CITATION], it was shown that [MATH] was monotone nondecreasing, However, [MATH] depends on the norm of [MATH].', '1312.0576-1-5-13': 'It does not give the optimal bound of the vanishing order.', '1312.0576-1-5-14': 'Kukavica considered almost the same frequency function in [CITATION].', '1312.0576-1-5-15': 'He was able to move the norm of [MATH] away from the exponential, but it only gave the aforementioned bound [MATH] due to the limitations of the method.', '1312.0576-1-5-16': 'Some of the limitations come from the fact that one can not explore [MATH] more because of its integration on the boundary of balls.', '1312.0576-1-5-17': 'Instead we consider a variant of ([REF]).', '1312.0576-1-5-18': 'See the frequency function of Schr[MATH]dinger equations in section 2 and the frequency functions of high order elliptic equations in section 3 for the details.', '1312.0576-1-5-19': 'First, we establish a monotonicity property of this new variant of frequency function.', '1312.0576-1-5-20': "Second, it leads to a [MATH] version of Hadamard's three-ball theorem, which further implies a [MATH] version of Hadamard's three-ball theorem.", '1312.0576-1-5-21': 'At last, by a propagation of smallness argument, we derive the vanishing order of solutions.', '1312.0576-1-6-0': 'A nature question is to study the quantitative uniqueness of higher order elliptic equations.', '1312.0576-1-6-1': 'Our second goal is to investigate the vanishing order for solutions of higher order elliptic equations.We consider this normalized model.', '1312.0576-1-6-2': '[EQUATION]', '1312.0576-1-6-3': 'We also assume that [MATH] and [MATH].', '1312.0576-1-6-4': 'To the best of our knowledge, the explicit vanishing order like Theorem [REF] is still unknown for higher order elliptic equations.', '1312.0576-1-6-5': 'By exploiting this variant of frequency function, we are able to obtain the following theorem.', '1312.0576-1-7-0': 'Assume that [MATH] and [MATH].', '1312.0576-1-7-1': 'Under the assumption in ([REF]), the maximal order of vanishing of [MATH] in ([REF]) is less than [MATH], where [MATH] depends on [MATH], [MATH] and [MATH].', '1312.0576-1-8-0': 'Unlike the Laplacian operator in ([REF]), new difficulties arise since some kind of "symmetry" is lost for higher order elliptic equations.', '1312.0576-1-8-1': 'Our idea is to break the higher order elliptic equations into a system of semilinear equations.', '1312.0576-1-8-2': 'However, it still does not give the most desirable result as Theorem [REF].', '1312.0576-1-8-3': "We also develop a [MATH] version of Hadamard's three-ball theorem by exploring [MATH] estimates for higher order elliptic equations (see section 3 for the details).", '1312.0576-1-8-4': 'Compared with the frequency function argument, it seems to more difficult to obtain the vanishing order of solutions for higher order elliptic equations by Carleman estimates.', '1312.0576-1-9-0': 'The vanishing order of solutions plays an important role in [CITATION] on Anderson localization for the Bernoulli model.', '1312.0576-1-9-1': 'Suppose that [MATH] is a solution of [EQUATION] where [MATH].', '1312.0576-1-9-2': 'Let [EQUATION] for [MATH] large.', '1312.0576-1-9-3': 'By using the result of ([REF]), Bourgain and Kenig in [CITATION] showed that [EQUATION] where [MATH] depends on [MATH] and [MATH].', '1312.0576-1-9-4': 'We also consider a similar quantitative unique continuation problem as ([REF]) for higher order elliptic equations.', '1312.0576-1-9-5': 'Suppose that [MATH] is a solution to [EQUATION] where [MATH].', '1312.0576-1-9-6': 'Theorem [REF] implies that following corollary for higher order elliptic equations.', '1312.0576-1-10-0': 'Let [MATH] be a solution to ([REF]) and [MATH].', '1312.0576-1-10-1': 'Then [EQUATION] where [MATH] depends on [MATH], [MATH] and [MATH].', '1312.0576-1-11-0': 'An easy consequence of Theorem [REF] is a strong unique continuation result for higher order elliptic equations when [MATH].', '1312.0576-1-11-1': 'Due to Theorem [REF], the solutions will not vanish of infinite order when [MATH].', '1312.0576-1-11-2': 'Without using the conclusion of Theorem [REF], we are also able to give a proof based on this variant of frequency function.', '1312.0576-1-11-3': 'It seems to be the first time for the frequency function to be applied in proving strong unique continuation results for higher order elliptic equations.', '1312.0576-1-11-4': 'We refer to [CITATION] for the strong unique continuation results of higher order elliptic equations by using Carleman estimates.', '1312.0576-1-11-5': 'Assume that [EQUATION] where [MATH].', '1312.0576-1-11-6': 'A function [MATH] is said to vanish of infinite order at [MATH] if for [MATH] sufficiently small, [EQUATION] for every positive integer [MATH].', '1312.0576-1-11-7': 'We are able to establish the following theorem.', '1312.0576-1-11-8': 'If [MATH] in ([REF]) vanishes of infinite order at [MATH], then [MATH] in [MATH].', '1312.0576-1-12-0': 'The outline of the paper is as follows.', '1312.0576-1-12-1': 'Section 2 is devoted to obtaining the vanishing order of Schr[MATH]dinger equations.', '1312.0576-1-12-2': 'In Section 3, the vanishing order of higher order elliptic equations is shown.', '1312.0576-1-12-3': 'In section 4, we obtain the strong unique continuation for higher order elliptic equations.', '1312.0576-1-12-4': 'In the whole paper, we will use various letters, such as [MATH], [MATH], [MATH], [MATH], to denote the positive constants which may depend [MATH] and [MATH], even if they are not explicitly stated.', '1312.0576-1-12-5': 'They may also vary from line to line.', '1312.0576-1-12-6': 'Especially the letters do not depend on [MATH], [MATH] in section 2 and section 3.', '1312.0576-1-13-0': '# Schr[MATH]dinger equations', '1312.0576-1-14-0': 'In this section, we focus on the maximal vanishing order of solutions in ([REF]).', '1312.0576-1-14-1': 'Let [MATH].', '1312.0576-1-14-2': 'We define [EQUATION]', '1312.0576-1-14-3': 'The value of [MATH] will be determined later on.', '1312.0576-1-14-4': 'Without loss of generality, we may assume [MATH] and denote [MATH] as [MATH], that is, [EQUATION].', '1312.0576-1-14-5': 'The advantage of the weight function [MATH] in the integration is that the boundary term will not appear whenever we use divergence theorem.', '1312.0576-1-14-6': 'Moreover, the value of [MATH] will help reduce the order of vanishing.', '1312.0576-1-14-7': 'The function in ([REF]) appeared in [CITATION] for the study of vortex of Ginzburg-Landau equations.', '1312.0576-1-14-8': 'We will also omit the integration on [MATH] when it is clear from the context.', '1312.0576-1-14-9': 'Taking the derivative with respect to [MATH] for [MATH], we get [EQUATION]', '1312.0576-1-14-10': 'Applying the divergence theorem for the second integral in the latter equality, [EQUATION] where [MATH].', '1312.0576-1-15-0': 'Using the divergence theorem again for [MATH], it follows that [EQUATION]', '1312.0576-1-15-1': 'We define the frequency function as [EQUATION]', '1312.0576-1-15-2': 'We are going to study the monotonicity properties of the frequency function [MATH].', '1312.0576-1-16-0': 'There exists a constant [MATH] depending only on [MATH] such that [EQUATION] is nondecreasing function of [MATH].', '1312.0576-1-17-0': 'By taking the derivative for [MATH] in ([REF]) with respect to [MATH] and using the divergence theorem twice, we have [EQUATION]', '1312.0576-1-17-1': 'Application of the divergence theorem on the second term with respect to [MATH]th derivative for the last inequality gives that [EQUATION]', '1312.0576-1-17-2': 'In view of ([REF]), we have [EQUATION]', '1312.0576-1-17-3': 'If one breaks down the second term in the last equality, one obtains [EQUATION]', '1312.0576-1-17-4': 'Applying the divergence theorem for the last term and the fact that [MATH], we arrive at [EQUATION]', '1312.0576-1-17-5': 'Hence, simple calculations lead to [EQUATION]', '1312.0576-1-17-6': 'Since [MATH], [EQUATION]', '1312.0576-1-17-7': 'In order to find the monotonicity of [MATH], it suffices to take the derivative for [MATH] with respect to t. Hence [EQUATION]', '1312.0576-1-17-8': 'By Cauchy-Schwarz inequality, we have [EQUATION] which implies the conclusion in the lemma.', '1312.0576-1-18-0': 'Unlike the monotonicity results in [CITATION], the function [MATH] is moved away from the exponential.', '1312.0576-1-18-1': 'More important, the positive position [MATH] and the radius [MATH] do not depend on [MATH] in Lemma [REF].', '1312.0576-1-18-2': 'The fact that [MATH] is independent of [MATH] is crucial in the propagation of smallness arguments in the proof of Theorem [REF].', '1312.0576-1-18-3': "With the help of monotonicity of [MATH], we are going to establish a [MATH] version of Hadamard's three-ball theorem.", '1312.0576-1-18-4': "For the variants of Hadamard's three-ball theorem, see e.g. [CITATION].", '1312.0576-1-18-5': 'However, we have to get rid of the weight function [MATH] in our function [MATH].', '1312.0576-1-18-6': 'Let [EQUATION].', '1312.0576-1-18-7': 'Without loss of generality, we may assume [MATH].', '1312.0576-1-18-8': 'We can easily check that [EQUATION] and [EQUATION] for any [MATH].', '1312.0576-1-19-0': 'Let [MATH], then [EQUATION] where [EQUATION] and [EQUATION] since [EQUATION]', '1312.0576-1-19-1': 'Taking integration from [MATH] to [MATH] in the last identity gives that [EQUATION].', '1312.0576-1-19-2': 'With the aid of Lemma [REF], it follows that [EQUATION] that is, [EQUATION]', '1312.0576-1-19-3': 'If we do similar calculations by integrating from [MATH] to [MATH], we deduce that [EQUATION] that is, [EQUATION]', '1312.0576-1-19-4': 'Combining the inequalities ([REF]) and ([REF]), note that [MATH], we conclude that [EQUATION]', '1312.0576-1-19-5': 'Thanks to ([REF]) and ([REF]), we have [EQUATION]', '1312.0576-1-19-6': 'Therefore, [EQUATION]', '1312.0576-1-19-7': 'We conduct the similar calculations as above for [MATH].', '1312.0576-1-19-8': 'Using ([REF]) and ([REF]) again, [EQUATION]', '1312.0576-1-19-9': 'So we obtain that [EQUATION]', '1312.0576-1-19-10': 'Taking ([REF]), ([REF]) and ([REF]) into account, we get [EQUATION].', '1312.0576-1-19-11': 'Namely, [EQUATION].', '1312.0576-1-19-12': 'Taking exponentials of both sides implies that [EQUATION].', '1312.0576-1-19-13': 'Note that [MATH].', '1312.0576-1-19-14': 'As we know, the minimum value of the exponential function in the last inequality will be achieved if we take [MATH].', '1312.0576-1-19-15': 'Hence [EQUATION] where [MATH] is a constant depending only on [MATH].', '1312.0576-1-20-0': 'From the above lemma, one sees that the appearance of [MATH] reduces the exponent of exponential in the [MATH] version of three-ball theorem.', '1312.0576-1-20-1': 'Thanks to Lemma [REF], we are able to establish a [MATH] version of three-ball theorem, which is crucial in the propagation of smallness argument.', '1312.0576-1-21-0': 'Let [MATH], [EQUATION] where [EQUATION] and [EQUATION].', '1312.0576-1-22-0': 'Using a standard elliptic theory for the solution in [MATH], [EQUATION] here [MATH] does not depends on [MATH].', '1312.0576-1-22-1': 'By some rescaling argument, [EQUATION] for [MATH].', '1312.0576-1-22-2': 'Then [EQUATION].', '1312.0576-1-22-3': 'Taking advantage of Lemma [REF], we deduce that [EQUATION].', '1312.0576-1-22-4': 'Then we arrive at the conclusion.', '1312.0576-1-23-0': 'Now we are ready to prove Theorem [REF].', '1312.0576-1-23-1': 'We apply the idea of propagation of smallness which is based on overlapping of three-ball argument.', '1312.0576-1-23-2': 'Similar arguments have been employed in [CITATION] and [CITATION].', '1312.0576-1-23-3': '[Proof of Theorem [REF]] We choose a small [MATH] such that [EQUATION].', '1312.0576-1-23-4': 'Obviously, [MATH].', '1312.0576-1-23-5': 'Since [MATH], there exists some [MATH] such that [MATH].', '1312.0576-1-23-6': 'We select a sequence of balls with radius [MATH] centered at [MATH] so that [MATH] and [MATH], where [MATH] depends on the radius [MATH] which we will fix later on.', '1312.0576-1-23-7': 'Employing Lemma [REF] with [MATH], [MATH], and [MATH] and the boundedness assumption of [MATH], we get [EQUATION] where [MATH] and [MATH] depends on the [MATH] norm of [MATH].', '1312.0576-1-24-0': 'Iterating the above argument with Lemma [REF] for balls centered at [MATH] and using the fact that [MATH], we have [EQUATION] for [MATH], where [MATH] is a constant depending on [MATH] and [MATH] norm of [MATH], and [MATH], [MATH] are constants depending on [MATH].', '1312.0576-1-24-1': 'By the fact that [MATH] and [MATH], we obtain [EQUATION] where [MATH] is a constant depending on [MATH] and [MATH], and [MATH] is a constant depending on [MATH].', '1312.0576-1-25-0': 'Applying the [MATH] type of three-ball lemma again centered at origin with [MATH] and [MATH], where [MATH] is sufficiently small, we have [EQUATION].', '1312.0576-1-25-1': 'Recall that [MATH] is the [MATH] norm of [MATH] in [MATH].', '1312.0576-1-25-2': 'Then [EQUATION] where [MATH] depends on [MATH] and [MATH] norm of [MATH], and [MATH] depends on [MATH], [EQUATION] with constant [MATH] depending on [MATH].', '1312.0576-1-25-3': 'Now we can fix the small [MATH].', '1312.0576-1-25-4': 'For instance, let [MATH].', '1312.0576-1-25-5': 'Then the number [MATH] is also determined.', '1312.0576-1-25-6': 'The inequality ([REF]) implies that [EQUATION] where the constants [MATH] depend on the dimension [MATH] and [MATH].', '1312.0576-1-25-7': 'Therefore Theorem 1 is completed.', '1312.0576-1-26-0': '# Higher order elliptic equations', '1312.0576-1-27-0': 'In this section, we consider the vanishing order of solutions for the higher order elliptic equations.', '1312.0576-1-27-1': 'As far as we know, it has not been investigated yet in the literature.', '1312.0576-1-27-2': 'Due to the higher derivatives of [MATH] which involve much calculations, we decompose the equation in ([REF]) into a system of [MATH] semilinear equations, that is, [EQUATION]', '1312.0576-1-27-3': 'Note that [MATH].', '1312.0576-1-27-4': 'It is nature to consider the following function for the system of semilinear equations in ([REF]).', '1312.0576-1-27-5': 'Let [EQUATION].', '1312.0576-1-27-6': 'As before, we may assume [MATH] and omit the integration on [MATH] if it is clear from the context, that is, [EQUATION].', '1312.0576-1-27-7': 'The value of [MATH] will be determined later on.', '1312.0576-1-27-8': 'If one takes derivative for [MATH] with respect to [MATH], one has [EQUATION]', '1312.0576-1-27-9': 'Performing the divergence theorem for the second term in the last equality, we obtain that [EQUATION] where [EQUATION].', '1312.0576-1-28-0': 'For the higher order elliptic equations, we define the frequency function as [EQUATION]', '1312.0576-1-28-1': 'We are able to obtain the following the monotonicity properties for the frequency function [MATH].', '1312.0576-1-29-0': 'There exists a constant [MATH] depending only on [MATH] such that [EQUATION] is nondecreasing function of [MATH].', '1312.0576-1-30-0': 'Applying the divergence theorem on [MATH], we have [EQUATION]', '1312.0576-1-30-1': 'With the help of ([REF]), we know that [EQUATION]', '1312.0576-1-30-2': 'Now differentiating [MATH] in ([REF]) with respect to [MATH], [EQUATION]', '1312.0576-1-30-3': 'Applying the divergence theorem for the second term, [EQUATION]', '1312.0576-1-30-4': 'If one performs divergence theorem with respect to [MATH]th derivative on the second term, one has [EQUATION]', '1312.0576-1-30-5': 'Using the equivalent system of equations in ([REF]), it follows that [EQUATION]', '1312.0576-1-30-6': 'Simplifying [MATH] with the aid of ([REF]), we have [EQUATION]', '1312.0576-1-30-7': 'Now we estimate each integration in the last equality.', '1312.0576-1-30-8': "Using H[MATH]lder's inequality, [EQUATION] and [EQUATION]", '1312.0576-1-30-9': "Similarly, by H[MATH]lder's inequality, [EQUATION]and [EQUATION]", '1312.0576-1-30-10': 'For ease of the notation, let [EQUATION].', '1312.0576-1-30-11': 'Combining the inequalities ([REF]) and ([REF]) and taking ([REF]) into considerations, we get [EQUATION]', '1312.0576-1-30-12': 'Therefore, together with ([REF]), ([REF]), and ([REF]), [EQUATION] where [MATH] depends only on [MATH] and [MATH].', '1312.0576-1-30-13': 'In order to get monotonicity of the frequency function, it suffices to differentiate [MATH].', '1312.0576-1-30-14': 'Hence [EQUATION]', '1312.0576-1-30-15': 'By Cauchy-Schwartz inequality, [EQUATION].', '1312.0576-1-30-16': 'Consequently, [EQUATION]', '1312.0576-1-30-17': 'As the conclusion in Lemma [REF], we move the function [MATH] away from the exponential and [MATH] does not depend on [MATH].', '1312.0576-1-30-18': 'We are ready to establish a [MATH] version of three-ball theorem for [MATH].', '1312.0576-1-30-19': 'For convenience, let [EQUATION].', '1312.0576-1-30-20': 'We also need to remove the weight function [MATH] in [MATH].', '1312.0576-1-30-21': 'Let [EQUATION].', '1312.0576-1-30-22': 'As usual, we will omit the dependent of the center of [MATH] for the ball.', '1312.0576-1-30-23': 'It is easy to check that [EQUATION] and [EQUATION] for any [MATH].', '1312.0576-1-31-0': 'Based on the monotonicity of [MATH] in the last lemma, we are able to establish the following [MATH] type of three-ball theorem.', '1312.0576-1-32-0': 'Let [MATH].', '1312.0576-1-32-1': 'Then [EQUATION] where [EQUATION].', '1312.0576-1-33-0': 'On one hand, integrating from [MATH] to [MATH] on the equality ([REF]) gives that [EQUATION] where we have used Lemma [REF] in the last inequality.', '1312.0576-1-33-1': 'Namely, [EQUATION]', '1312.0576-1-33-2': 'On the other hand, integrating from [MATH] to [MATH] on the equality([REF]) implies that [EQUATION] that is, [EQUATION]', '1312.0576-1-33-3': 'Taking ([REF]) and ([REF]) into considerations, we get [EQUATION]', '1312.0576-1-33-4': 'Thanks to ([REF]) and ([REF]), [EQUATION]', '1312.0576-1-33-5': 'Therefore, [EQUATION]', '1312.0576-1-33-6': 'We do the similar calculations for [MATH].', '1312.0576-1-33-7': 'Using ([REF]) and ([REF]) again, [EQUATION]', '1312.0576-1-33-8': 'Hence [EQUATION]', '1312.0576-1-33-9': 'Taking ([REF]), ([REF]) and ([REF]) into account, we have [EQUATION].', '1312.0576-1-33-10': 'Namely, [EQUATION].', '1312.0576-1-34-0': 'Taking exponentials of both sides and performing some calculations, we obtain [EQUATION].', '1312.0576-1-34-1': 'Since [EQUATION] we have [EQUATION].', '1312.0576-1-34-2': 'As we know, the minimum value of [MATH] is achieved if [MATH].', '1312.0576-1-34-3': 'Recall that [MATH].', '1312.0576-1-34-4': 'Therefore, the lemma is completed.', '1312.0576-1-35-0': 'Again we need to establish a [MATH] version of three-ball theorem.', '1312.0576-1-35-1': 'However, the classical elliptic estimates like ([REF]) does not seem to be known for higher order elliptic equations in the literature.', '1312.0576-1-35-2': 'We will deduce a similar estimate by Sobolev inequality and a [MATH] type estimate.', '1312.0576-1-35-3': 'We first present a [MATH] type estimates for higher order elliptic equations (see e.g. [CITATION]).', '1312.0576-1-35-4': 'Let [MATH] satisfy the following equation [EQUATION]', '1312.0576-1-35-5': 'Then we have', '1312.0576-1-36-0': 'Let [MATH].', '1312.0576-1-36-1': 'Suppose [MATH] satisfies ([REF]).', '1312.0576-1-36-2': 'Then there exits a constant [MATH] depending only on [MATH] such that for any [MATH], [EQUATION]', '1312.0576-1-36-3': 'Upon a rescaling argument, we have [EQUATION] for [MATH].', '1312.0576-1-37-0': 'Applying Lemma [REF], we are able to establish the [MATH] version of three-ball theorem for the solutions in ([REF]).', '1312.0576-1-38-0': 'Let [MATH] and [MATH], [EQUATION] where [EQUATION] and [EQUATION].', '1312.0576-1-39-0': 'Thanks to Lemma [REF] in the case of [MATH], we can estimate the solution in ([REF]) by the following [EQUATION].', '1312.0576-1-39-1': 'By Sobolev imbedding inequality, if [MATH], [EQUATION] where [MATH] depends on [MATH], [MATH] and [MATH].', '1312.0576-1-39-2': 'Applying Lemma [REF] again with [MATH] and the latter inequality, note that [MATH], [EQUATION]', '1312.0576-1-39-3': 'As we know [EQUATION].', '1312.0576-1-39-4': 'Employing the above bootstrap argument finite times, e.g. [MATH] times, which depends only on [MATH] and [MATH] and using the above Sobolev imbedding inequality, we get [EQUATION].', '1312.0576-1-39-5': 'Let [MATH], [EQUATION] where [MATH] depends on only [MATH] and [MATH].', '1312.0576-1-39-6': 'If [MATH], we will have the similar result by applying the bootstrap arguments twice.', '1312.0576-1-39-7': 'By a rescaling argument, we have [EQUATION] if [MATH].', '1312.0576-1-39-8': 'Furthermore, we get [EQUATION] for [MATH].', '1312.0576-1-39-9': 'Recall that [MATH].', '1312.0576-1-39-10': 'Then [EQUATION].', '1312.0576-1-39-11': 'Based on Lemma [REF] and the latter inequality, we deduce that [EQUATION] where [EQUATION] and [EQUATION].', '1312.0576-1-39-12': 'Taking Lemma [REF] and ([REF]) into account, we have [EQUATION] and [EQUATION].', '1312.0576-1-39-13': 'Since [EQUATION] if [MATH] and [MATH].', '1312.0576-1-39-14': 'From ([REF]), we obtain [EQUATION].', '1312.0576-1-39-15': 'By a rescaling argument, we arrive at the conclusion of the lemma.', '1312.0576-1-40-0': 'We begin to prove Theorem [REF].', '1312.0576-1-40-1': 'The idea is similar to the proof of Theorem [REF].', '1312.0576-1-40-2': 'We also use the propagation of smallness argument.', '1312.0576-1-40-3': '[Proof of Theorem [REF]] We choose a small [MATH] such that [EQUATION] where [MATH].', '1312.0576-1-40-4': 'Since [MATH], there should exist some [MATH] such that [MATH].', '1312.0576-1-40-5': 'We select a sequence of balls with radius [MATH] centered at [MATH] so that [MATH] and [MATH], where [MATH] depends on the radius [MATH] which is to be fixed.', '1312.0576-1-40-6': 'Employing the [MATH] version of three-ball lemma (i.e. Lemma [REF]) with [MATH], [MATH], and [MATH] and the boundedness assumption of [MATH], we get [EQUATION] where [MATH] and [MATH] depends on the [MATH] norm of [MATH], [MATH] and [MATH].', '1312.0576-1-41-0': 'Iterating the above argument with [MATH] version of three-ball lemma for ball centered at [MATH] and using the fact that [MATH], we have [EQUATION] for [MATH], where [MATH] is a constant depending on [MATH] and [MATH], and [MATH], [MATH], [MATH] are constants depending on [MATH].', '1312.0576-1-41-1': 'By the fact that [MATH] and [MATH], we obtain [EQUATION] where [MATH] is a constant depending on [MATH] and [MATH], and [MATH], [MATH] are constants depending on [MATH].', '1312.0576-1-42-0': 'Applying Lemma [REF] again centered at origin with [MATH] and [MATH], where [MATH] is sufficiently small, we have [EQUATION].', '1312.0576-1-42-1': 'Recall that [MATH] is the [MATH] norm for [MATH] in [MATH].', '1312.0576-1-42-2': 'Then [EQUATION] where [MATH] depends on [MATH] and [MATH] norm of [MATH], and [MATH], [MATH] depend on [MATH], [EQUATION] with constant [MATH] depending on [MATH].', '1312.0576-1-42-3': 'At this moment we fix the small value of [MATH].', '1312.0576-1-42-4': 'For instance, let [MATH].', '1312.0576-1-42-5': 'Then the value of [MATH] is determined too.', '1312.0576-1-42-6': 'The inequality ([REF]) implies that [EQUATION] where the constants [MATH] depend on the dimension [MATH], [MATH], and [MATH].', '1312.0576-1-42-7': 'The proof of Theorem [REF] is arrived.', '1312.0576-1-43-0': 'Relied on Theorem [REF], we are able to prove the following corollary for higher order elliptic equations in ([REF]).', '1312.0576-1-43-1': '[Proof of Corollary [REF]] We adapt the proof in [CITATION].', '1312.0576-1-43-2': 'Since [MATH] is continuous, we can find [MATH] so that [MATH].', '1312.0576-1-43-3': 'Let [EQUATION].', '1312.0576-1-43-4': 'Then [EQUATION] with [MATH] and [MATH].', '1312.0576-1-43-5': 'So [MATH] in the notation of Theorem [REF].', '1312.0576-1-43-6': 'If [MATH], then [MATH] and [MATH].', '1312.0576-1-43-7': 'Hence [MATH].', '1312.0576-1-43-8': 'Note that [MATH], where [MATH].', '1312.0576-1-43-9': 'The conclusion in Theorem [REF] leads to [EQUATION] where [MATH] depends on [MATH], [MATH] and [MATH].', '1312.0576-1-43-10': 'Hence the corollary follows.', '1312.0576-1-44-0': '# Strong unique continuation', '1312.0576-1-45-0': 'In the rest of the paper, we will show the strong unique continuation result for higher order elliptic equations by the monotonicity of frequency function.', '1312.0576-1-45-1': 'This variant of frequency function is also powerful in obtaining unique continuation results.', '1312.0576-1-45-2': 'For strong unique continuation results of semilinear equations using frequency function, we refer to the pioneering works in [CITATION] and [CITATION].', '1312.0576-1-45-3': 'Let [MATH] be the solution in ([REF]).', '1312.0576-1-45-4': 'Since we do not need to control the vanishing order of solutions, we assume [MATH] for [MATH], i.e. [EQUATION].', '1312.0576-1-45-5': 'We can check that [EQUATION] where [EQUATION].', '1312.0576-1-45-6': 'We consider the following frequency function [EQUATION]', '1312.0576-1-45-7': 'Similar arguments as the proof of Lemma [REF] lead to following conclusion.', '1312.0576-1-46-0': 'There exists a constant [MATH] depending only on [MATH] such that [EQUATION] is nondecreasing function of [MATH].', '1312.0576-1-47-0': 'Based on the monotonicity property in above lemma, we are able to show the proof of Theorem [REF].', '1312.0576-1-48-0': '[Proof of Theorem [REF]] By the equality ([REF]), we get [EQUATION] where [EQUATION].', '1312.0576-1-48-1': 'Recall that [MATH].', '1312.0576-1-48-2': 'Integrating from [MATH] to [MATH] for the equality ([REF]), we obtain [EQUATION] where [MATH] depends on [MATH] and [MATH].', '1312.0576-1-48-3': 'Taking exponential of both sides, [EQUATION] that is, [EQUATION]', '1312.0576-1-48-4': 'From the decomposition in ([REF]) and scaling arguments in ([REF]), we have [EQUATION] where [MATH] depends on [MATH].', '1312.0576-1-48-5': 'Therefore, with the aid of ([REF]), [EQUATION].', '1312.0576-1-48-6': 'Hence we get a doubling type estimate [EQUATION] where [MATH] depends on [MATH], [MATH], [MATH] and [MATH].', '1312.0576-1-48-7': 'Now we fix [MATH] and prove that [MATH] on [MATH] from ([REF]).', '1312.0576-1-48-8': 'The argument is standard.', '1312.0576-1-48-9': 'See e.g. [CITATION] on page 256-257.', '1312.0576-1-48-10': '[EQUATION] where [MATH] to be fixed.', '1312.0576-1-48-11': 'We choose [MATH] such that [MATH].', '1312.0576-1-48-12': 'It yields that [EQUATION] because of ([REF]).', '1312.0576-1-48-13': 'Then [MATH] in [MATH].', '1312.0576-1-48-14': 'Hence the proof of Theorem [REF] follows.'}

{'1312.0576-2-0-0': 'Based on a variant of frequency function, we improve the vanishing order of solutions for Schrodinger equations which describes quantitative behavior of strong uniqueness continuation property.', '1312.0576-2-0-1': 'For the first time, we investigate the quantitative uniqueness of higher order elliptic equations and show the vanishing order of solutions.', '1312.0576-2-0-2': 'Furthermore, strong unique continuation is established for higher order elliptic equations using this variant of frequency function.', '1312.0576-2-1-0': '# Introduction', '1312.0576-2-2-0': 'We say the vanishing order of solution at [MATH] is [MATH], if [MATH] is the largest integer such that [MATH] for all [MATH].', '1312.0576-2-2-1': 'It describes quantitative behavior of strong unique continuation property.', '1312.0576-2-2-2': 'It is well known that all zeros of nontrivial solutions of second order linear equations on smooth compact Riemannian manifolds are of finite order [CITATION].', '1312.0576-2-2-3': 'In the papers [CITATION] and [CITATION], Donnelly and Fefferman showed that if [MATH] is an eigenfunction on a compact smooth Riemannian manifold [MATH], that is, [EQUATION] for some [MATH], then the maximal vanishing order of [MATH] on [MATH] is less than [MATH], here [MATH] only depends on the manifold [MATH].', '1312.0576-2-2-4': 'Kukavica in [CITATION] considered the vanishing order of solutions of Schrodinger equation [EQUATION] where [MATH].', '1312.0576-2-2-5': 'He established that the vanishing order of solution in ([REF]) is everywhere less than [EQUATION] where [MATH], [MATH] and [MATH] only depends on the underlying domain [MATH].', '1312.0576-2-2-6': 'If [MATH], Kukavica was able to show that the upper bound of vanishing order is less than [MATH], where [MATH].', '1312.0576-2-2-7': "Based on the Donnelly and Fefferman's work in [CITATION], Kukavica conjectured that the rate of of vanishing order of [MATH] is less than [MATH] for the cases of [MATH] and [MATH].", '1312.0576-2-2-8': "For the upper bound in ([REF]), it agrees with Donnelly and Fefferman's results in the eigenvalue case [MATH].", '1312.0576-2-2-9': 'Recently, Kenig [CITATION] considered a similar problem which is motivated by his work with Bourgain in [CITATION] on Anderson localization for the Bernoulli model.', '1312.0576-2-2-10': 'Kenig investigated the following normalized model.', '1312.0576-2-2-11': 'Let [EQUATION] where [MATH] is a ball centered at origin with radius [MATH] in [MATH].', '1312.0576-2-2-12': 'Assume that [MATH] and [MATH].', '1312.0576-2-2-13': 'Kenig established that [EQUATION] where [MATH] depend only on [MATH], [MATH] and [MATH] depends on [MATH].', '1312.0576-2-2-14': 'By exploiting the Carleman estimates, Kenig proved the [MATH].', '1312.0576-2-2-15': "He also pointed out that the exponent [MATH] of [MATH] is sharp for complex valued [MATH] based on Meshkov's example in [CITATION].", '1312.0576-2-2-16': "On the basis of Donnelly and Fefferman's work, Kenig asked if [MATH] can be achieved for real [MATH].", '1312.0576-2-2-17': 'Very recently, Bakri in [CITATION] considered ([REF]) in the case of [MATH].', '1312.0576-2-2-18': 'He obtained that the vanishing order of solutions in ([REF]) is less than [MATH].', '1312.0576-2-2-19': 'His proof is an extension of the Carleman estimates in [CITATION].', '1312.0576-2-2-20': 'It is worthwhile to mention that the vanishing order of solutions is closely related to the study of eigenfunctions on manifolds.', '1312.0576-2-2-21': 'We refer to the survey [CITATION] for detailed account.', '1312.0576-2-3-0': 'We are especially interested in the model ([REF]).', '1312.0576-2-3-1': 'Our first goal in this paper is to address the above problems.', '1312.0576-2-3-2': 'Relied on a variant of frequency function, we are able to verify that [MATH] is indeed true for the case of [MATH] in ([REF]).', '1312.0576-2-3-3': 'In particular, our result also confirms that the vanishing order of solutions in ([REF]) is less than [MATH] if [MATH].', '1312.0576-2-4-0': 'Assume that [MATH].', '1312.0576-2-4-1': 'Under the assumptions in ([REF]) with [MATH], the maximal vanishing order of [MATH] in ([REF]) is less than [MATH], where [MATH] depends on [MATH] and [MATH].', '1312.0576-2-5-0': 'Generally speaking, the Carleman estimates and frequency function are two principal ways to obtain quantitative uniqueness and strong unique continuation results for solutions of partial differential equations.', '1312.0576-2-5-1': 'Carleman estimates were introduced by Carleman, when he studied the strong unique continuation property.', '1312.0576-2-5-2': 'Carleman estimates are weighted integral inequalities.', '1312.0576-2-5-3': 'See e.g. [CITATION], [CITATION], [CITATION], [CITATION], [CITATION], [CITATION], [CITATION], to just mention a few.', '1312.0576-2-5-4': "In order to obtain the quantitative uniqueness results for solutions, one uses the Carleman estimates with a special choice of weight functions to obtain a type of Hadamard's three-ball theorem, then doubling estimates follow.", '1312.0576-2-5-5': 'The vanishing order will come from the doubling estimates.', '1312.0576-2-5-6': 'The frequency function was first observed by Almgren [CITATION] for harmonic functions.', '1312.0576-2-5-7': 'The frequency function controls the local growth rate of [MATH] and is a local measure of its "degree" as a polynomial like function in [MATH].', '1312.0576-2-5-8': 'See e.g. [CITATION], [CITATION], [CITATION], [CITATION], [CITATION], [CITATION], etc.', '1312.0576-2-5-9': 'Garofalo and Lin in [CITATION], [CITATION] showed its powerful applications in strong unique continuation problem.', '1312.0576-2-5-10': 'The frequency function Garofalo and Lin investigated for equation ([REF]) is given by [EQUATION] where [MATH] and [MATH].', '1312.0576-2-5-11': 'After one proves certain monotonicity of [MATH], the doubling estimates will follow by a standard argument.', '1312.0576-2-5-12': 'In [CITATION], it was shown that [MATH] was monotone nondecreasing.', '1312.0576-2-5-13': 'However, [MATH] depends on the norm of [MATH].', '1312.0576-2-5-14': 'It can not give the optimal bound for the vanishing order of solutions.', '1312.0576-2-5-15': 'Kukavica considered almost the same frequency function in [CITATION].', '1312.0576-2-5-16': 'He was able to move the norm of [MATH] away from the exponential, but it only gave the aforementioned bound [MATH] due to the limitations of the method.', '1312.0576-2-5-17': 'Some of the limitations come from the fact that one can not explore [MATH] more because of its integration on the boundary of balls.', '1312.0576-2-5-18': 'Instead, we consider a variant of ([REF]).', '1312.0576-2-5-19': 'See our variant of frequency function for Schrodinger equations in section 2 and the frequency functions for high order elliptic equations in section 3 for the details.', '1312.0576-2-5-20': 'First, we establish a monotonicity property of this new variant of frequency function.', '1312.0576-2-5-21': "Second, based on the monotonicity results, it leads to a [MATH]-version of Hadamard's three-ball theorem, which further implies a [MATH]-version of Hadamard's three-ball theorem by elliptic estimates.", '1312.0576-2-5-22': 'At last, by a propagation of smallness argument, we derive the vanishing order of solutions.', '1312.0576-2-6-0': 'Higher order elliptic equations are also important models in the study of partial differential equations.', '1312.0576-2-6-1': 'A nature question is to study the quantitative uniqueness of higher order elliptic equations.', '1312.0576-2-6-2': 'Our second goal is to investigate the vanishing order for solutions of higher order elliptic equations.', '1312.0576-2-6-3': 'We consider this normalized model: [EQUATION]', '1312.0576-2-6-4': 'We also assume that [MATH] and [MATH].', '1312.0576-2-6-5': 'To the best of our knowledge, the explicit vanishing order as Theorem [REF] seems to be unknown for higher order elliptic equations.', '1312.0576-2-6-6': 'By exploiting this variant of frequency function, we are able to obtain the following theorem.', '1312.0576-2-7-0': 'Assume that [MATH] and [MATH].', '1312.0576-2-7-1': 'Under the assumption in ([REF]), the maximal order of vanishing of [MATH] in ([REF]) is less than [MATH], where [MATH] depends on [MATH], [MATH] and [MATH].', '1312.0576-2-8-0': 'Unlike the Laplacian operator in ([REF]), new difficulties arise since some kind of "symmetry" is lost for higher order elliptic equations.', '1312.0576-2-8-1': 'Our idea is to break the higher order elliptic equations into a system of semilinear equations.', '1312.0576-2-8-2': 'However, it still does not give the most desirable result as Theorem [REF].', '1312.0576-2-8-3': "We also develop a [MATH]-version of Hadamard's three-ball theorem by exploring [MATH] estimates for higher order elliptic equations (see section 3 for the details).", '1312.0576-2-8-4': 'Compared with the frequency function argument, it seems to more difficult to obtain the explicit vanishing order of solutions for higher order elliptic equations by Carleman estimates.', '1312.0576-2-9-0': 'The quantitative uniqueness has applications in mathematical physics.', '1312.0576-2-9-1': 'For instance, the vanishing order of solutions plays an important role in [CITATION] on Anderson localization for the Bernoulli model.', '1312.0576-2-9-2': 'Suppose that [MATH] is a solution of [EQUATION] where [MATH].', '1312.0576-2-9-3': 'Let [EQUATION] for [MATH] large.', '1312.0576-2-9-4': 'By using the result of ([REF]), Bourgain and Kenig in [CITATION] showed that [EQUATION] where [MATH] depends on [MATH] and [MATH].', '1312.0576-2-9-5': 'We can consider a similar quantitative unique continuation problem as ([REF]) for higher order elliptic equations.', '1312.0576-2-9-6': 'Suppose that [MATH] is a solution to [EQUATION] where [MATH].', '1312.0576-2-9-7': 'Theorem [REF] implies that following corollary for higher order elliptic equations.', '1312.0576-2-10-0': 'Let [MATH] be a solution to ([REF]) and [MATH].', '1312.0576-2-10-1': 'Then [EQUATION] where [MATH] depends on [MATH], [MATH] and [MATH].', '1312.0576-2-11-0': 'An easy consequence of Theorem [REF] is a strong unique continuation result for higher order elliptic equations when [MATH].', '1312.0576-2-11-1': 'Due to the conclusion in Theorem [REF], the solutions will not vanish of infinite order when [MATH].', '1312.0576-2-11-2': 'Without using the conclusion of Theorem [REF], we are also able to give a proof based on this variant of frequency function.', '1312.0576-2-11-3': 'We refer to, e.g. [CITATION], [CITATION], for the strong unique continuation results of higher order elliptic equations by using Carleman estimates.', '1312.0576-2-11-4': 'Assume that [EQUATION] where [MATH].', '1312.0576-2-11-5': 'A function [MATH] is said to vanish of infinite order at some point [MATH] if for [MATH] sufficiently small, [EQUATION] for every positive integer [MATH].', '1312.0576-2-11-6': 'We are able to establish the following theorem.', '1312.0576-2-11-7': 'If [MATH] in ([REF]) vanishes of infinite order at some point [MATH], then [MATH] in [MATH].', '1312.0576-2-12-0': 'The outline of the paper is as follows.', '1312.0576-2-12-1': 'Section 2 is devoted to obtaining the vanishing order of Schrodinger equations.', '1312.0576-2-12-2': 'In Section 3, the vanishing order of higher order elliptic equations is shown.', '1312.0576-2-12-3': 'In section 4, we obtain the strong unique continuation for higher order elliptic equations.', '1312.0576-2-12-4': 'In the whole paper, we will use various letters, such as [MATH], [MATH], [MATH], [MATH], to denote the positive constants which may depend [MATH] and [MATH], even if they are not explicitly stated.', '1312.0576-2-12-5': 'They may also vary from line to line.', '1312.0576-2-12-6': 'Especially, the letters do not depend on [MATH] in section 2 and [MATH] in section 3.', '1312.0576-2-13-0': '# Schro dinger equations', '1312.0576-2-14-0': 'In this section, we focus on the maximal vanishing order of solutions in ([REF]).', '1312.0576-2-14-1': 'Let [MATH].', '1312.0576-2-14-2': 'We define [EQUATION]', '1312.0576-2-14-3': 'The value of the constant [MATH] will be determined later on.', '1312.0576-2-14-4': 'We can assume that [MATH] by choosing [MATH] suitable small.', '1312.0576-2-14-5': 'Without loss of generality, we may assume [MATH] and denote [MATH] as [MATH], that is, [EQUATION].', '1312.0576-2-14-6': 'The advantage of the weight function [MATH] in the integration is that the boundary term will not appear whenever we use divergence theorem.', '1312.0576-2-14-7': 'Moreover, the value of [MATH] will help reduce the order of vanishing.', '1312.0576-2-14-8': 'The function in ([REF]) appeared in [CITATION] for the study of vortex of Ginzburg-Landau equations.', '1312.0576-2-14-9': 'We will also omit the integration on [MATH] when it is clear from the context.', '1312.0576-2-14-10': 'Taking the derivative with respect to [MATH] for [MATH], we get [EQUATION]', '1312.0576-2-14-11': 'Because of the presence of the weight function, as we mentioned before, there are no terms involving integration on the boundary.', '1312.0576-2-14-12': "One of it's advantage is that it simplifies our calculations in the following.", '1312.0576-2-14-13': 'Furthermore, [EQUATION]', '1312.0576-2-14-14': 'Applying the divergence theorem for the second term in the right hand side of the latter equality, we get [EQUATION] where [EQUATION]', '1312.0576-2-14-15': 'Using the divergence theorem again for [MATH], it follows that [EQUATION] where we perform integration by parts and use the equation ([REF]) in the last equality.', '1312.0576-2-15-0': 'We define our variant of frequency function as [EQUATION]', '1312.0576-2-15-1': 'Next we are going to study the monotonicity property of this special type of frequency function [MATH].', '1312.0576-2-15-2': 'We are able to obtain the following result.', '1312.0576-2-16-0': 'There exists a constant [MATH] depending only on [MATH] such that [EQUATION] is nondecreasing function of [MATH].', '1312.0576-2-17-0': 'To consider the monotonicity of [MATH], we shall consider the derivative of [MATH].', '1312.0576-2-17-1': 'By taking the derivative for [MATH] in ([REF]) with respect to [MATH], [EQUATION]', '1312.0576-2-17-2': 'We simply the first term in the right hand side of the latter equality.', '1312.0576-2-17-3': 'It yields that [EQUATION]', '1312.0576-2-17-4': 'Integrating by parts for the second term in the right hand side of the last equality gives that [EQUATION]', '1312.0576-2-17-5': 'We do further integration by parts for the second term in the right hand side of the last inequality with respect to [MATH]th derivative.', '1312.0576-2-17-6': 'It follows that [EQUATION] where we have used the equation ([REF]) in the latter equality.', '1312.0576-2-17-7': 'We want to interpret the first term in the the right hand side of the last equality in terms of [MATH].', '1312.0576-2-17-8': 'In view of ([REF]), we have [EQUATION]', '1312.0576-2-17-9': 'We breaks down the second term in the last equality as [EQUATION]', '1312.0576-2-17-10': 'Substituting the latter equality to [MATH], one obtains [EQUATION]', '1312.0576-2-17-11': 'Applying the divergence theorem for the last term in the right hand side of the latter equality and considering the fact that [MATH], we arrive at [EQUATION]', '1312.0576-2-17-12': 'Combining the third term and seventh term in the right hand side of the latter equality gives that [EQUATION]', '1312.0576-2-17-13': 'By the definition of [MATH] in ([REF]) and the assumption that [MATH], we obtain [EQUATION]', '1312.0576-2-17-14': 'In order to find the monotonicity of [MATH], it suffices to take the derivative for [MATH] with respect to [MATH].', '1312.0576-2-17-15': 'Taking [MATH] in ([REF]) and [MATH] in ([REF]) into consideration, we get [EQUATION] where we have used [MATH] in ([REF]) in the last inequality.', '1312.0576-2-17-16': 'By Cauchy-Schwarz inequality, we know [EQUATION].', '1312.0576-2-17-17': 'We finally arrive at [EQUATION] which implies the conclusion in the lemma.', '1312.0576-2-18-0': 'Let us compare more about our variant of frequency function and that in [CITATION].', '1312.0576-2-18-1': 'Both lead to monotonicity property.', '1312.0576-2-18-2': 'Unlike the monotonicity results in [CITATION], the function [MATH] is moved away from the exponential in Lemma [REF] .', '1312.0576-2-18-3': 'Our monotonicity result only relies on the polynomial growth of [MATH].', '1312.0576-2-18-4': 'More important, the positive position [MATH] and the radius [MATH] do not depend on [MATH] in Lemma [REF].', '1312.0576-2-18-5': 'The fact that [MATH] is independent of [MATH] is crucial in the propagation of smallness arguments in the proof of Theorem [REF].', '1312.0576-2-18-6': "With the help of monotonicity of [MATH], we are going to establish a [MATH]-version of Hadamard's three-ball theorem.", '1312.0576-2-18-7': "For the variants of Hadamard's three-ball theorem, see e.g. [CITATION] and [CITATION].", '1312.0576-2-18-8': 'We also want to get rid of the weight function [MATH] in our function [MATH].', '1312.0576-2-18-9': 'In this process, the value of [MATH] helps reduce the coefficient in the following three-ball theorem, which provides better vanishing order.', '1312.0576-2-18-10': 'This is another advantage we introduce the weight function.', '1312.0576-2-18-11': 'Let [EQUATION].', '1312.0576-2-18-12': 'Without loss of generality, we may assume [MATH].', '1312.0576-2-18-13': 'We can easily check that [EQUATION] and [EQUATION] for any [MATH].', '1312.0576-2-18-14': 'We are able to obtain the following three-ball theorem.', '1312.0576-2-19-0': 'Let [MATH].', '1312.0576-2-19-1': 'Then [EQUATION] where [EQUATION] and [EQUATION].', '1312.0576-2-20-0': 'From ([REF]), we have [EQUATION]', '1312.0576-2-20-1': 'Taking integration from [MATH] to [MATH] in the last identity gives that [EQUATION]', '1312.0576-2-20-2': 'By the monotonicity result in Lemma [REF], it follows that [EQUATION] that is, [EQUATION]', '1312.0576-2-20-3': 'If we perform similar calculations on ([REF]) by integrating from [MATH] to [MATH], we deduce that [EQUATION]', '1312.0576-2-20-4': 'Namely, [EQUATION]', '1312.0576-2-20-5': 'Combining the inequalities ([REF]) and ([REF]), note that [MATH], we conclude that [EQUATION]', '1312.0576-2-20-6': 'Thanks to ([REF]) and ([REF]), we have [EQUATION]', '1312.0576-2-20-7': 'Therefore, [EQUATION]', '1312.0576-2-20-8': 'We conduct the similar calculations as above for [MATH] in ([REF]).', '1312.0576-2-20-9': 'Using ([REF]) and ([REF]) again, [EQUATION]', '1312.0576-2-20-10': 'So we obtain that [EQUATION]', '1312.0576-2-20-11': 'Taking ([REF]), ([REF]) and ([REF]) into account, we get [EQUATION].', '1312.0576-2-20-12': 'Namely, [EQUATION].', '1312.0576-2-20-13': 'Taking exponentials of both sides implies that [EQUATION].', '1312.0576-2-20-14': 'Note that [MATH].', '1312.0576-2-20-15': 'As we know, the minimum value of the exponential function in the last inequality will be achieved if we take [MATH].', '1312.0576-2-20-16': 'Hence [EQUATION] where [MATH] is a constant depending only on [MATH].', '1312.0576-2-20-17': 'We are done with the [MATH]-version of three-ball theorem.', '1312.0576-2-21-0': 'From the above lemma, one can see that the appearance of [MATH] reduces the exponent of exponential in the [MATH]-version of three-ball theorem.', '1312.0576-2-21-1': 'Thanks to Lemma [REF], we are able to establish a [MATH]-version of three-ball theorem, which will be used in the propagation of smallness argument.', '1312.0576-2-22-0': 'Let [MATH].', '1312.0576-2-22-1': 'Then [EQUATION] where [EQUATION] and [EQUATION].', '1312.0576-2-23-0': 'Using the standard elliptic theory for the solution in [MATH], we have [EQUATION] here [MATH] does not depends on [MATH].', '1312.0576-2-23-1': 'By some rescaling argument, [EQUATION] for [MATH].', '1312.0576-2-23-2': 'Then [EQUATION].', '1312.0576-2-23-3': 'Taking advantage of Lemma [REF], we deduce that [EQUATION].', '1312.0576-2-23-4': 'Recall that [MATH].', '1312.0576-2-23-5': 'Thus, we arrive at the conclusion.', '1312.0576-2-24-0': 'Now we are ready to prove Theorem [REF].', '1312.0576-2-24-1': 'We apply the idea of propagation of smallness which is based on overlapping of three-ball argument.', '1312.0576-2-24-2': 'Similar arguments have been employed in [CITATION].', '1312.0576-2-24-3': '[Proof of Theorem [REF]] We choose a small [MATH] such that [EQUATION].', '1312.0576-2-24-4': 'Obviously, [MATH].', '1312.0576-2-24-5': 'Since [MATH], there exists some [MATH] such that [MATH].', '1312.0576-2-24-6': 'We select a sequence of balls with radius [MATH] centered at [MATH] so that [MATH] and [MATH], where [MATH] depends on the radius [MATH] which we will fix later on.', '1312.0576-2-24-7': 'Employing Lemma [REF] with [MATH], [MATH], and [MATH] and the boundedness assumption of [MATH], we get [EQUATION] where [MATH] and [MATH] depends on the [MATH]-norm of [MATH].', '1312.0576-2-25-0': 'Iterating the above argument with Lemma [REF] for balls centered at [MATH] and using the fact that [MATH], we have [EQUATION] for [MATH], where [MATH] is a constant depending on [MATH] and [MATH]-norm of [MATH], and [MATH], [MATH] are constants depending on [MATH].', '1312.0576-2-25-1': 'By the fact that [MATH] and [MATH], we obtain [EQUATION] where [MATH] is a constant depending on [MATH] and [MATH]-norm of [MATH], and [MATH] is a constant depending on [MATH].', '1312.0576-2-26-0': 'Applying the [MATH] type of three-ball lemma again centered at origin again with [MATH] and [MATH], where [MATH] is sufficiently small, we have [EQUATION].', '1312.0576-2-26-1': 'Recall that [MATH] is the [MATH] norm of [MATH] in [MATH].', '1312.0576-2-26-2': 'Then [EQUATION] where [MATH] depends on [MATH] and [MATH] norm of [MATH], [MATH] depends on [MATH], and [EQUATION] with constant [MATH] depending on [MATH].', '1312.0576-2-26-3': 'Now we can fix the small [MATH].', '1312.0576-2-26-4': 'For instance, let [MATH].', '1312.0576-2-26-5': 'Thus, the number [MATH] is also determined.', '1312.0576-2-26-6': 'The inequality ([REF]) implies that [EQUATION] where the constants [MATH] depend on the dimension [MATH] and [MATH].', '1312.0576-2-26-7': 'Therefore, Theorem 1 is completed.', '1312.0576-2-27-0': '# Higher order elliptic equations', '1312.0576-2-28-0': 'In this section, we consider the vanishing order of solutions for the higher order elliptic equations.', '1312.0576-2-28-1': 'As far as we know, the explicit vanishing order seems to be unknown in the literature.', '1312.0576-2-28-2': 'Due to the complexity of its structure, we decompose the model in ([REF]) into a system of [MATH] semilinear equations, that is, [EQUATION]', '1312.0576-2-28-3': 'Note that [MATH].', '1312.0576-2-28-4': 'Inspired by our frequency function in section 2, it is nature to consider the following function for the system of semilinear equations in ([REF]).', '1312.0576-2-28-5': 'Let [EQUATION]', '1312.0576-2-28-6': 'As before, we may assume [MATH] and omit the integration on [MATH] if it is clear from the context.', '1312.0576-2-28-7': 'Namely, [EQUATION].', '1312.0576-2-28-8': 'The value of the constant [MATH] will be determined later on.', '1312.0576-2-28-9': 'If one takes derivative for [MATH] with respect to [MATH], following the similar calculations in section 2, one has [EQUATION]', '1312.0576-2-28-10': 'Performing the divergence theorem for the second term in the right hand side of the last equality, we obtain that [EQUATION] where [EQUATION]', '1312.0576-2-28-11': 'Applying the divergence theorem on [MATH], we have [EQUATION]', '1312.0576-2-28-12': 'Considering the systems of equations ([REF]), it follows that [EQUATION]', '1312.0576-2-28-13': 'For the higher order elliptic equations, we define our variant of frequency function as [EQUATION]', '1312.0576-2-28-14': 'Since we are dealing with more complex structure, more careful calculations are devoted.', '1312.0576-2-28-15': 'Different from the semilinear equation case, higher regularity, i.e. [MATH] seems not be helpful.', '1312.0576-2-28-16': 'We consider the case that [MATH].', '1312.0576-2-28-17': 'We are able to obtain the following the monotonicity property for the frequency function [MATH].', '1312.0576-2-29-0': 'There exists a constant [MATH] depending only on [MATH] such that [EQUATION] is nondecreasing function of [MATH].', '1312.0576-2-30-0': 'To obtain the monotonicity result, we shall consider the derivative of [MATH].', '1312.0576-2-30-1': 'Now differentiating [MATH] in ([REF]) with respect to [MATH], [EQUATION]', '1312.0576-2-30-2': 'Integrating by parts for the second term in the right hand side of the latter equality, [EQUATION]', '1312.0576-2-30-3': 'If one performs the divergence theorem with respect to [MATH]th derivative on the second term in the right hand side of the last inequality, one has [EQUATION]', '1312.0576-2-30-4': 'Using the equivalent system of equations in ([REF]), it follows that [EQUATION]', '1312.0576-2-30-5': 'We want to transform the first term in the right hand side of the latter inequality in term of [MATH].', '1312.0576-2-30-6': 'Taking ([REF]) into consideration and performing some calculations, we have [EQUATION]', '1312.0576-2-30-7': 'Now we estimate each term in the right hand side of the last equality.', '1312.0576-2-30-8': "Using Holder's inequality and the definition of [MATH] in ([REF]), we obtain [EQUATION] and [EQUATION]", '1312.0576-2-30-9': "Similarly, by Holder's inequality, [EQUATION]and [EQUATION]", '1312.0576-2-30-10': 'For the ease of the notation, let [EQUATION].', '1312.0576-2-30-11': 'Combining the inequalities ([REF]) and ([REF]) and taking ([REF]) into account, we get [EQUATION]', '1312.0576-2-30-12': 'Therefore, together with ([REF]), ([REF]), and ([REF]), [EQUATION] where [MATH] depends only on [MATH] and [MATH].', '1312.0576-2-30-13': 'In order to get monotonicity of the frequency function, we differentiate [MATH].', '1312.0576-2-30-14': 'Recall [MATH] in ([REF]) and [MATH] in ([REF]).', '1312.0576-2-30-15': '[EQUATION] where we have used [MATH] in ([REF]) in the last inequality.', '1312.0576-2-30-16': 'By Cauchy-Schwartz inequality, [EQUATION].', '1312.0576-2-30-17': 'Consequently, [EQUATION]', '1312.0576-2-30-18': 'We complete the proof of the lemma.', '1312.0576-2-31-0': 'As the conclusion in Lemma [REF] indicates, the monotonicity property only relies on the polynomial growth of [MATH] and [MATH] does not depend on [MATH].', '1312.0576-2-31-1': 'We are going to establish a [MATH]-version of three-ball theorem.', '1312.0576-2-31-2': 'For convenience, let [EQUATION].', '1312.0576-2-31-3': 'We also need to remove the weight function [MATH] in [MATH].', '1312.0576-2-31-4': 'As in the section 2, let [EQUATION].', '1312.0576-2-31-5': 'As usual, we will omit the dependent of the center of [MATH] for the ball.', '1312.0576-2-31-6': 'It is easy to check that [EQUATION] and [EQUATION] for any [MATH].', '1312.0576-2-32-0': 'Based on the monotonicity of [MATH] in the last lemma, we are able to establish the following [MATH]-type of three-ball theorem.', '1312.0576-2-33-0': 'Let [MATH].', '1312.0576-2-33-1': 'Then [EQUATION] where [EQUATION] and [EQUATION] where [MATH] depends only on [MATH] and [MATH].', '1312.0576-2-34-0': 'From ([REF]), we deduce that [EQUATION]', '1312.0576-2-34-1': 'On one hand, integrating from [MATH] to [MATH] on the equality ([REF]) gives that [EQUATION] where we have used Lemma [REF] in the last inequality.', '1312.0576-2-34-2': 'Namely, [EQUATION]', '1312.0576-2-34-3': 'On the other hand, integrating from [MATH] to [MATH] on the equality ([REF]) implies that [EQUATION] that is, [EQUATION]', '1312.0576-2-34-4': 'Taking ([REF]) and ([REF]) into considerations, we get [EQUATION]', '1312.0576-2-34-5': 'Thanks to ([REF]) and ([REF]), [EQUATION]', '1312.0576-2-34-6': 'Therefore, [EQUATION]', '1312.0576-2-34-7': 'We do the similar calculations for [MATH].', '1312.0576-2-34-8': 'Using ([REF]) and ([REF]) again, [EQUATION]', '1312.0576-2-34-9': 'Thus, [EQUATION]', '1312.0576-2-34-10': 'Taking ([REF]), ([REF]) and ([REF]) into account, we have [EQUATION].', '1312.0576-2-34-11': 'Namely, [EQUATION].', '1312.0576-2-34-12': 'Taking exponentials of both sides and performing some simplications, we obtain [EQUATION].', '1312.0576-2-34-13': 'Since [EQUATION] we have [EQUATION].', '1312.0576-2-34-14': 'As we know, the minimum value of the function [MATH] is achieved in the case of [MATH].', '1312.0576-2-34-15': 'Recall that [MATH].', '1312.0576-2-34-16': 'Therefore, the lemma is completed.', '1312.0576-2-35-0': 'Again we need to establish a [MATH]-version of three-ball theorem.', '1312.0576-2-35-1': 'However, the classical elliptic estimates as ([REF]) does not seem to be known for higher order elliptic equations in the literature.', '1312.0576-2-35-2': 'We will deduce a similar estimate by Sobolev inequality and a [MATH] type estimate.', '1312.0576-2-35-3': 'We first present a [MATH] type estimates for higher order elliptic equations (see e.g. [CITATION]).', '1312.0576-2-35-4': 'Let [MATH] satisfy the following equation [EQUATION]', '1312.0576-2-35-5': 'Then we have', '1312.0576-2-36-0': 'Let [MATH].', '1312.0576-2-36-1': 'Suppose [MATH] satisfies ([REF]).', '1312.0576-2-36-2': 'Then there exits a constant [MATH] depending only on [MATH] such that for any [MATH], [EQUATION]', '1312.0576-2-36-3': 'Upon a rescaling argument, we have [EQUATION] for [MATH].', '1312.0576-2-37-0': 'Applying Lemma [REF], we are able to establish the [MATH]-version of three-ball theorem for the solutions in ([REF]).', '1312.0576-2-38-0': 'Let [MATH] and [MATH].', '1312.0576-2-38-1': 'Then [EQUATION] where [EQUATION] and [EQUATION].', '1312.0576-2-39-0': 'Thanks to Lemma [REF] in the case of [MATH], we can estimate the solution in ([REF]) by the following [EQUATION].', '1312.0576-2-39-1': 'By Sobolev imbedding inequality, if [MATH], [EQUATION] where [MATH] depends on [MATH], [MATH] and [MATH].', '1312.0576-2-39-2': 'Applying Lemma [REF] again with [MATH] and the latter inequality, note that [MATH], [EQUATION]', '1312.0576-2-39-3': 'As we know [EQUATION].', '1312.0576-2-39-4': 'Employing the above bootstrap argument finite times, e.g. [MATH] times, which depends only on [MATH] and [MATH] and using the above Sobolev imbedding inequality, we get [EQUATION].', '1312.0576-2-39-5': 'Let [MATH], [EQUATION] where [MATH] depends on only [MATH] and [MATH].', '1312.0576-2-39-6': 'If [MATH], we will have the similar result by applying the bootstrap arguments twice.', '1312.0576-2-39-7': 'By a rescaling argument, we have [EQUATION] if [MATH].', '1312.0576-2-39-8': 'Furthermore, we get [EQUATION] for [MATH].', '1312.0576-2-39-9': 'Recall that [MATH].', '1312.0576-2-39-10': 'Thus, [EQUATION].', '1312.0576-2-39-11': 'Based on Lemma [REF] and the latter inequality, we deduce that [EQUATION] where [EQUATION] and [EQUATION].', '1312.0576-2-39-12': 'Taking Lemma [REF] and ([REF]) into account with [MATH], we have [EQUATION] and [EQUATION].', '1312.0576-2-39-13': 'It is true that [EQUATION] if [MATH] and [MATH].', '1312.0576-2-39-14': 'From ([REF]) and the last three inequalities, we obtain [EQUATION].', '1312.0576-2-39-15': 'By a rescaling argument, we arrive at the conclusion of the lemma.', '1312.0576-2-40-0': 'We begin to prove Theorem [REF].', '1312.0576-2-40-1': 'The idea is similar to the proof of Theorem [REF].', '1312.0576-2-40-2': 'We also use the propagation of smallness argument.', '1312.0576-2-40-3': '[Proof of Theorem [REF]] We choose a small [MATH] such that [EQUATION] where [MATH].', '1312.0576-2-40-4': 'Since [MATH], there should exist some [MATH] such that [MATH].', '1312.0576-2-40-5': 'We select a sequence of balls with radius [MATH] centered at [MATH] so that [MATH] and [MATH], where [MATH] depends on the radius [MATH] which is to be fixed.', '1312.0576-2-40-6': 'Employing the [MATH]-version of three-ball lemma (i.e. Lemma [REF]) with [MATH], [MATH], and [MATH] and the boundedness assumption of [MATH], we get [EQUATION] where [MATH] and [MATH] depends on the [MATH]-norm of [MATH], [MATH] and [MATH].', '1312.0576-2-41-0': 'Iterating the above argument with [MATH]-version of three-ball lemma for ball centered at [MATH] and using the fact that [MATH], we have [EQUATION] for [MATH], where [MATH] is a constant depending on [MATH] and [MATH]-norm of [MATH], and [MATH], [MATH], [MATH] are constants depending on [MATH].', '1312.0576-2-41-1': 'By the fact that [MATH] and [MATH], we obtain [EQUATION] where [MATH] is a constant depending on [MATH] and [MATH]-norm of [MATH], and [MATH], [MATH] are constants depending on [MATH].', '1312.0576-2-42-0': 'Applying Lemma [REF] again centered at origin with [MATH] and [MATH], where [MATH] is sufficiently small, we have [EQUATION].', '1312.0576-2-42-1': 'Recall that [MATH] is the [MATH]-norm for [MATH] in [MATH].', '1312.0576-2-42-2': 'Then [EQUATION] where [MATH] depends on [MATH] and [MATH]-norm of [MATH], and [MATH], [MATH] depend on [MATH], [EQUATION] with constant [MATH] depending on [MATH].', '1312.0576-2-42-3': 'At this moment we fix the small value of [MATH].', '1312.0576-2-42-4': 'For instance, let [MATH].', '1312.0576-2-42-5': 'Then the value of [MATH] is determined too.', '1312.0576-2-42-6': 'The inequality ([REF]) implies that [EQUATION] where the constants [MATH] depend on the dimension [MATH], [MATH], and [MATH].', '1312.0576-2-42-7': 'The proof of Theorem [REF] is arrived.', '1312.0576-2-43-0': 'Thanks to Theorem [REF], we are able to prove the following corollary for higher order elliptic equations in ([REF]), which characterizes the asymptotic behavior of [MATH] at infinity.', '1312.0576-2-43-1': '[Proof of Corollary [REF]] We adapt the proof in [CITATION].', '1312.0576-2-43-2': 'Since [MATH] is continuous, we can find [MATH] so that [MATH].', '1312.0576-2-43-3': 'Let [EQUATION].', '1312.0576-2-43-4': 'Then [EQUATION] with [MATH] and [MATH].', '1312.0576-2-43-5': 'So [MATH] in the notation of Theorem [REF].', '1312.0576-2-43-6': 'If [MATH], then [MATH] and [MATH].', '1312.0576-2-43-7': 'Hence [MATH].', '1312.0576-2-43-8': 'Note that [MATH], where [MATH].', '1312.0576-2-43-9': 'The conclusion in Theorem [REF] leads to [EQUATION] where [MATH] depends on [MATH], [MATH] and [MATH].', '1312.0576-2-43-10': 'Thus, the corollary follows.', '1312.0576-2-44-0': '# Strong unique continuation', '1312.0576-2-45-0': 'In the rest of the paper, we will show the strong unique continuation result for higher order elliptic equations by the monotonicity of frequency function.', '1312.0576-2-45-1': 'This variant of frequency function is also powerful in obtaining unique continuation results.', '1312.0576-2-45-2': 'For strong unique continuation results of semilinear equations and system of equations using frequency function, we refer to [CITATION], [CITATION] and [CITATION] for the Lame system of elasticity.', '1312.0576-2-45-3': 'Let [MATH] be the solution in ([REF]).', '1312.0576-2-45-4': 'Since we do not need to control the vanishing order of solutions, we assume [MATH] for [MATH], i.e. [EQUATION].', '1312.0576-2-45-5': 'We can check that [EQUATION] where [EQUATION].', '1312.0576-2-45-6': 'We consider the following frequency function [EQUATION]', '1312.0576-2-45-7': 'Similar arguments as the proof of Lemma [REF] lead to following monotonicity.', '1312.0576-2-46-0': 'There exists a constant [MATH] depending only on [MATH] such that [EQUATION] is nondecreasing function of [MATH].', '1312.0576-2-47-0': 'Based on the monotonicity property in above lemma, we are able to show the proof of Theorem [REF].', '1312.0576-2-48-0': '[Proof of Theorem [REF]] By the equality ([REF]), we get [EQUATION] where [EQUATION].', '1312.0576-2-48-1': 'Recall that [MATH].', '1312.0576-2-48-2': 'Integrating from [MATH] to [MATH] for the equality ([REF]) yields that [EQUATION] where [MATH] is chosen to be small and [MATH] depends on [MATH] and [MATH].', '1312.0576-2-48-3': 'Taking exponential of both sides, [EQUATION] that is, [EQUATION] where [MATH] depends on [MATH].', '1312.0576-2-48-4': 'From the decomposition in ([REF]) and scaling arguments in ([REF]), we have [EQUATION]', '1312.0576-2-48-5': 'Therefore, with the aid of ([REF]), [EQUATION].', '1312.0576-2-48-6': 'Thus, we get a doubling type estimate [EQUATION] where [MATH] depends on [MATH], [MATH], [MATH] and [MATH].', '1312.0576-2-48-7': 'Now we fix [MATH] and prove that [MATH] on [MATH] from ([REF]).', '1312.0576-2-48-8': 'The argument is standard.', '1312.0576-2-48-9': 'See e.g. [CITATION] on page 256-257.', '1312.0576-2-48-10': '[EQUATION] where the constant [MATH] to be fixed.', '1312.0576-2-48-11': 'We choose [MATH] such that [MATH].', '1312.0576-2-48-12': 'It yields that [EQUATION] because of ([REF]).', '1312.0576-2-48-13': 'Then [MATH] in [MATH].', '1312.0576-2-48-14': 'Since we can choose [MATH] arbitrarily in [MATH], the proof of Theorem [REF] follows.'}

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'1312.0576-2-14-10'], ['1312.0576-1-29-0', '1312.0576-2-29-0'], ['1312.0576-1-48-0', '1312.0576-2-48-0'], ['1312.0576-1-48-5', '1312.0576-2-48-5'], ['1312.0576-1-48-7', '1312.0576-2-48-7'], ['1312.0576-1-48-8', '1312.0576-2-48-8'], ['1312.0576-1-48-11', '1312.0576-2-48-11'], ['1312.0576-1-48-12', '1312.0576-2-48-12'], ['1312.0576-1-40-0', '1312.0576-2-40-0'], ['1312.0576-1-40-1', '1312.0576-2-40-1'], ['1312.0576-1-40-2', '1312.0576-2-40-2'], ['1312.0576-1-40-3', '1312.0576-2-40-3'], ['1312.0576-1-40-4', '1312.0576-2-40-4'], ['1312.0576-1-40-5', '1312.0576-2-40-5'], ['1312.0576-1-22-1', '1312.0576-2-23-1'], ['1312.0576-1-22-3', '1312.0576-2-23-3'], ['1312.0576-1-39-0', '1312.0576-2-39-0'], ['1312.0576-1-39-1', '1312.0576-2-39-1'], ['1312.0576-1-39-2', '1312.0576-2-39-2'], ['1312.0576-1-39-3', '1312.0576-2-39-3'], ['1312.0576-1-39-4', '1312.0576-2-39-4'], ['1312.0576-1-39-5', '1312.0576-2-39-5'], ['1312.0576-1-39-6', '1312.0576-2-39-6'], ['1312.0576-1-39-7', '1312.0576-2-39-7'], ['1312.0576-1-39-8', '1312.0576-2-39-8'], ['1312.0576-1-39-11', '1312.0576-2-39-11'], ['1312.0576-1-39-15', '1312.0576-2-39-15'], ['1312.0576-1-0-1', '1312.0576-2-0-1'], ['1312.0576-1-0-2', '1312.0576-2-0-2'], ['1312.0576-1-5-1', '1312.0576-2-5-1'], ['1312.0576-1-5-2', '1312.0576-2-5-2'], ['1312.0576-1-5-4', '1312.0576-2-5-4'], ['1312.0576-1-5-5', '1312.0576-2-5-5'], ['1312.0576-1-5-6', '1312.0576-2-5-7'], ['1312.0576-1-5-9', '1312.0576-2-5-9'], ['1312.0576-1-5-14', '1312.0576-2-5-15'], ['1312.0576-1-5-15', '1312.0576-2-5-16'], ['1312.0576-1-5-16', '1312.0576-2-5-17'], ['1312.0576-1-5-19', '1312.0576-2-5-20'], ['1312.0576-1-5-21', '1312.0576-2-5-22'], ['1312.0576-1-8-0', '1312.0576-2-8-0'], ['1312.0576-1-8-1', '1312.0576-2-8-1'], ['1312.0576-1-8-2', '1312.0576-2-8-2'], ['1312.0576-1-30-2', '1312.0576-2-30-1'], ['1312.0576-1-30-5', '1312.0576-2-30-4'], ['1312.0576-1-30-12', '1312.0576-2-30-12'], ['1312.0576-1-30-15', '1312.0576-2-30-16'], ['1312.0576-1-30-19', '1312.0576-2-31-2'], ['1312.0576-1-30-20', '1312.0576-2-31-3'], ['1312.0576-1-30-22', '1312.0576-2-31-5'], ['1312.0576-1-30-23', '1312.0576-2-31-6'], ['1312.0576-1-2-0', '1312.0576-2-2-0'], ['1312.0576-1-2-1', '1312.0576-2-2-1'], ['1312.0576-1-2-7', '1312.0576-2-2-7'], ['1312.0576-1-2-10', '1312.0576-2-2-10'], ['1312.0576-1-2-11', '1312.0576-2-2-11'], ['1312.0576-1-2-12', '1312.0576-2-2-12'], ['1312.0576-1-2-14', '1312.0576-2-2-14'], ['1312.0576-1-2-15', '1312.0576-2-2-15'], ['1312.0576-1-2-16', '1312.0576-2-2-16'], ['1312.0576-1-2-17', '1312.0576-2-2-17'], ['1312.0576-1-2-18', '1312.0576-2-2-18'], ['1312.0576-1-2-19', '1312.0576-2-2-19'], ['1312.0576-1-2-21', '1312.0576-2-2-21'], ['1312.0576-1-33-0', '1312.0576-2-34-1'], ['1312.0576-1-33-3', '1312.0576-2-34-4'], ['1312.0576-1-33-6', '1312.0576-2-34-7'], ['1312.0576-1-34-1', '1312.0576-2-34-13'], ['1312.0576-1-34-4', '1312.0576-2-34-16'], ['1312.0576-1-27-0', '1312.0576-2-28-0']]

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[]

[['1312.0576-1-15-1', '1312.0576-2-15-0'], ['1312.0576-1-6-1', '1312.0576-2-6-2'], ['1312.0576-1-45-2', '1312.0576-2-45-2'], ['1312.0576-1-43-0', '1312.0576-2-43-0'], ['1312.0576-1-43-10', '1312.0576-2-43-10'], ['1312.0576-1-19-2', '1312.0576-2-20-2'], ['1312.0576-1-19-7', '1312.0576-2-20-8'], ['1312.0576-1-17-0', '1312.0576-2-17-1'], ['1312.0576-1-17-0', '1312.0576-2-17-11'], ['1312.0576-1-17-1', '1312.0576-2-17-5'], ['1312.0576-1-17-1', '1312.0576-2-17-11'], ['1312.0576-1-17-3', '1312.0576-2-17-9'], ['1312.0576-1-17-3', '1312.0576-2-17-10'], ['1312.0576-1-17-4', '1312.0576-2-17-11'], ['1312.0576-1-17-8', '1312.0576-2-17-16'], ['1312.0576-1-17-8', '1312.0576-2-17-17'], ['1312.0576-1-25-7', '1312.0576-2-26-7'], ['1312.0576-1-14-10', '1312.0576-2-14-14'], ['1312.0576-1-48-2', '1312.0576-2-48-2'], ['1312.0576-1-48-3', '1312.0576-2-48-3'], ['1312.0576-1-48-14', '1312.0576-2-48-14'], ['1312.0576-1-39-13', '1312.0576-2-39-13'], ['1312.0576-1-5-12', '1312.0576-2-5-12'], ['1312.0576-1-5-12', '1312.0576-2-5-13'], ['1312.0576-1-5-13', '1312.0576-2-5-14'], ['1312.0576-1-5-20', '1312.0576-2-5-21'], ['1312.0576-1-30-4', '1312.0576-2-30-3'], ['1312.0576-1-30-7', '1312.0576-2-30-7'], ['1312.0576-1-30-18', '1312.0576-2-31-1'], ['1312.0576-1-2-8', '1312.0576-2-2-8'], ['1312.0576-1-27-1', '1312.0576-2-28-1'], ['1312.0576-1-27-2', '1312.0576-2-28-2'], ['1312.0576-1-27-8', '1312.0576-2-28-9']]

[]

['1312.0576-1-4-0', '1312.0576-1-5-3', '1312.0576-1-5-7', '1312.0576-1-6-2', '1312.0576-1-7-0', '1312.0576-1-7-1', '1312.0576-1-10-0', '1312.0576-1-10-1', '1312.0576-1-14-1', '1312.0576-1-14-2', '1312.0576-1-17-6', '1312.0576-1-18-6', '1312.0576-1-19-5', '1312.0576-1-19-6', '1312.0576-1-19-8', '1312.0576-1-19-10', '1312.0576-1-19-11', '1312.0576-1-19-13', '1312.0576-1-21-0', '1312.0576-1-22-2', '1312.0576-1-23-4', '1312.0576-1-27-3', '1312.0576-1-27-5', '1312.0576-1-30-14', '1312.0576-1-30-16', '1312.0576-1-30-21', '1312.0576-1-32-0', '1312.0576-1-32-1', '1312.0576-1-33-1', '1312.0576-1-33-4', '1312.0576-1-33-5', '1312.0576-1-33-7', '1312.0576-1-33-8', '1312.0576-1-33-9', '1312.0576-1-33-10', '1312.0576-1-34-3', '1312.0576-1-35-5', '1312.0576-1-36-0', '1312.0576-1-38-0', '1312.0576-1-39-9', '1312.0576-1-39-10', '1312.0576-1-39-14', '1312.0576-1-43-3', '1312.0576-1-43-6', '1312.0576-1-43-7', '1312.0576-1-48-1', '1312.0576-1-48-9', '1312.0576-1-48-13', '1312.0576-2-4-0', '1312.0576-2-4-1', '1312.0576-2-5-3', '1312.0576-2-5-8', '1312.0576-2-7-0', '1312.0576-2-7-1', '1312.0576-2-10-0', '1312.0576-2-10-1', '1312.0576-2-14-1', '1312.0576-2-14-2', '1312.0576-2-14-13', '1312.0576-2-18-11', '1312.0576-2-19-0', '1312.0576-2-19-1', '1312.0576-2-20-0', '1312.0576-2-20-4', '1312.0576-2-20-6', '1312.0576-2-20-7', '1312.0576-2-20-9', '1312.0576-2-20-11', '1312.0576-2-20-12', '1312.0576-2-20-14', '1312.0576-2-22-0', '1312.0576-2-22-1', '1312.0576-2-23-2', '1312.0576-2-23-4', '1312.0576-2-24-4', '1312.0576-2-28-3', '1312.0576-2-28-5', '1312.0576-2-28-7', '1312.0576-2-30-14', '1312.0576-2-30-17', '1312.0576-2-33-0', '1312.0576-2-33-1', '1312.0576-2-34-2', '1312.0576-2-34-5', '1312.0576-2-34-6', '1312.0576-2-34-8', '1312.0576-2-34-9', '1312.0576-2-34-10', '1312.0576-2-34-11', '1312.0576-2-34-15', '1312.0576-2-35-5', '1312.0576-2-36-0', '1312.0576-2-38-0', '1312.0576-2-38-1', '1312.0576-2-39-9', '1312.0576-2-39-10', '1312.0576-2-41-0', '1312.0576-2-41-1', '1312.0576-2-43-3', '1312.0576-2-43-6', '1312.0576-2-43-7', '1312.0576-2-48-1', '1312.0576-2-48-9', '1312.0576-2-48-13']

{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1312.0576

hep-th-0102070

{'hep-th-0102070-1-0-0': 'The non-commutative version of the euclidean [MATH] theory is considered.', 'hep-th-0102070-1-0-1': 'By using Wilsonian flow equations the ultraviolet renormalizability can be proved to all orders in perturbation theory.', 'hep-th-0102070-1-0-2': 'On the other hand, the infrared sector cannot be treated perturbatively and requires a resummation of the leading divergences in the two-point function.', 'hep-th-0102070-1-0-3': 'This is somehow analogous to what is done in the Hard Thermal Loops resummation of finite temperature field theory.', 'hep-th-0102070-1-0-4': 'Next-to-leading order corrections to the self-energy are computed, resulting in [MATH] contributions in the massless case, and [MATH] in the massive one.', 'hep-th-0102070-1-1-0': ']', 'hep-th-0102070-1-2-0': 'Quantum field theories on non-commutative spaces have been the subject of intense investigation in the recent past, mainly motivated by their tight relation with string theories.', 'hep-th-0102070-1-2-1': 'Low energy excitations of a [MATH]-brane in a magnetic [MATH] background are indeed described by field theories with space non-commutativity [CITATION].', 'hep-th-0102070-1-2-2': 'In this limit the relevant description of dynamics is in term of massless open string states, while massive open string states and closed strings decouple: the full consistent string theory seems therefore truncated to the usual field theoretical degrees of freedom, suggesting the possibility that also the related quantum field theories are well defined.', 'hep-th-0102070-1-2-3': 'On the other hand their consistency is far from being obvious when examined from a purely field theoretical point of view: they are non-local (involving interactions with an arbitrarily high number of derivatives) and there is a new dimensionful parameter, other than the masses, taking into account the scale at which non-commutativity becomes relevant.', 'hep-th-0102070-1-2-4': 'It is then natural to ask whethe unitarity and renormalizability are implemented in these theories.', 'hep-th-0102070-1-2-5': "For what concernes unitarity, it was shown in [CITATION] that when the non-commutativity involves space and time the perturbative unitarity is indeed in trouble, whereas in the pure spatial case consistency with the Cutkoski's rules and unitary time evolution has been checked [CITATION].", 'hep-th-0102070-1-3-0': 'The issue of renormalizability is more subtle.', 'hep-th-0102070-1-3-1': 'Contrary to early suggestions it was shown in ref. [CITATION] that infinities appear when perturbative computations are performed in non-commutative scalar theories.', 'hep-th-0102070-1-3-2': 'Moreover, an higly non-trivial mixture between ultraviolet (UV) and infra-red (IR) behaviors [CITATION] makes a proof of perturbative renormalization along the usual lines quite cumbersome.', 'hep-th-0102070-1-3-3': 'Indeed, the one-loop self-energy in the non-commutative version of [MATH] theory gets a contribution of [MATH], where [MATH] is the scale of non-commutativity and [MATH] the external momentum.', 'hep-th-0102070-1-3-4': "This behavior is easily understood taking into account that the so called 'non-planar' graphs are effectively cut-off in the UV at a scale [MATH] and that the scalar self-energy is quadratically divergent in the commutative case [CITATION].", 'hep-th-0102070-1-3-5': 'When inserted in a higher order graph, the one-loop self-energy induces IR-divergences even in the case of a massive theory.', 'hep-th-0102070-1-3-6': 'For instance, the two-point function diverges quadratically in the IR for the massless theory at [MATH] and logarithmically for the massive one at [MATH], more tadpole insertions giving more and more IR-divergent behaviors.', 'hep-th-0102070-1-3-7': 'A similar behavior was discovered for gauge theories [CITATION].', 'hep-th-0102070-1-4-0': "Due to this problematic 'IR/UV connection' no complete calculation has been performed up to now in the scalar theory at next-to-leading order in perturbation theory.", 'hep-th-0102070-1-4-1': 'The possibility of absorbing UV divergences by means of local counterterms has been discussed at two-loops in refs. [CITATION], but no finite result could be obtained at that order due to the pathological behavior of the integrals in the IR.', 'hep-th-0102070-1-5-0': 'In this letter we present the result of a resummation of the IR divergences which allows a consistent computation of finite corrections beyond the leading perturbative order.', 'hep-th-0102070-1-5-1': 'The need of a resummation has been realized by different people, and discussed for instance in [CITATION].', 'hep-th-0102070-1-5-2': 'However, to our knowledge, no systematic approach has been formulated and therefore no explicit computation has been presented up to now.', 'hep-th-0102070-1-5-3': 'In the resummed perturbative expansion, the first correction to the leading [MATH] contribution to the self-energy of the massless theory arises at [MATH] instead of the naively expected [MATH].', 'hep-th-0102070-1-5-4': 'For the massive case, the resummation of all the problematic diagrams first arising at [MATH], results in a [MATH] behavior.', 'hep-th-0102070-1-6-0': 'The pattern of the IR problem in the non-commutative scalar theory presents some remarkable similarity with that arising in finite temperature field theory [CITATION].', 'hep-th-0102070-1-6-1': 'In that case, the UV divergences of the new thermal contributions are cut-off by the temperature [MATH], resulting in a [MATH] correction to the self-energy at one-loop.', 'hep-th-0102070-1-6-2': 'The ratio between the one-loop amplitude and the tree-level one, [MATH] becomes of [MATH] for soft external momenta, [MATH]<[MATH][MATH], so that a resummation must be performed in order to define a sensible perturbative expansion.', 'hep-th-0102070-1-6-3': "The program is accomplished by using a resummed propagator, in which a 'thermal mass' is included, [MATH].", 'hep-th-0102070-1-6-4': "Since the amplitude which has been resummed receives contributions mainly by the hard ([MATH]) momentum part of the one-loop integral, the resummed theory goes customarily under the name of 'hard thermal loops'.", 'hep-th-0102070-1-6-5': "Analogously, we will use propagators in which one-loop amplitudes, dominated by momenta [MATH], are resummed, and will consistently call this procedure 'hard non-commutative loops' resummation.", 'hep-th-0102070-1-7-0': 'Before giving the details and results of the resummation, we outline a formulation of the Wilsonian renormalization group (RG) a la Polchinski [CITATION] for the non-commutative scalar theory which will be presented in detail in a forthcoming publication.', 'hep-th-0102070-1-7-1': 'By studying perturbatively the Wilsonian flow equation we are able to prove UV renormalization to all orders.', 'hep-th-0102070-1-7-2': 'Then, turning to the IR regime, we will show how a resummation procedure emerges quite naturally in this context.', 'hep-th-0102070-1-8-0': '>From a field-theoretical point of view, the non-commutative version of [MATH] theory is again a scalar theory with the same tree-level propagator but a different vertex, which at tree-level is given by [EQUATION] where [MATH], [MATH] being the anti-symmetric matrix defining the commutation relations ([MATH]).', 'hep-th-0102070-1-8-1': 'As a consequence, the Wilsonian action and the RG equations can be derived in the same way as in refs. [CITATION] - they are actually the same equations, but with different boundary conditions.', 'hep-th-0102070-1-8-2': "Our proof of perturbative UV renormalization parallels quite closely the one given by Bonini, D'Attanasio and Marchesini for the commutative case [CITATION], so we only sketch here the main lines and stress only the differences emerging in the non-commutative theory.", 'hep-th-0102070-1-9-0': "A Wilsonian effective action can be defined, [MATH], as the generating functional of 1PI Green's functions obtained by integrating out loop momenta [MATH] such that [MATH].", 'hep-th-0102070-1-9-1': 'Our task is to prove that the double limit [MATH] (UV renormalizability) and [MATH] (IR finitness) can be taken.', 'hep-th-0102070-1-10-0': "[MATH] and the Green's function generated by [MATH]-derivating it, obey exact evolution equations in [MATH].", 'hep-th-0102070-1-10-1': "There is a simple recipe to obtain the RG equation for any [MATH]-point function; i) write the 1-loop expression for [MATH] obtained by using all the vertices up to [MATH], as if they were formally tree-level; ii) promote the tree-level vertices above to full, running, vertices, [MATH], and the tree-level propagator to the full, cut-off, propagator, [EQUATION] where [MATH] is the full, running, self-energy, and the cut-off function [MATH] is equal to one in the interval [MATH] and vanishes rapidly outside; iii) take the derivative with respect to [MATH] everywhere in the [MATH]'s but not in the [MATH]'s or [MATH]'s.", 'hep-th-0102070-1-11-0': 'So, the evolution equation for e.g. the self-energy is given by [EQUATION] where [EQUATION]', 'hep-th-0102070-1-11-1': 'The RG equations for higher point functions are obtained analogously, and they together form an infinite system of coupled ordinary differential equations which define the theory non-perturbatively.', 'hep-th-0102070-1-11-2': 'The renormalization conditions are imposed by properly chosing the boundary conditions.', 'hep-th-0102070-1-11-3': 'For the relevant vertices [EQUATION] (where [MATH] is the renormalization scale and the momenta [MATH] have been chosen such that [MATH]) the boundary conditions are given at the physical point [MATH], [EQUATION]', 'hep-th-0102070-1-11-4': "The boundary conditions for all the other -irrelevant- vertices (higher momentum derivatives in [MATH] and [MATH] and all the [MATH]'s with [MATH]) are instead fixed at the UV, [MATH], where the irrelevant vertices can be set equal to zero.", 'hep-th-0102070-1-12-0': 'With these boundary conditions, the relevant vertices at a generic [MATH] are given by integrals between [MATH] and [MATH], whereas the irrelevant ones are given by integrals between [MATH] and [MATH] [CITATION].', 'hep-th-0102070-1-12-1': 'The proof of UV renormalizability then amounts to proving that the integrals giving the irrelevant vertices are finite in the [MATH] limit.', 'hep-th-0102070-1-12-2': 'In order to disentangle the UV from the IR sector, at this stage we chose [MATH] much larger than any physical scale of the theory, that is [MATH].', 'hep-th-0102070-1-12-3': 'This simplifies enormously the power counting, and allows a inductive proof at any order in perturbation theory, essentially based on dimensional considerations.', 'hep-th-0102070-1-13-0': 'Without giving the details, we only remark here that the above mentioned [MATH] behavior of the 1-loop self-energy does not enter the discussion at this point, since the momenta in the relevant integrals is bounded from below by [MATH], and the one-loop self-energy is subdominant with respect to the tree-level [MATH] contribution in ([REF]).', 'hep-th-0102070-1-13-1': 'Essentially, the power-counting in the UV is the same as in the commutative case, and the proof of perturbative UV renormalization can be easily translated to the non-commutative case.', 'hep-th-0102070-1-14-0': 'When the IR regime comes under scrutiny, things change considerably.', 'hep-th-0102070-1-14-1': "In [CITATION] the IR finitness of Green's functions with non-exceptional external momenta (i.e [MATH]) was proved for the commutative massless theory at any order in perturbation theory.", 'hep-th-0102070-1-14-2': 'Crucial for that proof is the fact that [MATH] is at most logarithmically divergent as [MATH] at any finite order in the expansion.', 'hep-th-0102070-1-14-3': 'As we have repeatedly seen, this is not the case any more in the non-commutative case, where [MATH].', 'hep-th-0102070-1-14-4': 'Any perturbative computation is thus plagued by IR divergences which emerge sooner or later in the expansion in [MATH].', 'hep-th-0102070-1-15-0': 'A quick look at the exact form of the RG evolution equations, eq. ([REF]) and in particular at the kernel in eq. ([REF]), shows both what the problem is and how a solution can be found.', 'hep-th-0102070-1-15-1': 'Namely, the more dangerous IR divergences come out when -at any finite order in perturbation theory- one expands the full-propagator appearing in the kernel in powers of [MATH].', 'hep-th-0102070-1-15-2': 'Since [MATH] in the kernel, this ratio diverges as [MATH] in the massive theory and as [MATH] in the massless one.', 'hep-th-0102070-1-15-3': 'It is then clear that any Green function - even at non-exceptional momenta- will be divergent at a sufficiently high order in [MATH].', 'hep-th-0102070-1-16-0': 'The exact form of the RG kernel gives the solution as well.', 'hep-th-0102070-1-16-1': 'Since [MATH] comes in the denominator, it is clear that the full equations are indeed better behaved in the IR than any approximation to them computed at any finite order in [MATH].', 'hep-th-0102070-1-16-2': 'Actually, since the effective mass explodes as [MATH], they are even better behaved than those for the massive theory in the commutative case!', 'hep-th-0102070-1-17-0': 'It appears then clear that the IR pathologies are just an artifact of the perturbative expansion, which should disappear if this is properly reorganized.', 'hep-th-0102070-1-17-1': 'To this end, one can still pursue the RG framework, splitting the full two-point function as [EQUATION] where [MATH] is the leading IR contribution to the one-loop self-energy (we have defined [MATH]).', 'hep-th-0102070-1-17-2': "Eq. ([REF]) defines a new expansion in terms of [MATH] and the new 'tree-level' propagator [MATH].", 'hep-th-0102070-1-17-3': 'All this can be done consistently in the Wilsonian RG framework, where the evolution equation for [MATH] is easily obtained from that for the full self-energy.', 'hep-th-0102070-1-18-0': "In more common language, the resummation procedure simply amounts to adding and subtracting the term [EQUATION] to the tree-level lagrangian, so as to get the resummed propagator provided a new two-point 'interaction' in eq. ([REF]) is consistently taken into account, in very close analogy to what is done in the finite temperature theory case [CITATION].", 'hep-th-0102070-1-19-0': 'The interactions of the resummed theory give the Feynman rules in Fig. 1.', 'hep-th-0102070-1-20-0': 'Now we are ready to compute the next-to-leading order corrections to the self-energy, which are given by the two diagrams in Fig. 2, where the resummed propagator runs into the loop (of course also the graph with the UV counterterms has to be included, which is not shown in the figure).', 'hep-th-0102070-1-21-0': 'The tadpole diagram in the resummed theory gives [EQUATION]', 'hep-th-0102070-1-21-1': "In the UV, the integral has the same structure as for the non-resummed theory, with a quadratically divergent contribution from the 'planar' diagrams and a finite one from the 'non-planar' ones, giving the [MATH] term which is exactly cancelled by the new two-point interaction of the resummed theory.", 'hep-th-0102070-1-21-2': 'In the IR, the planar and non-planar contributions sum up.', 'hep-th-0102070-1-21-3': 'By writing [EQUATION] where [MATH] and [MATH] is a traceless symmetric matrix.', 'hep-th-0102070-1-21-4': 'The symmetry of the integrand in the IR regime, selects the [MATH] term as the dominant contribution.', 'hep-th-0102070-1-22-0': "In the massless case ([MATH]) we find the following contribution from the 'planar' graph [EQUATION] whereas from the 'non-planar' one we get [EQUATION] for [MATH] and [EQUATION] for [MATH], where we have defined [MATH].", 'hep-th-0102070-1-22-1': 'In the massive case we get (planar + non-planar) [EQUATION]', 'hep-th-0102070-1-22-2': 'As one could expect, the non-analiticity in the coupling [MATH] emerges at lower order in the massless case (where we find a [MATH] correction) compared to the massive one ([MATH]).', 'hep-th-0102070-1-22-3': 'This reflects the fact that, in ordinary perturbation theory, the self-energy is IR divergent at [MATH] in the former case and at [MATH] in the latter.', 'hep-th-0102070-1-23-0': 'In computing the next-to-next-to-leading order in the resummed perturbative expansion one must consistently take into account the two-point interaction in ([REF]).', 'hep-th-0102070-1-23-1': 'Indeed, the two-loop graph for the resummed [MATH] theory with one non-planar tadpole insertion (first graph in Fig. 3) gives a contribution of [MATH], the same as the corrections computed above.', 'hep-th-0102070-1-23-2': 'It is only when the graph containing the two-point interaction is added that the whole correction comes out [MATH].', 'hep-th-0102070-1-24-0': 'The [MATH] corrections that one gets at two-loop, coming from UV loop momenta, cannot modify the [MATH] term in eq. ([REF]).', 'hep-th-0102070-1-25-0': "The corrections computed above are really 'perturbatively small' compared to the leading two-point function [MATH] in any range of the momentum [MATH].", 'hep-th-0102070-1-25-1': 'Indeed, for large enough momenta, the [MATH] correction dominates over the [MATH] term, but in that regime the tree-level [MATH] term is leading.', 'hep-th-0102070-1-25-2': 'On the other hand in the IR the opposite happens, with [MATH] never dominating over [MATH].', 'hep-th-0102070-1-25-3': 'As a consequence, no tachyonic like behavior can be induced by the next-to-leading order corrections.', 'hep-th-0102070-1-26-0': 'We thank Marisa Bonini for useful discussions.'}

{'hep-th-0102070-2-0-0': 'The non-commutative version of the euclidean [MATH] theory is considered.', 'hep-th-0102070-2-0-1': 'By using Wilsonian flow equations the ultraviolet renormalizability can be proved to all orders in perturbation theory.', 'hep-th-0102070-2-0-2': 'On the other hand, the infrared sector cannot be treated perturbatively and requires a resummation of the leading divergences in the two-point function.', 'hep-th-0102070-2-0-3': 'This is somehow analogous to what is done in the Hard Thermal Loops resummation of finite temperature field theory.', 'hep-th-0102070-2-0-4': 'Next-to-leading order corrections to the self-energy are computed, resulting in [MATH] contributions in the massless case, and [MATH] in the massive one.', 'hep-th-0102070-2-1-0': ']', 'hep-th-0102070-2-2-0': 'Quantum field theories on non-commutative spaces have been the subject of intense investigation in the recent past, mainly motivated by their tight relation with string theories.', 'hep-th-0102070-2-2-1': 'Low energy excitations of a [MATH]-brane in a magnetic [MATH] background are indeed described by field theories with space non-commutativity [CITATION].', 'hep-th-0102070-2-2-2': 'In this limit the relevant description of dynamics is in term of massless open string states, while massive open string states and closed strings decouple: the full consistent string theory seems therefore truncated to the usual field theoretical degrees of freedom, suggesting the possibility that also the related quantum field theories are well defined.', 'hep-th-0102070-2-2-3': 'On the other hand their consistency is far from being obvious when examined from a purely field theoretical point of view: they are non-local (involving interactions with an arbitrarily high number of derivatives) and there is a new dimensionful parameter, other than the masses, taking into account the scale at which non-commutativity becomes relevant.', 'hep-th-0102070-2-3-0': 'It is then natural to ask whether these theories are renormalizable.', 'hep-th-0102070-2-3-1': 'Contrary to early suggestions it was shown in ref. [CITATION] that infinities appear when perturbative computations are performed in non-commutative scalar theories.', 'hep-th-0102070-2-3-2': 'Moreover, an higly non-trivial mixture between ultraviolet (UV) and infra-red (IR) behaviors [CITATION] makes a proof of perturbative renormalization along the usual lines quite cumbersome.', 'hep-th-0102070-2-3-3': 'Indeed, the one-loop self-energy in the non-commutative version of [MATH] theory gets a contribution of [MATH], where [MATH] is the scale of non-commutativity and [MATH] the external momentum.', 'hep-th-0102070-2-3-4': "This behavior is easily understood taking into account that the so called 'non-planar' graphs are effectively cut-off in the UV at a scale [MATH] and that the scalar self-energy is quadratically divergent in the commutative case [CITATION].", 'hep-th-0102070-2-3-5': 'When inserted in a higher order graph, the one-loop self-energy induces IR-divergences even in the case of a massive theory.', 'hep-th-0102070-2-3-6': 'For instance, the two-point function diverges quadratically in the IR for the massless theory at [MATH] and logarithmically for the massive one at [MATH], more tadpole insertions giving more and more IR-divergent behaviors.', 'hep-th-0102070-2-3-7': 'A similar behavior was discovered for gauge theories [CITATION].', 'hep-th-0102070-2-4-0': "Due to this problematic 'IR/UV connection' no complete calculation has been performed up to now in the scalar theory at next-to-leading order in perturbation theory.", 'hep-th-0102070-2-4-1': 'The possibility of absorbing UV divergences by means of local counterterms has been discussed at two-loops in refs. [CITATION], but no finite result could be obtained at that order due to the pathological behavior of the integrals in the IR.', 'hep-th-0102070-2-5-0': 'In this letter we will present the result of a resummation of the IR divergences which allows a consistent computation of finite corrections beyond the leading perturbative order.', 'hep-th-0102070-2-5-1': 'Before doing that, we will give the main lines of a proof of UV renormalizability to all orders in perturbation theory, which will be presented in full detail in [CITATION].', 'hep-th-0102070-2-6-0': 'The need of a resummation has been realized by different people, and discussed for instance in [CITATION].', 'hep-th-0102070-2-6-1': 'However, to our knowledge, no systematic approach has been formulated and therefore no explicit computation has been presented up to now.', 'hep-th-0102070-2-6-2': 'In the resummed perturbative expansion, the first correction to the leading [MATH] contribution to the self-energy of the massless theory arises at [MATH] instead of the naively expected [MATH].', 'hep-th-0102070-2-6-3': 'For the massive case, the resummation of all the problematic diagrams first arising at [MATH], results in a [MATH] behavior.', 'hep-th-0102070-2-7-0': 'The pattern of the IR problem in the non-commutative scalar theory presents some remarkable similarity with that arising in finite temperature field theory [CITATION].', 'hep-th-0102070-2-7-1': 'In that case, the UV divergences of the new thermal contributions are cut-off by the temperature [MATH], resulting in a [MATH] correction to the self-energy at one-loop.', 'hep-th-0102070-2-7-2': 'The ratio between the one-loop amplitude and the tree-level one, [MATH] becomes of [MATH] for soft external momenta, [MATH]<[MATH][MATH], so that a resummation must be performed in order to define a sensible perturbative expansion.', 'hep-th-0102070-2-7-3': "The program is accomplished by using a resummed propagator, in which a 'thermal mass' is included, [MATH].", 'hep-th-0102070-2-7-4': "Since the amplitude which has been resummed receives contributions mainly by the hard ([MATH]) momentum part of the one-loop integral, the resummed theory goes customarily under the name of 'hard thermal loops'.", 'hep-th-0102070-2-7-5': "Analogously, we will use propagators in which one-loop amplitudes, dominated by momenta [MATH], are resummed, and will consistently call this procedure 'hard non-commutative loops' resummation.", 'hep-th-0102070-2-8-0': 'Before giving the details and results of the resummation, we outline a formulation of the Wilsonian renormalization group (RG) a la Polchinski [CITATION] for the non-commutative scalar theory which will be presented in detail in a forthcoming publication.', 'hep-th-0102070-2-8-1': 'By studying perturbatively the Wilsonian flow equation we are able to prove UV renormalization to all orders.', 'hep-th-0102070-2-8-2': 'Then, turning to the IR regime, we will show how a resummation procedure emerges quite naturally in this context.', 'hep-th-0102070-2-9-0': 'From a field-theoretical point of view, the non-commutative version of [MATH] theory is again a scalar theory with the same tree-level propagator but a different vertex, which at tree-level is given by [EQUATION] where [MATH], [MATH] being the anti-symmetric matrix defining the commutation relations ([MATH]).', 'hep-th-0102070-2-9-1': 'As a consequence, the Wilsonian action and the RG equations can be derived in the same way as in refs. [CITATION] - they are actually the same equations, but with different boundary conditions.', 'hep-th-0102070-2-9-2': "Our proof of perturbative UV renormalization parallels quite closely the one given by Bonini, D'Attanasio and Marchesini for the commutative case [CITATION], so we only sketch here the main lines and stress only the differences emerging in the non-commutative theory.", 'hep-th-0102070-2-10-0': 'A Wilsonian effective action can be defined, [MATH], as the generating functional of 1PI Green functions obtained by integrating out loop momenta [MATH] such that [MATH].', 'hep-th-0102070-2-10-1': 'Our task is to prove that the double limit [MATH] (UV renormalizability) and [MATH] (IR finitness) can be taken.', 'hep-th-0102070-2-11-0': '[MATH] and the Green function generated by [MATH]-derivating it, obey exact evolution equations in [MATH].', 'hep-th-0102070-2-11-1': "There is a simple recipe to obtain the RG equation for any [MATH]-point function; i) write the 1-loop expression for [MATH] obtained by using all the vertices up to [MATH], as if they were formally tree-level; ii) promote the tree-level vertices above to full, running, vertices, [MATH], and the tree-level propagator to the full, cut-off, propagator, [EQUATION] where [MATH] is the full, running, self-energy, and the cut-off function [MATH] is equal to one in the interval [MATH] and vanishes rapidly outside; iii) take the derivative with respect to [MATH] everywhere in the [MATH]'s but not in the [MATH]'s or [MATH]'s.", 'hep-th-0102070-2-12-0': 'So, the evolution equation for e.g. the self-energy is given by [EQUATION] where [EQUATION]', 'hep-th-0102070-2-12-1': 'The RG equations for higher point functions are obtained analogously, and they together form an infinite system of coupled ordinary differential equations which define the theory non-perturbatively.', 'hep-th-0102070-2-12-2': 'The renormalization conditions are imposed by properly chosing the boundary conditions.', 'hep-th-0102070-2-12-3': 'For the relevant vertices [EQUATION] (where [MATH] is the renormalization scale and the momenta [MATH] have been chosen such that [MATH]) the boundary conditions are given at the physical point [MATH], [EQUATION]', 'hep-th-0102070-2-12-4': "The boundary conditions for all the other -irrelevant- vertices (higher momentum derivatives in [MATH] and [MATH] and all the [MATH]'s with [MATH]) are instead fixed at the UV, [MATH], where the irrelevant vertices can be set equal to zero.", 'hep-th-0102070-2-13-0': 'With these boundary conditions, the relevant vertices at a generic [MATH] are given by integrals between [MATH] and [MATH], whereas the irrelevant ones are given by integrals between [MATH] and [MATH] [CITATION].', 'hep-th-0102070-2-13-1': 'Then, one can disentangle the UV from the IR by taking the physical limit [MATH] and [MATH] in two successive steps.', 'hep-th-0102070-2-13-2': 'The proof of UV renormalizability follows quite closely the well known one given by Polchinski in the commutative case [CITATION], and discussed further in [CITATION].', 'hep-th-0102070-2-13-3': 'It exploits two remarkable features of the RG equations; i) the momentum ordering, by which a given irrelevant coupling evaluated at cut-off [MATH] receives contributions only from loop momenta [MATH], and ii) the one-loop structure of the exact equations, which make it possible to recover perturbation theory solving them iteratively.', 'hep-th-0102070-2-13-4': 'As a result, the proof is extremely simple, as it is based just on power counting arguments.', 'hep-th-0102070-2-13-5': 'In the UV regime, the non-commutative and the commutative theories exhibit essentially the same power counting.', 'hep-th-0102070-2-13-6': 'Indeed, by chosing [MATH] much larger than any physical scale of the theory, that is [MATH] the above mentioned [MATH] behavior of the 1-loop self-energy has not developed yet, as the momenta in the relevant integrals are bounded from below by [MATH], and the one-loop self-energy is subdominant with respect to the tree-level [MATH] contribution in ([REF]).', 'hep-th-0102070-2-13-7': 'As a consequence, the [MATH] limit can be shown to be finite at any perturbative order by a straightforward translation of the arguments given in [CITATION].', 'hep-th-0102070-2-13-8': 'All the details of the proof will be given in a separate paper [CITATION].', 'hep-th-0102070-2-14-0': 'When the IR regime comes under scrutiny, things change considerably.', 'hep-th-0102070-2-14-1': 'In [CITATION] the IR finitness of Green functions with non-exceptional external momenta (i.e [MATH]) was proved for the commutative massless theory at any order in perturbation theory.', 'hep-th-0102070-2-14-2': 'Crucial for that proof is the fact that [MATH] is at most logarithmically divergent as [MATH] at any finite order in the expansion.', 'hep-th-0102070-2-14-3': 'As we have repeatedly seen, this is not the case any more in the non-commutative case, where [MATH].', 'hep-th-0102070-2-14-4': 'Any perturbative computation is thus plagued by IR divergences which emerge sooner or later in the expansion in [MATH].', 'hep-th-0102070-2-15-0': 'A quick look at the exact form of the RG evolution equations, and in particular at the kernel in eq. ([REF]), shows both what the problem is and how a solution can be found.', 'hep-th-0102070-2-15-1': 'Namely, the more dangerous IR divergences come out when -at any finite order in perturbation theory- one expands the full-propagator appearing in the kernel in powers of [MATH].', 'hep-th-0102070-2-15-2': 'Since [MATH] in the kernel, this ratio diverges as [MATH] in the massive theory and as [MATH] in the massless one.', 'hep-th-0102070-2-15-3': 'It is then clear that any Green function - even at non-exceptional momenta- will be divergent at a sufficiently high order in [MATH].', 'hep-th-0102070-2-16-0': 'The exact form of the RG kernel gives the solution as well.', 'hep-th-0102070-2-16-1': 'Since [MATH] comes in the denominator, it is clear that the full equations are indeed better behaved in the IR than any approximation to them computed at any finite order in [MATH].', 'hep-th-0102070-2-16-2': 'Actually, since the effective mass explodes as [MATH], they are even better behaved than those for the massive theory in the commutative case!', 'hep-th-0102070-2-17-0': 'It appears then clear that the IR pathologies are just an artifact of the perturbative expansion, which should disappear if this is properly reorganized.', 'hep-th-0102070-2-17-1': 'To this end, one can still pursue the RG framework, splitting the full two-point function as [EQUATION] where [MATH] is the leading IR contribution to the one-loop self-energy (we have defined [MATH]).', 'hep-th-0102070-2-17-2': "Eq. ([REF]) defines a new expansion in terms of [MATH] and the new 'tree-level' propagator [MATH].", 'hep-th-0102070-2-17-3': 'All this can be done consistently in the Wilsonian RG framework, where the evolution equation for [MATH] is easily obtained from that for the full self-energy.', 'hep-th-0102070-2-18-0': "In more common language, the resummation procedure simply amounts to adding and subtracting the term [EQUATION] to the tree-level lagrangian, so as to get the resummed propagator provided a new two-point 'interaction' in eq. ([REF]) is consistently taken into account, in very close analogy to what is done in the finite temperature theory case [CITATION].", 'hep-th-0102070-2-19-0': 'The interactions of the resummed theory give the Feynman rules in Fig. 1.', 'hep-th-0102070-2-20-0': 'Now we are ready to compute the next-to-leading order corrections to the self-energy, which are given by the two diagrams in Fig. 2, where the resummed propagator runs into the loop (of course also the graph with the UV counterterms has to be included, which is not shown in the figure).', 'hep-th-0102070-2-21-0': 'The tadpole diagram in the resummed theory gives [EQUATION]', 'hep-th-0102070-2-21-1': "In the UV, the integral has the same structure as for the non-resummed theory, with a quadratically divergent contribution from the 'planar' diagrams and a finite one from the 'non-planar' ones, giving the [MATH] term which is exactly cancelled by the new two-point interaction of the resummed theory.", 'hep-th-0102070-2-21-2': 'In the IR, the planar and non-planar contributions sum up.', 'hep-th-0102070-2-21-3': 'By writing [EQUATION] where [MATH] and [MATH] is a traceless symmetric matrix.', 'hep-th-0102070-2-21-4': 'The symmetry of the integrand in the IR regime, selects the [MATH] term as the dominant contribution.', 'hep-th-0102070-2-22-0': "In the massless case ([MATH]) we find the following contribution from the 'planar' graph [EQUATION] whereas from the 'non-planar' one we get [EQUATION] for [MATH] and [MATH], for [MATH], where we have defined [MATH].", 'hep-th-0102070-2-22-1': 'In the massive case we get (planar + non-planar) [EQUATION]', 'hep-th-0102070-2-22-2': 'As one could expect, the non-analiticity in the coupling [MATH] emerges at lower order in the massless case (where we find a [MATH] correction) compared to the massive one ([MATH]).', 'hep-th-0102070-2-22-3': 'This reflects the fact that, in ordinary perturbation theory, the self-energy is IR divergent at [MATH] in the former case and at [MATH] in the latter.', 'hep-th-0102070-2-23-0': 'In computing the next-to-next-to-leading order in the resummed perturbative expansion one must consistently take into account the two-point interaction in ([REF]).', 'hep-th-0102070-2-23-1': 'Indeed, the two-loop graph for the resummed [MATH] theory with one non-planar tadpole insertion (first graph in Fig. 3) gives a contribution of [MATH], the same as the corrections computed above.', 'hep-th-0102070-2-23-2': 'It is only when the graph containing the two-point interaction is added that the whole correction comes out [MATH].', 'hep-th-0102070-2-24-0': 'The [MATH] corrections that one gets at two-loop, coming from UV loop momenta, cannot modify the [MATH] term in eq. ([REF]).', 'hep-th-0102070-2-24-1': "The corrections computed above are really 'perturbatively small' compared to the leading two-point function [MATH] in any range of the momentum [MATH].", 'hep-th-0102070-2-24-2': 'Indeed, for large enough momenta, the [MATH] correction dominates over the [MATH] term, but in that regime the tree-level [MATH] term is leading.', 'hep-th-0102070-2-24-3': 'On the other hand in the IR the opposite happens, with [MATH] never dominating over [MATH].', 'hep-th-0102070-2-24-4': 'Consequently, no tachyonic behavior can be induced by the next-to-leading order corrections.', 'hep-th-0102070-2-25-0': 'The resummed propagator was interpreted in ref. [CITATION] as originating from some high-energy degrees of freedom which, when integrated out, leave a commutative scalar theory with modified dispersion relations.', 'hep-th-0102070-2-25-1': 'In the RG language, the contributions to the two- and four-point functions induced by non-commutativity may be analogously seen as high-energy boundary conditions of an otherwise commutative theory valid up to some energy [MATH] [CITATION].', 'hep-th-0102070-2-26-0': 'It would be interesting to know if a resummation can be performed in gauge theories on the same spirit of this paper.', 'hep-th-0102070-2-26-1': 'Modifying only the two-point function is not a gauge invariant operation, so that an hypotetical resummation must necessarily involve all Green functions, as it is the case in hard-thermal-loop resummed QCD [CITATION].', 'hep-th-0102070-2-27-0': 'We thank Marisa Bonini for useful discussions.'}

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[]

[['hep-th-0102070-1-12-2', 'hep-th-0102070-2-13-1'], ['hep-th-0102070-1-13-0', 'hep-th-0102070-2-13-6'], ['hep-th-0102070-1-13-1', 'hep-th-0102070-2-13-5']]

[]

['hep-th-0102070-1-1-0', 'hep-th-0102070-1-26-0', 'hep-th-0102070-2-1-0', 'hep-th-0102070-2-27-0', 'hep-th-0102070-3-1-0', 'hep-th-0102070-3-27-0']

{'1': 'http://arxiv.org/licenses/assumed-1991-2003/', '2': 'http://arxiv.org/licenses/assumed-1991-2003/', '3': 'http://arxiv.org/licenses/assumed-1991-2003/'}

https://arxiv.org/abs/hep-th/0102070

{'hep-th-0102070-3-0-0': 'The non-commutative version of the euclidean [MATH] theory is considered.', 'hep-th-0102070-3-0-1': 'By using Wilsonian flow equations the ultraviolet renormalizability can be proved to all orders in perturbation theory.', 'hep-th-0102070-3-0-2': 'On the other hand, the infrared sector cannot be treated perturbatively and requires a resummation of the leading divergences in the two-point function.', 'hep-th-0102070-3-0-3': 'This is analogous to what is done in the Hard Thermal Loops resummation of finite temperature field theory.', 'hep-th-0102070-3-0-4': 'Next-to-leading order corrections to the self-energy are computed, resulting in [MATH] contributions in the massless case, and [MATH] in the massive one.', 'hep-th-0102070-3-1-0': ']', 'hep-th-0102070-3-2-0': 'Quantum field theories on non-commutative spaces have been the subject of intense investigation in the recent past, mainly motivated by their tight relation with string theories.', 'hep-th-0102070-3-2-1': 'Low energy excitations of a [MATH]-brane in a magnetic [MATH] background are indeed described by field theories with space non-commutativity [CITATION].', 'hep-th-0102070-3-2-2': 'In this limit the relevant description of dynamics is in term of massless open string states, while massive open string states and closed strings decouple: the full consistent string theory seems therefore truncated to the usual field theoretical degrees of freedom, suggesting the possibility that also the related quantum field theories are well defined.', 'hep-th-0102070-3-2-3': 'On the other hand their consistency is far from being obvious when examined from a purely field theoretical point of view: they are non-local (involving interactions with an arbitrarily high number of derivatives) and there is a new dimensionful parameter, other than the masses, taking into account the scale at which non-commutativity becomes relevant.', 'hep-th-0102070-3-3-0': 'It is then natural to ask whether these theories are renormalizable.', 'hep-th-0102070-3-3-1': 'Contrary to early suggestions it was shown in ref. [CITATION] that infinities appear when perturbative computations are performed in non-commutative scalar theories.', 'hep-th-0102070-3-3-2': 'Moreover, an higly non-trivial mixture between ultraviolet (UV) and infra-red (IR) behaviors [CITATION] makes a proof of perturbative renormalization along the usual lines quite cumbersome.', 'hep-th-0102070-3-3-3': 'Indeed, the one-loop self-energy in the non-commutative version of [MATH] theory gets a contribution of [MATH], where [MATH] is the scale of non-commutativity and [MATH] the external momentum.', 'hep-th-0102070-3-3-4': "This behavior is easily understood taking into account that the so called 'non-planar' graphs are effectively cut-off in the UV at a scale [MATH] and that the scalar self-energy is quadratically divergent in the commutative case [CITATION].", 'hep-th-0102070-3-3-5': 'When inserted in a higher order graph, the one-loop self-energy induces IR-divergences even in the case of a massive theory.', 'hep-th-0102070-3-3-6': 'For instance, the two-point function diverges quadratically in the IR for the massless theory at [MATH] and logarithmically for the massive one at [MATH], more tadpole insertions giving more and more IR-divergent behaviors.', 'hep-th-0102070-3-3-7': 'A similar behavior was discovered for gauge theories [CITATION].', 'hep-th-0102070-3-4-0': "Due to this problematic 'IR/UV connection' no complete calculation has been performed up to now in the scalar theory at next-to-leading order in perturbation theory.", 'hep-th-0102070-3-4-1': 'The possibility of absorbing UV divergences by means of local counterterms has been discussed at two-loops in refs. [CITATION], but no finite result could be obtained at that order due to the pathological behavior of the integrals in the IR.', 'hep-th-0102070-3-5-0': 'In this letter we will present the result of a resummation of the IR divergences which allows a consistent computation of finite corrections beyond the leading perturbative order.', 'hep-th-0102070-3-5-1': 'Before doing that, we will give the main lines of a proof of UV renormalizability to all orders in perturbation theory, which will be presented in full detail in [CITATION].', 'hep-th-0102070-3-6-0': 'The need of a resummation has been realized by different people, and discussed for instance in [CITATION].', 'hep-th-0102070-3-6-1': 'However, to our knowledge, no systematic approach has been formulated and therefore no explicit computation has been presented up to now.', 'hep-th-0102070-3-6-2': 'In the resummed perturbative expansion, the first correction to the leading [MATH] contribution to the self-energy of the massless theory arises at [MATH] instead of the naively expected [MATH].', 'hep-th-0102070-3-6-3': 'For the massive case, the resummation of all the problematic diagrams first arising at [MATH], results in a [MATH] behavior.', 'hep-th-0102070-3-7-0': 'The pattern of the IR problem in the non-commutative scalar theory presents some remarkable similarity with that arising in finite temperature field theory [CITATION].', 'hep-th-0102070-3-7-1': 'In that case, the UV divergences of the new thermal contributions are cut-off by the temperature [MATH], resulting in a [MATH] correction to the self-energy at one-loop.', 'hep-th-0102070-3-7-2': 'The ratio between the one-loop amplitude and the tree-level one, [MATH] becomes of [MATH] for soft external momenta, [MATH]<[MATH][MATH], so that a resummation must be performed in order to define a sensible perturbative expansion.', 'hep-th-0102070-3-7-3': "The program is accomplished by using a resummed propagator, in which a 'thermal mass' is included, [MATH].", 'hep-th-0102070-3-7-4': "Since the amplitude which has been resummed receives contributions mainly by the hard ([MATH]) momentum part of the one-loop integral, the resummed theory goes customarily under the name of 'hard thermal loops'.", 'hep-th-0102070-3-7-5': "Analogously, we will use propagators in which one-loop amplitudes, dominated by momenta [MATH], are resummed, and will consistently call this procedure 'hard non-commutative loops' resummation.", 'hep-th-0102070-3-8-0': 'Before giving the details and results of the resummation, we outline a formulation of the Wilsonian renormalization group (RG) a la Polchinski [CITATION] for the non-commutative scalar theory which will be presented in detail in a forthcoming publication.', 'hep-th-0102070-3-8-1': 'By studying perturbatively the Wilsonian flow equation we are able to prove UV renormalization to all orders.', 'hep-th-0102070-3-8-2': 'Then, turning to the IR regime, we will show how a resummation procedure emerges quite naturally in this context.', 'hep-th-0102070-3-9-0': '>From a field-theoretical point of view, the non-commutative version of [MATH] theory is again a scalar theory with the same tree-level propagator but a different vertex, which at tree-level is given by [EQUATION] where [MATH], [MATH] being the anti-symmetric matrix defining the commutation relations ([MATH]).', 'hep-th-0102070-3-9-1': 'As a consequence, the Wilsonian action and the RG equations can be derived in the same way as in refs. [CITATION] - they are actually the same equations, but with different boundary conditions.', 'hep-th-0102070-3-9-2': "Our proof of perturbative UV renormalization parallels quite closely the one given by Bonini, D'Attanasio and Marchesini for the commutative case [CITATION], so we only sketch here the main lines and stress only the differences emerging in the non-commutative theory.", 'hep-th-0102070-3-10-0': 'A Wilsonian effective action can be defined, [MATH], as the generating functional of 1PI Green functions obtained by integrating out loop momenta [MATH] such that [MATH].', 'hep-th-0102070-3-10-1': 'Our task is to prove that the double limit [MATH] (UV renormalizability) and [MATH] (IR finitness) can be taken.', 'hep-th-0102070-3-11-0': '[MATH] and the Green function generated by [MATH]-derivating it, obey exact evolution equations in [MATH].', 'hep-th-0102070-3-11-1': "There is a simple recipe to obtain the RG equation for any [MATH]-point function; i) write the 1-loop expression for [MATH] obtained by using all the vertices up to [MATH], as if they were formally tree-level; ii) promote the tree-level vertices above to full, running, vertices, [MATH], and the tree-level propagator to the full, cut-off, propagator, [EQUATION] where [MATH] is the full, running, self-energy, and the cut-off function [MATH] is equal to one in the interval [MATH] and vanishes rapidly outside; iii) take the derivative with respect to [MATH] everywhere in the [MATH]'s but not in the [MATH]'s or [MATH]'s.", 'hep-th-0102070-3-12-0': 'So, the evolution equation for e.g. the self-energy is given by [EQUATION] where [EQUATION]', 'hep-th-0102070-3-12-1': 'The RG equations for higher point functions are obtained analogously, and they together form an infinite system of coupled ordinary differential equations which define the theory non-perturbatively.', 'hep-th-0102070-3-12-2': 'The renormalization conditions are imposed by properly chosing the boundary conditions.', 'hep-th-0102070-3-12-3': 'For the relevant vertices [EQUATION] (where [MATH] is the renormalization scale and the momenta [MATH] have been chosen such that [MATH]) the boundary conditions are given at the physical point [MATH], [EQUATION]', 'hep-th-0102070-3-12-4': "The boundary conditions for all the other -irrelevant- vertices (higher momentum derivatives in [MATH] and [MATH] and all the [MATH]'s with [MATH]) are instead fixed at the UV, [MATH], where the irrelevant vertices can be set equal to zero.", 'hep-th-0102070-3-13-0': 'With these boundary conditions, the relevant vertices at a generic [MATH] are given by integrals between [MATH] and [MATH], whereas the irrelevant ones are given by integrals between [MATH] and [MATH] [CITATION].', 'hep-th-0102070-3-13-1': 'Then, one can disentangle the UV from the IR by taking the physical limit [MATH] and [MATH] in two successive steps.', 'hep-th-0102070-3-13-2': 'The proof of UV renormalizability follows quite closely the well known one given by Polchinski in the commutative case [CITATION], and discussed further in [CITATION].', 'hep-th-0102070-3-13-3': 'It exploits two remarkable features of the RG equations; i) the momentum ordering, by which a given irrelevant coupling evaluated at cut-off [MATH] receives contributions only from loop momenta [MATH], and ii) the one-loop structure of the exact equations, which make it possible to recover perturbation theory solving them iteratively.', 'hep-th-0102070-3-13-4': 'As a result, the proof is extremely simple, as it is based just on power counting arguments.', 'hep-th-0102070-3-13-5': 'In the UV regime, the non-commutative and the commutative theories exhibit essentially the same power counting.', 'hep-th-0102070-3-13-6': 'Indeed, by chosing [MATH] much larger than any physical scale of the theory, that is [MATH] the above mentioned [MATH] behavior of the 1-loop self-energy has not developed yet, as the momenta in the relevant integrals are bounded from below by [MATH], and the one-loop self-energy is subdominant with respect to the tree-level [MATH] contribution in ([REF]).', 'hep-th-0102070-3-13-7': 'As a consequence, the [MATH] limit can be shown to be finite at any perturbative order by a straightforward translation of the arguments given in [CITATION].', 'hep-th-0102070-3-13-8': 'All the details of the proof will be given in a separate paper [CITATION].', 'hep-th-0102070-3-14-0': 'When the IR regime comes under scrutiny, things change considerably.', 'hep-th-0102070-3-14-1': 'In [CITATION] the IR finitness of Green functions with non-exceptional external momenta (i.e [MATH]) was proved for the commutative massless theory at any order in perturbation theory.', 'hep-th-0102070-3-14-2': 'Crucial for that proof is the fact that [MATH] is at most logarithmically divergent as [MATH] at any finite order in the expansion.', 'hep-th-0102070-3-14-3': 'As we have repeatedly seen, this is not the case any more in the non-commutative case, where [MATH].', 'hep-th-0102070-3-14-4': 'Any perturbative computation is thus plagued by IR divergences which emerge sooner or later in the expansion in [MATH].', 'hep-th-0102070-3-15-0': 'A quick look at the exact form of the RG evolution equations, and in particular at the kernel in eq. ([REF]), shows both what the problem is and how a solution can be found.', 'hep-th-0102070-3-15-1': 'Namely, the more dangerous IR divergences come out when -at any finite order in perturbation theory- one expands the full-propagator appearing in the kernel in powers of [MATH].', 'hep-th-0102070-3-15-2': 'Since [MATH] in the kernel, this ratio diverges as [MATH] in the massive theory and as [MATH] in the massless one.', 'hep-th-0102070-3-15-3': 'It is then clear that any Green function - even at non-exceptional momenta- will be divergent at a sufficiently high order in [MATH].', 'hep-th-0102070-3-16-0': 'The exact form of the RG kernel gives the solution as well.', 'hep-th-0102070-3-16-1': 'Since [MATH] comes in the denominator, it is clear that the full equations are indeed better behaved in the IR than any approximation to them computed at any finite order in [MATH].', 'hep-th-0102070-3-16-2': 'Actually, since the effective mass explodes as [MATH], they are even better behaved than those for the massive theory in the commutative case!', 'hep-th-0102070-3-17-0': 'It appears then clear that the IR pathologies are just an artifact of the perturbative expansion, which should disappear if this is properly reorganized.', 'hep-th-0102070-3-17-1': 'To this end, one can still pursue the RG framework, splitting the full two-point function as [EQUATION] where [MATH] is the leading IR contribution the one-loop self-energy (we have defined [MATH], and [MATH] is the Bessel function).', 'hep-th-0102070-3-17-2': "Eq. ([REF]) defines a new expansion in terms of [MATH] and the new 'tree-level' propagator [MATH].", 'hep-th-0102070-3-17-3': 'All this can be done consistently in the Wilsonian RG framework, where the evolution equation for [MATH] is easily obtained from that for the full self-energy.', 'hep-th-0102070-3-17-4': 'Notice that, after resummation, the [MATH] and [MATH] limits can be interchanged.', 'hep-th-0102070-3-18-0': "In more common language, the resummation procedure simply amounts to adding and subtracting the term [EQUATION] to the tree-level lagrangian (we have taken the [MATH] limit), so as to get the resummed propagator provided a new two-point 'interaction' in eq. ([REF]) is consistently taken into account, in very close analogy to what is done in the finite temperature theory case [CITATION].", 'hep-th-0102070-3-19-0': 'The interactions of the resummed theory give the Feynman rules in Fig. 1.', 'hep-th-0102070-3-20-0': 'Now we are ready to compute the next-to-leading order corrections to the self-energy, which are given by the two diagrams in Fig. 2, where the resummed propagator runs into the loop (of course also the graph with the UV counterterms has to be included, which is not shown in the figure).', 'hep-th-0102070-3-21-0': 'The tadpole diagram in the resummed theory gives [EQUATION]', 'hep-th-0102070-3-21-1': "In the UV, the integral has the same structure as for the non-resummed theory, with a quadratically divergent contribution from the 'planar' diagrams and a finite one from the 'non-planar' ones, giving the [MATH] term which is exactly cancelled by the new two-point interaction of the resummed theory.", 'hep-th-0102070-3-21-2': 'In the IR, the planar and non-planar contributions sum up.', 'hep-th-0102070-3-21-3': 'By writing [EQUATION] where [MATH] and [MATH] is a traceless symmetric matrix.', 'hep-th-0102070-3-21-4': 'The symmetry of the integrand in the IR regime, selects the [MATH] term as the dominant contribution.', 'hep-th-0102070-3-22-0': "In the massless case ([MATH]) we find the following contribution from the 'planar' graph [EQUATION] whereas from the 'non-planar' one we get [EQUATION] for [MATH] and [MATH], for [MATH], where we have defined [MATH].", 'hep-th-0102070-3-22-1': 'In the massive case we get (planar + non-planar) [EQUATION]', 'hep-th-0102070-3-22-2': 'As one could expect, the non-analiticity in the coupling [MATH] emerges at lower order in the massless case (where we find a [MATH] correction) compared to the massive one ([MATH]).', 'hep-th-0102070-3-22-3': 'This reflects the fact that, in ordinary perturbation theory, the self-energy is IR divergent at [MATH] in the former case and at [MATH] in the latter.', 'hep-th-0102070-3-23-0': 'In computing the next-to-next-to-leading order in the resummed perturbative expansion one must consistently take into account the two-point interaction in ([REF]).', 'hep-th-0102070-3-23-1': 'Indeed, the two-loop graph for the resummed [MATH] theory with one non-planar tadpole insertion (first graph in Fig. 3) gives a contribution of [MATH], the same as the corrections computed above.', 'hep-th-0102070-3-23-2': 'It is only when the graph containing the two-point interaction is added that the whole correction comes out [MATH].', 'hep-th-0102070-3-24-0': 'The [MATH] corrections that one gets at two-loop, coming from UV loop momenta, cannot modify the [MATH] term in eq. ([REF]).', 'hep-th-0102070-3-24-1': "The corrections computed above are really 'perturbatively small' compared to the leading two-point function [MATH] in any range of the momentum [MATH].", 'hep-th-0102070-3-24-2': 'Indeed, for large enough momenta, the [MATH] correction dominates over the [MATH] term, but in that regime the tree-level [MATH] term is leading.', 'hep-th-0102070-3-24-3': 'On the other hand in the IR the opposite happens, with [MATH] never dominating over [MATH].', 'hep-th-0102070-3-24-4': 'Consequently, no tachyonic behavior can be induced by the next-to-leading order corrections.', 'hep-th-0102070-3-25-0': 'The resummed propagator was interpreted in ref. [CITATION] as originating from some high-energy degrees of freedom which, when integrated out, leave a commutative scalar theory with modified dispersion relations.', 'hep-th-0102070-3-25-1': 'In the RG language, the contributions to the two- and four-point functions induced by non-commutativity may be analogously seen as high-energy boundary conditions of an otherwise commutative theory valid up to some energy [MATH] [CITATION].', 'hep-th-0102070-3-26-0': 'It would be interesting to know if a resummation can be performed in gauge theories on the same spirit of this paper.', 'hep-th-0102070-3-26-1': 'Modifying only the two-point function is not a gauge invariant operation, so that an hypotetical resummation must necessarily involve all Green functions, as it is the case in hard-thermal-loop resummed QCD [CITATION].', 'hep-th-0102070-3-27-0': 'We thank Marisa Bonini for useful discussions.'}

1307.2982

{'1307.2982-1-0-0': 'There is growing interest in representing image data and feature descriptors using compact binary codes for fast near neighbor search.', '1307.2982-1-0-1': 'Although binary codes are motivated by their use as direct indices (addresses) into a hash table, codes longer than 32 bits are not being used as such, as it was thought to be ineffective.', '1307.2982-1-0-2': 'We introduce a rigorous way to build multiple hash tables on binary code substrings that enables exact k-nearest neighbor search in Hamming space.', '1307.2982-1-0-3': 'The approach is straightforward to implement and storage efficient.', '1307.2982-1-0-4': 'Theoretical analysis shows that the algorithm exhibits sub-linear run-time behavior for uniformly distributed codes.', '1307.2982-1-0-5': 'Empirical results show dramatic speed-ups over a linear scan baseline for datasets of up to one billion codes of 64, 128, or 256 bits.', '1307.2982-1-1-0': 'Binary codes, Hamming distance, nearest neighbor search, multi-index hashing.', '1307.2982-1-2-0': '# Introduction', '1307.2982-1-3-0': 'There has been growing interest in representing image data and feature descriptors in terms of compact binary codes, often to facilitate fast near neighbor search and feature matching in vision applications (e.g., [CITATION]).', '1307.2982-1-3-1': 'Binary codes are storage efficient and comparisons require just a small number of machine instructions.', '1307.2982-1-3-2': 'Millions of binary codes can be compared to a query in less than a second.', '1307.2982-1-3-3': 'But the most compelling reason for binary codes is their use as direct indices (addresses) into a hash table, yielding a dramatic increase in search speed compared to an exhaustive linear scan (e.g., [CITATION]).', '1307.2982-1-4-0': 'Nevertheless, using binary codes as direct hash indices is not necessarily efficient.', '1307.2982-1-4-1': 'To find near neighbors one needs to examine all hash table entries (or buckets) within some Hamming ball around the query.', '1307.2982-1-4-2': 'The problem is that the number of such buckets grows near-exponentially with the search radius.', '1307.2982-1-4-3': 'Even with a small search radius, the number of buckets to examine is often larger than the number of items in the database, and hence slower than linear scan.', '1307.2982-1-4-4': 'Recent papers on binary codes mention the use of hash tables, but resort to linear scan when codes are longer than [MATH] bits (e.g., [CITATION]).', '1307.2982-1-4-5': 'Not surprisingly, code lengths are often significantly longer than 32 bits in order to achieve satisfactory retrieval performance (e.g., see Fig. [REF]).', '1307.2982-1-5-0': 'This paper presents a new algorithm for exact [MATH]-nearest neighbor ([MATH]NN) search on binary codes that is dramatically faster than exhaustive linear scan.', '1307.2982-1-5-1': 'This has been an open problem since the introduction of hashing techniques with binary codes.', '1307.2982-1-5-2': 'Our new multi-index hashing algorithm exhibits sub-linear search times, is storage efficient, and straightforward to implement.', '1307.2982-1-5-3': 'Empirically, on databases of up to 1B codes we find that multi-index hashing is hundreds of times faster than linear scan.', '1307.2982-1-5-4': 'Extrapolation suggests that the speedup gain grows quickly with database size beyond 1B codes.', '1307.2982-1-6-0': '## Background: Problem and Related Work', '1307.2982-1-7-0': 'Nearest neighbor (NN) search on binary codes is used for image search [CITATION], matching local features [CITATION], image classification [CITATION], object segmentation [CITATION], and parameter estimation [CITATION].', '1307.2982-1-7-1': 'Sometimes the binary codes are generated directly as feature descriptors for images or image patches, such as BRIEF or FREAK [CITATION], and sometimes binary corpora are generated by discrete similarity-preserving mappings from high-dimensional data.', '1307.2982-1-7-2': 'Most such mappings are designed to preserve Euclidean distance (e.g., [CITATION]).', '1307.2982-1-7-3': 'Others focus on semantic similarity (e.g., [CITATION]).', '1307.2982-1-7-4': 'Our concern in this paper is not the algorithm used to generate the codes, but rather with fast search in Hamming space.', '1307.2982-1-8-0': 'We address two related search problems in Hamming space.', '1307.2982-1-8-1': 'Given a dataset of binary codes, [MATH], the first problem is to find the [MATH] codes in [MATH] that are closest in Hamming distance to a given query, i.e., [MATH]NN search in Hamming distance.', '1307.2982-1-8-2': 'The [MATH]-NN problem in Hamming space was called the Best Match problem by Minsky and Papert [CITATION].', '1307.2982-1-8-3': 'They observed that there are no obvious approaches significantly better than exhaustive search, and asked whether such approaches might exist.', '1307.2982-1-9-0': 'The second problem is to find all codes in a dataset [MATH] that are within a fixed Hamming distance of a query, sometimes called the Approximate Query problem [CITATION], or Point Location in Equal Balls (PLEB) [CITATION].', '1307.2982-1-9-1': 'A binary code is an [MATH]-neighbor of a query code, denoted [MATH], if it differs from [MATH] in [MATH] bits or less.', '1307.2982-1-9-2': 'We define the [MATH]-neighbor search problem as: find all [MATH]-neighbors of a query [MATH] from [MATH].', '1307.2982-1-10-0': 'One way to tackle [MATH]-neighbor search is to use a hash table populated with the binary codes [MATH], and examine all hash buckets whose indices are within [MATH] bits of a query [MATH] (e.g.,, [CITATION]).', '1307.2982-1-10-1': 'For binary codes of [MATH] bits, the number of distinct hash buckets to examine is [EQUATION]', '1307.2982-1-10-2': 'As shown in Fig. [REF] (top), [MATH] grows very rapidly with [MATH].', '1307.2982-1-10-3': 'Thus, this approach is only practical for small radii or short code lengths.', '1307.2982-1-10-4': 'Some vision applications restrict search to exact matches (i.e., [MATH]) or a small search radius (e.g., [CITATION] ), but in most cases of interest the desired search radius is larger than is currently feasible (e.g., see Fig. [REF] (bottom)).', '1307.2982-1-11-0': 'Our work is inspired in part by the multi-index hashing results of Greene, Parnas, and Yao [CITATION].', '1307.2982-1-11-1': 'Building on the classical Turan problem for hypergraphs, they construct a set of over-lapping binary substrings such that any two codes that differ by at most [MATH] bits are guaranteed to be identical in at least one of the constructed substrings.', '1307.2982-1-11-2': 'Accordingly, they propose an exact method for finding all [MATH]-neighbors of a query using multiple hash tables, one for each substring.', '1307.2982-1-11-3': 'At query time, candidate [MATH]-neighbors are found by using query substrings as indices into their corresponding hash tables.', '1307.2982-1-11-4': 'As explained below, while run-time efficient, the main drawback of their approach is the prohibitive storage required for the requisite number of hash tables.', '1307.2982-1-11-5': 'By comparison, the method we propose requires much less storage, and is only marginally slower in search performance.', '1307.2982-1-12-0': 'While we focus on exact search, there also exist algorithms for finding approximate [MATH]-neighbors ([MATH]-PLEB), or approximate nearest neighbors ([MATH]-NN) in Hamming distance.', '1307.2982-1-12-1': 'One example is Hamming Locality Sensitive Hashing [CITATION], which aims to solve the [MATH]-neighbors decision problem: determine whether there exists a binary code [MATH] such that [MATH], or whether all codes in [MATH] differ from [MATH] in [MATH] bits or more.', '1307.2982-1-12-2': 'Approximate methods are interesting, and the approach below could be made faster by allowing misses.', '1307.2982-1-12-3': 'Nonetheless, this paper will focus on the exact search problem.', '1307.2982-1-13-0': 'This paper proposes a data-structure that applies to both [MATH]NN and [MATH]-neighbor search in Hamming space.', '1307.2982-1-13-1': 'We prove that for uniformly distributed binary codes of [MATH] bits, and a search radius of [MATH] bits when [MATH] is small, our query time is sub-linear in the size of dataset.', '1307.2982-1-13-2': 'We also demonstrate impressive performance on real-world datasets.', '1307.2982-1-13-3': 'To our knowledge this is the first practical data-structure solving exact [MATH]NN in Hamming distance.', '1307.2982-1-14-0': '# Multi-Index Hashing', '1307.2982-1-15-0': 'Our approach is a form of multi-index hashing.', '1307.2982-1-15-1': 'Binary codes from the database are indexed [MATH] times into [MATH] different hash tables, based on [MATH] disjoint binary substrings.', '1307.2982-1-15-2': 'Given a query code, entries that fall close to the query in at least one such substring are considered neighbor candidates.', '1307.2982-1-15-3': 'Candidates are then checked for validity using the entire binary code, to remove any non-[MATH]-neighbors.', '1307.2982-1-15-4': 'To be practical for large-scale datasets, the substrings must be chosen so that the set of candidates is small, and storage requirements are reasonable.', '1307.2982-1-15-5': 'We also require that all true neighbors will be found.', '1307.2982-1-16-0': 'The key idea here stems from the fact that, with [MATH] binary codes of [MATH] bits, the vast majority of the [MATH] possible buckets in a full hash table will be empty, since [MATH].', '1307.2982-1-16-1': 'It seems expensive to examine all [MATH] buckets within [MATH] bits of a query, since most of them contain no items.', '1307.2982-1-16-2': 'Instead, we merge many buckets together (most of which are empty) by marginalizing over different dimensions of the Hamming space.', '1307.2982-1-16-3': 'The downside is that these larger buckets are not restricted to the Hamming volume of interest around the query.', '1307.2982-1-16-4': 'Hence not all items in the merged buckets are [MATH]-neighbors of the query, so we need to cull any candidate that is not a true [MATH]-neighbor.', '1307.2982-1-17-0': '## Substring Search Radii', '1307.2982-1-18-0': 'In more detail, each binary code [MATH], comprising [MATH] bits, is partitioned into [MATH] disjoint substrings, [MATH], each of length [MATH] or [MATH] bits.', '1307.2982-1-18-1': 'For convenience in what follows, we assume that [MATH] is divisible by [MATH], and that the substrings comprise contiguous bits.', '1307.2982-1-18-2': 'The key idea rests on the following statement: When two binary codes [MATH] and [MATH] differ by [MATH] bits or less, then, in at least one of their [MATH] substrings they must differ by at most [MATH] bits.', '1307.2982-1-18-3': 'This leads to the first proposition:', '1307.2982-1-19-0': 'Proposition 1: If [MATH], where [MATH] denotes the Hamming distance between [MATH] and [MATH], then [EQUATION] where [MATH].', '1307.2982-1-20-0': 'Proof of Proposition 1 follows straightforwardly from the Pigeonhole Principle.', '1307.2982-1-20-1': 'That is, suppose that the Hamming distance between each of the [MATH] substrings is strictly greater than [MATH].', '1307.2982-1-20-2': 'Then, [MATH].', '1307.2982-1-20-3': 'Clearly, [MATH], since [MATH] for some [MATH] where [MATH], which contradicts the premise.', '1307.2982-1-21-0': 'The significance of Proposition 1 derives from the fact that the substrings have only [MATH] bits, and that the required search radius in each substring is just [MATH].', '1307.2982-1-21-1': 'For example, if [MATH] and [MATH] differ by [MATH] bits or less, and [MATH], at least one of the 4 substrings must be identical.', '1307.2982-1-21-2': 'If they differ by at most [MATH] bits, then in at least one substring they differ by no more than [MATH] bit; i.e., we can search a Hamming radius of [MATH] bits by searching a radius of [MATH] bit on each of 4 substrings.', '1307.2982-1-21-3': 'More generally, instead of examining [MATH] hash buckets, it suffices to examine [MATH] buckets in each of [MATH] substring hash tables.', '1307.2982-1-22-0': 'While it suffices to examine all buckets within a radius of [MATH] in all [MATH] hash tables, we next show that it is not always necessary.', '1307.2982-1-22-1': 'Rather, it is often possible to use a radius of just [MATH] in some of the [MATH] substring hash tables while still guaranteeing that all [MATH]-neighbors of [MATH] will be found.', '1307.2982-1-22-2': 'In particular, with [MATH], where [MATH], to find any item within a radius of [MATH] on [MATH]-bit codes, it suffices to search [MATH] substring hash tables to a radius of [MATH], and the remaining [MATH] substring hash tables up to a radius of [MATH].', '1307.2982-1-22-3': 'Without loss of generality, since there is no order to the substring hash tables, we search the first [MATH] hash tables with radius [MATH], and all remaining hash tables with radius [MATH].', '1307.2982-1-23-0': 'Proposition 2: If [MATH], then [EQUATION]', '1307.2982-1-23-1': 'To prove Proposition 2, we show that when [REF] is false, ([REF]) must be true.', '1307.2982-1-23-2': 'If [REF] is false, then it must be that [MATH], since otherwise [MATH], in which case [REF] and Proposition 1 are equivalent.', '1307.2982-1-23-3': 'If ([REF]) is false, it also follows that [MATH] and [MATH] differ in each of their first [MATH] substrings by [MATH] or more bits.', '1307.2982-1-23-4': 'Thus, the total number of bits that differ in the first [MATH] substrings is at least [MATH].', '1307.2982-1-23-5': 'Because [MATH], it also follows that the total number of bits that differ in the remaining [MATH] substrings is at most [MATH].', '1307.2982-1-23-6': 'Then, using Proposition 1, the maximum search radius required in each of the remaining [MATH] substring hash tables is [EQUATION] and hence Proposition 2 is true.', '1307.2982-1-23-7': 'Because of the near exponential growth in the number of buckets for large search radii, the smaller substring search radius required by Proposition 2 is significant.', '1307.2982-1-24-0': '## Multi-Index Hashing for [MATH]-neighbor Search', '1307.2982-1-25-0': 'In a pre-processing step, given a dataset of binary codes, one hash table is built for each of the [MATH] substrings, as outlined in Algorithm 1.', '1307.2982-1-25-1': 'At query time, given a query [MATH] with substrings [MATH], we search the [MATH] substring hash table for entries that are within a Hamming distance of [MATH] or [MATH] of [MATH], as prescribed by ([REF]).', '1307.2982-1-25-2': 'By doing so we obtain a set of candidates from the [MATH] substring hash table, denoted [MATH].', '1307.2982-1-25-3': 'According to the propositions above, the union of the [MATH] sets, [MATH], is necessarily a superset of the [MATH]-neighbors of [MATH].', '1307.2982-1-25-4': 'The last step of the algorithm computes the full Hamming distance between [MATH] and each candidate in [MATH], retaining only those codes that are true [MATH]-neighbors of [MATH].', '1307.2982-1-25-5': 'Algorithm 2 outlines the [MATH]-neighbor retrieval procedure for a query [MATH].', '1307.2982-1-26-0': 'The search cost depends on the number of lookups (i.e., the number of buckets examined), and the number of candidates tested.', '1307.2982-1-26-1': 'Not surprisingly there is a natural trade-off between them.', '1307.2982-1-26-2': 'With a large number of lookups one can minimize the number of extraneous candidates.', '1307.2982-1-26-3': 'By merging many buckets to reduce the number of lookups, one obtains a large number of candidates to test.', '1307.2982-1-26-4': 'In the extreme case with [MATH], substrings are 1 bit long, so we can expect the candidate set to include almost the entire database.', '1307.2982-1-27-0': 'Note that the idea of building multiple hash tables is not novel in itself (e.g., see [CITATION]).', '1307.2982-1-27-1': 'However previous work relied heavily on exact matches in substrings.', '1307.2982-1-27-2': 'Relaxing this constraint is what leads to a more effective algorithm.', '1307.2982-1-28-0': '# Performance Analysis', '1307.2982-1-29-0': 'We next develop an analytical model of search performance to help address two key questions: (1) How does search cost depend on substring length, and hence the number of substrings?', '1307.2982-1-29-1': '(2) How do run-time and storage complexity depend on database size, code length, and search radius?', '1307.2982-1-30-0': 'To help answer these questions we exploit a well-known bound on the sum of binomial coefficients [CITATION]; i.e., for any [MATH] and [MATH].', '1307.2982-1-30-1': '[EQUATION] where [MATH] is the entropy of a Bernoulli distribution with probability [MATH].', '1307.2982-1-31-0': 'In what follows, [MATH] continues to denote the number of [MATH]-bit database codes, and [MATH] is the Hamming search radius.', '1307.2982-1-31-1': 'Let [MATH] denote the number of hash tables, and let [MATH] denote the substring length [MATH].', '1307.2982-1-31-2': 'Hence, the maximum substring search radius becomes [MATH].', '1307.2982-1-31-3': 'As above, for the sake of model simplicity, we assume [MATH] is divisible by [MATH].', '1307.2982-1-32-0': 'We begin by formulating an upper bound on the number of lookups.', '1307.2982-1-32-1': 'First, the number of lookups in Algorithm 3 is bounded above by the product of [MATH], the number of substring hash tables, and the number of hash buckets within a radius of [MATH] on substrings of length [MATH] bits.', '1307.2982-1-32-2': 'Accordingly, using ([REF]), if the search radius is less than half the code length, [MATH], then the total number of lookups is given by [EQUATION]', '1307.2982-1-32-3': 'Clearly, as we decrease the substring length [MATH], thereby increasing the number of substrings [MATH], exponentially fewer lookups are needed.', '1307.2982-1-33-0': 'To analyze the expected number of candidates per bucket, we consider the case in which the [MATH] binary codes are uniformly distributed over the Hamming space.', '1307.2982-1-33-1': 'In this case, for a substring of [MATH] bits, for which a substring hash table has [MATH] buckets, the expected number of items per bucket is [MATH].', '1307.2982-1-33-2': 'The expected size of the candidate set therefore equals the number of lookups times [MATH].', '1307.2982-1-34-0': 'The total search cost per query is the cost for lookups plus the cost for candidate tests.', '1307.2982-1-34-1': 'While these costs will vary with the code length [MATH] and the way the hash tables are implemented, we find that, to a reasonable approximation, the costs of a lookup and a candidate test are similar (when [MATH]).', '1307.2982-1-34-2': 'Accordingly, we model the total search cost per query, for retrieving all [MATH]-neighbors, in units of the time required for a single lookup, as [EQUATION]', '1307.2982-1-34-3': 'In practice, database codes will generally not be uniformly distributed, nor are uniformly distributed codes ideal for multi-index hashing.', '1307.2982-1-34-4': 'Indeed, the cost of search with uniformly distributed codes is relatively high since the search radius increases as the density of codes decreases.', '1307.2982-1-34-5': 'Rather, the uniform distribution is primarily a mathematical convenience that facilitates the analysis of run-time and storage complexity, thereby providing some insight into the effectiveness of the approach and how one might choose an effective substring length.', '1307.2982-1-35-0': '## Optimal Substring Length', '1307.2982-1-36-0': 'As noted above in Sec. [REF], finding a good substring length is critical to the efficiency of multi-index hashing.', '1307.2982-1-36-1': 'When the substring length is too large or too small the approach will not be effective.', '1307.2982-1-36-2': 'To find a good substring length, we first note that, dividing [MATH] in Eqn. ([REF]) by [MATH] has no effect on the optimal [MATH], denoted [MATH].', '1307.2982-1-36-3': 'Accordingly, one can view [MATH] as a function of two quantities, namely the number of items, [MATH], and the search ratio [MATH].', '1307.2982-1-37-0': 'Figure [REF] plots cost as a function of substring length [MATH], for [MATH]-bit codes, different database sizes [MATH], and different search radii (expressed as a fraction of the code length [MATH]).', '1307.2982-1-37-1': 'Dashed curves depict [MATH] in ([REF]) while solid curves of the same color depict the upper bound in ([REF]).', '1307.2982-1-37-2': 'The tightness of the bound is evident in the plots, as are the quantization effects of the upper range of the sum in ([REF]).', '1307.2982-1-37-3': 'The small circles in Fig. [REF] (top) depict cost when all quantization effects are included, and hence it is only shown at substring lengths that are integer divisors of the code length.', '1307.2982-1-38-0': 'Fig. [REF] (top) shows cost for search radii equal to [MATH], [MATH] and [MATH] of the code length, with [MATH] in all cases.', '1307.2982-1-38-1': 'One striking property of these curves is that the cost is persistently minimal in the vicinity of [MATH], indicated by the vertical line close to 30 bits.', '1307.2982-1-38-2': 'This behavior remains consistent over a wide range of database sizes.', '1307.2982-1-39-0': 'Fig. [REF] (bottom) shows the dependence of cost on [MATH] for databases with [MATH], [MATH], and [MATH], all with [MATH] and [MATH] bits.', '1307.2982-1-39-1': 'In this case we have laterally displaced each curve by [MATH]; notice how this aligns the minima close to [MATH].', '1307.2982-1-39-2': 'These curves suggest that, over a wide range of conditions, cost is minimal for [MATH].', '1307.2982-1-39-3': 'For this choice of the substring length, the expected number of items per substring bucket, i.e., [MATH], reduces to 1.', '1307.2982-1-39-4': 'As a consequence, the number of lookups is equal to the expected number of candidates.', '1307.2982-1-39-5': 'Interestingly, this choice of substring length is similar to that of Greene et al. [CITATION].', '1307.2982-1-40-0': '## Run-Time Complexity', '1307.2982-1-41-0': 'Choosing [MATH] in the vicinity of [MATH] provides a characterization of retrieval run-time complexity.', '1307.2982-1-41-1': 'When [MATH], the upper bound on the number of lookups ([REF]) also becomes a bound on the number candidates.', '1307.2982-1-41-2': 'In particular, if we substitute [MATH] for [MATH] in ([REF]), then we find the following upper bound on the cost, now as a function of database size, code length, and the search radius: [EQUATION]', '1307.2982-1-41-3': 'Thus, for a uniform distribution over binary codes, if we choose [MATH] such that [MATH], the expected query time complexity is [MATH].', '1307.2982-1-41-4': 'For a small ratio of [MATH] this is sub-linear in [MATH].', '1307.2982-1-41-5': 'For example, if [MATH], then [MATH], and the run-time complexity becomes [MATH].', '1307.2982-1-41-6': 'That is, the search time increases with the square root of the database size when the search radius is approximately 10% of the code length.', '1307.2982-1-41-7': 'For [MATH], this becomes [MATH].', '1307.2982-1-41-8': 'The time complexity with respect to [MATH] is not as important as that with respect to [MATH] since [MATH] is not expected to vary significantly in most applications.', '1307.2982-1-42-0': '## Storage Complexity', '1307.2982-1-43-0': 'The storage complexity of our multi-index hashing algorithm is asymptotically optimal.', '1307.2982-1-43-1': 'To store the full database of binary codes requires [MATH] bits.', '1307.2982-1-43-2': 'For each of [MATH] hash tables, we also need to store [MATH] unique identifiers to the database items.', '1307.2982-1-43-3': 'This allows one to identify the retrieved items and fetch their full codes; this requires an additional [MATH] bits.', '1307.2982-1-43-4': 'In sum, the storage required is [MATH].', '1307.2982-1-43-5': 'When [MATH], as is suggested above, this storage cost reduces to [MATH].', '1307.2982-1-43-6': 'Here, the [MATH] term does not cancel as [MATH], but in most interesting cases [MATH].', '1307.2982-1-44-0': 'While the storage cost for our multi-index hashing algorithm is linear in [MATH], the related multi-index hashing algorithm of Greene et al. [CITATION] entails storage complexity that is super-linear in [MATH].', '1307.2982-1-44-1': 'To find all [MATH]-neighbors, for a given search radius [MATH], they construct [MATH] substrings of length [MATH] bits per binary code.', '1307.2982-1-44-2': 'Their suggested substring length is also [MATH], so the number of substring hash tables becomes [MATH], each of which requires [MATH] in storage.', '1307.2982-1-44-3': 'As a consequence for large values of [MATH], even with small [MATH], this technique requires a prohibitive amount of memory to store the hash tables.', '1307.2982-1-45-0': 'Our approach is more memory-efficient than that of [CITATION] because we do not enforce exact equality in substring matching.', '1307.2982-1-45-1': 'In essence, instead of creating all of the hash tables off-line, and then having to store them, we flip bits of each substring at run-time and implicitly create some of the substring hash tables on-line.', '1307.2982-1-45-2': 'This increases run-time slightly, but greatly reduces storage costs.', '1307.2982-1-46-0': '# [MATH]-Nearest Neighbor Search', '1307.2982-1-47-0': 'To use the above multi-index hashing in practice, one must specify a Hamming search radius [MATH].', '1307.2982-1-47-1': 'For many tasks, the value of [MATH] is chosen such that queries will, on average, retrieve [MATH] near neighbors.', '1307.2982-1-47-2': 'Nevertheless, as expected, we find that for many hashing techniques and different sources of visual data, the distribution of binary codes is such that a single search radius for all queries will not produce similar numbers of neighbors.', '1307.2982-1-48-0': 'Figure [REF] depicts empirical distributions of search radii needed for [MATH]-NN and [MATH]-NN on three sets of binary codes obtained from 1B SIFT descriptors [CITATION].', '1307.2982-1-48-1': 'In all cases, for [MATH] and [MATH]-bit codes, and for hash functions based on LSH [CITATION] and MLH [CITATION], there is a substantial variance in the search radius.', '1307.2982-1-48-2': 'This suggests that binary codes are not uniformly distributed over the Hamming space.', '1307.2982-1-48-3': 'As an example, for [MATH]-NN in [MATH]-bit LSH codes, more than [MATH] of the queries require a search radius of [MATH] bits or larger, while for about [MATH] of the queries it can be [MATH] or smaller.', '1307.2982-1-48-4': 'Also evident from Fig. [REF] is the growth in the required search radius as one moves from [MATH]-bit codes to [MATH] bits, and from [MATH]-NN to [MATH]-NN.', '1307.2982-1-49-0': 'A fixed radius for all queries would produce too many neighbors for some queries, and too few for others.', '1307.2982-1-49-1': 'It is therefore more natural for many tasks to fix the number of required neighbors, i.e., [MATH], and let the search radius depend on the query.', '1307.2982-1-49-2': 'Fortunately, our multi-index hashing algorithm is easily adapted to accommodate query-dependent search radii.', '1307.2982-1-50-0': 'Given a query, one can progressively increase the Hamming search radius per substring, until a specified number of neighbors is found.', '1307.2982-1-50-1': "For example, if one examines all [MATH]-neighbors of a query's substrings, from which more than [MATH] candidates are found to be within a Hamming distance of [MATH] bits (using the full codes for validation), then it is guaranteed that [MATH]-nearest neighbors have been found.", '1307.2982-1-50-2': 'Indeed, if all [MATH]NNs of a query [MATH] differ from [MATH] in [MATH] bits or less, then Propositions 1 and 2 above provide guanantees all such neighbors will be found if one searches the substring hash tables with the prescribed radii.', '1307.2982-1-51-0': 'In our experiments, we follow this progressive increment of the search radius until we can find [MATH]NN in the guaranteed neighborhood of a query.', '1307.2982-1-51-1': 'This approach, given in Algorithm 3, is helpful because it uses a specific search radius for each query depending on the distribution of codes in that neighborhood.', '1307.2982-1-52-0': '# Experiments', '1307.2982-1-53-0': 'Experiments are run on two different architectures.', '1307.2982-1-53-1': 'The first is a mid- to low-end [MATH]Ghz dual quad-core AMD Opteron processor, with [MATH]MB of L2 cache, and [MATH]GB of RAM.', '1307.2982-1-53-2': 'The second is a high-end machine with a [MATH]Ghz dual quad-core Intel Xeon processor, [MATH]MB of L2 cache, and [MATH]B of RAM.', '1307.2982-1-53-3': 'The difference in the size of the L2 cache has a major impact on the run-time of linear scan, since the effectiveness of linear scan depends greatly on L2 cache lines.', '1307.2982-1-53-4': 'With roughly ten times the L2 cache, linear scan on the Intel platform is roughly twice as fast as on the AMD machines.', '1307.2982-1-53-5': 'By comparison, multi-index hashing does not have a serial memory access pattern and so the cache size does not have such a pronounced effect.', '1307.2982-1-53-6': 'Actual run-times for multi-index hashing on the Intel and AMD platforms are within 20% of one another.', '1307.2982-1-54-0': 'Both linear scan and multi-index hashing were implemented in C++ and compiled with identical compiler flags.', '1307.2982-1-54-1': 'To accommodate the large size of memory footprint required for 1B codes, we used the libhugetlbfs package and Linux Kernel [MATH] to allow the use of [MATH]MB page sizes.', '1307.2982-1-54-2': 'Further details about the implementations are given in the Appendix.', '1307.2982-1-54-3': 'Finally, despite the existence of multiple cores, all experiments are run on a single core to simplify run-time measurements.', '1307.2982-1-55-0': 'The memory requirements for multi-index hashing are described in detail in the Appendix.', '1307.2982-1-55-1': 'We currently require approximately [MATH]GB for multi-index hashing with 1B [MATH]-bit codes, and approximately twice that for [MATH]-bit codes.', '1307.2982-1-55-2': 'Fig. [REF] shows how the memory footprint depends on the database size for linear scan and multi-index hashing.', '1307.2982-1-55-3': 'As explained in the Appendix, and demonstrated in Fig. [REF] the memory requirements of multi-index hashing grow linearly in the database size, as does linear scan.', '1307.2982-1-55-4': 'While we use a single computer in our experiments, one could implement a distributed version of multi-index hashing on computers with much less memory by placing each substring hash table on a separate computer.', '1307.2982-1-56-0': '## Datasets', '1307.2982-1-57-0': 'We consider two well-known large-scale vision corpora: [MATH]M Gist descriptors from [MATH] million tiny images [CITATION] and [MATH]B SIFT features from the BIGANN dataset [CITATION].', '1307.2982-1-57-1': 'SIFT vectors [CITATION] are [MATH]D descriptors of local image structure in the vicinity of feature points.', '1307.2982-1-57-2': 'Gist features [CITATION] extracted from from [MATH] images capture global image structure in [MATH]D vectors.', '1307.2982-1-57-3': 'These two feature types cover a spectrum of NN search problems in vision from feature to image indexing.', '1307.2982-1-58-0': 'We use two similarity-preserving mappings to create datasets of binary codes, namely, binary angular Locality Sensitive Hashing (LSH) [CITATION], and Minimal Loss Hashing (MLH) [CITATION].', '1307.2982-1-58-1': 'LSH is considered a baseline random projection method, closely related to cosine similarity.', '1307.2982-1-58-2': 'MLH is a state-of-the-art learning algorithm that, given a set of similarity labels, finds an optimal mapping by minimizing a loss function over pairs or triplets of binary codes.', '1307.2982-1-59-0': 'Both the [MATH]M Gist and [MATH]B SIFT corpora comprise three disjoint sets, namely, a training set, a base set for populating the database, and a test query set.', '1307.2982-1-59-1': 'Using a random permutation, Gist descriptors are divided into a training set with [MATH] items, a base set of [MATH] million items, and a query set of size [MATH].', '1307.2982-1-59-2': 'The SIFT corpus comes with [MATH]M for training, [MATH] in the base set, and [MATH] test queries.', '1307.2982-1-60-0': 'For LSH we subtract the mean, and pick a set of coefficients from the standard normal density for a linear projection, followed by quantization.', '1307.2982-1-60-1': 'For MLH the training set is used to optimize hash function parameters [CITATION].', '1307.2982-1-60-2': 'After learning is complete, we remove the training data and use the resulting hash function with the base set to create the database of binary codes.', '1307.2982-1-60-3': 'With two image corpora (SIFT and Gist), up to three code lengths ([MATH], [MATH], and [MATH] bits), and two hashing methods (LSH and MLH), we obtain several datasets of binary codes with which to evaluate our multi-index hashing algorithm.', '1307.2982-1-60-4': 'Note that [MATH]-bit codes are only used with LSH and SIFT vectors.', '1307.2982-1-61-0': 'Figure [REF] shows Euclidean NN recall rates for [MATH]NN search on binary mappings of [MATH]M and [MATH]B SIFT descriptors.', '1307.2982-1-61-1': 'In particular, we plot the fraction of Euclidean [MATH] nearest neighbors found, by [MATH]NN in [MATH]-bit and [MATH]-bit LSH [CITATION] and MLH [CITATION] binary codes.', '1307.2982-1-61-2': 'As expected [MATH]-bit codes are more accurate, and MLH outperforms LSH.', '1307.2982-1-61-3': 'Since multi-index hashing solves exact [MATH]NN in Hamming distance, the approximation is due to quantization by the hash function.', '1307.2982-1-61-4': 'To preserve the similarity structure in the original SIFT descriptors, it seems useful to use longer codes, and exploit data-dependant hash functions such as MLH.', '1307.2982-1-61-5': 'Interestingly, as described below, the speedup factors of multi-index hashing on MLH codes are better than those for LSH.', '1307.2982-1-62-0': '## Results', '1307.2982-1-63-0': 'Each experiment below involves [MATH] queries, for which we report the average run-time.', '1307.2982-1-63-1': 'Our implementation of the linear scan baseline searches [MATH] million [MATH]-bit codes in just under one second on the AMD machine.', '1307.2982-1-63-2': 'On the Intel machine it examines over [MATH] million [MATH]-bit codes per second.', '1307.2982-1-63-3': 'This is remarkably fast compared to Euclidean NN search with [MATH]D SIFT vectors.', '1307.2982-1-63-4': 'The speed of linear scan is in part due to memory caching, without which it would be much slower.', '1307.2982-1-63-5': 'Run-times for linear scan on other datasets, on both architectures, are given in Tables [REF] and [REF].', '1307.2982-1-64-0': '## Multi-Index Hashing vs. Linear Scan', '1307.2982-1-65-0': 'Tables [REF] and [REF] shows run-time per query for the linear scan baseline, along with speed-up factors of multi-index hashing for different [MATH]NN problems and nine different datasets.', '1307.2982-1-65-1': 'Despite the remarkable speed of linear scan, the multi-index hashing implementation is hundreds of times faster.', '1307.2982-1-65-2': 'For example, the multi-index hashing method solves the exact [MATH]-NN for a dataset of 1B [MATH]-bit codes in about [MATH] ms, well over [MATH] times faster than linear scan (see Table [REF]).', '1307.2982-1-65-3': 'Performance on [MATH]-NN and [MATH]-NN are even more impressive.', '1307.2982-1-65-4': 'With [MATH]-bit MLH codes, multi-index hashing executes the 1NN search task over 1000 times faster than the linear scan baseline.', '1307.2982-1-66-0': 'The run-time of linear scan does not depend on the number of neighbors, nor on the underlying distribution of binary codes.', '1307.2982-1-66-1': 'The run-time for multi-index hashing, however, depends on both factors.', '1307.2982-1-66-2': 'In particular, as the desired number of NNs increases, the Hamming radius of the search also increases (e.g., see Fig. [REF]).', '1307.2982-1-66-3': 'This implies longer run-times for multi-index hashing.', '1307.2982-1-66-4': 'Indeed, notice that going from [MATH]-NN to [MATH]-NN on each row of the tables shows a decrease in the speed-up factors.', '1307.2982-1-67-0': 'The multi-index hashing run-time also depends on the distribution of binary codes.', '1307.2982-1-67-1': 'Indeed, one can see from Table [REF] that MLH code databases yield faster run times than the LSH codes; e.g., for [MATH]-NN in [MATH]B [MATH]-bit codes the speed-up for MLH is [MATH] vs [MATH] for LSH.', '1307.2982-1-67-2': 'Fig. [REF] depicts the histograms of search radii needed for [MATH]-NN with [MATH]B [MATH]-bit MLH and LSH codes.', '1307.2982-1-67-3': 'Interestingly, the mean of the search radii for MLH codes is [MATH] bits, while it is [MATH] for LSH.', '1307.2982-1-67-4': 'While the means are similar the variance are not; the standard deviations of the search radii for MLH and LSH are [MATH] and [MATH] respectively.', '1307.2982-1-67-5': 'The longer tail of the distribution of search radii for LSH plays an important role in the expected run-time.', '1307.2982-1-67-6': 'In fact, queries that require relatively large search radii tend to dominate the average query cost.', '1307.2982-1-68-0': 'It is also interesting to look at the multi-index hashing run-times as a function of [MATH], the number of binary codes in the database.', '1307.2982-1-68-1': 'To that end, Fig. [REF] and [REF] depict run-times for linear scan and multi-index [MATH]NN search on the AMD machine.', '1307.2982-1-68-2': 'The left two figures in each show different vertical scales (since the behavior of multi-index [MATH]NN and linear scan are hard to see at the same scale).', '1307.2982-1-68-3': 'The right-most panels show the same data on log-log axes.', '1307.2982-1-68-4': 'First, it is clear from these plots that multi-index hashing is much faster than linear scan for a wide range of dataset sizes and [MATH].', '1307.2982-1-68-5': 'Just as importantly, it is evident from the log-log plots that as we increase the database size, the speedup factors improve.', '1307.2982-1-68-6': 'The dashed lines on the log-log plots depict [MATH] (up to a scalar constant).', '1307.2982-1-68-7': 'The similar slope of multi-index hashing curves with the square root curves show that multi-index hashing exhibits sub-linear query time, even for the empirical, non-uniform distributions of codes.', '1307.2982-1-69-0': '## Direct lookups with a single hash table', '1307.2982-1-70-0': 'An alternative to linear scan and multi-index hashing is to hash the entire codes into a single hash table, and then use direct hashing with each query.', '1307.2982-1-70-1': 'As suggested in the introduction and Fig. [REF], although this approach avoids the need for any candidate checking, it may require a prohibitive number of lookups.', '1307.2982-1-70-2': 'Nevertheless, for sufficiently small code lengths or search radii, it may be effective in practice.', '1307.2982-1-71-0': 'Rather than implement this directly, given the complexity associated with efficiently implementing folded hash tables, we instead consider the empirical number of lookups one would need, as compared to the number of items in the database.', '1307.2982-1-71-1': 'If the number of lookups is vastly greater than the size of the dataset one can readily conclude that linear scan is likely to be as fast or faster than direct indexing into a single hash table.', '1307.2982-1-72-0': 'Fortunately, the statistics of neighborhood sizes and required search radii for [MATH]NN tasks are available from the multi-index hashing experiments reported above.', '1307.2982-1-72-1': 'One can use the substring search radii to compute the required radii on the full codes (e.g., see Fig. [REF]).', '1307.2982-1-72-2': 'For each radius one can compute the number of lookups that would be required to find all neighbors from a single hash table, for each query.', '1307.2982-1-72-3': 'Summed over a dataset, this provides an indication of the expected run-time.', '1307.2982-1-73-0': 'Figure [REF] shows the total number of lookups required for 1-NN and 1000-NN tasks on 64- and 128-bit codes (from LSH on SIFT) using a single hash table.', '1307.2982-1-73-1': 'They are plotted as a function of the size of the dataset, from [MATH] to [MATH] items.', '1307.2982-1-73-2': 'For comparison, the plots also show the number of database items, and the number of lookups that were needed for multi-index hashing (as reported in [REF] above for the AMD machine).', '1307.2982-1-73-3': 'Note that Fig. [REF] has logarithmic scales.', '1307.2982-1-74-0': 'It is evident that with a single hash table the number of lookups is almost always several orders of magnitude than there are items in the dataset.', '1307.2982-1-74-1': 'And not surprisingly, this is also several orders of magnitude more lookups than required for multi-index hashing.', '1307.2982-1-74-2': 'Although the relative speed of a lookup operation compared to a candidate check (as used in linear scan) is not known precisely, there are a few important considerations.', '1307.2982-1-74-3': 'Linear scan has an exactly serial memory access pattern and so can make very efficient use of cache, whereas lookups in a hash table are inherently random.', '1307.2982-1-74-4': 'Furthermore, in any plausible implementation of a single hash table for 64 bit or longer codes, there will be some penalty for collision detection.', '1307.2982-1-75-0': 'As illustrated in Fig. [REF], the only cases where a single hash table might potentially be more efficient than linear scan are with very small codes (64 bits or less), with a large dataset (1 billion items or more), and a small search distances (e.g., for 1-NN).', '1307.2982-1-75-1': 'In all other cases, linear scan requires orders of magnitude fewer operations.', '1307.2982-1-75-2': 'With any code length longer than 64 bits, a single hash table approach is completely infeasable to run, requiring upwards of 15 orders of magnitude more operations than linear scan for 128-bit codes.', '1307.2982-1-76-0': '## Substring Optimization', '1307.2982-1-77-0': 'The substring hash tables used above have been formed by simply dividing the full codes into disjoint but consecutive sequences bits.', '1307.2982-1-77-1': 'For LSH and MLH, this is equivalent to randomly assigning bits to substrings.', '1307.2982-1-78-0': 'It natural to ask whether further gains in efficiency are possible by optimizing this assignment.', '1307.2982-1-78-1': 'In particular, by carefully assigning bits to substrings one may be able to maximize the discriminability of the different substrings.', '1307.2982-1-78-2': 'In other words, while the radius of substring searches and hence the number of lookups is determined by the desired search radius on the full codes, and will remain fixed, by optimizing the assignment of bits to substrings one might be able to reduce the number of candidates one needs to validate.', '1307.2982-1-79-0': 'To explore this idea we considered a simple method for which substrings are assigned bits one at a time in a greedy fashion, based on correlations between bits.', '1307.2982-1-79-1': 'In particular, of those bits not yet assigned, the next substring is assigned the bit that minimizes the maximum correlation between that bit and all other bits already assigned to that substring.', '1307.2982-1-79-2': 'Initialization also occurs in a greedy manner: A random bit is assigned to the first substring, after which the first bit to substring [MATH] is that which is maximally correlated with the first bit of substring [MATH].', '1307.2982-1-79-3': 'This approach significantly decreases the correlation between bits within a single substring, which should make the distribution of substrings more uniform, and thereby lower the number of candidates within a given search radius.', '1307.2982-1-79-4': 'Arguably an even better approach would be to maximize the entropy of the entries within each substring hash table, thereby making the distribution of substrings as uniform as possible.', '1307.2982-1-79-5': 'This entropic approach is, however, left to future work.', '1307.2982-1-80-0': 'The results obtained with the correlation-based algorithm show that optimizing substrings can provide overall run-time reductions on the order of [MATH] against consecutive substrings for some cases.', '1307.2982-1-80-1': 'Table [REF] displays the improvements achieved by optimizing substrings for different codes lengths and different values of [MATH].', '1307.2982-1-80-2': 'Fig. [REF] shows the run-time performance of optimized substrings.', '1307.2982-1-81-0': '# Conclusion', '1307.2982-1-82-0': 'This paper describes a new algorithm for exact nearest neighbor search on large-scale datasets of binary codes.', '1307.2982-1-82-1': 'The algorithm is a form of multi-index hashing that has provably sub-linear run-time behavior for uniformly distributed codes.', '1307.2982-1-82-2': 'It is storage efficient and easy to implement.', '1307.2982-1-82-3': 'We show empirical performance on datasets of binary codes obtained from [MATH] billion SIFT, and [MATH] million Gist features.', '1307.2982-1-82-4': 'With these datasets we find that, for [MATH]-bit and [MATH]-bit codes, our new multi-index hashing implementation is often more than two orders of magnitude faster than a linear scan baseline.', '1307.2982-1-83-0': 'While the basic algorithm is developed in this paper there are several interesting avenues for future research.', '1307.2982-1-83-1': 'For example our preliminary research shows that [MATH] is a good choice for the substring length, and it should be possible to formulate a sound mathematical basis for this choice.', '1307.2982-1-83-2': 'The assignment of bits to substrings was shown to be important above, however the algorithm used for this assignment is clearly suboptimal.', '1307.2982-1-83-3': 'It is also likely that different substring lengths might be useful for the different hash tables.', '1307.2982-1-84-0': 'While the current paper concerns exact nearest-neighbor tasks, it would also be interesting to consider approximate methods based on the same multi-index hashing framework.', '1307.2982-1-84-1': 'Indeed there are several ways that one could find approximate rather than the exact nearest neighbors for a given query.', '1307.2982-1-84-2': 'For example, one could stop at a given radius of search, even though [MATH] items may not have been found.', '1307.2982-1-84-3': 'Alternatively, one might search until a fixed number of unique candidates have been found, even though all substring hash tables have not been inspected to the necessary radius, Such approximate algorithms have the potential for even greater efficiency, and would be the most natural methods to compare to most existing methods which are approximate, such as binary LSH.', '1307.2982-1-84-4': 'That said, such comparisons are more difficult than for exact methods since one must taken into account not only the storage and run-time costs, but also some measure of the cost of errors (usually in terms of recall and precision).', '1307.2982-1-85-0': 'Finally, recent results have shown that for many datasets in which the binary codes are the result of some form of vector quantization, an asymmetric Hamming distance is attractive [CITATION].', '1307.2982-1-85-1': 'In such methods, rather than converting the query into a binary code, one directly compares a real-valued query to the database of binary codes.', '1307.2982-1-85-2': 'The advantage is that the quantization noise entailed in converting the query to a binary string is avoided and one can more accurately using distances in the binary code space to approximate the desired distances in the feature space of the query.', '1307.2982-1-85-3': 'One simple way to do this is to use multi-index hashing and then only use an asymmetric distance when culling candidates.', '1307.2982-1-85-4': 'The potential for more interesting and effective methods is yet another promising anvenue for future work.', '1307.2982-1-86-0': '# Implementation Details', '1307.2982-1-87-0': '# Acknowledgments'}

{'1307.2982-2-0-0': 'There is growing interest in representing image data and feature descriptors using compact binary codes for fast near neighbor search.', '1307.2982-2-0-1': 'Although binary codes are motivated by their use as direct indices (addresses) into a hash table, codes longer than 32 bits are not being used as such, as it was thought to be ineffective.', '1307.2982-2-0-2': 'We introduce a rigorous way to build multiple hash tables on binary code substrings that enables exact k-nearest neighbor search in Hamming space.', '1307.2982-2-0-3': 'The approach is storage efficient and straight-forward to implement.', '1307.2982-2-0-4': 'Theoretical analysis shows that the algorithm exhibits sub-linear run-time behavior for uniformly distributed codes.', '1307.2982-2-0-5': 'Empirical results show dramatic speedups over a linear scan baseline for datasets of up to one billion codes of 64, 128, or 256 bits.', '1307.2982-2-1-0': '# Introduction', '1307.2982-2-2-0': 'There has been growing interest in representing image data and feature descriptors in terms of compact binary codes, often to facilitate fast near neighbor search and feature matching in vision applications (e.g., [CITATION]).', '1307.2982-2-2-1': 'Binary codes are storage efficient and comparisons require just a small number of machine instructions.', '1307.2982-2-2-2': 'Millions of binary codes can be compared to a query in less than a second.', '1307.2982-2-2-3': 'But the most compelling reason for binary codes, and discrete codes in general, is their use as direct indices (addresses) into a hash table, yielding a dramatic increase in search speed compared to an exhaustive linear scan (e.g., [CITATION]).', '1307.2982-2-3-0': 'Nevertheless, using binary codes as direct hash indices is not necessarily efficient.', '1307.2982-2-3-1': 'To find near neighbors one needs to examine all hash table entries (or buckets) within some Hamming ball around the query.', '1307.2982-2-3-2': 'The problem is that the number of such buckets grows near-exponentially with the search radius.', '1307.2982-2-3-3': 'Even with a small search radius, the number of buckets to examine is often larger than the number of items in the database, and hence slower than linear scan.', '1307.2982-2-3-4': 'Recent papers on binary codes mention the use of hash tables, but resort to linear scan when codes are longer than [MATH] bits (e.g., [CITATION]).', '1307.2982-2-3-5': 'Not surprisingly, code lengths are often significantly longer than 32 bits in order to achieve satisfactory retrieval performance (e.g., see Fig. [REF]).', '1307.2982-2-4-0': 'This paper presents a new algorithm for exact [MATH]-nearest neighbor ([MATH]NN) search on binary codes that is dramatically faster than exhaustive linear scan.', '1307.2982-2-4-1': 'This has been an open problem since the introduction of hashing techniques with binary codes.', '1307.2982-2-4-2': 'Our new multi-index hashing algorithm exhibits sub-linear search times, is storage efficient, and straightforward to implement.', '1307.2982-2-4-3': 'Empirically, on databases of up to 1B codes we find that multi-index hashing is hundreds of times faster than linear scan.', '1307.2982-2-4-4': 'Extrapolation suggests that the speedup gain grows quickly with database size beyond 1B codes.', '1307.2982-2-5-0': '## Background: Problem and Related Work', '1307.2982-2-6-0': 'Nearest neighbor (NN) search on binary codes is used for image search [CITATION], matching local features [CITATION], image classification [CITATION], object segmentation [CITATION], and parameter estimation [CITATION].', '1307.2982-2-6-1': 'Sometimes the binary codes are generated directly as feature descriptors for images or image patches, such as BRIEF or FREAK [CITATION], and sometimes binary corpora are generated by discrete similarity-preserving mappings from high-dimensional data.', '1307.2982-2-6-2': 'Most such mappings are designed to preserve Euclidean distance (e.g., [CITATION]).', '1307.2982-2-6-3': 'Others focus on semantic similarity (e.g., [CITATION]).', '1307.2982-2-6-4': 'Our concern in this paper is not the algorithm used to generate the codes, but rather with fast search in Hamming space.', '1307.2982-2-7-0': 'We address two related search problems in Hamming space.', '1307.2982-2-7-1': 'Given a dataset of binary codes, [MATH], the first problem is to find the [MATH] codes in [MATH] that are closest in Hamming distance to a given query, i.e., [MATH]NN search in Hamming distance.', '1307.2982-2-7-2': 'The [MATH]-NN problem in Hamming space was called the Best Match problem by Minsky and Papert [CITATION].', '1307.2982-2-7-3': 'They observed that there are no obvious approaches significantly better than exhaustive search, and asked whether such approaches might exist.', '1307.2982-2-8-0': 'The second problem is to find all codes in a dataset [MATH] that are within a fixed Hamming distance of a query, sometimes called the Approximate Query problem [CITATION], or Point Location in Equal Balls (PLEB) [CITATION].', '1307.2982-2-8-1': 'A binary code is an [MATH]-neighbor of a query code, denoted [MATH], if it differs from [MATH] in [MATH] bits or less.', '1307.2982-2-8-2': 'We define the [MATH]-neighbor search problem as: find all [MATH]-neighbors of a query [MATH] from [MATH].', '1307.2982-2-9-0': 'One way to tackle [MATH]-neighbor search is to use a hash table populated with the binary codes [MATH], and examine all hash buckets whose indices are within [MATH] bits of a query [MATH] (e.g.,, [CITATION]).', '1307.2982-2-9-1': 'For binary codes of [MATH] bits, the number of distinct hash buckets to examine is [EQUATION]', '1307.2982-2-9-2': 'As shown in Fig. [REF] (top), [MATH] grows very rapidly with [MATH].', '1307.2982-2-9-3': 'Thus, this approach is only practical for small radii or short code lengths.', '1307.2982-2-9-4': 'Some vision applications restrict search to exact matches (i.e., [MATH]) or a small search radius (e.g., [CITATION] ), but in most cases of interest the desired search radius is larger than is currently feasible (e.g., see Fig. [REF] (bottom)).', '1307.2982-2-10-0': 'Our work is inspired in part by the multi-index hashing results of Greene, Parnas, and Yao [CITATION].', '1307.2982-2-10-1': 'Building on the classical Turan problem for hypergraphs, they construct a set of over-lapping binary substrings such that any two codes that differ by at most [MATH] bits are guaranteed to be identical in at least one of the constructed substrings.', '1307.2982-2-10-2': 'Accordingly, they propose an exact method for finding all [MATH]-neighbors of a query using multiple hash tables, one for each substring.', '1307.2982-2-10-3': 'At query time, candidate [MATH]-neighbors are found by using query substrings as indices into their corresponding hash tables.', '1307.2982-2-10-4': 'As explained below, while run-time efficient, the main drawback of their approach is the prohibitive storage required for the requisite number of hash tables.', '1307.2982-2-10-5': 'By comparison, the method we propose requires much less storage, and is only marginally slower in search performance.', '1307.2982-2-11-0': 'While we focus on exact search, there also exist algorithms for finding approximate [MATH]-neighbors ([MATH]-PLEB), or approximate nearest neighbors ([MATH]-NN) in Hamming distance.', '1307.2982-2-11-1': 'One example is Hamming Locality Sensitive Hashing [CITATION], which aims to solve the [MATH]-neighbors decision problem: determine whether there exists a binary code [MATH] such that [MATH], or whether all codes in [MATH] differ from [MATH] in [MATH] bits or more.', '1307.2982-2-11-2': 'Approximate methods are interesting, and the approach below could be made faster by allowing misses.', '1307.2982-2-11-3': 'Nonetheless, this paper will focus on the exact search problem.', '1307.2982-2-12-0': 'This paper proposes a data-structure that applies to both [MATH]NN and [MATH]-neighbor search in Hamming space.', '1307.2982-2-12-1': 'We prove that for uniformly distributed binary codes of [MATH] bits, and a search radius of [MATH] bits when [MATH] is small, our query time is sub-linear in the size of dataset.', '1307.2982-2-12-2': 'We also demonstrate impressive performance on real-world datasets.', '1307.2982-2-12-3': 'To our knowledge this is the first practical data-structure solving exact [MATH]NN in Hamming distance.', '1307.2982-2-13-0': 'Section 2 describes a multi-index hashing algorithm for [MATH]-neighbor search in Hamming space, followed by run-time and memory analysis in Section [REF].', '1307.2982-2-13-1': 'Section Section [REF] describes our algorithm for [MATH]-nearest neighbor search, and Section Section [REF] reports results on empirical datasets.', '1307.2982-2-14-0': '# Multi-Index Hashing', '1307.2982-2-15-0': 'Our approach is a form of multi-index hashing.', '1307.2982-2-15-1': 'Binary codes from the database are indexed [MATH] times into [MATH] different hash tables, based on [MATH] disjoint binary substrings.', '1307.2982-2-15-2': 'Given a query code, entries that fall close to the query in at least one such substring are considered neighbor candidates.', '1307.2982-2-15-3': 'Candidates are then checked for validity using the entire binary code, to remove any non-[MATH]-neighbors.', '1307.2982-2-15-4': 'To be practical for large-scale datasets, the substrings must be chosen so that the set of candidates is small, and storage requirements are reasonable.', '1307.2982-2-15-5': 'We also require that all true neighbors will be found.', '1307.2982-2-16-0': 'The key idea here stems from the fact that, with [MATH] binary codes of [MATH] bits, the vast majority of the [MATH] possible buckets in a full hash table will be empty, since [MATH].', '1307.2982-2-16-1': 'It seems expensive to examine all [MATH] buckets within [MATH] bits of a query, since most of them contain no items.', '1307.2982-2-16-2': 'Instead, we merge many buckets together (most of which are empty) by marginalizing over different dimensions of the Hamming space.', '1307.2982-2-16-3': 'We do this by creating hash tables on substrings of the binary codes.', '1307.2982-2-16-4': 'The distribution of the code substring comprising the first [MATH] bits is the outcome of marginalizing the distribution of binary codes over the last [MATH] bits.', '1307.2982-2-16-5': 'As such, a given bucket of the substring hash table includes all codes with the same first [MATH] bits, but having any of the [MATH] values for the remaining [MATH] bits.', '1307.2982-2-16-6': 'Unfortunately these larger buckets are not restricted to the Hamming volume of interest around the query.', '1307.2982-2-16-7': 'Hence not all items in the merged buckets are [MATH]-neighbors of the query, so we then need to cull any candidate that is not a true [MATH]-neighbor.', '1307.2982-2-17-0': '## Substring Search Radii', '1307.2982-2-18-0': 'In more detail, each binary code [MATH], comprising [MATH] bits, is partitioned into [MATH] disjoint substrings, [MATH], each of length [MATH] or [MATH] bits.', '1307.2982-2-18-1': 'For convenience in what follows, we assume that [MATH] is divisible by [MATH], and that the substrings comprise contiguous bits.', '1307.2982-2-18-2': 'The key idea rests on the following statement: When two binary codes [MATH] and [MATH] differ by [MATH] bits or less, then, in at least one of their [MATH] substrings they must differ by at most [MATH] bits.', '1307.2982-2-18-3': 'This leads to the first proposition:', '1307.2982-2-19-0': 'Proposition 1: If [MATH], where [MATH] denotes the Hamming distance between [MATH] and [MATH], then [EQUATION] where [MATH].', '1307.2982-2-20-0': 'Proof of Proposition 1 follows straightforwardly from the Pigeonhole Principle.', '1307.2982-2-20-1': 'That is, suppose that the Hamming distance between each of the [MATH] substrings is strictly greater than [MATH].', '1307.2982-2-20-2': 'Then, [MATH].', '1307.2982-2-20-3': 'Clearly, [MATH], since [MATH] for some [MATH] where [MATH], which contradicts the premise.', '1307.2982-2-21-0': 'The significance of Proposition 1 derives from the fact that the substrings have only [MATH] bits, and that the required search radius in each substring is just [MATH].', '1307.2982-2-21-1': 'For example, if [MATH] and [MATH] differ by [MATH] bits or less, and [MATH], at least one of the 4 substrings must be identical.', '1307.2982-2-21-2': 'If they differ by at most [MATH] bits, then in at least one substring they differ by no more than [MATH] bit; i.e., we can search a Hamming radius of [MATH] bits by searching a radius of [MATH] bit on each of 4 substrings.', '1307.2982-2-21-3': 'More generally, instead of examining [MATH] hash buckets, it suffices to examine [MATH] buckets in each of [MATH] substring hash tables.', '1307.2982-2-22-0': 'While it suffices to examine all buckets within a radius of [MATH] in all [MATH] hash tables, we next show that it is not always necessary.', '1307.2982-2-22-1': 'Rather, it is often possible to use a radius of just [MATH] in some of the [MATH] substring hash tables while still guaranteeing that all [MATH]-neighbors of [MATH] will be found.', '1307.2982-2-22-2': 'In particular, with [MATH], where [MATH], to find any item within a radius of [MATH] on [MATH]-bit codes, it suffices to search [MATH] substring hash tables to a radius of [MATH], and the remaining [MATH] substring hash tables up to a radius of [MATH].', '1307.2982-2-22-3': 'Without loss of generality, since there is no order to the substring hash tables, we search the first [MATH] hash tables with radius [MATH], and all remaining hash tables with radius [MATH].', '1307.2982-2-23-0': 'Proposition 2: If [MATH], then [EQUATION]', '1307.2982-2-23-1': 'To prove Proposition 2, we show that when [REF] is false, ([REF]) must be true.', '1307.2982-2-23-2': 'If [REF] is false, then it must be that [MATH], since otherwise [MATH], in which case [REF] and Proposition 1 are equivalent.', '1307.2982-2-23-3': 'If ([REF]) is false, it also follows that [MATH] and [MATH] differ in each of their first [MATH] substrings by [MATH] or more bits.', '1307.2982-2-23-4': 'Thus, the total number of bits that differ in the first [MATH] substrings is at least [MATH].', '1307.2982-2-23-5': 'Because [MATH], it also follows that the total number of bits that differ in the remaining [MATH] substrings is at most [MATH].', '1307.2982-2-23-6': 'Then, using Proposition 1, the maximum search radius required in each of the remaining [MATH] substring hash tables is [EQUATION] and hence Proposition 2 is true.', '1307.2982-2-23-7': 'Because of the near exponential growth in the number of buckets for large search radii, the smaller substring search radius required by Proposition 2 is significant.', '1307.2982-2-24-0': 'A special case of Proposition 2 is when [MATH], hence [MATH] and [MATH].', '1307.2982-2-24-1': 'In this case, it suffices to search [MATH] substring hash tables for a radius of [MATH] (i.e., exact matches), and the remaining [MATH] substring hash tables can be ignored.', '1307.2982-2-24-2': 'Clearly, if a code does not match exactly with a query in any of the selected [MATH] substrings, then the code must differ from the query in at least [MATH] bits.', '1307.2982-2-25-0': '## Multi-Index Hashing for [MATH]-neighbor Search', '1307.2982-2-26-0': 'In a pre-processing step, given a dataset of binary codes, one hash table is built for each of the [MATH] substrings, as outlined in Algorithm 1.', '1307.2982-2-26-1': 'At query time, given a query [MATH] with substrings [MATH], we search the [MATH] substring hash table for entries that are within a Hamming distance of [MATH] or [MATH] of [MATH], as prescribed by ([REF]).', '1307.2982-2-26-2': 'By doing so we obtain a set of candidates from the [MATH] substring hash table, denoted [MATH].', '1307.2982-2-26-3': 'According to the propositions above, the union of the [MATH] sets, [MATH], is necessarily a superset of the [MATH]-neighbors of [MATH].', '1307.2982-2-26-4': 'The last step of the algorithm computes the full Hamming distance between [MATH] and each candidate in [MATH], retaining only those codes that are true [MATH]-neighbors of [MATH].', '1307.2982-2-26-5': 'Algorithm 2 outlines the [MATH]-neighbor retrieval procedure for a query [MATH].', '1307.2982-2-27-0': 'The search cost depends on the number of lookups (i.e., the number of buckets examined), and the number of candidates tested.', '1307.2982-2-27-1': 'Not surprisingly there is a natural trade-off between them.', '1307.2982-2-27-2': 'With a large number of lookups one can minimize the number of extraneous candidates.', '1307.2982-2-27-3': 'By merging many buckets to reduce the number of lookups, one obtains a large number of candidates to test.', '1307.2982-2-27-4': 'In the extreme case with [MATH], substrings are 1 bit long, so we can expect the candidate set to include almost the entire database.', '1307.2982-2-28-0': 'Note that the idea of building multiple hash tables is not novel in itself (e.g., see [CITATION]).', '1307.2982-2-28-1': 'However previous work relied heavily on exact matches in substrings.', '1307.2982-2-28-2': 'Relaxing this constraint is what leads to a more effective algorithm, especially in terms of the storage requirement.', '1307.2982-2-29-0': '# Performance Analysis', '1307.2982-2-30-0': 'We next develop an analytical model of search performance to help address two key questions: (1) How does search cost depend on substring length, and hence the number of substrings?', '1307.2982-2-30-1': '(2) How do run-time and storage complexity depend on database size, code length, and search radius?', '1307.2982-2-31-0': 'To help answer these questions we exploit a well-known bound on the sum of binomial coefficients [CITATION]; i.e., for any [MATH] and [MATH].', '1307.2982-2-31-1': '[EQUATION] where [MATH] is the entropy of a Bernoulli distribution with probability [MATH].', '1307.2982-2-32-0': 'In what follows, [MATH] continues to denote the number of [MATH]-bit database codes, and [MATH] is the Hamming search radius.', '1307.2982-2-32-1': 'Let [MATH] denote the number of hash tables, and let [MATH] denote the substring length [MATH].', '1307.2982-2-32-2': 'Hence, the maximum substring search radius becomes [MATH].', '1307.2982-2-32-3': 'As above, for the sake of model simplicity, we assume [MATH] is divisible by [MATH].', '1307.2982-2-33-0': 'We begin by formulating an upper bound on the number of lookups.', '1307.2982-2-33-1': 'First, the number of lookups in Algorithm 2 is bounded above by the product of [MATH], the number of substring hash tables, and the number of hash buckets within a radius of [MATH] on substrings of length [MATH] bits.', '1307.2982-2-33-2': 'Accordingly, using ([REF]), if the search radius is less than half the code length, [MATH], then the total number of lookups is given by [EQUATION]', '1307.2982-2-33-3': 'Clearly, as we decrease the substring length [MATH], thereby increasing the number of substrings [MATH], exponentially fewer lookups are needed.', '1307.2982-2-34-0': 'To analyze the expected number of candidates per bucket, we consider the case in which the [MATH] binary codes are uniformly distributed over the Hamming space.', '1307.2982-2-34-1': 'In this case, for a substring of [MATH] bits, for which a substring hash table has [MATH] buckets, the expected number of items per bucket is [MATH].', '1307.2982-2-34-2': 'The expected size of the candidate set therefore equals the number of lookups times [MATH].', '1307.2982-2-35-0': 'The total search cost per query is the cost for lookups plus the cost for candidate tests.', '1307.2982-2-35-1': 'While these costs will vary with the code length [MATH] and the way the hash tables are implemented, empirically we find that, to a reasonable approximation, the costs of a lookup and a candidate test are similar (when [MATH]).', '1307.2982-2-35-2': 'Accordingly, we model the total search cost per query, for retrieving all [MATH]-neighbors, in units of the time required for a single lookup, as [EQUATION]', '1307.2982-2-35-3': 'In practice, database codes will generally not be uniformly distributed, nor are uniformly distributed codes ideal for multi-index hashing.', '1307.2982-2-35-4': 'Indeed, the cost of search with uniformly distributed codes is relatively high since the search radius increases as the density of codes decreases.', '1307.2982-2-35-5': 'Rather, the uniform distribution is primarily a mathematical convenience that facilitates the analysis of run-time, thereby providing some insight into the effectiveness of the approach and how one might choose an effective substring length.', '1307.2982-2-36-0': '## Choosing an Effective Substring Length', '1307.2982-2-37-0': 'As noted above in Sec. [REF], finding a good substring length is central to the efficiency of multi-index hashing.', '1307.2982-2-37-1': 'When the substring length is too large or too small the approach will not be effective.', '1307.2982-2-37-2': 'In practice, an effective substring length for a given dataset can be determined by cross-validation.', '1307.2982-2-37-3': 'Nevertheless this can be expensive.', '1307.2982-2-38-0': 'In the case of uniformly distributed codes, one can instead use the analytic cost model in [REF] to find a near optimal substring length.', '1307.2982-2-38-1': 'As discussed below, we find that a substring length of [MATH] yields a near-optimal search cost.', '1307.2982-2-38-2': 'Further, with non-uniformly distributed codes in benchmark datasets, we find empirically that [MATH] is also a reasonable heuristic for choosing the substring length (e.g., see Table [REF] below).', '1307.2982-2-39-0': 'In more detail, to find a good substring length using the cost model above, assuming uniformly distributed binary codes, we first note that, dividing [MATH] in [REF] by [MATH] has no effect on the optimal [MATH].', '1307.2982-2-39-1': 'Accordingly, one can view the optimal [MATH] as a function of two quantities, namely the number of items, [MATH], and the search ratio [MATH].', '1307.2982-2-40-0': 'Figure [REF] plots cost as a function of substring length [MATH], for [MATH]-bit codes, different database sizes [MATH], and different search radii (expressed as a fraction of the code length [MATH]).', '1307.2982-2-40-1': 'Dashed curves depict [MATH] in ([REF]) while solid curves of the same color depict the upper bound in ([REF]).', '1307.2982-2-40-2': 'The tightness of the bound is evident in the plots, as are the quantization effects of the upper range of the sum in ([REF]).', '1307.2982-2-40-3': 'The small circles in Fig. [REF] (top) depict cost when all quantization effects are included, and hence it is only shown at substring lengths that are integer divisors of the code length.', '1307.2982-2-41-0': 'Fig. [REF] (top) shows cost for search radii equal to [MATH], [MATH] and [MATH] of the code length, with [MATH] in all cases.', '1307.2982-2-41-1': 'One striking property of these curves is that the cost is persistently minimal in the vicinity of [MATH], indicated by the vertical line close to 30 bits.', '1307.2982-2-41-2': 'This behavior is consistent over a wide range of database sizes.', '1307.2982-2-42-0': 'Fig. [REF] (bottom) shows the dependence of cost on [MATH] for databases with [MATH], [MATH], and [MATH], all with [MATH] and [MATH] bits.', '1307.2982-2-42-1': 'In this case we have laterally displaced each curve by [MATH]; notice how this aligns the minima close to [MATH].', '1307.2982-2-42-2': 'These curves suggest that, over a wide range of conditions, cost is minimal for [MATH].', '1307.2982-2-42-3': 'For this choice of the substring length, the expected number of items per substring bucket, i.e., [MATH], reduces to 1.', '1307.2982-2-42-4': 'As a consequence, the number of lookups is equal to the expected number of candidates.', '1307.2982-2-42-5': 'Interestingly, this choice of substring length is similar to that of Greene et al. [CITATION].', '1307.2982-2-42-6': "A somewhat involved theoretical analysis based on Stirling's approximation, omitted here, also suggests that as [MATH] goes to infinity, the optimal substring length converges asymptotically to [MATH].", '1307.2982-2-43-0': '## Run-Time Complexity', '1307.2982-2-44-0': 'Choosing [MATH] in the vicinity of [MATH] also permits a simple characterization of retrieval run-time complexity, for uniformly distributed binary codes.', '1307.2982-2-44-1': 'When [MATH], the upper bound on the number of lookups ([REF]) also becomes a bound on the number candidates.', '1307.2982-2-44-2': 'In particular, if we substitute [MATH] for [MATH] in ([REF]), then we find the following upper bound on the cost, now as a function of database size, code length, and the search radius: [EQUATION]', '1307.2982-2-44-3': 'Thus, for a uniform distribution over binary codes, if we choose [MATH] such that [MATH], the expected query time complexity is [MATH].', '1307.2982-2-44-4': 'For a small ratio of [MATH] this is sub-linear in [MATH].', '1307.2982-2-44-5': 'For example, if [MATH], then [MATH], and the run-time complexity becomes [MATH].', '1307.2982-2-44-6': 'That is, the search time increases with the square root of the database size when the search radius is approximately 10% of the code length.', '1307.2982-2-44-7': 'For [MATH], this becomes [MATH].', '1307.2982-2-44-8': 'The time complexity with respect to [MATH] is not as important as that with respect to [MATH] since [MATH] is not expected to vary significantly in most applications.', '1307.2982-2-45-0': '## Storage Complexity', '1307.2982-2-46-0': 'The storage complexity of our multi-index hashing algorithm is asymptotically optimal when [MATH].', '1307.2982-2-46-1': 'To store the full database of binary codes requires [MATH] bits.', '1307.2982-2-46-2': 'For each of [MATH] hash tables, we also need to store [MATH] unique identifiers to the database items.', '1307.2982-2-46-3': 'This allows one to identify the retrieved items and fetch their full codes; this requires an additional [MATH] bits.', '1307.2982-2-46-4': 'In sum, the storage required is [MATH].', '1307.2982-2-46-5': 'When [MATH], as is suggested above, this storage cost reduces to [MATH].', '1307.2982-2-46-6': 'Here, the [MATH] term does not cancel as [MATH], but in most interesting cases [MATH].', '1307.2982-2-47-0': 'While the storage cost for our multi-index hashing algorithm is linear in [MATH], the related multi-index hashing algorithm of Greene et al. [CITATION] entails storage complexity that is super-linear in [MATH].', '1307.2982-2-47-1': 'To find all [MATH]-neighbors, for a given search radius [MATH], they construct [MATH] substrings of length [MATH] bits per binary code.', '1307.2982-2-47-2': 'Their suggested substring length is also [MATH], so the number of substring hash tables becomes [MATH], each of which requires [MATH] in storage.', '1307.2982-2-47-3': 'As a consequence for large values of [MATH], even with small [MATH], this technique requires a prohibitive amount of memory to store the hash tables.', '1307.2982-2-48-0': 'Our approach is more memory-efficient than that of [CITATION] because we do not enforce exact equality in substring matching.', '1307.2982-2-48-1': 'In essence, instead of creating all of the hash tables off-line, and then having to store them, we flip bits of each substring at run-time and implicitly create some of the substring hash tables on-line.', '1307.2982-2-48-2': 'This increases run-time slightly, but greatly reduces storage costs.', '1307.2982-2-49-0': '# [MATH]-Nearest Neighbor Search', '1307.2982-2-50-0': 'To use the above multi-index hashing in practice, one must specify a Hamming search radius [MATH].', '1307.2982-2-50-1': 'For many tasks, the value of [MATH] is chosen such that queries will, on average, retrieve [MATH] near neighbors.', '1307.2982-2-50-2': 'Nevertheless, as expected, we find that for many hashing techniques and different sources of visual data, the distribution of binary codes is such that a single search radius for all queries will not produce similar numbers of neighbors.', '1307.2982-2-51-0': 'Figure [REF] depicts empirical distributions of search radii needed for [MATH]-NN and [MATH]-NN on three sets of binary codes obtained from 1B SIFT descriptors [CITATION].', '1307.2982-2-51-1': 'In all cases, for [MATH] and [MATH]-bit codes, and for hash functions based on LSH [CITATION] and MLH [CITATION], there is a substantial variance in the search radius.', '1307.2982-2-51-2': 'This suggests that binary codes are not uniformly distributed over the Hamming space.', '1307.2982-2-51-3': 'As an example, for [MATH]-NN in [MATH]-bit LSH codes, more than [MATH] of the queries require a search radius of [MATH] bits or larger, while for about [MATH] of the queries it can be [MATH] or smaller.', '1307.2982-2-51-4': 'Also evident from Fig. [REF] is the growth in the required search radius as one moves from [MATH]-bit codes to [MATH] bits, and from [MATH]-NN to [MATH]-NN.', '1307.2982-2-52-0': 'A fixed radius for all queries would produce too many neighbors for some queries, and too few for others.', '1307.2982-2-52-1': 'It is therefore more natural for many tasks to fix the number of required neighbors, i.e., [MATH], and let the search radius depend on the query.', '1307.2982-2-52-2': 'Fortunately, our multi-index hashing algorithm is easily adapted to accommodate query-dependent search radii.', '1307.2982-2-53-0': 'Given a query, one can progressively increase the Hamming search radius per substring, until a specified number of neighbors is found.', '1307.2982-2-53-1': "For example, if one examines all [MATH]-neighbors of a query's substrings, from which more than [MATH] candidates are found to be within a Hamming distance of [MATH] bits (using the full codes for validation), then it is guaranteed that [MATH]-nearest neighbors have been found.", '1307.2982-2-53-2': 'Indeed, if all [MATH]NNs of a query [MATH] differ from [MATH] in [MATH] bits or less, then Propositions 1 and 2 above provide guanantees all such neighbors will be found if one searches the substring hash tables with the prescribed radii.', '1307.2982-2-54-0': 'In our experiments, we follow this progressive increment of the search radius until we can find [MATH]NN in the guaranteed neighborhood of a query.', '1307.2982-2-54-1': 'This approach, outlined in Algorithm 3, is helpful because it uses a query-specific search radius depending on the distribution of codes in the neighborhood of the query.', '1307.2982-2-55-0': '# Experiments', '1307.2982-2-56-0': 'Our implementation of multi-index hashing is available at [CITATION].', '1307.2982-2-56-1': 'Experiments are run on two different architectures.', '1307.2982-2-56-2': 'The first is a mid- to low-end [MATH]Ghz dual quad-core AMD Opteron processor, with [MATH]MB of L2 cache, and [MATH]GB of RAM.', '1307.2982-2-56-3': 'The second is a high-end machine with a [MATH]Ghz dual quad-core Intel Xeon processor, [MATH]MB of L2 cache, and [MATH]B of RAM.', '1307.2982-2-56-4': 'The difference in the size of the L2 cache has a major impact on the run-time of linear scan, since the effectiveness of linear scan depends greatly on L2 cache lines.', '1307.2982-2-56-5': 'With roughly ten times the L2 cache, linear scan on the Intel platform is roughly twice as fast as on the AMD machines.', '1307.2982-2-56-6': 'By comparison, multi-index hashing does not have a serial memory access pattern and so the cache size does not have such a pronounced effect.', '1307.2982-2-56-7': 'Actual run-times for multi-index hashing on the Intel and AMD platforms are within 20% of one another.', '1307.2982-2-57-0': 'Both linear scan and multi-index hashing were implemented in C++ and compiled with identical compiler flags.', '1307.2982-2-57-1': 'To accommodate the large size of memory footprint required for 1B codes, we used the libhugetlbfs package and Linux Kernel [MATH] to allow the use of [MATH]MB page sizes.', '1307.2982-2-57-2': 'Further details about the implementations are given in Section [REF].', '1307.2982-2-57-3': 'Finally, despite the existence of multiple cores, all experiments are run on a single core to simplify run-time measurements.', '1307.2982-2-58-0': 'The memory requirements for multi-index hashing are described in detail in Section [REF].', '1307.2982-2-58-1': 'We currently require approximately [MATH]GB for multi-index hashing with 1B [MATH]-bit codes, and approximately twice that for [MATH]-bit codes.', '1307.2982-2-58-2': 'Figure [REF] shows how the memory footprint depends on the database size for linear scan and multi-index hashing.', '1307.2982-2-58-3': 'As explained in the Sec. [REF], and demonstrated in Figure [REF] the memory requirements of multi-index hashing grow linearly in the database size, as does linear scan.', '1307.2982-2-58-4': 'While we use a single computer in our experiments, one could implement a distributed version of multi-index hashing on computers with much less memory by placing each substring hash table on a separate computer.', '1307.2982-2-59-0': '## Datasets', '1307.2982-2-60-0': 'We consider two well-known large-scale vision corpora: [MATH]M Gist descriptors from [MATH] million tiny images [CITATION] and [MATH]B SIFT features from the BIGANN dataset [CITATION].', '1307.2982-2-60-1': 'SIFT vectors [CITATION] are [MATH]D descriptors of local image structure in the vicinity of feature points.', '1307.2982-2-60-2': 'Gist features [CITATION] extracted from from [MATH] images capture global image structure in [MATH]D vectors.', '1307.2982-2-60-3': 'These two feature types cover a spectrum of NN search problems in vision from feature to image indexing.', '1307.2982-2-61-0': 'We use two similarity-preserving mappings to create datasets of binary codes, namely, binary angular Locality Sensitive Hashing (LSH) [CITATION], and Minimal Loss Hashing (MLH) [CITATION].', '1307.2982-2-61-1': 'LSH is considered a baseline random projection method, closely related to cosine similarity.', '1307.2982-2-61-2': 'MLH is a state-of-the-art learning algorithm that, given a set of similarity labels, finds an optimal mapping by minimizing a loss function over pairs or triplets of binary codes.', '1307.2982-2-62-0': 'Both the [MATH]M Gist and [MATH]B SIFT corpora comprise three disjoint sets, namely, a training set, a base set for populating the database, and a test query set.', '1307.2982-2-62-1': 'Using a random permutation, Gist descriptors are divided into a training set with [MATH] items, a base set of [MATH] million items, and a query set of size [MATH].', '1307.2982-2-62-2': 'The SIFT corpus comes with [MATH]M for training, [MATH] in the base set, and [MATH] test queries.', '1307.2982-2-63-0': 'For LSH we subtract the mean, and pick a set of coefficients from the standard normal density for a linear projection, followed by quantization.', '1307.2982-2-63-1': 'For MLH the training set is used to optimize hash function parameters [CITATION].', '1307.2982-2-63-2': 'After learning is complete, we remove the training data and use the resulting hash function with the base set to create the database of binary codes.', '1307.2982-2-63-3': 'With two image corpora (SIFT and Gist), up to three code lengths ([MATH], [MATH], and [MATH] bits), and two hashing methods (LSH and MLH), we obtain several datasets of binary codes with which to evaluate our multi-index hashing algorithm.', '1307.2982-2-63-4': 'Note that [MATH]-bit codes are only used with LSH and SIFT vectors.', '1307.2982-2-64-0': 'Figure [REF] shows Euclidean NN recall rates for [MATH]NN search on binary codes generated from [MATH]M and [MATH]B SIFT descriptors.', '1307.2982-2-64-1': 'In particular, we plot the fraction of Euclidean [MATH] nearest neighbors found, by [MATH]NN in [MATH]-bit and [MATH]-bit LSH [CITATION] and MLH [CITATION] binary codes.', '1307.2982-2-64-2': 'As expected [MATH]-bit codes are more accurate, and MLH outperforms LSH.', '1307.2982-2-64-3': 'Note that the multi-index hashing algorithm solves exact [MATH]NN search in Hamming distance; the approximation that reduces recall is due to the mapping from the original Euclidean space to the Hamming space.', '1307.2982-2-64-4': 'To preserve the Euclidean structure in the original SIFT descriptors, it seems useful to use longer codes, and exploit data-dependant hash functions such as MLH.', '1307.2982-2-64-5': 'Interestingly, as described below, the speedup factors of multi-index hashing on MLH codes are better than those for LSH.', '1307.2982-2-65-0': 'Obviously, Hamming distance computed on q-bit binary codes is an integer between 0 and q. Thus, the nearest neighbors in Hamming distance can be divided into subsets of elements that have equal Hamming distance (at most q+1 subsets).', '1307.2982-2-65-1': 'Although Hamming distance does not provide a means to distinguish between equi-distant elements, often a re-ranking phase using Asymmetric Hamming distance [CITATION] or other distance measures is helpful in practice.', '1307.2982-2-65-2': 'Nevertheless, this paper is solely concerned with the exact Hamming [MATH]NN problem up to a selection of equi-distant elements in the top [MATH] elements.', '1307.2982-2-66-0': '## Results', '1307.2982-2-67-0': 'Each experiment below involves [MATH] queries, for which we report the average run-time.', '1307.2982-2-67-1': 'Our implementation of the linear scan baseline searches [MATH] million [MATH]-bit codes in just under one second on the AMD machine.', '1307.2982-2-67-2': 'On the Intel machine it examines over [MATH] million [MATH]-bit codes per second.', '1307.2982-2-67-3': 'This is remarkably fast compared to Euclidean NN search with [MATH]D SIFT vectors.', '1307.2982-2-67-4': 'The speed of linear scan is in part due to memory caching, without which it would be much slower.', '1307.2982-2-67-5': 'Run-times for linear scan on other datasets, on both architectures, are given in Tables [REF] and [REF].', '1307.2982-2-68-0': '## Multi-Index Hashing vs. Linear Scan', '1307.2982-2-69-0': 'Tables [REF] and [REF] shows run-time per query for the linear scan baseline, along with speedup factors of multi-index hashing for different [MATH]NN problems and nine different datasets.', '1307.2982-2-69-1': 'Despite the remarkable speed of linear scan, the multi-index hashing implementation is hundreds of times faster.', '1307.2982-2-69-2': 'For example, the multi-index hashing method solves the exact [MATH]-NN for a dataset of 1B [MATH]-bit codes in about [MATH] ms, well over [MATH] times faster than linear scan (see Table [REF]).', '1307.2982-2-69-3': 'Performance on [MATH]-NN and [MATH]-NN are even more impressive.', '1307.2982-2-69-4': 'With [MATH]-bit MLH codes, multi-index hashing executes the 1NN search task over 1000 times faster than the linear scan baseline.', '1307.2982-2-70-0': 'The run-time of linear scan does not depend on the number of neighbors, nor on the underlying distribution of binary codes.', '1307.2982-2-70-1': 'The run-time for multi-index hashing, however, depends on both factors.', '1307.2982-2-70-2': 'In particular, as the desired number of NNs increases, the Hamming radius of the search also increases (e.g., see Figure [REF]).', '1307.2982-2-70-3': 'This implies longer run-times for multi-index hashing.', '1307.2982-2-70-4': 'Indeed, notice that going from [MATH]-NN to [MATH]-NN on each row of the tables shows a decrease in the speedup factors.', '1307.2982-2-71-0': 'The multi-index hashing run-time also depends on the distribution of binary codes.', '1307.2982-2-71-1': 'Indeed, one can see from Table [REF] that MLH code databases yield faster run times than the LSH codes; e.g., for [MATH]-NN in [MATH]B [MATH]-bit codes the speedup for MLH is [MATH] vs [MATH] for LSH.', '1307.2982-2-71-2': 'Figure [REF] depicts the histograms of search radii needed for [MATH]-NN with [MATH]B [MATH]-bit MLH and LSH codes.', '1307.2982-2-71-3': 'Interestingly, the mean of the search radii for MLH codes is [MATH] bits, while it is [MATH] for LSH.', '1307.2982-2-71-4': 'While the means are similar the variances are not; the standard deviations of the search radii for MLH and LSH are [MATH] and [MATH] respectively.', '1307.2982-2-71-5': 'The longer tail of the distribution of search radii for LSH plays an important role in the expected run-time.', '1307.2982-2-71-6': 'In fact, queries that require relatively large search radii tend to dominate the average query cost.', '1307.2982-2-72-0': 'It is also interesting to look at the multi-index hashing run-times as a function of [MATH], the number of binary codes in the database.', '1307.2982-2-72-1': 'To that end, Figure [REF] and [REF] depict run-times for linear scan and multi-index [MATH]NN search on the AMD machine.', '1307.2982-2-72-2': 'The left two figures in each show different vertical scales (since the behavior of multi-index [MATH]NN and linear scan are hard to see at the same scale).', '1307.2982-2-72-3': 'The right-most panels show the same data on log-log axes.', '1307.2982-2-72-4': 'First, it is clear from these plots that multi-index hashing is much faster than linear scan for a wide range of dataset sizes and [MATH].', '1307.2982-2-72-5': 'Just as importantly, it is evident from the log-log plots that as we increase the database size, the speedup factors improve.', '1307.2982-2-72-6': 'The dashed lines on the log-log plots depict [MATH] (up to a scalar constant).', '1307.2982-2-72-7': 'The similar slope of multi-index hashing curves with the square root curves show that multi-index hashing exhibits sub-linear query time, even for the empirical, non-uniform distributions of codes.', '1307.2982-2-73-0': '## Direct lookups with a single hash table', '1307.2982-2-74-0': 'An alternative to linear scan and multi-index hashing is to hash the entire codes into a single hash table, and then use direct hashing with each query.', '1307.2982-2-74-1': 'As suggested in the introduction and Figure [REF], although this approach avoids the need for any candidate checking, it may require a prohibitive number of lookups.', '1307.2982-2-74-2': 'Nevertheless, for sufficiently small code lengths or search radii, it may be effective in practice.', '1307.2982-2-75-0': 'Given the complexity associated with efficiently implementing collision detection in large hash tables, we do not directly experiment with the single hash table approach.', '1307.2982-2-75-1': 'Instead, we consider the empirical number of lookups one would need, as compared to the number of items in the database.', '1307.2982-2-75-2': 'If the number of lookups is vastly greater than the size of the dataset one can readily conclude that linear scan is likely to be as fast or faster than direct indexing into a single hash table.', '1307.2982-2-76-0': 'Fortunately, the statistics of neighborhood sizes and required search radii for [MATH]NN tasks are available from the linear scan and multi-index hashing experiments reported above.', '1307.2982-2-76-1': "For a given query, one can use the [MATH] nearest neighbor's Hamming distance to compute the number of lookups from a single hash table that are required to find all of the query's [MATH] nearest neighbors.", '1307.2982-2-76-2': 'Summed over the set of queries, this provides an indication of the expected run-time.', '1307.2982-2-77-0': 'Figure [REF] shows the total number of lookups required for 1-NN and 1000-NN tasks on 64- and 128-bit codes (from LSH on SIFT) using a single hash table.', '1307.2982-2-77-1': 'They are plotted as a function of the size of the dataset, from [MATH] to [MATH] items.', '1307.2982-2-77-2': 'For comparison, the plots also show the number of database items, and the number of lookups that were needed for multi-index hashing.', '1307.2982-2-77-3': 'Note that Figure [REF] has logarithmic scales.', '1307.2982-2-78-0': 'It is evident that with a single hash table the number of lookups is almost always several orders of magnitude larger than the number of items in the dataset.', '1307.2982-2-78-1': 'And not surprisingly, this is also several orders of magnitude more lookups than required for multi-index hashing.', '1307.2982-2-78-2': 'Although the relative speed of a lookup operation compared to a candidate check, as used in linear scan, depends on the implementation, there are a few important considerations.', '1307.2982-2-78-3': 'Linear scan has an exactly serial memory access pattern and so can make very efficient use of cache, whereas lookups in a hash table are inherently random.', '1307.2982-2-78-4': 'Furthermore, in any plausible implementation of a single hash table for 64 bit or longer codes, there will be some penalty for collision detection.', '1307.2982-2-79-0': 'As illustrated in Figure [REF], the only cases where a single hash table might potentially be more efficient than linear scan are with very small codes (64 bits or less), with a large dataset (1 billion items or more), and a small search distances (e.g., for 1-NN).', '1307.2982-2-79-1': 'In all other cases, linear scan requires orders of magnitude fewer operations.', '1307.2982-2-79-2': 'With any code length longer than 64 bits, a single hash table approach is completely infeasible to run, requiring upwards of 15 orders of magnitude more operations than linear scan for 128-bit codes.', '1307.2982-2-80-0': '## Substring Optimization', '1307.2982-2-81-0': 'The substring hash tables used above have been formed by simply dividing the full codes into disjoint and consecutive sequences bits.', '1307.2982-2-81-1': 'For LSH and MLH, this is equivalent to randomly assigning bits to substrings.', '1307.2982-2-82-0': 'It natural to ask whether further gains in efficiency are possible by optimizing the assignment of bits to substrings.', '1307.2982-2-82-1': 'In particular, by careful substring optimization one may be able to maximize the discriminability of the different substrings.', '1307.2982-2-82-2': 'In other words, while the radius of substring searches and hence the number of lookups is determined by the desired search radius on the full codes, and will remain fixed, by optimizing the assignment of bits to substrings one might be able to reduce the number of candidates one needs to validate.', '1307.2982-2-83-0': 'To explore this idea we considered a simple method in which bits are assigned to substrings one at a time in a greedy fashion, based on correlations between bits.', '1307.2982-2-83-1': 'In particular, of those bits not yet assigned, the next substring is assigned the bit that minimizes the maximum correlation between that bit and all other bits already assigned to that substring.', '1307.2982-2-83-2': 'Initialization also occurs in a greedy manner: A random bit is assigned to the first substring, after which the first bit to substring [MATH] is that which is maximally correlated with the first bit of substring [MATH].', '1307.2982-2-83-3': 'This approach significantly decreases the correlation between bits within a single substring.', '1307.2982-2-83-4': 'This should make the distribution codes within substrings buckets more uniform, and thereby lower the number of candidates within a given search radius.', '1307.2982-2-83-5': 'Arguably an even better approach would be to maximize the entropy of the entries within each substring hash table, thereby making the distribution of substrings as uniform as possible.', '1307.2982-2-83-6': 'This entropic approach is, however, left to future work.', '1307.2982-2-84-0': 'The results obtained with the correlation-based greedy algorithm show that optimizing substrings can provide overall run-time reductions on the order of [MATH] against consecutive substrings for some cases.', '1307.2982-2-84-1': 'Table [REF] displays the improvements achieved by optimizing substrings for different codes lengths and different values of [MATH].', '1307.2982-2-84-2': 'Figure [REF] shows the run-time performance of optimized substrings.', '1307.2982-2-85-0': '# Implementation details', '1307.2982-2-86-0': 'Our implementation of multi-index hashing is publicly available at [CITATION].', '1307.2982-2-86-1': 'Nevertheless, for the interested reader we describe some of the important details here.', '1307.2982-2-87-0': 'As explained above, the algorithm hinges on hash tables built on disjoint [MATH]-bit substrings of the binary codes.', '1307.2982-2-87-1': 'We use direct address tables for the substring hash tables because the substrings are usually short ([MATH]).', '1307.2982-2-87-2': 'Direct address tables explicitly allocate memory for [MATH] buckets and store all data points associated with each substring in its corresponding bucket.', '1307.2982-2-87-3': 'There is a one-to-one mapping between buckets and substrings, so no time is spent on collision detection.', '1307.2982-2-88-0': 'One could implement direct address tables with an array of [MATH] pointers, some of which may be null (for empty buckets).', '1307.2982-2-88-1': 'On a [MATH]-bit machine, pointers are [MATH] bytes long, so just storing an empty address table for [MATH] requires [MATH]GB (as done in [CITATION]).', '1307.2982-2-88-2': 'For greater efficiency here, we use sparse direct address tables by grouping buckets into subsets of [MATH] elements.', '1307.2982-2-88-3': 'For each bucket group, a [MATH]-bit binary vector encodes whether each bucket in the group is empty or not.', '1307.2982-2-88-4': 'Then, a single pointer per group is use to point to a single resizable array that stores the data points associated with that bucket group.', '1307.2982-2-88-5': 'Data points within each array are ordered by their bucket index.', '1307.2982-2-88-6': 'To facilitate fast access, for each non-empty bucket we store the index of the beginning and the end of the corresponding segment of the array.', '1307.2982-2-88-7': 'Compared to the direct address tables in [CITATION], for [MATH], and bucket groups of size [MATH], an empty address table requires only [MATH]GB.', '1307.2982-2-88-8': 'Also note that accessing elements in any bucket of the sparse address table has a worst case run-time of [MATH].', '1307.2982-2-89-0': 'Memory Requirements: We store one [MATH]-bit pointer for each bucket group, and a [MATH]-bit binary vector to encode whether buckets in a group are empty; this entails [MATH] bytes for an empty [MATH]-bit hash table ([MATH]), or [MATH]GB when [MATH].', '1307.2982-2-89-1': 'Bookkeeping for each resizable array entails [MATH]-bit integers.', '1307.2982-2-89-2': 'In our experiments, most bucket groups have at least one non-empty bucket.', '1307.2982-2-89-3': 'Taking this into account, the total storage for an [MATH]-bit address table becomes [MATH] bytes ([MATH]GB for [MATH]).', '1307.2982-2-90-0': 'For each non-empty bucket within a bucket group, we store a [MATH]-bit integer to indicate the index of the beginning of the segment of the resizable array corresponding to that bucket.', '1307.2982-2-90-1': 'The number of non-empty buckets is at most [MATH], where [MATH] is the number of hash tables, and [MATH] is the number of codes.', '1307.2982-2-90-2': 'Thus we need an extra [MATH] bytes.', '1307.2982-2-90-3': 'For each data point per hash table we store an ID to reference the full binary code; each ID is [MATH] bytes since [MATH] for our datasets.', '1307.2982-2-90-4': 'This entails [MATH] bytes.', '1307.2982-2-90-5': 'Finally, storing the full binary codes themselves requires [MATH] bytes, since [MATH].', '1307.2982-2-91-0': 'The total memory cost is [MATH] bytes.', '1307.2982-2-91-1': 'For [MATH], this cost is [MATH].', '1307.2982-2-91-2': 'For 1B [MATH]-bit codes, and [MATH] hash tables ([MATH] bits each), the cost is [MATH]GB.', '1307.2982-2-91-3': 'For [MATH]-bit and [MATH]-bit codes our implementation requires [MATH]GB and [MATH]GB.', '1307.2982-2-91-4': 'Note that the last two terms in the memory cost for storing IDs and codes are irreducible, but the first terms can be reduced in a more memory efficient implementation.', '1307.2982-2-92-0': 'With [MATH] (unfolded) substring hash tables of length [MATH] bits, and a [MATH]-bit address per bucket, the empty hash tables requires [MATH] bytes.', '1307.2982-2-92-1': 'For each non-empty bucket a resizable array is allocated to store the associated data points.', '1307.2982-2-92-2': 'Resizable arrays are preferred over linked lists since they are more cache friendly.', '1307.2982-2-92-3': 'To store the size of the resizable arrays, at most [MATH] bytes are needed as the number of non-empty buckets is bounded by [MATH].', '1307.2982-2-92-4': 'For each data point per hash table we store an ID to reference the full binary code; each ID is [MATH] bytes as the size of datasets [MATH]; this yields a total of [MATH] bytes.', '1307.2982-2-92-5': 'Lastly, storing the full binary codes requires [MATH] bytes, because [MATH].', '1307.2982-2-93-0': 'In total, the memory cost is [MATH] bytes (for [MATH], this is [MATH]).', '1307.2982-2-93-1': 'For one billion [MATH]-bit codes, and two chunks ([MATH] bits each), this cost is [MATH]GB.', '1307.2982-2-93-2': 'Note that the last two terms (for the IDs and binary codes) are irreducible, but the first term can be reduced in a memory efficient implementation at least by a factor of two.', '1307.2982-2-93-3': 'The first term heavily dominates the storage cost.', '1307.2982-2-93-4': 'If we search [MATH]-bit binary codes instead of [MATH]-bit ones, then [MATH] and the storage cost drops to [MATH]GB.', '1307.2982-2-93-5': 'For [MATH]-bit codes our implementation requires [MATH]GB of storage, and for [MATH]-bit codes, [MATH]GB.', '1307.2982-2-94-0': 'For example, one could cope with [MATH] bit indices by using just the first 30 bits, thereby ignoring the last [MATH] bits of each substring.', '1307.2982-2-94-1': 'In this way, each bucket becomes the union of 4 buckets, the indices of which differ by at most two bits.', '1307.2982-2-94-2': 'This reduces storage costs by a factor of 4 but increases the size of the candidate set slightly.', '1307.2982-2-94-3': '[David: more here - see numbers in rebuttal.]', '1307.2982-2-95-0': 'Duplicate Candidates: When retrieving candidates from the [MATH] substring hash tables, some codes will be found multiple times.', '1307.2982-2-95-1': 'To detect duplicates, and discard them, we allocate one bit-string with [MATH] bits.', '1307.2982-2-95-2': 'When a candidate is found we check the corresponding bit and discard the candidate if it is marked as a duplicate.', '1307.2982-2-95-3': 'Before each query we initialize the bit-string to zero.', '1307.2982-2-95-4': 'In practice this has negligible run-time.', '1307.2982-2-95-5': 'In theory clearing an [MATH]-bit vector requires [MATH], but in theory there are more efficient ways to store an [MATH]-bit vector without explicit initialization.', '1307.2982-2-96-0': 'Hamming Distance: To compare a query and a candidate (for multi-index search or linear scan), we compute the Hamming distance on the full [MATH]-bit codes, with one xor operation for every 64 bits, followed by a pop count to tally the ones.', '1307.2982-2-96-1': 'We used the built-in GCC function __builtin_popcount for this purpose.', '1307.2982-2-97-0': 'Number of Substrings: The number of substring hash tables we use is determined with a hold-out validation set of database entries.', '1307.2982-2-97-1': 'From that set we estimate the running time of the algorithm for different choices of [MATH] near [MATH], and select the [MATH] that yields the minimum run-time.', '1307.2982-2-97-2': 'As shown in Table [REF] this empirical value for [MATH] is usually the closest integer to [MATH].', '1307.2982-2-98-0': 'Translation Lookaside Buffer and Huge Pages: Modern processors have an on-chip cache that holds a lookup table of memory addresses, for mapping virtual addresses to physical addresses for each running process.', '1307.2982-2-98-1': 'Typically, memory is split into 4KB pages, and a process that allocates memory is given pages by the operating system.', '1307.2982-2-98-2': 'The Translation Lookaside Buffer (TLB) keeps track of these pages.', '1307.2982-2-98-3': 'For processes that have large memory footprints (tens of GB), the number of pages quickly overtakes the size of the TLB (typically about 1500 entries).', '1307.2982-2-98-4': 'For processes using random memory access this means that almost every memory access produces a TLB miss - the requested address is in a page not cached in the TLB, hence the TLB entry must be fetched from slow RAM before the requested page can be accessed.', '1307.2982-2-98-5': 'This slows down memory access, and causes volatility in run-times for memory-access intensive processes.', '1307.2982-2-99-0': 'To avoid this problem, we use the libhugetlbfs Linux library.', '1307.2982-2-99-1': 'This allows the operating system to allocate Huge Pages (2MB each) rather than 4KB pages.', '1307.2982-2-99-2': 'This reduces the number of pages; hence it reduces the frequency of TLB misses, improves memory access speed, and reduces run-time volatility.', '1307.2982-2-99-3': 'The increase in speed of multi-index hashing results reported here compared to those in [CITATION] are attributed to the use of libhugetlbfs.', '1307.2982-2-100-0': '# Conclusion', '1307.2982-2-101-0': 'This paper describes a new algorithm for exact nearest neighbor search on large-scale datasets of binary codes.', '1307.2982-2-101-1': 'The algorithm is a form of multi-index hashing that has provably sub-linear run-time behavior for uniformly distributed codes.', '1307.2982-2-101-2': 'It is storage efficient and easy to implement.', '1307.2982-2-101-3': 'We show empirical performance on datasets of binary codes obtained from [MATH] billion SIFT, and [MATH] million Gist features.', '1307.2982-2-101-4': 'With these datasets we find that, for [MATH]-bit and [MATH]-bit codes, our new multi-index hashing implementation is often more than two orders of magnitude faster than a linear scan baseline.', '1307.2982-2-102-0': 'While the basic algorithm is developed in this paper there are several interesting avenues for future research.', '1307.2982-2-102-1': 'For example our preliminary research shows that [MATH] is a good choice for the substring length, and it should be possible to formulate a sound mathematical basis for this choice.', '1307.2982-2-102-2': 'The assignment of bits to substrings was shown to be important above, however the algorithm used for this assignment is clearly suboptimal.', '1307.2982-2-102-3': 'It is also likely that different substring lengths might be useful for the different hash tables.', '1307.2982-2-103-0': 'Our theoretical analysis proves sub-linear run-time behavior of the multi-index hashing algorithm on uniformly distributed codes, when search radius is small.', '1307.2982-2-103-1': 'Our experiments demonstrate sub-linear run-time behavior of the algorithm on real datasets, while the binary code in our experiments are clearly not uniformly distributed.', '1307.2982-2-103-2': 'Bridging the gap between theoretical analysis and empirical findings for the proposed algorithm remains an open problem.', '1307.2982-2-103-3': 'In particular, we are interested in more realistic assumptions on the binary codes, which still allow for theoretical analysis of the algorithm.', '1307.2982-2-104-0': 'While the current paper concerns exact nearest-neighbor tasks, it would also be interesting to consider approximate methods based on the same multi-index hashing framework.', '1307.2982-2-104-1': 'Indeed there are several ways that one could find approximate rather than the exact nearest neighbors for a given query.', '1307.2982-2-104-2': 'For example, one could stop at a given radius of search, even though [MATH] items may not have been found.', '1307.2982-2-104-3': 'Alternatively, one might search until a fixed number of unique candidates have been found, even though all substring hash tables have not been inspected to the necessary radius, Such approximate algorithms have the potential for even greater efficiency, and would be the most natural methods to compare to most existing methods which are approximate, such as binary LSH.', '1307.2982-2-104-4': 'That said, such comparisons are more difficult than for exact methods since one must taken into account not only the storage and run-time costs, but also some measure of the cost of errors (usually in terms of recall and precision).', '1307.2982-2-105-0': 'Finally, recent results have shown that for many datasets in which the binary codes are the result of some form of vector quantization, an asymmetric Hamming distance is attractive [CITATION].', '1307.2982-2-105-1': 'In such methods, rather than converting the query into a binary code, one directly compares a real-valued query to the database of binary codes.', '1307.2982-2-105-2': 'The advantage is that the quantization noise entailed in converting the query to a binary string is avoided and one can more accurately using distances in the binary code space to approximate the desired distances in the feature space of the query.', '1307.2982-2-105-3': 'One simple way to do this is to use multi-index hashing and then only use an asymmetric distance when culling candidates.', '1307.2982-2-105-4': 'The potential for more interesting and effective methods is yet another promising avenue for future work.', '1307.2982-2-106-0': '# Acknowledgments'}

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'1307.2982-3-75-1'], ['1307.2982-2-69-2', '1307.2982-3-75-2'], ['1307.2982-2-69-3', '1307.2982-3-75-3'], ['1307.2982-2-69-4', '1307.2982-3-75-4'], ['1307.2982-2-92-0', '1307.2982-3-98-0'], ['1307.2982-2-92-1', '1307.2982-3-98-1'], ['1307.2982-2-92-2', '1307.2982-3-98-2'], ['1307.2982-2-92-3', '1307.2982-3-98-3'], ['1307.2982-2-92-4', '1307.2982-3-98-4'], ['1307.2982-2-92-5', '1307.2982-3-98-5'], ['1307.2982-2-95-0', '1307.2982-3-101-0'], ['1307.2982-2-95-1', '1307.2982-3-101-1'], ['1307.2982-2-95-2', '1307.2982-3-101-2'], ['1307.2982-2-95-3', '1307.2982-3-101-3'], ['1307.2982-2-95-4', '1307.2982-3-101-4'], ['1307.2982-2-82-0', '1307.2982-3-88-0'], ['1307.2982-2-82-1', '1307.2982-3-88-1'], ['1307.2982-2-82-2', '1307.2982-3-88-2'], ['1307.2982-2-42-0', '1307.2982-3-46-0'], ['1307.2982-2-42-1', '1307.2982-3-46-1'], ['1307.2982-2-42-2', '1307.2982-3-46-2'], ['1307.2982-2-42-3', '1307.2982-3-46-3'], ['1307.2982-2-42-4', '1307.2982-3-46-4'], ['1307.2982-2-42-5', '1307.2982-3-46-5'], 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[]

[['1307.2982-1-71-0', '1307.2982-2-75-0'], ['1307.2982-1-79-3', '1307.2982-2-83-3'], ['1307.2982-1-79-3', '1307.2982-2-83-4'], ['1307.2982-1-78-0', '1307.2982-2-82-0'], ['1307.2982-1-72-2', '1307.2982-2-76-1'], ['1307.2982-1-73-2', '1307.2982-2-77-2'], ['1307.2982-1-27-2', '1307.2982-2-28-2'], ['1307.2982-1-61-3', '1307.2982-2-64-3'], ['1307.2982-1-41-0', '1307.2982-2-44-0'], ['1307.2982-1-74-2', '1307.2982-2-78-2'], ['1307.2982-1-0-3', '1307.2982-2-0-3'], ['1307.2982-2-83-1', '1307.2982-3-89-2'], ['1307.2982-2-83-1', '1307.2982-3-89-5'], ['1307.2982-2-83-2', '1307.2982-3-89-2'], ['1307.2982-2-83-2', '1307.2982-3-89-3'], ['1307.2982-2-83-2', '1307.2982-3-89-5'], ['1307.2982-2-83-5', '1307.2982-3-89-8'], ['1307.2982-2-46-5', '1307.2982-3-50-5'], ['1307.2982-2-84-2', '1307.2982-3-90-3'], ['1307.2982-2-95-5', '1307.2982-3-101-5'], ['1307.2982-1-36-2', '1307.2982-2-39-0']]

[]

['1307.2982-1-1-0', '1307.2982-1-18-3', '1307.2982-1-19-0', '1307.2982-1-20-2', '1307.2982-2-18-3', '1307.2982-2-19-0', '1307.2982-2-20-2', '1307.2982-3-18-3', '1307.2982-3-19-0', '1307.2982-3-20-2', '1307.2982-3-27-0', '1307.2982-3-28-0', '1307.2982-3-38-0', '1307.2982-3-39-0', '1307.2982-3-59-0']

{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '3': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1307.2982

{'1307.2982-3-0-0': 'There is growing interest in representing image data and feature descriptors using compact binary codes for fast near neighbor search.', '1307.2982-3-0-1': 'Although binary codes are motivated by their use as direct indices (addresses) into a hash table, codes longer than 32 bits are not being used as such, as it was thought to be ineffective.', '1307.2982-3-0-2': 'We introduce a rigorous way to build multiple hash tables on binary code substrings that enables exact k-nearest neighbor search in Hamming space.', '1307.2982-3-0-3': 'The approach is storage efficient and straight-forward to implement.', '1307.2982-3-0-4': 'Theoretical analysis shows that the algorithm exhibits sub-linear run-time behavior for uniformly distributed codes.', '1307.2982-3-0-5': 'Empirical results show dramatic speedups over a linear scan baseline for datasets of up to one billion codes of 64, 128, or 256 bits.', '1307.2982-3-1-0': '# Introduction', '1307.2982-3-2-0': 'There has been growing interest in representing image data and feature descriptors in terms of compact binary codes, often to facilitate fast near neighbor search and feature matching in vision applications (e.g., [CITATION]).', '1307.2982-3-2-1': 'Binary codes are storage efficient and comparisons require just a small number of machine instructions.', '1307.2982-3-2-2': 'Millions of binary codes can be compared to a query in less than a second.', '1307.2982-3-2-3': 'But the most compelling reason for binary codes, and discrete codes in general, is their use as direct indices (addresses) into a hash table, yielding a dramatic increase in search speed compared to an exhaustive linear scan (e.g., [CITATION]).', '1307.2982-3-3-0': 'Nevertheless, using binary codes as direct hash indices is not necessarily efficient.', '1307.2982-3-3-1': 'To find near neighbors one needs to examine all hash table entries (or buckets) within some Hamming ball around the query.', '1307.2982-3-3-2': 'The problem is that the number of such buckets grows near-exponentially with the search radius.', '1307.2982-3-3-3': 'Even with a small search radius, the number of buckets to examine is often larger than the number of items in the database, and hence slower than linear scan.', '1307.2982-3-3-4': 'Recent papers on binary codes mention the use of hash tables, but resort to linear scan when codes are longer than [MATH] bits (e.g., [CITATION]).', '1307.2982-3-3-5': 'Not surprisingly, code lengths are often significantly longer than 32 bits in order to achieve satisfactory retrieval performance (e.g., see Fig. [REF]).', '1307.2982-3-4-0': 'This paper presents a new algorithm for exact [MATH]-nearest neighbor ([MATH]NN) search on binary codes that is dramatically faster than exhaustive linear scan.', '1307.2982-3-4-1': 'This has been an open problem since the introduction of hashing techniques with binary codes.', '1307.2982-3-4-2': 'Our new multi-index hashing algorithm exhibits sub-linear search times, while it is storage efficient, and straightforward to implement.', '1307.2982-3-4-3': 'Empirically, on databases of up to 1B codes we find that multi-index hashing is hundreds of times faster than linear scan.', '1307.2982-3-4-4': 'Extrapolation suggests that the speedup gain grows quickly with database size beyond 1B codes.', '1307.2982-3-5-0': '## Background: Problem and Related Work', '1307.2982-3-6-0': 'Nearest neighbor (NN) search on binary codes is used for image search [CITATION], matching local features [CITATION], image classification [CITATION], object segmentation [CITATION], and parameter estimation [CITATION].', '1307.2982-3-6-1': 'Sometimes the binary codes are generated directly as feature descriptors for images or image patches, such as BRIEF [CITATION], FREAK [CITATION], [CITATION], or [CITATION], and sometimes binary corpora are generated by discrete similarity-preserving mappings from high-dimensional data.', '1307.2982-3-6-2': 'Most such mappings are designed to preserve Euclidean distance (e.g., [CITATION]).', '1307.2982-3-6-3': 'Others focus on semantic similarity (e.g., [CITATION]).', '1307.2982-3-6-4': 'Our concern in this paper is not the algorithm used to generate the codes, but rather with fast search in Hamming space.', '1307.2982-3-7-0': 'We address two related search problems in Hamming space.', '1307.2982-3-7-1': 'Given a dataset of binary codes, [MATH], the first problem is to find the [MATH] codes in [MATH] that are closest in Hamming distance to a given query, i.e., [MATH]NN search in Hamming distance.', '1307.2982-3-7-2': 'The [MATH]-NN problem in Hamming space was called the Best Match problem by Minsky and Papert [CITATION].', '1307.2982-3-7-3': 'They observed that there are no obvious approaches significantly better than exhaustive search, and asked whether such approaches might exist.', '1307.2982-3-8-0': 'The second problem is to find all codes in a dataset [MATH] that are within a fixed Hamming distance of a query, sometimes called the Approximate Query problem [CITATION], or Point Location in Equal Balls (PLEB) [CITATION].', '1307.2982-3-8-1': 'A binary code is an [MATH]-neighbor of a query code, denoted [MATH], if it differs from [MATH] in [MATH] bits or less.', '1307.2982-3-8-2': 'We define the [MATH]-neighbor search problem as: find all [MATH]-neighbors of a query [MATH] from [MATH].', '1307.2982-3-9-0': 'One way to tackle [MATH]-neighbor search is to use a hash table populated with the binary codes [MATH], and examine all hash buckets whose indices are within [MATH] bits of a query [MATH] (e.g.,, [CITATION]).', '1307.2982-3-9-1': 'For binary codes of [MATH] bits, the number of distinct hash buckets to examine is [EQUATION]', '1307.2982-3-9-2': 'As shown in Fig. [REF] (top), [MATH] grows very rapidly with [MATH].', '1307.2982-3-9-3': 'Thus, this approach is only practical for small radii or short code lengths.', '1307.2982-3-9-4': 'Some vision applications restrict search to exact matches (i.e., [MATH]) or a small search radius (e.g., [CITATION] ), but in most cases of interest the desired search radius is larger than is currently feasible (e.g., see Fig. [REF] (bottom)).', '1307.2982-3-10-0': 'Our work is inspired in part by the multi-index hashing results of Greene, Parnas, and Yao [CITATION].', '1307.2982-3-10-1': 'Building on the classical Turan problem for hypergraphs, they construct a set of over-lapping binary substrings such that any two codes that differ by at most [MATH] bits are guaranteed to be identical in at least one of the constructed substrings.', '1307.2982-3-10-2': 'Accordingly, they propose an exact method for finding all [MATH]-neighbors of a query using multiple hash tables, one for each substring.', '1307.2982-3-10-3': 'At query time, candidate [MATH]-neighbors are found by using query substrings as indices into their corresponding hash tables.', '1307.2982-3-10-4': 'As explained below, while run-time efficient, the main drawback of their approach is the prohibitive storage required for the requisite number of hash tables.', '1307.2982-3-10-5': 'By comparison, the method we propose requires much less storage, and is only marginally slower in search performance.', '1307.2982-3-11-0': 'While we focus on exact search, there also exist algorithms for finding approximate [MATH]-neighbors ([MATH]-PLEB), or approximate nearest neighbors ([MATH]-NN) in Hamming distance.', '1307.2982-3-11-1': 'One example is Hamming Locality Sensitive Hashing [CITATION], which aims to solve the [MATH]-neighbors decision problem: determine whether there exists a binary code [MATH] such that [MATH], or whether all codes in [MATH] differ from [MATH] in [MATH] bits or more.', '1307.2982-3-11-2': 'Approximate methods are interesting, and the approach below could be made faster by allowing misses.', '1307.2982-3-11-3': 'Nonetheless, this paper will focus on the exact search problem.', '1307.2982-3-12-0': 'This paper proposes a data-structure that applies to both [MATH]NN and [MATH]-neighbor search in Hamming space.', '1307.2982-3-12-1': 'We prove that for uniformly distributed binary codes of [MATH] bits, and a search radius of [MATH] bits when [MATH] is small, our query time is sub-linear in the size of dataset.', '1307.2982-3-12-2': 'We also demonstrate impressive performance on real-world datasets.', '1307.2982-3-12-3': 'To our knowledge this is the first practical data-structure solving exact [MATH]NN in Hamming distance.', '1307.2982-3-13-0': 'Section 2 describes a multi-index hashing algorithm for [MATH]-neighbor search in Hamming space, followed by run-time and memory analysis in Section [REF].', '1307.2982-3-13-1': 'Section Section [REF] describes our algorithm for [MATH]-nearest neighbor search, and Section Section [REF] reports results on empirical datasets.', '1307.2982-3-14-0': '# Multi-Index Hashing', '1307.2982-3-15-0': 'Our approach is called multi-index hashing, as binary codes from the database are indexed [MATH] times into [MATH] different hash tables, based on [MATH] disjoint substrings.', '1307.2982-3-15-1': 'Given a query code, entries that fall close to the query in at least one such substring are considered neighbor candidates.', '1307.2982-3-15-2': 'Candidates are then checked for validity using the entire binary code, to remove any non-[MATH]-neighbors.', '1307.2982-3-15-3': 'To be practical for large-scale datasets, the substrings must be chosen so that the set of candidates is small, and storage requirements are reasonable.', '1307.2982-3-15-4': 'We also require that all true neighbors will be found.', '1307.2982-3-16-0': 'The key idea here stems from the fact that, with [MATH] binary codes of [MATH] bits, the vast majority of the [MATH] possible buckets in a full hash table will be empty, since [MATH].', '1307.2982-3-16-1': 'It seems expensive to examine all [MATH] buckets within [MATH] bits of a query, since most of them contain no items.', '1307.2982-3-16-2': 'Instead, we merge many buckets together (most of which are empty) by marginalizing over different dimensions of the Hamming space.', '1307.2982-3-16-3': 'We do this by creating hash tables on substrings of the binary codes.', '1307.2982-3-16-4': 'The distribution of the code substring comprising the first [MATH] bits is the outcome of marginalizing the distribution of binary codes over the last [MATH] bits.', '1307.2982-3-16-5': 'As such, a given bucket of the substring hash table includes all codes with the same first [MATH] bits, but having any of the [MATH] values for the remaining [MATH] bits.', '1307.2982-3-16-6': 'Unfortunately these larger buckets are not restricted to the Hamming volume of interest around the query.', '1307.2982-3-16-7': 'Hence not all items in the merged buckets are [MATH]-neighbors of the query, so we then need to cull any candidate that is not a true [MATH]-neighbor.', '1307.2982-3-17-0': '## Substring Search Radii', '1307.2982-3-18-0': 'In more detail, each binary code [MATH], comprising [MATH] bits, is partitioned into [MATH] disjoint substrings, [MATH], each of length [MATH] bits.', '1307.2982-3-18-1': 'For convenience in what follows, we assume that [MATH] is divisible by [MATH], and that the substrings comprise contiguous bits.', '1307.2982-3-18-2': 'The key idea rests on the following statement: When two binary codes [MATH] and [MATH] differ by [MATH] bits or less, then, in at least one of their [MATH] substrings they must differ by at most [MATH] bits.', '1307.2982-3-18-3': 'This leads to the first proposition:', '1307.2982-3-19-0': 'Proposition 1: If [MATH], where [MATH] denotes the Hamming distance between [MATH] and [MATH], then [EQUATION] where [MATH].', '1307.2982-3-20-0': 'Proof of Proposition 1 follows straightforwardly from the Pigeonhole Principle.', '1307.2982-3-20-1': 'That is, suppose that the Hamming distance between each of the [MATH] substrings is strictly greater than [MATH].', '1307.2982-3-20-2': 'Then, [MATH].', '1307.2982-3-20-3': 'Clearly, [MATH], since [MATH] for some [MATH] where [MATH], which contradicts the premise.', '1307.2982-3-21-0': 'The significance of Proposition 1 derives from the fact that the substrings have only [MATH] bits, and that the required search radius in each substring is just [MATH].', '1307.2982-3-21-1': 'For example, if [MATH] and [MATH] differ by [MATH] bits or less, and [MATH], at least one of the 4 substrings must be identical.', '1307.2982-3-21-2': 'If they differ by at most [MATH] bits, then in at least one substring they differ by no more than [MATH] bit; i.e., we can search a Hamming radius of [MATH] bits by searching a radius of [MATH] bit on each of 4 substrings.', '1307.2982-3-21-3': 'More generally, instead of examining [MATH] hash buckets, it suffices to examine [MATH] buckets in each of [MATH] substring hash tables, i.e., a total of [MATH] buckets.', '1307.2982-3-22-0': 'While it suffices to examine all buckets within a radius of [MATH] in all [MATH] hash tables, we next show that it is not always necessary.', '1307.2982-3-22-1': 'Rather, it is often possible to use a radius of just [MATH] in some of the [MATH] substring hash tables while still guaranteeing that all [MATH]-neighbors of [MATH] will be found.', '1307.2982-3-22-2': 'In particular, with [MATH], where [MATH], to find any item within a radius of [MATH] on [MATH]-bit codes, it suffices to search [MATH] substring hash tables to a radius of [MATH], and the remaining [MATH] substring hash tables up to a radius of [MATH].', '1307.2982-3-22-3': 'Without loss of generality, since there is no order to the substring hash tables, we search the first [MATH] hash tables with radius [MATH], and all remaining hash tables with radius [MATH].', '1307.2982-3-23-0': 'Proposition 2: If [MATH], then [EQUATION]', '1307.2982-3-23-1': 'To prove Proposition 2, we show that when [REF] is false, ([REF]) must be true.', '1307.2982-3-23-2': 'If [REF] is false, then it must be that [MATH], since otherwise [MATH], in which case [REF] and Proposition 1 are equivalent.', '1307.2982-3-23-3': 'If ([REF]) is false, it also follows that [MATH] and [MATH] differ in each of their first [MATH] substrings by [MATH] or more bits.', '1307.2982-3-23-4': 'Thus, the total number of bits that differ in the first [MATH] substrings is at least [MATH].', '1307.2982-3-23-5': 'Because [MATH], it also follows that the total number of bits that differ in the remaining [MATH] substrings is at most [MATH].', '1307.2982-3-23-6': 'Then, using Proposition 1, the maximum search radius required in each of the remaining [MATH] substring hash tables is [EQUATION] and hence Proposition 2 is true.', '1307.2982-3-23-7': 'Because of the near exponential growth in the number of buckets for large search radii, the smaller substring search radius required by Proposition 2 is significant.', '1307.2982-3-24-0': 'A special case of Proposition 2 is when [MATH], hence [MATH] and [MATH].', '1307.2982-3-24-1': 'In this case, it suffices to search [MATH] substring hash tables for a radius of [MATH] (i.e., exact matches), and the remaining [MATH] substring hash tables can be ignored.', '1307.2982-3-24-2': 'Clearly, if a code does not match exactly with a query in any of the selected [MATH] substrings, then the code must differ from the query in at least [MATH] bits.', '1307.2982-3-25-0': '## Multi-Index Hashing for [MATH]-neighbor Search', '1307.2982-3-26-0': 'In a pre-processing step, given a dataset of binary codes, one hash table is built for each of the [MATH] substrings, as outlined in Algorithm [REF].', '1307.2982-3-26-1': 'At query time, given a query [MATH] with substrings [MATH], we search the [MATH] substring hash table for entries that are within a Hamming distance of [MATH] or [MATH] of [MATH], as prescribed by ([REF]).', '1307.2982-3-26-2': 'By doing so we obtain a set of candidates from the [MATH] substring hash table, denoted [MATH].', '1307.2982-3-26-3': 'According to the propositions above, the union of the [MATH] sets, [MATH], is necessarily a superset of the [MATH]-neighbors of [MATH].', '1307.2982-3-26-4': 'The last step of the algorithm computes the full Hamming distance between [MATH] and each candidate in [MATH], retaining only those codes that are true [MATH]-neighbors of [MATH].', '1307.2982-3-26-5': 'Algorithm [REF] outlines the [MATH]-neighbor retrieval procedure for a query [MATH].', '1307.2982-3-27-0': '[t] Building [MATH] substring hash tables.', '1307.2982-3-28-0': 'Binary code dataset: [MATH] to [MATH] Initialize [MATH] hash table [MATH] to [MATH] Insert [MATH] into [MATH] hash table', '1307.2982-3-29-0': 'The search cost depends on the number of lookups (i.e., the number of buckets examined), and the number of candidates tested.', '1307.2982-3-29-1': 'Not surprisingly there is a natural trade-off between them.', '1307.2982-3-29-2': 'With a large number of lookups one can minimize the number of extraneous candidates.', '1307.2982-3-29-3': 'By merging many buckets to reduce the number of lookups, one obtains a large number of candidates to test.', '1307.2982-3-29-4': 'In the extreme case with [MATH], substrings are 1 bit long, so we can expect the candidate set to include almost the entire database.', '1307.2982-3-30-0': 'Note that the idea of building multiple hash tables is not novel in itself (e.g., see [CITATION]).', '1307.2982-3-30-1': 'However previous work relied heavily on exact matches in substrings.', '1307.2982-3-30-2': 'Relaxing this constraint is what leads to a more effective algorithm, especially in terms of the storage requirement.', '1307.2982-3-31-0': '# Performance Analysis', '1307.2982-3-32-0': 'We next develop an analytical model of search performance to help address two key questions: (1) How does search cost depend on substring length, and hence the number of substrings?', '1307.2982-3-32-1': '(2) How do run-time and storage complexity depend on database size, code length, and search radius?', '1307.2982-3-33-0': 'To help answer these questions we exploit a well-known bound on the sum of binomial coefficients [CITATION]; i.e., for any [MATH] and [MATH].', '1307.2982-3-33-1': '[EQUATION] where [MATH] is the entropy of a Bernoulli distribution with probability [MATH].', '1307.2982-3-34-0': 'In what follows, [MATH] continues to denote the number of [MATH]-bit database codes, and [MATH] is the Hamming search radius.', '1307.2982-3-34-1': 'Let [MATH] denote the number of hash tables, and let [MATH] denote the substring length [MATH].', '1307.2982-3-34-2': 'Hence, the maximum substring search radius becomes [MATH].', '1307.2982-3-34-3': 'As above, for the sake of model simplicity, we assume [MATH] is divisible by [MATH].', '1307.2982-3-35-0': 'We begin by formulating an upper bound on the number of lookups.', '1307.2982-3-35-1': 'First, the number of lookups in Algorithm [REF] is bounded above by the product of [MATH], the number of substring hash tables, and the number of hash buckets within a radius of [MATH] on substrings of length [MATH] bits.', '1307.2982-3-35-2': 'Accordingly, using ([REF]), if the search radius is less than half the code length, [MATH], then the total number of lookups is given by [EQUATION]', '1307.2982-3-35-3': 'Clearly, as we decrease the substring length [MATH], thereby increasing the number of substrings [MATH], exponentially fewer lookups are needed.', '1307.2982-3-36-0': 'To analyze the expected number of candidates per bucket, we consider the case in which the [MATH] binary codes are uniformly distributed over the Hamming space.', '1307.2982-3-36-1': 'In this case, for a substring of [MATH] bits, for which a substring hash table has [MATH] buckets, the expected number of items per bucket is [MATH].', '1307.2982-3-36-2': 'The expected size of the candidate set therefore equals the number of lookups times [MATH].', '1307.2982-3-37-0': 'The total search cost per query is the cost for lookups plus the cost for candidate tests.', '1307.2982-3-37-1': 'While these costs will vary with the code length [MATH] and the way the hash tables are implemented, empirically we find that, to a reasonable approximation, the costs of a lookup and a candidate test are similar (when [MATH]).', '1307.2982-3-37-2': 'Accordingly, we model the total search cost per query, for retrieving all [MATH]-neighbors, in units of the time required for a single lookup, as [EQUATION]', '1307.2982-3-37-3': 'In practice, database codes will not be uniformly distributed, nor are uniformly distributed codes ideal for multi-index hashing.', '1307.2982-3-37-4': 'Indeed, the cost of search with uniformly distributed codes is relatively high since the search radius increases as the density of codes decreases.', '1307.2982-3-37-5': 'Rather, the uniform distribution is primarily a mathematical convenience that facilitates the analysis of run-time, thereby providing some insight into the effectiveness of the approach and how one might choose an effective substring length.', '1307.2982-3-38-0': '[t] [MATH]-Neighbor Search for Query [MATH].', '1307.2982-3-39-0': 'Query substrings: [MATH] Substring radius: [MATH], and [MATH] to [MATH] Lookup [MATH]-neighbors of [MATH] from [MATH] hash table [MATH] to [MATH] Lookup ([MATH]-[MATH])-neighbors of [MATH] from [MATH] hash table Remove all non [MATH]-neighbors from the candidate set.', '1307.2982-3-40-0': '## Choosing an Effective Substring Length', '1307.2982-3-41-0': 'As noted above in Sec. [REF], finding a good substring length is central to the efficiency of multi-index hashing.', '1307.2982-3-41-1': 'When the substring length is too large or too small the approach will not be effective.', '1307.2982-3-41-2': 'In practice, an effective substring length for a given dataset can be determined by cross-validation.', '1307.2982-3-41-3': 'Nevertheless this can be expensive.', '1307.2982-3-42-0': 'In the case of uniformly distributed codes, one can instead use the analytic cost model in [REF] to find a near optimal substring length.', '1307.2982-3-42-1': 'As discussed below, we find that a substring length of [MATH] yields a near-optimal search cost.', '1307.2982-3-42-2': 'Further, with non-uniformly distributed codes in benchmark datasets, we find empirically that [MATH] is also a reasonable heuristic for choosing the substring length (e.g., see Table [REF] below).', '1307.2982-3-43-0': 'In more detail, to find a good substring length using the cost model above, assuming uniformly distributed binary codes, we first note that, dividing [MATH] in [REF] by [MATH] has no effect on the optimal [MATH].', '1307.2982-3-43-1': 'Accordingly, one can view the optimal [MATH] as a function of two quantities, namely the number of items, [MATH], and the search ratio [MATH].', '1307.2982-3-44-0': 'Figure [REF] plots cost as a function of substring length [MATH], for [MATH]-bit codes, different database sizes [MATH], and different search radii (expressed as a fraction of the code length [MATH]).', '1307.2982-3-44-1': 'Dashed curves depict [MATH] in ([REF]) while solid curves of the same color depict the upper bound in ([REF]).', '1307.2982-3-44-2': 'The tightness of the bound is evident in the plots, as are the quantization effects of the upper range of the sum in ([REF]).', '1307.2982-3-44-3': 'The small circles in Fig. [REF] (top) depict cost when all quantization effects are included, and hence it is only shown at substring lengths that are integer divisors of the code length.', '1307.2982-3-45-0': 'Fig. [REF] (top) shows cost for search radii equal to [MATH], [MATH] and [MATH] of the code length, with [MATH] in all cases.', '1307.2982-3-45-1': 'One striking property of these curves is that the cost is persistently minimal in the vicinity of [MATH], indicated by the vertical line close to 30 bits.', '1307.2982-3-45-2': 'This behavior is consistent over a wide range of database sizes.', '1307.2982-3-46-0': 'Fig. [REF] (bottom) shows the dependence of cost on [MATH] for databases with [MATH], [MATH], and [MATH], all with [MATH] and [MATH] bits.', '1307.2982-3-46-1': 'In this case we have laterally displaced each curve by [MATH]; notice how this aligns the minima close to [MATH].', '1307.2982-3-46-2': 'These curves suggest that, over a wide range of conditions, cost is minimal for [MATH].', '1307.2982-3-46-3': 'For this choice of the substring length, the expected number of items per substring bucket, i.e., [MATH], reduces to 1.', '1307.2982-3-46-4': 'As a consequence, the number of lookups is equal to the expected number of candidates.', '1307.2982-3-46-5': 'Interestingly, this choice of substring length is similar to that of Greene et al. [CITATION].', '1307.2982-3-46-6': "A somewhat involved theoretical analysis based on Stirling's approximation, omitted here, also suggests that as [MATH] goes to infinity, the optimal substring length converges asymptotically to [MATH].", '1307.2982-3-47-0': '## Run-Time Complexity', '1307.2982-3-48-0': 'Choosing [MATH] in the vicinity of [MATH] also permits a simple characterization of retrieval run-time complexity, for uniformly distributed binary codes.', '1307.2982-3-48-1': 'When [MATH], the upper bound on the number of lookups ([REF]) also becomes a bound on the number candidates.', '1307.2982-3-48-2': 'In particular, if we substitute [MATH] for [MATH] in ([REF]), then we find the following upper bound on the cost, now as a function of database size, code length, and the search radius: [EQUATION]', '1307.2982-3-48-3': 'Thus, for a uniform distribution over binary codes, if we choose [MATH] such that [MATH], the expected query time complexity is [MATH].', '1307.2982-3-48-4': 'For a small ratio of [MATH] this is sub-linear in [MATH].', '1307.2982-3-48-5': 'For example, if [MATH], then [MATH], and the run-time complexity becomes [MATH].', '1307.2982-3-48-6': 'That is, the search time increases with the square root of the database size when the search radius is approximately 10% of the code length.', '1307.2982-3-48-7': 'For [MATH], this becomes [MATH].', '1307.2982-3-48-8': 'The time complexity with respect to [MATH] is not as important as that with respect to [MATH] since [MATH] is not expected to vary significantly in most applications.', '1307.2982-3-49-0': '## Storage Complexity', '1307.2982-3-50-0': 'The storage complexity of our multi-index hashing algorithm is asymptotically optimal when [MATH], as is suggested above.', '1307.2982-3-50-1': 'To store the full database of binary codes requires [MATH] bits.', '1307.2982-3-50-2': 'For each of [MATH] hash tables, we also need to store [MATH] unique identifiers to the database items.', '1307.2982-3-50-3': 'This allows one to identify the retrieved items and fetch their full codes; this requires an additional [MATH] bits.', '1307.2982-3-50-4': 'In sum, the storage required is [MATH].', '1307.2982-3-50-5': 'When [MATH], this storage cost reduces to [MATH].', '1307.2982-3-50-6': 'Here, the [MATH] term does not cancel as [MATH], but in most interesting cases [MATH], so the [MATH] term does not matter.', '1307.2982-3-51-0': 'While the storage cost for our multi-index hashing algorithm is linear in [MATH], the related multi-index hashing algorithm of Greene et al. [CITATION] entails storage complexity that is super-linear in [MATH].', '1307.2982-3-51-1': 'To find all [MATH]-neighbors, for a given search radius [MATH], they construct [MATH] substrings of length [MATH] bits per binary code.', '1307.2982-3-51-2': 'Their suggested substring length is also [MATH], so the number of substring hash tables becomes [MATH], each of which requires [MATH] in storage.', '1307.2982-3-51-3': 'As a consequence for large values of [MATH], even with small [MATH], this technique requires a prohibitive amount of memory to store the hash tables.', '1307.2982-3-52-0': 'Our approach is more memory-efficient than that of [CITATION] because we do not enforce exact equality in substring matching.', '1307.2982-3-52-1': 'In essence, instead of creating all of the hash tables off-line, and then having to store them, we flip bits of each substring at run-time and implicitly create some of the substring hash tables on-line.', '1307.2982-3-52-2': 'This increases run-time slightly, but greatly reduces storage costs.', '1307.2982-3-53-0': '# [MATH]-Nearest Neighbor Search', '1307.2982-3-54-0': 'To use the above multi-index hashing in practice, one must specify a Hamming search radius [MATH].', '1307.2982-3-54-1': 'For many tasks, the value of [MATH] is chosen such that queries will, on average, retrieve [MATH] near neighbors.', '1307.2982-3-54-2': 'Nevertheless, as expected, we find that for many hashing techniques and different sources of visual data, the distribution of binary codes is such that a single search radius for all queries will not produce similar numbers of neighbors.', '1307.2982-3-55-0': 'Figure [REF] depicts empirical distributions of search radii needed for [MATH]-NN and [MATH]-NN on three sets of binary codes obtained from 1B SIFT descriptors [CITATION].', '1307.2982-3-55-1': 'In all cases, for [MATH]- and [MATH]-bit codes, and for hash functions based on LSH [CITATION] and MLH [CITATION], there is a substantial variance in the search radius.', '1307.2982-3-55-2': 'This suggests that binary codes are not uniformly distributed over the Hamming space.', '1307.2982-3-55-3': 'As an example, for [MATH]-NN in [MATH]-bit LSH codes, about [MATH] of the queries require a search radius of [MATH] bits or larger, while about [MATH] of the queries need a search radius of [MATH] bits or smaller.', '1307.2982-3-55-4': 'Also evident from Fig. [REF] is the growth in the required search radius as one moves from [MATH]-bit codes to [MATH] bits, and from [MATH]-NN to [MATH]-NN.', '1307.2982-3-56-0': 'A fixed radius for all queries would produce too many neighbors for some queries, and too few for others.', '1307.2982-3-56-1': 'It is therefore more natural for many tasks to fix the number of required neighbors, i.e., [MATH], and let the search radius depend on the query.', '1307.2982-3-56-2': 'Fortunately, our multi-index hashing algorithm is easily adapted to accommodate query-dependent search radii.', '1307.2982-3-57-0': 'Given a query, one can progressively increase the Hamming search radius per substring, until a specified number of neighbors is found.', '1307.2982-3-57-1': "For example, if one examines all [MATH]-neighbors of a query's substrings, from which more than [MATH] candidates are found to be within a Hamming distance of [MATH] bits (using the full codes for validation), then it is guaranteed that [MATH]-nearest neighbors have been found.", '1307.2982-3-57-2': 'Indeed, if all [MATH]NNs of a query [MATH] differ from [MATH] in [MATH] bits or less, then Propositions 1 and 2 above provide guanantees all such neighbors will be found if one searches the substring hash tables with the prescribed radii.', '1307.2982-3-58-0': 'In our experiments, we follow this progressive increment of the search radius until we can find [MATH]NN in the guaranteed neighborhood of a query.', '1307.2982-3-58-1': 'This approach, outlined in Algorithm [REF], is helpful because it uses a query-specific search radius depending on the distribution of codes in the neighborhood of the query.', '1307.2982-3-59-0': '[t] [MATH]NN Search with Query [MATH].', '1307.2982-3-60-0': 'Query substrings: [MATH] Initialize sets: [MATH], for [MATH] Initialize integers: [MATH] Assertion: Full search radius should be [MATH].', '1307.2982-3-60-1': 'Lookup buckets in the [MATH] substring hash table that differ from [MATH] in exactly [MATH] bits.', '1307.2982-3-60-2': 'For each candidate found, measure full Hamming distance, and add items with distance [MATH] to [MATH].', '1307.2982-3-60-3': '[MATH] (i.e., [MATH]-neighbors are found)', '1307.2982-3-61-0': '# Experiments', '1307.2982-3-62-0': 'Our implementation of multi-index hashing is available on-line at [CITATION].', '1307.2982-3-62-1': 'Experiments are run on two different architectures.', '1307.2982-3-62-2': 'The first is a mid- to low-end [MATH]Ghz dual quad-core AMD Opteron processor, with [MATH]MB of L2 cache, and [MATH]GB of RAM.', '1307.2982-3-62-3': 'The second is a high-end machine with a [MATH]Ghz dual quad-core Intel Xeon processor, [MATH]MB of L2 cache, and [MATH]B of RAM.', '1307.2982-3-62-4': 'The difference in the size of the L2 cache has a major impact on the run-time of linear scan, since the effectiveness of linear scan depends greatly on L2 cache lines.', '1307.2982-3-62-5': 'With roughly ten times the L2 cache, linear scan on the Intel platform is roughly twice as fast as on the AMD machines.', '1307.2982-3-62-6': 'By comparison, multi-index hashing does not have a serial memory access pattern and so the cache size does not have such a pronounced effect.', '1307.2982-3-62-7': 'Actual run-times for multi-index hashing on the Intel and AMD platforms are within 20% of one another.', '1307.2982-3-63-0': 'Both linear scan and multi-index hashing were implemented in C++ and compiled with identical compiler flags.', '1307.2982-3-63-1': 'To accommodate the large size of memory footprint required for 1B codes, we used the libhugetlbfs package and Linux Kernel [MATH] to allow the use of [MATH]MB page sizes.', '1307.2982-3-63-2': 'Further details about the implementations are given in Section [REF].', '1307.2982-3-63-3': 'Despite the existence of multiple cores, all experiments are run on a single core to simplify run-time measurements.', '1307.2982-3-64-0': 'The memory requirements for multi-index hashing are described in detail in Section [REF].', '1307.2982-3-64-1': 'We currently require approximately [MATH]GB for multi-index hashing with 1B [MATH]-bit codes, and approximately twice that for [MATH]-bit codes.', '1307.2982-3-64-2': 'Figure [REF] shows how the memory footprint depends on the database size for linear scan and multi-index hashing.', '1307.2982-3-64-3': 'As explained in the Sec. [REF], and demonstrated in Figure [REF] the memory requirements of multi-index hashing grow linearly in the database size, as does linear scan.', '1307.2982-3-64-4': 'While we use a single computer in our experiments, one could implement a distributed version of multi-index hashing on computers with much less memory by placing each substring hash table on a separate computer.', '1307.2982-3-65-0': '## Datasets', '1307.2982-3-66-0': 'We consider two well-known large-scale vision corpora: [MATH]M Gist descriptors from [MATH] million tiny images [CITATION] and [MATH]B SIFT features from the BIGANN dataset [CITATION].', '1307.2982-3-66-1': 'SIFT vectors [CITATION] are [MATH]D descriptors of local image structure in the vicinity of feature points.', '1307.2982-3-66-2': 'Gist features [CITATION] extracted from [MATH] images capture global image structure in [MATH]D vectors.', '1307.2982-3-66-3': 'These two feature types cover a spectrum of NN search problems in vision from feature to image indexing.', '1307.2982-3-67-0': 'We use two similarity-preserving mappings to create datasets of binary codes, namely, binary angular Locality Sensitive Hashing (LSH) [CITATION], and Minimal Loss Hashing (MLH) [CITATION].', '1307.2982-3-67-1': 'LSH is considered a baseline random projection method, closely related to cosine similarity.', '1307.2982-3-67-2': 'MLH is a state-of-the-art learning algorithm that, given a set of similarity labels, optimizes a mapping by minimizing a loss function over pairs or triplets of binary codes.', '1307.2982-3-68-0': 'Both the [MATH]M Gist and [MATH]B SIFT corpora comprise three disjoint sets, namely, a training set, a base set for populating the database, and a test query set.', '1307.2982-3-68-1': 'Using a random permutation, Gist descriptors are divided into a training set with [MATH] items, a base set of [MATH] million items, and a query set of size [MATH].', '1307.2982-3-68-2': 'The SIFT corpus comes with [MATH]M for training, [MATH] in the base set, and [MATH] test queries.', '1307.2982-3-69-0': 'For LSH we subtract the mean, and pick a set of coefficients from the standard normal density for a linear projection, followed by quantization.', '1307.2982-3-69-1': 'For MLH the training set is used to optimize hash function parameters [CITATION].', '1307.2982-3-69-2': 'After learning is complete, we remove the training data and use the resulting hash function with the base set to create the database of binary codes.', '1307.2982-3-69-3': 'With two image corpora (SIFT and Gist), up to three code lengths ([MATH], [MATH], and [MATH] bits), and two hashing methods (LSH and MLH), we obtain several datasets of binary codes with which to evaluate our multi-index hashing algorithm.', '1307.2982-3-69-4': 'Note that [MATH]-bit codes are only used with LSH and SIFT vectors.', '1307.2982-3-70-0': 'Figure [REF] shows Euclidean NN recall rates for [MATH]NN search on binary codes generated from [MATH]M and [MATH]B SIFT descriptors.', '1307.2982-3-70-1': 'In particular, we plot the fraction of Euclidean [MATH] nearest neighbors found, by [MATH]NN in [MATH]- and [MATH]-bit LSH [CITATION] and MLH [CITATION] binary codes.', '1307.2982-3-70-2': 'As expected [MATH]-bit codes are more accurate, and MLH outperforms LSH.', '1307.2982-3-70-3': 'Note that the multi-index hashing algorithm solves exact [MATH]NN search in Hamming distance; the approximation that reduces recall is due to the mapping from the original Euclidean space to the Hamming space.', '1307.2982-3-70-4': 'To preserve the Euclidean structure in the original SIFT descriptors, it seems useful to use longer codes, and exploit data-dependant hash functions such as MLH.', '1307.2982-3-70-5': 'Interestingly, as described below, the speedup factors of multi-index hashing on MLH codes are better than those for LSH.', '1307.2982-3-71-0': 'Obviously, Hamming distance computed on q-bit binary codes is an integer between 0 and q. Thus, the nearest neighbors in Hamming distance can be divided into subsets of elements that have equal Hamming distance (at most q+1 subsets).', '1307.2982-3-71-1': 'Although Hamming distance does not provide a means to distinguish between equi-distant elements, often a re-ranking phase using Asymmetric Hamming distance [CITATION] or other distance measures is helpful in practice.', '1307.2982-3-71-2': 'Nevertheless, this paper is solely concerned with the exact Hamming [MATH]NN problem up to a selection of equi-distant elements in the top [MATH] elements.', '1307.2982-3-72-0': '## Results', '1307.2982-3-73-0': 'Each experiment below involves [MATH] queries, for which we report the average run-time.', '1307.2982-3-73-1': 'Our implementation of the linear scan baseline searches [MATH] million [MATH]-bit codes in just under one second on the AMD machine.', '1307.2982-3-73-2': 'On the Intel machine it examines over [MATH] million [MATH]-bit codes per second.', '1307.2982-3-73-3': 'This is remarkably fast compared to Euclidean NN search with [MATH]D SIFT vectors.', '1307.2982-3-73-4': 'The speed of linear scan is in part due to memory caching, without which it would be much slower.', '1307.2982-3-74-0': '## Multi-Index Hashing vs. Linear Scan', '1307.2982-3-75-0': 'Tables [REF] and [REF] shows run-time per query for the linear scan baseline, along with speedup factors of multi-index hashing for different [MATH]NN problems and nine different datasets.', '1307.2982-3-75-1': 'Despite the remarkable speed of linear scan, the multi-index hashing implementation is hundreds of times faster.', '1307.2982-3-75-2': 'For example, the multi-index hashing method solves the exact [MATH]-NN for a dataset of 1B [MATH]-bit codes in about [MATH] ms, well over [MATH] times faster than linear scan (see Table [REF]).', '1307.2982-3-75-3': 'Performance on [MATH]-NN and [MATH]-NN are even more impressive.', '1307.2982-3-75-4': 'With [MATH]-bit MLH codes, multi-index hashing executes the 1NN search task over 1000 times faster than the linear scan baseline.', '1307.2982-3-76-0': 'The run-time of linear scan does not depend on the number of neighbors, nor on the underlying distribution of binary codes.', '1307.2982-3-76-1': 'The run-time for multi-index hashing, however, depends on both factors.', '1307.2982-3-76-2': 'In particular, as the desired number of NNs increases, the Hamming radius of the search also increases (e.g., see Figure [REF]).', '1307.2982-3-76-3': 'This implies longer run-times for multi-index hashing.', '1307.2982-3-76-4': 'Indeed, notice that going from [MATH]-NN to [MATH]-NN on each row of the tables shows a decrease in the speedup factors.', '1307.2982-3-77-0': 'The multi-index hashing run-time also depends on the distribution of binary codes.', '1307.2982-3-77-1': 'Indeed, one can see from Table [REF] that MLH code databases yield faster run times than the LSH codes; e.g., for [MATH]-NN in [MATH]B [MATH]-bit codes the speedup for MLH is [MATH] vs [MATH] for LSH.', '1307.2982-3-77-2': 'Figure [REF] depicts the histograms of search radii needed for [MATH]-NN with [MATH]B [MATH]-bit MLH and LSH codes.', '1307.2982-3-77-3': 'Interestingly, the mean of the search radii for MLH codes is [MATH] bits, while it is [MATH] for LSH.', '1307.2982-3-77-4': 'While the means are similar the variances are not; the standard deviations of the search radii for MLH and LSH are [MATH] and [MATH] respectively.', '1307.2982-3-77-5': 'The longer tail of the distribution of search radii for LSH plays an important role in the expected run-time.', '1307.2982-3-77-6': 'In fact, queries that require relatively large search radii tend to dominate the average query cost.', '1307.2982-3-78-0': 'It is also interesting to look at the multi-index hashing run-times as a function of [MATH], the number of binary codes in the database.', '1307.2982-3-78-1': 'To that end, Figure [REF], [REF], and [REF] depict run-times for linear scan and multi-index [MATH]NN search on 64, 128, and 256-bit codes on the AMD machine.', '1307.2982-3-78-2': 'The left two panels in each figure show different vertical scales (since the behavior of multi-index [MATH]NN and linear scan are hard to see at the same scale).', '1307.2982-3-78-3': 'The right-most panels show the same data on log-log axes.', '1307.2982-3-78-4': 'First, it is clear from these plots that multi-index hashing is much faster than linear scan for a wide range of dataset sizes and [MATH].', '1307.2982-3-78-5': 'Just as importantly, it is evident from the log-log plots that as we increase the database size, the speedup factors improve.', '1307.2982-3-78-6': 'The dashed lines on the log-log plots depict [MATH] (up to a scalar constant).', '1307.2982-3-78-7': 'The similar slope of multi-index hashing curves with the square root curves show that multi-index hashing exhibits sub-linear query time, even for the empirical, non-uniform distributions of codes.', '1307.2982-3-79-0': '## Direct lookups with a single hash table', '1307.2982-3-80-0': 'An alternative to linear scan and multi-index hashing is to hash the entire codes into a single hash table (SHT), and then use direct hashing with each query.', '1307.2982-3-80-1': 'As suggested in the introduction and Figure [REF], although this approach avoids the need for any candidate checking, it may require a prohibitive number of lookups.', '1307.2982-3-80-2': 'Nevertheless, for sufficiently small code lengths or search radii, it may be effective in practice.', '1307.2982-3-81-0': 'Given the complexity associated with efficiently implementing collision detection in large hash tables, we do not directly experiment with the single hash table approach.', '1307.2982-3-81-1': 'Instead, we consider the empirical number of lookups one would need, as compared to the number of items in the database.', '1307.2982-3-81-2': 'If the number of lookups is vastly greater than the size of the dataset one can readily conclude that linear scan is likely to be as fast or faster than direct indexing into a single hash table.', '1307.2982-3-82-0': 'Fortunately, the statistics of neighborhood sizes and required search radii for [MATH]NN tasks are available from the linear scan and multi-index hashing experiments reported above.', '1307.2982-3-82-1': "For a given query, one can use the [MATH] nearest neighbor's Hamming distance to compute the number of lookups from a single hash table that are required to find all of the query's [MATH] nearest neighbors.", '1307.2982-3-82-2': 'Summed over the set of queries, this provides an indication of the expected run-time.', '1307.2982-3-83-0': 'Figure [REF] shows the average number of lookups required for 1-NN and 1000-NN tasks on [MATH]- and [MATH]-bit codes (from LSH on SIFT) using a single hash table.', '1307.2982-3-83-1': 'They are plotted as a function of the size of the dataset, from [MATH] to [MATH] items.', '1307.2982-3-83-2': 'For comparison, the plots also show the number of database items, and the number of lookups that were needed for multi-index hashing.', '1307.2982-3-83-3': 'Note that Figure [REF] has logarithmic scales.', '1307.2982-3-84-0': 'It is evident that with a single hash table the number of lookups is almost always several orders of magnitude larger than the number of items in the dataset.', '1307.2982-3-84-1': 'And not surprisingly, this is also several orders of magnitude more lookups than required for multi-index hashing.', '1307.2982-3-84-2': 'Although the relative speed of a lookup operation compared to a Hamming distance comparison, as used in linear scan, depends on the implementation, there are a few important considerations.', '1307.2982-3-84-3': 'Linear scan has an exactly serial memory access pattern and so can make very efficient use of cache, whereas lookups in a hash table are inherently random.', '1307.2982-3-84-4': 'Furthermore, in any plausible implementation of a single hash table for 64 bit or longer codes, there will be some penalty for collision detection.', '1307.2982-3-85-0': 'As illustrated in Figure [REF], the only cases where a single hash table might potentially be more efficient than linear scan are with very small codes (64 bits or less), with a large dataset (1 billion items or more), and a small search distances (e.g., for 1-NN).', '1307.2982-3-85-1': 'In all other cases, linear scan requires orders of magnitude fewer operations.', '1307.2982-3-85-2': 'With any code length longer than 64 bits, a single hash table approach is completely infeasible to run, requiring upwards of 15 orders of magnitude more operations than linear scan for 128-bit codes.', '1307.2982-3-86-0': '## Substring Optimization', '1307.2982-3-87-0': 'The substring hash tables used above have been formed by simply dividing the full codes into disjoint and consecutive sequences of bits.', '1307.2982-3-87-1': 'For LSH and MLH, this is equivalent to randomly assigning bits to substrings.', '1307.2982-3-88-0': 'It natural to ask whether further gains in efficiency are possible by optimizing the assignment of bits to substrings.', '1307.2982-3-88-1': 'In particular, by careful substring optimization one may be able to maximize the discriminability of the different substrings.', '1307.2982-3-88-2': 'In other words, while the radius of substring searches and hence the number of lookups is determined by the desired search radius on the full codes, and will remain fixed, by optimizing the assignment of bits to substrings one might be able to reduce the number of candidates one needs to validate.', '1307.2982-3-89-0': 'To explore this idea we considered a simple method in which bits are assigned to substrings one at a time in a greedy fashion based on the correlation between the bits.', '1307.2982-3-89-1': 'We initialize the substrings greedily.', '1307.2982-3-89-2': 'A random bit is assigned to the first substring.', '1307.2982-3-89-3': 'Then, a bit is assigned to substring [MATH], which is maximally correlated with the bit assigned to substring [MATH].', '1307.2982-3-89-4': 'Next, we iterate over the substrings, and assign more bits to them, one at a time.', '1307.2982-3-89-5': 'An unused bit is assigned to substring [MATH], if the maximum correlation between that bit and other bits already assigned to substring [MATH] is minimal.', '1307.2982-3-89-6': 'This approach significantly decreases the correlation between bits within a single substring.', '1307.2982-3-89-7': 'This should make the distribution of codes within substrings buckets more uniform, and thereby lower the number of candidates within a given search radius.', '1307.2982-3-89-8': 'Arguably, a better approach consists of maximizing the entropy of the entries within each substring hash table, thereby making the distribution of substrings as uniform as possible.', '1307.2982-3-89-9': 'However, this entropic approach is left to future work.', '1307.2982-3-90-0': 'The results obtained with the correlation-based greedy algorithm show that optimizing substrings can provide overall run-time reductions on the order of [MATH] against consecutive substrings for some cases.', '1307.2982-3-90-1': 'Table [REF] displays the improvements achieved by optimizing substrings for different codes lengths and different values of [MATH].', '1307.2982-3-90-2': 'Clearly, as the code length increases, substring optimization has a bigger impact.', '1307.2982-3-90-3': 'Figure [REF] shows the run-time behavior of optimized substrings as a function of dataset size.', '1307.2982-3-91-0': '# Implementation details', '1307.2982-3-92-0': 'Our implementation of multi-index hashing is publicly available at [CITATION].', '1307.2982-3-92-1': 'Nevertheless, for the interested reader we describe some of the important details here.', '1307.2982-3-93-0': 'As explained above, the algorithm hinges on hash tables built on disjoint [MATH]-bit substrings of the binary codes.', '1307.2982-3-93-1': 'We use direct address tables for the substring hash tables because the substrings are usually short ([MATH]).', '1307.2982-3-93-2': 'Direct address tables explicitly allocate memory for [MATH] buckets and store all data points associated with each substring in its corresponding bucket.', '1307.2982-3-93-3': 'There is a one-to-one mapping between buckets and substrings, so no time is spent on collision detection.', '1307.2982-3-94-0': 'One could implement direct address tables with an array of [MATH] pointers, some of which may be null (for empty buckets).', '1307.2982-3-94-1': 'On a [MATH]-bit machine, pointers are [MATH] bytes long, so just storing an empty address table for [MATH] requires [MATH]GB (as done in [CITATION]).', '1307.2982-3-94-2': 'For greater efficiency here, we use sparse direct address tables by grouping buckets into subsets of [MATH] elements.', '1307.2982-3-94-3': 'For each bucket group, a [MATH]-bit binary vector encodes whether each bucket in the group is empty or not.', '1307.2982-3-94-4': 'Then, a single pointer per group is used to point to a single resizable array that stores the data points associated with that bucket group.', '1307.2982-3-94-5': 'Data points within each array are ordered by their bucket index.', '1307.2982-3-94-6': 'To facilitate fast access, for each non-empty bucket we store the index of the beginning and the end of the corresponding segment of the array.', '1307.2982-3-94-7': 'Compared to the direct address tables in [CITATION], for [MATH], and bucket groups of size [MATH], an empty address table requires only [MATH]GB.', '1307.2982-3-94-8': 'Also note that accessing elements in any bucket of the sparse address table has a worst case run-time of [MATH].', '1307.2982-3-95-0': 'Memory Requirements: We store one [MATH]-bit pointer for each bucket group, and a [MATH]-bit binary vector to encode whether buckets in a group are empty; this entails [MATH] bytes for an empty [MATH]-bit hash table ([MATH]), or [MATH]GB when [MATH].', '1307.2982-3-95-1': 'Bookkeeping for each resizable array entails [MATH]-bit integers.', '1307.2982-3-95-2': 'In our experiments, most bucket groups have at least one non-empty bucket.', '1307.2982-3-95-3': 'Taking this into account, the total storage for an [MATH]-bit address table becomes [MATH] bytes ([MATH]GB for [MATH]).', '1307.2982-3-96-0': 'For each non-empty bucket within a bucket group, we store a [MATH]-bit integer to indicate the index of the beginning of the segment of the resizable array corresponding to that bucket.', '1307.2982-3-96-1': 'The number of non-empty buckets is at most [MATH], where [MATH] is the number of hash tables, and [MATH] is the number of codes.', '1307.2982-3-96-2': 'Thus we need an extra [MATH] bytes.', '1307.2982-3-96-3': 'For each data point per hash table we store an ID to reference the full binary code; each ID is [MATH] bytes since [MATH] for our datasets.', '1307.2982-3-96-4': 'This entails [MATH] bytes.', '1307.2982-3-96-5': 'Finally, storing the full binary codes themselves requires [MATH] bytes, since [MATH].', '1307.2982-3-97-0': 'The total memory cost is [MATH] bytes.', '1307.2982-3-97-1': 'For [MATH], this cost is [MATH].', '1307.2982-3-97-2': 'For 1B [MATH]-bit codes, and [MATH] hash tables ([MATH] bits each), the cost is [MATH]GB.', '1307.2982-3-97-3': 'For [MATH]-bit and [MATH]-bit codes our implementation requires [MATH]GB and [MATH]GB.', '1307.2982-3-97-4': 'Note that the last two terms in the memory cost for storing IDs and codes are irreducible, but the first terms can be reduced in a more memory efficient implementation.', '1307.2982-3-98-0': 'With [MATH] (unfolded) substring hash tables of length [MATH] bits, and a [MATH]-bit address per bucket, the empty hash tables requires [MATH] bytes.', '1307.2982-3-98-1': 'For each non-empty bucket a resizable array is allocated to store the associated data points.', '1307.2982-3-98-2': 'Resizable arrays are preferred over linked lists since they are more cache friendly.', '1307.2982-3-98-3': 'To store the size of the resizable arrays, at most [MATH] bytes are needed as the number of non-empty buckets is bounded by [MATH].', '1307.2982-3-98-4': 'For each data point per hash table we store an ID to reference the full binary code; each ID is [MATH] bytes as the size of datasets [MATH]; this yields a total of [MATH] bytes.', '1307.2982-3-98-5': 'Lastly, storing the full binary codes requires [MATH] bytes, because [MATH].', '1307.2982-3-99-0': 'In total, the memory cost is [MATH] bytes (for [MATH], this is [MATH]).', '1307.2982-3-99-1': 'For one billion [MATH]-bit codes, and two chunks ([MATH] bits each), this cost is [MATH]GB.', '1307.2982-3-99-2': 'Note that the last two terms (for the IDs and binary codes) are irreducible, but the first term can be reduced in a memory efficient implementation at least by a factor of two.', '1307.2982-3-99-3': 'The first term heavily dominates the storage cost.', '1307.2982-3-99-4': 'If we search [MATH]-bit binary codes instead of [MATH]-bit ones, then [MATH] and the storage cost drops to [MATH]GB.', '1307.2982-3-99-5': 'For [MATH]-bit codes our implementation requires [MATH]GB of storage, and for [MATH]-bit codes, [MATH]GB.', '1307.2982-3-100-0': 'For example, one could cope with [MATH] bit indices by using just the first 30 bits, thereby ignoring the last [MATH] bits of each substring.', '1307.2982-3-100-1': 'In this way, each bucket becomes the union of 4 buckets, the indices of which differ by at most two bits.', '1307.2982-3-100-2': 'This reduces storage costs by a factor of 4 but increases the size of the candidate set slightly.', '1307.2982-3-100-3': '[David: more here - see numbers in rebuttal.]', '1307.2982-3-101-0': 'Duplicate Candidates: When retrieving candidates from the [MATH] substring hash tables, some codes will be found multiple times.', '1307.2982-3-101-1': 'To detect duplicates, and discard them, we allocate one bit-string with [MATH] bits.', '1307.2982-3-101-2': 'When a candidate is found we check the corresponding bit and discard the candidate if it is marked as a duplicate.', '1307.2982-3-101-3': 'Before each query we initialize the bit-string to zero.', '1307.2982-3-101-4': 'In practice this has negligible run-time.', '1307.2982-3-101-5': 'In theory clearing an [MATH]-bit vector requires [MATH], but there exist ways to initialize an [MATH] array in constant time.', '1307.2982-3-102-0': 'Hamming Distance: To compare a query with a candidate (for multi-index search or linear scan), we compute Hamming distance on the full [MATH]-bit codes, with one xor operation for every 64 bits, followed by a pop count to tally the ones.', '1307.2982-3-102-1': 'We used the built-in GCC function __builtin_popcount for this purpose.', '1307.2982-3-103-0': 'Number of Substrings: The number of substring hash tables we use is determined with a hold-out validation set of database entries.', '1307.2982-3-103-1': 'From that set we estimate the running time of the algorithm for different choices of [MATH] near [MATH], and select the [MATH] that yields the minimum run-time.', '1307.2982-3-103-2': 'As shown in Table [REF] this empirical value for [MATH] is usually the closest integer to [MATH].', '1307.2982-3-104-0': 'Translation Lookaside Buffer and Huge Pages: Modern processors have an on-chip cache that holds a lookup table of memory addresses, for mapping virtual addresses to physical addresses for each running process.', '1307.2982-3-104-1': 'Typically, memory is split into 4KB pages, and a process that allocates memory is given pages by the operating system.', '1307.2982-3-104-2': 'The Translation Lookaside Buffer (TLB) keeps track of these pages.', '1307.2982-3-104-3': 'For processes that have large memory footprints (tens of GB), the number of pages quickly overtakes the size of the TLB (typically about 1500 entries).', '1307.2982-3-104-4': 'For processes using random memory access this means that almost every memory access produces a TLB miss - the requested address is in a page not cached in the TLB, hence the TLB entry must be fetched from slow RAM before the requested page can be accessed.', '1307.2982-3-104-5': 'This slows down memory access, and causes volatility in run-times for memory-access intensive processes.', '1307.2982-3-105-0': 'To avoid this problem, we use the libhugetlbfs Linux library.', '1307.2982-3-105-1': 'This allows the operating system to allocate Huge Pages (2MB each) rather than 4KB pages.', '1307.2982-3-105-2': 'This reduces the number of pages and the frequency of TLB misses, which improves memory access speed, and reduces run-time volatility.', '1307.2982-3-105-3': 'The increase in speed of multi-index hashing compared to the results reported in [CITATION] is partly attributed to the use of libhugetlbfs.', '1307.2982-3-106-0': '# Conclusion', '1307.2982-3-107-0': 'This paper describes a new algorithm for exact nearest neighbor search on large-scale datasets of binary codes.', '1307.2982-3-107-1': 'The algorithm is a form of multi-index hashing that has provably sub-linear run-time behavior for uniformly distributed codes.', '1307.2982-3-107-2': 'It is storage efficient and easy to implement.', '1307.2982-3-107-3': 'We show empirical performance on datasets of binary codes obtained from [MATH] billion SIFT, and [MATH] million Gist features.', '1307.2982-3-107-4': 'With these datasets we find that, for [MATH]-bit and [MATH]-bit codes, our new multi-index hashing implementation is often more than two orders of magnitude faster than a linear scan baseline.', '1307.2982-3-108-0': 'While the basic algorithm is developed in this paper there are several interesting avenues for future research.', '1307.2982-3-108-1': 'For example our preliminary research shows that [MATH] is a good choice for the substring length, and it should be possible to formulate a sound mathematical basis for this choice.', '1307.2982-3-108-2': 'The assignment of bits to substrings was shown to be important above, however the algorithm used for this assignment is clearly suboptimal.', '1307.2982-3-108-3': 'It is also likely that different substring lengths might be useful for the different hash tables.', '1307.2982-3-109-0': 'Our theoretical analysis proves sub-linear run-time behavior of the multi-index hashing algorithm on uniformly distributed codes, when search radius is small.', '1307.2982-3-109-1': 'Our experiments demonstrate sub-linear run-time behavior of the algorithm on real datasets, while the binary code in our experiments are clearly not uniformly distributed.', '1307.2982-3-109-2': 'Bridging the gap between theoretical analysis and empirical findings for the proposed algorithm remains an open problem.', '1307.2982-3-109-3': 'In particular, we are interested in more realistic assumptions on the binary codes, which still allow for theoretical analysis of the algorithm.', '1307.2982-3-110-0': 'While the current paper concerns exact nearest-neighbor tasks, it would also be interesting to consider approximate methods based on the same multi-index hashing framework.', '1307.2982-3-110-1': 'Indeed there are several ways that one could find approximate rather than the exact nearest neighbors for a given query.', '1307.2982-3-110-2': 'For example, one could stop at a given radius of search, even though [MATH] items may not have been found.', '1307.2982-3-110-3': 'Alternatively, one might search until a fixed number of unique candidates have been found, even though all substring hash tables have not been inspected to the necessary radius, Such approximate algorithms have the potential for even greater efficiency, and would be the most natural methods to compare to most existing methods which are approximate, such as binary LSH.', '1307.2982-3-110-4': 'That said, such comparisons are more difficult than for exact methods since one must taken into account not only the storage and run-time costs, but also some measure of the cost of errors (usually in terms of recall and precision).', '1307.2982-3-111-0': 'Finally, recent results have shown that for many datasets in which the binary codes are the result of some form of vector quantization, an asymmetric Hamming distance is attractive [CITATION].', '1307.2982-3-111-1': 'In such methods, rather than converting the query into a binary code, one directly compares a real-valued query to the database of binary codes.', '1307.2982-3-111-2': 'The advantage is that the quantization noise entailed in converting the query to a binary string is avoided and one can more accurately using distances in the binary code space to approximate the desired distances in the feature space of the query.', '1307.2982-3-111-3': 'One simple way to do this is to use multi-index hashing and then only use an asymmetric distance when culling candidates.', '1307.2982-3-111-4': 'The potential for more interesting and effective methods is yet another promising avenue for future work.', '1307.2982-3-112-0': '# Acknowledgments'}

1603.00688

{'1603.00688-1-0-0': 'We present a numerical study of collisional transport in a tokamak pedestal in the presence of non-trace impurities, using the radially global [MATH] neoclassical solver Perfect [M. Landreman et al. 2014 Plasma Phys.', '1603.00688-1-0-1': 'Control.', '1603.00688-1-0-2': 'Fusion 56 045005].', '1603.00688-1-0-3': 'It is known that in a tokamak core with non-trace impurities present the radial impurity flux opposes the bulk ion flux to provide an ambipolar particle transport, with the electron transport being negligibly small.', '1603.00688-1-0-4': 'However, in a sharp density pedestal with sub-sonic ion flows the electron transport can be comparable to the ion and impurity flows.', '1603.00688-1-0-5': 'Furthermore, the neoclassical particle transport is not intrinsically ambipolar, and the non-ambipolarity of the fluxes extends outside the pedestal region by the radial coupling of the perturbations.', '1603.00688-1-0-6': 'The neoclassical momentum transport, which is finite in the presence of ion orbit-width scale profile variations, is significantly enhanced when impurities are present in non-trace quantities, even if the total parallel mass flow is dominated by the bulk ions.', '1603.00688-1-1-0': '# Introduction', '1603.00688-1-2-0': 'The global confinement in tokamaks is strongly correlated with the performance indicators of the edge transport barrier [CITATION].', '1603.00688-1-2-1': 'Accordingly, having a good pedestal performance in a fusion reactor is considered to be crucial.', '1603.00688-1-2-2': 'In the pedestal, turbulent transport is dramatically reduced by decorrelation of turbulence by strong sheared flows [CITATION], thus the relative role of the (otherwise negligible) collisional transport becomes more important.', '1603.00688-1-3-0': 'Fusion reactors will need to operate with carbon-free and low-erosion plasma facing components to achieve reasonable lifetimes and for nuclear safety.', '1603.00688-1-3-1': 'To gain more operational experience in the proposed metallic wall devices, originally carbon walled tokamak experiments, such as JET and ASDEX have transitioned to operation with "ITER-like walls" (with Be and W components) in recent years.', '1603.00688-1-3-2': 'With this transition a global confinement degradation has been observed [CITATION], especially on JET.', '1603.00688-1-3-3': 'This is a serious concern, since the data on which experimental scaling relations (and projections for ITER) are based has been mostly collected on carbon wall experiments.', '1603.00688-1-3-4': 'This degradation is found to be primarily caused by reduced pedestal performance [CITATION].', '1603.00688-1-3-5': 'Impurity injection appears to have a beneficial effect in certain cases [CITATION].', '1603.00688-1-3-6': 'Over and above the general importance of collisional impurity transport, these observations motivate consideration of the effects of non-trace impurities in the pedestal.', '1603.00688-1-4-0': 'Modeling of neoclassical transport in the pedestal is difficult, since the very sharp profile variations make gradient scale lengths comparable to the radial extent of the ion drift orbits[CITATION], which renders the usual radially local modeling inadequate for the problem.', '1603.00688-1-4-1': 'It has been demonstrated analytically that finite orbit width effects can significantly modify neoclassical transport phenomena [CITATION], producing modified ion heat transport, flows, bootstrap current, and even zonal flow response.', '1603.00688-1-4-2': 'These studies are limited to large aspect-ratio, circular cross-section plasmas and describe profile variations mostly through local plasma parameters, and point at the importance of a short scale global variation of flows [CITATION].', '1603.00688-1-4-3': 'In this paper we go beyond the large aspect-ratio limit, and present a radially global numerical study of neoclassical transport in the presence of non-trace impurities.', '1603.00688-1-5-0': 'There are different possible modeling options of various sophistication and difficulty.', '1603.00688-1-5-1': 'The simplest option is the computationally cheap and usually adopted local [MATH] formalism, which assumes small orbit width compared to profile length scales and neglects the radial coupling of the perturbations.', '1603.00688-1-5-2': 'On the other end, the global full-[MATH] approach includes both neoclassical and turbulent transport (these transport channels cannot be completely decoupled in a pedestal, in contrast to the local limit) and allows for strong deviations from local thermodynamic equilibrium.', '1603.00688-1-5-3': 'This approach requires nonlinear collision operators to fully live up to its promises[CITATION], and is currently numerically too expensive to be used for exploratory studies[CITATION].', '1603.00688-1-5-4': 'In this work, we use the global [MATH] formalism [CITATION].', '1603.00688-1-5-5': 'This is a specific generalization of the local [MATH] formalism which includes global effects, while still allowing the distribution function to be linearized around an appropriately chosen lowest order Maxwellian.', '1603.00688-1-5-6': 'This linearization, which assumes sub-sonic parallel flows, imposes limitations on the profiles that can be considered.', '1603.00688-1-5-7': 'Accordingly, we restrict ourselves to using suitable model profiles, but with experimentally identifiable features.', '1603.00688-1-6-0': 'In core plasmas the neoclassical perturbations of the ion distribution are only weakly affected by the electrons.', '1603.00688-1-6-1': 'We find that neoclassical particle transport in a pedestal with subsonic ion flows can be very different from that in the plasma core ([REF]a-c): The electron particle flux can be comparable to the ion particle flux even in the presence of non-trace impurities, and the collisional particle transport is not ambipolar in general.', '1603.00688-1-6-2': 'Consequently, it can happen that ions and impurities are transported in the same direction.', '1603.00688-1-6-3': 'In the presence of sharp profile variations the neoclassical momentum transport is nonzero, and we observe that its magnitude is notably affected by non-trace impurities ([REF]a).', '1603.00688-1-7-0': 'The remainder of this paper is organized as follows.', '1603.00688-1-7-1': 'In [REF], we describe the global [MATH] method implemented in Perfect, and how this affects our choice of model profiles.', '1603.00688-1-7-2': 'In [REF] we first present the neoclassical fluxes, flows and poloidal density variations for our baseline case, and then compare the results between plasmas with trace and non-trace impurity content.', '1603.00688-1-7-3': 'Finally, in [REF] we discuss our results and conclude.', '1603.00688-1-8-0': '# Methods', '1603.00688-1-9-0': 'In a tokamak core the plasma parameter profiles often exhibit sufficiently slow radial variations that the departure of the distribution function from a Maxwellian remains small, and the collisional dynamics can be described in terms of local plasma parameters.', '1603.00688-1-9-1': 'In this situation neoclassical transport can be calculated using the local [MATH] formalism that yields a linear system for the perturbed distribution.', '1603.00688-1-9-2': 'If the driving radial gradients are strong enough to generate substantial parallel particle flows, and poloidal variation in plasma parameters, the problem becomes nonlinear [CITATION].', '1603.00688-1-9-3': 'Furthermore, if all plasma parameters are allowed to vary over a radial width of a typical ion orbit, the transport becomes radially non-local.', '1603.00688-1-9-4': 'To study such general situations a global full-[MATH] simulation code with a nonlinear collision operator would be necessary.', '1603.00688-1-9-5': 'While a limited number of such simulation codes exist [CITATION], their computational expense make them unfit for our exploratory purposes.', '1603.00688-1-9-6': 'To keep the problem tractable, we will only consider situations when the distribution functions are not far from Maxwell-Boltzmann distributions, but finite orbit width effects are still important.', '1603.00688-1-9-7': 'For this purpose, we use the radially global, [MATH], Eulerian neoclassical solver Perfect.', '1603.00688-1-10-0': 'The fact that the distribution functions must be close to Maxwellians puts constraints on the profiles, as will be discussed in the following section.', '1603.00688-1-10-1': 'These constraints might not typically be satisfied in an experiment.', '1603.00688-1-10-2': 'Therefore, we do not attempt to base our exploratory modeling on specific experimental profiles, instead we use model profiles chosen specifically to satisfy the assumptions in Perfect, while they are supposed to be representative of experimental profiles in some respects.', '1603.00688-1-10-3': 'The specific profiles we use are presented in Appendix [REF].', '1603.00688-1-10-4': 'To explain the origin of the constraints, the next section contains a brief summary of the equations solved in Perfect (for a more detailed description of the code, we refer the reader tonumlandreman2014).', '1603.00688-1-11-0': 'In addition to the constraints outlined below, Perfect does not capture the geometry of an X-point or the open field line region.', '1603.00688-1-11-1': 'Orbit losses [CITATION] and an influx of neutral atoms [CITATION] are expected to become important very close to the separatrix.', '1603.00688-1-11-2': 'For this reason we expect that our results are representative only of the inner part of the pedestal.', '1603.00688-1-12-0': '## The global [MATH] problem solved by PERFECT', '1603.00688-1-13-0': 'Perfect solves for the non-adiabatic perturbed distribution function [EQUATION] where [MATH] is the distribution function, [MATH] is the charge, and [MATH] is the temperature of species [MATH], [MATH] is the perturbed potential, with the unperturbed electrostatic potential [MATH] taken to be a flux function, [MATH].', '1603.00688-1-13-1': 'The flux surface average is defined as [MATH], where [MATH] is a [MATH]-periodic angle-like poloidal coordinate and [MATH] is the magnetic field.', '1603.00688-1-13-2': 'The perturbation [MATH] is required to be small compared to the lowest order distribution function, which is a Maxwell-Boltzmann distribution [EQUATION] where the radial coordinate [MATH] is [MATH] times the poloidal magnetic flux, [MATH] is the mass, [MATH] is the total unperturbed energy, and [MATH] is the pseudo-density.', '1603.00688-1-13-3': 'Henceforth, [MATH] refers to a first-order correction to the quantity [MATH], which is allowed to vary poloidally, while all the lowest order quantities are assumed to be flux functions.', '1603.00688-1-13-4': 'The linearized equation that Perfect solves is [EQUATION] where [MATH], [MATH] with the velocity [MATH], the lowest order drift velocity [MATH] contains the lowest order [MATH] drift and magnetic drifts [MATH], [MATH] is the linearized Fokker-Planck operator, and [MATH] is a source term, which will be explained shortly.', '1603.00688-1-13-5': 'The partial derivatives are taken at fixed magnetic moment [MATH] and unperturbed total energy [MATH].', '1603.00688-1-14-0': 'Note that boundary conditions in [MATH] are needed to fully specify [MATH] by [REF], in contrast to the local equation.', '1603.00688-1-14-1': 'Since the local theory should apply sufficiently far from the pedestal, the result of local simulations - in which the [MATH] term is dropped from [REF] - are imposed as boundary conditions where particles enter the domain [CITATION].', '1603.00688-1-15-0': 'As inputs, Perfect requires zeroth order (flux function) densities [MATH], temperature [MATH] and potential [MATH].', '1603.00688-1-15-1': 'Given these equilibrium profiles, [MATH] is calculated from [REF], and appropriate velocity moments of [MATH] provide the neoclassical flows and fluxes.', '1603.00688-1-15-2': 'The fluxes will in general not be divergence free and thus incompatible with the time-independent equilibrium profiles.', '1603.00688-1-15-3': 'It may seem instructive to restore the time derivative in [REF], and solve a time-dependent problem, in a hope to reach a steady state equilibrium.', '1603.00688-1-15-4': 'However, only exceptional profiles would lead to a steady state solution.', '1603.00688-1-15-5': 'More generally, particles and energy would accumulate in (or leave) the simulation domain until the [MATH] approach breaks down.', '1603.00688-1-15-6': 'Instead, the approach adopted is to add spatially varying sources [MATH] so that the zeroth-order profiles become consistent; these sources are solved for in the code alongside [MATH].', '1603.00688-1-15-7': 'These sources can be thought of as representing the effects of non-neoclassical transport needed to make the profiles consistent, and should also be present in a real pedestal.', '1603.00688-1-16-0': 'To guarantee that [MATH], the driving gradients in the right-hand side of [REF] should remain small.', '1603.00688-1-16-1': 'From [EQUATION] where prime denotes the [MATH]-derivative, we see that the [MATH] and temperature gradients set the size of [MATH], and thus drive the deviations from a Maxwellian.', '1603.00688-1-16-2': 'Hence the density and the electrostatic potential may have sharp gradients as long as they produce a slowly varying [MATH].', '1603.00688-1-16-3': 'To quantify what we mean by a sharp gradient, we may balance the [MATH] and [MATH] terms in [REF] to find that [EQUATION] should be satisfied by [MATH] and [MATH]; that is, these quantities should have a small relative change as experienced by a particle during its radial drift excursion.', '1603.00688-1-16-4': 'Here [MATH] is the poloidal Larmor radius of the species, with the thermal speed [MATH], and the poloidal magnetic field [MATH].', '1603.00688-1-17-0': '## Model profiles and magnetic geometry', '1603.00688-1-18-0': 'Although we would like to use pedestal profiles which are representative of experiments in some aspects, we require [REF] to be satisfied in the simulations for all species ([MATH] for electrons, ions and impurities, respectively).', '1603.00688-1-18-1': 'We consider pedestals where [MATH] and [MATH] are allowed to vary on the [MATH] scale, while the [MATH] and [MATH] gradients of the bulk and impurity ion species are constrained by [REF].', '1603.00688-1-19-0': 'As a starting point we considered profiles from typical JET discharges (Figure 16 ofnum0029-5515-55-11-113031), and modified them as necessary to satisfy our orderings, and to reduce the need for heat sources in the simulation domain.', '1603.00688-1-19-1': 'The specific choices made in constructing the model profiles are discussed in Appendix [REF].', '1603.00688-1-19-2': 'The resulting input profiles for our baseline case as functions of the normalized poloidal flux [MATH] are shown in [REF]a-d.', '1603.00688-1-19-3': 'Here, we introduced [MATH], where [MATH] is the poloidal flux at the last closed flux surface (LCFS).', '1603.00688-1-19-4': 'Note, that the density profile of impurities is much steeper than that of the bulk ions to make [MATH] vary slowly.', '1603.00688-1-19-5': 'We consider a deuterium plasma and fully ionized nitrogen impurities ([MATH]) with a concentration [MATH] in the core.', '1603.00688-1-19-6': 'The radii marking the beginning and the end of the pedestal are indicated by dotted vertical lines in the figures.', '1603.00688-1-20-0': 'In the simulations we use a local Miller model geometry [CITATION] (and neglect the radial variation of [MATH] and [MATH] in the domain), with elongation [MATH], [MATH], triangularity [MATH], [MATH], [MATH], [MATH], and inverse aspect ratio [MATH].', '1603.00688-1-20-1': 'These parameters were taken fromnumbelli2008.', '1603.00688-1-21-0': 'The numerical resolution and convergence tests are detailed in Appendix [REF].', '1603.00688-1-22-0': '## Units', '1603.00688-1-23-0': 'An input quantity [MATH] is supplied to Perfect in a normalized, dimensionless form [MATH], with the normalizing, dimensional quantity [MATH].', '1603.00688-1-23-1': 'We choose [MATH] and [MATH] to be the major radius and magnetic field at the magnetic axis.', '1603.00688-1-23-2': 'Furthermore we choose the following convenient units [MATH], [MATH], [MATH] (deuterium mass), and [MATH] is the elementary charge.', '1603.00688-1-23-3': 'We define the reference speed and collision frequency as [MATH] and [EQUATION] where [MATH] denotes the vacuum permittivity and [MATH] is the Coulomb logarithm.', '1603.00688-1-23-4': 'The ordinary same-species collision frequency [MATH] is defined as [REF] but with [MATH] and [MATH] replaced by [MATH] and [MATH], from which we may define the collisionality as [MATH].', '1603.00688-1-24-0': 'To quantify when the local approximation fails, it is useful to define a few additional quantities: The normalized electric field [MATH] measures how much the [MATH] drift competes with parallel streaming in terms of poloidal particle motion.', '1603.00688-1-24-1': 'The normalized logarithmic derivative [MATH] measures the variation of quantity [MATH] along a typical drift orbit, where [MATH] and [MATH].', '1603.00688-1-24-2': 'Local neoclassical theory is valid only when [MATH], [MATH], [MATH], and [MATH] are all much smaller than unity in magnitude.', '1603.00688-1-24-3': 'The global [MATH] model also requires that [MATH] and [MATH], but [MATH] and [MATH] can be [MATH].', '1603.00688-1-24-4': 'These derived quantities together with [MATH] are shown for the baseline profiles in [REF]e-h.', '1603.00688-1-25-0': '# Local and global simulation results', '1603.00688-1-26-0': 'To study the differences between local and global neoclassical transport we performed a number of Perfect simulations with the profiles and magnetic geometry described in [REF].', '1603.00688-1-26-1': 'Before presenting the simulation results for our baseline set of profiles, we discuss the input quantities.', '1603.00688-1-27-0': 'We define the following normalized output quantities: sources, [MATH], with [MATH]; [MATH], with [MATH]; particle flux, [MATH]; momentum flux (divided by mass), [MATH]; heat flux, [MATH]; conductive heat fluxes, [MATH]; parallel flow velocity, [MATH]; parallel current, [MATH].', '1603.00688-1-27-1': 'The neoclassical flow coefficient [MATH] is defined so that it reduces to the flux function poloidal flow coefficient for a single ion species plasma in the local limit, [EQUATION] where [MATH] is the pressure.', '1603.00688-1-27-2': 'Furthermore, we define the non-adiabatic density perturbation [MATH], the total density perturbation [MATH], and normalized scalar fluxes [EQUATION] with [MATH]) representing [MATH], [MATH] or [MATH], [MATH] or [MATH], respectively), and [MATH].', '1603.00688-1-27-3': 'Note that the flux normalizations are species independent.', '1603.00688-1-28-0': '## Results for the baseline case', '1603.00688-1-29-0': 'The scalar fluxes of [REF] divided by [MATH] are shown in [REF].', '1603.00688-1-29-1': 'Throughout [REF] solid lines represent global simulation results, and dashed lines represent local ones.', '1603.00688-1-30-0': 'When the impurity strength [MATH] is order unity - as in our baseline - the magnitude of the particle transport of electrons is typically [MATH] smaller than that of the the impurities, where [MATH] denotes the effective ion charge.', '1603.00688-1-30-1': 'In this case it is common to neglect [MATH] and calculate the ion particle transport from ambipolarity [MATH], leading to opposing ion and impurity particle fluxes.', '1603.00688-1-30-2': 'The local simulations in the core region obey these expectations.', '1603.00688-1-30-3': 'Since [MATH] is small in our baseline, temperature screening does not dominate, thus the local fluxes obey [MATH] and [MATH].', '1603.00688-1-30-4': 'In the pedestal the parallel ion-electron friction can be sufficiently large to compete with the ion-impurity friction, due to the high electron flow speeds.', '1603.00688-1-30-5': 'Therefore [MATH] cannot be neglected anymore in the ambipolarity condition: the strong outward electron flux means that both [MATH] and [MATH] are positive simultaneously.', '1603.00688-1-30-6': 'Thus the outward local [MATH] is not a result of temperature screening.', '1603.00688-1-31-0': 'As expected from the small [MATH] and [MATH] values, seen in [REF]e and f, the electron local and global fluxes are practically the same.', '1603.00688-1-31-1': 'However, finite orbit width effects strongly affect the ion and impurity dynamics.', '1603.00688-1-31-2': 'In the pedestal [MATH] is increased compared to the local value, which causes [MATH] to change sign compared to its local value.', '1603.00688-1-31-3': 'It is worth noting that the deviation between local and global results is not localized to the pedestal region only.', '1603.00688-1-31-4': 'For instance, the global and local [MATH] deviate well below [MATH]; the ion particle flux changes sign at [MATH] while the local result is positive everywhere.', '1603.00688-1-31-5': 'This is a result of radial coupling of the perturbations across the flux surfaces.', '1603.00688-1-31-6': 'The global particle fluxes are not ambipolar as seen at [MATH], where [MATH] and [MATH] are both inward, and [MATH] is small, very close to its local value.', '1603.00688-1-31-7': 'Note that while in a local simulation the radial current and momentum flux should vanish, it does not need to be so in global simulations, as pointed out innumlandreman2014.', '1603.00688-1-32-0': 'For both ions and impurities we observe that the conductive heat flux can significantly differ from the local value.', '1603.00688-1-32-1': 'A reduction compared to the the local value - observed around the pedestal top - may be explained by the shift of the trapped region towards the tail of the distribution at [MATH].', '1603.00688-1-32-2': 'However, we also find regions where the heat fluxes increase from their local values.', '1603.00688-1-32-3': 'Just as for the particle fluxes, we see a reduction inside the pedestal top: [MATH]) reaches a minimum at [MATH]).', '1603.00688-1-33-0': 'The corresponding sources are presented in [REF].', '1603.00688-1-33-1': 'We use poloidally symmetric sources with speed dependencies [MATH] for particle and heat sources, respectively, where [MATH].', '1603.00688-1-33-2': 'We see that to some degree the main ion particle and heat sources qualitatively mirror each other, and tend towards zero outside the pedestal.', '1603.00688-1-33-3': 'Recall that we specifically choose [MATH] to reduce the need for [MATH], while [MATH] also remains small, because [MATH] tends to be smaller than [MATH].', '1603.00688-1-33-4': 'The tremendous drop in [MATH] in the pedestal leads to a sharp peak in [MATH], and again we see an opposing trend for [MATH], but for impurities the combined sources are positive.', '1603.00688-1-33-5': 'The electron sources are localized to the pedestal and comparable in size to those of the main ions.', '1603.00688-1-34-0': 'The neoclassical flow coefficients, [MATH], are presented in [REF].', '1603.00688-1-34-1': 'The local (dashed lines) [MATH] is positive as expected in the banana regime, and exhibits a slight variation as a response to the radial variation of [MATH].', '1603.00688-1-34-2': 'In the local case [MATH] is a flux function, while globally it varies from the inboard side (darker curves) to the outboard side (lighter curves).', '1603.00688-1-34-3': 'These poloidal variations in the flow appear together with poloidal density variations, as will be discussed shortly.', '1603.00688-1-34-4': 'It has been shown analytically [CITATION] that [MATH] is affected by finite orbit width effects.', '1603.00688-1-34-5': 'As seen from the analytical results, where [MATH] is kept but the radial coupling is neglected, [MATH] is expected to decrease or become more negative in both the banana and plateau regimes [CITATION].', '1603.00688-1-34-6': 'However, the global modification to [MATH] is not a function of local plasma parameters (such as [MATH]) only, but even in the semi-global treatment ofnumcatto13 it satisfies a radial differential equation (i.e. [MATH] cannot be neglected).', '1603.00688-1-34-7': 'This is why [MATH] can differ in sign and magnitude from the local value well outside the pedestal, and can be larger than its local value inside the pedestal.', '1603.00688-1-34-8': 'It is interesting to note that there is a difference between global and local [MATH] even for the electrons.', '1603.00688-1-35-0': 'The density perturbations are shown in [REF].', '1603.00688-1-35-1': 'We first consider only the non-adiabatic contribution to the perturbations, [MATH], shown in [REF]a-f, where the global (local) results are plotted in the a-c (d-f) panels.', '1603.00688-1-35-2': 'The local simulations predict purely up-down density asymmetry for all species, which is weaker for bulk ions and electrons than for impurities.', '1603.00688-1-35-3': 'The global results show a more complicated poloidal density variation for both ions and impurities.', '1603.00688-1-35-4': 'For ions we see an in-out asymmetry (i.e. excess density around [MATH]) at the pedestal top, which transforms into an out-in asymmetry in the pedestal, and reverses again further out (similarly to the single species simulations ofnumpusztaiRFNEO).', '1603.00688-1-35-5': 'For impurities the most important difference compared to the local results is the weak in-out (instead of strong up-down) asymmetry in the pedestal.', '1603.00688-1-35-6': 'The electron density perturbation mostly follows its local behavior, exhibiting a large increase in the up-down asymmetry in the pedestal.', '1603.00688-1-35-7': 'To understand the total density perturbation [MATH] in global simulations we note that the potential perturbation [MATH] follows mostly the non-adiabatic ion density perturbation.', '1603.00688-1-35-8': 'The total electron density perturbation is dominated by the adiabatic response of electrons, thus it is very similar to the ion density perturbation.', '1603.00688-1-35-9': 'The impurity density variations show a competition between adiabatic response - especially in the pedestal where it tries to oppose the ion density perturbation - and non-adiabatic response.', '1603.00688-1-35-10': 'The relative impurity density variation stays below [MATH] everywhere, showing that the assumption of the density being nearly a flux function is not violated.', '1603.00688-1-35-11': 'However, for sufficiently high [MATH], nonlinearity from poloidal asymmetries can arise [CITATION].', '1603.00688-1-36-0': 'Finally, we consider the parallel flows and the bootstrap current for our baseline case; these are plotted in [REF].', '1603.00688-1-36-1': 'The global result for the parallel electron flow follows the local one, and it strongly increases in magnitude inside the pedestal, where the strong [MATH] and diamagnetic rotation contribute with the same sign for this species.', '1603.00688-1-36-2': 'For ions the flows remain small inside the pedestal, as their profiles were chosen specifically so that their diamagnetic and [MATH] flows mostly cancel.', '1603.00688-1-36-3': 'The ion and impurity flows deviate from the local results, being larger in magnitude from the middle of the pedestal inward.', '1603.00688-1-36-4': 'This is partly due to the reduction in [MATH] compared to the local value observed in [REF].', '1603.00688-1-36-5': 'Since we have a weak ion temperature pedestal, the modifications of the ion flows are not sufficient to cause an appreciable deviation of the bootstrap current from the local result inside the pedestal, as seen in [REF]a.', '1603.00688-1-36-6': 'If anything, outside the pedestal there is a slight difference between the local and global results (barely visible on the scale accommodating the huge bootstrap peak in the pedestal), since the electron flow is relatively small in those regions, while the global effects on [MATH] extend outside the pedestal.', '1603.00688-1-37-0': '## Non-trace impurities', '1603.00688-1-38-0': 'We want to assess whether, and how, non-trace impurities can affect collisional transport in the pedestal.', '1603.00688-1-38-1': 'To this end, we performed simulations with profiles similar to our baseline, except that we varied the impurity concentration by constant factors.', '1603.00688-1-38-2': 'The simulations shown in this section have impurity concentrations such that the impurity strength in the core is either [MATH] (trace impurities; shown with thin lines in the figures) or [MATH] (non-trace; thick lines).', '1603.00688-1-38-3': 'Note that [MATH] drops rapidly across the pedestal, thus the impurities eventually become trace even if they have a high core concentration (this is necessary in the presence of an electron density pedestal if we restrict profile variations to obey the orderings of [REF] for all ion species).', '1603.00688-1-39-0': 'The a-c panels of [REF] compare the local and global particle fluxes with trace and non-trace impurity concentration.', '1603.00688-1-39-1': 'As usual, the local and global results for [MATH] are virtually the same.', '1603.00688-1-39-2': 'At higher impurity content the increase in [MATH] around [MATH] reflects the increased [MATH] gradient due to the rapidly varying impurity concentration (note that [MATH] is kept fixed in the impurity scan, thus the [MATH] profile changes).', '1603.00688-1-39-3': 'The local [MATH] behaves as expected from the ambipolarity condition: it increases with impurity content in the core to balance the inward [MATH], and its core behavior is dominated by following the outward electron flux.', '1603.00688-1-39-4': 'It is interesting to note that while above [MATH] the local curves for different impurity concentration collapsed onto each other because of the low [MATH], the difference in the global [MATH] survives much further out in the pedestal.', '1603.00688-1-40-0': 'The d-f panels of [REF] show the heat fluxes.', '1603.00688-1-40-1': 'In the core and close to the pedestal top the global results for [MATH] are lower than the local values.', '1603.00688-1-40-2': 'This reduction is somewhat stronger in the presence of impurities.', '1603.00688-1-40-3': 'The local results for [MATH] and [MATH] are even higher in the pedestal at high impurity concentration, than would be expected simply due to the linear increase with [MATH].', '1603.00688-1-40-4': 'This extra increase is reduced by the global effects so that the global results at different concentrations are closer to each other.', '1603.00688-1-41-0': 'The momentum flux values for the different species (shown in [REF]g-i) vanish in the local limit, as they should, but are finite when global effects are considered.', '1603.00688-1-41-1': 'Unlike [MATH] and [MATH], the global results for [MATH] are different to the local ones, i.e. they are also finite, although the electrons do not transport an appreciable amount of momentum, as [MATH].', '1603.00688-1-41-2': 'The total momentum flux, [MATH], shown in [REF]a, is mostly negative in the studied radial range, and is strongly increase by the presence of non-trace impurities.', '1603.00688-1-41-3': 'The ions are responsible for most of the momentum transport for both impurity concentrations, thus the increase in [MATH] is not due to the increase in [MATH], but the modifications in [MATH] in the presence of impurities.', '1603.00688-1-42-0': 'The radial current [MATH] ([REF]b) - that in isolation from other transport channels would lead to charge separation and the evolution of the radial electric field - is also significantly increased in magnitude by the presence of the impurities over most of the studied radial domain.', '1603.00688-1-42-1': 'Although we observe a finite radial neoclassical current, we do not attempt to self-consistently calculate the radial electric field.', '1603.00688-1-42-2': 'For the interpretation of the momentum fluxes and the radial current it is useful to note that conservation of particle number and parallel momentum (in steady state, for a radially constant [MATH], and with sources even in [MATH]) imply that [MATH] and [MATH].', '1603.00688-1-42-3': 'The latter property is apparent from a comparison of [REF]a and b.', '1603.00688-1-42-4': 'The former relation states that non-ambipolar fluxes require particle sources for which [MATH] is not satisfied locally (the factor [MATH] comes from the assumed velocity space structure of [MATH]).', '1603.00688-1-43-0': 'In radially local formalisms momentum transport is often decomposed into diffusive ([MATH], where [MATH] is the normalized parallel mass flow), conductive ([MATH]) and intrinsic (independent of [MATH]) terms.', '1603.00688-1-43-1': 'Such a decomposition is not possible in our global formalism since the parallel mass flow is a non-local function of the various plasma parameter profiles, and so is the momentum transport.', '1603.00688-1-43-2': 'Nevertheless, it is instructive to compare the radial profile of the total momentum transport [MATH] to the radial variation of the total mass flow [MATH] shown in [REF]c (both quantities are dominated by the main ion contributions).', '1603.00688-1-43-3': 'By increasing the impurity content, [MATH] increases in the core region.', '1603.00688-1-43-4': 'This is caused by the higher [MATH] at higher [MATH] not being compensated by the slight reduction in [MATH] (shown in [REF]).', '1603.00688-1-43-5': 'As the impurity concentration drops radially across the pedestal, the effect from the reduction in the parallel ion flow becomes dominant.', '1603.00688-1-43-6': 'The radial drop in the global results for [MATH] across the pedestal is mostly due to the density variation in the pedestal.', '1603.00688-1-43-7': 'If the transport was local and purely diffusive this non-monotonic behavior of [MATH] would be accompanied by a sign change in the momentum transport.', '1603.00688-1-43-8': 'It is also interesting to note that at the point where we see the greatest relative increase in [MATH] between the different simulations ([MATH]), the global [MATH] becomes lower for higher core impurity concentration.', '1603.00688-1-44-0': 'The radial current is affected by the particle sources.', '1603.00688-1-44-1': 'The ion particle source profile shown in [REF]a is only weakly affected by the presence of non-trace impurities.', '1603.00688-1-44-2': 'In the meantime, the increase in impurity sources are approximately proportional to the increase in their concentration: the normalized particle source [MATH] is approximately the same in the two simulations, considering that the impurity content changes by a factor [MATH].', '1603.00688-1-45-0': 'Finally we would like to assess how much our profile choices affect the observation of increased momentum flux in the presence of non-trace impurities.', '1603.00688-1-45-1': 'One unnatural feature of our impurity density profile is its extreme steepness.', '1603.00688-1-45-2': 'Allowing [MATH] have a radial drop across the pedestal - within what is allowed by [REF] - to reduce the inward radial electric field, together with allowing [MATH] to have an increase in the same region, leads to a less sharp impurity pedestal.', '1603.00688-1-45-3': 'To further reduce the impurity density gradient in the pedestal we consider a fully ionized beryllium impurity ([MATH]).', '1603.00688-1-45-4': 'These changes make our pedestal less deeply sub-sonic, as seen from the increased mass flow in [REF]c, which shows the results of the modified input profiles.', '1603.00688-1-45-5': 'From [REF]a we can conclude that impurities also significantly increase the magnitude of the neoclassical momentum flux in the pedestal for more natural impurity profiles, when the inputs push the limitations imposed by the required orderings.', '1603.00688-1-46-0': '# Discussion and conclusions', '1603.00688-1-47-0': 'We have studied the differences in the collisional transport between radially local and global formalisms using the global [MATH] neoclassical solver Perfect, with a special emphasis on the effects of non-trace impurities.', '1603.00688-1-47-1': 'If the impurity density profile is arbitrary, impurities are likely to develop sonic flows and strong poloidal asymmetries in the pedestal.', '1603.00688-1-47-2': 'We use model profiles specifically chosen to guarantee that impurity flows remain sub-sonic, so that the assumptions of the [MATH] formalism are valid.', '1603.00688-1-47-3': 'In this way we can gain some insights into the effect of non-trace impurities without the need for a non-linear collision operator.', '1603.00688-1-48-0': 'In an impure plasma the electron particle transport is usually negligible, and the ion and impurity fluxes oppose each other to maintain ambipolarity.', '1603.00688-1-48-1': 'However, since in the pedestal the magnitude of the parallel electron flow can be much larger than the parallel ion and impurity flows, the friction of the various ion species on electrons can become non-negligible.', '1603.00688-1-48-2': 'Consequently, a substantial neoclassical electron particle transport can arise, which competes with the radial transport of other species.', '1603.00688-1-48-3': 'In particular, the ion and impurity fluxes can have the same sign.', '1603.00688-1-48-4': 'In the presence of strong radial profile variations on the [MATH] scale, the ambipolarity of fluxes is violated, as reported in previous studies [CITATION].', '1603.00688-1-48-5': 'We emphasize that the differences in neoclassical flows and fluxes between the local and global theory are not restricted to the pedestal region only but die off within a distance comparable with the ion orbit width.', '1603.00688-1-48-6': 'In our baseline from the outer core region to the pedestal top we observe that both the ion and impurity fluxes are inward due to global effects (see [REF]a-c).', '1603.00688-1-48-7': 'Inside the pedestal, local simulations predict both the impurity and the ion fluxes to be outward, due to the large outward electron flux, but in global simulations the impurity flux reverses to be inward.', '1603.00688-1-49-0': 'For our model profiles, chosen to give small parallel ion flows, the bootstrap current remains almost completely unaffected, while this need not be so if the ion temperature varies more rapidly than allowed by our orderings.', '1603.00688-1-49-1': 'Indeed, the neoclassical parallel flow coefficient [MATH] of ions and impurities is significantly affected by global effects.', '1603.00688-1-49-2': 'Due to the radial coupling even the sign of [MATH] is different from the local result well inside of the pedestal top.', '1603.00688-1-50-0': 'We observe that in the presence of global effects the poloidal variation of the density perturbation is not restricted to an up-down asymmetry, it can be more significant, and can develop rapid radial variations (in accordance with previous numerical results [CITATION]).', '1603.00688-1-50-1': 'The poloidal asymmetries observed here arise in the presence of finite inverse aspect ratio and finite orbit width effects, and are not closely related to those predicted by the analytical theories in [CITATION], since those require significant poloidal variations in the ion-impurity friction.', '1603.00688-1-50-2': 'The adiabatic response to the electrostatic perturbation generated by the ions competes with non-adiabatic contributions in determining the poloidal asymmetries developed by the impurities.', '1603.00688-1-50-3': 'For the moderate impurity charge considered here ([MATH]) the relative poloidal variation of the impurity density is still small so that the perturbative treatment remains valid.', '1603.00688-1-50-4': 'At sufficiently high [MATH] the relative poloidal variation of the impurity density is expected to become order unity and then - if the impurities are non-trace - nonlinear effects would start to play a role.', '1603.00688-1-50-5': 'The numerical investigation of that situation is left for a future study.', '1603.00688-1-51-0': 'In the pedestal, the neoclassical radial current does not vanish in general, and the momentum transport remains finite - in contrast to the local theory.', '1603.00688-1-51-1': 'In the vicinity of the pedestal the total neoclassical momentum transport is found to be negative in the studied case, which happens if the charge sources (resulting from the particle sources needed to sustain the pedestal) are mostly positive in that region.', '1603.00688-1-51-2': 'The radial variation of the parallel mass flow is non-monotonic with a sharply decreasing feature where the density drops in the pedestal.', '1603.00688-1-51-3': 'This non-monotonicity is not reflected in the radial momentum transport, which is now a non-local function of the mass flow.', '1603.00688-1-51-4': 'A simple decomposition of the momentum transport into diffusive, pinch and intrinsic terms is not possible in the global picture.', '1603.00688-1-51-5': 'Note that the same is also true for all the radial fluxes.', '1603.00688-1-52-0': 'We observe a strong effect of impurities on the total momentum transport, the magnitude of which increases significantly in the presence of non-trace impurities (see [REF]a).', '1603.00688-1-52-1': 'While it is not possible to disentangle the exact cause of this, it may be due to the sharp radial variation of the parallel impurity mass flow.', '1603.00688-1-52-2': 'The impurities represent only a minor fraction of the total mass flow, but their collisional coupling to the main ions is significant at an impurity strength of order unity.', '1603.00688-1-52-3': 'Pushing the limitations of our orderings we reduced the sharpness of the impurity density profile, in an attempt to demonstrate the robustness of the impact of impurities on neoclassical momentum transport.', '1603.00688-1-52-4': 'Although the impurity profile is chosen to be very specific, strong effects may occur when the impurities have more general density variations, especially because they then have much stronger relative flow speeds compared to the ions.', '1603.00688-1-53-0': 'Impurity seeding in tokamaks operating with ITER-like metallic walls has been experimentally found to have beneficial effects on the pedestal performance.', '1603.00688-1-53-1': 'It is then natural to raise the question of whether our results are consistent with this observation.', '1603.00688-1-53-2': 'Without taking into account other non-intrinsically ambipolar processes and turbulent transport it is not possible to evolve the profiles towards a steady state.', '1603.00688-1-53-3': 'However, we may speculate about possible consequences of the increased momentum transport in the presence of impurities.', '1603.00688-1-53-4': 'In steady state the neoclassical radial current and momentum transport should be balanced by opposing contributions of turbulent and other origin.', '1603.00688-1-53-5': 'An increased neoclassical momentum transport in the presence of impurities requires these contributions to increase as well.', '1603.00688-1-53-6': 'Sufficiently far from the open field line region the turbulent transport can dominate these opposing contributions.', '1603.00688-1-53-7': 'Impurities tend to reduce the turbulence level by dilution [CITATION].', '1603.00688-1-53-8': 'If the non-diffusive turbulent momentum transport is to be increased in spite of dilution effects, stronger deviations of the non-fluctuating distribution from a Maxwellian [CITATION] and stronger profile variations [CITATION] may be necessary; which could require a steepening of the pedestal to reach a new steady state.', '1603.00688-1-53-9': 'Whether this is indeed the case and, if so, its role in the observed confinement improvement in impurity seeded discharges, remains an open question and should be the basis of future investigation.', '1603.00688-1-54-0': 'The authors are grateful for J. Omotani, T. Fulop and S. Newton for fruitful discussions and instructive comments on the paper.', '1603.00688-1-54-1': 'IP and SB were supported by the International Career Grant of Vetenskapsraadet (Dnr.', '1603.00688-1-54-2': '330-2014-6313), and ML was supported by the U.S. DoE under award numbers DEFG0293ER54197 and DEFC0208ER54964.', '1603.00688-1-54-3': 'The simulations used computational resources of Hebbe at C3SE (project nr.', '1603.00688-1-54-4': 'C3SE2016-1-10).', '1603.00688-1-55-0': '# The construction of model pedestal profiles', '1603.00688-1-56-0': 'Here, we describe a method to construct appropriate densities given [MATH], [MATH], [MATH] profiles satisfying [REF], and an arbitrary [MATH].', '1603.00688-1-56-1': 'The orderings for the electrons are not a concern, since [MATH] is much smaller than the radial scale length of any profile in an experiment.', '1603.00688-1-57-0': 'The density profiles and potential discussed in this section are the inputs to Perfect and are thus, strictly speaking, only the zeroth order contributions to these quantities, assumed to be flux functions.', '1603.00688-1-58-0': 'Once we have specified both [MATH] and [MATH] (to be detailed shortly), the relation [MATH] gives the potential [EQUATION]', '1603.00688-1-58-1': 'A strong electric field from [MATH] compensates for the potentially large ion pressure drop to make the variation of [MATH] small (the situation of "electrostatic ion confinement," which is borne out in experimental results [CITATION]).', '1603.00688-1-58-2': 'In the radial force balance the ion pressure drop could in principle be balanced by a sonic ion flow, however we consider sub-sonic ion flows.', '1603.00688-1-58-3': 'Note that by allowing sonic ion flows, the relative flow speed of two ion species would in general also be sonic, which would severely complicate the treatment of collisions.', '1603.00688-1-58-4': 'Such a scenario is currently not supported by Perfect.', '1603.00688-1-58-5': 'We emphasize that [MATH] and [MATH] are fundamental inputs to the code, thus the relation [REF] is not an attempt to self-consistently calculate the potential.', '1603.00688-1-59-0': 'Then, given [MATH] from ([REF]) and an [MATH], we obtain the impurity density [EQUATION]', '1603.00688-1-59-1': 'Note that ([REF]) leads to impurity density profiles with a typical logarithmic density gradient in the pedestal [MATH] times larger than that of the main ions; this is unavoidable if [REF] is to be satisfied for all ion species ([MATH]).', '1603.00688-1-59-2': 'Finally, the electron density is obtained by demanding quasi-neutrality [EQUATION]', '1603.00688-1-59-3': 'The resulting [MATH] satisfies [REF] by virtue of the electron gyroradius being small.', '1603.00688-1-60-0': 'Although we have constraints and relations between profiles, we still have a large degree of freedom in specifying them.', '1603.00688-1-60-1': 'Here we discuss some specific choices we made for the profile set used as a baseline for the simulations.', '1603.00688-1-61-0': 'Since we use the local solution for the boundary condition, we also need to make sure that the assumptions of the local theory are satisfied at the boundary.', '1603.00688-1-61-1': 'This means that the sharp density or potential variations should be limited to the middle of the radial domain, sufficiently far from the boundaries.', '1603.00688-1-61-2': 'For weak electric fields the local and global results should agree, so we choose the [MATH] profile so that the potential calculated from [REF] is completely flat at the boundaries.', '1603.00688-1-61-3': 'We do this by letting [EQUATION] in the vicinity of the inner boundary, namely in the core up to the pedestal top where [MATH] (and so [MATH]) should be slowly varying.', '1603.00688-1-61-4': 'Thus [MATH] in this region, independent of the [MATH] profile.', '1603.00688-1-61-5': 'To achieve a flat potential in the vicinity of the outer boundary, we let [EQUATION] in that region, where [MATH] is a constant which fixes the value of [MATH].', '1603.00688-1-61-6': 'To remove the ambiguity of [MATH] far from the boundaries in a way that makes it a simple and smooth function, we linearly extrapolate [MATH] given by [REF] from the core region up to the bottom of the pedestal (where the sharp feature in [MATH] ends) and match it to the other expression [REF], choosing [MATH] from [REF] at the matching point.', '1603.00688-1-62-0': 'The [MATH] profile is chosen to be a linear function of [MATH] over the whole domain, with a logarithmic gradient matching that of [MATH] at the left boundary.', '1603.00688-1-63-0': 'Since the global [MATH] ordering does not allow an ion temperature pedestal, we consider [MATH] profiles with a gradient across the pedestal (and further out) equal to that of the electron temperature gradient in the core.', '1603.00688-1-63-1': 'Due to the density drop in the pedestal and the decreasing temperature profile, the ion heat flux is bound to be vastly different at the two boundaries, requiring large heat sources in the domain, except if a variation in [MATH] balances them.', '1603.00688-1-63-2': 'This can be avoided by setting a proxy for the heat fluxes, [MATH], to be equal at the boundaries.', '1603.00688-1-63-3': 'Thus, we artificially reduce [MATH] in the core to remove the need for sources close to the boundaries.', '1603.00688-1-64-0': 'Experimental density and temperature profiles can drop orders of magnitude across the pedestal, and a significant part of these variations occur in the open field line region.', '1603.00688-1-64-1': 'As an additional consideration, if we let the bulk densities and temperatures drop across the simulation domain as much as in a real pedestal, it would lead to difficulties related to large logarithmic gradients, and huge changes in collision frequency.', '1603.00688-1-64-2': 'Since the code does not capture the physics in the open field line region (where the pedestal foot would be in an experiment), the region outside the middle of the pedestal does not carry too much physical relevance and can be considered as a numerical buffer zone.', '1603.00688-1-64-3': 'To avoid the above mentioned complications we reduce the gradients starting from a point where crossing the separatrix would be expected in an experiment.', '1603.00688-1-64-4': 'We thus arbitrarily pick this reduced [MATH] to be [MATH] of its pedestal value, with [MATH] equal to its core value.', '1603.00688-1-65-0': 'Motivated by typical JET discharges (Figure 16 of 0029-5515-55-11-113031), for our baseline case we choose the [MATH] and [MATH] pedestal widths to be [MATH], with the values [MATH] and [MATH] at the pedestal top.', '1603.00688-1-65-1': 'These correspond to typical logarithmic gradients of about [MATH] and [MATH], where [MATH] is the minor radius defined as half of the width of the flux surface at the elevation of its centroid, and [MATH] at the last closed flux surface.', '1603.00688-1-65-2': 'The [MATH] and [MATH] profiles are generated by Bezier curve interpolation between three regions with linear profile variation: a core, a pedestal, and an outer "buffer" region.', '1603.00688-1-65-3': '(Using this type of smoothing ensures that the gradients transition smoothly and monotonically.)', '1603.00688-1-66-0': 'To transform the profiles from [MATH] to [MATH] space, [MATH] is needed.', '1603.00688-1-66-1': 'This we obtain from [MATH] where [MATH] is the toroidal magnetic flux which we calculate assuming simple elongated flux surfaces.', '1603.00688-1-66-2': 'Since we do not intend to model a specific experiment, we take model profiles for the safety factor [MATH] and the elongation [MATH], shown in [REF].', '1603.00688-1-66-3': 'The [MATH] was chosen as [MATH].', '1603.00688-1-66-4': 'For the on-axis toroidal magnetic fields we take [MATH] and neglect [MATH] corrections together with higher order shaping effects to get [MATH], where [MATH] denotes the major radius.', '1603.00688-1-66-5': 'The resulting [MATH] is taken to be constant across the entire pedestal.', '1603.00688-1-67-0': '# Numerical resolution', '1603.00688-1-68-0': 'The simulations used [MATH] radial grid points and cover a domain of [MATH] (the whole [MATH] domain is not shown in the figures).', '1603.00688-1-68-1': 'The number of poloidal grid points is [MATH].', '1603.00688-1-68-2': 'The number of expansion polynomials in the pitch angle cosine [MATH] is [MATH] for the distribution function and [MATH] for the Rosenbluth potentials (RP).', '1603.00688-1-68-3': 'The number of speed grid points is [MATH] and [MATH] for the RP.', '1603.00688-1-69-0': 'To demonstrate the degree of convergence, in [REF] we present the particle sources for the baseline simulation for the above mentioned resolution (red curve), together with three other cases, where we increase [MATH] to [MATH] (violet), [MATH] to [MATH] (cyan), and [MATH] to [MATH] (yellow), with all other parameters kept fixed.', '1603.00688-1-69-1': 'These three are the resolution parameters to which the accuracy of the solution is most sensitive.', '1603.00688-1-69-2': 'The results are almost identical except for slight differences near a few sharp features.', '1603.00688-1-70-0': 'To quantify the error, we define [MATH].', '1603.00688-1-70-1': 'Taking the [MATH] simulation as the reference [MATH], we find that the quantity with the highest error is the electron particle source, with [MATH].', '1603.00688-1-70-2': 'For other quantities, such as particle flows and radial fluxes, we observe errors below [MATH], thus [REF] presents the most stringent test for numerical convergence.', '1603.00688-1-70-3': 'The same convergence test was performed for all the simulations, with resulting errors (including those of the sources) all being below [MATH].'}

{'1603.00688-2-0-0': 'We present a numerical study of collisional transport in a tokamak pedestal in the presence of non-trace impurities, using the radially global [MATH] neoclassical solver Perfect [M. Landreman et al. 2014 Plasma Phys.', '1603.00688-2-0-1': 'Control.', '1603.00688-2-0-2': 'Fusion 56 045005].', '1603.00688-2-0-3': 'It is known that in a tokamak core with non-trace impurities present the radial impurity flux opposes the bulk ion flux to provide an ambipolar particle transport, with the electron transport being negligibly small.', '1603.00688-2-0-4': 'However, in a sharp density pedestal with sub-sonic ion flows the electron transport can be comparable to the ion and impurity flows.', '1603.00688-2-0-5': 'Furthermore, the neoclassical particle transport is not intrinsically ambipolar, and the non-ambipolarity of the fluxes extends outside the pedestal region by the radial coupling of the perturbations.', '1603.00688-2-0-6': 'The neoclassical momentum transport, which is finite in the presence of ion orbit-width scale profile variations, is significantly enhanced when impurities are present in non-trace quantities, even if the total parallel mass flow is dominated by the bulk ions.', '1603.00688-2-1-0': '# Introduction', '1603.00688-2-2-0': 'The global confinement in tokamaks is strongly correlated with the performance indicators of the edge transport barrier [CITATION].', '1603.00688-2-2-1': 'Accordingly, having a good pedestal performance in a fusion reactor is considered to be crucial.', '1603.00688-2-2-2': 'In the pedestal, turbulent transport is dramatically reduced by decorrelation of turbulence by strong sheared flows [CITATION], thus the relative role of the (otherwise negligible) collisional transport becomes more important.', '1603.00688-2-3-0': 'Fusion reactors will need to operate with carbon-free and low-erosion plasma facing components to achieve reasonable lifetimes and for nuclear safety.', '1603.00688-2-3-1': 'To gain more operational experience in the proposed metallic wall devices, originally carbon walled tokamak experiments, such as JET and ASDEX have transitioned to operation with "ITER-like walls" (with Be and W components) in recent years.', '1603.00688-2-3-2': 'With this transition a global confinement degradation has been observed [CITATION], especially on JET.', '1603.00688-2-3-3': 'This is a serious concern, since the data on which experimental scaling relations (and projections for ITER) are based has been mostly collected on carbon wall experiments.', '1603.00688-2-3-4': 'This degradation is found to be primarily caused by reduced pedestal performance [CITATION].', '1603.00688-2-3-5': 'Impurity injection appears to have a beneficial effect in certain cases [CITATION].', '1603.00688-2-3-6': 'Over and above the general importance of collisional impurity transport, these observations motivate consideration of the effects of non-trace impurities in the pedestal.', '1603.00688-2-4-0': 'Modeling of neoclassical transport in the pedestal is difficult, since the very sharp profile variations make gradient scale lengths comparable to the radial extent of the ion drift orbits[CITATION], which renders the usual radially local modeling inadequate for the problem.', '1603.00688-2-4-1': 'It has been demonstrated analytically that finite orbit width effects can significantly modify neoclassical transport phenomena [CITATION], producing modified ion heat transport, flows, bootstrap current, and even zonal flow response.', '1603.00688-2-4-2': 'These studies are limited to large aspect-ratio, circular cross-section plasmas and describe profile variations mostly through local plasma parameters, and point at the importance of a short scale global variation of flows [CITATION].', '1603.00688-2-4-3': 'In this paper we go beyond the large aspect-ratio limit, and present a radially global numerical study of neoclassical transport in the presence of non-trace impurities.', '1603.00688-2-5-0': 'There are different possible modeling options of various sophistication and difficulty.', '1603.00688-2-5-1': 'The simplest option is the computationally cheap and usually adopted local [MATH] formalism, which assumes small orbit width compared to profile length scales and neglects the radial coupling of the perturbations.', '1603.00688-2-5-2': 'On the other end, the global full-[MATH] approach includes both neoclassical and turbulent transport (these transport channels cannot be completely decoupled in a pedestal, in contrast to the local limit) and allows for strong deviations from local thermodynamic equilibrium.', '1603.00688-2-5-3': 'This approach requires nonlinear collision operators to fully live up to its promises[CITATION], and is currently numerically too expensive to be used for exploratory studies[CITATION].', '1603.00688-2-5-4': 'In this work, we use the global [MATH] formalism [CITATION].', '1603.00688-2-5-5': 'This is a specific generalization of the local [MATH] formalism which includes global effects, while still allowing the distribution function to be linearized around an appropriately chosen lowest order Maxwellian.', '1603.00688-2-5-6': 'This linearization, which assumes sub-sonic parallel flows, imposes limitations on the profiles that can be considered.', '1603.00688-2-5-7': 'Accordingly, we restrict ourselves to using suitable model profiles, but with experimentally identifiable features.', '1603.00688-2-6-0': 'In core plasmas the neoclassical perturbations of the ion distribution are only weakly affected by the electrons.', '1603.00688-2-6-1': 'We find that neoclassical particle transport in a pedestal with subsonic ion flows can be very different from that in the plasma core ([REF]a-c): The electron particle flux can be comparable to the ion particle flux even in the presence of non-trace impurities, and the collisional particle transport is not ambipolar in general.', '1603.00688-2-6-2': 'Consequently, it can happen that ions and impurities are transported in the same direction.', '1603.00688-2-6-3': 'In the presence of sharp profile variations the neoclassical momentum transport is nonzero, and we observe that its magnitude is notably affected by non-trace impurities ([REF]a).', '1603.00688-2-7-0': 'The remainder of this paper is organized as follows.', '1603.00688-2-7-1': 'In [REF], we describe the global [MATH] method implemented in Perfect, and how this affects our choice of model profiles.', '1603.00688-2-7-2': 'In [REF] we first present the neoclassical fluxes, flows and poloidal density variations for our baseline case, and then compare the results between plasmas with trace and non-trace impurity content.', '1603.00688-2-7-3': 'Finally, in [REF] we discuss our results and conclude.', '1603.00688-2-8-0': '# Methods', '1603.00688-2-9-0': 'In a tokamak core the plasma parameter profiles often exhibit sufficiently slow radial variations that the departure of the distribution function from a Maxwellian remains small, and the collisional dynamics can be described in terms of local plasma parameters.', '1603.00688-2-9-1': 'In this situation neoclassical transport can be calculated using the local [MATH] formalism that yields a linear system for the perturbed distribution.', '1603.00688-2-9-2': 'If the driving radial gradients are strong enough to generate substantial parallel particle flows, and poloidal variation in plasma parameters, the problem becomes nonlinear [CITATION].', '1603.00688-2-9-3': 'Furthermore, if all plasma parameters are allowed to vary over a radial width of a typical ion orbit, the transport becomes radially non-local.', '1603.00688-2-9-4': 'To study such general situations a global full-[MATH] simulation code with a nonlinear collision operator would be necessary.', '1603.00688-2-9-5': 'While a limited number of such simulation codes exist [CITATION], their computational expense make them unfit for our exploratory purposes.', '1603.00688-2-9-6': 'To keep the problem tractable, we will only consider situations when the distribution functions are not far from Maxwell-Boltzmann distributions, but finite orbit width effects are still important.', '1603.00688-2-9-7': 'For this purpose, we use the radially global, [MATH], Eulerian neoclassical solver Perfect.', '1603.00688-2-10-0': 'The fact that the distribution functions must be close to Maxwellians puts constraints on the profiles, as will be discussed in the following section.', '1603.00688-2-10-1': 'These constraints might not typically be satisfied in an experiment.', '1603.00688-2-10-2': 'Therefore, we do not attempt to base our exploratory modeling on specific experimental profiles, instead we use model profiles chosen specifically to satisfy the assumptions in Perfect, while they are supposed to be representative of experimental profiles in some respects.', '1603.00688-2-10-3': 'The specific profiles we use are presented in Appendix [REF].', '1603.00688-2-10-4': 'To explain the origin of the constraints, the next section contains a brief summary of the equations solved in Perfect (for a more detailed description of the code, we refer the reader tonumlandreman2014).', '1603.00688-2-11-0': 'In addition to the constraints outlined below, Perfect does not capture the geometry of an X-point or the open field line region.', '1603.00688-2-11-1': 'Orbit losses [CITATION] and an influx of neutral atoms [CITATION] are expected to become important very close to the separatrix.', '1603.00688-2-11-2': 'For this reason we expect that our results are representative only of the inner part of the pedestal.', '1603.00688-2-12-0': '## The global [MATH] problem solved by PERFECT', '1603.00688-2-13-0': 'Perfect solves for the non-adiabatic perturbed distribution function [EQUATION] where [MATH] is the distribution function, [MATH] is the charge, and [MATH] is the temperature of species [MATH], [MATH] is the perturbed potential, with the unperturbed electrostatic potential [MATH] taken to be a flux function, [MATH].', '1603.00688-2-13-1': 'The flux surface average is defined as [MATH], where [MATH] is a [MATH]-periodic angle-like poloidal coordinate and [MATH] is the magnetic field.', '1603.00688-2-13-2': 'The perturbation [MATH] is required to be small compared to the lowest order distribution function, which is a Maxwell-Boltzmann distribution [EQUATION] where the radial coordinate [MATH] is [MATH] times the poloidal magnetic flux, [MATH] is the mass, [MATH] is the total unperturbed energy, and [MATH] is the pseudo-density.', '1603.00688-2-13-3': 'Henceforth, [MATH] refers to a first-order correction to the quantity [MATH], which is allowed to vary poloidally, while all the lowest order quantities are assumed to be flux functions.', '1603.00688-2-13-4': 'The linearized equation that Perfect solves is [EQUATION] where [MATH], [MATH] with the velocity [MATH], the lowest order drift velocity [MATH] contains the lowest order [MATH] drift and magnetic drifts [MATH], [MATH] is the linearized Fokker-Planck operator, and [MATH] is a source term, which will be explained shortly.', '1603.00688-2-13-5': 'The partial derivatives are taken at fixed magnetic moment [MATH] and unperturbed total energy [MATH].', '1603.00688-2-14-0': 'Note that boundary conditions in [MATH] are needed to fully specify [MATH] by [REF], in contrast to the local equation.', '1603.00688-2-14-1': 'Since the local theory should apply sufficiently far from the pedestal, the result of local simulations - in which the [MATH] term is dropped from [REF] - are imposed as boundary conditions where particles enter the domain [CITATION].', '1603.00688-2-15-0': 'As inputs, Perfect requires zeroth order (flux function) densities [MATH], temperature [MATH] and potential [MATH].', '1603.00688-2-15-1': 'Given these equilibrium profiles, [MATH] is calculated from [REF], and appropriate velocity moments of [MATH] provide the neoclassical flows and fluxes.', '1603.00688-2-15-2': 'The fluxes will in general not be divergence free and thus incompatible with the time-independent equilibrium profiles.', '1603.00688-2-15-3': 'It may seem instructive to restore the time derivative in [REF], and solve a time-dependent problem, in a hope to reach a steady state equilibrium.', '1603.00688-2-15-4': 'However, only exceptional profiles would lead to a steady state solution.', '1603.00688-2-15-5': 'More generally, particles and energy would accumulate in (or leave) the simulation domain until the [MATH] approach breaks down.', '1603.00688-2-15-6': 'Instead, the approach adopted is to add spatially varying sources [MATH] so that the zeroth-order profiles become consistent; these sources are solved for in the code alongside [MATH].', '1603.00688-2-15-7': 'These sources can be thought of as representing the effects of non-neoclassical transport needed to make the profiles consistent, and should also be present in a real pedestal.', '1603.00688-2-16-0': 'To guarantee that [MATH], the driving gradients in the right-hand side of [REF] should remain small.', '1603.00688-2-16-1': 'From [EQUATION] where prime denotes the [MATH]-derivative, we see that the [MATH] and temperature gradients set the size of [MATH], and thus drive the deviations from a Maxwellian.', '1603.00688-2-16-2': 'Hence the density and the electrostatic potential may have sharp gradients as long as they produce a slowly varying [MATH].', '1603.00688-2-16-3': 'To quantify what we mean by a sharp gradient, we may balance the [MATH] and [MATH] terms in [REF] to find that [EQUATION] should be satisfied by [MATH] and [MATH]; that is, these quantities should have a small relative change as experienced by a particle during its radial drift excursion.', '1603.00688-2-16-4': 'Here [MATH] is the poloidal Larmor radius of the species, with the thermal speed [MATH], and the poloidal magnetic field [MATH].', '1603.00688-2-17-0': '## Model profiles and magnetic geometry', '1603.00688-2-18-0': 'Although we would like to use pedestal profiles which are representative of experiments in some aspects, we require [REF] to be satisfied in the simulations for all species ([MATH] for electrons, ions and impurities, respectively).', '1603.00688-2-18-1': 'We consider pedestals where [MATH] and [MATH] are allowed to vary on the [MATH] scale, while the [MATH] and [MATH] gradients of the bulk and impurity ion species are constrained by [REF].', '1603.00688-2-19-0': 'As a starting point we considered profiles from typical JET discharges (Figure 16 ofnum0029-5515-55-11-113031), and modified them as necessary to satisfy our orderings, and to reduce the need for heat sources in the simulation domain.', '1603.00688-2-19-1': 'The specific choices made in constructing the model profiles are discussed in Appendix [REF].', '1603.00688-2-19-2': 'The resulting input profiles for our baseline case as functions of the normalized poloidal flux [MATH] are shown in [REF]a-d.', '1603.00688-2-19-3': 'Here, we introduced [MATH], where [MATH] is the poloidal flux at the last closed flux surface (LCFS).', '1603.00688-2-19-4': 'Note, that the density profile of impurities is much steeper than that of the bulk ions to make [MATH] vary slowly.', '1603.00688-2-19-5': 'We consider a deuterium plasma and fully ionized nitrogen impurities ([MATH]) with a concentration [MATH] in the core.', '1603.00688-2-19-6': 'The radii marking the beginning and the end of the pedestal are indicated by dotted vertical lines in the figures.', '1603.00688-2-20-0': 'In the simulations we use a local Miller model geometry [CITATION] (and neglect the radial variation of [MATH] and [MATH] in the domain), with elongation [MATH], [MATH], triangularity [MATH], [MATH], [MATH], [MATH], and inverse aspect ratio [MATH].', '1603.00688-2-20-1': 'These parameters were taken fromnumbelli2008.', '1603.00688-2-21-0': 'The numerical resolution and convergence tests are detailed in Appendix [REF].', '1603.00688-2-22-0': '## Units', '1603.00688-2-23-0': 'An input quantity [MATH] is supplied to Perfect in a normalized, dimensionless form [MATH], with the normalizing, dimensional quantity [MATH].', '1603.00688-2-23-1': 'We choose [MATH] and [MATH] to be the major radius and magnetic field at the magnetic axis.', '1603.00688-2-23-2': 'Furthermore we choose the following convenient units [MATH], [MATH], [MATH] (deuterium mass), and [MATH] is the elementary charge.', '1603.00688-2-23-3': 'We define the reference speed and collision frequency as [MATH] and [EQUATION] where [MATH] denotes the vacuum permittivity and [MATH] is the Coulomb logarithm.', '1603.00688-2-23-4': 'The ordinary same-species collision frequency [MATH] is defined as [REF] but with [MATH] and [MATH] replaced by [MATH] and [MATH], from which we may define the collisionality as [MATH].', '1603.00688-2-24-0': 'To quantify when the local approximation fails, it is useful to define a few additional quantities: The normalized electric field [MATH] measures how much the [MATH] drift competes with parallel streaming in terms of poloidal particle motion.', '1603.00688-2-24-1': 'The normalized logarithmic derivative [MATH] measures the variation of quantity [MATH] along a typical drift orbit, where [MATH] and [MATH].', '1603.00688-2-24-2': 'Local neoclassical theory is valid only when [MATH], [MATH], [MATH], and [MATH] are all much smaller than unity in magnitude.', '1603.00688-2-24-3': 'The global [MATH] model also requires that [MATH] and [MATH], but [MATH] and [MATH] can be [MATH].', '1603.00688-2-24-4': 'These derived quantities together with [MATH] are shown for the baseline profiles in [REF]e-h.', '1603.00688-2-25-0': '# Local and global simulation results', '1603.00688-2-26-0': 'To study the differences between local and global neoclassical transport we performed a number of Perfect simulations with the profiles and magnetic geometry described in [REF].', '1603.00688-2-26-1': 'Before presenting the simulation results for our baseline set of profiles, we discuss the input quantities.', '1603.00688-2-27-0': 'We define the following normalized output quantities: sources, [MATH], with [MATH]; [MATH], with [MATH]; particle flux, [MATH]; momentum flux (divided by mass), [MATH]; heat flux, [MATH]; conductive heat fluxes, [MATH]; parallel flow velocity, [MATH]; parallel current, [MATH].', '1603.00688-2-27-1': 'The neoclassical flow coefficient [MATH] is defined so that it reduces to the flux function poloidal flow coefficient for a single ion species plasma in the local limit, [EQUATION] where [MATH] is the pressure.', '1603.00688-2-27-2': 'Furthermore, we define the non-adiabatic density perturbation [MATH], the total density perturbation [MATH], and normalized scalar fluxes [EQUATION] with [MATH]) representing [MATH], [MATH] or [MATH], [MATH] or [MATH], respectively), and [MATH].', '1603.00688-2-27-3': 'Note that the flux normalizations are species independent.', '1603.00688-2-28-0': '## Results for the baseline case', '1603.00688-2-29-0': 'The scalar fluxes of [REF] divided by [MATH] are shown in [REF].', '1603.00688-2-29-1': 'Throughout [REF] solid lines represent global simulation results, and dashed lines represent local ones.', '1603.00688-2-30-0': 'When the impurity strength [MATH] is order unity - as in our baseline - the magnitude of the particle transport of electrons is typically [MATH] smaller than that of the impurities, where [MATH] denotes the effective ion charge.', '1603.00688-2-30-1': 'In this case it is common to neglect [MATH] and calculate the ion particle transport from ambipolarity [MATH], leading to opposing ion and impurity particle fluxes.', '1603.00688-2-30-2': 'The local simulations in the core region obey these expectations.', '1603.00688-2-30-3': 'Since [MATH] is small in our baseline, temperature screening does not dominate, thus the local fluxes obey [MATH] and [MATH].', '1603.00688-2-30-4': 'In the pedestal the parallel ion-electron friction can be sufficiently large to compete with the ion-impurity friction, due to the high electron flow speeds.', '1603.00688-2-30-5': 'Therefore [MATH] cannot be neglected anymore in the ambipolarity condition: the strong outward electron flux means that both [MATH] and [MATH] are positive simultaneously.', '1603.00688-2-30-6': 'Thus the outward local [MATH] is not a result of temperature screening.', '1603.00688-2-31-0': 'As expected from the small [MATH] and [MATH] values, seen in [REF]e and f, the electron local and global fluxes are practically the same.', '1603.00688-2-31-1': 'However, finite orbit width effects strongly affect the ion and impurity dynamics.', '1603.00688-2-31-2': 'In the pedestal [MATH] is increased compared to the local value, which causes [MATH] to change sign compared to its local value.', '1603.00688-2-31-3': 'It is worth noting that the deviation between local and global results is not localized to the pedestal region only.', '1603.00688-2-31-4': 'For instance, the global and local [MATH] deviate well below [MATH]; the ion particle flux changes sign at [MATH] while the local result is positive everywhere.', '1603.00688-2-31-5': 'As a comparison we note, that the width of the large gradient region in [MATH] units is approximately [MATH], and the orbit width of a typical trapped ion at thermal speed is [MATH].', '1603.00688-2-32-0': 'The somewhat surprising observation that the finite orbit width effects extend outside the pedestal over several thermal ion orbit widths is worth a moment of thought.', '1603.00688-2-32-1': 'Since the existing analytical theories assume [MATH], which partly eliminates the radial coupling, they can only provide a limited guidance as to why this happens.', '1603.00688-2-32-2': 'The only radial coupling that cannot be completely eliminated from those theories is that due to the neoclassical parallel flow, [MATH]; indeed it is not a parameter in the theory, but it satisfies a radial differential equation (Eqs. (43) and (64) in Ref. catto13).', '1603.00688-2-32-3': 'As we will see, in our case the global result for the main ion [MATH] is very different from the local one, and it takes a rather long distance from the pedestal before it gets close to the local result.', '1603.00688-2-32-4': 'Also, in estimating the orbit width above, we considered particles at the thermal speed, while all the quantities of interest are dominated by super-thermal particles with wider orbits.', '1603.00688-2-32-5': 'The neoclassical drive and the radial coupling terms both include [MATH], the velocity space integration weight is [MATH], and the flow, particle flux, and heat flux contain an additional factor of [MATH], [MATH], or [MATH] respectively.', '1603.00688-2-33-0': 'The global particle fluxes are not ambipolar as seen at [MATH], where [MATH] and [MATH] are both inward, and [MATH] is small, very close to its local value.', '1603.00688-2-33-1': 'Note that while in a local simulation the radial current and momentum flux should vanish, it does not need to be so in global simulations, as pointed out innumlandreman2014.', '1603.00688-2-34-0': 'For both ions and impurities we observe that the conductive heat flux can significantly differ from the local value.', '1603.00688-2-34-1': 'A reduction compared to the the local value - observed around the pedestal top - may be explained by the shift of the trapped region towards the tail of the distribution at [MATH].', '1603.00688-2-34-2': 'However, we also find regions where the heat fluxes increase from their local values.', '1603.00688-2-34-3': 'Just as for the particle fluxes, we see a reduction inside the pedestal top: [MATH]) reaches a minimum at [MATH]).', '1603.00688-2-35-0': 'The corresponding sources are presented in [REF].', '1603.00688-2-35-1': 'We use poloidally symmetric sources with speed dependencies [MATH] for particle and heat sources, respectively, where [MATH].', '1603.00688-2-35-2': 'We see that to some degree the main ion particle and heat sources qualitatively mirror each other, and tend towards zero outside the pedestal.', '1603.00688-2-35-3': 'Recall that we specifically choose [MATH] to reduce the need for [MATH], while [MATH] also remains small, because [MATH] tends to be smaller than [MATH].', '1603.00688-2-35-4': 'The tremendous drop in [MATH] in the pedestal leads to a sharp peak in [MATH], and again we see an opposing trend for [MATH], but for impurities the combined sources are positive.', '1603.00688-2-35-5': 'The electron sources are localized to the pedestal and comparable in size to those of the main ions.', '1603.00688-2-36-0': 'The neoclassical flow coefficients, [MATH], are presented in [REF].', '1603.00688-2-36-1': 'The local (dashed lines) [MATH] is positive as expected in the banana regime, and exhibits a slight variation as a response to the radial variation of [MATH].', '1603.00688-2-36-2': 'In the local case [MATH] is a flux function, while globally it varies from the inboard side (darker curves) to the outboard side (lighter curves).', '1603.00688-2-36-3': 'These poloidal variations in the flow appear together with poloidal density variations, as will be discussed shortly.', '1603.00688-2-36-4': 'It has been shown analytically [CITATION] that [MATH] is affected by finite orbit width effects.', '1603.00688-2-36-5': 'As seen from the analytical results, where [MATH] is kept but the radial coupling is neglected, [MATH] is expected to decrease or become more negative in both the banana and plateau regimes [CITATION].', '1603.00688-2-36-6': 'However, the global modification to [MATH] is not a function of local plasma parameters (such as [MATH]) only, but even in the semi-global treatment ofnumcatto13 it satisfies a radial differential equation (i.e. [MATH] cannot be neglected).', '1603.00688-2-36-7': 'This is why [MATH] can differ in sign and magnitude from the local value well outside the pedestal, and can be larger than its local value inside the pedestal.', '1603.00688-2-36-8': 'It is interesting to note that there is a difference between global and local [MATH] even for the electrons.', '1603.00688-2-37-0': 'The density perturbations are shown in [REF].', '1603.00688-2-37-1': 'We first consider only the non-adiabatic contribution to the perturbations, [MATH], shown in [REF]a-f, where the global (local) results are plotted in the a-c (d-f) panels.', '1603.00688-2-37-2': 'The local simulations predict purely up-down density asymmetry for all species, which is weaker for bulk ions and electrons than for impurities.', '1603.00688-2-37-3': 'The global results show a more complicated poloidal density variation for both ions and impurities.', '1603.00688-2-37-4': 'For ions we see an in-out asymmetry (i.e. excess density around [MATH]) at the pedestal top, which transforms into an out-in asymmetry in the pedestal, and reverses again further out (similarly to the single species simulations ofnumpusztaiRFNEO).', '1603.00688-2-37-5': 'For impurities the most important difference compared to the local results is the weak in-out (instead of strong up-down) asymmetry in the pedestal.', '1603.00688-2-37-6': 'The electron density perturbation mostly follows its local behavior, exhibiting a large increase in the up-down asymmetry in the pedestal.', '1603.00688-2-37-7': 'To understand the total density perturbation [MATH] in global simulations we note that the potential perturbation [MATH] follows mostly the non-adiabatic ion density perturbation.', '1603.00688-2-37-8': 'The total electron density perturbation is dominated by the adiabatic response of electrons, thus it is very similar to the ion density perturbation.', '1603.00688-2-37-9': 'The impurity density variations show a competition between adiabatic response - especially in the pedestal where it tries to oppose the ion density perturbation - and non-adiabatic response.', '1603.00688-2-37-10': 'The relative impurity density variation stays below [MATH] everywhere, showing that the assumption of the density being nearly a flux function is not violated.', '1603.00688-2-37-11': 'However, for sufficiently high [MATH], nonlinearity from poloidal asymmetries can arise [CITATION].', '1603.00688-2-38-0': 'Finally, we consider the parallel flows and the bootstrap current for our baseline case; these are plotted in [REF].', '1603.00688-2-38-1': 'For the bulk and impurity ions the flows remain small inside the pedestal, as their profiles were chosen specifically so that their diamagnetic and [MATH] flows mostly cancel.', '1603.00688-2-38-2': 'For these the relative deviation between the local and global results is significant, the global results being larger in magnitude from the middle of the pedestal inward.', '1603.00688-2-38-3': 'This is partly due to the reduction in [MATH] compared to the local value observed in [REF].', '1603.00688-2-38-4': 'The parallel flow of electrons reaches a much higher magnitude inside the pedestal then that of the ions, where the strong [MATH] and diamagnetic rotation contribute with the same sign for this species.', '1603.00688-2-38-5': 'The scale is therefore different and the difference between the local and global results is less visible.', '1603.00688-2-38-6': 'Figure [REF]e shows this difference, which is comparable to to what is observed for ions.', '1603.00688-2-38-7': 'This is expected, because all the difference is due to a frictional coupling to the various ion species with modified flow speeds, as direct finite orbit width effects are negligible for electrons.', '1603.00688-2-38-8': 'Since we have a weak ion temperature pedestal, the modifications of the ion flows are not sufficient to cause an appreciable deviation of the bootstrap current from the local result inside the pedestal, as seen in [REF]a.', '1603.00688-2-38-9': 'If anything, outside the pedestal there is a slight difference between the local and global results (barely visible on the scale accommodating the huge bootstrap peak in the pedestal), since the electron flow is relatively small in those regions, while the global effects on [MATH] extend outside the pedestal.', '1603.00688-2-39-0': '## Non-trace impurities', '1603.00688-2-40-0': 'We want to assess whether, and how, non-trace impurities can affect collisional transport in the pedestal.', '1603.00688-2-40-1': 'To this end, we performed simulations with profiles similar to our baseline, except that we scaled the impurity concentration profiles.', '1603.00688-2-40-2': 'The simulations shown in this section have impurity concentrations such that the impurity strength in the core is either [MATH] (trace impurities; shown with thin lines in the figures) or [MATH] (non-trace; thick lines); as a comparison [MATH] for the baseline.', '1603.00688-2-40-3': 'Note that [MATH] drops rapidly across the pedestal, thus the impurities eventually become trace even if they have a high core concentration (this is necessary in the presence of an electron density pedestal if we restrict profile variations to obey the orderings of [REF] for all ion species).', '1603.00688-2-41-0': 'The a-c panels of [REF] compare the local and global particle fluxes with trace and non-trace impurity concentration.', '1603.00688-2-41-1': 'As usual, the local and global results for [MATH] are virtually the same.', '1603.00688-2-41-2': 'At higher impurity content the increase in [MATH] around [MATH] reflects the increased [MATH] gradient due to the rapidly varying impurity concentration (note that [MATH] is kept fixed in the impurity scan, thus the [MATH] profile changes).', '1603.00688-2-41-3': 'The local [MATH] behaves as expected from the ambipolarity condition: it increases with impurity content in the core to balance the inward [MATH], and its core behavior is dominated by following the outward electron flux.', '1603.00688-2-41-4': 'It is interesting to note that while above [MATH] the local curves for different impurity concentration collapsed onto each other because of the low [MATH], the difference in the global [MATH] survives much further out in the pedestal.', '1603.00688-2-42-0': 'The d-f panels of [REF] show the heat fluxes.', '1603.00688-2-42-1': 'In the core and close to the pedestal top the global results for [MATH] are lower than the local values.', '1603.00688-2-42-2': 'This reduction is somewhat stronger in the presence of impurities.', '1603.00688-2-42-3': 'The local results for [MATH] and [MATH] are even higher in the pedestal at high impurity concentration, than would be expected simply due to the linear increase with [MATH].', '1603.00688-2-42-4': 'This extra increase is reduced by the global effects so that the global results at different concentrations are closer to each other.', '1603.00688-2-43-0': 'The momentum flux values for the different species (shown in [REF]g-i) vanish in the local limit, as they should, but are finite when global effects are considered.', '1603.00688-2-43-1': 'Unlike [MATH] and [MATH], the global results for [MATH] are different to the local ones, i.e. they are also finite, although the electrons do not transport an appreciable amount of momentum, as [MATH].', '1603.00688-2-43-2': 'The total momentum flux, [MATH], shown in [REF]a, is mostly negative in the studied radial range, and is strongly increase by the presence of non-trace impurities.', '1603.00688-2-43-3': 'The ions are responsible for most of the momentum transport for both impurity concentrations, thus the increase in [MATH] is not due to the increase in [MATH], but the modifications in [MATH] in the presence of impurities.', '1603.00688-2-44-0': 'The radial current [MATH] ([REF]b) - that in isolation from other transport channels would lead to charge separation and the evolution of the radial electric field - is also significantly increased in magnitude by the presence of the impurities over most of the studied radial domain.', '1603.00688-2-44-1': 'Although we observe a finite radial neoclassical current, we do not attempt to self-consistently calculate the radial electric field.', '1603.00688-2-44-2': 'For the interpretation of the momentum fluxes and the radial current it is useful to note that conservation of particle number and parallel momentum (in steady state, for a radially constant [MATH], and with sources even in [MATH]) imply that [MATH] and [MATH].', '1603.00688-2-44-3': 'The latter property is apparent from a comparison of [REF]a and b.', '1603.00688-2-44-4': 'The former relation states that non-ambipolar fluxes require particle sources for which [MATH] is not satisfied locally (the factor [MATH] comes from the assumed velocity space structure of [MATH]).', '1603.00688-2-45-0': 'In radially local formalisms momentum transport is often decomposed into diffusive ([MATH], where [MATH] is the normalized parallel mass flow), conductive ([MATH]) and intrinsic (independent of [MATH]) terms.', '1603.00688-2-45-1': 'Such a decomposition is not possible in our global formalism since the parallel mass flow is a non-local function of the various plasma parameter profiles, and so is the momentum transport.', '1603.00688-2-45-2': 'Nevertheless, it is instructive to compare the radial profile of the total momentum transport [MATH] to the radial variation of the total mass flow [MATH] shown in [REF]c (both quantities are dominated by the main ion contributions).', '1603.00688-2-45-3': 'By increasing the impurity content, [MATH] increases in the core region.', '1603.00688-2-45-4': 'This is caused by the higher [MATH] at higher [MATH] not being compensated by the slight reduction in [MATH] (shown in [REF]).', '1603.00688-2-45-5': 'As the impurity concentration drops radially across the pedestal, the effect from the reduction in the parallel ion flow becomes dominant.', '1603.00688-2-45-6': 'The radial drop in the global results for [MATH] across the pedestal is mostly due to the density variation in the pedestal.', '1603.00688-2-45-7': 'If the transport was local and purely diffusive this non-monotonic behavior of [MATH] would be accompanied by a sign change in the momentum transport.', '1603.00688-2-45-8': 'It is also interesting to note that at the point where we see the greatest relative increase in [MATH] between the different simulations ([MATH]), the global [MATH] becomes lower for higher core impurity concentration.', '1603.00688-2-46-0': 'The ion particle source profile shown in [REF]a is only weakly affected by the presence of non-trace impurities.', '1603.00688-2-46-1': 'In the meantime, the increase in impurity sources are approximately proportional to the increase in their concentration: the normalized particle source [MATH] is approximately the same in the two simulations, considering that the impurity content changes by a factor [MATH].', '1603.00688-2-47-0': 'The radial current and the non-quasineutral particle sources [MATH] are consistent, that is, the divergence of the radial current is given by the charge source [MATH] (note, that the total charge source integrated across the pedestal is zero, due to the boundary conditions).', '1603.00688-2-47-1': 'However, the radial current is truly a consequence of the radial coupling in the global simulation, and not an artifact of the radially varying sources.', '1603.00688-2-47-2': 'Although it is not done in the code, source profiles could be calculated in the presence of the radially varying local particle and heat fluxes, for these to be consistent with the time independent plasma parameter profiles.', '1603.00688-2-47-3': 'Such sources would be quasineutral unlike those in the global simulations.', '1603.00688-2-48-0': 'In reality, the neoclassical radial current that we observe needs to be balanced by an opposing radial current, which represents transport processes not captured by our model (turbulence, atomic physics processes, orbit losses, magnetic ripple effects, etc.).', '1603.00688-2-48-1': 'Otherwise the system would not be steady state, because the radial electric field would vary in time and the [MATH] torque would change the plasma flows.', '1603.00688-2-48-2': 'Due to the construction of the code and the vanishing [MATH] moment of our sources, we observe the [MATH] and the radial current terms in the species-summed flux surface averaged angular momentum equation to exactly balance.', '1603.00688-2-48-3': 'Similarly, the finite neoclassical momentum transport predicted by the code should also be canceled by a momentum transport due to non-neoclassical processes, in steady state.', '1603.00688-2-49-0': 'Finally we would like to assess how much our profile choices affect the observation of increased momentum flux in the presence of non-trace impurities.', '1603.00688-2-49-1': 'One unnatural feature of our impurity density profile is its extreme steepness.', '1603.00688-2-49-2': 'Allowing [MATH] have a radial drop across the pedestal - within what is allowed by [REF] - to reduce the inward radial electric field, together with allowing [MATH] to have an increase in the same region, leads to a less sharp impurity pedestal.', '1603.00688-2-49-3': 'To further reduce the impurity density gradient in the pedestal we consider a fully ionized beryllium impurity ([MATH]).', '1603.00688-2-49-4': 'These changes make our pedestal less deeply sub-sonic, as seen from the increased mass flow in [REF]c, which shows the results of the modified input profiles.', '1603.00688-2-49-5': 'From [REF]a we can conclude that impurities also significantly increase the magnitude of the neoclassical momentum flux in the pedestal for more natural impurity profiles, when the inputs push the limitations imposed by the required orderings.', '1603.00688-2-50-0': '# Discussion and conclusions', '1603.00688-2-51-0': 'We have studied the differences in the collisional transport between radially local and global formalisms using the global [MATH] neoclassical solver Perfect, with a special emphasis on the effects of non-trace impurities.', '1603.00688-2-51-1': 'If the impurity density profile is arbitrary, impurities are likely to develop sonic flows and strong poloidal asymmetries in the pedestal.', '1603.00688-2-51-2': 'We use model profiles specifically chosen to guarantee that impurity flows remain sub-sonic, so that the assumptions of the [MATH] formalism are valid.', '1603.00688-2-51-3': 'In this way we can gain some insights into the effect of non-trace impurities without the need for a non-linear collision operator.', '1603.00688-2-52-0': 'In an impure plasma the electron particle transport is usually negligible, and the ion and impurity fluxes oppose each other to maintain ambipolarity.', '1603.00688-2-52-1': 'However, since in the pedestal the magnitude of the parallel electron flow can be much larger than the parallel ion and impurity flows, the friction of the various ion species on electrons can become non-negligible.', '1603.00688-2-52-2': 'Consequently, a substantial neoclassical electron particle transport can arise, which competes with the radial transport of other species.', '1603.00688-2-52-3': 'In particular, the ion and impurity fluxes can have the same sign.', '1603.00688-2-52-4': 'In the presence of strong radial profile variations on the [MATH] scale, the ambipolarity of fluxes is violated, as reported in previous studies [CITATION].', '1603.00688-2-52-5': 'We emphasize that the differences in neoclassical flows and fluxes between the local and global theory are not restricted to the pedestal region only but die off within a distance comparable with the ion orbit width.', '1603.00688-2-52-6': 'In our baseline from the outer core region to the pedestal top we observe that both the ion and impurity fluxes are inward due to global effects (see [REF]a-c).', '1603.00688-2-52-7': 'Inside the pedestal, local simulations predict both the impurity and the ion fluxes to be outward, due to the large outward electron flux, but in global simulations the impurity flux reverses to be inward.', '1603.00688-2-53-0': 'For our model profiles, chosen to give small parallel ion flows, the bootstrap current remains almost completely unaffected, while this need not be so if the ion temperature varies more rapidly than allowed by our orderings.', '1603.00688-2-53-1': 'Indeed, the neoclassical parallel flow coefficient [MATH] of ions and impurities is significantly affected by global effects.', '1603.00688-2-53-2': 'Due to the radial coupling even the sign of [MATH] is different from the local result well inside of the pedestal top.', '1603.00688-2-54-0': 'We observe that in the presence of global effects the poloidal variation of the density perturbation is not restricted to an up-down asymmetry, it can be more significant, and can develop rapid radial variations (in accordance with previous numerical results [CITATION]).', '1603.00688-2-54-1': 'The poloidal asymmetries observed here arise in the presence of finite inverse aspect ratio and finite orbit width effects, and are not closely related to those predicted by the analytical theories in [CITATION], since those require significant poloidal variations in the ion-impurity friction.', '1603.00688-2-54-2': 'The adiabatic response to the electrostatic perturbation generated by the ions competes with non-adiabatic contributions in determining the poloidal asymmetries developed by the impurities.', '1603.00688-2-54-3': 'For the moderate impurity charge considered here ([MATH]) the relative poloidal variation of the impurity density is still small so that the perturbative treatment remains valid.', '1603.00688-2-54-4': 'At sufficiently high [MATH] the relative poloidal variation of the impurity density is expected to become order unity and then - if the impurities are non-trace - nonlinear effects would start to play a role.', '1603.00688-2-54-5': 'The numerical investigation of that situation is left for a future study.', '1603.00688-2-55-0': 'In the pedestal, the neoclassical radial current does not vanish in general, and the momentum transport remains finite - in contrast to the local theory.', '1603.00688-2-55-1': 'In the vicinity of the pedestal the total neoclassical momentum transport is found to be negative in the studied case, which happens if the charge sources (resulting from the particle sources needed to sustain the pedestal) are mostly positive in that region.', '1603.00688-2-55-2': 'The radial variation of the parallel mass flow is non-monotonic with a sharply decreasing feature where the density drops in the pedestal.', '1603.00688-2-55-3': 'This non-monotonicity is not reflected in the radial momentum transport, which is now a non-local function of the mass flow.', '1603.00688-2-55-4': 'A simple decomposition of the momentum transport into diffusive, pinch and intrinsic terms is not possible in the global picture.', '1603.00688-2-55-5': 'Note that the same is also true for all the radial fluxes.', '1603.00688-2-56-0': 'We observe a strong effect of impurities on the total momentum transport, the magnitude of which increases significantly in the presence of non-trace impurities (see [REF]a).', '1603.00688-2-56-1': 'While it is not possible to disentangle the exact cause of this, it may be due to the sharp radial variation of the parallel impurity mass flow.', '1603.00688-2-56-2': 'The impurities represent only a minor fraction of the total mass flow, but their collisional coupling to the main ions is significant at an impurity strength of order unity.', '1603.00688-2-56-3': 'Pushing the limitations of our orderings we reduced the sharpness of the impurity density profile, in an attempt to demonstrate the robustness of the impact of impurities on neoclassical momentum transport.', '1603.00688-2-56-4': 'Although the impurity profile is chosen to be very specific, strong effects may occur when the impurities have more general density variations, especially because they then have much stronger relative flow speeds compared to the ions.', '1603.00688-2-57-0': 'Impurity seeding in tokamaks operating with ITER-like metallic walls has been experimentally found to have beneficial effects on the pedestal performance.', '1603.00688-2-57-1': 'It is then natural to raise the question of whether our results are consistent with this observation.', '1603.00688-2-57-2': 'Without taking into account other non-intrinsically ambipolar processes and turbulent transport it is not possible to evolve the profiles towards a steady state.', '1603.00688-2-57-3': 'However, we may speculate about possible consequences of the increased momentum transport in the presence of impurities.', '1603.00688-2-57-4': 'In steady state the neoclassical radial current and momentum transport should be balanced by opposing contributions of turbulent and other origin.', '1603.00688-2-57-5': 'An increased neoclassical momentum transport in the presence of impurities requires these contributions to increase as well.', '1603.00688-2-57-6': 'Sufficiently far from the open field line region the turbulent transport can dominate these opposing contributions.', '1603.00688-2-57-7': 'Impurities tend to reduce the turbulence level by dilution [CITATION].', '1603.00688-2-57-8': 'If the non-diffusive turbulent momentum transport is to be increased in spite of dilution effects, stronger deviations of the non-fluctuating distribution from a Maxwellian [CITATION] and stronger profile variations [CITATION] may be necessary; which could require a steepening of the pedestal to reach a new steady state.', '1603.00688-2-57-9': 'Whether this is indeed the case and, if so, its role in the observed confinement improvement in impurity seeded discharges, remains an open question and should be the basis of future investigation.', '1603.00688-2-58-0': 'The authors are grateful for J. Omotani, T. Fulop and S. Newton for fruitful discussions and instructive comments on the paper.', '1603.00688-2-58-1': 'IP and SB were supported by the International Career Grant of Vetenskapsraadet (Dnr.', '1603.00688-2-58-2': '330-2014-6313), and ML was supported by the U.S. DoE under award numbers DEFG0293ER54197 and DEFC0208ER54964.', '1603.00688-2-58-3': 'The simulations used computational resources of Hebbe at C3SE (project nr.', '1603.00688-2-58-4': 'C3SE2016-1-10).', '1603.00688-2-59-0': '# The construction of model pedestal profiles', '1603.00688-2-60-0': 'Here, we describe a method to construct appropriate densities given [MATH], [MATH], [MATH] profiles satisfying [REF], and an arbitrary [MATH].', '1603.00688-2-60-1': 'The orderings for the electrons are not a concern, since [MATH] is much smaller than the radial scale length of any profile in an experiment.', '1603.00688-2-61-0': 'The density profiles and potential discussed in this section are the inputs to Perfect and are thus, strictly speaking, only the zeroth order contributions to these quantities, assumed to be flux functions.', '1603.00688-2-62-0': 'Once we have specified both [MATH] and [MATH] (to be detailed shortly), the relation [MATH] gives the potential [EQUATION]', '1603.00688-2-62-1': 'A strong electric field from [MATH] compensates for the potentially large ion pressure drop to make the variation of [MATH] small (the situation of "electrostatic ion confinement," which is borne out in experimental results [CITATION]).', '1603.00688-2-62-2': 'In the radial force balance the ion pressure drop could in principle be balanced by a sonic ion flow, however we consider sub-sonic ion flows.', '1603.00688-2-62-3': 'Note that by allowing sonic ion flows, the relative flow speed of two ion species would in general also be sonic, which would severely complicate the treatment of collisions.', '1603.00688-2-62-4': 'Such a scenario is currently not supported by Perfect.', '1603.00688-2-62-5': 'We emphasize that [MATH] and [MATH] are fundamental inputs to the code, thus the relation [REF] is not an attempt to self-consistently calculate the potential.', '1603.00688-2-63-0': 'Then, given [MATH] from ([REF]) and an [MATH], we obtain the impurity density [EQUATION]', '1603.00688-2-63-1': 'Note that ([REF]) leads to impurity density profiles with a typical logarithmic density gradient in the pedestal [MATH] times larger than that of the main ions; this is unavoidable if [REF] is to be satisfied for all ion species ([MATH]).', '1603.00688-2-63-2': 'Finally, the electron density is obtained by demanding quasi-neutrality [EQUATION]', '1603.00688-2-63-3': 'The resulting [MATH] satisfies [REF] by virtue of the electron gyroradius being small.', '1603.00688-2-64-0': 'Although we have constraints and relations between profiles, we still have a large degree of freedom in specifying them.', '1603.00688-2-64-1': 'Here we discuss some specific choices we made for the profile set used as a baseline for the simulations.', '1603.00688-2-65-0': 'Since we use the local solution for the boundary condition, we also need to make sure that the assumptions of the local theory are satisfied at the boundary.', '1603.00688-2-65-1': 'This means that the sharp density or potential variations should be limited to the middle of the radial domain, sufficiently far from the boundaries.', '1603.00688-2-65-2': 'For weak electric fields the local and global results should agree, so we choose the [MATH] profile so that the potential calculated from [REF] is completely flat at the boundaries.', '1603.00688-2-65-3': 'We do this by letting [EQUATION] in the vicinity of the inner boundary, namely in the core up to the pedestal top where [MATH] (and so [MATH]) should be slowly varying.', '1603.00688-2-65-4': 'Thus [MATH] in this region, independent of the [MATH] profile.', '1603.00688-2-65-5': 'To achieve a flat potential in the vicinity of the outer boundary, we let [EQUATION] in that region, where [MATH] is a constant which fixes the value of [MATH].', '1603.00688-2-65-6': 'To remove the ambiguity of [MATH] far from the boundaries in a way that makes it a simple and smooth function, we linearly extrapolate [MATH] given by [REF] from the core region up to the bottom of the pedestal (where the sharp feature in [MATH] ends) and match it to the other expression [REF], choosing [MATH] from [REF] at the matching point.', '1603.00688-2-66-0': 'The [MATH] profile is chosen to be a linear function of [MATH] over the whole domain, with a logarithmic gradient matching that of [MATH] at the left boundary.', '1603.00688-2-67-0': 'Since the global [MATH] ordering does not allow an ion temperature pedestal, we consider [MATH] profiles with a gradient across the pedestal (and further out) equal to that of the electron temperature gradient in the core.', '1603.00688-2-67-1': 'Due to the density drop in the pedestal and the decreasing temperature profile, the ion heat flux is bound to be vastly different at the two boundaries, requiring large heat sources in the domain, except if a variation in [MATH] balances them.', '1603.00688-2-67-2': 'This can be avoided by setting a proxy for the heat fluxes, [MATH], to be equal at the boundaries.', '1603.00688-2-67-3': 'Thus, we artificially reduce [MATH] in the core to remove the need for sources close to the boundaries.', '1603.00688-2-68-0': 'Experimental density and temperature profiles can drop orders of magnitude across the pedestal, and a significant part of these variations occur in the open field line region.', '1603.00688-2-68-1': 'As an additional consideration, if we let the bulk densities and temperatures drop across the simulation domain as much as in a real pedestal, it would lead to difficulties related to large logarithmic gradients, and huge changes in collision frequency.', '1603.00688-2-68-2': 'Since the code does not capture the physics in the open field line region (where the pedestal foot would be in an experiment), the region outside the middle of the pedestal does not carry too much physical relevance and can be considered as a numerical buffer zone.', '1603.00688-2-68-3': 'To avoid the above mentioned complications we reduce the gradients starting from a point where crossing the separatrix would be expected in an experiment.', '1603.00688-2-68-4': 'We thus arbitrarily pick this reduced [MATH] to be [MATH] of its pedestal value, with [MATH] equal to its core value.', '1603.00688-2-69-0': 'Motivated by typical JET discharges (Figure 16 of 0029-5515-55-11-113031), for our baseline case we choose the [MATH] and [MATH] pedestal widths to be [MATH], with the values [MATH] and [MATH] at the pedestal top.', '1603.00688-2-69-1': 'These correspond to typical logarithmic gradients of about [MATH] and [MATH], where [MATH] is the minor radius defined as half of the width of the flux surface at the elevation of its centroid, and [MATH] at the last closed flux surface.', '1603.00688-2-69-2': 'The [MATH] and [MATH] profiles are generated by Bezier curve interpolation between three regions with linear profile variation: a core, a pedestal, and an outer "buffer" region.', '1603.00688-2-69-3': '(Using this type of smoothing ensures that the gradients transition smoothly and monotonically.)', '1603.00688-2-70-0': 'To transform the profiles from [MATH] to [MATH] space, [MATH] is needed.', '1603.00688-2-70-1': 'This we obtain from [MATH] where [MATH] is the toroidal magnetic flux which we calculate assuming simple elongated flux surfaces.', '1603.00688-2-70-2': 'Since we do not intend to model a specific experiment, we take model profiles for the safety factor [MATH] and the elongation [MATH], shown in [REF].', '1603.00688-2-70-3': 'The [MATH] was chosen as [MATH].', '1603.00688-2-70-4': 'For the on-axis toroidal magnetic fields we take [MATH] and neglect [MATH] corrections together with higher order shaping effects to get [MATH], where [MATH] denotes the major radius.', '1603.00688-2-70-5': 'The resulting [MATH] is taken to be constant across the entire pedestal.', '1603.00688-2-71-0': 'The ion temperature gradient, the ion and impurity [MATH] and the impurity strength profiles are shown in [REF], for the different impurity concentration simulations presented in this paper.', '1603.00688-2-72-0': '# Numerical resolution', '1603.00688-2-73-0': 'The simulations used [MATH] radial grid points and cover a domain of [MATH] (the whole [MATH] domain is not shown in the figures).', '1603.00688-2-73-1': 'The number of poloidal grid points is [MATH].', '1603.00688-2-73-2': 'The number of expansion polynomials in the pitch angle cosine [MATH] is [MATH] for the distribution function and [MATH] for the Rosenbluth potentials (RP).', '1603.00688-2-73-3': 'The number of speed grid points is [MATH] and [MATH] for the RP.', '1603.00688-2-74-0': 'To demonstrate the degree of convergence, in [REF] we present the particle sources for the baseline simulation for the above mentioned resolution (red curve), together with four other cases, where we increase [MATH] to [MATH] (violet), [MATH] to [MATH] (blue), [MATH] to [MATH] (cyan), and the radial domain size to [MATH] (yellow), with all other parameters kept fixed.', '1603.00688-2-74-1': 'These four are the resolution parameters to which the accuracy of the solution is most sensitive.', '1603.00688-2-74-2': 'The results are almost identical except for slight differences near a few sharp features.', '1603.00688-2-75-0': 'To quantify the error, we define [MATH].', '1603.00688-2-75-1': 'Taking the [MATH] simulation as the reference [MATH], we find that the quantity with the highest error is the electron particle source, with [MATH].', '1603.00688-2-75-2': 'For other quantities, such as particle flows and radial fluxes, we observe errors below [MATH], thus [REF] presents the most stringent test for numerical convergence.', '1603.00688-2-75-3': 'The same convergence test was performed for all the simulations, with resulting errors (including those of the sources) all being below [MATH].'}

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['1603.00688-2-15-4', '1603.00688-3-15-4'], ['1603.00688-2-15-5', '1603.00688-3-15-5'], ['1603.00688-2-15-6', '1603.00688-3-15-6'], ['1603.00688-2-15-7', '1603.00688-3-15-7'], ['1603.00688-2-73-0', '1603.00688-3-73-0'], ['1603.00688-2-73-1', '1603.00688-3-73-1'], ['1603.00688-2-73-2', '1603.00688-3-73-2'], ['1603.00688-2-73-3', '1603.00688-3-73-3'], ['1603.00688-2-41-0', '1603.00688-3-41-0'], ['1603.00688-2-41-1', '1603.00688-3-41-1'], ['1603.00688-2-41-2', '1603.00688-3-41-2'], ['1603.00688-2-41-3', '1603.00688-3-41-3'], ['1603.00688-2-6-0', '1603.00688-3-6-0'], ['1603.00688-2-6-1', '1603.00688-3-6-1'], ['1603.00688-2-6-2', '1603.00688-3-6-2'], ['1603.00688-2-6-3', '1603.00688-3-6-3'], ['1603.00688-2-51-0', '1603.00688-3-51-0'], ['1603.00688-2-51-1', '1603.00688-3-51-1'], ['1603.00688-2-51-2', '1603.00688-3-51-2'], ['1603.00688-2-51-3', '1603.00688-3-51-3'], ['1603.00688-2-53-0', '1603.00688-3-53-0'], ['1603.00688-2-53-1', '1603.00688-3-53-1'], ['1603.00688-2-53-2', '1603.00688-3-53-2'], ['1603.00688-2-16-0', '1603.00688-3-16-0'], ['1603.00688-2-16-1', '1603.00688-3-16-1'], ['1603.00688-2-16-2', '1603.00688-3-16-2'], ['1603.00688-2-16-3', '1603.00688-3-16-3'], ['1603.00688-2-16-4', '1603.00688-3-16-4'], ['1603.00688-2-30-0', '1603.00688-3-30-0'], ['1603.00688-2-30-1', '1603.00688-3-30-1'], ['1603.00688-2-30-2', '1603.00688-3-30-2'], ['1603.00688-2-30-3', '1603.00688-3-30-3'], ['1603.00688-2-30-4', '1603.00688-3-30-4'], ['1603.00688-2-30-5', '1603.00688-3-30-5'], ['1603.00688-2-30-6', '1603.00688-3-30-6'], ['1603.00688-2-14-0', '1603.00688-3-14-0'], ['1603.00688-2-14-1', '1603.00688-3-14-1'], ['1603.00688-2-34-0', '1603.00688-3-34-0'], ['1603.00688-2-34-1', '1603.00688-3-34-1'], ['1603.00688-2-34-2', '1603.00688-3-34-2'], ['1603.00688-2-34-3', '1603.00688-3-34-3'], ['1603.00688-2-74-0', '1603.00688-3-74-0'], ['1603.00688-2-74-1', '1603.00688-3-74-1'], ['1603.00688-2-19-0', '1603.00688-3-19-0'], ['1603.00688-2-19-1', '1603.00688-3-19-1'], ['1603.00688-2-19-2', '1603.00688-3-19-2'], ['1603.00688-2-19-3', '1603.00688-3-19-3'], ['1603.00688-2-19-6', '1603.00688-3-19-6'], ['1603.00688-2-24-0', '1603.00688-3-24-0'], ['1603.00688-2-24-1', '1603.00688-3-24-1'], ['1603.00688-2-24-2', '1603.00688-3-24-2'], ['1603.00688-2-24-3', '1603.00688-3-24-3'], ['1603.00688-2-24-4', '1603.00688-3-24-4'], ['1603.00688-2-9-0', '1603.00688-3-9-0'], ['1603.00688-2-9-1', '1603.00688-3-9-1'], ['1603.00688-2-9-2', '1603.00688-3-9-2'], ['1603.00688-2-9-3', '1603.00688-3-9-3'], ['1603.00688-2-9-4', '1603.00688-3-9-4'], ['1603.00688-2-9-5', '1603.00688-3-9-5'], ['1603.00688-2-9-6', '1603.00688-3-9-6'], ['1603.00688-2-9-7', '1603.00688-3-9-7'], ['1603.00688-2-75-0', '1603.00688-3-75-0'], ['1603.00688-2-75-1', '1603.00688-3-75-1'], ['1603.00688-2-75-2', '1603.00688-3-75-2'], ['1603.00688-2-75-3', '1603.00688-3-75-3'], ['1603.00688-2-56-0', '1603.00688-3-56-0'], ['1603.00688-2-56-1', '1603.00688-3-56-1'], ['1603.00688-2-56-2', '1603.00688-3-56-2'], ['1603.00688-2-56-3', '1603.00688-3-56-3'], ['1603.00688-2-56-4', '1603.00688-3-56-4'], ['1603.00688-2-26-0', '1603.00688-3-26-0'], ['1603.00688-2-61-0', '1603.00688-3-61-0'], ['1603.00688-1-31-0', '1603.00688-2-31-0'], ['1603.00688-1-31-1', '1603.00688-2-31-1'], ['1603.00688-1-31-2', '1603.00688-2-31-2'], ['1603.00688-1-31-3', '1603.00688-2-31-3'], ['1603.00688-1-31-4', '1603.00688-2-31-4'], ['1603.00688-1-31-6', '1603.00688-2-33-0'], ['1603.00688-1-31-7', '1603.00688-2-33-1']]

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[]

[['1603.00688-1-36-1', '1603.00688-2-38-4'], ['1603.00688-1-36-3', '1603.00688-2-38-2'], ['1603.00688-2-36-8', '1603.00688-3-36-8']]

[]

['1603.00688-1-0-1', '1603.00688-1-0-2', '1603.00688-1-54-2', '1603.00688-1-54-4', '1603.00688-2-0-1', '1603.00688-2-0-2', '1603.00688-2-58-2', '1603.00688-2-58-4', '1603.00688-3-0-1', '1603.00688-3-0-2', '1603.00688-3-58-2', '1603.00688-3-58-4']

{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '3': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1603.00688

{'1603.00688-3-0-0': 'We present a numerical study of collisional transport in a tokamak pedestal in the presence of non-trace impurities, using the radially global [MATH] neoclassical solver Perfect [M. Landreman et al. 2014 Plasma Phys.', '1603.00688-3-0-1': 'Control.', '1603.00688-3-0-2': 'Fusion 56 045005].', '1603.00688-3-0-3': 'It is known that in a tokamak core with non-trace impurities present the radial impurity flux opposes the bulk ion flux to provide an ambipolar particle transport, with the electron transport being negligibly small.', '1603.00688-3-0-4': 'However, in a sharp density pedestal with sub-sonic ion flows the electron transport can be comparable to the ion and impurity flows.', '1603.00688-3-0-5': 'Furthermore, the neoclassical particle transport is not intrinsically ambipolar, and the non-ambipolarity of the fluxes extends outside the pedestal region by the radial coupling of the perturbations.', '1603.00688-3-0-6': 'The neoclassical momentum transport, which is finite in the presence of ion orbit-width scale profile variations, is significantly enhanced when impurities are present in non-trace quantities, even if the total parallel mass flow is dominated by the bulk ions.', '1603.00688-3-1-0': '# Introduction', '1603.00688-3-2-0': 'The global confinement in tokamaks is strongly correlated with the performance indicators of the edge transport barrier [CITATION].', '1603.00688-3-2-1': 'Accordingly, having a good pedestal performance in a fusion reactor is considered to be crucial.', '1603.00688-3-2-2': 'In the pedestal, turbulent transport is dramatically reduced by decorrelation of turbulence by strong sheared flows [CITATION], thus the relative role of the (otherwise negligible) collisional transport becomes more important.', '1603.00688-3-3-0': 'Fusion reactors will need to operate with carbon-free and low-erosion plasma facing components to achieve reasonable lifetimes and for nuclear safety.', '1603.00688-3-3-1': 'To gain more operational experience in the proposed metallic wall devices, originally carbon walled tokamak experiments, such as JET and ASDEX have transitioned to operation with "ITER-like walls" (with Be and W components) in recent years.', '1603.00688-3-3-2': 'With this transition a global confinement degradation has been observed [CITATION], especially on JET.', '1603.00688-3-3-3': 'This is a serious concern, since the data on which experimental scaling relations (and projections for ITER) are based have been mostly collected on carbon wall experiments.', '1603.00688-3-3-4': 'This degradation is found to be primarily caused by reduced pedestal performance [CITATION].', '1603.00688-3-3-5': 'Impurity injection appears to have a beneficial effect in certain cases [CITATION].', '1603.00688-3-3-6': 'Over and above the general importance of collisional impurity transport, these observations motivate consideration of the effects of non-trace impurities in the pedestal.', '1603.00688-3-4-0': 'Modeling of neoclassical transport in the pedestal is difficult, since the very sharp profile variations make gradient scale lengths comparable to the radial extent of the ion drift orbits [CITATION], which renders the usual radially local modeling inadequate for the problem.', '1603.00688-3-4-1': 'It has been demonstrated analytically that finite orbit width effects can significantly modify neoclassical transport phenomena [CITATION], producing modified ion heat transport, flows, bootstrap current, and even zonal flow response.', '1603.00688-3-4-2': 'These studies are limited to large aspect-ratio, circular cross-section plasmas and describe profile variations mostly through local plasma parameters, and point at the importance of a short scale global variation of flows [CITATION].', '1603.00688-3-4-3': 'In this paper we go beyond the large aspect-ratio limit, and present a radially global numerical study of neoclassical transport in the presence of non-trace impurities.', '1603.00688-3-5-0': 'There are different possible modeling options of various sophistication and difficulty.', '1603.00688-3-5-1': 'The simplest option is the computationally cheap and usually adopted local [MATH] formalism, which assumes small orbit width compared to profile length scales and neglects the radial coupling of the perturbations.', '1603.00688-3-5-2': 'On the other end, the global full-[MATH] approach includes both neoclassical and turbulent transport (these transport channels cannot be completely decoupled in a pedestal, in contrast to the local limit) and allows for strong deviations from local thermodynamic equilibrium.', '1603.00688-3-5-3': 'This approach requires nonlinear collision operators to fully live up to its promises[CITATION], and is currently numerically too expensive to be used for exploratory studies[CITATION].', '1603.00688-3-5-4': 'In this work, we use the global [MATH] formalism [CITATION].', '1603.00688-3-5-5': 'This is a specific generalization of the local [MATH] formalism which includes global effects, while still allowing the distribution function to be linearized around an appropriately chosen lowest order Maxwellian.', '1603.00688-3-5-6': 'This linearization, which assumes sub-sonic parallel flows, imposes limitations on the profiles that can be considered.', '1603.00688-3-5-7': 'Accordingly, we restrict ourselves to using suitable model profiles, but with experimentally identifiable features.', '1603.00688-3-6-0': 'In core plasmas the neoclassical perturbations of the ion distribution are only weakly affected by the electrons.', '1603.00688-3-6-1': 'We find that neoclassical particle transport in a pedestal with subsonic ion flows can be very different from that in the plasma core ([REF]a-c): The electron particle flux can be comparable to the ion particle flux even in the presence of non-trace impurities, and the collisional particle transport is not ambipolar in general.', '1603.00688-3-6-2': 'Consequently, it can happen that ions and impurities are transported in the same direction.', '1603.00688-3-6-3': 'In the presence of sharp profile variations the neoclassical momentum transport is nonzero, and we observe that its magnitude is notably affected by non-trace impurities ([REF]a).', '1603.00688-3-7-0': 'The remainder of this paper is organized as follows.', '1603.00688-3-7-1': 'In [REF], we describe the global [MATH] method implemented in Perfect, and how this affects our choice of model profiles.', '1603.00688-3-7-2': 'In [REF] we first present the neoclassical fluxes, flows and poloidal density variations for our baseline case, and then compare the results between plasmas with trace and non-trace impurity content.', '1603.00688-3-7-3': 'Finally, in [REF] we discuss our results and conclude.', '1603.00688-3-8-0': '# Methods', '1603.00688-3-9-0': 'In a tokamak core the plasma parameter profiles often exhibit sufficiently slow radial variations that the departure of the distribution function from a Maxwellian remains small, and the collisional dynamics can be described in terms of local plasma parameters.', '1603.00688-3-9-1': 'In this situation neoclassical transport can be calculated using the local [MATH] formalism that yields a linear system for the perturbed distribution.', '1603.00688-3-9-2': 'If the driving radial gradients are strong enough to generate substantial parallel particle flows, and poloidal variation in plasma parameters, the problem becomes nonlinear [CITATION].', '1603.00688-3-9-3': 'Furthermore, if all plasma parameters are allowed to vary over a radial width of a typical ion orbit, the transport becomes radially non-local.', '1603.00688-3-9-4': 'To study such general situations a global full-[MATH] simulation code with a nonlinear collision operator would be necessary.', '1603.00688-3-9-5': 'While a limited number of such simulation codes exist [CITATION], their computational expense make them unfit for our exploratory purposes.', '1603.00688-3-9-6': 'To keep the problem tractable, we will only consider situations when the distribution functions are not far from Maxwell-Boltzmann distributions, but finite orbit width effects are still important.', '1603.00688-3-9-7': 'For this purpose, we use the radially global, [MATH], Eulerian neoclassical solver Perfect.', '1603.00688-3-10-0': 'The fact that the distribution functions must be close to Maxwellians puts constraints on the profiles, as will be discussed in the following section.', '1603.00688-3-10-1': 'These constraints might not typically be satisfied in an experiment.', '1603.00688-3-10-2': 'Therefore, we do not attempt to base our exploratory modeling on specific experimental profiles, instead we use model profiles chosen specifically to satisfy the assumptions in Perfect, while they are supposed to be representative of experimental profiles in some respects.', '1603.00688-3-10-3': 'The specific profiles we use are presented in Appendix [REF].', '1603.00688-3-10-4': 'To explain the origin of the constraints, the next section contains a brief summary of the equations solved in Perfect (for a more detailed description of the code, we refer the reader tonumlandreman2014).', '1603.00688-3-11-0': 'In addition to the constraints outlined below, Perfect does not capture the geometry of an X-point or the open field line region.', '1603.00688-3-11-1': 'Orbit losses [CITATION] and an influx of neutral atoms [CITATION] are expected to become important very close to the separatrix.', '1603.00688-3-11-2': 'For this reason we expect that our results are representative only of the inner part of the pedestal.', '1603.00688-3-12-0': '## The global [MATH] problem solved by PERFECT', '1603.00688-3-13-0': 'Perfect solves for the non-adiabatic perturbed distribution function [EQUATION] where [MATH] is the distribution function, [MATH] is the charge, and [MATH] is the temperature of species [MATH], [MATH] is the perturbed potential, with the unperturbed electrostatic potential [MATH] taken to be a flux function, [MATH].', '1603.00688-3-13-1': 'The flux surface average is defined as [MATH], where [MATH] is a [MATH]-periodic angle-like poloidal coordinate and [MATH] is the magnetic field.', '1603.00688-3-13-2': 'The perturbation [MATH] is required to be small compared to the lowest order distribution function, which is a Maxwell-Boltzmann distribution [EQUATION] where the radial coordinate [MATH] is [MATH] times the poloidal magnetic flux, [MATH] is the mass, [MATH] is the total unperturbed energy, and [MATH] is the pseudo-density, with the density [MATH].', '1603.00688-3-13-3': 'The linearized equation that Perfect solves is [EQUATION] where [MATH], [MATH] with the velocity [MATH], the lowest order drift velocity [MATH] contains the lowest order [MATH] drift and magnetic drifts [MATH], [MATH] is the linearized Fokker-Planck operator, and [MATH] is a source term, which will be explained shortly.', '1603.00688-3-13-4': 'The partial derivatives are taken at fixed magnetic moment [MATH] and unperturbed total energy [MATH].', '1603.00688-3-14-0': 'Note that boundary conditions in [MATH] are needed to fully specify [MATH] by [REF], in contrast to the local equation.', '1603.00688-3-14-1': 'Since the local theory should apply sufficiently far from the pedestal, the result of local simulations - in which the [MATH] term is dropped from [REF] - are imposed as boundary conditions where particles enter the domain [CITATION].', '1603.00688-3-15-0': 'As inputs, Perfect requires zeroth order (flux function) densities [MATH], temperature [MATH] and potential [MATH].', '1603.00688-3-15-1': 'Given these equilibrium profiles, [MATH] is calculated from [REF], and appropriate velocity moments of [MATH] provide the neoclassical flows and fluxes.', '1603.00688-3-15-2': 'The fluxes will in general not be divergence free and thus incompatible with the time-independent equilibrium profiles.', '1603.00688-3-15-3': 'It may seem instructive to restore the time derivative in [REF], and solve a time-dependent problem, in a hope to reach a steady state equilibrium.', '1603.00688-3-15-4': 'However, only exceptional profiles would lead to a steady state solution.', '1603.00688-3-15-5': 'More generally, particles and energy would accumulate in (or leave) the simulation domain until the [MATH] approach breaks down.', '1603.00688-3-15-6': 'Instead, the approach adopted is to add spatially varying sources [MATH] so that the zeroth-order profiles become consistent; these sources are solved for in the code alongside [MATH].', '1603.00688-3-15-7': 'These sources can be thought of as representing the effects of non-neoclassical transport needed to make the profiles consistent, and should also be present in a real pedestal.', '1603.00688-3-16-0': 'To guarantee that [MATH], the driving gradients in the right-hand side of [REF] should remain small.', '1603.00688-3-16-1': 'From [EQUATION] where prime denotes the [MATH]-derivative, we see that the [MATH] and temperature gradients set the size of [MATH], and thus drive the deviations from a Maxwellian.', '1603.00688-3-16-2': 'Hence the density and the electrostatic potential may have sharp gradients as long as they produce a slowly varying [MATH].', '1603.00688-3-16-3': 'To quantify what we mean by a sharp gradient, we may balance the [MATH] and [MATH] terms in [REF] to find that [EQUATION] should be satisfied by [MATH] and [MATH]; that is, these quantities should have a small relative change as experienced by a particle during its radial drift excursion.', '1603.00688-3-16-4': 'Here [MATH] is the poloidal Larmor radius of the species, with the thermal speed [MATH], and the poloidal magnetic field [MATH].', '1603.00688-3-17-0': '## Model profiles and magnetic geometry', '1603.00688-3-18-0': 'Although we would like to use pedestal profiles which are representative of experiments in some aspects, we require [REF] to be satisfied in the simulations for all species ([MATH] for electrons, ions and impurities, respectively).', '1603.00688-3-18-1': 'We consider pedestals where [MATH] and [MATH] are allowed to vary on the [MATH] scale, while the [MATH] and [MATH] gradients of the bulk and impurity ion species are constrained by [REF].', '1603.00688-3-19-0': 'As a starting point we considered profiles from typical JET discharges (Figure 16 ofnum0029-5515-55-11-113031), and modified them as necessary to satisfy our orderings, and to reduce the need for heat sources in the simulation domain.', '1603.00688-3-19-1': 'The specific choices made in constructing the model profiles are discussed in Appendix [REF].', '1603.00688-3-19-2': 'The resulting input profiles for our baseline case as functions of the normalized poloidal flux [MATH] are shown in [REF]a-d.', '1603.00688-3-19-3': 'Here, we introduced [MATH], where [MATH] is the poloidal flux at the last closed flux surface (LCFS).', '1603.00688-3-19-4': 'Note that the density profile of impurities is much steeper than that of the bulk ions to make [MATH] vary slowly.', '1603.00688-3-19-5': 'We consider a deuterium plasma and fully ionized nitrogen impurities ([MATH], where [MATH]) with a concentration [MATH] in the core.', '1603.00688-3-19-6': 'The radii marking the beginning and the end of the pedestal are indicated by dotted vertical lines in the figures.', '1603.00688-3-20-0': 'In the simulations we use a local Miller model geometry [CITATION] (and neglect the radial variation of [MATH] and [MATH] in the domain), with elongation [MATH], [MATH], triangularity [MATH], [MATH], [MATH], [MATH], and inverse aspect ratio [MATH], with [MATH] and [MATH] denoting the minor and major radii, respectively.', '1603.00688-3-20-1': 'These parameters were taken fromnumbelli2008.', '1603.00688-3-21-0': 'The numerical resolution and convergence tests are detailed in Appendix [REF].', '1603.00688-3-22-0': '## Units', '1603.00688-3-23-0': 'An input quantity [MATH] is supplied to Perfect in a normalized, dimensionless form [MATH], with the normalizing, dimensional quantity [MATH].', '1603.00688-3-23-1': 'We choose [MATH] and [MATH] to be the major radius and magnetic field at the magnetic axis.', '1603.00688-3-23-2': 'Furthermore we choose the following convenient units [MATH], [MATH], [MATH] (deuterium mass), and [MATH] is the elementary charge.', '1603.00688-3-23-3': 'We define the reference speed and collision frequency as [MATH] and [EQUATION] where [MATH] denotes the vacuum permittivity and [MATH] is the Coulomb logarithm.', '1603.00688-3-23-4': 'The ordinary same-species collision frequency [MATH] is defined as [REF] but with [MATH] and [MATH] replaced by [MATH] and [MATH], from which we may define the collisionality as [MATH].', '1603.00688-3-24-0': 'To quantify when the local approximation fails, it is useful to define a few additional quantities: The normalized electric field [MATH] measures how much the [MATH] drift competes with parallel streaming in terms of poloidal particle motion.', '1603.00688-3-24-1': 'The normalized logarithmic derivative [MATH] measures the variation of quantity [MATH] along a typical drift orbit, where [MATH] and [MATH].', '1603.00688-3-24-2': 'Local neoclassical theory is valid only when [MATH], [MATH], [MATH], and [MATH] are all much smaller than unity in magnitude.', '1603.00688-3-24-3': 'The global [MATH] model also requires that [MATH] and [MATH], but [MATH] and [MATH] can be [MATH].', '1603.00688-3-24-4': 'These derived quantities together with [MATH] are shown for the baseline profiles in [REF]e-h.', '1603.00688-3-25-0': '# Local and global simulation results', '1603.00688-3-26-0': 'To study the differences between local and global neoclassical transport we performed a number of Perfect simulations with the profiles and magnetic geometry described in [REF].', '1603.00688-3-26-1': 'Before presenting the simulation results for our baseline set of profiles, we discuss the output quantities.', '1603.00688-3-27-0': 'We define the following normalized output quantities: sources, [MATH], with [MATH]; [MATH], with [MATH]; particle flux, [MATH]; momentum flux (divided by mass), [MATH]; heat flux, [MATH]; conductive heat fluxes, [MATH]; parallel flow velocity, [MATH]; parallel current, [MATH].', '1603.00688-3-27-1': 'The neoclassical flow coefficient [MATH] is defined so that it reduces to the flux function poloidal flow coefficient for a single ion species plasma in the local limit, [EQUATION] where [MATH] is the pressure.', '1603.00688-3-27-2': 'Furthermore, we define the non-adiabatic density perturbation [MATH], the total density perturbation [MATH], and normalized scalar fluxes [EQUATION] with [MATH]) representing [MATH], [MATH] or [MATH], [MATH] or [MATH], respectively), and [MATH].', '1603.00688-3-27-3': 'Note that the flux normalizations are species independent.', '1603.00688-3-28-0': '## Results for the baseline case', '1603.00688-3-29-0': 'The scalar fluxes of [REF] divided by [MATH] are shown in [REF].', '1603.00688-3-29-1': 'Throughout [REF] solid lines represent global simulation results, and dashed lines represent local ones.', '1603.00688-3-30-0': 'When the impurity strength [MATH] is order unity - as in our baseline - the magnitude of the particle transport of electrons is typically [MATH] smaller than that of the impurities, where [MATH] denotes the effective ion charge.', '1603.00688-3-30-1': 'In this case it is common to neglect [MATH] and calculate the ion particle transport from ambipolarity [MATH], leading to opposing ion and impurity particle fluxes.', '1603.00688-3-30-2': 'The local simulations in the core region obey these expectations.', '1603.00688-3-30-3': 'Since [MATH] is small in our baseline, temperature screening does not dominate, thus the local fluxes obey [MATH] and [MATH].', '1603.00688-3-30-4': 'In the pedestal the parallel ion-electron friction can be sufficiently large to compete with the ion-impurity friction, due to the high electron flow speeds.', '1603.00688-3-30-5': 'Therefore [MATH] cannot be neglected anymore in the ambipolarity condition: the strong outward electron flux means that both [MATH] and [MATH] are positive simultaneously.', '1603.00688-3-30-6': 'Thus the outward local [MATH] is not a result of temperature screening.', '1603.00688-3-31-0': 'As expected from the small [MATH] and [MATH] values, seen in [REF]e and f, the electron local and global fluxes are practically the same.', '1603.00688-3-31-1': 'However, finite orbit width effects strongly affect the ion and impurity dynamics.', '1603.00688-3-31-2': 'In the pedestal [MATH] is increased compared to the local value, which causes [MATH] to change sign compared to its local value.', '1603.00688-3-31-3': 'It is worth noting that the deviation between local and global results is not localized to the pedestal region only.', '1603.00688-3-31-4': 'For instance, the global and local [MATH] deviate well below [MATH]; the ion particle flux changes sign at [MATH] while the local result is positive everywhere.', '1603.00688-3-31-5': 'As a comparison we note, that the width of the large gradient region in [MATH] units is approximately [MATH], and the orbit width of a typical trapped ion at thermal speed is [MATH].', '1603.00688-3-32-0': 'The somewhat surprising observation that the finite orbit width effects extend outside the pedestal over several thermal ion orbit widths is worth a moment of thought.', '1603.00688-3-32-1': 'Since the existing analytical theories assume [MATH], which partly eliminates the radial coupling, they can only provide a limited guidance as to why this happens.', '1603.00688-3-32-2': 'The only radial coupling that cannot be completely eliminated from those theories is that due to the neoclassical parallel flow, [MATH]; indeed it is not a parameter in the theory, but it satisfies a radial differential equation (Eqs. (43) and (64) in Ref. catto13).', '1603.00688-3-32-3': 'As we will see, in our case the global result for the main ion [MATH] is very different from the local one, and it takes a rather long distance from the pedestal before it gets close to the local result.', '1603.00688-3-32-4': 'Also, in estimating the orbit width above, we considered particles at the thermal speed, while all the quantities of interest are dominated by super-thermal particles with wider orbits.', '1603.00688-3-32-5': 'The neoclassical drive and the radial coupling terms both include [MATH], the velocity space integration weight is [MATH], and the flow, particle flux, and heat flux contain an additional factor of [MATH], [MATH], or [MATH] respectively.', '1603.00688-3-33-0': 'The global particle fluxes are not ambipolar as seen at [MATH], where [MATH] and [MATH] are both inward, and [MATH] is small, very close to its local value.', '1603.00688-3-33-1': 'Note that while in a local simulation the radial current and momentum flux should vanish, it does not need to be so in global simulations, as pointed out innumlandreman2014.', '1603.00688-3-34-0': 'For both ions and impurities we observe that the conductive heat flux can significantly differ from the local value.', '1603.00688-3-34-1': 'A reduction compared to the the local value - observed around the pedestal top - may be explained by the shift of the trapped region towards the tail of the distribution at [MATH].', '1603.00688-3-34-2': 'However, we also find regions where the heat fluxes increase from their local values.', '1603.00688-3-34-3': 'Just as for the particle fluxes, we see a reduction inside the pedestal top: [MATH]) reaches a minimum at [MATH]).', '1603.00688-3-35-0': 'The corresponding sources are presented in [REF].', '1603.00688-3-35-1': 'We use poloidally symmetric sources with speed dependencies [MATH] for particle and heat sources, respectively, where [MATH].', '1603.00688-3-35-2': 'We see that to some degree the main ion particle and heat sources qualitatively mirror each other, and tend towards zero outside the pedestal.', '1603.00688-3-35-3': 'We specifically choose [MATH] to reduce the need for [MATH], while [MATH] also remains small, because [MATH] tends to be smaller than [MATH].', '1603.00688-3-35-4': 'The tremendous drop in [MATH] in the pedestal leads to a sharp peak in [MATH], and again we see an opposing trend for [MATH], but for impurities the combined sources are positive.', '1603.00688-3-35-5': 'The electron sources are localized to the pedestal and comparable in size to those of the main ions.', '1603.00688-3-36-0': 'The neoclassical flow coefficients, [MATH], are presented in [REF].', '1603.00688-3-36-1': 'The local (dashed lines) [MATH] is positive as expected in the banana regime, and exhibits a slight variation as a response to the radial variation of [MATH].', '1603.00688-3-36-2': 'In the local case [MATH] is a flux function, while globally it varies from the inboard side (darker curves) to the outboard side (lighter curves).', '1603.00688-3-36-3': 'These poloidal variations in the flow appear together with poloidal density variations, as will be discussed shortly.', '1603.00688-3-36-4': 'It has been shown analytically [CITATION] that [MATH] is affected by finite orbit width effects.', '1603.00688-3-36-5': 'As seen from the analytical results, where [MATH] is kept but the radial coupling is neglected, [MATH] is expected to decrease or become more negative in both the banana and plateau regimes [CITATION].', '1603.00688-3-36-6': 'However, the global modification to [MATH] is not a function of local plasma parameters (such as [MATH]) only, but even in the semi-global treatment ofnumcatto13 it satisfies a radial differential equation (i.e. [MATH] cannot be neglected).', '1603.00688-3-36-7': 'This is why [MATH] can differ in sign and magnitude from the local value well outside the pedestal, and can be larger than its local value inside the pedestal.', '1603.00688-3-36-8': 'It is interesting to note that there is a difference between global and local [MATH] even for the electrons, which is due to the collisional coupling to the various ion species.', '1603.00688-3-37-0': 'The density perturbations are shown in [REF].', '1603.00688-3-37-1': 'We first consider only the non-adiabatic contribution to the perturbations, [MATH], shown in [REF]a-f, where the global (local) results are plotted in the a-c (d-f) panels.', '1603.00688-3-37-2': 'The local simulations predict purely up-down density asymmetry for all species, which is weaker for bulk ions and electrons than for impurities.', '1603.00688-3-37-3': 'The global results show a more complex poloidal density variation for both ions and impurities.', '1603.00688-3-37-4': 'For ions we see an in-out asymmetry (i.e. excess density around [MATH]) at the pedestal top, which transforms into an out-in asymmetry in the pedestal, and reverses again further out (similarly to the single species simulations ofnumpusztaiRFNEO).', '1603.00688-3-37-5': 'For impurities the most important difference compared to the local results is the weak in-out (instead of strong up-down) asymmetry in the pedestal.', '1603.00688-3-37-6': 'The electron density perturbation mostly follows its local behavior, exhibiting a large increase in the up-down asymmetry in the pedestal.', '1603.00688-3-37-7': 'To understand the total density perturbation [MATH] in global simulations we note that the potential perturbation [MATH] follows mostly the non-adiabatic ion density perturbation.', '1603.00688-3-37-8': 'The total electron density perturbation is dominated by the adiabatic response of electrons, thus it is very similar to the ion density perturbation.', '1603.00688-3-37-9': 'The impurity density variations show a competition between adiabatic response - especially in the pedestal where it tries to oppose the ion density perturbation - and non-adiabatic response.', '1603.00688-3-37-10': 'The relative impurity density variation stays below [MATH] everywhere, showing that the assumption of the density being nearly a flux function is not violated.', '1603.00688-3-37-11': 'However, for sufficiently high [MATH], nonlinearity from poloidal asymmetries can arise [CITATION].', '1603.00688-3-38-0': 'Finally, we consider the parallel flows and the bootstrap current for our baseline case; these are plotted in [REF].', '1603.00688-3-38-1': 'For the bulk and impurity ions the flows remain small inside the pedestal, as their profiles were chosen specifically so that their diamagnetic and [MATH] flows mostly cancel.', '1603.00688-3-38-2': 'For these the relative deviation between the local and global results is significant, the global results being larger in magnitude from the middle of the pedestal inward.', '1603.00688-3-38-3': 'This is partly due to the reduction in [MATH] compared to the local value observed in [REF].', '1603.00688-3-38-4': 'The parallel flow of electrons reaches a much higher magnitude inside the pedestal then that of the ions, where the strong [MATH] and diamagnetic rotation contribute with the same sign for this species.', '1603.00688-3-38-5': 'The scale is therefore different and the difference between the local and global results is less visible.', '1603.00688-3-38-6': 'Figure [REF]e shows this difference, which is comparable to what is observed for ions.', '1603.00688-3-38-7': 'This is expected, because all the difference is due to a frictional coupling to the various ion species with modified flow speeds, as direct finite orbit width effects are negligible for electrons.', '1603.00688-3-38-8': 'Since we have a weak ion temperature pedestal, the modifications of the ion flows are not sufficient to cause an appreciable deviation of the bootstrap current from the local result inside the pedestal, as seen in [REF]a.', '1603.00688-3-38-9': 'If anything, outside the pedestal there is a slight difference between the local and global results (barely visible on the scale accommodating the huge bootstrap peak in the pedestal), since the electron flow is relatively small in those regions, while the global effects on [MATH] extend outside the pedestal.', '1603.00688-3-39-0': '## Non-trace impurities', '1603.00688-3-40-0': 'We want to assess whether, and how, non-trace impurities can affect collisional transport in the pedestal.', '1603.00688-3-40-1': 'To this end, we performed simulations with profiles similar to our baseline, except that we scaled the impurity concentration profiles.', '1603.00688-3-40-2': 'The simulations shown in this section have impurity concentrations such that the impurity strength in the core is either [MATH] (trace impurities; shown with thin lines in the figures) or [MATH] (non-trace; thick lines); as a comparison [MATH] for the baseline.', '1603.00688-3-40-3': 'Note that [MATH] drops rapidly across the pedestal, thus the impurities eventually become trace even if they have a high core concentration (this is necessary in the presence of an electron density pedestal if we restrict profile variations to obey the orderings of [REF] for all ion species).', '1603.00688-3-41-0': 'The a-c panels of [REF] compare the local and global particle fluxes with trace and non-trace impurity concentration.', '1603.00688-3-41-1': 'As usual, the local and global results for [MATH] are virtually the same.', '1603.00688-3-41-2': 'At higher impurity content the increase in [MATH] around [MATH] reflects the increased [MATH] gradient due to the rapidly varying impurity concentration (note that [MATH] is kept fixed in the impurity scan, thus the [MATH] profile changes).', '1603.00688-3-41-3': 'The local [MATH] behaves as expected from the ambipolarity condition: it increases with impurity content in the core to balance the inward [MATH], and its core behavior is dominated by following the outward electron flux.', '1603.00688-3-41-4': 'It is interesting to note that while above [MATH] the local curves for different impurity concentration collapse onto each other because of the low [MATH], the difference in the global [MATH] survives much further out in the pedestal.', '1603.00688-3-42-0': 'The d-f panels of [REF] show the heat fluxes.', '1603.00688-3-42-1': 'In the core and close to the pedestal top the global results for [MATH] are lower than the local values.', '1603.00688-3-42-2': 'This reduction is somewhat stronger in the presence of impurities.', '1603.00688-3-42-3': 'The local results for [MATH] and [MATH] are even higher in the pedestal at high impurity concentration, than would be expected simply due to the linear increase with [MATH].', '1603.00688-3-42-4': 'This extra increase is reduced by the global effects so that the global results at different concentrations are closer to each other.', '1603.00688-3-43-0': 'The momentum flux values for the different species (shown in [REF]g-i) vanish in the local limit, as they should, but are finite when global effects are considered.', '1603.00688-3-43-1': 'Unlike [MATH] and [MATH], the global results for [MATH] are different to the local ones, i.e. they are also finite, although the electrons do not transport an appreciable amount of momentum, as [MATH].', '1603.00688-3-43-2': 'The total momentum flux, [MATH], shown in [REF]a, is mostly negative in the studied radial range, and is strongly increase by the presence of non-trace impurities.', '1603.00688-3-43-3': 'The ions are responsible for most of the momentum transport for both impurity concentrations, thus the increase in [MATH] is not due to the increase in [MATH], but the modifications in [MATH] in the presence of impurities.', '1603.00688-3-44-0': 'The radial current [MATH] ([REF]b) - that in isolation from other transport channels would lead to charge separation and the evolution of the radial electric field - is also significantly increased in magnitude by the presence of the impurities over most of the studied radial domain.', '1603.00688-3-44-1': 'Although we observe a finite radial neoclassical current, we do not attempt to self-consistently calculate the radial electric field.', '1603.00688-3-44-2': 'For the interpretation of the momentum fluxes and the radial current it is useful to note that conservation of particle number and parallel momentum (in steady state, for a radially constant [MATH], and with sources even in [MATH]) imply that [MATH] and [MATH].', '1603.00688-3-44-3': 'The latter property is apparent from a comparison of [REF]a and b.', '1603.00688-3-44-4': 'The former relation states that non-ambipolar fluxes require particle sources for which [MATH] is not satisfied locally (the factor [MATH] comes from the assumed velocity space structure of [MATH]).', '1603.00688-3-45-0': 'In radially local formalisms momentum transport is often decomposed into diffusive ([MATH], where [MATH] is the normalized parallel mass flow), conductive ([MATH]) and intrinsic (independent of [MATH]) terms.', '1603.00688-3-45-1': 'Such a decomposition is not possible in our global formalism since the parallel mass flow is a non-local function of the various plasma parameter profiles, and so is the momentum transport.', '1603.00688-3-45-2': 'Nevertheless, it is instructive to compare the radial profile of the total momentum transport [MATH] to the radial variation of the total mass flow [MATH] shown in [REF]c (both quantities are dominated by the main ion contributions).', '1603.00688-3-45-3': 'By increasing the impurity content, [MATH] increases in the core region.', '1603.00688-3-45-4': 'This is caused by the higher [MATH] at higher [MATH] not being compensated by the slight reduction in [MATH] (shown in [REF]).', '1603.00688-3-45-5': 'As the impurity concentration drops radially across the pedestal, the effect from the reduction in the parallel ion flow becomes dominant.', '1603.00688-3-45-6': 'The radial drop in the global results for [MATH] across the pedestal is mostly due to the density variation in the pedestal.', '1603.00688-3-45-7': 'If the transport was local and purely diffusive this non-monotonic behavior of [MATH] would be accompanied by a sign change in the momentum transport.', '1603.00688-3-45-8': 'It is also interesting to note that at the point where we see the greatest relative increase in [MATH] between the different simulations ([MATH]), the global [MATH] becomes lower for higher core impurity concentration.', '1603.00688-3-46-0': 'The ion particle source profile shown in [REF]a is only weakly affected by the presence of non-trace impurities.', '1603.00688-3-46-1': 'In the meantime, the increase in impurity sources are approximately proportional to the increase in their concentration: the normalized particle source [MATH] is approximately the same in the two simulations, considering that the impurity content changes by a factor [MATH].', '1603.00688-3-47-0': 'The radial current and the non-quasineutral particle sources [MATH] are consistent, that is, the divergence of the radial current is given by the charge source [MATH] (note, that the total charge source integrated across the pedestal is zero, due to the boundary conditions).', '1603.00688-3-47-1': 'However, the radial current is truly a consequence of the radial coupling in the global simulation, and not an artifact of the radially varying sources.', '1603.00688-3-47-2': 'Although it is not done in the code, source profiles could be calculated in the presence of the radially varying local particle and heat fluxes, for these to be consistent with the time independent plasma parameter profiles.', '1603.00688-3-47-3': 'Such sources would be quasineutral unlike those in the global simulations.', '1603.00688-3-48-0': 'In reality, the neoclassical radial current that we observe needs to be balanced by an opposing radial current, which represents transport processes not captured by our model (turbulence, atomic physics processes, orbit losses, magnetic ripple effects, etc.).', '1603.00688-3-48-1': 'Otherwise the system would not be steady state, because the radial electric field would vary in time and the [MATH] torque would change the plasma flows.', '1603.00688-3-48-2': 'Due to the construction of the code and the vanishing [MATH] moment of our sources, we observe the [MATH] and the radial current terms in the species-summed flux surface averaged angular momentum equation to exactly balance.', '1603.00688-3-48-3': 'Similarly, the finite neoclassical momentum transport predicted by the code should also be canceled by a momentum transport due to non-neoclassical processes, in steady state.', '1603.00688-3-49-0': 'Finally we would like to assess how much our profile choices affect the observation of increased momentum flux in the presence of non-trace impurities.', '1603.00688-3-49-1': 'One unnatural feature of our impurity density profile is its extreme steepness.', '1603.00688-3-49-2': 'Allowing [MATH] have a radial drop across the pedestal - within what is allowed by [REF] - to reduce the inward radial electric field, together with allowing [MATH] to have an increase in the same region, leads to a less sharp impurity pedestal.', '1603.00688-3-49-3': 'To further reduce the impurity density gradient in the pedestal we consider a fully ionized beryllium impurity ([MATH]).', '1603.00688-3-49-4': 'These changes make our pedestal less deeply sub-sonic, as seen from the increased mass flow in [REF]c that shows the results of the modified input profiles.', '1603.00688-3-49-5': 'From [REF]a we can conclude that impurities also significantly increase the magnitude of the neoclassical momentum flux in the pedestal for more natural impurity profiles, when the inputs push the limitations imposed by the required orderings.', '1603.00688-3-50-0': '# Discussion and conclusions', '1603.00688-3-51-0': 'We have studied the differences in the collisional transport between radially local and global formalisms using the global [MATH] neoclassical solver Perfect, with a special emphasis on the effects of non-trace impurities.', '1603.00688-3-51-1': 'If the impurity density profile is arbitrary, impurities are likely to develop sonic flows and strong poloidal asymmetries in the pedestal.', '1603.00688-3-51-2': 'We use model profiles specifically chosen to guarantee that impurity flows remain sub-sonic, so that the assumptions of the [MATH] formalism are valid.', '1603.00688-3-51-3': 'In this way we can gain some insights into the effect of non-trace impurities without the need for a non-linear collision operator.', '1603.00688-3-52-0': 'In an impure plasma the electron particle transport is usually negligible, and the ion and impurity fluxes oppose each other to maintain ambipolarity.', '1603.00688-3-52-1': 'However, since in the pedestal the magnitude of the parallel electron flow can be much larger than the parallel ion and impurity flows, the friction of the various ion species on electrons can become non-negligible.', '1603.00688-3-52-2': 'Consequently, a substantial neoclassical electron particle transport can arise, which competes with the radial transport of other species.', '1603.00688-3-52-3': 'In particular, the ion and impurity fluxes can have the same sign.', '1603.00688-3-52-4': 'In the presence of strong radial profile variations on the [MATH] scale, the ambipolarity of fluxes is violated, as reported in previous studies [CITATION].', '1603.00688-3-52-5': 'We emphasize that the differences in neoclassical flows and fluxes between the local and global theory are not restricted to the pedestal region only but die off within a distance comparable with the ion orbit width.', '1603.00688-3-52-6': 'In our baseline from the outer core region to the pedestal top we observe that both the ion and impurity fluxes are inward due to global effects (see [REF]a-c).', '1603.00688-3-52-7': 'Inside the pedestal, local simulations predict both the impurity and the ion fluxes to be outward, due to the large outward electron flux, but in global simulations the impurity flux reverses to be inward.', '1603.00688-3-53-0': 'For our model profiles, chosen to give small parallel ion flows, the bootstrap current remains almost completely unaffected, while this need not be so if the ion temperature varies more rapidly than allowed by our orderings.', '1603.00688-3-53-1': 'Indeed, the neoclassical parallel flow coefficient [MATH] of ions and impurities is significantly affected by global effects.', '1603.00688-3-53-2': 'Due to the radial coupling even the sign of [MATH] is different from the local result well inside of the pedestal top.', '1603.00688-3-54-0': 'We observe that in the presence of global effects the poloidal variation of the density perturbation is not restricted to an up-down asymmetry, it can be more significant, and can develop rapid radial variations (in accordance with previous numerical results [CITATION]).', '1603.00688-3-54-1': 'The poloidal asymmetries observed here arise in the presence of finite inverse aspect ratio and finite orbit width effects, and are not closely related to those predicted by the analytical theories in [CITATION], since those require significant poloidal variations in the ion-impurity friction.', '1603.00688-3-54-2': 'The adiabatic response to the electrostatic perturbation generated by the ions competes with non-adiabatic contributions in determining the poloidal asymmetries developed by the impurities.', '1603.00688-3-54-3': 'For the moderate impurity charge considered here ([MATH]) the relative poloidal variation of the impurity density is still small so that the perturbative treatment remains valid.', '1603.00688-3-54-4': 'At sufficiently high [MATH] the relative poloidal variation of the impurity density is expected to become order unity and then - if the impurities are non-trace - nonlinear effects would start to play a role.', '1603.00688-3-54-5': 'The numerical investigation of that situation is left for a future study.', '1603.00688-3-55-0': 'In the pedestal, the neoclassical radial current does not vanish in general, and the momentum transport remains finite - in contrast to the local theory.', '1603.00688-3-55-1': 'In the vicinity of the pedestal the total neoclassical momentum transport is found to be negative in the studied case, which happens if the charge sources (resulting from the particle sources needed to sustain the pedestal) are mostly positive in that region.', '1603.00688-3-55-2': 'The radial variation of the parallel mass flow is non-monotonic with a sharply decreasing feature where the density drops in the pedestal.', '1603.00688-3-55-3': 'This non-monotonicity is not reflected in the radial momentum transport, which is now a non-local function of the mass flow.', '1603.00688-3-55-4': 'A simple decomposition of the momentum transport into diffusive, conductive and intrinsic terms is not possible in the global picture.', '1603.00688-3-55-5': 'Note that the same is also true for all the radial fluxes.', '1603.00688-3-56-0': 'We observe a strong effect of impurities on the total momentum transport, the magnitude of which increases significantly in the presence of non-trace impurities (see [REF]a).', '1603.00688-3-56-1': 'While it is not possible to disentangle the exact cause of this, it may be due to the sharp radial variation of the parallel impurity mass flow.', '1603.00688-3-56-2': 'The impurities represent only a minor fraction of the total mass flow, but their collisional coupling to the main ions is significant at an impurity strength of order unity.', '1603.00688-3-56-3': 'Pushing the limitations of our orderings we reduced the sharpness of the impurity density profile, in an attempt to demonstrate the robustness of the impact of impurities on neoclassical momentum transport.', '1603.00688-3-56-4': 'Although the impurity profile is chosen to be very specific, strong effects may occur when the impurities have more general density variations, especially because they then have much stronger relative flow speeds compared to the ions.', '1603.00688-3-57-0': 'Impurity seeding in tokamaks operating with ITER-like metallic walls has been experimentally found to have beneficial effects on the pedestal performance.', '1603.00688-3-57-1': 'It is then natural to raise the question of whether our results are consistent with this observation.', '1603.00688-3-57-2': 'Without taking into account other non-intrinsically ambipolar processes and turbulent transport it is not possible to evolve the profiles towards a steady state.', '1603.00688-3-57-3': 'However, we may speculate about possible consequences of the increased momentum transport in the presence of impurities.', '1603.00688-3-57-4': 'In steady state the neoclassical radial current and momentum transport should be balanced by opposing contributions of turbulent and other origin.', '1603.00688-3-57-5': 'An increased neoclassical momentum transport in the presence of impurities requires these contributions to increase as well.', '1603.00688-3-57-6': 'Sufficiently far from the open field line region the turbulent transport can dominate these opposing contributions.', '1603.00688-3-57-7': 'Impurities tend to reduce the turbulence level by dilution [CITATION].', '1603.00688-3-57-8': 'If the non-diffusive turbulent momentum transport is to be increased in spite of dilution effects, stronger deviations of the non-fluctuating distribution from a Maxwellian [CITATION] and stronger profile variations [CITATION] may be necessary; which could require a steepening of the pedestal to reach a new steady state.', '1603.00688-3-57-9': 'Whether this is indeed the case and, if so, its role in the observed confinement improvement in impurity seeded discharges, remains an open question and should be the basis of future investigation.', '1603.00688-3-58-0': 'The authors are grateful for J. Omotani, T. Fulop and S. Newton for fruitful discussions and instructive comments on the paper.', '1603.00688-3-58-1': 'IP and SB were supported by the International Career Grant of Vetenskapsraadet (Dnr.', '1603.00688-3-58-2': '330-2014-6313), and ML was supported by the U.S. DoE under award numbers DEFG0293ER54197 and DEFC0208ER54964.', '1603.00688-3-58-3': 'The simulations used computational resources of Hebbe at C3SE (project nr.', '1603.00688-3-58-4': 'C3SE2016-1-10 and SNIC2016-1-161).', '1603.00688-3-59-0': '# The construction of model pedestal profiles', '1603.00688-3-60-0': 'Here, we describe a method to construct appropriate densities given [MATH], [MATH], [MATH] profiles satisfying [REF], and an arbitrary [MATH].', '1603.00688-3-60-1': 'The orderings for the electrons are not a concern, since [MATH] is much smaller than the radial scale length of any profile in an experiment.', '1603.00688-3-61-0': 'The density profiles and potential discussed in this section are the inputs to Perfect and are thus, strictly speaking, only the zeroth order contributions to these quantities, assumed to be flux functions.', '1603.00688-3-62-0': 'Once we have specified both [MATH] and [MATH] (to be detailed shortly), the relation [MATH] gives the potential [EQUATION]', '1603.00688-3-62-1': 'A strong electric field from [MATH] compensates for the potentially large ion pressure drop to make the variation of [MATH] small (the situation of "electrostatic ion confinement," which is borne out in experimental results [CITATION]).', '1603.00688-3-62-2': 'In the radial force balance the ion pressure drop could in principle be balanced by a sonic ion flow, however we consider sub-sonic ion flows.', '1603.00688-3-62-3': 'Note that by allowing sonic ion flows, the relative flow speed of two ion species would in general also be sonic, which would severely complicate the treatment of collisions.', '1603.00688-3-62-4': 'Such a scenario is currently not supported by Perfect.', '1603.00688-3-62-5': 'We emphasize that [MATH] and [MATH] are fundamental inputs to the code, thus the relation [REF] is not an attempt to self-consistently calculate the potential.', '1603.00688-3-63-0': 'Then, given [MATH] from ([REF]) and an [MATH], we obtain the impurity density [EQUATION]', '1603.00688-3-63-1': 'Note that ([REF]) leads to impurity density profiles with a typical logarithmic density gradient in the pedestal [MATH] times larger than that of the main ions; this is unavoidable if [REF] is to be satisfied for all ion species.', '1603.00688-3-63-2': 'Finally, the electron density is obtained by demanding quasi-neutrality [EQUATION]', '1603.00688-3-63-3': 'The resulting [MATH] satisfies [REF] by virtue of the electron gyroradius being small.', '1603.00688-3-64-0': 'Although we have constraints and relations between profiles, we still have a large degree of freedom in specifying them.', '1603.00688-3-64-1': 'Here we discuss some specific choices we made for the profile set used as a baseline for the simulations.', '1603.00688-3-65-0': 'Since we use the local solution for the boundary condition, we also need to make sure that the assumptions of the local theory are satisfied at the boundary.', '1603.00688-3-65-1': 'This means that the sharp density or potential variations should be limited to the middle of the radial domain, sufficiently far from the boundaries.', '1603.00688-3-65-2': 'For weak electric fields the local and global results should agree, so we choose the [MATH] profile so that the potential calculated from [REF] is completely flat at the boundaries.', '1603.00688-3-65-3': 'We do this by letting [EQUATION] in the vicinity of the inner boundary, namely in the core up to the pedestal top where [MATH] (and so [MATH]) should be slowly varying.', '1603.00688-3-65-4': 'Thus [MATH] in this region, independent of the [MATH] profile.', '1603.00688-3-65-5': 'To achieve a flat potential in the vicinity of the outer boundary, we let [EQUATION] in that region, where [MATH] is a constant which fixes the value of [MATH].', '1603.00688-3-65-6': 'To remove the ambiguity of [MATH] far from the boundaries in a way that makes it a simple and smooth function, we linearly extrapolate [MATH] given by [REF] from the core region up to the bottom of the pedestal (where the sharp feature in [MATH] ends) and match it to the other expression [REF], choosing [MATH] from [REF] at the matching point.', '1603.00688-3-66-0': 'The [MATH] profile is chosen to be a linear function of [MATH] over the whole domain, with a logarithmic gradient matching that of [MATH] at the left boundary.', '1603.00688-3-67-0': 'Since the global [MATH] ordering does not allow an ion temperature pedestal, we consider [MATH] profiles with a gradient across the pedestal (and further out) equal to that of the electron temperature gradient in the core.', '1603.00688-3-67-1': 'Due to the density drop in the pedestal and the decreasing temperature profile, the ion heat flux is bound to be vastly different at the two boundaries, requiring large heat sources in the domain, except if a variation in [MATH] balances them.', '1603.00688-3-67-2': 'This can be avoided by setting a proxy for the heat fluxes, [MATH], to be equal at the boundaries.', '1603.00688-3-67-3': 'Thus, we artificially reduce [MATH] in the core to remove the need for sources close to the boundaries.', '1603.00688-3-68-0': 'Experimental density and temperature profiles can drop orders of magnitude across the pedestal, and a significant part of these variations occur in the open field line region.', '1603.00688-3-68-1': 'As an additional consideration, if we let the bulk densities and temperatures drop across the simulation domain as much as in a real pedestal, it would lead to difficulties related to large logarithmic gradients, and huge changes in collision frequency.', '1603.00688-3-68-2': 'Since the code does not capture the physics in the open field line region (where the pedestal foot would be in an experiment), the region outside the middle of the pedestal does not carry too much physical relevance and can be considered as a numerical buffer zone.', '1603.00688-3-68-3': 'To avoid the above mentioned complications we reduce the gradients starting from a point where crossing the separatrix would be expected in an experiment.', '1603.00688-3-68-4': 'We thus arbitrarily pick this reduced [MATH] to be [MATH] of its pedestal value, with [MATH] equal to its core value.', '1603.00688-3-69-0': 'Motivated by typical JET discharges (Figure 16 of 0029-5515-55-11-113031), for our baseline case we choose the [MATH] and [MATH] pedestal widths to be [MATH], with the values [MATH] and [MATH] at the pedestal top.', '1603.00688-3-69-1': 'These correspond to typical logarithmic gradients of about [MATH] and [MATH], where [MATH] is the minor radius defined as half of the width of the flux surface at the elevation of its centroid, and [MATH] at the last closed flux surface.', '1603.00688-3-69-2': 'The [MATH] and [MATH] profiles are generated by Bezier curve interpolation between three regions with linear profile variation: a core, a pedestal, and an outer "buffer" region.', '1603.00688-3-69-3': '(Using this type of smoothing ensures that the gradients transition smoothly and monotonically.)', '1603.00688-3-70-0': 'To transform the profiles from [MATH] to [MATH] space, [MATH] is needed.', '1603.00688-3-70-1': 'This we obtain from [MATH] where [MATH] is the toroidal magnetic flux which we calculate assuming simple elongated flux surfaces.', '1603.00688-3-70-2': 'Since we do not intend to model a specific experiment, we take model profiles for the safety factor [MATH] and the elongation [MATH], shown in [REF].', '1603.00688-3-70-3': 'The [MATH] was chosen as [MATH].', '1603.00688-3-70-4': 'For the on-axis toroidal magnetic fields we take [MATH] and neglect [MATH] corrections together with higher order shaping effects to get [MATH].', '1603.00688-3-70-5': 'The resulting [MATH] is taken to be constant across the entire pedestal.', '1603.00688-3-71-0': 'The ion temperature gradient, the ion and impurity [MATH] and the impurity strength profiles are shown in [REF], for the different impurity concentration simulations presented in this paper.', '1603.00688-3-72-0': '# Numerical resolution', '1603.00688-3-73-0': 'The simulations used [MATH] radial grid points and cover a domain of [MATH] (the whole [MATH] domain is not shown in the figures).', '1603.00688-3-73-1': 'The number of poloidal grid points is [MATH].', '1603.00688-3-73-2': 'The number of expansion polynomials in the pitch angle cosine [MATH] is [MATH] for the distribution function and [MATH] for the Rosenbluth potentials (RP).', '1603.00688-3-73-3': 'The number of speed grid points is [MATH] and [MATH] for the RP.', '1603.00688-3-74-0': 'To demonstrate the degree of convergence, in [REF] we present the particle sources for the baseline simulation for the above mentioned resolution (red curve), together with four other cases, where we increase [MATH] to [MATH] (violet), [MATH] to [MATH] (blue), [MATH] to [MATH] (cyan), and the radial domain size to [MATH] (yellow), with all other parameters kept fixed.', '1603.00688-3-74-1': 'These four are the resolution parameters to which the accuracy of the solution is most sensitive.', '1603.00688-3-74-2': 'The results are almost identical except for slight differences near sharp features.', '1603.00688-3-75-0': 'To quantify the error, we define [MATH].', '1603.00688-3-75-1': 'Taking the [MATH] simulation as the reference [MATH], we find that the quantity with the highest error is the electron particle source, with [MATH].', '1603.00688-3-75-2': 'For other quantities, such as particle flows and radial fluxes, we observe errors below [MATH], thus [REF] presents the most stringent test for numerical convergence.', '1603.00688-3-75-3': 'The same convergence test was performed for all the simulations, with resulting errors (including those of the sources) all being below [MATH].'}

1305.4894

{'1305.4894-1-0-0': "We prove an asymptotic version of a conjecture by Varagnolo and Vasserot on an equivalence between the category [MATH] for a cyclotomic Rational Cherednik algebra and a suitable truncation of an affine parabolic category [MATH] that, in particular, implies Rouquier's conjecture on the decomposition numbers in the former.", '1305.4894-1-0-1': "Our proof uses multiple ingredients: a ramification of Rouquier's deformation approach, categorical actions on highest weight categories and related combinatorics, and derived equivalences produced from Procesi bundles on symplectic resolutions of quotient singularities.", '1305.4894-1-1-0': '# Introduction', '1305.4894-1-2-0': 'Rational Cherednik algebras were introduced by Etingof and Ginzburg, [CITATION].', '1305.4894-1-2-1': 'These are associative algebras over [MATH] constructed from a complex reflection group, say [MATH], and depending on some parameters, say [MATH].', '1305.4894-1-2-2': 'They have many things in common with the universal enveloping algebras of semisimple Lie algebras, in particular, they have a triangular decomposition.', '1305.4894-1-2-3': 'This allows to define the categories [MATH] for such algebras, this was done in [CITATION].', '1305.4894-1-2-4': 'There are analogs of Verma modules, parameterized by irreducible [MATH]-modules, and an ordering on the set of simples in a Cherednik category [MATH] making it into a highest weight category.', '1305.4894-1-2-5': 'So there is a basic question one can ask: compute the multiplicity of a given simple module in a given standard (=Verma) module.', '1305.4894-1-3-0': 'The nicest and, perhaps, most important family of complex reflection groups is [MATH], where [MATH] are positive integers.', '1305.4894-1-3-1': 'This group acts on [MATH] by permutations of coordinates followed by multiplications by [MATH]th root of [MATH].', '1305.4894-1-3-2': 'There are more general infinite families, the groups [MATH], where [MATH] is a divisor of [MATH], but the study of the corresponding categories [MATH] can be, to some extent, reduced to the case of [MATH] and this is one of the reasons why our case is important.', '1305.4894-1-3-3': 'The other reason is that the corresponding category has an additional interesting structure that is not present in the other cases, a categorical Kac-Moody action to be recalled below.', '1305.4894-1-4-0': 'A significant progress in determining the multiplicities was made by Rouquier in [CITATION], where he determined the multiplicities in the case [MATH] and made a conjecture for all [MATH] (the conjecture was made for some special, but, in a sense, the most interesting and "non-degenerate" values of [MATH]).', '1305.4894-1-4-1': 'The conjecture was that the multiplicities are given by certain parabolic Kazhdan-Lusztig polynomials.', '1305.4894-1-4-2': 'We will recall the conjecture below.', '1305.4894-1-4-3': 'The techniques used in the proof for [MATH] were roughly as follows.', '1305.4894-1-4-4': 'In [CITATION] the authors introduced a so called KZ functor from the Cherednik category [MATH] to the category of modules over the Hecke algebra [MATH] of [MATH] with parameters recovered from [MATH].', '1305.4894-1-4-5': 'This is a quotient functor.', '1305.4894-1-4-6': 'Rouquier developed techniques that allow to check when two highest weight categories admitting quotient functors to [MATH]-[MATH] are equivalent.', '1305.4894-1-4-7': 'For [MATH] there is another category with a required quotient functor, the category of modules over an appropriate [MATH]-Schur algebra that was shown to be equivalent to the Cherednik category [MATH].', '1305.4894-1-5-0': 'For [MATH], the situation is more complicated.', '1305.4894-1-5-1': 'For certain, so to say, "dominant" and "faithful", values of [MATH] Rouquier proved in [CITATION] that the Cherednik category [MATH] is equivalent to the category of modules over a suitable cyclotomic [MATH]-Schur algebra of Dipper, James and Mathas.', '1305.4894-1-5-2': 'The multiplicities for the latter categories were recently computed by Stroppel and Webster, [CITATION].', '1305.4894-1-6-0': 'On the other hand, Varagnolo and Vasserot in [CITATION] produced another category with required multiplicities, where they are actually known.', '1305.4894-1-6-1': 'Their category is a certain truncation of an affine parabolic category [MATH].', '1305.4894-1-6-2': 'They conjectured an equivalence of that category with the Cherednik category [MATH].', '1305.4894-1-7-0': 'The goal of this paper is to prove that conjecture (in a somewhat weaker form that is still sufficient for checking the Rouquier conjecture).', '1305.4894-1-7-1': 'Together with earlier results of Shan, Varagnolo and Vasserot, our result also implies a conjecture of Chuang and Miyashi, [CITATION], claiming that the Cherednik category [MATH] is Koszul and describing the Koszul dual.', '1305.4894-1-8-0': '## Ideas of proof', '1305.4894-1-9-0': 'Our proof of the Varagnolo-Vasserot conjecture uses a variety of ideas.', '1305.4894-1-9-1': 'First, we use deformation ideas initially due to Rouquier, [CITATION] with further ramifications some of them due to Rouquier, Shan, Varagnolo, Vasserot and some to be developed in the present paper.', '1305.4894-1-9-2': 'Second, to properly implement these ideas we need categorical actions on highest weight categories, a topic initiated by the author in [CITATION],[CITATION] and further developed by the author and Webster in [CITATION] and in the present paper.', '1305.4894-1-9-3': 'The third crucial tool is certain derived equivalences for the Cherednik categories constructed in [CITATION] using Procesi bundles on resolutions of symplectic quotient singularities.', '1305.4894-1-10-0': 'Let us describe the deformation ideas.', '1305.4894-1-10-1': 'The Cherednik category [MATH] admits a quotient functor (the KZ functor of [CITATION]) to the category of modules over a cyclotomic quotient of the affine Hecke algebra.', '1305.4894-1-10-2': 'This functor is fully faithful on certain subcategories: for example, on the categories of tilting and of projective objects.', '1305.4894-1-10-3': "Also it is fully faithful on the whole category of standardly filtered objects ([MATH]-faithful in Rouquier's terminology) under some restrictions on the parameters for the Cherednik algebra.", '1305.4894-1-10-4': 'As Rouquier checked this implies that after a generic one-parametric deformation of the categories of interest the KZ functor becomes 1-faithful (i.e., an isomorphism on [MATH] and [MATH] between standardly filtered objects).', '1305.4894-1-10-5': 'It is then not difficult to see that if we have two highest weight categories over [MATH] with the same order that admit 1-faithful quotient functors that are equivalences generically to the same category, then the highest weight categories are equivalent.', '1305.4894-1-10-6': 'So the problem is to establish such a functor for a truncated affine parabolic category [MATH].', '1305.4894-1-10-7': 'To produce a functor is not difficult, see below, what is much harder is to prove faithfulness properties.', '1305.4894-1-10-8': 'Recently, Rouquier, Shan, Varagnolo and Vasserot proposed to consider 2-parametric deformations and announced that 0-faithfulness in points of codimension 1 yields 1-faithfulness in deformation.', '1305.4894-1-10-9': 'This is one ramification of the original technique of Roquier that we will use.', '1305.4894-1-10-10': 'More precisely, we reduce the equivalence conjecture to the claims that the quotient functor is [MATH]-faithful in codimension [MATH] and [MATH]-faithful (i.e., faithful - but a priori not full - on standard objects) in codimension [MATH] - the latter happens to be a much harder thing to check (but is sufficient to be checked for [MATH]).', '1305.4894-1-10-11': "There are two more ramifications of Rouquier's technique that we use.", '1305.4894-1-10-12': 'One is to check faithfulness only only on standardly filtered objects with prescribed standard filtration terms.', '1305.4894-1-10-13': 'The other is more significant: to bypass the problem that sometimes the quotient functors to the cyclotomic Hecke categories are not 0-faithful by considering larger quotients.', '1305.4894-1-10-14': 'One of the consequences of the deformation techniques is that (after the multi-deformation) the category of standardly filtered objects in the truncated affine parabolic category is included into the corresponding Cherednik category.', '1305.4894-1-10-15': 'A problem is to show that this embedding is an equivalence.', '1305.4894-1-10-16': 'Of course, it is enough to prove that the projectives in the Cherednik category lie in the parabolic affine category.', '1305.4894-1-11-0': 'Let us now explain how the theory of categorical actions on highest weight categories comes into play.', '1305.4894-1-11-1': 'Results of Rouquier, [CITATION], show that one can produce a quotient functor to a cyclotomic Hecke category from a category [MATH] if one equips [MATH] with a categorical action of [MATH] that categorifies a highest weight [MATH]-module.', '1305.4894-1-11-2': 'There is a categorical action on the affine parabolic category [MATH] before the truncation: this is provided by the Kazhdan-Lusztig tensor products but this action does not restrict to the truncated category in a straightforward way (as the truncated category is not stable under the categorification functors).', '1305.4894-1-11-3': 'However one can still define a ("restricted" but this is not of importance) categorical action on the truncated category using the categorical splitting techniques from [CITATION].', '1305.4894-1-11-4': 'This produces a required quotient functor.', '1305.4894-1-11-5': 'Further, one it turns out that using structural results obtained in [CITATION] one can reduce the study of faithfulness of this functor to some purely combinatorial questions concerning crystal structures on the multipartitions: there are combinatorial conditions that guarantee [MATH]-faithfulness and [MATH]-faithfulness of the quotient functor.', '1305.4894-1-11-6': 'However, it is very rare that these conditions can be checked.', '1305.4894-1-11-7': "It is possible when [MATH] (here we recover Rouquier's results) and partially when [MATH].", '1305.4894-1-11-8': 'Also this combinatorial approach allows to check 1-faithfulness in codimension 1.', '1305.4894-1-11-9': 'Unfortunately, the when [MATH], the combinatorics is already too complicated to prove the required results.', '1305.4894-1-12-0': 'Or one can try to use the ideology used in [CITATION], that is, to check that the quotient functor from the truncated affine parabolic category [MATH] is fully faithful on the projective objects and that the images of the indecomposable projectives coincides with analogous image under [MATH].', '1305.4894-1-12-1': 'This approach works when [MATH] and in the special ("dominant") case when [MATH] but one needs to include some other functors in addition to the Kac-Moody ones (getting what we call a categorical Schur action).', '1305.4894-1-12-2': 'Also using these ideas combined with those explained in the previous paragraph one can fix the situation when the KZ functor is not [MATH]-faithful.', '1305.4894-1-12-3': 'Here one needs to consider larger quotients, and the technical problem to solve is to show that the corresponding quotient categories are the same.', '1305.4894-1-12-4': 'Still, it seems that the general case of [MATH] cannot be approached just by those techniques.', '1305.4894-1-13-0': 'This is where we need the third group of ideas: derived equivalences coming from Procesi bundles.', '1305.4894-1-13-1': 'Recall that we basically have an inclusion between the categories of standardly stratified objects.', '1305.4894-1-13-2': 'It turns out that there are derived equivalences between the Cherednik categories corresponding to different values of parameters.', '1305.4894-1-13-3': 'They were constructed in [CITATION] using Procesi bundles on symplectic resolutions of the corresponding quotient singularity.', '1305.4894-1-13-4': 'Using properties of Procesi bundles we will see that there is an equivalence from the Cherednik category with dominant parameters to a given one that maps projectives to standardly filtered objects (called pseudo-projective).', '1305.4894-1-13-5': "The equivalence is nicely compatible with categorical actions and gives the identity map between [MATH]'s.", '1305.4894-1-13-6': 'Using that we will be able to describe pseudo-projective objects pretty explicitly.', '1305.4894-1-13-7': 'Using a partial information about [MATH]-faithfulness in the [MATH] case that we have we will see that all pseudo-projective objects actually lie in the truncated affine parabolic category.', '1305.4894-1-13-8': 'This (together with some standard techniques) will essentially complete the proof.', '1305.4894-1-14-0': 'We want to indicate the dependence of the present paper on a related work.', '1305.4894-1-14-1': 'We use an idea due to Rouquier, Shan, Varagnolo and Vasserot explained before.', '1305.4894-1-14-2': "This idea was mentioned in Shan's talk in Luminy in July 2012 without explanations on how to make it to work, and the paper, [CITATION], has just appeared.", '1305.4894-1-14-3': 'There is also a work of Webster, [CITATION], where he proves an equivalence between the Cherednik category and a certain diagrammatic category, where one can compute the multiplicities.', '1305.4894-1-14-4': 'In order to compute the multiplicities in the diagrammatic category, Webster also uses a derived equivalence relating given parameters with dominant ones.', '1305.4894-1-14-5': 'We have arrived at an idea to use such an equivalence independently and the idea is realized in very different ways for very different purposes (we do not need to compute the multiplicities in the affine parabolic category, which in our case are known, we need to check the faithfulness properties).', '1305.4894-1-15-0': '## Structure of the paper', '1305.4894-1-16-0': 'In Section [REF] we describe the highest weight categories we consider: the categories [MATH] for cyclotomic Rational Cherednik algebra and affine parabolic categories [MATH] both in the undeformed and deformed settings.', '1305.4894-1-16-1': 'We also recall basic combinatorics of these categories.', '1305.4894-1-16-2': 'This section contains no new results.', '1305.4894-1-17-0': 'In Section [REF] we provide general results on faithfulness properties of quotient functors from highest weight categories.', '1305.4894-1-17-1': 'The section follows [CITATION], its main result is new but is not really original.', '1305.4894-1-18-0': 'Section [REF] deals with categorical Kac-Moody actions on highest weight categories.', '1305.4894-1-18-1': 'It defines categorical type A Kac-Moody actions and recalls results from [CITATION],[CITATION].', '1305.4894-1-18-2': 'There are no new results there.', '1305.4894-1-19-0': 'Section [REF] is new.', '1305.4894-1-19-1': 'There we equip the truncated affine parabolic category [MATH] with a restricted type A categorical Kac-Moody action and so produce a functor to the cyclotomic Hecke category.', '1305.4894-1-20-0': 'In Section [REF] we study an interplay between the faithfulness properties of quotient morphisms and combinatorial properties of crystals.', '1305.4894-1-20-1': 'Namely, we state two combinatorial conditions that guarantee vanishing of [MATH] and [MATH] from a suitable simple to a suitable tilting.', '1305.4894-1-20-2': 'We relate the faithfulness properties to those vanishing properties.', '1305.4894-1-20-3': 'Finally, we check that our combinatorial conditions hold in certain special cases.', '1305.4894-1-21-0': 'In Section [REF] we introduce the notion of a Schur category.', '1305.4894-1-21-1': 'An example is provided by the Cherednik category [MATH] for [MATH].', '1305.4894-1-21-2': 'Then we show that any Schur category is equivalent to the direct sum of modules over the level [MATH]-Schur algebras (for all parameters).', '1305.4894-1-21-3': 'Next, we define a Schur categorical action that happens to be a special case of a categorical Kac-Moody action of type A.', '1305.4894-1-22-0': 'Section [REF] we define new quotient functors that are "larger" then the functors considered (our old functors factor through new ones).', '1305.4894-1-22-1': 'Then we show that the target categories for our new functors in the Cherednik and in the parabolic setting are equivalent.', '1305.4894-1-23-0': 'In Section [REF] we recall derived equivalences between Cherednik categories [MATH] and study their properties.', '1305.4894-1-24-0': 'Finally, in the last section of this paper, we complete the proof of the main equivalence theorem.'}

{'1305.4894-2-0-0': "We prove an asymptotic version of a conjecture by Varagnolo and Vasserot on an equivalence between the category [MATH] for a cyclotomic Rational Cherednik algebra and a suitable truncation of an affine parabolic category [MATH] that, in particular, implies Rouquier's conjecture on the decomposition numbers in the former.", '1305.4894-2-0-1': "Our proof uses multiple ingredients: a ramification of Rouquier's deformation approach, categorical actions on highest weight categories and related combinatorics, and derived equivalences produced from Procesi bundles on symplectic resolutions of quotient singularities.", '1305.4894-2-1-0': '# Introduction', '1305.4894-2-2-0': 'Rational Cherednik algebras were introduced by Etingof and Ginzburg, [CITATION].', '1305.4894-2-2-1': 'These are associative algebras over [MATH] constructed from a complex reflection group, say [MATH], and depending on a parameter, say [MATH], that is a collection of complex numbers.', '1305.4894-2-2-2': 'They have many things in common with the universal enveloping algebras of semisimple Lie algebras, in particular, they have a triangular decomposition.', '1305.4894-2-2-3': 'This allows one to define the categories [MATH] for such algebras, this was done in [CITATION].', '1305.4894-2-2-4': 'There are analogs of Verma modules, parameterized by irreducible [MATH]-modules, and an ordering on the set of simples in the GGOR category [MATH] making it into a highest weight category.', '1305.4894-2-2-5': 'So there is a basic question one can ask: compute the multiplicity of a given simple module in a given standard (=Verma) module.', '1305.4894-2-3-0': 'The nicest and, perhaps, most important family of complex reflection groups is [MATH], where [MATH] are positive integers.', '1305.4894-2-3-1': 'This group acts on [MATH] by permutations of coordinates followed by multiplications by a root of [MATH] of order [MATH].', '1305.4894-2-3-2': 'There are more general infinite families, the groups [MATH], where [MATH] is a divisor of [MATH], but the study of the corresponding categories [MATH] can be, to some extent, reduced to the case of [MATH] and this is one of the reasons why our case is important.', '1305.4894-2-3-3': 'Another reason is that the corresponding category has an additional interesting structure that is not present in the other cases, a categorical Kac-Moody action to be recalled below.', '1305.4894-2-3-4': 'Yet another reason is a connection to the geometry of symplectic resolutions of quotient singularities.', '1305.4894-2-4-0': 'A significant progress in determining the multiplicities was made by Rouquier in [CITATION], where he determined the multiplicities in the case [MATH] and made a conjecture for all [MATH] (the conjecture was made for some special, but, in a sense, the most interesting and "non-degenerate" values of [MATH]).', '1305.4894-2-4-1': 'The conjecture says that the multiplicities are given by certain parabolic Kazhdan-Lusztig polynomials.', '1305.4894-2-4-2': 'The techniques used in the proof for [MATH] were roughly as follows.', '1305.4894-2-4-3': 'In [CITATION] the authors introduced a so called KZ functor from the Cherednik category [MATH] to the category of modules over the Hecke algebra [MATH] of [MATH] with parameters recovered from [MATH].', '1305.4894-2-4-4': 'This is a quotient functor.', '1305.4894-2-4-5': 'Rouquier developed techniques that allow to check when two highest weight categories admitting quotient functors to [MATH]-[MATH] are equivalent.', '1305.4894-2-4-6': 'For [MATH] there is another category with a nice quotient functor, the category of modules over an appropriate [MATH]-Schur algebra that was shown to be equivalent to the Cherednik category [MATH] (under a certain "faithfulness" condition on the parameters).', '1305.4894-2-5-0': 'For [MATH], the situation is more complicated.', '1305.4894-2-5-1': 'For certain, so to say, "dominant" and "faithful", values of [MATH] Rouquier proved in [CITATION] that the Cherednik category [MATH] is equivalent to the category of modules over a suitable cyclotomic [MATH]-Schur algebra of Dipper, James and Mathas.', '1305.4894-2-5-2': 'The multiplicities for the latter categories were recently computed by Stroppel and Webster, [CITATION].', '1305.4894-2-6-0': 'On the other hand, Varagnolo and Vasserot in [CITATION] produced another category, where the multiplicities were shown to have a required form.', '1305.4894-2-6-1': 'Their category is a certain truncation of an affine parabolic category [MATH].', '1305.4894-2-6-2': 'They conjectured an equivalence of that category with the Cherednik category [MATH].', '1305.4894-2-7-0': 'The goal of this paper is to prove that conjecture (in a somewhat weaker form that is still sufficient for checking the Rouquier conjecture).', '1305.4894-2-7-1': 'Together with earlier results of Shan, Varagnolo and Vasserot, our result also implies a conjecture of Chuang and Miyashi, [CITATION], claiming that the Cherednik category [MATH] is Koszul and describing the Koszul dual.', '1305.4894-2-8-0': '## Ideas of proof', '1305.4894-2-9-0': 'Our proof of the Varagnolo-Vasserot conjecture uses a variety of ideas.', '1305.4894-2-9-1': 'First, we use deformation ideas initially due to Rouquier, [CITATION] with further ramifications.', '1305.4894-2-9-2': 'Some of them are due to Rouquier, Shan, Varagnolo, Vasserot and some are to be developed in the present paper.', '1305.4894-2-9-3': 'Second, to properly implement these ideas we need categorical actions on highest weight categories, a topic initiated by the author in [CITATION],[CITATION] and further developed by the author and Webster in [CITATION] and in the present paper.', '1305.4894-2-9-4': 'The third crucial tool is certain derived equivalences for the GGOR categories constructed in [CITATION] using Procesi bundles on resolutions of symplectic quotient singularities.', '1305.4894-2-10-0': 'Let us describe the deformation ideas.', '1305.4894-2-10-1': 'The GGOR category [MATH] admits a quotient functor (the KZ functor of [CITATION]) to the category of modules over a cyclotomic quotient of the affine Hecke algebra.', '1305.4894-2-10-2': 'This functor is fully faithful on certain subcategories: for example, on the categories of tilting and of projective objects.', '1305.4894-2-10-3': "Also it is fully faithful on the whole category of standardly filtered objects ([MATH]-faithful in Rouquier's terminology) under some restrictions on the parameters for the Cherednik algebra.", '1305.4894-2-10-4': 'As Rouquier checked this implies that after a generic one-parametrer deformation of the categories of interest the KZ functor becomes 1-faithful (i.e., an isomorphism on [MATH] and [MATH] between standardly filtered objects).', '1305.4894-2-10-5': 'It is not difficult to see that two highest weight categories over [MATH] with quotient functors to the same category are equivalent provided their orders are the same, the quotient functors are 1-faithful and are equivalences over [MATH].', '1305.4894-2-10-6': 'So the problem is to establish an analog of the KZ functor for a truncated affine parabolic category [MATH].', '1305.4894-2-10-7': 'To produce a functor is not difficult, this is done using categorical Kac-Moody actions, what is much harder is to prove faithfulness properties.', '1305.4894-2-10-8': 'Recently, Rouquier, Shan, Varagnolo and Vasserot proposed to consider 2-parametric deformations and announced that 0-faithfulness in points of codimension 1 yields 1-faithfulness for the deformed categories.', '1305.4894-2-10-9': 'This is one ramification of the original technique of Roquier that we will use.', '1305.4894-2-10-10': 'More precisely, we reduce the equivalence conjecture to the following two claims.', '1305.4894-2-10-11': 'First, the quotient functor is [MATH]-faithful in codimension [MATH] (roughly speaking, for a general parameter on a hyperplane in a parameter space), the category [MATH] significantly simplifies for such parameters.', '1305.4894-2-10-12': 'Second, the quotient functor is [MATH]-faithful (i.e., faithful - but a priori not full - on standard objects) in codimension [MATH].', '1305.4894-2-10-13': 'The second claim happens to be a much harder thing to check - but it is, basically, sufficient to check it for [MATH].', '1305.4894-2-11-0': "There are two more ramifications of Rouquier's technique that we use.", '1305.4894-2-11-1': 'One is to check faithfulness only only on standardly filtered objects with prescribed standard filtration terms.', '1305.4894-2-11-2': 'The other is more significant: to bypass the problem that sometimes the quotient functors to the cyclotomic Hecke categories are not 0-faithful by considering larger quotients.', '1305.4894-2-12-0': 'One of the consequences of the deformation techniques is that the category of standardly filtered objects in the fully deformed truncated affine parabolic category is included into the corresponding GGOR category.', '1305.4894-2-12-1': 'A problem is to show that this embedding is an equivalence.', '1305.4894-2-12-2': 'Of course, it is enough to prove that the projectives in the GGOR category lie in the parabolic affine category.', '1305.4894-2-12-3': 'This finishes our discussion of the deformation techniques.', '1305.4894-2-13-0': 'Let us now explain how the theory of categorical actions on highest weight categories comes into play.', '1305.4894-2-13-1': 'Results of Rouquier, [CITATION], suggests a way to produce a quotient functor to a cyclotomic Hecke category (i.e., the direct sum over all [MATH] of the module categories over cyclotomic Hecke algebra with fixed parameters and [MATH] variables) from some category [MATH].', '1305.4894-2-13-2': 'Namely, one gets such a functor if one equips [MATH] with a categorical action of [MATH] that categorifies a highest weight [MATH]-module.', '1305.4894-2-13-3': 'There is a categorical action on the affine parabolic category [MATH] before the truncation: this is provided by the Kazhdan-Lusztig tensor products.', '1305.4894-2-13-4': 'This action does not restrict to the truncated category in a straightforward way (as the truncated category is not stable under the categorification functors).', '1305.4894-2-13-5': 'However one can still define a ("restricted", but this is not of importance) categorical action on the truncated category using the categorical splitting techniques from [CITATION].', '1305.4894-2-13-6': 'This produces a required quotient functor.', '1305.4894-2-13-7': 'Further, using structural results obtained in [CITATION], one can reduce the study of the faithfulness properties for this functor to some purely combinatorial questions concerning crystal structures on the multipartitions.', '1305.4894-2-13-8': 'More precisely, there are combinatorial conditions that guarantee [MATH]-faithfulness and [MATH]-faithfulness of the quotient functor.', '1305.4894-2-13-9': 'However, it is very rare that these conditions can be checked.', '1305.4894-2-13-10': 'It is possible when [MATH] and partially when [MATH].', '1305.4894-2-13-11': 'Also this combinatorial approach allows to check 1-faithfulness in codimension 1.', '1305.4894-2-13-12': 'Unfortunately, when [MATH], the combinatorics is already too complicated.', '1305.4894-2-14-0': 'Or one can try to use the ideology used in [CITATION], that is, to check that the quotient functor from the truncated affine parabolic category [MATH] is fully faithful on the projective objects and that the images of the indecomposable projectives coincides with analogous images under [MATH].', '1305.4894-2-14-1': 'This approach works when [MATH] and for the special choice of parameters ("dominant" parameters) when [MATH].', '1305.4894-2-14-2': 'In order to make this work, one needs to include some other functors in addition to the Kac-Moody ones (getting what we call a categorical Schur action).', '1305.4894-2-14-3': 'Also using these ideas combined with those explained above one can fix the situation when the KZ functor is not [MATH]-faithful.', '1305.4894-2-14-4': 'Still, we were not able to approach the general parameters in the [MATH] case just by using those techniques.', '1305.4894-2-15-0': 'This is where we need the third group of ideas: derived equivalences coming from Procesi bundles.', '1305.4894-2-15-1': 'Recall that we basically have an inclusion from the category of standardly filtered objects in the deformed truncated parabolic affine category [MATH] to an analogous GGOR category.', '1305.4894-2-15-2': 'It turns out that there are derived equivalences between the GGOR categories corresponding to different values of parameters.', '1305.4894-2-15-3': 'They were constructed in [CITATION] using Procesi bundles on symplectic resolutions of the corresponding quotient singularity.', '1305.4894-2-15-4': 'Using properties of Procesi bundles we will see that there is an equivalence from the GGOR category with dominant parameters to a given one that maps projectives to standardly filtered objects (called pseudo-projective).', '1305.4894-2-15-5': "The equivalence is nicely compatible with categorical actions and gives the identity map between [MATH]'s.", '1305.4894-2-15-6': 'This will allow us to describe pseudo-projective objects pretty explicitly.', '1305.4894-2-15-7': 'Then, using a partial information about [MATH]-faithfulness in the [MATH] case that we can extract from the combinatorics, we will see that all pseudo-projective objects actually lie in the truncated affine parabolic category.', '1305.4894-2-15-8': 'This (together with some standard techniques) will essentially complete the proof.', '1305.4894-2-16-0': 'We want to indicate the dependence of the present paper on a related work.', '1305.4894-2-16-1': 'We use an idea due to Rouquier, Shan, Varagnolo and Vasserot explained before.', '1305.4894-2-16-2': "This idea was mentioned in Shan's talk in Luminy in July 2012 without explanations on how to make it to work, and the paper, [CITATION], appeared when our paper was ready.", '1305.4894-2-16-3': 'There is also a work of Webster, [CITATION], where he proves an equivalence between the GGOR category and a certain diagrammatic category, where one can compute the multiplicities.', '1305.4894-2-16-4': 'In order to compute the multiplicities in the diagrammatic category, Webster also uses a derived equivalence relating given parameters with dominant ones.', '1305.4894-2-16-5': 'We have arrived at an idea to use such an equivalence independently and the idea is realized in very different ways for very different purposes (we do not need to compute the multiplicities in the affine parabolic category, which in our case are known, we need to check the faithfulness properties).', '1305.4894-2-17-0': '## Structure of the paper', '1305.4894-2-18-0': 'In Section [REF] we describe the highest weight categories we consider: the categories [MATH] for cyclotomic Rational Cherednik algebra and affine parabolic categories [MATH] both in the undeformed and deformed settings.', '1305.4894-2-18-1': 'We also recall basic combinatorics of these categories.', '1305.4894-2-18-2': 'This section contains no new results.', '1305.4894-2-19-0': 'In Section [REF] we provide general results on faithfulness properties of quotient functors from highest weight categories.', '1305.4894-2-19-1': 'The section follows [CITATION], its main result is new but is not really original.', '1305.4894-2-20-0': 'Section [REF] deals with categorical Kac-Moody actions on highest weight categories.', '1305.4894-2-20-1': 'It defines categorical type A Kac-Moody actions and recalls results from [CITATION],[CITATION].', '1305.4894-2-20-2': 'There are no new results there.', '1305.4894-2-21-0': 'Section [REF] is new.', '1305.4894-2-21-1': 'There we equip the truncated affine parabolic category [MATH] with a restricted type A categorical Kac-Moody action and so produce a functor to the cyclotomic Hecke category.', '1305.4894-2-22-0': 'In Section [REF] we study an interplay between the faithfulness properties of quotient morphisms and combinatorial properties of crystals.', '1305.4894-2-22-1': 'Namely, we state two combinatorial conditions that guarantee vanishing of [MATH] and [MATH] from a suitable simple to a suitable tilting.', '1305.4894-2-22-2': 'We relate the faithfulness properties to those vanishing properties.', '1305.4894-2-22-3': 'Finally, we check that our combinatorial conditions hold in certain special cases.', '1305.4894-2-23-0': 'In Section [REF] we introduce the notion of a Schur category.', '1305.4894-2-23-1': 'An example is provided by the GGOR category [MATH] for [MATH].', '1305.4894-2-23-2': 'Then we show that any Schur category is equivalent to the direct sum of modules over the level [MATH]-Schur algebras (for all parameters).', '1305.4894-2-23-3': 'Next, we define a Schur categorical action that happens to be a special case of a categorical Kac-Moody action of type A.', '1305.4894-2-24-0': 'Section [REF] we define new quotient functors that are "larger" then the functors considered (our old functors factor through new ones).', '1305.4894-2-24-1': 'Then we show that the target categories for our new functors in the GGOR and in the parabolic setting are equivalent.', '1305.4894-2-25-0': 'In Section [REF] we recall derived equivalences between GGOR categories [MATH] and study their properties.', '1305.4894-2-26-0': 'Finally, in the last section of this paper, we complete the proof of the main equivalence theorem.'}

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['1305.4894-1-16-1', '1305.4894-2-18-1'], ['1305.4894-1-16-2', '1305.4894-2-18-2'], ['1305.4894-1-19-0', '1305.4894-2-21-0'], ['1305.4894-1-19-1', '1305.4894-2-21-1'], ['1305.4894-2-16-0', '1305.4894-3-13-0'], ['1305.4894-2-16-1', '1305.4894-3-13-1'], ['1305.4894-2-16-2', '1305.4894-3-13-2'], ['1305.4894-2-18-0', '1305.4894-3-16-0'], ['1305.4894-2-18-1', '1305.4894-3-16-1'], ['1305.4894-2-18-2', '1305.4894-3-16-2'], ['1305.4894-2-5-0', '1305.4894-3-5-0'], ['1305.4894-2-5-1', '1305.4894-3-5-1'], ['1305.4894-2-5-2', '1305.4894-3-5-2'], ['1305.4894-2-21-0', '1305.4894-3-19-0'], ['1305.4894-2-21-1', '1305.4894-3-19-1'], ['1305.4894-2-2-0', '1305.4894-3-2-0'], ['1305.4894-2-2-1', '1305.4894-3-2-1'], ['1305.4894-2-2-2', '1305.4894-3-2-2'], ['1305.4894-2-2-3', '1305.4894-3-2-3'], ['1305.4894-2-2-4', '1305.4894-3-2-4'], ['1305.4894-2-2-5', '1305.4894-3-2-5'], ['1305.4894-2-6-1', '1305.4894-3-6-1'], ['1305.4894-2-6-2', '1305.4894-3-6-2'], ['1305.4894-2-19-0', '1305.4894-3-17-0'], ['1305.4894-2-9-2', '1305.4894-3-9-2'], ['1305.4894-2-9-3', '1305.4894-3-9-3'], ['1305.4894-2-0-0', '1305.4894-3-0-0'], ['1305.4894-2-3-0', '1305.4894-3-3-0'], ['1305.4894-2-3-1', '1305.4894-3-3-1'], ['1305.4894-2-3-2', '1305.4894-3-3-2'], ['1305.4894-2-3-3', '1305.4894-3-3-3'], ['1305.4894-2-3-4', '1305.4894-3-3-4'], ['1305.4894-2-7-0', '1305.4894-3-7-0'], ['1305.4894-2-7-1', '1305.4894-3-7-1'], ['1305.4894-2-10-0', '1305.4894-3-10-0'], ['1305.4894-2-10-6', '1305.4894-3-10-6'], ['1305.4894-2-10-8', '1305.4894-3-10-9'], ['1305.4894-2-24-1', '1305.4894-3-21-1'], ['1305.4894-2-20-0', '1305.4894-3-18-0'], ['1305.4894-2-20-1', '1305.4894-3-18-1'], ['1305.4894-2-20-2', '1305.4894-3-18-2'], ['1305.4894-2-22-0', '1305.4894-3-20-0'], ['1305.4894-2-4-1', '1305.4894-3-4-1'], ['1305.4894-2-4-2', '1305.4894-3-4-2'], ['1305.4894-2-4-3', '1305.4894-3-4-3'], ['1305.4894-2-4-4', '1305.4894-3-4-4'], ['1305.4894-2-4-5', '1305.4894-3-4-5'], ['1305.4894-2-4-6', '1305.4894-3-4-6'], ['1305.4894-2-13-3', '1305.4894-3-12-3'], ['1305.4894-2-13-4', '1305.4894-3-12-4'], ['1305.4894-2-13-7', '1305.4894-3-12-7'], ['1305.4894-3-16-0', '1305.4894-4-16-0'], ['1305.4894-3-16-1', '1305.4894-4-16-1'], ['1305.4894-3-16-2', '1305.4894-4-16-2'], ['1305.4894-3-10-0', '1305.4894-4-10-0'], ['1305.4894-3-10-1', '1305.4894-4-10-1'], ['1305.4894-3-10-2', '1305.4894-4-10-2'], ['1305.4894-3-10-3', '1305.4894-4-10-3'], ['1305.4894-3-10-4', '1305.4894-4-10-4'], ['1305.4894-3-10-5', '1305.4894-4-10-5'], ['1305.4894-3-10-6', '1305.4894-4-10-6'], ['1305.4894-3-10-7', '1305.4894-4-10-7'], ['1305.4894-3-10-8', '1305.4894-4-10-8'], ['1305.4894-3-10-9', '1305.4894-4-10-9'], ['1305.4894-3-10-10', '1305.4894-4-10-10'], ['1305.4894-3-10-11', '1305.4894-4-10-11'], ['1305.4894-3-10-12', '1305.4894-4-10-12'], ['1305.4894-3-19-0', '1305.4894-4-19-0'], ['1305.4894-3-19-1', '1305.4894-4-19-1'], ['1305.4894-3-23-0', '1305.4894-4-23-0'], ['1305.4894-3-13-0', '1305.4894-4-13-0'], ['1305.4894-3-13-1', '1305.4894-4-13-1'], ['1305.4894-3-13-2', '1305.4894-4-13-2'], ['1305.4894-3-13-3', '1305.4894-4-13-3'], ['1305.4894-3-22-0', '1305.4894-4-22-0'], ['1305.4894-3-11-0', '1305.4894-4-11-0'], ['1305.4894-3-11-1', '1305.4894-4-11-1'], ['1305.4894-3-11-2', '1305.4894-4-11-2'], ['1305.4894-3-11-3', '1305.4894-4-11-3'], ['1305.4894-3-7-0', '1305.4894-4-7-0'], ['1305.4894-3-7-1', '1305.4894-4-7-1'], ['1305.4894-3-20-0', '1305.4894-4-20-0'], ['1305.4894-3-20-1', '1305.4894-4-20-1'], ['1305.4894-3-20-2', '1305.4894-4-20-2'], ['1305.4894-3-6-0', '1305.4894-4-6-0'], ['1305.4894-3-6-1', '1305.4894-4-6-1'], ['1305.4894-3-6-2', '1305.4894-4-6-2'], ['1305.4894-3-9-0', '1305.4894-4-9-0'], ['1305.4894-3-9-1', '1305.4894-4-9-1'], ['1305.4894-3-9-2', '1305.4894-4-9-2'], ['1305.4894-3-9-3', '1305.4894-4-9-3'], ['1305.4894-3-21-1', '1305.4894-4-21-1'], ['1305.4894-3-2-0', '1305.4894-4-2-0'], ['1305.4894-3-2-1', '1305.4894-4-2-1'], ['1305.4894-3-2-2', '1305.4894-4-2-2'], ['1305.4894-3-2-3', '1305.4894-4-2-3'], ['1305.4894-3-2-4', '1305.4894-4-2-4'], ['1305.4894-3-2-5', '1305.4894-4-2-5'], ['1305.4894-3-17-0', '1305.4894-4-17-0'], ['1305.4894-3-17-1', '1305.4894-4-17-1'], ['1305.4894-3-12-0', '1305.4894-4-12-0'], ['1305.4894-3-12-1', '1305.4894-4-12-1'], ['1305.4894-3-12-3', '1305.4894-4-12-3'], ['1305.4894-3-12-4', '1305.4894-4-12-4'], ['1305.4894-3-12-5', '1305.4894-4-12-5'], ['1305.4894-3-12-6', '1305.4894-4-12-6'], ['1305.4894-3-12-7', '1305.4894-4-12-7'], ['1305.4894-3-12-8', '1305.4894-4-12-8'], ['1305.4894-3-12-9', '1305.4894-4-12-9'], ['1305.4894-3-12-10', '1305.4894-4-12-10'], ['1305.4894-3-18-0', '1305.4894-4-18-0'], ['1305.4894-3-18-1', '1305.4894-4-18-1'], ['1305.4894-3-18-2', '1305.4894-4-18-2'], ['1305.4894-3-0-0', '1305.4894-4-0-0'], ['1305.4894-3-0-1', '1305.4894-4-0-1'], ['1305.4894-3-4-0', '1305.4894-4-4-0'], ['1305.4894-3-4-1', '1305.4894-4-4-1'], ['1305.4894-3-4-2', '1305.4894-4-4-2'], ['1305.4894-3-4-3', '1305.4894-4-4-3'], ['1305.4894-3-4-4', '1305.4894-4-4-4'], ['1305.4894-3-4-5', '1305.4894-4-4-5'], ['1305.4894-3-4-6', '1305.4894-4-4-6'], ['1305.4894-3-14-0', '1305.4894-4-14-0'], ['1305.4894-3-14-1', '1305.4894-4-14-1'], ['1305.4894-3-14-2', '1305.4894-4-14-2'], ['1305.4894-3-3-0', '1305.4894-4-3-0'], ['1305.4894-3-3-2', '1305.4894-4-3-2'], ['1305.4894-3-3-3', '1305.4894-4-3-3'], ['1305.4894-3-3-4', '1305.4894-4-3-4'], ['1305.4894-3-5-0', '1305.4894-4-5-0'], ['1305.4894-3-5-1', '1305.4894-4-5-1'], ['1305.4894-3-5-2', '1305.4894-4-5-2'], ['1305.4894-1-10-0', '1305.4894-2-10-0'], ['1305.4894-1-10-2', '1305.4894-2-10-2'], ['1305.4894-1-10-3', '1305.4894-2-10-3'], ['1305.4894-1-10-9', '1305.4894-2-10-9'], ['1305.4894-1-10-11', '1305.4894-2-11-0'], ['1305.4894-1-10-12', '1305.4894-2-11-1'], ['1305.4894-1-10-13', '1305.4894-2-11-2'], ['1305.4894-1-10-15', '1305.4894-2-12-1']]

[['1305.4894-1-4-1', '1305.4894-2-4-1'], ['1305.4894-1-4-7', '1305.4894-2-4-6'], ['1305.4894-1-12-0', '1305.4894-2-14-0'], ['1305.4894-1-12-2', '1305.4894-2-14-3'], ['1305.4894-1-11-3', '1305.4894-2-13-5'], ['1305.4894-1-21-1', '1305.4894-2-23-1'], ['1305.4894-1-23-0', '1305.4894-2-25-0'], ['1305.4894-1-9-3', '1305.4894-2-9-4'], ['1305.4894-1-3-1', '1305.4894-2-3-1'], ['1305.4894-1-3-3', '1305.4894-2-3-3'], ['1305.4894-1-22-1', '1305.4894-2-24-1'], ['1305.4894-1-14-2', '1305.4894-2-16-2'], ['1305.4894-1-14-3', '1305.4894-2-16-3'], ['1305.4894-1-2-3', '1305.4894-2-2-3'], ['1305.4894-1-2-4', '1305.4894-2-2-4'], ['1305.4894-1-13-2', '1305.4894-2-15-2'], ['1305.4894-1-13-4', '1305.4894-2-15-4'], ['1305.4894-1-13-7', '1305.4894-2-15-7'], ['1305.4894-2-16-3', '1305.4894-3-13-3'], ['1305.4894-2-6-0', '1305.4894-3-6-0'], ['1305.4894-2-9-1', '1305.4894-3-9-1'], ['1305.4894-2-10-1', '1305.4894-3-10-1'], ['1305.4894-2-10-2', '1305.4894-3-10-2'], ['1305.4894-2-10-3', '1305.4894-3-10-3'], ['1305.4894-2-10-4', '1305.4894-3-10-4'], ['1305.4894-2-10-5', '1305.4894-3-10-5'], ['1305.4894-2-24-0', '1305.4894-3-21-0'], ['1305.4894-2-22-1', '1305.4894-3-20-1'], ['1305.4894-2-22-3', '1305.4894-3-20-2'], ['1305.4894-2-4-0', '1305.4894-3-4-0'], ['1305.4894-2-13-0', '1305.4894-3-12-0'], ['1305.4894-2-13-1', '1305.4894-3-12-1'], ['1305.4894-2-13-5', '1305.4894-3-12-5'], ['1305.4894-3-21-0', '1305.4894-4-21-0'], ['1305.4894-3-12-2', '1305.4894-4-12-2'], ['1305.4894-3-3-1', '1305.4894-4-3-1'], ['1305.4894-1-10-1', '1305.4894-2-10-1'], ['1305.4894-1-10-4', '1305.4894-2-10-4'], ['1305.4894-1-10-6', '1305.4894-2-10-6'], ['1305.4894-1-10-8', '1305.4894-2-10-8'], ['1305.4894-1-10-14', '1305.4894-2-12-0'], ['1305.4894-1-10-16', '1305.4894-2-12-2']]

[]

[['1305.4894-1-12-1', '1305.4894-2-14-1'], ['1305.4894-1-12-1', '1305.4894-2-14-2'], ['1305.4894-1-12-4', '1305.4894-2-14-4'], ['1305.4894-1-11-1', '1305.4894-2-13-1'], ['1305.4894-1-11-2', '1305.4894-2-13-3'], ['1305.4894-1-11-2', '1305.4894-2-13-4'], ['1305.4894-1-11-5', '1305.4894-2-13-7'], ['1305.4894-1-11-5', '1305.4894-2-13-8'], ['1305.4894-1-11-7', '1305.4894-2-13-10'], ['1305.4894-1-11-9', '1305.4894-2-13-12'], ['1305.4894-1-9-1', '1305.4894-2-9-1'], ['1305.4894-1-6-0', '1305.4894-2-6-0'], ['1305.4894-1-2-1', '1305.4894-2-2-1'], ['1305.4894-1-13-1', '1305.4894-2-15-1'], ['1305.4894-1-13-6', '1305.4894-2-15-6'], ['1305.4894-2-26-0', '1305.4894-3-22-0'], ['1305.4894-2-9-0', '1305.4894-3-9-0'], ['1305.4894-2-0-1', '1305.4894-3-0-1'], ['1305.4894-2-11-0', '1305.4894-3-11-0'], ['1305.4894-2-11-2', '1305.4894-3-11-1'], ['1305.4894-2-10-7', '1305.4894-3-10-7'], ['1305.4894-2-10-7', '1305.4894-3-10-8'], ['1305.4894-2-10-9', '1305.4894-3-10-10'], ['1305.4894-2-13-2', '1305.4894-3-12-2'], ['1305.4894-2-13-6', '1305.4894-3-12-6'], ['1305.4894-2-13-8', '1305.4894-3-12-8'], ['1305.4894-2-13-11', '1305.4894-3-12-9'], ['1305.4894-1-10-5', '1305.4894-2-10-5'], ['1305.4894-1-10-7', '1305.4894-2-10-7'], ['1305.4894-1-10-10', '1305.4894-2-10-12']]

[]

[]

{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '3': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '4': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1305.4894

{'1305.4894-3-0-0': "We prove an asymptotic version of a conjecture by Varagnolo and Vasserot on an equivalence between the category [MATH] for a cyclotomic Rational Cherednik algebra and a suitable truncation of an affine parabolic category [MATH] that, in particular, implies Rouquier's conjecture on the decomposition numbers in the former.", '1305.4894-3-0-1': "Our proof uses two ingredients: an extension of Rouquier's deformation approach as well as categorical actions on highest weight categories and related combinatorics.", '1305.4894-3-1-0': '# Introduction', '1305.4894-3-2-0': 'Rational Cherednik algebras were introduced by Etingof and Ginzburg, [CITATION].', '1305.4894-3-2-1': 'These are associative algebras over [MATH] constructed from a complex reflection group, say [MATH], and depending on a parameter, say [MATH], that is a collection of complex numbers.', '1305.4894-3-2-2': 'They have many things in common with the universal enveloping algebras of semisimple Lie algebras, in particular, they have a triangular decomposition.', '1305.4894-3-2-3': 'This allows one to define the categories [MATH] for such algebras, this was done in [CITATION].', '1305.4894-3-2-4': 'There are analogs of Verma modules, parameterized by irreducible [MATH]-modules, and an ordering on the set of simples in the GGOR category [MATH] making it into a highest weight category.', '1305.4894-3-2-5': 'So there is a basic question one can ask: compute the multiplicity of a given simple module in a given standard (=Verma) module.', '1305.4894-3-3-0': 'The nicest and, perhaps, most important family of complex reflection groups is [MATH], where [MATH] are positive integers.', '1305.4894-3-3-1': 'This group acts on [MATH] by permutations of coordinates followed by multiplications by a root of [MATH] of order [MATH].', '1305.4894-3-3-2': 'There are more general infinite families, the groups [MATH], where [MATH] is a divisor of [MATH], but the study of the corresponding categories [MATH] can be, to some extent, reduced to the case of [MATH] and this is one of the reasons why our case is important.', '1305.4894-3-3-3': 'Another reason is that the corresponding category has an additional interesting structure that is not present in the other cases, a categorical Kac-Moody action to be recalled below.', '1305.4894-3-3-4': 'Yet another reason is a connection to the geometry of symplectic resolutions of quotient singularities.', '1305.4894-3-4-0': 'A significant progress in determining the multiplicities was made by Rouquier in [CITATION], where he computed the multiplicities in the case [MATH] and made a conjecture for all [MATH] (the conjecture was made for some special, but, in a sense, the most interesting and "non-degenerate" values of [MATH]).', '1305.4894-3-4-1': 'The conjecture says that the multiplicities are given by certain parabolic Kazhdan-Lusztig polynomials.', '1305.4894-3-4-2': 'The techniques used in the proof for [MATH] were roughly as follows.', '1305.4894-3-4-3': 'In [CITATION] the authors introduced a so called KZ functor from the Cherednik category [MATH] to the category of modules over the Hecke algebra [MATH] of [MATH] with parameters recovered from [MATH].', '1305.4894-3-4-4': 'This is a quotient functor.', '1305.4894-3-4-5': 'Rouquier developed techniques that allow to check when two highest weight categories admitting quotient functors to [MATH]-[MATH] are equivalent.', '1305.4894-3-4-6': 'For [MATH] there is another category with a nice quotient functor, the category of modules over an appropriate [MATH]-Schur algebra that was shown to be equivalent to the Cherednik category [MATH] (under a certain "faithfulness" condition on the parameters).', '1305.4894-3-5-0': 'For [MATH], the situation is more complicated.', '1305.4894-3-5-1': 'For certain, so to say, "dominant" and "faithful", values of [MATH] Rouquier proved in [CITATION] that the Cherednik category [MATH] is equivalent to the category of modules over a suitable cyclotomic [MATH]-Schur algebra of Dipper, James and Mathas.', '1305.4894-3-5-2': 'The multiplicities for the latter categories were recently computed by Stroppel and Webster, [CITATION].', '1305.4894-3-6-0': 'On the other hand, Varagnolo and Vasserot in [CITATION] produced another category, where the multiplicities were shown to be as required by the Rouquier conjecture.', '1305.4894-3-6-1': 'Their category is a certain truncation of an affine parabolic category [MATH].', '1305.4894-3-6-2': 'They conjectured an equivalence of that category with the Cherednik category [MATH].', '1305.4894-3-7-0': 'The goal of this paper is to prove that conjecture (in a somewhat weaker form that is still sufficient for checking the Rouquier conjecture).', '1305.4894-3-7-1': 'Together with earlier results of Shan, Varagnolo and Vasserot, our result also implies a conjecture of Chuang and Miyashi, [CITATION], claiming that the Cherednik category [MATH] is Koszul and describing the Koszul dual.', '1305.4894-3-8-0': '## Ideas of proof', '1305.4894-3-9-0': 'Our proof of the Varagnolo-Vasserot conjecture uses two groups of ideas.', '1305.4894-3-9-1': 'First, we use deformation ideas initially due to Rouquier, [CITATION], with further extensions.', '1305.4894-3-9-2': 'Some of them are due to Rouquier, Shan, Varagnolo, Vasserot and some are to be developed in the present paper.', '1305.4894-3-9-3': 'Second, to properly implement these ideas we need categorical actions on highest weight categories, a topic initiated by the author in [CITATION],[CITATION] and further developed by the author and Webster in [CITATION] and in the present paper.', '1305.4894-3-10-0': 'Let us describe the deformation ideas.', '1305.4894-3-10-1': 'The GGOR category [MATH] admits a quotient functor (the KZ functor of [CITATION] to be reviewed in Section [REF]) to the category of modules over a cyclotomic quotient of the affine Hecke algebra.', '1305.4894-3-10-2': 'This functor is fully faithful on certain subcategories: for example, on the categories of tilting and of projective objects, [CITATION].', '1305.4894-3-10-3': "Also it is fully faithful on the whole category of standardly filtered objects ([MATH]-faithful in Rouquier's terminology) under some restrictions on the parameters for the Cherednik algebra, see [CITATION].", '1305.4894-3-10-4': 'As Rouquier checked in [CITATION] this implies that, after a generic one-parameter deformation of the categories of interest, the KZ functor becomes 1-faithful (i.e., an isomorphism on [MATH] and [MATH] between standardly filtered objects).', '1305.4894-3-10-5': 'Two highest weight categories over [MATH] with quotient functors to the same category are equivalent provided their orders are the same, the quotient functors are 1-faithful and are equivalences over [MATH], [CITATION].', '1305.4894-3-10-6': 'So the problem is to establish an analog of the KZ functor for a truncated affine parabolic category [MATH].', '1305.4894-3-10-7': 'To produce a functor is not difficult, this is done using categorical Kac-Moody actions, we define a projective object representing the functor in Proposition [REF] below.', '1305.4894-3-10-8': 'What is much harder is to prove faithfulness properties.', '1305.4894-3-10-9': 'Recently, Rouquier, Shan, Varagnolo and Vasserot proposed to consider 2-parametric deformations and announced that 0-faithfulness in points of codimension 1 yields 1-faithfulness for the deformed categories.', '1305.4894-3-10-10': 'Considering 2-parametric deformations is one extension of the original technique of Rouquier that we will use.', '1305.4894-3-10-11': 'We will see, Theorem [REF] and Proposition [REF], that it is enough to show that the quotient functor from the deformed affine category [MATH] is only 0-faithful.', '1305.4894-3-10-12': 'The 0-faithfulness condition follows from checking (-1)-faithfulness in codimension [MATH], Proposition [REF].', '1305.4894-3-11-0': "There is one more significant extension of Rouquier's technique that we use.", '1305.4894-3-11-1': 'We bypass the problem that sometimes the quotient functors to the cyclotomic Hecke categories are not 0-faithful by considering larger quotients described in Section [REF].', '1305.4894-3-11-2': 'The main result of that section is that the larger quotients of the truncated affine category and of the Cherednik category are equivalent.', '1305.4894-3-11-3': 'Modulo checking the faithfulness properties of the quotient functor from the affine category [MATH], this yields a proof of the Varagnolo-Vasserot conjecture.', '1305.4894-3-12-0': 'Let us explain how the theory of categorical actions on highest weight categories comes into play.', '1305.4894-3-12-1': 'Results of Rouquier, [CITATION], see, in particular, Corollary 5.7 there, suggest a way to produce a quotient functor to a cyclotomic Hecke category (i.e., the direct sum over all [MATH] of the module categories over cyclotomic Hecke algebra with fixed parameters and [MATH] variables) from some category [MATH].', '1305.4894-3-12-2': 'Namely, one gets such a functor if [MATH] is equipped with a categorical action of [MATH] that categorifies an integrable [MATH]-module generated by a highest weight vector.', '1305.4894-3-12-3': 'There is a categorical action on the affine parabolic category [MATH] before the truncation: this is provided by the Kazhdan-Lusztig tensor products.', '1305.4894-3-12-4': 'This action does not restrict to the truncated category in a straightforward way (as the truncated category is not stable under the categorification functors).', '1305.4894-3-12-5': 'However one can still define a ("restricted", but this is not of importance) categorical action on the truncated category using the categorical splitting techniques from [CITATION], this will be done in Section [REF].', '1305.4894-3-12-6': 'This produces a required quotient functor, Section [REF].', '1305.4894-3-12-7': 'Further, using structural results obtained in [CITATION], one can reduce the study of the faithfulness properties for this functor to some purely combinatorial questions concerning crystal structures on the multipartitions.', '1305.4894-3-12-8': 'More precisely, there is a combinatorial condition that guarantees [MATH]-faithfulness of the quotient functor, see Section [REF].', '1305.4894-3-12-9': 'The combinatorial condition is already sufficient to checking the (-1)-faithfulness in codimension [MATH] for the affine parabolic category.', '1305.4894-3-12-10': 'This completes the proof of the Varagnolo-Vasserot conjecture.', '1305.4894-3-13-0': 'We want to indicate the dependence of the present paper on a related work.', '1305.4894-3-13-1': 'We use an idea due to Rouquier, Shan, Varagnolo and Vasserot explained before.', '1305.4894-3-13-2': "This idea was mentioned in Shan's talk in Luminy in July 2012 without explanations on how to make it to work, and the paper, [CITATION], appeared when our paper was ready.", '1305.4894-3-13-3': 'There is also a related work of Webster, [CITATION], where he proves an equivalence between the GGOR category and a certain diagrammatic category.', '1305.4894-3-14-0': 'The version of this paper that appeared in 2013 had a serious gap.', '1305.4894-3-14-1': 'Presumably, the gap can be fixed using Zuckerman functors for affine parabolic categories [MATH], however the fix is by no means easy.', '1305.4894-3-14-2': 'In July 2015, we have discovered Theorem [REF] that allows to significantly simplify the original proof.', '1305.4894-3-15-0': '## Structure of the paper', '1305.4894-3-16-0': 'In Section [REF] we describe the highest weight categories we consider: the categories [MATH] for cyclotomic Rational Cherednik algebra and affine parabolic categories [MATH] both in the undeformed and deformed settings.', '1305.4894-3-16-1': 'We also recall basic combinatorics of these categories.', '1305.4894-3-16-2': 'This section contains no new results.', '1305.4894-3-17-0': 'In Section [REF] we provide general results on faithfulness properties of quotient functors from highest weight categories.', '1305.4894-3-17-1': 'The main results of this section are Proposition [REF] (that is a version of [CITATION]), Theorem [REF] and Proposition [REF].', '1305.4894-3-18-0': 'Section [REF] deals with categorical Kac-Moody actions on highest weight categories.', '1305.4894-3-18-1': 'It defines categorical type A Kac-Moody actions and recalls results from [CITATION],[CITATION].', '1305.4894-3-18-2': 'There are no new results there.', '1305.4894-3-19-0': 'Section [REF] is new.', '1305.4894-3-19-1': 'There we equip the truncated affine parabolic category [MATH] with a restricted type A categorical Kac-Moody action and so produce a functor to the cyclotomic Hecke category.', '1305.4894-3-20-0': 'In Section [REF] we study an interplay between the faithfulness properties of quotient morphisms and combinatorial properties of crystals.', '1305.4894-3-20-1': 'Namely, we state a combinatorial condition that guarantees vanishing of [MATH] from a suitable simple to a suitable tilting.', '1305.4894-3-20-2': 'Finally, we check that our combinatorial condition holds in a certain special case.', '1305.4894-3-21-0': 'Section [REF] we define new quotient functors that are "larger" than the functors considered (our old functors factor through new ones).', '1305.4894-3-21-1': 'Then we show that the target categories for our new functors in the GGOR and in the parabolic setting are equivalent.', '1305.4894-3-22-0': 'Finally, in the last section of this paper, we complete the proof of the main equivalence theorem that yields an asymptotic version of the Varagnolo-Vasserot conjecture.', '1305.4894-3-23-0': 'The paper contains an appendix that provides an independent proof for [MATH].'}

{'1305.4894-4-0-0': "We prove an asymptotic version of a conjecture by Varagnolo and Vasserot on an equivalence between the category [MATH] for a cyclotomic Rational Cherednik algebra and a suitable truncation of an affine parabolic category [MATH] that, in particular, implies Rouquier's conjecture on the decomposition numbers in the former.", '1305.4894-4-0-1': "Our proof uses two ingredients: an extension of Rouquier's deformation approach as well as categorical actions on highest weight categories and related combinatorics.", '1305.4894-4-1-0': '# Introduction', '1305.4894-4-2-0': 'Rational Cherednik algebras were introduced by Etingof and Ginzburg, [CITATION].', '1305.4894-4-2-1': 'These are associative algebras over [MATH] constructed from a complex reflection group, say [MATH], and depending on a parameter, say [MATH], that is a collection of complex numbers.', '1305.4894-4-2-2': 'They have many things in common with the universal enveloping algebras of semisimple Lie algebras, in particular, they have a triangular decomposition.', '1305.4894-4-2-3': 'This allows one to define the categories [MATH] for such algebras, this was done in [CITATION].', '1305.4894-4-2-4': 'There are analogs of Verma modules, parameterized by irreducible [MATH]-modules, and an ordering on the set of simples in the GGOR category [MATH] making it into a highest weight category.', '1305.4894-4-2-5': 'So there is a basic question one can ask: compute the multiplicity of a given simple module in a given standard (=Verma) module.', '1305.4894-4-3-0': 'The nicest and, perhaps, most important family of complex reflection groups is [MATH], where [MATH] are positive integers.', '1305.4894-4-3-1': 'This group acts on [MATH] by permutations of coordinates followed by multiplications by roots of [MATH] of order [MATH].', '1305.4894-4-3-2': 'There are more general infinite families, the groups [MATH], where [MATH] is a divisor of [MATH], but the study of the corresponding categories [MATH] can be, to some extent, reduced to the case of [MATH] and this is one of the reasons why our case is important.', '1305.4894-4-3-3': 'Another reason is that the corresponding category has an additional interesting structure that is not present in the other cases, a categorical Kac-Moody action to be recalled below.', '1305.4894-4-3-4': 'Yet another reason is a connection to the geometry of symplectic resolutions of quotient singularities.', '1305.4894-4-4-0': 'A significant progress in determining the multiplicities was made by Rouquier in [CITATION], where he computed the multiplicities in the case [MATH] and made a conjecture for all [MATH] (the conjecture was made for some special, but, in a sense, the most interesting and "non-degenerate" values of [MATH]).', '1305.4894-4-4-1': 'The conjecture says that the multiplicities are given by certain parabolic Kazhdan-Lusztig polynomials.', '1305.4894-4-4-2': 'The techniques used in the proof for [MATH] were roughly as follows.', '1305.4894-4-4-3': 'In [CITATION] the authors introduced a so called KZ functor from the Cherednik category [MATH] to the category of modules over the Hecke algebra [MATH] of [MATH] with parameters recovered from [MATH].', '1305.4894-4-4-4': 'This is a quotient functor.', '1305.4894-4-4-5': 'Rouquier developed techniques that allow to check when two highest weight categories admitting quotient functors to [MATH]-[MATH] are equivalent.', '1305.4894-4-4-6': 'For [MATH] there is another category with a nice quotient functor, the category of modules over an appropriate [MATH]-Schur algebra that was shown to be equivalent to the Cherednik category [MATH] (under a certain "faithfulness" condition on the parameters).', '1305.4894-4-5-0': 'For [MATH], the situation is more complicated.', '1305.4894-4-5-1': 'For certain, so to say, "dominant" and "faithful", values of [MATH] Rouquier proved in [CITATION] that the Cherednik category [MATH] is equivalent to the category of modules over a suitable cyclotomic [MATH]-Schur algebra of Dipper, James and Mathas.', '1305.4894-4-5-2': 'The multiplicities for the latter categories were recently computed by Stroppel and Webster, [CITATION].', '1305.4894-4-6-0': 'On the other hand, Varagnolo and Vasserot in [CITATION] produced another category, where the multiplicities were shown to be as required by the Rouquier conjecture.', '1305.4894-4-6-1': 'Their category is a certain truncation of an affine parabolic category [MATH].', '1305.4894-4-6-2': 'They conjectured an equivalence of that category with the Cherednik category [MATH].', '1305.4894-4-7-0': 'The goal of this paper is to prove that conjecture (in a somewhat weaker form that is still sufficient for checking the Rouquier conjecture).', '1305.4894-4-7-1': 'Together with earlier results of Shan, Varagnolo and Vasserot, our result also implies a conjecture of Chuang and Miyashi, [CITATION], claiming that the Cherednik category [MATH] is Koszul and describing the Koszul dual.', '1305.4894-4-8-0': '## Ideas of proof', '1305.4894-4-9-0': 'Our proof of the Varagnolo-Vasserot conjecture uses two groups of ideas.', '1305.4894-4-9-1': 'First, we use deformation ideas initially due to Rouquier, [CITATION], with further extensions.', '1305.4894-4-9-2': 'Some of them are due to Rouquier, Shan, Varagnolo, Vasserot and some are to be developed in the present paper.', '1305.4894-4-9-3': 'Second, to properly implement these ideas we need categorical actions on highest weight categories, a topic initiated by the author in [CITATION],[CITATION] and further developed by the author and Webster in [CITATION] and in the present paper.', '1305.4894-4-10-0': 'Let us describe the deformation ideas.', '1305.4894-4-10-1': 'The GGOR category [MATH] admits a quotient functor (the KZ functor of [CITATION] to be reviewed in Section [REF]) to the category of modules over a cyclotomic quotient of the affine Hecke algebra.', '1305.4894-4-10-2': 'This functor is fully faithful on certain subcategories: for example, on the categories of tilting and of projective objects, [CITATION].', '1305.4894-4-10-3': "Also it is fully faithful on the whole category of standardly filtered objects ([MATH]-faithful in Rouquier's terminology) under some restrictions on the parameters for the Cherednik algebra, see [CITATION].", '1305.4894-4-10-4': 'As Rouquier checked in [CITATION] this implies that, after a generic one-parameter deformation of the categories of interest, the KZ functor becomes 1-faithful (i.e., an isomorphism on [MATH] and [MATH] between standardly filtered objects).', '1305.4894-4-10-5': 'Two highest weight categories over [MATH] with quotient functors to the same category are equivalent provided their orders are the same, the quotient functors are 1-faithful and are equivalences over [MATH], [CITATION].', '1305.4894-4-10-6': 'So the problem is to establish an analog of the KZ functor for a truncated affine parabolic category [MATH].', '1305.4894-4-10-7': 'To produce a functor is not difficult, this is done using categorical Kac-Moody actions, we define a projective object representing the functor in Proposition [REF] below.', '1305.4894-4-10-8': 'What is much harder is to prove faithfulness properties.', '1305.4894-4-10-9': 'Recently, Rouquier, Shan, Varagnolo and Vasserot proposed to consider 2-parametric deformations and announced that 0-faithfulness in points of codimension 1 yields 1-faithfulness for the deformed categories.', '1305.4894-4-10-10': 'Considering 2-parametric deformations is one extension of the original technique of Rouquier that we will use.', '1305.4894-4-10-11': 'We will see, Theorem [REF] and Proposition [REF], that it is enough to show that the quotient functor from the deformed affine category [MATH] is only 0-faithful.', '1305.4894-4-10-12': 'The 0-faithfulness condition follows from checking (-1)-faithfulness in codimension [MATH], Proposition [REF].', '1305.4894-4-11-0': "There is one more significant extension of Rouquier's technique that we use.", '1305.4894-4-11-1': 'We bypass the problem that sometimes the quotient functors to the cyclotomic Hecke categories are not 0-faithful by considering larger quotients described in Section [REF].', '1305.4894-4-11-2': 'The main result of that section is that the larger quotients of the truncated affine category and of the Cherednik category are equivalent.', '1305.4894-4-11-3': 'Modulo checking the faithfulness properties of the quotient functor from the affine category [MATH], this yields a proof of the Varagnolo-Vasserot conjecture.', '1305.4894-4-12-0': 'Let us explain how the theory of categorical actions on highest weight categories comes into play.', '1305.4894-4-12-1': 'Results of Rouquier, [CITATION], see, in particular, Corollary 5.7 there, suggest a way to produce a quotient functor to a cyclotomic Hecke category (i.e., the direct sum over all [MATH] of the module categories over cyclotomic Hecke algebra with fixed parameters and [MATH] variables) from some category [MATH].', '1305.4894-4-12-2': 'Namely, one gets such a functor if [MATH] is equipped with a categorical action of [MATH] that categorifies an integrable [MATH]-module with weights bounded from above.', '1305.4894-4-12-3': 'There is a categorical action on the affine parabolic category [MATH] before the truncation: this is provided by the Kazhdan-Lusztig tensor products.', '1305.4894-4-12-4': 'This action does not restrict to the truncated category in a straightforward way (as the truncated category is not stable under the categorification functors).', '1305.4894-4-12-5': 'However one can still define a ("restricted", but this is not of importance) categorical action on the truncated category using the categorical splitting techniques from [CITATION], this will be done in Section [REF].', '1305.4894-4-12-6': 'This produces a required quotient functor, Section [REF].', '1305.4894-4-12-7': 'Further, using structural results obtained in [CITATION], one can reduce the study of the faithfulness properties for this functor to some purely combinatorial questions concerning crystal structures on the multipartitions.', '1305.4894-4-12-8': 'More precisely, there is a combinatorial condition that guarantees [MATH]-faithfulness of the quotient functor, see Section [REF].', '1305.4894-4-12-9': 'The combinatorial condition is already sufficient to checking the (-1)-faithfulness in codimension [MATH] for the affine parabolic category.', '1305.4894-4-12-10': 'This completes the proof of the Varagnolo-Vasserot conjecture.', '1305.4894-4-13-0': 'We want to indicate the dependence of the present paper on a related work.', '1305.4894-4-13-1': 'We use an idea due to Rouquier, Shan, Varagnolo and Vasserot explained before.', '1305.4894-4-13-2': "This idea was mentioned in Shan's talk in Luminy in July 2012 without explanations on how to make it to work, and the paper, [CITATION], appeared when our paper was ready.", '1305.4894-4-13-3': 'There is also a related work of Webster, [CITATION], where he proves an equivalence between the GGOR category and a certain diagrammatic category.', '1305.4894-4-14-0': 'The version of this paper that appeared in 2013 had a serious gap.', '1305.4894-4-14-1': 'Presumably, the gap can be fixed using Zuckerman functors for affine parabolic categories [MATH], however the fix is by no means easy.', '1305.4894-4-14-2': 'In July 2015, we have discovered Theorem [REF] that allows to significantly simplify the original proof.', '1305.4894-4-15-0': '## Structure of the paper', '1305.4894-4-16-0': 'In Section [REF] we describe the highest weight categories we consider: the categories [MATH] for cyclotomic Rational Cherednik algebra and affine parabolic categories [MATH] both in the undeformed and deformed settings.', '1305.4894-4-16-1': 'We also recall basic combinatorics of these categories.', '1305.4894-4-16-2': 'This section contains no new results.', '1305.4894-4-17-0': 'In Section [REF] we provide general results on faithfulness properties of quotient functors from highest weight categories.', '1305.4894-4-17-1': 'The main results of this section are Proposition [REF] (that is a version of [CITATION]), Theorem [REF] and Proposition [REF].', '1305.4894-4-18-0': 'Section [REF] deals with categorical Kac-Moody actions on highest weight categories.', '1305.4894-4-18-1': 'It defines categorical type A Kac-Moody actions and recalls results from [CITATION],[CITATION].', '1305.4894-4-18-2': 'There are no new results there.', '1305.4894-4-19-0': 'Section [REF] is new.', '1305.4894-4-19-1': 'There we equip the truncated affine parabolic category [MATH] with a restricted type A categorical Kac-Moody action and so produce a functor to the cyclotomic Hecke category.', '1305.4894-4-20-0': 'In Section [REF] we study an interplay between the faithfulness properties of quotient morphisms and combinatorial properties of crystals.', '1305.4894-4-20-1': 'Namely, we state a combinatorial condition that guarantees vanishing of [MATH] from a suitable simple to a suitable tilting.', '1305.4894-4-20-2': 'Finally, we check that our combinatorial condition holds in a certain special case.', '1305.4894-4-21-0': 'Section [REF] we define new quotient functors that are "larger" than the functors considered before (our old functors factor through new ones).', '1305.4894-4-21-1': 'Then we show that the target categories for our new functors in the GGOR and in the parabolic setting are equivalent.', '1305.4894-4-22-0': 'Finally, in the last section of this paper, we complete the proof of the main equivalence theorem that yields an asymptotic version of the Varagnolo-Vasserot conjecture.', '1305.4894-4-23-0': 'The paper contains an appendix that provides an independent proof for [MATH].'}

1907.00781

{'1907.00781-1-0-0': 'After introducing the notion of weak Berwald spacetime with a maximal proper acceleration, it is shown that such spacetimes are free of divergent singularities.', '1907.00781-1-0-1': 'The upper bounds in the curvature that a maximal proper acceleration implies are analyzed in two cases: for conformally flat spacetimes and for Ricci flat spacetimes.', '1907.00781-1-1-0': '# Introduction', '1907.00781-1-2-0': 'The appearance of spacetime singularities in general relativity is one of the main motivations to search for a new theory of gravity that, superseding the current paradigm, should resolve the singularities.', '1907.00781-1-2-1': 'It is also believed that new phenomenological and theoretical clues for understanding gravity will be found in regimes and domains where the gravitational interaction manifests strongly.', '1907.00781-1-3-0': 'The strength of a classical gravitational field is characterized by the curvature tensor of a Levi-Civita connection.', '1907.00781-1-3-1': 'Hence strong gravity implies strong curvature.', '1907.00781-1-3-2': 'Curvature is also a measure of the relative separation of nearby geodesics.', '1907.00781-1-3-3': 'In particular, the Riemann curvature endomorphism is related with the relative acceleration between neighboring geodesics.', '1907.00781-1-3-4': 'Therefore, the strength of a gravitational field is related with the values of relative accelerations between geodesics.', '1907.00781-1-4-0': 'On the other hand, it has been speculated for a long time with the idea of a maximal proper acceleration [CITATION].', '1907.00781-1-4-1': 'For a recent review of some aspects of maximal acceleration, the reader is referred to ref. [CITATION].', '1907.00781-1-4-2': 'The intuitive implication of a maximal acceleration for curvature is an uniform upper bounds on the curvature(s).', '1907.00781-1-4-3': 'In this paper we confirm this insight, showing that certain type of spacetimes with metrics of maximal proper acceleration are free of divergent spacetime singularities, the ones that have associated a divergent curvature tensor.', '1907.00781-1-5-0': 'The present paper is not the first time claims where maximal acceleration resolves singularities have been investigated and well-known studies of particular examples are [CITATION] and [CITATION].', '1907.00781-1-5-1': 'However, our treatment is the first to show that this is the generic feature in a particular framework of spacetimes with metrics of maximal acceleration, namely, in the framework of classical higher order jet geometries [CITATION].', '1907.00781-1-6-0': 'The structure of this paper is the following.', '1907.00781-1-6-1': 'In section 2, the notion of spacetime with a metric of maximal acceleration is reviewed from [CITATION].', '1907.00781-1-6-2': 'In section 3, the Jacobi equation and the notion of weak Berwald spacetime is introduced in the above framework.', '1907.00781-1-6-3': 'These are examples of spacetimes of higher order jet geometry such that along the geodesics of a limit Lorentzian structure, the metric of maximal acceleration structure is geodesically equivalent to the limit Lorentzian structure.', '1907.00781-1-6-4': 'The notion of weak Berwald spacetimes allows us to implement the weak equivalence principle in a natural way in the context of higher order jet geometries [CITATION].', '1907.00781-1-6-5': 'Then we discuss the general implications of the existence of a maximal proper acceleration for the Riemann curvature tensor.', '1907.00781-1-6-6': 'We show in section 4 that the spacetimes under consideration are free of divergent singularities.', '1907.00781-1-6-7': 'The meaning of the bounds on the curvature discussed in more detail in section 5.', '1907.00781-1-6-8': 'First, we investigate spacetimes where the conformal Weyl tensor is zero.', '1907.00781-1-6-9': 'Second, we consider spacetimes where the Ricci tensor is zero.', '1907.00781-1-6-10': 'In both cases, quadratic combinations of the remaining curvatures appear to be upper bounded.', '1907.00781-1-6-11': 'Few general remarks are highlighted in the discussion section, where also a very brief outlook for future developments is presented.', '1907.00781-1-7-0': '# Metrics of maximal acceleration', '1907.00781-1-8-0': 'Let us first introduce the geometric framework for this work.', '1907.00781-1-8-1': 'The spacetime manifold [MATH] will be a [MATH]-dimensional smooth manifold.', '1907.00781-1-8-2': 'The metric structures that we will consider are called metrics of maximal acceleration [CITATION].', '1907.00781-1-8-3': 'Basically, they are metric structures that depend upon the second jet of smooth world lines [MATH].', '1907.00781-1-8-4': 'These curves are interpreted as the world lines of test particles probing the structure of a classical spacetime, that is, a spacetime not subjected to quantum fluctuations.', '1907.00781-1-8-5': 'The metric structure determines the time that an observer will measure along its world line [MATH].', '1907.00781-1-8-6': 'If the curve [MATH] is (piecewise)-smooth causal, which means [MATH], the proper time is of the form [EQUATION] where the integration parameter will be specified later, since in general the integrands that we will consider are not homogeneous under changes of integration parameter.', '1907.00781-1-9-0': "If we adopt Synge's chronodynamics postulates [CITATION] except for Einstein's clock hypothesis [CITATION] and we complete such postulates with the assumption of the existence of maximal acceleration scale [MATH] embedded in the kinematical properties of the spacetime, then it is natural to adopt the following form for the integrand [MATH] in the expression for the proper time [REF], [EQUATION] where [MATH] is the second jet at [MATH] of the curve [MATH].", '1907.00781-1-10-0': 'The expression [REF] introduces a family of structures [EQUATION] parameterized by the value of what it will be identified with the maximal proper acceleration, namely, the parameter [MATH].', '1907.00781-1-10-1': 'Thus we can consider limits when [MATH] in the family of metrics [MATH].', '1907.00781-1-10-2': 'We require that the metric structure obtained by the limit [EQUATION] to be compatible with the clock hypothesis.', '1907.00781-1-10-3': 'Hence the natural form for the components of [MATH] are analytical in [MATH], [EQUATION]', '1907.00781-1-10-4': 'Compatibility with the clock hypothesis in the above mentioned limit implies, [EQUATION]', '1907.00781-1-10-5': "A weaker form of Synge's chronodynamics postulates implies that the metric structure [MATH] is a generalized Finsler structure, but to simplify the treatment, it will be a Lorentzian metric [MATH].", '1907.00781-1-11-0': 'The Lorentzian spacetime [MATH] can be thought as a limit of a spacetime endowed with a metric of maximal acceleration, associated to the proper time [REF] whose integrand is of the form [EQUATION] and where [MATH] is the covariant derivative of the Levi-Civita connection of [MATH].', '1907.00781-1-12-0': 'We specify the parameter respect to which we are taking derivatives and parameterizing curves to be the proper time parameter associated to the Lorentzian metric [MATH].', '1907.00781-1-12-1': 'Then the action of the metric of maximal acceleration on two arbitrary vector fields [MATH] along [MATH] is given by [EQUATION]', '1907.00781-1-12-2': 'Originally, the form [REF] appeared as a covariant version [CITATION] of the metric developed by E. Caianiello and co-workers [CITATION], but it has a more natural interpretation as a higher order jet geometry [CITATION].', '1907.00781-1-12-3': "From this second point of view, the metric structure of the spacetime assigns to each test particle probing the structure of the classical spacetime a line element which depends upon higher order derivatives of the test particle's world line.", '1907.00781-1-12-4': 'Let us consider a world line [MATH] with [MATH], a causal curve respect to the metric [MATH].', '1907.00781-1-12-5': 'Let us assume that at each point [MATH] the spacetime is probed by a test particle with identical [MATH]-jet than [MATH].', '1907.00781-1-12-6': 'Then the proper time functional is of the form [EQUATION] from where we can extract the expression [REF] and the expression [REF].', '1907.00781-1-13-0': 'It follows that the proper acceleration respect to the limit metric [MATH] is bounded, in the sense that for any causal curve such that [MATH] implies the bound [EQUATION]', '1907.00781-1-13-1': 'This is the reason to call the structures given by [REF] or by the form [REF] metrics of maximal acceleration.', '1907.00781-1-13-2': 'In the context of a given classical theory or model, the relation [REF] must be adopted as of universal validity, for any type of motion described from any physical frame.', '1907.00781-1-14-0': 'The theory of metric of maximal acceleration can be given a global formulation, where the metrics are sections of a particular kind of fibre bundles over jet bundles.', '1907.00781-1-14-1': 'This should make possible to define a relevant connection theory with the corresponding notion of geodesic and geodesic deviation [CITATION].', '1907.00781-1-14-2': 'However, such a tour de force is not necessary for the purposes of this paper, as we will show in the next section.', '1907.00781-1-15-0': '# Jacobi equation and bounds on the Lorentzian curvature', '1907.00781-1-16-0': 'Let us consider a regular point [MATH].', '1907.00781-1-16-1': 'That is, at [MATH] it is possible to choose arbitrary initial conditions for the geodesics of the Levi-Civita connection [MATH].', '1907.00781-1-16-2': 'The point [MATH] can be set in an open set [MATH] such that all the points at [MATH] are regular in the above sense.', '1907.00781-1-16-3': 'Then we can consider a geodesic tubular neighborhood containing [MATH], constructed as follows.', '1907.00781-1-16-4': 'Given the initial condition [MATH] with [MATH] a causal tangent vector, then there is a geodesic [MATH] with initial conditions [MATH].', '1907.00781-1-16-5': 'Let us consider [MATH] orthogonal to [MATH] with respect to the metric [MATH].', '1907.00781-1-16-6': 'We construct the parallel transport [MATH] of [MATH] along the geodesic with initial conditions [MATH] from the initial point [MATH] to a final point [MATH].', '1907.00781-1-16-7': 'For close enough points [MATH], this construction can be done always, as consequence of elementary ordinary differential equations theory.', '1907.00781-1-16-8': 'We can consider the initial conditions [MATH] for constructing a geodesic of the connection [MATH], extending the geodesic to the reverse direction at [MATH], in order that [MATH] is in the interior of the two geodesics with initial conditions [MATH] and [MATH].', '1907.00781-1-17-0': 'Repeating this procedure for each pair [MATH] with [MATH] fixed and [MATH] orthogonal to [MATH] but otherwise arbitrary, we can embed [MATH] in an open tube set [MATH] filled by the image of a aggregate of geodesics [MATH] and such that [MATH] also contains the central geodesic [MATH].', '1907.00781-1-17-1': 'Moreover, by reducing enough the size of the tube, the geodesics will not cross to each other.', '1907.00781-1-17-2': 'Since the central geodesic [MATH] is causal, then the geodesics [MATH] are also causal and of the same type.', '1907.00781-1-18-0': 'For our purposes, the above geodesic tube [MATH] containing the point [MATH] is useful because its fundamental role in the proof of the following result,', '1907.00781-1-19-0': 'If [MATH] is point of [MATH] a geodesic tube of [MATH], then the maximal acceleration spacetime metric [MATH] is equivalent to [MATH], when probed by geodesic curves of [MATH].', '1907.00781-1-20-0': 'For each regular point [MATH] there is a geodesic [MATH] which pass through [MATH] and which image is contained in [MATH].', '1907.00781-1-20-1': 'When we particularize the expression [REF] for the case when [MATH] for geodesics [MATH], then [MATH] in the whole tubular neighborhood [MATH].', '1907.00781-1-21-0': 'Assume the existence of a connection which preserves [MATH].', '1907.00781-1-21-1': 'We can say that the connection is associated to [MATH].', '1907.00781-1-21-2': 'Then the physical significance of this Lemma becomes clear if we consider the following notion,', '1907.00781-1-22-0': 'A spacetime of maximal acceleration is weak Berwald if there is a connection associated to [MATH] which is also the Levi-Civita connection of the limit metric [MATH] along its geodesics.', '1907.00781-1-23-0': 'This notion is related with the corresponding notion of Berwald spacetime in the theory of Finsler spacetimes.', '1907.00781-1-23-1': "Generalized Berwald spacetimes are general structures supporting Synge's assumptions of chronodynamics and being compatible with the weak equivalence principle [CITATION].", '1907.00781-1-24-0': 'Let us note that the restriction to the geodesic support in Definition [REF] is fundamental for speaking of universal free fall.', '1907.00781-1-24-1': 'If the process of probing the metric structure of the spacetime does not perturb greatly enough the spacetime geometry, as by universal free fall is required, then the geodesics of [MATH] must be the geodesics of [MATH], since [MATH] is the limit of [MATH] when the probing action does not implies back reaction.', '1907.00781-1-24-2': 'In addition, free fall motion is equivalent to select an aggregate of world lines as being the geodesics of an affine connection.', '1907.00781-1-24-3': 'Therefore, it is then natural to adopt as definition of geodesics the solutions of the system of differential equations [EQUATION]', '1907.00781-1-24-4': 'This argument shows that the condition of being weak Berwald spacetime, in the framework of spacetimes with a metric of maximal acceleration, is the natural condition to accommodate the weak equivalence principle.', '1907.00781-1-25-0': 'After the above considerations, it is direct the following', '1907.00781-1-26-0': 'A spacetime of maximal acceleration is a weak Berwald spacetime.', '1907.00781-1-27-0': 'Several consequences of Lemma [REF] are of physical significance, since it allows to translate results from Lorentzian geometry to spacetimes with metrics of maximal acceleration in relevant physical situations:', '1907.00781-1-28-0': 'Let [MATH] be in a geodesic tube [MATH] of [MATH].', '1907.00781-1-28-1': 'When the spacetime structure is probed by geodesic curves of [MATH], then the following properties hold true:', '1907.00781-1-29-0': 'In order to prove that the first point holds, just note that by Lemma [REF], when the spacetime structure is tested by geodesics in [MATH], then [MATH] and hence [MATH].', '1907.00781-1-30-0': "Similarly and for the same reason, in order to prove the second point, we can follow the standard derivation of Jacobi's equation in Lorentzian geometry [CITATION].", '1907.00781-1-30-1': 'Note that here [MATH] is a Jacobi field of the Levi-Civita connection [MATH].', '1907.00781-1-31-0': 'Proposition [REF] has relevant physical consequences.', '1907.00781-1-31-1': 'Let us first emphasize that the upper bound [REF] in the maximal acceleration should apply to the relative accelerations of the displacement vector [MATH] (a Jacobi field of D).', '1907.00781-1-31-2': "In the case that an observer has associated the timelike geodesic [MATH], this interpretation follows from application of the principle of general covariance in the spirit of Einstein's original theory of relativity [CITATION].", '1907.00781-1-31-3': 'The interpretation of the second covariant derivative [MATH] as the relative acceleration between geodesics is also valid to the case of null geodesics, although then the central geodesic [MATH] does not have associated a physical observer, but it can be seen as a limit case of a continuous series of timelike observers.', '1907.00781-1-32-0': 'As a result of the universality of the maximal acceleration and the properties discussed in Proposition [REF] we have that [EQUATION]', '1907.00781-1-32-1': 'The relation [REF] is the general form of a pointwise upper bound on the curvature of the Lorentzian metric [MATH].', '1907.00781-1-32-2': 'In order to understand its meaning, it is useful to consider geodesic congruences containing a given point [MATH].', '1907.00781-1-32-3': 'Let us consider a particular initial condition [MATH], where [MATH] and [MATH] is a causal tangent vector.', '1907.00781-1-32-4': 'For concreteness, let us assume that [MATH] is a time-like vector and three supplementary initial conditions [MATH] orthogonal Jacobi fields, that without loss of generality, satisfy the relations [EQUATION]', '1907.00781-1-32-5': 'But for the curvature endomorphisms we have [EQUATION] where the index [MATH] is not summed and where [MATH] forms an orthogonal frame of vector fields along [MATH].', '1907.00781-1-32-6': 'However, the frame [MATH] can be taken as the frame [MATH] and this will be the case from now on.', '1907.00781-1-33-0': 'In a spacetime geometry of maximal acceleration [MATH], the following bounds on the Riemannian curvature of the limit Lorentzian curvature [MATH] must be satisfied: [EQUATION]', '1907.00781-1-33-1': 'This is a direct consequence of the conditions [REF] and the previous considerations.', '1907.00781-1-34-0': 'In the relation [REF], the repeated indexes [MATH] are not summed.', '1907.00781-1-34-1': 'Their meaning refers in terms of the orthonormal frame [MATH].', '1907.00781-1-35-0': '# General consequences of maximal acceleration for singularities', '1907.00781-1-36-0': 'The Jacobi equation [REF] and the relation [REF] can be applied at the points of the spacetime that can be used as initial conditions for geodesics.', '1907.00781-1-36-1': 'Hence, singularity points of the spacetime are excluded in the above construction.', '1907.00781-1-36-2': 'The set [MATH] of singularities of the spacetime is the complement of an open set (this follows by general considerations of ODE theory).', '1907.00781-1-36-3': 'Therefore, [MATH] is a close set.', '1907.00781-1-37-0': 'Let us further assume that the topological boundary of [MATH] is not empty.', '1907.00781-1-37-1': 'Then one of the following four alternatives must hold:', '1907.00781-1-38-0': 'Hence we have proved the following,', '1907.00781-1-39-0': 'In a spacetime [MATH] of maximal acceleration of weak Berwald type there are no spacetime singularities where the curvature tensor is divergent.', '1907.00781-1-40-0': 'Note that the arguments discussed in the first three points are independent of the existence of a non-trivial topological boundary in [MATH].', '1907.00781-1-40-1': 'Also, let us remark that we have proved this result under the above mentioned restrictions, namely, that the spacetime is of weak Berwald type, that the singularity set has a non-trivial topological boundary and that the curvature diverges at the singularities.', '1907.00781-1-40-2': 'The second condition can be modified, for instance, assuming that singularities are isolated or that they form a regular sub-manifold of [MATH], to obtain analogous results.', '1907.00781-1-40-3': 'The third condition possibly can be substituted by a weaker condition indicating that gravity is strong enough.', '1907.00781-1-40-4': 'The condition of being weak Berwald type is essential for these results to hold.', '1907.00781-1-40-5': 'From now on, we will assume the weak Berwald condition.', '1907.00781-1-41-0': '# Implications of maximal acceleration for spacetime curvature', '1907.00781-1-42-0': 'In order to clarify the significance of the upper bound [REF], let us consider the Ricci decomposition of the Riemann curvature tensor [CITATION], that for a spacetime of dimension four is given by the expression [EQUATION]', '1907.00781-1-42-1': 'This expression relates the Riemann tensor with the Ricci tensor [MATH], the Ricci curvature [MATH] and the Weyl conformal tensor [MATH].', '1907.00781-1-42-2': 'In particular, we gave [EQUATION] since [MATH].', '1907.00781-1-42-3': 'In the derivation of this relation we have used orthonormal frames where the metric [MATH] is diagonal and of the form [MATH] at a given point of the central geodesic [MATH].', '1907.00781-1-43-0': 'By direct computation, using the Ricci decomposition for the geometric framework that we have discussed above, one obtains the following bounds in the curvature components respect to the frame [MATH], [EQUATION] where in this expression, we have imposed already that [MATH].', '1907.00781-1-44-0': 'There are two situations where the relation [REF] has clear physical implications:', '1907.00781-1-45-0': '1.', '1907.00781-1-45-1': 'Conformally flat spacetimes.', '1907.00781-1-45-2': 'In the first one, one makes the assumption that the conformal tensor is zero, [EQUATION]', '1907.00781-1-45-3': 'Then the relation [REF] implies [EQUATION]', '1907.00781-1-45-4': 'By a re-ordering of the first line, this expression can be re-written as [EQUATION]', '1907.00781-1-45-5': 'Since [EQUATION] we have the expression [EQUATION]', '1907.00781-1-45-6': 'From this relation, it follows the following result:', '1907.00781-1-46-0': 'For a conformally flat spacetime of maximal acceleration of weak Berwald type, we have a general bound of the form [EQUATION] where [MATH].', '1907.00781-1-47-0': 'There are several consequences of Proposition [REF],', '1907.00781-1-48-0': 'Assume for the Ricci scalar [MATH] in a spacetime of maximal acceleration of weak Berwald type which is conformally flat.', '1907.00781-1-48-1': 'If in addition the relation [EQUATION] holds good in the orthonormal frames considered before, then [EQUATION]', '1907.00781-1-48-2': 'The condition [MATH] is illustrated by the value of the curvature of a Robertson-Walker spacetime with [MATH], where [MATH] is valid for dust mater source and for radiation described by perfect fluids.', '1907.00781-1-48-3': 'Therefore, the condition [MATH] appears as a reasonable assumption.', '1907.00781-1-48-4': 'This remark can be extrapolated to the case when [MATH].', '1907.00781-1-49-0': 'We can appreciate the meaning of the condition [MATH] by considering Einstein equations without the cosmological constant.', '1907.00781-1-49-1': 'In this case, the condition [REF] follows from the dominant energy condition in an orthonormal frame, [EQUATION]', '1907.00781-1-49-2': 'It is natural to think that the condition [MATH] holds at least for classical theories that quantitatively do not differ significatively from general relativity.', '1907.00781-1-50-0': '2.', '1907.00781-1-50-1': 'Ricci-flat spacetimes.', '1907.00781-1-50-2': 'In this case, the Ricci-flat condition [MATH] and the general form of the bound [REF] implies [EQUATION]', '1907.00781-1-50-3': 'The conformal tensor is characterized by having all traces null and by having the same symmetries than the Riemann tensor.', '1907.00781-1-50-4': 'The conditions of null trace are [EQUATION]', '1907.00781-1-50-5': 'For spacetimes of dimension four, these trace-free conditions imply [MATH] constrains, reducing the number of independent components of the conformal tensor to [MATH] functions: since the conformal tensor has the same symmetries than the Riemann tensor, in dimension [MATH] the conformal tensor has [MATH] components, which are subjected to the [MATH] trace-free conditions.', '1907.00781-1-50-6': 'On the other hand, the conditions [REF] imply three additional constraints.', '1907.00781-1-50-7': 'Hence, the remaining four components independent of [MATH] are constrained by four quadratic inequalities, the result of applying coordinate changes to the relations [REF].', '1907.00781-1-51-0': 'For a spacetime with a maximal acceleration of weak Berwald type and Ricci flat, the components of the conformal tensor respect to orthonormal frame [MATH] are bounded.', '1907.00781-1-52-0': '# Discussion', '1907.00781-1-53-0': 'We have shown that if the weak equivalence principle is accepted and one assumes the existence of an uniform or universal bound for the proper acceleration of test particles in the form of a weak Berwald spacetime, then there are no spacetime singularities whose curvature diverges.', '1907.00781-1-54-0': 'The notion of spacetimes of maximal acceleration that we have considered are not necessarily originated by corrections to the classical general relativity due to a quantum origin of gravity.', '1907.00781-1-54-1': 'Indeed, we have discussed the upper bounds in the Riemann curvature in the contest of classical geometries of maximal acceleration, where the notion of higher order geometry is fundamental.', '1907.00781-1-54-2': 'Hence, our proposed resolution for the incompleteness of general relativity, namely, Theorem [REF], is a classical result, in contrast with arguments involving quantum gravity [CITATION].', '1907.00781-1-54-3': 'The main argument in favour of a metric of maximal acceleration has been discussed elsewhere [CITATION].', '1907.00781-1-54-4': 'It is based upon considerations of the mathematical treatment of radiation reaction together with an analysis of the limits of applicability of the clock hypothesis in the foundations of the theory of relativity [CITATION].', '1907.00781-1-54-5': 'Indeed, we have adopted the viewpoint that the existence of a maximal proper acceleration should be imprinted in the geometry of the spacetime, in a similar way as the existence of a maximal speed is imprinted in the causal structure causal structure of the spacetime.', '1907.00781-1-55-0': 'Although the hypothesis of a maximal acceleration is fertile in consequences, there is no empirical evidence of it, However, a new approach to observe effects of maximal acceleration in laser-plasma dynamics has been recently discussed [CITATION].', '1907.00781-1-55-1': 'In that paper it was argued that a modification of the Lorentz force due to maximal acceleration is potentially testable in near future laser-plasma acceleration facilities.', '1907.00781-1-56-0': 'The general result of the absence of divergent singularities in spacetimes of maximal acceleration, Theorem [REF], applies under rather general conditions.', '1907.00781-1-56-1': 'It does not involve particular field equations or energy conditions.', '1907.00781-1-56-2': 'This shows that the assumption of a maximal proper acceleration and the validity of the weak equivalence principle and the principle of general covariance, are strong hypotheses when they are considered together.', '1907.00781-1-56-3': 'However, Theorem [REF] provides few details about the origin and physical significance of the maximal proper acceleration.', '1907.00781-1-57-0': 'The results of section 5 illustrate the meaning of the bound of maximal acceleration in our framework, although these results are not formulated in a covariant way, since they involve conditions satisfied in specific orthonormal frames.', '1907.00781-1-57-1': 'One is tented to relate Proposition [REF] and Proposition [REF] with Weyl curvature hypothesis [CITATION] extrapolated to frameworks of spacetimes with metrics of maximal acceleration.', '1907.00781-1-57-2': "The Weyl curvature hypothesis states that at an initial singularity, Weyl's curvature was zero (or very small), meanwhile the Ricci curvature was large but finite.", '1907.00781-1-57-3': 'Then we observe that maximal acceleration constrains the appearence of singular Ricci tensor for initial singularities.', '1907.00781-1-58-0': 'A fundamental point missing in this paper is an example of complete gravitational model with maximal acceleration in the framework of higher order jet classical spacetime geometries.', '1907.00781-1-58-1': 'One can partially address this issue by considering the models of maximal acceleration of Caianiello and co-workers [CITATION], that although based upon a mathematical formalism which is no general covariant, it could be considered a first step towards a consistent theory of spacetimes with metrics of maximal acceleration.', '1907.00781-1-58-2': 'However, a more sophisticated and rigourous line of research that all these issues raise is to developed a theory of gravity in the framework of higher order jet geometry [CITATION].', '1907.00781-1-59-0': 'The case when the singularities are not associated to the divergence of the curvature tensor is not directly considered in this papers.', '1907.00781-1-59-1': 'It seems to me that the methods to investigate the effect of maximal acceleration for such singularities are intrinsically different from the methods considered in this paper.', '1907.00781-1-60-0': 'Finally, we have not considered in this paper the aspects of the thermodynamics of black holes that should be modified by the existence of a maximal acceleration.', '1907.00781-1-60-1': 'This fundamental and interesting question will be developed in future work.'}

{'1907.00781-2-0-0': 'A rigourous geometric framework for metrics of maximal acceleration is discussed, including a theory of connections associated with the geometry of maximal acceleration.', '1907.00781-2-0-1': 'In this context, it is shown that the uniform bound on proper maximal acceleration implies an uniform bound for curvature in the whole spacetime.', '1907.00781-2-1-0': '# Introduction', '1907.00781-2-2-0': 'The conjecture on the existence of universal or uniform bounds on proper acceleration has attracted the attention of researches for a long time [CITATION].', '1907.00781-2-2-1': 'The hypothesis of maximal proper acceleration was first discussed by E. Caianiello [CITATION] in the context of a geometric approach to the foundations of the quantum theory [CITATION].', '1907.00781-2-2-2': 'As a consistence requirement for the positiveness in the mass spectra of quantum particles and the existence of a maximal speed, Caianiello found a positiveness condition for a Sasaki-type metric in the phase space description of quantum mechanics.', '1907.00781-2-2-3': 'Such condition leaded to the existence of a maximal proper acceleration depending on the mass of the particle.', '1907.00781-2-2-4': 'In classical models of gravity, the consequences of the existence of a maximal proper acceleration have been studied extensively.', '1907.00781-2-2-5': 'Let us mention for instance the investigation of maximal proper acceleration for Rindler spaces [CITATION], Schwarzschild [CITATION], Reissner-Nordst[MATH]m [CITATION], Kerr-Newman [CITATION] and Friedman-Lemaitre metrics [CITATION], among other investigations.', '1907.00781-2-2-6': 'Independently, in the theory developed by H. Brandt, the starting point is the energy-time uncertainty principle combined with the usual argument on the breakdown of the topological and smooth structure of spacetime at the Planck scale [CITATION].', '1907.00781-2-2-7': 'This approach developed further into a effective tangent spacetime geometry [CITATION].', '1907.00781-2-2-8': 'These two theories, although speculative, have interesting consequences and initiated the study of other theories where maximal proper acceleration is upper bounded [CITATION].', '1907.00781-2-3-0': "Caianiello's and from Brandt's theories could be classified as quantum mechanical motivated frameworks.", '1907.00781-2-3-1': 'In a different research line, maximal proper acceleration has been related with the foundations of the theory of special relativity as follows [CITATION].', '1907.00781-2-3-2': 'It is well known that when developing the geometric framework for relativistic theories, an hypothesis on the characteristics of ideal clocks, namely, the clock hypothesis, is usually adopted.', '1907.00781-2-3-3': 'It affirms that ideal clocks do not depend upon the acceleration suffered by the clock [CITATION].', '1907.00781-2-3-4': 'The clock hypothesis, in conjunction with the rest of principles of relativistic theories, lead to the conclusion that spacetime geometry must be of Finslerian type ([CITATION], chapter I), among which Lorentzian geometry is a prominent example.', '1907.00781-2-3-5': 'However, the adoption of the clock hypothesis is logically unjustified, even if it leads to a simplification in the theoretical treatment of the properties and behavior of ideal clocks in accelerated motion [CITATION].', '1907.00781-2-3-6': 'Furthermore, it has been convincingly argued that in situations where radiation reaction is not negligible, the clock hypothesis is not applicable [CITATION], leading to the idea that spacetime geometry must be described by a geometric structure depending also on the acceleration of the probe particles world line curves.', '1907.00781-2-4-0': "Motivated by the problem of finding a rigourous mathematical formalism for Caianiello's theory of metrics with maximal acceleration [CITATION] and the problem of radiation reaction in classical electrodynamics, an approach to spacetime metric structures compatible with maximal proper acceleration was proposed in a geometric framework of higher order jet geometry [CITATION].", '1907.00781-2-4-1': 'This axiomatic approach assumes directly that the clock hypothesis does not hold in certain relevant dynamical domains and therefore, the spacetime metrics must be a higher order spacetime jet metric.', '1907.00781-2-4-2': 'It is also assumed that when the effects due to acceleration are negligible, the spacetime metric must respect the clock hypothesis with high accuracy, which leads in the simplest case, to Lorentzian models of spacetime.', '1907.00781-2-4-3': 'For higher order jet geometry models the spacetime metric structure depends on how spacetime is probed by test point particles.', '1907.00781-2-4-4': 'This implies that the spacetime metric models depend at least upon the second jet bundle of the world lines of test particles.', '1907.00781-2-4-5': 'We restrict the models to depend upon second derivatives only, since this is enough to provide consistent classical models of radiation-reaction systems [CITATION].', '1907.00781-2-4-6': 'In contrast with others approaches to the foundations of maximal acceleration geometry [CITATION], our approach has the advantage of not imposing a duplicity of metric structures (the metric of maximal acceleration and the usual Lorentzian spacetime), either explicitly or implicitly through a dynamical mechanism generating the maximal acceleration.', '1907.00781-2-4-7': 'The only fundamental structure is the higher order jet metric.', '1907.00781-2-4-8': 'When the model is made explicit as a power series on the inverse of the maximal acceleration [CITATION], fundamental physical principles implies that the metric has a particular form, that we called metric of maximal acceleration, because these class of metrics imply the existence of an uniform bound on maximal proper accelerations.', '1907.00781-2-5-0': 'The intuitive implication of a maximal proper acceleration for spacetime curvature is the existence of an uniform upper bound on the Riemannian curvature tensor of the Lorentzian metric limit and that such a bound is valid in the region of the spacetime where the curvature is finite.', '1907.00781-2-5-1': "Precedents of this result are found, for instance, in the context of Caianiello's quantum geometry, where it has been argued how maximal acceleration implies the regularization of big bang singularity [CITATION].", '1907.00781-2-5-2': 'Recently, it has been argued in the context of loop quantum gravity that maximal acceleration implies the resolution of singularities [CITATION].', '1907.00781-2-5-3': 'In this paper we confirm these insights, showing how in spacetimes with metrics of maximal proper acceleration as discussed in [CITATION] and developed with higher clarity and deepness in the present paper, there are uniform bounds in certain combinations in the components of Riemannian curvature endomorphims of the associated Lorentzian limit metric.', '1907.00781-2-5-4': 'We achieve this results by extending the uniform of maximal proper accelerations from the perturbative regime discussed in [CITATION] to large accelerations and by extending the uniform bound on the proper accelerations to certain relative accelerations associated to Jacobi fields.', '1907.00781-2-6-0': 'The structure of this paper is the following.', '1907.00781-2-6-1': 'In section 2, the notion of spacetime with a metric of maximal acceleration is introduced in a more general framework of higher order jet geometries.', '1907.00781-2-6-2': 'The treatment presented here is based upon fundamental principles and supersedes the perturbative approach discussed in previous works [CITATION].', '1907.00781-2-6-3': 'We give a precise definition of the notion of maximal proper acceleration.', '1907.00781-2-6-4': 'In section 3 it is recalled how curvature and relative acceleration along a geodesic are related through the Jacobi field and Jacobi geodesic deviation.', '1907.00781-2-6-5': 'The proper acceleration is an acceleration along a causal curve, while the notion of relative acceleration is a vector field along a given geodesic.', '1907.00781-2-6-6': 'However, it is shown how maximal acceleration can also be applied to these notion of relative acceleration.', '1907.00781-2-6-7': 'Such extension is achieved by means of considering world lines whose acceleration coincide pointwise with the relative acceleration associated to the solutions of the Jacobi equation.', '1907.00781-2-6-8': 'Then it is shown that the relative acceleration field can be associated with an aggregate of causal world line curves where the uniform bound on maximal proper acceleration as discussed in section 2 can be applied.', '1907.00781-2-6-9': 'As a consequence, an uniform bound on the spacetime curvature must hold in the whole spacetime.', '1907.00781-2-7-0': 'The bound on the curvature applies to the Riemannian curvature of the Lorentzian limit associated to the metric of maximal acceleration.', '1907.00781-2-7-1': 'The significance for the metric of maximal acceleration is clarified in section 4 and section 5, where the foundations for a geometric theory for spacetimes with maximal acceleration are investigated.', '1907.00781-2-7-2': 'Starting with a discussion of a new version of the equivalence principle in spacetimes of metrics with maximal acceleration, we develop a theory of connection compatible with that form of the equivalence principle and with the notion of spacetime with maximal acceleration.', '1907.00781-2-7-3': 'We then describe an axiomatic characterization of the connection in terms of structure equations.', '1907.00781-2-8-0': 'In the discussion section, several general remarks on the theory developed in this paper are highlighted, as well as the relation of the present work with other investigations by the author on maximal proper acceleration.', '1907.00781-2-8-1': 'Several future research lines are briefly indicated.', '1907.00781-2-8-2': 'They include the search for a complete gravitational model in the framework of metrics with maximal acceleration, the investigation of the possible regularization of singularities and the consequences of maximal proper acceleration for black hole physics.', '1907.00781-2-9-0': '# Metrics of maximal acceleration', '1907.00781-2-10-0': 'Let us first introduce the geometric framework for this work.', '1907.00781-2-10-1': 'The spacetime manifold [MATH] will be a [MATH]-dimensional smooth manifold.', '1907.00781-2-10-2': 'We assume that spacetime is classical, that is, the spacetime models are not allowed to be in superposition of spacetime structures, implying that there are quantum fluctuations affecting the spacetime structure.', '1907.00781-2-10-3': 'Then the theoretical standard procedure for testing the structure of the spacetime is by observing the smooth world lines [MATH] associated to point probe particles.', '1907.00781-2-10-4': 'The metric structures that we will consider were named metrics of maximal proper acceleration or metrics of maximal acceleration [CITATION].', '1907.00781-2-11-0': 'A metric of maximal proper acceleration [MATH] on [MATH] is a spacetime structure that depends upon the second jet of smooth world lines giving the proper time that an ideal observer will measure along its world line [MATH].', '1907.00781-2-11-1': 'Therefore, we develop first the fundamental concepts for spacetimes with higher order jet geometry.', '1907.00781-2-11-2': 'If the curve [MATH] is (piecewise)-smooth causal, which means [MATH], then the proper time that an ideal observer with world line [MATH] will measure is given by the expression [EQUATION]', '1907.00781-2-11-3': 'The integrand [MATH] is not homogeneous under changes of the integration parameter [MATH].', '1907.00781-2-11-4': 'The higher order jet metric requires a precise specification of the parameter of integration in the definition of the proper time functional [REF], as we will do bellow.', '1907.00781-2-12-0': 'For the spacetimes that we will consider in this paper, Einstein clock hypothesis [CITATION] does not hold.', '1907.00781-2-12-1': 'The proper time functional [MATH] depends upon acceleration of the test particle world line [MATH].', '1907.00781-2-12-2': 'A fundamental motivation to adopt spacetime of metrics with maximal acceleration has been discussed by the author in [CITATION] and concerns the problem of radiation reaction in classical electrodynamics, were it was argued that clock hypothesis must be violated [CITATION].', '1907.00781-2-12-3': 'On the other hand, the existence of a maximal acceleration can be motivated by the problem of run-away solutions in electrodynamics.', '1907.00781-2-12-4': 'Indeed, the existence of a maximal proper acceleration forbids such solutions [CITATION].', '1907.00781-2-13-0': 'In the framework of higher order jet order fields, it is natural to adopt the following form for the integrand [MATH] in the expression for the proper time [REF], [EQUATION] where the time parameter calculating the derivatives is the proper time associated to the metric structure [MATH].', '1907.00781-2-13-1': '[MATH] is the second jet at [MATH] of the curve [MATH].', '1907.00781-2-13-2': 'In local coordinates, [MATH] is represented by [EQUATION]', '1907.00781-2-13-3': 'The expressions [REF]-[REF] determine the general form of a proper time compatible with our assumptions, namely:', '1907.00781-2-14-0': 'Dependence of the spacetime metric structure [MATH] on the second jet bundle coordinates [MATH] in a way that Einstein clock hypothesis does not hold in general,', '1907.00781-2-15-0': 'The compatibility with Einstein clock hypothesis is recovered in limit of the spacetime structure when the acceleration dependence is negligible.', '1907.00781-2-15-1': 'Hence it is required that the metric structure obtained by the limit [EQUATION] has to be compatible with the clock hypothesis.', '1907.00781-2-16-0': 'Compatibility with the clock hypothesis in the limit [MATH] implies that [MATH] lives on the first jet bundle [MATH].', '1907.00781-2-16-1': 'That the general form of a classical spacetime structure compatible with the fundamental principles of relativity needs to be of Finsler type was discussed already by Synge [CITATION], Ch.', '1907.00781-2-16-2': 'I. However, in order to simplify our treatment, we will consider the case when [MATH] is a Lorentzian metric [MATH].', '1907.00781-2-16-3': 'In the expression [REF], [MATH] determines a spacetime metric compatible with Einstein clock hypothesis, while [MATH] is the part of the metric [MATH] that explicitly violates the clock hypothesis.', '1907.00781-2-17-0': 'The existence of this metric [MATH] is justified in the framework of spacetime with metrics of maximal acceleration, since one can probe the spacetime structure using geodesics, in which case, the covariant condition [MATH] holds, in accordance with the general philosophy of spacetimes with higher order jet metrics.', '1907.00781-2-17-1': 'In this case, the spacetime structure found is the Lorentzian metric [MATH], where [MATH] is the covariant derivative of an affine connection.', '1907.00781-2-17-2': 'Since it is already at our disposition the Lorentzian [MATH] structure, [MATH] is interpreted as the Levi-Civita connection of [MATH].', '1907.00781-2-18-0': 'The explicit violation of the Einstein clock hypothesis is parameterized by a function defined along the lift [MATH] [EQUATION] introduces the parameter [MATH].', '1907.00781-2-18-1': 'Since [MATH] must be negligible when the effects of acceleration are small, as we know from the precise validity of classical relativistic dynamics, a natural way to quantify the effects of acceleration on the spacetime structure is through the comparison of the relevant notion of acceleration with respect to the deformation parameter [MATH].', '1907.00781-2-18-2': 'Hence the formal dependence of [MATH] on the acceleration must be through the quotient [MATH].', '1907.00781-2-19-0': 'According to these considerations, the general expression for the higher order spacetime structure is of the form [EQUATION]', '1907.00781-2-19-1': 'It is also natural to assume that the components of [MATH] are analytical in [MATH], [EQUATION] with [MATH] being spacetime functions.', '1907.00781-2-20-0': 'According to our motivations to consider higher order metrics of the form [MATH], when radiation reaction effects are of relevance, Einstein clock hypothesis must be violated.', '1907.00781-2-20-1': 'This implies that [MATH] must be monotonic on the argument [MATH].', '1907.00781-2-20-2': 'Otherwise, there will be local domains of acceleration, apart from when the acceleration is small, where the metric [MATH] effectively does not depend upon [MATH] and hence, clock hypothesis hold.', '1907.00781-2-20-3': 'But this implies special scales of acceleration, apart from the small acceleration, where radiation reaction effects are negligible, which seems physically unjustified.', '1907.00781-2-20-4': 'That [MATH] is monotonic implies that each of terms [EQUATION] must be either zero or all the remain non-trivial terms must have the same sign.', '1907.00781-2-21-0': 'Furthermore, if the functions [MATH] are not constant, then the notion of small acceleration needs to be compared not only against [MATH], but also against each of the functions [MATH].', '1907.00781-2-21-1': 'Such comparisons imply the introduction of additional scales.', '1907.00781-2-21-2': 'Also, if [MATH] are not constant, they will need to have a very definite dynamics, with an addition of a large conceptual and technical complication.', '1907.00781-2-21-3': 'These difficulties are overcome if we assume that [MATH] are all positive constants.', '1907.00781-2-21-4': 'Hence it is another reason to consider them constants.', '1907.00781-2-22-0': 'According with the above arguments, the expression for [MATH] must be of the form [EQUATION] where each of the functions [MATH] is constant and such that [MATH] for all [MATH].', '1907.00781-2-23-0': '## Causal structure of the higher order jet metrics', '1907.00781-2-24-0': 'The null bundle associated to the Lorentzian metric [MATH] is [EQUATION] and to the metric of maximal acceleration [MATH] is [EQUATION]', '1907.00781-2-24-1': 'The projections are [MATH] such that if [MATH], then [MATH] for a given [MATH] and [MATH].', '1907.00781-2-24-2': 'Then [MATH].', '1907.00781-2-24-3': 'The projection [MATH] is defined similarly.', '1907.00781-2-24-4': 'Consider a section [MATH].', '1907.00781-2-24-5': 'Then [MATH] too, for any smooth function [MATH].', '1907.00781-2-24-6': 'The analogous property for sections of [MATH] is not true: if [MATH], then in general [MATH].', '1907.00781-2-24-7': 'Hence we have that [EQUATION] since their sections do not coincide.', '1907.00781-2-24-8': 'However, if we only consider light-like curves which are geodesics of the Levi-Civita connection [MATH], then we have the following result,', '1907.00781-2-25-0': 'Let [MATH] be a geodesic of the Levi-Civita connection [MATH].', '1907.00781-2-25-1': 'Then [MATH] iff [MATH].', '1907.00781-2-26-0': 'Thus when restricted to geodesic motion, the null structure of [MATH] coincides with the null structure of [MATH].', '1907.00781-2-26-1': 'If light rays follow geodesics of [MATH], then in spacetimes with a higher order jet metric of the form [REF], there is an unique light cone structure.', '1907.00781-2-27-0': 'The following converse of Proposition [REF] also holds good:', '1907.00781-2-28-0': 'In a spacetime [MATH] such that [MATH] is given by [REF], the condition [MATH] and [MATH] implies [MATH].', '1907.00781-2-29-0': 'A time-like curve of the higher order jet metric [REF] is a curve [MATH] such that [EQUATION]', '1907.00781-2-29-1': 'Analogously, space-like curve of the higher order jet metric [REF] is a curve [MATH] such that [EQUATION]', '1907.00781-2-29-2': 'The following lines develop an heuristic argument fixing the signs of all the non-zero constants [MATH].', '1907.00781-2-29-3': 'We have showed that in spacetimes with higher order jet metrics geodesic curves [MATH] satisfying the null condition [MATH] full-fill also the condition [MATH], a result that applies directly to the propagation of light rays propagating in vacuum according to General Relativity.', '1907.00781-2-29-4': 'However, the situation is different for non-geodesic motion and in particular, for the propagation of gravitational field perturbations.', '1907.00781-2-29-5': 'In a treatment where the full field equations (beyond linearization) are considered, there must be self-interactions of the gravitational field perturbation with itself.', '1907.00781-2-29-6': 'Such interactions modify the equation of motion of high frequency waves from plane waves, as it happens in General Relativity (see for instance [CITATION], section IV.7 or reference [CITATION], paragraph 35.13) which implies a modification of the associated rays from geodesic motion.', '1907.00781-2-29-7': 'Hence rays corresponding to high frequency gravitational waves should be such that [MATH].', '1907.00781-2-29-8': 'As consequence, in spacetimes with higher order jet metrics of the form [REF], gravitational rays must be either space-like or time-like respect to the higher order jet metric [MATH], depending on the sign of the non-zero constants [MATH].', '1907.00781-2-30-0': 'On the other hand, let us choose the parametrization of the curves in the initial definition of the functional [MATH] given by the expression [REF] to be the proper time of the metric [MATH].', '1907.00781-2-30-1': 'Since [MATH] is the covariant derivative of the Levi-Civita connection of [MATH], then the condition [MATH] implies [MATH].', '1907.00781-2-30-2': 'Hence we have by a standard argument that [MATH].', '1907.00781-2-31-0': 'Finally, if we further assume that physical rays are represented by causal world lines of the metric [MATH] and since [MATH], then the constants [MATH] must be non-negative, in order that gravitational rays associated to short wave propagation are described by time-like curves of [MATH].', '1907.00781-2-32-0': 'Note that the above argument is based upon the assumption that field equations for the gravitational field in spacetimes of higher order jet metrics must be non-linear and that short wave propagating gravitational waves must suffer, as in with General Relativity, of non-linear terms, deviating from vacuum linear wave equations.', '1907.00781-2-32-1': 'These general assumptions in the above argumentation are reasonable for theories reproducing General Relativity in the limit [MATH].', '1907.00781-2-32-2': 'It is remarkable that the metric [MATH] does not hold the so called orthonormal condition [MATH], since [MATH] does not necessarily preserves [MATH].', '1907.00781-2-32-3': 'The orthonormality condition is of fundamental relevance in the derivation of the Lorentz-Dirac equation in classical electrodynamics.', '1907.00781-2-32-4': 'However, it fails in the case of higher order jet geometry allows the formulation of a second order differential equation for the electron [CITATION].', '1907.00781-2-33-0': '## The metric of maximal proper acceleration as a special case of a higher order jet metric', '1907.00781-2-34-0': 'With parametrization of the curve [MATH] such that [MATH] and causal in the sense that [MATH], it holds that [EQUATION] in the range of validity of the analytical expression [REF].', '1907.00781-2-34-1': 'According to the argument given above, let us assume that constant [MATH] is positive.', '1907.00781-2-34-2': 'Then [MATH] can be re-absorbed within the metric [MATH].', '1907.00781-2-34-3': 'Therefore, we can take without loss of generality the condition [MATH].', '1907.00781-2-34-4': 'If we denote by [EQUATION] then the constrain [REF] on [MATH] is of the form [EQUATION]', '1907.00781-2-34-5': 'It is reasonable that for dynamical systems where interactions are strong, the value of [MATH] can reach arbitrary close values to [MATH].', '1907.00781-2-34-6': 'Then we assume that the validity of [REF] is on the physical range [MATH], or at least, in a domain where [MATH].', '1907.00781-2-34-7': 'Since all the terms in the left side of the expression [REF] are positive, then the condition [MATH] implies the limit conditions [EQUATION]', '1907.00781-2-34-8': 'Since [MATH] for [MATH], this condition can only holds if [EQUATION]', '1907.00781-2-34-9': 'In this way, we arrive to a compact expression for the higher order jet metric of the type described by the expressions [REF], namely, [EQUATION] where a parametrization of the curve [MATH] such that [MATH] has been used.', '1907.00781-2-35-0': 'The form [REF] is a covariant version [CITATION] of the metric thoroughly investigated by E. Caianiello and co-workers [CITATION], but it has a natural interpretation as a higher order jet geometry defined by the relation [REF] [CITATION].', '1907.00781-2-35-1': "From this second point of view, the metric structure of the spacetime assigns to each test particle probing the structure of the classical spacetime a line element which depends upon the second order time coordinates derivatives of the test particle's world line.", '1907.00781-2-35-2': 'Furthermore, it is applied to general dynamical systems where a notion of maximal acceleration emerges.', '1907.00781-2-35-3': "This includes, the original proposal of Caianiello's quantum geometry [CITATION], the proposal of Brandt of maximal acceleration from vacuum fluctuations [CITATION] and the proposal from the author of higher order classical electrodynamics [CITATION].", '1907.00781-2-36-0': 'Generalizing the expression [REF], the metric of maximal acceleration acting on two arbitrary vector fields [MATH] along [MATH] is a tensor field of order [MATH] living on the second jet [MATH] is given by the expression [EQUATION]', '1907.00781-2-36-1': 'The proper time functional in a spacetime with a metric of maximal acceleration is of the form [EQUATION] where [MATH] is the proper time functional parameter calculated with [MATH] along [MATH] and is defined by the functional [EQUATION]', '1907.00781-2-36-2': 'The functional [MATH] is not re-parametrization invariant in the usual sense, since the proper time parameter [MATH] has been fixed to be the proper time of the metric [MATH] and the integrand in [MATH] is not homogeneous.', '1907.00781-2-36-3': 'However, the functional [MATH] is invariant under re-parameterizations [MATH].', '1907.00781-2-36-4': 'This implies that in physical terms, also the functional [MATH] is re-parametrization invariant as long as re-parametrization are mediated by the intermediate parameter given by the proper time functional [MATH]: if the proper time is first evaluated by using an arbitrary parameter [MATH], then used to evaluate the proper parameter [MATH] and then this is only the parameter used to obtain [MATH], then [MATH] is indeed invariant under re-parametrization of [MATH].', '1907.00781-2-36-5': 'This is not the usual notion of re-parametrization invariance, but it is good as long as the definition of [MATH] is given in terms of [MATH].', '1907.00781-2-37-0': 'The reality condition [MATH] implies that the proper acceleration respect to the limit metric [MATH] must be bounded in the following sense,', '1907.00781-2-38-0': 'In a spacetime [MATH] where [MATH] is a higher order jet metric given by the expression [REF], for any time-like curve with [MATH], the reality of [MATH] implies the uniform bound on the proper acceleration, [EQUATION]', '1907.00781-2-38-1': 'For a time-like curve [MATH] and by the expression [REF], then we have that the condition [REF] holds good.', '1907.00781-2-39-0': 'The choice of the proper time of [MATH] as parameter is of relevance because of the following argument: if initially we have a time parameter such that [MATH], a change of the time parameter [MATH] could make the normalization condition [MATH] not valid.', '1907.00781-2-40-0': 'The following interpretation of the maximal acceleration is in order.', '1907.00781-2-40-1': 'The existence of a maximal proper acceleration [MATH] is equivalent to the existence of a minimal proper time laps [MATH] such that [MATH], where [MATH] is the speed of light in vacuum.', '1907.00781-2-40-2': 'An argument for this property was provided by P. Caldirola in the context of a classical model for the electron [CITATION].', '1907.00781-2-40-3': 'We here extend the argument to any fundamental interaction compatible with an uniform maximal acceleration.', '1907.00781-2-41-0': 'Because proposition [REF] we call the structures given by [REF] or by the expression [REF], spacetime metrics of maximal acceleration or simply metrics of maximal acceleration.', '1907.00781-2-41-1': 'Remarkably, in a spacetime with a metric of maximal acceleration, the bound condition [REF] must hold for any time-like world line [MATH], independently from the association of the time-like curves with physical world lines or when such association is absente.', '1907.00781-2-42-0': 'Also, note that the condition of maximal acceleration implies a bound on the physical domain of [MATH]:', '1907.00781-2-43-0': 'In a spacetime with a metric of maximal acceleration, the maximal domain of definition of [MATH] is given by [MATH].', '1907.00781-2-44-0': 'Note that in principle there is no physical reason to preclude world lines of maximal acceleration, [MATH], similarly as in relativistic theories, there are world lines of maximal local speed.', '1907.00781-2-44-1': 'In fact, from [REF] we have that maximal acceleration curves, characterized by [EQUATION] are null curves of [MATH].', '1907.00781-2-44-2': 'Furthermore, since for curves of maximal acceleration [EQUATION] if [MATH] then [MATH] are time like curves of [MATH] that are parameterized by the proper time of [MATH].', '1907.00781-2-45-0': 'The metric of maximal acceleration [REF] has been obtained in the framework of higher order jet metrics under very general assumptions.', '1907.00781-2-45-1': 'From one side, there are conditions of physical nature, like the existence of the term [MATH] violating Einstein clock hypothesis, the assumption that physical rays are described by causal curves of [MATH] and the existence of the limit metric [MATH] as a good representation of the spacetime geometric arena when the effects of the acceleration are negligible.', '1907.00781-2-45-2': 'On the other hand, there is the technical hypothesis as the analytical dependence of [MATH] in the parameter [MATH] and the properties of constancy and non-negativeness of the factors [MATH], that although motivated on formal logical grounds, maybe could be proved from more clear principles.', '1907.00781-2-46-0': '# Jacobi equation and bounds on the Lorentzian curvature', '1907.00781-2-47-0': 'Given a time-like geodesic [MATH] of the Levi-Civita connection [MATH] of the Lorentzian metric [MATH], the Jacobi equation is the linear equation [EQUATION] where [MATH] is the curvature endomorphisms of [MATH] associated to [MATH] vector fields along [MATH].', '1907.00781-2-47-1': 'The second covariant derivative [MATH] admits an heuristic interpretation as relative acceleration between nearby geodesics of the congruence associated to the Jacobi field [CITATION].', '1907.00781-2-47-2': 'As such, the relative acceleration [MATH] is a vector field along [MATH].', '1907.00781-2-47-3': 'Although [MATH] is spacelike, it is not obvious that it can be interpreted as the proper acceleration field of a time-like curve [MATH], which is the type of acceleration bounded by a metric of maximal acceleration [REF].', '1907.00781-2-47-4': 'However, the following results show that such interpretation is possible and that one can extend the applicability of the bound [REF] to the relative acceleration [MATH] associated to a Jacobi field.', '1907.00781-2-48-0': 'Let [MATH] be a spacetime with a metric of maximal acceleration.', '1907.00781-2-48-1': 'A point [MATH] will be called regular if there is a compact set [MATH] containing [MATH] such that the Riemann curvature of [MATH] has bound components on [MATH].', '1907.00781-2-48-2': 'The aggregate of all regular points of [MATH] is denoted by [MATH].', '1907.00781-2-49-0': 'We assume than around each point [MATH] we can take the usual derivative and differential operations.', '1907.00781-2-49-1': 'This is the case when [MATH] is a manifold or a manifold with boundary.', '1907.00781-2-50-0': 'Let [MATH] be a spacetime with a metric of maximal acceleration.', '1907.00781-2-50-1': 'Let [MATH] be a time-like geodesic of [MATH] parameterized by proper time, [MATH], and [MATH] let be a regular point.', '1907.00781-2-50-2': 'Then there is a Jacobi field [MATH] along [MATH] such that the uniform bound holds good: [EQUATION]', '1907.00781-2-50-3': 'The assumption of bound for the components of the Riemann tensor at [MATH] in Lemma [REF] is not an uniform bound condition.', '1907.00781-2-50-4': 'It is required in order to apply Peano theorem on the existence of solutions of ordinary differential equations.', '1907.00781-2-50-5': 'The thesis, given by the expression [REF], states the existence of an uniform condition for spacetimes with a maximal acceleration metric, since [MATH] does not depend on the spacetime point [MATH].', '1907.00781-2-51-0': 'Let us consider a geodesic [MATH] such that [MATH] and parameterized by the proper time, [MATH].', '1907.00781-2-51-1': 'We will consider differential equations of the form, [EQUATION] where the dot-derivatives are taken respect to proper parameter of [MATH] along [MATH].', '1907.00781-2-51-2': 'The initial values for the condition [REF] are [EQUATION]', '1907.00781-2-51-3': 'Therefore, we have the constraint [EQUATION]', '1907.00781-2-51-4': 'Since the model that we will propose for [MATH] is a continuous model, we can assume that the curve [MATH] is time-like.', '1907.00781-2-51-5': 'On the other hand, if [MATH] is to interpreted as a proper acceleration, [MATH] needs to be space-like vector field along [MATH].', '1907.00781-2-51-6': 'Hence the orthogonality condition [EQUATION] must hold.', '1907.00781-2-52-0': 'Let us consider the ansatz [EQUATION]', '1907.00781-2-52-1': 'Then the orthogonal initial conditions [REF] holds good.', '1907.00781-2-52-2': 'Furthermore, we need to impose a constrain to determine [MATH] along [MATH].', '1907.00781-2-52-3': 'For this, we impose [EQUATION]', '1907.00781-2-52-4': 'Taking the derivative of this constraint along [MATH], [EQUATION]', '1907.00781-2-52-5': 'This constraint can be satisfied if we declare the vector field [MATH] is a solution of the differential equation [EQUATION]', '1907.00781-2-52-6': 'This equation is subjected to the initial condition [EQUATION]', '1907.00781-2-52-7': 'If [MATH] is a Jacobi field along [MATH], then [MATH] is spacelike and the initial condition [REF] ensures the orthogonality condition [REF] at the initial time [MATH] and the equation [REF] ensures that such condition is preserved along [MATH].', '1907.00781-2-53-0': 'In a compact way, the ordinary differential equations that determine the above construction are [EQUATION]', '1907.00781-2-53-1': 'Given the curvature endormorphism [MATH], the system of differential equations [REF] together with the initial conditions [EQUATION] determine a time-like curve whose initial acceleration are the specified one: [MATH], [MATH] and the initial proper acceleration [MATH] is the relative acceleration [MATH] at [MATH] of the central geodesic [MATH].', '1907.00781-2-54-0': 'The system of equations [REF] can be expressed in local components by the equations [EQUATION] subjected to the initial conditions [REF].', '1907.00781-2-54-1': 'Since the right hand sides of the equations [REF] are [MATH]-smooth functions on [MATH], Peano theorem (see for instance [CITATION], pg.', '1907.00781-2-54-2': '10) can be applied as follows.', '1907.00781-2-54-3': 'First note that the curvature endomorphism [MATH] with [MATH], acts on the vector fields [MATH] and [MATH] in a [MATH]-smoothly way.', '1907.00781-2-54-4': 'Let us denote by [MATH] the triple [MATH] and consider the bound [EQUATION] where [MATH] is a constant depending on the initial conditions.', '1907.00781-2-54-5': 'In a compact domain [MATH] containing the image [MATH], we have define a distance function [MATH], which depends upon de local coordinate system where we are applying Peano theorem.', '1907.00781-2-54-6': 'Let us fix a local coordinate system.', '1907.00781-2-54-7': 'Then the constant [MATH] is constructed as [CITATION] [EQUATION]', '1907.00781-2-54-8': 'By the hypothesis of the Lemma, [MATH].', '1907.00781-2-54-9': 'This can be achieved easily by a change in the initial conditions [MATH], that makes the quotient [MATH], where [MATH] is the minimum laps of proper time associated to the maximal acceleration [MATH].', '1907.00781-2-55-0': 'The relative acceleration [MATH] is interpreted as the initial proper acceleration of a curve [MATH].', '1907.00781-2-55-1': 'In order to apply the relation [REF], it is necessary to assure that the Jacobi field [MATH] is compatible with the parametrization condition [MATH].', '1907.00781-2-55-2': 'Then the relation [REF] follows.', '1907.00781-2-56-0': 'Not all the values of the initial condition [MATH] are compatible with the condition [MATH].', '1907.00781-2-56-1': 'Indeed, under the change of initial condition [MATH], for asymptotically large [MATH] we have that [EQUATION] in contradiction with the hypothesis.', '1907.00781-2-56-2': 'Hence not all the Jacobi vector fields [MATH] are compatible with existence of a minimal value for the parameter [MATH] and by the above proof, with the thesis of the Lemma [REF].', '1907.00781-2-57-0': 'Lemma [REF] can be applied at each [MATH], Consider the spacetime with a metric of maximal acceleration [MATH].', '1907.00781-2-57-1': 'Then there is a Jacobi field [MATH] along the time-like geodesic [MATH] compatible with the condition [MATH] such that the uniform bound [EQUATION] holds good along the geodesic, where [MATH] is the curvature of the Lorentzian metric [MATH].', '1907.00781-2-58-0': 'Let us consider a particular initial condition [MATH], where [MATH] and [MATH] is a time-like tangent vector such that [MATH].', '1907.00781-2-58-1': 'Let [MATH] be a tangent basis for [MATH] and assume that [MATH] is a time-like vector and [MATH] are supplementary tangent vectors at [MATH], that without loss of generality, satisfy the relations [EQUATION]', '1907.00781-2-58-2': 'Let [MATH] be a Jacobi field satisfying the conditions of proposition [REF].', '1907.00781-2-58-3': 'In terms of the above basis, we have [EQUATION] since [MATH] in the above basis and [MATH] because the symmetries of the curvature endomorphism of the Riemann curvature.', '1907.00781-2-58-4': 'Using the orthonormal relations, we obtain for every admissible [MATH] such that it is compatible with the minimal time laps that [EQUATION]', '1907.00781-2-58-5': 'Every spacetime direction [MATH] at [MATH] has associated an unique admissible spacelike vector [MATH] such that one can apply Lemma [REF].', '1907.00781-2-58-6': '[EQUATION] where [MATH] is the spacelike vector proportional to [MATH] compatible with the minimum time laps [MATH] and of maximal modulus with [MATH].', '1907.00781-2-58-7': 'Then we have the following', '1907.00781-2-59-0': 'In a spacetime geometry of maximal acceleration [MATH] and in the basis [MATH], the Riemannian curvature components are uniformly bounded in the sense that [EQUATION] at any regular point [MATH] and maximal admissible spacelike vector [MATH].', '1907.00781-2-60-0': 'Since the tangent vector [MATH] has different from zero, finite norm, the components of the curvature endomorphism [MATH] and the components of the curvature tensor [MATH] remain finite for any regular point [MATH].', '1907.00781-2-61-0': '# The equivalence principle for spacetimes of maximal acceleration', '1907.00781-2-62-0': 'Einstein equivalence principle is on the basis of our current theories for the gravitational interaction.', '1907.00781-2-62-1': 'Following Thorne, Lee and Lightman [CITATION], the principle can be stated as follows:', '1907.00781-2-63-0': 'Einstein equivalence principle.', '1907.00781-2-63-1': 'The following postulates hold good:', '1907.00781-2-64-0': 'The weak equivalence principle or universality of free falling holds: if an un-charged test body is placed at an initial event in spacetime and is given an initial velocity there, then its subsequent world line will be independent of its internal structure and composition.', '1907.00781-2-65-0': '1.', '1907.00781-2-65-1': 'The outcome of any local, non-gravitational test experiment is independent of where and when in the universe is performed and 2.', '1907.00781-2-65-2': 'It is independent of the velocity of the free falling experimental apparatus where the experiment is realized.', '1907.00781-2-66-0': 'The first postulate in Einstein equivalence principle is full-filed if there is a connection [MATH] whose auto-parallel curves are in one to one correspondence with the world lines of free falling bodies.', '1907.00781-2-66-1': 'Then the connection [MATH] must be independent of the composition and structure of the free falling test bodies.', '1907.00781-2-66-2': 'This form of the postulate is not difficult to accomplish in the category of Finslerian spacetimes, as discussed for example in [CITATION].', '1907.00781-2-66-3': 'In such a framework, the Lorentzian models of spacetime are a sub-category and is where general relativity finds its mathematical formulation, but also note that in the Finslerian category there are connections violating the postulate.', '1907.00781-2-66-4': 'Hence the weak equivalence principle is already a restriction on the mathematical formalism.', '1907.00781-2-67-0': 'The second postulate in Einstein equivalence principle presumes the existence of smooth free falling local coordinate systems where the experimental set up can be constructed and where all the gravitational effects can locally be eliminated.', '1907.00781-2-67-1': 'These free falling coordinates are usually mathematically implemented as the Fermi coordinate system along [MATH] representing the world line of the laboratory system.', '1907.00781-2-67-2': 'In such free falling coordinate systems, the outcomes of any experiment showing deviation from the free evolution of the studied physical indicates the presence of a non-gravitational field.', '1907.00781-2-67-3': 'This way of identifying non-gravitational fields fails when the dynamical effects of the connection cannot be eliminated locally in the free falling local coordinate system.', '1907.00781-2-68-0': 'The second postulate in Einstein equivalence principle also introduces a strong constrain on the possible physical laws: the outcome of non-gravitational experiments must be independent of the state of motion of the free falling local coordinate systems.', '1907.00781-2-68-1': 'This part of the postulates implies that in local free falling coordinate frames, the laws describing physical phenomena are consistent with a theory of relativity, either Galilean relativity or Einstein special relativity.', '1907.00781-2-68-2': 'Examples of gravitational theories obeying Einstein equivalence principle are Newton-Cartan theory and general relativity [CITATION].', '1907.00781-2-69-0': 'The first postulate and the first part of the second postulate in Einstein equivalence principle can be implemented geometrically and rather independently of the second half of the second postualte if the connection [MATH] defined by free fall motion is either an affine connection defined over the spacetime [MATH] or it is determined in a direct way by an affine connection on [MATH].', '1907.00781-2-69-1': 'The second part formulated in the second postulate is full-filed if the spacetime geometry is consistent with a principle of relativity, either Galilean or Einstein special relativity.', '1907.00781-2-69-2': 'These class geometries are to be found in Finsler spacetimes geometries, from which Lorentzian spacetime geometry is the standard case.', '1907.00781-2-70-0': 'Theories of gravitational interaction are based upon the principle that the effects due to gravity solely can be locally suppressed in local free falling coordinate systems.', '1907.00781-2-70-1': 'This suggests an alternative formulation of the equivalence principle, that although is not as strong in the constraints that Einstein principle imposes, it still captures this essential characterization of gravity.', '1907.00781-2-70-2': 'Modified in the following way, the new version of the equivalence principle is put to work under similar purposes than the original Einstein equivalence principle, but we will see that this new version can be accommodated to spacetimes with a maximal acceleration, where neither the Galilean relativity nor special relativity hold.', '1907.00781-2-70-3': 'In this context, we propose to consider the following:', '1907.00781-2-71-0': 'New version of the equivalence principle.', '1907.00781-2-71-1': 'The following two conditions hold good:', '1907.00781-2-72-0': 'The weak equivalence principle or universality of free falling holds: if an un-charged test body is placed at an initial event in spacetime and is given an initial velocity there, then its subsequent world line will be independent of its internal structure and composition.', '1907.00781-2-73-0': 'Existence of smooth free falling local coordinate systems, where the experimental set up is constructed and where gravitational effect integrated in the connection can be locally eliminated.', '1907.00781-2-74-0': 'In this new version of the equivalence principle, gravitational interactions are characterized exactly as the ones that for point non-charged test particle, the given interaction can be switch-off locally in local free falling coordinate systems.', '1907.00781-2-74-1': 'By switch-off locally we mean that the effect on the dynamics is trivialized for world lines close enough to the given free fall coordinate system.', '1907.00781-2-74-2': 'A detailed specification of the concept is discussed in [CITATION].', '1907.00781-2-75-0': 'This provides a strong restriction on the possible geometric models, as we will see in the next section.', '1907.00781-2-75-1': 'However, let us note that the principle here considered can be extended in the form of a generalization or modification of the Einstein equivalence principle, a necessary step towards field equations involving spacetimes with metrics of maximal acceleration.', '1907.00781-2-75-2': 'Probably, this could imply deviation and phenomenological deviations from Einstein equivalence principle.', '1907.00781-2-75-3': 'We will not consider this problem in this work, leaving it for future research.', '1907.00781-2-76-0': '# A geometric framework for spacetimes with a metric of maximal acceleration', '1907.00781-2-77-0': 'We consider now a geometric framework for spacetimes with maximal acceleration where the two conditions of the new version of the equivalence principle are satisfied.', '1907.00781-2-77-1': 'The connection [MATH] will be a connection on a fibre bundle [MATH].', '1907.00781-2-77-2': 'For instance, in the case of models of gravitational interaction based upon the theory of Finsler spacetimes [CITATION], the connections are defined on vector bundles over the slit tangent bundle [MATH].', '1907.00781-2-77-3': 'Similarly, in the case of spacetimes of maximal acceleration, [MATH] will be a connection defined on vector bundles over the [MATH]-jet bundle [MATH].', '1907.00781-2-78-0': '## The equivalence principle and the constrains on the connection', '1907.00781-2-79-0': 'Let us consider the pull-back bundle [MATH], defined by the commutative diagram [EQUATION] [MATH] is a vector bundle, where the fiber [MATH] with [MATH], is diffeomorphic to the vector space [MATH] for each [MATH].', '1907.00781-2-79-1': 'Note that since [MATH] is a projection onto [MATH], then [EQUATION]', '1907.00781-2-79-2': 'The connections that we will consider on the vector bundle [MATH] are inspired by analogous Finslerian constructions [CITATION].', '1907.00781-2-79-3': 'In a similar way as for Finslerian spacetime models of gravity [CITATION], the new form of the equivalence principle as stated in the previous section imposes on the spacetime metric structure non-trivial constrains on the connection [MATH] defined on the vector bundle [MATH].', '1907.00781-2-79-4': 'In particular, the connection [MATH] must be compatible with the existence of smooth free falling local coordinate systems.', '1907.00781-2-79-5': 'A candidate for [MATH] with such property is the pull-back connection [MATH] of the Levi-Civita connection of the Lorentzian metric [MATH], which is defined by the relation [EQUATION]', '1907.00781-2-79-6': 'The introduction of free falling coordinate systems associated to the connection [MATH] requires first the discussion of several notions and results.', '1907.00781-2-79-7': 'Let us first introduce the following notion of auto-parallel condition in [MATH],', '1907.00781-2-80-0': 'An auto-parallel curve in [MATH] is a curve [MATH] whose tangent vector field [MATH] is such that [MATH] satisfies the auto-parallel condition of the Levi-Civita connection, [MATH].', '1907.00781-2-81-0': '[MATH] is a canonical projection and [MATH] is an affine connection on [MATH].', '1907.00781-2-81-1': 'Hence we have the following', '1907.00781-2-82-0': 'The following properties of the connection [MATH] of the bundle [MATH] hold good:', '1907.00781-2-83-0': 'The connection [MATH] is a linear connection.', '1907.00781-2-84-0': 'The connection coefficients of [MATH] live on the spacetime manifold [MATH].', '1907.00781-2-85-0': 'The auto-parallel condition [REF] is equivalent to the auto-parallel condition of the Levi-Civita connection [MATH].', '1907.00781-2-86-0': 'The first property is direct from the construction of the pull-back connection.', '1907.00781-2-87-0': 'The second property is proved using local natural coordinate systems [MATH] in [MATH] and the definition [REF] of [MATH].', '1907.00781-2-88-0': 'The third property is direct from the definition of [MATH] and the definition of [REF] of the pull-back connection.', '1907.00781-2-88-1': 'Indeed, if [MATH], then [EQUATION]', '1907.00781-2-88-2': 'Conversely, since [MATH] iff [MATH], then [MATH] implies [MATH].', '1907.00781-2-89-0': 'Clearly, we can adopt as connection [MATH] in our theory of spacetimes with metrics of maximal acceleration the connection [MATH], [EQUATION]', '1907.00781-2-89-1': 'To understand how this connection [MATH] admits free falling coordinate systems in the form of Fermi-like coordinates, we need to investigate some of its further fundamental properties.', '1907.00781-2-90-0': '## Properties of the connection [MATH]', '1907.00781-2-91-0': 'We study now some basic properties of the connection [MATH].', '1907.00781-2-92-0': '## The Torsion tensor of the pull-back connection [MATH]', '1907.00781-2-93-0': 'There is no defined a conventional torsion tensor for a given connection [MATH] in [MATH].', '1907.00781-2-93-1': 'However, a close analogous tensor field along [MATH] that one can define is defined as follows.', '1907.00781-2-93-2': 'If [MATH] are local vector fields on [MATH], then [EQUATION]', '1907.00781-2-93-3': 'We can call [MATH] the pull-back torsion tensor of the connection [MATH].', '1907.00781-2-93-4': 'Then we have the following', '1907.00781-2-94-0': 'The pull-back torsion tensor of [MATH] is zero, [EQUATION]', '1907.00781-2-94-1': 'For [MATH] we have that [EQUATION] since for the Levi-Civita connection [MATH].', '1907.00781-2-95-0': '## Fermi coordinates for the pull-back connection [MATH]', '1907.00781-2-96-0': 'The theory of connections developed above accommodate the new version of the equivalence principle.', '1907.00781-2-96-1': 'Proposition [REF] implies that auto-parallel curves of [MATH] are the geodesics of the Levi-Civita connection [MATH], full-filling the first condition of the new version of the equivalence principle.', '1907.00781-2-96-2': 'Moreover, the existence of local coordinates where all the gravitational effects encoded in the connection can be eliminated in small enough neighborhood, is realized by the existence of Fermi frames for [MATH] along arbitrary geodesics [MATH].', '1907.00781-2-96-3': 'Since [MATH], this implies local frames of [MATH] along the corresponding lift [MATH] where the connection coefficients of the pull-back connection [MATH] are zero.', '1907.00781-2-97-0': '## Curvature and Jacobi fields of the pull-back connection [MATH]', '1907.00781-2-98-0': 'One can consider a generalization of the Jacobi equation for the connection [MATH].', '1907.00781-2-98-1': 'The curvature endomorphism of [MATH] is defined by the expression [EQUATION]', '1907.00781-2-98-2': 'Let us consider [MATH] such that [MATH] with [MATH] and [MATH].', '1907.00781-2-98-3': 'Then [MATH] holds good.', '1907.00781-2-98-4': 'By the torsion-free property [REF] [EQUATION]', '1907.00781-2-98-5': 'If the auto-parallel condition [EQUATION] holds good, then we have [EQUATION]', '1907.00781-2-98-6': 'Equation [REF] is the Jacobi equation for the pull-back connection [MATH].', '1907.00781-2-99-0': 'The Jacobi fields of [MATH] and the Jacobi fields of [MATH] are related.', '1907.00781-2-99-1': 'First note that if [MATH], then from the definition of [MATH] it follows that [EQUATION] and as consequence [EQUATION]', '1907.00781-2-99-2': 'By direct application of these two relations, we can prove the following', '1907.00781-2-100-0': 'The Jacobi fields of [MATH] and [MATH] are in one to one correspondence.', '1907.00781-2-101-0': 'If [MATH] is a Jacobi field of [MATH] along [MATH] with tangent vector field [MATH] and such that [MATH], then by the relations [REF]-[REF], the Jacobi equation [REF] and the definition of auto-parallel condition [MATH], the field [MATH] is a Jacobi field [MATH] along the curve [MATH].', '1907.00781-2-102-0': 'By the same arguments, but reversing the deductions, it is shown that for any Jacobi field [MATH] along a geodesic of [MATH], there is a Jacobi field of [MATH] along the lift [MATH] to the second jet bundle.', '1907.00781-2-103-0': '## Relation between the metric of maximal acceleration and the pull-back connection [MATH]', '1907.00781-2-104-0': 'According to the geometric framework discussed above, the properties of [MATH] and [MATH] are intimately related.', '1907.00781-2-104-1': 'It has also been convincingly showed that Jacobi fields of [MATH] are determined by the Jacobi fields of [MATH], the Levi-Civita connection of the Lorentzian metric [MATH].', '1907.00781-2-104-2': 'We show now the relation between the metric of maximal acceleration [MATH] and the pull-back connection [MATH].', '1907.00781-2-105-0': 'Instead of considering directly the metric [MATH], we will work with a deformed pull-back fiber metric defined as acting on sections of [MATH].', '1907.00781-2-105-1': 'This deformation is defined by the expression [EQUATION] where the pull-back metric [MATH] is defined by the relation [EQUATION]', '1907.00781-2-105-2': 'A direct computation of the covariant derivative of the pull-back metric [REF] leads to [EQUATION] where [MATH] is the differential of the function [MATH].', '1907.00781-2-105-3': 'The covariant derivative of the pull-back metric is [EQUATION] by the metric condition [MATH] of the Levi-Civita connection.', '1907.00781-2-105-4': 'Let us introduce the [MATH]-form [MATH] by [EQUATION]', '1907.00781-2-105-5': 'Hence we have the relation between [MATH] and [MATH] [EQUATION]', '1907.00781-2-105-6': 'If [MATH] is a local frame in [MATH], then we have [EQUATION]', '1907.00781-2-105-7': 'In dimension [MATH], the number of connection coefficients [EQUATION] of an arbitrary connection on [MATH] is [MATH].', '1907.00781-2-105-8': 'However, there are constraints arising from the definition of the pull-back connection [MATH].', '1907.00781-2-105-9': 'In particular, we have the conditions [EQUATION]', '1907.00781-2-105-10': 'Since [MATH], these constraints impose [MATH] pointwise conditions on the connection coefficients of the pull-back connection [MATH].', '1907.00781-2-106-0': 'The non-metricity condition implies a number of [MATH] independent constraints.', '1907.00781-2-107-0': 'The number of independent constrains on the introduced by the torsion free relations [MATH] is [MATH].', '1907.00781-2-107-1': 'Thus the total number of constraints is [MATH], which is enough and sufficient to determine completely the connection [MATH].', '1907.00781-2-107-2': 'Therefore, the constraints [REF] and [REF] are enough to determine the pull-back connection [MATH].', '1907.00781-2-108-0': 'The pull-back connection [MATH] is the unique linear connection [MATH] on the bundle [MATH] determined by the conditions', '1907.00781-2-109-0': 'We would like to remark the similarity of this construction with Chern connection in Finsler geometry [CITATION].', '1907.00781-2-109-1': 'However, we did not require the introduction of a non-linear connection in [MATH] in order to determine the condition.', '1907.00781-2-109-2': 'Instead, we introduce the conditions [REF], than together with torsion free and metric non-compatibility conditions, are enough to determine the connection consistently.', '1907.00781-2-110-0': '# Discussion', '1907.00781-2-111-0': 'We have shown that the existence of an universal bound for the proper acceleration of test particles and that such a bound implies that curvature must remain uniformly bound in certain sense in the whole region of the spacetime where it is defined and finite.', '1907.00781-2-111-1': 'This result applies rather generally.', '1907.00781-2-111-2': 'In particular, it applies to the theory developed by Caianiello and collaborators [CITATION], to the theory of metrics of maximal acceleration developed by the author [CITATION] and also could be applied to other theories of maximal acceleration [CITATION].', '1907.00781-2-112-0': 'We have developed a geometric framework for spacetimes endowed with a metric of maximal acceleration.', '1907.00781-2-112-1': 'The driving principle has been a new version of the equivalence principle, that, although less strong than Einstein equivalence principle, is naturally fitted for spacetimes with a high order jet metric, including Finsler spacetimes [CITATION] and the spacetimes considered in this paper.', '1907.00781-2-112-2': 'The introduction of a connection and its determination by natural conditions and the form of the metric of maximal acceleration.', '1907.00781-2-113-0': 'The spacetimes of maximal acceleration that we have considered are not necessarily originated by corrections to the classical general relativity due to a quantum origin of gravity or quantum mechanical frameworks.', '1907.00781-2-113-1': 'Indeed, we have discussed uniform upper bounds in the Riemann curvature in the contest of classical geometries of maximal acceleration, where the fundamental notion of metric of maximal acceleration is a classical object, in contrast with arguments involving quantum gravity [CITATION] or quantum mechanics [CITATION].', '1907.00781-2-113-2': 'We have adopted the viewpoint that the existence of a maximal proper acceleration should be imprinted in the geometry of the spacetime, in a similar way as the existence of a maximal speed is imprinted in the causal structure causal structure of the spacetime.', '1907.00781-2-113-3': 'The consequence of this is that spacetime more malleable mathematical arena than in current models of classical physics and quantum field theory, since the metric of maximal acceleration depend upon the way the spacetime is tested.', '1907.00781-2-114-0': 'A common criticism for maximal acceleration hypothesis is that, although the idea is very fertile in consequences, there is no empirical evidence for it.', '1907.00781-2-114-1': 'However, a new approach to observe effects of maximal acceleration in laser-plasma dynamics has been recently discussed [CITATION].', '1907.00781-2-114-2': 'In that paper it was argued that modifications of the Lorentz force due to maximal acceleration is potentially testable in near future laser-plasma acceleration facilities.', '1907.00781-2-114-3': 'Confirmation of such qualitative dynamical effect in modern laser-plasma facilities could show for first time deviations from a local Lorentzian structure of spacetime.', '1907.00781-2-115-0': 'A fundamental point missing in this paper is an example of complete gravitational model in the framework of spacetimes with a maximal acceleration.', '1907.00781-2-115-1': 'One can partially address this issue by considering models of maximal acceleration already developed by E. Caianiello and co-workers [CITATION], that although based upon a mathematical formalism which is not general covariant, it could be considered a starting point towards a consistent theory of spacetimes with metrics of maximal acceleration.', '1907.00781-2-115-2': 'However, we think that as happened with general relativity, only the combination of a powerful set of physical principles with mathematical rigurous methods, could bring light on the new field equations of motion for spacetimes with metrics of maximal acceleration.', '1907.00781-2-116-0': 'Further topics to be investigatedinclude a comprehensive treatment of singularities and its eventual resolution in the framework of metrics with maximal acceleration.', '1907.00781-2-116-1': 'Also, the aspects of the thermodynamics of black holes that should be modified by the existence of a maximal acceleration have not been considered in this paper.', '1907.00781-2-116-2': 'These interesting problems will be investigated in future work.'}

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{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1907.00781

cond-mat-0509004

{'cond-mat-0509004-1-0-0': 'The single-site dynamical mean field theory approximation to the double exchange model is found to exhibit a previously unnoticed instability, in which a well-defined ground state which is stable against small perturbations is found to be unstable to large-amplitude but purely local fluctuations.', 'cond-mat-0509004-1-0-1': 'The instability is shown to arise either from phase separation or, in a narrow parameter regime, from the presence of a competing phase.', 'cond-mat-0509004-1-0-2': 'The instability is therefore suggested as a computationally inexpensive means of locating regimes of parameter space in which phase separation occurs.', 'cond-mat-0509004-1-1-0': '# Introduction', 'cond-mat-0509004-1-2-0': 'Dynamical mean field theory (DMFT) has been widely applied to many strongly correlated electron systems[CITATION].', 'cond-mat-0509004-1-2-1': 'Since DMFT takes local quantum fluctuations into account, it is especially successful for models whose many body effect comes from the on-site interaction, like Hubbard[CITATION] or Kondo[CITATION][CITATION] (double exchange) model.', 'cond-mat-0509004-1-2-2': 'Correlated systems often exhibit different phases which are quite close in energy, and this proximity can lead to phase separation, which is often important for electronic physics[CITATION].', 'cond-mat-0509004-1-2-3': 'Phase separation is in principle a "global" property of the phase diagram and requiring substantial effort to establish: one must compute the free energy over a wide parameter range, and then perform a Maxwell construction.', 'cond-mat-0509004-1-2-4': 'In this paper we show that within the single-site DMFT formalism a straightforward calculation at a fixed parameter value can reveal the presence of phase separation.', 'cond-mat-0509004-1-2-5': 'Specifically, we find that at zero temperature, the DMFT can give a ground state which is stable against small perturbations but is unstable to a large amplitude local perturbation; at non-zero temperature the standard methods simply fail to converge to a stable solution.', 'cond-mat-0509004-1-2-6': 'By computing the free energy and performing a Maxwell construction we show that for wide parameter ranges this instability occurs in the regions in which phase separation exists.', 'cond-mat-0509004-1-2-7': 'In a narrow parameter regime it signals instead the onset of a different, but apparently uniform, phase.', 'cond-mat-0509004-1-2-8': 'We therefore propose that the instability of the DMFT equations can be used as an approximate, computationally convenient estimator for the boundaries of the regimes in which phase separation occurs.', 'cond-mat-0509004-1-3-0': 'The balance of this paper is organized as follows.', 'cond-mat-0509004-1-3-1': 'We first present the model and then a zero temperature dynamical mean field analysis explicitly showing the instability.', 'cond-mat-0509004-1-3-2': 'In section III we calculate the full T=0 phase diagram in the energy-density plane, establish the regime of phase separation via the usual Maxwell construction, and extend the treatment to [MATH].', 'cond-mat-0509004-1-3-3': 'In section IV we discuss the implications of this instability.', 'cond-mat-0509004-1-3-4': 'Finally in section V we present a brief conclusion.', 'cond-mat-0509004-1-4-0': '# Double Exchange Model and Dynamical Mean Field Approximation', 'cond-mat-0509004-1-5-0': '## Double Exchange Model', 'cond-mat-0509004-1-6-0': 'In this paper, we consider the single orbital "double exchange" or Kondo lattice model of carriers hopping between sites on a lattice and coupled to an array of spins.', 'cond-mat-0509004-1-6-1': 'This model has been studied by many authors and contains important aspects of the physics of the "colossal"[CITATION] magnetoresistance manganites and is also solvable in a variety of approximations, permitting detailed examination of its behavior.', 'cond-mat-0509004-1-6-2': 'Here we use it to investigate the physical meaning of a previously unnoticed instability of the dynamical mean field equations.', 'cond-mat-0509004-1-7-0': 'The model is defined by the Hamiltonian [EQUATION] with [MATH] labeling the sites and the [MATH] denoting the spins.', 'cond-mat-0509004-1-7-1': 'We assume the spins are classical ([MATH]) and are of fixed length.', 'cond-mat-0509004-1-7-2': 'We choose the convention [MATH].', 'cond-mat-0509004-1-8-0': 'The hopping [MATH] defines an energy dispersion [MATH] and thus a density of states [MATH].', 'cond-mat-0509004-1-8-1': 'In our actual computations we specialize to the [MATH] limit of the Bethe lattice, for which [MATH] because the availability of convenient analytical expressions allows us to accurately compute the small difference between free energies of different states.', 'cond-mat-0509004-1-8-2': 'We choose energy units such that [MATH].', 'cond-mat-0509004-1-9-0': 'We also note that the ground state properties of the model may be straightforwardly obtained, because for any fixed configuration of the spins the model is quadratic in the fermions and easily diagonalizable.', 'cond-mat-0509004-1-10-0': '## Dymanical Mean Field Method', 'cond-mat-0509004-1-11-0': 'We now present the dynamical mean field analysis of this model.', 'cond-mat-0509004-1-11-1': 'In the single site dynamical mean field method[CITATION], one neglects the momentum dependence of the self energy.', 'cond-mat-0509004-1-11-2': 'The properties of the model may then be calculated by solving an auxiliary quantum impurity model, along with a self consistency condition.', 'cond-mat-0509004-1-11-3': 'The quantum impurity model corresponding to Eqn([REF]) is specified by the partition function [EQUATION] with [MATH] where the trace is over frequency, and [MATH].', 'cond-mat-0509004-1-11-4': '[MATH] is determined by a spin-dependent mean field function [MATH].', 'cond-mat-0509004-1-11-5': 'In a magnetic phase, [MATH].', 'cond-mat-0509004-1-11-6': 'Note that the assumption of classical core spins means that [MATH] denotes a simple scalar integral over directions of the core spin [MATH], and that no Berry phase term occurs in the argument of the exponential.', 'cond-mat-0509004-1-12-0': "The Green function [MATH] and self energy [MATH] of the impurity model are given by [EQUATION] [MATH] is fixed by requiring the impurity Green's function [MATH] equals to the local Green's function of the lattice problem.", 'cond-mat-0509004-1-12-1': 'The form of the self consistency equation depends on the state which is studied.', 'cond-mat-0509004-1-12-2': 'For a ferromagnetic (FM) state, it is [EQUATION] while for a 2 sublattice antiferromagnetic (AF) state, [EQUATION] where [MATH] and the [MATH] -sum is over tje reduced Brillouin zone (RBZ).', 'cond-mat-0509004-1-12-3': 'The two equations become equivalent in the paramagnetic (PM) state where [MATH]', 'cond-mat-0509004-1-13-0': 'The solution of Eqs [[REF] to [REF]] determines the magnetic phase, the single particle properties, and the free energy.', 'cond-mat-0509004-1-13-1': "In particular, in the dynamical mean field approximation the Gibb's free energy is [CITATION][CITATION] [EQUATION] where the trace is over the spin and lattice degree of freedom.", 'cond-mat-0509004-1-13-2': 'The Helmholtz free energy is [MATH].', 'cond-mat-0509004-1-13-3': 'At zero temperature, the ground state energy (Helmholtz free energy) is [EQUATION]', 'cond-mat-0509004-1-13-4': 'We also note that the solution [MATH] defines an effective potential for the core spin, which depends on the angle [MATH] between the core spin and local magnetization direction, so that [EQUATION] with [EQUATION]', 'cond-mat-0509004-1-14-0': '## Phase Boundaries and Maxwell Construction', 'cond-mat-0509004-1-15-0': 'The model is known to exhibit ferromagnetic, spiral[CITATION] and commensurate antiferromagnetic phases.', 'cond-mat-0509004-1-15-1': 'For our subsequent analysis, an accurate determination of phase boundaries will be important.', 'cond-mat-0509004-1-15-2': 'We therefore present here a few calculational details.', 'cond-mat-0509004-1-16-0': 'We require the [MATH] phase boundary separating the ferromagnetic and spiral phases[CITATION][CITATION].', 'cond-mat-0509004-1-16-1': 'The energy of a spiral state may most easily be found by performing a site-dependent spin rotation to a basis in which the spin quantization axis is parallel to the local spin orientation.', 'cond-mat-0509004-1-16-2': 'The problem may then be easily diagonalized.', 'cond-mat-0509004-1-16-3': 'For the infinite dimensional Bethe lattice one finds, for a diagonal spiral of pitch [MATH], that in the rotated basis, the local Green function is given by[CITATION] [EQUATION] where [MATH] is the angle between two nearest neighbor magnetization, and tilde is used for the spiral states.', 'cond-mat-0509004-1-17-0': 'To locate the [MATH] FM/spiral second order phase boundary, it suffices to expand ground state energy [MATH] (Eqn([REF])) to second order in [MATH].', 'cond-mat-0509004-1-17-1': 'The energy difference between FM and spiral states is [EQUATION] with [MATH] equaling to [EQUATION] where [MATH].', 'cond-mat-0509004-1-17-2': 'The FM/spiral phase boundary is determined by [MATH].', 'cond-mat-0509004-1-18-0': 'The model also exhibits phase separation in some regimes.', 'cond-mat-0509004-1-18-1': 'To determine the boundaries of the regime where phase separation occurs, we use the DMFT method to compute the Helmholtz free energy as a function of occupation number [MATH] (Eqn([REF])) and then perform the Maxwell construction.', 'cond-mat-0509004-1-18-2': 'An example is shown in Fig [REF].', 'cond-mat-0509004-1-18-3': 'We find that in fact that over much of the phase diagram a phase separation between FM and [MATH] AF states preempts the formation of spiral or [MATH] AF state.', 'cond-mat-0509004-1-18-4': 'The general structure of our phase diagram agrees with earlier work[CITATION], but the precise locations of phase boundaries differ by roughly [MATH].', 'cond-mat-0509004-1-19-0': '# DMFT Instability', 'cond-mat-0509004-1-20-0': 'In this section we show that the DMFT equations exhibit an apparently previously unnoticed instability.', 'cond-mat-0509004-1-20-1': 'We begin with [MATH].', 'cond-mat-0509004-1-20-2': 'From Eqn([REF]), the [MATH] is [EQUATION] where [MATH] means the angular average with respect to the weight function [MATH], with [MATH] defined in Eqn([REF]) .', 'cond-mat-0509004-1-20-3': 'At zero temperature, the only contribution of the angular average is from the absolute minimum of [MATH].', 'cond-mat-0509004-1-20-4': 'To find the DMFT solution at [MATH], one first assumes the absolute minimum of [MATH] occurs at a fixed value, for example [MATH], obtains [MATH] from Eqn([REF]), and gets the self energy [MATH] from Eqn([REF]).', 'cond-mat-0509004-1-20-5': 'Finally, one uses the [MATH] obtained by the above procedure to calculate [MATH] to see if the minimum is located at the point originally assumed.', 'cond-mat-0509004-1-20-6': 'Note that different ground states (FM, AF.', 'cond-mat-0509004-1-20-7': '.)', 'cond-mat-0509004-1-20-8': 'enter the above procedure only via the self consistent equation Eqn([REF]) (or Eqn([REF])).', 'cond-mat-0509004-1-21-0': 'Fig [REF] shows that as density is increased at fixed large [MATH], the self consistency breaks down, in an unusual manner: [MATH] remains locally stable (slope around the assumed minimum [MATH] remains positive) but the global minimum of [MATH] moves to [MATH].', 'cond-mat-0509004-1-21-1': 'In the regime where this phenomenon occurs, no solution of the DMFT equations exists.', 'cond-mat-0509004-1-21-2': 'Any initial solution we have considered leads to a similar inconsistency (as is shown in panel b of Fig[REF] for the case of anitferromagnetism).', 'cond-mat-0509004-1-22-0': 'This instability is also manifest at [MATH].', 'cond-mat-0509004-1-22-1': 'As [MATH] is decreased at fixed [MATH], the convergence becomes slower and below some temperature [MATH], no stable solution can be found for a [MATH]-dependent range of [MATH].', 'cond-mat-0509004-1-22-2': 'The absence of a solution for some range of [MATH] can be seen in a different way by solving the model as a function of chemical potential [MATH] Fig [REF](a) shows that as [MATH] is increasedat fixed low [MATH], a first order transition occurs to a paramagnetic state, with a corresponding jump in [MATH].', 'cond-mat-0509004-1-22-3': 'Associated with the first order transition is a coexisting region in which two solutions are locally stable (FM with lower [MATH] and PM with higher [MATH]); the DMFT equations correspondingly have two solutions, which one is found depends on the initial seed.', 'cond-mat-0509004-1-22-4': 'The solid and dashed lines in Fig [REF](a) are obtained from initial seeds close to FM and PM states respectively.', 'cond-mat-0509004-1-23-0': 'The absence of convergence may be understood from the density dependent effective potential, shown e.g. in Fig [REF](b).', 'cond-mat-0509004-1-23-1': 'One sees that as [MATH] is increased, [MATH] decreases; this is a precursor of the effect shown in Fig [REF](a).', 'cond-mat-0509004-1-23-2': 'Indeed, the curve [MATH] is reduced by an [MATH]-dependent scale factor.', 'cond-mat-0509004-1-23-3': 'For [MATH] larger than a critical value (here [MATH]), [MATH] is small enough relative to the temperature that this region begins to contribute to [MATH], lowering the maximum [MATH] that can be sustained and destabilizing the ferromagnetic solution.', 'cond-mat-0509004-1-24-0': '# Interpretation', 'cond-mat-0509004-1-25-0': 'We argue in this section that the DMFT instability documented in the previous section is a manifestation of competing instabilities (primarily phase separation) in the original model.', 'cond-mat-0509004-1-25-1': 'To establish this we show in the panel (a) of Fig [REF] a [MATH] phase diagram in the density-coupling plane.', 'cond-mat-0509004-1-25-2': 'The dash-dot line shows the phase separation boundary obtained from the global energy computation; for [MATH] above this line the model phase separates into an [MATH] AF and an [MATH] FM state.', 'cond-mat-0509004-1-25-3': 'The dotted line shows the phase boundary between uniform FM and spiral states.', 'cond-mat-0509004-1-25-4': 'Finally, the heavy solid line shows the region above which the FM DMFT solution is unstable at [MATH].', 'cond-mat-0509004-1-25-5': 'When [MATH] is large enough that the FM state is fully polarized ([MATH]), we see that the DMFT instability line follows the phase separation line, but is inside the region of phase separation.', 'cond-mat-0509004-1-25-6': 'We therefore suggest that in this region the DMFT instability is a consequence of phase separation and this DMFT instability line can be used as a rough estimate of the real phase separation boundary.', 'cond-mat-0509004-1-26-0': 'When [MATH], the DMFT instability indicates the presence of a spiral state with lower energy than the ferromagnetic state.', 'cond-mat-0509004-1-26-1': 'For [MATH] (Fig [REF](b)), there exists a narrow region of [MATH] where none of the uniform phases we considered solve the DMFT equations and the Maxwell constructions seem not to indecate phase separation.', 'cond-mat-0509004-1-26-2': 'We believe that in this region there exists a uniform non FM/AF/spiral/paramagnetic state (either the ground state or the phase separation beteen FM and that state) which we do not know yet.', 'cond-mat-0509004-1-27-0': 'At [MATH] the situation is similar.', 'cond-mat-0509004-1-27-1': 'The DMFT instability is contained inside the regime of phase separation.', 'cond-mat-0509004-1-27-2': 'For example, we show in Fig [REF](a) the phase diagram and the range of DMFT instability in the density-temperature plane for [MATH].', 'cond-mat-0509004-1-27-3': 'The heavy line shows the boundary of the regime of phase separation obtained by Maxwell construction: for [MATH], the phase separation is between FM and AF([MATH]); for [MATH], the phase separation is between PM and AF([MATH]); for [MATH], the phase separation is either PM-AF([MATH]) or FM-PM[CITATION] according to the location [MATH] relative to the homogenous Curie temperature (dotted line).', 'cond-mat-0509004-1-27-4': 'The dashed line shows the region where the DMFT solution fails to converge at that given density [MATH] (the DMFT equation has stable solution for all [MATH], see Fig [REF](a)).', 'cond-mat-0509004-1-27-5': 'For [MATH], the DMFT instability line denotes the temperature below which (a) the paramagnetic state is linearly unstable to antiferromagnetic and (b) no stable antiferromagnetic solution exists (except [MATH]).', 'cond-mat-0509004-1-28-0': 'Fig [REF](b) shows the results of a Maxwell construction for [MATH] and that at [MATH], presented as the difference between calculated free energy [MATH] and the interpolating line [MATH] with [MATH].', 'cond-mat-0509004-1-28-1': 'Phase separation is seen to occur for [MATH], while the DMFT instability range is [MATH] and [MATH].', 'cond-mat-0509004-1-29-0': '# Conclusion', 'cond-mat-0509004-1-30-0': 'We have found an instability in the ferromagnetic DMFT equation for the single site double exchange model and shown that this instability corresponds to the FM/AF phase separation when the coupling [MATH] is larger than half bandwidth (2[MATH]) and to another ground state (spiral) in the small coupling region.', 'cond-mat-0509004-1-30-1': 'There exists a small window, around intermediate [MATH], where no stable FM DMFT solutions exist while the spiral or phase separation is not the ground state, and we believe there is a non FM/AF/Spiral/Para ground state existing in this region.', 'cond-mat-0509004-1-30-2': 'We have presented evidence that the instability is a signal, obtained from a calculation at a fixed parameter value, of the existence of an instability (typically phase separation) which normally is established via a global computation, comparing free energies at many different parameter values.', 'cond-mat-0509004-1-30-3': 'We therefore propose that the DMFT instability is a computationally convenient way to estimate the boundary of phase separation.', 'cond-mat-0509004-1-31-0': 'We thank Dr. Satoshi Okamoto for many helpful discussions.', 'cond-mat-0509004-1-31-1': 'This work is supported by DOE ER46169 and Columbia University MRSEC.'}

{'cond-mat-0509004-2-0-0': 'The single-site dynamical mean field theory approximation to the double exchange model is found to exhibit a previously unnoticed instability, in which a well-defined ground state which is stable against small perturbations is found to be unstable to large-amplitude but purely local fluctuations.', 'cond-mat-0509004-2-0-1': 'The instability is shown to arise either from phase separation or, in a narrow parameter regime, from the presence of a competing phase.', 'cond-mat-0509004-2-0-2': 'The instability is therefore suggested as a computationally inexpensive means of locating regimes of parameter space in which phase separation occurs.', 'cond-mat-0509004-2-1-0': '# Introduction', 'cond-mat-0509004-2-2-0': 'Dynamical mean field theory (DMFT) has been widely applied to many strongly correlated electron systems[CITATION].', 'cond-mat-0509004-2-2-1': 'Since DMFT takes local quantum fluctuations into account, it is especially successful for models whose many body effect comes from the on-site interaction, like Hubbard[CITATION] or Kondo[CITATION][CITATION] (double exchange) model.', 'cond-mat-0509004-2-2-2': 'Correlated systems often exhibit different phases which are quite close in energy, and this proximity can lead to phase separation, which is often important for electronic physics[CITATION].', 'cond-mat-0509004-2-2-3': 'Phase separation is in principle a "global" property of the phase diagram and requiring substantial effort to establish: one must compute the free energy over a wide parameter range, and then perform a Maxwell construction.', 'cond-mat-0509004-2-2-4': 'In this paper we show that within the single-site DMFT formalism a straightforward calculation at a fixed parameter value can reveal the presence of phase separation.', 'cond-mat-0509004-2-2-5': 'Specifically, we find that at zero temperature, the DMFT can give a ground state which is stable against small perturbations but is unstable to a large amplitude local perturbation; at non-zero temperature the standard methods simply fail to converge to a stable solution.', 'cond-mat-0509004-2-2-6': 'By computing the free energy and performing a Maxwell construction we show that for wide parameter ranges this instability occurs in the regions in which phase separation exists.', 'cond-mat-0509004-2-2-7': 'In a narrow parameter regime it signals instead the onset of a different, but apparently uniform, phase.', 'cond-mat-0509004-2-2-8': 'We therefore propose that the instability of the DMFT equations can be used as an approximate, computationally convenient estimator for the boundaries of the regimes in which phase separation occurs.', 'cond-mat-0509004-2-3-0': 'The balance of this paper is organized as follows.', 'cond-mat-0509004-2-3-1': 'We first present the model and then a zero temperature dynamical mean field analysis explicitly showing the instability.', 'cond-mat-0509004-2-3-2': 'In section III we calculate the full T=0 phase diagram in the energy-density plane, establish the regime of phase separation via the usual Maxwell construction, and extend the treatment to [MATH].', 'cond-mat-0509004-2-3-3': 'In section IV we discuss the implications of this instability.', 'cond-mat-0509004-2-3-4': 'Finally in section V we present a brief conclusion.', 'cond-mat-0509004-2-4-0': '# Double Exchange Model and Dynamical Mean Field Approximation', 'cond-mat-0509004-2-5-0': '## Double Exchange Model', 'cond-mat-0509004-2-6-0': 'In this paper, we consider the single orbital "double exchange" or Kondo lattice model of carriers hopping between sites on a lattice and coupled to an array of spins.', 'cond-mat-0509004-2-6-1': 'This model has been studied by many authors and contains important aspects of the physics of the "colossal"[CITATION] magnetoresistance manganites and is also solvable in a variety of approximations, permitting detailed examination of its behavior.', 'cond-mat-0509004-2-6-2': 'Here we use it to investigate the physical meaning of a previously unnoticed instability of the dynamical mean field equations.', 'cond-mat-0509004-2-7-0': 'The model is defined by the Hamiltonian [EQUATION] with [MATH] labeling the sites and the [MATH] denoting the spins.', 'cond-mat-0509004-2-7-1': 'We assume the spins are classical ([MATH]) and are of fixed length.', 'cond-mat-0509004-2-7-2': 'We choose the convention [MATH].', 'cond-mat-0509004-2-8-0': 'The hopping [MATH] defines an energy dispersion [MATH] and thus a density of states [MATH].', 'cond-mat-0509004-2-8-1': 'In our actual computations we specialize to the [MATH] limit of the Bethe lattice, for which [MATH] because the availability of convenient analytical expressions allows us to accurately compute the small difference between free energies of different states.', 'cond-mat-0509004-2-8-2': 'We choose energy units such that [MATH].', 'cond-mat-0509004-2-9-0': 'We also note that the ground state properties of the model may be straightforwardly obtained, because for any fixed configuration of the spins the model is quadratic in the fermions and easily diagonalizable.', 'cond-mat-0509004-2-10-0': '## Dymanical Mean Field Method', 'cond-mat-0509004-2-11-0': 'We now present the dynamical mean field analysis of this model.', 'cond-mat-0509004-2-11-1': 'In the single site dynamical mean field method[CITATION], one neglects the momentum dependence of the self energy.', 'cond-mat-0509004-2-11-2': 'The properties of the model may then be calculated by solving an auxiliary quantum impurity model, along with a self consistency condition.', 'cond-mat-0509004-2-11-3': 'The quantum impurity model corresponding to Eqn([REF]) is specified by the partition function [EQUATION] with [MATH] where the trace is over frequency, and [MATH].', 'cond-mat-0509004-2-11-4': '[MATH] is determined by a spin-dependent mean field function [MATH].', 'cond-mat-0509004-2-11-5': 'In a magnetic phase, [MATH].', 'cond-mat-0509004-2-11-6': 'Note that the assumption of classical core spins means that [MATH] denotes a simple scalar integral over directions of the core spin [MATH], and that no Berry phase term occurs in the argument of the exponential.', 'cond-mat-0509004-2-12-0': "The Green function [MATH] and self energy [MATH] of the impurity model are given by [EQUATION] [MATH] is fixed by requiring the impurity Green's function [MATH] equals to the local Green's function of the lattice problem.", 'cond-mat-0509004-2-12-1': 'The form of the self consistency equation depends on the state which is studied.', 'cond-mat-0509004-2-12-2': 'For a ferromagnetic (FM) state, it is [EQUATION] while for a 2 sublattice antiferromagnetic (AF) state, [EQUATION] where [MATH] and the [MATH] -sum is over tje reduced Brillouin zone (RBZ).', 'cond-mat-0509004-2-12-3': 'The two equations become equivalent in the paramagnetic (PM) state where [MATH]', 'cond-mat-0509004-2-13-0': 'The solution of Eqs [[REF] to [REF]] determines the magnetic phase, the single particle properties, and the free energy.', 'cond-mat-0509004-2-13-1': "In particular, in the dynamical mean field approximation the Gibb's free energy is [CITATION][CITATION] [EQUATION] where the trace is over the spin and lattice degree of freedom.", 'cond-mat-0509004-2-13-2': 'The Helmholtz free energy is [MATH].', 'cond-mat-0509004-2-13-3': 'At zero temperature, the ground state energy (Helmholtz free energy) is [EQUATION]', 'cond-mat-0509004-2-13-4': 'We also note that the solution [MATH] defines an effective potential for the core spin, which depends on the angle [MATH] between the core spin and local magnetization direction, so that [EQUATION] with [EQUATION]', 'cond-mat-0509004-2-14-0': '## Phase Boundaries and Maxwell Construction', 'cond-mat-0509004-2-15-0': 'The model is known to exhibit ferromagnetic, spiral[CITATION] and commensurate antiferromagnetic phases.', 'cond-mat-0509004-2-15-1': 'For our subsequent analysis, an accurate determination of phase boundaries will be important.', 'cond-mat-0509004-2-15-2': 'We therefore present here a few calculational details.', 'cond-mat-0509004-2-16-0': 'We require the [MATH] phase boundary separating the ferromagnetic and spiral phases[CITATION][CITATION].', 'cond-mat-0509004-2-16-1': 'The energy of a spiral state may most easily be found by performing a site-dependent spin rotation to a basis in which the spin quantization axis is parallel to the local spin orientation.', 'cond-mat-0509004-2-16-2': 'The problem may then be easily diagonalized.', 'cond-mat-0509004-2-16-3': 'For the infinite dimensional Bethe lattice one finds, for a diagonal spiral of pitch [MATH], that in the rotated basis, the local Green function is given by[CITATION] [EQUATION] where [MATH] is the angle between two nearest neighbor magnetization, and tilde is used for the spiral states.', 'cond-mat-0509004-2-17-0': 'To locate the [MATH] FM/spiral second order phase boundary, it suffices to expand ground state energy [MATH] (Eqn([REF])) to second order in [MATH].', 'cond-mat-0509004-2-17-1': 'The energy difference between FM and spiral states is [EQUATION] with [MATH] equaling to [EQUATION] where [MATH].', 'cond-mat-0509004-2-17-2': 'The FM/spiral phase boundary is determined by [MATH].', 'cond-mat-0509004-2-18-0': 'The model also exhibits phase separation in some regimes.', 'cond-mat-0509004-2-18-1': 'To determine the boundaries of the regime where phase separation occurs, we use the DMFT method to compute the Helmholtz free energy as a function of occupation number [MATH] (Eqn([REF])) and then perform the Maxwell construction.', 'cond-mat-0509004-2-18-2': 'An example is shown in Fig [REF].', 'cond-mat-0509004-2-18-3': 'We find that in fact that over much of the phase diagram a phase separation between FM and [MATH] AF states preempts the formation of spiral or [MATH] AF state.', 'cond-mat-0509004-2-18-4': 'The general structure of our phase diagram agrees with earlier work[CITATION][CITATION], but the precise locations of phase boundaries differ by roughly [MATH].', 'cond-mat-0509004-2-19-0': '# DMFT Instability', 'cond-mat-0509004-2-20-0': 'In this section we show that the DMFT equations exhibit an apparently previously unnoticed instability.', 'cond-mat-0509004-2-20-1': 'We begin with [MATH].', 'cond-mat-0509004-2-20-2': 'From Eqn([REF]), the [MATH] is [EQUATION] where [MATH] means the angular average with respect to the weight function [MATH], with [MATH] defined in Eqn([REF]) .', 'cond-mat-0509004-2-20-3': 'At zero temperature, the only contribution of the angular average is from the absolute minimum of [MATH].', 'cond-mat-0509004-2-20-4': 'To find the DMFT solution at [MATH], one first assumes the absolute minimum of [MATH] occurs at a fixed value, for example [MATH], obtains [MATH] from Eqn([REF]), and gets the self energy [MATH] from Eqn([REF]).', 'cond-mat-0509004-2-20-5': 'Finally, one uses the [MATH] obtained by the above procedure to calculate [MATH] to see if the minimum is located at the point originally assumed.', 'cond-mat-0509004-2-20-6': 'Note that different ground states (FM, AF.', 'cond-mat-0509004-2-20-7': '.)', 'cond-mat-0509004-2-20-8': 'enter the above procedure only via the self consistent equation Eqn([REF]) (or Eqn([REF])).', 'cond-mat-0509004-2-21-0': 'Fig [REF] shows that as density is increased at fixed large [MATH], the self consistency breaks down, in an unusual manner: [MATH] remains locally stable (slope around the assumed minimum [MATH] remains positive) but the global minimum of [MATH] moves to [MATH].', 'cond-mat-0509004-2-21-1': 'In the regime where this phenomenon occurs, no solution of the DMFT equations exists.', 'cond-mat-0509004-2-21-2': 'Any initial solution we have considered leads to a similar inconsistency (as is shown in panel b of Fig[REF] for the case of anitferromagnetism).', 'cond-mat-0509004-2-22-0': 'This instability is also manifest at [MATH].', 'cond-mat-0509004-2-22-1': 'As [MATH] is decreased at fixed [MATH], the convergence becomes slower and below some temperature [MATH], no stable solution can be found for a [MATH]-dependent range of [MATH].', 'cond-mat-0509004-2-22-2': 'The absence of a solution for some range of [MATH] can be seen in a different way by solving the model as a function of chemical potential [MATH] Fig [REF](a) shows that as [MATH] is increasedat fixed low [MATH], a first order transition occurs to a paramagnetic state, with a corresponding jump in [MATH].', 'cond-mat-0509004-2-22-3': 'Associated with the first order transition is a coexisting region in which two solutions are locally stable (FM with lower [MATH] and PM with higher [MATH]); the DMFT equations correspondingly have two solutions, which one is found depends on the initial seed.', 'cond-mat-0509004-2-22-4': 'The solid and dashed lines in Fig [REF](a) are obtained from initial seeds close to FM and PM states respectively.', 'cond-mat-0509004-2-23-0': 'The absence of convergence may be understood from the density dependent effective potential, shown e.g. in Fig [REF](b).', 'cond-mat-0509004-2-23-1': 'One sees that as [MATH] is increased, [MATH] decreases; this is a precursor of the effect shown in Fig [REF](a).', 'cond-mat-0509004-2-23-2': 'Indeed, the curve [MATH] is reduced by an [MATH]-dependent scale factor.', 'cond-mat-0509004-2-23-3': 'For [MATH] larger than a critical value (here [MATH]), [MATH] is small enough relative to the temperature that this region begins to contribute to [MATH], lowering the maximum [MATH] that can be sustained and destabilizing the ferromagnetic solution.', 'cond-mat-0509004-2-24-0': '# Interpretation', 'cond-mat-0509004-2-25-0': 'We argue in this section that the DMFT instability documented in the previous section is a manifestation of competing instabilities (primarily phase separation) in the original model.', 'cond-mat-0509004-2-25-1': 'To establish this we show in the panel (a) of Fig [REF] a [MATH] phase diagram in the density-coupling plane.', 'cond-mat-0509004-2-25-2': 'The dash-dot line shows the phase separation boundary obtained from the global energy computation; for [MATH] above this line the model phase separates into an [MATH] AF and an [MATH] FM state.', 'cond-mat-0509004-2-25-3': 'The dotted line shows the phase boundary between uniform FM and spiral states.', 'cond-mat-0509004-2-25-4': 'Finally, the heavy solid line shows the region above which the FM DMFT solution is unstable at [MATH].', 'cond-mat-0509004-2-25-5': 'When [MATH] is large enough that the FM state is fully polarized ([MATH]), we see that the DMFT instability line follows the phase separation line, but is inside the region of phase separation.', 'cond-mat-0509004-2-25-6': 'We therefore suggest that in this region the DMFT instability is a consequence of phase separation and this DMFT instability line can be used as a rough estimate of the real phase separation boundary.', 'cond-mat-0509004-2-26-0': 'When [MATH], the DMFT instability indicates the presence of a spiral state with lower energy than the ferromagnetic state.', 'cond-mat-0509004-2-26-1': 'For [MATH] (Fig [REF](b)), there exists a narrow region of [MATH] where none of the uniform phases we considered solve the DMFT equations and the Maxwell constructions seem not to indecate phase separation.', 'cond-mat-0509004-2-26-2': 'We believe that in this region there exists a uniform non FM/AF/spiral/paramagnetic state (either the ground state or the phase separation beteen FM and that state) which we do not know yet.', 'cond-mat-0509004-2-27-0': 'At [MATH] the situation is similar.', 'cond-mat-0509004-2-27-1': 'The DMFT instability is contained inside the regime of phase separation.', 'cond-mat-0509004-2-27-2': 'For example, we show in Fig [REF](a) the phase diagram and the range of DMFT instability in the density-temperature plane for [MATH].', 'cond-mat-0509004-2-27-3': 'The heavy line shows the boundary of the regime of phase separation obtained by Maxwell construction: for [MATH], the phase separation is between FM and AF([MATH]); for [MATH], the phase separation is between PM and AF([MATH]); for [MATH], the phase separation is either PM-AF([MATH]) or FM-PM[CITATION] according to the location [MATH] relative to the homogenous Curie temperature (dotted line).', 'cond-mat-0509004-2-27-4': 'The dashed line shows the region where the DMFT solution fails to converge at that given density [MATH] (the DMFT equation has stable solution for all [MATH], see Fig [REF](a)).', 'cond-mat-0509004-2-27-5': 'For [MATH], the DMFT instability line denotes the temperature below which (a) the paramagnetic state is linearly unstable to antiferromagnetic and (b) no stable antiferromagnetic solution exists (except [MATH]).', 'cond-mat-0509004-2-28-0': 'Fig [REF](b) shows the results of a Maxwell construction for [MATH] and that at [MATH], presented as the difference between calculated free energy [MATH] and the interpolating line [MATH] with [MATH].', 'cond-mat-0509004-2-28-1': 'Phase separation is seen to occur for [MATH], while the DMFT instability range is [MATH] and [MATH].', 'cond-mat-0509004-2-29-0': '# Conclusion', 'cond-mat-0509004-2-30-0': 'We have found an instability in the ferromagnetic DMFT equation for the single site double exchange model and shown that this instability corresponds to the FM/AF phase separation when the coupling [MATH] is larger than half bandwidth (2[MATH]) and to another ground state (spiral) in the small coupling region.', 'cond-mat-0509004-2-30-1': 'There exists a small window, around intermediate [MATH], where no stable FM DMFT solutions exist while the spiral or phase separation is not the ground state, and we believe there is a non FM/AF/Spiral/Para ground state existing in this region.', 'cond-mat-0509004-2-30-2': 'We have presented evidence that the instability is a signal, obtained from a calculation at a fixed parameter value, of the existence of an instability (typically phase separation) which normally is established via a global computation, comparing free energies at many different parameter values.', 'cond-mat-0509004-2-30-3': 'We therefore propose that the DMFT instability is a computationally convenient way to estimate the boundary of phase separation.', 'cond-mat-0509004-2-31-0': 'We thank Dr. Satoshi Okamoto for many helpful discussions.', 'cond-mat-0509004-2-31-1': 'This work is supported by DOE ER46169 and Columbia University MRSEC.'}

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[]

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[]

['cond-mat-0509004-1-20-7', 'cond-mat-0509004-2-20-7']

{'1': 'http://arxiv.org/licenses/assumed-1991-2003/', '2': 'http://arxiv.org/licenses/assumed-1991-2003/'}

https://arxiv.org/abs/cond-mat/0509004

math-0607050

{'math-0607050-1-0-0': 'Stochastic partial differential equations arise as mathematical models of complex multiscale systems under random influences.', 'math-0607050-1-0-1': 'Invariant manifolds often provide geometric structure for understanding stochastic dynamics.', 'math-0607050-1-0-2': 'In this paper, a random invariant manifold reduction principle is proved for a class of stochastic partial differential equations.', 'math-0607050-1-0-3': 'The dynamical behavior is shown to be described by a stochastic ordinary differential equation on an invariant manifold, under suitable conditions.', 'math-0607050-1-0-4': 'The change of dynamical structures for the stochastic partial differential equations is thus obtained by investigating the stochastic ordinary differential equation.', 'math-0607050-1-0-5': 'The random cone invariant property is used in the approach.', 'math-0607050-1-0-6': 'Moreover, the invariant manifold reduction principle is applied to detect bifurcation phenomena and stationary states in stochastic parabolic and hyperbolic partial differential equations.', 'math-0607050-1-1-0': '# Introduction', 'math-0607050-1-2-0': 'Stochastic partial differential equations (SPDEs) have drawn more attention for their importance as macroscopic mathematical models of complex multiscale systems under random influences.', 'math-0607050-1-2-1': 'With the development of random dynamical systems (RDS) [CITATION] recently, SPDEs have been investigated in the context of random dynamical systems, e.g., [CITATION], among others.', 'math-0607050-1-2-2': 'At this time, many SPDEs are known to generate random dynamical systems.', 'math-0607050-1-2-3': 'Despite the rapid development of random dynamical systems theory, invariant manifolds and bifurcations in random systems are still far from being understood.', 'math-0607050-1-2-4': 'There are few rigorous general results and criteria, and in fact, some results are obtained by numerical simulations or hold only for special models.', 'math-0607050-1-3-0': 'Invariant manifolds are special invariant sets represented by graphs in function phase spaces where solution processes of SPDEs live.', 'math-0607050-1-3-1': 'A random invariant manifold provides a geometric structure to reduce stochastic dynamics.', 'math-0607050-1-3-2': 'Stochastic bifurcation, in a sense, is about the changes in invariant structures for random dynamical systems, including the qualitative changes of invariant manifolds, random attractors, and invariant measures.', 'math-0607050-1-4-0': 'The qualitative property of invariant measures for dissipative SPDEs with additive white noises may be investigated by a multiscale method as in [CITATION].', 'math-0607050-1-4-1': 'Stochastic amplitude equations, which are stochastic ordinary differential equations (SODEs) for systems on bounded domains, are used to describe the dynamics of the amplitudes of dominant modes evolving on some slow time scale, near the deterministic bifurcation point.', 'math-0607050-1-4-2': 'And the unique invariant measure of the corresponding random dynamical system can also be approximated by that of the amplitude equation.', 'math-0607050-1-4-3': 'The study of the qualitative property of an invariant measure appears to be difficult in the case of non-unique invariant measures for SPDEs with multiplicative noises.', 'math-0607050-1-5-0': 'Caraballo et.', 'math-0607050-1-5-1': 'al.[CITATION] studied the stochastic bifurcation by detecting the change in random attractors for SPDEs.', 'math-0607050-1-5-2': 'They considered the following Chafee-Infante reaction-diffusion equation perturbed by a multiplicative white noise in the sense of Stratonovich [EQUATION] with the zero Dirichlet boundary condition on an open interval [MATH] in [MATH].', 'math-0607050-1-5-3': 'It is proved that, for [MATH] (the first eigenvalue of negative Laplace on [MATH] with Dirichlet boundary), the random attractor consists of the fixed point [MATH].', 'math-0607050-1-5-4': 'If [MATH], two new random fixed points, [MATH], appear.', 'math-0607050-1-5-5': 'The random attractor now consists of the interval [MATH].', 'math-0607050-1-5-6': 'This phenomena is called the stochastic pitchfork bifurcation in [CITATION], accompanied by the birth of two new random fixed points or equilibrium states.', 'math-0607050-1-5-7': 'But, as [CITATION] pointed out, there is no criteria of the stability for the new fixed points [MATH].', 'math-0607050-1-5-8': 'The monotonic random dynamical system theory, e.g., [CITATION], is used in their approach.', 'math-0607050-1-6-0': 'In this paper we consider invariant manifold reduction of SPDEs and thus reduce the bifurcation problem for SPDEs to that of SODEs.', 'math-0607050-1-6-1': 'As a by product, we obtain the existence of a stochastic pitchfork bifurcation for the parabolic SPDE ([REF]) above, together with the criteria of the stability for the accompanying new equilibrium states [MATH]; see 5.1 below.', 'math-0607050-1-6-2': 'Moreover, in 5.2, we apply this random invariant manifold reduction to investigate existence of the stationary solution of a hyperbolic SPDE with large diffusivity and highly damped term.', 'math-0607050-1-7-0': 'Our main tool is the random invariant manifold theory and random invariant cone property.', 'math-0607050-1-7-1': 'It is well known that invariant manifold is an important tool to study the structure of attractor [CITATION].', 'math-0607050-1-7-2': 'In this paper we first consider invariant manifolds for a class infinite dimensional RDS defined by SPDEs, then reduce the random dynamics to the invariant manifolds, and finally obtain local bifurcation results of the SPDEs.', 'math-0607050-1-7-3': 'The basic idea is to reduce an infinite dimensional random system onto a finite dimensional, asymptotically complete, random invariant manifold (see section [REF]), and the reduced system on the invariant manifold is in fact equivalent to a stochastic ordinary differential system.', 'math-0607050-1-7-4': 'Then we study the bifurcation of the SPDEs through the stochastic ordinary differential equations (SODEs).', 'math-0607050-1-7-5': 'In our approach the random cone invariance concept [REF], is used to prove asymptotic completeness property of the random invariant manifold.', 'math-0607050-1-7-6': 'Cone invariance is also an important property in the study of the inertial manifold for infinite dimensional dynamical systems; see [CITATION].', 'math-0607050-1-8-0': 'This paper is organized as follows.', 'math-0607050-1-8-1': 'We state the main result on random invariant manifold reduction principle for a class of SPDEs, and some background materials in random dynamical systems in section [REF].', 'math-0607050-1-8-2': 'In section [REF], we outline basic setup of the random invariant manifold theory and obtain the existence of random invariant manifolds.', 'math-0607050-1-8-3': 'The main result is proved in section [REF], and applications in stochastic bifurcation and stationary solution are discussed in the final section [REF].', 'math-0607050-1-9-0': '# Main results', 'math-0607050-1-10-0': 'We study the stochastic evolutionary equations in the form [EQUATION] where [MATH] is the generator of a [MATH] semigroup on Hilbert space [MATH] with inner product [MATH], [MATH] is a continuous nonlinear function which Lipschitz constant [MATH], [MATH] and [MATH] is a standard real valued Wiener process, [MATH], [MATH].', 'math-0607050-1-10-1': 'Moreover, [MATH] denotes the stochastic differential in the sense of Stratonovich.', 'math-0607050-1-10-2': 'And [MATH] is a small parameter so that [MATH] can be seen as a small perturbation, that is, we have [MATH] as [MATH].', 'math-0607050-1-10-3': 'Suppose that the operator [MATH] has eigenvalues [EQUATION] with corresponding eigenvectors [MATH].', 'math-0607050-1-10-4': 'Denote the kernel of [MATH] by [MATH].', 'math-0607050-1-10-5': 'Then [MATH] and let [MATH].', 'math-0607050-1-10-6': 'Then for [MATH] we have [MATH] with [MATH] and [MATH].', 'math-0607050-1-10-7': 'Define the projections [MATH] and [MATH] as [MATH] and [MATH].', 'math-0607050-1-10-8': 'Further we assume [MATH] and [MATH] commute with [MATH].', 'math-0607050-1-10-9': 'Define the nonlinear map on [MATH] as [EQUATION] where [MATH] is the zero element in [MATH].', 'math-0607050-1-10-10': 'Then we have the following main result of the paper.', 'math-0607050-1-11-0': 'Theorem A (Invariant manifold reduction)', 'math-0607050-1-12-0': 'Consider the following stochastic evolutionary equation [EQUATION]', 'math-0607050-1-12-1': 'Assume that this evolutionary equation generates a dissipative random dynamical system, that is, the system has a random absorbing set, whose diameter or radius is a tempered random variable.', 'math-0607050-1-12-2': 'Then for small [MATH], the system has an n dimensional invariant manifold [MATH] such that for any solution [MATH] of ([REF]) there is a flow [MATH] on [MATH] [EQUATION]', 'math-0607050-1-12-3': 'Moreover the flow on [MATH] is topologically equivalent to the following stochastic ordinary differential equation in [MATH] [EQUATION] provided ([REF]) is structurally stable for [MATH].', 'math-0607050-1-13-0': 'The random dynamical system [MATH] generated by ([REF]) is called structurally stable, if for any small perturbation (small in the sense of the usual metric in the space of continuous functions) to [MATH], the perturbed random dynamical system [MATH] is topologically equivalent to [MATH].', 'math-0607050-1-13-1': 'Namely, there exists a random homeomorphism [MATH] so that [MATH].', 'math-0607050-1-13-2': 'For more detail, we refer to [CITATION].', 'math-0607050-1-14-0': 'Note that if [MATH] is large enough, the above result still holds even when [MATH] is not small.', 'math-0607050-1-14-1': 'This can be seen from the proof of the main result.', 'math-0607050-1-15-0': 'Having the above result, we can study the local bifurcation of ([REF]) near [MATH].', 'math-0607050-1-15-1': 'In fact the dynamical behavior of the system ([REF]) is determined by the restricted system of ([REF]) on [MATH].', 'math-0607050-1-15-2': 'Furthermore the flow on the invariant manifold is topologically equivalent to the SODE ([REF]) which is structurally stable if [MATH].', 'math-0607050-1-15-3': 'So we can study the dynamics of ([REF]) as [MATH] passes through 0 by the system ([REF]).', 'math-0607050-1-15-4': 'And if ([REF]) undergoes a stochastic bifurcation as [MATH] passes through 0, so does ([REF]).', 'math-0607050-1-15-5': 'We will demonstrate this bifurcation analysis method for a stochastic parabolic partial differential equation ([REF]) mentioned above; see [REF].1.', 'math-0607050-1-16-0': 'Another application of the main result is to study the existence of stationary solution of stochastic dynamical systems.', 'math-0607050-1-16-1': 'For this issue, we will consider the following stochastic hyperbolic partial differential equation, in [REF].2, with large diffusivity and highly damped term on [MATH] [EQUATION] with [EQUATION] where [MATH], [MATH].', 'math-0607050-1-16-2': 'And [MATH] is a bounded globally Lipschitzian nonlinearity.', 'math-0607050-1-16-3': 'Note that the stochastic Sine-Gordon equation ([MATH]) and stochastic Klein-Gordon equation ([MATH] being a polynomial in [MATH]) are in our scope.', 'math-0607050-1-16-4': 'By the same approach for the proof of theorem A, the dynamical behavior of the system ([REF]) restricted on a finite dimensional invariant manifold, which attracts all the solution of ([REF]), is determined by an SODE.', 'math-0607050-1-16-5': 'So the existence of the stationary solution for ([REF]) can be obtained by that of the SODE.', 'math-0607050-1-17-0': 'Now we present the basic setup and some materials in random dynamical systems.', 'math-0607050-1-18-0': 'Let [MATH] be a separable Hilbert space with norm [MATH] and inner product [MATH].', 'math-0607050-1-18-1': '[MATH] is dense in [MATH] and [MATH] is a linear operator which generates a strong continuous semigroup [MATH] on [MATH].', 'math-0607050-1-18-2': 'Further we suppose [MATH] satisfy the following exponential dichotomy.', 'math-0607050-1-18-3': 'There are two projections on [MATH] [EQUATION] such that [MATH] and [MATH] and there exists constants [MATH], [MATH], such that [EQUATION] for [MATH].', 'math-0607050-1-18-4': 'We write [MATH].', 'math-0607050-1-18-5': 'Here we consider the general case that [MATH] may be bigger than the kernel of [MATH].', 'math-0607050-1-18-6': 'In fact if [MATH] is the kernel of [MATH], the exponent [MATH] can be taken as [MATH].', 'math-0607050-1-18-7': 'Further, [MATH] and [MATH] are assumed to be commuting with [MATH].', 'math-0607050-1-19-0': 'For our purpose we use the canonical probability space [MATH] which consists of the sample paths of [MATH], for more see [CITATION].', 'math-0607050-1-19-1': 'Consider the following stochastic evolutionary equations [EQUATION]', 'math-0607050-1-19-2': 'This equation can be written in the following mild integral form [EQUATION]', 'math-0607050-1-19-3': 'By the assumption of [MATH] and [MATH] we know that equation ([REF]) has a unique local solution [MATH] for some [MATH] in the sense of probability.', 'math-0607050-1-19-4': 'For more about the solution of SPDEs we refer to [CITATION].', 'math-0607050-1-20-0': 'For the future reference we give some knowledge of random dynamical systems.', 'math-0607050-1-20-1': 'First we are given a driven dynamical system (also called a metric dynamical system) [MATH], [MATH] satisfied', 'math-0607050-1-21-0': 'A random dynamical system (RDS) on a Polish space [MATH] with Borel [MATH]- algebra [MATH] over [MATH] on [MATH] is a measurable map [EQUATION] such that', 'math-0607050-1-22-0': 'A RDS [MATH] is continuous or differentiable if [MATH] is continuous or differentiable (see [CITATION] for more details on RDS).', 'math-0607050-1-22-1': 'As we consider the canonical probability space the metric dynamical system can be defined as [EQUATION] where [MATH] is a sample path of W(t).', 'math-0607050-1-23-0': 'For a continuous random dynamical system [MATH] over [MATH], we need the following notions to describe its dynamical behavior.', 'math-0607050-1-24-0': 'A multifunction [MATH] of nonempty closed sets [MATH], [MATH], contained in [MATH] is called a random set if [EQUATION] is a random variable for any [MATH].', 'math-0607050-1-25-0': 'A random set [MATH] is called an tempered absorbing set of [MATH] if for any bounded set [MATH] there exists [MATH] such that [MATH] [EQUATION] and for all [MATH] [EQUATION] where [MATH], with [MATH], is the diameter of [MATH].', 'math-0607050-1-26-0': 'For more about random set we refer to [CITATION].', 'math-0607050-1-27-0': 'A random set [MATH] is called an invariant set for a random dynamical system [MATH] if we have [EQUATION]', 'math-0607050-1-27-1': 'If [MATH] can be written as a graph of a Lipschitz mapping [EQUATION] where [MATH], such that [EQUATION]', 'math-0607050-1-27-2': 'Then [MATH] is called a Lipschitz invariant manifold of [MATH].', 'math-0607050-1-28-0': 'For more about the random invariant manifold, see [CITATION].', 'math-0607050-1-29-0': 'The following definition is important in the study of the random invariant manifold.', 'math-0607050-1-30-0': 'Let [MATH] is an invariant manifold of RDS [MATH].', 'math-0607050-1-30-1': 'The invariant manifold [MATH] is called to have the asymptotic completeness property if [MATH], there is [MATH] such that [EQUATION] where [MATH] and [MATH] are some positive constants.', 'math-0607050-1-31-0': 'Asymptotic completeness property describes the attracting property of [MATH] for RDS [MATH].', 'math-0607050-1-31-1': 'Then [MATH] can be reduced onto [MATH], and the dynamical behavior of [MATH] is determined by the reduced system on [MATH].', 'math-0607050-1-32-0': 'By the assumption that system ([REF]) generates a dissipative random dynamical system.', 'math-0607050-1-32-1': 'We suppose the tempered random set [MATH] is an absorbing set for system ([REF]).', 'math-0607050-1-32-2': 'Then we may only consider the system restricted on [MATH].', 'math-0607050-1-32-3': 'For convenience we still write the restricted system as ([REF]).', 'math-0607050-1-32-4': 'Then [MATH] is still Lipschitz with constant [MATH] and is bounded by [MATH] which is a tempered random variables.', 'math-0607050-1-32-5': 'And it is easy to see that [MATH] a.e. as [MATH].', 'math-0607050-1-33-0': 'In order to use the random dynamical theory we introduce the following stationary process [MATH] which solves [EQUATION]', 'math-0607050-1-33-1': 'Solving equation ([REF]) with initial value [EQUATION] we have a unique stationary process for ([REF]) [EQUATION]', 'math-0607050-1-33-2': 'The mapping [MATH] is continuous.', 'math-0607050-1-33-3': 'Moreover [EQUATION]', 'math-0607050-1-33-4': 'We should point out that above property hold in a full probability [MATH] invariant set [MATH].', 'math-0607050-1-33-5': 'For the proof see [CITATION].', 'math-0607050-1-33-6': 'In the following part to the end we consider the probability space [MATH] where [EQUATION].', 'math-0607050-1-34-0': 'We consider the following random partial differential equation on [MATH] [EQUATION] where [MATH].', 'math-0607050-1-34-1': 'It is easy to verify that [MATH] has the same Lipschitz constant as [MATH] and is also bounded by [MATH] since [MATH].', 'math-0607050-1-34-2': 'By the classical evolutionary equation theory equation ([REF]) has a unique solution [MATH] which is continuous about [MATH] for every [MATH].', 'math-0607050-1-34-3': 'Then [EQUATION] defines a continuous random dynamical system.', 'math-0607050-1-34-4': 'We now introduce the transform [EQUATION] and its inverse transform [EQUATION] for [MATH].', 'math-0607050-1-34-5': 'Then for the random dynamical system [MATH] generated by ([REF]), [EQUATION] is the random dynamical system generated by ([REF]).', 'math-0607050-1-34-6': 'For more the relation between ([REF]) and ([REF]) we refer to [CITATION].', 'math-0607050-1-35-0': '# Random invariant manifolds', 'math-0607050-1-36-0': 'In this section we prove the existence of a random invariant manifold for ([REF]).', 'math-0607050-1-37-0': 'Let [MATH] be the metric induced by the norm [MATH].', 'math-0607050-1-37-1': 'Then [MATH] is a complete separable metric space.', 'math-0607050-1-38-0': 'Now we briefly give an approach to obtain the random invariant manifold for system ([REF]) by the Lyapunov-Perron method; for detail see [CITATION].', 'math-0607050-1-39-0': 'Projecting the system ([REF]) onto [MATH] and [MATH] respectively we have [EQUATION] where [EQUATION] and [EQUATION].', 'math-0607050-1-40-0': 'Define the following Banach space for [MATH], [MATH], [EQUATION] with norm [EQUATION].', 'math-0607050-1-41-0': 'Define the nonlinear operator [MATH] on [MATH] as [EQUATION] where [MATH].', 'math-0607050-1-41-1': 'Then for each [MATH], [MATH], we have [EQUATION]', 'math-0607050-1-41-2': 'If [EQUATION] then by the fixed point theorem [EQUATION] has a unique solution [MATH].', 'math-0607050-1-41-3': 'Then we have the following result about the existence of random invariant manifold for the random dynamical system [MATH] generated by ([REF]).', 'math-0607050-1-41-4': 'For the detailed proof see [CITATION].', 'math-0607050-1-42-0': 'Then by the transform [MATH] we have', 'math-0607050-1-43-0': '[MATH] is the unique n dimensional invariant manifold of ([REF]).', 'math-0607050-1-44-0': 'It is easy to see that [MATH] is independent of the choice of [MATH].', 'math-0607050-1-45-0': '# Reduction to stochastic ordinary differential equations', 'math-0607050-1-46-0': 'In this section we prove that the invariant manifold obtained in the last section has the asymptotic completeness property.', 'math-0607050-1-46-1': 'Then the dynamical behavior of ([REF]) is determined by the system restricted on the invariant manifold.', 'math-0607050-1-46-2': 'Furthermore the system restricted on the invariant manifold is in fact topologically equivalent to a random ordinary differential equation provided the random ordinary differential equation is structurally stable.', 'math-0607050-1-46-3': 'This will complete the proof of main result Theorem A stated in [REF].', 'math-0607050-1-47-0': 'First we introduce the cone invariance in the random case.', 'math-0607050-1-47-1': 'For a positive random variable [MATH], define the following random set [EQUATION]', 'math-0607050-1-47-2': 'And the fiber [MATH] is called random cone.', 'math-0607050-1-47-3': 'For a given random dynamical system [MATH] we give the following definition.', 'math-0607050-1-48-0': '[MATH]Cone invariance[MATH] For a random cone [MATH], there is a random variable [MATH] almost surely such that for all [MATH], [MATH], [EQUATION] implies [EQUATION].', 'math-0607050-1-48-1': 'Then the random dynamical system [MATH] is called possessing the cone invariance property for the cone [MATH].', 'math-0607050-1-49-0': 'Cone invariance is an important property to study the inertial manifold of infinite dimensional system, see [CITATION].', 'math-0607050-1-49-1': 'In [CITATION] the authors introduced the nonautonomous cone to study the inertial manifold for nonautonomous evolutionary systems.', 'math-0607050-1-50-0': 'For the random dynamical system [MATH] generated by ([REF]), we have the following result.', 'math-0607050-1-51-0': 'Let [MATH], [MATH] be two solutions of ([REF]) and [MATH], [MATH], then [EQUATION]', 'math-0607050-1-51-1': 'From ([REF]), ([REF]) and by the property of [MATH] and [MATH] we have [EQUATION] and [EQUATION]', 'math-0607050-1-51-2': 'Then ([REF])[MATH] ([REF]), we have [EQUATION]', 'math-0607050-1-51-3': 'It is easy to see that if [MATH] (the boundary of the cone [MATH]), then [MATH] and [EQUATION]', 'math-0607050-1-51-4': 'Note that if [MATH] is so small that [EQUATION] then [MATH] is decreasing on [MATH].', 'math-0607050-1-51-5': 'Thus it is obvious that whenever [MATH], [MATH] can not leave [MATH].', 'math-0607050-1-52-0': 'We now prove the second claim.', 'math-0607050-1-52-1': 'If there is [MATH] such that [MATH], the cone invariance yields [EQUATION] that is [EQUATION].', 'math-0607050-1-52-2': 'Then by ([REF]) we have [EQUATION].', 'math-0607050-1-52-3': 'Hence [EQUATION].', 'math-0607050-1-52-4': 'So we have [EQUATION].', 'math-0607050-1-53-0': 'This completes the proof of the proposition.', 'math-0607050-1-54-0': 'Now we complete the proof of the main result of this paper.', 'math-0607050-1-55-0': 'Proof of the main result: Theorem A in [REF]', 'math-0607050-1-56-0': 'It remains to prove the asymptotic completeness of [MATH].', 'math-0607050-1-56-1': 'We fix a [MATH].', 'math-0607050-1-56-2': 'Consider a solution [EQUATION] of ([REF]).', 'math-0607050-1-56-3': 'Since [MATH] is invariant manifold for ([REF]), there is a solution [MATH] of ([REF]) on [MATH], such that [EQUATION] with [EQUATION].', 'math-0607050-1-56-4': 'Let [EQUATION] and [EQUATION]', 'math-0607050-1-56-5': 'By the construction of [MATH] [EQUATION]', 'math-0607050-1-56-6': 'And [MATH] is almost surely finite by the tempered property of [MATH] and [MATH], see section [REF].', 'math-0607050-1-56-7': 'In fact it is also easy to see that [MATH] as [MATH] and [MATH] as [MATH].', 'math-0607050-1-57-0': 'Since [MATH], by the cone invariance [EQUATION].', 'math-0607050-1-58-0': 'Let [MATH].', 'math-0607050-1-58-1': 'Since [EQUATION] [MATH] is a bounded random set in finite dimensional space.', 'math-0607050-1-58-2': 'So we can pick out a sequence [MATH] such that [EQUATION].', 'math-0607050-1-58-3': 'Let [MATH] be a solution of ([REF]) with [MATH].', 'math-0607050-1-58-4': 'Then it is easy to check by contradiction that [EQUATION] which means the asymptotic completeness of [MATH].', 'math-0607050-1-58-5': 'So the dynamical behavior of ([REF]) is determined by [EQUATION]', 'math-0607050-1-58-6': 'Furthermore since [MATH] as [MATH], if random ordinary differential equation [EQUATION] is structurally stable, the dynamical behavior of the system ([REF]) is topologically equivalent to that of the system ([REF]) provided [MATH] is small enough.', 'math-0607050-1-58-7': 'Notice that [MATH] with [MATH].', 'math-0607050-1-58-8': 'That is the dynamical behavior of the infinite dimensional system ([REF]) restricted on [MATH] can be obtained from the following stochastic ordinary differential equation [EQUATION] provided ([REF]) is structurally stable.', 'math-0607050-1-58-9': 'Then the condition [MATH] yields the main result with ([REF]) becoming ([REF]).', 'math-0607050-1-59-0': 'From the proof of the Theorem A, we see that the theorem is really proved for the random partial differential equation ([REF]).', 'math-0607050-1-59-1': 'In other words, we may apply Theorem A to ([REF]) as long as appropriate conditions are satisfied.', 'math-0607050-1-59-2': 'Moreover, if the stochastic partial differential equation ([REF]), or its transformed form, the random partial differential equation ([REF]), has the linear part with finite number of eigenvalues of zero real part and/or positive real part, we can similarly prove a theorem on reduction to the random invariant center-unstable manifold.', 'math-0607050-1-59-3': 'In fact, we will apply this approach to a stochastic wave type equation in [REF].2.', 'math-0607050-1-60-0': '# Applications: Stochastic bifurcation and stationary states', 'math-0607050-1-61-0': 'In this final section, we consider stochastic bifurcation and stationary solution for SPDEs, via a couple of examples.', 'math-0607050-1-61-1': 'We use the random invariant manifold reduction principle established in the previous sections.', 'math-0607050-1-62-0': '## Parabolic partial differential equations driven by multiplicative white noise', 'math-0607050-1-63-0': 'We study the stochastic bifurcation in the Chafee-Infante reaction diffusion perturbed by a multiplicative white noise in the sense of Stratonovich [EQUATION] where [MATH] is an open interval, [MATH] is a real parameter, and [MATH] is a real valued Wiener process.', 'math-0607050-1-63-1': 'Let [MATH] on [MATH] with Dirichlet boundary condition and [MATH] be the spectrum of [MATH] with eigenvectors [MATH] respectively.', 'math-0607050-1-63-2': 'Then we have the following evolutionary system on [MATH] [EQUATION]', 'math-0607050-1-63-3': 'It is easy to prove that if [MATH], [MATH] is the globally exponentially stable stationary solution, see [CITATION].', 'math-0607050-1-63-4': 'And if [MATH] then [MATH] is unstable.', 'math-0607050-1-63-5': 'We will prove that system ([REF]) undergoes stochastic bifurcation when [MATH] passes through [MATH].', 'math-0607050-1-63-6': 'In fact we will apply our result to system ([REF]) by truncating the nonlinear term [MATH].', 'math-0607050-1-63-7': 'Let [EQUATION] where [MATH] is a smooth function with [EQUATION] and [MATH] with [MATH].', 'math-0607050-1-63-8': 'It is easy to verify that [MATH] is globally bounded, [EQUATION] and globally Lipschitz, [EQUATION] with [MATH] and [MATH] for constants [MATH], [MATH].', 'math-0607050-1-64-0': 'Let us consider the following truncated system [EQUATION] with [MATH].', 'math-0607050-1-64-1': 'The semigroup generated by the operator [MATH] satisfies ([REF]) and ([REF]) with [MATH], [MATH] for some positive constant [MATH] and [MATH].', 'math-0607050-1-64-2': 'Then if [MATH] is small, by Theorem A in [REF], the system ([REF]) has a one dimensional invariant manifold [MATH] which is asymptotically complete and the dynamical behavior of ([REF]) restricted on [MATH] is topologically equivalent to the following one dimensional SODE [EQUATION] with [MATH] and [EQUATION].', 'math-0607050-1-64-3': 'Near [MATH], by the definition of [MATH], [MATH].', 'math-0607050-1-64-4': 'Then [MATH] with [MATH] and ([REF]) is [EQUATION]', 'math-0607050-1-64-5': 'It is well known, see [CITATION] or [CITATION], that near the solution [MATH], system ([REF]) undergoes a stochastic pitchfork bifurcation when [MATH] passes through [MATH].', 'math-0607050-1-64-6': 'If [MATH], [MATH] is a stable stationary solution of system ([REF]); if [MATH], [MATH] is unstable and two new stable stationary solutions appear.', 'math-0607050-1-64-7': 'Then by Theorem A, near the stationary solution [MATH], system ([REF]) also undergoes a stochastic bifurcation when [MATH] passes through [MATH].', 'math-0607050-1-64-8': 'And the bifurcation diagram also can be described by that of the system ([REF]).', 'math-0607050-1-65-0': 'The amplitude equations, which are SODEs, for stochastic partial differential equations also give some information to the dynamical behavior of the system near a change of stability, see [CITATION] for the amplitude equations of the stochastic partial differential equations with small noise on a bounded domain.', 'math-0607050-1-66-0': '## Hyperbolic partial differential equations with large diffusivity and highly damped term driven by multiplicative white noise', 'math-0607050-1-67-0': 'We consider the following hyperbolic equation with Dirichlet boundary condition driven by multiplicative white noise, [EQUATION] with [EQUATION] where [MATH] is taken as the interval [MATH] for simplicity and [MATH].', 'math-0607050-1-67-1': 'Here [MATH] is bounded with global Lipschitz constant [MATH].', 'math-0607050-1-67-2': 'For example [MATH], which yields the Sine-Gordon equation.', 'math-0607050-1-67-3': 'Here [MATH] is a real valued Wiener process.', 'math-0607050-1-68-0': 'We study the existence of the stationary solution of ([REF]) by reducing the system to a finite dimensional system on the invariant manifold, which is asymptotically complete.', 'math-0607050-1-68-1': 'In fact by the same proof of Theorem A in [REF], the dynamical behavior of ([REF]) restricted on the invariant manifold, is determined by a stochastic ordinary differential equation.', 'math-0607050-1-68-2': 'Then the existence and stability of the stationary solution of the stochastic hyperbolic equation can be obtained from that of the stochastic ordinary differential equation.', 'math-0607050-1-69-0': 'Let [MATH].', 'math-0607050-1-69-1': 'Rewrite the system ([REF]) as the following one order stochastic evolutionary equation in [MATH] [EQUATION] where [MATH] with Dirichlet boundary condition on [MATH].', 'math-0607050-1-69-2': 'Note that Theorem A can not be applied to the system ([REF])-([REF]) directly.', 'math-0607050-1-69-3': 'But by Remark [REF], we can still have the same result of the Theorem A for the stochastic hyperbolic system ([REF]).', 'math-0607050-1-69-4': 'It will be clear after we transform ([REF])-([REF]) into the form of ([REF]).', 'math-0607050-1-70-0': 'First we prove that the system ([REF])-([REF]) generates a continuous random dynamical system in [MATH].', 'math-0607050-1-70-1': 'Let [MATH], [MATH] where [MATH] is the stationary solution of ([REF]).', 'math-0607050-1-70-2': 'Then we have the following random evolutionary equation [EQUATION]', 'math-0607050-1-70-3': 'By a standard Galerkin approximation procedure as in [CITATION], the system ([REF])-([REF]) is wellposed.', 'math-0607050-1-70-4': 'In fact we give a prior estimates.', 'math-0607050-1-70-5': 'Multiplying ([REF]) with [MATH] in [MATH] and ([REF]) with [MATH] in [MATH].', 'math-0607050-1-70-6': 'Since [MATH] is bounded by a simple calculation we have [EQUATION] for appropriate constant [MATH].', 'math-0607050-1-70-7': 'Then for any [MATH], [MATH] is bounded in [MATH] which ensures the weak-star convergence by the Lipschitz property of [MATH], for the detail see [CITATION].', 'math-0607050-1-70-8': 'Let [MATH], then [MATH] defines a continuous random dynamical system in [MATH].', 'math-0607050-1-70-9': 'Notice that the stochastic system ([REF])-([REF]) is conjugated to the random system ([REF])-([REF]) by the homeomorphism [EQUATION] with inverse [EQUATION].', 'math-0607050-1-70-10': 'Then [MATH] is the random dynamical system generated by ([REF])-([REF]).', 'math-0607050-1-70-11': 'For more relation about the two systems see [CITATION].', 'math-0607050-1-71-0': 'Define [EQUATION] where [MATH] is the identity operator on the Hilbert space [MATH].', 'math-0607050-1-72-0': 'Then ([REF])-([REF]) can be written as [EQUATION] which is in the form of ([REF]).', 'math-0607050-1-72-1': 'Thus by Remark [REF], we can still apply Theorem A here.', 'math-0607050-1-73-0': 'The eigenvalues of the operator [MATH] are [MATH] with corresponding eigenvectors [MATH], [MATH].', 'math-0607050-1-73-1': 'Then the operator [MATH] has the eigenvalues [EQUATION] and corresponding eigenvectors are [MATH].', 'math-0607050-1-73-2': 'Define subspace of [MATH] [EQUATION] and [MATH].', 'math-0607050-1-73-3': 'Write the projections from [MATH] to [MATH], [MATH] and [MATH] as [MATH], [MATH] and [MATH] respectively.', 'math-0607050-1-73-4': 'We also use the subspaces [EQUATION] with the projections [MATH] and [MATH] from [MATH] to [MATH] and [MATH] respectively.', 'math-0607050-1-73-5': 'Let [MATH] and [MATH].', 'math-0607050-1-74-0': 'Here we consider a special case that [MATH] and [MATH].', 'math-0607050-1-74-1': 'Then the operator [MATH] has one zero eigenvalue [MATH], one negative eigenvalue [MATH] and the others are all complex numbers with positive real part.', 'math-0607050-1-74-2': 'Since [MATH] are not orthogonal, we introduce a new inner product (see [CITATION]), which defines an equivalent norm on [MATH].', 'math-0607050-1-74-3': 'For [MATH], [MATH], define [EQUATION] and for [MATH], [MATH] define [EQUATION] where [MATH] is usual inner product in [MATH].', 'math-0607050-1-74-4': 'Then we introduce the following new inner product in [MATH] defined by [EQUATION] for [MATH], [MATH].', 'math-0607050-1-74-5': 'And the new norm [EQUATION] is equivalent to the usual norm [MATH].', 'math-0607050-1-74-6': 'Moreover [MATH] and [MATH] is orthogonal.', 'math-0607050-1-74-7': 'By the definition of the new norm a simple calculation yields that', 'math-0607050-1-75-0': 'For the detail proof we refer to [CITATION].', 'math-0607050-1-76-0': 'A simple calculation yields that in terms of the new norm the semigroup [MATH] generated by [MATH] satisfies the exponential dichotomy ([REF])-([REF]) with [MATH], [MATH] and [MATH].', 'math-0607050-1-76-1': 'Taking [MATH], condition ([REF]) for ([REF]) becomes [EQUATION] which holds if [MATH] is large enough.', 'math-0607050-1-77-0': 'Define the space [MATH] with [MATH] replaced by [MATH].', 'math-0607050-1-77-1': 'Then by the random invariant manifold theory in 3 above, we know the system ([REF]) has a two dimensional random invariant manifold [MATH] in [MATH] provided [MATH] is large enough.', 'math-0607050-1-77-2': 'Furthermore by the same proof of Theorem A, [MATH] is asymptotically complete.', 'math-0607050-1-77-3': 'Then the dynamical behavior of ([REF]), that is, ([REF]), restricted on [MATH] is determined by following reduced system which is a finite dimensional random system [EQUATION] provided [MATH] is large enough.', 'math-0607050-1-77-4': 'To see the reduced system more clear, projecting the system ([REF]) to span[MATH] and span[MATH] by the projection [MATH] and [MATH] respectively we have the following system [EQUATION]', 'math-0607050-1-77-5': 'By the new inner product in [MATH], define the following new inner product in [MATH] as [EQUATION].', 'math-0607050-1-77-6': 'Then projecting the above system ([REF])-([REF]) to [MATH] by the new inner product in [MATH] and noticing [MATH] is taken as 0 in the reduced system ([REF]) we have the final reduced system [EQUATION] which is a two dimensional stochastic ordinary differential equation.', 'math-0607050-1-77-7': 'And the dynamical behavior of the system ([REF]) restricted on the manifold [MATH] is determined by that of the system ([REF])-([REF]) if [MATH] is large.', 'math-0607050-1-77-8': 'So if system ([REF])-([REF]) has a stationary solution, so does the system ([REF]).', 'math-0607050-1-77-9': 'And the stability property of the stationary solution is also determined by that of ([REF])-([REF]).'}

{'math-0607050-2-0-0': 'Random invariant manifolds often provide geometric structures for understanding stochastic dynamics.', 'math-0607050-2-0-1': 'In this paper, a dynamical approximation estimate is derived for a class of stochastic partial differential equations, by showing that the random invariant manifold is almost surely asymptotically complete.', 'math-0607050-2-0-2': 'The asymptotic dynamical behavior is thus described by a stochastic ordinary differential system on the random invariant manifold, under suitable conditions.', 'math-0607050-2-0-3': 'As an application, stationary states (invariant measures) is considered for one example of stochastic partial differential equations.', 'math-0607050-2-1-0': '# Introduction', 'math-0607050-2-2-0': 'Stochastic partial differential equations (SPDEs or stochastic PDEs) arise as macroscopic mathematical models of complex systems under random influences.', 'math-0607050-2-2-1': 'There have been rapid progresses in this area [CITATION].', 'math-0607050-2-2-2': 'More recently, SPDEs have been investigated in the context of random dynamical systems (RDS) [CITATION]; see for example, [CITATION], among others.', 'math-0607050-2-3-0': 'Invariant manifolds are special invariant sets represented by graphs in state spaces (function spaces) where solution processes of SPDEs live.', 'math-0607050-2-3-1': 'A random invariant manifold provides a geometric structure to reduce stochastic dynamics.', 'math-0607050-2-3-2': 'Stochastic bifurcation, in a sense, is about the changes in invariant structures for random dynamical systems.', 'math-0607050-2-3-3': 'This includes qualitative changes of invariant manifolds, random attractors, and invariant measures or stationary states.', 'math-0607050-2-4-0': 'Duan et al. [CITATION] have recently proved results on existence and smoothness of random invariant manifolds for a class of stochastic partial differential equations.', 'math-0607050-2-4-1': 'In this paper, we further derive a dynamical approximation estimate between the solutions of stochastic partial differential equations and the orbits on the random invariant manifolds.', 'math-0607050-2-4-2': 'This is achieved by showing that the random invariant manifold is almost surely asymptotically complete (see Definition [REF]).', 'math-0607050-2-4-3': 'The asymptotic dynamical behavior thus can be described by a stochastic ordinary differential system on the random invariant manifold, under suitable conditions.', 'math-0607050-2-4-4': 'In this approach one key assumption is that the global Lipschitz constant of nonlinear term is small enough.', 'math-0607050-2-5-0': 'If the invariant manifold is almost surely asymptotically complete we can approximate the infinite dimensional system by a system restricted on the random invariant manifold which is in fact finite dimensional.', 'math-0607050-2-5-1': 'That is the infinite dimensional system is reduced to a finite dimensional system, which is useful for understanding asymptotic behavior of the original stochastic system [CITATION].', 'math-0607050-2-6-0': 'As a application, in 5, we study the existence of stationary solutions of a hyperbolic SPDE.', 'math-0607050-2-6-1': 'Specifically, we will consider the following stochastic hyperbolic partial differential equation, in [REF], with large diffusivity and highly damped term, on the space-time domain [MATH] [EQUATION] with [EQUATION] where [MATH] and [MATH] are both positive.', 'math-0607050-2-6-2': 'And [MATH] is a bounded globally Lipschitz nonlinearity.', 'math-0607050-2-6-3': 'Note that the stochastic Sine-Gordon equation ([MATH]) is an example.', 'math-0607050-2-6-4': 'When the damping is large enough the existence of the stationary solutions for ([REF]) is obtained by considering the stochastic system on the random invariant manifold.', 'math-0607050-2-7-0': 'Invariant manifolds are often used as a tool to study the structure of attractors [CITATION].', 'math-0607050-2-7-1': 'In this paper we first consider invariant manifolds for a class infinite dimensional RDS defined by SPDEs, then reduce the random dynamics to the invariant manifolds.', 'math-0607050-2-7-2': 'When the invariant manifolds are shown to be almost surely asymptotically complete (see Definition [REF]), we obtain dynamical approximations of the solutions of stochastic PDEs by orbits on the invariant manifolds.', 'math-0607050-2-7-3': 'Almost sure cone invariance concept (see Definition [REF]) is used to prove almost sure asymptotic completeness property of the random invariant manifolds.', 'math-0607050-2-8-0': 'This paper is organized as follows.', 'math-0607050-2-8-1': 'We state the main result on dynamical approximation for a class of SPDEs in section [REF].', 'math-0607050-2-8-2': 'Then we recall background materials in random dynamical systems and the existence result of invariant manifolds [CITATION] in section [REF].', 'math-0607050-2-8-3': 'The main result is proved in section [REF], and applications in detecting stationary states are discussed in the final section [REF].', 'math-0607050-2-9-0': '# Main result', 'math-0607050-2-10-0': 'We consider the stochastic evolutionary system [EQUATION] where [MATH] is the generator of a [MATH]-semigroup [MATH] on real valued separable Hilbert space [MATH] with inner product [MATH]; [MATH] is a continuous nonlinear function with [MATH] and Lipschitz constant [MATH] is assumed to be small; and [MATH] is a standard real valued Wiener process.', 'math-0607050-2-10-1': 'Moreover, [MATH] denotes the stochastic differential in the sense of Stratonovich.', 'math-0607050-2-10-2': 'Suppose that [MATH], the spectrum of operator [MATH], splits as [EQUATION] with [EQUATION] where [MATH] denotes the complex numbers set.', 'math-0607050-2-10-3': '[MATH] is assumed to be a finite set.', 'math-0607050-2-10-4': 'Denote the corresponding eigenvectors to [MATH] by [MATH].', 'math-0607050-2-11-0': 'By the above assumptions, there is an [MATH]-invariant decomposition [MATH] such that for the restrictions [MATH], [MATH] one has [MATH] and [MATH].', 'math-0607050-2-11-1': 'Moreover [MATH] is a group on [MATH] and there exist projections [MATH] and [MATH] such that [MATH], [MATH] and [MATH].', 'math-0607050-2-11-2': 'We also suppose that there are', 'math-0607050-2-12-0': 'postive constants [MATH] and [MATH] with property [MATH], such that [EQUATION]', 'math-0607050-2-12-1': 'For instance, [MATH] may be a strongly elliptic and symmetric second order differential operator on a smooth domain with zero Dirichlet boundary condition.', 'math-0607050-2-13-0': 'Since [MATH] is a finite set, [MATH] is a finite dimensional space of dimension, say, [MATH].', 'math-0607050-2-13-1': 'For [MATH] we have [MATH] with [MATH] and [MATH].', 'math-0607050-2-13-2': 'Furthermore, we assume that the projections [MATH] and [MATH] commute with [MATH].', 'math-0607050-2-13-3': 'Define the nonlinear map on [MATH] as [EQUATION] where [MATH] is the zero element in the vector space [MATH].', 'math-0607050-2-13-4': 'The concept of random invariant manifolds will be introduced in the next section.', 'math-0607050-2-13-5': 'Let [MATH] be the metric dynamical system generated by Wiener process [MATH], see ([REF]).', 'math-0607050-2-13-6': 'We will obtain the following main result.', 'math-0607050-2-14-0': '(Dynamical approximations)', 'math-0607050-2-15-0': 'Consider the following stochastic evolutionary system [EQUATION] where the linear operator [MATH] and the nonlinearity [MATH] satisfy the conditions listed above.', 'math-0607050-2-15-1': 'If the Lipschitz constant of the nonlinearity [MATH] is small enough, then this stochastic system has an n dimensional invariant manifold [MATH] and there exists a positive random variable [MATH] and a positive constant [MATH] such that: For any solution [MATH] of ([REF]), there is an orbit [MATH] on the invariant manifold [MATH], with the following approximation property: [EQUATION]', 'math-0607050-2-15-2': 'The above result can be seen as a dynamical approximation for the system ([REF]).', 'math-0607050-2-15-3': 'Any solution [MATH] of ([REF]) can be approximated by an orbit [MATH] on the manifold [MATH].', 'math-0607050-2-15-4': 'In fact the function [MATH] can be represented as [MATH], where [MATH] satisfies the following stochastic equation [EQUATION] and, moreover, [EQUATION] is a random Lipschitz map; see [REF].', 'math-0607050-2-15-5': 'Here we also remark that [MATH] depends on [MATH], so ([REF]) in fact is a non-autonomous stochastic differential equation on [MATH].', 'math-0607050-2-16-0': '# Random invariant manifolds', 'math-0607050-2-17-0': 'Now we recall the basic concepts in random dynamical systems, and the basic result on the existence of random invariant manifolds for stochastic PDEs, from Duan et al. [CITATION].', 'math-0607050-2-18-0': 'For our purpose we work on the canonical probability space [MATH], where the sample space [MATH] consists of the sample paths of [MATH], that is [EQUATION] for more see [CITATION].', 'math-0607050-2-18-1': 'Consider the following stochastic evolutionary equations [EQUATION]', 'math-0607050-2-18-2': 'This equation can be written in the following mild integral form [EQUATION]', 'math-0607050-2-18-3': 'By the assumption of [MATH] and [MATH] we know that equation ([REF]) has a unique solution [MATH] for any [MATH] in the sense of probability.', 'math-0607050-2-18-4': 'For more about the solution of SPDEs we refer to [CITATION].', 'math-0607050-2-19-0': 'We now present some basics of random dynamical systems.', 'math-0607050-2-19-1': 'First we start with a driven dynamical system which models white noise: [MATH], [MATH] that satisfies the usual definition for a (deterministic) dynamical system', 'math-0607050-2-20-0': 'A random dynamical system (RDS) on a metric space [MATH] with Borel [MATH]- algebra [MATH] over [MATH] on [MATH] is a measurable map [EQUATION] such that', 'math-0607050-2-21-0': 'A RDS [MATH] is continuous or differentiable if [MATH] is continuous or differentiable (see [CITATION] for more details on RDS).', 'math-0607050-2-21-1': 'As we consider the canonical probability space the driven dynamical system [MATH] can be defined as [EQUATION] where [MATH] is a sample path of the Wiener process or Brownian motion [MATH].', 'math-0607050-2-22-0': 'For a continuous random dynamical system [MATH] over [MATH], we need the following notions to describe its dynamical behavior.', 'math-0607050-2-23-0': 'A collection [MATH], of nonempty closed sets [MATH], [MATH], contained in [MATH], is called a random set if [EQUATION] is a real valued random variable for any [MATH].', 'math-0607050-2-24-0': 'A random set [MATH] is called a tempered absorbing set for a random dynamical system [MATH] if for any bounded set [MATH] there exists [MATH] such that [MATH] [EQUATION] and for all [MATH] [EQUATION] where [MATH], with [MATH], is the diameter of [MATH].', 'math-0607050-2-25-0': 'For more about random set we refer to [CITATION].', 'math-0607050-2-26-0': 'A random set [MATH] is called a positive invariant set for a random dynamical system [MATH] if [EQUATION]', 'math-0607050-2-26-1': 'If [MATH] can be written as a graph of a Lipschitz mapping [EQUATION] where [MATH], such that [EQUATION]', 'math-0607050-2-26-2': 'Then [MATH] is called a Lipschitz invariant manifold of [MATH].', 'math-0607050-2-27-0': 'For more about the random invariant manifold theory, see [CITATION].', 'math-0607050-2-28-0': 'In order to apply the random dynamical systems framework, we transform the stochastic PDE ([REF]) into a random partial differential equation (random PDE).', 'math-0607050-2-28-1': 'To this end, we introduce the following stationary process [MATH] which solves [EQUATION]', 'math-0607050-2-28-2': 'Solving equation ([REF]) with initial value [EQUATION] we have a unique stationary process for ([REF]) [EQUATION]', 'math-0607050-2-28-3': 'The mapping [MATH] is continuous.', 'math-0607050-2-28-4': 'Moreover [EQUATION]', 'math-0607050-2-28-5': 'We should point out that above properties hold in a [MATH] invariant set [MATH] of full probability.', 'math-0607050-2-28-6': 'For the proof see [CITATION].', 'math-0607050-2-28-7': 'In the following part to the end we consider the probability space [MATH] where [EQUATION]', 'math-0607050-2-29-0': 'The following random PDE on the probability space [MATH] is a transformed version of the original stochastic PDE ([REF]).', 'math-0607050-2-29-1': '[EQUATION] where [MATH].', 'math-0607050-2-29-2': 'It is easy to verify that [MATH] has the same Lipschitz constant as [MATH] and [MATH].', 'math-0607050-2-29-3': 'By the classical evolutionary equation theory equation ([REF]) has a unique solution [MATH] which is continuous in [MATH] for every [MATH].', 'math-0607050-2-29-4': 'Then [EQUATION] defines a continuous random dynamical system.', 'math-0607050-2-29-5': 'We now introduce the transform [EQUATION] and its inverse transform [EQUATION] for [MATH].', 'math-0607050-2-29-6': 'Then for the random dynamical system [MATH] generated by ([REF]), [EQUATION] is the random dynamical system generated by ([REF]).', 'math-0607050-2-29-7': 'For more about the relation between ([REF]) and ([REF]) we refer to [CITATION].', 'math-0607050-2-30-0': 'We now prove the existence of a random invariant manifold for ([REF]) as in [CITATION].', 'math-0607050-2-31-0': 'Let [MATH] be the metric induced by the norm [MATH].', 'math-0607050-2-31-1': 'Then [MATH] is a complete separable metric space.', 'math-0607050-2-32-0': 'Now we briefly give an approach to obtain the random invariant manifold for system ([REF]) by the Lyapunov-Perron method; for detail see [CITATION].', 'math-0607050-2-33-0': 'Projecting the system ([REF]) onto [MATH] and [MATH] respectively we have [EQUATION] where [EQUATION] and [EQUATION].', 'math-0607050-2-34-0': 'Define the following Banach space for [MATH], [MATH], [EQUATION] with norm [EQUATION].', 'math-0607050-2-35-0': 'Define the nonlinear operator [MATH] on [MATH] as [EQUATION] where [MATH].', 'math-0607050-2-35-1': 'Then for any given [MATH] and each [MATH], [MATH], we have [EQUATION]', 'math-0607050-2-35-2': 'If [EQUATION] then by the fixed point argument [EQUATION] has a unique solution [MATH].', 'math-0607050-2-35-3': 'Let [MATH].', 'math-0607050-2-35-4': 'Then [EQUATION] [MATH] and [MATH] is Lipschitz continuous with Lipschitz constant [MATH], [MATH] is a positive constant.', 'math-0607050-2-35-5': 'Then we have the following result about the existence of random invariant manifold for the random dynamical system [MATH] generated by ([REF]).', 'math-0607050-2-35-6': 'For the detailed proof see [CITATION].', 'math-0607050-2-36-0': 'Suppose the assumptions on [MATH] and [MATH] in section [REF] and condition ([REF]) hold.', 'math-0607050-2-36-1': 'Then there exists a Lipschitz continuous random invariant manifold [MATH] for [MATH] which is given by [MATH].', 'math-0607050-2-37-0': 'Then by the transform [MATH], as defined in ([REF]), we have the following conclusion.', 'math-0607050-2-37-1': '(Invariant manifold for stochastic PDE)', 'math-0607050-2-38-0': '[MATH] is a Lipschitz continuous random invariant manifold for the stochastic PDE ([REF]).', 'math-0607050-2-39-0': 'Note that [MATH] is independent of the choice of [MATH].', 'math-0607050-2-40-0': '# Dynamical approximations', 'math-0607050-2-41-0': 'In this section we prove Theorem [REF], by showing that the invariant manifold obtained in the last section has the almost sure asymptotic completeness property.', 'math-0607050-2-41-1': 'Then the dynamical behavior of ([REF]) is determined by the system restricted on the invariant manifold.', 'math-0607050-2-42-0': 'The following concept is important in the study of the dynamical approximations of stochastic PDEs.', 'math-0607050-2-43-0': '(Almost sure asymptotic completeness)', 'math-0607050-2-44-0': 'Let [MATH] be an invariant manifold for a random dynamical system [MATH].', 'math-0607050-2-44-1': 'The invariant manifold [MATH] is called almost surely asymptotically complete if for every [MATH], there exists [MATH] such that [EQUATION] for almost all [MATH], where [MATH] is some positive constant and [MATH] is a positive random variable.', 'math-0607050-2-45-0': 'Now we introduce the almost sure cone invariance concept.', 'math-0607050-2-45-1': 'For a positive random variable [MATH], define the following random set [EQUATION]', 'math-0607050-2-45-2': 'And the fiber [MATH] is called random cone.', 'math-0607050-2-45-3': 'For a given random dynamical system [MATH] we give the following definition.', 'math-0607050-2-46-0': '(Almost sure cone invariance)', 'math-0607050-2-47-0': 'For a random cone [MATH], there is a random variable [MATH] almost surely such that for all [MATH], [MATH], [EQUATION] implies [EQUATION].', 'math-0607050-2-47-1': 'Then the random dynamical system [MATH] is called have the cone invariance property for the cone [MATH].', 'math-0607050-2-48-0': 'Both asymptotic completeness and cone invariance are important tools to study the inertial manifold of deterministic infinite dimensional systems [CITATION].', 'math-0607050-2-48-1': 'Here we modified both concepts for random systems.', 'math-0607050-2-49-0': 'Almost sure asymptotic completeness describes the attracting property of [MATH] for RDS [MATH].', 'math-0607050-2-49-1': 'When this property holds, the infinite dimensional system [MATH] can be reduced to a finite dimensional system on [MATH], and the asymptotic behavior of [MATH] can be determined by that of the reduced system on [MATH].', 'math-0607050-2-50-0': 'For the random dynamical system [MATH] generated by the random PDE ([REF]), we have the following result.', 'math-0607050-2-51-0': 'For small Lipschitz constant [MATH] random dynamical system [MATH] possesses the cone invariance property for a cone with a deterministic positive constant [MATH].', 'math-0607050-2-51-1': 'Moreover if there exists [MATH] such that for [MATH], [MATH] and [EQUATION] then [EQUATION] where [MATH] is a positive tempered random variable and [MATH].', 'math-0607050-2-52-0': 'Note that the smallness condition on the Lipschitz constant [MATH] is specifically defined in ([REF]) below.', 'math-0607050-2-53-0': 'Let [MATH], [MATH] be two solutions of ([REF]) and [MATH], [MATH], then [EQUATION]', 'math-0607050-2-53-1': 'From ([REF]), ([REF]) and by the property of [MATH] and [MATH] we have [EQUATION] and [EQUATION]', 'math-0607050-2-53-2': 'Then ([REF])[MATH] ([REF]), we have [EQUATION]', 'math-0607050-2-53-3': 'Note that if [MATH] (the boundary of the cone [MATH]), then [MATH] and [EQUATION]', 'math-0607050-2-53-4': 'If [MATH] is small enough such that [EQUATION] then [MATH] is decreasing on [MATH].', 'math-0607050-2-53-5': 'Thus it is obvious that whenever [MATH], [MATH] can not leave [MATH].', 'math-0607050-2-54-0': 'We now prove the second claim.', 'math-0607050-2-54-1': 'If there is [MATH] such that [MATH], the cone invariance yields [EQUATION] that is [EQUATION].', 'math-0607050-2-54-2': 'Then by ([REF]) we have [EQUATION].', 'math-0607050-2-54-3': 'Hence [EQUATION].', 'math-0607050-2-54-4': 'Then by the property of [MATH] there is a tempered random variable [MATH] such that [EQUATION].', 'math-0607050-2-55-0': 'This completes the proof of the lemma.', 'math-0607050-2-56-0': 'Before we prove Theorem [REF], we need the following lemma, which implies the backward solvability of the system ([REF]) restricted on the invariant manifold [MATH].', 'math-0607050-2-57-0': 'For any given final time [MATH], consider the following system for [MATH] [EQUATION] where [MATH] is defined as ([REF]).', 'math-0607050-2-57-1': 'Rewrite the above problem in the following equivalent integral form [EQUATION] [MATH].', 'math-0607050-2-58-0': 'Let ([REF]) hold.', 'math-0607050-2-58-1': 'Then for any [MATH], ([REF]), ([REF]) has a unique solution [MATH].', 'math-0607050-2-58-2': 'Moreover for any [MATH], [MATH] for almost all [MATH].', 'math-0607050-2-59-0': 'The existence and uniqueness on small time interval can be obtained by a contraction argument, as in Lemma 3.3 of [CITATION].', 'math-0607050-2-59-1': 'Then the solution can be extended to any time interval; see Theorem 3.8 of [CITATION].', 'math-0607050-2-60-0': 'Now we complete the proof of the main result of this paper.', 'math-0607050-2-61-0': 'Proof of the main result: Theorem [REF] in [REF]', 'math-0607050-2-62-0': 'It remains only to prove the almost sure asymptotic completeness of [MATH].', 'math-0607050-2-62-1': 'We fix a [MATH].', 'math-0607050-2-62-2': 'Consider a solution [EQUATION] of ([REF]).', 'math-0607050-2-62-3': 'For any [MATH] by Lemma [REF] we can find a solution of ([REF]) [MATH], lying on [MATH] such that [EQUATION].', 'math-0607050-2-62-4': 'Then [MATH] depends on [MATH].', 'math-0607050-2-62-5': 'Write [EQUATION] and [EQUATION]', 'math-0607050-2-62-6': 'By the construction of [MATH] [EQUATION]', 'math-0607050-2-62-7': 'It is easy to see that [MATH] is a finite tempered random variable and [MATH] almost surely.', 'math-0607050-2-62-8': 'And since [MATH], by the cone invariance [EQUATION].', 'math-0607050-2-63-0': 'Let [MATH].', 'math-0607050-2-63-1': 'Notice that [EQUATION]', 'math-0607050-2-63-2': 'Then [MATH] is a random bounded set in finite dimensional space, that is for almost all [MATH], [MATH] is a bounded set in [MATH] and the bound may not be uniform to [MATH].', 'math-0607050-2-63-3': 'But for almost all [MATH] we can pick out a sequence [MATH] such that [EQUATION].', 'math-0607050-2-63-4': 'Moreover [MATH] is measurable with respect to [MATH].', 'math-0607050-2-63-5': 'Define [MATH] is large.', 'math-0607050-2-63-6': 'So if system ([REF])-([REF]) has a stationary solution, so does system ([REF]).', 'math-0607050-2-63-7': 'And the stability property of the stationary solution is also determined by that of ([REF])-([REF]).'}

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[]

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{'1': 'http://arxiv.org/licenses/assumed-1991-2003/', '2': 'http://arxiv.org/licenses/assumed-1991-2003/'}

https://arxiv.org/abs/math/0607050

hep-ph-0604013

{'hep-ph-0604013-1-0-0': 'The chiral quark model combined with unitarity and impact parameter picture provides simple semiclassical mechanism for generation of the single-spin asymmetry [MATH] in the [MATH]-production in the polarized proton collisions at RHIC.', 'hep-ph-0604013-1-0-1': 'We derive its linear [MATH]-dependence in polarized proton fragmentation region along with the energy and transverse momentum independence at large [MATH] values.', 'hep-ph-0604013-1-1-0': '# Introduction', 'hep-ph-0604013-1-2-0': 'Single-spin asymmetry (SSA) is a sensitive tool to probe QCD at small and large distances.', 'hep-ph-0604013-1-2-1': 'Experimentally significant SSA was observed in various processes of elastic scattering and inclusive hadron production.', 'hep-ph-0604013-1-2-2': 'Study of transverse single-spin asymmetries in deep-inelastic processes (DIS) observed a significant progress during last years; it has been shown that asymmetry can be related to an account of the final-state interactions due to gluon exchange [CITATION] - coherent effect not suppressed in the Bjorken limit.', 'hep-ph-0604013-1-2-3': 'Another step in this direction is the nonperturbative, instanton-induced mechanism of SSA generation [CITATION].', 'hep-ph-0604013-1-3-0': 'The processes of hadron-hadron interactions are more complicated and origin of SSA in these reactions is not clarified despite a lot of theoretical efforts devoted to this problem.', 'hep-ph-0604013-1-3-1': 'The most recent and widespread approaches in this field are based on the assumed extended factorization in QCD with account for the internal parton transverse momenta in structure functions [CITATION] or in the fragmentation function [CITATION].', 'hep-ph-0604013-1-3-2': 'An account for the direct higher twists contributions to parton scattering subprocesses also leads to a nonzero asymmetry [CITATION].', 'hep-ph-0604013-1-4-0': 'A unified picture of SSA generation in Drell-Yan processes which combines Sivers mechanism with account for the higher twists contributions was proposed recently in [CITATION].', 'hep-ph-0604013-1-5-0': 'Combination of perturbative quark-quark scattering due to gluon exchange with pion emission by quark in the instanton vertex was proposed as driving mechanism for SSA in [CITATION].', 'hep-ph-0604013-1-5-1': 'It leads to a rather significant magnitude of [MATH] at the quark level at moderate [MATH] and to its decrease at higher transverse momenta as well as the approaches mentioned above do.', 'hep-ph-0604013-1-6-0': 'Decreasing dependence of SSA with [MATH] has not yet been observed experimentally.', 'hep-ph-0604013-1-6-1': 'Another important point regarding unpolarized inclusive cross-section of [MATH]-production was discussed in [CITATION]: it is unclear are the above approaches able to describe unpolarized inclusive cross-section dependence on transverse momenta.', 'hep-ph-0604013-1-6-2': 'It has also been shown in the above paper that the description of the inclusive cross-section for [MATH]-production, at the energies lower than the RHIC energies meets difficulties in the framework of the perturbative QCD.', 'hep-ph-0604013-1-6-3': 'Recent analysis of pQCD scaling of inclusive cross-section at large [MATH] and its experimental status was given in [CITATION] where the role of higher twist effects was treated.', 'hep-ph-0604013-1-6-4': 'Deviation from the pQCD scaling is mostly noticeable in the forward region where the most significant asymmetry in the [MATH] production in [MATH] has also been observed by the STAR collaboration at RHIC [CITATION] at [MATH] GeV (in the fragmentation region of the polarized proton).', 'hep-ph-0604013-1-6-5': 'At the same time [MATH] in the neutral pion production in the backward and midrapidity regions [CITATION].', 'hep-ph-0604013-1-6-6': 'SSA has also a zero value in the [MATH], while [MATH] in the [MATH] [CITATION] in the polarized proton fragmentation region.', 'hep-ph-0604013-1-7-0': 'In this note we show that the gross features of SSA measurements at RHIC can be explained and qualitatively described in the framework of the simple semiclassical mechanism based on the further development of the specific chiral quark model [CITATION] and results of its adaptation for the treatment of the polarized and unpolarized inclusive cross-sections [CITATION].', 'hep-ph-0604013-1-7-1': 'The new data of STAR collaboration [CITATION] for the unpolarized inclusive cross-section can simultaneously be described.', 'hep-ph-0604013-1-8-0': '# Semiclassical mechanism of SSA generation', 'hep-ph-0604013-1-9-0': 'It might happen that the SSA originates from the nonperturbative sector of QCD and is related to the mechanism of spontaneous chiral symmetry breaking ([MATH]SB) in QCD [CITATION], which leads to generation of quark masses and appearance of quark condensates.', 'hep-ph-0604013-1-9-1': 'This mechanism describes transition of current into constituent quarks, which are the quasiparticles with masses comparable to a hadron mass scale.', 'hep-ph-0604013-1-9-2': 'The other well known direct result of [MATH]SB is appearance of the Goldstone bosons.', 'hep-ph-0604013-1-9-3': 'Thus constituent quarks and Goldstone bosons are the effective degrees of freedom in the chiral quark model.', 'hep-ph-0604013-1-10-0': 'Thus we consider a hadron as an extended object consisting of the valence constituent quarks located in the central core which is embedded into a quark condensate.', 'hep-ph-0604013-1-10-1': 'Collective excitations of the condensate are the Goldstone bosons and the constituent quarks interact via exchange of Goldstone bosons; this interaction is mainly due to a pion field and of the spin-flip nature [CITATION].', 'hep-ph-0604013-1-11-0': 'At the first stage of hadron interaction common effective self-consistent field appears.', 'hep-ph-0604013-1-11-1': 'This field is generated by [MATH] pairs and pions interacting with quarks.', 'hep-ph-0604013-1-11-2': 'The time of generation of the effective field [MATH] [EQUATION] where [MATH] is the total interaction time.', 'hep-ph-0604013-1-11-3': 'This assumption on the almost instantaneous generation of the effective field has some support in the very short thermalization time revealed in heavy-ion collisions at RHIC [CITATION].', 'hep-ph-0604013-1-12-0': 'Valence constituent quarks are scattered simultaneously (due to strong coupling with Goldstone bosons) and in a quasi-independent way by this effective strong field.', 'hep-ph-0604013-1-12-1': 'Such ideas were used in the model [CITATION] which has been applied to description of elastic scattering and hadron production [CITATION].', 'hep-ph-0604013-1-13-0': 'In the initial state of the reaction [MATH] the proton is polarized and can be represented in the simple SU(6) model as following: [EQUATION]', 'hep-ph-0604013-1-13-1': 'We will exploit the common feature of chiral quark models: the constituent quark [MATH] with transverse spin in up-direction can fluctuate into Goldstone boson and another constituent quark [MATH] with opposite spin direction, i. e. perform a spin-flip transition [CITATION]: [EQUATION]', 'hep-ph-0604013-1-13-2': 'The [MATH]-fluctuations of quarks do not change the quark flavor and assuming they have equal probabilities in the processes: [EQUATION] the production of [MATH] by the polarized proton [MATH] in this simple [MATH] picture can be regarded as a result of the fluctuation of the constituent quark [MATH] or [MATH]) in the effective field into the system [MATH] (Fig. 1).', 'hep-ph-0604013-1-14-0': 'The contributions to the cross-sections difference of the quarks polarized in opposite directions compensate each other (as it will be clear in what follows), and it is not the case for the [MATH]-production in the unpolarized case.', 'hep-ph-0604013-1-14-1': 'Therefore the asymmetry [MATH] should obey the inequality [MATH].', 'hep-ph-0604013-1-15-0': 'To compensate quark spin flip [MATH], an orbital angular momentum [MATH] should be generated in the final state of reaction ([REF]).', 'hep-ph-0604013-1-15-1': 'The presence of this orbital momentum [MATH] in its turn means shift in the impact parameter value of the Goldstone boson [MATH]: [EQUATION].', 'hep-ph-0604013-1-15-2': 'Due to different strengths of interaction at the different impact distances, i.e. [EQUATION] the processes of transition [MATH] and [MATH] to [MATH] will have different probabilities which leads eventually to nonzero asymmetry [MATH].', 'hep-ph-0604013-1-15-3': 'Eqs. ([REF]) clarify mechanism of the SSA generation: when shift in impact parameter is [MATH] the interaction is stronger than when the shift is [MATH], and the asymmetry [MATH] is positive.', 'hep-ph-0604013-1-15-4': 'It is important to note here that the shift of [MATH] (the impact parameter of final pion) is equivalent to the shift of the impact parameter of the initial proton according to the relation between impact parameters in the multiparticle production[CITATION]: [EQUATION]', 'hep-ph-0604013-1-15-5': 'The variable [MATH] is conjugated to the transverse momentum of [MATH], but relations between functions depending on the impact parameters [MATH], which will be used further for the calculation of asymmetry, are nonlinear and therefore we are using the semiclassical correspondence between small and large values of transverse momentum and impact parameter: [EQUATION]', 'hep-ph-0604013-1-15-6': 'We consider production of [MATH] in the fragmentation region, i.e. at large [MATH] and therefore use the approximate relation [EQUATION] which results from Eq. ([REF]) with an additional assumption on the small values of Feynman [MATH] for other particles.', 'hep-ph-0604013-1-15-7': 'In the symmetrical case of [MATH]-interactions the model assumes equal mean multiplicities in the forward and backward hemispheres and small momentum transfer between two sides.', 'hep-ph-0604013-1-15-8': 'In that sense the model is similar the approach of Chou and Yang [CITATION].', 'hep-ph-0604013-1-16-0': 'We apply chiral quark semiclassical mechanism which takes into account unitarity in the direct channel to obtain qualitative conclusions on asymmetry dependence on the kinematical variables.', 'hep-ph-0604013-1-17-0': '# Asymmetry and inclusive cross-section', 'hep-ph-0604013-1-18-0': 'The main feature of the mechanism is an account of unitarity in the direct channel of reaction.', 'hep-ph-0604013-1-18-1': 'The corresponding formulas for inclusive cross-sections of the process [EQUATION] where hadron [MATH] in this particular case is [MATH] meson and [MATH], [MATH] are protons, were obtained in [CITATION] and have the following form [EQUATION] where [MATH] is the impact parameter of the initial protons.', 'hep-ph-0604013-1-18-2': 'Here the function [MATH] is the generalized reaction matrix (averaged over initial spin states) which is determined by the basic dynamics of elastic scattering.', 'hep-ph-0604013-1-18-3': 'The elastic scattering amplitude in the impact parameter representation [MATH] is then given [CITATION] by the relation: [EQUATION]', 'hep-ph-0604013-1-18-4': 'This equation allows one to obey unitarity provided inequality [MATH] is fulfilled.', 'hep-ph-0604013-1-18-5': 'The functions [MATH] in Eq. ([REF]) are related to the functions [MATH] - the multiparticle analogs of the function [MATH] [CITATION] in the polarized case.', 'hep-ph-0604013-1-18-6': 'The kinematical variables [MATH] and [MATH] for example) describe the state of the produced particle [MATH].', 'hep-ph-0604013-1-18-7': 'Arrows [MATH] and [MATH] denote transverse spin directions of the polarized proton [MATH].', 'hep-ph-0604013-1-19-0': 'Asymmetry [MATH] can be expressed in terms of the functions [MATH], [MATH] and [MATH]: [EQUATION] where [MATH] and [MATH] and [MATH] obey the sum rule [EQUATION] here [MATH] stands for the mean multiplicity in the impact parameter representation.', 'hep-ph-0604013-1-20-0': 'On the basis of the described mechanism we can postulate that the functions [MATH] and [MATH] are related to the functions [MATH] and [MATH], respectively, i.e. [EQUATION]', 'hep-ph-0604013-1-20-1': 'We can connect [MATH] with the radius of quark interaction [MATH] responsible for the transition changing quark spin: [EQUATION].', 'hep-ph-0604013-1-21-0': 'Using the above relations and, in particular, ([REF]), we can write the following expression for asymmetry [MATH] [EQUATION] where [MATH].', 'hep-ph-0604013-1-21-1': 'In ([REF]) we made replacement according to relation ([REF]): [EQUATION].', 'hep-ph-0604013-1-21-2': 'It is clear that [MATH] ([REF]) should be positive because [MATH].', 'hep-ph-0604013-1-22-0': 'The function [MATH] is chosen as a product of the averaged quark amplitudes in accordance with the quasi-independence of valence constituent quark scattering in the self-consistent mean field [CITATION].', 'hep-ph-0604013-1-22-1': 'The generalized reaction matrix [MATH] (in a pure imaginary case, which we consider here for simplicity) is the following [EQUATION] where the function [MATH] power like increases at large values of [MATH] [EQUATION] [MATH] is the total mass of [MATH] constituent quarks with mass [MATH] in the initial hadrons and parameter [MATH] determines a universal scale for the quark interaction radius in the model, i.e. [MATH].', 'hep-ph-0604013-1-23-0': 'To evaluate asymmetry dependence on [MATH] and [MATH] we use semiclassical correspondence between transverse momentum and impact parameter values, ([REF]).', 'hep-ph-0604013-1-23-1': 'Performing integration by parts and choosing the region of small [MATH] we select the large values of impact parameter and we obtain [EQUATION] where [MATH] is the hadron interaction radius, which serve as a scale separating large and small impact parameter regions.', 'hep-ph-0604013-1-23-2': 'In the large impact parameter region: [MATH] for [MATH] and therefore we have a small dynamically suppressed asymmetry [MATH] in the region of small and moderate values of [MATH], i.e. [MATH].', 'hep-ph-0604013-1-24-0': 'But at small values of [MATH] the value of [MATH]-matrix is large, [MATH], and we can neglect unity in the denominators of the integrands.', 'hep-ph-0604013-1-25-0': 'Thus the ratio of two integrals (after integration by parts of nominator in Eq. ([REF])) is of order of unity, i.e. the energy and [MATH]-independent behavior of asymmetry [MATH] takes place at the values of transverse momentum [MATH]: [EQUATION]', 'hep-ph-0604013-1-25-1': 'This flat transverse momentum dependence of asymmetry results from the similar rescattering effects for the different spin states, i.e. spin-flip and spin-nonflip interactions undergo similar absorption at short distances and the relative magnitude of this absorption does not depend on energy.', 'hep-ph-0604013-1-25-2': 'It is one of the manifestations of the unitarity.', 'hep-ph-0604013-1-25-3': 'The numeric value of polarization [MATH] can be significant; there are no small factors in ([REF]).', 'hep-ph-0604013-1-25-4': 'In Eq. ([REF]) [MATH] is proportional to two nucleon masses, the value of parameter [MATH].', 'hep-ph-0604013-1-25-5': 'We expect that [MATH] fm on the basis of the model estimate [CITATION].', 'hep-ph-0604013-1-25-6': 'The above qualitative features of asymmetry dependence on [MATH], [MATH] and energy are in a good agreement with the experimentally observed trends [CITATION].', 'hep-ph-0604013-1-25-7': 'For example, Fig. 2 demonstrates that the linear [MATH] dependence is in a good agreement with the experimental data of STAR Collaboration at RHIC [CITATION] in the fragmentation region ([MATH]) where the model should be applicable.', 'hep-ph-0604013-1-25-8': 'Of course, the conclusion on the [MATH]-independence of polarization is a qualitative one and small deviations from such behavior cannot be excluded.', 'hep-ph-0604013-1-26-0': 'Similar mechanism should generate SSA in the production of charged pions.', 'hep-ph-0604013-1-26-1': 'The relevant process for [MATH]-production in polarized [MATH] interactions [EQUATION] leads to a negative shift in the impact parameter and consequently to the positive asymmetry [MATH], while the corresponding process for the [MATH]-production [EQUATION] leads to the positive shift in impact parameter and, respectively, to the negative asymmetry [MATH].', 'hep-ph-0604013-1-26-2': 'Asymmetry [MATH] in the [MATH]-production in the fragmentation region of polarized proton should have linear [MATH]-dependence at [MATH] and flat [MATH] dependence at large [MATH].', 'hep-ph-0604013-1-26-3': 'Those dependencies are similar to the ones depicted on Fig. 2 for [MATH]-production.', 'hep-ph-0604013-1-27-0': 'The reversed mechanism (chiral quark spin filtering) was used for the explanation of the hyperon polarization [CITATION].', 'hep-ph-0604013-1-27-1': 'Note that polarization of [MATH] - hyperon has the same generic dependence on [MATH] and [MATH].', 'hep-ph-0604013-1-28-0': 'To demonstrate the model self-consistency it should be noted that it is able to describe the unpolarized cross-section of [MATH]-production also.', 'hep-ph-0604013-1-28-1': 'In the fragmentation region it was shown [CITATION] that at small [MATH] the poles in impact parameter plane at [MATH] lead to the exponential [MATH]-dependence of inclusive cross-section.', 'hep-ph-0604013-1-29-0': 'At high [MATH] the power-like dependence [MATH] with [MATH] should take place.', 'hep-ph-0604013-1-29-1': 'It originates from the singularity at zero impact parameter [MATH].', 'hep-ph-0604013-1-29-2': 'The exponent [MATH] does not depend on [MATH].', 'hep-ph-0604013-1-29-3': 'Experimental data are in a good agreement with the [MATH]-dependence of the unpolarized inclusive cross-section (Fig. 3).', 'hep-ph-0604013-1-30-0': 'Thus, in the approach with effective degrees of freedom - constituent quarks and Goldstone bosons - differential cross-section at high transverse momenta has a generic power-like dependencies on [MATH] at large transverse momenta.', 'hep-ph-0604013-1-30-1': 'The differential cross-section and the asymmetry [MATH] are in agreement with the experimental data at highest available energy at RHIC.', 'hep-ph-0604013-1-31-0': '# Conclusion', 'hep-ph-0604013-1-32-0': 'The proposed semiclassical mechanism of SSA generation considers the effective degrees of freedom and takes into account collective aspects of QCD dynamics.', 'hep-ph-0604013-1-32-1': 'Together with unitarity, which is an essential part of this approach, it leads to linear dependence on [MATH] and flat dependence of SSA on transverse momentum at large [MATH] in the polarized proton fragmentation region.', 'hep-ph-0604013-1-32-2': 'Such dependencies with the energy independent behavior of asymmetry at large transverse momenta are the direct phenomenological consequences of the proposed mechanism.', 'hep-ph-0604013-1-33-0': 'The chiral quark fluctuation mechanism in effective field with spin flip is relatively suppressed when compared to direct elastic scattering of quarks in effective field and therefore does not play a role in the reaction [MATH] in the fragmentation region, but it should not be suppressed in [MATH].', 'hep-ph-0604013-1-33-1': 'These features really take place in the experimental data set: asymmetry [MATH] is zero for proton production and deviates from zero for neutron production in the forward region.', 'hep-ph-0604013-1-34-0': 'We discussed here particle production in the fragmentation region of polarized proton.', 'hep-ph-0604013-1-34-1': 'In the symmetrical case of [MATH]-interactions the model leads to the equal mean multiplicities in the forward and backward hemispheres and very small momentum transfer is expected between the two sides.', 'hep-ph-0604013-1-34-2': 'In the central and backward regions where correlations between impact parameter of the initial and impact parameters of the final particles are weak or completely degraded, the asymmetry cannot be generated due to this chiral quark semiclassical mechanism.', 'hep-ph-0604013-1-34-3': 'The vanishing asymmetries in the central and backward regions observed experimentally provide the indirect evidences in favor of this mechanism.'}

{'hep-ph-0604013-2-0-0': 'The chiral quark model combined with unitarity and impact parameter picture provides simple semiclassical mechanism for generation of the single-spin asymmetry [MATH] in the [MATH]-production in the polarized proton collisions at RHIC.', 'hep-ph-0604013-2-0-1': 'We derive its linear [MATH]-dependence in polarized proton fragmentation region along with the energy and transverse momentum independence at large [MATH] values.', 'hep-ph-0604013-2-1-0': '# Introduction', 'hep-ph-0604013-2-2-0': 'Single-spin asymmetry (SSA) is a sensitive tool to probe QCD at small and large distances.', 'hep-ph-0604013-2-2-1': 'Experimentally significant SSA was observed in various processes of elastic scattering and inclusive hadron production.', 'hep-ph-0604013-2-2-2': 'Study of transverse single-spin asymmetries in deep-inelastic processes (DIS) observed a significant progress during last years; it has been shown that asymmetry can be related to a rescattering in the final-state interactions due to gluon exchange [CITATION] - coherent effect not suppressed in the Bjorken limit.', 'hep-ph-0604013-2-2-3': 'Another step in this direction is the nonperturbative, instanton-induced mechanism of SSA generation [CITATION].', 'hep-ph-0604013-2-2-4': 'Instantons lead to appearance of the quark anomalous magnetic moment [CITATION] and soft rescattering can provide a source for a leading twist SSA in semi-inclusive DIS [CITATION].', 'hep-ph-0604013-2-2-5': 'Moreover, as it was noted there it would also affect integrated parton distributions and may jeopardize QCD factorization.', 'hep-ph-0604013-2-2-6': 'Pauli coupling have an important consequences opening possibility of transversity studies in DIS processes [CITATION].', 'hep-ph-0604013-2-2-7': 'A unified picture of SSA generation in Drell-Yan processes which combines Sivers mechanism with account for the higher twists contributions was proposed recently in [CITATION].', 'hep-ph-0604013-2-2-8': 'Thus the significant role of non-perturbative effects in the mechanism of SSA generation is becoming more and more evident nowadays.', 'hep-ph-0604013-2-3-0': 'The processes of hadron-hadron interactions are even more complicated than DIS and origin of SSA in these reactions is not fully clarified.', 'hep-ph-0604013-2-3-1': 'Despite a great progress in theoretical studies devoted to this problem, the phenomenological success is rather limited; a comprehensive approach able to get description of the existing set of the experimental data on polarization, asymmetries, spin correlation parameters and cross-sections is still absent at the moment.', 'hep-ph-0604013-2-4-0': 'The most widespread approaches in the field of hadronic processes are based on the assumed extended factorization in QCD with account for the internal parton transverse momenta in structure functions [CITATION] or in the fragmentation function [CITATION].', 'hep-ph-0604013-2-4-1': 'An account for the direct higher twists contributions to parton scattering subprocesses also can lead to a nonzero asymmetry [CITATION].', 'hep-ph-0604013-2-5-0': 'However, as it was shown recently the Collins frgmentation mechanism is suppressed in hadronic pion production [CITATION].', 'hep-ph-0604013-2-5-1': 'It was shown in this work that the Sivers effect gives dominating contribution to asymmetry and the second relevant contribution gives a convolution of transversity with Collins fragmentation while other contributions to asymmetry can be neglected.', 'hep-ph-0604013-2-5-2': 'It can lead to a significant contribution to [MATH] for the quark Sivers mechanism at moderate [MATH] and to its decrease at higher transverse momenta.', 'hep-ph-0604013-2-5-3': 'Note, that the asymmetry is predicted to be non-zero at [MATH] for the both quark and gluon Sivers mechanisms and the same is valid for the asymmetry [MATH] for the gluon Sivers mechanism at [MATH].', 'hep-ph-0604013-2-6-0': 'Decreasing dependence of SSA with [MATH] has not yet been observed experimentally, most experimental data are consistent with a flat transverse momentum dependence at [MATH] GeV/c.', 'hep-ph-0604013-2-6-1': 'Another important point regarding unpolarized inclusive cross-section of [MATH]-production was discussed in [CITATION]: it is unclear are the above approaches able to describe unpolarized inclusive cross-section dependence on transverse momenta.', 'hep-ph-0604013-2-6-2': 'It has also been shown in the above paper that the description of the inclusive cross-section for [MATH]-production, at the energies lower than the RHIC energies meets difficulties in the framework of the perturbative QCD.', 'hep-ph-0604013-2-6-3': 'Role of higher twist contributions was studied in the recent analysis of pQCD scaling of inclusive cross-section at large [MATH] and its experimental status was given in [CITATION].', 'hep-ph-0604013-2-6-4': 'Deviation from the pQCD scaling is mostly noticeable in the forward region where the most significant asymmetry in the [MATH] production in [MATH] has also been observed by the STAR collaboration at RHIC [CITATION] at [MATH] GeV (in the fragmentation region of the polarized proton).', 'hep-ph-0604013-2-6-5': 'At the same time [MATH] in the neutral pion production in the backward and midrapidity regions [CITATION].', 'hep-ph-0604013-2-6-6': 'SSA has also a zero value in the [MATH], while [MATH] in the [MATH] [CITATION] in the polarized proton fragmentation region.', 'hep-ph-0604013-2-6-7': 'The experimental features observed at RHIC represent a difficulty for the explanation in the above mentioned theoretical approaches, in particular, those based on Sivers mechanism (is at variance with zero asymmetry at [MATH]) or account for the anomalous magnetic moment of quarks (is at variance with zero asymmetry in the process [MATH]).', 'hep-ph-0604013-2-6-8': 'Of course, more experimental data are needed to perform a conclusive test of various theoretical predictions and those predictions should be more specified and elaborated for the observables at the hadronic level.', 'hep-ph-0604013-2-7-0': 'Keeping in mind the experimental and theoretical situation in the field of SSA studies we show in this note that the gross features of SSA measurements at RHIC and FNAL (linear increase of asymmetry with [MATH] and flat transverse momentum dependence at [MATH] GeV/c) can be explained and qualitatively described in the framework of the simple semiclassical mechanism based on the further development of the specific chiral quark model [CITATION] and results of its adaptation for the treatment of the polarized and unpolarized inclusive cross-sections [CITATION].', 'hep-ph-0604013-2-7-1': 'The data of STAR collaboration [CITATION] for the unpolarized inclusive cross-section can simultaneously be described.', 'hep-ph-0604013-2-7-2': 'This mechanism is consistent with other new experimental facts found at RHIC.', 'hep-ph-0604013-2-8-0': '# Semiclassical mechanism of SSA generation', 'hep-ph-0604013-2-9-0': 'It might happen that the SSA originates from the nonperturbative sector of QCD and is related to the mechanism of spontaneous chiral symmetry breaking ([MATH]SB) in QCD [CITATION], which leads to generation of quark masses and appearance of quark condensates.', 'hep-ph-0604013-2-9-1': 'This mechanism describes transition of current into constituent quarks, which are the quasiparticles with masses comparable to a hadron mass scale.', 'hep-ph-0604013-2-9-2': 'The other well known direct result of [MATH]SB is appearance of the Goldstone bosons.', 'hep-ph-0604013-2-9-3': 'Thus constituent quarks and Goldstone bosons are the effective degrees of freedom in the chiral quark model.', 'hep-ph-0604013-2-10-0': 'Thus we consider a hadron as an extended object consisting of the valence constituent quarks located in the central core which is embedded into a quark condensate.', 'hep-ph-0604013-2-10-1': 'Collective excitations of the condensate are the Goldstone bosons and the constituent quarks interact via exchange of Goldstone bosons; this interaction is mainly due to a pion field and of the spin-flip nature [CITATION].', 'hep-ph-0604013-2-11-0': 'At the first stage of hadron interaction common effective self-consistent field appears.', 'hep-ph-0604013-2-11-1': 'This field is generated by [MATH] pairs and pions interacting with quarks.', 'hep-ph-0604013-2-11-2': 'The time of generation of the effective field [MATH] [EQUATION] where [MATH] is the total interaction time.', 'hep-ph-0604013-2-11-3': 'This assumption on the almost instantaneous generation of the effective field has some support in the very short thermalization time revealed in heavy-ion collisions at RHIC [CITATION].', 'hep-ph-0604013-2-12-0': 'Valence constituent quarks are scattered simultaneously (due to strong coupling with Goldstone bosons) and in a quasi-independent way by this effective strong field.', 'hep-ph-0604013-2-12-1': 'Such ideas were used in the model [CITATION] which has been applied to description of elastic scattering and hadron production [CITATION].', 'hep-ph-0604013-2-13-0': 'In the initial state of the reaction [MATH] the proton is polarized and can be represented in the simple SU(6) model as following: [EQUATION]', 'hep-ph-0604013-2-13-1': 'We will exploit the common feature of chiral quark models: the constituent quark [MATH] with transverse spin in up-direction can fluctuate into Goldstone boson and another constituent quark [MATH] with opposite spin direction, i. e. perform a spin-flip transition [CITATION]: [EQUATION]', 'hep-ph-0604013-2-13-2': 'The [MATH]-fluctuations of quarks do not change the quark flavor and assuming they have equal probabilities in the processes: [EQUATION] the production of [MATH] by the polarized proton [MATH] in this simple [MATH] picture can be regarded as a result of the fluctuation of the constituent quark [MATH] or [MATH]) in the effective field into the system [MATH] (Fig. 1).', 'hep-ph-0604013-2-14-0': 'The contributions to the cross-sections difference of the quarks polarized in opposite directions compensate each other (as it will be clear in what follows), and it is not the case for the [MATH]-production in the unpolarized case.', 'hep-ph-0604013-2-14-1': 'Therefore the asymmetry [MATH] should obey the inequality [MATH].', 'hep-ph-0604013-2-15-0': 'To compensate quark spin flip [MATH], an orbital angular momentum [MATH] should be generated in the final state of reaction ([REF]).', 'hep-ph-0604013-2-15-1': 'The presence of this orbital momentum [MATH] in its turn means shift in the impact parameter value of the Goldstone boson [MATH]: [EQUATION].', 'hep-ph-0604013-2-15-2': 'Due to different strengths of interaction at the different impact distances, i.e. [EQUATION] the processes of transition [MATH] and [MATH] to [MATH] will have different probabilities which leads eventually to nonzero asymmetry [MATH].', 'hep-ph-0604013-2-15-3': 'Eqs. ([REF]) clarify mechanism of the SSA generation: when shift in impact parameter is [MATH] the interaction is stronger than when the shift is [MATH], and the asymmetry [MATH] is positive.', 'hep-ph-0604013-2-15-4': 'It is important to note here that the shift of [MATH] (the impact parameter of final pion) is equivalent to the shift of the impact parameter of the initial proton according to the relation between impact parameters in the multiparticle production[CITATION]: [EQUATION]', 'hep-ph-0604013-2-15-5': 'The variable [MATH] is conjugated to the transverse momentum of [MATH], but relations between functions depending on the impact parameters [MATH], which will be used further for the calculation of asymmetry, are nonlinear and therefore we are using the semiclassical correspondence between small and large values of transverse momentum and impact parameter: [EQUATION]', 'hep-ph-0604013-2-15-6': 'We consider production of [MATH] in the fragmentation region, i.e. at large [MATH] and therefore use the approximate relation [EQUATION] which results from Eq. ([REF]) with an additional assumption on the small values of Feynman [MATH] for other particles.', 'hep-ph-0604013-2-15-7': 'In the symmetrical case of [MATH]-interactions the model assumes equal mean multiplicities in the forward and backward hemispheres and small momentum transfer between two sides.', 'hep-ph-0604013-2-15-8': 'In that sense the model is similar the approach of Chou and Yang [CITATION].', 'hep-ph-0604013-2-16-0': 'We apply chiral quark semiclassical mechanism which takes into account unitarity in the direct channel to obtain qualitative conclusions on asymmetry dependence on the kinematical variables.', 'hep-ph-0604013-2-17-0': '# Asymmetry and inclusive cross-section', 'hep-ph-0604013-2-18-0': 'The main feature of the mechanism is an account of unitarity in the direct channel of reaction.', 'hep-ph-0604013-2-18-1': 'The corresponding formulas for inclusive cross-sections of the process [EQUATION] where hadron [MATH] in this particular case is [MATH] meson and [MATH], [MATH] are protons, were obtained in [CITATION] and have the following form [EQUATION] where [MATH] is the impact parameter of the initial protons.', 'hep-ph-0604013-2-18-2': 'Here the function [MATH] is the generalized reaction matrix (averaged over initial spin states) which is determined by the basic dynamics of elastic scattering.', 'hep-ph-0604013-2-18-3': 'The elastic scattering amplitude in the impact parameter representation [MATH] is then given [CITATION] by the relation: [EQUATION]', 'hep-ph-0604013-2-18-4': 'This equation allows one to obey unitarity provided inequality [MATH] is fulfilled.', 'hep-ph-0604013-2-18-5': 'The functions [MATH] in Eq. ([REF]) are related to the functions [MATH] - the multiparticle analogs of the function [MATH] [CITATION] in the polarized case.', 'hep-ph-0604013-2-18-6': 'The kinematical variables [MATH] and [MATH] for example) describe the state of the produced particle [MATH].', 'hep-ph-0604013-2-18-7': 'Arrows [MATH] and [MATH] denote transverse spin directions of the polarized proton [MATH].', 'hep-ph-0604013-2-19-0': 'Asymmetry [MATH] can be expressed in terms of the functions [MATH], [MATH] and [MATH]: [EQUATION] where [MATH] and [MATH] and [MATH] obey the sum rule [EQUATION] here [MATH] stands for the mean multiplicity in the impact parameter representation.', 'hep-ph-0604013-2-20-0': 'On the basis of the described mechanism we can postulate that the functions [MATH] and [MATH] are related to the functions [MATH] and [MATH], respectively, i.e. [EQUATION]', 'hep-ph-0604013-2-20-1': 'We can connect [MATH] with the radius of quark interaction [MATH] responsible for the transition changing quark spin: [EQUATION].', 'hep-ph-0604013-2-21-0': 'Using the above relations and, in particular, ([REF]), we can write the following expression for asymmetry [MATH] [EQUATION] where [MATH].', 'hep-ph-0604013-2-21-1': 'In ([REF]) we made replacement according to relation ([REF]): [EQUATION].', 'hep-ph-0604013-2-21-2': 'It is clear that [MATH] ([REF]) should be positive because [MATH].', 'hep-ph-0604013-2-22-0': 'The function [MATH] is chosen as a product of the averaged quark amplitudes in accordance with the quasi-independence of valence constituent quark scattering in the self-consistent mean field [CITATION].', 'hep-ph-0604013-2-22-1': 'The generalized reaction matrix [MATH] (in a pure imaginary case, which we consider here for simplicity) is the following [EQUATION] where the function [MATH] power like increases at large values of [MATH] [EQUATION] [MATH] is the total mass of [MATH] constituent quarks with mass [MATH] in the initial hadrons and parameter [MATH] determines a universal scale for the quark interaction radius in the model, i.e. [MATH].', 'hep-ph-0604013-2-23-0': 'To evaluate asymmetry dependence on [MATH] and [MATH] we use semiclassical correspondence between transverse momentum and impact parameter values, ([REF]).', 'hep-ph-0604013-2-23-1': 'Performing integration by parts and choosing the region of small [MATH] we select the large values of impact parameter and we obtain [EQUATION] where [MATH] is the hadron interaction radius, which serve as a scale separating large and small impact parameter regions.', 'hep-ph-0604013-2-23-2': 'In the large impact parameter region: [MATH] for [MATH] and therefore we have a small dynamically suppressed asymmetry [MATH] in the region of small and moderate values of [MATH], i.e. [MATH].', 'hep-ph-0604013-2-24-0': 'But at small values of [MATH] the value of [MATH]-matrix is large, [MATH], and we can neglect unity in the denominators of the integrands.', 'hep-ph-0604013-2-25-0': 'Thus the ratio of two integrals (after integration by parts of nominator in Eq. ([REF])) is of order of unity, i.e. the energy and [MATH]-independent behavior of asymmetry [MATH] takes place at the values of transverse momentum [MATH]: [EQUATION]', 'hep-ph-0604013-2-25-1': 'This flat transverse momentum dependence of asymmetry results from the similar rescattering effects for the different spin states, i.e. spin-flip and spin-nonflip interactions undergo similar absorption at short distances and the relative magnitude of this absorption does not depend on energy.', 'hep-ph-0604013-2-25-2': 'It is one of the manifestations of the unitarity.', 'hep-ph-0604013-2-25-3': 'The numeric value of polarization [MATH] can be significant; there are no small factors in ([REF]).', 'hep-ph-0604013-2-25-4': 'In Eq. ([REF]) [MATH] is proportional to two nucleon masses, the value of parameter [MATH].', 'hep-ph-0604013-2-25-5': 'We expect that [MATH] fm on the basis of the model estimate [CITATION].', 'hep-ph-0604013-2-25-6': 'The above qualitative features of asymmetry dependence on [MATH], [MATH] and energy are in a good agreement with the experimentally observed trends [CITATION].', 'hep-ph-0604013-2-25-7': 'For example, Fig. 2 demonstrates that the linear [MATH] dependence is in a good agreement with the experimental data of STAR Collaboration at RHIC [CITATION] in the fragmentation region ([MATH]) where the model should be applicable.', 'hep-ph-0604013-2-25-8': 'Of course, the conclusion on the [MATH]-independence of polarization is a qualitative one and small deviations from such behavior cannot be excluded.', 'hep-ph-0604013-2-25-9': 'The same dependencies are compared with the FNAL E704 data [CITATION] (Fig.3).', 'hep-ph-0604013-2-25-10': 'Those dependencies as it is clear from their derivations are valid in high-energy approximation and therefore have been compared with FNAL and RHIC data only.', 'hep-ph-0604013-2-25-11': 'However, they are in qualitative agreement with the lower energy data also [CITATION].', 'hep-ph-0604013-2-26-0': 'Similar mechanism should generate SSA in the production of charged pions.', 'hep-ph-0604013-2-26-1': 'The relevant process for [MATH]-production in polarized [MATH] interactions [EQUATION] leads to a negative shift in the impact parameter and consequently to the positive asymmetry [MATH], while the corresponding process for the [MATH]-production [EQUATION] leads to the positive shift in impact parameter and, respectively, to the negative asymmetry [MATH].', 'hep-ph-0604013-2-26-2': 'Asymmetry [MATH] in the [MATH]-production in the fragmentation region of polarized proton should have linear [MATH]-dependence at [MATH] and flat [MATH] dependence at large [MATH].', 'hep-ph-0604013-2-26-3': 'Those dependencies are similar to the ones depicted on Fig. 2 for [MATH]-production.', 'hep-ph-0604013-2-27-0': 'The reversed mechanism (chiral quark spin filtering) was used for the explanation of the hyperon polarization [CITATION].', 'hep-ph-0604013-2-27-1': 'Note that polarization of [MATH] - hyperon has the same generic dependence on [MATH] and [MATH].', 'hep-ph-0604013-2-28-0': 'To demonstrate the model self-consistency it should be noted that it is able to describe the unpolarized cross-section of [MATH]-production also.', 'hep-ph-0604013-2-28-1': 'In the fragmentation region it was shown [CITATION] that at small [MATH] the poles in impact parameter plane at [MATH] lead to the exponential [MATH]-dependence of inclusive cross-section.', 'hep-ph-0604013-2-29-0': 'At high [MATH] the power-like dependence [MATH] with [MATH] should take place.', 'hep-ph-0604013-2-29-1': 'It originates from the singularity at zero impact parameter [MATH].', 'hep-ph-0604013-2-29-2': 'The exponent [MATH] does not depend on [MATH].', 'hep-ph-0604013-2-29-3': 'Experimental data are in a good agreement with the [MATH]-dependence of the unpolarized inclusive cross-section (Fig. 3).', 'hep-ph-0604013-2-29-4': 'Recently a similar [MATH]-dependence has been obtained for the soft contribution to quark-quark scattering induced by an anomalous chromomagnetic interaction due to instanton mechanism [CITATION].', 'hep-ph-0604013-2-30-0': 'Thus, in the approach with effective degrees of freedom - constituent quarks and Goldstone bosons - differential cross-section at high transverse momenta has a generic power-like dependencies on [MATH] at large transverse momenta.', 'hep-ph-0604013-2-30-1': 'The differential cross-section and the asymmetry [MATH] are in agreement with the experimental data at the highest available energy at RHIC and asymmetry is in agreement with FNAL data also.', 'hep-ph-0604013-2-31-0': '# Conclusion', 'hep-ph-0604013-2-32-0': 'The proposed semiclassical mechanism of SSA generation considers the effective degrees of freedom and takes into account collective aspects of QCD dynamics.', 'hep-ph-0604013-2-32-1': 'Together with unitarity, which is an essential part of this approach, it leads to linear dependence on [MATH] and flat dependence of SSA on transverse momentum at large [MATH] in the polarized proton fragmentation region.', 'hep-ph-0604013-2-32-2': 'Such dependencies with the energy independent behavior of asymmetry at large transverse momenta are the direct phenomenological consequences of the proposed mechanism.', 'hep-ph-0604013-2-33-0': 'The chiral quark fluctuation mechanism in effective field with spin flip is relatively suppressed when compared to direct elastic scattering of quarks in effective field and therefore does not play a role in the reaction [MATH] in the fragmentation region, but it should not be suppressed in [MATH].', 'hep-ph-0604013-2-33-1': 'These features really take place in the experimental data set: asymmetry [MATH] is zero for proton production and deviates from zero for neutron production in the forward region.', 'hep-ph-0604013-2-34-0': 'We discussed here particle production in the fragmentation region of polarized proton.', 'hep-ph-0604013-2-34-1': 'In the symmetrical case of [MATH]-interactions the model leads to the equal mean multiplicities in the forward and backward hemispheres and very small momentum transfer is expected between the two sides.', 'hep-ph-0604013-2-34-2': 'In the central and backward regions where correlations between impact parameter of the initial and impact parameters of the final particles are weak or completely degraded, the asymmetry cannot be generated due to this chiral quark semiclassical mechanism.', 'hep-ph-0604013-2-34-3': 'The vanishing asymmetries in the central and backward regions observed experimentally provide the indirect evidences in favor of this mechanism.'}

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'hep-ph-0604013-2-2-2'], ['hep-ph-0604013-1-6-0', 'hep-ph-0604013-2-6-0'], ['hep-ph-0604013-1-3-0', 'hep-ph-0604013-2-3-0']]

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[['hep-ph-0604013-1-7-0', 'hep-ph-0604013-2-7-0'], ['hep-ph-0604013-1-7-1', 'hep-ph-0604013-2-7-1'], ['hep-ph-0604013-1-30-1', 'hep-ph-0604013-2-30-1'], ['hep-ph-0604013-1-6-3', 'hep-ph-0604013-2-6-3'], ['hep-ph-0604013-1-3-1', 'hep-ph-0604013-2-4-0'], ['hep-ph-0604013-1-3-2', 'hep-ph-0604013-2-4-1'], ['hep-ph-0604013-1-2-2', 'hep-ph-0604013-2-2-2']]

[]

[['hep-ph-0604013-1-6-0', 'hep-ph-0604013-2-6-0'], ['hep-ph-0604013-1-3-0', 'hep-ph-0604013-2-3-0']]

[]

[]

{'1': 'http://arxiv.org/licenses/assumed-1991-2003/', '2': 'http://arxiv.org/licenses/assumed-1991-2003/'}

https://arxiv.org/abs/hep-ph/0604013

1404.1006

{'1404.1006-1-0-0': 'Most real networks are characterized by connectivity patterns that evolve in time following complex, non-Markovian, dynamics.', '1404.1006-1-0-1': 'Here we investigate the impact of this ubiquitous feature by studying the Susceptible-Infected-Recovered (SIR) and Susceptible-Infected-Susceptible (SIS) epidemic models on activity driven networks with and without memory (i.e., Markovian and non-Markovian).', '1404.1006-1-0-2': 'We show that while memory inhibits the spreading process in SIR models, where the epidemic threshold is moved to larger values, it plays the opposite effect in the case of the SIS, where the threshold is lowered.', '1404.1006-1-0-3': 'The heterogeneity in tie strengths, and the frequent repetition of connections that it entails, allows in fact less virulent SIS-like diseases to survive in tightly connected local clusters that serve as reservoir for the virus.', '1404.1006-1-0-4': 'We validate this picture by evaluating the threshold of both processes in a real temporal network.', '1404.1006-1-0-5': 'Our findings confirm the important role played by non-Markovian network dynamics on dynamical processes.', '1404.1006-1-1-0': 'Virtually any system can be represented as a network whose basic units are described as nodes and its interactions as links between them [CITATION].', '1404.1006-1-1-1': 'In general, the connections are not static, but evolve in time subject to nontrivial dynamics [CITATION].', '1404.1006-1-1-2': 'Consider for example face to face or online interactions networks where individuals talk and exchange informations through evolving contacts [CITATION].', '1404.1006-1-1-3': 'The recent advances in technology have allowed to collect, monitor and probe such interactions generating an unprecedented amount of time-resolved high resolution datasets [CITATION].', '1404.1006-1-1-4': "The analysis of such real systems exposed the limits of canonical static and annealed networks representations [CITATION] calling for the development of a new theory to model, understand and account for network's temporal properties.", '1404.1006-1-1-5': 'In particular, the recent data deluge has allowed to start identifying the effects that time varying topologies have on dynamical processes taking place in their fabric [CITATION].', '1404.1006-1-1-6': 'Prototypical examples are the spreading of memes, ideas, and infectious diseases.', '1404.1006-1-1-7': "All these phenomena can be successfully described as diffusion processes on contact networks and are affected by the ordering, concurrence, duration, and heterogeneity in nodes' activities and connectivity patterns [CITATION].", '1404.1006-1-2-0': 'One of most distinctive property of social networks is the heterogeneity of interactions strengths [CITATION].', '1404.1006-1-2-1': 'Individuals have memory of their inner circle of friends and most important connections activating some links more often then others, thus building up strong and weak ties with their peers.', '1404.1006-1-2-2': 'In other words, links creation is not a Markovian process [CITATION].', '1404.1006-1-2-3': 'While this property has been observed, studied, and modeled in details in static networks [CITATION], its understanding in the context of time-varying graphs is still far to be complete.', '1404.1006-1-2-4': 'Indeed, few studies have tackled this subject with different approaches [CITATION] uncovering a rich phenomenology.', '1404.1006-1-2-5': 'In particular, the non-Markovian link dynamics has been proven to be responsible either for slow-down/inhibit or, more surprisingly, speed-up/facilitate diffusion processes [CITATION].', '1404.1006-1-3-0': 'Here we study the effects of memory on two different classes of epidemic spreading models, namely the Susceptible-Infected-Recovered (SIR) and the Susceptible-Infected-Susceptible (SIS) models [CITATION].', '1404.1006-1-3-1': 'We consider a recently proposed class of time-varying networks called activity driven models [CITATION], based on the observation that the propensity of nodes to initiate a connection, the activity, is heterogeneously distributed.', '1404.1006-1-3-2': 'In its basic formulation node activities are modeled with accuracy but the link creation is assumed to be Markovian.', '1404.1006-1-3-3': 'While such approximation allows analytical treatments [CITATION], it does not capture many real properties of time-varying networks as the memory of individuals.', '1404.1006-1-3-4': 'Recently, this limitation has been overcome with the introduction of a non-Markovian generalization of the modeling framework based on a simple reinforcement mechanism that allows reproducing with accuracy the evolution of individuals contacts [CITATION].', '1404.1006-1-3-5': 'We study the dynamical properties of SIR and SIS models in activity driven networks with and without memory.', '1404.1006-1-3-6': 'In particular, we focus on one of the most important dynamical property of epidemic diffusion process, namely the epidemic threshold, defining the conditions necessary for the spreading of the disease to a macroscopic fraction of the population [CITATION].', '1404.1006-1-4-0': 'We find that memory acts in opposite ways on SIR and SIS models.', '1404.1006-1-4-1': 'In SIR models the epidemic threshold is shifted to larger values, making the spreading of the disease more difficult, while in SIS it moves to smaller values.', '1404.1006-1-4-2': 'Thus, non-Markovian dynamics facilitate the spreading of SIS-like diseases, like sexual transmitted illnesses, that can survive reaching an endemic state, reservoir, in tightly connected clusters.', '1404.1006-1-4-3': 'This difference between the two models is due to the fundamentally different nature the two processes that induce distinct behaviors also in the case of static networks [CITATION].', '1404.1006-1-4-4': 'In order to confirm the picture emerging in activity-driven networks, we consider also the real-world network built using messages exchanged between users on Twitter.', '1404.1006-1-4-5': 'Here each user is described as node and a link between two nodes is created if they interacted at least one time via message.', '1404.1006-1-4-6': 'Remarkably, the results are qualitatively similar to what observed in synthetic networks.', '1404.1006-1-5-0': 'The paper is structured as follow.', '1404.1006-1-5-1': 'In section [REF] we introduced activity driven networks without and with memory.', '1404.1006-1-5-2': 'In section [REF] we study numerically the threshold behavior of SIR and SIS epidemic models in activity driven networks.', '1404.1006-1-5-3': 'In section [REF] we analyze numerically the threshold of the two processes in the Twitter dataset.', '1404.1006-1-5-4': 'In section [REF] we draw our final remarks and conclusions.', '1404.1006-1-6-0': '# Activity driven models', '1404.1006-1-7-0': 'In this section we describe the modeling framework used to produce the synthetic time-varying networks considered.', '1404.1006-1-8-0': '## Memoryless activity driven models (ML)', '1404.1006-1-9-0': 'In their basic formulation activity driven models are Markovian and memoryless.', '1404.1006-1-9-1': 'Each node is characterized by an activity rate [MATH], extracted from a distribution [MATH], describing its probability per unit time to establish links.', '1404.1006-1-9-2': 'To account for the observation that human behaviors are characterized by broad activity distributions we will consider power-law distributions of activity [MATH]), if not specified differently.', '1404.1006-1-9-3': 'In this setting, the generative process of the network is defined according to the following rules (see Figure [REF]):', '1404.1006-1-10-0': 'Thus, all interactions have a constant duration [MATH], that without loss of generality we fix to one, i.e. [MATH].', '1404.1006-1-11-0': 'At each time step the network [MATH] is a simple random graph with low average connectivity.', '1404.1006-1-11-1': 'Indeed, on average the number of active nodes per time step is [MATH], corresponding to an average number of edges equal to [MATH], and an average degree [MATH].', '1404.1006-1-11-2': 'However, integrating the links over [MATH] time steps, so that [MATH], induces networks whose degree distribution follows the activity functional form [CITATION] so that, for example, broad distributions of activity will generate broad degree distributions.', '1404.1006-1-11-3': 'The creation of hubs (highly connected nodes) results from the presence of nodes with high activity rate, which are more prone to repeatedly engage in interactions.', '1404.1006-1-12-0': '## Activity driven models with memory (WM)', '1404.1006-1-13-0': 'It has long been acknowledge that links in real-world networks can be grouped in (at least) two classes, namely strong and weak ties [CITATION].', '1404.1006-1-13-1': 'The first represent connections that are activated often and describe the inner social circle of each node.', '1404.1006-1-13-2': 'The latter describe occasional contacts that are activated sporadically.', '1404.1006-1-13-3': 'The reason for these different classes of links is that individuals are non-Markovian, and the evolution of their ego-centered networks is deeply influenced by their memory.', '1404.1006-1-13-4': 'Interestingly, empirical observations indicate that the probability for an individual that had interacted with [MATH] people to initiate a connection towards a [MATH]th individual is a function of [MATH].', '1404.1006-1-13-5': 'More precisely, the analysis of a large-scale mobile phone dataset [CITATION] identified the relation [EQUATION] where [MATH] is a constant function of the number of nodes already contacted, the degree.', '1404.1006-1-13-6': "Thus, setting for simplicity [MATH], it is possible to generalize the activity driven framework accounting for individuals' memory [CITATION].", '1404.1006-1-13-7': 'Given, as for the ML case, [MATH] nodes each characterized by an activity rate [MATH] extracted from a distribution [MATH], the generative process of the WM network is defined according to the following rules (see Figure [REF]):', '1404.1006-1-14-0': 'The structural properties of time-aggregated ML and WM activity driven networks are fundamentally different.', '1404.1006-1-14-1': "As clear from Figure [REF]-ML networks show a heavy-tailed cumulative degree and a homogeneous weight distribution, where the weights measuring the number of times each link is activated reflect the Markovian links' creation dynamics (see [REF]-B).", '1404.1006-1-14-2': 'On the other hand, WM networks show a broad degree distribution, steeper than one observed in ML systems, (see Figure [REF]-A) and a heavy-tailed weight distribution indicating the heterogeneity of tie strengths (see Figure [REF]-B).', '1404.1006-1-14-3': 'In Figure [REF]-C we also compare the behavior of the largest connected component (LCC) integrating the links as function of time.', '1404.1006-1-14-4': 'Interestingly, in ML networks the LCC appears earlier.', '1404.1006-1-14-5': 'Memory slows down the growth of the connected component as individuals are more likely to activate previous connections.', '1404.1006-1-15-0': '# SIR and SIS models in activity driven networks', '1404.1006-1-16-0': 'We consider two classic epidemic models, namely the SIR and SIS model [CITATION].', '1404.1006-1-16-1': 'In both cases the population is divided in compartments indicating the health status of individuals.', '1404.1006-1-16-2': 'In the SIR model nodes can be in the susceptible (S), infected (I) or recovered (R) compartments.', '1404.1006-1-16-3': 'Susceptible nodes are healthy individuals.', '1404.1006-1-16-4': 'Infected nodes instead contracted the illness and can spread it.', '1404.1006-1-16-5': 'Recovered are nodes that healed from the disease becoming immune.', '1404.1006-1-16-6': 'The model is described by the following reaction scheme: [EQUATION]', '1404.1006-1-16-7': 'The first transition indicates the contagion process.', '1404.1006-1-16-8': 'Susceptible nodes in contact with infected individuals become infected with rate [MATH].', '1404.1006-1-16-9': 'In particular, [MATH] takes into account the average contacts per node [MATH] and the per contact probability of transmission [MATH], i.e [MATH].', '1404.1006-1-16-10': 'The second transition instead, describes the recovery process.', '1404.1006-1-16-11': 'Infected individuals recover permanently with rate [MATH].', '1404.1006-1-17-0': 'Whether the disease is able to spread affecting a macroscopic fraction of the network or not depends on the value of infection rate, recovery rate and networks dynamics.', '1404.1006-1-17-1': 'In particular, in ML networks the SIR contagion process is able to spread if [EQUATION]', '1404.1006-1-17-2': 'See Refs. [CITATION] for the derivation details.', '1404.1006-1-17-3': 'The quantity [MATH] defines the epidemic threshold of the process.', '1404.1006-1-17-4': 'For value of [MATH] instead the disease will die out.', '1404.1006-1-17-5': 'Interestingly, the threshold is function of the first and second moments of the activity distribution, and completely neglects any time-integrated network representation.', '1404.1006-1-18-0': 'In the SIS model nodes can be either in the susceptible (S) or infected (I) compartment.', '1404.1006-1-18-1': 'The model is described by the following reaction scheme: [EQUATION]', '1404.1006-1-18-2': 'The first transition is the same of SIR models.', '1404.1006-1-18-3': 'In the second transition instead infected individuals heal spontaneously but instead of becoming immune to the disease move back to the susceptible compartment with rate [MATH].', '1404.1006-1-18-4': 'In ML networks the epidemic threshold of SIS contagion process, [MATH], is: [EQUATION]', '1404.1006-1-19-0': 'See Refs. [CITATION].', '1404.1006-1-19-1': 'Interestingly, the threshold is the same also for the SIR model, i.e. [MATH].', '1404.1006-1-19-2': "This is a characteristic of ML activity driven networks and is due to the Markovian links' creation dynamics [CITATION].", '1404.1006-1-20-0': 'In this paper we investigate numerically the epidemic dynamics occurring on WM networks.', '1404.1006-1-21-0': '### The SIR process on ML and WM networks', '1404.1006-1-22-0': 'We consider a SIR model and start the epidemic setting at [MATH] with a fraction [MATH] randomly selected nodes as seeds.', '1404.1006-1-22-1': 'SIR models reach the so called disease-free equilibrium in which the population is divided in: [EQUATION]', '1404.1006-1-22-2': 'All the variables refer to the density of individuals in the population.', '1404.1006-1-22-3': 'The infected individuals will always disappear from the population, as each one of them will eventually recovered becoming immune.', '1404.1006-1-22-4': 'Below the threshold, in the thermodynamic limit, [MATH].', '1404.1006-1-22-5': 'Above the threshold instead [MATH] reaches a macroscopic value, i.e. [MATH].', '1404.1006-1-22-6': 'The transition between the two regimes is continuous and the behavior of [MATH] can be studied as a second order phase transition with control parameter [MATH] [CITATION].', '1404.1006-1-23-0': 'In Figure [REF] we show the results obtained measuring [MATH] in ML and WM networks for different values of [MATH].', '1404.1006-1-23-1': 'Without loss of generality we fix [MATH] and use [MATH] as free parameter.', '1404.1006-1-23-2': 'The epidemic threshold in WN networks is clearly larger than in ML systems.', '1404.1006-1-23-3': 'The memory of individuals shifts the threshold to larger values, making the systems less vulnerable to disease spreading, as the repetition of interactions within strong ties inhibits the spreading potential of the disease.', '1404.1006-1-23-4': 'Indeed, infected individuals will contact with higher probability their inner circle of ties infecting possibly some of them.', '1404.1006-1-23-5': 'However, the newly infected nodes will be prone to keep contacting back the initial seeds and eventually recover.', '1404.1006-1-23-6': 'On the contrary, in ML networks nodes initiate random connections at each time step increasing their probability of interacting with infected individuals.', '1404.1006-1-23-7': 'It is also interesting to notice that in WM networks the final fraction of individuals affected by the disease is smaller.', '1404.1006-1-23-8': 'For example, for values of [MATH], [MATH] reaches a fraction [MATH] in ML networks, while just [MATH] in WM graphs.', '1404.1006-1-23-9': 'In summary, memory roughly doubles the epidemic threshold of a SIR process making the system more resilient to the spreading, and above threshold reduces the size of affected population.', '1404.1006-1-24-0': '### The SIS process on ML and WM networks', '1404.1006-1-25-0': 'We now turn our attention to SIS processes.', '1404.1006-1-25-1': 'Also in this case we start the epidemic setting at [MATH] a fraction [MATH] of randomly selected nodes as seeds.', '1404.1006-1-25-2': 'The nature of this epidemic model is fundamentally different respect to the SIR.', '1404.1006-1-25-3': 'Indeed, above threshold SIS processes show an endemic state characterized by a constant fraction of nodes, [MATH], in the infected compartment.', '1404.1006-1-25-4': 'Below threshold instead, the process reaches a disease-free equilibrium, i.e. [MATH].', '1404.1006-1-25-5': 'In order to estimate the critical value of [MATH] we proceed as seen for the SIR model, considering in this case the behavior of [MATH].', '1404.1006-1-25-6': 'In Figure [REF]-A we show the results of SIS processes evolving on top of ML and WM activity-driven networks.', '1404.1006-1-25-7': 'While the threshold of SIS coincides with the one of SIR in ML networks, it becomes smaller in WM networks.', '1404.1006-1-26-0': 'In general, in SIS processes the numerical estimation of the threshold is more prone to size and noise effects, due to the subtleties relative to the identification of the endemic state and the fact that [MATH] is not a monotonically increasing quantity as in the case of [MATH].', '1404.1006-1-26-1': 'Therefore, we consider also the life time [MATH] and the coverage [MATH] of the process as a function of [MATH] [CITATION], defining the duration of the process and the fraction of nodes that acquire the infection, respectively.', '1404.1006-1-26-2': 'In SIS processes for values of [MATH] above threshold the life time is infinite (endemic state) and the coverage reaches [MATH].', '1404.1006-1-26-3': 'Instead, below threshold both [MATH] and [MATH] vanish in the thermodynamic limit.', '1404.1006-1-26-4': 'Interestingly, the life time obtained by averaging over many realizations is equivalent to the susceptibility [MATH] in standard percolation theory.', '1404.1006-1-26-5': 'This method allows us to detect the threshold precisely [CITATION].', '1404.1006-1-26-6': 'Indeed, following Ref. [CITATION] we can consider as above threshold any realization that reaches a macroscopic coverage [MATH].', '1404.1006-1-26-7': 'Without loss of generality we set [MATH], fix [MATH] and evaluate [MATH] as a function of [MATH].', '1404.1006-1-26-8': 'For small values of the contagion rate the disease dies out quickly and the coverage remain below the threshold [MATH], while for very large values of [MATH] the disease will be able to spread quickly reaching a fraction [MATH].', '1404.1006-1-26-9': 'For intermediate values of [MATH], [MATH] will increase showing a peak close to the actual epidemic threshold.', '1404.1006-1-26-10': 'Figure [REF]-B shows that the estimation of the threshold performed considering the life time of the process produces results very similar to what observed in Figure [REF]-A.', '1404.1006-1-26-11': 'We can therefore conclude that the threshold of a SIS process unfolding in WM networks is smaller than in ML systems.', '1404.1006-1-26-12': 'This behavior is quite surprising and opposite to what is observed in the case of SIR models.', '1404.1006-1-26-13': 'The repeated connections in the ego-centered networks of each node allow the disease to survive in local and small clusters of strong ties making the system more fragile to the disease spreading.', '1404.1006-1-26-14': 'Such a behavior is not observed in SIR processes due to the presence of recovered individuals that become immune to the disease and are unable to sustain the spreading with multiple reinfections.', '1404.1006-1-27-0': 'In order to further substantiate the picture emerging from these analysis we study the surviving probability, [MATH], of SIS processes in the two networks [CITATION], defined as the probability that the process survives, [MATH], up to time [MATH].', '1404.1006-1-27-1': 'Above threshold the disease reaches an endemic state implying a constant [MATH].', '1404.1006-1-27-2': 'Below threshold instead [MATH] decays to zero [CITATION].', '1404.1006-1-27-3': 'In Figure [REF]-C we show the results in ML and WM activity driven networks considering for both cases values of [MATH] above and below the threshold estimated in Figure [REF]-B.', '1404.1006-1-27-4': 'The survival probability is constant and equal to one for the two values of [MATH] estimated to be above threshold .', '1404.1006-1-27-5': 'Instead, it decays for the other two values below the threshold.', '1404.1006-1-27-6': 'This analysis confirms that the threshold of SIS processes in activity driven networks with memory is smaller than the threshold in memoryless networks.', '1404.1006-1-28-0': '# SIS and SIR models in real time varying networks', '1404.1006-1-29-0': 'In order to validate the results obtained on synthetic time-varying networks we study the dynamical properties of SIR and SIS processes on a real temporal system.', '1404.1006-1-29-1': 'We consider the interactions between [MATH] Twitter users in the first tree months of [MATH] via [MATH] messages and coarse-grain the data adopting a time resolution of one day.', '1404.1006-1-29-2': 'Each user is represented as a node, and an undirected link is drawn between two nodes if they exchanged at least one message.', '1404.1006-1-29-3': 'Arguably such network is driven by non-Markovian human dynamics as many users tend to interact several times with the same circle of accounts.', '1404.1006-1-30-0': 'In order to single out the effects of memory we consider also a randomized version of the considered Twitter network, where non-Markovian dynamics are eliminated.', '1404.1006-1-30-1': 'The randomization is performed by reshuffling the interactions for each time stamp, so that memory effects are removed while the sequence of activation times for each node, the final time integrated degree distribution, and the degree distribution at each time step are preserved [CITATION].', '1404.1006-1-30-2': 'In Figure [REF]-A we plot the behavior of [MATH] as a function of [MATH] for the original Twitter dataset and for the reshuffled version of it.', '1404.1006-1-30-3': 'The threshold is a bit smaller in the randomized version than in the original network, although the difference is small.', '1404.1006-1-30-4': 'Furthermore the growth of the number of recovered is much faster.', '1404.1006-1-30-5': 'These findings support the qualitative picture discussed in the previous sections.', '1404.1006-1-30-6': 'Moreover, it is easy to understand why in synthetic networks the difference between the two thresholds was more clear and evident.', '1404.1006-1-30-7': 'The reason is that our conservative randomization strategy preserves the degree distribution, while the degree distributions of the two synthetic networks are quite different as clear from Figure [REF]-A.', '1404.1006-1-30-8': 'In Figure [REF]-B we plot the behavior of the life time, [MATH], of a SIS process in both networks.', '1404.1006-1-30-9': 'In this case the threshold in the original network is smaller than in the randomized one, analogously to what is observed in synthetic time-varying networks.', '1404.1006-1-30-10': 'Thus, also in real networks memory moves the threshold of SIS processes to smaller values facilitating the survival of the disease.', '1404.1006-1-31-0': '# Conclusions', '1404.1006-1-32-0': 'In general, real networks are characterized by temporal and non-Markovian dynamics.', '1404.1006-1-32-1': 'For example, in social networks, individuals interact more frequently with a small set of strong ties.', '1404.1006-1-32-2': 'In other words, people keep memory of their past connections.', '1404.1006-1-32-3': "While this crucial aspect has been analyzed in details in static networks' representations, very little attention has been devoted to its characterization on temporal networks.", '1404.1006-1-32-4': 'Here we studied the dynamical properties of SIR and SIS models in activity driven networks with and without memory.', '1404.1006-1-32-5': 'In order to single out the effects of non-Markovian dynamics we studied the epidemic threshold in basic activity driven models that by construction are Markovian and memoryless, and in a recent generalization of this modeling framework that explicitly consider non-Markovian link dynamics.', '1404.1006-1-32-6': 'We found that memory acts on SIR processes shifting the epidemic threshold to larger values making the system more resilient to the disease spreading.', '1404.1006-1-32-7': 'On the contrary, memory acts on SIS processes by lowering the epidemic threshold to smaller values thus making the systems more prone to the disease invasion.', '1404.1006-1-32-8': "In fact, the heterogeneity in ties' strength induces frequent repetition of contacts that allow the survival of SIS-like diseases in local groups of tightly connected individuals.", '1404.1006-1-32-9': 'The illness reaches its endemic state in small clusters that act as reservoir for the virus.', '1404.1006-1-33-0': 'Although activity driven models with memory capture fundamental aspects of real time varying networks, they do not account for other important features as appearance of new nodes, disappearance of old ones, and bursty behaviors just to name a few.', '1404.1006-1-33-1': 'While the introduction of these ingredients is left for future work, here we validated the picture obtained from synthetic networks by considering a real time-varying system, namely the network of communications in Twitter.', '1404.1006-1-33-2': 'Interestingly, the results obtained in this case confirm qualitatively the findings observed in activity driven networks.', '1404.1006-1-34-0': 'In conclusion, the results here presented show that memory can have opposite effects on different classes of spreading processes, and corroborate the important role played by non-Markovian dynamics on the dynamical processes unfolding on realistic networks [CITATION].'}

{'1404.1006-2-0-0': 'Most real networks are characterized by connectivity patterns that evolve in time following complex, non-Markovian, dynamics.', '1404.1006-2-0-1': 'Here we investigate the impact of this ubiquitous feature by studying the Susceptible-Infected-Recovered (SIR) and Susceptible-Infected-Susceptible (SIS) epidemic models on activity driven networks with and without memory (i.e., Markovian and non-Markovian).', '1404.1006-2-0-2': 'We find that memory inhibits the spreading process in SIR models by shifting the epidemic threshold to larger values and reducing the final fraction of recovered nodes.', '1404.1006-2-0-3': "In SIS processes instead, memory reduces the epidemic threshold and, for a wide range of diseases' parameters, increases the fraction of nodes affected by the disease in the endemic state.", '1404.1006-2-0-4': 'The heterogeneity in tie strengths, and the frequent repetition of strong ties it entails, allows in fact less virulent SIS-like diseases to survive in tightly connected local clusters that serve as reservoir for the virus.', '1404.1006-2-0-5': 'We validate this picture by studying both processes on two real temporal networks.', '1404.1006-2-1-0': 'Virtually any system can be represented as a network whose basic units are described as nodes and its interactions as links between them [CITATION].', '1404.1006-2-1-1': 'In general, connections are not static, but evolve in time subject to nontrivial dynamics [CITATION].', '1404.1006-2-1-2': 'Consider for example face to face or online interaction networks where individuals talk and exchange information through evolving contacts [CITATION].', '1404.1006-2-1-3': 'Recent advances in technology have allowed researchers to collect, monitor and probe such interactions generating an unprecedented amount of time-resolved high resolution data [CITATION].', '1404.1006-2-1-4': "The analysis of such real systems has exposed the limits of canonical static and annealed network representations [CITATION] calling for the development of a new theory to understand network's temporal properties.", '1404.1006-2-1-5': 'In particular, the recent data deluge has allowed researchers to start identifying the effects that time varying topologies have on dynamical processes taking place on them [CITATION].', '1404.1006-2-1-6': 'Prototypical examples are the spreading of memes, ideas, and infectious diseases.', '1404.1006-2-1-7': "All of these phenomena can be described as diffusion processes on contact networks and are affected by the ordering, concurrence, duration, and heterogeneity in nodes' activities and connectivity patterns [CITATION].", '1404.1006-2-2-0': 'One of most distinctive properties of social networks is the heterogeneity of interaction strength [CITATION].', '1404.1006-2-2-1': 'Individuals remember their inner circle of friends and most important connections, activating some links more often than others, thus building up strong and weak ties with their peers.', '1404.1006-2-2-2': 'In other words, the creation of links is not a Markov process [CITATION].', '1404.1006-2-2-3': 'While this property has been studied in detail in static networks [CITATION], its understanding in the context of time-varying graphs is still far from complete.', '1404.1006-2-2-4': 'Indeed, only a few studies have tackled this subject, and each with a different approach [CITATION].', '1404.1006-2-2-5': 'Nonetheless, these studies have revealed a rich phenomenology.', '1404.1006-2-2-6': 'In particular, non-Markovian link dynamics has been shown to be responsible for changing the spreading rate of diffusion processes, either slowing them down or, perhaps surprisingly, speeding them up [CITATION].', '1404.1006-2-3-0': 'Here we study the effects of memory on two different classes of epidemic spreading models, namely the Susceptible Infected Recovered (SIR) and the Susceptible Infected Susceptible (SIS) models [CITATION].', '1404.1006-2-3-1': 'We consider a recently proposed class of time-varying networks called activity driven models [CITATION], based on the observation that the propensity of nodes to initiate a connection (the activity) is heterogeneously distributed.', '1404.1006-2-3-2': 'In its basic formulation node activities are modeled with accuracy but the link creation is assumed to be Markovian.', '1404.1006-2-3-3': 'While such an approximation allows analytical treatments [CITATION], it does not capture many real properties of time-varying networks such as the memory of individuals.', '1404.1006-2-3-4': "Recently, this limitation has been overcome with the introduction of a non-Markovian generalization of the modeling framework based on a simple reinforcement mechanism that allows one to reproduce with accuracy the evolution of individual's contacts [CITATION].", '1404.1006-2-4-0': 'We study the dynamical properties of SIR and SIS models on activity driven networks with and without memory.', '1404.1006-2-4-1': 'In particular, we consider one of the most important dynamical properties of epidemic diffusion process, namely the epidemic threshold, defining the conditions necessary for the spreading of the disease to a macroscopic fraction of the population [CITATION].', '1404.1006-2-4-2': 'We also consider the effect of the disease on the population evaluating the final fraction of recovered nodes, in SIR processes, and the fraction of infected nodes in the endemic state, reached above threshold in SIS dynamics.', '1404.1006-2-5-0': 'We find that memory acts in different ways on SIR and SIS models.', '1404.1006-2-5-1': 'In SIR processes the epidemic threshold is shifted to larger values, making the spreading of the disease more difficult.', '1404.1006-2-5-2': 'Also, the final fraction of recovered nodes is significantly reduced.', '1404.1006-2-5-3': "In SIS dynamics memory moves the epidemic threshold to smaller values and shifts the endemic state, for a wide range of disease's parameters, to larger values.", '1404.1006-2-5-4': 'Thus, non-Markovian dynamics might facilitate the spreading of SIS-like diseases, like sexual transmitted illnesses, that can survive reaching an endemic state, in tightly connected clusters.', '1404.1006-2-5-5': 'The difference between the two models is due to the fundamentally different natures of the two processes that induce distinct behaviors also in the case of static networks [CITATION].', '1404.1006-2-6-0': 'Finally, we consider two real-world networks built using messages exchanged between users on Twitter and co-authorships of papers in a scientific journal.', '1404.1006-2-6-1': 'To isolate the role of non-Markovian dynamics in this case, we compare the spreading of SIR and SIS processes unfolding on real networks with the same dynamics unfolding on a randomized version of them.', '1404.1006-2-6-2': 'Interestingly, in the case of SIS processes the results are qualitatively similar to what is observed in synthetic networks.', '1404.1006-2-6-3': 'In the case of SIR dynamics we do not observe a significant change in the epidemic threshold.', '1404.1006-2-6-4': 'However, consistently to what observed in synthetic networks, the real non-Markovian dynamics hampers the disease spreading reducing significantly the final fraction of recovered nodes.', '1404.1006-2-7-0': 'In this section we describe the modeling framework used to produce the considered synthetic time-varying networks.', '1404.1006-2-8-0': '## Memoryless activity driven models (ML)', '1404.1006-2-9-0': 'In their basic formulation activity driven models are memoryless.', '1404.1006-2-9-1': 'Each node is characterized by an activity rate [MATH], extracted from a distribution [MATH], describing its probability per unit time to establish links.', '1404.1006-2-9-2': 'To account for the observation that human behaviors are characterized by broad activity distributions we will consider power-law distributions of activity [MATH]), unless specified differently.', '1404.1006-2-9-3': 'In this setting, the generative process of the network is defined according to the following rules (see Figure [REF]):', '1404.1006-2-10-0': 'At each discrete time step [MATH] the network [MATH] starts with [MATH] disconnected vertices; With probability [MATH] each vertex [MATH] becomes active and generates [MATH] links that are connected to [MATH] other randomly selected vertices.', '1404.1006-2-10-1': 'Non-active nodes can still receive connections from other active vertices; At the next time step [MATH], all the edges in the network [MATH] are deleted.', '1404.1006-2-11-0': 'Thus, all interactions have a constant duration [MATH], that without loss of generality we fix to one, i.e. [MATH].', '1404.1006-2-12-0': 'At each time step the network [MATH] is a simple random graph with low average connectivity.', '1404.1006-2-12-1': 'Indeed, on average the number of active nodes per time step is [MATH], corresponding to an average number of edges equal to [MATH], and an average degree [MATH].', '1404.1006-2-12-2': 'However, integrating the links over [MATH] time steps, so that [MATH], induces networks whose degree distribution follows the activity functional form [CITATION] so that, for example, broad distributions of activity will generate broad degree distributions.', '1404.1006-2-12-3': 'The creation of hubs (highly connected nodes) results from the presence of nodes with high activity rate, which are more prone to repeatedly engage in interactions.', '1404.1006-2-13-0': '## Activity driven models with memory (WM)', '1404.1006-2-14-0': 'It has long been acknowledged that links in real-world networks can be grouped in (at least) two classes, namely strong and weak ties [CITATION].', '1404.1006-2-14-1': 'The first represent connections that are activated often and describe, for example, the inner social circle of each node.', '1404.1006-2-14-2': 'The latter describe occasional contacts that are activated sporadically.', '1404.1006-2-14-3': 'From a modeling standpoint these different classes of links can be described considering individuals as non-Markovian.', '1404.1006-2-14-4': 'Indeed, the evolution of their ego-centered networks is deeply influenced by their social memory.', '1404.1006-2-14-5': 'Interestingly, empirical observations indicate that the probability for an individual that had interacted with [MATH] people to initiate a connection towards a [MATH]th individual is a function of [MATH].', '1404.1006-2-14-6': 'More precisely, the analysis of a large-scale mobile phone dataset [CITATION] identified the relation [EQUATION] where [MATH] is the total number of other nodes contacted measured at the end of the datasets, and [MATH] is a constant mildly dependent on the degree.', '1404.1006-2-14-7': "Thus, setting for simplicity [MATH], it is possible to generalize the activity driven framework accounting for individuals' memory [CITATION].", '1404.1006-2-14-8': 'Given, as for the ML case, [MATH] nodes each characterized by an activity rate [MATH] extracted from a distribution [MATH], the generative process of the WM network is defined according to the following rules (see Figure [REF]):', '1404.1006-2-15-0': 'At each discrete time step [MATH] the network [MATH] starts with [MATH] disconnected vertices; With probability [MATH] each vertex [MATH] becomes active and generates [MATH] links; Each link is established with probability [MATH] at random, and with probability [MATH] towards one of the [MATH] previously connected nodes.', '1404.1006-2-15-1': 'Non-active nodes can still receive connections from other active vertices; At the next time step [MATH], the memory of each node is updated and all the edges in the network [MATH] are deleted.', '1404.1006-2-16-0': 'The structural properties of time-aggregated ML and WM activity driven networks are fundamentally different.', '1404.1006-2-16-1': "As is clear from Figure [REF] ML networks show a heavy-tailed cumulative degree and a homogeneous weight distribution, where the weights measuring the number of times each link is activated reflect the Markovian links' creation dynamics (see Figure [REF]-B).", '1404.1006-2-16-2': 'On the other hand, WM networks show a broad degree distribution, steeper than the one observed in ML systems, (see Figure [REF]-A) and a heavy-tailed weight distribution indicating the heterogeneity of tie strengths (see Figure [REF]-B).', '1404.1006-2-16-3': 'In Figure [REF]-C we also compare the behavior of the largest connected component (LCC) integrating the links as a function of time.', '1404.1006-2-16-4': 'Interestingly, in ML networks the LCC appears earlier.', '1404.1006-2-16-5': 'Memory slows down the growth of the connected component as individuals are more likely to activate previous connections.', '1404.1006-2-17-0': '# SIR and SIS models in activity driven networks', '1404.1006-2-18-0': 'We consider two classic epidemic models, namely the SIR and SIS model [CITATION].', '1404.1006-2-18-1': 'In both cases the population is divided in compartments indicating the health status of individuals.', '1404.1006-2-18-2': 'In the SIR model nodes can be in the susceptible (S), infected (I) or recovered (R) compartments.', '1404.1006-2-18-3': 'Susceptible nodes are healthy individuals that never experienced the illness.', '1404.1006-2-18-4': 'Infected nodes have contracted the illness and can spread it.', '1404.1006-2-18-5': 'Recovered nodes have been cured of the disease and are immune.', '1404.1006-2-18-6': 'The model is described by the following reaction scheme: [EQUATION]', '1404.1006-2-18-7': 'The first transition indicates the contagion process.', '1404.1006-2-18-8': 'Susceptible nodes in contact with infected individuals become infected with rate [MATH].', '1404.1006-2-18-9': 'In particular, [MATH] takes into account the average contacts per node, [MATH], and the per contact probability of transmission [MATH], i.e [MATH].', '1404.1006-2-18-10': 'The second transition describes the recovery process.', '1404.1006-2-18-11': 'Infected individuals recover permanently with rate [MATH].', '1404.1006-2-19-0': 'Whether the disease is able to spread affecting a macroscopic fraction of the network or not depends on the value of the infection rate, the recovery rate and the networks dynamics.', '1404.1006-2-19-1': 'In particular, in ML networks the SIR contagion process is able to spread if [EQUATION]', '1404.1006-2-19-2': 'See Refs. [CITATION] for the derivation details.', '1404.1006-2-19-3': 'The quantity [MATH] defines the epidemic threshold of the process.', '1404.1006-2-19-4': 'For value of [MATH] the disease will die out.', '1404.1006-2-19-5': 'Interestingly, the threshold as a function of the first and second moments of the activity distribution, and completely neglects any time-integrated network representation.', '1404.1006-2-20-0': 'In the SIS model nodes can be either in the susceptible (S) or infected (I) compartment.', '1404.1006-2-20-1': 'The model is described by the following reaction scheme: [EQUATION]', '1404.1006-2-20-2': 'The first transition is the same as SIR models.', '1404.1006-2-20-3': 'In the second transition infected individuals heal spontaneously but instead of becoming immune to the disease move back to the susceptible compartment with rate [MATH].', '1404.1006-2-20-4': 'In ML networks the epidemic threshold of an SIS contagion process, [MATH], is: [EQUATION]', '1404.1006-2-21-0': 'See Refs. [CITATION].', '1404.1006-2-21-1': 'Interestingly, the threshold is the same as for the SIR model, i.e. [MATH].', '1404.1006-2-21-2': 'This is a characteristic of ML activity driven networks and is due to the Markovian link creation dynamics [CITATION].', '1404.1006-2-22-0': 'In this paper we investigate numerically the epidemic dynamics occurring on WM networks.', '1404.1006-2-23-0': '### The SIR process on ML and WM networks', '1404.1006-2-24-0': 'We consider a SIR model and start the epidemic at [MATH] with a fraction [MATH] randomly selected nodes as seeds.', '1404.1006-2-24-1': 'SIR models reach the so called disease-free equilibrium in which the population is divided in: [EQUATION]', '1404.1006-2-24-2': 'All the variables refer to the density of individuals in the population.', '1404.1006-2-24-3': 'The infected individuals will always disappear from the population, as each one of them will eventually recover becoming immune.', '1404.1006-2-24-4': 'Below the threshold, in the thermodynamic limit, [MATH].', '1404.1006-2-24-5': 'Above the threshold instead [MATH] reaches a macroscopic value, i.e. [MATH].', '1404.1006-2-24-6': 'The transition between the two regimes is continuous and the behavior of [MATH] can be studied as a second order phase transition with control parameter [MATH] [CITATION].', '1404.1006-2-25-0': 'In Figure [REF] we show the results obtained by measuring [MATH] in ML and WM networks for different values of [MATH].', '1404.1006-2-25-1': 'Without loss of generality we fix [MATH] and [MATH] (inset) and use [MATH] as free parameter.', '1404.1006-2-25-2': 'The epidemic threshold in WM networks is clearly larger than in ML systems.', '1404.1006-2-25-3': 'The memory of individuals shifts the threshold to larger values, making the systems less vulnerable to disease spreading.', '1404.1006-2-25-4': 'The repetition of interactions within strong ties inhibits the spreading potential of the disease.', '1404.1006-2-25-5': 'Indeed, infected individuals will be more likely to contact their inner circle of ties infecting possibly some of them.', '1404.1006-2-25-6': 'However, the newly infected nodes will be prone to keep contacting back the initial seeds and eventually recover.', '1404.1006-2-25-7': 'On the contrary, in ML networks these nodes initiate random connections at each time step increasing their probability of interacting with susceptible individuals.', '1404.1006-2-25-8': 'Furthermore, for all the values of [MATH] sampled, the final fraction of infected nodes in WM networks is significantly reduced.', '1404.1006-2-25-9': 'In summary, memory roughly doubles the epidemic threshold of a SIR process and reduces [MATH] making the system more resilient to the spreading.', '1404.1006-2-26-0': '### The SIS process on ML and WM networks', '1404.1006-2-27-0': 'We now turn our attention to SIS processes.', '1404.1006-2-27-1': 'Also in this case we start the epidemic at [MATH] with a fraction [MATH] of randomly selected nodes as seeds.', '1404.1006-2-27-2': 'The nature of this epidemic model is fundamentally different than the SIR.', '1404.1006-2-27-3': 'Indeed, above threshold SIS processes show an endemic state characterized by a constant fraction of nodes, [MATH], in the infected compartment.', '1404.1006-2-27-4': 'Below the threshold instead, the process reaches a disease-free equilibrium, i.e. [MATH].', '1404.1006-2-27-5': 'In general, in SIS processes the numerical estimation of the threshold is more prone to size and noise effects, due to the subtleties related to the identification of the endemic state and the fact that [MATH] is not a monotonically increasing quantity as [MATH].', '1404.1006-2-27-6': 'Therefore, we also consider the life time [MATH] and the coverage [MATH] of the process as a function of [MATH] [CITATION], defining the duration of the process and the fraction of nodes that acquire the infection, respectively.', '1404.1006-2-27-7': 'In SIS processes for values of [MATH] above threshold the life time is infinite (endemic state) and the coverage reaches [MATH].', '1404.1006-2-27-8': 'Below threshold both [MATH] and [MATH] vanish in the thermodynamic limit.', '1404.1006-2-27-9': 'Interestingly, the life time obtained by averaging over many realizations is equivalent to the susceptibility [MATH] in standard percolation theory.', '1404.1006-2-27-10': 'This method allows us to detect the threshold precisely [CITATION].', '1404.1006-2-27-11': 'Indeed, following Ref. [CITATION] we can consider as above threshold any realization that reaches a macroscopic coverage [MATH].', '1404.1006-2-27-12': 'For small values of the contagion rate the disease dies out quickly and the coverage remains below the threshold [MATH], while for very large values of [MATH] the disease will be able to spread quickly reaching a fraction [MATH].', '1404.1006-2-27-13': 'For intermediate values of [MATH], [MATH] will increase showing a peak close to the actual epidemic threshold.', '1404.1006-2-27-14': 'Figure [REF] shows that the estimation of the threshold performed considering both [MATH] (panel A) and the life time of the process (panel B) using [MATH].', '1404.1006-2-27-15': 'We fix [MATH] in the inset) and evaluate [MATH] and [MATH] as a function of [MATH].', '1404.1006-2-28-0': 'From the two plots we can conclude that the threshold of an SIS process unfolding in WM networks is smaller than in ML systems.', '1404.1006-2-28-1': 'This behavior is quite surprising and opposite to what is observed in the case of SIR models.', '1404.1006-2-28-2': 'The repeated connections in the ego-centered networks of each node allow the disease to survive in local and small clusters of strong ties making the system more fragile to the disease spreading.', '1404.1006-2-28-3': 'Such a behavior is not observed in SIR processes due to the presence of recovered individuals that become immune to the disease and are unable to sustain the spreading with multiple reinfections.', '1404.1006-2-28-4': 'Furthermore, in WM networks, for a wide range of [MATH] values above threshold, [MATH] is shifted to larger values.', '1404.1006-2-28-5': 'In this region the disease, due to the repetition of contacts, is able to reach an endemic state that involves a larger fraction of the population.', '1404.1006-2-28-6': 'As [MATH] increases the difference between WM and ML networks reduces and eventually reverses.', '1404.1006-2-28-7': 'Indeed, for very large values of the infection rate the disease spreading is favorited by Markovian link dynamics: at each time step active infectious nodes interact with new vertices that, in this regime, can be easily infected.', '1404.1006-2-29-0': 'From this observation we can better understand the effects of memory on the spreading dynamics of SIS processes.', '1404.1006-2-29-1': 'The repetition of contacts it entails might counterbalance the effects of small [MATH] values helping the diffusion.', '1404.1006-2-29-2': 'However, for large values of infection rate, memory might hamper the spreading reducing the impact of the disease.', '1404.1006-2-29-3': 'In this regime random connections are more efficient.', '1404.1006-2-29-4': 'In summary, memory shifts the threshold of SIS processes to smaller values, and for a wide range of infection rates, induces a larger values of [MATH].', '1404.1006-2-30-0': '# SIS and SIR models in real time varying networks', '1404.1006-2-31-0': 'In order to validate the results obtained on synthetic time-varying networks we study the dynamical properties of SIR and SIS processes on two real temporal datasets.', '1404.1006-2-31-1': 'We consider the interactions between [MATH] Twitter users via [MATH] messages and coarse-grain the data adopting a time resolution of a day.', '1404.1006-2-31-2': 'Each user is represented as a node, and at each time step an undirected link is drawn between two nodes if they exchanged at least one message in that time window.', '1404.1006-2-31-3': 'The second real dataset is a co-authorship network built considering [MATH] papers published by [MATH] researches in Physical Review Letters (PRL).', '1404.1006-2-31-4': 'We adopted the time resolution of one year.', '1404.1006-2-31-5': 'Each author is described as a node, and at each time step an undirected link is drawn between two nodes if they co-authored at least one paper in that time window.', '1404.1006-2-31-6': 'Arguably such networks are driven by non-Markovian human dynamics as many users and authors tend to interact several times with the same circle of accounts and collaborators.', '1404.1006-2-32-0': 'In order to single out the effects of memory we consider also two randomized versions of the real networks, where non-Markovian dynamics are washed out.', '1404.1006-2-32-1': 'The randomization is performed by reshuffling the interactions for each time stamp, so that memory effects are removed while the sequence of activation times for each node and the degree distribution at each time step are preserved [CITATION].', '1404.1006-2-32-2': 'In Figure [REF]-A we plot the behavior of [MATH] as a function of [MATH] for the original Twitter dataset and for the reshuffled version of it considering two values of [MATH].', '1404.1006-2-32-3': 'We do not observe a clear difference between the two epidemiological thresholds.', '1404.1006-2-32-4': 'The effects of memory are visible just on the growth of the number of recovered nodes.', '1404.1006-2-32-5': 'Indeed, [MATH] increases faster in the randomized network.', '1404.1006-2-32-6': 'Thus the repetition of contacts that memory entails hampers SIR spreading processes also on this real network.', '1404.1006-2-33-0': 'In Figure [REF]-B we plot the behavior of the life time, [MATH], of an SIS process in the original Twitter network and in its randomized version considering two values of [MATH].', '1404.1006-2-33-1': 'In this case the threshold in the original network is smaller than in the randomized one, analogously to what is observed in synthetic time-varying networks.', '1404.1006-2-33-2': 'Interestingly, also in real networks memory moves the threshold of SIS processes to smaller values facilitating the survival of the disease.', '1404.1006-2-33-3': 'In Figure [REF] we show the results of the same simulations considering the PRL collaboration network.', '1404.1006-2-33-4': 'Also in this real dataset memory does not change the epidemic threshold of SIR dynamics acting just reducing the final epidemic size [MATH].', '1404.1006-2-33-5': 'Furthermore, in the case of SIS spreading, memory shifts the epidemic thresholds to smaller values.', '1404.1006-2-34-0': 'Overall, these observations on two real temporal networks confirm qualitatively the picture emerging from synthetic time-varying graphs.', '1404.1006-2-35-0': '# Conclusions', '1404.1006-2-36-0': 'In general, real networks are characterized by temporal and non-Markovian dynamics.', '1404.1006-2-36-1': 'For example, in social networks, individuals interact more frequently with a small set of strong ties keeping memory of their past connections.', '1404.1006-2-36-2': "While this crucial aspect has been analyzed in detail in static networks' representations, very little attention has been devoted to its characterization on temporal networks.", '1404.1006-2-36-3': 'Here we studied the dynamical properties of SIR and SIS models in activity driven networks with and without memory.', '1404.1006-2-36-4': 'In order to single out the effects of non-Markovian dynamics we studied the epidemic threshold in basic activity driven models that by construction are Markovian and memoryless, and in a recent generalization of this modeling framework that explicitly consider non-Markovian link dynamics.', '1404.1006-2-36-5': 'We found that memory acts on SIR processes making the system more resilient to the disease spreading.', '1404.1006-2-36-6': "On the contrary, memory acts on SIS processes by lowering the epidemic threshold to smaller values and increasing the fraction of infected nodes in the endemic state (for a wide range of disease's parameters) thus possibly making the systems more prone to the disease invasion.", '1404.1006-2-36-7': "In fact, the heterogeneity in ties' strength induces frequent repetition of contacts that allow the survival of SIS-like diseases in local groups of tightly connected individuals.", '1404.1006-2-36-8': 'The illness reaches its endemic state in small clusters that act as reservoir for the virus.', '1404.1006-2-37-0': 'Although activity driven models with memory capture fundamental aspects of real time varying networks, they do not account for other important features as appearance of new nodes, disappearance of old ones, and bursty behaviors just to name a few.', '1404.1006-2-37-1': 'While the introduction of these ingredients in the modeling framework is left for future work, here we validated the picture obtained from synthetic networks by considering two real time-varying systems, namely the network of communications in Twitter, and a co-authorship network.', '1404.1006-2-37-2': 'Interestingly, the results obtained in this case confirm qualitatively the findings observed in activity driven networks for SIS dynamics.', '1404.1006-2-37-3': 'In the case of SIR spreading memory does not change the threshold.', '1404.1006-2-37-4': 'However, it reduces significantly the final fraction of nodes affected by the disease thus hampering its spread.', '1404.1006-2-38-0': 'In conclusion, the results here presented show that memory can have opposite effects on different classes of spreading processes, and corroborate the important role played by non-Markovian dynamics on the dynamical processes unfolding on temporal networks [CITATION].'}

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[]

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[]

['1404.1006-1-9-3', '1404.1006-1-13-7', '1404.1006-1-19-0', '1404.1006-2-9-3', '1404.1006-2-14-8', '1404.1006-2-21-0']

{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1404.1006

1906.06206

{'1906.06206-1-0-0': 'Quantum devices, such as quantum simulators, quantum annealers, and quantum computers, may be exploited to solve problems beyond what is tractable with classical computers.', '1906.06206-1-0-1': "This may be achieved as the Hilbert space available to perform such 'calculations' is far larger than that which may be classically simulated.", '1906.06206-1-0-2': 'In practice, however, quantum devices have imperfections, which may limit the accessibility to the whole Hilbert space.', '1906.06206-1-0-3': 'Actually, the dimension of the space of quantum states that are available to a quantum device is a meaningful measure of its functionality, but unfortunately this quantity cannot be directly experimentally determined.', '1906.06206-1-0-4': 'Here we outline an experimentally realisable approach to obtaining the scaling of the required Hilbert space of such a device to compute such evolution, by exploiting the thermalization dynamics of a probe qubit.', '1906.06206-1-0-5': 'This is achieved by obtaining a fluctuation-dissipation theorem for high-temperature chaotic quantum systems, which facilitates the extraction of information on the Hilbert space dimension via measurements of the decay rate, and time-fluctuations.', '1906.06206-1-1-0': '# Introduction', '1906.06206-1-2-0': 'The ability to control and manipulate microscopic systems at the single particle level is an essential requirement for many quantum technologies.', '1906.06206-1-2-1': 'Experimental setups where atoms or qubits can be arranged in ordered structures and studied in quantum non-equilibrium states include neutral atoms in optical lattices [CITATION], trapped ions [CITATION], Rydberg atoms [CITATION], and superconducting circuits[CITATION].', '1906.06206-1-2-2': 'These systems can be used for the quantum simulation of many-body models, or different forms of digital or adiabatic quantum computing.', '1906.06206-1-2-3': 'Most of these physical setups have limitations in the accessibility to certain observables.', '1906.06206-1-2-4': 'Thus, having extra tools to characterize quantum systems in a simple an efficient way can be useful in the diagnosis and certification of quantum devices.', '1906.06206-1-3-0': 'One of the most prominent properties of a quantum device is its size in terms of the dimension of the associated Hilbert space.', '1906.06206-1-3-1': 'The size of a quantum computer or simulator is often given in terms of number of qubits, a measure that ignores the effect of disorder or the possible lack of connectivity between different zones in the device.', '1906.06206-1-3-2': 'A more useful quantity would be the number of eigenstates of the Hamiltonian that take part in the quantum dynamics.', '1906.06206-1-3-3': 'This is however an elusive measure in realistic experimental situations.', '1906.06206-1-4-0': 'In this work we show that the equilibration dynamics [CITATION] of a quantum system can be used to extract information on the dimension of the Hilbert space.', '1906.06206-1-4-1': 'Indeed, advancements in quantum technologies described above have inspired a bounty of theoretical work in the field of quantum thermalization [CITATION].', '1906.06206-1-4-2': "In the following, we aim to help 'bridge the gap' between theoretical and experimental work in this field [CITATION].", '1906.06206-1-5-0': 'We assume a quantum quench scenario [CITATION] in which a quantum system is initialized in a fully-decohered, infinite temperature state, except for a subsystem that acts as a sensor and is prepared in a pure state.', '1906.06206-1-5-1': "For simplicity, we assume that this subsystem is a single qubit, which we refer to as the 'probe' qubit.", '1906.06206-1-5-2': 'The relaxation dynamics of the probe qubit depends on the details of the underlying structure of the Hamiltonian, however, in the most generic case of non-integrable systems, an appropriate description can be given in terms of random matrix theory (RMT) [CITATION].', '1906.06206-1-5-3': 'We show that the time-fluctuations of the probe in the long-time limit contain information about the Hilbert space dimension of the device.', '1906.06206-1-6-0': 'Our article is structured as follows.', '1906.06206-1-6-1': 'Firstly, we present the basic scheme and summarize our main result, which relies on an infinite-temperature fluctuation dissipation theorem (FDT) [CITATION] for the dynamics of the probe qubit.', '1906.06206-1-6-2': 'We continue, presenting numerical calculations that validate our predictions via exact diagonalization of a spin chain Hamiltonian.', '1906.06206-1-6-3': 'We then summarize our key findings, before presenting a more detailed derivation of our analytical results in terms of RMT.', '1906.06206-1-7-0': '# Results', '1906.06206-1-8-0': '## Set Up', '1906.06206-1-9-0': 'We assume that we have a quantum system (from here on, "quantum device") that is initially in an infinite temperature state.', '1906.06206-1-9-1': 'This is the case, for example, of a quantum computer device that has not been properly initialized.', '1906.06206-1-9-2': "A single qubit in the device, the 'probe' qubit, is prepared in a pure state.", '1906.06206-1-9-3': 'We then let this qubit evolve in time and reach an equilibrium state.', '1906.06206-1-9-4': 'We assume that the initial state is given by: [EQUATION] with [MATH] the density matrix of the quantum device, which we will assume to be in a fully decohered state, [MATH].', '1906.06206-1-9-5': 'Later on this condition will be relaxed, allowing for high but finite temperatures.', '1906.06206-1-9-6': 'The system evolves under the interacting Hamiltonian, [EQUATION] where [MATH] is just the Hamiltonian of the device and we assume that the qubit does not evolve at all in the absence of coupling to the device, which is given by the operator [MATH].', '1906.06206-1-10-0': 'The quantity under study, the long-time fluctuations, are defined by [EQUATION] where [MATH].', '1906.06206-1-10-1': 'Assuming only that the many-body eigenenergies are non-degenerate, and that also their energy gaps are non-degenerate, we may express the time fluctuations in terms of matrix elements between eigenstates of the coupled qubit-device system [CITATION], [MATH], [EQUATION] where [MATH], and [MATH].', '1906.06206-1-11-0': 'We assume that [MATH] is well approximated by a random matrix and build on a statistical theory for the many-body wave-functions [CITATION], [EQUATION] where summations are understood to be taken from [MATH] to [MATH], the dimension of the probe + device Hilbert space.', '1906.06206-1-11-1': 'Here we have [MATH] diagonalized by the non-interacting basis [MATH] and [MATH] is approximated by a random matrix from the Gaussian Orthogonal Ensemble (GOE), appropriately scaled by a coupling strength [MATH].', '1906.06206-1-11-2': 'Later on we will relax these approximations.', '1906.06206-1-11-3': 'In Ref. [CITATION] we showed that this model can be solved and it allows us to calculate matrix elements in the interacting basis.', '1906.06206-1-12-0': 'The initial state can be written in terms of eigenstates of the joint qubit-device system, [EQUATION] where the joint qubit-device Hamiltonian eigenbasis is built by ordering product states such that [MATH] odd) , [MATH] even).', '1906.06206-1-13-0': '## Main Result', '1906.06206-1-14-0': 'We prove that, assuming that the qubit-device dynamics are ergodic, the following relation is satisfied [EQUATION]', '1906.06206-1-14-1': 'The quantity [MATH], with [MATH] the total number of qubits in the total probe plus device system, depends on the size of the system in the following way, [EQUATION]', '1906.06206-1-14-2': 'The quantity [MATH] is the average inverse decay rate of the qubit, and where [MATH] is the average density of states (DOS) of the system.', '1906.06206-1-14-3': 'Both are defined below (see Eq. [REF]) in a more precise manner.', '1906.06206-1-14-4': '[MATH] is the device Hilbert space dimension, and finally, [MATH] is a constant of order 1 that does not depend on the size of the system or coupling strength.', '1906.06206-1-14-5': 'Eq. ([REF]) can be thus understood as a fluctuation-dissipation relation, which relates the time-fluctuations in the steady-state with the decay rate after a quantum quench.', '1906.06206-1-15-0': 'Eq. [REF] assumes that the system is ergodic, that is, [MATH] couples the qubit to the whole spectrum of the quantum device.', '1906.06206-1-15-1': 'In that case the function [MATH], where [MATH] is the number of sites in the device.', '1906.06206-1-16-0': 'Our approach relies on the calculation of correlation functions from a statistical theory of random wave-functions [MATH].', '1906.06206-1-16-1': 'Here we summarize the essential ingredients to our model, and give details in the Methods section below.', '1906.06206-1-16-2': "Our theory, developed in Ref. [CITATION] by extending Deutch's RMT model [CITATION], can be used to obtain arbitrary correlation functions [MATH], where [MATH] denotes the ensemble average over an ensemble of random matrix perturbations, [MATH], for a [MATH] Hamiltonian of the form [REF], with [MATH], with [MATH] and [MATH] a random matrix selected from the GOE, with [MATH] .", '1906.06206-1-16-3': 'We showed that these may be expressed as sums of products of four-point correlation functions, given by (for [MATH]), [EQUATION] where [MATH] is defined as [EQUATION] with [MATH].', '1906.06206-1-16-4': '[MATH] is defined similarly to Eq. ([REF]b), with [MATH].', '1906.06206-1-17-0': 'In the case applied here, where the observable is diagonal in the non-interacting basis, we require that the microcanonical average of [MATH] varies slowly over the width [MATH] of the many-body eigenstates.', '1906.06206-1-17-1': 'This is a reasonable assumption for such systems, where the observable equilibrium value is not sensitive to energy on the microscopic level.', '1906.06206-1-17-2': 'Further, in our specific case here, we have in-fact that this average is approximately constant (zero) over the entire bulk of the spectrum, as the probe qubit has no additional field acting to bias a particular state.', '1906.06206-1-18-0': 'In order to evaluate Eq. [REF], we thus use Eq. [REF] to write [MATH] in terms of the random wave-functions [MATH], and non-interacting matrix elements [MATH] and [MATH].', '1906.06206-1-18-1': 'We then use the self-averaging property of random matrices, which here may be written [MATH], and obtain the relevant correlation functions [MATH] (note that matrix elements in the non-interacting basis are not affected by the ensemble average).', '1906.06206-1-18-2': 'We thereby obtain, [EQUATION] where [MATH] depends only on the choice of probe observable.', '1906.06206-1-18-3': 'We see that this is a particular simple case of Eq. [REF], valid for the random matrix toy model where the DOS, [MATH] and [MATH] are constant in energy over the entire bulk of the spectrum.', '1906.06206-1-19-0': 'In order to relate this to the time-decay of the probe observable [MATH], we must further calculate the time evolution [MATH], which can be approached by the same recipe to obtain for a generic observable [MATH], [EQUATION] where [MATH] is the diagonal ensemble of the observable [MATH], which we find to be equal to the long-time average of [MATH] as required.', '1906.06206-1-19-1': 'We derive Eq. [REF] in the supplemental material.', '1906.06206-1-19-2': 'This is analogous to the result in Reference [CITATION] for pure-states.', '1906.06206-1-19-3': 'We note that the same Equation has similarly been obtained in Ref. [CITATION], which allows also for the perturbation matrix [MATH] to be inhomogeneous.', '1906.06206-1-19-4': 'In the specific case of interest here, Eq. [REF] becomes, [EQUATION] as [MATH] and [MATH].', '1906.06206-1-19-5': 'We note that for the random matrix case here, [MATH] is constant in energy over the bulk states, in the sense that an initial eigenstate [MATH] of [MATH] will decay at the same rate for all [MATH] in the bulk of the spectrum, and thus we have [MATH] in Eq. [REF].', '1906.06206-1-19-6': 'We thus observe the emergence of a classical fluctuation-dissipation theorem, relating the time-fluctuations and decay rate of our probe observable [MATH].', '1906.06206-1-19-7': 'The susceptibility [MATH] in Eq. [REF] can be seen to be related to the Hilbert space dimension of the device, [MATH], and thus measurements of the decay rate, [MATH], and fluctuations [MATH], which are both obtainable from the time evolution, can be exploited to infer [MATH].', '1906.06206-1-20-0': '## Application to Real Systems', '1906.06206-1-21-0': 'In this section, we show the application to a spin-chain system using exact diagonalization [CITATION].', '1906.06206-1-21-1': 'We note that some care must be taken in doing so; rather than a constant density of states (DOS) [MATH], and decay rate [MATH], these quantities may in principle change with energy, and thus we have [MATH], and [MATH].', '1906.06206-1-22-0': 'We instead make the much weaker assumption that the DOS, and decay rate vary slowly with respect to the energy width of a single random wave-function, [MATH].', '1906.06206-1-22-1': 'More concretely, we assume [MATH], and [MATH].', '1906.06206-1-22-2': 'We show in the Methods section, these assumptions lead to our main result, Eq. [REF].', '1906.06206-1-23-0': 'In Fig. [REF], we show the manifestation of Eq. [REF] in a spin-chain system described by the Hamiltonian [MATH], where [MATH] is the system Hamiltonian (acting as our probe), [MATH] is our device Hamiltonian, given by [EQUATION] which acts on sites with index [MATH], which is the probe index.', '1906.06206-1-23-1': 'The probe and device are coupled by the interaction Hamiltonian, [EQUATION]', '1906.06206-1-23-2': 'Here [MATH] is the device site where the probe is coupled, which we set as 2 throughout.', '1906.06206-1-23-3': 'This spin-chain model may be related to the random matrix toy model, [MATH], via the prescription [MATH], and [MATH].', '1906.06206-1-24-0': 'In particular, we see that, as [MATH], we expect that if all of the available Hilbert space is being utilized in the unitary dynamics we will observe the following scaling: [EQUATION]', '1906.06206-1-24-1': 'This is the relation that we test in Fig. [REF].', '1906.06206-1-25-0': 'It is important to note that this exponential scaling of [MATH], Eq. [REF], is expected from not only the contribution of [MATH], but also from the average DOS [MATH].', '1906.06206-1-25-1': 'This average is often trivially obtained, as for example, for an ensemble of [MATH] two-level systems [MATH], where [MATH] is the range of energies available [MATH], regardless of the microscopic properties of the DOS.', '1906.06206-1-25-2': 'We thus also study the quantity [MATH], as this quantity has no dependence on the DOS, and an observation of the exponential scaling in system size is confirmation that, indeed, [MATH].', '1906.06206-1-25-3': 'This is shown in Fig. [REF], where we observe an exponential scaling of the Hilbert space dimension, with [MATH], compared to [MATH] if the entire Hilbert space were explored in the dynamics.', '1906.06206-1-26-0': 'We further observe in Figs. [REF] and [REF], that the FDT similarly applies at finite temperatures [MATH].', '1906.06206-1-26-1': 'The extension of our theoretical approach to this case is discussed below, with details given in the supplemental material.', '1906.06206-1-26-2': 'Indeed, we can show that for high temperatures, such that [MATH], we obtain an FDT of the same form as Eq. [REF], by employing a high energy cut-off [MATH] to the bath state occupation.', '1906.06206-1-27-0': 'For finite temperatures, we show in the Methods section below, that the FDT depends on the partition function [MATH] itself, rather than the Hilbert space dimension.', '1906.06206-1-27-1': 'Indeed, one can see that in the infinite temperature limit [MATH].', '1906.06206-1-28-0': 'We note that in Ref. [CITATION], the current authors obtained a FDT for pure states, which can be seen to be recovered in the low temperature limit, [MATH], for which [MATH] does not depend explicitly on the Hilbert space dimension [MATH].', '1906.06206-1-28-1': 'This can also be analytically seen to be the same as the low temperature limit of our treatment below, which indicates that there is a smooth transition between these two cases.', '1906.06206-1-28-2': 'This is indeed observed in the numerics of Figs [REF] and [REF].', '1906.06206-1-29-0': '# Discussion', '1906.06206-1-30-0': 'The results shown above demonstrate how the chaotic dynamics of thermalization may be exploited in order to gain information on the complexity of the unitary quantum dynamics of a system.', '1906.06206-1-30-1': 'We have proposed an experimentally viable protocol, by which measurements of a local observable of a probe qubit may be exploited to measure the Hilbert space dimension of an ergodic quantum device, initialized in an infinite temperature state.', '1906.06206-1-30-2': 'We note that this measures the dimension of the states directly involved in dynamics only, and thus provides a more accurate measure of the complexity of the dynamics than a simple estimate of the Hilbert space dimension from the number of qubits.', '1906.06206-1-30-3': "In this sense, such a measurement of a large enough quantum device, if shown to be ergodic in the sense outlined above, would be a convincing indicator of the so called 'quantum supremacy' of the quantum device.", '1906.06206-1-31-0': 'On a practical level, our results may be observed in two ways: observation of a probe observable for (i) changing the number of qubits/ions/... in the quantum device (as in Figs. [REF] and [REF]) , or (ii) changing the probe-device coupling (as in Fig. [REF]).', '1906.06206-1-31-1': 'The latter is perhaps the simplest experimental methodology, which we show can confirm the ergodic behaviour of a system, that is, that the unitary dynamics requires an extensive proportion of the Hilbert space, by showing a linear relationship between the long-time fluctuations and decay rate.', '1906.06206-1-31-2': 'For a model where the device size may be altered, our FDT provides even deeper insight, allowing also for the experimental observation of the scaling of the Hilbert space dimension with system size.'}

{'1906.06206-2-0-0': 'Quantum devices, such as quantum simulators, quantum annealers, and quantum computers, may be exploited to solve problems beyond what is tractable with classical computers.', '1906.06206-2-0-1': "This may be achieved as the Hilbert space available to perform such 'calculations' is far larger than that which may be classically simulated.", '1906.06206-2-0-2': 'In practice, however, quantum devices have imperfections, which may limit the accessibility to the whole Hilbert space.', '1906.06206-2-0-3': 'We thus determine that the dimension of the space of quantum states that are available to a quantum device is a meaningful measure of its functionality, though unfortunately this quantity cannot be directly experimentally determined.', '1906.06206-2-0-4': 'Here we outline an experimentally realisable approach to obtaining the required Hilbert space dimension of such a device to compute its time evolution, by exploiting the thermalization dynamics of a probe qubit.', '1906.06206-2-0-5': 'This is achieved by obtaining a fluctuation-dissipation theorem for high-temperature chaotic quantum systems, which facilitates the extraction of information on the Hilbert space dimension via measurements of the decay rate, and time-fluctuations.', '1906.06206-2-1-0': '# Introduction', '1906.06206-2-2-0': 'The ability to control and manipulate microscopic systems at the single particle level is an essential requirement for many quantum technologies.', '1906.06206-2-2-1': 'Experimental setups where atoms or qubits can be arranged in ordered structures and studied in quantum non-equilibrium states include neutral atoms in optical lattices [CITATION], trapped ions [CITATION], Rydberg atoms [CITATION], and superconducting circuits[CITATION].', '1906.06206-2-2-2': 'These systems can be used for the quantum simulation of many-body models, or different forms of digital or adiabatic quantum computing.', '1906.06206-2-2-3': 'Most of these physical setups have limitations in the accessibility to certain observables.', '1906.06206-2-2-4': 'Thus, having extra tools to characterize quantum systems in a simple and efficient way can be useful in the diagnosis and certification of quantum devices.', '1906.06206-2-3-0': 'One of the most prominent properties of a quantum device is its size in terms of the dimension of the associated Hilbert space.', '1906.06206-2-3-1': 'The size of a quantum computer or simulator is often given in terms of number of qubits, such that the Hilbert space dimension is [MATH] for [MATH] qubits.', '1906.06206-2-3-2': 'This is a measure that ignores the effect of disorder or the possible lack of connectivity between different zones in the device.', '1906.06206-2-4-0': 'A more useful quantity would be the number of eigenstates of the Hamiltonian that take part in the quantum dynamics, which is bounded in this case by [MATH], but would exclude the degrees of freedom that do not contribute to the evolution of the initial state.', '1906.06206-2-4-1': 'Thereby establishing an effective Hilbert space dimension that more accurately describes the complexity of the system, in terms of some effective fully connected Hamiltonian.', '1906.06206-2-4-2': 'This Hilbert space dimension is, however, an elusive measure in realistic experimental situations.', '1906.06206-2-5-0': 'In this work we show that the equilibration dynamics [CITATION] of a quantum system can be used to extract such information on the dimension of the Hilbert space of a quantum device, in terms of the effective number of states that contribute to the dynamics of a local observable.', '1906.06206-2-5-1': 'Indeed, advancements in quantum technologies described above have inspired a bounty of theoretical work in the field of quantum thermalization [CITATION].', '1906.06206-2-5-2': "In the following, we aim to help 'bridge the gap' between theoretical and experimental work in this field [CITATION].", '1906.06206-2-6-0': 'We assume a quantum quench scenario [CITATION] in which a quantum system is initialized in a fully-decohered, infinite temperature state, except for a subsystem that acts as a sensor and is prepared in a pure state.', '1906.06206-2-6-1': "For simplicity, we assume that this subsystem is a single qubit, which we refer to as the 'probe' qubit.", '1906.06206-2-6-2': 'The relaxation dynamics of the probe qubit depends on the details of the underlying structure of the Hamiltonian, however, in the most generic case of non-integrable systems, an appropriate description can be given in terms of random matrix theory (RMT) [CITATION].', '1906.06206-2-6-3': 'We show that the time-fluctuations of the probe in the long-time limit contain information about the Hilbert space dimension of the device.', '1906.06206-2-7-0': 'Our article is structured as follows.', '1906.06206-2-7-1': 'Firstly, in Section [REF] we present the basic scheme and summarize our main result, which relies on an infinite-temperature fluctuation dissipation theorem (FDT) [CITATION] for the dynamics of the probe qubit in order to extract information on the scaling of the effective Hilbert space of a quantum device.', '1906.06206-2-7-2': 'We continue, presenting numerical calculations that validate our predictions via exact diagonalization of a spin chain Hamiltonian in Section [REF].', '1906.06206-2-7-3': 'We then outline in more detail our RMT model in Section [REF].', '1906.06206-2-7-4': 'In Section [REF] we derive an expression for the time dependence of generic observables in chaotic quantum systems, and discuss how this can be exploited for our measurements.', '1906.06206-2-7-5': 'We then derive the FDT, and extensions from the RMT model, and to finite temperatures, in Section [REF].', '1906.06206-2-7-6': 'Finally, we summarise our findings in Section [REF].', '1906.06206-2-7-7': 'Further details and proofs are included in Appendices.', '1906.06206-2-7-8': 'We note that this is arranged such that our key findings can be understood from sections [REF] and [REF], with the detailed calculations presented later in the text.', '1906.06206-2-8-0': '# Setup and main results', '1906.06206-2-9-0': '## Proposed setup', '1906.06206-2-10-0': "We assume that we have a quantum system (from here on, 'quantum device') made up of a single 'probe' qubit, initialized in a pure state, which at [MATH] is coupled to the rest of the quantum device (referred to as the 'bath').", '1906.06206-2-10-1': 'We initialize the bath in an infinite temperature state.', '1906.06206-2-10-2': 'This is sketched in Figure [REF].', '1906.06206-2-10-3': 'This can be routinely achieved, for example, in a quantum computer device that has not been properly initialized.', '1906.06206-2-10-4': 'Since quantum devices always suffer from some kind of decoherence, creating an infinite temperature (or fully decohered) state is typically a simple task.', '1906.06206-2-10-5': 'We then let this qubit evolve in time and reach an equilibrium state.', '1906.06206-2-10-6': 'The initial state is thus, [EQUATION] [MATH] is the density matrix of the bath in the fully decohered state, [EQUATION] where [MATH] is a set of orthonormal wave functions in the bath, which we take as the eigenstates of the bath Hamiltonian [MATH].', '1906.06206-2-10-7': '[MATH] is the Hilbert space dimension of the bath, and is the parameter on which we wish to infer information via measurements of the probe qubit.', '1906.06206-2-10-8': 'Later on the fully decohered state condition will be relaxed, allowing for finite temperatures.', '1906.06206-2-11-0': 'The system evolves under the interacting Hamiltonian, [EQUATION] where [MATH] is the Hamiltonian of the uncoupled probe + bath device, and we assume that the probe qubit does not evolve at all in the absence of coupling to the bath, which is given by the operator [MATH].', '1906.06206-2-11-1': 'A prominent role in the following discussions will be played by the eigensystems of the interacting Hamiltonian, [EQUATION] and the eigensystem of the uncoupled probe-bath system [EQUATION]', '1906.06206-2-11-2': 'All throughout this work we will use the convention that indices [MATH], [MATH] refer to summations over eigenstates of [MATH], whereas [MATH] refer to eigenstates of [MATH].', '1906.06206-2-12-0': 'We focus on the quantum dynamics of the expectation value of a probe observable, which we take for concreteness to be [MATH].', '1906.06206-2-12-1': 'Note that, however, the results should not depend on the particular choice for the probe observable, since [MATH] has no probe energy term.', '1906.06206-2-12-2': 'The probe-bath coupling [MATH], may introduce a dependence on the chosen observable, but the results below should hold in the general case of quantum chaotic or non-integrable systems.', '1906.06206-2-13-0': 'The quantity under study, the time-averaged fluctuations of the probe observable, is defined by [EQUATION] where [MATH].', '1906.06206-2-13-1': 'Assuming only that the many-body eigenenergies are non-degenerate, and that also their energy gaps are non-degenerate, we may express the time fluctuations in terms of matrix elements between eigenstates of the coupled probe-bath system [CITATION], [EQUATION] where [MATH], and [MATH].', '1906.06206-2-14-0': '## Summary of main results of this work', '1906.06206-2-15-0': 'In many practical situations the quantum device is a non-integrable, quantum chaotic system.', '1906.06206-2-15-1': 'We expect that in such cases a qualitative description can be obtained by assuming that [MATH] is a random matrix.', '1906.06206-2-15-2': 'This assumption leads to a statistical theory for the many body wave functions in Eqs. ([REF], [REF]) [CITATION], [EQUATION] where summations are understood to be taken from [MATH] to [MATH], the dimension of the Hilbert space of the device.', '1906.06206-2-15-3': 'We will refer to [MATH] as quantum chaotic or random wave functions.', '1906.06206-2-15-4': 'We assume that [MATH] belongs to the Gaussian Orthogonal Ensemble (GOE), appropriately scaled by a coupling strength [MATH].', '1906.06206-2-15-5': 'In Ref. [CITATION] we showed that this model can be solved and it allows us to calculate matrix elements in the interacting basis.', '1906.06206-2-16-0': 'Special care must be taken to translate the probe initial state defined in Eq. ([REF]) in the random matrix formalism.', '1906.06206-2-16-1': 'Without loss of generality we can assume that non-interacting eigenfunctions are ordered such that odd (even) values of [MATH] correspond to the probe qubit in state [MATH]).', '1906.06206-2-16-2': 'This leads to the initial condition [EQUATION]', '1906.06206-2-16-3': 'In the following sections we prove that random matrix theory leads to a relation between the time-fluctuations and the typical decay rate of the probe qubit.', '1906.06206-2-16-4': 'This relation is the main result of this work: [EQUATION] where [MATH] is the average inverse decay rate of the qubit.', '1906.06206-2-16-5': 'This average is an unbiased average over all initially populated initial state energies, which spans the entire energy range [MATH] of the device due to the initial infinite temperature bath state.', '1906.06206-2-16-6': 'We will see that this unbiased average originates from the decay to equilibrium of each state [MATH] contributing to the initial state [REF].', '1906.06206-2-16-7': 'In Sections [REF] and [REF] we will see that this can be related to the decay rate obtained from a fit to the decay to equilibrium of a local probe observable in our current set up.', '1906.06206-2-16-8': 'This quantity is thus simply an average of the decay rates experienced by the probe qubit.', '1906.06206-2-17-0': 'The quantity [MATH], with [MATH] the total number of qubits, depends on the size of the system in the following way, [EQUATION] [MATH] is the average density of states (DOS) of the system, defined analogously to the average decay rate above (see also Eq. [REF]).', '1906.06206-2-17-1': '[MATH] is the bath Hilbert space dimension, and finally, [MATH] is a constant of order one that does not depend on the size of the system or coupling strength.', '1906.06206-2-18-0': 'Eq. ([REF]) can be understood as a fluctuation-dissipation relation, which relates the time-fluctuations in the steady-state with the decay rate after a quantum quench.', '1906.06206-2-18-1': 'The ratio between fluctuations and average decay time allows us to quantify the dimension of the Hilbert space over which the ergodic quantum dynamics takes place.', '1906.06206-2-18-2': 'In a physical system if [MATH] couples the probe qubit to the whole spectrum of the quantum device, we expect that the function [MATH], where [MATH] is the number of qubits in the part of the device that acts as a bath.', '1906.06206-2-19-0': 'Our main result, Eq. ([REF]), can be obtained by following these steps:', '1906.06206-2-20-0': '(i) We consider the random matrix model with an homogeneous coupling matrix [MATH], and calculate [MATH] by extending the formalism developed in [CITATION] to mixed states (see Section [REF]).', '1906.06206-2-20-1': 'To carry out this calculation we assume a constant DOS, [MATH], with [MATH] the mean energy separation.', '1906.06206-2-21-0': 'This calculation allows us to predict an exponential decay for a probe observable of the form: [EQUATION] where [MATH] is the diagonal ensemble of the observable [MATH], which we find to be equal to the long-time average of [MATH] as required, and [MATH] refers to the free evolution of the observable [MATH] under the Hamiltonian [MATH].', '1906.06206-2-21-1': 'This result is derived in Section [REF].', '1906.06206-2-21-2': 'This is analogous to the result in Reference [CITATION] for pure-states.', '1906.06206-2-21-3': 'We note that the same Equation has similarly been obtained in Ref. [CITATION], which allows also for the perturbation matrix [MATH] to be inhomogeneous.', '1906.06206-2-21-4': 'In the specific case of interest here, Eq. [REF] becomes, [EQUATION] as [MATH] and [MATH].', '1906.06206-2-22-0': 'Using this random matrix model we also obtain a preliminary version of the infinite temperature fluctuation-dissipation theorem, Eq. ([REF]), with [MATH].', '1906.06206-2-23-0': '(ii) In a realistic system both the DOS and the coupling strength will depend on the energy.', '1906.06206-2-23-1': 'This leads to energy-dependent qubit decay rates, [MATH], and DOS, [MATH].', '1906.06206-2-23-2': 'Due to the infinite-temperature initial condition, it is not possible a priori to approximate those quantities to any single value, since the quantum dynamics of the probe qubit result from contributions from all possible initial states.', '1906.06206-2-23-3': 'It is thus necessary to extend the RMT formalism to allow for variations in both the DOS and the decay rate in energy over the width of the initial mixed state.', '1906.06206-2-24-0': 'It is also important to note that when applied to a realistic system [MATH] should be thought of as the number of eigenstates that may contribute to the dynamics of a local observable - it is thus a measure of the effective Hilbert space dimension in the sense of the number of degrees of freedom that may be explored from a given initial state.', '1906.06206-2-24-1': 'This measure accounts for effects such as locality of interactions, and disorder, which will reduce the Hilbert space available for evolution of the system.', '1906.06206-2-24-2': 'As such, [MATH] will in general be bounded by the total Hilbert space dimension, but is a more realistic measure of the size of the explored Hilbert space in the thermalization dynamics of the device.', '1906.06206-2-25-0': 'In Section [REF] we show that the decay rate observed from such an initial state, defined with an energy dependent coupling strength, is the thermal average decay rate over the energy width defined by the initial state.', '1906.06206-2-26-0': 'We formulate the generalization of the FDT in Section [REF], however a brief summary of the approach is as follows: To account for the energy variation of the DOS and decay rate we instead make the assumption that both change slowly in energy with respect to the energy width of a single eigenstate.', '1906.06206-2-26-1': 'In this case, one can reformulate the theory such that the random wave functions have a smoothly varying width.', '1906.06206-2-26-2': 'In effect this is the statement that a given random wave function contributes as if it was the eigenstate of a random matrix model with a constant DOS and decay rate, as over the width of the wave function itself these parameters do not change appreciably.', '1906.06206-2-27-0': 'In this case, we see that the average DOS and decay rate over the initially populated initial states contributes to the FDT, as in Eq. [REF].', '1906.06206-2-27-1': 'This is extended to finite temperature systems, where instead a thermal average can be seen to contribute, in Section [REF].', '1906.06206-2-28-0': 'Finally, we may relate the two decay rate averages, as the thermal average observed from the decay differs slightly from that appearing in Eq. [REF].', '1906.06206-2-28-1': 'These can be seen to be related by a constant factor that does not depend on the coupling strength, or bath size, and thus scaling information of the Hilbert space dimension may be recovered.', '1906.06206-2-28-2': 'This is shown in Section [REF], and Appendix [REF], where the FDT is recast in terms of thermal averages.', '1906.06206-2-29-0': 'In Section [REF] we show that when an exponential decay of the form Eq. [REF] is observed, this indicates that [MATH] is approximately constant over the bulk of the initially populated states.', '1906.06206-2-29-1': 'In this case we are able to extract the numerical value of the Hilbert space dimension directly, as the averages ove [MATH] occurring in the FDT, and observed from the decay, are equal.', '1906.06206-2-30-0': 'In summary, we observe the emergence of a classical fluctuation-dissipation theorem, relating the time-fluctuations and decay rate of our probe observable [MATH].', '1906.06206-2-30-1': 'The susceptibility [MATH] in Eq. [REF] can be seen to be related to the Hilbert space dimension of the bath, [MATH], and thus measurements of the decay rate, [MATH], and fluctuations [MATH], which are both obtainable from the time evolution, can be exploited to obtain information on the device Hilbert space dimension, [MATH].', '1906.06206-2-31-0': '# Numerical Experiments', '1906.06206-2-32-0': 'Before going into the technical details of our derivation, we present numerical evidence that confirms the validity of the random matrix approach and our main results.', '1906.06206-2-32-1': 'In this section, we show the application to a spin-chain system using exact diagonalization [CITATION].', '1906.06206-2-32-2': 'From this, we observe the FDT numerically using a realistic experimentally observable model.', '1906.06206-2-33-0': 'In Fig. [REF], we show the manifestation of Eq. [REF] in a spin-chain system described by the Hamiltonian [MATH], where [MATH] is the system Hamiltonian (acting as our probe), [MATH] is our bath Hamiltonian, given by [EQUATION] which acts on sites with index [MATH], which is the probe index.', '1906.06206-2-33-1': 'The probe and bath are coupled by the interaction Hamiltonian, [EQUATION]', '1906.06206-2-33-2': 'Here [MATH] is the device site where the probe is coupled, which we set as 2 throughout.', '1906.06206-2-33-3': 'This spin-chain model may be related to the random matrix toy model, [MATH], via the prescription [MATH], and [MATH].', '1906.06206-2-34-0': 'In particular, we see that, as [MATH], we expect that if all of the available Hilbert space is being utilized in the unitary dynamics we will observe the following scaling: [EQUATION]', '1906.06206-2-34-1': 'This is the relation that we test in Fig. [REF].', '1906.06206-2-35-0': 'It is important to note that this exponential scaling of [MATH], Eq. [REF], is expected from not only the contribution of [MATH], but also from the average DOS, [MATH].', '1906.06206-2-35-1': 'This average is often trivially obtained, as for example, for an ensemble of [MATH] two-level systems [MATH], where [MATH] is the range of energies available [MATH] (which may itself change with [MATH]), regardless of the microscopic properties of the DOS.', '1906.06206-2-35-2': 'We thus also study the quantity [MATH], as this quantity has no dependence on the DOS, and an observation of the exponential scaling in system size is confirmation that, indeed, [MATH].', '1906.06206-2-35-3': 'This is shown in Fig. [REF], where we observe an exponential scaling of the Hilbert space dimension, with [MATH], compared to [MATH] if the entire Hilbert space were explored in the dynamics.', '1906.06206-2-36-0': 'We further observe in Figs. [REF] and [REF], that the FDT similarly applies at finite temperatures [MATH].', '1906.06206-2-36-1': 'The extension of our theoretical approach to this case is discussed below, with additional details given in Appendix [REF].', '1906.06206-2-36-2': 'Indeed, we can show that for high temperatures, such that [MATH], we obtain an FDT of the same form as Eq. [REF], by employing a high energy cut-off [MATH] to the bath state occupation.', '1906.06206-2-37-0': 'For finite temperatures we show below that the FDT depends on the partition function [MATH] itself, rather than the Hilbert space dimension.', '1906.06206-2-37-1': 'Indeed, one can see that in the infinite temperature limit [MATH].', '1906.06206-2-38-0': 'Finally, we see that when the observable is found to decay exponentially to its equilibrium value, this indicates that the decay rate is approximately constant over the bulk of the initially occupied states.', '1906.06206-2-38-1': 'This is shown in Section [REF].', '1906.06206-2-38-2': 'Exploiting this observation, we are able to directly obtain the Hilbert space dimension, as for an infinite temperature initial bath state the average [MATH] is equal to the measured decay rate.', '1906.06206-2-38-3': 'The bath Hilbert space dimension [MATH], as calculated from Eq. [REF], is plotted for varying device sizes in Fig. [REF].', '1906.06206-2-38-4': 'Here we observe that [MATH] indeed increases exponentially with systems size, yet is somewhat smaller than its maximum possible value [MATH], which is expected to the locality of interactions within the chain.', '1906.06206-2-39-0': 'We again note that in Fig. [REF] the measurement of [MATH] is a measurement of the explored Hilbert space, or the total number of eigenstates that contribute to the evolution of the initial state.', '1906.06206-2-39-1': 'Thus, for a maximally connected device this would be [MATH], whereas locality of interactions in this case restricts some areas of the Hilbert space.', '1906.06206-2-40-0': 'We note that for models where [MATH] is not approximately constant in energy, which would be marked by a deviation from the exponential decay in Eq. [REF], one would still have access to Figs. [REF]-[REF], and thus scaling information of the Hilbert space dimension, yet the numerical value [MATH] would be obscured.', '1906.06206-2-40-1': 'This is explained in more detail in Section [REF], where we see how information on [MATH] is extracted from the decay, and in Section [REF] and Appendix [REF], where we see how this relates to the observed FDT.', '1906.06206-2-40-2': 'The key detail is that when the FDT is expressed in terms of the same value measured from the decay it differs by a constant, thereby leaving the scaling of [MATH] with system size or decay rate accessible, yet obscuring the numerical value.', '1906.06206-2-41-0': 'We note that in Ref. [CITATION], the current authors obtained a FDT for pure states, which can be seen to be recovered in the low temperature limit, [MATH], for which [MATH] does not depend explicitly on the Hilbert space dimension [MATH].', '1906.06206-2-41-1': 'This can also be analytically seen to be the same as the low temperature limit of our treatment below, which indicates that there is a smooth transition between these two cases.', '1906.06206-2-41-2': 'This is indeed observed in the numerics of Figs [REF] and [REF].', '1906.06206-2-42-0': '# Model', '1906.06206-2-43-0': '## RMT Approach', '1906.06206-2-44-0': 'Our approach relies on the calculation of correlation functions from a statistical theory of random wave functions [MATH].', '1906.06206-2-44-1': 'Here we summarize the essential ingredients to our model, and give details on the calculations in the sections below.', '1906.06206-2-45-0': "Our theory, developed in Ref. [CITATION] by extending Deutch's RMT model [CITATION], can be used to obtain arbitrary correlation functions [MATH], where [MATH] denotes the ensemble average over an ensemble of random matrix perturbations, [MATH], for a [MATH] Hamiltonian of the form [REF], with [MATH], with [MATH] and [MATH] a random matrix selected from the GOE, with [MATH].", '1906.06206-2-45-1': 'In practice, those correlations allow us to calculate any dynamical quantity of interest within the RMT formalism.', '1906.06206-2-46-0': 'To illustrate the use of such correlation functions we briefly consider the simple example of the diagonal observable matrix elements [MATH].', '1906.06206-2-46-1': 'These can be written as [EQUATION]', '1906.06206-2-46-2': 'Now, using the self-averaging property of random matrices, which we prove for this model in Appendix [REF], we see that [EQUATION]', '1906.06206-2-46-3': 'Note that only the random wave functions [MATH] remain inside the ensemble average, as all other factors are in the non-interacting basis and thus do not depend on [MATH].', '1906.06206-2-46-4': 'We thus observe that the diagonal observable matrix elements depend on the correlation function [MATH] (note that we will see how to deal with the summation and non-interacting observable elements in Section [REF] below).', '1906.06206-2-47-0': 'More generally, when calculating more complicated quantities, we have that there is also a non-trivial contribution of a four-point correlation function of the form [MATH] which is given by (for [MATH]), [EQUATION] where [MATH] is defined as [EQUATION] with [MATH].', '1906.06206-2-47-1': '[MATH] is defined similarly to Eq. ([REF]b), with [MATH].', '1906.06206-2-47-2': 'The Lorentzian form of Eq. [REF] is found for a homogeneous perturbation [MATH] [CITATION], selected from the GOE.', '1906.06206-2-47-3': 'We obtain that the four-point correlation function, Eq. [REF], can be described in terms of product of two-point correlators [MATH], as if the random wave function distribution was purely Gaussian, plus a correction term originating from the effective interaction due to the mutual orthogonality of random wave functions.', '1906.06206-2-47-4': 'We will see below that an approach in terms of Gaussian and non-Gaussian contractions can be formulated to describe more general correlation functions.', '1906.06206-2-48-0': 'We note that our theory can be extended to account for different forms of the quantum chaotic wave function [MATH].', '1906.06206-2-48-1': 'This may appear, for example, for non-homogeneous [MATH].', '1906.06206-2-48-2': 'In this case, the form of the function [MATH] would change, however the algebraic structure of our theory would remain.', '1906.06206-2-49-0': 'In order to evaluate Eq. [REF], we thus use Eq. [REF] to write [MATH] in terms of the random wave functions [MATH], and non-interacting matrix elements [MATH] and [MATH].', '1906.06206-2-49-1': 'We then use the self-averaging property of random matrices (which we discuss in [REF], and prove in Appendix [REF]), and obtain the relevant correlation functions [MATH].', '1906.06206-2-50-0': '## Computing Correlation Functions', '1906.06206-2-51-0': 'As we have seen, it is important to have a systematic approach to obtaining correlation functions for this model.', '1906.06206-2-51-1': 'This is a non-trivial task as the random wave functions of our theory are not Gaussian independent variables, but include an effective interaction due to the orthogonality condition [MATH] [CITATION].', '1906.06206-2-51-2': 'Our approach to the statistical theory of random wave functions is summarized in Appendix [REF].', '1906.06206-2-52-0': 'Below, we present such a systematic approach to obtaining arbitrary correlation functions in terms of contractions representing the Gaussian and non-Gaussian terms in the four-point correlator (which is the largest non-factorizable correlation function of our theory).', '1906.06206-2-53-0': 'The four-point correlation function of Eq. [REF] may be understood in terms of the contractions of non-interacting indices, indeed it can be seen to be the sum of a Gaussian contraction [MATH] and non-Gaussian contractions, given by [EQUATION] where [EQUATION]', '1906.06206-2-53-1': 'We reserve the double line contraction notation of Eq. [REF] for the non-Gaussian case.', '1906.06206-2-53-2': 'Note that these must occur in pairs of contractions between different interacting indices [MATH].', '1906.06206-2-54-0': 'Now, we can see from Eq. [REF] that each contracted pair of indices contributes a Kronecker-[MATH] symbol, and thus, when the correlation function is summed over its non-interacting indices, the number of summations is reduced.', '1906.06206-2-54-1': 'We see that as each [MATH] contributes a factor on the order [MATH], and a summation on the order [MATH], a reduced summation will act to render a term negligible in comparison to a term with no such reduction.', '1906.06206-2-54-2': 'Further, we see that the contribution of the non-Gaussian term Eq. [REF] is of order [MATH], whereas that of the Gaussian term is [MATH], and thus [MATH], and as such, one can see that for the non-Gaussian contractions to contribute, they must be acted on my an extra summation.', '1906.06206-2-54-3': 'Indeed, one can see that this occurs for one of the two non-Gaussian terms when one has repeated summations, i.e. [MATH] in Eq. [REF].', '1906.06206-2-55-0': 'For further details we refer the reader to Ref. [CITATION], and the calculations in Appendices [REF] and [REF].', '1906.06206-2-55-1': 'Here we have seen the key intuition, however: that repeated indices in correlation functions leads to the dominant contribution of contractions that would otherwise have contracted the pair of equal indices.', '1906.06206-2-56-0': '## Assumptions on Observables', '1906.06206-2-57-0': 'After obtaining the relevant correlation function, one needs to perform the summations over remaining indices.', '1906.06206-2-57-1': 'See, for example, the simple case of Eq. [REF] above.', '1906.06206-2-57-2': 'To perform the summations, certain assumptions on the form of observable matrix elements in the non-interacting basis must be made.', '1906.06206-2-57-3': 'We note that in this basis, local system observables are usually of a known form.', '1906.06206-2-58-0': 'Our theory relies on assumptions that we expect to be satisfied for such local observables.', '1906.06206-2-58-1': 'The key assumption is related to the behaviour of matrix elements, which must have a well defined average, that does not vary pathologically with energy.', '1906.06206-2-58-2': 'A more formal definition of our assumption can be written in terms of the function [MATH], as, [EQUATION] with [MATH], and [EQUATION]', '1906.06206-2-58-3': 'We will see that this assumption will be necessary in order to compute summations over the non-interacting indices.', '1906.06206-2-58-4': 'In this section we explain in more detail the requirements on the form of [MATH] for Eq. [REF] to be valid, as well as the physical interpretation of the assumption.', '1906.06206-2-59-0': 'The essential assumption here, which we label smoothness of [MATH], as in Ref. [CITATION], is that the microcanonical average [MATH] changes slowly over the width [MATH] of the function [MATH].', '1906.06206-2-59-1': 'We showed in Ref. [CITATION] that this is the case under the assumptions, [EQUATION] which thus leads us to two reasonable conditions,', '1906.06206-2-60-0': 'There are many states in the energy width [MATH] The microcanonical average changes slowly over the width [MATH].', '1906.06206-2-61-0': "We note that the latter condition, combined with the fact that the microcanonical average and time average are equal (which is shown below), is equivalent to the statement that the time-average of the observable is not sensitive to the particular initial state (microstate), rather, it's macroscopic energy.", '1906.06206-2-61-1': 'In fact, one can see that the conditions [REF] are precisely those required in order to define a microcanonical average that does not vary pathologically with small changes in the energy window.', '1906.06206-2-61-2': 'In this sense, this assumption is the minimal assumption one would expect to require for thermalization to a microcanonical average that changes smoothly with initial state energy to occur.', '1906.06206-2-62-0': 'We further note that in the consideration of time evolution below, we will consider more general observables that are not necessarily diagonal in the non-interacting basis, but fulfil a sparsity condition.', '1906.06206-2-62-1': 'This can be written as [MATH], where for a given observable there is a non-extensive number [MATH] of groups of non-zero matrix elements at given energy widths, such that after the course graining procedure the observable matrix elements are non-zero for energy gaps [MATH] that are the possible energy gaps of [MATH].', '1906.06206-2-62-2': 'This form can be seen [CITATION] to be reasonable for local observables.', '1906.06206-2-62-3': 'We note that it is of course possible to find observables that do not fulfil this assumption, although it is easily seen to be true for e.g. local Pauli operator observables.', '1906.06206-2-62-4': 'We will see below that our treatment of time evolution could potentially also capture a wider range of observables as well, if the form in the non-interacting basis is known.', '1906.06206-2-62-5': "In the following we refer to observables fulfilling the above assumptions as 'generic' observables.", '1906.06206-2-63-0': '# Equilibration Dynamics', '1906.06206-2-64-0': "In this section we present a description of the time dependence of 'generic' observables as defined above, from an arbitrary initial condition [EQUATION]", '1906.06206-2-64-1': 'This calculation may be performed by exploiting the methods outlined in Section [REF].', '1906.06206-2-64-2': 'The general approach may be summarized in three steps: i) Writing the observable time dependence in terms of parameters in the non-interacting basis, ii) computing the relevant correlation functions (see Section [REF]), and iii) performing summations using the assumptions on observables (see Section [REF]).', '1906.06206-2-65-0': 'Proceeding as such, we write the time dependent density operator in the form, [EQUATION] which may be used to obtain the time evolved observable expectation value by [MATH].', '1906.06206-2-65-1': 'By taking the trace over the interacting basis [MATH], we thus obtain [EQUATION]', '1906.06206-2-65-2': 'Noting the so-called diagonal ensemble contribution is defined by, [EQUATION] which can be seen to be equal to the long-time average value of the observable [EQUATION] assuming no degenerate energy levels, we thus define [EQUATION]', '1906.06206-2-65-3': 'Using that [MATH], we have [EQUATION]', '1906.06206-2-65-4': 'We see that, using the self-averaging property, this depends on the four-point correlation function given by Eq. [REF], such that [EQUATION]', '1906.06206-2-65-5': 'The third term can be shown to be negligible, proof of which is given in Appendix [REF].', '1906.06206-2-65-6': 'We note that it is this bound that requires the sparsity assumption above.', '1906.06206-2-65-7': 'We now use the smoothness assumption, exploiting Eqs. [REF] and [REF], we obtain, [EQUATION] where to obtain the first term one may note that [MATH], and make the change of variables [MATH] to perform the integrals over the new variables.', '1906.06206-2-65-8': 'Here [MATH] is the free evolution of the observable under the Hamiltonian [MATH].', '1906.06206-2-65-9': 'The second term may be re-expressed by defining [MATH], to obtain, [EQUATION] where [MATH].', '1906.06206-2-65-10': 'Noting, then, that as [MATH] is peaked around [MATH], and that [MATH] changes slowly over the width [MATH] of the function [MATH], we can make the replacement [MATH].', '1906.06206-2-65-11': 'This allows the summations over [MATH] to be performed, which become Fourier transforms of the Lorentzian functions [MATH] in the continuum limit [MATH].', '1906.06206-2-65-12': 'We thus find, [EQUATION]', '1906.06206-2-65-13': 'Noting that at [MATH] we by definition have [MATH], we obtain that [MATH].', '1906.06206-2-65-14': 'Noting Eq. [REF], we see that the equality of the time and microcanonical averages is derived from our RMT approach.', '1906.06206-2-65-15': 'Thus, using the definition in Eq. [REF], we obtain [EQUATION]', '1906.06206-2-65-16': 'This is the same as that obtained in Reference [CITATION] for pure-states.', '1906.06206-2-66-0': 'The approach outlined above is valid assuming that the decay rate [MATH] is constant in energy.', '1906.06206-2-66-1': 'In fact, for the system under consideration, we have to allow [MATH] to change with the initial state energy, such that [MATH] (note that here the change in DOS does not affect the calculation).', '1906.06206-2-66-2': 'Accounting for this, rather than Eq. [REF], we obtain [EQUATION]', '1906.06206-2-66-3': 'Now, for our system we have that, as [MATH], the microcanonical average [MATH] for all [MATH] in the bulk of the spectrum.', '1906.06206-2-66-4': 'Also, using that for our proposed experimental protocol, both the initial state and observable are diagonal in the non-interacting basis, we have [EQUATION]', '1906.06206-2-66-5': 'We have that, as the initial state [MATH] for all initial device states, [MATH] for all non-zero [MATH], and thus [EQUATION]', '1906.06206-2-66-6': 'We thus wish to obtain the value [MATH] that will be obtained when measuring the decay of an observable.', '1906.06206-2-66-7': 'To find this, one may simply consider the time integration of the evolution obtained above from the initial state, [EQUATION] describing our probe-bath model, with an initial finite temperature bath state at inverse temperature [MATH].', '1906.06206-2-66-8': 'The time integration is then, [EQUATION] where we have used in the second line that [MATH], and defined the thermal average [MATH] at inverse temperature [MATH], and we have defined [MATH] as the decay rate of the initial state [MATH].', '1906.06206-2-66-9': 'We thus see that it is the thermal average of the inverse decay rate that is measured by a fit to the time dependence of an observable.', '1906.06206-2-67-0': 'The integral form of the thermal average of a function [MATH], is given by, [EQUATION] with [MATH].', '1906.06206-2-67-1': 'We will see below that the FDT will be initially expressed in terms of a different average over the values [MATH].', '1906.06206-2-67-2': 'This difference is resolved in Appendix [REF], where we re-express our FDT in terms of the thermal average above.', '1906.06206-2-67-3': 'We show that the form differs only by a constant that is independent of [MATH] and the coupling strength, and thus the scaling with Hilbert space dimension remains the same, and this difference is not important for our application.', '1906.06206-2-68-0': 'Finally, we note that when [MATH] is approximately constant across the bulk of the initially populated initial states, the thermal average above is approximately equal to the unbiased average appearing in [REF].', '1906.06206-2-68-1': 'We also see that in this case, from Eq. [REF], one expects to observe an exponential decay at the rate [MATH], as in Eq. [REF].', '1906.06206-2-68-2': 'Indeed, this is what we observe in our numerical example in Section [REF] above, and thus we are able to recover the Hilbert space dimension directly in Fig. [REF].', '1906.06206-2-68-3': 'If a non-exponential decay is observed, then the average [MATH] is obtainable via integration as above, and the scaling of the Hilbert space dimension is still obtainable as in Fig. [REF].', '1906.06206-2-69-0': '# Fluctuation-Dissipation Theorem', '1906.06206-2-70-0': '## Derivation from RMT', '1906.06206-2-71-0': 'Here we perform the full derivation of the FDT for the random matrix model described above.', '1906.06206-2-71-1': 'We initially focus on the case of a diagonal initial bath state [MATH].', '1906.06206-2-71-2': 'We then restrict the treatment to the specific protocol outlined in Section [REF], where the initial state is the product of a single probe qubit in a pure state, and a bath in an infinite temperature state, see Eq. [REF].', '1906.06206-2-71-3': 'We will follow a very similar steps as those outlined in the previous section, however we will see here that the correlation function calculation is somewhat more complicated.', '1906.06206-2-72-0': 'The RMT model here is limited to the case of constant decay rate and DOS, we will thus extend the treatment to more realistic cases in the next section.', '1906.06206-2-73-0': 'We are interested in the calculation of the long-time fluctuations, defined by the diagonal ensemble result, [EQUATION]', '1906.06206-2-73-1': 'We begin be writing the initial density operator matrix elements as, [EQUATION] then using Eqs. [REF], and that [MATH], we may write the time fluctuations as, [EQUATION] where coefficients of the initial state are labelled as unprimed indices [MATH], and coefficients of the observable are labelled by primed indices.', '1906.06206-2-74-0': 'Using the self-averaging property of random matrices (see Appendices [REF] and [REF]), we may replace the product of coefficients [MATH] by their ensemble average [MATH]; the above expression may then be written in terms of a sum over 8-point correlation functions, weighted by the initial state and observable coefficients [MATH] and [MATH]: [EQUATION]', '1906.06206-2-74-1': 'Now, using the method of contractions outlined in Section [REF], we see that this 8-point correlation function may be split up into to four-point correlation functions, each consisting of both Gaussian and non-Gaussian contractions.', '1906.06206-2-74-2': 'These are computed explicitly in Appendix [REF], in which we see that there are three dominating contributions to the fluctuations, given by, [EQUATION] and, [EQUATION]', '1906.06206-2-74-3': 'These three terms can be seen as the contributions to the 8-point correlation function arising due to products of Gaussian, non-Gaussian, and mixed Gaussian and non-Gaussian 4-point correlation functions respectively.', '1906.06206-2-74-4': 'In the above, in order to perform the summations over non-interacting indices in Eq. [REF] we define course grained averages of observable elements [MATH] as in Eq. [REF], as well as the mixed averages, [EQUATION]', '1906.06206-2-74-5': 'We thus define [MATH] by [EQUATION] with, [EQUATION]', '1906.06206-2-74-6': 'We now take our bath to be in an initial infinite temperature state, such that [MATH], and [MATH].', '1906.06206-2-74-7': 'As such, [MATH] is in fact energy independent, as the probe Hamiltonian [MATH], so microcanonical averages of probe observables are also energy independent.', '1906.06206-2-74-8': 'Now, as [MATH], all terms in [MATH] are [MATH], we define, [EQUATION] where [MATH], and we have used that [MATH].', '1906.06206-2-74-9': 'We see that [MATH] is a constant of the order of unity that depends only on the observable (e.g. [MATH] for [MATH]).', '1906.06206-2-74-10': 'Finally, taking the thermodynamic limit, such that [MATH] (not that the diagonal terms in the summation can be seen to be negligible, as they contribute to a higher order in [MATH]), we have [EQUATION] which may be evaluated using, [EQUATION] where in the last line we have used that [MATH], such that [MATH].', '1906.06206-2-74-11': 'We then obtain, [EQUATION] where we have used that [MATH].', '1906.06206-2-75-0': 'We can see, then, that Eq. [REF] is of the form of our main result, Eq. (3) of the main text, where [MATH].', '1906.06206-2-75-1': 'What follows is to generalize this relation, allowing the DOS and [MATH] to vary in energy, and for finite temperatures.', '1906.06206-2-76-0': '## Extension to Realistic Systems', '1906.06206-2-77-0': 'The key issue with directly applying the RMT results to realistic models is that in general the DOS, and decay rate, are energy dependent, and thus change over the width of the initial state distribution (this is especially important for the high/infinite temperatures considered here).', '1906.06206-2-77-1': 'In order to account for this, we must then go back to the evaluation of the integrals over energy, in Eq. [REF], and substitute [MATH], and [MATH].', '1906.06206-2-77-2': 'This is justified under the assumption that neither [MATH], nor [MATH], vary appreciably over the width [MATH].', '1906.06206-2-77-3': 'i.e. [MATH], and [MATH].', '1906.06206-2-78-0': 'We see then, that the integral in Eq. [REF] is now [EQUATION] where we have used that, [EQUATION] with [MATH].', '1906.06206-2-78-1': 'This can be seen to be valid as long as the above conditions on [MATH] and [MATH] hold, that is, as long as they change sufficiently slowly over the energy width [MATH].', '1906.06206-2-78-2': 'The contribution of each eigenstate [MATH] to the fluctuations at a given energy is then that of a local (in energy) random matrix model, with a constant DOS and decay rate.', '1906.06206-2-78-3': '[MATH] and [MATH] can then be allowed to change over an energy much wider than a the width of [MATH], as over such energy widths the contributions of relevant eigenstates are independent.', '1906.06206-2-79-0': 'Now, we further define [MATH], and make the change of variables [MATH], and thus obtain [EQUATION] where in the second line we have assumed that [MATH] and [MATH] is approximately constant over the width [MATH].', '1906.06206-2-79-1': 'Now, we define the unbiased average of a function [MATH] as, [EQUATION] (not to be confused with the average [MATH] above) and see that, noting [MATH], [EQUATION] where [MATH] depends only on the choice of observable.', '1906.06206-2-79-2': 'We note that for the random matrix model, as the DOS and [MATH] are both constant in energy, the average [MATH] is equal to the thermal average [MATH] obtained from a fit to the decay of an observable (see Section [REF] above).', '1906.06206-2-79-3': 'In the case above, however, where the DOS and [MATH] change in energy, the unbiased average decay rate is not necessarily the same as that obtained from a fit to the decay.', '1906.06206-2-79-4': 'We may fix this problem directly, as we do in the last section, where we see that the unbiased thermal averages may be replaced by regular thermal averages weighted by the DOS at the expense of a constant that depends on the functional form of [MATH] and [MATH] (but importantly, not on [MATH], or the coupling strength).', '1906.06206-2-79-5': 'We can also see, that if [MATH] is approximately constant over the width of the DOS, which is often the case in such systems (in fact from Eq. [REF] this can be seen to be the case if an exponential decay of the observable is observed), then the biased and unbiased thermal averages of [MATH] are approximately equal for [MATH], and Eq. [REF] may be directly experimentally confirmed as in Fig. [REF].', '1906.06206-2-80-0': '## Finite Temperature FDT', '1906.06206-2-81-0': 'In this section we extend the above approach to finite temperature initial bath states, where the initial state is described by [EQUATION] where the joint probe-bath Hamiltonian eigenbasis is built by ordering product states such that [MATH] odd) , [MATH] even).', '1906.06206-2-81-1': 'In this case, we have [EQUATION] where [MATH], when the bath is initially a finite temperature state at inverse temperature [MATH], and the probe qubit is initially in state [MATH].', '1906.06206-2-81-2': 'We thus obtain for the microcanonical averages of [MATH], assuming that [MATH], [EQUATION] and [EQUATION] such that [MATH].', '1906.06206-2-81-3': 'Now, our most general form for the long-time fluctuations (which assumes only the ability to define the required microcanonical averages that vary smoothly over a width [MATH]) is [EQUATION] where [MATH], and [MATH] is written in Eq. [REF].', '1906.06206-2-81-4': 'Indeed, noting that the mixed average [MATH], where [MATH], and that as each term in [MATH] is [MATH] (see Eq. [REF]), we may define [EQUATION] so [EQUATION]', '1906.06206-2-81-5': 'This may be evaluated, including a variable DOS [MATH], as in the main text for the infinite temperature case, via [EQUATION] where in the second line we have made the change of variables [MATH] with [MATH] and [MATH], and used that [MATH] as in the main text.', '1906.06206-2-81-6': 'We now define the unbiased thermal average of the function [MATH] as, [EQUATION] where [MATH].', '1906.06206-2-81-7': 'Now, we have [EQUATION]', '1906.06206-2-81-8': 'Noting, then, that [MATH], and [MATH], we recover the infinite temperature case as required, [EQUATION]', '1906.06206-2-81-9': 'We note that, unlike in the RMT case above, the average [MATH] is not equal to the thermal average [MATH], which is that obtained from a fit to the decay.', '1906.06206-2-81-10': 'In Appendix [REF] we show how the FDT may be defined in terms of this thermal average.', '1906.06206-2-81-11': 'Importantly for our proposed application, we obtain that the FDT in this form is related simply by a constant [MATH], defined in Appendix [REF], that does not depend on the the size of the device or coupling strength (within the weak coupling regime).', '1906.06206-2-81-12': 'For infinite temperatures we thus have [EQUATION]', '1906.06206-2-81-13': 'Therefore, we can directly relate the measured inverse decay rate [MATH] to the time-averaged fluctuations, and from measurement of each for changing device size or coupling strength, as shown in the numerical experiments of Section [REF], yields information on the scaling of the Hilbert space dimension.', '1906.06206-2-81-14': 'We finally note that it is simply the lack of direct knowledge of the constant [MATH] which prevents measurements where there is a non-negligible change in the decay rate with energy (a non-exponential decay to equilibrium) from constituting a direct measurement of the value of the Hilbert space dimension.', '1906.06206-2-81-15': 'This constant depends on the functional form [MATH] and [MATH], and thus, if these are known, inference of the Hilbert space dimension itself is thus obtainable.', '1906.06206-2-82-0': 'Finally, we note that the finite temperature approach above can be extended to the low temperature regime, as shown in Appendix [REF], from which we can recover the pure state FDT found in Ref. [CITATION] in the low temperature limit.', '1906.06206-2-83-0': '# Discussion', '1906.06206-2-84-0': 'The results shown above demonstrate how the chaotic dynamics of thermalization may be exploited in order to gain information on the complexity of the unitary quantum dynamics of a system.', '1906.06206-2-84-1': 'We have proposed an experimentally viable protocol, by which measurements of a local observable of a probe qubit may be exploited to measure the Hilbert space dimension of an ergodic quantum device, initialized in an infinite temperature state.', '1906.06206-2-84-2': 'We note that this measures the dimension of the states directly involved in dynamics only, and thus provides a more accurate measure of the complexity of the dynamics than a simple estimate of the Hilbert space dimension from the number of qubits.', '1906.06206-2-84-3': "In this sense, such a measurement of a large enough quantum device, if shown to be ergodic in the sense outlined above, would be a convincing indicator of the so called 'quantum supremacy' of the quantum device.", '1906.06206-2-85-0': 'On a practical level, for a generic non-integrable Hamiltonian, our results may be observed in two ways: measurement of a probe observable for (i) changing the number of qubits/ions/... in the quantum device (as in Figs. [REF] and [REF]) , or (ii) changing the probe-bath coupling (as in Fig. [REF]).', '1906.06206-2-85-1': 'The latter is perhaps the simplest experimental methodology, which we show can confirm the ergodic behaviour of a system, that is, that the unitary dynamics requires an extensive proportion of the Hilbert space, by showing a linear relationship between the long-time fluctuations and decay rate.', '1906.06206-2-85-2': 'For a model where the device size may be altered, our FDT provides even deeper insight, allowing also for the experimental observation of the scaling of the Hilbert space dimension with system size.', '1906.06206-2-86-0': 'For cases where an exponential decay to equilibrium is observed, which we show implies that the decay rate is constant over a large range of energies, our method allows the experimenter to access the numerical value of the Hilbert space dimension itself, not simply its scaling with size or coupling strength.', '1906.06206-2-86-1': 'This can be obtained from a single time trace of the decay to equilibrium of the observable, from the measurement of the decay rate, and fluctuations around equilibrium.'}

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['1906.06206-2-64-2', '1906.06206-3-64-2'], ['1906.06206-2-54-0', '1906.06206-3-54-0'], ['1906.06206-2-54-1', '1906.06206-3-54-1'], ['1906.06206-2-54-2', '1906.06206-3-54-2'], ['1906.06206-2-54-3', '1906.06206-3-54-3'], ['1906.06206-2-39-0', '1906.06206-3-39-0'], ['1906.06206-2-39-1', '1906.06206-3-39-1'], ['1906.06206-2-75-0', '1906.06206-3-75-0'], ['1906.06206-2-75-1', '1906.06206-3-75-1'], ['1906.06206-2-32-0', '1906.06206-3-32-0'], ['1906.06206-2-32-1', '1906.06206-3-32-1'], ['1906.06206-2-32-2', '1906.06206-3-32-2'], ['1906.06206-2-3-0', '1906.06206-3-3-0'], ['1906.06206-2-3-1', '1906.06206-3-3-1'], ['1906.06206-2-3-2', '1906.06206-3-3-2'], ['1906.06206-2-67-0', '1906.06206-3-67-0'], ['1906.06206-2-67-1', '1906.06206-3-67-1'], ['1906.06206-2-67-2', '1906.06206-3-67-2'], ['1906.06206-2-67-3', '1906.06206-3-67-3'], ['1906.06206-2-4-0', '1906.06206-3-4-0'], ['1906.06206-2-4-1', '1906.06206-3-4-1'], ['1906.06206-2-4-2', '1906.06206-3-4-2'], ['1906.06206-2-58-0', '1906.06206-3-58-0'], 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[]

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{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://creativecommons.org/licenses/by/4.0/', '3': 'http://creativecommons.org/licenses/by/4.0/'}

https://arxiv.org/abs/1906.06206

{'1906.06206-3-0-0': 'Quantum devices, such as quantum simulators, quantum annealers, and quantum computers, may be exploited to solve problems beyond what is tractable with classical computers.', '1906.06206-3-0-1': "This may be achieved as the Hilbert space available to perform such 'calculations' is far larger than that which may be classically simulated.", '1906.06206-3-0-2': 'In practice, however, quantum devices have imperfections, which may limit the accessibility to the whole Hilbert space.', '1906.06206-3-0-3': 'We thus determine that the dimension of the space of quantum states that are available to a quantum device is a meaningful measure of its functionality, though unfortunately this quantity cannot be directly experimentally determined.', '1906.06206-3-0-4': 'Here we outline an experimentally realisable approach to obtaining the required Hilbert space dimension of such a device to compute its time evolution, by exploiting the thermalization dynamics of a probe qubit.', '1906.06206-3-0-5': 'This is achieved by obtaining a fluctuation-dissipation theorem for high-temperature chaotic quantum systems, which facilitates the extraction of information on the Hilbert space dimension via measurements of the decay rate, and time-fluctuations.', '1906.06206-3-1-0': '# Introduction', '1906.06206-3-2-0': 'The ability to control and manipulate microscopic systems at the single particle level is an essential requirement for many quantum technologies.', '1906.06206-3-2-1': 'Experimental setups where atoms or qubits can be arranged in ordered structures and studied in quantum non-equilibrium states include neutral atoms in optical lattices [CITATION], trapped ions [CITATION], Rydberg atoms [CITATION], and superconducting circuits[CITATION].', '1906.06206-3-2-2': 'These systems can be used for the quantum simulation of many-body models, or different forms of digital or adiabatic quantum computing.', '1906.06206-3-2-3': 'Most of these physical setups have limitations in the accessibility to certain observables.', '1906.06206-3-2-4': 'Thus, having extra tools to characterize quantum systems in a simple and efficient way can be useful in the diagnosis and certification of quantum devices.', '1906.06206-3-3-0': 'One of the most prominent properties of a quantum device is its size in terms of the dimension of the associated Hilbert space.', '1906.06206-3-3-1': 'The size of a quantum computer or simulator is often given in terms of number of qubits, such that the Hilbert space dimension is [MATH] for [MATH] qubits.', '1906.06206-3-3-2': 'This is a measure that ignores the effect of disorder or the possible lack of connectivity between different zones in the device.', '1906.06206-3-4-0': 'A more useful quantity would be the number of eigenstates of the Hamiltonian that take part in the quantum dynamics, which is bounded in this case by [MATH], but would exclude the degrees of freedom that do not contribute to the evolution of the initial state.', '1906.06206-3-4-1': 'Thereby establishing an effective Hilbert space dimension that more accurately describes the complexity of the system, in terms of some effective fully connected Hamiltonian.', '1906.06206-3-4-2': 'This Hilbert space dimension is, however, an elusive measure in realistic experimental situations.', '1906.06206-3-5-0': 'In this work we show that the equilibration dynamics [CITATION] of a quantum system can be used to extract such information on the dimension of the Hilbert space of a quantum device, in terms of the effective number of states that contribute to the dynamics of a local observable.', '1906.06206-3-5-1': 'Indeed, advancements in quantum technologies described above have inspired a bounty of theoretical work in the field of quantum thermalization [CITATION].', '1906.06206-3-5-2': "In the following, we aim to help 'bridge the gap' between theoretical and experimental work in this field [CITATION].", '1906.06206-3-6-0': 'We assume a quantum quench scenario [CITATION] in which a quantum system is initialized in a fully-decohered, infinite temperature state, except for a subsystem that acts as a sensor and is prepared in a pure state.', '1906.06206-3-6-1': "For simplicity, we assume that this subsystem is a single qubit, which we refer to as the 'probe' qubit.", '1906.06206-3-6-2': 'The relaxation dynamics of the probe qubit depends on the details of the underlying structure of the Hamiltonian, however, in the most generic case of non-integrable systems, an appropriate description can be given in terms of random matrix theory (RMT) [CITATION].', '1906.06206-3-6-3': 'We show that the time-fluctuations of the probe in the long-time limit contain information about the Hilbert space dimension of the device.', '1906.06206-3-7-0': 'Our article is structured as follows.', '1906.06206-3-7-1': 'Firstly, in Section [REF] we present the basic scheme and summarize our main result, which relies on an infinite-temperature fluctuation dissipation theorem (FDT) [CITATION] for the dynamics of the probe qubit in order to extract information on the scaling of the effective Hilbert space of a quantum device.', '1906.06206-3-7-2': 'We continue, presenting numerical calculations that validate our predictions via exact diagonalization of a spin chain Hamiltonian in Section [REF].', '1906.06206-3-7-3': 'We then outline in more detail our RMT model in Section [REF].', '1906.06206-3-7-4': 'In Section [REF] we derive an expression for the time dependence of generic observables in chaotic quantum systems, and discuss how this can be exploited for our measurements.', '1906.06206-3-7-5': 'We then derive the FDT, and extensions from the RMT model, and to finite temperatures, in Section [REF].', '1906.06206-3-7-6': 'Finally, we summarise our findings in Section [REF].', '1906.06206-3-7-7': 'Further details and proofs are included in Appendices.', '1906.06206-3-7-8': 'We note that this is arranged such that our key findings can be understood from sections [REF] and [REF], with the detailed calculations presented later in the text.', '1906.06206-3-8-0': '# Setup and main results', '1906.06206-3-9-0': '## Proposed setup', '1906.06206-3-10-0': "We assume that we have a quantum system (from here on, 'quantum device') made up of a single 'probe' qubit, initialized in a pure state, which at [MATH] is coupled to the rest of the quantum device (referred to as the 'bath').", '1906.06206-3-10-1': 'We initialize the bath in an infinite temperature state.', '1906.06206-3-10-2': 'This is sketched in Figure [REF].', '1906.06206-3-10-3': 'This can be routinely achieved, for example, in a quantum computer device that has not been properly initialized.', '1906.06206-3-10-4': 'Since quantum devices always suffer from some kind of decoherence, creating an infinite temperature (or fully decohered) state is typically a simple task.', '1906.06206-3-10-5': 'We then let this qubit evolve in time and reach an equilibrium state.', '1906.06206-3-10-6': 'The initial state is thus, [EQUATION] [MATH] is the density matrix of the bath in the fully decohered state, [EQUATION] where [MATH] is a set of orthonormal wave functions in the bath, which we take as the eigenstates of the bath Hamiltonian [MATH].', '1906.06206-3-10-7': '[MATH] is the Hilbert space dimension of the bath, and is the parameter on which we wish to infer information via measurements of the probe qubit.', '1906.06206-3-10-8': 'Later on the fully decohered state condition will be relaxed, allowing for finite temperatures.', '1906.06206-3-11-0': 'The system evolves under the interacting Hamiltonian, [EQUATION] where [MATH] is the Hamiltonian of the uncoupled probe + bath device, and we assume that the probe qubit does not evolve at all in the absence of coupling to the bath, which is given by the operator [MATH].', '1906.06206-3-11-1': 'A prominent role in the following discussions will be played by the eigensystems of the interacting Hamiltonian, [EQUATION] and the eigensystem of the uncoupled probe-bath system [EQUATION]', '1906.06206-3-11-2': 'All throughout this work we will use the convention that indices [MATH], [MATH] refer to summations over eigenstates of [MATH], whereas [MATH] refer to eigenstates of [MATH].', '1906.06206-3-12-0': 'We focus on the quantum dynamics of the expectation value of a probe observable, which we take for concreteness to be [MATH].', '1906.06206-3-12-1': 'Note that, however, the results should not depend on the particular choice for the probe observable, since [MATH] has no probe energy term.', '1906.06206-3-12-2': 'The probe-bath coupling [MATH], may introduce a dependence on the chosen observable, but the results below should hold in the general case of quantum chaotic or non-integrable systems.', '1906.06206-3-13-0': 'The quantity under study, the time-averaged fluctuations of the probe observable, is defined by [EQUATION] where [MATH].', '1906.06206-3-13-1': 'Assuming only that the many-body eigenenergies are non-degenerate, and that also their energy gaps are non-degenerate, we may express the time fluctuations in terms of matrix elements between eigenstates of the coupled probe-bath system [CITATION], [EQUATION] where [MATH], and [MATH].', '1906.06206-3-14-0': '## Summary of main results of this work', '1906.06206-3-15-0': 'In many practical situations the quantum device is a non-integrable, quantum chaotic system.', '1906.06206-3-15-1': 'We expect that in such cases a qualitative description can be obtained by assuming that [MATH] is a random matrix.', '1906.06206-3-15-2': 'This assumption leads to a statistical theory for the many body wave functions in Eqs. ([REF], [REF]) [CITATION], [EQUATION] where summations are understood to be taken from [MATH] to [MATH], the dimension of the Hilbert space of the device.', '1906.06206-3-15-3': 'We will refer to [MATH] as quantum chaotic or random wave functions.', '1906.06206-3-15-4': 'We assume that [MATH] belongs to the Gaussian Orthogonal Ensemble (GOE), appropriately scaled by a coupling strength [MATH].', '1906.06206-3-15-5': 'In Ref. [CITATION] we showed that this model can be solved and it allows us to calculate matrix elements in the interacting basis.', '1906.06206-3-16-0': 'Special care must be taken to translate the probe initial state defined in Eq. ([REF]) in the random matrix formalism.', '1906.06206-3-16-1': 'Without loss of generality we can assume that non-interacting eigenfunctions are ordered such that odd (even) values of [MATH] correspond to the probe qubit in state [MATH]).', '1906.06206-3-16-2': 'This leads to the initial condition [EQUATION]', '1906.06206-3-16-3': 'In the following sections we prove that random matrix theory leads to a relation between the time-fluctuations and the typical decay rate of the probe qubit.', '1906.06206-3-16-4': 'This relation is the main result of this work: [EQUATION] where [MATH] is the average inverse decay rate of the qubit.', '1906.06206-3-16-5': 'This average is an unbiased average over all initially populated initial state energies, which spans the entire energy range [MATH] of the device due to the initial infinite temperature bath state.', '1906.06206-3-16-6': 'We will see that this unbiased average originates from the decay to equilibrium of each state [MATH] contributing to the initial state [REF].', '1906.06206-3-16-7': 'In Sections [REF] and [REF] we will see that this can be related to the decay rate obtained from a fit to the decay to equilibrium of a local probe observable in our current set up.', '1906.06206-3-16-8': 'This quantity is thus simply an average of the decay rates experienced by the probe qubit.', '1906.06206-3-17-0': 'The quantity [MATH], with [MATH] the total number of qubits, depends on the size of the system in the following way, [EQUATION] [MATH] is the average density of states (DOS) of the system, defined analogously to the average decay rate above (see also Eq. [REF]).', '1906.06206-3-17-1': '[MATH] is the bath Hilbert space dimension, and finally, [MATH] is a constant of order one that does not depend on the size of the system or coupling strength.', '1906.06206-3-18-0': 'Eq. ([REF]) can be understood as a fluctuation-dissipation relation, which relates the time-fluctuations in the steady-state with the decay rate after a quantum quench.', '1906.06206-3-18-1': 'The ratio between fluctuations and average decay time allows us to quantify the dimension of the Hilbert space over which the ergodic quantum dynamics takes place.', '1906.06206-3-18-2': 'In a physical system if [MATH] couples the probe qubit to the whole spectrum of the quantum device, we expect that the function [MATH], where [MATH] is the number of qubits in the part of the device that acts as a bath.', '1906.06206-3-19-0': 'Our main result, Eq. ([REF]), can be obtained by following these steps:', '1906.06206-3-20-0': '(i) We consider the random matrix model with an homogeneous coupling matrix [MATH], and calculate [MATH] by extending the formalism developed in [CITATION] to mixed states (see Section [REF]).', '1906.06206-3-20-1': 'To carry out this calculation we assume a constant DOS, [MATH], with [MATH] the mean energy separation.', '1906.06206-3-21-0': 'This calculation allows us to predict an exponential decay for a probe observable of the form: [EQUATION] where [MATH] is the diagonal ensemble of the observable [MATH], which we find to be equal to the long-time average of [MATH] as required, and [MATH] refers to the free evolution of the observable [MATH] under the Hamiltonian [MATH].', '1906.06206-3-21-1': 'This result is derived in Section [REF].', '1906.06206-3-21-2': 'This is analogous to the result in Reference [CITATION] for pure-states.', '1906.06206-3-21-3': 'We note that the same Equation has similarly been obtained in Ref. [CITATION], which allows also for the perturbation matrix [MATH] to be inhomogeneous.', '1906.06206-3-21-4': 'In the specific case of interest here, Eq. [REF] becomes, [EQUATION] as [MATH] and [MATH].', '1906.06206-3-22-0': 'Using this random matrix model we also obtain a preliminary version of the infinite temperature fluctuation-dissipation theorem, Eq. ([REF]), with [MATH].', '1906.06206-3-23-0': '(ii) In a realistic system both the DOS and the coupling strength will depend on the energy.', '1906.06206-3-23-1': 'This leads to energy-dependent qubit decay rates, [MATH], and DOS, [MATH].', '1906.06206-3-23-2': 'Due to the infinite-temperature initial condition, it is not possible a priori to approximate those quantities to any single value, since the quantum dynamics of the probe qubit result from contributions from all possible initial states.', '1906.06206-3-23-3': 'It is thus necessary to extend the RMT formalism to allow for variations in both the DOS and the decay rate in energy over the width of the initial mixed state.', '1906.06206-3-24-0': 'It is also important to note that when applied to a realistic system [MATH] should be thought of as the number of eigenstates that may contribute to the dynamics of a local observable - it is thus a measure of the effective Hilbert space dimension in the sense of the number of degrees of freedom that may be explored from a given initial state.', '1906.06206-3-24-1': 'This measure accounts for effects such as locality of interactions, and disorder, which will reduce the Hilbert space available for evolution of the system.', '1906.06206-3-24-2': 'As such, [MATH] will in general be bounded by the total Hilbert space dimension, but is a more realistic measure of the size of the explored Hilbert space in the thermalization dynamics of the device.', '1906.06206-3-25-0': 'In Section [REF] we show that the decay rate observed from such an initial state, defined with an energy dependent coupling strength, is the thermal average decay rate over the energy width defined by the initial state.', '1906.06206-3-26-0': 'We formulate the generalization of the FDT in Section [REF], however a brief summary of the approach is as follows: To account for the energy variation of the DOS and decay rate we instead make the assumption that both change slowly in energy with respect to the energy width of a single eigenstate.', '1906.06206-3-26-1': 'In this case, one can reformulate the theory such that the random wave functions have a smoothly varying width.', '1906.06206-3-26-2': 'In effect this is the statement that a given random wave function contributes as if it was the eigenstate of a random matrix model with a constant DOS and decay rate, as over the width of the wave function itself these parameters do not change appreciably.', '1906.06206-3-27-0': 'In this case, we see that the average DOS and decay rate over the initially populated initial states contributes to the FDT, as in Eq. [REF].', '1906.06206-3-27-1': 'This is extended to finite temperature systems, where instead a thermal average can be seen to contribute, in Section [REF].', '1906.06206-3-28-0': 'Finally, we may relate the two decay rate averages, as the thermal average observed from the decay differs slightly from that appearing in Eq. [REF].', '1906.06206-3-28-1': 'These can be seen to be related by a constant factor that does not depend on the coupling strength, or bath size, and thus scaling information of the Hilbert space dimension may be recovered.', '1906.06206-3-28-2': 'This is shown in Section [REF], and Appendix [REF], where the FDT is recast in terms of thermal averages.', '1906.06206-3-29-0': 'In Section [REF] we show that when an exponential decay of the form Eq. [REF] is observed, this indicates that [MATH] is approximately constant over the bulk of the initially populated states.', '1906.06206-3-29-1': 'In this case we are able to extract the numerical value of the Hilbert space dimension directly, as the averages ove [MATH] occurring in the FDT, and observed from the decay, are equal.', '1906.06206-3-30-0': 'In summary, we observe the emergence of a classical fluctuation-dissipation theorem, relating the time-fluctuations and decay rate of our probe observable [MATH].', '1906.06206-3-30-1': 'The susceptibility [MATH] in Eq. [REF] can be seen to be related to the Hilbert space dimension of the bath, [MATH], and thus measurements of the decay rate, [MATH], and fluctuations [MATH], which are both obtainable from the time evolution, can be exploited to obtain information on the device Hilbert space dimension, [MATH].', '1906.06206-3-31-0': '# Numerical Experiments', '1906.06206-3-32-0': 'Before going into the technical details of our derivation, we present numerical evidence that confirms the validity of the random matrix approach and our main results.', '1906.06206-3-32-1': 'In this section, we show the application to a spin-chain system using exact diagonalization [CITATION].', '1906.06206-3-32-2': 'From this, we observe the FDT numerically using a realistic experimentally observable model.', '1906.06206-3-33-0': 'In Fig. [REF], we show the manifestation of Eq. [REF] in a spin-chain system described by the Hamiltonian [MATH], where [MATH] is the system Hamiltonian (acting as our probe), [MATH] is our bath Hamiltonian, given by [EQUATION] which acts on sites with index [MATH], which is the probe index.', '1906.06206-3-33-1': 'The probe and bath are coupled by the interaction Hamiltonian, [EQUATION]', '1906.06206-3-33-2': 'Here [MATH] is the device site where the probe is coupled, which we set as 2 throughout.', '1906.06206-3-33-3': 'This spin-chain model may be related to the random matrix toy model, [MATH], via the prescription [MATH], and [MATH].', '1906.06206-3-34-0': 'In particular, we see that, as [MATH], we expect that if all of the available Hilbert space is being utilized in the unitary dynamics we will observe the following scaling: [EQUATION]', '1906.06206-3-34-1': 'This is the relation that we test in Fig. [REF].', '1906.06206-3-35-0': 'It is important to note that this exponential scaling of [MATH], Eq. [REF], is expected from not only the contribution of [MATH], but also from the average DOS, [MATH].', '1906.06206-3-35-1': 'This average is often trivially obtained, as for example, for an ensemble of [MATH] two-level systems [MATH], where [MATH] is the range of energies available [MATH] (which may itself change with [MATH]), regardless of the microscopic properties of the DOS.', '1906.06206-3-35-2': 'We thus also study the quantity [MATH], as this quantity has no dependence on the DOS, and an observation of the exponential scaling in system size is confirmation that, indeed, [MATH].', '1906.06206-3-35-3': 'This is shown in Fig. [REF], where we observe an exponential scaling of the Hilbert space dimension, with [MATH], compared to [MATH] if the entire Hilbert space were explored in the dynamics.', '1906.06206-3-36-0': 'We further observe in Figs. [REF] and [REF], that the FDT similarly applies at finite temperatures [MATH].', '1906.06206-3-36-1': 'The extension of our theoretical approach to this case is discussed below, with additional details given in Appendix [REF].', '1906.06206-3-36-2': 'Indeed, we can show that for high temperatures, such that [MATH], we obtain an FDT of the same form as Eq. [REF], by employing a high energy cut-off [MATH] to the bath state occupation.', '1906.06206-3-37-0': 'For finite temperatures we show below that the FDT depends on the partition function [MATH] itself, rather than the Hilbert space dimension.', '1906.06206-3-37-1': 'Indeed, one can see that in the infinite temperature limit [MATH].', '1906.06206-3-38-0': 'Finally, we see that when the observable is found to decay exponentially to its equilibrium value, this indicates that the decay rate is approximately constant over the bulk of the initially occupied states.', '1906.06206-3-38-1': 'This is shown in Section [REF].', '1906.06206-3-38-2': 'Exploiting this observation, we are able to directly obtain the Hilbert space dimension, as for an infinite temperature initial bath state the average [MATH] is equal to the measured decay rate.', '1906.06206-3-38-3': 'The bath Hilbert space dimension [MATH], as calculated from Eq. [REF], is plotted for varying device sizes in Fig. [REF].', '1906.06206-3-38-4': 'Here we observe that [MATH] indeed increases exponentially with systems size, yet is somewhat smaller than its maximum possible value [MATH], which is expected to the locality of interactions within the chain.', '1906.06206-3-39-0': 'We again note that in Fig. [REF] the measurement of [MATH] is a measurement of the explored Hilbert space, or the total number of eigenstates that contribute to the evolution of the initial state.', '1906.06206-3-39-1': 'Thus, for a maximally connected device this would be [MATH], whereas locality of interactions in this case restricts some areas of the Hilbert space.', '1906.06206-3-40-0': 'We note that for models where [MATH] is not approximately constant in energy, which would be marked by a deviation from the exponential decay in Eq. [REF], one would still have access to Figs. [REF]-[REF], and thus scaling information of the Hilbert space dimension, yet the numerical value [MATH] would be obscured.', '1906.06206-3-40-1': 'This is explained in more detail in Section [REF], where we see how information on [MATH] is extracted from the decay, and in Section [REF] and Appendix [REF], where we see how this relates to the observed FDT.', '1906.06206-3-40-2': 'The key detail is that when the FDT is expressed in terms of the same value measured from the decay it differs by a constant, thereby leaving the scaling of [MATH] with system size or decay rate accessible, yet obscuring the numerical value.', '1906.06206-3-41-0': 'We note that in Ref. [CITATION], the current authors obtained a FDT for pure states, which can be seen to be recovered in the low temperature limit, [MATH], for which [MATH] does not depend explicitly on the Hilbert space dimension [MATH].', '1906.06206-3-41-1': 'This can also be analytically seen to be the same as the low temperature limit of our treatment below, which indicates that there is a smooth transition between these two cases.', '1906.06206-3-41-2': 'This is indeed observed in the numerics of Figs [REF] and [REF].', '1906.06206-3-42-0': '# Model', '1906.06206-3-43-0': '## RMT Approach', '1906.06206-3-44-0': 'Our approach relies on the calculation of correlation functions from a statistical theory of random wave functions [MATH].', '1906.06206-3-44-1': 'Here we summarize the essential ingredients to our model, and give details on the calculations in the sections below.', '1906.06206-3-45-0': "Our theory, developed in Ref. [CITATION] by extending Deutch's RMT model [CITATION], can be used to obtain arbitrary correlation functions [MATH], where [MATH] denotes the ensemble average over an ensemble of random matrix perturbations, [MATH], for a [MATH] Hamiltonian of the form [REF], with [MATH], with [MATH] and [MATH] a random matrix selected from the GOE, with [MATH].", '1906.06206-3-45-1': 'In practice, those correlations allow us to calculate any dynamical quantity of interest within the RMT formalism.', '1906.06206-3-46-0': 'To illustrate the use of such correlation functions we briefly consider the simple example of the diagonal observable matrix elements [MATH].', '1906.06206-3-46-1': 'These can be written as [EQUATION]', '1906.06206-3-46-2': 'Now, using the self-averaging property of random matrices, which we prove for this model in Appendix [REF], we see that [EQUATION]', '1906.06206-3-46-3': 'Note that only the random wave functions [MATH] remain inside the ensemble average, as all other factors are in the non-interacting basis and thus do not depend on [MATH].', '1906.06206-3-46-4': 'We thus observe that the diagonal observable matrix elements depend on the correlation function [MATH] (note that we will see how to deal with the summation and non-interacting observable elements in Section [REF] below).', '1906.06206-3-47-0': 'More generally, when calculating more complicated quantities, we have that there is also a non-trivial contribution of a four-point correlation function of the form [MATH] which is given by (for [MATH]), [EQUATION] where [MATH] is defined as [EQUATION] with [MATH].', '1906.06206-3-47-1': '[MATH] is defined similarly to Eq. ([REF]b), with [MATH].', '1906.06206-3-47-2': 'The Lorentzian form of Eq. [REF] is found for a homogeneous perturbation [MATH] [CITATION], selected from the GOE.', '1906.06206-3-47-3': 'We obtain that the four-point correlation function, Eq. [REF], can be described in terms of product of two-point correlators [MATH], as if the random wave function distribution was purely Gaussian, plus a correction term originating from the effective interaction due to the mutual orthogonality of random wave functions.', '1906.06206-3-47-4': 'We will see below that an approach in terms of Gaussian and non-Gaussian contractions can be formulated to describe more general correlation functions.', '1906.06206-3-48-0': 'We note that our theory can be extended to account for different forms of the quantum chaotic wave function [MATH].', '1906.06206-3-48-1': 'This may appear, for example, for non-homogeneous [MATH].', '1906.06206-3-48-2': 'In this case, the form of the function [MATH] would change, however the algebraic structure of our theory would remain.', '1906.06206-3-49-0': 'In order to evaluate Eq. [REF], we thus use Eq. [REF] to write [MATH] in terms of the random wave functions [MATH], and non-interacting matrix elements [MATH] and [MATH].', '1906.06206-3-49-1': 'We then use the self-averaging property of random matrices (which we discuss in [REF], and prove in Appendix [REF]), and obtain the relevant correlation functions [MATH].', '1906.06206-3-50-0': '## Computing Correlation Functions', '1906.06206-3-51-0': 'As we have seen, it is important to have a systematic approach to obtaining correlation functions for this model.', '1906.06206-3-51-1': 'This is a non-trivial task as the random wave functions of our theory are not Gaussian independent variables, but include an effective interaction due to the orthogonality condition [MATH] [CITATION].', '1906.06206-3-51-2': 'Our approach to the statistical theory of random wave functions is summarized in Appendix [REF].', '1906.06206-3-52-0': 'Below, we present such a systematic approach to obtaining arbitrary correlation functions in terms of contractions representing the Gaussian and non-Gaussian terms in the four-point correlator (which is the largest non-factorizable correlation function of our theory).', '1906.06206-3-53-0': 'The four-point correlation function of Eq. [REF] may be understood in terms of the contractions of non-interacting indices, indeed it can be seen to be the sum of a Gaussian contraction [MATH] and non-Gaussian contractions, given by [EQUATION] where [EQUATION]', '1906.06206-3-53-1': 'We reserve the double line contraction notation of Eq. [REF] for the non-Gaussian case.', '1906.06206-3-53-2': 'Note that these must occur in pairs of contractions between different interacting indices [MATH].', '1906.06206-3-54-0': 'Now, we can see from Eq. [REF] that each contracted pair of indices contributes a Kronecker-[MATH] symbol, and thus, when the correlation function is summed over its non-interacting indices, the number of summations is reduced.', '1906.06206-3-54-1': 'We see that as each [MATH] contributes a factor on the order [MATH], and a summation on the order [MATH], a reduced summation will act to render a term negligible in comparison to a term with no such reduction.', '1906.06206-3-54-2': 'Further, we see that the contribution of the non-Gaussian term Eq. [REF] is of order [MATH], whereas that of the Gaussian term is [MATH], and thus [MATH], and as such, one can see that for the non-Gaussian contractions to contribute, they must be acted on my an extra summation.', '1906.06206-3-54-3': 'Indeed, one can see that this occurs for one of the two non-Gaussian terms when one has repeated summations, i.e. [MATH] in Eq. [REF].', '1906.06206-3-55-0': 'For further details we refer the reader to Ref. [CITATION], and the calculations in Appendices [REF] and [REF].', '1906.06206-3-55-1': 'Here we have seen the key intuition, however: that repeated indices in correlation functions leads to the dominant contribution of contractions that would otherwise have contracted the pair of equal indices.', '1906.06206-3-56-0': '## Assumptions on Observables', '1906.06206-3-57-0': 'After obtaining the relevant correlation function, one needs to perform the summations over remaining indices.', '1906.06206-3-57-1': 'See, for example, the simple case of Eq. [REF] above.', '1906.06206-3-57-2': 'To perform the summations, certain assumptions on the form of observable matrix elements in the non-interacting basis must be made.', '1906.06206-3-57-3': 'We note that in this basis, local system observables are usually of a known form.', '1906.06206-3-58-0': 'Our theory relies on assumptions that we expect to be satisfied for such local observables.', '1906.06206-3-58-1': 'The key assumption is related to the behaviour of matrix elements, which must have a well defined average, that does not vary pathologically with energy.', '1906.06206-3-58-2': 'A more formal definition of our assumption can be written in terms of the function [MATH], as, [EQUATION] with [MATH], and [EQUATION]', '1906.06206-3-58-3': 'We will see that this assumption will be necessary in order to compute summations over the non-interacting indices.', '1906.06206-3-58-4': 'In this section we explain in more detail the requirements on the form of [MATH] for Eq. [REF] to be valid, as well as the physical interpretation of the assumption.', '1906.06206-3-59-0': 'The essential assumption here, which we label smoothness of [MATH], as in Ref. [CITATION], is that the microcanonical average [MATH] changes slowly over the width [MATH] of the function [MATH].', '1906.06206-3-59-1': 'We showed in Ref. [CITATION] that this is the case under the assumptions, [EQUATION] which thus leads us to two reasonable conditions,', '1906.06206-3-60-0': 'There are many states in the energy width [MATH] The microcanonical average changes slowly over the width [MATH].', '1906.06206-3-61-0': "We note that the latter condition, combined with the fact that the microcanonical average and time average are equal (which is shown below), is equivalent to the statement that the time-average of the observable is not sensitive to the particular initial state (microstate), rather, it's macroscopic energy.", '1906.06206-3-61-1': 'In fact, one can see that the conditions [REF] are precisely those required in order to define a microcanonical average that does not vary pathologically with small changes in the energy window.', '1906.06206-3-61-2': 'In this sense, this assumption is the minimal assumption one would expect to require for thermalization to a microcanonical average that changes smoothly with initial state energy to occur.', '1906.06206-3-62-0': 'We further note that in the consideration of time evolution below, we will consider more general observables that are not necessarily diagonal in the non-interacting basis, but fulfil a sparsity condition.', '1906.06206-3-62-1': 'This can be written as [MATH], where for a given observable there is a non-extensive number [MATH] of groups of non-zero matrix elements at given energy widths, such that after the course graining procedure the observable matrix elements are non-zero for energy gaps [MATH] that are the possible energy gaps of [MATH].', '1906.06206-3-62-2': 'This form can be seen [CITATION] to be reasonable for local observables.', '1906.06206-3-62-3': 'We note that it is of course possible to find observables that do not fulfil this assumption, although it is easily seen to be true for e.g. local Pauli operator observables.', '1906.06206-3-62-4': 'We will see below that our treatment of time evolution could potentially also capture a wider range of observables as well, if the form in the non-interacting basis is known.', '1906.06206-3-62-5': "In the following we refer to observables fulfilling the above assumptions as 'generic' observables.", '1906.06206-3-63-0': '# Equilibration Dynamics', '1906.06206-3-64-0': "In this section we present a description of the time dependence of 'generic' observables as defined above, from an arbitrary initial condition [EQUATION]", '1906.06206-3-64-1': 'This calculation may be performed by exploiting the methods outlined in Section [REF].', '1906.06206-3-64-2': 'The general approach may be summarized in three steps: i) Writing the observable time dependence in terms of parameters in the non-interacting basis, ii) computing the relevant correlation functions (see Section [REF]), and iii) performing summations using the assumptions on observables (see Section [REF]).', '1906.06206-3-65-0': 'Proceeding as such, we write the time dependent density operator in the form, [EQUATION] which may be used to obtain the time evolved observable expectation value by [MATH].', '1906.06206-3-65-1': 'By taking the trace over the interacting basis [MATH], we thus obtain [EQUATION]', '1906.06206-3-65-2': 'Noting the so-called diagonal ensemble contribution is defined by, [EQUATION] which can be seen to be equal to the long-time average value of the observable [EQUATION] assuming no degenerate energy levels, we thus define [EQUATION]', '1906.06206-3-65-3': 'Using that [MATH], we have [EQUATION]', '1906.06206-3-65-4': 'We see that, using the self-averaging property, this depends on the four-point correlation function given by Eq. [REF], such that [EQUATION]', '1906.06206-3-65-5': 'The third term can be shown to be negligible, proof of which is given in Appendix [REF].', '1906.06206-3-65-6': 'We note that it is this bound that requires the sparsity assumption above.', '1906.06206-3-65-7': 'We now use the smoothness assumption, exploiting Eqs. [REF] and [REF], we obtain, [EQUATION] where to obtain the first term one may note that [MATH], and make the change of variables [MATH] to perform the integrals over the new variables.', '1906.06206-3-65-8': 'Here [MATH] is the free evolution of the observable under the Hamiltonian [MATH].', '1906.06206-3-65-9': 'The second term may be re-expressed by defining [MATH], to obtain, [EQUATION] where [MATH].', '1906.06206-3-65-10': 'Noting, then, that as [MATH] is peaked around [MATH], and that [MATH] changes slowly over the width [MATH] of the function [MATH], we can make the replacement [MATH].', '1906.06206-3-65-11': 'This allows the summations over [MATH] to be performed, which become Fourier transforms of the Lorentzian functions [MATH] in the continuum limit [MATH].', '1906.06206-3-65-12': 'We thus find, [EQUATION]', '1906.06206-3-65-13': 'Noting that at [MATH] we by definition have [MATH], we obtain that [MATH].', '1906.06206-3-65-14': 'Noting Eq. [REF], we see that the equality of the time and microcanonical averages is derived from our RMT approach.', '1906.06206-3-65-15': 'Thus, using the definition in Eq. [REF], we obtain [EQUATION]', '1906.06206-3-65-16': 'This is the same as that obtained in Reference [CITATION] for pure-states.', '1906.06206-3-66-0': 'The approach outlined above is valid assuming that the decay rate [MATH] is constant in energy.', '1906.06206-3-66-1': 'In fact, for the system under consideration, we have to allow [MATH] to change with the initial state energy, such that [MATH] (note that here the change in DOS does not affect the calculation).', '1906.06206-3-66-2': 'Accounting for this, rather than Eq. [REF], we obtain [EQUATION]', '1906.06206-3-66-3': 'Now, for our system we have that, as [MATH], the microcanonical average [MATH] for all [MATH] in the bulk of the spectrum.', '1906.06206-3-66-4': 'Also, using that for our proposed experimental protocol, both the initial state and observable are diagonal in the non-interacting basis, we have [EQUATION]', '1906.06206-3-66-5': 'We have that, as the initial state [MATH] for all initial device states, [MATH] for all non-zero [MATH], and thus [EQUATION]', '1906.06206-3-66-6': 'We thus wish to obtain the value [MATH] that will be obtained when measuring the decay of an observable.', '1906.06206-3-66-7': 'To find this, one may simply consider the time integration of the evolution obtained above from the initial state, [EQUATION] describing our probe-bath model, with an initial finite temperature bath state at inverse temperature [MATH].', '1906.06206-3-66-8': 'The time integration is then, [EQUATION] where we have used in the second line that [MATH], and defined the thermal average [MATH] at inverse temperature [MATH], and we have defined [MATH] as the decay rate of the initial state [MATH].', '1906.06206-3-66-9': 'We thus see that it is the thermal average of the inverse decay rate that is measured by a fit to the time dependence of an observable.', '1906.06206-3-67-0': 'The integral form of the thermal average of a function [MATH], is given by, [EQUATION] with [MATH].', '1906.06206-3-67-1': 'We will see below that the FDT will be initially expressed in terms of a different average over the values [MATH].', '1906.06206-3-67-2': 'This difference is resolved in Appendix [REF], where we re-express our FDT in terms of the thermal average above.', '1906.06206-3-67-3': 'We show that the form differs only by a constant that is independent of [MATH] and the coupling strength, and thus the scaling with Hilbert space dimension remains the same, and this difference is not important for our application.', '1906.06206-3-68-0': 'Finally, we note that when [MATH] is approximately constant across the bulk of the initially populated initial states, the thermal average above is approximately equal to the unbiased average appearing in [REF].', '1906.06206-3-68-1': 'We also see that in this case, from Eq. [REF], one expects to observe an exponential decay at the rate [MATH], as in Eq. [REF].', '1906.06206-3-68-2': 'Indeed, this is what we observe in our numerical example in Section [REF] above, and thus we are able to recover the Hilbert space dimension directly in Fig. [REF].', '1906.06206-3-68-3': 'If a non-exponential decay is observed, then the average [MATH] is obtainable via integration as above, and the scaling of the Hilbert space dimension is still obtainable as in Fig. [REF].', '1906.06206-3-69-0': '# Fluctuation-Dissipation Theorem', '1906.06206-3-70-0': '## Derivation from RMT', '1906.06206-3-71-0': 'Here we perform the full derivation of the FDT for the random matrix model described above.', '1906.06206-3-71-1': 'We initially focus on the case of a diagonal initial bath state [MATH].', '1906.06206-3-71-2': 'We then restrict the treatment to the specific protocol outlined in Section [REF], where the initial state is the product of a single probe qubit in a pure state, and a bath in an infinite temperature state, see Eq. [REF].', '1906.06206-3-71-3': 'We will follow a very similar steps as those outlined in the previous section, however we will see here that the correlation function calculation is somewhat more complicated.', '1906.06206-3-72-0': 'The RMT model here is limited to the case of constant decay rate and DOS, we will thus extend the treatment to more realistic cases in the next section.', '1906.06206-3-73-0': 'We are interested in the calculation of the long-time fluctuations, defined by the diagonal ensemble result, [EQUATION]', '1906.06206-3-73-1': 'We begin be writing the initial density operator matrix elements as, [EQUATION] then using Eqs. [REF], and that [MATH], we may write the time fluctuations as, [EQUATION] where coefficients of the initial state are labelled as unprimed indices [MATH], and coefficients of the observable are labelled by primed indices.', '1906.06206-3-74-0': 'Using the self-averaging property of random matrices (see Appendices [REF] and [REF]), we may replace the product of coefficients [MATH] by their ensemble average [MATH]; the above expression may then be written in terms of a sum over 8-point correlation functions, weighted by the initial state and observable coefficients [MATH] and [MATH]: [EQUATION]', '1906.06206-3-74-1': 'Now, using the method of contractions outlined in Section [REF], we see that this 8-point correlation function may be split up into to four-point correlation functions, each consisting of both Gaussian and non-Gaussian contractions.', '1906.06206-3-74-2': 'These are computed explicitly in Appendix [REF], in which we see that there are three dominating contributions to the fluctuations, given by, [EQUATION] and, [EQUATION]', '1906.06206-3-74-3': 'These three terms can be seen as the contributions to the 8-point correlation function arising due to products of Gaussian, non-Gaussian, and mixed Gaussian and non-Gaussian 4-point correlation functions respectively.', '1906.06206-3-74-4': 'In the above, in order to perform the summations over non-interacting indices in Eq. [REF] we define course grained averages of observable elements [MATH] as in Eq. [REF], as well as the mixed averages, [EQUATION]', '1906.06206-3-74-5': 'We thus define [MATH] by [EQUATION] with, [EQUATION]', '1906.06206-3-74-6': 'We now take our bath to be in an initial infinite temperature state, such that [MATH], and [MATH].', '1906.06206-3-74-7': 'As such, [MATH] is in fact energy independent, as the probe Hamiltonian [MATH], so microcanonical averages of probe observables are also energy independent.', '1906.06206-3-74-8': 'Now, as [MATH], all terms in [MATH] are [MATH], we define, [EQUATION] where [MATH], and we have used that [MATH].', '1906.06206-3-74-9': 'We see that [MATH] is a constant of the order of unity that depends only on the observable (e.g. [MATH] for [MATH]).', '1906.06206-3-74-10': 'Finally, taking the thermodynamic limit, such that [MATH] (not that the diagonal terms in the summation can be seen to be negligible, as they contribute to a higher order in [MATH]), we have [EQUATION] which may be evaluated using, [EQUATION] where in the last line we have used that [MATH], such that [MATH].', '1906.06206-3-74-11': 'We then obtain, [EQUATION] where we have used that [MATH].', '1906.06206-3-75-0': 'We can see, then, that Eq. [REF] is of the form of our main result, Eq. (3) of the main text, where [MATH].', '1906.06206-3-75-1': 'What follows is to generalize this relation, allowing the DOS and [MATH] to vary in energy, and for finite temperatures.', '1906.06206-3-76-0': '## Extension to Realistic Systems', '1906.06206-3-77-0': 'The key issue with directly applying the RMT results to realistic models is that in general the DOS, and decay rate, are energy dependent, and thus change over the width of the initial state distribution (this is especially important for the high/infinite temperatures considered here).', '1906.06206-3-77-1': 'In order to account for this, we must then go back to the evaluation of the integrals over energy, in Eq. [REF], and substitute [MATH], and [MATH].', '1906.06206-3-77-2': 'This is justified under the assumption that neither [MATH], nor [MATH], vary appreciably over the width [MATH].', '1906.06206-3-77-3': 'i.e. [MATH], and [MATH].', '1906.06206-3-78-0': 'We see then, that the integral in Eq. [REF] is now [EQUATION] where we have used that, [EQUATION] with [MATH].', '1906.06206-3-78-1': 'This can be seen to be valid as long as the above conditions on [MATH] and [MATH] hold, that is, as long as they change sufficiently slowly over the energy width [MATH].', '1906.06206-3-78-2': 'The contribution of each eigenstate [MATH] to the fluctuations at a given energy is then that of a local (in energy) random matrix model, with a constant DOS and decay rate.', '1906.06206-3-78-3': '[MATH] and [MATH] can then be allowed to change over an energy much wider than a the width of [MATH], as over such energy widths the contributions of relevant eigenstates are independent.', '1906.06206-3-79-0': 'Now, we further define [MATH], and make the change of variables [MATH], and thus obtain [EQUATION] where in the second line we have assumed that [MATH] and [MATH] is approximately constant over the width [MATH].', '1906.06206-3-79-1': 'Now, we define the unbiased average of a function [MATH] as, [EQUATION] (not to be confused with the average [MATH] above) and see that, noting [MATH], [EQUATION] where [MATH] depends only on the choice of observable.', '1906.06206-3-79-2': 'We note that for the random matrix model, as the DOS and [MATH] are both constant in energy, the average [MATH] is equal to the thermal average [MATH] obtained from a fit to the decay of an observable (see Section [REF] above).', '1906.06206-3-79-3': 'In the case above, however, where the DOS and [MATH] change in energy, the unbiased average decay rate is not necessarily the same as that obtained from a fit to the decay.', '1906.06206-3-79-4': 'We may fix this problem directly, as we do in the last section, where we see that the unbiased thermal averages may be replaced by regular thermal averages weighted by the DOS at the expense of a constant that depends on the functional form of [MATH] and [MATH] (but importantly, not on [MATH], or the coupling strength).', '1906.06206-3-79-5': 'We can also see, that if [MATH] is approximately constant over the width of the DOS, which is often the case in such systems (in fact from Eq. [REF] this can be seen to be the case if an exponential decay of the observable is observed), then the biased and unbiased thermal averages of [MATH] are approximately equal for [MATH], and Eq. [REF] may be directly experimentally confirmed as in Fig. [REF].', '1906.06206-3-80-0': '## Finite Temperature FDT', '1906.06206-3-81-0': 'In this section we extend the above approach to finite temperature initial bath states, where the initial state is described by [EQUATION] where the joint probe-bath Hamiltonian eigenbasis is built by ordering product states such that [MATH] odd) , [MATH] even).', '1906.06206-3-81-1': 'In this case, we have [EQUATION] where [MATH], when the bath is initially a finite temperature state at inverse temperature [MATH], and the probe qubit is initially in state [MATH].', '1906.06206-3-81-2': 'We thus obtain for the microcanonical averages of [MATH], assuming that [MATH], [EQUATION] and [EQUATION] such that [MATH].', '1906.06206-3-81-3': 'Now, our most general form for the long-time fluctuations (which assumes only the ability to define the required microcanonical averages that vary smoothly over a width [MATH]) is [EQUATION] where [MATH], and [MATH] is written in Eq. [REF].', '1906.06206-3-81-4': 'Indeed, noting that the mixed average [MATH], where [MATH], and that as each term in [MATH] is [MATH] (see Eq. [REF]), we may define [EQUATION] so [EQUATION]', '1906.06206-3-81-5': 'This may be evaluated, including a variable DOS [MATH], as in the main text for the infinite temperature case, via [EQUATION] where in the second line we have made the change of variables [MATH] with [MATH] and [MATH], and used that [MATH] as in the main text.', '1906.06206-3-81-6': 'We now define the unbiased thermal average of the function [MATH] as, [EQUATION] where [MATH].', '1906.06206-3-81-7': 'Now, we have [EQUATION]', '1906.06206-3-81-8': 'Noting, then, that [MATH], and [MATH], we recover the infinite temperature case as required, [EQUATION]', '1906.06206-3-81-9': 'We note that, unlike in the RMT case above, the average [MATH] is not equal to the thermal average [MATH], which is that obtained from a fit to the decay.', '1906.06206-3-81-10': 'In Appendix [REF] we show how the FDT may be defined in terms of this thermal average.', '1906.06206-3-81-11': 'Importantly for our proposed application, we obtain that the FDT in this form is related simply by a constant [MATH], defined in Appendix [REF], that does not depend on the the size of the device or coupling strength (within the weak coupling regime).', '1906.06206-3-81-12': 'For infinite temperatures we thus have [EQUATION]', '1906.06206-3-81-13': 'Therefore, we can directly relate the measured inverse decay rate [MATH] to the time-averaged fluctuations, and from measurement of each for changing device size or coupling strength, as shown in the numerical experiments of Section [REF], yields information on the scaling of the Hilbert space dimension.', '1906.06206-3-81-14': 'We finally note that it is simply the lack of direct knowledge of the constant [MATH] which prevents measurements where there is a non-negligible change in the decay rate with energy (a non-exponential decay to equilibrium) from constituting a direct measurement of the value of the Hilbert space dimension.', '1906.06206-3-81-15': 'This constant depends on the functional form [MATH] and [MATH], and thus, if these are known, inference of the Hilbert space dimension itself is thus obtainable.', '1906.06206-3-82-0': 'Finally, we note that the finite temperature approach above can be extended to the low temperature regime, as shown in Appendix [REF], from which we can recover the pure state FDT found in Ref. [CITATION] in the low temperature limit.', '1906.06206-3-83-0': '# Discussion', '1906.06206-3-84-0': 'The results shown above demonstrate how the chaotic dynamics of thermalization may be exploited in order to gain information on the complexity of the unitary quantum dynamics of a system.', '1906.06206-3-84-1': 'We have proposed an experimentally viable protocol, by which measurements of a local observable of a probe qubit may be exploited to measure the Hilbert space dimension of an ergodic quantum device, initialized in an infinite temperature state.', '1906.06206-3-84-2': 'We note that this measures the dimension of the states directly involved in dynamics only, and thus provides a more accurate measure of the complexity of the dynamics than a simple estimate of the Hilbert space dimension from the number of qubits.', '1906.06206-3-84-3': "In this sense, such a measurement of a large enough quantum device, if shown to be ergodic in the sense outlined above, would be a convincing indicator of the so called 'quantum supremacy' of the quantum device.", '1906.06206-3-85-0': 'On a practical level, for a generic non-integrable Hamiltonian, our results may be observed in two ways: measurement of a probe observable for (i) changing the number of qubits/ions/... in the quantum device (as in Figs. [REF] and [REF]) , or (ii) changing the probe-bath coupling (as in Fig. [REF]).', '1906.06206-3-85-1': 'The latter is perhaps the simplest experimental methodology, which we show can confirm the ergodic behaviour of a system, that is, that the unitary dynamics requires an extensive proportion of the Hilbert space, by showing a linear relationship between the long-time fluctuations and decay rate.', '1906.06206-3-85-2': 'For a model where the device size may be altered, our FDT provides even deeper insight, allowing also for the experimental observation of the scaling of the Hilbert space dimension with system size.', '1906.06206-3-86-0': 'For cases where an exponential decay to equilibrium is observed, which we show implies that the decay rate is constant over a large range of energies, our method allows the experimenter to access the numerical value of the Hilbert space dimension itself, not simply its scaling with size or coupling strength.', '1906.06206-3-86-1': 'This can be obtained from a single time trace of the decay to equilibrium of the observable, from the measurement of the decay rate, and fluctuations around equilibrium.'}

hep-th-0505053

{'hep-th-0505053-1-0-0': 'Infinite-dimensional Lie algebras are introduced, which are only partially graded, and are specified by indices lying on cyclotomic rings.', 'hep-th-0505053-1-0-1': 'They may be thought of as generalizations of the Onsager algebra, but unlike it, or its [MATH] generalizations, they are not subalgebras of the loop algebras associated with [MATH].', 'hep-th-0505053-1-0-2': 'In a particular interesting case associated with [MATH], their indices lie on the Eisenstein integer triangular lattice, and these algebras are expected to underlie vertex operator combinations in CFT, brane physics, and graphite monolayers.', 'hep-th-0505053-1-1-0': '# The New Algebras', 'hep-th-0505053-1-2-0': 'We briefly introduce a class of infinite-dimensional vertex-operator Lie algebras.', 'hep-th-0505053-1-2-1': 'They have two indices, one of which lacks conventional grading.', 'hep-th-0505053-1-2-2': 'Instead, its composition motivates placing it on a cyclotomic ring, which thus makes it effectively equivalent to a multiplet of integers.', 'hep-th-0505053-1-2-3': 'We expect these algebras to feature in CFT and other areas of physics with enhanced symmetry.', 'hep-th-0505053-1-3-0': 'Consider the Lie algebras [EQUATION] where the indices [MATH] and the parameter [MATH] may be arbitrary, in general.', 'hep-th-0505053-1-4-0': 'However, as will become evident, the choice of [MATH] as an [MATH]-th root of unity, [MATH], hence [MATH], and [MATH] integers, [MATH] proportional to integers, yields by far the most interesting family.', 'hep-th-0505053-1-4-1': 'In that case, the upper indices are only distinct mod[MATH], and the lower indices take values in the cyclotomic integer ring Z[MATH], namely, [MATH].', 'hep-th-0505053-1-4-2': 'Note the grading of the upper indices, but lack of conventional grading for the lower indices, in contrast to conventional maximally graded two-index infinite Lie algebras such as [CITATION].', 'hep-th-0505053-1-4-3': '(The Lie algebras introduced here appear distinct from those based on affine quasicrystals, associated with [MATH]-th roots of unity, Coxeter groups, and Penrose pentilings [CITATION] - but some intriguing connection to these algebras should not be excluded.)', 'hep-th-0505053-1-5-0': 'This algebra satisfies the Jacobi identity, and possesses the central element [MATH].', 'hep-th-0505053-1-5-1': 'For the cyclotomic family, "Casimir invariants" may be written as [EQUATION] provided [MATH] if [MATH].', 'hep-th-0505053-1-6-0': 'In fact, the above Lie algebra might be constructed from the group algebra of associative operators [EQUATION] which satisfy [EQUATION]', 'hep-th-0505053-1-6-1': 'It would be customary in such cases [CITATION] to also consider the anticommutator of these operators, to produce a partner graded Lie algebra, [EQUATION]', 'hep-th-0505053-1-6-2': 'A simple operator realization of this algebra is [EQUATION] as may be checked by virtue of the translation action of [MATH].', 'hep-th-0505053-1-6-3': 'It is easy to see in this realization that the scale of the [MATH] is fixed, but that of the [MATH] is labile, as they can be rescaled with no change to the structure of the algebra.', 'hep-th-0505053-1-7-0': 'A variant rewriting of this realization results from the simplifying Campbell-Baker-Hausdorff expansion for the particular operators involved, [EQUATION]', 'hep-th-0505053-1-7-1': 'Equivalently, given oscillator operators, [MATH], the above realizations may be written in a form evocative of vertex operators, [EQUATION]', 'hep-th-0505053-1-7-2': 'In the cyclotomic case, [MATH], [MATH] are equivalent mod[MATH], so [MATH].', 'hep-th-0505053-1-7-3': 'The [MATH] case, [MATH], is trivial, as the corresponding lower index ring is that of the conventional integers, and the resulting algebra is essentially the Onsager algebra, a subalgebra of the [MATH] loop algebra, discussed in the next section.', 'hep-th-0505053-1-8-0': 'As an aside, a less "asymmetric", albeit more cumbersome rewriting of eqn ([REF]) might be [EQUATION] so that eqn ([REF]) reads [EQUATION]', 'hep-th-0505053-1-8-1': 'Antisymmetrization leads to the corresponding notation for the Lie algebra ([REF]), [EQUATION]', 'hep-th-0505053-1-9-0': '# [MATH] Degenerate Case and the Onsager Algebra', 'hep-th-0505053-1-10-0': 'Onsager, in his celebrated solution of the two-dimensional Ising model [CITATION], introduced the integer-indexed infinite-dimensional Lie algebra, [EQUATION] (Also see [CITATION].)', 'hep-th-0505053-1-10-1': 'Evidently, [MATH].', 'hep-th-0505053-1-10-2': 'A potential central element, [MATH], is not generated on the r.h.s. of the algebra.', 'hep-th-0505053-1-10-3': 'Onsager also recognized that his algebra is effectively a subalgebra of the [MATH] loop algebra ([MATH] centerless Kac-Moody algebra in modern conventions).', 'hep-th-0505053-1-10-4': 'The loop Lie algebra consists of three integer-indexed "towers" of elements, with [EQUATION]', 'hep-th-0505053-1-10-5': 'Given the linear involutive automorphism of this algebra, [EQUATION] the Onsager algebra is identifiable with the (fixed-point) subalgebra [CITATION], ie, the subalgebra invariant under the automorphism, consisting of two "towers", [EQUATION]', 'hep-th-0505053-1-10-6': 'It can be checked that for [MATH], thus [MATH], the above algebra ([REF]) also contains the Onsager algebra as a subalgebra, [EQUATION]', 'hep-th-0505053-1-10-7': 'It can then be seen that a graded extension of the Onsager algebra of the type ([REF]) is trivial, since [EQUATION] check to be central, ie, they commute with all elements, [MATH].', 'hep-th-0505053-1-11-0': 'Thus, [MATH]constant; hence, conversely, requiring a trivial graded extension of the Onsager algebra essentially amounts to ([REF]).', 'hep-th-0505053-1-11-1': '(Note from eqn ([REF]) that [MATH] is not an invariant of the Onsager algebra per se, but only upon this further condition, [MATH].)', 'hep-th-0505053-1-12-0': 'The realization ([REF]) reduces here to [EQUATION]', 'hep-th-0505053-1-12-1': 'In this realization, the potential candidate for a graded extension, [EQUATION] manifestly commutes with all elements, [MATH] .', 'hep-th-0505053-1-13-0': 'An alternate realization in terms of Pauli matrices is [EQUATION] similarly illustrating the triviality of [MATH] 1 1.', 'hep-th-0505053-1-14-0': '# [MATH] and the Eisenstein Integer Lattice', 'hep-th-0505053-1-15-0': 'For [MATH], the resulting algebra appears to be new, since, for [MATH], the lower indices are of the form [MATH] (with integer [MATH]), closing under addition, subtraction, and multiplication.', 'hep-th-0505053-1-15-1': 'These comprise the Euclidean ring Z[MATH] of Eisenstein-Jacobi integers [CITATION], which define a triangular 2-d lattice with hexagonal rotational symmetry: there are three lines at 60[MATH] to each other going through each such integer and connecting it to its six nearest neighbors, forming honeycomb hexagons.', 'hep-th-0505053-1-16-0': '(380,210) (110,10)*6 (185,10)*6 (260,10)*6 (335,10)*6 (75,75)*6 (150,75)*6 (225,75)*6 (300,75)*6 (375,75)*6 (110,140)*6 (185,140)*6 (260,140)*6 (335,140)*6 (75,205)*6 (150,205)*6 (225,205)*6 (300,205)*6 (375,205)*6', 'hep-th-0505053-1-17-0': 'This lattice is of utility in cohesive energy calculations for monolayer graphite [CITATION], 3-state-Potts models associated with WZW CFT models [CITATION], and, perhaps more provocatively, complexifies [CITATION] to define the complex Leech lattice, of significance in string theory, and Z[MATH] orbifolds in CFT [CITATION].', 'hep-th-0505053-1-18-0': 'Each point on the lattice may be connected to the origin by shifts along the [MATH] root and along the [MATH]-axis.', 'hep-th-0505053-1-18-1': 'A 60[MATH] rotation [MATH], on [MATH], for integer coordinates [MATH], may be represented by [EQUATION] for [MATH], and [MATH].', 'hep-th-0505053-1-18-2': 'Thus, the lower indices of the algebra may be considered as a doublet of integers composing through this rule.', 'hep-th-0505053-1-19-0': 'We care to illustrate this case explicitly to stress the differences from conventional loop algebras and [MATH] generalizations of the Onsager algebra.', 'hep-th-0505053-1-19-1': 'Instead of the differential realization ([REF]), consider a faithful representation in terms of [MATH] matrices.', 'hep-th-0505053-1-19-2': 'Sylvester\'s "nonion" basis for [MATH] groups [CITATION], is built out of his standard clock and shift unitary unimodular matrices, [EQUATION] so that [MATH].', 'hep-th-0505053-1-19-3': 'These obey the braiding identity [MATH] [CITATION].', 'hep-th-0505053-1-19-4': 'For integer indices adding mod 3, the complete set of nine unitary unimodular [MATH] matrices [EQUATION] where [MATH], and Tr[MATH], except for [MATH] mod3, suffice to span the group algebra of [MATH].', 'hep-th-0505053-1-19-5': 'Since [EQUATION] where [MATH], they further satisfy the Lie algebra of [MATH] [CITATION], [EQUATION]', 'hep-th-0505053-1-19-6': 'It is then simple to realize ([REF],[REF]) in the unimodular [MATH] matrix representation, [EQUATION] ie, the three "towers", [EQUATION]', 'hep-th-0505053-1-19-7': 'One may contrast this Lie algebra to not only [MATH] loop algebra, but also to its subalgebras, such as the the [MATH] generalization of the Onsager algebra, introduced by ref [CITATION] and consisting of five towers.', 'hep-th-0505053-1-19-8': 'Specifically, the relevant involutive automorphism of [MATH] loop algebra, in standard Chevalley notation, is [EQUATION]', 'hep-th-0505053-1-19-9': 'The subalgebra left invariant under this automorphism consists of the five towers [CITATION], [EQUATION] or, explicitly, [EQUATION]', 'hep-th-0505053-1-20-0': '# General Case: [MATH], and Quasicrystals', 'hep-th-0505053-1-21-0': 'For higher N, the cyclotomic integer rings Z[MATH] are less compelling, and are only linked to quasicrystals.', 'hep-th-0505053-1-21-1': 'Specifically, the 2-dimensional complex plane quasilattice fills up densely with the set of indices, which fail to close to a "sparse" periodic structure analogous to the Eisenstein lattice.', 'hep-th-0505053-1-21-2': 'A quasicrystal is a higher-dimensional deterministic discrete periodic structure whose projection to an embedded "external space" (in our case, the complex plane) yields nonperiodic structures of enhanced regularity [CITATION].', 'hep-th-0505053-1-22-0': 'For example, for [MATH], motions are symmetric on a 4-dimensional periodic lattice, [MATH], and [MATH], with [EQUATION] so lower indices may be effectively regarded as a quartet of integers-and, likewise, an [MATH]-tuplet of integers for higher [MATH].', 'hep-th-0505053-1-22-1': 'However, projected on the actual complex plane, nearby numbers are not necessarily represented by contiguous points on the 4-d lattice.', 'hep-th-0505053-1-23-0': 'As indicated at the beginning, there may be links between the present algebras over cyclotomic fields and those on quasicrystals which exhibit a five-fold symmetry [CITATION].', 'hep-th-0505053-1-23-1': 'For [MATH] and the golden ratio, [MATH], which satisfies [MATH], one sees that [MATH], since then [MATH] follows.', 'hep-th-0505053-1-23-2': 'There is considerable work [CITATION] on algebras defined over such quadratic number fields, Z[MATH] Z[MATH]Z[MATH], while the associated geometric constructions of quasicrystal lattices are available in textbooks [CITATION].', 'hep-th-0505053-1-23-3': 'Possibly, detailed investigations of the connection with algebras defined over the cyclotomic fields will be a fruitful source of insight.', 'hep-th-0505053-1-23-4': 'Given the vertex operator realization of the Lie algebras introduced here and its evocation of coherent states, useful applications in CFT and brane physics appear likely.', 'hep-th-0505053-1-24-0': 'This work was supported by the US Department of Energy, Division of High Energy Physics, Contract W-31-109-ENG-38.', 'hep-th-0505053-1-24-1': 'Discussions with V Armitage and J Hoppe are acknowledged.'}

{'hep-th-0505053-2-0-0': 'Infinite-dimensional Lie algebras are introduced, which are only partially graded, and are specified by indices lying on cyclotomic rings.', 'hep-th-0505053-2-0-1': 'They may be thought of as generalizations of the Onsager algebra, but unlike it, or its [MATH] generalizations, they are not subalgebras of the loop algebras associated with [MATH].', 'hep-th-0505053-2-0-2': 'In a particular interesting case associated with [MATH], their indices lie on the Eisenstein integer triangular lattice, and these algebras are expected to underlie vertex operator combinations in CFT, brane physics, and graphite monolayers.', 'hep-th-0505053-2-1-0': '# The New Algebras', 'hep-th-0505053-2-2-0': 'We briefly introduce a class of infinite-dimensional vertex-operator Lie algebras.', 'hep-th-0505053-2-2-1': 'They have two indices, one of which lacks conventional grading.', 'hep-th-0505053-2-2-2': 'Instead, its composition motivates placing it on a cyclotomic ring, which thus makes it effectively equivalent to a multiplet of integers.', 'hep-th-0505053-2-2-3': 'We expect these algebras to feature in CFT and other areas of physics with enhanced symmetry.', 'hep-th-0505053-2-3-0': 'Consider the Lie algebras [EQUATION] where the indices [MATH] and the parameter [MATH] may be arbitrary, in general.', 'hep-th-0505053-2-4-0': 'However, as will become evident, the choice of [MATH] as an [MATH]-th root of unity, [MATH], hence [MATH], and [MATH] integers, [MATH] proportional to integers, yields by far the most interesting family.', 'hep-th-0505053-2-4-1': 'In that case, the upper indices are only distinct mod[MATH], and the lower indices take values in the cyclotomic integer ring Z[MATH], namely, [MATH].', 'hep-th-0505053-2-4-2': 'Note the grading of the upper indices, but lack of conventional grading for the lower indices, in contrast to conventional maximally graded two-index infinite Lie algebras such as [CITATION].', 'hep-th-0505053-2-4-3': '(The Lie algebras introduced here appear distinct from those based on affine quasicrystals, associated with [MATH]-th roots of unity, Coxeter groups, and Penrose pentilings [CITATION] - but some intriguing connection to these algebras should not be excluded.)', 'hep-th-0505053-2-5-0': 'This algebra satisfies the Jacobi identity, and possesses the central element [MATH].', 'hep-th-0505053-2-5-1': 'For the cyclotomic family, "Casimir invariants" may be written as [EQUATION] provided [MATH] if [MATH].', 'hep-th-0505053-2-6-0': 'In fact, the above Lie algebra might be constructed from the group algebra of associative operators [EQUATION] which satisfy [EQUATION]', 'hep-th-0505053-2-6-1': 'It would be customary in such cases [CITATION] to also consider the anticommutator of these operators, to produce a partner graded Lie algebra, [EQUATION]', 'hep-th-0505053-2-6-2': 'A simple operator realization of this algebra is [EQUATION] as may be checked by virtue of the translation action of [MATH].', 'hep-th-0505053-2-6-3': 'It is easy to see in this realization that the scale of the [MATH] is fixed, but that of the [MATH] is labile, as they can be rescaled with no change to the structure of the algebra.', 'hep-th-0505053-2-7-0': 'A variant rewriting of this realization results from the simplifying Campbell-Baker-Hausdorff expansion for the particular operators involved, [EQUATION]', 'hep-th-0505053-2-7-1': 'Equivalently, given oscillator operators, [MATH], the above realizations may be written in a form evocative of vertex operators, [EQUATION]', 'hep-th-0505053-2-7-2': 'In the cyclotomic case, [MATH], [MATH] are equivalent mod[MATH], so [MATH].', 'hep-th-0505053-2-7-3': 'The [MATH] case, [MATH], is trivial, as the corresponding lower index ring is that of the conventional integers, and the resulting algebra is essentially the Onsager algebra, a subalgebra of the [MATH] loop algebra, discussed in the next section.', 'hep-th-0505053-2-8-0': 'As an aside, a less "asymmetric", albeit more cumbersome rewriting of eqn ([REF]) might be [EQUATION] so that eqn ([REF]) reads [EQUATION]', 'hep-th-0505053-2-8-1': 'Antisymmetrization leads to the corresponding notation for the Lie algebra ([REF]), [EQUATION]', 'hep-th-0505053-2-9-0': '# [MATH] Degenerate Case and the Onsager Algebra', 'hep-th-0505053-2-10-0': 'Onsager, in his celebrated solution of the two-dimensional Ising model [CITATION], introduced the integer-indexed infinite-dimensional Lie algebra, [EQUATION] (Also see [CITATION].)', 'hep-th-0505053-2-10-1': 'Evidently, [MATH].', 'hep-th-0505053-2-10-2': 'A potential central element, [MATH], is not generated on the r.h.s. of the algebra.', 'hep-th-0505053-2-10-3': 'Onsager also recognized that his algebra is effectively a subalgebra of the [MATH] loop algebra ([MATH] centerless Kac-Moody algebra in modern conventions).', 'hep-th-0505053-2-10-4': 'The loop Lie algebra consists of three integer-indexed "towers" of elements, with [EQUATION]', 'hep-th-0505053-2-10-5': 'Given the linear involutive automorphism of this algebra, [EQUATION] the Onsager algebra is identifiable with the fixed-point subalgebra [CITATION], ie, the subalgebra invariant under the automorphism, consisting of two "towers", [EQUATION]', 'hep-th-0505053-2-10-6': 'It can be checked that for [MATH], thus [MATH], the above algebra ([REF]) also contains the Onsager algebra as a subalgebra, [EQUATION]', 'hep-th-0505053-2-10-7': 'It can then be seen that a graded extension of the Onsager algebra of the type ([REF]) is trivial, since [EQUATION] check to be central, ie, they commute with all elements, [MATH].', 'hep-th-0505053-2-11-0': 'Thus, [MATH]constant; hence, conversely, requiring a trivial graded extension of the Onsager algebra essentially amounts to ([REF]).', 'hep-th-0505053-2-11-1': '(Note from eqn ([REF]) that [MATH] is not an invariant of the Onsager algebra per se, but only upon this further condition, [MATH].)', 'hep-th-0505053-2-12-0': 'The realization ([REF]) reduces here to [EQUATION]', 'hep-th-0505053-2-12-1': 'In this realization, the potential candidate for a graded extension, [EQUATION] manifestly commutes with all elements, [MATH] .', 'hep-th-0505053-2-13-0': 'An alternate realization in terms of Pauli matrices is [EQUATION] similarly illustrating the triviality of [MATH] 1 1.', 'hep-th-0505053-2-14-0': '# [MATH] and the Eisenstein Integer Lattice', 'hep-th-0505053-2-15-0': 'For [MATH], the resulting algebra appears to be new, since, for [MATH], the lower indices are of the form [MATH] (with integer [MATH]), closing under addition, subtraction, and multiplication.', 'hep-th-0505053-2-15-1': 'These comprise the Euclidean ring Z[MATH] of Eisenstein-Jacobi integers [CITATION], which define a triangular 2-d lattice with hexagonal rotational symmetry: there are three lines at 60[MATH] to each other going through each such integer and connecting it to its six nearest neighbors, forming honeycomb hexagons.', 'hep-th-0505053-2-16-0': '(380,210) (110,10)*6 (185,10)*6 (260,10)*6 (335,10)*6 (75,75)*6 (150,75)*6 (225,75)*6 (300,75)*6 (375,75)*6 (110,140)*6 (185,140)*6 (260,140)*6 (335,140)*6 (75,205)*6 (150,205)*6 (225,205)*6 (300,205)*6 (375,205)*6', 'hep-th-0505053-2-17-0': 'This lattice is of utility in cohesive energy calculations for monolayer graphite [CITATION], 3-state-Potts models associated with WZW CFT models [CITATION], and, perhaps more provocatively, complexifies [CITATION] to define the complex Leech lattice, of significance in string theory, and Z[MATH] orbifolds in CFT [CITATION].', 'hep-th-0505053-2-18-0': 'Each point on the lattice may be connected to the origin by shifts along the [MATH] root and along the [MATH]-axis.', 'hep-th-0505053-2-18-1': 'A 60[MATH] rotation [MATH], on [MATH], for integer coordinates [MATH], may be represented by [EQUATION] for [MATH], and [MATH].', 'hep-th-0505053-2-18-2': 'Thus, the lower indices of the algebra may be considered as a doublet of integers composing through this rule.', 'hep-th-0505053-2-19-0': 'We care to illustrate this case explicitly to stress the differences from conventional loop algebras and [MATH] generalizations of the Onsager algebra.', 'hep-th-0505053-2-19-1': 'Instead of the differential realization ([REF]), consider a faithful representation in terms of [MATH] matrices.', 'hep-th-0505053-2-19-2': 'Sylvester\'s "nonion" basis for [MATH] groups [CITATION], is built out of his standard clock and shift unitary unimodular matrices, [EQUATION] so that [MATH].', 'hep-th-0505053-2-19-3': 'These obey the braiding identity [MATH] [CITATION].', 'hep-th-0505053-2-19-4': 'For integer indices adding mod 3, the complete set of nine unitary unimodular [MATH] matrices [EQUATION] where [MATH], and Tr[MATH], except for [MATH] mod3, suffice to span the group algebra of [MATH].', 'hep-th-0505053-2-19-5': 'Since [EQUATION] where [MATH], they further satisfy the Lie algebra of [MATH] [CITATION], [EQUATION]', 'hep-th-0505053-2-19-6': 'It is then simple to realize ([REF],[REF]) in the unimodular [MATH] matrix representation, [EQUATION] ie, the three "towers", [EQUATION]', 'hep-th-0505053-2-19-7': 'One may contrast this Lie algebra to not only [MATH] loop algebra, but also to its subalgebras, such as the the [MATH] generalization of the Onsager algebra, introduced by ref [CITATION] and consisting of five towers.', 'hep-th-0505053-2-19-8': 'Specifically, the relevant involutive automorphism of [MATH] loop algebra, in standard Chevalley notation, is [EQUATION]', 'hep-th-0505053-2-19-9': 'The subalgebra left invariant under this automorphism consists of the five towers [CITATION], [EQUATION] or, explicitly, [EQUATION]', 'hep-th-0505053-2-20-0': '# General Case: [MATH], and Quasicrystals', 'hep-th-0505053-2-21-0': 'For higher N, the cyclotomic integer rings Z[MATH] are less compelling, and are only linked to quasicrystals.', 'hep-th-0505053-2-21-1': 'Specifically, the 2-dimensional complex plane quasilattice fills up densely with the set of indices, which fail to close to a "sparse" periodic structure analogous to the Eisenstein lattice.', 'hep-th-0505053-2-21-2': 'A quasicrystal is a higher-dimensional deterministic discrete periodic structure whose projection to an embedded "external space" (in our case, the complex plane) yields nonperiodic structures of enhanced regularity [CITATION].', 'hep-th-0505053-2-22-0': 'For example, for [MATH], motions are symmetric on a 4-dimensional periodic lattice, [MATH], and [MATH], with [EQUATION] so lower indices may be effectively regarded as a quartet of integers-and, likewise, an [MATH]-tuplet of integers for higher [MATH].', 'hep-th-0505053-2-22-1': 'However, projected on the actual complex plane, nearby numbers are not necessarily represented by contiguous points on the 4-d lattice.', 'hep-th-0505053-2-23-0': 'As indicated at the beginning, there may be links between the present algebras over cyclotomic fields and those on quasicrystals which exhibit a five-fold symmetry [CITATION].', 'hep-th-0505053-2-23-1': 'For [MATH] and the golden ratio, [MATH], which satisfies [MATH], one sees that [MATH], since then [MATH] follows.', 'hep-th-0505053-2-23-2': 'There is considerable work [CITATION] on algebras defined over such quadratic number fields, Z[MATH] Z[MATH]Z[MATH], while the associated geometric constructions of quasicrystal lattices are available in textbooks [CITATION].', 'hep-th-0505053-2-23-3': 'Possibly, detailed investigations of the connection with algebras defined over the cyclotomic fields will be a fruitful source of insight.', 'hep-th-0505053-2-23-4': 'Given the vertex operator realization of the Lie algebras introduced here and its evocation of coherent states, useful applications in CFT and brane physics appear likely.', 'hep-th-0505053-2-24-0': 'This work was supported by the US Department of Energy, Division of High Energy Physics, Contract W-31-109-ENG-38.', 'hep-th-0505053-2-24-1': 'Discussions with V Armitage and J Hoppe are acknowledged.'}

[['hep-th-0505053-1-5-0', 'hep-th-0505053-2-5-0'], ['hep-th-0505053-1-5-1', 'hep-th-0505053-2-5-1'], ['hep-th-0505053-1-4-0', 'hep-th-0505053-2-4-0'], ['hep-th-0505053-1-4-1', 'hep-th-0505053-2-4-1'], ['hep-th-0505053-1-4-2', 'hep-th-0505053-2-4-2'], ['hep-th-0505053-1-4-3', 'hep-th-0505053-2-4-3'], ['hep-th-0505053-1-10-0', 'hep-th-0505053-2-10-0'], ['hep-th-0505053-1-10-2', 'hep-th-0505053-2-10-2'], ['hep-th-0505053-1-10-3', 'hep-th-0505053-2-10-3'], ['hep-th-0505053-1-10-4', 'hep-th-0505053-2-10-4'], ['hep-th-0505053-1-10-6', 'hep-th-0505053-2-10-6'], ['hep-th-0505053-1-10-7', 'hep-th-0505053-2-10-7'], ['hep-th-0505053-1-6-0', 'hep-th-0505053-2-6-0'], ['hep-th-0505053-1-6-1', 'hep-th-0505053-2-6-1'], ['hep-th-0505053-1-6-2', 'hep-th-0505053-2-6-2'], ['hep-th-0505053-1-6-3', 'hep-th-0505053-2-6-3'], ['hep-th-0505053-1-22-0', 'hep-th-0505053-2-22-0'], ['hep-th-0505053-1-22-1', 'hep-th-0505053-2-22-1'], ['hep-th-0505053-1-3-0', 'hep-th-0505053-2-3-0'], ['hep-th-0505053-1-21-0', 'hep-th-0505053-2-21-0'], ['hep-th-0505053-1-21-1', 'hep-th-0505053-2-21-1'], ['hep-th-0505053-1-21-2', 'hep-th-0505053-2-21-2'], ['hep-th-0505053-1-7-0', 'hep-th-0505053-2-7-0'], ['hep-th-0505053-1-7-1', 'hep-th-0505053-2-7-1'], ['hep-th-0505053-1-7-2', 'hep-th-0505053-2-7-2'], ['hep-th-0505053-1-7-3', 'hep-th-0505053-2-7-3'], ['hep-th-0505053-1-19-0', 'hep-th-0505053-2-19-0'], ['hep-th-0505053-1-19-1', 'hep-th-0505053-2-19-1'], ['hep-th-0505053-1-19-2', 'hep-th-0505053-2-19-2'], ['hep-th-0505053-1-19-3', 'hep-th-0505053-2-19-3'], ['hep-th-0505053-1-19-4', 'hep-th-0505053-2-19-4'], ['hep-th-0505053-1-19-5', 'hep-th-0505053-2-19-5'], ['hep-th-0505053-1-19-6', 'hep-th-0505053-2-19-6'], ['hep-th-0505053-1-19-7', 'hep-th-0505053-2-19-7'], ['hep-th-0505053-1-19-8', 'hep-th-0505053-2-19-8'], ['hep-th-0505053-1-19-9', 'hep-th-0505053-2-19-9'], ['hep-th-0505053-1-24-0', 'hep-th-0505053-2-24-0'], ['hep-th-0505053-1-24-1', 'hep-th-0505053-2-24-1'], ['hep-th-0505053-1-2-0', 'hep-th-0505053-2-2-0'], ['hep-th-0505053-1-2-1', 'hep-th-0505053-2-2-1'], ['hep-th-0505053-1-2-2', 'hep-th-0505053-2-2-2'], ['hep-th-0505053-1-2-3', 'hep-th-0505053-2-2-3'], ['hep-th-0505053-1-12-0', 'hep-th-0505053-2-12-0'], ['hep-th-0505053-1-12-1', 'hep-th-0505053-2-12-1'], ['hep-th-0505053-1-15-0', 'hep-th-0505053-2-15-0'], ['hep-th-0505053-1-15-1', 'hep-th-0505053-2-15-1'], ['hep-th-0505053-1-8-0', 'hep-th-0505053-2-8-0'], ['hep-th-0505053-1-8-1', 'hep-th-0505053-2-8-1'], ['hep-th-0505053-1-13-0', 'hep-th-0505053-2-13-0'], ['hep-th-0505053-1-23-0', 'hep-th-0505053-2-23-0'], ['hep-th-0505053-1-23-1', 'hep-th-0505053-2-23-1'], ['hep-th-0505053-1-23-2', 'hep-th-0505053-2-23-2'], ['hep-th-0505053-1-23-3', 'hep-th-0505053-2-23-3'], ['hep-th-0505053-1-23-4', 'hep-th-0505053-2-23-4'], ['hep-th-0505053-1-17-0', 'hep-th-0505053-2-17-0'], ['hep-th-0505053-1-11-0', 'hep-th-0505053-2-11-0'], ['hep-th-0505053-1-11-1', 'hep-th-0505053-2-11-1'], ['hep-th-0505053-1-0-0', 'hep-th-0505053-2-0-0'], ['hep-th-0505053-1-0-1', 'hep-th-0505053-2-0-1'], ['hep-th-0505053-1-0-2', 'hep-th-0505053-2-0-2'], ['hep-th-0505053-1-18-0', 'hep-th-0505053-2-18-0'], ['hep-th-0505053-1-18-1', 'hep-th-0505053-2-18-1'], ['hep-th-0505053-1-18-2', 'hep-th-0505053-2-18-2'], ['hep-th-0505053-1-10-5', 'hep-th-0505053-2-10-5']]

[['hep-th-0505053-1-5-0', 'hep-th-0505053-2-5-0'], ['hep-th-0505053-1-5-1', 'hep-th-0505053-2-5-1'], ['hep-th-0505053-1-4-0', 'hep-th-0505053-2-4-0'], ['hep-th-0505053-1-4-1', 'hep-th-0505053-2-4-1'], ['hep-th-0505053-1-4-2', 'hep-th-0505053-2-4-2'], ['hep-th-0505053-1-4-3', 'hep-th-0505053-2-4-3'], ['hep-th-0505053-1-10-0', 'hep-th-0505053-2-10-0'], ['hep-th-0505053-1-10-2', 'hep-th-0505053-2-10-2'], ['hep-th-0505053-1-10-3', 'hep-th-0505053-2-10-3'], ['hep-th-0505053-1-10-4', 'hep-th-0505053-2-10-4'], ['hep-th-0505053-1-10-6', 'hep-th-0505053-2-10-6'], ['hep-th-0505053-1-10-7', 'hep-th-0505053-2-10-7'], ['hep-th-0505053-1-6-0', 'hep-th-0505053-2-6-0'], ['hep-th-0505053-1-6-1', 'hep-th-0505053-2-6-1'], ['hep-th-0505053-1-6-2', 'hep-th-0505053-2-6-2'], ['hep-th-0505053-1-6-3', 'hep-th-0505053-2-6-3'], ['hep-th-0505053-1-22-0', 'hep-th-0505053-2-22-0'], ['hep-th-0505053-1-22-1', 'hep-th-0505053-2-22-1'], ['hep-th-0505053-1-3-0', 'hep-th-0505053-2-3-0'], ['hep-th-0505053-1-21-0', 'hep-th-0505053-2-21-0'], ['hep-th-0505053-1-21-1', 'hep-th-0505053-2-21-1'], ['hep-th-0505053-1-21-2', 'hep-th-0505053-2-21-2'], ['hep-th-0505053-1-7-0', 'hep-th-0505053-2-7-0'], ['hep-th-0505053-1-7-1', 'hep-th-0505053-2-7-1'], ['hep-th-0505053-1-7-2', 'hep-th-0505053-2-7-2'], ['hep-th-0505053-1-7-3', 'hep-th-0505053-2-7-3'], ['hep-th-0505053-1-19-0', 'hep-th-0505053-2-19-0'], ['hep-th-0505053-1-19-1', 'hep-th-0505053-2-19-1'], ['hep-th-0505053-1-19-2', 'hep-th-0505053-2-19-2'], ['hep-th-0505053-1-19-3', 'hep-th-0505053-2-19-3'], ['hep-th-0505053-1-19-4', 'hep-th-0505053-2-19-4'], ['hep-th-0505053-1-19-5', 'hep-th-0505053-2-19-5'], ['hep-th-0505053-1-19-6', 'hep-th-0505053-2-19-6'], ['hep-th-0505053-1-19-7', 'hep-th-0505053-2-19-7'], ['hep-th-0505053-1-19-8', 'hep-th-0505053-2-19-8'], ['hep-th-0505053-1-19-9', 'hep-th-0505053-2-19-9'], ['hep-th-0505053-1-24-0', 'hep-th-0505053-2-24-0'], ['hep-th-0505053-1-24-1', 'hep-th-0505053-2-24-1'], ['hep-th-0505053-1-2-0', 'hep-th-0505053-2-2-0'], ['hep-th-0505053-1-2-1', 'hep-th-0505053-2-2-1'], ['hep-th-0505053-1-2-2', 'hep-th-0505053-2-2-2'], ['hep-th-0505053-1-2-3', 'hep-th-0505053-2-2-3'], ['hep-th-0505053-1-12-0', 'hep-th-0505053-2-12-0'], ['hep-th-0505053-1-12-1', 'hep-th-0505053-2-12-1'], ['hep-th-0505053-1-15-0', 'hep-th-0505053-2-15-0'], ['hep-th-0505053-1-15-1', 'hep-th-0505053-2-15-1'], ['hep-th-0505053-1-8-0', 'hep-th-0505053-2-8-0'], ['hep-th-0505053-1-8-1', 'hep-th-0505053-2-8-1'], ['hep-th-0505053-1-13-0', 'hep-th-0505053-2-13-0'], ['hep-th-0505053-1-23-0', 'hep-th-0505053-2-23-0'], ['hep-th-0505053-1-23-1', 'hep-th-0505053-2-23-1'], ['hep-th-0505053-1-23-2', 'hep-th-0505053-2-23-2'], ['hep-th-0505053-1-23-3', 'hep-th-0505053-2-23-3'], ['hep-th-0505053-1-23-4', 'hep-th-0505053-2-23-4'], ['hep-th-0505053-1-17-0', 'hep-th-0505053-2-17-0'], ['hep-th-0505053-1-11-0', 'hep-th-0505053-2-11-0'], ['hep-th-0505053-1-11-1', 'hep-th-0505053-2-11-1'], ['hep-th-0505053-1-0-0', 'hep-th-0505053-2-0-0'], ['hep-th-0505053-1-0-1', 'hep-th-0505053-2-0-1'], ['hep-th-0505053-1-0-2', 'hep-th-0505053-2-0-2'], ['hep-th-0505053-1-18-0', 'hep-th-0505053-2-18-0'], ['hep-th-0505053-1-18-1', 'hep-th-0505053-2-18-1'], ['hep-th-0505053-1-18-2', 'hep-th-0505053-2-18-2']]

[['hep-th-0505053-1-10-5', 'hep-th-0505053-2-10-5']]

[]

[]

[]

['hep-th-0505053-1-10-1', 'hep-th-0505053-1-16-0', 'hep-th-0505053-2-10-1', 'hep-th-0505053-2-16-0']

{'1': 'http://arxiv.org/licenses/assumed-1991-2003/', '2': 'http://arxiv.org/licenses/assumed-1991-2003/'}

https://arxiv.org/abs/hep-th/0505053

1907.06409

{'1907.06409-1-0-0': 'The Barzilai-Borwein (BB) method is a popular and efficient tool for solving large-scale unconstrained optimization problems.', '1907.06409-1-0-1': 'Its search direction is the same as for the steepest descent (Cauchy) method, but its stepsize rule is different.', '1907.06409-1-0-2': 'Owing to this, it converges much faster than the Cauchy method.', '1907.06409-1-0-3': 'A feature of the BB method is that it may generate too long steps, which throw the iterates too far away from the solution.', '1907.06409-1-0-4': 'Moreover, it may not converge, even when the objective function is strongly convex.', '1907.06409-1-0-5': 'In this paper, a stabilization technique is introduced.', '1907.06409-1-0-6': 'It consists in bounding the distance between each pair of successive iterates, which often allows for decreasing the number of BB iterations.', '1907.06409-1-0-7': 'When the BB method does not converge, our simple modification of this method makes it convergent.', '1907.06409-1-0-8': 'Under suitable assumptions, we prove its global convergence, despite the fact that no line search is involved, and only gradient values are used.', '1907.06409-1-0-9': 'Since the number of stabilization steps is proved to be finite, the stabilized version inherits the fast local convergence of the BB method.', '1907.06409-1-0-10': 'The presented results of extensive numerical experiments show that our stabilization technique often allows the BB method to solve problems in a fewer iterations, or even to solve problems where the latter fails.', '1907.06409-1-1-0': '65K05, 90C06, 90C30.', '1907.06409-1-2-0': '# Introduction', '1907.06409-1-3-0': 'In this paper, we consider spectral gradient methods for solving the unconstrained optimization problem [EQUATION] where [MATH] is a sufficiently smooth function.', '1907.06409-1-3-1': 'Its minimizer is denoted by [MATH].', '1907.06409-1-3-2': 'Gradient-type iterative methods used for solving problem [REF] have the form [EQUATION] where [MATH] and [MATH] is a stepsize.', '1907.06409-1-3-3': 'Methods of this type differ in the stepsize rules which they follow.', '1907.06409-1-4-0': 'We focus here on the two choices of [MATH] proposed in 1988 by Barzilai and Borwein [CITATION], usually refereed to as the BB method.', '1907.06409-1-4-1': 'The rationale behind these choices is related to viewing the gradient-type methods as quasi-Newton methods, where [MATH] in [REF] is replaced by the matrix [MATH].', '1907.06409-1-4-2': 'This matrix is served as an approximation of the inverse Hessian matrix.', '1907.06409-1-4-3': 'Following the quasi-Newton approach, the stepsize is calculated by forcing either [MATH] (BB1 method) or [MATH] (BB2 method) to satisfy the secant equation in the least squares sense.', '1907.06409-1-4-4': 'The corresponding two problems are formulated as [EQUATION] where [MATH] and [MATH].', '1907.06409-1-4-5': 'The solutions to these problems are [EQUATION] respectively.', '1907.06409-1-4-6': 'Here and in what follows, [MATH] denotes the Euclidean vector norm and the induced matrix norm.', '1907.06409-1-4-7': 'Other norms used in this paper will be denoted in a different way.', '1907.06409-1-5-0': 'Barzilai and Borwein [CITATION] proved that their method converges [MATH]-superlinearly for two-dimensional strictly convex quadratics.', '1907.06409-1-5-1': 'Dai and Fletcher [CITATION] analyzed the asymptotic behavior of BB-like methods, and they obtained [MATH]-superlinear convergence of the BB method for the three-dimensional case.', '1907.06409-1-5-2': 'Global convergence of the BB method for the [MATH]-dimensional case was established by Raydan [CITATION] and further refined by Dai and Liao [CITATION] for obtaining the R-linear rate.', '1907.06409-1-5-3': 'For nonquadratic functions, local convergence proof of the BB method with R-linear rate was, first, sketched in some detail by Liu and Dai [CITATION], and then it was later rigorously proved by Dai et al. [CITATION].', '1907.06409-1-5-4': 'Extensive numerical experiments show that the two BB stepsize rules significantly improve the performance of gradient methods (see, e.g., [CITATION]), both in quadratic and nonquadratic cases.', '1907.06409-1-6-0': 'A variety of modifications and extensions have been developed, such as gradient methods with retards [CITATION], alternate BB method [CITATION], cyclic BB method [CITATION], limited memory gradient method [CITATION] etc.', '1907.06409-1-6-1': 'Several approaches were proposed for dealing with nonconvex objective functions, in which case the BB stepsize [REF] may become negative.', '1907.06409-1-6-2': 'In our numerical experiments, we use the one proposed in [CITATION].', '1907.06409-1-6-3': 'The BB method has been extended to solving symmetric and nonsymmetric linear equations [CITATION].', '1907.06409-1-6-4': 'Furthermore, by incorporating the nonmontone line search by Grippo et al. [CITATION], Raydan [CITATION] and Grippo et al. [CITATION] developed the global BB method for general unconstrained optimization problems.', '1907.06409-1-6-5': "Later, Birgin et al. [CITATION] proposed the so-called spectral projected gradient method which extends Raydan's method to smooth convex constrained problems.", '1907.06409-1-6-6': 'For more works on BB-like methods, see [CITATION] and references therein.', '1907.06409-1-7-0': 'As it was observed by many authors, the BB method may generate too long steps, which throw the iterates too far away from the solution.', '1907.06409-1-7-1': 'In practice, it may not converge even for strongly convex functions (see, e.g., [CITATION]).', '1907.06409-1-7-2': 'The purpose of this paper is to introduce a simple stabilization technique and to justify its efficiency both theoretically and practically.', '1907.06409-1-7-3': 'Our stabilization does not assume any objective function evaluations.', '1907.06409-1-7-4': 'It consists in uniformly bounding [MATH], the distance between each pair of successive iterates.', '1907.06409-1-7-5': 'It should be emphasized that, if the BB method safely converges for a given function, then there is no necessity in stabilizing it.', '1907.06409-1-7-6': 'In such cases, the stabilization may increase the number of iterations.', '1907.06409-1-7-7': 'In other cases, as it will be demonstrated by results of our numerical experiments, the stabilization may allow for decreasing the number of iterations or even to make the BB method convergent.', '1907.06409-1-8-0': 'Although we focus here on stabilizing the conventional BB method, our approach can directly be combined with the existing modifications of the BB method, where a nonmonotone line search is used.', '1907.06409-1-9-0': 'The paper is organized as follows.', '1907.06409-1-9-1': 'In the next section, we present an example of a strictly convex function and show that the BB method does not converge in this case.', '1907.06409-1-9-2': 'This contributes to a motivation for stabilizing this method.', '1907.06409-1-9-3': 'In the same section, its stabilized version is introduced.', '1907.06409-1-9-4': 'In Section [REF], a global convergence of our stabilized BB algorithm as well as its R-linear rate of convergence are proved under suitable assumptions.', '1907.06409-1-9-5': 'Results of numerical experiments are reported and discussed in Section [REF].', '1907.06409-1-9-6': 'Finally, some conclusions are included in the last section of the paper.', '1907.06409-1-10-0': '# Stabilized Algorithm', '1907.06409-1-11-0': 'Before formulating our stabilized algorithm, we wish to begin with a motivation based on presenting an example of a strongly convex function for which we theoretically prove that neither of the BB methods converge.', '1907.06409-1-11-1': 'To the best of our knowledge, no theoretical evidence of this kind is available in the literature.', '1907.06409-1-12-0': 'In the review paper by Fletcher [CITATION], it is claimed that the BB method diverges in practice for certain initial points in the test problem referred to as Strictly Convex 2 by Raydan [CITATION], in which [EQUATION]', '1907.06409-1-12-1': 'This strongly convex function will be used in Section [REF] for illustrating the efficiency of the stabilized algorithm.', '1907.06409-1-12-2': 'Our numerical experiments show that, in this specific case, the failure of the BB method is related to the underflow and overflow effects in the computer arithmetic.', '1907.06409-1-12-3': 'We are not acquainted with any theoretical justification of the divergence of the BB method for this or any other functions.', '1907.06409-1-13-0': 'We will present now an instance of a function for which the BB method does not converge in the exact arithmetic.', '1907.06409-1-13-1': 'For this purpose, the notation [EQUATION] will be used.', '1907.06409-1-13-2': 'Consider the univariate function [EQUATION]', '1907.06409-1-13-3': 'Its first derivative [EQUATION] is continuously differentiable, and [MATH] is an odd monotonically increasing function (see Fugure [REF]).', '1907.06409-1-14-0': 'It can be easily verified that the function [MATH] is twice continuously differentiable with [EQUATION].', '1907.06409-1-14-1': 'This means that this function is strongly convex, and its first derivative is Lipschitz-continuous.', '1907.06409-1-15-0': 'For any univariate objective function, there is no difference between BB1 and BB2 versions, and they are equivalent to the secant method applied to the first derivative.', '1907.06409-1-15-1': 'For function [REF], if to initiate the BB method with [MATH] and [MATH], then the subsequent iterates are [EQUATION]', '1907.06409-1-15-2': 'This clearly shows that the BB method cycles between four points (see Fugure [REF]).', '1907.06409-1-15-3': 'The presented counter-example can be easily extended to [MATH]-dimensional case.', '1907.06409-1-16-0': 'After motivating the necessity of stabilizing the BB method, we can now proceed to presenting the basic idea of our stabilized BB algorithm, where [MATH] is a parameter.', '1907.06409-1-16-1': 'It consists in choosing the stepsize in [REF] in the way that [MATH], whenever [MATH].', '1907.06409-1-16-2': 'Thus, denoting [EQUATION] we propose to choose [EQUATION]', '1907.06409-1-16-3': 'Here [MATH] or [MATH], depending on the specific BB method in [REF].', '1907.06409-1-16-4': 'A formal description of our stabilized BB algorithm follows.', '1907.06409-1-17-0': 'BBstab.', '1907.06409-1-18-0': '1px', '1907.06409-1-19-0': 'Given: initial points [MATH] such that [MATH], and scalar [MATH].', '1907.06409-1-20-0': '0.5px', '1907.06409-1-21-0': 'Evaluate [MATH] and [MATH].', '1907.06409-1-22-0': 'for [MATH] do', '1907.06409-1-23-0': 'if [MATH] then stop.', '1907.06409-1-24-0': 'Set [MATH] and [MATH].', '1907.06409-1-25-0': 'Compute [MATH] by formula [REF].', '1907.06409-1-26-0': 'Set [MATH] and evaluate [MATH].', '1907.06409-1-27-0': 'end (for)', '1907.06409-1-28-0': '0.5px', '1907.06409-1-29-0': 'This algorithm will be refereed to as BB1stab or BB2stab depending on the corresponding choice of [MATH] in [REF].', '1907.06409-1-29-1': 'Note that, for [MATH], it reduces to the underlying standard BB algorithm.', '1907.06409-1-30-0': '# Convergence Analysis', '1907.06409-1-31-0': 'In this section, global convergence of the BBstab algorithm will be proved.', '1907.06409-1-31-1': 'Whenever iterates [MATH] are mentioned, they are assumed to be generated by BBstab, where it is required that [MATH].', '1907.06409-1-32-0': 'Throughout this section, the objective function is assumed to comply with the following requirement.', '1907.06409-1-33-0': '[A1.]', '1907.06409-1-33-1': 'The function [MATH] is twice continuously differentiable, and there exist positive constants [MATH] such that [EQUATION]', '1907.06409-1-33-2': 'This assumption implies that [EQUATION]', '1907.06409-1-33-3': 'Extra assumptions are introduced below in proper places.', '1907.06409-1-34-0': 'We shall use the following notation: [EQUATION] which will be motivated later.', '1907.06409-1-34-1': 'Here [EQUATION].', '1907.06409-1-34-2': 'Obviously, [MATH], and [MATH], where [MATH].', '1907.06409-1-34-3': 'We shall use similar notation for other unions of sets [MATH].', '1907.06409-1-35-0': 'Inequalities [REF] ensure that [EQUATION] which in turn means that [EQUATION] and [EQUATION]', '1907.06409-1-35-1': 'These bounds justify the implications [EQUATION]', '1907.06409-1-35-2': 'We can now prove the following result.', '1907.06409-1-36-0': 'Let [MATH] be arbitrary starting points.', '1907.06409-1-36-1': 'Then for any [MATH], the iterates [MATH] have the property that [EQUATION] where [EQUATION].', '1907.06409-1-37-0': 'Using Assumption A1, we get [EQUATION] where the matrix [MATH] is symmetric, and it fulfills the relations [EQUATION].', '1907.06409-1-37-1': 'Clearly, [EQUATION]', '1907.06409-1-37-2': 'Consider, first, the case when [MATH].', '1907.06409-1-37-3': 'Using the inequality [MATH] and relations [REF], we can derive for [REF] the following upper bound [EQUATION].', '1907.06409-1-37-4': 'This proves the upper inequality in [REF].', '1907.06409-1-38-0': 'Suppose now that [MATH], i.e., [MATH].', '1907.06409-1-38-1': 'Then, using [REF], we get the bounds [MATH], which together with the inequalities [MATH] yield [EQUATION].', '1907.06409-1-38-2': 'By combining this estimate with [REF], we finally prove the lower inequality in [REF].', '1907.06409-1-38-3': '[MATH]', '1907.06409-1-39-0': 'Lemma [REF] implies that the stabilization steps have the following properties [EQUATION]', '1907.06409-1-39-1': 'Next, we prove that, after a finite number of iterations, all iterates belong to the bounded set [MATH].', '1907.06409-1-40-0': 'For any [MATH] and [MATH], there exists an integer [MATH] such that the inequality [EQUATION] holds, that is [MATH], for all [MATH].', '1907.06409-1-40-1': 'Moreover, [MATH] is the iteration number corresponding to the first iterate [MATH] that belongs to [MATH].', '1907.06409-1-41-0': 'Notice that [REF] is satisfied if and only if [MATH].', '1907.06409-1-41-1': 'We first show that if [MATH], then so does the next iterate.', '1907.06409-1-41-2': 'Indeed, in view of [REF] and [REF], if [MATH], then [MATH].', '1907.06409-1-41-3': 'On the other hand, if [MATH], i.e. [MATH], then, by Lemma [REF], we have [MATH].', '1907.06409-1-42-0': 'Suppose now that [MATH].', '1907.06409-1-42-1': 'Then it immediately follows from relations [REF] and [REF], that there exists [MATH] such that [MATH].', '1907.06409-1-42-2': 'As it was shown above, this means that [MATH] for all [MATH].', '1907.06409-1-42-3': '[MATH]', '1907.06409-1-43-0': 'It follows from [REF] that, when iterates belong to the set [MATH], the value [MATH] monotonically decreases as indicated by [REF].', '1907.06409-1-43-1': 'Furthermore, the actual decrease may speed-up in accordance with [REF].', '1907.06409-1-43-2': 'When the iterates reach [MATH], the decrease is naturally expected to slow down, and this is followed by a non-monotonic behavior of [MATH], which is a typical feature of the BB steps.', '1907.06409-1-44-0': 'One can observe all these stages in the behavior of BBstab in Figure [REF].', '1907.06409-1-44-1': 'It presents changes of [MATH] with [MATH] in the process of minimizing Raydan function [REF].', '1907.06409-1-44-2': 'Details of these runs are discussed in Section [REF].', '1907.06409-1-44-3': 'Note that both BB1 and BB2 fail to solve this problem starting from the same points.', '1907.06409-1-44-4': 'The figure illustrates the role of stabilization in providing convergence of BBstab.', '1907.06409-1-44-5': 'One can clearly recognize the first stage of the process when the stabilization steps ensure a monotonic decrease of [MATH].', '1907.06409-1-44-6': 'For the BB1stab and BB2stab, the iteration when the standard BB step was used for the first time is 228 and 226, respectively.', '1907.06409-1-44-7': 'For them, the last stabilization step was used in iteration 379 and 353, respectively.', '1907.06409-1-44-8': 'Observe that the spikes of [MATH] produced by BB1 is much larger than those for BB2.', '1907.06409-1-45-0': 'Lemma [REF] indicates that a proper choice of [MATH] allows for BBstab to reach any neighborhood of [MATH].', '1907.06409-1-45-1': 'We use the notation [EQUATION] in the following formulation of this useful feature of BBstab.', '1907.06409-1-46-0': 'Let [MATH] be any starting points.', '1907.06409-1-46-1': 'Then for any [MATH] and positive [MATH], there exists [MATH] such that the iterates [MATH] satisfy the condition [EQUATION].', '1907.06409-1-47-0': 'Combining [REF] and Lemma [REF], we get the relations [EQUATION] which are satisfied for all sufficiently large [MATH].', '1907.06409-1-47-1': 'This completes the proof.', '1907.06409-1-47-2': '[MATH]', '1907.06409-1-48-0': 'We shall make use of Lemma [REF] for proving global convergence result for BBstab.', '1907.06409-1-48-1': 'We show also that its local rate of convergence is R-linear, which means that there exist positive [MATH] and [MATH] such that [EQUATION]', '1907.06409-1-48-2': 'These convergence results are based on our convergence analysis presented in the next sub-section for convex quadratic functions.', '1907.06409-1-49-0': '## Convergence in Quadratic Case', '1907.06409-1-50-0': 'In this sub-section, we focus on minimizing convex quadratic functions of the form [EQUATION] where the matrix [MATH] is positive definite, and [MATH].', '1907.06409-1-50-1': 'For these functions, we have the following convergence result.', '1907.06409-1-51-0': 'Let [MATH] be arbitrary starting points.', '1907.06409-1-51-1': 'Then for any [MATH], the sequence [MATH] converges to [MATH] with R-linear rate of convergence.', '1907.06409-1-52-0': 'It is well known that the BB method is invariant under orthogonal transformation of the variables and, as it can be easily seen, so does its stabilized version.', '1907.06409-1-52-1': 'Hence, we can assume without loss of generality that the matrix [MATH] is of the form [EQUATION] where [MATH].', '1907.06409-1-52-2': 'Here, like it is often done for the gradient methods (see, e.g., [CITATION]), it is assumed without loss of generality that the matrix [MATH] has distinct eigenvalues.', '1907.06409-1-53-0': 'Let [MATH] denote the [MATH]th component of [MATH].', '1907.06409-1-53-1': 'Clearly, [EQUATION]', '1907.06409-1-53-2': 'By Lemma [REF], for all sufficiently large [MATH], iterates [MATH] belong to [MATH], which is a compact set.', '1907.06409-1-53-3': 'Hence, there exists at least one limit point of the sequence [MATH].', '1907.06409-1-53-4': 'If [MATH] is a limit point, then it is the only limit point, i.e., the iterates [MATH] converge to [MATH].', '1907.06409-1-53-5': 'This follows from the fact that [MATH], whenever [MATH], which holds for all sufficiently large [MATH].', '1907.06409-1-54-0': 'To prove convergence, suppose for contradiction that there exists [MATH] such that [EQUATION]', '1907.06409-1-54-1': 'We use induction in [MATH] to show that there exists a limit point [MATH] in which [EQUATION]', '1907.06409-1-54-2': 'Let [MATH] be a limit point characterized by the minimal value of [MATH] among all stationary points.', '1907.06409-1-54-3': 'To show that [REF] holds for [MATH], we assume the contrary, i.e., that [MATH] and consider a sub-sequence [MATH] which converges to [MATH].', '1907.06409-1-54-4': 'Combining [REF] and [REF], we get [EQUATION] where [MATH].', '1907.06409-1-54-5': 'This inequality, in the limit as [MATH], yields [MATH], where [MATH] is a limit point for the sub-sequence [MATH].', '1907.06409-1-54-6': 'This contradicts the assumption that [MATH] is minimal and proves that [MATH].', '1907.06409-1-55-0': 'Now, assume that [REF] holds for [MATH].', '1907.06409-1-55-1': 'Let [MATH] be a limit point with the minimal value of [MATH] among all stationary points satisfying [REF].', '1907.06409-1-55-2': 'Suppose for the moment that [MATH] and consider any sub-sequence [MATH] converging to [MATH].', '1907.06409-1-55-3': 'Let us derive an upper bound for the stepsize [MATH].', '1907.06409-1-55-4': 'This will be done separately for BB1- and BB2-based iterates.', '1907.06409-1-56-0': 'For the BB1 case, we have [EQUATION] where [EQUATION].', '1907.06409-1-57-0': 'For BB2, denote [MATH].', '1907.06409-1-57-1': 'Then [EQUATION] where [EQUATION].', '1907.06409-1-58-0': 'Relations [REF] and [REF] show that [MATH] and [MATH] when [MATH].', '1907.06409-1-58-1': 'For the BB1 iterates, the bounds [EQUATION] ensure that the inequality [EQUATION] is satisfied for all sufficiently large [MATH].', '1907.06409-1-58-2': 'The same inequality is obviously satisfied for the BB2 iterates.', '1907.06409-1-58-3': 'This, together with [REF], means that inequality [REF] holds for [MATH] and all sufficiently large [MATH].', '1907.06409-1-58-4': 'As in the case of [MATH], we get a contradiction, which proves that [MATH].', '1907.06409-1-58-5': 'Thus, it was shown by induction that [MATH] is a limit point or, equivalently, that iterates [MATH] converge to [MATH].', '1907.06409-1-59-0': 'The fact that, for all sufficiently large [MATH], [MATH] allows us to conclude that, for all these iterates, [MATH].', '1907.06409-1-59-1': 'This finally proves that the rate of convergence is R-linear.', '1907.06409-1-59-2': '[MATH]', '1907.06409-1-60-0': 'The following result plays an important role in our proof of R-linear convergence for nonquadratic functions.', '1907.06409-1-61-0': 'There exists a positive integer [MATH], such that, for any [MATH] and [MATH], the inequality [EQUATION] holds for all [MATH].', '1907.06409-1-62-0': 'Like in the poof of Theorema [REF], we assume here without loss of generality that [MATH] has the form of [REF], and we shall use the notation [MATH] for the [MATH]th component of [MATH].', '1907.06409-1-62-1': 'We will also make use of the following definition, where [EQUATION].', '1907.06409-1-63-0': 'We say that the choice of the stepsize in [REF] has property P if there exist an integer [MATH] and positive constants [MATH] and [MATH] such that, for all [MATH],', '1907.06409-1-64-0': '(i) [MATH];', '1907.06409-1-65-0': '(ii) for any integer [MATH] and real number [MATH], if [MATH] and [MATH] hold for [MATH], then [MATH].', '1907.06409-1-66-0': 'Firstly, we prove that the step size [MATH] has property P. Lemma [REF] ensures that [MATH] for all [MATH].', '1907.06409-1-66-1': 'Then the bounds [REF] show that [MATH] has property P(i) with [MATH].', '1907.06409-1-67-0': 'Next, we will show, for any integer [MATH] and real number [MATH], that the inequality [MATH] is satisfied, whenever [MATH] and [MATH].', '1907.06409-1-67-1': 'This will be done separately for BB1- and BB2-based iterates.', '1907.06409-1-68-0': 'For the BB1 case, we have [EQUATION]', '1907.06409-1-68-1': 'For BB2, we obtain [EQUATION]', '1907.06409-1-68-2': 'Thus, P(ii) holds for [MATH] and [MATH].', '1907.06409-1-68-3': 'This implies that BBstab stepsize [MATH] satisfies P.', '1907.06409-1-68-4': 'Then we can skip the rest of the proof because it is similar to the proof of Theorem 4.1 in [CITATION].', '1907.06409-1-68-5': '[MATH]', '1907.06409-1-69-0': 'It should be emphasized that, in this lemma, the value of [MATH] depends only on [MATH] and [MATH].', '1907.06409-1-70-0': '## Convergence in General Case', '1907.06409-1-71-0': 'For nonquadratic functions, we shall first prove local R-linear convergence of BBstab.', '1907.06409-1-71-1': 'This result will then be used for showing that it converges from any starting point.', '1907.06409-1-72-0': 'Throughout this sub-section, we need to additionally assume that the Hessian matrix [MATH] is Lipschitz-continuous at [MATH].', '1907.06409-1-72-1': 'In what follows, we use the notation [MATH].', '1907.06409-1-73-0': '[A2.]', '1907.06409-1-73-1': 'There exist a radius [MATH] and a Lipschitz constant [MATH] such that [EQUATION].', '1907.06409-1-74-0': 'This assumption implies that [EQUATION]', '1907.06409-1-74-1': 'The second-order Taylor approximation to [MATH] around [MATH] is given by the quadratic function [EQUATION]', '1907.06409-1-74-2': 'Define new iterates [MATH] associated with [MATH] as follows: [EQUATION] where [EQUATION]', '1907.06409-1-74-3': 'Here [MATH] or [MATH] and [MATH] with [EQUATION] [MATH], [MATH] and [MATH].', '1907.06409-1-74-4': 'In what follows, whenever we mention [MATH] and [MATH], they are assumed to be generated as defined above.', '1907.06409-1-75-0': 'The next result follows immediately from Lemma [REF].', '1907.06409-1-76-0': 'Let [MATH] be any scalar, such that [MATH].', '1907.06409-1-76-1': 'Then there exists a positive integer [MATH], dependent only on [MATH] and [MATH], such that, for any [MATH], the inequality holds [EQUATION].', '1907.06409-1-77-0': 'It can be easily seen that if [MATH], then all corresponding [MATH].', '1907.06409-1-77-1': 'In this case, BBstab stepsize [MATH] satisfies the bounds [REF], and similarly for [MATH], we have the bounds [EQUATION]', '1907.06409-1-77-2': 'The following result will be used for proving local R-linear convergence.', '1907.06409-1-78-0': 'Let integer [MATH] be arbitrary.', '1907.06409-1-78-1': 'Then there exist positive scalars [MATH] and [MATH] with the following property: for any [MATH], [MATH] and [MATH], for which [EQUATION] we have the inequality [EQUATION] satisfied for all [MATH].', '1907.06409-1-79-0': 'Throughout the proof, let [MATH] denote a generic positive constant, which may depend on some of fixed constants, such as [MATH], [MATH], [MATH], [MATH] or [MATH], but not on the choice of [MATH] or [MATH].', '1907.06409-1-79-1': 'For brevity, we will use the same notation in all inequalities, even though every specific value of [MATH] depends on the one, where it is used.', '1907.06409-1-79-2': 'Important is that the number of these inequalities is finite.', '1907.06409-1-80-0': 'We first notice that, by Lemma [REF], the relation [MATH] holds for all [MATH].', '1907.06409-1-80-1': 'The process of proving [REF] will be combined with showing that the inequalities [EQUATION] are satisfied for all [MATH].', '1907.06409-1-81-0': 'The proof of [REF]-[REF] is by induction on [MATH].', '1907.06409-1-81-1': 'For [MATH], noticing that [MATH], [MATH] and [MATH], by [REF], [REF] and [REF], we can immediately get [REF]-[REF] satisfied for [MATH].', '1907.06409-1-82-0': 'Suppose that there exist [MATH] and [MATH] with the property that if [REF] holds for any [MATH], then [REF]-[REF] are satisfied for all [MATH].', '1907.06409-1-82-1': 'Next, we shall show that for a smaller choice of [MATH], we can replace [MATH] by [MATH].', '1907.06409-1-82-2': 'Hence, we suppose that [REF] holds for all [MATH].', '1907.06409-1-82-3': 'Since [REF] holds for all [MATH], it follows from the induction hypothesis and [REF] that [EQUATION]', '1907.06409-1-82-4': 'By analogy with the proof of Lemma 2.2 in [CITATION], we derive from [REF], [REF], [REF], [REF], [REF] and the induction hypothesis that [REF]-[REF] hold for [MATH].', '1907.06409-1-82-5': 'Then we just need to show that [EQUATION]', '1907.06409-1-82-6': 'It follows from [REF] that [EQUATION].', '1907.06409-1-82-7': 'Then by choosing any [MATH], using relations [REF], [REF], [REF]-[REF], [REF] and the same reasoning as in the proof of Lemma 2.2 in [CITATION], we obtain [EQUATION]', '1907.06409-1-82-8': 'In the following, the proof of [REF] will be done by separately considering four different cases.', '1907.06409-1-83-0': 'Case I: [MATH] and [MATH].', '1907.06409-1-84-0': 'Then [REF] directly leads to [EQUATION].', '1907.06409-1-85-0': 'Case II: [MATH] and [MATH].', '1907.06409-1-86-0': 'If [MATH], then [REF] implies [EQUATION].', '1907.06409-1-86-1': 'Suppose now that [MATH].', '1907.06409-1-86-2': 'Then we have [EQUATION]', '1907.06409-1-86-3': 'It follows from [REF] and [REF] that [EQUATION]', '1907.06409-1-86-4': 'By [REF] and [REF], we get [EQUATION].', '1907.06409-1-86-5': 'This along with [REF] leads to [EQUATION] where [MATH] whenever [MATH].', '1907.06409-1-86-6': 'Then we obtain [EQUATION]', '1907.06409-1-86-7': 'This together with [REF] shows that [REF] holds.', '1907.06409-1-87-0': 'Case III: [MATH] and [MATH].', '1907.06409-1-88-0': 'If [MATH], then by [REF], we have [EQUATION].', '1907.06409-1-88-1': 'Suppose now that [MATH].', '1907.06409-1-88-2': 'Then we get [EQUATION].', '1907.06409-1-88-3': 'To use the same reasoning as in Case II, we need to have lower bounds for [MATH] and [MATH].', '1907.06409-1-88-4': 'To this end, applying [REF] and [REF], we obtain [EQUATION]', '1907.06409-1-88-5': 'Furthermore, [REF], [REF] and [REF] yield [EQUATION].', '1907.06409-1-88-6': 'This lower bound is positive whenever [MATH].', '1907.06409-1-88-7': 'The two lower bounds allows us to conclude, by analogy with Case II, that [REF] holds.', '1907.06409-1-89-0': 'Case IV: [MATH] and [MATH].', '1907.06409-1-90-0': 'It follows from [REF], [REF], [REF] and [REF] that [EQUATION]', '1907.06409-1-90-1': 'Collecting the results in the considered four cases, one can see that [REF] is satisfied for any [EQUATION].', '1907.06409-1-90-2': 'This completes the induction and finally proves that inequalities [REF]-[REF] hold for all [MATH].', '1907.06409-1-90-3': '[MATH]', '1907.06409-1-91-0': 'Next we will establish the local convergence property of BBstab for nonquadratic functions.', '1907.06409-1-92-0': 'There exists positive [MATH] such that, for any positive [MATH] and any starting points [MATH], the sequence [MATH] converges to [MATH] R-linearly.', '1907.06409-1-93-0': 'Lemma [REF] allows us to skip the proof of this theorem because the reasoning is similar to the proof of Theorem 2.3 in [CITATION].', '1907.06409-1-94-0': 'We complete the analysis by presenting the following global convergence result.', '1907.06409-1-95-0': 'There exists positive [MATH] such that, for any positive [MATH] and any starting points [MATH], the sequence [MATH] converges to [MATH] with R-linear rate of convergence.', '1907.06409-1-96-0': 'Let [MATH] be given by Theorem [REF], which ensures local convergence to [MATH].', '1907.06409-1-96-1': 'According to Lemma [REF], after a finite number of BBstab iterations, all iterates will belong to [MATH].', '1907.06409-1-96-2': 'This finally proves global convergence with R-linear rate.', '1907.06409-1-96-3': '[MATH]', '1907.06409-1-97-0': '# Numerical Results', '1907.06409-1-98-0': 'Our algorithms were implemented in MATLAB.', '1907.06409-1-98-1': 'They terminate when either the number of iterations exceeds [MATH], or [EQUATION].', '1907.06409-1-98-2': 'In the next two subsections, results of numerical experiments are presented separately for quadratic and nonquadratic test functions.', '1907.06409-1-99-0': 'A successful value of [MATH] is obviously problem dependent.', '1907.06409-1-99-1': 'In our implementation, we try to estimate its order of magnitude by setting [MATH] for the first few iterations and making use of [MATH] produced at these iterations by the standard BB algorithm.', '1907.06409-1-99-2': 'At the subsequent iterations, the constant value [EQUATION] is applied, where [MATH] is a parameter.', '1907.06409-1-99-3': 'It turns out that this adaptive choice of [MATH] is less problem dependent.', '1907.06409-1-100-0': 'It is necessary to emphasize that the stabilization was designed not to speed-up the BB method when it safely converges.', '1907.06409-1-100-1': 'In such cases, it may increase the number of iterations, which is a negative outcome.', '1907.06409-1-100-2': 'The main purpose of the stabilization is to prevent the BB method from making too long steps.', '1907.06409-1-100-3': 'This serves for decreasing the number of BB iterations in case of its poor convergence or even making the method convergent when it fails, which is a positive outcome.', '1907.06409-1-100-4': 'Outcomes of all these aforementioned types were observed in our numerical experiments with stabilizing the BB method.', '1907.06409-1-100-5': 'One can easily recognize them in the tables presented below.', '1907.06409-1-101-0': 'We focus here on demonstrating the potentials of improving convergence for the BB method.', '1907.06409-1-101-1': 'Therefore, our stabilized version is not checked here against another optimization algorithms.', '1907.06409-1-101-2': 'Since the computational cost of one iteration for the BB algorithms are practically the same as for their stabilized versions, only the number of iterations are compared.', '1907.06409-1-101-3': 'Notice that the number of iterations is the same as the number of gradient evaluations.', '1907.06409-1-102-0': 'In our numerical experiments, the BB1 algorithm was generating too long steps more frequently than its counterpart.', '1907.06409-1-102-1': 'This is often caused by relatively too small values of the scalar product [MATH] in the denominator of [MATH].', '1907.06409-1-102-2': 'This explains why the stabilization is, in general, more important for the BB1 stepsize choice than for its counterpart.', '1907.06409-1-102-3': 'Therefore, the presented here numerical results refer mainly to the BB1.', '1907.06409-1-103-0': '## Quadratic test functions', '1907.06409-1-104-0': 'A part of the numerical experiments was related to minimizing convex quadratic functions [REF].', '1907.06409-1-104-1': 'This problem is equivalent to solving the system of linear equations [EQUATION].', '1907.06409-1-104-2': 'The matrices in our set of test problems come from the SuiteSparse Matrix Collection [CITATION].', '1907.06409-1-104-3': 'For generating the vector [MATH], we assumed that the solution [MATH], i.e., [MATH], where [MATH].', '1907.06409-1-104-4': 'The total number of problems in our test set is [MATH], where the problem size [MATH] varies from thousands to millions.', '1907.06409-1-105-0': 'For the adaptive selection of [MATH] by formula [REF], we tried just a few values of the parameter [MATH], namely, [MATH], [MATH] and [MATH].', '1907.06409-1-105-1': 'In Tables [REF] and [REF], the number of iterations are reported for algorithms BB1 and BB1stab.', '1907.06409-1-105-2': 'For the latter, the best of the three results is presented along with the corresponding value of [MATH].', '1907.06409-1-105-3': 'If the reported result is the same as for the BB1 algorithm, then it is obvious that the number of iterations remains the same for all values of c larger than the indicated one.', '1907.06409-1-105-4': 'The number of iterations, which is not worse than for the BB1 algorithm, are highlighted in this and other tables in this paper.', '1907.06409-1-105-5': 'One can see that, comparing with the BB1, its stabilized version was faster in solving [MATH] problems, while it was slower in [MATH] problems.', '1907.06409-1-105-6': 'Furthermore, the reduction in the number of iterations obtained by virtue of the stabilization was often substantial.', '1907.06409-1-106-0': '## Nonquadratic test functions', '1907.06409-1-107-0': 'For general functions, it is more difficult than for quadratic ones to avoid the cases, when [MATH] is chosen too close to [MATH] or too far away of it.', '1907.06409-1-107-1': 'In order to avoid such poor choices of these two points, our BBstab algorithms are initialized with only one point, namely, [MATH].', '1907.06409-1-107-2': 'Point [MATH] is produced in the algorithms by checking if the inequality [MATH] is satisfied for [MATH], where [MATH].', '1907.06409-1-107-3': 'Otherwise, a number, typically few, of backtracking steps are performed by dividing the current vector [MATH] by [MATH], while the required inequality is violated.', '1907.06409-1-108-0': 'We begin here by comparing the performance of the BB algorithms and their stabilized versions on the strongly convex Raydan function [REF] for [MATH].', '1907.06409-1-108-1': 'Point [MATH] was used for starting the algorithms.', '1907.06409-1-108-2': 'The standard BB1 algorithm failed to solve the problem.', '1907.06409-1-108-3': 'After two iterations, an overflow in computing [MATH] was reported.', '1907.06409-1-108-4': 'If to introduce the bounds [MATH] for [MATH], like it is often done in practice, then it also fails, although after a larger number of iterations.', '1907.06409-1-108-5': 'Namely, at iteration 123 and all subsequent iterations, an underflow was observed in calculating [MATH] for [MATH].', '1907.06409-1-108-6': 'In these two cases, the standard BB2 also failed.', '1907.06409-1-108-7': 'However, the same test problem for the same [MATH] was successfully solved by BB1stab and BB2stab with [MATH] in [MATH] and [MATH] iterations, respectively.', '1907.06409-1-108-8': 'No bounds, like [MATH], are used in our implementation of the BB algorithms and their stabilized versions.', '1907.06409-1-109-0': 'Figure [REF] illustrates the stabilization effect.', '1907.06409-1-109-1': 'One can see that the BB1 was generating too long steps more frequently than the BB2.', '1907.06409-1-109-2': 'This observation is in general agreement with the other numerical experiments that we performed and also with the theory, which says that [MATH].', '1907.06409-1-110-0': 'The performance of our algorithms was compared also for unconstrained minimization problems from the CUTEst collection [CITATION], which provides a standard starting point [MATH] for each of them.', '1907.06409-1-110-1': 'We excluded from our comparison quadratic problems and those, in which the BB1 algorithm converged in less than [MATH] iterations.', '1907.06409-1-110-2': 'The results reported here concern only the problems, where at least one of the compared algorithms converged, and also those, where the both algorithms converged to the same point.', '1907.06409-1-111-0': 'Recall that the BB method was originally designed for solving convex problems in which case it is guaranteed that [MATH] is nonnegative.', '1907.06409-1-111-1': 'Since the most of the unconstrained minimization test problems in the CUTEst collection are nonconvex, we had to adapt the BB method to solving this kind of problems.', '1907.06409-1-111-2': 'In our implementation of the BB method and its stabilized version, we follow paper [CITATION] in setting [EQUATION] whenever [MATH].', '1907.06409-1-111-3': 'This makes our algorithms much more robust.', '1907.06409-1-111-4': 'Figure [REF] presents results of solving [MATH] problems from the CUTEst collection.', '1907.06409-1-111-5': 'The BB1 and BB2 algorithms failed in [MATH] and [MATH] cases, respectively.', '1907.06409-1-111-6': 'The plots of the performance profiles introduced in [CITATION] indicate that the BB2 algorithm is more robust than the BB1.', '1907.06409-1-111-7': 'Furthermore, the former algorithm required, on average, fewer iterations for solving problems.', '1907.06409-1-111-8': 'The main reason is that the BB1 algorithm generates too long steps more frequently.', '1907.06409-1-111-9': 'In what follows, we focus on presenting here results of stabilizing the BB1 algorithm, because it gains more from the stabilization than the BB2 algorithm.', '1907.06409-1-112-0': 'Table [REF] presents results of solving [MATH] nonquadratic test problems from the CUTEst collection.', '1907.06409-1-112-1': 'We tried only three values of the parameter [MATH] in the adaptive choice of [MATH] using [REF], namely, [MATH], [MATH] and [MATH].', '1907.06409-1-112-2': 'The BB1 and BB1stab algorithms were not able to solve problems during [MATH] iterations in [MATH] and [MATH] cases, respectively.', '1907.06409-1-112-3': 'The BB1stab required less number of iterations in [MATH] cases, while the BB1 performed better only in [MATH] cases.', '1907.06409-1-112-4': 'In [MATH] cases, the BB1stab with the indicated values of [MATH] required the same number of iterations as the BB1.', '1907.06409-1-113-0': 'We made experiments also with directly setting a certain value of [MATH] in the BB1stab.', '1907.06409-1-113-1': 'The trial values were [MATH], [MATH] and [MATH].', '1907.06409-1-113-2': 'For a few test problems, the results were better than for the aforementioned adaptive choice with [MATH], [MATH] and [MATH].', '1907.06409-1-113-3': 'For [MATH] of [MATH] problems, the number of iterations is smaller than in case of the BB1.', '1907.06409-1-114-0': 'These results are reported in Table [REF].', '1907.06409-1-114-1': 'The preselected values of [MATH] allowed the BB1stab to solve five problems of those not solved by the BB1, including problems MOREBV and TQUARTIC, in which the adaptive choice of [MATH] failed.', '1907.06409-1-114-2': 'In case of TQUARTIC, the BB1 terminated because of producing NaN (Not a Number) in Matlab.', '1907.06409-1-114-3': 'The experiments with the preselected values of [MATH] indicate that there is plenty of room for improving the very simple adaptive strategy proposed in this paper.', '1907.06409-1-115-0': '# Conclusions', '1907.06409-1-116-0': 'In the present paper, it was proposed to stabilize the conventional BB method by virtue of bounding the distance between sequential iterates.', '1907.06409-1-116-1': 'The purpose was to improve its convergence, when it is affected by too long steps [MATH], and also to make the BB method convergent, when it fails to converge.', '1907.06409-1-116-2': 'Both a theoretical and numerical study of the stabilized version was conducted.', '1907.06409-1-116-3': 'We have proved that the stabilization provides the BB method with a global convergence without recourse to using any line search.', '1907.06409-1-116-4': 'The numerical results presented here are highly encouraging.', '1907.06409-1-116-5': 'The proposed very simple adaptive selection of [MATH] was able to successfully trap a value which is appropriate for each specific problem.', '1907.06409-1-116-6': 'However, we hope that this paper will stimulate development of more efficient algorithms for adaptive selection of [MATH].'}

{'1907.06409-2-0-0': 'The Barzilai-Borwein (BB) method is a popular and efficient tool for solving large-scale unconstrained optimization problems.', '1907.06409-2-0-1': 'Its search direction is the same as for the steepest descent (Cauchy) method, but its stepsize rule is different.', '1907.06409-2-0-2': 'Owing to this, it converges much faster than the Cauchy method.', '1907.06409-2-0-3': 'A feature of the BB method is that it may generate too long steps, which throw the iterates too far away from the solution.', '1907.06409-2-0-4': 'Moreover, it may not converge, even when the objective function is strongly convex.', '1907.06409-2-0-5': 'In this paper, a stabilization technique is introduced.', '1907.06409-2-0-6': 'It consists in bounding the distance between each pair of successive iterates, which often allows for decreasing the number of BB iterations.', '1907.06409-2-0-7': 'When the BB method does not converge, our simple modification of this method makes it convergent.', '1907.06409-2-0-8': 'For strongly convex functions with Lipschits gradients, we prove its global convergence, despite the fact that no line search is involved, and only gradient values are used.', '1907.06409-2-0-9': 'Since the number of stabilization steps is proved to be finite, the stabilized version inherits the fast local convergence of the BB method.', '1907.06409-2-0-10': 'The presented results of extensive numerical experiments show that our stabilization technique often allows the BB method to solve problems in a fewer iterations, or even to solve problems where the latter fails.', '1907.06409-2-1-0': '65K05, 90C06, 90C30.', '1907.06409-2-2-0': '# Introduction', '1907.06409-2-3-0': 'In this paper, we consider spectral gradient methods for solving the unconstrained optimization problem [EQUATION] where [MATH] is a sufficiently smooth function.', '1907.06409-2-3-1': 'Its minimizer is denoted by [MATH].', '1907.06409-2-3-2': 'Gradient-type iterative methods used for solving problem [REF] have the form [EQUATION] where [MATH] and [MATH] is a stepsize.', '1907.06409-2-3-3': 'Methods of this type differ in the stepsize rules which they follow.', '1907.06409-2-4-0': 'We focus here on the two choices of [MATH] proposed in 1988 by Barzilai and Borwein [CITATION], usually refereed to as the BB method.', '1907.06409-2-4-1': 'The rationale behind these choices is related to viewing the gradient-type methods as quasi-Newton methods, where [MATH] in [REF] is replaced by the matrix [MATH].', '1907.06409-2-4-2': 'This matrix is served as an approximation of the inverse Hessian matrix.', '1907.06409-2-4-3': 'Following the quasi-Newton approach, the stepsize is calculated by forcing either [MATH] (BB1 method) or [MATH] (BB2 method) to satisfy the secant equation in the least squares sense.', '1907.06409-2-4-4': 'The corresponding two problems are formulated as [EQUATION] where [MATH] and [MATH].', '1907.06409-2-4-5': 'The solutions to these problems are [EQUATION] respectively.', '1907.06409-2-4-6': 'Here and in what follows, [MATH] denotes the Euclidean vector norm and the induced matrix norm.', '1907.06409-2-4-7': 'Other norms used in this paper will be denoted in a different way.', '1907.06409-2-5-0': 'Barzilai and Borwein [CITATION] proved that their method converges [MATH]-superlinearly for two-dimensional strictly convex quadratics.', '1907.06409-2-5-1': 'Dai and Fletcher [CITATION] analyzed the asymptotic behavior of BB-like methods, and they obtained [MATH]-superlinear convergence of the BB method for the three-dimensional case.', '1907.06409-2-5-2': 'Global convergence of the BB method for the [MATH]-dimensional case was established by Raydan [CITATION] and further refined by Dai and Liao [CITATION] for obtaining the R-linear rate.', '1907.06409-2-5-3': 'For nonquadratic functions, local convergence proof of the BB method with R-linear rate was, first, sketched in some detail by Liu and Dai [CITATION], and then it was later rigorously proved by Dai et al. [CITATION].', '1907.06409-2-5-4': 'Extensive numerical experiments show that the two BB stepsize rules significantly improve the performance of gradient methods (see, e.g., [CITATION]), both in quadratic and nonquadratic cases.', '1907.06409-2-6-0': 'A variety of modifications and extensions have been developed, such as gradient methods with retards [CITATION], alternate BB method [CITATION], cyclic BB method [CITATION], limited memory gradient method [CITATION] etc.', '1907.06409-2-6-1': 'Several approaches were proposed for dealing with nonconvex objective functions, in which case the BB stepsize [REF] may become negative.', '1907.06409-2-6-2': 'In our numerical experiments, we use the one proposed in [CITATION].', '1907.06409-2-6-3': 'The BB method has been extended to solving symmetric and nonsymmetric linear equations [CITATION].', '1907.06409-2-6-4': 'Furthermore, by incorporating the nonmontone line search by Grippo et al. [CITATION], Raydan [CITATION] and Grippo et al. [CITATION] developed the global BB method for general unconstrained optimization problems.', '1907.06409-2-6-5': "Later, Birgin et al. [CITATION] proposed the so-called spectral projected gradient method which extends Raydan's method to smooth convex constrained problems.", '1907.06409-2-6-6': 'For more works on BB-like methods, see [CITATION] and references therein.', '1907.06409-2-7-0': 'As it was observed by many authors, the BB method may generate too long steps, which throw the iterates too far away from the solution.', '1907.06409-2-7-1': 'In practice, it may not converge even for strongly convex functions (see, e.g., [CITATION]).', '1907.06409-2-7-2': 'The purpose of this paper is to introduce a simple stabilization technique and to justify its efficiency both theoretically and practically.', '1907.06409-2-7-3': 'Our stabilization does not assume any objective function evaluations.', '1907.06409-2-7-4': 'It consists in uniformly bounding [MATH], the distance between each pair of successive iterates.', '1907.06409-2-7-5': 'It should be emphasized that, if the BB method safely converges for a given function, then there is no necessity in stabilizing it.', '1907.06409-2-7-6': 'In such cases, the stabilization may increase the number of iterations.', '1907.06409-2-7-7': 'In other cases, as it will be demonstrated by results of our numerical experiments, the stabilization may allow for decreasing the number of iterations or even to make the BB method convergent.', '1907.06409-2-8-0': 'Although we focus here on stabilizing the conventional BB method, our approach can directly be combined with the existing modifications of the BB method, where a nonmonotone line search is used.', '1907.06409-2-9-0': 'The paper is organized as follows.', '1907.06409-2-9-1': 'In the next section, we present an example of a strictly convex function and show that the BB method does not converge in this case.', '1907.06409-2-9-2': 'This contributes to a motivation for stabilizing this method.', '1907.06409-2-9-3': 'In the same section, its stabilized version is introduced.', '1907.06409-2-9-4': 'In Section [REF], a global convergence of our stabilized BB algorithm as well as its R-linear rate of convergence are proved under suitable assumptions.', '1907.06409-2-9-5': 'Results of numerical experiments are reported and discussed in Section [REF].', '1907.06409-2-9-6': 'Finally, some conclusions are included in the last section of the paper.', '1907.06409-2-10-0': '# Stabilized Algorithm', '1907.06409-2-11-0': 'Before formulating our stabilized algorithm, we wish to begin with a motivation based on presenting an example of a strongly convex function for which we theoretically prove that neither of the BB methods converge.', '1907.06409-2-11-1': 'To the best of our knowledge, no theoretical evidence of this kind is available in the literature.', '1907.06409-2-12-0': 'In the review paper by Fletcher [CITATION], it is claimed that the BB method diverges in practice for certain initial points in the test problem referred to as Strictly Convex 2 by Raydan [CITATION], in which [EQUATION]', '1907.06409-2-12-1': 'This strongly convex function will be used in Section [REF] for illustrating the efficiency of the stabilized algorithm.', '1907.06409-2-12-2': 'Our numerical experiments show that, in this specific case, the failure of the BB method is related to the underflow and overflow effects in the computer arithmetic.', '1907.06409-2-12-3': 'We are not acquainted with any theoretical justification of the divergence of the BB method for this or any other functions.', '1907.06409-2-13-0': 'We will present now an instance of a function for which the BB method does not converge in the exact arithmetic.', '1907.06409-2-13-1': 'For this purpose, the notation [EQUATION] will be used.', '1907.06409-2-13-2': 'Consider the univariate function [EQUATION]', '1907.06409-2-13-3': 'Its first derivative [EQUATION] is continuously differentiable, and [MATH] is an odd monotonically increasing function (see Fugure [REF]).', '1907.06409-2-14-0': 'It can be easily verified that the function [MATH] is twice continuously differentiable with [EQUATION].', '1907.06409-2-14-1': 'This means that this function is strongly convex, and its first derivative is Lipschitz-continuous.', '1907.06409-2-15-0': 'For any univariate objective function, there is no difference between BB1 and BB2 versions, and they are equivalent to the secant method applied to the first derivative.', '1907.06409-2-15-1': 'For function [REF], if to initiate the BB method with [MATH] and [MATH], then the subsequent iterates are [EQUATION]', '1907.06409-2-15-2': 'This clearly shows that the BB method cycles between four points (see Fugure [REF]).', '1907.06409-2-15-3': 'The presented counter-example can be easily extended to [MATH]-dimensional case.', '1907.06409-2-16-0': 'After motivating the necessity of stabilizing the BB method, we can now proceed to presenting the basic idea of our stabilized BB algorithm, where [MATH] is a parameter.', '1907.06409-2-16-1': 'It consists in choosing the stepsize in [REF] in the way that [MATH], whenever [MATH].', '1907.06409-2-16-2': 'Thus, denoting [EQUATION] we propose to choose [EQUATION]', '1907.06409-2-16-3': 'Here [MATH] or [MATH], depending on the specific BB method in [REF].', '1907.06409-2-16-4': 'A formal description of our stabilized BB algorithm follows.', '1907.06409-2-17-0': 'BBstab.', '1907.06409-2-18-0': '1px', '1907.06409-2-19-0': 'Given: initial points [MATH] such that [MATH], and scalar [MATH].', '1907.06409-2-20-0': '0.5px', '1907.06409-2-21-0': 'Evaluate [MATH] and [MATH].', '1907.06409-2-22-0': 'for [MATH] do', '1907.06409-2-23-0': 'if [MATH] then stop.', '1907.06409-2-24-0': 'Set [MATH] and [MATH].', '1907.06409-2-25-0': 'Compute [MATH] by formula [REF].', '1907.06409-2-26-0': 'Set [MATH] and evaluate [MATH].', '1907.06409-2-27-0': 'end (for)', '1907.06409-2-28-0': '0.5px', '1907.06409-2-29-0': 'This algorithm will be refereed to as BB1stab or BB2stab depending on the corresponding choice of [MATH] in [REF].', '1907.06409-2-29-1': 'Note that, for [MATH], it reduces to the underlying standard BB algorithm.', '1907.06409-2-30-0': '# Convergence Analysis', '1907.06409-2-31-0': 'In this section, global convergence of the BBstab algorithm will be proved.', '1907.06409-2-31-1': 'Whenever iterates [MATH] are mentioned, they are assumed to be generated by BBstab, where it is required that [MATH].', '1907.06409-2-32-0': 'Throughout this section, the objective function is assumed to comply with the following requirement.', '1907.06409-2-33-0': '[A1.]', '1907.06409-2-33-1': 'The function [MATH] is twice continuously differentiable, and there exist positive constants [MATH] such that [EQUATION]', '1907.06409-2-33-2': 'This assumption implies that [EQUATION]', '1907.06409-2-33-3': 'Extra assumptions are introduced below in proper places.', '1907.06409-2-34-0': 'We shall use the following notation: [EQUATION] which will be motivated later.', '1907.06409-2-34-1': 'Here [EQUATION].', '1907.06409-2-34-2': 'Obviously, [MATH], and [MATH], where [MATH].', '1907.06409-2-34-3': 'We shall use similar notation for other unions of sets [MATH].', '1907.06409-2-35-0': 'Inequalities [REF] ensure that [EQUATION] which in turn means that [EQUATION] and [EQUATION]', '1907.06409-2-35-1': 'These bounds justify the implications [EQUATION]', '1907.06409-2-35-2': 'We can now prove the following result.', '1907.06409-2-36-0': 'Let [MATH] be arbitrary starting points.', '1907.06409-2-36-1': 'Then for any [MATH], the iterates [MATH] have the property that [EQUATION] where [EQUATION].', '1907.06409-2-37-0': 'Using Assumption A1, we get [EQUATION] where the matrix [MATH] is symmetric, and it fulfills the relations [EQUATION].', '1907.06409-2-37-1': 'Clearly, [EQUATION]', '1907.06409-2-37-2': 'Consider, first, the case when [MATH].', '1907.06409-2-37-3': 'Using the inequality [MATH] and relations [REF], we can derive for [REF] the following upper bound [EQUATION].', '1907.06409-2-37-4': 'This proves the upper inequality in [REF].', '1907.06409-2-38-0': 'Suppose now that [MATH], i.e., [MATH].', '1907.06409-2-38-1': 'Then, using [REF], we get the bounds [MATH], which together with the inequalities [MATH] yield [EQUATION].', '1907.06409-2-38-2': 'By combining this estimate with [REF], we finally prove the lower inequality in [REF].', '1907.06409-2-38-3': '[MATH]', '1907.06409-2-39-0': 'Lemma [REF] implies that the stabilization steps have the following properties [EQUATION]', '1907.06409-2-39-1': 'Next, we prove that, after a finite number of iterations, all iterates belong to the bounded set [MATH].', '1907.06409-2-40-0': 'For any [MATH] and [MATH], there exists an integer [MATH] such that the inequality [EQUATION] holds, that is [MATH], for all [MATH].', '1907.06409-2-40-1': 'Moreover, [MATH] is the iteration number corresponding to the first iterate [MATH] that belongs to [MATH].', '1907.06409-2-41-0': 'Notice that [REF] is satisfied if and only if [MATH].', '1907.06409-2-41-1': 'We first show that if [MATH], then so does the next iterate.', '1907.06409-2-41-2': 'Indeed, in view of [REF] and [REF], if [MATH], then [MATH].', '1907.06409-2-41-3': 'On the other hand, if [MATH], i.e. [MATH], then, by Lemma [REF], we have [MATH].', '1907.06409-2-42-0': 'Suppose now that [MATH].', '1907.06409-2-42-1': 'Then it immediately follows from relations [REF] and [REF], that there exists [MATH] such that [MATH].', '1907.06409-2-42-2': 'As it was shown above, this means that [MATH] for all [MATH].', '1907.06409-2-42-3': '[MATH]', '1907.06409-2-43-0': 'It follows from [REF] that, when iterates belong to the set [MATH], the value [MATH] monotonically decreases as indicated by [REF].', '1907.06409-2-43-1': 'Furthermore, the actual decrease may speed-up in accordance with [REF].', '1907.06409-2-43-2': 'When the iterates reach [MATH], the decrease is naturally expected to slow down, and this is followed by a non-monotonic behavior of [MATH], which is a typical feature of the BB steps.', '1907.06409-2-44-0': 'One can observe all these stages in the behavior of BBstab in Figure [REF].', '1907.06409-2-44-1': 'It presents changes of [MATH] with [MATH] in the process of minimizing Raydan function [REF].', '1907.06409-2-44-2': 'Details of these runs are discussed in Section [REF].', '1907.06409-2-44-3': 'Note that both BB1 and BB2 fail to solve this problem starting from the same points.', '1907.06409-2-44-4': 'The figure illustrates the role of stabilization in providing convergence of BBstab.', '1907.06409-2-44-5': 'One can clearly recognize the first stage of the process when the stabilization steps ensure a monotonic decrease of [MATH].', '1907.06409-2-44-6': 'For the BB1stab and BB2stab, the iteration when the standard BB step was used for the first time is 228 and 226, respectively.', '1907.06409-2-44-7': 'For them, the last stabilization step was used in iteration 379 and 353, respectively.', '1907.06409-2-44-8': 'Observe that the spikes of [MATH] produced by BB1 is much larger than those for BB2.', '1907.06409-2-45-0': 'Lemma [REF] allows us to deduce that, if the BB method generates bounded steps, it cannot generate unbounded iterates because one can choose a sufficiently large [MATH], which is not binding.', '1907.06409-2-45-1': 'The same lemma indicates that a proper choice of [MATH] allows for BBstab to reach any neighborhood of [MATH].', '1907.06409-2-45-2': 'We use the notation [EQUATION] in the following formulation of this useful feature of BBstab.', '1907.06409-2-46-0': 'Let [MATH] be any starting points.', '1907.06409-2-46-1': 'Then for any [MATH] and positive [MATH], there exists [MATH] such that the iterates [MATH] satisfy the condition [EQUATION].', '1907.06409-2-47-0': 'Combining [REF] and Lemma [REF], we get the relations [EQUATION] which are satisfied for all sufficiently large [MATH].', '1907.06409-2-47-1': 'This completes the proof.', '1907.06409-2-47-2': '[MATH]', '1907.06409-2-48-0': 'We shall make use of Lemma [REF] for proving global convergence result for BBstab.', '1907.06409-2-48-1': 'We show also that its local rate of convergence is R-linear, which means that there exist positive [MATH] and [MATH] such that [EQUATION]', '1907.06409-2-48-2': 'These convergence results are based on our convergence analysis presented in the next sub-section for convex quadratic functions.', '1907.06409-2-49-0': '## Convergence in Quadratic Case', '1907.06409-2-50-0': 'In this sub-section, we focus on minimizing convex quadratic functions of the form [EQUATION] where the matrix [MATH] is positive definite, and [MATH].', '1907.06409-2-50-1': 'For these functions, we have the following convergence result.', '1907.06409-2-51-0': 'Let [MATH] be arbitrary starting points.', '1907.06409-2-51-1': 'Then for any [MATH], the sequence [MATH] converges to [MATH] with R-linear rate of convergence.', '1907.06409-2-52-0': 'It is well known that the BB method is invariant under orthogonal transformation of the variables and, as it can be easily seen, so does its stabilized version.', '1907.06409-2-52-1': 'Hence, we can assume without loss of generality that the matrix [MATH] is of the form [EQUATION] where [MATH].', '1907.06409-2-52-2': 'Here, like it is often done for the gradient methods (see, e.g., [CITATION]), it is assumed without loss of generality that the matrix [MATH] has distinct eigenvalues.', '1907.06409-2-52-3': 'Then denoting the [MATH]-th component of [MATH] by [MATH], we have [EQUATION]', '1907.06409-2-52-4': 'To prove convergence, suppose for contradiction that there exists [MATH] such that [EQUATION]', '1907.06409-2-52-5': 'Based on this assumption, we now use induction in [MATH] to show that [EQUATION]', '1907.06409-2-52-6': 'By Lemma [REF], for all sufficiently large [MATH], iterates [MATH] belong to [MATH].', '1907.06409-2-52-7': 'Therefore, in proving [REF], we assume that [MATH].', '1907.06409-2-52-8': 'Then combining [REF] and [REF], we get the inequality [EQUATION] which holds for [MATH].', '1907.06409-2-52-9': 'This proves [REF] for [MATH].', '1907.06409-2-53-0': 'Now, assume that [REF] holds for [MATH].', '1907.06409-2-53-1': 'Let us, first, derive an upper bound for the stepsize [MATH].', '1907.06409-2-53-2': 'This will be done separately for BB1- and BB2-based iterates.', '1907.06409-2-54-0': 'For the BB1 case, we have [EQUATION] where [EQUATION].', '1907.06409-2-55-0': 'For BB2, denote [MATH].', '1907.06409-2-55-1': 'Then [EQUATION] where [EQUATION].', '1907.06409-2-56-0': 'The assumed low bound [REF] and the induction hypothesis [REF] show that [MATH] and [MATH] when [MATH].', '1907.06409-2-56-1': 'For the BB1 iterates, the bounds [EQUATION] ensure that the inequality [EQUATION] is satisfied for all sufficiently large [MATH].', '1907.06409-2-56-2': 'The same inequality is obviously satisfied for the BB2 iterates.', '1907.06409-2-56-3': 'Then it follows from [REF] that inequality [REF] holds for [MATH] and all sufficiently large [MATH].', '1907.06409-2-56-4': 'Consequently, [EQUATION].', '1907.06409-2-56-5': 'The induction is now complete.', '1907.06409-2-57-0': 'Thus, [REF] holds for [MATH], which contradicts the assumption [REF].', '1907.06409-2-57-1': 'This means that [MATH] is a limit point.', '1907.06409-2-57-2': 'Then it is the only limit point, i.e., the iterates [MATH] converge to [MATH].', '1907.06409-2-57-3': 'This follows from the fact that [MATH], whenever [MATH], which holds for all sufficiently large [MATH].', '1907.06409-2-57-4': 'The same fact allows us to finally conclude that the rate of convergence is R-linear.', '1907.06409-2-57-5': '[MATH]', '1907.06409-2-58-0': 'The following result plays an important role in our proof of R-linear convergence for nonquadratic functions.', '1907.06409-2-59-0': 'There exists a positive integer [MATH], such that, for any [MATH] and [MATH], the inequality [EQUATION] holds for all [MATH].', '1907.06409-2-60-0': 'Like in the poof of Theorema [REF], we assume here without loss of generality that [MATH] has the form of [REF], and we shall use the notation [MATH] for the [MATH]th component of [MATH].', '1907.06409-2-60-1': 'We will also make use of the following definition, where [EQUATION].', '1907.06409-2-61-0': 'We say that the choice of the stepsize in [REF] has property P if there exist an integer [MATH] and positive constants [MATH] and [MATH] such that, for all [MATH],', '1907.06409-2-62-0': '(i) [MATH];', '1907.06409-2-63-0': '(ii) for any integer [MATH] and real number [MATH], if [MATH] and [MATH] hold for [MATH], then [MATH].', '1907.06409-2-64-0': 'Firstly, we prove that the step size [MATH] has property P. Lemma [REF] ensures that [MATH] for all [MATH].', '1907.06409-2-64-1': 'Then the bounds [REF] show that [MATH] has property P(i) with [MATH].', '1907.06409-2-65-0': 'Next, we will show, for any integer [MATH] and real number [MATH], that the inequality [MATH] is satisfied, whenever [MATH] and [MATH].', '1907.06409-2-65-1': 'This will be done separately for BB1- and BB2-based iterates.', '1907.06409-2-66-0': 'For the BB1 case, we have [EQUATION]', '1907.06409-2-66-1': 'For BB2, we obtain [EQUATION]', '1907.06409-2-66-2': 'Thus, P(ii) holds for [MATH] and [MATH].', '1907.06409-2-66-3': 'This implies that BBstab stepsize [MATH] satisfies P.', '1907.06409-2-66-4': 'Then we can skip the rest of the proof because it is similar to the proof of Theorem 4.1 in [CITATION].', '1907.06409-2-66-5': '[MATH]', '1907.06409-2-67-0': 'It should be emphasized that, in this lemma, the value of [MATH] depends only on [MATH] and [MATH].', '1907.06409-2-68-0': '## Convergence in General Case', '1907.06409-2-69-0': 'For nonquadratic functions, we shall first prove local R-linear convergence of BBstab.', '1907.06409-2-69-1': 'This result will then be used for showing that it converges from any starting point.', '1907.06409-2-70-0': 'Throughout this sub-section, we need to additionally assume that the Hessian matrix [MATH] is Lipschitz-continuous at [MATH].', '1907.06409-2-70-1': 'In what follows, we use the notation [MATH].', '1907.06409-2-71-0': '[A2.]', '1907.06409-2-71-1': 'There exist a radius [MATH] and a Lipschitz constant [MATH] such that [EQUATION].', '1907.06409-2-72-0': 'This assumption implies that [EQUATION]', '1907.06409-2-72-1': 'The second-order Taylor approximation to [MATH] around [MATH] is given by the quadratic function [EQUATION]', '1907.06409-2-72-2': 'Define new iterates [MATH] associated with [MATH] as follows: [EQUATION] where [EQUATION]', '1907.06409-2-72-3': 'Here [MATH] or [MATH] and [MATH] with [EQUATION] [MATH], [MATH] and [MATH].', '1907.06409-2-72-4': 'In what follows, whenever we mention [MATH] and [MATH], they are assumed to be generated as defined above.', '1907.06409-2-73-0': 'The next result follows immediately from Lemma [REF].', '1907.06409-2-74-0': 'Let [MATH] be any scalar, such that [MATH].', '1907.06409-2-74-1': 'Then there exists a positive integer [MATH], dependent only on [MATH] and [MATH], such that, for any [MATH], the inequality holds [EQUATION].', '1907.06409-2-75-0': 'It can be easily seen that if [MATH], then all corresponding [MATH].', '1907.06409-2-75-1': 'In this case, BBstab stepsize [MATH] satisfies the bounds [REF], and similarly for [MATH], we have the bounds [EQUATION]', '1907.06409-2-75-2': 'The following result will be used for proving local R-linear convergence.', '1907.06409-2-76-0': 'Let integer [MATH] be arbitrary.', '1907.06409-2-76-1': 'Then there exist positive scalars [MATH] and [MATH] with the following property: for any [MATH], [MATH] and [MATH], for which [EQUATION] we have the inequality [EQUATION] satisfied for all [MATH].', '1907.06409-2-77-0': 'Throughout the proof, let [MATH] denote a generic positive constant, which may depend on some of fixed constants, such as [MATH], [MATH], [MATH], [MATH] or [MATH], but not on the choice of [MATH] or [MATH].', '1907.06409-2-77-1': 'For brevity, we will use the same notation in all inequalities, even though every specific value of [MATH] depends on the one, where it is used.', '1907.06409-2-77-2': 'Important is that the number of these inequalities is finite.', '1907.06409-2-78-0': 'We first notice that, by Lemma [REF], the relation [MATH] holds for all [MATH].', '1907.06409-2-78-1': 'The process of proving [REF] will be combined with showing that the inequalities [EQUATION] are satisfied for all [MATH].', '1907.06409-2-79-0': 'The proof of [REF]-[REF] is by induction on [MATH].', '1907.06409-2-79-1': 'For [MATH], noticing that [MATH], [MATH] and [MATH], by [REF], [REF] and [REF], we can immediately get [REF]-[REF] satisfied for [MATH].', '1907.06409-2-80-0': 'Suppose that there exist [MATH] and [MATH] with the property that if [REF] holds for any [MATH], then [REF]-[REF] are satisfied for all [MATH].', '1907.06409-2-80-1': 'Next, we shall show that for a smaller choice of [MATH], we can replace [MATH] by [MATH].', '1907.06409-2-80-2': 'Hence, we suppose that [REF] holds for all [MATH].', '1907.06409-2-80-3': 'Since [REF] holds for all [MATH], it follows from the induction hypothesis and [REF] that [EQUATION]', '1907.06409-2-80-4': 'By analogy with the proof of Lemma 2.2 in [CITATION], we derive from [REF], [REF], [REF], [REF], [REF] and the induction hypothesis that [REF]-[REF] hold for [MATH].', '1907.06409-2-80-5': 'Then we just need to show that [EQUATION]', '1907.06409-2-80-6': 'It follows from [REF] that [EQUATION].', '1907.06409-2-80-7': 'Then by choosing any [MATH], using relations [REF], [REF], [REF]-[REF], [REF] and the same reasoning as in the proof of Lemma 2.2 in [CITATION], we obtain [EQUATION]', '1907.06409-2-80-8': 'In the following, the proof of [REF] will be done by separately considering four different cases.', '1907.06409-2-81-0': 'Case I: [MATH] and [MATH].', '1907.06409-2-82-0': 'Then [REF] directly leads to [EQUATION].', '1907.06409-2-83-0': 'Case II: [MATH] and [MATH].', '1907.06409-2-84-0': 'If [MATH], then [REF] implies [EQUATION].', '1907.06409-2-84-1': 'Suppose now that [MATH].', '1907.06409-2-84-2': 'Then we have [EQUATION]', '1907.06409-2-84-3': 'It follows from [REF] and [REF] that [EQUATION]', '1907.06409-2-84-4': 'By [REF] and [REF], we get [EQUATION].', '1907.06409-2-84-5': 'This along with [REF] leads to [EQUATION] where [MATH] whenever [MATH].', '1907.06409-2-84-6': 'Then we obtain [EQUATION]', '1907.06409-2-84-7': 'This together with [REF] shows that [REF] holds.', '1907.06409-2-85-0': 'Case III: [MATH] and [MATH].', '1907.06409-2-86-0': 'If [MATH], then by [REF], we have [EQUATION].', '1907.06409-2-86-1': 'Suppose now that [MATH].', '1907.06409-2-86-2': 'Then we get [EQUATION].', '1907.06409-2-86-3': 'To use the same reasoning as in Case II, we need to have lower bounds for [MATH] and [MATH].', '1907.06409-2-86-4': 'To this end, applying [REF] and [REF], we obtain [EQUATION]', '1907.06409-2-86-5': 'Furthermore, [REF], [REF] and [REF] yield [EQUATION].', '1907.06409-2-86-6': 'This lower bound is positive whenever [MATH].', '1907.06409-2-86-7': 'The two lower bounds allows us to conclude, by analogy with Case II, that [REF] holds.', '1907.06409-2-87-0': 'Case IV: [MATH] and [MATH].', '1907.06409-2-88-0': 'It follows from [REF], [REF], [REF] and [REF] that [EQUATION]', '1907.06409-2-88-1': 'Collecting the results in the considered four cases, one can see that [REF] is satisfied for any [EQUATION].', '1907.06409-2-88-2': 'This completes the induction and finally proves that inequalities [REF]-[REF] hold for all [MATH].', '1907.06409-2-88-3': '[MATH]', '1907.06409-2-89-0': 'Next we will establish the local convergence property of BBstab for nonquadratic functions.', '1907.06409-2-90-0': 'There exists positive [MATH] such that, for any positive [MATH] and any starting points [MATH], the sequence [MATH] converges to [MATH] R-linearly.', '1907.06409-2-91-0': 'Lemma [REF] allows us to skip the proof of this theorem because the reasoning is similar to the proof of Theorem 2.3 in [CITATION].', '1907.06409-2-92-0': 'We complete the analysis by presenting the following global convergence result.', '1907.06409-2-93-0': 'There exists positive [MATH] such that, for any positive [MATH] and any starting points [MATH], the sequence [MATH] converges to [MATH] with R-linear rate of convergence.', '1907.06409-2-94-0': 'Let [MATH] be given by Theorem [REF], which ensures local convergence to [MATH].', '1907.06409-2-94-1': 'According to Lemma [REF], after a finite number of BBstab iterations, all iterates will belong to [MATH].', '1907.06409-2-94-2': 'This finally proves global convergence with R-linear rate.', '1907.06409-2-94-3': '[MATH]', '1907.06409-2-95-0': '# Numerical Results', '1907.06409-2-96-0': 'Our algorithms were implemented in MATLAB.', '1907.06409-2-96-1': 'They terminate when either the number of iterations exceeds [MATH], or [EQUATION].', '1907.06409-2-96-2': 'In the next two subsections, results of numerical experiments are presented separately for quadratic and nonquadratic test functions.', '1907.06409-2-97-0': 'A successful value of [MATH] is obviously problem dependent.', '1907.06409-2-97-1': 'In our implementation, we try to estimate its order of magnitude by setting [MATH] for the first few iterations and making use of [MATH] produced at these iterations by the standard BB algorithm.', '1907.06409-2-97-2': 'At the subsequent iterations, the constant value [EQUATION] is applied, where [MATH] is a parameter.', '1907.06409-2-97-3': 'It turns out that this adaptive choice of [MATH] is less problem dependent.', '1907.06409-2-98-0': 'It is necessary to emphasize that the stabilization was designed not to speed-up the BB method when it safely converges.', '1907.06409-2-98-1': 'In such cases, it may increase the number of iterations, which is a negative outcome.', '1907.06409-2-98-2': 'The main purpose of the stabilization is to prevent the BB method from making too long steps.', '1907.06409-2-98-3': 'This serves for decreasing the number of BB iterations in case of its poor convergence or even making the method convergent when it fails, which is a positive outcome.', '1907.06409-2-98-4': 'Outcomes of all these aforementioned types were observed in our numerical experiments with stabilizing the BB method.', '1907.06409-2-98-5': 'One can easily recognize them in the tables presented below.', '1907.06409-2-99-0': 'We focus here on demonstrating the potentials of improving convergence for the BB method.', '1907.06409-2-99-1': 'Therefore, our stabilized version is not checked here against another optimization algorithms.', '1907.06409-2-99-2': 'Since the computational cost of one iteration for the BB algorithms are practically the same as for their stabilized versions, only the number of iterations are compared.', '1907.06409-2-99-3': 'Notice that the number of iterations is the same as the number of gradient evaluations.', '1907.06409-2-100-0': 'In our numerical experiments, the BB1 algorithm was generating too long steps more frequently than its counterpart.', '1907.06409-2-100-1': 'This is often caused by relatively too small values of the scalar product [MATH] in the denominator of [MATH].', '1907.06409-2-100-2': 'This explains why the stabilization is, in general, more important for the BB1 stepsize choice than for its counterpart.', '1907.06409-2-100-3': 'Therefore, the presented here numerical results refer mainly to the BB1.', '1907.06409-2-101-0': '## Quadratic test functions', '1907.06409-2-102-0': 'A part of the numerical experiments was related to minimizing convex quadratic functions [REF].', '1907.06409-2-102-1': 'This problem is equivalent to solving the system of linear equations [EQUATION].', '1907.06409-2-102-2': 'The matrices in our set of test problems come from the SuiteSparse Matrix Collection [CITATION].', '1907.06409-2-102-3': 'For generating the vector [MATH], we assumed that the solution [MATH], i.e., [MATH], where [MATH].', '1907.06409-2-102-4': 'The total number of problems in our test set is [MATH], where the problem size [MATH] varies from thousands to millions.', '1907.06409-2-103-0': 'For the adaptive selection of [MATH] by formula [REF], we tried just a few values of the parameter [MATH], namely, [MATH], [MATH] and [MATH].', '1907.06409-2-103-1': 'In Tables [REF] and [REF], the number of iterations are reported for algorithms BB1 and BB1stab.', '1907.06409-2-103-2': 'For the latter, the best of the three results is presented along with the corresponding value of [MATH].', '1907.06409-2-103-3': 'If the reported result is the same as for the BB1 algorithm, then it is obvious that the number of iterations remains the same for all values of c larger than the indicated one.', '1907.06409-2-103-4': 'The number of iterations, which is not worse than for the BB1 algorithm, are highlighted in this and other tables in this paper.', '1907.06409-2-103-5': 'One can see that, comparing with the BB1, its stabilized version was faster in solving [MATH] problems, while it was slower in [MATH] problems.', '1907.06409-2-103-6': 'Furthermore, the reduction in the number of iterations obtained by virtue of the stabilization was often substantial.', '1907.06409-2-104-0': '## Nonquadratic test functions', '1907.06409-2-105-0': 'For general functions, it is more difficult than for quadratic ones to avoid the cases, when [MATH] is chosen too close to [MATH] or too far away of it.', '1907.06409-2-105-1': 'In order to avoid such poor choices of these two points, our BBstab algorithms are initialized with only one point, namely, [MATH].', '1907.06409-2-105-2': 'Point [MATH] is produced in the algorithms by checking if the inequality [MATH] is satisfied for [MATH], where [MATH].', '1907.06409-2-105-3': 'Otherwise, a number, typically few, of backtracking steps are performed by dividing the current vector [MATH] by [MATH], while the required inequality is violated.', '1907.06409-2-106-0': 'We begin here by comparing the performance of the BB algorithms and their stabilized versions on the strongly convex Raydan function [REF] for [MATH].', '1907.06409-2-106-1': 'Point [MATH] was used for starting the algorithms.', '1907.06409-2-106-2': 'The standard BB1 algorithm failed to solve the problem.', '1907.06409-2-106-3': 'After two iterations, an overflow in computing [MATH] was reported.', '1907.06409-2-106-4': 'If to introduce the bounds [MATH] for [MATH], like it is often done in practice, then it also fails, although after a larger number of iterations.', '1907.06409-2-106-5': 'Namely, at iteration 123 and all subsequent iterations, an underflow was observed in calculating [MATH] for [MATH].', '1907.06409-2-106-6': 'In these two cases, the standard BB2 also failed.', '1907.06409-2-106-7': 'However, the same test problem for the same [MATH] was successfully solved by BB1stab and BB2stab with [MATH] in [MATH] and [MATH] iterations, respectively.', '1907.06409-2-106-8': 'No bounds, like [MATH], are used in our implementation of the BB algorithms and their stabilized versions.', '1907.06409-2-107-0': 'Figure [REF] illustrates the stabilization effect.', '1907.06409-2-107-1': 'One can see that the BB1 was generating too long steps more frequently than the BB2.', '1907.06409-2-107-2': 'This observation is in general agreement with the other numerical experiments that we performed and also with the theory, which says that [MATH].', '1907.06409-2-108-0': 'The performance of our algorithms was compared also for unconstrained minimization problems from the CUTEst collection [CITATION], which provides a standard starting point [MATH] for each of them.', '1907.06409-2-108-1': 'We excluded from our comparison quadratic problems and those, in which the BB1 algorithm converged in less than [MATH] iterations.', '1907.06409-2-108-2': 'The results reported here concern only the problems, where at least one of the compared algorithms converged, and also those, where the both algorithms converged to the same point.', '1907.06409-2-109-0': 'Recall that the BB method was originally designed for solving convex problems in which case it is guaranteed that [MATH] is nonnegative.', '1907.06409-2-109-1': 'Since the most of the unconstrained minimization test problems in the CUTEst collection are nonconvex, we had to adapt the BB method to solving this kind of problems.', '1907.06409-2-109-2': 'In our implementation of the BB method and its stabilized version, we follow paper [CITATION] in setting [EQUATION] whenever [MATH].', '1907.06409-2-109-3': 'This makes our algorithms much more robust.', '1907.06409-2-109-4': 'Figure [REF] presents results of solving [MATH] problems from the CUTEst collection.', '1907.06409-2-109-5': 'The BB1 and BB2 algorithms failed in [MATH] and [MATH] cases, respectively.', '1907.06409-2-109-6': 'The plots of the performance profiles introduced in [CITATION] indicate that the BB2 algorithm is more robust than the BB1.', '1907.06409-2-109-7': 'Furthermore, the former algorithm required, on average, fewer iterations for solving problems.', '1907.06409-2-109-8': 'The main reason is that the BB1 algorithm generates too long steps more frequently.', '1907.06409-2-109-9': 'In what follows, we focus on presenting here results of stabilizing the BB1 algorithm, because it gains more from the stabilization than the BB2 algorithm.', '1907.06409-2-110-0': 'Table [REF] presents results of solving [MATH] nonquadratic test problems from the CUTEst collection.', '1907.06409-2-110-1': 'We tried only three values of the parameter [MATH] in the adaptive choice of [MATH] using [REF], namely, [MATH], [MATH] and [MATH].', '1907.06409-2-110-2': 'The BB1 and BB1stab algorithms were not able to solve problems during [MATH] iterations in [MATH] and [MATH] cases, respectively.', '1907.06409-2-110-3': 'The BB1stab required less number of iterations in [MATH] cases, while the BB1 performed better only in [MATH] cases.', '1907.06409-2-110-4': 'In [MATH] cases, the BB1stab with the indicated values of [MATH] required the same number of iterations as the BB1.', '1907.06409-2-111-0': 'We made experiments also with directly setting a certain value of [MATH] in the BB1stab.', '1907.06409-2-111-1': 'The trial values were [MATH], [MATH] and [MATH].', '1907.06409-2-111-2': 'For a few test problems, the results were better than for the aforementioned adaptive choice with [MATH], [MATH] and [MATH].', '1907.06409-2-111-3': 'For [MATH] of [MATH] problems, the number of iterations is smaller than in case of the BB1.', '1907.06409-2-112-0': 'These results are reported in Table [REF].', '1907.06409-2-112-1': 'The preselected values of [MATH] allowed the BB1stab to solve five problems of those not solved by the BB1, including problems MOREBV and TQUARTIC, in which the adaptive choice of [MATH] failed.', '1907.06409-2-112-2': 'In case of TQUARTIC, the BB1 terminated because of producing NaN (Not a Number) in Matlab.', '1907.06409-2-112-3': 'The experiments with the preselected values of [MATH] indicate that there is plenty of room for improving the very simple adaptive strategy proposed in this paper.', '1907.06409-2-113-0': '# Conclusions', '1907.06409-2-114-0': 'In the present paper, it was proposed to stabilize the conventional BB method by virtue of bounding the distance between sequential iterates.', '1907.06409-2-114-1': 'The purpose was to improve its convergence, when it is affected by too long steps [MATH], and also to make the BB method convergent, when it fails to converge.', '1907.06409-2-114-2': 'Both a theoretical and numerical study of the stabilized version was conducted.', '1907.06409-2-114-3': 'We have proved that the stabilization provides the BB method with a global convergence without recourse to using any line search.', '1907.06409-2-114-4': 'The numerical results presented here are highly encouraging.', '1907.06409-2-114-5': 'The proposed very simple adaptive selection of [MATH] was able to successfully trap a value which is appropriate for each specific problem.', '1907.06409-2-114-6': 'However, we hope that this paper will stimulate development of more efficient algorithms for adaptive selection of [MATH].'}

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['1907.06409-1-12-2', '1907.06409-2-12-2'], ['1907.06409-1-12-3', '1907.06409-2-12-3'], ['1907.06409-1-95-0', '1907.06409-2-93-0'], ['1907.06409-1-109-0', '1907.06409-2-107-0'], ['1907.06409-1-109-1', '1907.06409-2-107-1'], ['1907.06409-1-109-2', '1907.06409-2-107-2'], ['1907.06409-1-108-0', '1907.06409-2-106-0'], ['1907.06409-1-108-1', '1907.06409-2-106-1'], ['1907.06409-1-108-2', '1907.06409-2-106-2'], ['1907.06409-1-108-3', '1907.06409-2-106-3'], ['1907.06409-1-108-4', '1907.06409-2-106-4'], ['1907.06409-1-108-5', '1907.06409-2-106-5'], ['1907.06409-1-108-6', '1907.06409-2-106-6'], ['1907.06409-1-108-7', '1907.06409-2-106-7'], ['1907.06409-1-108-8', '1907.06409-2-106-8'], ['1907.06409-1-112-0', '1907.06409-2-110-0'], ['1907.06409-1-112-1', '1907.06409-2-110-1'], ['1907.06409-1-112-2', '1907.06409-2-110-2'], ['1907.06409-1-112-3', '1907.06409-2-110-3'], ['1907.06409-1-112-4', '1907.06409-2-110-4'], ['1907.06409-1-71-0', '1907.06409-2-69-0'], ['1907.06409-1-71-1', '1907.06409-2-69-1'], ['1907.06409-1-100-0', '1907.06409-2-98-0'], ['1907.06409-1-100-1', '1907.06409-2-98-1'], ['1907.06409-1-100-2', '1907.06409-2-98-2'], ['1907.06409-1-100-3', '1907.06409-2-98-3'], ['1907.06409-1-100-4', '1907.06409-2-98-4'], ['1907.06409-1-100-5', '1907.06409-2-98-5'], ['1907.06409-1-0-0', '1907.06409-2-0-0'], ['1907.06409-1-0-1', '1907.06409-2-0-1'], ['1907.06409-1-0-2', '1907.06409-2-0-2'], ['1907.06409-1-0-3', '1907.06409-2-0-3'], ['1907.06409-1-0-4', '1907.06409-2-0-4'], ['1907.06409-1-0-5', '1907.06409-2-0-5'], ['1907.06409-1-0-6', '1907.06409-2-0-6'], ['1907.06409-1-0-7', '1907.06409-2-0-7'], ['1907.06409-1-0-9', '1907.06409-2-0-9'], ['1907.06409-1-0-10', '1907.06409-2-0-10'], ['1907.06409-1-41-0', '1907.06409-2-41-0'], ['1907.06409-1-41-1', '1907.06409-2-41-1'], ['1907.06409-1-41-2', '1907.06409-2-41-2'], ['1907.06409-1-41-3', '1907.06409-2-41-3'], ['1907.06409-1-3-0', '1907.06409-2-3-0'], ['1907.06409-1-3-1', '1907.06409-2-3-1'], ['1907.06409-1-3-2', 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['1907.06409-1-68-4', '1907.06409-2-66-4'], ['1907.06409-1-101-0', '1907.06409-2-99-0'], ['1907.06409-1-101-1', '1907.06409-2-99-1'], ['1907.06409-1-101-2', '1907.06409-2-99-2'], ['1907.06409-1-101-3', '1907.06409-2-99-3'], ['1907.06409-1-67-0', '1907.06409-2-65-0'], ['1907.06409-1-67-1', '1907.06409-2-65-1'], ['1907.06409-1-9-0', '1907.06409-2-9-0'], ['1907.06409-1-9-1', '1907.06409-2-9-1'], ['1907.06409-1-9-2', '1907.06409-2-9-2'], ['1907.06409-1-9-3', '1907.06409-2-9-3'], ['1907.06409-1-9-4', '1907.06409-2-9-4'], ['1907.06409-1-9-5', '1907.06409-2-9-5'], ['1907.06409-1-9-6', '1907.06409-2-9-6'], ['1907.06409-1-16-0', '1907.06409-2-16-0'], ['1907.06409-1-16-1', '1907.06409-2-16-1'], ['1907.06409-1-16-2', '1907.06409-2-16-2'], ['1907.06409-1-16-3', '1907.06409-2-16-3'], ['1907.06409-1-16-4', '1907.06409-2-16-4'], ['1907.06409-1-61-0', '1907.06409-2-59-0'], ['1907.06409-1-73-1', '1907.06409-2-71-1'], ['1907.06409-1-116-0', '1907.06409-2-114-0'], ['1907.06409-1-116-1', 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[]

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[]

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{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '3': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1907.06409

{'1907.06409-3-0-0': 'The Barzilai-Borwein (BB) method is a popular and efficient tool for solving large-scale unconstrained optimization problems.', '1907.06409-3-0-1': 'Its search direction is the same as for the steepest descent (Cauchy) method, but its stepsize rule is different.', '1907.06409-3-0-2': 'Owing to this, it converges much faster than the Cauchy method.', '1907.06409-3-0-3': 'A feature of the BB method is that it may generate too long steps, which throw the iterates too far away from the solution.', '1907.06409-3-0-4': 'Moreover, it may not converge, even when the objective function is strongly convex.', '1907.06409-3-0-5': 'In this paper, a stabilization technique is introduced.', '1907.06409-3-0-6': 'It consists in bounding the distance between each pair of successive iterates, which often allows for decreasing the number of BB iterations.', '1907.06409-3-0-7': 'When the BB method does not converge, our simple modification of this method makes it convergent.', '1907.06409-3-0-8': 'For strongly convex functions with Lipschits gradients, we prove its global convergence, despite the fact that no line search is involved, and only gradient values are used.', '1907.06409-3-0-9': 'Since the number of stabilization steps is proved to be finite, the stabilized version inherits the fast local convergence of the BB method.', '1907.06409-3-0-10': 'The presented results of extensive numerical experiments show that our stabilization technique often allows the BB method to solve problems in a fewer iterations, or even to solve problems where the latter fails.', '1907.06409-3-1-0': '65K05, 90C06, 90C30.', '1907.06409-3-2-0': '# Introduction', '1907.06409-3-3-0': 'In this paper, we consider spectral gradient methods for solving the unconstrained optimization problem [EQUATION] where [MATH] is a sufficiently smooth function.', '1907.06409-3-3-1': 'Its minimizer is denoted by [MATH].', '1907.06409-3-3-2': 'Gradient-type iterative methods used for solving problem [REF] have the form [EQUATION] where [MATH] and [MATH] is a stepsize.', '1907.06409-3-3-3': 'Methods of this type differ in the stepsize rules which they follow.', '1907.06409-3-4-0': 'We focus here on the two choices of [MATH] proposed in 1988 by Barzilai and Borwein [CITATION], usually refereed to as the BB method.', '1907.06409-3-4-1': 'The rationale behind these choices is related to viewing the gradient-type methods as quasi-Newton methods, where [MATH] in [REF] is replaced by the matrix [MATH].', '1907.06409-3-4-2': 'This matrix is served as an approximation of the inverse Hessian matrix.', '1907.06409-3-4-3': 'Following the quasi-Newton approach, the stepsize is calculated by forcing either [MATH] (BB1 method) or [MATH] (BB2 method) to satisfy the secant equation in the least squares sense.', '1907.06409-3-4-4': 'The corresponding two problems are formulated as [EQUATION] where [MATH] and [MATH].', '1907.06409-3-4-5': 'The solutions to these problems are [EQUATION] respectively.', '1907.06409-3-4-6': 'Here and in what follows, [MATH] denotes the Euclidean vector norm and the induced matrix norm.', '1907.06409-3-4-7': 'Other norms used in this paper will be denoted in a different way.', '1907.06409-3-5-0': 'Barzilai and Borwein [CITATION] proved that their method converges [MATH]-superlinearly for two-dimensional strictly convex quadratics.', '1907.06409-3-5-1': 'Dai and Fletcher [CITATION] analyzed the asymptotic behavior of BB-like methods, and they obtained [MATH]-superlinear convergence of the BB method for the three-dimensional case.', '1907.06409-3-5-2': 'Global convergence of the BB method for the [MATH]-dimensional case was established by Raydan [CITATION] and further refined by Dai and Liao [CITATION] for obtaining the R-linear rate.', '1907.06409-3-5-3': 'For nonquadratic functions, local convergence proof of the BB method with R-linear rate was, first, sketched in some detail by Liu and Dai [CITATION], and then it was later rigorously proved by Dai et al. [CITATION].', '1907.06409-3-5-4': 'Extensive numerical experiments show that the two BB stepsize rules significantly improve the performance of gradient methods (see, e.g., [CITATION]), both in quadratic and nonquadratic cases.', '1907.06409-3-6-0': 'A variety of modifications and extensions have been developed, such as gradient methods with retards [CITATION], alternate BB method [CITATION], cyclic BB method [CITATION], limited memory gradient method [CITATION] etc.', '1907.06409-3-6-1': 'Several approaches were proposed for dealing with nonconvex objective functions, in which case the BB stepsize [REF] may become negative.', '1907.06409-3-6-2': 'In our numerical experiments, we use the one proposed in [CITATION].', '1907.06409-3-6-3': 'The BB method has been extended to solving symmetric and nonsymmetric linear equations [CITATION].', '1907.06409-3-6-4': 'Furthermore, by incorporating the nonmontone line search by Grippo et al. [CITATION], Raydan [CITATION] and Grippo et al. [CITATION] developed the global BB method for general unconstrained optimization problems.', '1907.06409-3-6-5': "Later, Birgin et al. [CITATION] proposed the so-called spectral projected gradient method which extends Raydan's method to smooth convex constrained problems.", '1907.06409-3-6-6': 'For more works on BB-like methods, see [CITATION] and references therein.', '1907.06409-3-7-0': 'As it was observed by many authors, the BB method may generate too long steps, which throw the iterates too far away from the solution.', '1907.06409-3-7-1': 'In practice, it may not converge even for strongly convex functions (see, e.g., [CITATION]).', '1907.06409-3-7-2': 'The purpose of this paper is to introduce a simple stabilization technique and to justify its efficiency both theoretically and practically.', '1907.06409-3-7-3': 'Our stabilization does not assume any objective function evaluations.', '1907.06409-3-7-4': 'It consists in uniformly bounding [MATH], the distance between each pair of successive iterates.', '1907.06409-3-7-5': 'It should be emphasized that, if the BB method safely converges for a given function, then there is no necessity in stabilizing it.', '1907.06409-3-7-6': 'In such cases, the stabilization may increase the number of iterations.', '1907.06409-3-7-7': 'In other cases, as it will be demonstrated by results of our numerical experiments, the stabilization may allow for decreasing the number of iterations or even to make the BB method convergent.', '1907.06409-3-8-0': 'Although we focus here on stabilizing the conventional BB method, our approach can directly be combined with the existing modifications of the BB method, where a nonmonotone line search is used.', '1907.06409-3-9-0': 'The paper is organized as follows.', '1907.06409-3-9-1': 'In the next section, we present an example of a strictly convex function and show that the BB method does not converge in this case.', '1907.06409-3-9-2': 'This contributes to a motivation for stabilizing this method.', '1907.06409-3-9-3': 'In the same section, its stabilized version is introduced.', '1907.06409-3-9-4': 'In Section [REF], a global convergence of our stabilized BB algorithm as well as its R-linear rate of convergence are proved under suitable assumptions.', '1907.06409-3-9-5': 'Results of numerical experiments are reported and discussed in Section [REF].', '1907.06409-3-9-6': 'Finally, some conclusions are included in the last section of the paper.', '1907.06409-3-10-0': '# Stabilized Algorithm', '1907.06409-3-11-0': 'Before formulating our stabilized algorithm, we wish to begin with a motivation based on presenting an example of a strongly convex function for which we theoretically prove that neither of the BB methods converge.', '1907.06409-3-11-1': 'To the best of our knowledge, no theoretical evidence of BB methods being divergent is available in the literature.', '1907.06409-3-12-0': 'In the review paper by Fletcher [CITATION], it is claimed that the BB method diverges in practice for certain initial points in the test problem referred to as Strictly Convex 2 by Raydan [CITATION], in which [EQUATION]', '1907.06409-3-12-1': 'This strongly convex function will be used in Section [REF] for illustrating the efficiency of the stabilized algorithm.', '1907.06409-3-12-2': 'Our numerical experiments show that, in this specific case, the failure of the BB method is related to the underflow and overflow effects in the computer arithmetic.', '1907.06409-3-12-3': 'We are not acquainted with any theoretical justification of the divergence of the BB method for this or any other functions.', '1907.06409-3-13-0': 'We will present now an instance of a function for which the BB method does not converge in the exact arithmetic.', '1907.06409-3-13-1': 'For this purpose, the notation [EQUATION] will be used.', '1907.06409-3-13-2': 'Consider the univariate function [EQUATION]', '1907.06409-3-13-3': 'Its first derivative [EQUATION] is continuously differentiable, and [MATH] is an odd monotonically increasing function (see Figure [REF]).', '1907.06409-3-14-0': 'It can be easily verified that the function [MATH] is twice continuously differentiable with [EQUATION].', '1907.06409-3-14-1': 'This means that this function is strongly convex, and its first derivative is Lipschitz-continuous.', '1907.06409-3-15-0': 'For any univariate objective function, there is no difference between BB1 and BB2 versions, and they are equivalent to the secant method applied to the first derivative.', '1907.06409-3-15-1': 'For function [REF], if to initiate the BB method with [MATH] and [MATH], then the subsequent iterates are [EQUATION]', '1907.06409-3-15-2': 'This clearly shows that the BB method cycles between four points (see Figure [REF]).', '1907.06409-3-15-3': 'The presented counter-example can be easily extended to [MATH]-dimensional case.', '1907.06409-3-15-4': 'As an example, one can consider a separable objective function equal to the sum of any number of functions of the form [REF], where no variable appears in more than one of these functions.', '1907.06409-3-16-0': 'After motivating the necessity of stabilizing the BB method, we can now proceed to presenting the basic idea of our stabilized BB algorithm, where [MATH] is a parameter.', '1907.06409-3-16-1': 'It consists in choosing the stepsize in [REF] in the way that [MATH], whenever [MATH], i.e. [MATH].', '1907.06409-3-16-2': 'In other cases, we choose [MATH], which results in [MATH].', '1907.06409-3-16-3': 'Thus, denoting [EQUATION] we propose to choose [EQUATION]', '1907.06409-3-16-4': 'Here [MATH] or [MATH], depending on the specific BB method in [REF].', '1907.06409-3-16-5': 'A formal description of our stabilized BB algorithm follows.', '1907.06409-3-17-0': 'BBstab.', '1907.06409-3-18-0': '1px', '1907.06409-3-19-0': 'Given: initial points [MATH] such that [MATH], and scalar [MATH].', '1907.06409-3-20-0': '0.5px', '1907.06409-3-21-0': 'Evaluate [MATH] and [MATH].', '1907.06409-3-22-0': 'for [MATH] do', '1907.06409-3-23-0': 'if [MATH] then stop.', '1907.06409-3-24-0': 'Set [MATH] and [MATH].', '1907.06409-3-25-0': 'Compute [MATH] by formula [REF].', '1907.06409-3-26-0': 'Set [MATH] and evaluate [MATH].', '1907.06409-3-27-0': 'end (for)', '1907.06409-3-28-0': '0.5px', '1907.06409-3-29-0': 'This algorithm will be refereed to as BB1stab or BB2stab depending on the corresponding choice of [MATH] in [REF].', '1907.06409-3-29-1': 'Note that, for [MATH], it reduces to the underlying standard BB algorithm.', '1907.06409-3-30-0': '# Convergence Analysis', '1907.06409-3-31-0': 'In this section, global convergence of the BBstab algorithm will be proved.', '1907.06409-3-31-1': 'Whenever iterates [MATH] are mentioned, they are assumed to be generated by BBstab, where it is required that [MATH].', '1907.06409-3-32-0': 'Throughout this section, the objective function is assumed to comply with the following requirement.', '1907.06409-3-33-0': '[A1.]', '1907.06409-3-33-1': 'The function [MATH] is twice continuously differentiable, and there exist positive constants [MATH] such that [EQUATION]', '1907.06409-3-33-2': 'This assumption implies that [EQUATION]', '1907.06409-3-33-3': 'Extra assumptions are introduced below in proper places.', '1907.06409-3-34-0': 'We shall use the following notation: [EQUATION] which will be motivated later.', '1907.06409-3-34-1': 'Here [EQUATION].', '1907.06409-3-34-2': 'Obviously, [MATH], and [MATH], where [MATH].', '1907.06409-3-34-3': 'We shall use similar notation for other unions of sets [MATH].', '1907.06409-3-35-0': 'Inequalities [REF] ensure that [EQUATION] which in turn means that [EQUATION] and [EQUATION]', '1907.06409-3-35-1': 'These bounds justify the implications [EQUATION]', '1907.06409-3-35-2': 'We can now prove the following result.', '1907.06409-3-36-0': 'Let [MATH] be arbitrary starting points.', '1907.06409-3-36-1': 'Then for any [MATH], the iterates [MATH] have the property that [EQUATION] where [EQUATION].', '1907.06409-3-37-0': 'Using Assumption A1, we get [EQUATION] where the matrix [MATH] is symmetric, and it fulfills the relations [EQUATION].', '1907.06409-3-37-1': 'Clearly, [EQUATION]', '1907.06409-3-37-2': 'Consider, first, the case when [MATH].', '1907.06409-3-37-3': 'Using the inequality [MATH] and relations [REF], we can derive for [REF] the following upper bound [EQUATION].', '1907.06409-3-37-4': 'This proves the upper inequality in [REF].', '1907.06409-3-38-0': 'Suppose now that [MATH], i.e., [MATH].', '1907.06409-3-38-1': 'Then, using [REF], we get the bounds [MATH], which together with the inequalities [MATH] yield [EQUATION].', '1907.06409-3-38-2': 'By combining this estimate with [REF], we finally prove the lower inequality in [REF].', '1907.06409-3-38-3': '[MATH]', '1907.06409-3-39-0': 'Lemma [REF] implies that the stabilization steps have the following properties [EQUATION]', '1907.06409-3-39-1': 'Next, we prove that, after a finite number of iterations, all iterates belong to the bounded set [MATH].', '1907.06409-3-40-0': 'For any [MATH] and [MATH], there exists an integer [MATH] such that the inequality [EQUATION] holds, that is [MATH], for all [MATH].', '1907.06409-3-40-1': 'Moreover, [MATH] is the iteration number corresponding to the first iterate [MATH] that belongs to [MATH].', '1907.06409-3-41-0': 'Notice that [REF] is satisfied if and only if [MATH].', '1907.06409-3-41-1': 'We first show that if [MATH], then so does the next iterate.', '1907.06409-3-41-2': 'Indeed, in view of [REF] and [REF], if [MATH], then [MATH].', '1907.06409-3-41-3': 'On the other hand, if [MATH], i.e. [MATH], then, by Lemma [REF], we have [MATH].', '1907.06409-3-42-0': 'Suppose now that [MATH].', '1907.06409-3-42-1': 'Then it immediately follows from relations [REF] and [REF], that there exists [MATH] such that [MATH].', '1907.06409-3-42-2': 'As it was shown above, this means that [MATH] for all [MATH].', '1907.06409-3-42-3': '[MATH]', '1907.06409-3-43-0': 'It follows from [REF] that, when iterates belong to the set [MATH], the value [MATH] monotonically decreases as indicated by [REF].', '1907.06409-3-43-1': 'Furthermore, the actual decrease may speed-up in accordance with [REF].', '1907.06409-3-43-2': 'When the iterates reach [MATH], the decrease is naturally expected to slow down, and this is followed by a non-monotonic behavior of [MATH], which is a typical feature of the BB steps.', '1907.06409-3-44-0': 'One can observe all these stages in the behavior of BBstab in Figure [REF].', '1907.06409-3-44-1': 'It presents changes of [MATH] with [MATH] in the process of minimizing Raydan function [REF].', '1907.06409-3-44-2': 'Details of these runs are discussed in Section [REF].', '1907.06409-3-44-3': 'Note that both BB1 and BB2 fail to solve this problem starting from the same points.', '1907.06409-3-44-4': 'The figure illustrates the role of stabilization in providing convergence of BBstab.', '1907.06409-3-44-5': 'One can clearly recognize the first stage of the process when the stabilization steps ensure a monotonic decrease of [MATH].', '1907.06409-3-44-6': 'For the BB1stab and BB2stab, the iteration when the standard BB step was used for the first time is 228 and 226, respectively.', '1907.06409-3-44-7': 'For them, the last stabilization step was used in iteration 379 and 353, respectively.', '1907.06409-3-44-8': 'Observe that the spikes of [MATH] produced by BB1 is much larger than those for BB2.', '1907.06409-3-45-0': 'Lemma [REF] allows us to deduce an interesting property of the BB method, namely, that if it generates bounded steps, it cannot generate unbounded iterates because one can choose a sufficiently large [MATH], which is not binding.', '1907.06409-3-45-1': 'The same lemma indicates that a proper choice of [MATH] allows for BBstab to reach any neighborhood of [MATH].', '1907.06409-3-45-2': 'We use the notation [EQUATION] in the following formulation of this useful feature of BBstab.', '1907.06409-3-46-0': 'Let [MATH] be any starting points.', '1907.06409-3-46-1': 'Then for any [MATH] and positive [MATH], there exists [MATH] such that the iterates [MATH] satisfy the condition [EQUATION].', '1907.06409-3-47-0': 'Combining [REF] and Lemma [REF], we get the relations [EQUATION] which are satisfied for all sufficiently large [MATH].', '1907.06409-3-47-1': 'This completes the proof.', '1907.06409-3-47-2': '[MATH]', '1907.06409-3-48-0': 'We shall make use of Lemma [REF] for proving global convergence result for BBstab.', '1907.06409-3-48-1': 'We show also that its local rate of convergence is R-linear, which means that there exist positive [MATH] and [MATH] such that [EQUATION]', '1907.06409-3-48-2': 'These convergence results are based on our convergence analysis presented in the next sub-section for convex quadratic functions.', '1907.06409-3-49-0': '## Convergence in Quadratic Case', '1907.06409-3-50-0': 'In this sub-section, we focus on minimizing convex quadratic functions of the form [EQUATION] where the matrix [MATH] is positive definite, and [MATH].', '1907.06409-3-50-1': 'For these functions, we derive the convergence with R-linear rate.', '1907.06409-3-50-2': 'To this end, we will make use of the following property which is the same as Property A in [CITATION].', '1907.06409-3-51-0': 'We say that the choice of the stepsize in [REF] has property P if there exist an integer [MATH] and positive constants [MATH] and [MATH] such that, for all [MATH],', '1907.06409-3-52-0': '(i) [MATH];', '1907.06409-3-53-0': '(ii) for any integer [MATH] and real number [MATH], if [MATH] and [MATH] hold for [MATH], then [MATH].', '1907.06409-3-54-0': 'Let [MATH] be arbitrary starting points.', '1907.06409-3-54-1': 'Then for any [MATH], the sequence [MATH] converges to [MATH] with R-linear rate.', '1907.06409-3-54-2': 'Moreover, there exists a positive integer [MATH], such that, for any [MATH], [MATH] and [MATH], the inequality [EQUATION] holds for all [MATH].', '1907.06409-3-55-0': 'It is well known that the BB method is invariant under orthogonal transformation of the variables and, as it can be easily seen, so does its stabilized version.', '1907.06409-3-55-1': 'Hence, we can assume without loss of generality that the matrix [MATH] is of the form [EQUATION] where [MATH].', '1907.06409-3-55-2': 'Here, like it is often done for the gradient methods (see, e.g., [CITATION]), it is assumed without loss of generality that the matrix [MATH] has distinct eigenvalues.', '1907.06409-3-55-3': 'Then denoting the [MATH]-th component of [MATH] by [MATH], we have [EQUATION]', '1907.06409-3-55-4': 'We will also make use of the following notation: [EQUATION].', '1907.06409-3-56-0': 'Firstly, we prove that the step size [MATH] has property P. Lemma [REF] ensures that [MATH] for all [MATH].', '1907.06409-3-56-1': 'Then the bounds [REF] show that [MATH] has property P(i) with [MATH].', '1907.06409-3-57-0': 'Next, we will show, for any integer [MATH] and real number [MATH], that the inequality [MATH] is satisfied, whenever [MATH] and [MATH].', '1907.06409-3-57-1': 'This will be done separately for BB1- and BB2-based iterates.', '1907.06409-3-58-0': 'For the BB1 case, we have [EQUATION]', '1907.06409-3-58-1': 'For BB2, we obtain [EQUATION]', '1907.06409-3-58-2': 'Thus, P(ii) holds for [MATH] and [MATH].', '1907.06409-3-58-3': 'This implies that BBstab stepsize [MATH] satisfies P.', '1907.06409-3-58-4': 'Then we can skip the rest of the proof because it is similar to the proof of Theorem 4.1 in [CITATION].', '1907.06409-3-58-5': '[MATH]', '1907.06409-3-59-0': 'It should be emphasized that, in this lemma, the value of [MATH] depends only on [MATH] and [MATH].', '1907.06409-3-60-0': '## Convergence in General Case', '1907.06409-3-61-0': 'For nonquadratic functions, we shall first prove local R-linear convergence of BBstab.', '1907.06409-3-61-1': 'This result will then be used for showing that it converges from any starting point.', '1907.06409-3-62-0': 'Throughout this sub-section, we need to additionally assume that the Hessian matrix [MATH] is Lipschitz-continuous at [MATH].', '1907.06409-3-62-1': 'In what follows, we use the notation [MATH].', '1907.06409-3-63-0': '[A2.]', '1907.06409-3-63-1': 'There exist a radius [MATH] and a Lipschitz constant [MATH] such that [EQUATION].', '1907.06409-3-64-0': 'This assumption implies that [EQUATION]', '1907.06409-3-64-1': 'The second-order Taylor approximation to [MATH] around [MATH] is given by the quadratic function [EQUATION]', '1907.06409-3-64-2': 'Define new iterates [MATH] associated with [MATH] as follows: [EQUATION] where [EQUATION]', '1907.06409-3-64-3': 'Here [MATH] or [MATH] and [MATH] with [EQUATION] [MATH], [MATH] and [MATH].', '1907.06409-3-64-4': 'In what follows, whenever we mention [MATH] and [MATH], they are assumed to be generated as defined above.', '1907.06409-3-65-0': 'The next result follows immediately from Theorem [REF].', '1907.06409-3-66-0': 'Let [MATH] be any scalar, such that [MATH].', '1907.06409-3-66-1': 'Then there exists a positive integer [MATH], dependent only on [MATH] and [MATH], such that, for any [MATH] and [MATH], the inequality holds [EQUATION].', '1907.06409-3-67-0': 'It can be easily seen that if [MATH], then all corresponding [MATH].', '1907.06409-3-67-1': 'In this case, BBstab stepsize [MATH] satisfies the bounds [REF], and similarly for [MATH], we have the bounds [EQUATION]', '1907.06409-3-67-2': 'The following result will be used for proving local R-linear convergence.', '1907.06409-3-68-0': 'Let integer [MATH] be arbitrary.', '1907.06409-3-68-1': 'Then there exist positive scalars [MATH] and [MATH] with the following property: for any [MATH], [MATH], [MATH] and [MATH], for which [EQUATION] we have the inequality [EQUATION] satisfied for all [MATH].', '1907.06409-3-69-0': 'Throughout the proof, let [MATH] denote a generic positive constant, which may depend on some of fixed constants, such as [MATH], [MATH], [MATH], [MATH] or [MATH], but not on the choice of [MATH] or [MATH].', '1907.06409-3-69-1': 'For brevity, we will use the same notation in all inequalities, even though every specific value of [MATH] depends on the one, where it is used.', '1907.06409-3-69-2': 'What is important is that the number of these inequalities is finite.', '1907.06409-3-70-0': 'We first notice that, by Lemma [REF], the relation [MATH] holds for all [MATH].', '1907.06409-3-70-1': 'The process of proving [REF] will be combined with showing that the inequalities [EQUATION] are satisfied for all [MATH].', '1907.06409-3-71-0': 'The proof of [REF]-[REF] is by induction on [MATH].', '1907.06409-3-71-1': 'For [MATH], noticing that [MATH], [MATH] and [MATH], by [REF], [REF] and [REF], we can immediately get [REF]-[REF] satisfied for [MATH].', '1907.06409-3-72-0': 'Suppose that there exist [MATH] and [MATH] with the property that if [REF] holds for any [MATH], then [REF]-[REF] are satisfied for all [MATH].', '1907.06409-3-72-1': 'Next, we shall show that for a smaller choice of [MATH], we can replace [MATH] by [MATH].', '1907.06409-3-72-2': 'Hence, we suppose that [REF] holds for all [MATH].', '1907.06409-3-72-3': 'Since [REF] holds for all [MATH], it follows from the induction hypothesis and [REF] that [EQUATION]', '1907.06409-3-72-4': 'By analogy with the proof of Lemma 2.2 in [CITATION], we derive from [REF], [REF], [REF], [REF], [REF] and the induction hypothesis that [REF]-[REF] hold for [MATH].', '1907.06409-3-72-5': 'Then we just need to show that [EQUATION]', '1907.06409-3-72-6': 'It follows from [REF] that [EQUATION].', '1907.06409-3-72-7': 'Then by choosing any [MATH], using relations [REF], [REF], [REF]-[REF], [REF] and the same reasoning as in the proof of Lemma 2.2 in [CITATION], we obtain [EQUATION]', '1907.06409-3-72-8': 'In the following, the proof of [REF] will be done by separately considering four different cases.', '1907.06409-3-73-0': 'Case I: [MATH] and [MATH].', '1907.06409-3-74-0': 'Then [REF] directly leads to [EQUATION].', '1907.06409-3-75-0': 'Case II: [MATH] and [MATH].', '1907.06409-3-76-0': 'If [MATH], then [REF] implies [EQUATION].', '1907.06409-3-76-1': 'Suppose now that [MATH].', '1907.06409-3-76-2': 'Then we have [EQUATION]', '1907.06409-3-76-3': 'It follows from [REF] and [REF] that [EQUATION]', '1907.06409-3-76-4': 'By [REF] and [REF], we get [EQUATION].', '1907.06409-3-76-5': 'This along with [REF] leads to [EQUATION] where [MATH] whenever [MATH].', '1907.06409-3-76-6': 'Then we obtain [EQUATION]', '1907.06409-3-76-7': 'This together with [REF] shows that [REF] holds.', '1907.06409-3-77-0': 'Case III: [MATH] and [MATH].', '1907.06409-3-78-0': 'If [MATH], then by [REF], we have [EQUATION].', '1907.06409-3-78-1': 'Suppose now that [MATH].', '1907.06409-3-78-2': 'Then we get [EQUATION].', '1907.06409-3-78-3': 'To use the same reasoning as in Case II, we need to have lower bounds for [MATH] and [MATH].', '1907.06409-3-78-4': 'To this end, applying [REF] and [REF], we obtain [EQUATION]', '1907.06409-3-78-5': 'Furthermore, [REF], [REF] and [REF] yield [EQUATION].', '1907.06409-3-78-6': 'This lower bound is positive whenever [MATH].', '1907.06409-3-78-7': 'The two lower bounds allows us to conclude, by analogy with Case II, that [REF] holds.', '1907.06409-3-79-0': 'Case IV: [MATH] and [MATH].', '1907.06409-3-80-0': 'It follows from [REF], [REF], [REF] and [REF] that [EQUATION]', '1907.06409-3-80-1': 'Collecting the results in the considered four cases, one can see that [REF] is satisfied for any [EQUATION].', '1907.06409-3-80-2': 'This completes the induction and finally proves that inequalities [REF]-[REF] hold for all [MATH].', '1907.06409-3-80-3': '[MATH]', '1907.06409-3-81-0': 'Next we will establish the local convergence property of BBstab for nonquadratic functions.', '1907.06409-3-82-0': 'There exists positive [MATH] such that, for any positive [MATH] and any starting points [MATH], the sequence [MATH] converges to [MATH] with R-linear rate.', '1907.06409-3-83-0': 'Lemma [REF] allows us to skip the proof of this theorem because the reasoning is similar to the proof of Theorem 2.3 in [CITATION].', '1907.06409-3-84-0': 'We complete the analysis by presenting the following global convergence result.', '1907.06409-3-85-0': 'There exists positive [MATH] such that, for any positive [MATH] and any starting points [MATH], the sequence [MATH] converges to [MATH] with R-linear rate.', '1907.06409-3-86-0': 'Let [MATH] be given by Theorem [REF], which ensures local convergence to [MATH].', '1907.06409-3-86-1': 'According to Lemma [REF], after a finite number of BBstab iterations, all iterates will belong to [MATH].', '1907.06409-3-86-2': 'This finally proves global convergence with R-linear rate.', '1907.06409-3-86-3': '[MATH]', '1907.06409-3-87-0': '# Numerical Results', '1907.06409-3-88-0': 'Our algorithms were implemented in MATLAB.', '1907.06409-3-88-1': 'The algorithms are terminated when either the number of iterations exceeds [MATH], or [EQUATION].', '1907.06409-3-88-2': 'In the next two subsections, results of numerical experiments are presented separately for quadratic and nonquadratic test functions.', '1907.06409-3-89-0': 'A successful value of [MATH] is obviously problem dependent.', '1907.06409-3-89-1': 'In our implementation, we try to estimate its order of magnitude by setting [MATH] for the first few iterations and making use of [MATH] produced at these iterations by the standard BB algorithm.', '1907.06409-3-89-2': 'At the subsequent iterations, the constant value [EQUATION] is applied, where [MATH] is a parameter.', '1907.06409-3-89-3': 'It turns out that this adaptive choice of [MATH] is less problem dependent.', '1907.06409-3-90-0': 'It is necessary to emphasize that the stabilization was designed not to speed-up the BB method when it safely converges.', '1907.06409-3-90-1': 'In such cases, it may increase the number of iterations, which is a negative outcome.', '1907.06409-3-90-2': 'The main purpose of the stabilization is to prevent the BB method from making too long steps.', '1907.06409-3-90-3': 'This serves for decreasing the number of BB iterations in case of its poor convergence or even making the method convergent when it fails, which is a positive outcome.', '1907.06409-3-90-4': 'Outcomes of all these aforementioned types were observed in our numerical experiments with stabilizing the BB method.', '1907.06409-3-90-5': 'One can easily recognize them in the tables presented below.', '1907.06409-3-91-0': 'We focus here on demonstrating the potentials of improving convergence for the BB method.', '1907.06409-3-91-1': 'Therefore, our stabilized version is not checked here against another optimization algorithms.', '1907.06409-3-91-2': 'Since the computational cost of one iteration for the BB algorithms are practically the same as for their stabilized versions, only the number of iterations are compared.', '1907.06409-3-91-3': 'Notice that the number of iterations is the same as the number of gradient evaluations.', '1907.06409-3-92-0': 'In our numerical experiments, the BB1 algorithm was generating too long steps more frequently than the BB2 algorithm.', '1907.06409-3-92-1': 'This is often caused by relatively too small values of the scalar product [MATH] in the denominator of [MATH].', '1907.06409-3-92-2': 'This explains why the stabilization is, in general, more important for the BB1 stepsize choice than for the BB2.', '1907.06409-3-92-3': 'Therefore, the numerical results presented here refer mainly to the BB1.', '1907.06409-3-93-0': '## Quadratic test functions', '1907.06409-3-94-0': 'A part of the numerical experiments was related to minimizing convex quadratic functions [REF].', '1907.06409-3-94-1': 'This problem is equivalent to solving the system of linear equations [EQUATION].', '1907.06409-3-94-2': 'The matrices in our set of test problems come from the SuiteSparse Matrix Collection [CITATION].', '1907.06409-3-94-3': 'For generating the vector [MATH], we assumed that the solution [MATH], i.e., [MATH], where [MATH].', '1907.06409-3-94-4': 'The total number of problems in our test set is [MATH], where the problem size [MATH] varies from thousands to millions.', '1907.06409-3-95-0': 'For the adaptive selection of [MATH] by formula [REF], we tried just a few values of the parameter [MATH], namely, [MATH], [MATH] and [MATH].', '1907.06409-3-95-1': 'In Tables [REF] and [REF], the number of iterations are reported for algorithms BB1 and BB1stab.', '1907.06409-3-95-2': 'For the latter, the best of the three results is presented along with the corresponding value of [MATH].', '1907.06409-3-95-3': 'If the reported result is the same as for the BB1 algorithm, then it is obvious that the number of iterations remains the same for all values of c larger than the indicated one.', '1907.06409-3-95-4': 'The number of iterations, which is not worse than for the BB1 algorithm, are highlighted in this and other tables in this paper.', '1907.06409-3-95-5': 'One can see that, comparing with the BB1, its stabilized version is faster in solving [MATH] problems, while it is slower in [MATH] problems.', '1907.06409-3-95-6': 'Furthermore, the reduction in the number of iterations obtained by virtue of the stabilization was often substantial.', '1907.06409-3-95-7': 'We also tested the BB2 and BB2stab algorithms for these same 141 problems.', '1907.06409-3-95-8': 'We tried [MATH], [MATH], [MATH] and [MATH] in the adaptive selection of [MATH] by formula [REF].', '1907.06409-3-95-9': 'Comparing with the BB2, BB2stab is faster in solving [MATH] problems, while for the given values of [MATH], the stabilization is unable to decrease the number of BB2 iterations in [MATH] problems.', '1907.06409-3-96-0': '## Nonquadratic test functions', '1907.06409-3-97-0': 'For general functions, it is more difficult than for quadratic ones to avoid the cases, when [MATH] is chosen too close to [MATH] or too far away of it.', '1907.06409-3-97-1': 'In order to avoid such poor choices of these two points, our BBstab algorithms are initialized with only one point, namely, [MATH].', '1907.06409-3-97-2': 'The point [MATH] is produced in the algorithms by checking if the inequality [MATH] is satisfied for [MATH], where [MATH].', '1907.06409-3-97-3': 'Otherwise, a number, typically few, of backtracking steps are performed by dividing the current vector [MATH] by [MATH], while the required inequality is violated.', '1907.06409-3-98-0': 'We begin here by comparing the performance of the BB algorithms and their stabilized versions on the strongly convex Raydan function [REF] for [MATH].', '1907.06409-3-98-1': 'The point [MATH] was used for starting the algorithms.', '1907.06409-3-98-2': 'The standard BB1 algorithm failed to solve the problem.', '1907.06409-3-98-3': 'After two iterations, an overflow in computing [MATH] was reported.', '1907.06409-3-98-4': 'If to introduce the bounds [MATH] for [MATH], like it is often done in practice, then it also fails, although after a larger number of iterations.', '1907.06409-3-98-5': 'Namely, at iteration 123 and all subsequent iterations, an underflow was observed in calculating [MATH] for [MATH].', '1907.06409-3-98-6': 'In these two cases, the standard BB2 also failed.', '1907.06409-3-98-7': 'However, the same test problem for the same [MATH] was successfully solved by BB1stab and BB2stab with [MATH] in [MATH] and [MATH] iterations, respectively.', '1907.06409-3-98-8': 'No bounds, like [MATH], are used in our implementation of the BB algorithms and their stabilized versions.', '1907.06409-3-99-0': 'Figure [REF] illustrates the stabilization effect.', '1907.06409-3-99-1': 'One can see that the BB1 was generating too long steps more frequently than the BB2.', '1907.06409-3-99-2': 'This observation is in general agreement with the other numerical experiments that we performed and also with the theory, which says that [MATH].', '1907.06409-3-100-0': 'The performance of our algorithms was compared also for unconstrained minimization problems from the CUTEst collection [CITATION], which provides a standard starting point [MATH] for each of them.', '1907.06409-3-100-1': 'We excluded from our comparison quadratic problems and those, in which the BB1/BB2 algorithm converged in less than [MATH] iterations.', '1907.06409-3-100-2': 'The results reported here concern only the problems, where at least one of the compared algorithms converged, and also those, where the both algorithms converged to the same point.', '1907.06409-3-101-0': 'Recall that the BB method was originally designed for solving convex problems in which case it is guaranteed that [MATH] is nonnegative.', '1907.06409-3-101-1': 'Since the most of the unconstrained minimization test problems in the CUTEst collection are nonconvex, we had to adapt the BB method to solving this kind of problems.', '1907.06409-3-101-2': 'In our implementation of the BB method and its stabilized version, we follow paper [CITATION] in setting [EQUATION] whenever [MATH].', '1907.06409-3-101-3': 'This makes our algorithms much more robust.', '1907.06409-3-101-4': 'Figure [REF] presents results of solving [MATH] problems from the CUTEst collection.', '1907.06409-3-101-5': 'The BB1 and BB2 algorithms failed in [MATH] and [MATH] cases, respectively.', '1907.06409-3-101-6': 'The plots of the performance profiles introduced in [CITATION] indicate that the BB2 algorithm is more robust than the BB1.', '1907.06409-3-101-7': 'Furthermore, the former algorithm required, on average, fewer iterations for solving problems.', '1907.06409-3-101-8': 'The main reason is that the BB1 algorithm generates too long steps more frequently.', '1907.06409-3-101-9': 'In what follows, we focus on presenting here results of stabilizing the BB1 algorithm, because it gains more from the stabilization than the BB2 algorithm.', '1907.06409-3-102-0': 'Table [REF] presents results of solving [MATH] nonquadratic test problems from the CUTEst collection.', '1907.06409-3-102-1': 'We tried only three values of the parameter [MATH] in the adaptive choice of [MATH] using [REF], namely, [MATH], [MATH] and [MATH].', '1907.06409-3-102-2': 'The BB1 and BB1stab algorithms were not able to solve problems during [MATH] iterations in [MATH] and [MATH] cases, respectively.', '1907.06409-3-102-3': 'The BB1stab requires fewer number of iterations in [MATH] cases, while the BB1 performs better only in [MATH] cases.', '1907.06409-3-102-4': 'In [MATH] cases, the BB1stab with the indicated values of [MATH] requires the same number of iterations as the BB1.', '1907.06409-3-103-0': 'We made experiments also with directly setting a certain value of [MATH] in the BB1stab.', '1907.06409-3-103-1': 'The trial values were [MATH], [MATH] and [MATH].', '1907.06409-3-103-2': 'For a few test problems, the results are better than for the aforementioned adaptive choice with [MATH], [MATH] and [MATH].', '1907.06409-3-103-3': 'For [MATH] of [MATH] problems, the number of iterations is smaller than in case of the BB1.', '1907.06409-3-104-0': 'These results are reported in Table [REF].', '1907.06409-3-104-1': 'The preselected values of [MATH] allowed the BB1stab to solve five problems of those not solved by the BB1, including problems MOREBV and TQUARTIC, in which the adaptive choice of [MATH] failed.', '1907.06409-3-104-2': 'In case of TQUARTIC, the BB1 terminated because of producing NaN (Not a Number) in Matlab.', '1907.06409-3-104-3': 'The experiments with the preselected values of [MATH] indicate that there is plenty of room for improving the very simple adaptive strategy proposed in this paper.', '1907.06409-3-105-0': 'For the BB2stab algorithm, we still tried the same three values of the parameter [MATH] in the adaptive choice of [MATH] using [REF] as for BB1stab.', '1907.06409-3-105-1': 'In [MATH] test problems, the BB2stab performs better in [MATH] cases, while the BB2 performs better only in [MATH] cases.', '1907.06409-3-105-2': 'Table [REF] presents results for all the cases when the BB2stab requires fewer number of iterations.', '1907.06409-3-106-0': '# Conclusions', '1907.06409-3-107-0': 'In the present paper, it was proposed to stabilize the conventional BB method by virtue of bounding the distance between sequential iterates.', '1907.06409-3-107-1': 'The purpose was to improve its convergence, when it is affected by too long steps [MATH], and also to make the BB method convergent, when it fails to converge.', '1907.06409-3-107-2': 'Both a theoretical and numerical study of the stabilized version was conducted.', '1907.06409-3-107-3': 'We have proved that the stabilization provides the BB method with a global convergence without recourse to using any line search.', '1907.06409-3-107-4': 'The numerical results presented here are highly encouraging.', '1907.06409-3-107-5': 'The proposed very simple adaptive selection of [MATH] was able to successfully trap a value which is appropriate for each specific problem.', '1907.06409-3-107-6': 'However, we hope that this paper will stimulate development of more efficient algorithms for adaptive selection of [MATH].'}

1601.03918

{'1601.03918-1-0-0': 'We construct the six-point NMHV one-loop amplitude in [MATH] supergravity using unitarity and recursion.', '1601.03918-1-0-1': 'The use of recursion requires the introduction of rational descendants of the cut-constructible pieces of the amplitude and the computation of the non-standard factorisation terms arising from the loop integrals.', '1601.03918-1-1-0': '# Introduction', '1601.03918-1-2-0': 'Despite perturbative Quantum Gravity being a mature subject [CITATION], it is a very challenging area computationally.', '1601.03918-1-2-1': 'Although great strides have been made in computing tree amplitudes [CITATION], there remain a very limited number of loop calculations available to study.', '1601.03918-1-2-2': 'For the four and five point amplitudes all one-loop graviton scattering amplitudes have now been calculated [CITATION].', '1601.03918-1-2-3': 'For Maximal supergravity great progress has also be made at multi-loop level for the four point amplitude [CITATION].', '1601.03918-1-2-4': 'These computations are by necessity amplitudes which are "Maximally-Helicity-Violating" (MHV).', '1601.03918-1-2-5': 'MHV amplitudes are very special and have many features not shared by non-MHV amplitudes.', '1601.03918-1-2-6': 'In this article we compute the six graviton "Next-to-MHV" (NMHV) scattering amplitude for [MATH] supergravity.', '1601.03918-1-2-7': '(The first NMHV amplitude appears at six-points.)', '1601.03918-1-2-8': 'The six-graviton scattering amplitude has been computed for [MATH] and [MATH] supergravity.', '1601.03918-1-3-0': 'This amplitude has considerable algebraic complexity relative to the more supersymmetric cases including the appearance of rational terms.', '1601.03918-1-3-1': 'We construct the [MATH] amplitude using unitarity and recursion augmented by limited off-shell behaviour.', '1601.03918-1-4-0': 'One-loop amplitudes in a massless theory can be expressed as [CITATION] [EQUATION] where the [MATH] are [MATH]-point scalar integral functions and the [MATH] etc. are rational coefficients.', '1601.03918-1-4-1': '[MATH] is a purely rational term.', '1601.03918-1-4-2': 'The box, triangle and bubble coefficients can be determined via unitarity methods [CITATION] using four dimensional on-shell tree amplitudes.', '1601.03918-1-4-3': 'These contributions are termed cut-constructible.', '1601.03918-1-4-4': 'Progress has been made both via the two-particle cuts [CITATION] and using generalisations of unitarity [CITATION] where, for example, triple [CITATION] and quadruple cuts [CITATION] are utilised to identify the triangle and box coefficients respectively.', '1601.03918-1-4-5': 'The remaining purely rational term, [MATH], may in principle be obtained using unitarity but this requires a knowledge of [MATH] dimensional tree amplitudes [CITATION].', '1601.03918-1-4-6': 'In this paper a recursive approach is adopted that generates the rational term from four dimensional amplitudes.', '1601.03918-1-5-0': 'An important technique for computing tree amplitudes is Britto-Cachazo-Feng-Witten (BCFW) [CITATION] recursion which applies complex analysis to amplitudes.', '1601.03918-1-5-1': "By shifting the momenta [EQUATION] the resultant amplitude [MATH] may be computed via Cauchy's theorem.", '1601.03918-1-5-2': 'Loop amplitudes, as functions of complexified momentum, contain both poles and cuts so BCFW does not immediately apply to these.', '1601.03918-1-5-3': 'However by defining [EQUATION] where [MATH] is the cut-constructible part of the amplitude, we can compute the purely rational [MATH] from a knowledge of its singularities.', '1601.03918-1-5-4': '[MATH] has singularities corresponding to the poles of the amplitude but also induced singularities arising because [MATH] has singularities which are not present in the amplitude and which must be cancelled by equal and opposite singularities in [MATH].', '1601.03918-1-5-5': 'We refer to these contributions as the rational descendants of [MATH].', '1601.03918-1-6-0': 'The particle content of the [MATH] graviton and matter multiplets are shown in table [REF].', '1601.03918-1-7-0': 'For convenience, we will calculate the one-loop amplitude using the [MATH] matter multiplet, which is related to the amplitude with the graviton multiplet circulating in the loop by [EQUATION]', '1601.03918-1-7-1': 'The [MATH] and [MATH] components are considerably simpler and given in [CITATION].', '1601.03918-1-7-2': 'In particular [MATH] for these two components.', '1601.03918-1-8-0': '# Cut-Constructible Pieces', '1601.03918-1-9-0': 'The cut-constructable pieces consist of box-functions, triangle functions and bubble integral functions.', '1601.03918-1-9-1': 'The analytic form of these depends upon how many of the external legs have non-null momentum: these are referred to as massive legs although non-null is more correct.', '1601.03918-1-9-2': 'For the one-mass box the fourth leg is conventionally the massive leg and the integral function depends upon [MATH], [MATH] and [MATH].', '1601.03918-1-9-3': 'For the two-mass boxes there are two types: "two-mass-easy" ([MATH]) where legs [MATH] and [MATH] are massive and "two-mass-hard" ([MATH]) where legs [MATH] and [MATH] are massive.', '1601.03918-1-9-4': 'For six-point amplitudes the three and four mass boxes do not appear and for the NMHV [MATH] amplitude there are no two-mass-easy boxes.', '1601.03918-1-10-0': 'The various box, triangle and bubble contributions present in the six-point NMHV are shown in fig. [REF] together with the labelling of helicities which yield a non-zero coefficient.', '1601.03918-1-10-1': 'Permuting the positive and negative helicity legs separately gives eighteen one-mass boxes, thirty-six two-mass hard boxes, nine 2:4 bubbles, nine 3:3 bubbles and six three-mass triangles.', '1601.03918-1-11-0': '2.0 truecm', '1601.03918-1-12-0': '## IR consistency and Choice of Integral Function Basis', '1601.03918-1-13-0': 'For one-loop amplitudes Infra-Red (IR) consistency imposes a system of constraints on the rational coefficients of the integral functions.', '1601.03918-1-13-1': 'For the matter multiplets [CITATION] there are in fact no IR singular terms in the amplitude, so the singular terms in the individual integral functions cancel.', '1601.03918-1-13-2': 'This gives enough information to fix the coefficients of the one- and two-mass triangles in terms of the box coefficients.', '1601.03918-1-13-3': 'The three-mass triangle is IR finite, so its coefficient is not determined by these constraints.', '1601.03918-1-13-4': 'It is convenient to combine the boxes and triangles in such a way that these infinities are manifestly absent.', '1601.03918-1-13-5': 'There are several ways to do this [CITATION], here we choose to work with truncated box functions, [EQUATION] where the [MATH] and [MATH] are chosen to make [MATH] IR finite.', '1601.03918-1-13-6': 'This effectively incorporates the one- and two-mass triangle contributions into the box contributions.', '1601.03918-1-13-7': 'Using these truncated boxes the amplitudes can be written as [EQUATION]', '1601.03918-1-13-8': 'There is one remaining IR consistency constraint: [MATH].', '1601.03918-1-14-0': 'The one-mass and two-mass-hard truncated integral functions are [EQUATION] where [MATH] and [MATH].', '1601.03918-1-15-0': '## Boxes', '1601.03918-1-16-0': 'The six-pt NMHV amplitude contains both one-mass and two-mass hard boxes.', '1601.03918-1-16-1': 'The coefficients of these boxes are readily obtained using quadruple cuts [CITATION].', '1601.03918-1-17-0': 'The coefficient of the first one-mass box in fig. [REF] is [EQUATION] where [MATH].', '1601.03918-1-17-1': 'The other one-mass box coefficients can be obtained from this by conjugation and relabelling.', '1601.03918-1-17-2': 'The coefficient of the two-mass-hard box is [EQUATION] and the other coefficients are obtained by relabelling.', '1601.03918-1-18-0': '## Canonical basis approach for triangle and bubble coefficients', '1601.03918-1-19-0': 'The canonical basis approach [CITATION] is a systematic method to determine the coefficients of triangle and bubble integral functions from the three and two-particle cuts.', '1601.03918-1-19-1': 'The two particle cut is shown in fig. [REF] and is [EQUATION]', '1601.03918-1-19-2': 'The product of tree amplitudes appearing in the two-particle cut can be decomposed in terms of canonical forms [MATH], [EQUATION] where the [MATH] are coefficients independent of [MATH].', '1601.03918-1-19-3': 'We then use substitution rules to replace the [MATH] by the rational functions [MATH] to obtain the coefficient of the bubble integral function as [EQUATION]', '1601.03918-1-19-4': 'Similarly, we can obtain the coefficient of the triangle functions from the triple cut [CITATION] as shown in figure [REF], [EQUATION]', '1601.03918-1-19-5': 'The product of tree amplitudes can, again, be expressed in terms of standard forms of [MATH], [EQUATION] and substitution rules used to replace the [MATH] by the functions [MATH] to obtain the triangle coefficient as [EQUATION]', '1601.03918-1-19-6': 'In general the integrands are rational functions of [MATH] of degree [MATH].', '1601.03918-1-19-7': 'The simplest canonical forms have [MATH] or [MATH].', '1601.03918-1-19-8': 'More complex denominators can be expressed in terms of the simplest forms by partial fractioning.', '1601.03918-1-19-9': 'Terms in the integrand with [MATH] only contribute to higher point integral functions.', '1601.03918-1-19-10': 'The degree generally decreases with increasing supersymmetry and for maximally supersymmetric Yang-Mills and supergravity there are no triangles.', '1601.03918-1-19-11': 'With increasing [MATH] the canonical forms become increasingly complex.', '1601.03918-1-20-0': '## Triangles', '1601.03918-1-21-0': 'Using the truncated box functions, we only need to compute the coefficient of the (IR finite) three-mass triangle: [MATH].', '1601.03918-1-21-1': 'Using the Kawai-Lewellen-Tye (KLT) relations [CITATION], with a scalar circulating in the loop the cut integrand is [EQUATION]', '1601.03918-1-21-2': 'The four-point Yang-Mills amplitudes above are simultaneously MHV and [MATH] amplitudes, so there is a choice of two expressions for each.', '1601.03918-1-21-3': 'For algebraic convenience we take a mixed form for the first ordering: [EQUATION] and a purely MHV form for the second: [EQUATION]', '1601.03918-1-21-4': 'The contribution to the cut from a particle of helicity [MATH] circulating in the loop is given by [EQUATION] so that summing over the [MATH] matter multiplet gives [EQUATION]', '1601.03918-1-21-5': 'The [MATH]-factor can be written in several ways.', '1601.03918-1-21-6': 'The natural choices given [REF] and [REF] are [EQUATION] with the corresponding [MATH]-factors being [EQUATION] where [EQUATION]', '1601.03918-1-21-7': 'The cut is then [EQUATION]', '1601.03918-1-21-8': 'Using [EQUATION] and [EQUATION] we find [EQUATION] where [EQUATION]', '1601.03918-1-21-9': 'Now we may partial fraction the expression, i.e. use the identity [EQUATION] with [EQUATION] so that the cut is written as a sum of canonical forms: [EQUATION] which have corresponding canonical functions [EQUATION] and [EQUATION] where [EQUATION] and [MATH] is the set of [MATH] permutations of [MATH] necessary to generate symmetry in these variables.', '1601.03918-1-22-0': 'We obtain the coefficient [EQUATION]', '1601.03918-1-22-1': 'The corresponding three mass triangle integral function is [CITATION] [EQUATION] where [EQUATION]', '1601.03918-1-23-0': '# Bubbles', '1601.03918-1-24-0': 'The final cut constructible pieces are the bubbles integral functions [MATH].', '1601.03918-1-24-1': 'There are two distinct types of bubbles coefficients depending upon whether [MATH] is a two particle momentum, [MATH], or three particle, [MATH].', '1601.03918-1-24-2': 'In the three particle case the cut only involves MHV tree amplitudes whereas the two particle case requires the NMHV tree amplitude.', '1601.03918-1-25-0': 'For the [MATH] bubble the cut integrand with a scalar circulating in the loop is [EQUATION] where [EQUATION]', '1601.03918-1-25-1': 'As in the three-mass triangle, the contribution from a particle of helicity [MATH] is [MATH].', '1601.03918-1-25-2': 'Summing over the multiplet has the effect of multiplying by [MATH].', '1601.03918-1-25-3': 'For this cut [EQUATION]', '1601.03918-1-25-4': 'Multiplying [MATH] by [MATH] gives [EQUATION]', '1601.03918-1-25-5': 'Using [EQUATION] we find [EQUATION] where [EQUATION]', '1601.03918-1-25-6': 'Partial fractioning on [MATH] and [MATH] yields [EQUATION] where [EQUATION]', '1601.03918-1-25-7': 'The cut is now expressed in terms of the canonical form [EQUATION] which has the corresponding canonical function [EQUATION]', '1601.03918-1-25-8': 'The contribution of [REF] to the bubble coefficient is then [EQUATION] leading to the full bubble coefficient [EQUATION]', '1601.03918-1-25-9': 'The coefficient of the bubble [MATH] is obtained from the cut [EQUATION]', '1601.03918-1-25-10': 'This cut can also be decomposed in canonical forms.', '1601.03918-1-25-11': 'However the six-point NMHV tree amplitude in the cut is a sum of fourteen terms [CITATION] which leads to a lengthy expression for this bubble coefficient.', '1601.03918-1-25-12': 'Full expressions for the bubble coefficients of this type are given in appendix [REF].', '1601.03918-1-26-0': '# Cancellation Webs and Rational Descendants', '1601.03918-1-27-0': 'Although we may split the amplitude into cut-constructible pieces and rational terms, when we examine the singularities in the amplitude there is a mixing between the two which is important when we reconstruct [MATH] from its singularities.', '1601.03918-1-27-1': 'The cut-constructible pieces of the amplitude introduce a number of singularities that cannot be present in the full amplitude.', '1601.03918-1-27-2': 'These can be spurious singularities that occur at kinematic points where the full amplitude should be finite or higher order singularities occurring at points where the amplitude has a simple pole.', '1601.03918-1-27-3': 'If these poles are of sufficiently high order, they generate rational descendants.', '1601.03918-1-28-0': '## Higher Order Poles', '1601.03918-1-29-0': 'As an example of a higher order pole consider the behaviour of the one-mass box contribution [MATH] as [MATH].', '1601.03918-1-29-1': 'The box coefficient [MATH] in eq. ([REF]) contains a factor of [MATH] and the expansion of the box integral function around [MATH] is [CITATION] [EQUATION] where [EQUATION]', '1601.03918-1-29-2': 'As the cut constructible terms contain all of the logarithms and dilogarithms in the amplitude, the logarithmic pieces of this expansion must combine with the bubbles to give an effective coefficients that are linearly divergent as [MATH].', '1601.03918-1-29-3': 'We have confirmed numerically that the relevant cancellations between the one-mass box contributions and the bubble contributions occur.', '1601.03918-1-29-4': 'The quadratic divergence in the rational descendant, [MATH], must be cancelled by the rational piece of the amplitude.', '1601.03918-1-29-5': 'The full rational part of the amplitude will ultimately be obtained by recursion and one contribution to it will arise from this rational descendant if the shift excites the [MATH] pole.', '1601.03918-1-30-0': 'There are corresponding [MATH] poles in [MATH] which are obtained from those above by conjugation.', '1601.03918-1-31-0': '## [MATH] Spurious Singularity', '1601.03918-1-32-0': 'The coefficients of the two mass hard boxes have singularities of the form [MATH].', '1601.03918-1-32-1': 'These singularities also occur in the three mass triangle contributions: three powers of the pole are explicit in the leading term of the canonical form and a fourth arises in the partial fractioning that splits the cut integrand into canonical forms.', '1601.03918-1-33-0': 'In terms of kinematic variables, [MATH] corresponds to [MATH].', '1601.03918-1-33-1': 'The two mass hard integral functions depend on [MATH], [MATH], [MATH] and [MATH], while the three mass triangle depends on [MATH], [MATH] and [MATH] and there is the kinematic constraint: [MATH].', '1601.03918-1-33-2': 'For given [MATH], [MATH] and [MATH] it is possible find [MATH] and [MATH]) so that [MATH].', '1601.03918-1-33-3': 'Close to the pole the two mass hard box integral functions can be expanded as a series in terms of [MATH].', '1601.03918-1-33-4': 'In this context it is convenient to work with dimensionless box integral functions [MATH] defined by [EQUATION]', '1601.03918-1-33-5': 'The expansion of this dimensionless box function is [EQUATION] where [EQUATION] and [EQUATION] with [EQUATION]', '1601.03918-1-33-6': 'The same pole appears in the two boxes with integral function: [MATH] and [MATH].', '1601.03918-1-33-7': 'On the pole the coefficients of these two boxes are not equal and neither integral function vanishes.', '1601.03918-1-33-8': 'However, the sum of the dimensionless integral functions vanishes, i.e. [EQUATION]', '1601.03918-1-33-9': 'On this pole the dilogarithms in the individual boxes and triangles survive, but cancel between them.', '1601.03918-1-33-10': 'Setting [EQUATION] the integral functions are related by [EQUATION] where the sign ambiguity is associated with the choice of sign for [MATH].', '1601.03918-1-33-11': 'Schematically, expressing the box coefficients in terms of their sum and difference, [MATH], [MATH], the box and triangle contributions are [EQUATION]', '1601.03918-1-33-12': 'Expanding about [MATH], thanks to [REF] there is no dilogarithm component in the first term for any [MATH].', '1601.03918-1-33-13': 'However, we can only use [REF] if [MATH] and [MATH] are equal.', '1601.03918-1-33-14': 'In fact [EQUATION]', '1601.03918-1-33-15': 'Hence the dilogarithms vanish from any term that is singular as [MATH], as required by the factorisation theorems [CITATION].', '1601.03918-1-34-0': 'As in the one-mass case, there are subleading singularities at cubic order multiplying logarithms.', '1601.03918-1-34-1': 'These combine with the bubble contributions and cancel up to and including order [MATH], leaving no spurious singularity in the logarithms.', '1601.03918-1-35-0': 'The rational descendant of this combination of boxes and triangle contain both [MATH] and [MATH] singularities.', '1601.03918-1-35-1': 'Both of these singularities must be cancelled by the rational piece of the amplitude [MATH].', '1601.03918-1-35-2': 'As the expansion has been performed about a singularity specified in terms of [MATH], [MATH] and [MATH], there is no need to expand the three mass triangle integral function when determining this rational descendant.', '1601.03918-1-36-0': '## [MATH] Spurious Singularity', '1601.03918-1-37-0': 'The three mass triangle contributions have [MATH] poles which can be seen explicitly in the canonical forms.', '1601.03918-1-37-1': 'Around the [MATH] pole the integral function has the expansion, [EQUATION] where [EQUATION]', '1601.03918-1-37-2': 'The logarithmic terms in this expansion combine with the bubble contributions to yield a finite contribution on the pole.', '1601.03918-1-37-3': 'The rational piece in the expansion must cancel with the rational part of the amplitude.', '1601.03918-1-38-0': '# Obtaining [MATH] By Recursion', '1601.03918-1-39-0': 'BCFW [CITATION] recursion applies complex analysis to amplitudes.', '1601.03918-1-39-1': "Using Cauchy's theorem, if a complex function is analytic except at simple poles [MATH] (all non-zero) and [MATH] as [MATH] then by considering the integral [EQUATION] where the contour [MATH] is the circle at infinity, we obtain [EQUATION]", '1601.03918-1-39-2': 'We wish to apply this with [MATH], where [MATH] has been complexified by a BCFW shift of momenta.', '1601.03918-1-39-3': 'Since [EQUATION] the singularities and residues of [MATH] are both those arising from the physical factorisations of [MATH] and those induced by the necessity to cancel the spurious singularities of the cut-constructible pieces.', '1601.03918-1-40-0': '## Choice of Shift', '1601.03918-1-41-0': 'The rational part of an amplitude can be obtained recursively if the factorisation properties of the amplitude are understood at all of the relevant poles.', '1601.03918-1-41-1': 'There are three main obstacles to this: quadratic poles in the amplitude, non-standard factorisations for complex momenta and contributions for large shifted momenta.', '1601.03918-1-41-2': 'Quadratic poles in the amplitude lead to recursive contributions that depend on the off-shell behaviour of the factorised currents.', '1601.03918-1-41-3': 'This can be addressed using augmented recursion [CITATION].', '1601.03918-1-41-4': 'For non-supersymmetric theories there are double poles of the form [EQUATION]', '1601.03918-1-41-5': 'For the six-point NMHV amplitude the tree amplitude vanishes since it has a single positive helicity leg.', '1601.03918-1-41-6': '(This is no longer the case for seven and higher point NMHV amplitudes.)', '1601.03918-1-42-0': 'Non-standard factorisations for complex momenta are unavoidable and are considered in detail below.', '1601.03918-1-42-1': 'The final obstacle is the possibility of contributions from asymptotically large shifted momenta.', '1601.03918-1-42-2': "The amplitude doesn't factorise in this limit, so the residue is undetermined.", '1601.03918-1-42-3': 'This issue may be avoided if the shift employed causes the amplitude to vanish for asymptotically large shifted momenta.', '1601.03918-1-42-4': 'As the amplitude is as yet undetermined, its behaviour under any shift is unknown.', '1601.03918-1-42-5': "However, if the cut constructible pieces don't vanish for asymptotically large shifted momenta there is little hope that the rational pieces would.", '1601.03918-1-43-0': 'For example under a shift involving two negative helicity legs, [EQUATION] the cut constructible pieces of the amplitude are divergent for large [MATH].', '1601.03918-1-44-0': 'However, for a shift involving one negative helicity leg and one positive helicity leg, [EQUATION] the cut-constructible pieces all vanish at large [MATH], at least suggesting that the rational piece is also well behaved there.', '1601.03918-1-44-1': 'The shift [REF] will be used to obtain [MATH].', '1601.03918-1-45-0': 'The contributions to [MATH] can be grouped into three classes: standard factorisations, non-factorising contributions and rational descendants of the cut-constructible pieces : [EQUATION]', '1601.03918-1-46-0': '## Standard Factorisations', '1601.03918-1-47-0': 'The standard factorisations of a six-point one-loop amplitude have the forms: [EQUATION]', '1601.03918-1-47-1': 'In a supersymmetric theory the 3-point loop amplitudes vanish and so the third class are absent in this case.', '1601.03918-1-47-2': 'With the shift [REF] the factorisations of the first type are: [EQUATION]', '1601.03918-1-47-3': 'While the factorisations of the second type are: [EQUATION]', '1601.03918-1-47-4': 'For generic six-point kinematics, the kinematic points at which the 4- and 5-point loop amplitudes appearing in these factorisations are evaluated are in no way special, hence the rational contribution to the residue comes solely from the rational part of the 4- and 5-point loop amplitudes.', '1601.03918-1-47-5': 'Each factorisation therefore gives a contribution to [MATH] of [EQUATION] with [CITATION] [EQUATION] where [EQUATION] and [MATH] denotes a sum over the six distinct permutations of [MATH] and [MATH] noting the symmetry of [MATH] under [MATH].', '1601.03918-1-47-6': 'The full contribution of the standard factorisations is then [EQUATION] where the sum is over all of the standard factorisation channels given in [REF] and [REF].', '1601.03918-1-48-0': '## Contribution Of Rational Descendants', '1601.03918-1-49-0': 'As discussed above, higher order poles in the coefficients of the box and triangle contributions to the amplitude can generate rational descendants when those poles are excited.', '1601.03918-1-49-1': 'The shift [REF] excites some poles of each type.', '1601.03918-1-49-2': 'Specifically we have the various singularities listed in table [REF] (with [MATH], [MATH]).', '1601.03918-1-50-0': 'Denoting the rational descendant in each case by [MATH], the corresponding coefficient by [MATH] and the value of [MATH] on the pole by [MATH], the contribution on each of these poles is [EQUATION] so that [EQUATION] where the sum is over all of the poles listed above.', '1601.03918-1-51-0': 'Individual terms in the bubble coefficients contain a range of other higher order poles.', '1601.03918-1-51-1': 'In principle these could also generate rational descendants, however in the full bubble coefficients these are at most simple poles and so do not generate further rational descendants: [EQUATION]', '1601.03918-1-52-0': '## Non-standard Factorisations for Complex Momenta', '1601.03918-1-53-0': 'Factorisations of the amplitude occur when propagators go on shell.', '1601.03918-1-53-1': 'The standard factorisation channels arise when the on-shell propagator is not in the loop and is explicit in, for example a Feynman diagram approach.', '1601.03918-1-54-0': 'The loop momentum integral may also generate poles in the amplitude [CITATION] particularly for complex momenta.', '1601.03918-1-54-1': 'Since we are computing the amplitude by recursion in complex momenta we must determine these complex factorisations.', '1601.03918-1-54-2': '2.0truecm', '1601.03918-1-55-0': 'Poles can arise when two adjacent massless legs on a loop became collinear as illustrated in fig. [REF].', '1601.03918-1-55-1': 'This case has been discussed in the context of amplitudes with a single negative helicity leg [CITATION].', '1601.03918-1-55-2': 'In the integration region [MATH] the three propagators connected to [MATH] and [MATH] all become on shell when [MATH] and [MATH] are collinear.', '1601.03918-1-55-3': 'The diagrams of interest can be grouped together to form a one mass triangle in the integral reduction sense (i.e. the massive corner represents a sum of all possible tree diagrams).', '1601.03918-1-55-4': 'The integration region of interest has all three propagators on shell and so the pole may be determined by the triple cut of this triangle.', '1601.03918-1-55-5': "This triple cut wouldn't normally exist, but opens up when [MATH] and [MATH] are collinear.", '1601.03918-1-56-0': 'Using an axial gauge with reference spinor [MATH] [CITATION], the contribution of fig. [REF] with a scalar particle circulating in the loop is [EQUATION] where [MATH] represents the sum of all possible tree diagrams.', '1601.03918-1-56-1': '[MATH] and [MATH] are given by [EQUATION] and satisfy [MATH].', '1601.03918-1-56-2': 'The integrands for other particle types in the loop are related to the scalar contribution by an [MATH]-factor: [EQUATION] and the [MATH] contribution is obtained by multiplying the integrand by a factor of [MATH] where, [EQUATION]', '1601.03918-1-56-3': 'For [MATH] the integrand of [REF] contains [MATH] factors in its numerator, leaving a pole in [MATH] in the integral.', '1601.03918-1-56-4': 'As [MATH] is finite as [MATH], [MATH] in the region of interest.', '1601.03918-1-56-5': 'For a scalar particle circulating in the loop the KLT relations [CITATION] give [EQUATION]', '1601.03918-1-56-6': "Of the six terms in the permutation sum in [REF], the two which don't permute [MATH] can be neglected due to the explicit [MATH] factor.", '1601.03918-1-56-7': 'The remaining four form two pairs with the members of each pair being related by interchange of legs [MATH] and [MATH].', '1601.03918-1-56-8': 'The [MATH] contributions of one member of each of these pairs are [EQUATION] and [EQUATION]', '1601.03918-1-56-9': 'Partial fractioning the integrand of [MATH] using the [MATH] factor in the numerator yields six terms whose integrands have loop momentum dependence [EQUATION] with [MATH] and [MATH].', '1601.03918-1-56-10': 'In the integration region of interest [MATH] and [MATH] are negligible allowing the integrands to be rewritten as quartic pentagon integrands [EQUATION]', '1601.03918-1-56-11': 'For [MATH], using [EQUATION] splits each of these quartic pentagons into a pair of cubic one-mass boxes and a cubic pentagon which can be neglected.', '1601.03918-1-56-12': 'As a box with two adjacent corners attached to single external legs of the same helicity has a vanishing quadruple cut, these cubic one-mass box integrals reduce to bubble and rational contributions only.', '1601.03918-1-56-13': 'The bubble coefficients can be evaluated by direct parametrisation.', '1601.03918-1-56-14': 'For example the box integral [REF] which is illustrated in fig. [REF] has bubbles associated with its [MATH] and [MATH] cuts.', '1601.03918-1-56-15': 'The [MATH] cut of [REF] gives the bubble coefficient [EQUATION] where terms of order [MATH] have been extracted from the leading term to simplify its denominator as far as possible.', '1601.03918-1-56-16': 'The remaining [MATH] singularity in this bubble coefficient is spurious and must cancel with the [MATH] bubble as [MATH].', '1601.03918-1-56-17': 'So that this singularity is not present in the logarithmic part of the integral, the sum of the two bubble coefficients must be finite.', '1601.03918-1-56-18': 'The sum of the two bubble contributions then involves the singular parts of the [MATH] bubble coefficient multiplied by the difference of the integral functions of the two bubbles.', '1601.03918-1-56-19': 'With [MATH] and [MATH] both being small, the rational descendant of the bubbles on the [MATH] pole is [EQUATION]', '1601.03918-1-56-20': 'The leading term of the rational descendant has a [MATH] spurious pole.', '1601.03918-1-56-21': 'This must be cancelled by the rational piece of the integral, allowing the rational term to be identified as, [EQUATION]', '1601.03918-1-56-22': 'For [MATH] in [REF] the reduction to boxes uses the identity [EQUATION] which yields a pair of quartic box integrals whose rational pieces are evaluated using the approach described above.', '1601.03918-1-56-23': 'The full rational contribution of [MATH] is [EQUATION] where [EQUATION]', '1601.03918-1-56-24': 'The [MATH] contributions involve both quintic and quartic pentagon integrals, but their rational pieces can be obtained in a similar fashion to the [MATH] contributions.', '1601.03918-1-56-25': 'Separating the quintic and quartic pentagon integrals, [EQUATION] where [EQUATION] with [EQUATION] and [EQUATION] with [EQUATION]', '1601.03918-1-56-26': 'The contribution of these non-standard factorisations to the rational part of the 6-pt amplitude is obtained by recursion: [EQUATION]', '1601.03918-1-56-27': 'The contributions arising from the conjugate poles, e.g. [MATH], can be obtained using the flip-conjugation symmetry of the amplitude.', '1601.03918-1-56-28': 'Defining [EQUATION] the full non-factorising contribution to [MATH] is [EQUATION]', '1601.03918-1-56-29': 'We have computed [MATH] systematically using its pole structure.', '1601.03918-1-56-30': 'Underlying this is the assumption that the amplitude vanishes for large shifts.', '1601.03918-1-56-31': 'This is difficult to justify a priori.', '1601.03918-1-56-32': 'However the expression obtained has the correct symmetries and collinear limits (checked numerically).', '1601.03918-1-56-33': 'Generically a BCFW recursion produces terms which are not manifestly symmetric and the restoration of symmetry is a good indicator that the amplitude has been correctly determined.', '1601.03918-1-57-0': '# Conclusions', '1601.03918-1-58-0': 'Graviton scattering amplitudes have a rich structure.', '1601.03918-1-58-1': 'In particular [MATH] supergravity has proven to have a much softer UV behaviour then previously expected with the underlying symmetry reason still unclear.', '1601.03918-1-58-2': 'It is important to understand which structures of [MATH] survive in theories with lower supersymmetry.', '1601.03918-1-58-3': 'It is also important to study amplitudes beyond MHV since this can often have a misleadingly simple structure.', '1601.03918-1-58-4': 'In this article we have constructed the six-point NMHV amplitude in [MATH] supergravity.', '1601.03918-1-58-5': 'Of particular interest is the rational term since in the MHV case a particularly simple and suggestive structure appears [CITATION].', '1601.03918-1-58-6': 'The rational terms in the NMHV case do not appear to have any such simple structure although this may be hiding given the algebraic complexity of the amplitude.', '1601.03918-1-59-0': 'Computing the rational terms has required a blending of techniques including obtaining the rational descendants of the cut-constructible pieces.', '1601.03918-1-59-1': 'Amongst the cut-constructible pieces the coefficients of the bubble integral functions have been particularly cumbersome although, fortunately, these do not generate any rational descendants in this amplitude.'}

{'1601.03918-2-0-0': 'We construct the six-point NMHV one-loop amplitude in [MATH] supergravity using unitarity and recursion.', '1601.03918-2-0-1': 'The use of recursion requires the introduction of rational descendants of the cut-constructible pieces of the amplitude and the computation of the non-standard factorisation terms arising from the loop integrals.', '1601.03918-2-1-0': '# Introduction', '1601.03918-2-2-0': 'Despite perturbative Quantum Gravity being a mature subject [CITATION], it is a very challenging area computationally.', '1601.03918-2-2-1': 'Although great strides have been made in computing tree amplitudes [CITATION], there remain a very limited number of loop calculations available to study.', '1601.03918-2-2-2': 'For the four and five point amplitudes all one-loop graviton scattering amplitudes have now been calculated [CITATION].', '1601.03918-2-2-3': 'For Maximal supergravity great progress has also be made at multi-loop level for the four point amplitude [CITATION].', '1601.03918-2-2-4': 'These computations are by necessity amplitudes which are "Maximally-Helicity-Violating" (MHV).', '1601.03918-2-2-5': 'MHV amplitudes are very special and have many features not shared by non-MHV amplitudes.', '1601.03918-2-2-6': 'In this article we compute the six graviton "Next-to-MHV" (NMHV) scattering amplitude for [MATH] supergravity.', '1601.03918-2-2-7': '(The first NMHV amplitude appears at six-points.)', '1601.03918-2-2-8': 'The six-graviton scattering amplitude has been computed for [MATH] and [MATH] supergravity.', '1601.03918-2-3-0': 'This amplitude has considerable algebraic complexity relative to the more supersymmetric cases including the appearance of rational terms.', '1601.03918-2-3-1': 'We construct the [MATH] amplitude using unitarity and recursion augmented by limited off-shell behaviour.', '1601.03918-2-4-0': 'One-loop amplitudes in a massless theory can be expressed as [CITATION] [EQUATION] where the [MATH] are [MATH]-point scalar integral functions and the [MATH] etc. are rational coefficients.', '1601.03918-2-4-1': '[MATH] is a purely rational term.', '1601.03918-2-4-2': 'The box, triangle and bubble coefficients can be determined via unitarity methods [CITATION] using four dimensional on-shell tree amplitudes.', '1601.03918-2-4-3': 'These contributions are termed cut-constructible.', '1601.03918-2-4-4': 'Progress has been made both via the two-particle cuts [CITATION] and using generalisations of unitarity [CITATION] where, for example, triple [CITATION] and quadruple cuts [CITATION] are utilised to identify the triangle and box coefficients respectively.', '1601.03918-2-4-5': 'The remaining purely rational term, [MATH], may in principle be obtained using unitarity but this requires a knowledge of [MATH] dimensional tree amplitudes [CITATION].', '1601.03918-2-4-6': 'In this paper a recursive approach is adopted that generates the rational term from four dimensional amplitudes.', '1601.03918-2-5-0': 'An important technique for computing tree amplitudes is Britto-Cachazo-Feng-Witten (BCFW) [CITATION] recursion which applies complex analysis to amplitudes.', '1601.03918-2-5-1': "By shifting the momenta [EQUATION] the resultant amplitude [MATH] may be computed via Cauchy's theorem.", '1601.03918-2-5-2': 'Loop amplitudes, as functions of complexified momentum, contain both poles and cuts so BCFW does not immediately apply to these.', '1601.03918-2-5-3': 'However by defining [EQUATION] where [MATH] is the cut-constructible part of the amplitude, we can compute the purely rational [MATH] from a knowledge of its singularities.', '1601.03918-2-5-4': '[MATH] has singularities corresponding to the poles of the amplitude but also induced singularities arising because [MATH] has singularities which are not present in the amplitude and which must be cancelled by equal and opposite singularities in [MATH].', '1601.03918-2-5-5': 'We refer to these contributions as the rational descendants of [MATH].', '1601.03918-2-6-0': 'The particle content of the [MATH] graviton and matter multiplets are shown in table [REF].', '1601.03918-2-7-0': 'For convenience, we will calculate the one-loop amplitude using the [MATH] matter multiplet, which is related to the amplitude with the graviton multiplet circulating in the loop by [EQUATION]', '1601.03918-2-7-1': 'The [MATH] and [MATH] components are considerably simpler and given in [CITATION].', '1601.03918-2-7-2': 'In particular [MATH] for these two components.', '1601.03918-2-8-0': '# Cut-Constructible Pieces', '1601.03918-2-9-0': 'The cut-constructable pieces consist of box-functions, triangle functions and bubble integral functions.', '1601.03918-2-9-1': 'The analytic form of these depends upon how many of the external legs have non-null momentum: these are referred to as massive legs although non-null is more correct.', '1601.03918-2-9-2': 'For the one-mass box the fourth leg is conventionally the massive leg and the integral function depends upon [MATH], [MATH] and [MATH].', '1601.03918-2-9-3': 'For the two-mass boxes there are two types: "two-mass-easy" ([MATH]) where legs [MATH] and [MATH] are massive and "two-mass-hard" ([MATH]) where legs [MATH] and [MATH] are massive.', '1601.03918-2-9-4': 'For six-point amplitudes the three and four mass boxes do not appear and for the NMHV [MATH] amplitude there are no two-mass-easy boxes.', '1601.03918-2-10-0': 'The various box, triangle and bubble contributions present in the six-point NMHV are shown in fig. [REF] together with the labelling of helicities which yield a non-zero coefficient.', '1601.03918-2-10-1': 'Permuting the positive and negative helicity legs separately gives eighteen one-mass boxes, thirty-six two-mass hard boxes, nine 2:4 bubbles, nine 3:3 bubbles and six three-mass triangles.', '1601.03918-2-11-0': '2.0 truecm', '1601.03918-2-12-0': '## IR consistency and Choice of Integral Function Basis', '1601.03918-2-13-0': 'For one-loop amplitudes Infra-Red (IR) consistency imposes a system of constraints on the rational coefficients of the integral functions.', '1601.03918-2-13-1': 'For the matter multiplets [CITATION] there are in fact no IR singular terms in the amplitude, so the singular terms in the individual integral functions cancel.', '1601.03918-2-13-2': 'This gives enough information to fix the coefficients of the one- and two-mass triangles in terms of the box coefficients.', '1601.03918-2-13-3': 'The three-mass triangle is IR finite, so its coefficient is not determined by these constraints.', '1601.03918-2-13-4': 'It is convenient to combine the boxes and triangles in such a way that these infinities are manifestly absent.', '1601.03918-2-13-5': 'There are several ways to do this [CITATION], here we choose to work with truncated box functions, [EQUATION] where the [MATH] and [MATH] are chosen to make [MATH] IR finite.', '1601.03918-2-13-6': 'This effectively incorporates the one- and two-mass triangle contributions into the box contributions.', '1601.03918-2-13-7': 'Using these truncated boxes the amplitudes can be written as [EQUATION]', '1601.03918-2-13-8': 'There is one remaining IR consistency constraint: [MATH].', '1601.03918-2-14-0': 'The one-mass and two-mass-hard truncated integral functions are [EQUATION] where [MATH] and [MATH].', '1601.03918-2-15-0': '## Boxes', '1601.03918-2-16-0': 'The six-pt NMHV amplitude contains both one-mass and two-mass hard boxes.', '1601.03918-2-16-1': 'The coefficients of these boxes are readily obtained using quadruple cuts [CITATION].', '1601.03918-2-17-0': 'The coefficient of the first one-mass box in fig. [REF] is [EQUATION] where [MATH].', '1601.03918-2-17-1': 'The other one-mass box coefficients can be obtained from this by conjugation and relabelling.', '1601.03918-2-17-2': 'The coefficient of the two-mass-hard box is [EQUATION] and the other coefficients are obtained by relabelling.', '1601.03918-2-18-0': '## Canonical basis approach for triangle and bubble coefficients', '1601.03918-2-19-0': 'The canonical basis approach [CITATION] is a systematic method to determine the coefficients of triangle and bubble integral functions from the three and two-particle cuts.', '1601.03918-2-19-1': 'The two particle cut is shown in fig. [REF] and is [EQUATION]', '1601.03918-2-19-2': 'The product of tree amplitudes appearing in the two-particle cut can be decomposed in terms of canonical forms [MATH], [EQUATION] where the [MATH] are coefficients independent of [MATH].', '1601.03918-2-19-3': 'We then use substitution rules to replace the [MATH] by the rational functions [MATH] to obtain the coefficient of the bubble integral function as [EQUATION]', '1601.03918-2-19-4': 'Similarly, we can obtain the coefficient of the triangle functions from the triple cut [CITATION] as shown in figure [REF], [EQUATION]', '1601.03918-2-19-5': 'The product of tree amplitudes can, again, be expressed in terms of standard forms of [MATH], [EQUATION] and substitution rules used to replace the [MATH] by the functions [MATH] to obtain the triangle coefficient as [EQUATION]', '1601.03918-2-19-6': 'In general the integrands are rational functions of [MATH] of degree [MATH].', '1601.03918-2-19-7': 'The simplest canonical forms have [MATH] or [MATH].', '1601.03918-2-19-8': 'More complex denominators can be expressed in terms of the simplest forms by partial fractioning.', '1601.03918-2-19-9': 'Terms in the integrand with [MATH] only contribute to higher point integral functions.', '1601.03918-2-19-10': 'The degree generally decreases with increasing supersymmetry and for maximally supersymmetric Yang-Mills and supergravity there are no triangles.', '1601.03918-2-19-11': 'With increasing [MATH] the canonical forms become increasingly complex.', '1601.03918-2-20-0': '## Triangles', '1601.03918-2-21-0': 'Using the truncated box functions, we only need to compute the coefficient of the (IR finite) three-mass triangle: [MATH].', '1601.03918-2-21-1': 'Using the Kawai-Lewellen-Tye (KLT) relations [CITATION], with a scalar circulating in the loop the cut integrand is [EQUATION]', '1601.03918-2-21-2': 'The four-point Yang-Mills amplitudes above are simultaneously MHV and [MATH] amplitudes, so there is a choice of two expressions for each.', '1601.03918-2-21-3': 'For algebraic convenience we take a mixed form for the first ordering: [EQUATION] and a purely MHV form for the second: [EQUATION]', '1601.03918-2-21-4': 'The contribution to the cut from a particle of helicity [MATH] circulating in the loop is given by [EQUATION] so that summing over the [MATH] matter multiplet gives [EQUATION]', '1601.03918-2-21-5': 'The [MATH]-factor can be written in several ways.', '1601.03918-2-21-6': 'The natural choices given [REF] and [REF] are [EQUATION] with the corresponding [MATH]-factors being [EQUATION] where [EQUATION]', '1601.03918-2-21-7': 'The cut is then [EQUATION]', '1601.03918-2-21-8': 'Using [EQUATION] and [EQUATION] we find [EQUATION] where [EQUATION]', '1601.03918-2-21-9': 'Now we may partial fraction the expression, i.e. use the identity [EQUATION] with [EQUATION] so that the cut is written as a sum of canonical forms: [EQUATION] which have corresponding canonical functions [EQUATION] and [EQUATION] where [EQUATION] and [MATH] is the set of [MATH] permutations of [MATH] necessary to generate symmetry in these variables.', '1601.03918-2-22-0': 'We obtain the coefficient [EQUATION]', '1601.03918-2-22-1': 'The corresponding three mass triangle integral function is [CITATION] [EQUATION] where [EQUATION]', '1601.03918-2-23-0': '# Bubbles', '1601.03918-2-24-0': 'The final cut constructible pieces are the bubbles integral functions [MATH].', '1601.03918-2-24-1': 'There are two distinct types of bubbles coefficients depending upon whether [MATH] is a two particle momentum, [MATH], or three particle, [MATH].', '1601.03918-2-24-2': 'In the three particle case the cut only involves MHV tree amplitudes whereas the two particle case requires the NMHV tree amplitude.', '1601.03918-2-25-0': 'For the [MATH] bubble the cut integrand with a scalar circulating in the loop is [EQUATION] where [EQUATION]', '1601.03918-2-25-1': 'As in the three-mass triangle, the contribution from a particle of helicity [MATH] is [MATH].', '1601.03918-2-25-2': 'Summing over the multiplet has the effect of multiplying by [MATH].', '1601.03918-2-25-3': 'For this cut [EQUATION]', '1601.03918-2-25-4': 'Multiplying [MATH] by [MATH] gives [EQUATION]', '1601.03918-2-25-5': 'Using [EQUATION] we find [EQUATION] where [EQUATION]', '1601.03918-2-25-6': 'Partial fractioning on [MATH] and [MATH] yields [EQUATION] where [EQUATION]', '1601.03918-2-25-7': 'The cut is now expressed in terms of the canonical form [EQUATION] which has the corresponding canonical function [EQUATION]', '1601.03918-2-25-8': 'The contribution of [REF] to the bubble coefficient is then [EQUATION] leading to the full bubble coefficient [EQUATION]', '1601.03918-2-25-9': 'The coefficient of the bubble [MATH] is obtained from the cut [EQUATION]', '1601.03918-2-25-10': 'This cut can also be decomposed in canonical forms.', '1601.03918-2-25-11': 'However the six-point NMHV tree amplitude in the cut is a sum of fourteen terms [CITATION] which leads to a lengthy expression for this bubble coefficient.', '1601.03918-2-25-12': 'Full expressions for the bubble coefficients of this type are given in appendix [REF].', '1601.03918-2-26-0': '# Cancellation Webs and Rational Descendants', '1601.03918-2-27-0': 'Although we may split the amplitude into cut-constructible pieces and rational terms, when we examine the singularities in the amplitude there is a mixing between the two which is important when we reconstruct [MATH] from its singularities.', '1601.03918-2-27-1': 'The cut-constructible pieces of the amplitude introduce a number of singularities that cannot be present in the full amplitude.', '1601.03918-2-27-2': 'These can be spurious singularities that occur at kinematic points where the full amplitude should be finite or higher order singularities occurring at points where the amplitude has a simple pole.', '1601.03918-2-27-3': 'If these poles are of sufficiently high order, they generate rational descendants.', '1601.03918-2-28-0': '## Higher Order Poles', '1601.03918-2-29-0': 'As an example of a higher order pole consider the behaviour of the one-mass box contribution [MATH] as [MATH].', '1601.03918-2-29-1': 'The box coefficient [MATH] in eq. ([REF]) contains a factor of [MATH] and the expansion of the box integral function around [MATH] is [CITATION] [EQUATION] where [EQUATION]', '1601.03918-2-29-2': 'As the cut constructible terms contain all of the logarithms and dilogarithms in the amplitude, the logarithmic pieces of this expansion must combine with the bubbles to give an effective coefficients that are linearly divergent as [MATH].', '1601.03918-2-29-3': 'We have confirmed numerically that the relevant cancellations between the one-mass box contributions and the bubble contributions occur.', '1601.03918-2-29-4': 'The quadratic divergence in the rational descendant, [MATH], must be cancelled by the rational piece of the amplitude.', '1601.03918-2-29-5': 'The full rational part of the amplitude will ultimately be obtained by recursion and one contribution to it will arise from this rational descendant if the shift excites the [MATH] pole.', '1601.03918-2-30-0': 'There are corresponding [MATH] poles in [MATH] which are obtained from those above by conjugation.', '1601.03918-2-31-0': '## [MATH] Spurious Singularity', '1601.03918-2-32-0': 'The coefficients of the two mass hard boxes have singularities of the form [MATH].', '1601.03918-2-32-1': 'These singularities also occur in the three mass triangle contributions: three powers of the pole are explicit in the leading term of the canonical form and a fourth arises in the partial fractioning that splits the cut integrand into canonical forms.', '1601.03918-2-33-0': 'In terms of kinematic variables, [MATH] corresponds to [MATH].', '1601.03918-2-33-1': 'The two mass hard integral functions depend on [MATH], [MATH], [MATH] and [MATH], while the three mass triangle depends on [MATH], [MATH] and [MATH] and there is the kinematic constraint: [MATH].', '1601.03918-2-33-2': 'For given [MATH], [MATH] and [MATH] it is possible find [MATH] and [MATH]) so that [MATH].', '1601.03918-2-33-3': 'Close to the pole the two mass hard box integral functions can be expanded as a series in terms of [MATH].', '1601.03918-2-33-4': 'In this context it is convenient to work with dimensionless box integral functions [MATH] defined by [EQUATION]', '1601.03918-2-33-5': 'The expansion of this dimensionless box function is [EQUATION] where [EQUATION] and [EQUATION] with [EQUATION]', '1601.03918-2-33-6': 'The same pole appears in the two boxes with integral function: [MATH] and [MATH].', '1601.03918-2-33-7': 'On the pole the coefficients of these two boxes are not equal and neither integral function vanishes.', '1601.03918-2-33-8': 'However, the sum of the dimensionless integral functions vanishes, i.e. [EQUATION]', '1601.03918-2-33-9': 'On this pole the dilogarithms in the individual boxes and triangles survive, but cancel between them.', '1601.03918-2-33-10': 'Setting [EQUATION] the integral functions are related by [EQUATION] where the sign ambiguity is associated with the choice of sign for [MATH].', '1601.03918-2-33-11': 'Schematically, expressing the box coefficients in terms of their sum and difference, [MATH], [MATH], the box and triangle contributions are [EQUATION]', '1601.03918-2-33-12': 'Expanding about [MATH], thanks to [REF] there is no dilogarithm component in the first term for any [MATH].', '1601.03918-2-33-13': 'However, we can only use [REF] if [MATH] and [MATH] are equal.', '1601.03918-2-33-14': 'In fact [EQUATION]', '1601.03918-2-33-15': 'Hence the dilogarithms vanish from any term that is singular as [MATH], as required by the factorisation theorems [CITATION].', '1601.03918-2-34-0': 'As in the one-mass case, there are subleading singularities at cubic order multiplying logarithms.', '1601.03918-2-34-1': 'These combine with the bubble contributions and cancel up to and including order [MATH], leaving no spurious singularity in the logarithms.', '1601.03918-2-35-0': 'The rational descendant of this combination of boxes and triangle contain both [MATH] and [MATH] singularities.', '1601.03918-2-35-1': 'Both of these singularities must be cancelled by the rational piece of the amplitude [MATH].', '1601.03918-2-35-2': 'As the expansion has been performed about a singularity specified in terms of [MATH], [MATH] and [MATH], there is no need to expand the three mass triangle integral function when determining this rational descendant.', '1601.03918-2-36-0': '## [MATH] Spurious Singularity', '1601.03918-2-37-0': 'The three mass triangle contributions have [MATH] poles which can be seen explicitly in the canonical forms.', '1601.03918-2-37-1': 'Around the [MATH] pole the integral function has the expansion, [EQUATION] where [EQUATION]', '1601.03918-2-37-2': 'The logarithmic terms in this expansion combine with the bubble contributions to yield a finite contribution on the pole.', '1601.03918-2-37-3': 'The rational piece in the expansion must cancel with the rational part of the amplitude.', '1601.03918-2-38-0': '# Obtaining [MATH] By Recursion', '1601.03918-2-39-0': 'BCFW [CITATION] recursion applies complex analysis to amplitudes.', '1601.03918-2-39-1': "Using Cauchy's theorem, if a complex function is analytic except at simple poles [MATH] (all non-zero) and [MATH] as [MATH] then by considering the integral [EQUATION] where the contour [MATH] is the circle at infinity, we obtain [EQUATION]", '1601.03918-2-39-2': 'We wish to apply this with [MATH], where [MATH] has been complexified by a BCFW shift of momenta.', '1601.03918-2-39-3': 'Since [EQUATION] the singularities and residues of [MATH] are both those arising from the physical factorisations of [MATH] and those induced by the necessity to cancel the spurious singularities of the cut-constructible pieces.', '1601.03918-2-40-0': '## Choice of Shift', '1601.03918-2-41-0': 'The rational part of an amplitude can be obtained recursively if the factorisation properties of the amplitude are understood at all of the relevant poles.', '1601.03918-2-41-1': 'There are three main obstacles to this: quadratic poles in the amplitude, non-standard factorisations for complex momenta and contributions for large shifted momenta.', '1601.03918-2-41-2': 'Quadratic poles in the amplitude lead to recursive contributions that depend on the off-shell behaviour of the factorised currents.', '1601.03918-2-41-3': 'This can be addressed using augmented recursion [CITATION].', '1601.03918-2-41-4': 'For non-supersymmetric theories there are double poles of the form [EQUATION]', '1601.03918-2-41-5': 'For the six-point NMHV amplitude the tree amplitude vanishes since it has a single positive helicity leg.', '1601.03918-2-41-6': '(This is no longer the case for seven and higher point NMHV amplitudes.)', '1601.03918-2-42-0': 'Non-standard factorisations for complex momenta are unavoidable and are considered in detail below.', '1601.03918-2-42-1': 'The final obstacle is the possibility of contributions from asymptotically large shifted momenta.', '1601.03918-2-42-2': "The amplitude doesn't factorise in this limit, so the residue is undetermined.", '1601.03918-2-42-3': 'This issue may be avoided if the shift employed causes the amplitude to vanish for asymptotically large shifted momenta.', '1601.03918-2-42-4': 'As the amplitude is as yet undetermined, its behaviour under any shift is unknown.', '1601.03918-2-42-5': "However, if the cut constructible pieces don't vanish for asymptotically large shifted momenta there is little hope that the rational pieces would.", '1601.03918-2-43-0': 'For example under a shift involving two negative helicity legs, [EQUATION] the cut constructible pieces of the amplitude are divergent for large [MATH].', '1601.03918-2-44-0': 'However, for a shift involving one negative helicity leg and one positive helicity leg, [EQUATION] the cut-constructible pieces all vanish at large [MATH], at least suggesting that the rational piece is also well behaved there.', '1601.03918-2-44-1': 'The shift [REF] will be used to obtain [MATH].', '1601.03918-2-45-0': 'The contributions to [MATH] can be grouped into three classes: standard factorisations, non-factorising contributions and rational descendants of the cut-constructible pieces : [EQUATION]', '1601.03918-2-46-0': '## Standard Factorisations', '1601.03918-2-47-0': 'The standard factorisations of a six-point one-loop amplitude have the forms: [EQUATION]', '1601.03918-2-47-1': 'In a supersymmetric theory the 3-point loop amplitudes vanish and so the third class are absent in this case.', '1601.03918-2-47-2': 'With the shift [REF] the factorisations of the first type are: [EQUATION]', '1601.03918-2-47-3': 'While the factorisations of the second type are: [EQUATION]', '1601.03918-2-47-4': 'For generic six-point kinematics, the kinematic points at which the 4- and 5-point loop amplitudes appearing in these factorisations are evaluated are in no way special, hence the rational contribution to the residue comes solely from the rational part of the 4- and 5-point loop amplitudes.', '1601.03918-2-47-5': 'Each factorisation therefore gives a contribution to [MATH] of [EQUATION] with [CITATION] [EQUATION] where [EQUATION] and [MATH] denotes a sum over the six distinct permutations of [MATH] and [MATH] noting the symmetry of [MATH] under [MATH].', '1601.03918-2-47-6': 'The full contribution of the standard factorisations is then [EQUATION] where the sum is over all of the standard factorisation channels given in [REF] and [REF].', '1601.03918-2-48-0': '## Contribution Of Rational Descendants', '1601.03918-2-49-0': 'As discussed above, higher order poles in the coefficients of the box and triangle contributions to the amplitude can generate rational descendants when those poles are excited.', '1601.03918-2-49-1': 'The shift [REF] excites some poles of each type.', '1601.03918-2-49-2': 'Specifically we have the various singularities listed in table [REF] (with [MATH], [MATH]).', '1601.03918-2-50-0': 'Denoting the rational descendant in each case by [MATH], the corresponding coefficient by [MATH] and the value of [MATH] on the pole by [MATH], the contribution on each of these poles is [EQUATION] so that [EQUATION] where the sum is over all of the poles listed above.', '1601.03918-2-51-0': 'Individual terms in the bubble coefficients contain a range of other higher order poles.', '1601.03918-2-51-1': 'In principle these could also generate rational descendants, however in the full bubble coefficients these are at most simple poles and so do not generate further rational descendants: [EQUATION]', '1601.03918-2-52-0': '## Non-standard Factorisations for Complex Momenta', '1601.03918-2-53-0': 'Factorisations of the amplitude occur when propagators go on shell.', '1601.03918-2-53-1': 'The standard factorisation channels arise when the on-shell propagator is not in the loop and is explicit in, for example a Feynman diagram approach.', '1601.03918-2-54-0': 'The loop momentum integral may also generate poles in the amplitude [CITATION] particularly for complex momenta.', '1601.03918-2-54-1': 'Since we are computing the amplitude by recursion in complex momenta we must determine these complex factorisations.', '1601.03918-2-54-2': '2.0truecm', '1601.03918-2-55-0': 'Poles can arise when two adjacent massless legs on a loop became collinear as illustrated in fig. [REF].', '1601.03918-2-55-1': 'This case has been discussed in the context of amplitudes with a single negative helicity leg [CITATION].', '1601.03918-2-55-2': 'In the integration region [MATH] the three propagators connected to [MATH] and [MATH] all become on shell when [MATH] and [MATH] are collinear.', '1601.03918-2-55-3': 'The diagrams of interest can be grouped together to form a one mass triangle in the integral reduction sense (i.e. the massive corner represents a sum of all possible tree diagrams).', '1601.03918-2-55-4': 'The integration region of interest has all three propagators on shell and so the pole may be determined by the triple cut of this triangle.', '1601.03918-2-55-5': "This triple cut wouldn't normally exist, but opens up when [MATH] and [MATH] are collinear.", '1601.03918-2-56-0': 'Using an axial gauge with reference spinor [MATH] [CITATION], the contribution of fig. [REF] with a scalar particle circulating in the loop is [EQUATION] where [MATH] represents the sum of all possible tree diagrams.', '1601.03918-2-56-1': '[MATH] and [MATH] are given by [EQUATION] and satisfy [MATH].', '1601.03918-2-56-2': 'The integrands for other particle types in the loop are related to the scalar contribution by an [MATH]-factor: [EQUATION] and the [MATH] contribution is obtained by multiplying the integrand by a factor of [MATH] where, [EQUATION]', '1601.03918-2-56-3': 'For [MATH] the integrand of [REF] contains [MATH] factors in its numerator, leaving a pole in [MATH] in the integral.', '1601.03918-2-56-4': 'As [MATH] is finite as [MATH], [MATH] in the region of interest.', '1601.03918-2-56-5': 'For a scalar particle circulating in the loop the KLT relations [CITATION] give [EQUATION]', '1601.03918-2-56-6': "Of the six terms in the permutation sum in [REF], the two which don't permute [MATH] can be neglected due to the explicit [MATH] factor.", '1601.03918-2-56-7': 'The remaining four form two pairs with the members of each pair being related by interchange of legs [MATH] and [MATH].', '1601.03918-2-56-8': 'The [MATH] contributions of one member of each of these pairs are [EQUATION] and [EQUATION]', '1601.03918-2-56-9': 'Partial fractioning the integrand of [MATH] using the [MATH] factor in the numerator yields six terms whose integrands have loop momentum dependence [EQUATION] with [MATH] and [MATH].', '1601.03918-2-56-10': 'In the integration region of interest [MATH] and [MATH] are negligible allowing the integrands to be rewritten as quartic pentagon integrands [EQUATION]', '1601.03918-2-56-11': 'For [MATH], using [EQUATION] splits each of these quartic pentagons into a pair of cubic one-mass boxes and a cubic pentagon which can be neglected.', '1601.03918-2-56-12': 'As a box with two adjacent corners attached to single external legs of the same helicity has a vanishing quadruple cut, these cubic one-mass box integrals reduce to bubble and rational contributions only.', '1601.03918-2-56-13': 'The bubble coefficients can be evaluated by direct parametrisation.', '1601.03918-2-56-14': 'For example the box integral [REF] which is illustrated in fig. [REF] has bubbles associated with its [MATH] and [MATH] cuts.', '1601.03918-2-56-15': 'The [MATH] cut of [REF] gives the bubble coefficient [EQUATION] where terms of order [MATH] have been extracted from the leading term to simplify its denominator as far as possible.', '1601.03918-2-56-16': 'The remaining [MATH] singularity in this bubble coefficient is spurious and must cancel with the [MATH] bubble as [MATH].', '1601.03918-2-56-17': 'So that this singularity is not present in the logarithmic part of the integral, the sum of the two bubble coefficients must be finite.', '1601.03918-2-56-18': 'The sum of the two bubble contributions then involves the singular parts of the [MATH] bubble coefficient multiplied by the difference of the integral functions of the two bubbles.', '1601.03918-2-56-19': 'With [MATH] and [MATH] both being small, the rational descendant of the bubbles on the [MATH] pole is [EQUATION]', '1601.03918-2-56-20': 'The leading term of the rational descendant has a [MATH] spurious pole.', '1601.03918-2-56-21': 'This must be cancelled by the rational piece of the integral, allowing the rational term to be identified as, [EQUATION]', '1601.03918-2-56-22': 'For [MATH] in [REF] the reduction to boxes uses the identity [EQUATION] which yields a pair of quartic box integrals whose rational pieces are evaluated using the approach described above.', '1601.03918-2-56-23': 'The full rational contribution of [MATH] is [EQUATION] where [EQUATION]', '1601.03918-2-56-24': 'The [MATH] contributions involve both quintic and quartic pentagon integrals, but their rational pieces can be obtained in a similar fashion to the [MATH] contributions.', '1601.03918-2-56-25': 'Separating the quintic and quartic pentagon integrals, [EQUATION] where [EQUATION] with [EQUATION] and [EQUATION] with [EQUATION]', '1601.03918-2-56-26': 'The contribution of these non-standard factorisations to the rational part of the 6-pt amplitude is obtained by recursion: [EQUATION]', '1601.03918-2-56-27': 'The contributions arising from the conjugate poles, e.g. [MATH], can be obtained using the flip-conjugation symmetry of the amplitude.', '1601.03918-2-56-28': 'Defining [EQUATION] the full non-factorising contribution to [MATH] is [EQUATION]', '1601.03918-2-56-29': 'We have computed [MATH] systematically using its pole structure.', '1601.03918-2-56-30': 'Underlying this is the assumption that the amplitude vanishes for large shifts.', '1601.03918-2-56-31': 'This is difficult to justify a priori.', '1601.03918-2-56-32': 'However the expression obtained has the correct symmetries and collinear limits (checked numerically).', '1601.03918-2-56-33': 'Generically a BCFW recursion produces terms which are not manifestly symmetric and the restoration of symmetry is a good indicator that the amplitude has been correctly determined.', '1601.03918-2-57-0': 'An explicit form of [MATH] is available in Mathematica format at http://pyweb.swan.ac.uk/ dunbar/sixgraviton/R6.html.', '1601.03918-2-58-0': '# Conclusions', '1601.03918-2-59-0': 'Graviton scattering amplitudes have a rich structure.', '1601.03918-2-59-1': 'In particular [MATH] supergravity has proven to have a much softer UV behaviour then previously expected with the underlying symmetry reason still unclear.', '1601.03918-2-59-2': 'It is important to understand which structures of [MATH] survive in theories with lower supersymmetry.', '1601.03918-2-59-3': 'It is also important to study amplitudes beyond MHV since this can often have a misleadingly simple structure.', '1601.03918-2-59-4': 'In this article we have constructed the six-point NMHV amplitude in [MATH] supergravity.', '1601.03918-2-59-5': 'Of particular interest is the rational term since in the MHV case a particularly simple and suggestive structure appears [CITATION].', '1601.03918-2-59-6': 'The rational terms in the NMHV case do not appear to have any such simple structure although this may be hiding given the algebraic complexity of the amplitude.', '1601.03918-2-60-0': 'Computing the rational terms has required a blending of techniques including obtaining the rational descendants of the cut-constructible pieces.', '1601.03918-2-60-1': 'Amongst the cut-constructible pieces the coefficients of the bubble integral functions have been particularly cumbersome although, fortunately, these do not generate any rational descendants in this amplitude.'}

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'1601.03918-2-42-4'], ['1601.03918-1-42-5', '1601.03918-2-42-5'], ['1601.03918-1-45-0', '1601.03918-2-45-0'], ['1601.03918-1-29-0', '1601.03918-2-29-0'], ['1601.03918-1-29-1', '1601.03918-2-29-1'], ['1601.03918-1-29-2', '1601.03918-2-29-2'], ['1601.03918-1-29-3', '1601.03918-2-29-3'], ['1601.03918-1-29-4', '1601.03918-2-29-4'], ['1601.03918-1-29-5', '1601.03918-2-29-5'], ['1601.03918-1-25-0', '1601.03918-2-25-0'], ['1601.03918-1-25-1', '1601.03918-2-25-1'], ['1601.03918-1-25-2', '1601.03918-2-25-2'], ['1601.03918-1-25-3', '1601.03918-2-25-3'], ['1601.03918-1-25-4', '1601.03918-2-25-4'], ['1601.03918-1-25-5', '1601.03918-2-25-5'], ['1601.03918-1-25-6', '1601.03918-2-25-6'], ['1601.03918-1-25-7', '1601.03918-2-25-7'], ['1601.03918-1-25-8', '1601.03918-2-25-8'], ['1601.03918-1-25-9', '1601.03918-2-25-9'], ['1601.03918-1-25-10', '1601.03918-2-25-10'], ['1601.03918-1-25-11', '1601.03918-2-25-11'], ['1601.03918-1-25-12', '1601.03918-2-25-12'], ['1601.03918-1-22-0', '1601.03918-2-22-0'], ['1601.03918-1-22-1', '1601.03918-2-22-1'], ['1601.03918-1-44-0', '1601.03918-2-44-0'], ['1601.03918-1-44-1', '1601.03918-2-44-1'], ['1601.03918-1-53-0', '1601.03918-2-53-0'], ['1601.03918-1-53-1', '1601.03918-2-53-1'], ['1601.03918-1-21-0', '1601.03918-2-21-0'], ['1601.03918-1-21-1', '1601.03918-2-21-1'], ['1601.03918-1-21-2', '1601.03918-2-21-2'], ['1601.03918-1-21-3', '1601.03918-2-21-3'], ['1601.03918-1-21-4', '1601.03918-2-21-4'], ['1601.03918-1-21-5', '1601.03918-2-21-5'], ['1601.03918-1-21-6', '1601.03918-2-21-6'], ['1601.03918-1-21-7', '1601.03918-2-21-7'], ['1601.03918-1-21-8', '1601.03918-2-21-8'], ['1601.03918-1-21-9', '1601.03918-2-21-9'], ['1601.03918-1-59-0', '1601.03918-2-60-0'], ['1601.03918-1-59-1', '1601.03918-2-60-1'], ['1601.03918-1-4-0', '1601.03918-2-4-0'], ['1601.03918-1-4-1', '1601.03918-2-4-1'], ['1601.03918-1-4-2', '1601.03918-2-4-2'], ['1601.03918-1-4-3', '1601.03918-2-4-3'], ['1601.03918-1-4-4', '1601.03918-2-4-4'], ['1601.03918-1-4-5', '1601.03918-2-4-5'], ['1601.03918-1-4-6', '1601.03918-2-4-6'], ['1601.03918-1-41-0', '1601.03918-2-41-0'], ['1601.03918-1-41-1', '1601.03918-2-41-1'], ['1601.03918-1-41-2', '1601.03918-2-41-2'], ['1601.03918-1-41-3', '1601.03918-2-41-3'], ['1601.03918-1-41-4', '1601.03918-2-41-4'], ['1601.03918-1-41-5', '1601.03918-2-41-5'], ['1601.03918-1-41-6', '1601.03918-2-41-6'], ['1601.03918-1-5-0', '1601.03918-2-5-0'], ['1601.03918-1-5-1', '1601.03918-2-5-1'], ['1601.03918-1-5-2', '1601.03918-2-5-2'], ['1601.03918-1-5-3', '1601.03918-2-5-3'], ['1601.03918-1-5-4', '1601.03918-2-5-4'], ['1601.03918-1-5-5', '1601.03918-2-5-5'], ['1601.03918-1-14-0', '1601.03918-2-14-0'], ['1601.03918-1-33-0', '1601.03918-2-33-0'], ['1601.03918-1-33-1', '1601.03918-2-33-1'], ['1601.03918-1-33-2', '1601.03918-2-33-2'], ['1601.03918-1-33-3', '1601.03918-2-33-3'], ['1601.03918-1-33-4', '1601.03918-2-33-4'], ['1601.03918-1-33-5', '1601.03918-2-33-5'], ['1601.03918-1-33-6', '1601.03918-2-33-6'], ['1601.03918-1-33-7', '1601.03918-2-33-7'], ['1601.03918-1-33-8', '1601.03918-2-33-8'], ['1601.03918-1-33-9', '1601.03918-2-33-9'], ['1601.03918-1-33-10', '1601.03918-2-33-10'], ['1601.03918-1-33-11', '1601.03918-2-33-11'], ['1601.03918-1-33-12', '1601.03918-2-33-12'], ['1601.03918-1-33-13', '1601.03918-2-33-13'], ['1601.03918-1-33-15', '1601.03918-2-33-15'], ['1601.03918-1-7-0', '1601.03918-2-7-0'], ['1601.03918-1-7-1', '1601.03918-2-7-1'], ['1601.03918-1-7-2', '1601.03918-2-7-2'], ['1601.03918-1-34-0', '1601.03918-2-34-0'], ['1601.03918-1-34-1', '1601.03918-2-34-1'], ['1601.03918-1-30-0', '1601.03918-2-30-0'], ['1601.03918-1-47-0', '1601.03918-2-47-0'], ['1601.03918-1-47-1', '1601.03918-2-47-1'], ['1601.03918-1-47-2', '1601.03918-2-47-2'], ['1601.03918-1-47-3', '1601.03918-2-47-3'], ['1601.03918-1-47-4', '1601.03918-2-47-4'], ['1601.03918-1-47-5', '1601.03918-2-47-5'], ['1601.03918-1-47-6', '1601.03918-2-47-6'], ['1601.03918-1-58-0', '1601.03918-2-59-0'], ['1601.03918-1-58-1', '1601.03918-2-59-1'], ['1601.03918-1-58-2', '1601.03918-2-59-2'], ['1601.03918-1-58-3', '1601.03918-2-59-3'], ['1601.03918-1-58-4', '1601.03918-2-59-4'], ['1601.03918-1-58-5', '1601.03918-2-59-5'], ['1601.03918-1-58-6', '1601.03918-2-59-6'], ['1601.03918-1-2-0', '1601.03918-2-2-0'], ['1601.03918-1-2-1', '1601.03918-2-2-1'], ['1601.03918-1-2-2', '1601.03918-2-2-2'], ['1601.03918-1-2-3', '1601.03918-2-2-3'], ['1601.03918-1-2-4', '1601.03918-2-2-4'], ['1601.03918-1-2-5', '1601.03918-2-2-5'], ['1601.03918-1-2-6', '1601.03918-2-2-6'], ['1601.03918-1-2-7', '1601.03918-2-2-7'], ['1601.03918-1-2-8', '1601.03918-2-2-8'], ['1601.03918-1-32-0', '1601.03918-2-32-0'], ['1601.03918-1-32-1', '1601.03918-2-32-1'], ['1601.03918-1-13-0', '1601.03918-2-13-0'], ['1601.03918-1-13-1', '1601.03918-2-13-1'], ['1601.03918-1-13-2', '1601.03918-2-13-2'], ['1601.03918-1-13-3', '1601.03918-2-13-3'], ['1601.03918-1-13-4', '1601.03918-2-13-4'], ['1601.03918-1-13-5', '1601.03918-2-13-5'], ['1601.03918-1-13-6', '1601.03918-2-13-6'], ['1601.03918-1-13-7', '1601.03918-2-13-7'], ['1601.03918-1-13-8', '1601.03918-2-13-8'], ['1601.03918-1-24-0', '1601.03918-2-24-0'], ['1601.03918-1-24-1', '1601.03918-2-24-1'], ['1601.03918-1-24-2', '1601.03918-2-24-2'], ['1601.03918-1-35-0', '1601.03918-2-35-0'], ['1601.03918-1-35-1', '1601.03918-2-35-1'], ['1601.03918-1-35-2', '1601.03918-2-35-2'], ['1601.03918-1-50-0', '1601.03918-2-50-0'], ['1601.03918-1-56-0', '1601.03918-2-56-0'], ['1601.03918-1-56-1', '1601.03918-2-56-1'], ['1601.03918-1-56-2', '1601.03918-2-56-2'], ['1601.03918-1-56-3', '1601.03918-2-56-3'], ['1601.03918-1-56-4', '1601.03918-2-56-4'], ['1601.03918-1-56-5', '1601.03918-2-56-5'], ['1601.03918-1-56-6', '1601.03918-2-56-6'], ['1601.03918-1-56-7', '1601.03918-2-56-7'], ['1601.03918-1-56-8', '1601.03918-2-56-8'], ['1601.03918-1-56-9', '1601.03918-2-56-9'], ['1601.03918-1-56-10', '1601.03918-2-56-10'], ['1601.03918-1-56-11', '1601.03918-2-56-11'], ['1601.03918-1-56-12', '1601.03918-2-56-12'], ['1601.03918-1-56-13', '1601.03918-2-56-13'], ['1601.03918-1-56-14', '1601.03918-2-56-14'], ['1601.03918-1-56-15', '1601.03918-2-56-15'], ['1601.03918-1-56-16', '1601.03918-2-56-16'], ['1601.03918-1-56-17', '1601.03918-2-56-17'], ['1601.03918-1-56-18', '1601.03918-2-56-18'], ['1601.03918-1-56-19', '1601.03918-2-56-19'], ['1601.03918-1-56-20', '1601.03918-2-56-20'], ['1601.03918-1-56-21', '1601.03918-2-56-21'], ['1601.03918-1-56-22', '1601.03918-2-56-22'], ['1601.03918-1-56-23', '1601.03918-2-56-23'], ['1601.03918-1-56-24', '1601.03918-2-56-24'], ['1601.03918-1-56-25', '1601.03918-2-56-25'], ['1601.03918-1-56-26', '1601.03918-2-56-26'], ['1601.03918-1-56-27', '1601.03918-2-56-27'], ['1601.03918-1-56-28', '1601.03918-2-56-28'], ['1601.03918-1-56-29', '1601.03918-2-56-29'], ['1601.03918-1-56-30', '1601.03918-2-56-30'], ['1601.03918-1-56-31', '1601.03918-2-56-31'], ['1601.03918-1-56-32', '1601.03918-2-56-32'], ['1601.03918-1-56-33', '1601.03918-2-56-33'], ['1601.03918-1-9-0', '1601.03918-2-9-0'], ['1601.03918-1-9-1', '1601.03918-2-9-1'], ['1601.03918-1-9-2', '1601.03918-2-9-2'], ['1601.03918-1-9-3', '1601.03918-2-9-3'], ['1601.03918-1-9-4', '1601.03918-2-9-4'], ['1601.03918-1-6-0', '1601.03918-2-6-0'], ['1601.03918-1-16-0', '1601.03918-2-16-0'], ['1601.03918-1-16-1', '1601.03918-2-16-1'], ['1601.03918-1-27-0', '1601.03918-2-27-0'], ['1601.03918-1-27-1', '1601.03918-2-27-1'], ['1601.03918-1-27-2', '1601.03918-2-27-2'], ['1601.03918-1-27-3', '1601.03918-2-27-3'], ['1601.03918-1-49-0', '1601.03918-2-49-0'], ['1601.03918-1-49-1', '1601.03918-2-49-1'], ['1601.03918-1-49-2', '1601.03918-2-49-2'], ['1601.03918-1-10-0', '1601.03918-2-10-0'], ['1601.03918-1-10-1', '1601.03918-2-10-1'], ['1601.03918-1-17-0', '1601.03918-2-17-0'], ['1601.03918-1-17-1', '1601.03918-2-17-1'], ['1601.03918-1-17-2', '1601.03918-2-17-2'], ['1601.03918-1-39-0', '1601.03918-2-39-0'], ['1601.03918-1-39-1', '1601.03918-2-39-1'], ['1601.03918-1-39-2', '1601.03918-2-39-2'], ['1601.03918-1-39-3', '1601.03918-2-39-3'], ['1601.03918-1-54-0', '1601.03918-2-54-0'], ['1601.03918-1-54-1', '1601.03918-2-54-1'], ['1601.03918-1-43-0', '1601.03918-2-43-0'], ['1601.03918-1-55-0', '1601.03918-2-55-0'], ['1601.03918-1-55-1', '1601.03918-2-55-1'], ['1601.03918-1-55-2', '1601.03918-2-55-2'], ['1601.03918-1-55-3', '1601.03918-2-55-3'], ['1601.03918-1-55-4', '1601.03918-2-55-4'], ['1601.03918-1-55-5', '1601.03918-2-55-5'], ['1601.03918-1-3-0', '1601.03918-2-3-0'], ['1601.03918-1-3-1', '1601.03918-2-3-1'], ['1601.03918-1-0-0', '1601.03918-2-0-0'], ['1601.03918-1-0-1', '1601.03918-2-0-1'], ['1601.03918-2-6-0', '1601.03918-3-6-0'], ['1601.03918-2-21-0', '1601.03918-3-21-0'], ['1601.03918-2-21-1', '1601.03918-3-21-1'], ['1601.03918-2-21-2', '1601.03918-3-21-2'], ['1601.03918-2-21-3', '1601.03918-3-21-3'], ['1601.03918-2-21-4', '1601.03918-3-21-4'], ['1601.03918-2-21-5', '1601.03918-3-21-5'], ['1601.03918-2-21-6', '1601.03918-3-21-6'], ['1601.03918-2-21-7', '1601.03918-3-21-7'], ['1601.03918-2-21-8', '1601.03918-3-21-8'], ['1601.03918-2-21-9', '1601.03918-3-21-9'], ['1601.03918-2-24-0', '1601.03918-3-24-0'], ['1601.03918-2-24-1', '1601.03918-3-24-1'], ['1601.03918-2-24-2', '1601.03918-3-24-2'], ['1601.03918-2-39-0', '1601.03918-3-39-0'], ['1601.03918-2-39-1', '1601.03918-3-39-1'], ['1601.03918-2-39-2', '1601.03918-3-39-2'], ['1601.03918-2-39-3', '1601.03918-3-39-3'], ['1601.03918-2-29-0', '1601.03918-3-29-0'], ['1601.03918-2-29-1', '1601.03918-3-29-1'], ['1601.03918-2-29-2', '1601.03918-3-29-2'], ['1601.03918-2-29-3', '1601.03918-3-29-3'], ['1601.03918-2-29-4', '1601.03918-3-29-4'], ['1601.03918-2-29-5', '1601.03918-3-29-5'], ['1601.03918-2-37-0', '1601.03918-3-37-0'], ['1601.03918-2-37-2', '1601.03918-3-37-2'], ['1601.03918-2-37-3', '1601.03918-3-37-3'], ['1601.03918-2-0-0', '1601.03918-3-0-0'], ['1601.03918-2-0-1', '1601.03918-3-0-1'], ['1601.03918-2-5-0', '1601.03918-3-5-0'], ['1601.03918-2-5-1', '1601.03918-3-5-1'], ['1601.03918-2-5-2', '1601.03918-3-5-2'], ['1601.03918-2-5-3', 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'1601.03918-3-56-12'], ['1601.03918-2-56-13', '1601.03918-3-56-13'], ['1601.03918-2-56-14', '1601.03918-3-56-14'], ['1601.03918-2-56-15', '1601.03918-3-56-15'], ['1601.03918-2-56-16', '1601.03918-3-56-16'], ['1601.03918-2-56-17', '1601.03918-3-56-17'], ['1601.03918-2-56-18', '1601.03918-3-56-18'], ['1601.03918-2-56-19', '1601.03918-3-56-19'], ['1601.03918-2-56-20', '1601.03918-3-56-20'], ['1601.03918-2-56-21', '1601.03918-3-56-21'], ['1601.03918-2-56-22', '1601.03918-3-56-22'], ['1601.03918-2-56-23', '1601.03918-3-56-23'], ['1601.03918-2-56-24', '1601.03918-3-56-24'], ['1601.03918-2-56-25', '1601.03918-3-56-25'], ['1601.03918-2-56-26', '1601.03918-3-56-26'], ['1601.03918-2-56-27', '1601.03918-3-56-27'], ['1601.03918-2-56-28', '1601.03918-3-56-28'], ['1601.03918-2-56-29', '1601.03918-3-56-29'], ['1601.03918-2-56-30', '1601.03918-3-56-30'], ['1601.03918-2-56-31', '1601.03918-3-56-31'], ['1601.03918-2-56-32', '1601.03918-3-56-32'], ['1601.03918-2-56-33', '1601.03918-3-56-33'], ['1601.03918-2-42-0', '1601.03918-3-42-0'], ['1601.03918-2-42-1', '1601.03918-3-42-1'], ['1601.03918-2-42-2', '1601.03918-3-42-2'], ['1601.03918-2-42-3', '1601.03918-3-42-3'], ['1601.03918-2-42-4', '1601.03918-3-42-4'], ['1601.03918-2-42-5', '1601.03918-3-42-5'], ['1601.03918-2-59-0', '1601.03918-3-59-0'], ['1601.03918-2-59-1', '1601.03918-3-59-1'], ['1601.03918-2-59-2', '1601.03918-3-59-2'], ['1601.03918-2-59-3', '1601.03918-3-59-3'], ['1601.03918-2-59-4', '1601.03918-3-59-4'], ['1601.03918-2-59-5', '1601.03918-3-59-5'], ['1601.03918-2-59-6', '1601.03918-3-59-6'], ['1601.03918-2-50-0', '1601.03918-3-50-0'], ['1601.03918-2-35-0', '1601.03918-3-35-0'], ['1601.03918-2-35-1', '1601.03918-3-35-1'], ['1601.03918-2-35-2', '1601.03918-3-35-2'], ['1601.03918-2-53-0', '1601.03918-3-53-0'], ['1601.03918-2-53-1', '1601.03918-3-53-1'], ['1601.03918-2-55-0', '1601.03918-3-55-0'], ['1601.03918-2-55-1', '1601.03918-3-55-1'], ['1601.03918-2-55-2', '1601.03918-3-55-2'], ['1601.03918-2-55-3', '1601.03918-3-55-3'], ['1601.03918-2-55-4', '1601.03918-3-55-4'], ['1601.03918-2-55-5', '1601.03918-3-55-5'], ['1601.03918-2-60-0', '1601.03918-3-60-0'], ['1601.03918-2-60-1', '1601.03918-3-60-1'], ['1601.03918-2-7-0', '1601.03918-3-7-0'], ['1601.03918-2-7-1', '1601.03918-3-7-1'], ['1601.03918-2-7-2', '1601.03918-3-7-2'], ['1601.03918-2-44-0', '1601.03918-3-44-0'], ['1601.03918-2-44-1', '1601.03918-3-44-1'], ['1601.03918-2-54-0', '1601.03918-3-54-0'], ['1601.03918-2-54-1', '1601.03918-3-54-1'], ['1601.03918-2-57-0', '1601.03918-3-57-0'], ['1601.03918-2-41-0', '1601.03918-3-41-0'], ['1601.03918-2-41-1', '1601.03918-3-41-1'], ['1601.03918-2-41-2', '1601.03918-3-41-2'], ['1601.03918-2-41-3', '1601.03918-3-41-3'], ['1601.03918-2-41-4', '1601.03918-3-41-4'], ['1601.03918-2-41-5', '1601.03918-3-41-5'], ['1601.03918-2-41-6', '1601.03918-3-41-6'], ['1601.03918-2-13-0', '1601.03918-3-13-0'], ['1601.03918-2-13-1', '1601.03918-3-13-1'], ['1601.03918-2-13-2', '1601.03918-3-13-2'], ['1601.03918-2-13-3', '1601.03918-3-13-3'], ['1601.03918-2-13-4', '1601.03918-3-13-4'], ['1601.03918-2-13-5', '1601.03918-3-13-5'], ['1601.03918-2-13-6', '1601.03918-3-13-6'], ['1601.03918-2-13-7', '1601.03918-3-13-7'], ['1601.03918-2-13-8', '1601.03918-3-13-8'], ['1601.03918-2-9-0', '1601.03918-3-9-0'], ['1601.03918-2-9-1', '1601.03918-3-9-1'], ['1601.03918-2-9-2', '1601.03918-3-9-2'], ['1601.03918-2-9-3', '1601.03918-3-9-3'], ['1601.03918-2-9-4', '1601.03918-3-9-4'], ['1601.03918-2-51-0', '1601.03918-3-51-0'], ['1601.03918-2-51-1', '1601.03918-3-51-1'], ['1601.03918-2-3-0', '1601.03918-3-3-0'], ['1601.03918-2-3-1', '1601.03918-3-3-1'], ['1601.03918-2-14-0', '1601.03918-3-14-0'], ['1601.03918-2-17-0', '1601.03918-3-17-0'], ['1601.03918-2-17-1', '1601.03918-3-17-1'], ['1601.03918-2-17-2', '1601.03918-3-17-2'], ['1601.03918-2-34-0', '1601.03918-3-34-0'], ['1601.03918-2-34-1', '1601.03918-3-34-1'], ['1601.03918-2-19-0', '1601.03918-3-19-0'], ['1601.03918-2-19-1', '1601.03918-3-19-1'], ['1601.03918-2-19-2', '1601.03918-3-19-2'], ['1601.03918-2-19-3', '1601.03918-3-19-3'], ['1601.03918-2-19-4', '1601.03918-3-19-4'], ['1601.03918-2-19-5', '1601.03918-3-19-5'], ['1601.03918-2-19-6', '1601.03918-3-19-6'], ['1601.03918-2-19-7', '1601.03918-3-19-7'], ['1601.03918-2-19-8', '1601.03918-3-19-8'], ['1601.03918-2-19-9', '1601.03918-3-19-9'], ['1601.03918-2-19-10', '1601.03918-3-19-10'], ['1601.03918-2-19-11', '1601.03918-3-19-11'], ['1601.03918-2-30-0', '1601.03918-3-30-0'], ['1601.03918-2-4-0', '1601.03918-3-4-0'], ['1601.03918-2-4-1', '1601.03918-3-4-1'], ['1601.03918-2-4-2', '1601.03918-3-4-2'], ['1601.03918-2-4-3', '1601.03918-3-4-3'], ['1601.03918-2-4-4', '1601.03918-3-4-4'], ['1601.03918-2-4-5', '1601.03918-3-4-5'], ['1601.03918-2-4-6', '1601.03918-3-4-6'], ['1601.03918-2-2-0', '1601.03918-3-2-0'], ['1601.03918-2-2-1', '1601.03918-3-2-1'], ['1601.03918-2-2-2', '1601.03918-3-2-2'], ['1601.03918-2-2-3', '1601.03918-3-2-3'], ['1601.03918-2-2-4', '1601.03918-3-2-4'], ['1601.03918-2-2-5', '1601.03918-3-2-5'], ['1601.03918-2-2-6', '1601.03918-3-2-6'], ['1601.03918-2-2-7', '1601.03918-3-2-7'], ['1601.03918-2-2-8', '1601.03918-3-2-8']]

[['1601.03918-2-37-1', '1601.03918-3-37-1']]

[]

[]

[]

['1601.03918-1-11-0', '1601.03918-1-33-14', '1601.03918-1-54-2', '1601.03918-2-11-0', '1601.03918-2-33-14', '1601.03918-2-54-2', '1601.03918-3-11-0', '1601.03918-3-33-14', '1601.03918-3-54-2']

{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '3': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1601.03918

{'1601.03918-3-0-0': 'We construct the six-point NMHV one-loop amplitude in [MATH] supergravity using unitarity and recursion.', '1601.03918-3-0-1': 'The use of recursion requires the introduction of rational descendants of the cut-constructible pieces of the amplitude and the computation of the non-standard factorisation terms arising from the loop integrals.', '1601.03918-3-1-0': '# Introduction', '1601.03918-3-2-0': 'Despite perturbative Quantum Gravity being a mature subject [CITATION], it is a very challenging area computationally.', '1601.03918-3-2-1': 'Although great strides have been made in computing tree amplitudes [CITATION], there remain a very limited number of loop calculations available to study.', '1601.03918-3-2-2': 'For the four and five point amplitudes all one-loop graviton scattering amplitudes have now been calculated [CITATION].', '1601.03918-3-2-3': 'For Maximal supergravity great progress has also be made at multi-loop level for the four point amplitude [CITATION].', '1601.03918-3-2-4': 'These computations are by necessity amplitudes which are "Maximally-Helicity-Violating" (MHV).', '1601.03918-3-2-5': 'MHV amplitudes are very special and have many features not shared by non-MHV amplitudes.', '1601.03918-3-2-6': 'In this article we compute the six graviton "Next-to-MHV" (NMHV) scattering amplitude for [MATH] supergravity.', '1601.03918-3-2-7': '(The first NMHV amplitude appears at six-points.)', '1601.03918-3-2-8': 'The six-graviton scattering amplitude has been computed for [MATH] and [MATH] supergravity.', '1601.03918-3-3-0': 'This amplitude has considerable algebraic complexity relative to the more supersymmetric cases including the appearance of rational terms.', '1601.03918-3-3-1': 'We construct the [MATH] amplitude using unitarity and recursion augmented by limited off-shell behaviour.', '1601.03918-3-4-0': 'One-loop amplitudes in a massless theory can be expressed as [CITATION] [EQUATION] where the [MATH] are [MATH]-point scalar integral functions and the [MATH] etc. are rational coefficients.', '1601.03918-3-4-1': '[MATH] is a purely rational term.', '1601.03918-3-4-2': 'The box, triangle and bubble coefficients can be determined via unitarity methods [CITATION] using four dimensional on-shell tree amplitudes.', '1601.03918-3-4-3': 'These contributions are termed cut-constructible.', '1601.03918-3-4-4': 'Progress has been made both via the two-particle cuts [CITATION] and using generalisations of unitarity [CITATION] where, for example, triple [CITATION] and quadruple cuts [CITATION] are utilised to identify the triangle and box coefficients respectively.', '1601.03918-3-4-5': 'The remaining purely rational term, [MATH], may in principle be obtained using unitarity but this requires a knowledge of [MATH] dimensional tree amplitudes [CITATION].', '1601.03918-3-4-6': 'In this paper a recursive approach is adopted that generates the rational term from four dimensional amplitudes.', '1601.03918-3-5-0': 'An important technique for computing tree amplitudes is Britto-Cachazo-Feng-Witten (BCFW) [CITATION] recursion which applies complex analysis to amplitudes.', '1601.03918-3-5-1': "By shifting the momenta [EQUATION] the resultant amplitude [MATH] may be computed via Cauchy's theorem.", '1601.03918-3-5-2': 'Loop amplitudes, as functions of complexified momentum, contain both poles and cuts so BCFW does not immediately apply to these.', '1601.03918-3-5-3': 'However by defining [EQUATION] where [MATH] is the cut-constructible part of the amplitude, we can compute the purely rational [MATH] from a knowledge of its singularities.', '1601.03918-3-5-4': '[MATH] has singularities corresponding to the poles of the amplitude but also induced singularities arising because [MATH] has singularities which are not present in the amplitude and which must be cancelled by equal and opposite singularities in [MATH].', '1601.03918-3-5-5': 'We refer to these contributions as the rational descendants of [MATH].', '1601.03918-3-6-0': 'The particle content of the [MATH] graviton and matter multiplets are shown in table [REF].', '1601.03918-3-7-0': 'For convenience, we will calculate the one-loop amplitude using the [MATH] matter multiplet, which is related to the amplitude with the graviton multiplet circulating in the loop by [EQUATION]', '1601.03918-3-7-1': 'The [MATH] and [MATH] components are considerably simpler and given in [CITATION].', '1601.03918-3-7-2': 'In particular [MATH] for these two components.', '1601.03918-3-8-0': '# Cut-Constructible Pieces', '1601.03918-3-9-0': 'The cut-constructable pieces consist of box-functions, triangle functions and bubble integral functions.', '1601.03918-3-9-1': 'The analytic form of these depends upon how many of the external legs have non-null momentum: these are referred to as massive legs although non-null is more correct.', '1601.03918-3-9-2': 'For the one-mass box the fourth leg is conventionally the massive leg and the integral function depends upon [MATH], [MATH] and [MATH].', '1601.03918-3-9-3': 'For the two-mass boxes there are two types: "two-mass-easy" ([MATH]) where legs [MATH] and [MATH] are massive and "two-mass-hard" ([MATH]) where legs [MATH] and [MATH] are massive.', '1601.03918-3-9-4': 'For six-point amplitudes the three and four mass boxes do not appear and for the NMHV [MATH] amplitude there are no two-mass-easy boxes.', '1601.03918-3-10-0': 'The various box, triangle and bubble contributions present in the six-point NMHV are shown in fig. [REF] together with the labelling of helicities which yield a non-zero coefficient.', '1601.03918-3-10-1': 'Permuting the positive and negative helicity legs separately gives eighteen one-mass boxes, thirty-six two-mass hard boxes, nine 2:4 bubbles, nine 3:3 bubbles and six three-mass triangles.', '1601.03918-3-11-0': '2.0 truecm', '1601.03918-3-12-0': '## IR consistency and Choice of Integral Function Basis', '1601.03918-3-13-0': 'For one-loop amplitudes Infra-Red (IR) consistency imposes a system of constraints on the rational coefficients of the integral functions.', '1601.03918-3-13-1': 'For the matter multiplets [CITATION] there are in fact no IR singular terms in the amplitude, so the singular terms in the individual integral functions cancel.', '1601.03918-3-13-2': 'This gives enough information to fix the coefficients of the one- and two-mass triangles in terms of the box coefficients.', '1601.03918-3-13-3': 'The three-mass triangle is IR finite, so its coefficient is not determined by these constraints.', '1601.03918-3-13-4': 'It is convenient to combine the boxes and triangles in such a way that these infinities are manifestly absent.', '1601.03918-3-13-5': 'There are several ways to do this [CITATION], here we choose to work with truncated box functions, [EQUATION] where the [MATH] and [MATH] are chosen to make [MATH] IR finite.', '1601.03918-3-13-6': 'This effectively incorporates the one- and two-mass triangle contributions into the box contributions.', '1601.03918-3-13-7': 'Using these truncated boxes the amplitudes can be written as [EQUATION]', '1601.03918-3-13-8': 'There is one remaining IR consistency constraint: [MATH].', '1601.03918-3-14-0': 'The one-mass and two-mass-hard truncated integral functions are [EQUATION] where [MATH] and [MATH].', '1601.03918-3-15-0': '## Boxes', '1601.03918-3-16-0': 'The six-pt NMHV amplitude contains both one-mass and two-mass hard boxes.', '1601.03918-3-16-1': 'The coefficients of these boxes are readily obtained using quadruple cuts [CITATION].', '1601.03918-3-17-0': 'The coefficient of the first one-mass box in fig. [REF] is [EQUATION] where [MATH].', '1601.03918-3-17-1': 'The other one-mass box coefficients can be obtained from this by conjugation and relabelling.', '1601.03918-3-17-2': 'The coefficient of the two-mass-hard box is [EQUATION] and the other coefficients are obtained by relabelling.', '1601.03918-3-17-3': 'The expressions for the box coefficients have the appropriate symmetries under exchange of external legs.', '1601.03918-3-17-4': 'Specifically the two-mass hard is invariant under the joint operation of [MATH] and conjugation.', '1601.03918-3-17-5': 'The one-mass coefficient is invariant under [MATH] although this is not manifest (and similarly under [MATH]).', '1601.03918-3-18-0': '## Canonical basis approach for triangle and bubble coefficients', '1601.03918-3-19-0': 'The canonical basis approach [CITATION] is a systematic method to determine the coefficients of triangle and bubble integral functions from the three and two-particle cuts.', '1601.03918-3-19-1': 'The two particle cut is shown in fig. [REF] and is [EQUATION]', '1601.03918-3-19-2': 'The product of tree amplitudes appearing in the two-particle cut can be decomposed in terms of canonical forms [MATH], [EQUATION] where the [MATH] are coefficients independent of [MATH].', '1601.03918-3-19-3': 'We then use substitution rules to replace the [MATH] by the rational functions [MATH] to obtain the coefficient of the bubble integral function as [EQUATION]', '1601.03918-3-19-4': 'Similarly, we can obtain the coefficient of the triangle functions from the triple cut [CITATION] as shown in figure [REF], [EQUATION]', '1601.03918-3-19-5': 'The product of tree amplitudes can, again, be expressed in terms of standard forms of [MATH], [EQUATION] and substitution rules used to replace the [MATH] by the functions [MATH] to obtain the triangle coefficient as [EQUATION]', '1601.03918-3-19-6': 'In general the integrands are rational functions of [MATH] of degree [MATH].', '1601.03918-3-19-7': 'The simplest canonical forms have [MATH] or [MATH].', '1601.03918-3-19-8': 'More complex denominators can be expressed in terms of the simplest forms by partial fractioning.', '1601.03918-3-19-9': 'Terms in the integrand with [MATH] only contribute to higher point integral functions.', '1601.03918-3-19-10': 'The degree generally decreases with increasing supersymmetry and for maximally supersymmetric Yang-Mills and supergravity there are no triangles.', '1601.03918-3-19-11': 'With increasing [MATH] the canonical forms become increasingly complex.', '1601.03918-3-20-0': '## Triangles', '1601.03918-3-21-0': 'Using the truncated box functions, we only need to compute the coefficient of the (IR finite) three-mass triangle: [MATH].', '1601.03918-3-21-1': 'Using the Kawai-Lewellen-Tye (KLT) relations [CITATION], with a scalar circulating in the loop the cut integrand is [EQUATION]', '1601.03918-3-21-2': 'The four-point Yang-Mills amplitudes above are simultaneously MHV and [MATH] amplitudes, so there is a choice of two expressions for each.', '1601.03918-3-21-3': 'For algebraic convenience we take a mixed form for the first ordering: [EQUATION] and a purely MHV form for the second: [EQUATION]', '1601.03918-3-21-4': 'The contribution to the cut from a particle of helicity [MATH] circulating in the loop is given by [EQUATION] so that summing over the [MATH] matter multiplet gives [EQUATION]', '1601.03918-3-21-5': 'The [MATH]-factor can be written in several ways.', '1601.03918-3-21-6': 'The natural choices given [REF] and [REF] are [EQUATION] with the corresponding [MATH]-factors being [EQUATION] where [EQUATION]', '1601.03918-3-21-7': 'The cut is then [EQUATION]', '1601.03918-3-21-8': 'Using [EQUATION] and [EQUATION] we find [EQUATION] where [EQUATION]', '1601.03918-3-21-9': 'Now we may partial fraction the expression, i.e. use the identity [EQUATION] with [EQUATION] so that the cut is written as a sum of canonical forms: [EQUATION] which have corresponding canonical functions [EQUATION] and [EQUATION] where [EQUATION] and [MATH] is the set of [MATH] permutations of [MATH] necessary to generate symmetry in these variables.', '1601.03918-3-22-0': 'We obtain the coefficient [EQUATION]', '1601.03918-3-22-1': 'The corresponding three mass triangle integral function is [CITATION] [EQUATION] where [EQUATION]', '1601.03918-3-23-0': '# Bubbles', '1601.03918-3-24-0': 'The final cut constructible pieces are the bubbles integral functions [MATH].', '1601.03918-3-24-1': 'There are two distinct types of bubbles coefficients depending upon whether [MATH] is a two particle momentum, [MATH], or three particle, [MATH].', '1601.03918-3-24-2': 'In the three particle case the cut only involves MHV tree amplitudes whereas the two particle case requires the NMHV tree amplitude.', '1601.03918-3-25-0': 'For the [MATH] bubble the cut integrand with a scalar circulating in the loop is [EQUATION] where [EQUATION]', '1601.03918-3-25-1': 'As in the three-mass triangle, the contribution from a particle of helicity [MATH] is [MATH].', '1601.03918-3-25-2': 'Summing over the multiplet has the effect of multiplying by [MATH].', '1601.03918-3-25-3': 'For this cut [EQUATION]', '1601.03918-3-25-4': 'Multiplying [MATH] by [MATH] gives [EQUATION]', '1601.03918-3-25-5': 'Using [EQUATION] we find [EQUATION] where [EQUATION]', '1601.03918-3-25-6': 'Partial fractioning on [MATH] and [MATH] yields [EQUATION] where [EQUATION]', '1601.03918-3-25-7': 'The cut is now expressed in terms of the canonical form [EQUATION] which has the corresponding canonical function [EQUATION]', '1601.03918-3-25-8': 'The contribution of [REF] to the bubble coefficient is then [EQUATION] leading to the full bubble coefficient [EQUATION]', '1601.03918-3-25-9': 'The coefficient of the bubble [MATH] is obtained from the cut [EQUATION]', '1601.03918-3-25-10': 'This cut can also be decomposed in canonical forms.', '1601.03918-3-25-11': 'However the six-point NMHV tree amplitude in the cut is a sum of fourteen terms [CITATION] which leads to a lengthy expression for this bubble coefficient.', '1601.03918-3-25-12': 'Full expressions for the bubble coefficients of this type are given in appendix [REF].', '1601.03918-3-25-13': 'An explicit form of the bubble coefficients is available in Mathematica format at http://pyweb.swan.ac.uk/ dunbar/sixgraviton/R6.html.', '1601.03918-3-26-0': '# Cancellation Webs and Rational Descendants', '1601.03918-3-27-0': 'Although we may split the amplitude into cut-constructible pieces and rational terms, when we examine the singularities in the amplitude there is a mixing between the two which is important when we reconstruct [MATH] from its singularities.', '1601.03918-3-27-1': 'This has proven useful in the context of QCD amplitudes [CITATION].', '1601.03918-3-27-2': 'The cut-constructible pieces of the amplitude introduce a number of singularities that cannot be present in the full amplitude.', '1601.03918-3-27-3': 'These can be spurious singularities that occur at kinematic points where the full amplitude should be finite or higher order singularities occurring at points where the amplitude has a simple pole.', '1601.03918-3-27-4': 'If these poles are of sufficiently high order, they generate rational descendants.', '1601.03918-3-28-0': '## Higher Order Poles', '1601.03918-3-29-0': 'As an example of a higher order pole consider the behaviour of the one-mass box contribution [MATH] as [MATH].', '1601.03918-3-29-1': 'The box coefficient [MATH] in eq. ([REF]) contains a factor of [MATH] and the expansion of the box integral function around [MATH] is [CITATION] [EQUATION] where [EQUATION]', '1601.03918-3-29-2': 'As the cut constructible terms contain all of the logarithms and dilogarithms in the amplitude, the logarithmic pieces of this expansion must combine with the bubbles to give an effective coefficients that are linearly divergent as [MATH].', '1601.03918-3-29-3': 'We have confirmed numerically that the relevant cancellations between the one-mass box contributions and the bubble contributions occur.', '1601.03918-3-29-4': 'The quadratic divergence in the rational descendant, [MATH], must be cancelled by the rational piece of the amplitude.', '1601.03918-3-29-5': 'The full rational part of the amplitude will ultimately be obtained by recursion and one contribution to it will arise from this rational descendant if the shift excites the [MATH] pole.', '1601.03918-3-30-0': 'There are corresponding [MATH] poles in [MATH] which are obtained from those above by conjugation.', '1601.03918-3-31-0': '## [MATH] Spurious Singularity', '1601.03918-3-32-0': 'The coefficients of the two mass hard boxes have singularities of the form [MATH].', '1601.03918-3-32-1': 'These singularities also occur in the three mass triangle contributions: three powers of the pole are explicit in the leading term of the canonical form and a fourth arises in the partial fractioning that splits the cut integrand into canonical forms.', '1601.03918-3-33-0': 'In terms of kinematic variables, [MATH] corresponds to [MATH].', '1601.03918-3-33-1': 'The two mass hard integral functions depend on [MATH], [MATH], [MATH] and [MATH], while the three mass triangle depends on [MATH], [MATH] and [MATH] and there is the kinematic constraint: [MATH].', '1601.03918-3-33-2': 'For given [MATH], [MATH] and [MATH] it is possible find [MATH] and [MATH]) so that [MATH].', '1601.03918-3-33-3': 'Close to the pole the two mass hard box integral functions can be expanded as a series in terms of [MATH].', '1601.03918-3-33-4': 'In this context it is convenient to work with dimensionless box integral functions [MATH] defined by [EQUATION]', '1601.03918-3-33-5': 'The expansion of this dimensionless box function is [EQUATION] where [EQUATION] and [EQUATION] with [EQUATION]', '1601.03918-3-33-6': 'The same pole appears in the two boxes with integral function: [MATH] and [MATH].', '1601.03918-3-33-7': 'On the pole the coefficients of these two boxes are not equal and neither integral function vanishes.', '1601.03918-3-33-8': 'However, the sum of the dimensionless integral functions vanishes, i.e. [EQUATION]', '1601.03918-3-33-9': 'On this pole the dilogarithms in the individual boxes and triangles survive, but cancel between them.', '1601.03918-3-33-10': 'Setting [EQUATION] the integral functions are related by [EQUATION] where the sign ambiguity is associated with the choice of sign for [MATH].', '1601.03918-3-33-11': 'Schematically, expressing the box coefficients in terms of their sum and difference, [MATH], [MATH], the box and triangle contributions are [EQUATION]', '1601.03918-3-33-12': 'Expanding about [MATH], thanks to [REF] there is no dilogarithm component in the first term for any [MATH].', '1601.03918-3-33-13': 'However, we can only use [REF] if [MATH] and [MATH] are equal.', '1601.03918-3-33-14': 'In fact [EQUATION]', '1601.03918-3-33-15': 'Hence the dilogarithms vanish from any term that is singular as [MATH], as required by the factorisation theorems [CITATION].', '1601.03918-3-34-0': 'As in the one-mass case, there are subleading singularities at cubic order multiplying logarithms.', '1601.03918-3-34-1': 'These combine with the bubble contributions and cancel up to and including order [MATH], leaving no spurious singularity in the logarithms.', '1601.03918-3-35-0': 'The rational descendant of this combination of boxes and triangle contain both [MATH] and [MATH] singularities.', '1601.03918-3-35-1': 'Both of these singularities must be cancelled by the rational piece of the amplitude [MATH].', '1601.03918-3-35-2': 'As the expansion has been performed about a singularity specified in terms of [MATH], [MATH] and [MATH], there is no need to expand the three mass triangle integral function when determining this rational descendant.', '1601.03918-3-36-0': '## [MATH] Spurious Singularity', '1601.03918-3-37-0': 'The three mass triangle contributions have [MATH] poles which can be seen explicitly in the canonical forms.', '1601.03918-3-37-1': 'Around the [MATH] pole the integral function has the expansion [CITATION], [EQUATION] where [EQUATION]', '1601.03918-3-37-2': 'The logarithmic terms in this expansion combine with the bubble contributions to yield a finite contribution on the pole.', '1601.03918-3-37-3': 'The rational piece in the expansion must cancel with the rational part of the amplitude.', '1601.03918-3-38-0': '# Obtaining [MATH] By Recursion', '1601.03918-3-39-0': 'BCFW [CITATION] recursion applies complex analysis to amplitudes.', '1601.03918-3-39-1': "Using Cauchy's theorem, if a complex function is analytic except at simple poles [MATH] (all non-zero) and [MATH] as [MATH] then by considering the integral [EQUATION] where the contour [MATH] is the circle at infinity, we obtain [EQUATION]", '1601.03918-3-39-2': 'We wish to apply this with [MATH], where [MATH] has been complexified by a BCFW shift of momenta.', '1601.03918-3-39-3': 'Since [EQUATION] the singularities and residues of [MATH] are both those arising from the physical factorisations of [MATH] and those induced by the necessity to cancel the spurious singularities of the cut-constructible pieces.', '1601.03918-3-40-0': '## Choice of Shift', '1601.03918-3-41-0': 'The rational part of an amplitude can be obtained recursively if the factorisation properties of the amplitude are understood at all of the relevant poles.', '1601.03918-3-41-1': 'There are three main obstacles to this: quadratic poles in the amplitude, non-standard factorisations for complex momenta and contributions for large shifted momenta.', '1601.03918-3-41-2': 'Quadratic poles in the amplitude lead to recursive contributions that depend on the off-shell behaviour of the factorised currents.', '1601.03918-3-41-3': 'This can be addressed using augmented recursion [CITATION].', '1601.03918-3-41-4': 'For non-supersymmetric theories there are double poles of the form [EQUATION]', '1601.03918-3-41-5': 'For the six-point NMHV amplitude the tree amplitude vanishes since it has a single positive helicity leg.', '1601.03918-3-41-6': '(This is no longer the case for seven and higher point NMHV amplitudes.)', '1601.03918-3-42-0': 'Non-standard factorisations for complex momenta are unavoidable and are considered in detail below.', '1601.03918-3-42-1': 'The final obstacle is the possibility of contributions from asymptotically large shifted momenta.', '1601.03918-3-42-2': "The amplitude doesn't factorise in this limit, so the residue is undetermined.", '1601.03918-3-42-3': 'This issue may be avoided if the shift employed causes the amplitude to vanish for asymptotically large shifted momenta.', '1601.03918-3-42-4': 'As the amplitude is as yet undetermined, its behaviour under any shift is unknown.', '1601.03918-3-42-5': "However, if the cut constructible pieces don't vanish for asymptotically large shifted momenta there is little hope that the rational pieces would.", '1601.03918-3-43-0': 'For example under a shift involving two negative helicity legs, [EQUATION] the cut constructible pieces of the amplitude are divergent for large [MATH].', '1601.03918-3-44-0': 'However, for a shift involving one negative helicity leg and one positive helicity leg, [EQUATION] the cut-constructible pieces all vanish at large [MATH], at least suggesting that the rational piece is also well behaved there.', '1601.03918-3-44-1': 'The shift [REF] will be used to obtain [MATH].', '1601.03918-3-45-0': 'The contributions to [MATH] can be grouped into three classes: standard factorisations, non-factorising contributions and rational descendants of the cut-constructible pieces : [EQUATION]', '1601.03918-3-46-0': '## Standard Factorisations', '1601.03918-3-47-0': 'The standard factorisations of a six-point one-loop amplitude have the forms: [EQUATION]', '1601.03918-3-47-1': 'In a supersymmetric theory the 3-point loop amplitudes vanish and so the third class are absent in this case.', '1601.03918-3-47-2': 'With the shift [REF] the factorisations of the first type are: [EQUATION]', '1601.03918-3-47-3': 'While the factorisations of the second type are: [EQUATION]', '1601.03918-3-47-4': 'For generic six-point kinematics, the kinematic points at which the 4- and 5-point loop amplitudes appearing in these factorisations are evaluated are in no way special, hence the rational contribution to the residue comes solely from the rational part of the 4- and 5-point loop amplitudes.', '1601.03918-3-47-5': 'Each factorisation therefore gives a contribution to [MATH] of [EQUATION] with [CITATION] [EQUATION] where [EQUATION] and [MATH] denotes a sum over the six distinct permutations of [MATH] and [MATH] noting the symmetry of [MATH] under [MATH].', '1601.03918-3-47-6': 'The full contribution of the standard factorisations is then [EQUATION] where the sum is over all of the standard factorisation channels given in [REF] and [REF].', '1601.03918-3-48-0': '## Contribution Of Rational Descendants', '1601.03918-3-49-0': 'As discussed above, higher order poles in the coefficients of the box and triangle contributions to the amplitude can generate rational descendants when those poles are excited.', '1601.03918-3-49-1': 'The shift [REF] excites some poles of each type.', '1601.03918-3-49-2': 'Specifically we have the various singularities listed in table [REF] (with [MATH], [MATH]).', '1601.03918-3-50-0': 'Denoting the rational descendant in each case by [MATH], the corresponding coefficient by [MATH] and the value of [MATH] on the pole by [MATH], the contribution on each of these poles is [EQUATION] so that [EQUATION] where the sum is over all of the poles listed above.', '1601.03918-3-51-0': 'Individual terms in the bubble coefficients contain a range of other higher order poles.', '1601.03918-3-51-1': 'In principle these could also generate rational descendants, however in the full bubble coefficients these are at most simple poles and so do not generate further rational descendants: [EQUATION]', '1601.03918-3-52-0': '## Non-standard Factorisations for Complex Momenta', '1601.03918-3-53-0': 'Factorisations of the amplitude occur when propagators go on shell.', '1601.03918-3-53-1': 'The standard factorisation channels arise when the on-shell propagator is not in the loop and is explicit in, for example a Feynman diagram approach.', '1601.03918-3-54-0': 'The loop momentum integral may also generate poles in the amplitude [CITATION] particularly for complex momenta.', '1601.03918-3-54-1': 'Since we are computing the amplitude by recursion in complex momenta we must determine these complex factorisations.', '1601.03918-3-54-2': '2.0truecm', '1601.03918-3-55-0': 'Poles can arise when two adjacent massless legs on a loop became collinear as illustrated in fig. [REF].', '1601.03918-3-55-1': 'This case has been discussed in the context of amplitudes with a single negative helicity leg [CITATION].', '1601.03918-3-55-2': 'In the integration region [MATH] the three propagators connected to [MATH] and [MATH] all become on shell when [MATH] and [MATH] are collinear.', '1601.03918-3-55-3': 'The diagrams of interest can be grouped together to form a one mass triangle in the integral reduction sense (i.e. the massive corner represents a sum of all possible tree diagrams).', '1601.03918-3-55-4': 'The integration region of interest has all three propagators on shell and so the pole may be determined by the triple cut of this triangle.', '1601.03918-3-55-5': "This triple cut wouldn't normally exist, but opens up when [MATH] and [MATH] are collinear.", '1601.03918-3-56-0': 'Using an axial gauge with reference spinor [MATH] [CITATION], the contribution of fig. [REF] with a scalar particle circulating in the loop is [EQUATION] where [MATH] represents the sum of all possible tree diagrams.', '1601.03918-3-56-1': '[MATH] and [MATH] are given by [EQUATION] and satisfy [MATH].', '1601.03918-3-56-2': 'The integrands for other particle types in the loop are related to the scalar contribution by an [MATH]-factor: [EQUATION] and the [MATH] contribution is obtained by multiplying the integrand by a factor of [MATH] where, [EQUATION]', '1601.03918-3-56-3': 'For [MATH] the integrand of [REF] contains [MATH] factors in its numerator, leaving a pole in [MATH] in the integral.', '1601.03918-3-56-4': 'As [MATH] is finite as [MATH], [MATH] in the region of interest.', '1601.03918-3-56-5': 'For a scalar particle circulating in the loop the KLT relations [CITATION] give [EQUATION]', '1601.03918-3-56-6': "Of the six terms in the permutation sum in [REF], the two which don't permute [MATH] can be neglected due to the explicit [MATH] factor.", '1601.03918-3-56-7': 'The remaining four form two pairs with the members of each pair being related by interchange of legs [MATH] and [MATH].', '1601.03918-3-56-8': 'The [MATH] contributions of one member of each of these pairs are [EQUATION] and [EQUATION]', '1601.03918-3-56-9': 'Partial fractioning the integrand of [MATH] using the [MATH] factor in the numerator yields six terms whose integrands have loop momentum dependence [EQUATION] with [MATH] and [MATH].', '1601.03918-3-56-10': 'In the integration region of interest [MATH] and [MATH] are negligible allowing the integrands to be rewritten as quartic pentagon integrands [EQUATION]', '1601.03918-3-56-11': 'For [MATH], using [EQUATION] splits each of these quartic pentagons into a pair of cubic one-mass boxes and a cubic pentagon which can be neglected.', '1601.03918-3-56-12': 'As a box with two adjacent corners attached to single external legs of the same helicity has a vanishing quadruple cut, these cubic one-mass box integrals reduce to bubble and rational contributions only.', '1601.03918-3-56-13': 'The bubble coefficients can be evaluated by direct parametrisation.', '1601.03918-3-56-14': 'For example the box integral [REF] which is illustrated in fig. [REF] has bubbles associated with its [MATH] and [MATH] cuts.', '1601.03918-3-56-15': 'The [MATH] cut of [REF] gives the bubble coefficient [EQUATION] where terms of order [MATH] have been extracted from the leading term to simplify its denominator as far as possible.', '1601.03918-3-56-16': 'The remaining [MATH] singularity in this bubble coefficient is spurious and must cancel with the [MATH] bubble as [MATH].', '1601.03918-3-56-17': 'So that this singularity is not present in the logarithmic part of the integral, the sum of the two bubble coefficients must be finite.', '1601.03918-3-56-18': 'The sum of the two bubble contributions then involves the singular parts of the [MATH] bubble coefficient multiplied by the difference of the integral functions of the two bubbles.', '1601.03918-3-56-19': 'With [MATH] and [MATH] both being small, the rational descendant of the bubbles on the [MATH] pole is [EQUATION]', '1601.03918-3-56-20': 'The leading term of the rational descendant has a [MATH] spurious pole.', '1601.03918-3-56-21': 'This must be cancelled by the rational piece of the integral, allowing the rational term to be identified as, [EQUATION]', '1601.03918-3-56-22': 'For [MATH] in [REF] the reduction to boxes uses the identity [EQUATION] which yields a pair of quartic box integrals whose rational pieces are evaluated using the approach described above.', '1601.03918-3-56-23': 'The full rational contribution of [MATH] is [EQUATION] where [EQUATION]', '1601.03918-3-56-24': 'The [MATH] contributions involve both quintic and quartic pentagon integrals, but their rational pieces can be obtained in a similar fashion to the [MATH] contributions.', '1601.03918-3-56-25': 'Separating the quintic and quartic pentagon integrals, [EQUATION] where [EQUATION] with [EQUATION] and [EQUATION] with [EQUATION]', '1601.03918-3-56-26': 'The contribution of these non-standard factorisations to the rational part of the 6-pt amplitude is obtained by recursion: [EQUATION]', '1601.03918-3-56-27': 'The contributions arising from the conjugate poles, e.g. [MATH], can be obtained using the flip-conjugation symmetry of the amplitude.', '1601.03918-3-56-28': 'Defining [EQUATION] the full non-factorising contribution to [MATH] is [EQUATION]', '1601.03918-3-56-29': 'We have computed [MATH] systematically using its pole structure.', '1601.03918-3-56-30': 'Underlying this is the assumption that the amplitude vanishes for large shifts.', '1601.03918-3-56-31': 'This is difficult to justify a priori.', '1601.03918-3-56-32': 'However the expression obtained has the correct symmetries and collinear limits (checked numerically).', '1601.03918-3-56-33': 'Generically a BCFW recursion produces terms which are not manifestly symmetric and the restoration of symmetry is a good indicator that the amplitude has been correctly determined.', '1601.03918-3-57-0': 'An explicit form of [MATH] is available in Mathematica format at http://pyweb.swan.ac.uk/ dunbar/sixgraviton/R6.html.', '1601.03918-3-58-0': '# Conclusions', '1601.03918-3-59-0': 'Graviton scattering amplitudes have a rich structure.', '1601.03918-3-59-1': 'In particular [MATH] supergravity has proven to have a much softer UV behaviour then previously expected with the underlying symmetry reason still unclear.', '1601.03918-3-59-2': 'It is important to understand which structures of [MATH] survive in theories with lower supersymmetry.', '1601.03918-3-59-3': 'It is also important to study amplitudes beyond MHV since this can often have a misleadingly simple structure.', '1601.03918-3-59-4': 'In this article we have constructed the six-point NMHV amplitude in [MATH] supergravity.', '1601.03918-3-59-5': 'Of particular interest is the rational term since in the MHV case a particularly simple and suggestive structure appears [CITATION].', '1601.03918-3-59-6': 'The rational terms in the NMHV case do not appear to have any such simple structure although this may be hiding given the algebraic complexity of the amplitude.', '1601.03918-3-60-0': 'Computing the rational terms has required a blending of techniques including obtaining the rational descendants of the cut-constructible pieces.', '1601.03918-3-60-1': 'Amongst the cut-constructible pieces the coefficients of the bubble integral functions have been particularly cumbersome although, fortunately, these do not generate any rational descendants in this amplitude.'}

1806.07309

{'1806.07309-1-0-0': 'The amount of available videos in the Web has significantly increased not only for entertainment etc., but also to convey educational or scientific information in an effective way.', '1806.07309-1-0-1': 'There are several web portals that offer access to the latter kind of video material.', '1806.07309-1-0-2': 'One of them is the TIB AV-Portal of the Leibniz Information Centre for Science and Technology (TIB), which hosts scientific and educational video content.', '1806.07309-1-0-3': 'In contrast to other video portals, automatic audiovisual analysis (visual concept classification, optical character recognition, speech recognition) is utilized to enhance metadata information and semantic search.', '1806.07309-1-0-4': 'In this paper, we propose to further exploit and enrich this automatically generated information by linking it to the Integrated Authority File (GND) of the German National Library.', '1806.07309-1-0-5': 'This information is used to derive a measure to compare the similarity of two videos which serves as a basis for recommending semantically similar videos.', '1806.07309-1-0-6': 'A user study demonstrates the feasibility of the proposed approach.', '1806.07309-1-1-0': '# Introduction', '1806.07309-1-2-0': 'Videos hold a great potential to communicate educational and scientific information.', '1806.07309-1-2-1': 'This is, for instance, reflected by e-Learning platforms such as Udacity (https://udacity.com) or Coursera (http://www.coursera.org).', '1806.07309-1-2-2': 'Another type of Web portals offers also access to scientific videos, one of them is the TIB AV-Portal (https://av.tib.eu) of the Leibniz Information Centre for Science and Technology (TIB).', '1806.07309-1-2-3': 'Researchers can provide, search, and access scientific and educational audiovisual material, while benefiting from a number of advantages compared to other portals.', '1806.07309-1-2-4': 'First, submitted videos are reviewed to check whether they contain scientific or educational content.', '1806.07309-1-2-5': 'Second, videos are represented in a persistent way using DOIs (digital object identifier), potentially even at the segment and frame level, making it easy and reliable to reference them.', '1806.07309-1-2-6': 'Finally, audiovisual content analysis is applied in order to allow the user to not only search for terms in descriptive metadata (e.g., title, manually annotated keywords), but also in the audiovisual content, i.e., in the speech transcript, in the recognized ovrelaid or scene text through video OCR (optical character recognition), and keywords derived from visual concept and scene classification.', '1806.07309-1-3-0': 'Usually, recommender systems in online shopping platforms or video portals mainly rely on user-based information such as the viewing history [CITATION] or current trends [CITATION].', '1806.07309-1-3-1': 'In this paper, we investigate the question how similar videos can be recommended based on their metadata, in particular, by additionally making use of automatically extracted metadata from audiovisual content analysis.', '1806.07309-1-3-2': 'This is relevant, for example, when users do not agree to track their search behavior or sufficient amount of user data is not available.', '1806.07309-1-3-3': 'Particularly, we propose to further exploit and enrich the entire set of available metadata, be it created manually or extracted automatically, in order to improve recommendations of semantically similar videos.', '1806.07309-1-3-4': 'In a first step, we utilize a Word2Vec approach [CITATION] to make the semantic content of two videos comparable based on title, tags, and abstract.', '1806.07309-1-3-5': 'Then, the automatically extracted metadata about the audiovisual content is enriched by linking it to the Integrated Authority File (GND: Gemeinsame Normdatei) of the German National Library (DNB: Deutsche Nationalbibliothek).', '1806.07309-1-3-6': 'These two kinds of information are used to derive a measure to compare the content of two videos which serves as a basis for recommending similar video.', '1806.07309-1-3-7': 'A user study demonstrates the feasibility of the proposed approach.', '1806.07309-1-4-0': 'The paper is structured as follows.', '1806.07309-1-4-1': 'First, we give a brief overview of related work in Section 2.', '1806.07309-1-4-2': 'The proposed approach to generate video recommendations is presented in Section 3.', '1806.07309-1-4-3': 'Section 4 describes the conducted user study to evaluate the proposed approach, while Section 5 concludes the paper.', '1806.07309-1-5-0': '# Related Work', '1806.07309-1-6-0': 'Scientific Video Portals: Yovisto is a scientific video portal that allows the user to search for information via text-based metadata [CITATION].', '1806.07309-1-6-1': 'The users can reduce the number of search results by refining their query via additional criteria and grouping videos by language, organisation, or category.', '1806.07309-1-6-2': 'On the contrary, to increase the scope of possible results, a tool for explorative search reveals interrelations between different types of videos in order to present a broader spectrum of results to the user.', '1806.07309-1-6-3': 'This is done by exploiting an ontology structure, which is part of every video element and Linked Open Data (LOD) resources, namely DBpedia (http://wiki.dbpedia.org).', '1806.07309-1-6-4': 'Another similar portal is described by Marchionini [CITATION], where the uploaded content is automatically fed into an automatic data analysis chain.', '1806.07309-1-6-5': 'Semantic entities are automatically assigned to each video segment resulting in a storyboard comprising the video content.', '1806.07309-1-6-6': 'In contrast to the AV-Portal, this information is hidden from the user.', '1806.07309-1-6-7': "Marchionini's approach focuses on providing a good explorative search tool, i.e., a user should be able to find what s/he is looking for even when being unsure about the correct phrasing.", '1806.07309-1-6-8': 'Recommendation Systems for Scientific Videos: Clustering semantically similar videos is a possible approach to provide video recommendations based on a given, currently watched video.', '1806.07309-1-6-9': 'A fundamental problem of this research is the semantic gap between low-level features and high-level semantics portrayed in visual content [CITATION].', '1806.07309-1-6-10': 'To circumvent this problem, textual cues can be used in addition to visual content.', '1806.07309-1-6-11': 'These can be manually added tags by the author of the video or automatically extracted keywords by machine learning algorithms.', '1806.07309-1-6-12': 'Either way, they are often superficial, noisy, incomplete or ambiguous which makes the process of clustering a challenge.', '1806.07309-1-6-13': 'Vahdat et al. [CITATION] enrich the set of tags by modelling them from visual features and correct the existing ones by checking their agreement with the visual content.', '1806.07309-1-6-14': 'They are able to show that this method outperforms existing ones that use either modality and even the naive combination.', '1806.07309-1-6-15': 'Wang et al. [CITATION] discover that by incorporating hierarchical information - instead of considering a "flat" tag taxonomy - the semantics of a video can be described even better.', '1806.07309-1-6-16': 'Despite only using two levels of abstraction in their hierarchical multi-label random forest model, strong correlations between ambiguous visual features and sparse, incomplete tags could be found.', '1806.07309-1-7-0': '# Enriching Video Metadata through Linked Open Data', '1806.07309-1-8-0': 'In this section, we present our approach to enrich metadata with open data sources.', '1806.07309-1-8-1': 'First, the set of available metadata is described before the acquisition of additional information from an open data source is explained in Section [REF].', '1806.07309-1-8-2': 'Second, a similaritiy measure to compare videos based on a Word2Vec representation and enriched metadata is derived in Section [REF].', '1806.07309-1-8-3': 'The overall workflow is displayed in Figure [REF].', '1806.07309-1-8-4': 'The input of our system consists of manually generated and automatically extracted information, where the former comprises abstract and title.', '1806.07309-1-8-5': 'Additional inputs are the following automatically extracted Tags (see Figure [REF]) derived from: 1) Transcript based on speech recognition, 2) Results of video OCR, and 3) results of visual concept and scene classification.', '1806.07309-1-8-6': 'All of them have a representation in the German National Library, which is the key requirement for the enrichment process.', '1806.07309-1-9-0': '## Acquiring Additional Information from Open Data Source', '1806.07309-1-10-0': 'Automatically generated tags usually contain a certain amount of errors and noise.', '1806.07309-1-10-1': 'Although state-of-the-art algorithms can achieve human performance [CITATION] in specific tasks and settings, issues with audio quality in lecture rooms or hardly legible handwritings can cause errors.', '1806.07309-1-10-2': 'We try to circumvent this problem by evaluating additional information provided by the German National Library.', '1806.07309-1-10-3': 'Besides information such as synonyms and related scientific publications, the Dewey Decimal Classification (DDC) for every tag is provided.', '1806.07309-1-10-4': 'The DDC is a library classification system, which categorizes technical terms into ten classes via three-digit arabic numerals [CITATION].', '1806.07309-1-10-5': 'These main classes are then further divided into subcategories denoted by the decimals after these three digits, where additional decimals depict a more specific subject.', '1806.07309-1-10-6': 'For instance, SPARQL is contained in 006.74 - Markup Language, 005.74 - Data files and Databases and 005.133 - Individual Programming Languages, which yields valuable contextual information.', '1806.07309-1-11-0': '## Defining a Similarity Measure for Scientific Videos', '1806.07309-1-12-0': 'Simply comparing two videos for mutual tags is not sufficient to determine semantic similarity.', '1806.07309-1-12-1': 'Even if two sets of tags have little to no overlap they might be highly correlated when their context is considered.', '1806.07309-1-12-2': 'We address this issue by utilizing fastText [CITATION] to generate word embeddings, which has several advantages for this task.', '1806.07309-1-12-3': 'First, semantically similar words are modeled closer to one another so that a simple distance measure indicates the correlation of two words.', '1806.07309-1-12-4': 'Second, since fastText works on substrings rather than whole words it is able to produce valuable features even for misspelled or words unknown to the word embedding.', '1806.07309-1-12-5': 'Finally, a pre-trained model is available for a large number of languages.', '1806.07309-1-12-6': 'Title, tags, and abstract are taken from the metadata and processed via fastText.', '1806.07309-1-12-7': 'It generates a 300-dimensional feature vector for every word in the metadata.', '1806.07309-1-12-8': 'The average of these vectors is our representation for a particular video.', '1806.07309-1-12-9': 'This approach is our baseline and denoted as method without LOD in the sequel.', '1806.07309-1-13-0': 'The improvement of this already powerful feature extraction method is the main contribution of this paper.', '1806.07309-1-13-1': 'It is achieved by incorporating the information provided by the DDC notation in addition to the fastText embeddings.', '1806.07309-1-13-2': 'As a preprocessing step we need to create a vector [MATH], which consists of all DDC tags that occur in our dataset and which will be assigned to every video entry [MATH].', '1806.07309-1-13-3': 'Since the upper level classes of the notation are also encoded in the codes of the classes at lower levels, we divide them accordingly.', '1806.07309-1-13-4': 'Therefore, the length of [MATH] equals the total number of these tag fragments.', '1806.07309-1-13-5': 'For instance, if the video corpus would only contain the tags 005.74 and 005.133, we would split them into [MATH] (indices mark the level in the hierarchy) resulting in a vector [MATH] of length [MATH].', '1806.07309-1-13-6': 'If a particular tag fragment occurs in a video, we set the corresponding bin in [MATH] to the term frequency - inverse document frequency (tf-idf), or zero otherwise.', '1806.07309-1-13-7': 'This assures that the more specific, and therefore more informative, DDC classes have more influence on the result.', '1806.07309-1-13-8': 'For example, if two tags share the main DDC class Science and Mathematics, it does not mean that they are necessarily closely correlated, but if both share the class Data Compression they most likely cover a similar topic.', '1806.07309-1-13-9': 'For the "method with LOD" the two vectors [MATH] and [MATH] of video [MATH] and [MATH] are compared via cosine similarity.', '1806.07309-1-13-10': 'It is important to note that this method also uses the fastText features of the method without LOD.', '1806.07309-1-13-11': 'In order to compute the overall similarity, both methods are applied and the average is used to form [MATH] (see Figure [REF]).', '1806.07309-1-14-0': '# Experimental Results', '1806.07309-1-15-0': 'Videos of the TIB AV-Portal were used in the experiment.', '1806.07309-1-15-1': 'The complete stock of metadata that falls under the Creative Commons License CC0 1.0 Universal is made available by the TIB (https://av.tib.eu/opendata) as Resource Description Framework (RDF) triples.', '1806.07309-1-15-2': 'To extract the necessary annotations we utilized SPARQL.', '1806.07309-1-15-3': 'In a first step, it was necessary to keep only videos that allowed "derivate works" in addition to the CC0 1.0 license, since content analysis is applied.', '1806.07309-1-15-4': '[MATH] samples satisfied these conditions.', '1806.07309-1-15-5': 'Unfortunately, word embeddings of two different languages cannot be directly compared forcing us to use a subset of videos with the same language (German in this case, 1430 videos).', '1806.07309-1-15-6': 'Annotations are represented in JSON format to make them easily accessible for future tasks without rebuilding the RDF graph.', '1806.07309-1-15-7': 'After gathering all tags of an entry, we employed another SPARQL query assigning a GND (German: Gemeinsame Normdatei, English: Integrated Authority File) link to each tag, which is the key part of linking it to the data of the German National Library (DNB) and retrieving the corresponding DDC notations.', '1806.07309-1-16-0': 'We evaluated the quality of our similarity measure by conducting a user study with eight participants, five men and three women.', '1806.07309-1-16-1': 'A random selection of 50 videos was presented to every participant along with ten video recommendations, randomly either completely provided by the method without LOD or the method with LOD.', '1806.07309-1-16-2': 'The results were integrated by a Greasemonkey script in the Firefox browser.', '1806.07309-1-16-3': 'Every participant had to rate each of the ten recommendations from [MATH], i.e., 0: not relevant; 1: low relevance; 2: medium relevance; 3: highly relevant.', '1806.07309-1-16-4': 'The results are displayed in Figure [REF].', '1806.07309-1-17-0': 'The results show that the method with LOD increases the number of video recommendations with medium ([MATH]) and low relevance ([MATH]), while the effect is small ([MATH]) for the highly relevant recommendations.', '1806.07309-1-17-1': 'However, the method with LOD significantly decreases the number of irrelevant recommendations (by [MATH]).', '1806.07309-1-17-2': 'This indicates that this method is superior to the text-based method, most likely due to the hierarchical nature of the DDC notation.', '1806.07309-1-17-3': 'We assume that the rather small improvement for the very relevant recommendations is a result of the restrictions we had to oblige to (license and language), i.e., the relatively small set of remaining videos ([MATH]) does not contain more highly relevant samples.', '1806.07309-1-17-4': 'A chi-square test shows that the method with LOD is significantly better than our baseline (Chi-Square=[MATH], p-value=[MATH]).', '1806.07309-1-18-0': '# Conclusions', '1806.07309-1-19-0': 'In this paper, we have proposed a method to generate recommendations for scientific videos based on noisy, automatically extracted tags by utilizing linked open data to weave in hierarchical semantic metadata.', '1806.07309-1-19-1': 'This enables users to find relevant information more quickly improving their overall learning experience.', '1806.07309-1-19-2': 'In future work, we plan to incorporate recommendations for scientific papers or definitions of technical terms through linked open data.', '1806.07309-1-20-0': 'compat=1.14 compatibility=false', '1806.07309-1-21-0': 'Recommending Scientific Videos based on Metadata Enrichment using Linked Open Data Recommending Scientific Videos using Linked Open Data', '1806.07309-1-22-0': '# Abstract The amount of available videos in the Web has significantly increased not only for entertainment etc., but also to convey educational or scientific information in an effective way.', '1806.07309-1-22-1': 'There are several web portals that offer access to the latter kind of video material.', '1806.07309-1-22-2': 'One of them is the TIB AV-Portal of the Leibniz Information Centre for Science and Technology (TIB), which hosts scientific and educational video content.', '1806.07309-1-22-3': 'In contrast to other video portals, automatic audiovisual analysis (visual concept classification, optical character recognition, speech recognition) is utilized to enhance metadata information and semantic search.', '1806.07309-1-22-4': 'In this paper, we propose to further exploit and enrich this automatically generated information by linking it to the Integrated Authority File (GND) of the German National Library.', '1806.07309-1-22-5': 'This information is used to derive a measure to compare the similarity of two videos which serves as a basis for recommending semantically similar videos.', '1806.07309-1-22-6': 'A user study demonstrates the feasibility of the proposed approach.', '1806.07309-1-23-0': '# Introduction', '1806.07309-1-24-0': 'Videos hold a great potential to communicate educational and scientific information.', '1806.07309-1-24-1': 'This is, for instance, reflected by e-Learning platforms such as Udacity (https://udacity.com) or Coursera (http://www.coursera.org).', '1806.07309-1-24-2': 'Another type of Web portals offers also access to scientific videos, one of them is the TIB AV-Portal (https://av.tib.eu) of the Leibniz Information Centre for Science and Technology (TIB).', '1806.07309-1-24-3': 'Researchers can provide, search, and access scientific and educational audiovisual material, while benefiting from a number of advantages compared to other portals.', '1806.07309-1-24-4': 'First, submitted videos are reviewed to check whether they contain scientific or educational content.', '1806.07309-1-24-5': 'Second, videos are represented in a persistent way using DOIs (digital object identifier), potentially even at the segment and frame level, making it easy and reliable to reference them.', '1806.07309-1-24-6': 'Finally, audiovisual content analysis is applied in order to allow the user to not only search for terms in descriptive metadata (e.g., title, manually annotated keywords), but also in the audiovisual content, i.e., in the speech transcript, in the recognized ovrelaid or scene text through video OCR (optical character recognition), and keywords derived from visual concept and scene classification.', '1806.07309-1-25-0': 'Usually, recommender systems in online shopping platforms or video portals mainly rely on user-based information such as the viewing history [CITATION] or current trends [CITATION].', '1806.07309-1-25-1': 'In this paper, we investigate the question how similar videos can be recommended based on their metadata, in particular, by additionally making use of automatically extracted metadata from audiovisual content analysis.', '1806.07309-1-25-2': 'This is relevant, for example, when users do not agree to track their search behavior or sufficient amount of user data is not available.', '1806.07309-1-25-3': 'Particularly, we propose to further exploit and enrich the entire set of available metadata, be it created manually or extracted automatically, in order to improve recommendations of semantically similar videos.', '1806.07309-1-25-4': 'In a first step, we utilize a Word2Vec approach [CITATION] to make the semantic content of two videos comparable based on title, tags, and abstract.', '1806.07309-1-25-5': 'Then, the automatically extracted metadata about the audiovisual content is enriched by linking it to the Integrated Authority File (GND: Gemeinsame Normdatei) of the German National Library (DNB: Deutsche Nationalbibliothek).', '1806.07309-1-25-6': 'These two kinds of information are used to derive a measure to compare the content of two videos which serves as a basis for recommending similar video.', '1806.07309-1-25-7': 'A user study demonstrates the feasibility of the proposed approach.', '1806.07309-1-26-0': 'The paper is structured as follows.', '1806.07309-1-26-1': 'First, we give a brief overview of related work in Section 2.', '1806.07309-1-26-2': 'The proposed approach to generate video recommendations is presented in Section 3.', '1806.07309-1-26-3': 'Section 4 describes the conducted user study to evaluate the proposed approach, while Section 5 concludes the paper.', '1806.07309-1-27-0': '# Related Work', '1806.07309-1-28-0': 'Scientific Video Portals: Yovisto is a scientific video portal that allows the user to search for information via text-based metadata [CITATION].', '1806.07309-1-28-1': 'The users can reduce the number of search results by refining their query via additional criteria and grouping videos by language, organisation, or category.', '1806.07309-1-28-2': 'On the contrary, to increase the scope of possible results, a tool for explorative search reveals interrelations between different types of videos in order to present a broader spectrum of results to the user.', '1806.07309-1-28-3': 'This is done by exploiting an ontology structure, which is part of every video element and Linked Open Data (LOD) resources, namely DBpedia (http://wiki.dbpedia.org).', '1806.07309-1-28-4': 'Another similar portal is described by Marchionini [CITATION], where the uploaded content is automatically fed into an automatic data analysis chain.', '1806.07309-1-28-5': 'Semantic entities are automatically assigned to each video segment resulting in a storyboard comprising the video content.', '1806.07309-1-28-6': 'In contrast to the AV-Portal, this information is hidden from the user.', '1806.07309-1-28-7': "Marchionini's approach focuses on providing a good explorative search tool, i.e., a user should be able to find what s/he is looking for even when being unsure about the correct phrasing.", '1806.07309-1-28-8': 'Recommendation Systems for Scientific Videos: Clustering semantically similar videos is a possible approach to provide video recommendations based on a given, currently watched video.', '1806.07309-1-28-9': 'A fundamental problem of this research is the semantic gap between low-level features and high-level semantics portrayed in visual content [CITATION].', '1806.07309-1-28-10': 'To circumvent this problem, textual cues can be used in addition to visual content.', '1806.07309-1-28-11': 'These can be manually added tags by the author of the video or automatically extracted keywords by machine learning algorithms.', '1806.07309-1-28-12': 'Either way, they are often superficial, noisy, incomplete or ambiguous which makes the process of clustering a challenge.', '1806.07309-1-28-13': 'Vahdat et al. [CITATION] enrich the set of tags by modelling them from visual features and correct the existing ones by checking their agreement with the visual content.', '1806.07309-1-28-14': 'They are able to show that this method outperforms existing ones that use either modality and even the naive combination.', '1806.07309-1-28-15': 'Wang et al. [CITATION] discover that by incorporating hierarchical information - instead of considering a "flat" tag taxonomy - the semantics of a video can be described even better.', '1806.07309-1-28-16': 'Despite only using two levels of abstraction in their hierarchical multi-label random forest model, strong correlations between ambiguous visual features and sparse, incomplete tags could be found.', '1806.07309-1-29-0': '# Enriching Video Metadata through Linked Open Data', '1806.07309-1-30-0': 'In this section, we present our approach to enrich metadata with open data sources.', '1806.07309-1-30-1': 'First, the set of available metadata is described before the acquisition of additional information from an open data source is explained in Section [REF].', '1806.07309-1-30-2': 'Second, a similaritiy measure to compare videos based on a Word2Vec representation and enriched metadata is derived in Section [REF].', '1806.07309-1-30-3': 'The overall workflow is displayed in Figure [REF].', '1806.07309-1-30-4': 'The input of our system consists of manually generated and automatically extracted information, where the former comprises abstract and title.', '1806.07309-1-30-5': 'Additional inputs are the following automatically extracted Tags (see Figure [REF]) derived from: 1) Transcript based on speech recognition, 2) Results of video OCR, and 3) results of visual concept and scene classification.', '1806.07309-1-30-6': 'All of them have a representation in the German National Library, which is the key requirement for the enrichment process.', '1806.07309-1-31-0': '## Acquiring Additional Information from Open Data Source', '1806.07309-1-32-0': 'Automatically generated tags usually contain a certain amount of errors and noise.', '1806.07309-1-32-1': 'Although state-of-the-art algorithms can achieve human performance [CITATION] in specific tasks and settings, issues with audio quality in lecture rooms or hardly legible handwritings can cause errors.', '1806.07309-1-32-2': 'We try to circumvent this problem by evaluating additional information provided by the German National Library.', '1806.07309-1-32-3': 'Besides information such as synonyms and related scientific publications, the Dewey Decimal Classification (DDC) for every tag is provided.', '1806.07309-1-32-4': 'The DDC is a library classification system, which categorizes technical terms into ten classes via three-digit arabic numerals [CITATION].', '1806.07309-1-32-5': 'These main classes are then further divided into subcategories denoted by the decimals after these three digits, where additional decimals depict a more specific subject.', '1806.07309-1-32-6': 'For instance, SPARQL is contained in 006.74 - Markup Language, 005.74 - Data files and Databases and 005.133 - Individual Programming Languages, which yields valuable contextual information.', '1806.07309-1-33-0': '## Defining a Similarity Measure for Scientific Videos', '1806.07309-1-34-0': 'Simply comparing two videos for mutual tags is not sufficient to determine semantic similarity.', '1806.07309-1-34-1': 'Even if two sets of tags have little to no overlap they might be highly correlated when their context is considered.', '1806.07309-1-34-2': 'We address this issue by utilizing fastText [CITATION] to generate word embeddings, which has several advantages for this task.', '1806.07309-1-34-3': 'First, semantically similar words are modeled closer to one another so that a simple distance measure indicates the correlation of two words.', '1806.07309-1-34-4': 'Second, since fastText works on substrings rather than whole words it is able to produce valuable features even for misspelled or words unknown to the word embedding.', '1806.07309-1-34-5': 'Finally, a pre-trained model is available for a large number of languages.', '1806.07309-1-34-6': 'Title, tags, and abstract are taken from the metadata and processed via fastText.', '1806.07309-1-34-7': 'It generates a 300-dimensional feature vector for every word in the metadata.', '1806.07309-1-34-8': 'The average of these vectors is our representation for a particular video.', '1806.07309-1-34-9': 'This approach is our baseline and denoted as method without LOD in the sequel.', '1806.07309-1-35-0': 'The improvement of this already powerful feature extraction method is the main contribution of this paper.', '1806.07309-1-35-1': 'It is achieved by incorporating the information provided by the DDC notation in addition to the fastText embeddings.', '1806.07309-1-35-2': 'As a preprocessing step we need to create a vector [MATH], which consists of all DDC tags that occur in our dataset and which will be assigned to every video entry [MATH].', '1806.07309-1-35-3': 'Since the upper level classes of the notation are also encoded in the codes of the classes at lower levels, we divide them accordingly.', '1806.07309-1-35-4': 'Therefore, the length of [MATH] equals the total number of these tag fragments.', '1806.07309-1-35-5': 'For instance, if the video corpus would only contain the tags 005.74 and 005.133, we would split them into [MATH] (indices mark the level in the hierarchy) resulting in a vector [MATH] of length [MATH].', '1806.07309-1-35-6': 'If a particular tag fragment occurs in a video, we set the corresponding bin in [MATH] to the term frequency - inverse document frequency (tf-idf), or zero otherwise.', '1806.07309-1-35-7': 'This assures that the more specific, and therefore more informative, DDC classes have more influence on the result.', '1806.07309-1-35-8': 'For example, if two tags share the main DDC class Science and Mathematics, it does not mean that they are necessarily closely correlated, but if both share the class Data Compression they most likely cover a similar topic.', '1806.07309-1-35-9': 'For the "method with LOD" the two vectors [MATH] and [MATH] of video [MATH] and [MATH] are compared via cosine similarity.', '1806.07309-1-35-10': 'It is important to note that this method also uses the fastText features of the method without LOD.', '1806.07309-1-35-11': 'In order to compute the overall similarity, both methods are applied and the average is used to form [MATH] (see Figure [REF]).', '1806.07309-1-36-0': '# Experimental Results', '1806.07309-1-37-0': 'Videos of the TIB AV-Portal were used in the experiment.', '1806.07309-1-37-1': 'The complete stock of metadata that falls under the Creative Commons License CC0 1.0 Universal is made available by the TIB (https://av.tib.eu/opendata) as Resource Description Framework (RDF) triples.', '1806.07309-1-37-2': 'To extract the necessary annotations we utilized SPARQL.', '1806.07309-1-37-3': 'In a first step, it was necessary to keep only videos that allowed "derivate works" in addition to the CC0 1.0 license, since content analysis is applied.', '1806.07309-1-37-4': '[MATH] samples satisfied these conditions.', '1806.07309-1-37-5': 'Unfortunately, word embeddings of two different languages cannot be directly compared forcing us to use a subset of videos with the same language (German in this case, 1430 videos).', '1806.07309-1-37-6': 'Annotations are represented in JSON format to make them easily accessible for future tasks without rebuilding the RDF graph.', '1806.07309-1-37-7': 'After gathering all tags of an entry, we employed another SPARQL query assigning a GND (German: Gemeinsame Normdatei, English: Integrated Authority File) link to each tag, which is the key part of linking it to the data of the German National Library (DNB) and retrieving the corresponding DDC notations.', '1806.07309-1-38-0': 'We evaluated the quality of our similarity measure by conducting a user study with eight participants, five men and three women.', '1806.07309-1-38-1': 'A random selection of 50 videos was presented to every participant along with ten video recommendations, randomly either completely provided by the method without LOD or the method with LOD.', '1806.07309-1-38-2': 'The results were integrated by a Greasemonkey script in the Firefox browser.', '1806.07309-1-38-3': 'Every participant had to rate each of the ten recommendations from [MATH], i.e., 0: not relevant; 1: low relevance; 2: medium relevance; 3: highly relevant.', '1806.07309-1-38-4': 'The results are displayed in Figure [REF].', '1806.07309-1-39-0': 'The results show that the method with LOD increases the number of video recommendations with medium ([MATH]) and low relevance ([MATH]), while the effect is small ([MATH]) for the highly relevant recommendations.', '1806.07309-1-39-1': 'However, the method with LOD significantly decreases the number of irrelevant recommendations (by [MATH]).', '1806.07309-1-39-2': 'This indicates that this method is superior to the text-based method, most likely due to the hierarchical nature of the DDC notation.', '1806.07309-1-39-3': 'We assume that the rather small improvement for the very relevant recommendations is a result of the restrictions we had to oblige to (license and language), i.e., the relatively small set of remaining videos ([MATH]) does not contain more highly relevant samples.', '1806.07309-1-39-4': 'A chi-square test shows that the method with LOD is significantly better than our baseline (Chi-Square=[MATH], p-value=[MATH]).', '1806.07309-1-40-0': '# Conclusions', '1806.07309-1-41-0': 'In this paper, we have proposed a method to generate recommendations for scientific videos based on noisy, automatically extracted tags by utilizing linked open data to weave in hierarchical semantic metadata.', '1806.07309-1-41-1': 'This enables users to find relevant information more quickly improving their overall learning experience.', '1806.07309-1-41-2': 'In future work, we plan to incorporate recommendations for scientific papers or definitions of technical terms through linked open data.'}

{'1806.07309-2-0-0': 'The amount of available videos in the Web has significantly increased not only for entertainment etc., but also to convey educational or scientific information in an effective way.', '1806.07309-2-0-1': 'There are several web portals that offer access to the latter kind of video material.', '1806.07309-2-0-2': 'One of them is the TIB AV-Portal of the Leibniz Information Centre for Science and Technology (TIB), which hosts scientific and educational video content.', '1806.07309-2-0-3': 'In contrast to other video portals, automatic audiovisual analysis (visual concept classification, optical character recognition, speech recognition) is utilized to enhance metadata information and semantic search.', '1806.07309-2-0-4': 'In this paper, we propose to further exploit and enrich this automatically generated information by linking it to the Integrated Authority File (GND) of the German National Library.', '1806.07309-2-0-5': 'This information is used to derive a measure to compare the similarity of two videos which serves as a basis for recommending semantically similar videos.', '1806.07309-2-0-6': 'A user study demonstrates the feasibility of the proposed approach.', '1806.07309-2-1-0': '# Introduction', '1806.07309-2-2-0': 'Videos hold a great potential to communicate educational and scientific information.', '1806.07309-2-2-1': 'This is, for instance, reflected by e-Learning platforms such as Udacity (https://udacity.com) or Coursera (http://www.coursera.org).', '1806.07309-2-2-2': 'Another type of Web portals offers also access to scientific videos, one of them is the TIB AV-Portal (https://av.tib.eu) of the Leibniz Information Centre for Science and Technology (TIB).', '1806.07309-2-2-3': 'Researchers can provide, search, and access scientific and educational audiovisual material, while benefiting from a number of advantages compared to other portals.', '1806.07309-2-2-4': 'First, submitted videos are reviewed to check whether they contain scientific or educational content.', '1806.07309-2-2-5': 'Second, videos are represented in a persistent way using DOIs (digital object identifier), potentially even at the segment and frame level, making it easy and reliable to reference them.', '1806.07309-2-2-6': 'Finally, audiovisual content analysis is applied in order to allow the user to not only search for terms in descriptive metadata (e.g., title, manually annotated keywords), but also in the audiovisual content, i.e., in the speech transcript, in the recognized overlaid or scene text through video OCR (optical character recognition), and keywords derived from visual concept and scene classification.', '1806.07309-2-3-0': 'Usually, recommender systems in online shopping platforms or video portals mainly rely on user-based information such as the viewing history [CITATION] or current trends [CITATION].', '1806.07309-2-3-1': 'In this paper, we investigate the question how similar videos can be recommended based on their metadata, in particular, by additionally making use of automatically extracted metadata from audiovisual content analysis.', '1806.07309-2-3-2': 'This is relevant, for example, when users do not agree to track their search behavior or sufficient amount of user data is not available.', '1806.07309-2-3-3': 'Particularly, we propose to further exploit and enrich the entire set of available metadata, be it created manually or extracted automatically, in order to improve recommendations of semantically similar videos.', '1806.07309-2-3-4': 'In a first step, we utilize a Word2Vec approach [CITATION] to make the semantic content of two videos comparable based on title, tags, and abstract.', '1806.07309-2-3-5': 'Then, the automatically extracted metadata about the audiovisual content is enriched by linking it to the Integrated Authority File (GND: Gemeinsame Normdatei) of the German National Library (DNB: Deutsche Nationalbibliothek).', '1806.07309-2-3-6': 'These two kinds of information are used to derive a measure to compare the content of two videos which serves as a basis for recommending similar video.', '1806.07309-2-3-7': 'A user study demonstrates the feasibility of the proposed approach.', '1806.07309-2-4-0': 'The paper is structured as follows.', '1806.07309-2-4-1': 'First, we give a brief overview of related work in Section 2.', '1806.07309-2-4-2': 'The proposed approach to generate video recommendations is presented in Section 3.', '1806.07309-2-4-3': 'Section 4 describes the conducted user study to evaluate the proposed approach, while Section 5 concludes the paper.', '1806.07309-2-5-0': '# Related Work', '1806.07309-2-6-0': 'Scientific Video Portals: Yovisto is a scientific video portal that allows the user to search for information via text-based metadata [CITATION].', '1806.07309-2-6-1': 'The users can reduce the number of search results by refining their query via additional criteria and grouping videos by language, organization, or category.', '1806.07309-2-6-2': 'On the contrary, to increase the scope of possible results, a tool for explorative search reveals interrelations between different types of videos in order to present a broader spectrum of results to the user.', '1806.07309-2-6-3': 'This is done by exploiting an ontology structure, which is part of every video element and Linked Open Data (LOD) resources, namely DBpedia (http://wiki.dbpedia.org).', '1806.07309-2-7-0': 'Another similar portal is described by Marchionini [CITATION], where the uploaded content is automatically fed into an automatic data analysis chain.', '1806.07309-2-7-1': 'Semantic entities are automatically assigned to each video segment resulting in a storyboard comprising the video content.', '1806.07309-2-7-2': 'In contrast to the AV-Portal, this information is hidden from the user.', '1806.07309-2-7-3': "Marchionini's approach focuses on providing a good explorative search tool, i.e., a user should be able to find what s/he is looking for even when being unsure about the correct phrasing.", '1806.07309-2-7-4': 'Recommendation Systems for Scientific Videos: Clustering semantically similar videos is a possible approach to provide video recommendations based on a given, currently watched video.', '1806.07309-2-7-5': 'A fundamental problem of this research is the semantic gap between low-level features and high-level semantics portrayed in visual content [CITATION].', '1806.07309-2-7-6': 'To circumvent this problem, textual cues can be used in addition to visual content.', '1806.07309-2-7-7': 'These can be manually added tags by the author of the video or automatically extracted keywords by machine learning algorithms.', '1806.07309-2-7-8': 'Either way, they are often superficial, noisy, incomplete or ambiguous which makes the process of clustering a challenge.', '1806.07309-2-7-9': 'Vahdat et al. [CITATION] enrich the set of tags by modeling them from visual features and correct the existing ones by checking their agreement with the visual content.', '1806.07309-2-7-10': 'They are able to show that this method outperforms existing ones that use either modality and even the naive combination.', '1806.07309-2-7-11': 'Wang et al. [CITATION] discover that by incorporating hierarchical information - instead of considering a "flat" tag taxonomy - the semantics of a video can be described even better.', '1806.07309-2-7-12': 'Despite only using two levels of abstraction in their hierarchical multi-label random forest model, strong correlations between ambiguous visual features and sparse, incomplete tags could be found.', '1806.07309-2-8-0': '# Enriching Video Metadata through Linked Open Data', '1806.07309-2-9-0': 'In this section, we present our approach to enrich metadata with open data sources.', '1806.07309-2-9-1': 'First, the set of available metadata is described before the acquisition of additional information from an open data source is explained in Section [REF].', '1806.07309-2-9-2': 'Second, a similaritiy measure to compare videos based on a Word2Vec representation and enriched metadata is derived in Section [REF].', '1806.07309-2-9-3': 'The overall workflow is displayed in Figure [REF].', '1806.07309-2-9-4': 'The input of our system consists of manually generated and automatically extracted information, where the former comprises abstract and title.', '1806.07309-2-9-5': 'Additional inputs are the following automatically extracted Tags (see Figure [REF]) derived from: 1) Transcript based on speech recognition, 2) Results of video OCR, and 3) results of visual concept and scene classification.', '1806.07309-2-9-6': 'All of them have a representation in the German National Library, which is the key requirement for the enrichment process.', '1806.07309-2-10-0': '## Acquiring Additional Information from Open Data Source', '1806.07309-2-11-0': 'Automatically generated tags usually contain a certain amount of errors and noise.', '1806.07309-2-11-1': 'Although state-of-the-art algorithms can achieve human performance [CITATION] in specific tasks and settings, issues with audio quality in lecture rooms or hardly legible handwritings can cause errors.', '1806.07309-2-11-2': 'We try to circumvent this problem by evaluating additional information provided by the German National Library.', '1806.07309-2-11-3': 'Besides information such as synonyms and related scientific publications, the Dewey Decimal Classification (DDC) for every tag is provided.', '1806.07309-2-11-4': 'The DDC is a library classification system, which categorizes technical terms into ten classes via three-digit arabic numerals [CITATION].', '1806.07309-2-11-5': 'These main classes are then further divided into subcategories denoted by the decimals after these three digits, where additional decimals depict a more specific subject.', '1806.07309-2-11-6': 'For instance, SPARQL is contained in 006.74 - Markup Language, 005.74 - Data files and Databases and 005.133 - Individual Programming Languages, which yields valuable contextual information.', '1806.07309-2-12-0': '## Defining a Similarity Measure for Scientific Videos', '1806.07309-2-13-0': 'Simply comparing two videos for mutual tags is not sufficient to determine semantic similarity.', '1806.07309-2-13-1': 'Even if two sets of tags have little to no overlap they might be highly correlated when their context is considered.', '1806.07309-2-13-2': 'We address this issue by utilizing fastText [CITATION] to generate word embeddings, which has several advantages for this task.', '1806.07309-2-13-3': 'First, semantically similar words are modeled closer to one another so that a simple distance measure indicates the correlation of two words.', '1806.07309-2-13-4': 'Second, since fastText works on substrings rather than whole words it is able to produce valuable features even for misspelled or words unknown to the word embedding.', '1806.07309-2-13-5': 'Finally, a pre-trained model is available for a large number of languages.', '1806.07309-2-13-6': 'Title, tags, and abstract are taken from the metadata and processed via fastText.', '1806.07309-2-13-7': 'It generates a 300-dimensional feature vector for every word in the metadata.', '1806.07309-2-13-8': 'The average of these vectors is our representation for a particular video.', '1806.07309-2-13-9': 'This approach is our baseline and denoted as method without LOD in the sequel.', '1806.07309-2-14-0': 'The improvement of this already powerful feature extraction method is the main contribution of this paper.', '1806.07309-2-14-1': 'It is achieved by incorporating the information provided by the DDC notation in addition to the fastText embeddings.', '1806.07309-2-14-2': 'As a preprocessing step we need to create a vector [MATH], which consists of all DDC tags that occur in our dataset and which will be assigned to every video entry [MATH].', '1806.07309-2-14-3': 'Since the upper level classes of the notation are also encoded in the codes of the classes at lower levels, we divide them accordingly.', '1806.07309-2-14-4': 'Therefore, the length of [MATH] equals the total number of these tag fragments.', '1806.07309-2-14-5': 'For instance, if the video corpus would only contain the tags 005.74 and 005.133, we would split them into [MATH] (indices mark the level in the hierarchy) resulting in a vector [MATH] of length [MATH].', '1806.07309-2-14-6': 'If a particular tag fragment occurs in a video, we set the corresponding bin in [MATH] to the term frequency - inverse document frequency (tf-idf), or zero otherwise.', '1806.07309-2-14-7': 'This assures that the more specific, and therefore more informative, DDC classes have more influence on the result.', '1806.07309-2-14-8': 'For example, if two tags share the main DDC class Science and Mathematics, it does not mean that they are necessarily closely correlated, but if both share the class Data Compression they most likely cover a similar topic.', '1806.07309-2-14-9': 'For the "method with LOD" the two vectors [MATH] and [MATH] of video [MATH] and [MATH] are compared via cosine similarity.', '1806.07309-2-14-10': 'It is important to note that this method also uses the fastText features of the method without LOD.', '1806.07309-2-14-11': 'In order to compute the overall similarity, both methods are applied and the average is used to form [MATH] (see Figure [REF]).', '1806.07309-2-15-0': '# Experimental Results', '1806.07309-2-16-0': 'Videos of the TIB AV-Portal were used in the experiment.', '1806.07309-2-16-1': 'The complete stock of metadata that falls under the Creative Commons License CC0 1.0 Universal is made available by the TIB (https://av.tib.eu/opendata) as Resource Description Framework (RDF) triples.', '1806.07309-2-16-2': 'To extract the necessary annotations we utilized SPARQL.', '1806.07309-2-16-3': 'In a first step, it was necessary to keep only videos that allowed "derivate works" in addition to the CC0 1.0 license, since content analysis is applied.', '1806.07309-2-16-4': '[MATH] samples satisfied these conditions.', '1806.07309-2-16-5': 'Unfortunately, word embeddings of two different languages cannot be directly compared forcing us to use a subset of videos with the same language (German in this case, 1430 videos).', '1806.07309-2-16-6': 'Annotations are represented in JSON format to make them easily accessible for future tasks without rebuilding the RDF graph.', '1806.07309-2-16-7': 'After gathering all tags of an entry, we employed another SPARQL query assigning a GND (German: Gemeinsame Normdatei, English: Integrated Authority File) link to each tag, which is the key part of linking it to the data of the German National Library (DNB) and retrieving the corresponding DDC notations.', '1806.07309-2-17-0': 'We evaluated the quality of our similarity measure by conducting a user study with eight participants, five men and three women.', '1806.07309-2-17-1': 'A random selection of 50 videos was presented to every participant along with ten video recommendations, randomly either completely provided by the method without LOD or the method with LOD.', '1806.07309-2-17-2': 'The results were integrated by a Greasemonkey script in the Firefox browser.', '1806.07309-2-17-3': 'Every participant had to rate each of the ten recommendations from [MATH], i.e., 0: not relevant; 1: low relevance; 2: medium relevance; 3: highly relevant.', '1806.07309-2-17-4': 'The results are displayed in Figure [REF].', '1806.07309-2-18-0': 'The results show that the method with LOD increases the number of video recommendations with medium ([MATH]) and low relevance ([MATH]), while the effect is small ([MATH]) for the highly relevant recommendations.', '1806.07309-2-18-1': 'However, the method with LOD significantly decreases the number of irrelevant recommendations (by [MATH]).', '1806.07309-2-18-2': 'This indicates that this method is superior to the text-based method, most likely due to the hierarchical nature of the DDC notation.', '1806.07309-2-18-3': 'We assume that the rather small improvement for the very relevant recommendations is a result of the restrictions we had to oblige to (license and language), i.e., the relatively small set of remaining videos ([MATH]) does not contain more highly relevant samples.', '1806.07309-2-18-4': 'A chi-square test shows that the method with LOD is significantly better than our baseline (Chi-Square=[MATH], p-value=[MATH]).', '1806.07309-2-19-0': '# Conclusions', '1806.07309-2-20-0': 'In this paper, we have proposed a method to generate recommendations for scientific videos based on noisy, automatically extracted tags by utilizing linked open data to weave in hierarchical semantic metadata.', '1806.07309-2-20-1': 'This enables users to find relevant information more quickly improving their overall learning experience.', '1806.07309-2-20-2': 'In future work, we plan to incorporate recommendations for scientific papers or definitions of technical terms through linked open data.', '1806.07309-2-21-0': 'compat=1.14 compatibility=false', '1806.07309-2-22-0': 'Recommending Scientific Videos based on Metadata Enrichment using Linked Open Data Recommending Scientific Videos using Linked Open Data', '1806.07309-2-23-0': '# Abstract The amount of available videos in the Web has significantly increased not only for entertainment etc., but also to convey educational or scientific information in an effective way.', '1806.07309-2-23-1': 'There are several web portals that offer access to the latter kind of video material.', '1806.07309-2-23-2': 'One of them is the TIB AV-Portal of the Leibniz Information Centre for Science and Technology (TIB), which hosts scientific and educational video content.', '1806.07309-2-23-3': 'In contrast to other video portals, automatic audiovisual analysis (visual concept classification, optical character recognition, speech recognition) is utilized to enhance metadata information and semantic search.', '1806.07309-2-23-4': 'In this paper, we propose to further exploit and enrich this automatically generated information by linking it to the Integrated Authority File (GND) of the German National Library.', '1806.07309-2-23-5': 'This information is used to derive a measure to compare the similarity of two videos which serves as a basis for recommending semantically similar videos.', '1806.07309-2-23-6': 'A user study demonstrates the feasibility of the proposed approach.', '1806.07309-2-24-0': '# Introduction', '1806.07309-2-25-0': 'Videos hold a great potential to communicate educational and scientific information.', '1806.07309-2-25-1': 'This is, for instance, reflected by e-Learning platforms such as Udacity (https://udacity.com) or Coursera (http://www.coursera.org).', '1806.07309-2-25-2': 'Another type of Web portals offers also access to scientific videos, one of them is the TIB AV-Portal (https://av.tib.eu) of the Leibniz Information Centre for Science and Technology (TIB).', '1806.07309-2-25-3': 'Researchers can provide, search, and access scientific and educational audiovisual material, while benefiting from a number of advantages compared to other portals.', '1806.07309-2-25-4': 'First, submitted videos are reviewed to check whether they contain scientific or educational content.', '1806.07309-2-25-5': 'Second, videos are represented in a persistent way using DOIs (digital object identifier), potentially even at the segment and frame level, making it easy and reliable to reference them.', '1806.07309-2-25-6': 'Finally, audiovisual content analysis is applied in order to allow the user to not only search for terms in descriptive metadata (e.g., title, manually annotated keywords), but also in the audiovisual content, i.e., in the speech transcript, in the recognized overlaid or scene text through video OCR (optical character recognition), and keywords derived from visual concept and scene classification.', '1806.07309-2-26-0': 'Usually, recommender systems in online shopping platforms or video portals mainly rely on user-based information such as the viewing history [CITATION] or current trends [CITATION].', '1806.07309-2-26-1': 'In this paper, we investigate the question how similar videos can be recommended based on their metadata, in particular, by additionally making use of automatically extracted metadata from audiovisual content analysis.', '1806.07309-2-26-2': 'This is relevant, for example, when users do not agree to track their search behavior or sufficient amount of user data is not available.', '1806.07309-2-26-3': 'Particularly, we propose to further exploit and enrich the entire set of available metadata, be it created manually or extracted automatically, in order to improve recommendations of semantically similar videos.', '1806.07309-2-26-4': 'In a first step, we utilize a Word2Vec approach [CITATION] to make the semantic content of two videos comparable based on title, tags, and abstract.', '1806.07309-2-26-5': 'Then, the automatically extracted metadata about the audiovisual content is enriched by linking it to the Integrated Authority File (GND: Gemeinsame Normdatei) of the German National Library (DNB: Deutsche Nationalbibliothek).', '1806.07309-2-26-6': 'These two kinds of information are used to derive a measure to compare the content of two videos which serves as a basis for recommending similar video.', '1806.07309-2-26-7': 'A user study demonstrates the feasibility of the proposed approach.', '1806.07309-2-27-0': 'The paper is structured as follows.', '1806.07309-2-27-1': 'First, we give a brief overview of related work in Section 2.', '1806.07309-2-27-2': 'The proposed approach to generate video recommendations is presented in Section 3.', '1806.07309-2-27-3': 'Section 4 describes the conducted user study to evaluate the proposed approach, while Section 5 concludes the paper.', '1806.07309-2-28-0': '# Related Work', '1806.07309-2-29-0': 'Scientific Video Portals: Yovisto is a scientific video portal that allows the user to search for information via text-based metadata [CITATION].', '1806.07309-2-29-1': 'The users can reduce the number of search results by refining their query via additional criteria and grouping videos by language, organization, or category.', '1806.07309-2-29-2': 'On the contrary, to increase the scope of possible results, a tool for explorative search reveals interrelations between different types of videos in order to present a broader spectrum of results to the user.', '1806.07309-2-29-3': 'This is done by exploiting an ontology structure, which is part of every video element and Linked Open Data (LOD) resources, namely DBpedia (http://wiki.dbpedia.org).', '1806.07309-2-30-0': 'Another similar portal is described by Marchionini [CITATION], where the uploaded content is automatically fed into an automatic data analysis chain.', '1806.07309-2-30-1': 'Semantic entities are automatically assigned to each video segment resulting in a storyboard comprising the video content.', '1806.07309-2-30-2': 'In contrast to the AV-Portal, this information is hidden from the user.', '1806.07309-2-30-3': "Marchionini's approach focuses on providing a good explorative search tool, i.e., a user should be able to find what s/he is looking for even when being unsure about the correct phrasing.", '1806.07309-2-30-4': 'Recommendation Systems for Scientific Videos: Clustering semantically similar videos is a possible approach to provide video recommendations based on a given, currently watched video.', '1806.07309-2-30-5': 'A fundamental problem of this research is the semantic gap between low-level features and high-level semantics portrayed in visual content [CITATION].', '1806.07309-2-30-6': 'To circumvent this problem, textual cues can be used in addition to visual content.', '1806.07309-2-30-7': 'These can be manually added tags by the author of the video or automatically extracted keywords by machine learning algorithms.', '1806.07309-2-30-8': 'Either way, they are often superficial, noisy, incomplete or ambiguous which makes the process of clustering a challenge.', '1806.07309-2-30-9': 'Vahdat et al. [CITATION] enrich the set of tags by modeling them from visual features and correct the existing ones by checking their agreement with the visual content.', '1806.07309-2-30-10': 'They are able to show that this method outperforms existing ones that use either modality and even the naive combination.', '1806.07309-2-30-11': 'Wang et al. [CITATION] discover that by incorporating hierarchical information - instead of considering a "flat" tag taxonomy - the semantics of a video can be described even better.', '1806.07309-2-30-12': 'Despite only using two levels of abstraction in their hierarchical multi-label random forest model, strong correlations between ambiguous visual features and sparse, incomplete tags could be found.', '1806.07309-2-31-0': '# Enriching Video Metadata through Linked Open Data', '1806.07309-2-32-0': 'In this section, we present our approach to enrich metadata with open data sources.', '1806.07309-2-32-1': 'First, the set of available metadata is described before the acquisition of additional information from an open data source is explained in Section [REF].', '1806.07309-2-32-2': 'Second, a similaritiy measure to compare videos based on a Word2Vec representation and enriched metadata is derived in Section [REF].', '1806.07309-2-32-3': 'The overall workflow is displayed in Figure [REF].', '1806.07309-2-32-4': 'The input of our system consists of manually generated and automatically extracted information, where the former comprises abstract and title.', '1806.07309-2-32-5': 'Additional inputs are the following automatically extracted Tags (see Figure [REF]) derived from: 1) Transcript based on speech recognition, 2) Results of video OCR, and 3) results of visual concept and scene classification.', '1806.07309-2-32-6': 'All of them have a representation in the German National Library, which is the key requirement for the enrichment process.', '1806.07309-2-33-0': '## Acquiring Additional Information from Open Data Source', '1806.07309-2-34-0': 'Automatically generated tags usually contain a certain amount of errors and noise.', '1806.07309-2-34-1': 'Although state-of-the-art algorithms can achieve human performance [CITATION] in specific tasks and settings, issues with audio quality in lecture rooms or hardly legible handwritings can cause errors.', '1806.07309-2-34-2': 'We try to circumvent this problem by evaluating additional information provided by the German National Library.', '1806.07309-2-34-3': 'Besides information such as synonyms and related scientific publications, the Dewey Decimal Classification (DDC) for every tag is provided.', '1806.07309-2-34-4': 'The DDC is a library classification system, which categorizes technical terms into ten classes via three-digit arabic numerals [CITATION].', '1806.07309-2-34-5': 'These main classes are then further divided into subcategories denoted by the decimals after these three digits, where additional decimals depict a more specific subject.', '1806.07309-2-34-6': 'For instance, SPARQL is contained in 006.74 - Markup Language, 005.74 - Data files and Databases and 005.133 - Individual Programming Languages, which yields valuable contextual information.', '1806.07309-2-35-0': '## Defining a Similarity Measure for Scientific Videos', '1806.07309-2-36-0': 'Simply comparing two videos for mutual tags is not sufficient to determine semantic similarity.', '1806.07309-2-36-1': 'Even if two sets of tags have little to no overlap they might be highly correlated when their context is considered.', '1806.07309-2-36-2': 'We address this issue by utilizing fastText [CITATION] to generate word embeddings, which has several advantages for this task.', '1806.07309-2-36-3': 'First, semantically similar words are modeled closer to one another so that a simple distance measure indicates the correlation of two words.', '1806.07309-2-36-4': 'Second, since fastText works on substrings rather than whole words it is able to produce valuable features even for misspelled or words unknown to the word embedding.', '1806.07309-2-36-5': 'Finally, a pre-trained model is available for a large number of languages.', '1806.07309-2-36-6': 'Title, tags, and abstract are taken from the metadata and processed via fastText.', '1806.07309-2-36-7': 'It generates a 300-dimensional feature vector for every word in the metadata.', '1806.07309-2-36-8': 'The average of these vectors is our representation for a particular video.', '1806.07309-2-36-9': 'This approach is our baseline and denoted as method without LOD in the sequel.', '1806.07309-2-37-0': 'The improvement of this already powerful feature extraction method is the main contribution of this paper.', '1806.07309-2-37-1': 'It is achieved by incorporating the information provided by the DDC notation in addition to the fastText embeddings.', '1806.07309-2-37-2': 'As a preprocessing step we need to create a vector [MATH], which consists of all DDC tags that occur in our dataset and which will be assigned to every video entry [MATH].', '1806.07309-2-37-3': 'Since the upper level classes of the notation are also encoded in the codes of the classes at lower levels, we divide them accordingly.', '1806.07309-2-37-4': 'Therefore, the length of [MATH] equals the total number of these tag fragments.', '1806.07309-2-37-5': 'For instance, if the video corpus would only contain the tags 005.74 and 005.133, we would split them into [MATH] (indices mark the level in the hierarchy) resulting in a vector [MATH] of length [MATH].', '1806.07309-2-37-6': 'If a particular tag fragment occurs in a video, we set the corresponding bin in [MATH] to the term frequency - inverse document frequency (tf-idf), or zero otherwise.', '1806.07309-2-37-7': 'This assures that the more specific, and therefore more informative, DDC classes have more influence on the result.', '1806.07309-2-37-8': 'For example, if two tags share the main DDC class Science and Mathematics, it does not mean that they are necessarily closely correlated, but if both share the class Data Compression they most likely cover a similar topic.', '1806.07309-2-37-9': 'For the "method with LOD" the two vectors [MATH] and [MATH] of video [MATH] and [MATH] are compared via cosine similarity.', '1806.07309-2-37-10': 'It is important to note that this method also uses the fastText features of the method without LOD.', '1806.07309-2-37-11': 'In order to compute the overall similarity, both methods are applied and the average is used to form [MATH] (see Figure [REF]).', '1806.07309-2-38-0': '# Experimental Results', '1806.07309-2-39-0': 'Videos of the TIB AV-Portal were used in the experiment.', '1806.07309-2-39-1': 'The complete stock of metadata that falls under the Creative Commons License CC0 1.0 Universal is made available by the TIB (https://av.tib.eu/opendata) as Resource Description Framework (RDF) triples.', '1806.07309-2-39-2': 'To extract the necessary annotations we utilized SPARQL.', '1806.07309-2-39-3': 'In a first step, it was necessary to keep only videos that allowed "derivate works" in addition to the CC0 1.0 license, since content analysis is applied.', '1806.07309-2-39-4': '[MATH] samples satisfied these conditions.', '1806.07309-2-39-5': 'Unfortunately, word embeddings of two different languages cannot be directly compared forcing us to use a subset of videos with the same language (German in this case, 1430 videos).', '1806.07309-2-39-6': 'Annotations are represented in JSON format to make them easily accessible for future tasks without rebuilding the RDF graph.', '1806.07309-2-39-7': 'After gathering all tags of an entry, we employed another SPARQL query assigning a GND (German: Gemeinsame Normdatei, English: Integrated Authority File) link to each tag, which is the key part of linking it to the data of the German National Library (DNB) and retrieving the corresponding DDC notations.', '1806.07309-2-40-0': 'We evaluated the quality of our similarity measure by conducting a user study with eight participants, five men and three women.', '1806.07309-2-40-1': 'A random selection of 50 videos was presented to every participant along with ten video recommendations, randomly either completely provided by the method without LOD or the method with LOD.', '1806.07309-2-40-2': 'The results were integrated by a Greasemonkey script in the Firefox browser.', '1806.07309-2-40-3': 'Every participant had to rate each of the ten recommendations from [MATH], i.e., 0: not relevant; 1: low relevance; 2: medium relevance; 3: highly relevant.', '1806.07309-2-40-4': 'The results are displayed in Figure [REF].', '1806.07309-2-41-0': 'The results show that the method with LOD increases the number of video recommendations with medium ([MATH]) and low relevance ([MATH]), while the effect is small ([MATH]) for the highly relevant recommendations.', '1806.07309-2-41-1': 'However, the method with LOD significantly decreases the number of irrelevant recommendations (by [MATH]).', '1806.07309-2-41-2': 'This indicates that this method is superior to the text-based method, most likely due to the hierarchical nature of the DDC notation.', '1806.07309-2-41-3': 'We assume that the rather small improvement for the very relevant recommendations is a result of the restrictions we had to oblige to (license and language), i.e., the relatively small set of remaining videos ([MATH]) does not contain more highly relevant samples.', '1806.07309-2-41-4': 'A chi-square test shows that the method with LOD is significantly better than our baseline (Chi-Square=[MATH], p-value=[MATH]).', '1806.07309-2-42-0': '# Conclusions', '1806.07309-2-43-0': 'In this paper, we have proposed a method to generate recommendations for scientific videos based on noisy, automatically extracted tags by utilizing linked open data to weave in hierarchical semantic metadata.', '1806.07309-2-43-1': 'This enables users to find relevant information more quickly improving their overall learning experience.', '1806.07309-2-43-2': 'In future work, we plan to incorporate recommendations for scientific papers or definitions of technical terms through linked open data.'}

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'1806.07309-2-30-0'], ['1806.07309-1-28-5', '1806.07309-2-30-1'], ['1806.07309-1-28-6', '1806.07309-2-30-2'], ['1806.07309-1-28-7', '1806.07309-2-30-3'], ['1806.07309-1-28-8', '1806.07309-2-30-4'], ['1806.07309-1-28-9', '1806.07309-2-30-5'], ['1806.07309-1-28-10', '1806.07309-2-30-6'], ['1806.07309-1-28-11', '1806.07309-2-30-7'], ['1806.07309-1-28-12', '1806.07309-2-30-8'], ['1806.07309-1-28-14', '1806.07309-2-30-10'], ['1806.07309-1-28-15', '1806.07309-2-30-11'], ['1806.07309-1-28-16', '1806.07309-2-30-12']]

[['1806.07309-1-2-6', '1806.07309-2-2-6'], ['1806.07309-1-24-6', '1806.07309-2-25-6'], ['1806.07309-1-6-1', '1806.07309-2-6-1'], ['1806.07309-1-6-13', '1806.07309-2-7-9'], ['1806.07309-1-28-1', '1806.07309-2-29-1'], ['1806.07309-1-28-13', '1806.07309-2-30-9']]

[]

[]

[]

['1806.07309-1-20-0', '1806.07309-2-21-0']

{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1806.07309

1506.03158

{'1506.03158-1-0-0': 'The elliptic flow ([MATH]) at [MATH] 11.5, 39, and 200 GeV and triangular flow ([MATH]) at [MATH] 200 GeV of identified particles ([MATH] and [MATH]) from 0%-80% central Au+Au collisions are analyzed using a multiphase transport (AMPT) model.', '1506.03158-1-0-1': 'It is shown that the experimental results from the [MATH]-sub event plane method can be reproduced with a parton scattering cross section between 1.5 and 3 mb.', '1506.03158-1-0-2': 'We also studied the differential and integrated [MATH] ratios and conclude that they are anti-corrected with the parton scattering cross section.', '1506.03158-1-1-0': '# Introduction', '1506.03158-1-2-0': 'One of the main goals of heavy-ion collision experiments at the Relativistic Heavy Ion Collider (RHIC) is to study the properties and formation of the Quark Gluon Plasma (QGP).', '1506.03158-1-2-1': 'The study of the azimuthal anisotropy of emitted particles, based on Fourier decomposition, is considered to be one of the most important tools to investigate the hot and dense matter created in heavy-ion collisions [CITATION].', '1506.03158-1-2-2': 'Several interesting observations of the second harmonic Fourier coefficient [MATH], also called elliptic flow, have been reported during the past decade by using the data from the top RHIC heavy-ion energy of [MATH] 200 GeV [CITATION].', '1506.03158-1-2-3': 'At low transverse momenta ([MATH] 2.0 GeV/[MATH]), a mass ordering of [MATH] was observed [CITATION], which can be understood within a hydrodynamic framework [CITATION].', '1506.03158-1-2-4': 'At intermediate transverse momenta (2.0 [MATH] 6.0 GeV/[MATH]), a Number-of-Constituent Quark (NCQ) scaling [CITATION] was observed [CITATION].', '1506.03158-1-2-5': 'The NCQ scaling was used to conclude that the relevant degrees of freedom in the created matter are quarks and gluons.', '1506.03158-1-2-6': 'In addition to measurements at top RHIC energies, a Beam Energy Scan (BES) program has been carried out in the years 2010, 2011 and 2014 at to study the structure of the QCD phase diagram.', '1506.03158-1-2-7': 'Au+Au collisions were recorded at [MATH] 7.7, 11.5, 14.5, 19.6, 27, 39, and 62.4 GeV.', '1506.03158-1-2-8': 'The mass ordering in the low [MATH] region and NCQ scaling at intermediate transverse momenta were also observed BES energies [CITATION].', '1506.03158-1-3-0': 'In the recent years, the third harmonic Fourier coefficient [MATH], called triangular flow, has attracted more attention and was studied intensively [CITATION].', '1506.03158-1-3-1': 'It is thought to be related to the near-side ridge structure observed in two particle correlation measurements [CITATION].', '1506.03158-1-3-2': 'The triangular flow seems to be more sensitive to the viscous effects than [MATH] [CITATION], but less sensitive to the collision centrality [CITATION].', '1506.03158-1-3-3': 'Thus, [MATH] is a natural choice to study initial fluctuations and viscosity effects.', '1506.03158-1-3-4': 'Similar flow patterns as for [MATH] were recently observed in [MATH] measurements [CITATION].', '1506.03158-1-4-0': 'Viscous relativistic hydrodynamic calculations [CITATION] suggest a combined analysis of elliptic and triangular flow data to put tight constraints on the initial states of nucleus-nucleus collisions.', '1506.03158-1-4-1': 'Other hydrodynamic model calculations [CITATION] predict a stronger damping of the higher harmonic flow coefficients (n [MATH] 2) relative to [MATH] at high transverse momenta.', '1506.03158-1-4-2': 'A constant ratio of [MATH] at high [MATH] is predicted, where high means [MATH] and [MATH] is the highest flow velocity.', '1506.03158-1-4-3': 'Thus, by combining elliptic and triangular flow, more information about the initial state and the influence of the viscous effect can be collected.', '1506.03158-1-5-0': 'In this paper, the differential and integrated ratio of [MATH] of identified particles ([MATH] and [MATH]) are studied with the AMPT model.', '1506.03158-1-5-1': 'This paper is organized as follows.', '1506.03158-1-5-2': 'Section [REF] gives a brief introduction to the [MATH]-sub event plane method and the AMPT model.', '1506.03158-1-5-3': 'In section [REF], the collective flow calculations ([MATH] and [MATH]) from the AMPT model are presented.', '1506.03158-1-5-4': 'The differential and integrated [MATH] ratios are discussed in Sec. [REF] and Sec. [REF].', '1506.03158-1-5-5': 'A summary is given in Sec. [REF].', '1506.03158-1-6-0': '# [MATH]-sub Event Plane Method and the AMPT Model', '1506.03158-1-7-0': '## [MATH]-sub Event Method', '1506.03158-1-8-0': 'The event plane method is one of the most widely used methods to analyze anisotropic flow in heavy-ion collisions [CITATION].', '1506.03158-1-8-1': 'In non-central Au+Au collision, the overlap region has an almond shape in the coordinate space.', '1506.03158-1-8-2': 'As the system evolves, the pressure gradient pushes the anisotropy from coordinate space to momentum space, therefore the produced particles have an anisotropic distribution in momentum space.', '1506.03158-1-8-3': 'The azimuthal distribution of the produced particles can be written as [CITATION]: [EQUATION] where [MATH] is the azimuthal angle of a particle, [MATH] is the observed n-th harmonic flow., and [MATH] is the n-th harmonic event plane angle reconstructed by the produced particles, defined as: [EQUATION]', '1506.03158-1-8-4': 'The sum goes over all charged particles in the event, [MATH] is a weight applied to optimize the event plane resolution, see [CITATION] for details.', '1506.03158-1-8-5': 'It is worth to note that the range of n-th harmonic event plane angle is 0 [MATH]/n.', '1506.03158-1-9-0': 'In this method the n-th observed harmonic flow can be corrected for statistical effects by the n-th harmonic event plane resolution: [EQUATION] with [EQUATION]', '1506.03158-1-9-1': 'Here [MATH] denotes the average over all particles in all events and [MATH] is the n-th real event plane angle [CITATION].', '1506.03158-1-9-2': 'The real event plane means the participant plane [CITATION], which can not be achieved due to finite multiplicity of an event.', '1506.03158-1-10-0': 'To reduce the short range "non-flow" effects such as HBT correlations, a [MATH]-sub event plane method was introduced [CITATION].', '1506.03158-1-10-1': 'The [MATH]-sub event plane method divides the event into two independent sub-events with different pseudo rapidity ([MATH]) ranges.', '1506.03158-1-10-2': 'An additional gap of 0.1 in pseudo rapidity is added to further avoid short range correlations.', '1506.03158-1-10-3': 'The procedure results in a positive and negative [MATH]-sub event plane.', '1506.03158-1-10-4': 'The n-th observed harmonic flow can now be calculated in respect to the [MATH]-sub event planes.', '1506.03158-1-10-5': 'As a consequence, the [MATH]-sub event plane resolution [CITATION] is used instead of full event plane resolution: [EQUATION] where the [MATH] and the [MATH] are the n-th harmonic event plane angles for the positive and negative [MATH]-sub ranges.', '1506.03158-1-11-0': '## AMPT Model', '1506.03158-1-12-0': 'AMPT is a transport model which consists of four main components: the initial conditions, partonic interactions, conversion from partonic to hadronic matter, and hadronic interactions [CITATION].', '1506.03158-1-12-1': 'It has two versions to deal with different scenarios which are default AMPT (AMPT Def in following figures) and string melting AMPT (AMPT Str in following figures).', '1506.03158-1-12-2': 'The initial conditions are generated by the HIJING (Heavy Ion Jet Interaction Generator) model [CITATION].', '1506.03158-1-12-3': 'HIJING includes only two body nucleon-nucleon interactions and generates mini jets and excited strings through hard processes and soft processes separately.', '1506.03158-1-13-0': 'Excited strings are treated differently in the two AMPT versions.', '1506.03158-1-13-1': 'In default AMPT, excited strings are combined to hadrons according to the Lund string fragmentation model they further go through a hadronic interaction stage [CITATION].', '1506.03158-1-13-2': 'In string melting AMPT, excited strings first convert to partons, i.e. melting, and then go through a partonic interaction stage with original soft partons.', '1506.03158-1-13-3': "The partonic interactions for both default AMPT and string melting AMPT are described by ZPC (Zhang's Parton Cascade) model [CITATION].", '1506.03158-1-13-4': 'In the final stage of the ZPC model, partons in default AMPT are recombined with parent strings and hadronize via the Lund string fragmentation model.', '1506.03158-1-13-5': 'However, in string melting AMPT, the hadronization of partons is described by a coalescence model.', '1506.03158-1-13-6': 'After hadronization, the hadronic interactions are modelled by ART (A Relativistic Transport) model [CITATION].', '1506.03158-1-14-0': 'In previous studies it was found that a large parton scattering cross section (approximately 6 mb or 10 mb) is needed to reproduce the observed elliptic flow of charged hadrons [CITATION].', '1506.03158-1-14-1': 'In these studies, the elliptic flow was calculated with respect to the reaction plane (determined by the beam axis and impact parameter direction), which is 0 in the AMPT model, and this might underestimate the elliptic flow observed in the experiment data, which measured relative to the event plane but not the reaction plane [CITATION].', '1506.03158-1-14-2': 'As shown in Ref. [CITATION], both the charged particle multiplicity and elliptic flow can be reproduced with a smaller parton scattering cross section (1.5 mb) by using the default values for the parameters in the Lund string fragmentation function.', '1506.03158-1-15-0': 'In this paper, approximately 20 million events for 0%-80% central Au+Au collisions at [MATH] 11.5, 39, and 200 GeV with default AMPT (v1.25) and string melting AMPT (v2.25) were generated.', '1506.03158-1-15-1': 'Three different parton scattering cross sections (1.5 mb, 3 mb, and 6 mb) are used in the string melting version of AMPT.', '1506.03158-1-15-2': 'The parameters in the Lund string fragmentation function are set to the default values for AMPT with 1.5 mb parton scattering cross section (the values can be found at Table I in Ref. [CITATION]).', '1506.03158-1-15-3': 'Furthermore the [MATH]-sub event plane method is used to calculate elliptic and triangular flow.', '1506.03158-1-16-0': '# Results', '1506.03158-1-17-0': 'Identical analysis steps are used for simulation and data collected from the experiments [CITATION].', '1506.03158-1-17-1': 'Finial state charged particles with 0.2 [MATH] 2.0 GeV/c and [MATH] 1.0, are used to reconstruct the event plane.', '1506.03158-1-17-2': 'The [MATH] range of the two [MATH]sub events are -1.0 [MATH] -0.05 and 0.05 [MATH] 1.0, respectively.', '1506.03158-1-17-3': 'The elliptic and triangular flow are calculated within -1.0 [MATH] 1.0, and the auto-correlation is naturally avoided by the [MATH]-sub event plane method.', '1506.03158-1-17-4': 'For all the plots in this section, only statistical errors are shown.', '1506.03158-1-18-0': '## Elliptic Flow and Triangular Flow at RHIC', '1506.03158-1-19-0': 'Figure [REF] presents the second and third harmonic event plane resolutions from the AMPT string melting model for parton scattering cross sections of 1.5 mb, 3 mb and 6 mb from 0%-80% central Au+Au collisions at [MATH] 11.5, 39 and 200 GeV by using the [MATH]-sub event plane method.', '1506.03158-1-19-1': 'The event plane resolution shows a clear parton scattering cross section and energy dependence: both second and third harmonic event plane resolution increases with increasing parton scattering cross section and energy.', '1506.03158-1-20-0': 'The third harmonic event plane resolution is about two times smaller and peaks more central than the second harmonic event plane resolution, which is comparable to the one from experiment.', '1506.03158-1-20-1': 'For [MATH] 11.5 and 39 GeV, [MATH] can only be calculated up to 0-60 (0-70) for some of the parton scattering cross sections, since we only have event plane resolution up to these centrality.', '1506.03158-1-21-0': 'Figure [REF], Fig. [REF] and Fig. [REF] show the comparison between data from experiment and from the AMPT model for [MATH] of identified particles ([MATH] and [MATH]) at [MATH] 11.5, 39 and 200 GeV from 0-80 central Au+Au collisions.', '1506.03158-1-21-1': 'The data from experiment are taken from Ref. [CITATION].', '1506.03158-1-21-2': 'For all three energies it is observed that the elliptic flow increases with increasing parton scattering cross section.', '1506.03158-1-21-3': 'The elliptic flow from default AMPT is lower than the corresponding results from experiment, which indicates that the parton scattering process is important to produce a large elliptic flow as observed in experiment data.', '1506.03158-1-21-4': 'Previous studies show that a 6-10 mb parton scattering cross section is needed to reproduce the elliptic flow observed in experiment data [CITATION].', '1506.03158-1-21-5': 'In those studies the elliptic flow was calculated relative to the reaction plane.', '1506.03158-1-22-0': 'By using the [MATH]-sub event plane method most of the experiment data points at low transverse momenta are between AMPT default and AMPT string melting with a 3 mb parton scattering cross section.', '1506.03158-1-22-1': 'At intermediate [MATH] some particle species, e.g. [MATH], are systematically above AMPT string melting with a 3 mb parton scattering cross section.', '1506.03158-1-22-2': 'That means previous studies have underestimated the elliptic flow by using the reaction plane instead of the event plane.', '1506.03158-1-22-3': 'Latter one fluctuates event-by-event around the reaction plane.', '1506.03158-1-22-4': 'The collective flow measured in experiment are not mean values, but closer to a root-mean-sqaure [CITATION], therefore, the event-by-event fluctuations did not cancel each other but give a positive contribution to the measured collective flow, which makes the collective flow relative to the event plane is always larger compare to the collective flow relative to the reaction plane.', '1506.03158-1-23-0': 'Figure [REF] shows the comparison between experiment data [CITATION] and AMPT model calculations for [MATH] at [MATH] 200 GeV from 0-80 central Au+Au collisions of identified particles ([MATH] and [MATH]).', '1506.03158-1-23-1': 'A similar parton scattering cross section dependence is observed as for the elliptic flow, but here the dependence is stronger: [MATH] is about 40 lower than [MATH] with 1.5 mb parton scattering cross section but comparable with [MATH] with a 6 mb parton scattering cross section.', '1506.03158-1-23-2': 'The driving force behind [MATH] is the almond shape of initial nuclei overlap region, therefore, [MATH] has a relative large value compare to [MATH] which is mainly generated by initial state fluctuations [CITATION].', '1506.03158-1-23-3': 'This effect can be observed by comparing the [MATH] and [MATH] values from AMPT default model calculations in Fig. [REF] and Fig. [REF].', '1506.03158-1-23-4': 'The higher parton scattering cross section makes the transportation from the initial coordinate space to the final momentum space more efficient, which increases the values for both [MATH] and [MATH].', '1506.03158-1-23-5': 'The relative increase of [MATH] is larger than for [MATH], since the effects of the parton scattering is more important for [MATH].', '1506.03158-1-23-6': 'The triangular and elliptic flow from experiment can be well reproduced by AMPT default and AMPT string melting with a 1.5 mb parton scattering cross section, see Fig. [REF] and Fig. [REF].', '1506.03158-1-24-0': '## Differential [MATH] Ratio from AMPT Model', '1506.03158-1-25-0': 'Figure [REF] depicts AMPT model calculations for the transverse momentum dependent differential ratio [MATH] for identified particles ([MATH] and [MATH]) from 0%-80% central Au+Au collisions at [MATH] 200 GeV.', '1506.03158-1-25-1': 'The differential ratio decreases significantly at low transverse momenta ([MATH] 1.5 GeV/c) and becomes flat at intermediate transverse momenta (1.5 [MATH] 3.0 GeV/c).', '1506.03158-1-25-2': 'The [MATH] dependent differential ratio shows an anti-correlation with parton scattering cross section for all particle species: the higher the parton scattering cross section, the lower the [MATH] ratio.', '1506.03158-1-25-3': 'This is in agreement with a larger increase of [MATH] with increasing parton scattering cross section compare to [MATH] discussed in Fig. [REF] and Fig. [REF].', '1506.03158-1-25-4': 'This effect can also be understood in a hydrodynamic frame work.', '1506.03158-1-25-5': 'The higher parton scattering cross section in the AMPT string melting model is equivalent to a lower viscosity in viscous hydrodynamics [CITATION].', '1506.03158-1-25-6': 'From this study it is known that [MATH] is more sensitive to viscosity than [MATH] [CITATION], therefore a lower viscosity (higher parton scattering cross section) leads to a lower [MATH] ratio.', '1506.03158-1-26-0': '## Integrated [MATH] Ratio from AMPT Model', '1506.03158-1-27-0': 'Figure [REF] shows the AMPT calculation for the integrated [MATH] ratio for identified particles from 0%-80% central Au+Au collisions at [MATH] 39 and 200 GeV.', '1506.03158-1-27-1': 'The integration region is limited to 1.5 [MATH] 2.8 GeV/[MATH] to avoid the steep decrease in the ratio and the large error region.', '1506.03158-1-27-2': 'Please note, the triangular flow used to calculate the [MATH] ratio for 1.5 mb and 3 mb parton scattering cross section are taken from 0-60 and 0-70 due to the event plane resolution limits, as shown in Fig. [REF].', '1506.03158-1-27-3': 'The other of [MATH] values are taken from 0-80 results.', '1506.03158-1-28-0': 'A similar anti-correlation between the integrated [MATH] ratio and the parton scattering cross section is observed as in the differential [MATH] ratio: the larger the parton scattering cross section, the smaller the integrated [MATH] ratio.', '1506.03158-1-28-1': 'In addition, for the same parton scattering cross section, an energy dependence is observed: the lower the energy the larger the integrated [MATH] ratio.', '1506.03158-1-28-2': 'We further observe that the integrated [MATH] ratio calculated by string melting AMPT shows no mass (particle type) dependence, the ratio between integrated [MATH] of all particle species to that of [MATH] are consistent with 1 within 10%.', '1506.03158-1-28-3': 'On the other hand the ratios of [MATH] from default AMPT are lower than the rest of the particle species at both [MATH] 39 and 200 GeV.', '1506.03158-1-28-4': 'This indicates that the quark coalescence mechanism might play an important role in flattening the [MATH] ratio.', '1506.03158-1-28-5': 'The no mass (particle type) dependence of the [MATH] ratio was predicted by viscous hydrodynamics, for fast particles (particles with large [MATH]).', '1506.03158-1-28-6': 'The [MATH] ratio is constant and does not depend on the mass of the particle [CITATION].', '1506.03158-1-28-7': 'Based on the viscous hydrodynamics in Ref. [CITATION], the collective flow depends on dissipative contributions and the Fourier coefficient ([MATH]) of the flow velocity distribution relative to the event plane, the dissipative contributions cancelled each other in [MATH] ratio calculation and only [MATH] left which does not depend on the mass of different particle species.', '1506.03158-1-28-8': 'Since the parton scattering cross section is related to the viscosity in hydrodynamics, the magnitude of the [MATH] ratio might be usable to quantify the viscosity of the system.', '1506.03158-1-29-0': '# Summary', '1506.03158-1-30-0': 'The AMPT calculations of elliptic flow at [MATH] 11.5, 39 and 200 GeV and triangular flow at [MATH] 200 GeV of identified particles ([MATH] and [MATH]) from 0%-80% central Au+Au collisions were presented.', '1506.03158-1-30-1': 'Most of results from experiments can be reproduced with a parton scattering cross section between 1.5 and 3 mb by following the same analysis method as used in the experiments.', '1506.03158-1-30-2': 'An anti-correlation between the parton scattering cross section for both the differential and integrated [MATH] ratio is observed.', '1506.03158-1-30-3': 'This is due to a different sensitivity of [MATH] and [MATH] to the parton scattering cross section.', '1506.03158-1-30-4': 'A mass (particle type) independence of the integrated [MATH] ratio is observed from string melting AMPT, but not for the default AMPT.', '1506.03158-1-30-5': 'The no mass (particle type) dependence of the [MATH] ratio also can be understood in the viscous hydrodynamic framework.', '1506.03158-1-30-6': 'Latter one suggests that the magnitude of the ratio might be used to quantify the viscosity of the system.', '1506.03158-1-31-0': 'We thank Dr. Nu Xu for his great idea and important discussions.', '1506.03158-1-31-1': 'This work was supported by the National Natural Science Foundation of China under grant No.11475070 and U1332125 and the Program for Innovation Research of Science in Harbin Institute of Technology (PIRS OF HIT B201408).'}

{'1506.03158-2-0-0': 'The elliptic flow ([MATH]) at [MATH] 11.5, 39, and 200 GeV and triangular flow ([MATH]) at [MATH] 200 GeV of identified particles ([MATH] and [MATH]) from 0%-80% central Au+Au collisions are analyzed using a multiphase transport (AMPT) model.', '1506.03158-2-0-1': 'It is shown that the experimental results from the [MATH]-sub event plane method can be reproduced with a parton scattering cross section between 1.5 and 3 mb.', '1506.03158-2-0-2': 'We also studied the differential and integrated [MATH] ratios and conclude that they are anti-correlated with the parton scattering cross section.', '1506.03158-2-1-0': '# Introduction', '1506.03158-2-2-0': 'One of the main goals of heavy-ion collision experiments at the Relativistic Heavy Ion Collider (RHIC) is to study the properties and formation of the Quark Gluon Plasma (QGP).', '1506.03158-2-2-1': 'The study of the azimuthal anisotropy of emitted particles, based on Fourier decomposition, is considered to be one of the most important tools to investigate the hot and dense matter created in heavy-ion collisions [CITATION].', '1506.03158-2-2-2': 'Several interesting observations of the second harmonic Fourier coefficient [MATH], also called elliptic flow, have been reported during the past decade by using the data from the top RHIC heavy-ion energy of [MATH] 200 GeV [CITATION].', '1506.03158-2-2-3': 'At low transverse momenta ([MATH] 2.0 GeV/[MATH]), a mass ordering of [MATH] was observed [CITATION], which can be understood within a hydrodynamic framework [CITATION].', '1506.03158-2-2-4': 'At intermediate transverse momenta (2.0 [MATH] 6.0 GeV/[MATH]), a Number-of-Constituent Quark (NCQ) scaling [CITATION] was observed [CITATION].', '1506.03158-2-2-5': 'The NCQ scaling was used to conclude that the relevant degrees of freedom in the created matter are quarks and gluons.', '1506.03158-2-2-6': 'In addition to measurements at top RHIC energies, a Beam Energy Scan (BES) program has been carried out in the years 2010, 2011 and 2014 at to study the structure of the QCD phase diagram.', '1506.03158-2-2-7': 'Au+Au collisions were recorded at [MATH] 7.7, 11.5, 14.5, 19.6, 27, 39, and 62.4 GeV.', '1506.03158-2-2-8': 'The mass ordering in the low [MATH] region and NCQ scaling at intermediate transverse momenta were also observed BES energies [CITATION].', '1506.03158-2-3-0': 'In the recent years, the third harmonic Fourier coefficient [MATH], called triangular flow, has attracted more attention and was studied intensively [CITATION].', '1506.03158-2-3-1': 'It is thought to be related to the near-side ridge structure observed in two particle correlation measurements [CITATION].', '1506.03158-2-3-2': 'The triangular flow seems to be more sensitive to the viscous effects than [MATH] [CITATION], but less sensitive to the collision centrality [CITATION].', '1506.03158-2-3-3': 'Thus, [MATH] is a natural choice to study initial fluctuations and viscosity effects.', '1506.03158-2-3-4': 'Similar flow patterns as for [MATH] were recently observed in [MATH] measurements [CITATION].', '1506.03158-2-4-0': 'Viscous relativistic hydrodynamic calculations [CITATION] suggest a combined analysis of elliptic and triangular flow data to put tight constraints on the initial states of nucleus-nucleus collisions.', '1506.03158-2-4-1': 'Other hydrodynamic model calculations [CITATION] predict a stronger damping of the higher harmonic flow coefficients (n [MATH] 2) relative to [MATH] at high transverse momenta.', '1506.03158-2-4-2': 'A constant ratio of [MATH] at high [MATH] is predicted, where high means [MATH] and [MATH] is the highest flow velocity.', '1506.03158-2-4-3': 'Thus, by combining elliptic and triangular flow, more information about the initial state and the influence of the viscous effect can be collected.', '1506.03158-2-5-0': 'In this paper, the differential and integrated ratio of [MATH] of identified particles ([MATH] and [MATH]) are studied with the AMPT model.', '1506.03158-2-5-1': 'This paper is organized as follows.', '1506.03158-2-5-2': 'Section [REF] gives a brief introduction to the [MATH]-sub event plane method and the AMPT model.', '1506.03158-2-5-3': 'In section [REF], the collective flow calculations ([MATH] and [MATH]) from the AMPT model are presented.', '1506.03158-2-5-4': 'The differential and integrated [MATH] ratios are discussed in Sec. [REF] and Sec. [REF].', '1506.03158-2-5-5': 'A summary is given in Sec. [REF].', '1506.03158-2-6-0': '# [MATH]-sub Event Plane Method and the AMPT Model', '1506.03158-2-7-0': '## [MATH]-sub Event Method', '1506.03158-2-8-0': 'The event plane method is one of the most widely used methods to analyze anisotropic flow in heavy-ion collisions [CITATION].', '1506.03158-2-8-1': 'In non-central Au+Au collision, the overlap region has an almond shape in the coordinate space.', '1506.03158-2-8-2': 'As the system evolves, the pressure gradient pushes the anisotropy from coordinate space to momentum space, therefore the produced particles have an anisotropic distribution in momentum space.', '1506.03158-2-8-3': 'The azimuthal distribution of the produced particles can be written as [CITATION]: [EQUATION] where [MATH] is the azimuthal angle of a particle, [MATH] is the observed n-th harmonic flow., and [MATH] is the n-th harmonic event plane angle reconstructed by the produced particles, defined as: [EQUATION]', '1506.03158-2-8-4': 'The sum goes over all charged particles in the event, [MATH] is a weight applied to optimize the event plane resolution, see [CITATION] for details.', '1506.03158-2-8-5': 'It is worth to note that the range of n-th harmonic event plane angle is 0 [MATH]/n.', '1506.03158-2-9-0': 'In this method the n-th observed harmonic flow can be corrected for statistical effects by the n-th harmonic event plane resolution: [EQUATION] with [EQUATION]', '1506.03158-2-9-1': 'Here [MATH] denotes the average over all particles in all events and [MATH] is the n-th real event plane angle [CITATION].', '1506.03158-2-9-2': 'The real event plane means the participant plane [CITATION], which can not be achieved due to finite multiplicity of an event.', '1506.03158-2-10-0': 'To reduce the short range "non-flow" effects such as HBT correlations, a [MATH]-sub event plane method was introduced [CITATION].', '1506.03158-2-10-1': 'The [MATH]-sub event plane method divides the event into two independent sub-events with different pseudo rapidity ([MATH]) ranges.', '1506.03158-2-10-2': 'An additional gap of 0.1 in pseudo rapidity is added to further avoid short range correlations.', '1506.03158-2-10-3': 'The procedure results in a positive and negative [MATH]-sub event plane.', '1506.03158-2-10-4': 'The n-th observed harmonic flow can now be calculated in respect to the [MATH]-sub event planes.', '1506.03158-2-10-5': 'As a consequence, the [MATH]-sub event plane resolution [CITATION] is used instead of full event plane resolution: [EQUATION] where the [MATH] and the [MATH] are the n-th harmonic event plane angles for the positive and negative [MATH]-sub ranges.', '1506.03158-2-11-0': '## AMPT Model', '1506.03158-2-12-0': 'AMPT is a transport model which consists of four main components: the initial conditions, partonic interactions, conversion from partonic to hadronic matter, and hadronic interactions [CITATION].', '1506.03158-2-12-1': 'It has two versions to deal with different scenarios which are default AMPT (AMPT Def in following figures) and string melting AMPT (AMPT Str in following figures).', '1506.03158-2-12-2': 'The initial conditions are generated by the HIJING (Heavy Ion Jet Interaction Generator) model [CITATION].', '1506.03158-2-12-3': 'HIJING includes only two body nucleon-nucleon interactions and generates mini jets and excited strings through hard processes and soft processes separately.', '1506.03158-2-13-0': 'Excited strings are treated differently in the two AMPT versions.', '1506.03158-2-13-1': 'In default AMPT, excited strings are combined to hadrons according to the Lund string fragmentation model they further go through a hadronic interaction stage [CITATION].', '1506.03158-2-13-2': 'In string melting AMPT, excited strings first convert to partons, i.e. melting, and then go through a partonic interaction stage with original soft partons.', '1506.03158-2-13-3': "The partonic interactions for both default AMPT and string melting AMPT are described by ZPC (Zhang's Parton Cascade) model [CITATION].", '1506.03158-2-13-4': 'In the final stage of the ZPC model, partons in default AMPT are recombined with parent strings and hadronize via the Lund string fragmentation model.', '1506.03158-2-13-5': 'However, in string melting AMPT, the hadronization of partons is described by a coalescence model.', '1506.03158-2-13-6': 'After hadronization, the hadronic interactions are modelled by ART (A Relativistic Transport) model [CITATION].', '1506.03158-2-14-0': 'In previous studies it was found that a large parton scattering cross section (approximately 6 mb or 10 mb) is needed to reproduce the observed elliptic flow of charged hadrons [CITATION].', '1506.03158-2-14-1': 'In these studies, the elliptic flow was calculated with respect to the reaction plane (determined by the beam axis and impact parameter direction), which is 0 in the AMPT model, and this might underestimate the elliptic flow observed in the experiment data, which measured relative to the event plane but not the reaction plane [CITATION].', '1506.03158-2-14-2': 'As shown in Ref. [CITATION], both the charged particle multiplicity and elliptic flow can be reproduced with a smaller parton scattering cross section (1.5 mb) by using the default values for the parameters in the Lund string fragmentation function.', '1506.03158-2-15-0': 'In this paper, approximately 20 million events for 0%-80% central Au+Au collisions at [MATH] 11.5, 39, and 200 GeV with default AMPT (v1.25) and string melting AMPT (v2.25) were generated.', '1506.03158-2-15-1': 'Three different parton scattering cross sections (1.5 mb, 3 mb, and 6 mb) are used in the string melting version of AMPT.', '1506.03158-2-15-2': 'The parameters in the Lund string fragmentation function are set to the default values for AMPT with 1.5 mb parton scattering cross section (the values can be found at Table I in Ref. [CITATION]).', '1506.03158-2-15-3': 'Furthermore the [MATH]-sub event plane method is used to calculate elliptic and triangular flow.', '1506.03158-2-16-0': '# Results', '1506.03158-2-17-0': 'Identical analysis steps are used for simulation and data collected from the experiments [CITATION].', '1506.03158-2-17-1': 'Finial state charged particles with 0.2 [MATH] 2.0 GeV/c and [MATH] 1.0, are used to reconstruct the event plane.', '1506.03158-2-17-2': 'The [MATH] range of the two [MATH]sub events are -1.0 [MATH] -0.05 and 0.05 [MATH] 1.0, respectively.', '1506.03158-2-17-3': 'The elliptic and triangular flow are calculated within -1.0 [MATH] 1.0, and the auto-correlation is naturally avoided by the [MATH]-sub event plane method.', '1506.03158-2-17-4': 'For all the plots in this section, only statistical errors are shown.', '1506.03158-2-18-0': '## Elliptic Flow and Triangular Flow at RHIC', '1506.03158-2-19-0': 'Figure [REF] presents the second and third harmonic event plane resolutions from the AMPT string melting model for parton scattering cross sections of 1.5 mb, 3 mb and 6 mb from 0%-80% central Au+Au collisions at [MATH] 11.5, 39 and 200 GeV by using the [MATH]-sub event plane method.', '1506.03158-2-19-1': 'The event plane resolution shows a clear parton scattering cross section and energy dependence: both second and third harmonic event plane resolution increases with increasing parton scattering cross section and energy.', '1506.03158-2-20-0': 'The third harmonic event plane resolution is about two times smaller and peaks more central than the second harmonic event plane resolution, which is comparable to the one from experiment.', '1506.03158-2-20-1': 'For [MATH] 11.5 and 39 GeV, [MATH] can only be calculated up to 0-60 (0-70) for some of the parton scattering cross sections, since we only have event plane resolution up to these centrality.', '1506.03158-2-21-0': 'Figure [REF], Fig. [REF] and Fig. [REF] show the comparison between data from experiment and from the AMPT model for [MATH] of identified particles ([MATH] and [MATH]) at [MATH] 11.5, 39 and 200 GeV from 0-80 central Au+Au collisions.', '1506.03158-2-21-1': 'The data from experiment are taken from Ref. [CITATION].', '1506.03158-2-21-2': 'For all three energies it is observed that the elliptic flow increases with increasing parton scattering cross section.', '1506.03158-2-21-3': 'The elliptic flow from default AMPT is lower than the corresponding results from experiment, which indicates that the parton scattering process is important to produce a large elliptic flow as observed in experiment data.', '1506.03158-2-21-4': 'Previous studies show that a 6-10 mb parton scattering cross section is needed to reproduce the elliptic flow observed in experiment data [CITATION].', '1506.03158-2-21-5': 'In those studies the elliptic flow was calculated relative to the reaction plane.', '1506.03158-2-22-0': 'By using the [MATH]-sub event plane method most of the experiment data points at low transverse momenta are between AMPT default and AMPT string melting with a 3 mb parton scattering cross section.', '1506.03158-2-22-1': 'At intermediate [MATH] some particle species, e.g. [MATH], are systematically above AMPT string melting with a 3 mb parton scattering cross section.', '1506.03158-2-22-2': 'That means previous studies have underestimated the elliptic flow by using the reaction plane instead of the event plane.', '1506.03158-2-22-3': 'Latter one fluctuates event-by-event around the reaction plane.', '1506.03158-2-22-4': 'The collective flow measured in experiment are not mean values, but closer to a root-mean-sqaure [CITATION], therefore, the event-by-event fluctuations did not cancel each other but give a positive contribution to the measured collective flow, which makes the collective flow relative to the event plane is always larger compare to the collective flow relative to the reaction plane.', '1506.03158-2-23-0': 'Figure [REF] shows the comparison between experiment data [CITATION] and AMPT model calculations for [MATH] at [MATH] 200 GeV from 0-80 central Au+Au collisions of identified particles ([MATH] and [MATH]).', '1506.03158-2-23-1': 'A similar parton scattering cross section dependence is observed as for the elliptic flow, but here the dependence is stronger: [MATH] is about 40 lower than [MATH] with 1.5 mb parton scattering cross section but comparable with [MATH] with a 6 mb parton scattering cross section.', '1506.03158-2-23-2': 'The driving force behind [MATH] is the almond shape of initial nuclei overlap region, therefore, [MATH] has a relative large value compare to [MATH] which is mainly generated by initial state fluctuations [CITATION].', '1506.03158-2-23-3': 'This effect can be observed by comparing the [MATH] and [MATH] values from AMPT default model calculations in Fig. [REF] and Fig. [REF].', '1506.03158-2-23-4': 'The higher parton scattering cross section makes the transportation from the initial coordinate space to the final momentum space more efficient, which increases the values for both [MATH] and [MATH].', '1506.03158-2-23-5': 'The relative increase of [MATH] is larger than for [MATH], since the effects of the parton scattering is more important for [MATH].', '1506.03158-2-23-6': 'The triangular and elliptic flow from experiment can be well reproduced by AMPT default and AMPT string melting with a 1.5 mb parton scattering cross section, see Fig. [REF] and Fig. [REF].', '1506.03158-2-23-7': 'It is worth to be noticed here that we did not find out a single parton scattering cross section to describe all particle species.', '1506.03158-2-23-8': 'This is because the collective flow results from AMPT model depend on both the magnitude and the distribution of parton scattering cross section.', '1506.03158-2-23-9': 'Recent analysis [CITATION] showed that the experiment elliptic flow results can be described by the string melting AMPT model with a smaller but more isotropic parton scattering cross section.', '1506.03158-2-23-10': 'Thus, by carefully tuning the parton scattering cross section and its distribution, we might achieve a roughly description of all parton species, but this is beyond the goal of this paper.', '1506.03158-2-24-0': '## Differential [MATH] Ratio from AMPT Model', '1506.03158-2-25-0': 'Figure [REF] depicts AMPT model calculations for the transverse momentum dependent differential ratio [MATH] for identified particles ([MATH] and [MATH]) from 0%-80% central Au+Au collisions at [MATH] 200 GeV.', '1506.03158-2-25-1': 'The differential ratio decreases significantly at low transverse momenta ([MATH] 1.5 GeV/c) and becomes flat at intermediate transverse momenta (1.5 [MATH] 3.0 GeV/c).', '1506.03158-2-25-2': 'The [MATH] dependent differential ratio shows an anti-correlation with parton scattering cross section for all particle species: the higher the parton scattering cross section, the lower the [MATH] ratio.', '1506.03158-2-25-3': 'This is in agreement with a larger increase of [MATH] with increasing parton scattering cross section compare to [MATH] discussed in Fig. [REF] and Fig. [REF].', '1506.03158-2-25-4': 'This effect can also be understood in a hydrodynamic frame work.', '1506.03158-2-25-5': 'The higher parton scattering cross section in the AMPT string melting model is equivalent to a lower viscosity in viscous hydrodynamics [CITATION].', '1506.03158-2-25-6': 'From this study it is known that [MATH] is more sensitive to viscosity than [MATH] [CITATION], therefore a lower viscosity (higher parton scattering cross section) leads to a lower [MATH] ratio.', '1506.03158-2-26-0': '## Integrated [MATH] Ratio from AMPT Model', '1506.03158-2-27-0': 'Figure [REF] shows the AMPT calculation for the integrated [MATH] ratio for identified particles from 0%-80% central Au+Au collisions at [MATH] 39 and 200 GeV.', '1506.03158-2-27-1': 'The integration region is limited to 1.5 [MATH] 2.8 GeV/[MATH] to avoid the steep decrease in the differential ratio and the large error region.', '1506.03158-2-27-2': 'Please note, the triangular flow used to calculate the [MATH] ratio for 1.5 mb and 3 mb parton scattering cross section are taken from 0-60 and 0-70 due to the event plane resolution limits, as shown in Fig. [REF].', '1506.03158-2-27-3': 'The other of [MATH] values are taken from 0-80 results.', '1506.03158-2-28-0': 'A similar anti-correlation between the integrated [MATH] ratio and the parton scattering cross section is observed as in the differential [MATH] ratio: the larger the parton scattering cross section, the smaller the integrated [MATH] ratio.', '1506.03158-2-28-1': 'In addition, for the same parton scattering cross section, an energy dependence is observed: the lower the energy the larger the integrated [MATH] ratio.', '1506.03158-2-28-2': 'We further observe that the integrated [MATH] ratio calculated by string melting AMPT shows no mass (particle type) dependence, the ratio between integrated [MATH] of all particle species to that of [MATH] are consistent with 1 within 10%.', '1506.03158-2-28-3': 'On the other hand the ratios of [MATH] from default AMPT are lower than the rest of the particle species at both [MATH] 39 and 200 GeV.', '1506.03158-2-28-4': 'This indicates that the quark coalescence mechanism might play an important role in flattening the [MATH] ratio.', '1506.03158-2-28-5': 'The no mass (particle type) dependence of the [MATH] ratio was predicted by viscous hydrodynamics, for fast particles (particles with large [MATH]).', '1506.03158-2-28-6': 'The [MATH] ratio is constant and does not depend on the mass of the particle [CITATION].', '1506.03158-2-28-7': 'Based on the viscous hydrodynamics in Ref. [CITATION], the collective flow depends on dissipative contributions and the Fourier coefficient ([MATH]) of the flow velocity distribution relative to the event plane, the dissipative contributions cancelled each other in [MATH] ratio calculation and only [MATH] left which does not depend on the mass of different particle species.', '1506.03158-2-28-8': 'Since the parton scattering cross section is related to the viscosity in hydrodynamics, the magnitude of the [MATH] ratio might be usable to quantify the viscosity of the system.', '1506.03158-2-29-0': '# Summary', '1506.03158-2-30-0': 'The AMPT calculations of elliptic flow at [MATH] 11.5, 39 and 200 GeV and triangular flow at [MATH] 200 GeV of identified particles ([MATH] and [MATH]) from 0%-80% central Au+Au collisions were presented.', '1506.03158-2-30-1': 'Most of results from experiments can be reproduced with a parton scattering cross section between 1.5 and 3 mb by following the same analysis method as used in the experiments.', '1506.03158-2-30-2': 'An anti-correlation between the parton scattering cross section for both the differential and integrated [MATH] ratio is observed.', '1506.03158-2-30-3': 'This is due to a different sensitivity of [MATH] and [MATH] to the parton scattering cross section.', '1506.03158-2-30-4': 'A mass (particle type) independence of the integrated [MATH] ratio is observed from string melting AMPT, but not for the default AMPT.', '1506.03158-2-30-5': 'The no mass (particle type) dependence of the [MATH] ratio also can be understood in the viscous hydrodynamic framework.', '1506.03158-2-30-6': 'Latter one suggests that the magnitude of the ratio might be used to quantify the viscosity of the system.', '1506.03158-2-31-0': 'We thank Dr. Nu Xu for his great idea and important discussions and Dr. Guoliang Ma for his help on AMPT model calculations.', '1506.03158-2-31-1': 'This work was supported by the National Natural Science Foundation of China under grant No.11475070 and U1332125 and the Program for Innovation Research of Science in Harbin Institute of Technology (PIRS OF HIT B201408).'}

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[]

[['1506.03158-1-31-0', '1506.03158-2-31-0']]

[]

['1506.03158-1-2-7', '1506.03158-1-17-2', '1506.03158-2-2-7', '1506.03158-2-17-2']

{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1506.03158

hep-th-9701046

{'hep-th-9701046-1-0-0': '0.3cm We review the fundamentals of Jahn-Teller interactions and their field theoretical modelings and show that a 2+1 dimensional gauge theory where the gauge field couples to flavored fermions arises in a natural way from a two-band model describing the dynamical Jahn-Teller effect.', 'hep-th-9701046-1-0-1': 'The theory exhibits a second order phase transition to novel finite-temperature superconductivity.', 'hep-th-9701046-1-0-2': '0.6cm 1.', 'hep-th-9701046-1-0-3': 'Introduction 0.4cm', 'hep-th-9701046-1-1-0': 'The discovery of the cuprate high-[MATH] superconductors [CITATION] together with the fact that there is to date no generally accepted theory for the relevant mechanism [CITATION], gives physicists reasons to search for novel scenarios responsible for macroscopic quantum coherence phenomena in condensed matter physics.', 'hep-th-9701046-1-1-1': 'One of the main problems is the dependence of the nature of superconducting phase transitions on dimensionality.', 'hep-th-9701046-1-1-2': 'Of course, one has to be careful to make statements like high-[MATH] superconductivity is essentially a two-dimensional problem since it is not a priori clear how the very nature of the superconducting phase transition in the cuprates is related to or affected by some kind of interplane coupling.', 'hep-th-9701046-1-1-3': 'Nevertheless, there is no doubt that a general (i.e. quantum field theoretical) study of the possibility of idealized two-dimensional superconductivity at finite temperature is a challenging task.', 'hep-th-9701046-1-1-4': 'There is an old celebrated theorem due to Hohenberg, Mermin and Wagner, and Coleman [CITATION] stating that conventional off-diagonal long range order (ODLRO) is suppressed at any finite temperature in 2+1 dimensional quantum systems.', 'hep-th-9701046-1-1-5': 'At this point the fundamental question of the very existence of loopholes arises - as often in theoretical physics.', 'hep-th-9701046-1-1-6': 'In the following we present a proposal for a microscopic mechanism which may provide such a loophole and possibly explain high-[MATH] superconductivity or may open the door to even more interesting superconducting materials.', 'hep-th-9701046-1-1-7': 'It is based on an effective electron-electron (resp. hole-hole) interaction induced by a generalized dynamical Jahn-Teller effect [CITATION].', 'hep-th-9701046-1-1-8': 'Akin to fractional statistics this interaction is a phasing interaction, i.e. one exitation modulates the quantum mechanical phase of the other exitation giving rise to a net attractive interaction for the relevant fermions.', 'hep-th-9701046-1-1-9': 'Alternatively, a phasing interaction may be viewed as a renormalization of the statistical properties of the quanta changing their statistical identity.', 'hep-th-9701046-1-1-10': 'It is our objective to make plausible that it is the phasing aspect overlooked in more conventional treatments of the Jahn-Teller interaction which may play a crucial role in the mechanism of high-[MATH] superconductivity.', 'hep-th-9701046-1-1-11': 'Our discussion is organized as follows: We start by recalling the early history of the Jahn-Teller theory in molecular physics.', 'hep-th-9701046-1-1-12': 'We proceed by reviewing the concept of the geometrical phase (now called Berry phase and Aharonov-Anandan phase in the adiabatic and in the non-adiabatic cases, respectively) [CITATION] first analyzed systematically in the context of quantum chemistry in the pioneering work of Mead who dubbed this phenomenon molecular Aharonov-Bohm effect [CITATION].', 'hep-th-9701046-1-1-13': 'Then we show how to set up a theory of strongly correlated electrons interacting via a generalized dynamical Jahn-Teller interaction.', 'hep-th-9701046-1-1-14': 'Motivated by the work of Yu and Anderson [CITATION] we show that our ansatz can be expressed in terms of a double-well tunneling event which we christened solid state instanton.', 'hep-th-9701046-1-1-15': 'It is related to what is known as a polaron in solid state physics, but it is not exactly the same thing.', 'hep-th-9701046-1-1-16': 'The microscopic double-well mechanism allows to break parity and time-reversal spontaneously thus giving a microscopic reason for the existence of anyons without being restricted to them.', 'hep-th-9701046-1-1-17': "This double-well idea is intimately related to T. D. Lee's early 70's work on CP violation in elementary particle physics [CITATION].", 'hep-th-9701046-1-1-18': 'It is very amusing to observe that already T. D. Lee relates his own concept to geometrical ideas associated with abstract distortive deformations.', 'hep-th-9701046-1-1-19': 'Notice that the actual breaking and its magnitude of these discrete symmetries depend on a fine tuning of the coupling parameters.', 'hep-th-9701046-1-1-20': 'The effective Lagrangean describing such a polaronic interaction is by no means unique and adding or discarding terms determines whether P and T are broken or not.', 'hep-th-9701046-1-1-21': 'Moreover, there may exist the possibility of a phase transition between an anyonic and a Berezinskii-Kosterlitz-Thouless-type phase [CITATION] within the high-temperature conducting phase.', 'hep-th-9701046-1-1-22': 'Finally we show how to model the generalized dynamical Jahn-Teller interaction field theoretically and give an explicit example for a quasi two-dimensional finite temperature ansatz - firstly written down and analyzed by Joe Kapusta and his collaborators (without any reference to the generalized dynamical Jahn-Teller mechanism, however).', 'hep-th-9701046-1-1-23': 'This model indeed shows up a finite temperature phase transition to novel superconductivity within a finite window of parameters [CITATION].', 'hep-th-9701046-1-1-24': 'Its phase strucuture is very reminiscent of the phase diagram proposed by Chakraverty in 1979 [CITATION] which led Bednorz and Muller [CITATION] to their discovery of novel superconductivity.', 'hep-th-9701046-1-1-25': 'This model should be seen in a wider context including two-dimensional scenarios without parity and time-reversal violation.', 'hep-th-9701046-1-1-26': 'Finally, we give a heuristical argument for macroscopic quantum coherence induced by Jahn-Teller systems.', 'hep-th-9701046-1-1-27': 'The field theoretical ideas presented here are based on joint work with Heinz [CITATION].', 'hep-th-9701046-1-1-28': '0.6cm 2.', 'hep-th-9701046-1-1-29': 'What is the Jahn-Teller effect?', 'hep-th-9701046-1-1-30': '0.4cm As early as 1929 v.Neumann and Wigner [CITATION] asked a very interesting question: Given a hermitean [MATH] matrix whose entries are depending on a sequence of parameters [MATH] - how many parameters have to be changed in order to get a collision of two eigenvalues of the matrix.', 'hep-th-9701046-1-1-31': 'The general answer is at least three.', 'hep-th-9701046-1-1-32': 'Let us make this explicit for the most simplest case [MATH].', 'hep-th-9701046-1-1-33': 'Any hermitean [MATH] matrix [MATH] may be expanded with respect to the Pauli basis: [EQUATION]', 'hep-th-9701046-1-1-34': 'Solving the eigenvalue equation [MATH] gives [EQUATION]', 'hep-th-9701046-1-1-35': "Clearly we have to send all three space-like [MATH]'s to zero in order to get a collision!", 'hep-th-9701046-1-1-36': 'Evidently, in case of a real hermitean (i.e. symmetric) matrix this reduces to two parameters, in case of a real diagonal matrix to one parameter.', 'hep-th-9701046-1-1-37': 'Interpreting the space of independent parameters as a mechanical configuration space and cutting out the point of conicidence, i.e. the level crossing point, we obtain manifolds with non-trivial topological structures.', 'hep-th-9701046-1-1-38': 'This suggests that in all three cases, once physically realized, we may expect topological quantization effects giving rise to very interesting physics.', 'hep-th-9701046-1-1-39': 'To get real physics from this mathematical observations we must find a Hamiltonian having the structure of [MATH] and a mechanism which prevents the level crossing.', 'hep-th-9701046-1-1-40': 'This was done in the pioneering work of Jahn and Teller in 1937 who studied the stability of polyatomic molecules in degenerate electronic states [CITATION].', 'hep-th-9701046-1-1-41': 'Essentially the famous Jahn-Teller theorem states the following: A configuration of a polyatomic molecule for an electronic state having orbital degeneracy cannot be stable with respect to all displacements of the nuclei unless the nuclei all lie on a straight line.', 'hep-th-9701046-1-1-42': 'The original proof of Jahn and Teller [CITATION] is based on a detailed discussion of particular symmetries and their realizations.', 'hep-th-9701046-1-1-43': 'A more general proof within the framework of induced representations of finite groups was given by Ruch and Schonhofer in 1965 [CITATION].', 'hep-th-9701046-1-1-44': 'It is interesting to note that - by the side of the Weiss theory of ferromagnetism - the Jahn-Teller effect is the archetype of what is commonly called spontaneously symmetry breaking (SSB), a fact recently recalled by Nambu [CITATION].', 'hep-th-9701046-1-1-45': 'In modern terminology a dynamics determined by a sensible matching of vibration modes and electronic excitations is called a vibronic interaction [CITATION].', 'hep-th-9701046-1-1-46': 'There is no doubt that a phonon induced fermion-fermion interaction incorporating Jahn-Teller-type effects may exhibit new pecularities beyond the marks of the conventional electron-phonon interaction.', 'hep-th-9701046-1-1-47': 'Due to the non-trivial topological structure of the configuration spaces involved, we may expect highly non-trivial quantization and coherence phenomena, and it is the aim of this discussion to push forward the thesis that the understanding of high-[MATH] superconductivity is at least related to the generalized dynamics of vibronic interactions.', 'hep-th-9701046-1-1-48': '0.6cm 3.', 'hep-th-9701046-1-1-49': 'The geometric phase in Jahn-Teller systems 0.4cm To sum up, the (static) Jahn-Teller effect is an electronic symmetry breaking phenomenon associated with a spontaneous distortion.', 'hep-th-9701046-1-1-50': 'To be concrete, let us consider a vibronic interaction of a doubly degenerate electronic state [MATH] with a doubly degenerate vibrational mode [MATH] [CITATION].', 'hep-th-9701046-1-1-51': 'According to the Jahn-Teller theorem the nuclear motion lifts the electronic degeneracy, i.e. there are nuclear configurations of lower energy than the symmetric state.', 'hep-th-9701046-1-1-52': "This [MATH] Jahn-Teller effect is the archetype of Berry's phase: A quantum-mechanical phase shift of purely geometrical origin associated to an adiabatic cycle starting and ending at the same pure state [CITATION].", 'hep-th-9701046-1-1-53': 'In our case we start by writing down the quantum-mechanical two-dimensional harmonic oscillator Hamiltonian for a double degenerated vibrational mode [EQUATION] with [MATH] and [MATH].', 'hep-th-9701046-1-1-54': 'This Hamiltonian is thought to be acting on two-component (Pauli) wave functions [MATH] (i=1,2).', 'hep-th-9701046-1-1-55': 'The vibronic coupling is given by adding a term [MATH][MATH] to [MATH], whereby [MATH] denote the Pauli matrices and k is a coupling constant.', 'hep-th-9701046-1-1-56': '(We use the letter [MATH] to avoid confusion with spin degrees of freedom).', 'hep-th-9701046-1-1-57': 'In analogy to elementary particle physics we call the associated internal quantum number of the electron a Jahn-Teller isospin or flavor.', 'hep-th-9701046-1-1-58': 'Explicitely we have [EQUATION] with [MATH] and [MATH].', 'hep-th-9701046-1-1-59': 'Let us rewrite the Hamiltonian in cylinder coordinates: [EQUATION]', 'hep-th-9701046-1-1-60': 'In the adiabatic or Born-Oppenheimer approximation we neglegt the kinetic energy term and diagonalize the remainder.', 'hep-th-9701046-1-1-61': 'We obtain [EQUATION] with [EQUATION]', 'hep-th-9701046-1-1-62': 'Note that the matrix [EQUATION] is a double-valued function in the polar angle [MATH].', 'hep-th-9701046-1-1-63': 'The associated energy eigenvalues are [EQUATION] (where we have written [MATH] for [MATH]) and correspond to two sheets (the upper cone-like, the lower sombrero-like) coinciding at a point of degeneracy at the origin.', 'hep-th-9701046-1-1-64': 'This point of zero distortion defines a conical intersection.', 'hep-th-9701046-1-1-65': 'Because of the double-valued character of the diagonalizing similarity transformation the eigenstates [MATH] corresponding to [MATH] are double-valued in [MATH].', 'hep-th-9701046-1-1-66': 'The multiple-valuedness may be compensated for by an appropriate local gauge transformation [EQUATION] which in turn induces the change [EQUATION] with [EQUATION] in the nuclear energy operator.', 'hep-th-9701046-1-1-67': 'The vector potential [MATH] corresponds to a fictitious flux tube with strength [MATH] confined to the origin.', 'hep-th-9701046-1-1-68': 'In order to visualize the circuit in [MATH] inducing the quasi-spinorial sign change of the electronic states one takes a look on a typical example for the [MATH] Jahn-Teller effect: the trimer.', 'hep-th-9701046-1-1-69': 'A circuit of distortions - corresponding to the natural motions of the nuclei - avoids the point of symmetry, at which the trimer looks like an equilateral triangle.', 'hep-th-9701046-1-1-70': 'Of course, while it is true, that we generally assume that the complete system must be described by a single-valued wave function, it is the splitting between the subsystem and its relative environment which introduces the geometrical phase factor: Hence the multiple-valuedness of the electronic wavefunction is compensated for by the multiple-valuedness of the nuclear wavefunctions.', 'hep-th-9701046-1-1-71': '0.6cm 4.', 'hep-th-9701046-1-1-72': 'Beyond the adiabatic approximation 0.4cm In the last section we observed that in a Jahn-Teller system (hereafter designated by JT) the electronic wave functions are in general multiple-valued functions in the slow nuclear coordinates; in particular they are double-valued in the nuclear polar angle in the case of a trimer.', 'hep-th-9701046-1-1-73': 'Our claim is that this could give rise to a novel quality of an electron-electron interaction mediated by an oscillating JT ionic configuration.', 'hep-th-9701046-1-1-74': 'In the following - generalizing this classical dynamical yet adiabatic JT approach somewhat - we will go beyond the approximation of Born and Oppenheimer and write down an ansatz for an effective field theory describing a dynamical non-adiabatic JT interaction.', 'hep-th-9701046-1-1-75': 'One reason for this proceeding lies in the fact that e.g. for the case of the octahedron in the [MATH] superconductor we encounter neither an appropriate degeneracy nor an appropriate configuration space of distortions justifying the applicability of the naive JT theorem and the Mead model described above [CITATION].', 'hep-th-9701046-1-1-76': 'Conversely, what we expect is that the point of degeneracy in the relevant JT-like system is smeared-out (just like a smeared-out Aharonov-Bohm flux line or a regularized anyon), such that the adiabatic transformation definitely breaks down and, in addition, the oscillating ionic arrangement does not simply sweep through all configurations classically possible, but tunnels between some of them instead.', 'hep-th-9701046-1-1-77': 'We propose a scenario in which both delicate aspects appear as natural consequences of the same fundamental mechanism, but nevertheless, the topological quality of the interaction, reminiscent of Meads molecular Aharonov-Bohm effect [CITATION] will survive.', 'hep-th-9701046-1-1-78': 'To convert the nuclear Hamiltonian [EQUATION] into a field theoretical Hamiltonian describing an electron-electron interaction mediated by an oscillating ionic configuration we simply make the replacement [EQUATION] and introduce phonon field operators such that we get a two-band field theory with electron-phonon vertices.', 'hep-th-9701046-1-1-79': 'A more general Hamiltonian may be obtained by introducing weights for the different modes and adding a conventional term.', 'hep-th-9701046-1-1-80': 'Integrating out phonons in a standard way we obtain a four-fermion BCS-like Hamiltonian representing interband-intraband interactions with some wrong-sign couplings.', 'hep-th-9701046-1-1-81': 'Englman, Halperin, and Weger proposed a JT theory for the high-[MATH] superconductivity of the cuprates, in which the coupling between the copper [MATH] and [MATH] states leads to a pairing mechanism of the same form, but opposite sign to that of the BCS theory [CITATION].', 'hep-th-9701046-1-1-82': 'They argued that their own ansatz is well-supported by band structure calculations and experiments indicating the involvement of both [MATH] and [MATH] type states in the carrier states of cuprate superconductors.', 'hep-th-9701046-1-1-83': 'Furthermore they showed that the proposed pairing mechanism is stable against lattice distortion even for strong coupling.', 'hep-th-9701046-1-1-84': 'It is noteworthy to remark that there are a number of proposals how the JT effect comes into play in superconductors, in particular in high-[MATH] superconductors.', 'hep-th-9701046-1-1-85': 'Some classical papers can be found in Ref. [CITATION] and more current contributions are listed in Refs. [CITATION].', 'hep-th-9701046-1-1-86': 'Especially interesting is the recent work of K. H. Johnson et al. [CITATION] who argued that the observed superconductivity at 18 K in potassium-doped fullerene is induced by a cooperative JT coupling leading to a BCS-like mechanism.', 'hep-th-9701046-1-1-87': 'Topological aspects relating the JT phenomenon and superconductivity have been almost ignored up until now.', 'hep-th-9701046-1-1-88': 'The only paper, to our knowledge, relating topological quantization effects (especially fractional quantization) to the JT effect and superconductivity was written by Kuratsuji [CITATION].', 'hep-th-9701046-1-1-89': 'Appel pointed out to me that if high-[MATH] superconductivity is due to a JT-like scenario then the description of the relevant mechanism surely has to go far beyond the adiabatic approximation [CITATION].', 'hep-th-9701046-1-1-90': 'In particular he was inspired by the work of Cohen and collaborators [CITATION] who emphasized that - due to the fundamental instability of the oxygen ion which causes its motions influencing the charge density between the copper and oxygen - anharmonic double-well potentials for normal modes may give larger coupling then expected from harmonic phonons and are less sensitive to the mass.', 'hep-th-9701046-1-1-91': 'Appel argued that the local phonon ansatz by Yu and Anderson [CITATION] - considered as the non-adiabatic extension of the dynamical JT effect - provides a suitable framework to describe the fundamental interaction.', 'hep-th-9701046-1-1-92': 'Note that we are not interested in the exact details of the interaction (e.g. apex in-plane charge interaction, apex positional splitting, out-of-plane motions etc.); we only assume that the essential dynamics is governed by an inharmonic potential.', 'hep-th-9701046-1-1-93': 'In a Yu-Anderson-type model we restrict ourselves to a one mode description replacing the double sheeted sombrero by a double sheeted double well.', 'hep-th-9701046-1-1-94': 'Explicitely we have [EQUATION]', 'hep-th-9701046-1-1-95': 'To diagonalize the phonon matrix we introduce the chiral - i.e. left-handed and right-handed - linear combinations [EQUATION] and get [EQUATION] i.e. an interaction term proportional to [MATH].', 'hep-th-9701046-1-1-96': 'Yu and Anderson proceed by integrating out the electron degrees of freedom and calculate the dynamical modification of the harmonic oscillator potential giving a dynamical double well[MATH] replacing the more singular [MATH] term.', 'hep-th-9701046-1-1-97': 'The calculation is very involved and relies heavily of path integral techniques reminiscent of instanton calculations in quantum field theory.', 'hep-th-9701046-1-1-98': 'An effective non-relativistic Lagrangean for this model may have the form [EQUATION] where [MATH] is a quartic term and [MATH][MATH] is a two-component Schrodinger field.', 'hep-th-9701046-1-1-99': 'That anharmonicity modifies the mass-frequency relation of a quantum mechanical oscillator and hence the isotope effect is due to the non-analytic character of the solution of the double-well tunneling problem - a rather general feature.', 'hep-th-9701046-1-1-100': 'In field theory tunneling events are called instantons and it is, to our opinion, appropriate to name the Yu-Anderson local phonon a solid state instanton.', 'hep-th-9701046-1-1-101': 'Summarizing, complementary to the BCS-like four fermion interaction a la Engelman, Halperin, and Weger [CITATION] which is obtained by integrating out the phonons, we get an effective anharmonic phonon potential by integrating out the fermions.', 'hep-th-9701046-1-1-102': 'A very interesting point lies in the fact that double-well system is intimately related to a two-level system in that the lowest states of the former are to be identified with the only states spanning the latter.', 'hep-th-9701046-1-1-103': 'Now the dynamics of the two-level system considered as an abstract spin-[MATH] system is driven by an abstract external magnetic field - self-consistently generated through the local phonon tunneling dynamics.', 'hep-th-9701046-1-1-104': 'In case of a real spin-[MATH] system the driving external magnetic field introduces definitely an oddness under time reversal.', 'hep-th-9701046-1-1-105': 'We do not really know under which conditions this oddness under time reversal carries over to the abstract case, but at least as a possibility it remains.', 'hep-th-9701046-1-1-106': 'The oddness under T is also suggested by the fact that a hidden parity violation is already present in the model due to the interference of odd and even modes and due to the fact that PT should be a good symmetry in solid state physics.', 'hep-th-9701046-1-1-107': 'Hence a relativistic ansatz for a T, P, and C invariant effective Lagrangean based on pseudoscalar anharmonic phonons may be written [EQUATION] with [EQUATION] where we demand [MATH], and the relativistic spin degrees of freedom are identified with the two bands of the generalized JT interaction in the non-relativistic limit.', 'hep-th-9701046-1-1-108': '0.6cm 5.', 'hep-th-9701046-1-1-109': "A phase transition towards two-dimensional superconductivity 0.4cm Our relativistic Lagrangean is identical to the one studied by T. D. Lee in the early 70's as a simple example for spontaneous T violation [CITATION], a phenomenon discussed in the framework of anyon physics [CITATION].", 'hep-th-9701046-1-1-110': 'Unfortunately, the experimental situation is compatible with the absence of anyons in high-[MATH] materials rather than with their presence [CITATION].', 'hep-th-9701046-1-1-111': 'Nevertheless, if high-[MATH] superconductivity is still a really two-dimensional phenomenon, then we should continue to study anyon superconductivity because it is a nice toy model possibly exhibiting features of the true theory.', 'hep-th-9701046-1-1-112': 'Therefore our stategy is to develop a theory in which anyons appear as a consequence of a microscopic mechanism and induce finite temperature superconductivity.', 'hep-th-9701046-1-1-113': 'At the end we will try to find out how to modify the model in order to preserve P and T. Assuming they exist, anyons are never elementary particles like electrons such that the question remains: How can we get anyons and hence spontaneously T violation from fundamental electronic interactions?', 'hep-th-9701046-1-1-114': 'We think it is near at hand and much more natural to reverse the standard argumentation and consider the violation of time reversal and not necessarily the validity of the two-dimensional description as the main problem.', 'hep-th-9701046-1-1-115': "This view is also supported by Wilczek's axion electrodynamics [CITATION] and the chiron model by Chaplin and Yagamishi [CITATION].", 'hep-th-9701046-1-1-116': 'Both models go beyond two dimensions while preserving the P and T violating character.', 'hep-th-9701046-1-1-117': 'Indeed, we should find the reason for anyons - departing from a microscopic picture.', 'hep-th-9701046-1-1-118': 'Our picture is that it is a background of tunneling impurities which does generate an effective four fermion interaction.', 'hep-th-9701046-1-1-119': "In particular, we expect that it should be possible to derive anyon physics as a consequence of this ansatz casted in the form of T. D. Lee's Lagrangean.", 'hep-th-9701046-1-1-120': 'Let us briefly sketch the original Lee mechanism.', 'hep-th-9701046-1-1-121': 'Though the Langrangean is invariant under time reversal and parity, the vacuum expectation value [EQUATION] of the [MATH] field is not: It changes its sign under P and T.', 'hep-th-9701046-1-1-122': 'In the tree graph approximation [MATH] we have [MATH].', 'hep-th-9701046-1-1-123': 'Quantum fluctuations [EQUATION] yield an effective potential [EQUATION] with [MATH] and [MATH].', 'hep-th-9701046-1-1-124': 'The mass of the [MATH] quantum does not vanish: Since T is a discrete symmetry, we have no Goldstone modes here.', 'hep-th-9701046-1-1-125': 'To conclude, we just have described an effective field theory of a generalized dynamical JT effect incorporating spontaneous T violation.', 'hep-th-9701046-1-1-126': 'The main input are the two flavors and the double-well, i.e. the breaking and restoring of a symmetry associated to a microscopic degeneracy.', 'hep-th-9701046-1-1-127': 'How can we derive an effective two-dimensional theory from this picture?', 'hep-th-9701046-1-1-128': 'Evidently, there must exist a description of this scenario in terms of a four-fermion interaction.', 'hep-th-9701046-1-1-129': "In the spirit of Bjorken's work who - motivated by the BCS theory - investigated the general possibility of constructing a gauge field from fundamental fermionic interactions [CITATION], we are looking for a relativistic ansatz incorporating a non-propagating gauge coupling similar to our local phonon.", 'hep-th-9701046-1-1-130': 'It was found by Ogievetskii and Polubarinov (OP), who have shown that it works in field theories with antisymmetric tensor gauge bosons [CITATION].', 'hep-th-9701046-1-1-131': 'It is amusing that the latter have been called phonon modes by Balachandran et al. in a different context [CITATION].', 'hep-th-9701046-1-1-132': 'Opposed to the massive Maxwell-Dirac Lagrangean [EQUATION] the most simple massive OP phonon quantum electrodynamics is described by [EQUATION]', 'hep-th-9701046-1-1-133': 'Here the massless limit [MATH] is taken at the end of the computation.', 'hep-th-9701046-1-2-0': 'Since in this gauge interaction picture the double well is no longer present, we have to input the two flavors by hand and obtain as the final effective Lagangean [EQUATION]', 'hep-th-9701046-1-2-1': 'And now comes the point of the story: Reducing this Lagrangean down to a non-relativistic 2+1 dimensional situation what is done by cancelling the third row and column of the antisymmetric tensor field [MATH] and identifiying the rest with the dual of a 2+1 dimensional vector potential [MATH] such that the interaction term must have the form field strength times a current diagonal in flavor we get a Lagrangean of the form [EQUATION] where [MATH] and [MATH].', 'hep-th-9701046-1-2-2': 'Note that the coefficient of the Chern-Simons term is a priori undetermined, it is a posteriori fixed by the correspondence of the number of flavors and the statistics parameter according to Mavromatos et al. and others [CITATION].', 'hep-th-9701046-1-2-3': 'The final 2+1 dimensional Lagrangean coincides with the Lagrangean of Kapusta et al. who suggested that the internal degree of freedom may be identified with the spin, though they did not forbid other interpretations [CITATION].', 'hep-th-9701046-1-2-4': 'From this we get a statistical magnetic field [EQUATION] such that a breaking of the chiral invariance in the generalized JT model at low temperatures gives [MATH] a finite value.', 'hep-th-9701046-1-2-5': 'In a study of the finite temperature Meissner-Ochsenfeld effect Kapusta et al. arrived at a set of four coupled integro-differential equations, indicating that superconductivity terminates at [MATH].', 'hep-th-9701046-1-2-6': "With a mean field approximation and certain values for the coupling constants and effective mass Kapusta et al. arrived at reasonable [MATH]'s.", 'hep-th-9701046-1-2-7': 'Thus Kapusta and collaborators have shown that there exist 2+1 dimensional gauge theories exhibiting a true second order superconducting phase transition at finite temperature.', 'hep-th-9701046-1-2-8': '0.6cm 6.', 'hep-th-9701046-1-2-9': 'Conclusions 0.4cm', 'hep-th-9701046-1-3-0': 'Let us take a look at the familiar conventional 3+1 dimensional superconducting phase transition from a quantum field theoretical point of view: The electric local gauge symmetry is spontaneously broken in the superconducting phase giving rise to a would-be Goldstone boson absorbed into the Meissner-Ochsenfeld effect.', 'hep-th-9701046-1-3-1': '(We use the quotation marks indicating that from a rigorous point of view local gauge symmentries are never spontaneously broken according to the Elitzur-Luscher theorem [CITATION].)', 'hep-th-9701046-1-3-2': 'Complementarily, we may view the transition from the superconducting state to the normal state as a spontaneously symmetry breaking of a magnetic gauge symmetry whose generator is a magnetic charge quantum number (vorticity) carried by fictitious infinite long fluxlines of infinite energy [CITATION].', 'hep-th-9701046-1-3-3': 'In this framework the Goldstone bosons are the photons or, physically speaking, they manifest themselves as the absence of the Meissner-Ochsenfeld effect [CITATION].', 'hep-th-9701046-1-3-4': 'Conversely, in two space dimensions conventional superconductivity does not exist at any finite temperature because fluctuations overcome energy in destroying off-diagonal long range order.', 'hep-th-9701046-1-3-5': 'This is the essential conclusion of the Hohenberg-Mermin-Wagner-Coleman theorem [CITATION], and one reason to consider anyon superconductivity was the possibility of an evasion of this theorem.', 'hep-th-9701046-1-3-6': 'There might exist other ways to evade the assumptions underlying this theorem and according our table borrowed from Ref. [CITATION] the Berezinskii-Kosterlitz-Thouless transition is an obvious choice [CITATION].', 'hep-th-9701046-1-3-7': 'Kovner and Rosenstein motivated this choice by the growing body of experimental data that points to the KT nature of the superconducting phase transition in [MATH] materials [CITATION].', 'hep-th-9701046-1-3-8': 'They propose a Lagrangean in which a doublet of two-component complex Dirac spinors couples to a vector field.', 'hep-th-9701046-1-3-9': 'In this theory the electric gauge symmetry is implemented in a BKT mode and the vector field reminscent of our OP phonon degree of freedom represents the corresponding BKT boson.', 'hep-th-9701046-1-3-10': 'Contrary to the case of thin superconducting metal films where we observe an vortex-antivortex unbinding transition [CITATION] the Kovner-Rosenstein vortices are charged.', 'hep-th-9701046-1-3-11': 'This is very reminiscent of anyonic superconductivity where vortices and quasi-particles are one and the same entities.', 'hep-th-9701046-1-3-12': 'From a conservative point of view, one may argue that BKT exitations tend to suppress two-dimensional superconductivity and this suppression will be probably damped by interlayer interactions enhancing three-dimensional superconductivity.', 'hep-th-9701046-1-3-13': 'In the case of the Kovner-Rosenstein vortices the interplane coupling shifts the values obtained in a two-dimensional theory only by a certain amount [CITATION] thus preserving the overall two-dimensional character.', 'hep-th-9701046-1-3-14': 'This seems to be compatible with our JT anyon toy model [CITATION].', 'hep-th-9701046-1-3-15': 'We think there is still a lot of experimental work to be done to find out what is really happening microscopically.', 'hep-th-9701046-1-3-16': 'But there is one point which cannot be overemphasized: It is by no means sufficient to confine ourselves to a discussion of the critical behavior of the systems we are interested in.', 'hep-th-9701046-1-3-17': 'Critical phenomena encode a wide category of specifications into the same universality class (e.g. 3D XY).', 'hep-th-9701046-1-3-18': 'Critical behavior is distinguished by the equal importance of all length and time scales at a certain point in the phase diagram.', 'hep-th-9701046-1-3-19': 'Thus there is no logic which allows us to deduce statements about the microscopic mechanism from critical behavior [CITATION].', 'hep-th-9701046-1-3-20': 'Even the dimensionality and the internal symmetry of a system cannot be read off from the critical behavior of a system (e.g. asymptotic freedom of two dimensional sigma models vs. asymptotic freedom four dimensional Yang-Mills theories).', 'hep-th-9701046-1-3-21': 'Expressed in different words, the question whether the superconducting phase transition in the cuprates is of a novel type cannot be answered from the study of critical indices alone.', 'hep-th-9701046-1-3-22': 'In conclusion, we have formulated a theoretical model, in which the fermions are interacting via a generalized JT interaction leading to a superconducting phase transition violating parity and time reversal.', 'hep-th-9701046-1-3-23': 'It is an example for a quantum mechanical distinction between left- and right-handedness different from other interesting proposals doing this [CITATION].', 'hep-th-9701046-1-3-24': 'Our model may be regarded as a variant of a more general BKT mechanism for two-dimensional superconductivity including scenarios without P and T violation.', 'hep-th-9701046-1-3-25': 'Moreover, a phase transition between a T preserving and T violating phase within the superconducting state seems to be possible.', 'hep-th-9701046-1-4-0': '0.6cm 7.'}

{'hep-th-9701046-2-0-0': '0.3cm We review the fundamentals of Jahn-Teller interactions and their field theoretical modelings and show that a 2+1 dimensional gauge theory where the gauge field couples to flavored fermions arises in a natural way from a two-band model describing the dynamical Jahn-Teller effect.', 'hep-th-9701046-2-0-1': 'The theory exhibits a second order phase transition to novel finite-temperature superconductivity.', 'hep-th-9701046-2-0-2': '0.6cm 1.', 'hep-th-9701046-2-0-3': 'Introduction 0.4cm', 'hep-th-9701046-2-1-0': 'The discovery of the cuprate high-[MATH] superconductors [CITATION] together with the fact that there is to date no generally accepted theory for the relevant mechanism [CITATION], gives physicists reasons to search for novel scenarios responsible for macroscopic quantum coherence phenomena in condensed matter physics.', 'hep-th-9701046-2-1-1': 'One of the main problems is the dependence of the nature of superconducting phase transitions on dimensionality.', 'hep-th-9701046-2-1-2': 'Of course, one has to be careful to make statements like high-[MATH] superconductivity is essentially a two-dimensional problem since it is not a priori clear how the very nature of the superconducting phase transition in the cuprates is related to or affected by some kind of interplane coupling.', 'hep-th-9701046-2-1-3': 'Nevertheless, there is no doubt that a general (i.e. quantum field theoretical) study of the possibility of idealized two-dimensional superconductivity at finite temperature is a challenging task.', 'hep-th-9701046-2-1-4': 'There is an old celebrated theorem due to Hohenberg, Mermin and Wagner, and Coleman [CITATION] stating that conventional off-diagonal long range order (ODLRO) is suppressed at any finite temperature in 2+1 dimensional quantum systems.', 'hep-th-9701046-2-1-5': 'At this point the fundamental question of the very existence of loopholes arises - as often in theoretical physics.', 'hep-th-9701046-2-1-6': 'In the following we present a proposal for a microscopic mechanism which may provide such a loophole and possibly explain high-[MATH] superconductivity or may open the door to even more interesting superconducting materials.', 'hep-th-9701046-2-1-7': 'It is based on an effective electron-electron (resp. hole-hole) interaction induced by a generalized dynamical Jahn-Teller effect [CITATION].', 'hep-th-9701046-2-1-8': 'Akin to fractional statistics this interaction is a phasing interaction, i.e. one exitation modulates the quantum mechanical phase of the other exitation giving rise to a net attractive interaction for the relevant fermions.', 'hep-th-9701046-2-1-9': 'Alternatively, a phasing interaction may be viewed as a renormalization of the statistical properties of the quanta changing their statistical identity.', 'hep-th-9701046-2-1-10': 'It is our objective to make plausible that it is the phasing aspect overlooked in more conventional treatments of the Jahn-Teller interaction which may play a crucial role in the mechanism of high-[MATH] superconductivity.', 'hep-th-9701046-2-1-11': 'Our discussion is organized as follows: We start by recalling the early history of the Jahn-Teller theory in molecular physics.', 'hep-th-9701046-2-1-12': 'We proceed by reviewing the concept of the geometrical phase (now called Berry phase and Aharonov-Anandan phase in the adiabatic and in the non-adiabatic cases, respectively) [CITATION] first analyzed systematically in the context of quantum chemistry in the pioneering work of Mead who dubbed this phenomenon molecular Aharonov-Bohm effect [CITATION].', 'hep-th-9701046-2-1-13': 'Then we show how to set up a theory of strongly correlated electrons interacting via a generalized dynamical Jahn-Teller interaction.', 'hep-th-9701046-2-1-14': 'Motivated by the work of Yu and Anderson [CITATION] we show that our ansatz can be expressed in terms of a double-well tunneling event which we christened solid state instanton.', 'hep-th-9701046-2-1-15': 'It is related to what is known as a polaron in solid state physics, but it is not exactly the same thing.', 'hep-th-9701046-2-1-16': 'The microscopic double-well mechanism allows to break parity and time-reversal spontaneously thus giving a microscopic reason for the existence of anyons without being restricted to them.', 'hep-th-9701046-2-1-17': "This double-well idea is intimately related to T. D. Lee's early 70's work on CP violation in elementary particle physics [CITATION].", 'hep-th-9701046-2-1-18': 'It is very amusing to observe that already T. D. Lee relates his own concept to geometrical ideas associated with abstract distortive deformations.', 'hep-th-9701046-2-1-19': 'Notice that the actual breaking and its magnitude of these discrete symmetries depend on a fine tuning of the coupling parameters.', 'hep-th-9701046-2-1-20': 'The effective Lagrangean describing such a polaronic interaction is by no means unique and adding or discarding terms determines whether P and T are broken or not.', 'hep-th-9701046-2-1-21': 'Moreover, there may exist the possibility of a phase transition between an anyonic and a Berezinskii-Kosterlitz-Thouless-type phase [CITATION] within the high-temperature conducting phase.', 'hep-th-9701046-2-1-22': 'Finally we show how to model the generalized dynamical Jahn-Teller interaction field theoretically and give an explicit example for a quasi two-dimensional finite temperature ansatz - firstly written down and analyzed by Joe Kapusta and his collaborators (without any reference to the generalized dynamical Jahn-Teller mechanism, however).', 'hep-th-9701046-2-1-23': 'This model indeed shows up a finite temperature phase transition to novel superconductivity within a finite window of parameters [CITATION].', 'hep-th-9701046-2-1-24': 'Its phase structure is very reminiscent of the phase diagram proposed by Chakraverty in 1979 [CITATION] which led Bednorz and Muller [CITATION] to their discovery of novel superconductivity.', 'hep-th-9701046-2-1-25': 'This model should be seen in a wider context including two-dimensional scenarios without parity and time-reversal violation.', 'hep-th-9701046-2-1-26': 'Finally, we give a heuristical argument for macroscopic quantum coherence induced by Jahn-Teller systems.', 'hep-th-9701046-2-1-27': 'The field theoretical ideas presented here are based on joint work with Heinz [CITATION].', 'hep-th-9701046-2-1-28': '0.6cm 2.', 'hep-th-9701046-2-1-29': 'What is the Jahn-Teller effect?', 'hep-th-9701046-2-1-30': '0.4cm As early as 1929 v.Neumann and Wigner [CITATION] asked a very interesting question: Given a hermitean [MATH] matrix whose entries are depending on a sequence of parameters [MATH] - how many parameters have to be changed in order to get a collision of two eigenvalues of the matrix.', 'hep-th-9701046-2-1-31': 'The general answer is at least three.', 'hep-th-9701046-2-1-32': 'Let us make this explicit for the most simplest case [MATH].', 'hep-th-9701046-2-1-33': 'Any hermitean [MATH] matrix [MATH] may be expanded with respect to the Pauli basis: [EQUATION]', 'hep-th-9701046-2-1-34': 'Solving the eigenvalue equation [MATH] gives [EQUATION]', 'hep-th-9701046-2-1-35': "Clearly we have to send all three space-like [MATH]'s to zero in order to get a collision!", 'hep-th-9701046-2-1-36': 'Evidently, in case of a real hermitean (i.e. symmetric) matrix this reduces to two parameters, in case of a real diagonal matrix to one parameter.', 'hep-th-9701046-2-1-37': 'Interpreting the space of independent parameters as a mechanical configuration space and cutting out the point of conicidence, i.e. the level crossing point, we obtain manifolds with non-trivial topological structures.', 'hep-th-9701046-2-1-38': 'This suggests that in all three cases, once physically realized, we may expect topological quantization effects giving rise to very interesting physics.', 'hep-th-9701046-2-1-39': 'To get real physics from this mathematical observations we must find a Hamiltonian having the structure of [MATH] and a mechanism which prevents the level crossing.', 'hep-th-9701046-2-1-40': 'This was done in the pioneering work of Jahn and Teller in 1937 who studied the stability of polyatomic molecules in degenerate electronic states [CITATION].', 'hep-th-9701046-2-1-41': 'Essentially the famous Jahn-Teller theorem states the following: A configuration of a polyatomic molecule for an electronic state having orbital degeneracy cannot be stable with respect to all displacements of the nuclei unless the nuclei all lie on a straight line.', 'hep-th-9701046-2-1-42': 'The original proof of Jahn and Teller [CITATION] is based on a detailed discussion of particular symmetries and their realizations.', 'hep-th-9701046-2-1-43': 'A more general proof within the framework of induced representations of finite groups was given by Ruch and Schonhofer in 1965 [CITATION].', 'hep-th-9701046-2-1-44': 'It is interesting to note that - by the side of the Weiss theory of ferromagnetism - the Jahn-Teller effect is the archetype of what is commonly called spontaneously symmetry breaking (SSB), a fact recently recalled by Nambu [CITATION].', 'hep-th-9701046-2-1-45': 'In modern terminology a dynamics determined by a sensible matching of vibration modes and electronic excitations is called a vibronic interaction [CITATION].', 'hep-th-9701046-2-1-46': 'There is no doubt that a phonon induced fermion-fermion interaction incorporating Jahn-Teller-type effects may exhibit new pecularities beyond the marks of the conventional electron-phonon interaction.', 'hep-th-9701046-2-1-47': 'Due to the non-trivial topological structure of the configuration spaces involved, we may expect highly non-trivial quantization and coherence phenomena, and it is the aim of this discussion to push forward the thesis that the understanding of high-[MATH] superconductivity is at least related to the generalized dynamics of vibronic interactions.', 'hep-th-9701046-2-1-48': '0.6cm 3.', 'hep-th-9701046-2-1-49': 'The geometric phase in Jahn-Teller systems 0.4cm To sum up, the (static) Jahn-Teller effect is an electronic symmetry breaking phenomenon associated with a spontaneous distortion.', 'hep-th-9701046-2-1-50': 'To be concrete, let us consider a vibronic interaction of a doubly degenerate electronic state [MATH] with a doubly degenerate vibrational mode [MATH] [CITATION].', 'hep-th-9701046-2-1-51': 'According to the Jahn-Teller theorem the nuclear motion lifts the electronic degeneracy, i.e. there are nuclear configurations of lower energy than the symmetric state.', 'hep-th-9701046-2-1-52': "This [MATH] Jahn-Teller effect is the archetype of Berry's phase: A quantum-mechanical phase shift of purely geometrical origin associated to an adiabatic cycle starting and ending at the same pure state [CITATION].", 'hep-th-9701046-2-1-53': 'In our case we start by writing down the quantum-mechanical two-dimensional harmonic oscillator Hamiltonian for a double degenerated vibrational mode [EQUATION] with [MATH] and [MATH].', 'hep-th-9701046-2-1-54': 'This Hamiltonian is thought to be acting on two-component (Pauli) wave functions [MATH] (i=1,2).', 'hep-th-9701046-2-1-55': 'The vibronic coupling is given by adding a term [MATH][MATH] to [MATH], whereby [MATH] denote the Pauli matrices and k is a coupling constant.', 'hep-th-9701046-2-1-56': '(We use the letter [MATH] to avoid confusion with spin degrees of freedom).', 'hep-th-9701046-2-1-57': 'In analogy to elementary particle physics we call the associated internal quantum number of the electron a Jahn-Teller isospin or flavor.', 'hep-th-9701046-2-1-58': 'Explicitely we have [EQUATION] with [MATH] and [MATH].', 'hep-th-9701046-2-1-59': 'Let us rewrite the Hamiltonian in cylinder coordinates: [EQUATION]', 'hep-th-9701046-2-1-60': 'In the adiabatic or Born-Oppenheimer approximation we neglegt the kinetic energy term and diagonalize the remainder.', 'hep-th-9701046-2-1-61': 'We obtain [EQUATION] with [EQUATION]', 'hep-th-9701046-2-1-62': 'Note that the matrix [EQUATION] is a double-valued function in the polar angle [MATH].', 'hep-th-9701046-2-1-63': 'The associated energy eigenvalues are [EQUATION] (where we have written [MATH] for [MATH]) and correspond to two sheets (the upper cone-like, the lower sombrero-like) coinciding at a point of degeneracy at the origin.', 'hep-th-9701046-2-1-64': 'This point of zero distortion defines a conical intersection.', 'hep-th-9701046-2-1-65': 'Because of the double-valued character of the diagonalizing similarity transformation the eigenstates [MATH] corresponding to [MATH] are double-valued in [MATH].', 'hep-th-9701046-2-1-66': 'The multiple-valuedness may be compensated for by an appropriate local gauge transformation [EQUATION] which in turn induces the change [EQUATION] with [EQUATION] in the nuclear energy operator.', 'hep-th-9701046-2-1-67': 'The vector potential [MATH] corresponds to a fictitious flux tube with strength [MATH] confined to the origin.', 'hep-th-9701046-2-1-68': 'In order to visualize the circuit in [MATH] inducing the quasi-spinorial sign change of the electronic states one takes a look on a typical example for the [MATH] Jahn-Teller effect: the trimer.', 'hep-th-9701046-2-1-69': 'A circuit of distortions - corresponding to the natural motions of the nuclei - avoids the point of symmetry, at which the trimer looks like an equilateral triangle.', 'hep-th-9701046-2-1-70': 'Of course, while it is true, that we generally assume that the complete system must be described by a single-valued wave function, it is the splitting between the subsystem and its relative environment which introduces the geometrical phase factor: Hence the multiple-valuedness of the electronic wavefunction is compensated for by the multiple-valuedness of the nuclear wavefunctions.', 'hep-th-9701046-2-1-71': '0.6cm 4.', 'hep-th-9701046-2-1-72': 'Beyond the adiabatic approximation 0.4cm In the last section we observed that in a Jahn-Teller system (hereafter designated by JT) the electronic wave functions are in general multiple-valued functions in the slow nuclear coordinates; in particular they are double-valued in the nuclear polar angle in the case of a trimer.', 'hep-th-9701046-2-1-73': 'Our claim is that this could give rise to a novel quality of an electron-electron interaction mediated by an oscillating JT ionic configuration.', 'hep-th-9701046-2-1-74': 'In the following - generalizing this classical dynamical yet adiabatic JT approach somewhat - we will go beyond the approximation of Born and Oppenheimer and write down an ansatz for an effective field theory describing a dynamical non-adiabatic JT interaction.', 'hep-th-9701046-2-1-75': 'One reason for this proceeding lies in the fact that e.g. for the case of the octahedron in the [MATH] superconductor we encounter neither an appropriate degeneracy nor an appropriate configuration space of distortions justifying the applicability of the naive JT theorem and the Mead model described above [CITATION].', 'hep-th-9701046-2-1-76': 'Conversely, what we expect is that the point of degeneracy in the relevant JT-like system is smeared-out (just like a smeared-out Aharonov-Bohm flux line or a regularized anyon), such that the adiabatic transformation definitely breaks down and, in addition, the oscillating ionic arrangement does not simply sweep through all configurations classically possible, but tunnels between some of them instead.', 'hep-th-9701046-2-1-77': 'We propose a scenario in which both delicate aspects appear as natural consequences of the same fundamental mechanism, but nevertheless, the topological quality of the interaction, reminiscent of Meads molecular Aharonov-Bohm effect [CITATION] will survive.', 'hep-th-9701046-2-1-78': 'To convert the nuclear Hamiltonian [EQUATION] into a field theoretical Hamiltonian describing an electron-electron interaction mediated by an oscillating ionic configuration we simply make the replacement [EQUATION] and introduce phonon field operators such that we get a two-band field theory with electron-phonon vertices.', 'hep-th-9701046-2-1-79': 'A more general Hamiltonian may be obtained by introducing weights for the different modes and adding a conventional term.', 'hep-th-9701046-2-1-80': 'Integrating out phonons in a standard way we obtain a four-fermion BCS-like Hamiltonian representing interband-intraband interactions with some wrong-sign couplings.', 'hep-th-9701046-2-1-81': 'Englman, Halperin, and Weger proposed a JT theory for the high-[MATH] superconductivity of the cuprates, in which the coupling between the copper [MATH] and [MATH] states leads to a pairing mechanism of the same form, but opposite sign to that of the BCS theory [CITATION].', 'hep-th-9701046-2-1-82': 'They argued that their own ansatz is well-supported by band structure calculations and experiments indicating the involvement of both [MATH] and [MATH] type states in the carrier states of cuprate superconductors.', 'hep-th-9701046-2-1-83': 'Furthermore they showed that the proposed pairing mechanism is stable against lattice distortion even for strong coupling.', 'hep-th-9701046-2-1-84': 'It is noteworthy to remark that there are a number of proposals how the JT effect comes into play in superconductors, in particular in high-[MATH] superconductors.', 'hep-th-9701046-2-1-85': 'Some classical papers can be found in Ref. [CITATION] and more current contributions are listed in Refs. [CITATION].', 'hep-th-9701046-2-1-86': 'Especially interesting is the recent work of K. H. Johnson et al. [CITATION] who argued that the observed superconductivity at 18 K in potassium-doped fullerene is induced by a cooperative JT coupling leading to a BCS-like mechanism.', 'hep-th-9701046-2-1-87': 'Topological aspects relating the JT phenomenon and superconductivity have been almost ignored up until now.', 'hep-th-9701046-2-1-88': 'The only paper, to our knowledge, relating topological quantization effects (especially fractional quantization) to the JT effect and superconductivity was written by Kuratsuji [CITATION].', 'hep-th-9701046-2-1-89': 'Appel pointed out to me that if high-[MATH] superconductivity is due to a JT-like scenario then the description of the relevant mechanism surely has to go far beyond the adiabatic approximation [CITATION].', 'hep-th-9701046-2-1-90': 'In particular he was inspired by the work of Cohen and collaborators [CITATION] who emphasized that - due to the fundamental instability of the oxygen ion which causes its motions influencing the charge density between the copper and oxygen - anharmonic double-well potentials for normal modes may give larger coupling then expected from harmonic phonons and are less sensitive to the mass.', 'hep-th-9701046-2-1-91': 'Appel argued that the local phonon ansatz by Yu and Anderson [CITATION] - considered as the non-adiabatic extension of the dynamical JT effect - provides a suitable framework to describe the fundamental interaction.', 'hep-th-9701046-2-1-92': 'Note that we are not interested in the exact details of the interaction (e.g. apex in-plane charge interaction, apex positional splitting, out-of-plane motions etc.); we only assume that the essential dynamics is governed by an inharmonic potential.', 'hep-th-9701046-2-1-93': 'In a Yu-Anderson-type model we restrict ourselves to a one mode description replacing the double sheeted sombrero by a double sheeted double well.', 'hep-th-9701046-2-1-94': 'Explicitely we have [EQUATION]', 'hep-th-9701046-2-1-95': 'To diagonalize the phonon matrix we introduce the chiral - i.e. left-handed and right-handed - linear combinations [EQUATION] and get [EQUATION] i.e. an interaction term proportional to [MATH].', 'hep-th-9701046-2-1-96': 'Yu and Anderson proceed by integrating out the electron degrees of freedom and calculate the dynamical modification of the harmonic oscillator potential giving a dynamical double well[MATH] replacing the more singular [MATH] term.', 'hep-th-9701046-2-1-97': 'The calculation is very involved and relies heavily of path integral techniques reminiscent of instanton calculations in quantum field theory.', 'hep-th-9701046-2-1-98': 'An effective non-relativistic Lagrangean for this model may have the form [EQUATION] where [MATH] is a quartic term and [MATH][MATH] is a two-component Schrodinger field.', 'hep-th-9701046-2-1-99': 'That anharmonicity modifies the mass-frequency relation of a quantum mechanical oscillator and hence the isotope effect is due to the non-analytic character of the solution of the double-well tunneling problem - a rather general feature.', 'hep-th-9701046-2-1-100': 'In field theory tunneling events are called instantons and it is, to our opinion, appropriate to name the Yu-Anderson local phonon a solid state instanton.', 'hep-th-9701046-2-1-101': 'Summarizing, complementary to the BCS-like four fermion interaction a la Engelman, Halperin, and Weger [CITATION] which is obtained by integrating out the phonons, we get an effective anharmonic phonon potential by integrating out the fermions.', 'hep-th-9701046-2-1-102': 'A very interesting point lies in the fact that double-well system is intimately related to a two-level system in that the lowest states of the former are to be identified with the only states spanning the latter.', 'hep-th-9701046-2-1-103': 'Now the dynamics of the two-level system considered as an abstract spin-[MATH] system is driven by an abstract external magnetic field - self-consistently generated through the local phonon tunneling dynamics.', 'hep-th-9701046-2-1-104': 'In case of a real spin-[MATH] system the driving external magnetic field introduces definitely an oddness under time reversal.', 'hep-th-9701046-2-1-105': 'We do not really know under which conditions this oddness under time reversal carries over to the abstract case, but at least as a possibility it remains.', 'hep-th-9701046-2-1-106': 'The oddness under T is also suggested by the fact that a hidden parity violation is already present in the model due to the interference of odd and even modes and due to the fact that PT should be a good symmetry in solid state physics.', 'hep-th-9701046-2-1-107': 'Hence a relativistic ansatz for a T, P, and C invariant effective Lagrangean based on pseudoscalar anharmonic phonons may be written [EQUATION] with [EQUATION] where we demand [MATH], and the relativistic spin degrees of freedom are identified with the two bands of the generalized JT interaction in the non-relativistic limit.', 'hep-th-9701046-2-1-108': '0.6cm 5.', 'hep-th-9701046-2-1-109': "A phase transition towards two-dimensional superconductivity 0.4cm Our relativistic Lagrangean is identical to the one studied by T. D. Lee in the early 70's as a simple example for spontaneous T violation [CITATION], a phenomenon discussed in the framework of anyon physics [CITATION].", 'hep-th-9701046-2-1-110': 'Unfortunately, the experimental situation is compatible with the absence of anyons in high-[MATH] materials rather than with their presence [CITATION].', 'hep-th-9701046-2-1-111': 'Nevertheless, if high-[MATH] superconductivity is still a really two-dimensional phenomenon, then we should continue to study anyon superconductivity because it is a nice toy model possibly exhibiting features of the true theory.', 'hep-th-9701046-2-1-112': 'Therefore our stategy is to develop a theory in which anyons appear as a consequence of a microscopic mechanism and induce finite temperature superconductivity.', 'hep-th-9701046-2-1-113': 'At the end we will try to find out how to modify the model in order to preserve P and T. Assuming they exist, anyons are never elementary particles like electrons such that the question remains: How can we get anyons and hence spontaneously T violation from fundamental electronic interactions?', 'hep-th-9701046-2-1-114': 'We think it is near at hand and much more natural to reverse the standard argumentation and consider the violation of time reversal and not necessarily the validity of the two-dimensional description as the main problem.', 'hep-th-9701046-2-1-115': "This view is also supported by Wilczek's axion electrodynamics [CITATION] and the chiron model by Chaplin and Yagamishi [CITATION].", 'hep-th-9701046-2-1-116': 'Both models go beyond two dimensions while preserving the P and T violating character.', 'hep-th-9701046-2-1-117': 'Indeed, we should find the reason for anyons - departing from a microscopic picture.', 'hep-th-9701046-2-1-118': 'Our picture is that it is a background of tunneling impurities which does generate an effective four fermion interaction.', 'hep-th-9701046-2-1-119': "In particular, we expect that it should be possible to derive anyon physics as a consequence of this ansatz casted in the form of T. D. Lee's Lagrangean.", 'hep-th-9701046-2-1-120': 'Let us briefly sketch the original Lee mechanism.', 'hep-th-9701046-2-1-121': 'Though the Langrangean is invariant under time reversal and parity, the vacuum expectation value [EQUATION] of the [MATH] field is not: It changes its sign under P and T.', 'hep-th-9701046-2-1-122': 'In the tree graph approximation [MATH] we have [MATH].', 'hep-th-9701046-2-1-123': 'Quantum fluctuations [EQUATION] yield an effective potential [EQUATION] with [MATH] and [MATH].', 'hep-th-9701046-2-1-124': 'The mass of the [MATH] quantum does not vanish: Since T is a discrete symmetry, we have no Goldstone modes here.', 'hep-th-9701046-2-1-125': 'To conclude, we just have described an effective field theory of a generalized dynamical JT effect incorporating spontaneous T violation.', 'hep-th-9701046-2-1-126': 'The main input are the two flavors and the double-well, i.e. the breaking and restoring of a symmetry associated to a microscopic degeneracy.', 'hep-th-9701046-2-1-127': 'How can we derive an effective two-dimensional theory from this picture?', 'hep-th-9701046-2-1-128': 'Evidently, there must exist a description of this scenario in terms of a four-fermion interaction.', 'hep-th-9701046-2-1-129': "In the spirit of Bjorken's work who - motivated by the BCS theory - investigated the general possibility of constructing a gauge field from fundamental fermionic interactions [CITATION], we are looking for a relativistic ansatz incorporating a non-propagating gauge coupling similar to our local phonon.", 'hep-th-9701046-2-1-130': 'It was found by Ogievetskii and Polubarinov (OP), who have shown that it works in field theories with antisymmetric tensor gauge bosons [CITATION].', 'hep-th-9701046-2-1-131': 'It is amusing that the latter have been called phonon modes by Balachandran et al. in an entirely different context [CITATION].', 'hep-th-9701046-2-1-132': 'Opposed to the massive Maxwell-Dirac Lagrangean [EQUATION] the most simple massive OP phonon quantum electrodynamics is described by [EQUATION]', 'hep-th-9701046-2-1-133': 'Here the massless limit [MATH] is taken at the end of the computation.', 'hep-th-9701046-2-2-0': 'Since in this gauge interaction picture the double well is no longer present, we have to input the two flavors by hand and obtain as the final effective Lagangean [EQUATION]', 'hep-th-9701046-2-2-1': 'And now comes the point of the story: Reducing this Lagrangean down to a non-relativistic 2+1 dimensional situation what is done by cancelling the third row and column of the antisymmetric tensor field [MATH] and identifiying the rest with the dual of a 2+1 dimensional vector potential [MATH] such that the interaction term must have the form field strength times a current diagonal in flavor we get a Lagrangean of the form [EQUATION] where [MATH] and [MATH].', 'hep-th-9701046-2-2-2': 'Note that the coefficient of the Chern-Simons term is a priori undetermined, it is a posteriori fixed by the correspondence of the number of flavors and the statistics parameter according to Mavromatos et al. and others [CITATION].', 'hep-th-9701046-2-2-3': 'The final 2+1 dimensional Lagrangean coincides with the Lagrangean of Kapusta et al. who suggested that the internal degree of freedom may be identified with the spin, though they did not forbid other interpretations [CITATION].', 'hep-th-9701046-2-2-4': 'From this we get a statistical magnetic field [EQUATION] such that a breaking of the chiral invariance in the generalized JT model at low temperatures gives [MATH] a finite value.', 'hep-th-9701046-2-2-5': 'In a study of the finite temperature Meissner-Ochsenfeld effect Kapusta et al. arrived at a set of four coupled integro-differential equations, indicating that superconductivity terminates at [MATH].', 'hep-th-9701046-2-2-6': "With a mean field approximation and certain values for the coupling constants and effective mass Kapusta et al. arrived at reasonable [MATH]'s.", 'hep-th-9701046-2-2-7': 'Thus Kapusta and collaborators have shown that there exist 2+1 dimensional gauge theories exhibiting a true second order superconducting phase transition at finite temperature.', 'hep-th-9701046-2-2-8': '0.6cm 6.', 'hep-th-9701046-2-2-9': 'Conclusions 0.4cm', 'hep-th-9701046-2-3-0': 'Let us take a look at the familiar conventional 3+1 dimensional superconducting phase transition from a quantum field theoretical point of view: The electric local gauge symmetry is spontaneously broken in the superconducting phase giving rise to a would-be Goldstone boson absorbed into the Meissner-Ochsenfeld effect.', 'hep-th-9701046-2-3-1': '(We use the quotation marks indicating that from a rigorous point of view local gauge symmentries are never spontaneously broken according to the Elitzur-Luscher theorem [CITATION].)', 'hep-th-9701046-2-3-2': 'Complementarily, we may view the transition from the superconducting state to the normal state as a spontaneously symmetry breaking of a magnetic gauge symmetry whose generator is a magnetic charge quantum number (vorticity) carried by fictitious infinite long fluxlines of infinite energy [CITATION].', 'hep-th-9701046-2-3-3': 'In this framework the Goldstone bosons are the photons or, physically speaking, they manifest themselves as the absence of the Meissner-Ochsenfeld effect [CITATION].', 'hep-th-9701046-2-3-4': 'Conversely, in two space dimensions conventional superconductivity does not exist at any finite temperature because fluctuations overcome energy in destroying off-diagonal long range order.', 'hep-th-9701046-2-3-5': 'This is the essential conclusion of the Hohenberg-Mermin-Wagner-Coleman theorem [CITATION], and one reason to consider anyon superconductivity was the possibility of an evasion of this theorem.', 'hep-th-9701046-2-3-6': 'There might exist other ways to evade the assumptions underlying this theorem and according our table borrowed from Ref. [CITATION] the Berezinskii-Kosterlitz-Thouless transition is an obvious choice [CITATION].', 'hep-th-9701046-2-3-7': 'Kovner and Rosenstein motivated this choice by the growing body of experimental data that points to the KT nature of the superconducting phase transition in [MATH] materials [CITATION].', 'hep-th-9701046-2-3-8': 'They propose a Lagrangean in which a doublet of two-component complex Dirac spinors couples to a vector field.', 'hep-th-9701046-2-3-9': 'In this theory the electric gauge symmetry is implemented in a BKT mode and the vector field reminscent of our OP phonon degree of freedom represents the corresponding BKT boson.', 'hep-th-9701046-2-3-10': 'Contrary to the case of thin superconducting metal films where we observe an vortex-antivortex unbinding transition [CITATION] the Kovner-Rosenstein vortices are charged.', 'hep-th-9701046-2-3-11': 'This is very reminiscent of anyonic superconductivity where vortices and quasi-particles are one and the same entities.', 'hep-th-9701046-2-3-12': 'From a conservative point of view, one may argue that BKT exitations tend to suppress two-dimensional superconductivity and this suppression will be probably damped by interlayer interactions enhancing three-dimensional superconductivity.', 'hep-th-9701046-2-3-13': 'In the case of the Kovner-Rosenstein vortices the interplane coupling shifts the values obtained in a two-dimensional theory only by a certain amount [CITATION] thus preserving the overall two-dimensional character.', 'hep-th-9701046-2-3-14': 'This seems to be compatible with our JT anyon toy model [CITATION].', 'hep-th-9701046-2-3-15': 'We think there is still a lot of experimental work to be done to find out what is really happening microscopically.', 'hep-th-9701046-2-3-16': 'But there is one point which cannot be overemphasized: It is by no means sufficient to confine ourselves to a discussion of the critical behavior of the systems we are interested in.', 'hep-th-9701046-2-3-17': 'Critical phenomena encode a wide category of specifications into the same universality class (e.g. 3D XY).', 'hep-th-9701046-2-3-18': 'Critical behavior is distinguished by the equal importance of all length and time scales at a certain point in the phase diagram.', 'hep-th-9701046-2-3-19': 'Thus there is no logic which allows us to deduce statements about the microscopic mechanism from critical behavior [CITATION].', 'hep-th-9701046-2-3-20': 'Even the dimensionality and the internal symmetry of a system cannot be read off from the critical behavior of a system (e.g. asymptotic freedom of two dimensional sigma models vs. asymptotic freedom four dimensional Yang-Mills theories).', 'hep-th-9701046-2-3-21': 'Expressed in different words, the question whether the superconducting phase transition in the cuprates is of a novel type cannot be answered from the study of critical indices alone.', 'hep-th-9701046-2-3-22': 'In conclusion, we have formulated a theoretical model, in which the fermions are interacting via a generalized JT interaction leading to a superconducting phase transition violating parity and time reversal.', 'hep-th-9701046-2-3-23': 'It is an example for a quantum mechanical distinction between left- and right-handedness different from other interesting proposals doing this [CITATION].', 'hep-th-9701046-2-3-24': 'Our model may be regarded as a variant of a more general BKT mechanism for two-dimensional superconductivity including scenarios without P and T violation.', 'hep-th-9701046-2-3-25': 'Moreover, a phase transition between a T preserving and T violating phase within the superconducting state seems to be possible.', 'hep-th-9701046-2-4-0': '0.6cm 7.'}

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['hep-th-9701046-1-3-16', 'hep-th-9701046-2-3-16'], ['hep-th-9701046-1-3-17', 'hep-th-9701046-2-3-17'], ['hep-th-9701046-1-3-18', 'hep-th-9701046-2-3-18'], ['hep-th-9701046-1-3-19', 'hep-th-9701046-2-3-19'], ['hep-th-9701046-1-3-20', 'hep-th-9701046-2-3-20'], ['hep-th-9701046-1-3-21', 'hep-th-9701046-2-3-21'], ['hep-th-9701046-1-3-22', 'hep-th-9701046-2-3-22'], ['hep-th-9701046-1-3-23', 'hep-th-9701046-2-3-23'], ['hep-th-9701046-1-3-24', 'hep-th-9701046-2-3-24'], ['hep-th-9701046-1-3-25', 'hep-th-9701046-2-3-25'], ['hep-th-9701046-1-0-0', 'hep-th-9701046-2-0-0'], ['hep-th-9701046-1-0-1', 'hep-th-9701046-2-0-1']]

[['hep-th-9701046-1-1-24', 'hep-th-9701046-2-1-24'], ['hep-th-9701046-1-1-131', 'hep-th-9701046-2-1-131']]

[]

[]

[]

['hep-th-9701046-1-0-2', 'hep-th-9701046-1-0-3', 'hep-th-9701046-1-1-28', 'hep-th-9701046-1-1-48', 'hep-th-9701046-1-1-71', 'hep-th-9701046-1-1-108', 'hep-th-9701046-1-2-8', 'hep-th-9701046-1-2-9', 'hep-th-9701046-1-4-0', 'hep-th-9701046-2-0-2', 'hep-th-9701046-2-0-3', 'hep-th-9701046-2-1-28', 'hep-th-9701046-2-1-48', 'hep-th-9701046-2-1-71', 'hep-th-9701046-2-1-108', 'hep-th-9701046-2-2-8', 'hep-th-9701046-2-2-9', 'hep-th-9701046-2-4-0']

{'1': 'http://arxiv.org/licenses/assumed-1991-2003/', '2': 'http://arxiv.org/licenses/assumed-1991-2003/'}

https://arxiv.org/abs/hep-th/9701046

1310.6479

{'1310.6479-1-0-0': 'The [MATH] singlet algebra is a vertex operator algebra that is strongly generated by a Virasoro field of central charge [MATH] and a single Virasoro primary field of conformal weight [MATH].', '1310.6479-1-0-1': 'Here, the modular properties of the characters of the uncountably many simple modules of each singlet algebra are investigated and the results used as the input to a continuous analogue of the Verlinde formula to obtain the "fusion rules" of the singlet modules.', '1310.6479-1-0-2': 'The effect of the failure of fusion to be exact in general is studied at the level of Verlinde products and the rules derived are lifted to the [MATH] triplet algebras by regarding these algebras as simple current extensions of their singlet cousins.', '1310.6479-1-0-3': 'The result is a relatively effortless derivation of the triplet "fusion rules" that agrees with those previously proposed in the literature.', '1310.6479-1-1-0': '# Introduction', '1310.6479-1-2-0': 'The [MATH] singlet and triplet models (for [MATH] a positive integer) are perhaps the most basic known examples of logarithmic conformal field theories.', '1310.6479-1-2-1': 'Introduced in [CITATION], their logarithmic nature was exposed, at least for [MATH], through a connection to symplectic fermions and [MATH]-ghosts [CITATION].', '1310.6479-1-2-2': 'Further investigations, for example [CITATION], addressed the issue of generalising the tools familiar from rational conformal field theory to these models and they have remained popular objects of study ever since.', '1310.6479-1-3-0': 'Generalisations of these triplet models, called [MATH]-models, for [MATH] and [MATH] coprime and positive, were introduced in [CITATION].', '1310.6479-1-3-1': 'Because their central charges match those of the Virasoro [MATH] minimal models when [MATH] and [MATH] are greater than [MATH], they are sometimes referred to as logarithmic minimal models.', '1310.6479-1-3-2': 'One might hope that these logarithmic models could capture the universal features of critical lattice models that are missed by the minimal models (crossing probabilities, fractal dimensions and so on), but this is still contentious.', '1310.6479-1-3-3': 'Nevertheless, there has been persistent interest in these models from both theoretical physicists and mathematicians.', '1310.6479-1-3-4': 'One reason for this interest is that the underlying vertex operator algebras are not simple, so these models allow one to explore the consequences of this non-simplicity in a tractable, though still very challenging, setting.', '1310.6479-1-3-5': 'We remark that the simple quotients are precisely the minimal model vertex operator algebras.', '1310.6479-1-4-0': 'As with other logarithmic models, one of the main difficulties to surmount is that of obtaining a detailed structural understanding of the reducible but indecomposable modules which appear in the spectrum.', '1310.6479-1-4-1': 'While a complete classification of the indecomposables may well be infeasible, a first aim would be to identify the spectrum of simple modules and their projective covers.', '1310.6479-1-4-2': 'This is expected to be sufficient to construct bulk state spaces with modular invariant partition functions, for example.', '1310.6479-1-4-3': 'However, the current state of knowledge regarding projectives in non-semisimple module categories over vertex operator algebras is still in its infancy, so much of our intuition stems from examples like the [MATH]-models.', '1310.6479-1-5-0': 'But even in examples, the rigorous identification of projectives remains a formidable task.', '1310.6479-1-5-1': 'Indeed, this has only been achieved for [MATH] [CITATION].', '1310.6479-1-5-2': 'However, there is no shortage of proposals and conjectures for general [MATH] and [MATH] (with varying degrees of structural detail), see [CITATION] for example.', '1310.6479-1-5-3': 'These proposals rely on conjectured equivalences of categories, numerical computations within integrable lattice discretisations, and explicit construction, the latter giving the most information (but requiring the most effort).', '1310.6479-1-5-4': 'In this direction, a powerful tool for structural investigations of indecomposables is the celebrated Nahm-Gaberdiel-Kausch algorithm [CITATION] that explicitly constructs (filtered quotients of) the fusion product of two modules.', '1310.6479-1-6-0': 'Determining fusion rules is, of course, another of the main difficulties one would like to overcome along the path to understanding a given logarithmic conformal field theory.', '1310.6479-1-6-1': 'While the Nahm-Gaberdiel-Kausch fusion algorithm allows one to construct enough of a fusion product to identify it completely in principle, the calculations are too computationally intensive for all but the smallest theories, even when performed by computer.', '1310.6479-1-6-2': 'Another issue is that the algorithm in practice only provides an "upper bound" on the fusion product in the sense that the true result could be, in principle, a proper subspace of what has been deduced.', '1310.6479-1-6-3': 'However, both of these issues can be circumvented by generalising another standard tool from rational conformal field theory to the logarithmic setting: the Verlinde formula.', '1310.6479-1-7-0': 'The Verlinde formula [CITATION] for rational conformal field theories computes the fusion product of two modules from the modular transformation properties of their characters.', '1310.6479-1-7-1': 'As this formula may be shown to follow from the internal consistency conditions that must be satisfied by any conformally-invariant quantum field theory [CITATION], one expects that it should remain valid in some form for more general classes of non-rational theories.', '1310.6479-1-7-2': 'In the logarithmic setting, characters cannot distinguish between reducible but indecomposable modules and the direct sum of their simple composition factors, hence the Verlinde formula cannot be expected to compute the true fusion rules, but only tell us which composition factors appear, and with what multiplicity, in a given fusion product.', '1310.6479-1-7-3': 'However, this is already very valuable information.', '1310.6479-1-7-4': 'In many cases, one can easily rule out the possibility that the simple factors combine to form an indecomposable and then the Verlinde formula gives the fusion rules as in rational theories.', '1310.6479-1-7-5': 'If an indecomposable can be formed, then this formula provides the character of the indecomposable effortlessly, thus solving the "upper bound" problem.', '1310.6479-1-7-6': 'Moreover, it also tells us which fusion products need to be checked for indecomposability, thus potentially saving computational resources.', '1310.6479-1-8-0': 'Unfortunately, the modular properties of the triplet [MATH]-models (with [MATH]) are not as nice as one could have hoped for.', '1310.6479-1-8-1': 'In particular, the character of the vacuum module transforms under [MATH] into a linear combination of of other characters, but the coefficients depend non-trivially on [MATH] [CITATION].', '1310.6479-1-8-2': 'This would appear to invalidate a naive application of the Verlinde formula.', '1310.6479-1-8-3': 'Nevertheless, one can arrive at a [MATH]-independent [MATH]-transformation by postulating a non-standard automorphy matrix and a generalised Verlinde formula exploiting this has been demonstrated for the triplet models with [MATH] [CITATION].', '1310.6479-1-8-4': 'This proposed recipe does produce non-negative integer structure constants which agree with the known (Grothendieck) fusion coefficients.', '1310.6479-1-8-5': 'However, the generalised Verlinde formula itself is significantly more unwieldy than the original and we are not aware of any attempts to derive its analogues for other logarithmic conformal field theories.', '1310.6479-1-8-6': 'Another way of obtaining [MATH] -independent coefficients is to enlarge the space of characters to the space of torus 1-point functions, that is, to add certain linear combinations of characters multiplied by appropriate powers of [MATH] [CITATION], see also [CITATION].', '1310.6479-1-9-0': 'Here, we follow a different path to the Verlinde formula.', '1310.6479-1-9-1': 'Instead of working directly with the triplet [MATH]-models that have been so thoroughly studied in the literature, we focus our attention on the relatively unexplored singlet [MATH]-models.', '1310.6479-1-9-2': 'Whereas the triplet vertex operator algebras are known to possess a finite number of inequivalent simple modules [CITATION], the singlet algebras admit an uncountable infinity of them.', '1310.6479-1-9-3': 'However, this is not a bug, but a feature!', '1310.6479-1-9-4': 'We will see that the modular transformation properties of the characters of these simple singlet modules are very well behaved.', '1310.6479-1-9-5': 'Moreover, applying the standard Verlinde formula (but with an integral replacing the sum) leads again to non-negative integer structure constants.', '1310.6479-1-9-6': 'Finally, these results can be lifted from the singlet algebra to its triplet cousin using the technology of simple current extensions.', '1310.6479-1-9-7': 'In particular, our results provide an effortless derivation of the [MATH] triplet (Grothendieck) fusion rules without the need for non-standard automorphy factors and complicated generalisations of the Verlinde formula.', '1310.6479-1-10-0': 'This path to the Verlinde formula is actually a special case of a rather more general formalism that has been proposed for non-rational conformal field theories in [CITATION].', '1310.6479-1-10-1': 'There, one starts with a continuous spectrum of so-called standard modules which are typically simple and whose characters have good modular properties.', '1310.6479-1-10-2': 'In logarithmic theories, the atypical standard modules are reducible but indecomposable and the not-so-good modular properties of the characters of the simple subquotients may be determined using standard methods of homological algebra.', '1310.6479-1-10-3': 'This approach was developed for logarithmic models based on affine (super)algebras [CITATION] where the natural spectrum is continuous.', '1310.6479-1-10-4': 'One of the successes of this approach is the complete resolution of the longstanding problem of negative fusion coefficients in fractional level Wess-Zumino-Witten models [CITATION].', '1310.6479-1-11-0': 'The application of this general formalism to the [MATH] singlet models is relatively straightforward (see [CITATION] for the case [MATH] and [CITATION] for more general [MATH]).', '1310.6479-1-11-1': 'However, the generalisation to all [MATH] singlet algebras is rather more interesting because, in the case where [MATH] and [MATH] are both greater than [MATH], the fusion product is no longer expected to define exact functors on the (natural) category of vertex operator algebra modules.', '1310.6479-1-11-2': 'This non-exactness was first noted in [CITATION] for the [MATH] triplet model.', '1310.6479-1-11-3': 'Consequently, the Grothendieck group spanned by the (equivalence classes of) simple modules does not inherit a ring structure from the fusion product.', '1310.6479-1-11-4': 'One therefore cannot expect that the ring structure defined by the Verlinde formula on the span of the characters of the simple modules - we call the resulting object the Verlinde ring - will coincide with a Grothendieck ring of fusion.', '1310.6479-1-11-5': 'The natural question of how the non-exactness of fusion is manifested in the Verlinde ring is what motivated our work on this problem.', '1310.6479-1-11-6': 'As we will see, the answer is natural and satisfying, though there are subtleties worth remarking upon.', '1310.6479-1-12-0': 'We begin in Section [REF] with notations and conventions, reviewing the definitions of the singlet and triplet [MATH]-algebras in terms of the Heisenberg algebra and its simple current extensions.', '1310.6479-1-12-1': 'The irreducible modules of both vertex operator algebras are constructed and their classifications are quoted with the necessary structural aspects of these modules being deferred to an appendix.', '1310.6479-1-12-2': 'This material has many sources, for example [CITATION].', '1310.6479-1-12-3': 'Here, we mostly follow the notation of [CITATION].', '1310.6479-1-13-0': 'In Section [REF], the modular properties of the characters of the singlet modules are derived.', '1310.6479-1-13-1': 'We begin with the standard modules which are of Feigin-Fuchs type, adding a Heisenberg charge to their characters so that all non-isomorphic simple modules have distinct characters.', '1310.6479-1-13-2': 'The [MATH]-transformations of these characters are deduced in the usual way.', '1310.6479-1-13-3': 'The algebraic definitions of the remaining (atypical) simple modules then lead to resolutions for each atypical simple in terms of standards.', '1310.6479-1-13-4': 'The resulting character formulae then give the [MATH]-transformations of the atypical characters directly.', '1310.6479-1-13-5': 'Of note here is that when [MATH] and [MATH] are both greater than [MATH], there exist atypical simple modules [MATH] whose [MATH]-matrix entries cannot be expressed as functions, but only as distributions.', '1310.6479-1-13-6': 'Indeed, this is also the case for the (non-simple) vacuum module.', '1310.6479-1-14-0': 'We then turn to the Verlinde formula and the Verlinde product that it induces in Section [REF].', '1310.6479-1-14-1': 'Most importantly, we show that the [MATH] completely decouple in the Verlinde ring and may be consistently set to zero.', '1310.6479-1-14-2': 'This lets us replace, when [MATH], the [MATH]-matrix entries involving the vacuum module by those involving its maximal submodule, which happens to be simple.', '1310.6479-1-14-3': 'The Verlinde formula is then well-defined, because we no longer need to divide by a distribution, and direct computation ensues.', '1310.6479-1-14-4': 'We thereby obtain a completely explicit description in Equation [REF] of the Verlinde product of the characters of any two simple singlet modules, excepting the [MATH] whose characters have been set to [MATH].', '1310.6479-1-15-0': 'This result is then lifted to the triplet models through their realisations as simple current extensions of the corresponding singlet models.', '1310.6479-1-15-1': 'Actually, these realisations remain conjectural in general because we can only verify the simple current property at the level of the Verlinde rings, not the fusion rings themselves.', '1310.6479-1-15-2': 'Nevertheless, we apply standard simple current technology to deduce the triplet analogues of Equation [REF].', '1310.6479-1-15-3': 'The resulting Verlinde product rules, reported in Equation [REF], are then compared favourably with the rules that have been proposed elsewhere in the literature.', '1310.6479-1-15-4': 'We close with a conclusion and discussion of our results.', '1310.6479-1-16-0': '# The [MATH] Singlet and Triplet Models', '1310.6479-1-17-0': 'In this section, we introduce and fix our notation for the singlet and triplet models.', '1310.6479-1-17-1': 'These conformal field theories are parametrised by two coprime positive integers [MATH] and [MATH].', '1310.6479-1-17-2': 'As one would expect, many of the important quantities that we will study take a somewhat unwieldy form when expressed in terms of these parameters, so to partially alleviate this, we introduce the following quantities: [EQUATION]', '1310.6479-1-17-3': 'Here, [MATH], [MATH] and [MATH] will always be assumed to be integers.', '1310.6479-1-17-4': 'Note that the [MATH] so-defined satisfy [EQUATION]', '1310.6479-1-17-5': 'We may therefore choose [MATH] so that [MATH] and [MATH], when convenient.', '1310.6479-1-18-0': '## Feigin-Fuchs Modules', '1310.6479-1-19-0': 'Consider the Fock module [MATH] of the Heisenberg algebra [MATH] with highest weight [MATH].', '1310.6479-1-19-1': 'As is well known, the vacuum module [MATH] carries the structure of a vertex operator algebra.', '1310.6479-1-19-2': 'There exists a continuous family of conformal structures for this vertex operator algebra and we will choose the corresponding Virasoro algebra so that the central charge is [EQUATION]', '1310.6479-1-19-3': 'Restricting to the action of this Virasoro algebra, the Fock modules [MATH] become Virasoro modules which we shall also denote by [MATH].', '1310.6479-1-19-4': 'When considering the [MATH] as Virasoro modules, we shall refer to them as Feigin-Fuchs modules.', '1310.6479-1-20-0': 'The structure of these Feigin-Fuchs modules was determined by Feigin and Fuchs in [CITATION] (see also [CITATION] for a comprehensive treatment).', '1310.6479-1-20-1': 'If [MATH] is not of the form [MATH] for some [MATH], then [MATH] is simple as a Virasoro module.', '1310.6479-1-20-2': 'If we choose [MATH] in what follows so that [MATH] and [MATH], then the structure depends only upon whether [MATH] and [MATH] are [MATH] and [MATH], respectively, and upon the sign of [MATH].', '1310.6479-1-20-3': 'We defer the explicit structural details of these Virasoro modules, in the form of socle filtrations, to Appendix [REF].', '1310.6479-1-21-0': 'The structure of the Feigin-Fuchs modules may be used to derive the Felder complexes [CITATION] [EQUATION] where we have simplified our notation by setting [MATH].', '1310.6479-1-21-1': 'Indeed, the Virasoro homomorphisms defining these complexes may be identified with (suitably regularised) powers of screening operators [CITATION].', '1310.6479-1-21-2': 'The complex [REF] may be checked to be exact when [MATH].', '1310.6479-1-21-3': 'Moreover, it only fails to be exact when [MATH] at the [MATH] term, in which case the homology is the simple Virasoro module [MATH] whose highest weight vector has conformal weight [MATH].', '1310.6479-1-21-4': 'Similarly, [REF] is exact for [MATH] and otherwise only has non-zero homology, again given by [MATH], at [MATH].', '1310.6479-1-22-0': '## The Singlet Algebra and its Modules', '1310.6479-1-23-0': 'We define the following (Virasoro) submodules of [MATH] for [MATH] and [MATH]: [EQUATION]', '1310.6479-1-23-1': 'If [MATH], the [MATH] are not defined by [REF].', '1310.6479-1-23-2': 'We will therefore set [MATH] in this case.', '1310.6479-1-23-3': 'Similarly, if [MATH], we set [MATH].', '1310.6479-1-23-4': 'Because of the exactness of the Felder complexes, we have the identifications ([MATH], [MATH]) [EQUATION] where the superscript "[MATH]" stands for [MATH], [MATH], or is empty.', '1310.6479-1-23-5': 'Working out the Virasoro module structures of the [MATH] modules using the socle series of the Feigin-Fuchs modules (Appendix [REF]), one arrives at the (non-split) short exact sequences [EQUATION]', '1310.6479-1-23-6': 'Finally, the definitions [REF] immediately imply the exact sequences [EQUATION] for all [MATH], and further contemplation of Virasoro structures (see Appendix [REF]) leads to [EQUATION] again for all [MATH], which are likewise exact.', '1310.6479-1-24-0': 'Recall that [MATH] carries the structure of a vertex operator algebra.', '1310.6479-1-24-1': 'As [MATH] and [MATH] are both kernels of screening operators acting on this vertex operator algebra, they define vertex operator subalgebras, as does their intersection [MATH].', '1310.6479-1-24-2': 'The vertex operator algebra corresponding to [MATH] is called the singlet algebra [MATH].', '1310.6479-1-24-3': 'It is simple if and only if [MATH] or [MATH] is 1.', '1310.6479-1-24-4': 'We remark that if [MATH], then [MATH] and we see that the singlet algebra [MATH] is nothing but the Heisenberg algebra (with central charge [MATH]).', '1310.6479-1-24-5': 'In general, the singlet algebra is strongly generated by the energy-momentum tensor and a single Virasoro primary of dimension [MATH] [CITATION].', '1310.6479-1-25-0': 'Each of the [MATH], [MATH] and [MATH], as well as the [MATH], become modules for the singlet vertex operator algebra.', '1310.6479-1-25-1': 'A complete list of simple [MATH]-modules is given by', '1310.6479-1-26-0': 'This is a straightforward corollary of [CITATION], though the case when [MATH] or [MATH] is [MATH] was settled previously in [CITATION].', '1310.6479-1-26-1': 'We remark that when [MATH], the sets of [MATH], [MATH] and [MATH] are empty - the only [MATH]-simples are the [MATH], the [MATH] and the [MATH].', '1310.6479-1-26-2': 'The story when [MATH] is similar.', '1310.6479-1-27-0': 'Notice that the [MATH] are simple for generic [MATH].', '1310.6479-1-27-1': 'In the formalism proposed in [CITATION] for general (logarithmic) conformal field theories, the [MATH] may be identified as the standard singlet modules.', '1310.6479-1-27-2': 'The simple standard modules, those with [MATH] or with [MATH], are called typical in this formalism and the remaining simple modules, the [MATH] and [MATH], are examples of atypical singlet modules.', '1310.6479-1-27-3': 'We will use this terminology freely in what follows.', '1310.6479-1-28-0': '## The Triplet Algebra and its Modules', '1310.6479-1-29-0': 'Just like the singlet algebra [MATH] W ( p_+,p_- )) can be defined as a vertex operator subalgebra of a lattice vertex operator algebra [MATH] F_1,1;0) by the group of simple currents generated by [MATH] F_1,1;-2)).', '1310.6479-1-29-1': 'In terms of Feigin-Fuchs modules, we therefore have the decomposition [EQUATION]', '1310.6479-1-29-2': 'This lattice vertex operator algebra is known to be rational, meaning that all of its modules are semisimple and that there are only finitely many inequivalent simple modules.', '1310.6479-1-29-3': 'The number of inequivalent simple modules is, in this case, exactly [MATH] and they can be parametrised by two integers, [MATH] and [MATH] , and a label "[MATH]".', '1310.6479-1-29-4': 'We denote these simple [MATH]F_r,s^[MATH]F_r,s^[MATH][MATH]+[MATH]-[MATH] W ( p_+,p_- )) may also be defined as an intersection of kernels: [EQUATION]', '1310.6479-1-29-5': 'Again, when [MATH] or [MATH], at least one of the Felder complexes is not defined and its corresponding kernel is replaced by the lattice module [MATH].', '1310.6479-1-29-6': 'In particular, it follows that [MATH] which is well known to be isomorphic to the level [MATH] vertex operator algebra [MATH].', '1310.6479-1-29-7': 'In general, the triplet vertex operator algebra [MATH] is strongly generated by the energy momentum tensor and three Virasoro primaries of dimension [MATH] [CITATION].', '1310.6479-1-29-8': 'The singlet algebra [MATH] is naturally generated as a vertex operator subalgebra by removing two of these Virasoro primaries.', '1310.6479-1-30-0': 'Unlike the lattice algebra [MATH], the triplet algebra [MATH] W ( p_+,p_- ))-modules is finite and, in fact, this number is [MATH] [CITATION].', '1310.6479-1-30-1': 'We give a complete list of these simples along with their decompositions into singlet modules (for later convenience):', '1310.6479-1-31-0': 'Again, this is an easy consequence of the results of [CITATION].', '1310.6479-1-31-1': 'Note that when [MATH], there are no [MATH], [MATH] or [MATH] in this list - the simples are exhausted by the [MATH] (with [MATH] allowed).', '1310.6479-1-31-2': 'Again, the story is similar for [MATH].', '1310.6479-1-32-0': 'In the terminology of [CITATION], the [MATH] are the typical triplet modules, being direct sums of typical singlet modules.', '1310.6479-1-32-1': 'The [MATH], with either [MATH] or [MATH], and the [MATH] are then atypical triplet modules.', '1310.6479-1-32-2': 'We remark that the decompositions of the triplet simples into singlet simples suggest that the triplet algebra is just a simple current extension of the singlet algebra.', '1310.6479-1-32-3': 'Indeed, the decomposition of [MATH] shows that the simple currents responsible for this conjectured extension are the [MATH].', '1310.6479-1-32-4': 'We will verify that this conjecture is consistent with our Verlinde formula computations in Section [REF], though we will see that there are interesting subtleties which prevent the evidence from being conclusive.', '1310.6479-1-33-0': '# Characters and Modular Transformations for Singlet Models', '1310.6479-1-34-0': 'This section details the derivation of the modular [MATH]-transformations of the characters of the simple [MATH]-modules.', '1310.6479-1-34-1': 'The methodology follows the approach proposed in [CITATION] for general non-rational conformal field theories.', '1310.6479-1-34-2': 'Specifically, the characters of the standard [MATH]-modules, simple and non-simple, are taken as a (topological) basis for a vector space which is shown to be preserved by the natural action of [MATH].', '1310.6479-1-34-3': 'This space then carries a representation of [MATH] of uncountably-infinite dimension.', '1310.6479-1-34-4': 'Resolutions are then derived for the atypical simple modules in terms of the non-simple standard modules and this gives expressions for the characters of the former as infinite alternating sums of the characters of the latter (the basis characters).', '1310.6479-1-34-5': 'In this way, we arrive at [MATH]-transformations for all simple [MATH]-modules.', '1310.6479-1-35-0': 'The computations are straightforward when [MATH] or [MATH] is 1.', '1310.6479-1-35-1': 'However, the case where [MATH] is more interesting (as expected) because the "[MATH]-matrix entries", that describe the decomposition of the [MATH]-transformed character of [MATH] into standard characters, are no longer functions of the parameters, but must instead be regarded as distributions.', '1310.6479-1-35-2': 'Consequently, the same is true for the [MATH]-matrix entries of the vacuum module [MATH], leading to conceptual difficulties in applying the Verlinde formula.', '1310.6479-1-36-0': '## Characters of Feigin-Fuchs modules and their Modular Transformations', '1310.6479-1-37-0': 'For any Virasoro module [MATH] at central charge [MATH] , one defines its character to be the following power series in [MATH] : [EQUATION]', '1310.6479-1-37-1': 'For example, the characters of the Feigin-Fuchs modules [MATH] (q) [MATH] F_) and [MATH] z=(2izeta) [MATH]z[MATH]-_0 / 2[MATH] S)-transformation of characters is the map [EQUATION]', '1310.6479-1-37-2': 'The characters of the Feigin-Fuchs modules satisfy the transformation formulae [EQUATION] where the [MATH] [ F_ F_ ]) are given by [EQUATION]', '1310.6479-1-37-3': 'These coefficients follow from a straightforward gaussian integral, convergent for [MATH] hence [MATH].', '1310.6479-1-37-4': 'In particular, for [MATH] , the [MATH]I ( p_+,p_- )[MATH] K_r,s;n^[MATH] I_r,s;n^[MATH] L_r,s) in terms of Feigin-Fuchs modules.', '1310.6479-1-37-5': 'These, in turn, allow us to calculate the characters of the atypical singlet modules in terms of characters of the standard (Feigin-Fuchs) singlet modules.', '1310.6479-1-38-0': 'It is important to note that the maps defining the Felder complexes and exact sequences are not Heisenberg algebra homomorphisms in general.', '1310.6479-1-38-1': 'The interpretation of [MATH] in Feigin-Fuchs characters as tracking the eigenvalue of the Heisenberg zero-mode therefore does not lift to the character formulae for the atypical singlet modules that we shall derive.', '1310.6479-1-38-2': 'There is no Heisenberg zero-mode in the singlet algebra, so the singlet characters should in fact be computed with [MATH].', '1310.6479-1-38-3': 'However, if one does this, one immediately encounters the problem that non-isomorphic singlet modules can have identical characters.', '1310.6479-1-38-4': 'We will therefore keep [MATH] as a formal variable in the singlet character formulae that follow.', '1310.6479-1-38-5': 'Its function remains to naturally facilitate the distinguishing of characters of non-isomorphic modules, though it no longer appears to have any (obvious) interpretation in terms of eigenvalues of zero-modes.', '1310.6479-1-39-0': '### The [MATH] modules', '1310.6479-1-40-0': 'As long as we avoid the non-exact parts of the Felder complexes [REF], we can use them to give (co)resolutions of the [MATH] in terms of Feigin-Fuchs modules, which in turn allow us to derive character formulae and [MATH]n 0[MATH]I_r,s;n^+[MATH]n<0[MATH] 1rp_+-1 [MATH] 1sp_- [MATH]I_r,s;n^+[MATH] nZ [MATH] s=p_- [MATH] S)-matrix coefficients are obtained by adding and subtracting the [MATH] [ F_p_+ - r,s;n-2k-1 F_ ]) and [MATH]n 0[MATH] [ F_r,s;n+2k F_ ]) and [MATH]n 1[MATH] S)-matrix coefficients simplify to the common form (for all [MATH]) [EQUATION] where we have used the identity [MATH] in the denominator.', '1310.6479-1-41-0': '### The [MATH] modules', '1310.6479-1-42-0': 'The derivation of (co)resolutions, character formulae and [MATH] I_r,s;n^- [MATH] I_r,s;n^+ [MATH] 1rp_+ [MATH] 1sp_-1 [MATH] I_r,s;n^- [MATH] S)-matrix coefficients [EQUATION] for general [MATH].', '1310.6479-1-42-1': 'Here, we have used the identity [MATH] in the denominator as well as the analytic continuations [REF].', '1310.6479-1-43-0': '### The [MATH] 1rp_+-1 [MATH] 1sp_-1 [MATH] I _r,s;n) can be resolved by iteratively splicing the exact sequences [REF] to obtain [EQUATION]', '1310.6479-1-43-1': 'We therefore arrive at two seemingly different character formulae: [EQUATION]', '1310.6479-1-43-2': 'However, the resulting [MATH]I_r,s;n[MATH]I_r,s;n^[MATH]L_r,s[MATH] 1rp_+-1 [MATH] 1sp_-1 [MATH]n=0[MATH] L_r,s).', '1310.6479-1-43-3': 'By the Euler-Poincare principle, the Felder complex [REF] implies that the character of the simple singlet module [MATH]z[MATH]z=1[MATH]z[MATH]I_r,s;n[MATH] S)-matrix coefficients: [EQUATION]', '1310.6479-1-43-4': 'The superscript "[MATH]" serves to remind us which Felder complex was used in the derivation.', '1310.6479-1-43-5': 'We note that the sums in these formulae do not define functions but must be interpreted as distributions (see Section [REF]).', '1310.6479-1-43-6': 'We note in addition that the character formulae [REF] also do not appear to respect the isomorphism [MATH].', '1310.6479-1-43-7': 'Again, the disambiguation variable [MATH] is to blame.', '1310.6479-1-43-8': 'We shall see shortly that this non-uniqueness problem solves itself rather naturally in the setting of the Verlinde algebra, though there is of course a price that has to be paid.', '1310.6479-1-44-0': '### The [MATH] 1rp_+-1 [MATH] 1sp_-1 [MATH] K _r,s;0) singlet modules: [EQUATION]', '1310.6479-1-44-1': 'These in turn yield two distinct [MATH]K_1,1;0[MATH]K_1,1;0[MATH]p_+ = 1[MATH]p_- = 1[MATH]K_1,1;0 I_1,1;0[MATH]S[MATH]I_r,s;n^[MATH]L_r,s[MATH]I_r,s;n^+[MATH]r p_+[MATH]n 1[MATH]n[MATH]K_r,s;0^+[MATH]I_r,s;0^+[MATH]*[MATH]r[MATH]n[MATH]p_+ - r[MATH]n[MATH]I_r,s;n^+[MATH]n 1[MATH]L_r,s[MATH]I_r,s;n^+[MATH]n 0[MATH]n>0[MATH](-1)^n [ L_*,s ][MATH]K_r,s;0^+[MATH]I_r,s;0^+[MATH]L_r,s[MATH]n 1[MATH]V [ p_+,p_- ][MATH]S[MATH]I ( 1,1 )[MATH]V [ 1,1 ][MATH]I ( 1,p_- )[MATH]I ( p_+,1 )[MATH]W ( 1,p_- )[MATH]W ( p_+,1 )[MATH]p_+,p_- > 1[MATH] I_r,s;n) to include [MATH] and [MATH] .', '1310.6479-1-44-2': 'Let [MATH] and [MATH] and then define [EQUATION]', '1310.6479-1-44-3': 'For [MATH] and [MATH] , the [MATH] L_r,s)', '1310.6479-1-45-0': 'For two singlet modules [MATH] and [MATH] , the Verlinde formula for "fusion products" states that there is a product (on certain equivalence classes of modules) given by [EQUATION]', '1310.6479-1-45-1': 'The "fusion coefficents" [MATH] S)-matrix coefficients by [EQUATION] where "[MATH]" denotes complex conjugation.', '1310.6479-1-45-2': 'Consider the denominator of the above integrand.', '1310.6479-1-45-3': 'According to Equation [REF], this [MATH] S)-matrix coefficients of [MATH] I_1,1;0) can be evaluated using [REF] and [REF]: [EQUATION]', '1310.6479-1-45-4': 'The product of the last two factors on the right-hand-side can be identified with zero because [EQUATION]', '1310.6479-1-45-5': 'A similar calculation gives the same conclusion for [MATH] .', '1310.6479-1-45-6': 'We may therefore replace the vacuum [MATH] [ K_1,1;0 F_ ]^[MATH] [ I_1,1;0 F_ ]), the [MATH] I_1,1;0).', '1310.6479-1-45-7': 'In this way, the non-uniqueness of the vacuum [MATH] S)-matrix coefficients [MATH] , and hence the [MATH] as well, can only be understood as distributions: [EQUATION]', '1310.6479-1-45-8': 'From this point of view, the vanishing of the quotient [REF] and its "[MATH]" version is manifest because the factors [MATH] may be replaced by [MATH] when brought into the sum over [MATH], so the coefficient of each delta function is zero.', '1310.6479-1-46-0': 'We next argue that at the level of the Verlinde products, the minimal model modules [MATH] N [MATH] N_L_r,s N^L_r,s N =0 [MATH](r,s)=(1,1)[MATH] [ K_1,1;0 F_ ]^[ I_1,1;0 F_ ]) that [MATH] .', '1310.6479-1-46-1': 'Summarising, we have seen that the alarming non-uniqueness and distributional nature of the [MATH]L_r,s[MATH]L_r,s[MATH][ L_r,s ] = 0[MATH] [ F_ ]mathbin [ F_ ]) of standard singlet module characters for [MATH].', '1310.6479-1-46-2': 'The Verlinde coefficient we seek is [EQUATION] which, by means of the trigonometric identity [EQUATION] simplifies to [EQUATION]', '1310.6479-1-46-3': 'The Verlinde formula [REF] therefore yields the following product: [EQUATION]', '1310.6479-1-46-4': 'We next turn to the product [MATH] 1rp_+ [MATH] 1sp_+ [MATH] nZ [MATH] R [MATH] 1rp_+-1 [MATH] 1sp_-1 [MATH]r=p_+[MATH]s=p_-[MATH] I_1,1;m), [MATH] I_1,2;0): [EQUATION]', '1310.6479-1-46-5': 'We note for future use that the linear [MATH]-span of the [MATH] is closed under the Verlinde product.', '1310.6479-1-47-0': 'To illustrate how to apply [REF] to (co)resolutions, we present the details of the derivation of the product [MATH], with [MATH].', '1310.6479-1-47-1': 'First, we recall that [MATH], so the resolution [REF] (or rather the corresponding character formula [REF]) allows us to write [EQUATION] the last equality following from [REF].', '1310.6479-1-47-2': 'The remaining Verlinde products of [REF] are similarly derived.', '1310.6479-1-48-0': '## Presentations and the Verlinde Ring', '1310.6479-1-49-0': 'It is clear from Equation [REF] that [MATH] [ I_2,1;0 ]) and [MATH] [ I_r,s;n ]), for [MATH] , [MATH] and [MATH] , by repeatedly taking Verlinde products with one another.', '1310.6479-1-49-1': 'Mathematically, this implies that there is a ring homomorphism [MATH] from the polynomial ring [MATH] to the subring of the singlet Verlinde ring [MATH] [ I_r,s;n ]).', '1310.6479-1-49-2': 'We will denote this subring by [MATH]I_p_+,p_-;n F_p_+,p_-;n[MATH][MATH] [ I_2,1;0 ]) with [MATH] [ I_1,2;0 ]) with [MATH]sl ( 2 )[MATH]r<p_+[MATH]s<p_-[MATH][MATH]r=p_+[MATH]s=p_-[MATH] I_r,s;n) modules: [EQUATION]', '1310.6479-1-49-3': 'Finally, we remark that the sums involving [MATH] in this result should be understood to vanish whenever [MATH].', '1310.6479-1-49-4': 'This formula follows directly from the multiplication formulae [REF] and the easily derived relations [EQUATION]', '1310.6479-1-50-0': '# The Verlinde Ring for Triplet Models', '1310.6479-1-51-0': 'Having determined explicit formulae for the Verlinde products of simple singlet modules, we now consider analogous formulae for the triplet modules.', '1310.6479-1-51-1': 'We note that there is almost nothing in the literature devoted to fusion rules for singlet models, but that there are many sources where triplet fusion rules have been conjectured or computed [CITATION].', '1310.6479-1-51-2': 'Comparing these results with the triplet Verlinde products that we will deduce therefore gives very strong consistency checks of both our results and those in the literature.', '1310.6479-1-52-0': '## Simple Currents for the Singlet Verlinde Ring', '1310.6479-1-53-0': 'As previewed in Section [REF], the [MATH] are simple currents in the Verlinde ring [MATH], that is, they are units of the Verlinde ring that act as permutations on the set of all simple modules.', '1310.6479-1-53-1': 'Indeed, the Verlinde products [REF] and [REF] give [EQUATION]', '1310.6479-1-53-2': 'In particular, [MATH] is the inverse, with respect to the Verlinde product, of [MATH] (because [MATH] is the identity).', '1310.6479-1-53-3': 'We note that when [MATH] and [MATH] and [MATH]), these simple currents are identified with the [MATH]).', '1310.6479-1-53-4': 'For [MATH], the identification is rather with the [MATH].', '1310.6479-1-54-0': 'When [MATH], so the singlet algebra coincides with the Heisenberg algebra, the [MATH] are well known to be simple currents in the fusion ring.', '1310.6479-1-54-1': 'Indeed, extending [MATH] by [MATH] leads to [MATH].', '1310.6479-1-54-2': 'We conjecture that this generalises so that the [MATH] define simple currents, in a sense that we will shortly describe, with respect to the fusion product of the singlet algebra [MATH].', '1310.6479-1-54-3': 'As remarked in Section [REF], this conjecture is already suggested by the decomposition of the simple triplet algebra modules into singlet modules, at least for [MATH] even.', '1310.6479-1-55-0': 'Let us consider the case where [MATH] and [MATH] (the case [MATH] and [MATH] is analogous).', '1310.6479-1-55-1': 'As we have noted above, there is then no Felder complex [REF], hence the list of simple [MATH]-modules given in Section [REF] truncates to the typical [MATH] and the atypicals [MATH] with [MATH] and [MATH] (the analogous modules with [MATH] are typical: [MATH]).', '1310.6479-1-55-2': 'In particular, there are no problematic modules [MATH] to worry about and we have a bijective correspondence between the simple [MATH]-modules and their (linearly independent) representatives in the Verlinde ring.', '1310.6479-1-55-3': 'We therefore claim that the Verlinde products [REF] lift to fusion products as follows: [EQUATION]', '1310.6479-1-55-4': 'Note that [MATH] is the vacuum module of [MATH].', '1310.6479-1-55-5': 'These singlet fusion products are consistent with the triplet fusion products that have appeared in the literature, though we will only verify this here at the level of the Verlinde ring.', '1310.6479-1-55-6': 'Essentially, we claim that the Verlinde products guarantee that these fusion products are simple, hence that there are no possible ambiguities concerning their structure.', '1310.6479-1-55-7': 'This is equivalent to fusion being exact on [MATH]-modules and the Verlinde ring, as defined above, coinciding with the Grothendieck ring of fusion.', '1310.6479-1-56-0': 'If we accept these arguments supporting the [MATH] being simple currents, then it is a simple matter to determine the (simple) spectrum of the simple current extension.', '1310.6479-1-56-1': 'We will do this for the group of simple currents corresponding to [MATH] in order to compare with the known spectrum of the triplet algebra [MATH].', '1310.6479-1-56-2': 'As the simple currents act freely (under the fusion product) on the simple singlet modules, the simple extended algebra modules are realised by summing over the orbits of the simple current group: [EQUATION]', '1310.6479-1-56-3': "Note that [MATH] 0 < /2 [MATH]I_1,1;n^-[MATH]n[MATH]W_1,s^[MATH]F_[]^[MATH]= _1,p_-[MATH]I ( 1,p_- )[MATH]I_1,1;2^-[MATH]W ( 1,p_- )[MATH]I_1,1;2^-[MATH]p_+, p_- > 1[MATH]I ( p_+,p_- )[MATH]L_r,s[MATH]L_r,s[MATH]I ( p_+,p_- )[MATH]V [ p_+,p_- ][MATH]I ( p_+,p_- )[MATH]L_r,s[MATH]L_r,s[MATH][MATH]+[MATH]-[MATH]*[MATH]n+n'=0[MATH]W ( 2,3 )[MATH]W ( p_+,p_- )[MATH]I_1,1;n[MATH]( K_1,1;0^ )^*[MATH]W ( p_+,p_- )[MATH]p_+, p_- > 1[MATH]I ( p_+,p_- )[MATH]I_1,1;2[MATH]p_+[MATH]p_-[MATH] W ( p_+,p_- )) is the simple current extension of the singlet algebra [MATH] V [ p_+,p_- ]) can be computed in terms of singlet Verlinde products by regarding each triplet module as a direct sum over an orbit of singlet modules under the action of the simple current (semi)group, choosing arbitrary representatives of each orbit, computing the Verlinde product of the representatives and, finally, determining the orbit of the resulting product.", '1310.6479-1-56-4': 'It is not hard to see that this general procedure reduces to the following simple rules: [EQUATION]', '1310.6479-1-56-5': 'Here, we distinguish the Verlinde product ([MATH]) of the triplet Verlinde ring [MATH][MATH] V [ p_+,p_- ]).', '1310.6479-1-56-6': 'It is clear now that the triplet Verlinde products can be directly read off from those of the singlet Verlinde ring [MATH][ W_1,1^- ][MATH]I_1,1;2[MATH] V [ p_+,p_- ]) now follow directly from the decompositions of Section [REF] and the Verlinde products of the singlet Verlinde ring given in [REF].', '1310.6479-1-56-7': 'The orbit of the singlet module [MATH] gives rise to the identity of [MATH]: [MATH].', '1310.6479-1-56-8': 'As the orbits of the singlet generators [MATH], [MATH] and [MATH] define the triplet modules [MATH], [MATH] and [MATH], respectively, Equation [REF] implies their Verlinde products: [EQUATION]', '1310.6479-1-56-9': 'The [MATH], [MATH] and [MATH] therefore generate [MATH].', '1310.6479-1-57-0': 'The general formula for the Verlinde product in the triplet Verlinde ring is similarly obtained from [REF]: [EQUATION] where [MATH] and [MATH] were defined in [REF].', '1310.6479-1-57-1': 'These formulae reproduce the Grothendieck fusion rules conjectured in [CITATION] from a Kazhdan-Lusztig-like correspondence with a certain quantum group, and are consistent with the fusion rules proposed in [CITATION] from lattice considerations, and those computed for certain small values of [MATH] and [MATH] in [CITATION] using the Nahm-Gaberdiel-Kausch algorithm.', '1310.6479-1-58-0': '# Discussion and Conclusions', '1310.6479-1-59-0': 'We have seen above that the Verlinde ring of the singlet algebra [MATH] may be straightforwardly derived from the modular transformation properties of the simple singlet modules and a continuous version of the Verlinde formula.', '1310.6479-1-59-1': 'Moreover, the Verlinde ring of the triplet algebra [MATH] then follows from some basic simple current technology and the result compares favourably with what is known of the triplet fusion ring.', '1310.6479-1-59-2': 'Indeed, it appears that this rather effortless approach captures pretty much all the information about the fusion ring that can be divined from the simple characters alone.', '1310.6479-1-59-3': 'The most difficult step was, in a sense, understanding the representation theory of the singlet algebra in the first place.', '1310.6479-1-60-0': 'On a heuristic level, we can understand the good behaviour of the modular properties of the singlet characters, as compared with those of the triplet characters, as stemming from the uncountable nature of the spectrum of simple singlet modules.', '1310.6479-1-60-1': 'For the singlet, the parametrisation of standard modules defines a countable set of points - points of atypicality let us say - at which the standard singlet modules become reducible.', '1310.6479-1-60-2': 'For the triplet, with its finite spectrum of simple modules, one finds instead that the majority correspond to atypical points (the exception is [MATH] of course).', '1310.6479-1-60-3': 'Now, observe that the [MATH]-matrix elements derived for the atypical simple singlet modules have poles at atypical parameter values.', '1310.6479-1-60-4': 'Consequently, we see that these poles need not cause problems when integrating over a continuous spectrum as we do for the singlet (the poles form a set of measure zero after all) but that they will definitely cause problems if one tries to perform a discrete sum as one would like to do for the triplet.', '1310.6479-1-61-0': 'It was first discovered in [CITATION] that the fusion product of the triplet algebra [MATH], with [MATH] and [MATH] greater than [MATH], does not necessarily map exact sequences to exact sequences.', '1310.6479-1-61-1': 'However, this failure of exactness was always observed to involve the simple modules [MATH] L_r,s) form an ideal with respect to the fusion product in the category of [MATH] W ( p_+,p_- ))-modules by the ideal generated by the [MATH] I ( p_+,p_- )).', '1310.6479-1-61-2': 'This quotient appears to be exactly what the singlet Verlinde ring [MATH] L_r,s) to zero.', '1310.6479-1-61-3': 'We therefore conjecture that the [MATH](p_+,p_-)[MATH](1,p)[MATH][MATH]z[MATH](1,p)[MATH](p_+,p_-)[MATH] R [MATH] _[MATH] r,s;n [MATH] __r,s;n [MATH] c = 1 - 3 _0^2 [MATH] L()).', '1310.6479-1-62-0': 'The socle of a Virasoro module [MATH] is its maximal semisimple submodule.', '1310.6479-1-62-1': 'The socle series of [MATH] is the ascending series [EQUATION] of submodules of [MATH] for which [MATH] (for [MATH] ) is the socle of [MATH] .', '1310.6479-1-62-2': 'Socle series are unique if they exist.', '1310.6479-1-62-3': 'Recalling the notation [MATH], there are five different possibilities for the socle series factors [MATH] of the Feigin-Fuchs modules [MATH] 1r< p_+ [MATH] 1s<p_- [MATH] nZ [MATH] a=0 [MATH] n0 [MATH] a=1 [MATH] n<0 [MATH] 1s<p_- [MATH] nZ [MATH] a=0 [MATH] n1 [MATH] a=1 [MATH] n<1 [MATH] 1r< p_+ [MATH] nZ [MATH] a=1 [MATH] n1 [MATH] a=0 [MATH] n<1 [MATH] nZ [MATH] F_p_+,p_-;n) is semisimple as a Virasoro module: [EQUATION]', '1310.6479-1-62-4': 'Finally, for [MATH] , with [MATH] for any [MATH] , [MATH] and [MATH] , the Feigin-Fuchs module [MATH]I ( p_+,p_- )[MATH] F_), with [MATH] for any [MATH] , [MATH] and [MATH] , are simple singlet modules, as are the [MATH]L_r,s[MATH]I_r,s;n^[MATH] 1r< p_+ [MATH] 1s<p_- [MATH] nZ [MATH] I_r,s;n) is therefore [MATH] .', '1310.6479-1-62-5': 'For [MATH] and [MATH] , [EQUATION]', '1310.6479-1-62-6': 'The minimal conformal weight of the states of [MATH] is therefore [MATH] .', '1310.6479-1-62-7': 'For [MATH] and [MATH] , [EQUATION]', '1310.6479-1-62-8': 'The minimal conformal weight of the states of [MATH] is therefore [MATH] .', '1310.6479-1-63-0': 'Of course, the minimal conformal weight of the states of [MATH] _p_+,p_-;n [MATH] F_) is [MATH] .'}

{'1310.6479-2-0-0': 'The [MATH] singlet algebra is a vertex operator algebra that is strongly generated by a Virasoro field of central charge [MATH] and a single Virasoro primary field of conformal weight [MATH].', '1310.6479-2-0-1': 'Here, the modular properties of the characters of the uncountably many simple modules of each singlet algebra are investigated and the results used as the input to a continuous analogue of the Verlinde formula to obtain the "fusion rules" of the singlet modules.', '1310.6479-2-0-2': 'The effect of the failure of fusion to be exact in general is studied at the level of Verlinde products and the rules derived are lifted to the [MATH] triplet algebras by regarding these algebras as simple current extensions of their singlet cousins.', '1310.6479-2-0-3': 'The result is a relatively effortless derivation of the triplet "fusion rules" that agrees with those previously proposed in the literature.', '1310.6479-2-1-0': '# Introduction', '1310.6479-2-2-0': 'The [MATH] singlet and triplet models (for [MATH] a positive integer) are perhaps the most basic known examples of logarithmic conformal field theories.', '1310.6479-2-2-1': 'Introduced in [CITATION], their logarithmic nature was exposed, at least for [MATH], through a connection to symplectic fermions and [MATH]-ghosts [CITATION].', '1310.6479-2-2-2': 'Further investigations, for example [CITATION], addressed the issue of generalising the tools familiar from rational conformal field theory to these models and they have remained popular objects of study ever since.', '1310.6479-2-3-0': 'Generalisations of these triplet models, called [MATH]-models, for [MATH] and [MATH] coprime and positive, were introduced in [CITATION].', '1310.6479-2-3-1': 'Because their central charges match those of the Virasoro [MATH] minimal models when [MATH] and [MATH] are greater than [MATH], they are sometimes referred to as logarithmic minimal models.', '1310.6479-2-3-2': 'One might hope that these logarithmic models could capture the universal features of critical lattice models that are missed by the minimal models (crossing probabilities, fractal dimensions and so on), but this is still contentious.', '1310.6479-2-3-3': 'Nevertheless, there has been persistent interest in these models from both theoretical physicists and mathematicians.', '1310.6479-2-3-4': 'One reason for this interest is that the underlying vertex operator algebras are not simple, so these models allow one to explore the consequences of this non-simplicity in a tractable, though still very challenging, setting.', '1310.6479-2-3-5': 'We remark that the simple quotients are precisely the minimal model vertex operator algebras.', '1310.6479-2-4-0': 'As with other logarithmic models, one of the main difficulties to surmount is that of obtaining a detailed structural understanding of the reducible but indecomposable modules which appear in the spectrum.', '1310.6479-2-4-1': 'While a complete classification of the indecomposables may well be infeasible, a first aim would be to identify the spectrum of simple modules and their projective covers.', '1310.6479-2-4-2': 'This is expected to be sufficient to construct bulk state spaces with modular invariant partition functions, for example.', '1310.6479-2-4-3': 'However, the current state of knowledge regarding projectives in non-semisimple module categories over vertex operator algebras is still in its infancy, so much of our intuition stems from examples like the [MATH]-models.', '1310.6479-2-5-0': 'But even in examples, the rigorous identification of projectives remains a formidable task.', '1310.6479-2-5-1': 'Indeed, this has only been achieved for [MATH] [CITATION].', '1310.6479-2-5-2': 'However, the literature contains many proposals and conjectures for general [MATH] and [MATH] (with varying degrees of structural detail), see [CITATION] for example.', '1310.6479-2-5-3': 'These proposals rely on conjectured equivalences of categories, numerical computations within integrable lattice discretisations, and explicit construction, the latter giving the most information (but requiring the most effort).', '1310.6479-2-5-4': 'In this direction, a powerful tool for structural investigations of indecomposables is the celebrated Nahm-Gaberdiel-Kausch algorithm [CITATION] that explicitly constructs (filtered quotients of) the fusion product of two modules.', '1310.6479-2-6-0': 'Determining fusion rules is, of course, another of the main difficulties one would like to overcome along the path to understanding a given logarithmic conformal field theory.', '1310.6479-2-6-1': 'While the Nahm-Gaberdiel-Kausch fusion algorithm allows one to construct enough of a fusion product to identify it completely in principle, the calculations are too computationally intensive for all but the smallest theories, even when performed by computer.', '1310.6479-2-6-2': 'Another issue is that the algorithm in practice only provides an "upper bound" on the fusion product in the sense that the true result could be, in principle, a proper subspace of what has been deduced.', '1310.6479-2-6-3': 'However, both of these issues can be circumvented by generalising another standard tool from rational conformal field theory to the logarithmic setting: the Verlinde formula.', '1310.6479-2-7-0': 'The Verlinde formula [CITATION] for rational conformal field theories computes the fusion product of two modules from the modular transformation properties of their characters.', '1310.6479-2-7-1': 'As this formula may be shown to follow from the internal consistency conditions that must be satisfied by any conformally-invariant quantum field theory [CITATION], one expects that it should remain valid in some form for more general classes of non-rational theories.', '1310.6479-2-7-2': 'In the logarithmic setting, characters cannot distinguish between reducible but indecomposable modules and the direct sum of their simple composition factors, hence the Verlinde formula cannot be expected to compute the true fusion rules, but only tell us which composition factors appear, and with what multiplicity, in a given fusion product.', '1310.6479-2-7-3': 'However, this is already very valuable information.', '1310.6479-2-7-4': 'In many cases, one can easily rule out the possibility that the simple factors combine to form an indecomposable and then the Verlinde formula gives the fusion rules as in rational theories.', '1310.6479-2-7-5': 'If an indecomposable can be formed, then this formula provides the character of the indecomposable effortlessly, thus solving the "upper bound" problem.', '1310.6479-2-7-6': 'Moreover, it also tells us which fusion products need to be checked for indecomposability, thus potentially saving computational resources.', '1310.6479-2-8-0': 'Unfortunately, the modular properties of the triplet [MATH]-models (with [MATH]) are not as nice as one could have hoped for.', '1310.6479-2-8-1': 'In particular, the character of the vacuum module transforms under [MATH] into a linear combination of other characters, but the coefficients depend non-trivially on [MATH] [CITATION].', '1310.6479-2-8-2': 'This would appear to invalidate a naive application of the Verlinde formula.', '1310.6479-2-8-3': 'Nevertheless, one can arrive at a [MATH]-independent [MATH]-transformation by postulating a non-standard automorphy matrix and a generalised Verlinde formula exploiting this has been demonstrated for the triplet models with [MATH] [CITATION].', '1310.6479-2-8-4': 'This proposed recipe does produce non-negative integer structure constants which agree with the known (Grothendieck) fusion coefficients.', '1310.6479-2-8-5': 'However, the generalised Verlinde formula itself is significantly more unwieldy than the original and we are not aware of any attempts to derive its analogues for other logarithmic conformal field theories.', '1310.6479-2-8-6': 'Another way of obtaining [MATH] -independent coefficients is to enlarge the space of characters to the space of torus 1-point functions, that is, to add certain linear combinations of characters multiplied by appropriate powers of [MATH] [CITATION], see also [CITATION].', '1310.6479-2-8-7': 'Other proposals for triplet Verlinde formulae may be found in [CITATION].', '1310.6479-2-9-0': 'Here, we follow a different path to the Verlinde formula.', '1310.6479-2-9-1': 'Instead of working directly with the triplet [MATH]-models that have received so much attention in the literature, we focus our attention on the relatively unexplored singlet [MATH]-models.', '1310.6479-2-9-2': 'Whereas the triplet vertex operator algebras are known to possess a finite number of inequivalent simple modules [CITATION], the singlet algebras admit an uncountable infinity of them.', '1310.6479-2-9-3': 'However, this is not a bug, but a feature!', '1310.6479-2-9-4': 'We will see that the modular transformation properties of the characters of these simple singlet modules are very well behaved.', '1310.6479-2-9-5': 'Moreover, applying the standard Verlinde formula (but with an integral replacing the sum) leads again to non-negative integer structure constants.', '1310.6479-2-9-6': 'Finally, these results can be lifted from the singlet algebra to its triplet cousin using the technology of simple current extensions.', '1310.6479-2-9-7': 'In particular, our results provide an effortless derivation of the [MATH] triplet (Grothendieck) fusion rules without the need for non-standard automorphy factors and complicated generalisations of the Verlinde formula.', '1310.6479-2-10-0': 'This path to the Verlinde formula is actually a special case of a rather more general formalism that has been proposed for non-rational conformal field theories in [CITATION].', '1310.6479-2-10-1': 'There, one starts with a continuous spectrum of so-called standard modules which are typically simple and whose characters have good modular properties.', '1310.6479-2-10-2': 'In logarithmic theories, the atypical standard modules are reducible but indecomposable and the not-so-good modular properties of the characters of the simple subquotients may be determined using standard methods of homological algebra.', '1310.6479-2-10-3': 'This approach was developed for logarithmic models based on affine (super)algebras [CITATION] where the natural spectrum is continuous.', '1310.6479-2-10-4': 'One of the successes of this approach is the complete resolution of the longstanding problem of negative fusion coefficients in fractional level Wess-Zumino-Witten models [CITATION].', '1310.6479-2-11-0': 'The application of this general formalism to the [MATH] singlet models is relatively straightforward (see [CITATION] for the case [MATH] and [CITATION] for more general [MATH]).', '1310.6479-2-11-1': 'However, the generalisation to all [MATH] singlet algebras is rather more interesting because, in the case where [MATH] and [MATH] are both greater than [MATH], the fusion product is no longer expected to define exact functors on the (natural) category of vertex operator algebra modules.', '1310.6479-2-11-2': 'This non-exactness was first noted in [CITATION] for the [MATH] triplet model.', '1310.6479-2-11-3': 'Consequently, the Grothendieck group spanned by the (equivalence classes of) simple modules does not inherit a ring structure from the fusion product.', '1310.6479-2-11-4': 'One therefore cannot expect that the ring structure defined by the Verlinde formula on the span of the characters of the simple modules - we call the resulting object the Verlinde ring - will coincide with a Grothendieck ring of fusion.', '1310.6479-2-11-5': 'The natural question of how the non-exactness of fusion is manifested in the Verlinde ring is what motivated our work on this problem.', '1310.6479-2-11-6': 'As we will see, the answer is natural and satisfying, though there are subtleties worth remarking upon.', '1310.6479-2-12-0': 'We begin in Section [REF] with notations and conventions, reviewing the definitions of the singlet and triplet [MATH]-algebras in terms of the Heisenberg algebra and its simple current extensions.', '1310.6479-2-12-1': 'The irreducible modules of both vertex operator algebras are constructed and their classifications are quoted with the necessary structural aspects of these modules being deferred to an appendix.', '1310.6479-2-12-2': 'This material has many sources, for example [CITATION].', '1310.6479-2-12-3': 'Here, we mostly follow the notation of [CITATION].', '1310.6479-2-13-0': 'In Section [REF], the modular properties of the characters of the singlet modules are derived.', '1310.6479-2-13-1': 'We begin with the standard modules which are of Feigin-Fuchs type, adding a Heisenberg charge to their characters so that all non-isomorphic simple modules have distinct characters.', '1310.6479-2-13-2': 'The [MATH]-transformations of these characters are deduced in the usual way.', '1310.6479-2-13-3': 'The algebraic definitions of the remaining (atypical) simple modules then lead to resolutions for each atypical simple in terms of standards.', '1310.6479-2-13-4': 'The resulting character formulae then give the [MATH]-transformations of the atypical characters directly.', '1310.6479-2-13-5': 'Of note here is that when [MATH] and [MATH] are both greater than [MATH], there exist atypical simple modules [MATH] whose [MATH]-matrix entries cannot be expressed as functions, but only as distributions.', '1310.6479-2-13-6': 'Indeed, this is also the case for the (non-simple) vacuum module.', '1310.6479-2-14-0': 'We then turn to the Verlinde formula and the Verlinde product that it induces in Section [REF].', '1310.6479-2-14-1': 'Most importantly, we show that the [MATH] completely decouple in the Verlinde ring and may be consistently set to zero.', '1310.6479-2-14-2': 'This lets us replace, when [MATH], the [MATH]-matrix entries involving the vacuum module by those involving its maximal submodule, which happens to be simple.', '1310.6479-2-14-3': 'The Verlinde formula is then well-defined, because we no longer need to divide by a distribution, and direct computation ensues.', '1310.6479-2-14-4': 'We thereby obtain a completely explicit description in Equation [REF] of the Verlinde product of the characters of any two simple singlet modules, excepting the [MATH] whose characters have been set to [MATH].', '1310.6479-2-15-0': 'This result is then lifted to the triplet models through their realisations as simple current extensions of the corresponding singlet models.', '1310.6479-2-15-1': 'Actually, these realisations remain conjectural in general because we can only verify the simple current property at the level of the Verlinde rings, not the fusion rings themselves.', '1310.6479-2-15-2': 'Nevertheless, we apply standard simple current technology to deduce the triplet analogues of Equation [REF].', '1310.6479-2-15-3': 'The resulting Verlinde product rules, reported in Equation [REF], are then compared favourably with the rules that have been proposed elsewhere in the literature.', '1310.6479-2-15-4': 'We close with a conclusion and discussion of our results.', '1310.6479-2-16-0': '# The [MATH] Singlet and Triplet Models', '1310.6479-2-17-0': 'In this section, we introduce and fix our notation for the singlet and triplet models.', '1310.6479-2-17-1': 'These conformal field theories are parametrised by two coprime positive integers [MATH] and [MATH].', '1310.6479-2-17-2': 'As one would expect, many of the important quantities that we will study take a somewhat unwieldy form when expressed in terms of these parameters, so to partially alleviate this, we introduce the following quantities: [EQUATION]', '1310.6479-2-17-3': 'Here, [MATH], [MATH] and [MATH] will always be assumed to be integers.', '1310.6479-2-17-4': 'Note that the [MATH] so-defined satisfy [EQUATION]', '1310.6479-2-17-5': 'We may therefore choose [MATH] so that [MATH] and [MATH], when convenient.', '1310.6479-2-18-0': '## Feigin-Fuchs Modules', '1310.6479-2-19-0': 'Consider the Fock module [MATH] of the Heisenberg algebra [MATH] with highest weight [MATH].', '1310.6479-2-19-1': 'As is well known, the vacuum module [MATH] carries the structure of a vertex operator algebra.', '1310.6479-2-19-2': 'There exists a continuous family of conformal structures for this vertex operator algebra and we will choose the corresponding Virasoro algebra so that the central charge is [EQUATION]', '1310.6479-2-19-3': 'Restricting to the action of this Virasoro algebra, the Fock modules [MATH] become Virasoro modules which we shall also denote by [MATH].', '1310.6479-2-19-4': 'When considering the [MATH] as Virasoro modules, we shall refer to them as Feigin-Fuchs modules.', '1310.6479-2-20-0': 'The structure of these Feigin-Fuchs modules was determined by Feigin and Fuchs in [CITATION] (see also [CITATION] for a comprehensive treatment).', '1310.6479-2-20-1': 'If [MATH] is not of the form [MATH] for some [MATH], then [MATH] is simple as a Virasoro module.', '1310.6479-2-20-2': 'If we choose [MATH] in what follows so that [MATH] and [MATH], then the structure depends only upon whether [MATH] and [MATH] are [MATH] and [MATH], respectively, and upon the sign of [MATH].', '1310.6479-2-20-3': 'We defer the explicit structural details of these Virasoro modules, in the form of socle filtrations, to Appendix [REF].', '1310.6479-2-21-0': 'The structure of the Feigin-Fuchs modules may be used to derive the Felder complexes [CITATION] [EQUATION] where we have simplified our notation by setting [MATH].', '1310.6479-2-21-1': 'Indeed, the Virasoro homomorphisms defining these complexes may be identified with (suitably regularised) powers of screening operators [CITATION].', '1310.6479-2-21-2': 'The complex [REF] may be checked to be exact when [MATH].', '1310.6479-2-21-3': 'Moreover, it only fails to be exact when [MATH] at the [MATH] term, in which case the homology is the simple Virasoro module [MATH] whose highest weight vector has conformal weight [MATH].', '1310.6479-2-21-4': 'Similarly, [REF] is exact for [MATH] and otherwise only has non-zero homology, again given by [MATH], at [MATH].', '1310.6479-2-22-0': '## The Singlet Algebra and its Modules', '1310.6479-2-23-0': 'We define the following (Virasoro) submodules of [MATH] for [MATH] and [MATH]: [EQUATION]', '1310.6479-2-23-1': 'If [MATH], the [MATH] are not defined by [REF].', '1310.6479-2-23-2': 'We will therefore set [MATH] in this case.', '1310.6479-2-23-3': 'Similarly, if [MATH], we set [MATH].', '1310.6479-2-23-4': 'Because of the exactness of the Felder complexes, we have the identifications ([MATH], [MATH]) [EQUATION] where the superscript "[MATH]" stands for [MATH], [MATH], or is empty.', '1310.6479-2-23-5': 'Working out the Virasoro module structures of the [MATH] modules using the socle series of the Feigin-Fuchs modules (Appendix [REF]), one arrives at the (non-split) short exact sequences [EQUATION]', '1310.6479-2-23-6': 'Finally, the definitions [REF] immediately imply the exact sequences [EQUATION] for all [MATH], and further contemplation of Virasoro structures (see Appendix [REF]) leads to [EQUATION] again for all [MATH], which are likewise exact.', '1310.6479-2-24-0': 'Recall that [MATH] carries the structure of a vertex operator algebra.', '1310.6479-2-24-1': 'As [MATH] and [MATH] are both kernels of screening operators acting on this vertex operator algebra, they define vertex operator subalgebras, as does their intersection [MATH].', '1310.6479-2-24-2': 'The vertex operator algebra corresponding to [MATH] is called the singlet algebra [MATH].', '1310.6479-2-24-3': 'It is simple if and only if [MATH] or [MATH] is 1.', '1310.6479-2-24-4': 'We remark that if [MATH], then [MATH] and we see that the singlet algebra [MATH] is nothing but the Heisenberg algebra (with central charge [MATH]).', '1310.6479-2-24-5': 'In general, the singlet algebra is strongly generated by the energy-momentum tensor and a single Virasoro primary of dimension [MATH] [CITATION].', '1310.6479-2-25-0': 'Each of the [MATH], [MATH] and [MATH], as well as the [MATH], become modules for the singlet vertex operator algebra.', '1310.6479-2-25-1': 'A complete list of simple [MATH]-modules is given by', '1310.6479-2-26-0': 'The case when [MATH] or [MATH] is [MATH] was settled in [CITATION].', '1310.6479-2-26-1': 'When [MATH], then the completeness of the above list is a straightforward corollary of [CITATION] using the same arguments as in [CITATION].', '1310.6479-2-26-2': 'We remark that when [MATH], the sets of [MATH], [MATH] and [MATH] are empty - the only [MATH]-simples are the [MATH], the [MATH] and the [MATH].', '1310.6479-2-26-3': 'The story when [MATH] is similar.', '1310.6479-2-27-0': 'Notice that the [MATH] are simple for generic [MATH].', '1310.6479-2-27-1': 'In the formalism proposed in [CITATION] for general (logarithmic) conformal field theories, the [MATH] may be identified as the standard singlet modules.', '1310.6479-2-27-2': 'The simple standard modules, those with [MATH] or with [MATH], are called typical in this formalism and the remaining simple modules, the [MATH] and [MATH], are examples of atypical singlet modules.', '1310.6479-2-27-3': 'We will use this terminology freely in what follows.', '1310.6479-2-28-0': '## The Triplet Algebra and its Modules', '1310.6479-2-29-0': 'Just like the singlet algebra [MATH] W ( p_+,p_- )) can be defined as a vertex operator subalgebra of a lattice vertex operator algebra [MATH] F_1,1;0) by the group of simple currents generated by [MATH] F_1,1;-2)).', '1310.6479-2-29-1': 'In terms of Feigin-Fuchs modules, we therefore have the decomposition [EQUATION]', '1310.6479-2-29-2': 'This lattice vertex operator algebra is known to be rational, meaning that all of its modules are semisimple and that there are only finitely many inequivalent simple modules.', '1310.6479-2-29-3': 'The number of inequivalent simple modules is, in this case, exactly [MATH] and they can be parametrised by two integers, [MATH] and [MATH] , and a label "[MATH]".', '1310.6479-2-29-4': 'We denote these simple [MATH]F_r,s^[MATH]F_r,s^[MATH][MATH]+[MATH]-[MATH] W ( p_+,p_- )) may also be defined as an intersection of kernels: [EQUATION]', '1310.6479-2-29-5': 'Again, when [MATH] or [MATH], at least one of the Felder complexes is not defined and its corresponding kernel is replaced by the lattice module [MATH].', '1310.6479-2-29-6': 'In particular, it follows that [MATH] which is well known to be isomorphic to the level [MATH] vertex operator algebra [MATH].', '1310.6479-2-29-7': 'In general, the triplet vertex operator algebra [MATH] is strongly generated by the energy momentum tensor and three Virasoro primaries of dimension [MATH] [CITATION].', '1310.6479-2-29-8': 'The singlet algebra [MATH] is naturally generated as a vertex operator subalgebra by removing two of these Virasoro primaries.', '1310.6479-2-30-0': 'Unlike the lattice algebra [MATH], the triplet algebra [MATH] W ( p_+,p_- ))-modules is finite and, in fact, this number is [MATH] [CITATION].', '1310.6479-2-30-1': 'We give a complete list of these simples along with their decompositions into singlet modules (for later convenience):', '1310.6479-2-31-0': 'Again, this is an easy consequence of the results of [CITATION].', '1310.6479-2-31-1': 'Note that when [MATH], there are no [MATH], [MATH] or [MATH] in this list - the simples are exhausted by the [MATH] (with [MATH] allowed).', '1310.6479-2-31-2': 'Again, the story is similar for [MATH].', '1310.6479-2-32-0': 'In the terminology of [CITATION], the [MATH] are the typical triplet modules, being direct sums of typical singlet modules.', '1310.6479-2-32-1': 'The [MATH], with either [MATH] or [MATH], and the [MATH] are then atypical triplet modules.', '1310.6479-2-32-2': 'We remark that the decompositions of the triplet simples into singlet simples suggest that the triplet algebra is just a simple current extension of the singlet algebra.', '1310.6479-2-32-3': 'Indeed, the decomposition of [MATH] shows that the simple currents responsible for this conjectured extension are the [MATH].', '1310.6479-2-32-4': 'We will verify that this conjecture is consistent with our Verlinde formula computations in Section [REF], though we will see that there are interesting subtleties which prevent the evidence from being conclusive.', '1310.6479-2-33-0': '# Characters and Modular Transformations for Singlet Models', '1310.6479-2-34-0': 'This section details the derivation of the modular [MATH]-transformations of the characters of the simple [MATH]-modules.', '1310.6479-2-34-1': 'The methodology follows the approach proposed in [CITATION] for general non-rational conformal field theories.', '1310.6479-2-34-2': 'Specifically, the characters of the standard [MATH]-modules, simple and non-simple, are taken as a (topological) basis for a vector space which is shown to be preserved by the natural action of [MATH].', '1310.6479-2-34-3': 'This space then carries a representation of [MATH] of uncountably-infinite dimension.', '1310.6479-2-34-4': 'Resolutions are then derived for the atypical simple modules in terms of the non-simple standard modules and this gives expressions for the characters of the former as infinite alternating sums of the characters of the latter (the basis characters).', '1310.6479-2-34-5': 'In this way, we arrive at [MATH]-transformations for all simple [MATH]-modules.', '1310.6479-2-35-0': 'The computations are straightforward when [MATH] or [MATH] is 1.', '1310.6479-2-35-1': 'However, the case where [MATH] is more interesting (as expected) because the "[MATH]-matrix entries", that describe the decomposition of the [MATH]-transformed character of [MATH] into standard characters, are no longer functions of the parameters, but must instead be regarded as distributions.', '1310.6479-2-35-2': 'Consequently, the same is true for the [MATH]-matrix entries of the vacuum module [MATH], leading to conceptual difficulties in applying the Verlinde formula.', '1310.6479-2-36-0': '## Characters of Feigin-Fuchs modules and their Modular Transformations', '1310.6479-2-37-0': 'For any Virasoro module [MATH] at central charge [MATH] , one defines its character to be the following power series in [MATH] : [EQUATION]', '1310.6479-2-37-1': 'For example, the characters of the Feigin-Fuchs modules [MATH] (q) [MATH] F_) and [MATH] z=(2izeta) [MATH]z[MATH]-_0 / 2[MATH] S)-transformation of characters is the map [EQUATION]', '1310.6479-2-37-2': 'The characters of the Feigin-Fuchs modules satisfy the transformation formulae [EQUATION] where the [MATH] [ F_ F_ ]) are given by [EQUATION]', '1310.6479-2-37-3': 'These coefficients follow from a straightforward gaussian integral, convergent for [MATH] hence [MATH].', '1310.6479-2-37-4': 'In particular, for [MATH] , the [MATH]I ( p_+,p_- )[MATH] K_r,s;n^[MATH] I_r,s;n^[MATH] L_r,s) in terms of Feigin-Fuchs modules.', '1310.6479-2-37-5': 'These, in turn, allow us to calculate the characters of the atypical singlet modules in terms of characters of the standard (Feigin-Fuchs) singlet modules.', '1310.6479-2-38-0': 'It is important to note that the maps defining the Felder complexes and exact sequences are not Heisenberg algebra homomorphisms in general.', '1310.6479-2-38-1': 'The interpretation of [MATH] in Feigin-Fuchs characters as tracking the eigenvalue of the Heisenberg zero-mode therefore does not lift to the character formulae for the atypical singlet modules that we shall derive.', '1310.6479-2-38-2': 'There is no Heisenberg zero-mode in the singlet algebra, so the singlet characters should in fact be computed with [MATH].', '1310.6479-2-38-3': 'However, if one does this, one immediately encounters the problem that non-isomorphic singlet modules can have identical characters.', '1310.6479-2-38-4': 'We will therefore keep [MATH] as a formal variable in the singlet character formulae that follow.', '1310.6479-2-38-5': 'Its function remains to naturally facilitate the distinguishing of characters of non-isomorphic modules, though it no longer appears to have any (obvious) interpretation in terms of eigenvalues of zero-modes.', '1310.6479-2-39-0': '### The [MATH] modules', '1310.6479-2-40-0': 'As long as we avoid the non-exact parts of the Felder complexes [REF], we can use them to give (co)resolutions of the [MATH] in terms of Feigin-Fuchs modules, which in turn allow us to derive character formulae and [MATH]n 0[MATH]I_r,s;n^+[MATH]n<0[MATH] 1rp_+-1 [MATH] 1sp_- [MATH]I_r,s;n^+[MATH] nZ [MATH] s=p_- [MATH] S)-matrix coefficients are obtained by adding and subtracting the [MATH] [ F_p_+ - r,s;n-2k-1 F_ ]) and [MATH]n 0[MATH] [ F_r,s;n+2k F_ ]) and [MATH]n 1[MATH] S)-matrix coefficients simplify to the common form (for all [MATH]) [EQUATION] where we have used the identity [MATH] in the denominator.', '1310.6479-2-41-0': '### The [MATH] modules', '1310.6479-2-42-0': 'The derivation of (co)resolutions, character formulae and [MATH] I_r,s;n^- [MATH] I_r,s;n^+ [MATH] 1rp_+ [MATH] 1sp_-1 [MATH] I_r,s;n^- [MATH] S)-matrix coefficients [EQUATION] for general [MATH].', '1310.6479-2-42-1': 'Here, we have used the identity [MATH] in the denominator as well as the analytic continuations [REF].', '1310.6479-2-43-0': '### The [MATH] 1rp_+-1 [MATH] 1sp_-1 [MATH] I _r,s;n) can be resolved by iteratively splicing the exact sequences [REF] to obtain [EQUATION]', '1310.6479-2-43-1': 'We therefore arrive at two seemingly different character formulae: [EQUATION]', '1310.6479-2-43-2': 'However, the resulting [MATH]I_r,s;n[MATH]I_r,s;n^[MATH]L_r,s[MATH] 1rp_+-1 [MATH] 1sp_-1 [MATH]n=0[MATH] L_r,s).', '1310.6479-2-43-3': 'By the Euler-Poincare principle, the Felder complex [REF] implies that the character of the simple singlet module [MATH]z[MATH]z[MATH]z=1[MATH]_p_+-r,s;2n+1 - _0/2 = -( _r,p_-s;-2n-1 - _0/2 )[MATH]I_r,s;n[MATH] S)-matrix coefficients: [EQUATION]', '1310.6479-2-43-4': 'The superscript "[MATH]" serves to remind us which Felder complex was used in the derivation.', '1310.6479-2-43-5': 'We note that the sums in these formulae do not define functions but must be interpreted as distributions (see Section [REF]).', '1310.6479-2-43-6': 'We note in addition that the character formulae [REF] also do not appear to respect the isomorphism [MATH].', '1310.6479-2-43-7': 'Again, the disambiguation variable [MATH] is to blame.', '1310.6479-2-43-8': 'We shall see shortly that this non-uniqueness problem solves itself rather naturally in the setting of the Verlinde algebra, though there is of course a price that has to be paid.', '1310.6479-2-44-0': '### The [MATH] 1rp_+-1 [MATH] 1sp_-1 [MATH] K _r,s;0) singlet modules: [EQUATION]', '1310.6479-2-44-1': 'These in turn yield two distinct [MATH]K_1,1;0[MATH]K_1,1;0[MATH]p_+ = 1[MATH]p_- = 1[MATH]K_1,1;0 I_1,1;0[MATH]S[MATH]I_r,s;n^[MATH]L_r,s[MATH]I_r,s;n^+[MATH]r p_+[MATH]n 1[MATH]n[MATH]K_r,s;0^+[MATH]I_r,s;0^+[MATH]*[MATH]r[MATH]n[MATH]p_+ - r[MATH]n[MATH]I_r,s;n^+[MATH]n 1[MATH]L_r,s[MATH]I_r,s;n^+[MATH]n 0[MATH]n>0[MATH](-1)^n [ L_*,s ][MATH]K_r,s;0^+[MATH]I_r,s;0^+[MATH]L_r,s[MATH]n 1[MATH]V [ p_+,p_- ][MATH]S[MATH]I ( 1,1 )[MATH]V [ 1,1 ][MATH]I ( 1,p_- )[MATH]I ( p_+,1 )[MATH]W ( 1,p_- )[MATH]W ( p_+,1 )[MATH]p_+,p_- > 1[MATH]L_r,s[MATH] I_r,s;n) to include [MATH] and [MATH] .', '1310.6479-2-44-2': 'Let [MATH] and [MATH] and then define [EQUATION]', '1310.6479-2-44-3': 'For [MATH] and [MATH] , the [MATH] L_r,s)', '1310.6479-2-45-0': 'For two singlet modules [MATH] and [MATH] , the Verlinde formula for "fusion products" states that there is a product (on certain equivalence classes of modules) given by [EQUATION]', '1310.6479-2-45-1': 'The "fusion coefficents" [MATH] S)-matrix coefficients by [EQUATION] where "[MATH]" denotes complex conjugation.', '1310.6479-2-45-2': 'Consider the denominator of the above integrand.', '1310.6479-2-45-3': 'According to Equation [REF], this [MATH] S)-matrix coefficients of [MATH] I_1,1;0) can be evaluated using [REF] and [REF]: [EQUATION]', '1310.6479-2-45-4': 'The product of the last two factors on the right-hand-side can be identified with zero because [EQUATION]', '1310.6479-2-45-5': 'A similar calculation gives the same conclusion for [MATH] .', '1310.6479-2-45-6': 'We may therefore replace the vacuum [MATH] [ K_1,1;0 F_ ]^[MATH] [ I_1,1;0 F_ ]), the [MATH] I_1,1;0).', '1310.6479-2-45-7': 'In this way, the non-uniqueness of the vacuum [MATH] S)-matrix coefficients [MATH] , and hence the [MATH] as well, can only be understood as distributions: [EQUATION]', '1310.6479-2-45-8': 'From this point of view, the vanishing of the quotient [REF] and its "[MATH]" version is manifest because the factors [MATH] may be replaced by [MATH] when brought into the sum over [MATH], so the coefficient of each delta function is zero.', '1310.6479-2-46-0': 'We next argue that at the level of the Verlinde products, the minimal model modules [MATH] N [MATH] N_L_r,s N^L_r,s N =0 [MATH](r,s)=(1,1)[MATH] [ K_1,1;0 F_ ]^[ I_1,1;0 F_ ]) that [MATH] .', '1310.6479-2-46-1': 'Summarising, we have seen that the alarming non-uniqueness and distributional nature of the [MATH]L_r,s[MATH]L_r,s[MATH][ L_r,s ] = 0[MATH] [ F_ ]mathbin [ F_ ]) of standard singlet module characters for [MATH].', '1310.6479-2-46-2': 'The Verlinde coefficient we seek is [EQUATION] which, by means of the trigonometric identity [EQUATION] simplifies to [EQUATION]', '1310.6479-2-46-3': 'The Verlinde formula [REF] therefore yields the following product: [EQUATION]', '1310.6479-2-46-4': 'We next turn to the product [MATH] 1rp_+ [MATH] 1sp_+ [MATH] nZ [MATH] R [MATH] 1rp_+-1 [MATH] 1sp_-1 [MATH]r=p_+[MATH]s=p_-[MATH] I_1,1;m), [MATH] I_1,2;0): [EQUATION]', '1310.6479-2-46-5': 'We note for future use that the linear [MATH]-span of the [MATH] is closed under the Verlinde product.', '1310.6479-2-47-0': 'To illustrate how to apply [REF] to (co)resolutions, we present the details of the derivation of the product [MATH], with [MATH].', '1310.6479-2-47-1': 'First, we recall that [MATH], so the resolution [REF] (or rather the corresponding character formula [REF]) allows us to write [EQUATION] the last equality following from [REF].', '1310.6479-2-47-2': 'The remaining Verlinde products of [REF] are similarly derived.', '1310.6479-2-48-0': '## Presentations and the Verlinde Ring', '1310.6479-2-49-0': 'It is clear from Equation [REF] that [MATH] [ I_2,1;0 ]) and [MATH] [ I_r,s;n ]), for [MATH] , [MATH] and [MATH] , by repeatedly taking Verlinde products with one another.', '1310.6479-2-49-1': 'Mathematically, this implies that there is a ring homomorphism [MATH] from the polynomial ring [MATH] to the subring of the singlet Verlinde ring [MATH] [ I_r,s;n ]).', '1310.6479-2-49-2': 'We will denote this subring by [MATH]I_p_+,p_-;n F_p_+,p_-;n[MATH][MATH] [ I_2,1;0 ]) with [MATH] [ I_1,2;0 ]) with [MATH]sl ( 2 )[MATH]r<p_+[MATH]s<p_-[MATH][MATH]r=p_+[MATH]s=p_-[MATH] I_r,s;n) modules: [EQUATION]', '1310.6479-2-49-3': 'Finally, we remark that the sums involving [MATH] in this result should be understood to vanish whenever [MATH].', '1310.6479-2-49-4': 'This formula follows directly from the multiplication formulae [REF] and the easily derived relations [EQUATION]', '1310.6479-2-50-0': '# The Verlinde Ring for Triplet Models', '1310.6479-2-51-0': 'Having determined explicit formulae for the Verlinde products of simple singlet modules, we now consider analogous formulae for the triplet modules.', '1310.6479-2-51-1': 'We note that there is almost nothing in the literature devoted to fusion rules for singlet models, but that there are many sources where triplet fusion rules have been conjectured or computed [CITATION].', '1310.6479-2-51-2': 'Comparing these results with the triplet Verlinde products that we will deduce therefore gives very strong consistency checks of both our results and those in the literature.', '1310.6479-2-52-0': '## Simple Currents for the Singlet Verlinde Ring', '1310.6479-2-53-0': 'As previewed in Section [REF], the [MATH] are simple currents in the Verlinde ring [MATH], that is, they are units of the Verlinde ring that act as permutations on the set of all simple modules.', '1310.6479-2-53-1': 'Indeed, the Verlinde products [REF] and [REF] give [EQUATION]', '1310.6479-2-53-2': 'In particular, [MATH] is the inverse, with respect to the Verlinde product, of [MATH] (because [MATH] is the identity).', '1310.6479-2-53-3': 'We note that when [MATH] and [MATH] and [MATH]), these simple currents are identified with the [MATH]).', '1310.6479-2-53-4': 'For [MATH], the identification is rather with the [MATH].', '1310.6479-2-54-0': 'When [MATH], so the singlet algebra coincides with the Heisenberg algebra, the [MATH] are well known to be simple currents in the fusion ring.', '1310.6479-2-54-1': 'Indeed, extending [MATH] by [MATH] leads to [MATH].', '1310.6479-2-54-2': 'We conjecture that this generalises so that the [MATH] define simple currents, in a sense that we will shortly describe, with respect to the fusion product of the singlet algebra [MATH].', '1310.6479-2-54-3': 'As remarked in Section [REF], this conjecture is already suggested by the decomposition of the simple triplet algebra modules into singlet modules, at least for [MATH] even.', '1310.6479-2-55-0': 'Let us consider the case where [MATH] and [MATH] (the case [MATH] and [MATH] is analogous).', '1310.6479-2-55-1': 'As we have noted above, there is then no Felder complex [REF], hence the list of simple [MATH]-modules given in Section [REF] truncates to the typical [MATH] and the atypicals [MATH] with [MATH] and [MATH] (the analogous modules with [MATH] are typical: [MATH]).', '1310.6479-2-55-2': 'In particular, there are no problematic modules [MATH] to worry about and we have a bijective correspondence between the simple [MATH]-modules and their (linearly independent) representatives in the Verlinde ring.', '1310.6479-2-55-3': 'We therefore claim that the Verlinde products [REF] lift to fusion products as follows: [EQUATION]', '1310.6479-2-55-4': 'Note that [MATH] is the vacuum module of [MATH].', '1310.6479-2-55-5': 'These singlet fusion products are consistent with the triplet fusion products that have appeared in the literature, though we will only verify this here at the level of the Verlinde ring.', '1310.6479-2-55-6': 'Essentially, we claim that the Verlinde products guarantee that these fusion products are simple, hence that there are no possible ambiguities concerning their structure.', '1310.6479-2-55-7': 'This is equivalent to fusion being exact on [MATH]-modules and the Verlinde ring, as defined above, coinciding with the Grothendieck ring of fusion.', '1310.6479-2-56-0': 'If we accept these arguments supporting the [MATH] being simple currents, then it is a simple matter to determine the (simple) spectrum of the simple current extension.', '1310.6479-2-56-1': 'We will do this for the group of simple currents corresponding to [MATH] in order to compare with the known spectrum of the triplet algebra [MATH].', '1310.6479-2-56-2': 'As the simple currents act freely (under the fusion product) on the simple singlet modules, the simple extended algebra modules are realised by summing over the orbits of the simple current group: [EQUATION]', '1310.6479-2-56-3': "Note that [MATH] 0 < /2 [MATH]I_1,1;n^-[MATH]n[MATH]W_1,s^[MATH]F_[]^[MATH]= _1,p_-[MATH]I ( 1,p_- )[MATH]I_1,1;2^-[MATH]W ( 1,p_- )[MATH]I_1,1;2^-[MATH]p_+, p_- > 1[MATH]I ( p_+,p_- )[MATH]L_r,s[MATH]L_r,s[MATH]I ( p_+,p_- )[MATH]V [ p_+,p_- ][MATH]I ( p_+,p_- )[MATH]L_r,s[MATH]L_r,s[MATH][MATH]+[MATH]-[MATH]*[MATH]n+n'=0[MATH]W ( 2,3 )[MATH]W ( p_+,p_- )[MATH]I_1,1;n[MATH]( K_1,1;0^ )^*[MATH]W ( p_+,p_- )[MATH]p_+, p_- > 1[MATH]I ( p_+,p_- )[MATH]I_1,1;2[MATH]p_+[MATH]p_-[MATH] W ( p_+,p_- )) is the simple current extension of the singlet algebra [MATH] V [ p_+,p_- ]) can be computed in terms of singlet Verlinde products by regarding each triplet module as a direct sum over an orbit of singlet modules under the action of the simple current (semi)group, choosing arbitrary representatives of each orbit, computing the Verlinde product of the representatives and, finally, determining the orbit of the resulting product.", '1310.6479-2-56-4': 'It is not hard to see that this general procedure reduces to the following simple rules: [EQUATION]', '1310.6479-2-56-5': 'Here, we distinguish the Verlinde product ([MATH]) of the triplet Verlinde ring [MATH][MATH] V [ p_+,p_- ]).', '1310.6479-2-56-6': 'It is clear now that the triplet Verlinde products can be directly read off from those of the singlet Verlinde ring [MATH][ W_1,1^- ][MATH]I_1,1;2[MATH] V [ p_+,p_- ]) now follow directly from the decompositions of Section [REF] and the Verlinde products of the singlet Verlinde ring given in [REF].', '1310.6479-2-56-7': 'The orbit of the singlet module [MATH] gives rise to the identity of [MATH]: [MATH].', '1310.6479-2-56-8': 'As the orbits of the singlet generators [MATH], [MATH] and [MATH] define the triplet modules [MATH], [MATH] and [MATH], respectively, Equation [REF] implies their Verlinde products: [EQUATION]', '1310.6479-2-56-9': 'The [MATH], [MATH] and [MATH] therefore generate [MATH].', '1310.6479-2-57-0': 'The general formula for the Verlinde product in the triplet Verlinde ring is similarly obtained from [REF]: [EQUATION] where [MATH] and [MATH] were defined in [REF].', '1310.6479-2-57-1': 'These formulae reproduce the Grothendieck fusion rules conjectured in [CITATION] from a Kazhdan-Lusztig-like correspondence with a certain quantum group, and are consistent with the fusion rules proposed in [CITATION] from lattice considerations, and those computed for certain small values of [MATH] and [MATH] in [CITATION] using the Nahm-Gaberdiel-Kausch algorithm, once the [MATH] have been set to zero.', '1310.6479-2-58-0': '# Discussion and Conclusions', '1310.6479-2-59-0': 'We have seen above that the Verlinde ring of the singlet algebra [MATH] may be straightforwardly derived from the modular transformation properties of the simple singlet modules and a continuous version of the Verlinde formula.', '1310.6479-2-59-1': 'Moreover, the Verlinde ring of the triplet algebra [MATH] then follows from some basic simple current technology and the result compares favourably with what is known of the triplet fusion ring.', '1310.6479-2-59-2': 'Indeed, it appears that this rather effortless approach captures pretty much all the information about the fusion ring that can be divined from the simple characters alone.', '1310.6479-2-59-3': 'The most difficult step was, in a sense, understanding the representation theory of the singlet algebra in the first place.', '1310.6479-2-60-0': 'On a heuristic level, we can understand the good behaviour of the modular properties of the singlet characters, as compared with those of the triplet characters, as stemming from the uncountable nature of the spectrum of simple singlet modules.', '1310.6479-2-60-1': 'For the singlet, the parametrisation of standard modules defines a countable set of points - points of atypicality let us say - at which the standard singlet modules become reducible.', '1310.6479-2-60-2': 'For the triplet, with its finite spectrum of simple modules, one finds instead that the majority correspond to atypical points (the exception is [MATH] of course).', '1310.6479-2-60-3': 'Now, observe that the [MATH]-matrix elements derived for the atypical simple singlet modules have poles at atypical parameter values.', '1310.6479-2-60-4': 'Consequently, we see that these poles need not cause problems when integrating over a continuous spectrum as we do for the singlet (the poles form a set of measure zero after all) but that they will definitely cause problems if one tries to perform a discrete sum as one would like to do for the triplet.', '1310.6479-2-61-0': 'It was first discovered in [CITATION] that the fusion product of the triplet algebra [MATH], with [MATH] and [MATH] greater than [MATH], does not necessarily map exact sequences to exact sequences.', '1310.6479-2-61-1': 'However, this failure of exactness was always observed to involve the simple modules [MATH] L_r,s) form an ideal with respect to the fusion product in the category of [MATH] W ( p_+,p_- ))-modules by the ideal generated by the [MATH] I ( p_+,p_- )).', '1310.6479-2-61-2': 'This quotient appears to be exactly what the singlet Verlinde ring [MATH] L_r,s) to zero.', '1310.6479-2-61-3': 'We therefore conjecture that the [MATH](p_+,p_-)[MATH](1,p)[MATH][MATH]z[MATH](1,p)[MATH](p_+,p_-)[MATH] R [MATH] _[MATH] r,s;n [MATH] __r,s;n [MATH] c = 1 - 3 _0^2 [MATH] L()).', '1310.6479-2-62-0': 'The socle of a Virasoro module [MATH] is its maximal semisimple submodule.', '1310.6479-2-62-1': 'The socle series of [MATH] is the ascending series [EQUATION] of submodules of [MATH] for which [MATH] (for [MATH] ) is the socle of [MATH] .', '1310.6479-2-62-2': 'Socle series are unique if they exist.', '1310.6479-2-62-3': 'Recalling the notation [MATH], there are five different possibilities for the socle series factors [MATH] of the Feigin-Fuchs modules [MATH] 1r< p_+ [MATH] 1s<p_- [MATH] nZ [MATH] a=0 [MATH] n0 [MATH] a=1 [MATH] n<0 [MATH] 1s<p_- [MATH] nZ [MATH] a=0 [MATH] n1 [MATH] a=1 [MATH] n<1 [MATH] 1r< p_+ [MATH] nZ [MATH] a=1 [MATH] n1 [MATH] a=0 [MATH] n<1 [MATH] nZ [MATH] F_p_+,p_-;n) is semisimple as a Virasoro module: [EQUATION]', '1310.6479-2-62-4': 'Finally, for [MATH] , with [MATH] for any [MATH] , [MATH] and [MATH] , the Feigin-Fuchs module [MATH]I ( p_+,p_- )[MATH] F_), with [MATH] for any [MATH] , [MATH] and [MATH] , are simple singlet modules, as are the [MATH]L_r,s[MATH]I_r,s;n^[MATH] 1r< p_+ [MATH] 1s<p_- [MATH] nZ [MATH] I_r,s;n) is therefore [MATH] .', '1310.6479-2-62-5': 'For [MATH] and [MATH] , [EQUATION]', '1310.6479-2-62-6': 'The minimal conformal weight of the states of [MATH] is therefore [MATH] .', '1310.6479-2-62-7': 'For [MATH] and [MATH] , [EQUATION]', '1310.6479-2-62-8': 'The minimal conformal weight of the states of [MATH] is therefore [MATH] .', '1310.6479-2-63-0': 'Of course, the minimal conformal weight of the states of [MATH] _p_+,p_-;n [MATH] F_) is [MATH] .'}

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[]

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[]

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{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1310.6479

cond-mat-0010329

{'cond-mat-0010329-1-0-0': 'Bilayer quantum Hall states support a flow of nearly dissipationless staggered current which can only decay through collective channels.', 'cond-mat-0010329-1-0-1': 'We study the dominant finite-temperature dissipation mechanism which in narrow bars is driven by thermal nucleation of pseudospin solitons.', 'cond-mat-0010329-1-0-2': 'We find the finite-temperature resistivity, predict the resulting staggered current-voltage characteristics, and calculate the associated zero-temperature critical staggered current and gate voltage.', 'cond-mat-0010329-1-1-0': ']', 'cond-mat-0010329-1-2-0': 'A 2D electron gas bilayer, subjected to a strong perpendicular magnetic field, can exhibit incompressible quantum Hall (QH) states even for filling fractions corresponding to compressible states of noninteracting layers [CITATION].', 'cond-mat-0010329-1-2-1': 'The nontrivial, strongly interacting nature of these QH states lies in the fact that they survive the limit of vanishing interlayer tunneling [CITATION].', 'cond-mat-0010329-1-2-2': 'They are stabilized by the exchange part of the Coulomb interaction, which, in the limit of vanishing single-particle tunneling, sets the scale of the gap and leads to macroscopic interlayer phase coherence.', 'cond-mat-0010329-1-3-0': 'In addition to exhibiting the QHE for a uniform current, these states support persistent currents that are counter-propagating in the two layers with [MATH].', 'cond-mat-0010329-1-3-1': 'In this Letter, we study a thermally-driven decay mechanism of [MATH] which controls the current-voltage characteristics for staggered currents smaller than the critical current [MATH].', 'cond-mat-0010329-1-3-2': 'Because the bilayer system displays a quantum Hall gap [MATH] in the phase-coherent ground state, dissipation via single-particle mechanisms is strongly suppressed for [MATH].', 'cond-mat-0010329-1-3-3': 'Therefore, as with supercurrents in superconductors [CITATION], the staggered-current decay rate is dominated, for a range of parameters, by the collective mechanism of soliton nucleation.', 'cond-mat-0010329-1-4-0': 'A convenient language for describing this strongly correlated quantum-coherent gapped state is in terms of a pseudospin unit vector field [MATH] [CITATION], with [MATH] giving the electron charge-density difference between top and bottom layers and [MATH] characterizing the relative phase [MATH] of electrons in two layers.', 'cond-mat-0010329-1-4-1': 'The energy functional describing long length scale (larger than the magnetic length [MATH]) variations of [MATH] is given by [CITATION] [EQUATION] where electron Coulomb interaction is the origin of the effective exchange constants [MATH] that drive the transition into the pseudo-ferromagnetic ground state, corresponding to the interlayer phase coherent QH state.', 'cond-mat-0010329-1-4-2': 'The electrostatic capacitive energy [MATH] introduces a hard-axis anisotropy, which forces the pseudo-magnetization to lie in the [MATH] plane ([MATH]) and thereby reduces the full SU(2) pseudospin symmetry to U(1) [CITATION].', 'cond-mat-0010329-1-4-3': 'A combination of the external gate voltage [MATH] and the single-electron interlayer tunneling [MATH] acts as an external pseudo-magnetic field [MATH].', 'cond-mat-0010329-1-4-4': 'Because the tunneling [MATH] can be tuned independently of [MATH] to be quite small, the low-energy physics of this anisotropic QH pseudo-ferromagnet, described by the Goldstone mode [MATH], can be fully explored experimentally.', 'cond-mat-0010329-1-5-0': 'In the limit of vanishing tunneling [MATH], an essentially exact analytical treatment of narrow (1D limit) QH bars is possible and leads to the following results.', 'cond-mat-0010329-1-5-1': 'The bilayer QH phase exhibits staggered current-carrying states that are metastable and therefore supports staggered persistent currents for [MATH], where the critical current density is given by [EQUATION] where [MATH] is the critical current at zero gate voltage, which vanishes in the SU(2) invariant [MATH] limit [CITATION], [MATH] is a reduced measure of the gate voltage [MATH], and [MATH].', 'cond-mat-0010329-1-5-2': 'At finite temperature, [MATH], plotted in Fig. [REF], therefore delineates a low resistivity regime, where Ohmic dissipation is dominated by slow thermal soliton nucleation, from a highly resistive state dominated by quasi-particle dissipation.', 'cond-mat-0010329-1-6-0': 'For [MATH], the staggered IV characteristics are plotted in Fig. [REF] and given by [EQUATION] where [MATH] is the microscopic attempt frequency, [MATH] is the reduced (dimensionless) current density, [MATH], with [MATH] the narrow sample dimension [MATH], and [MATH] is the saddle-point energy barrier separating two different current-carrying states.', 'cond-mat-0010329-1-6-1': 'The barrier is plotted in the inset of Fig. [REF] and is explicitly given by [EQUATION] where the limits of integration are [MATH] and [MATH], with [MATH], and the dimensionless wavevector [MATH] is defined implicitly through the current [MATH].', 'cond-mat-0010329-1-6-2': 'The analytic expression for the barrier simplifies considerably when one of its arguments vanishes.', 'cond-mat-0010329-1-6-3': 'We find that [MATH] depends weakly on [MATH] and for [MATH] is given by', 'cond-mat-0010329-1-7-0': '[EQUATION] and [MATH].', 'cond-mat-0010329-1-8-0': 'For such narrow Hall bars the staggered linear resistivity [MATH] is always finite at finite temperature and is given by [EQUATION]', 'cond-mat-0010329-1-8-1': 'From exact diagonalization studies [CITATION] at [MATH], we have [MATH] and [MATH].', 'cond-mat-0010329-1-8-2': 'Taking in addition [MATH], we obtain [EQUATION] where [MATH].', 'cond-mat-0010329-1-8-3': 'Setting [MATH] respectively gives [MATH].', 'cond-mat-0010329-1-9-0': 'For a realistic system, there is a limited range of validity of the above results, with other effects dominating outside of this range.', 'cond-mat-0010329-1-9-1': 'The constraint of quasi-equilibrium, which implies low decay rate, together with the requirement that the thermal collective dissipation mechanism dominates over single-particle current decay requires [MATH].', 'cond-mat-0010329-1-9-2': 'At the same time, however, [MATH] must be sufficiently high so that thermal nucleation dominates over quantum tunneling of phase slips.', 'cond-mat-0010329-1-9-3': '[CITATION] Furthermore, in order for the bulk nucleation rate to be experimentally observable, it is necessary that it dominates over phase slips nucleated at surfaces, contacts, and sample inhomogeneities.', 'cond-mat-0010329-1-9-4': 'Since the bulk nucleation rate scales with the Hall bar length [MATH], we expect that the bulk mechanism dominates over surface nucleation for [MATH].', 'cond-mat-0010329-1-9-5': 'Also, for the staggered current decay rate to be dominated by the 1D line solitons studied here, the saddle-point energy barrier given in Eq. ([REF]) must be lower than barriers for the competing mechanism of [MATH] vortex pair nucleation.', 'cond-mat-0010329-1-9-6': 'For a sufficiently wide Hall bar, the latter scenario will dominate, with the crossover occurring for [MATH].', 'cond-mat-0010329-1-10-0': 'We now present the highlights of calculations that lead to these results.', 'cond-mat-0010329-1-10-1': 'Although quite distinct in detail, the spirit of our analysis follows the classic work of Langer and Ambegaokar [CITATION].', 'cond-mat-0010329-1-11-0': 'The Euler-Lagrange (EL) equations for Eq. ([REF]) admit the uniform current-carrying solutions', 'cond-mat-0010329-1-12-0': '[EQUATION] where [MATH] is the dimensionless wave vector and we have taken our sample to lie in the [MATH]-[MATH] plane with dimensions [MATH].', 'cond-mat-0010329-1-12-1': 'Equation ([REF]) is valid in the region [MATH] and [MATH].', 'cond-mat-0010329-1-12-2': '(We consider [MATH] to be non-negative.)', 'cond-mat-0010329-1-12-3': 'The staggered current for this solution is [MATH], or, equivalently, [MATH].', 'cond-mat-0010329-1-13-0': 'For nonuniform solutions, the EL equations can be combined into a single equation which, after some manipulation, can be written as [EQUATION] for some constant [MATH], with', 'cond-mat-0010329-1-14-0': '[EQUATION]', 'cond-mat-0010329-1-14-1': 'In the usual mechanical analogy, Eq. ([REF]) represents the energy [MATH] of a particle at "position" [MATH] and "time" [MATH], moving in a potential [MATH] and with a space-dependent mass [MATH].', 'cond-mat-0010329-1-14-2': 'This potential is plotted in Fig. [REF] for [MATH]) and [MATH].', 'cond-mat-0010329-1-15-0': 'The "conservation of energy," Eq. ([REF]), immediately implies the existence of two extended solutions [CITATION].', 'cond-mat-0010329-1-15-1': 'A uniform current-carrying solution is given by Eq. ([REF]), corresponding to the particle forever remaining on top of the hill at [MATH] with angular velocity [MATH].', 'cond-mat-0010329-1-15-2': 'For this solution, [MATH] winds azimuthally at a constant rate as a function of [MATH] with a constant amplitude [MATH] deviating from the equatorial plane ([MATH]) with increasing [MATH].', 'cond-mat-0010329-1-15-3': 'For fixed winding [MATH], this uniform current-carrying solution is a local minimum of [MATH] and is hence metastable.', 'cond-mat-0010329-1-15-4': 'Since the energy is lowered with decreasing [MATH], we expect at finite temperature the system will thermally activate down to the [MATH] zero-current ground state via successive thermal transitions [MATH].', 'cond-mat-0010329-1-15-5': 'To calculate the rate of such transitions we need to compute the saddle-point barrier separating two "neighboring" current-carrying metastable states.', 'cond-mat-0010329-1-15-6': 'Brief reflection on the mechanical analogy shows that the second solution is in fact a saddle-point solution corresponding to the particle starting at "position" [MATH] and "time" [MATH], spiraling down hill to [MATH] at [MATH] while conserving angular momentum but increasing its angular velocity [MATH], and finally bouncing back out to [MATH] for [MATH].', 'cond-mat-0010329-1-15-7': 'The resulting "energy conservation" equation allows an exact determination of the saddle-point solution, written in terms of a 1D integral, [EQUATION] for an appropriate value of [MATH], and is shown in the inset of Fig. [REF].', 'cond-mat-0010329-1-16-0': 'The saddle-point solution is the nucleation site for the eventual singularity where [MATH]), and whereby the system can slip a loop reducing the total phase winding [MATH] by [MATH], and therefore leading to staggered-current decay.', 'cond-mat-0010329-1-16-1': 'Contrary to what is often tacitly assumed, the barrier for defect nucleation is not determined by the energy of the defect-which may be either singular, as it is in superconductors, or non-singular, as it is in the present case of the full SU(2) space-although we expect them to be close in energy.', 'cond-mat-0010329-1-16-2': 'Instead, the barrier is controlled by a non-singular saddle-point field configuration (inset of Fig. [REF]), which is in the same topological sector as the current-carrying metastable minimum.', 'cond-mat-0010329-1-16-3': 'It is also important to note that for sufficiently small [MATH], the energy for such phase slips is much smaller than the quantum Hall gap [MATH] and that therefore the entire system remains in the fully gapped quantum Hall state throughout the phase-slip process.', 'cond-mat-0010329-1-16-4': 'This is to be contrasted with phase slips inside a superconductor, where the order parameter, and therefore the bulk gap, are suppressed within the vortex core and one in principle has to take into account the core-confined low energy (normal) quasi-particle degrees of freedom.', 'cond-mat-0010329-1-17-0': 'The energy barrier is defined as the difference in energy between the saddle-point solution, Eq. ([REF]) and the uniform current-carrying solution, Eq. ([REF]).', 'cond-mat-0010329-1-17-1': 'By exploiting the mechanical analogy, and in particular the conservation law, Eq. ([REF]), we obtain the expression quoted in Eq. ([REF]).', 'cond-mat-0010329-1-18-0': 'In the steady-state regime, the staggered voltage [MATH] is proportional to the net rate of phase slips, [MATH], which itself is the difference between the rate of current-decreasing transitions ([MATH]) and the rate of current-increasing transitions ([MATH]).', 'cond-mat-0010329-1-18-1': 'In a sample of length [MATH], there are approximately [MATH] possible nucleation sites.', 'cond-mat-0010329-1-18-2': 'These considerations lead directly to the expression in Eq. ([REF]).', 'cond-mat-0010329-1-19-0': 'In contrast to the energy barrier [MATH], which is a static quantity, we must consider pseudospin dynamics in order to compute the attempt frequency [MATH] appearing in Eq. ([REF]).', 'cond-mat-0010329-1-19-1': 'Within the microscopic dynamical model valid at low temperatures, [MATH], for finite interlayer tunneling, [MATH] is given by the ratio of the curvature of the metastable well ([MATH]) and the dynamical mass term ([MATH]), leading to [MATH].', 'cond-mat-0010329-1-19-2': 'In contrast, for [MATH], we expect classical Langevin dynamics[CITATION], characterized by a kinetic "drag" coefficient [MATH], and which in the simplest estimate gives [MATH].', 'cond-mat-0010329-1-20-0': 'When the transverse (narrow) dimension [MATH] of the sample becomes sufficiently large, the energy barrier [MATH] for nucleating a line-soliton defect becomes comparable to the energy of nucleating a [MATH] vortex pair-a competing mechanism for inducing phase-slips.', 'cond-mat-0010329-1-20-1': 'Up to weak logarithmic corrections, the energy of such a vortex pair is [MATH], where [MATH] is the core size.', 'cond-mat-0010329-1-20-2': 'Vortex nucleation should therefore dominate the 1D soliton nucleation considered here for [MATH], which can be tuned independently of the QH gap [MATH].', 'cond-mat-0010329-1-20-3': 'This is in contrast with superconductors, where the corresponding 1D to 2D crossover scale is the Ginzburg-Landau correlation length, controlled by the superconducting gap.', 'cond-mat-0010329-1-21-0': 'Although it is difficult to extend our exact 1D analysis to the 2D limit, we can estimate the staggered current decay rate using simple scaling arguments.', 'cond-mat-0010329-1-21-1': 'In 2D, the phase slip rate is controlled by [MATH] vortex pair nucleation, analogously to superfluids and superconductors.', 'cond-mat-0010329-1-21-2': 'However, in contrast to those more familiar systems, here vortices (half-skyrmion, i.e., merons) carry [MATH] electromagnetic charge in addition to their [MATH] topological charge, and there are therefore four elementary vortex defects [MATH], [MATH], [MATH], and [MATH], with [MATH] respectively corresponding to [MATH] circulation of [MATH].', 'cond-mat-0010329-1-21-3': 'Correspondingly, within the interlayer phase ([MATH]) coherent state, the [MATH] and [MATH] vortices are bound into [MATH] types of topologically neutral pairs: (i) electromagnetically neutral pairs [MATH]-[MATH] and (ii) electromagnetically charged pairs [MATH]-[MATH] [CITATION].', 'cond-mat-0010329-1-21-4': 'Nevertheless, we do not expect the Coulomb interaction, which is subdominant to the topological-charge confining potential, to play a role in staggered-current-induced vortex ionization processes.', 'cond-mat-0010329-1-21-5': 'Hence, in the limit of vanishing [MATH], standard nucleation analysis for dissipation in superconducting films[CITATION] can be easily extended to our system.', 'cond-mat-0010329-1-21-6': 'It predicts a highly nonlinear power-law staggered IV, [MATH], with [MATH] in the interlayer coherent state, a result that contrasts strikingly with the linear staggered IV found in the 1D limit.', 'cond-mat-0010329-1-22-0': 'Nonvanishing interlayer tunneling, [MATH] explicitly breaks [MATH] symmetry and leads to nonuniform staggered current-carrying states composed of a lattice of solitons of width [MATH] and density [MATH] akin to a periodic array of Rayleigh-Benard current rolls.', 'cond-mat-0010329-1-22-1': '[CITATION].', 'cond-mat-0010329-1-23-0': 'In the dense soliton limit, [MATH], (relevant for small [MATH] and large [MATH]) the current [MATH] is nearly uniform and our 1D results directly apply.', 'cond-mat-0010329-1-23-1': 'In the dilute limit, [MATH] (which is always reached for sufficiently low [MATH][CITATION]), for [MATH], phase slips are confined to a single soliton, inside which [MATH] is uniform and the tunneling energy is on average zero.', 'cond-mat-0010329-1-23-2': 'Hence, our 1D, [MATH] analysis again applies with the [MATH]-independent wavevector [MATH].', 'cond-mat-0010329-1-23-3': 'Furthermore, scaling analysis[CITATION] suggests that in the opposite limit [MATH], this solution is still valid, but with the nucleation width set by [MATH], rather than [MATH].', 'cond-mat-0010329-1-24-0': 'In the 2D limit, the staggered current decays by ionization of [MATH]-[MATH] vortex pairs, whose energy for [MATH] grows linearly with the separation [MATH] as [MATH].', 'cond-mat-0010329-1-24-1': 'This therefore suggests the existence of a true staggered critical current [MATH], with [MATH] even at finite [MATH] (up to corrections that vanish in the thermodynamic limit), and for [MATH], [MATH].', 'cond-mat-0010329-1-25-0': 'We thank Anton Andreev, Ramin Abolfath and Allan MacDonald for discussions.', 'cond-mat-0010329-1-25-1': 'This work was supported by the NSF Grant DMR-9625111, and by the A. P. Sloan and Packard Foundations.'}

{'cond-mat-0010329-2-0-0': 'Bilayer quantum Hall states support a flow of nearly dissipationless staggered current which can only decay through collective channels.', 'cond-mat-0010329-2-0-1': 'We study the dominant finite-temperature dissipation mechanism which in narrow bars is driven by thermal nucleation of pseudospin solitons.', 'cond-mat-0010329-2-0-2': 'We find the finite-temperature resistivity, predict the resulting staggered current-voltage characteristics, and calculate the associated zero-temperature critical staggered current and gate voltage.', 'cond-mat-0010329-2-1-0': ']', 'cond-mat-0010329-2-2-0': 'A 2D electron gas bilayer, subjected to a strong perpendicular magnetic field, can exhibit incompressible quantum Hall (QH) states even for filling fractions corresponding to compressible states of noninteracting layers [CITATION].', 'cond-mat-0010329-2-2-1': 'The nontrivial, strongly interacting nature of these QH states lies in the fact that they survive the limit of vanishing interlayer tunneling [CITATION].', 'cond-mat-0010329-2-2-2': 'They are stabilized by the exchange part of the Coulomb interaction, which, in the limit of vanishing single-particle tunneling, sets the scale of the gap and leads to macroscopic interlayer phase coherence.', 'cond-mat-0010329-2-3-0': 'In addition to exhibiting the QHE for a uniform current, these states support persistent currents that are counter-propagating in the two layers with [MATH].', 'cond-mat-0010329-2-3-1': 'In this Letter, we study a thermally-driven decay mechanism of [MATH] which controls the current-voltage characteristics for staggered currents smaller than the critical current [MATH].', 'cond-mat-0010329-2-3-2': 'Because the bilayer system displays a quantum Hall gap [MATH] in the phase-coherent ground state, dissipation via single-particle mechanisms is strongly suppressed for [MATH].', 'cond-mat-0010329-2-3-3': 'Therefore, as with supercurrents in superconductors [CITATION], the staggered-current decay rate is dominated, for a range of parameters, by the collective mechanism of soliton nucleation.', 'cond-mat-0010329-2-4-0': 'A convenient language for describing this strongly correlated quantum-coherent gapped state is in terms of a pseudospin unit vector field [MATH] [CITATION], with [MATH] giving the electron charge-density difference between top and bottom layers and [MATH] characterizing the relative phase [MATH] of electrons in two layers.', 'cond-mat-0010329-2-4-1': 'The energy functional describing long length scale (larger than the magnetic length [MATH]) variations of [MATH] is given by [CITATION] [EQUATION] where electron Coulomb interaction is the origin of the effective exchange constants [MATH] that drive the transition into the pseudo-ferromagnetic ground state, corresponding to the interlayer phase coherent QH state.', 'cond-mat-0010329-2-4-2': 'The electrostatic capacitive energy [MATH] introduces a hard-axis anisotropy, which forces the pseudo-magnetization to lie in the [MATH] plane ([MATH]) and thereby reduces the full SU(2) pseudospin symmetry to U(1) [CITATION].', 'cond-mat-0010329-2-4-3': 'A combination of the external gate voltage [MATH] and the single-electron interlayer tunneling [MATH] acts as an external pseudo-magnetic field [MATH].', 'cond-mat-0010329-2-4-4': 'Because the tunneling [MATH] can be tuned independently of [MATH] to be quite small, the low-energy physics of this anisotropic QH pseudo-ferromagnet, described by the Goldstone mode [MATH], can be fully explored experimentally.', 'cond-mat-0010329-2-5-0': 'In the limit of vanishing tunneling [MATH], an essentially exact analytical treatment of narrow (1D limit) QH bars is possible and leads to the following results.', 'cond-mat-0010329-2-5-1': 'The bilayer QH phase exhibits staggered current-carrying states that are metastable and therefore supports staggered persistent currents for [MATH], where the critical current density is given by [EQUATION] where [MATH] is the critical current at zero gate voltage, which vanishes in the SU(2) invariant [MATH] limit [CITATION], [MATH] is a reduced measure of the gate voltage [MATH], and [MATH].', 'cond-mat-0010329-2-5-2': 'At finite temperature, [MATH], plotted in Fig. [REF], therefore delineates a low resistivity regime, where Ohmic dissipation is dominated by slow thermal soliton nucleation, from a highly resistive state dominated by quasi-particle dissipation.', 'cond-mat-0010329-2-6-0': 'For [MATH], the staggered IV characteristics are plotted in Fig. [REF] and given by [EQUATION] where [MATH] is the microscopic attempt frequency, [MATH] is the reduced (dimensionless) current density, [MATH], with [MATH] the narrow sample dimension [MATH], and [MATH] is the saddle-point energy barrier separating two different current-carrying states.', 'cond-mat-0010329-2-6-1': 'The barrier is plotted in the inset of Fig. [REF] and is explicitly given by [EQUATION] where the limits of integration are [MATH] and [MATH], with [MATH], and the dimensionless wavevector [MATH] is defined implicitly through the current [MATH].', 'cond-mat-0010329-2-6-2': 'The analytic expression for the barrier simplifies considerably when one of its arguments vanishes.', 'cond-mat-0010329-2-6-3': 'We find that [MATH] depends weakly on [MATH] and for [MATH] is given by', 'cond-mat-0010329-2-7-0': '[EQUATION] and [MATH].', 'cond-mat-0010329-2-8-0': 'For such narrow Hall bars the staggered linear resistivity [MATH] is always finite at finite temperature and is given by [EQUATION]', 'cond-mat-0010329-2-8-1': 'From exact diagonalization studies [CITATION] at [MATH], we have [MATH] and [MATH].', 'cond-mat-0010329-2-8-2': 'Taking in addition [MATH], we obtain [EQUATION] where [MATH].', 'cond-mat-0010329-2-8-3': 'Setting [MATH] respectively gives [MATH].', 'cond-mat-0010329-2-9-0': 'For a realistic system, there is a limited range of validity of the above results, with other effects dominating outside of this range.', 'cond-mat-0010329-2-9-1': 'The constraint of quasi-equilibrium, which implies low decay rate, together with the requirement that the thermal collective dissipation mechanism dominates over single-particle current decay requires [MATH].', 'cond-mat-0010329-2-9-2': 'At the same time, however, [MATH] must be sufficiently high so that thermal nucleation dominates over quantum tunneling of phase slips.', 'cond-mat-0010329-2-9-3': '[CITATION] Furthermore, in order for the bulk nucleation rate to be experimentally observable, it is necessary that it dominates over phase slips nucleated at surfaces, contacts, and sample inhomogeneities.', 'cond-mat-0010329-2-9-4': 'Since the bulk nucleation rate scales with the Hall bar length [MATH], we expect that the bulk mechanism dominates over surface nucleation for [MATH].', 'cond-mat-0010329-2-9-5': 'Also, for the staggered current decay rate to be dominated by the 1D line solitons studied here, the saddle-point energy barrier given in Eq. ([REF]) must be lower than barriers for the competing mechanism of [MATH] vortex pair nucleation.', 'cond-mat-0010329-2-9-6': 'For a sufficiently wide Hall bar, the latter scenario will dominate, with the crossover occurring for [MATH].', 'cond-mat-0010329-2-10-0': 'We now present the highlights of calculations that lead to these results.', 'cond-mat-0010329-2-10-1': 'Although quite distinct in detail, the spirit of our analysis follows the classic work of Langer and Ambegaokar [CITATION].', 'cond-mat-0010329-2-11-0': 'The Euler-Lagrange (EL) equations for Eq. ([REF]) admit the uniform current-carrying solutions', 'cond-mat-0010329-2-12-0': '[EQUATION] where [MATH] is the dimensionless wave vector and we have taken our sample to lie in the [MATH]-[MATH] plane with dimensions [MATH].', 'cond-mat-0010329-2-12-1': 'Equation ([REF]) is valid in the region [MATH] and [MATH].', 'cond-mat-0010329-2-12-2': '(We consider [MATH] to be non-negative.)', 'cond-mat-0010329-2-12-3': 'The staggered current for this solution is [MATH], or, equivalently, [MATH].', 'cond-mat-0010329-2-13-0': 'For nonuniform solutions, the EL equations can be combined into a single equation which, after some manipulation, can be written as [EQUATION] for some constant [MATH], with', 'cond-mat-0010329-2-14-0': '[EQUATION]', 'cond-mat-0010329-2-14-1': 'In the usual mechanical analogy, Eq. ([REF]) represents the energy [MATH] of a particle at "position" [MATH] and "time" [MATH], moving in a potential [MATH] and with a space-dependent mass [MATH].', 'cond-mat-0010329-2-14-2': 'This potential is plotted in Fig. [REF] for [MATH]) and [MATH].', 'cond-mat-0010329-2-15-0': 'The "conservation of energy," Eq. ([REF]), immediately implies the existence of two extended solutions [CITATION].', 'cond-mat-0010329-2-15-1': 'A uniform current-carrying solution is given by Eq. ([REF]), corresponding to the particle forever remaining on top of the hill at [MATH] with angular velocity [MATH].', 'cond-mat-0010329-2-15-2': 'For this solution, [MATH] winds azimuthally at a constant rate as a function of [MATH] with a constant amplitude [MATH] deviating from the equatorial plane ([MATH]) with increasing [MATH].', 'cond-mat-0010329-2-15-3': 'For fixed winding [MATH], this uniform current-carrying solution is a local minimum of [MATH] and is hence metastable.', 'cond-mat-0010329-2-15-4': 'Since the energy is lowered with decreasing [MATH], we expect at finite temperature the system will thermally activate down to the [MATH] zero-current ground state via successive thermal transitions [MATH].', 'cond-mat-0010329-2-15-5': 'To calculate the rate of such transitions we need to compute the saddle-point barrier separating two "neighboring" current-carrying metastable states.', 'cond-mat-0010329-2-15-6': 'Brief reflection on the mechanical analogy shows that the second solution is in fact a saddle-point solution corresponding to the particle starting at "position" [MATH] and "time" [MATH], spiraling down hill to [MATH] at [MATH] while conserving angular momentum but increasing its angular velocity [MATH], and finally bouncing back out to [MATH] for [MATH].', 'cond-mat-0010329-2-15-7': 'The resulting "energy conservation" equation allows an exact determination of the saddle-point solution, written in terms of a 1D integral, [EQUATION] for an appropriate value of [MATH], and is shown in the inset of Fig. [REF].', 'cond-mat-0010329-2-16-0': 'The saddle-point solution is the nucleation site for the eventual singularity where [MATH]), and whereby the system can slip a loop reducing the total phase winding [MATH] by [MATH], and therefore leading to staggered-current decay.', 'cond-mat-0010329-2-16-1': 'Contrary to what is often tacitly assumed, the barrier for defect nucleation is not determined by the energy of the defect-which may be either singular, as it is in superconductors, or non-singular, as it is in the present case of the full SU(2) space-although we expect them to be close in energy.', 'cond-mat-0010329-2-16-2': 'Instead, the barrier is controlled by a non-singular saddle-point field configuration (inset of Fig. [REF]), which is in the same topological sector as the current-carrying metastable minimum.', 'cond-mat-0010329-2-16-3': 'It is also important to note that for sufficiently small [MATH], the energy for such phase slips is much smaller than the quantum Hall gap [MATH] and that therefore the entire system remains in the fully gapped quantum Hall state throughout the phase-slip process.', 'cond-mat-0010329-2-16-4': 'This is to be contrasted with phase slips inside a superconductor, where the order parameter, and therefore the bulk gap, are suppressed within the vortex core and one in principle has to take into account the core-confined low energy (normal) quasi-particle degrees of freedom.', 'cond-mat-0010329-2-17-0': 'The energy barrier is defined as the difference in energy between the saddle-point solution, Eq. ([REF]) and the uniform current-carrying solution, Eq. ([REF]).', 'cond-mat-0010329-2-17-1': 'By exploiting the mechanical analogy, and in particular the conservation law, Eq. ([REF]), we obtain the expression quoted in Eq. ([REF]).', 'cond-mat-0010329-2-18-0': 'In the steady-state regime, the staggered voltage [MATH] is proportional to the net rate of phase slips, [MATH], which itself is the difference between the rate of current-decreasing transitions ([MATH]) and the rate of current-increasing transitions ([MATH]).', 'cond-mat-0010329-2-18-1': 'In a sample of length [MATH], there are approximately [MATH] possible nucleation sites.', 'cond-mat-0010329-2-18-2': 'These considerations lead directly to the expression in Eq. ([REF]).', 'cond-mat-0010329-2-19-0': 'In contrast to the energy barrier [MATH], which is a static quantity, we must consider pseudospin dynamics in order to compute the attempt frequency [MATH] appearing in Eq. ([REF]).', 'cond-mat-0010329-2-19-1': 'Within the microscopic dynamical model valid at low temperatures, [MATH], for finite interlayer tunneling, [MATH] is given by the ratio of the curvature of the metastable well ([MATH]) and the dynamical mass term ([MATH]), leading to [MATH].', 'cond-mat-0010329-2-19-2': 'In contrast, for [MATH], we expect classical Langevin dynamics[CITATION], characterized by a kinetic "drag" coefficient [MATH], and which in the simplest estimate gives [MATH].', 'cond-mat-0010329-2-20-0': 'When the transverse (narrow) dimension [MATH] of the sample becomes sufficiently large, the energy barrier [MATH] for nucleating a line-soliton defect becomes comparable to the energy of nucleating a [MATH] vortex pair-a competing mechanism for inducing phase-slips.', 'cond-mat-0010329-2-20-1': 'Up to weak logarithmic corrections, the energy of such a vortex pair is [MATH], where [MATH] is the core size.', 'cond-mat-0010329-2-20-2': 'Vortex nucleation should therefore dominate the 1D soliton nucleation considered here for [MATH], which can be tuned independently of the QH gap [MATH].', 'cond-mat-0010329-2-20-3': 'This is in contrast with superconductors, where the corresponding 1D to 2D crossover scale is the Ginzburg-Landau correlation length, controlled by the superconducting gap.', 'cond-mat-0010329-2-21-0': 'Although it is difficult to extend our exact 1D analysis to the 2D limit, we can estimate the staggered current decay rate using simple scaling arguments.', 'cond-mat-0010329-2-21-1': 'In 2D, the phase slip rate is controlled by [MATH] vortex pair nucleation, analogously to superfluids and superconductors.', 'cond-mat-0010329-2-21-2': 'However, in contrast to those more familiar systems, here vortices (half-skyrmion, i.e., merons) carry [MATH] electromagnetic charge in addition to their [MATH] topological charge, and there are therefore four elementary vortex defects [MATH], [MATH], [MATH], and [MATH], with [MATH] respectively corresponding to [MATH] circulation of [MATH].', 'cond-mat-0010329-2-21-3': 'Correspondingly, within the interlayer phase ([MATH]) coherent state, the [MATH] and [MATH] vortices are bound into [MATH] types of topologically neutral pairs: (i) electromagnetically neutral pairs [MATH]-[MATH] and (ii) electromagnetically charged pairs [MATH]-[MATH] [CITATION].', 'cond-mat-0010329-2-21-4': 'Nevertheless, we do not expect the Coulomb interaction, which is subdominant to the topological-charge confining potential, to play a role in staggered-current-induced vortex ionization processes.', 'cond-mat-0010329-2-21-5': 'Hence, in the limit of vanishing [MATH], standard nucleation analysis for dissipation in superconducting films[CITATION] can be easily extended to our system.', 'cond-mat-0010329-2-21-6': 'It predicts a highly nonlinear power-law staggered IV, [MATH], with [MATH] in the interlayer coherent state, a result that contrasts strikingly with the linear staggered IV found in the 1D limit.', 'cond-mat-0010329-2-22-0': 'Nonvanishing interlayer tunneling, [MATH] explicitly breaks [MATH] symmetry and leads to nonuniform staggered current-carrying states composed of a lattice of solitons of width [MATH] and density [MATH] akin to a periodic array of Rayleigh-Benard current rolls.', 'cond-mat-0010329-2-22-1': '[CITATION].', 'cond-mat-0010329-2-23-0': 'In the dense soliton limit, [MATH], (relevant for small [MATH] and large [MATH]) the current [MATH] is nearly uniform and our 1D results directly apply.', 'cond-mat-0010329-2-23-1': 'In the dilute limit, [MATH] (which is always reached for sufficiently low [MATH][CITATION]), for [MATH], phase slips are confined to a single soliton, inside which [MATH] is uniform and the tunneling energy is on average zero.', 'cond-mat-0010329-2-23-2': 'Hence, our 1D, [MATH] analysis again applies with the [MATH]-independent wavevector [MATH].', 'cond-mat-0010329-2-23-3': 'Furthermore, scaling analysis[CITATION] suggests that in the opposite limit [MATH], this solution is still valid, but with the nucleation width set by [MATH], rather than [MATH].', 'cond-mat-0010329-2-24-0': 'In the 2D limit, the staggered current decays by ionization of [MATH]-[MATH] vortex pairs, whose energy for [MATH] grows linearly with the separation [MATH] as [MATH].', 'cond-mat-0010329-2-24-1': 'This therefore suggests the existence of a true staggered critical current [MATH], with [MATH] even at finite [MATH] (up to corrections that vanish in the thermodynamic limit), and for [MATH], [MATH].', 'cond-mat-0010329-2-25-0': 'We thank Anton Andreev, Ramin Abolfath and Allan MacDonald for discussions.', 'cond-mat-0010329-2-25-1': 'This work was supported by the NSF Grant DMR-9625111, and by the A. P. Sloan and Packard Foundations.'}

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[]

[]

[]

[]

['cond-mat-0010329-1-1-0', 'cond-mat-0010329-1-7-0', 'cond-mat-0010329-1-14-0', 'cond-mat-0010329-1-22-1', 'cond-mat-0010329-2-1-0', 'cond-mat-0010329-2-7-0', 'cond-mat-0010329-2-14-0', 'cond-mat-0010329-2-22-1']

{'1': 'http://arxiv.org/licenses/assumed-1991-2003/', '2': 'http://arxiv.org/licenses/assumed-1991-2003/'}

https://arxiv.org/abs/cond-mat/0010329

1809.05576

{'1809.05576-1-0-0': '# Introduction', '1809.05576-1-1-0': 'Identifying events from a given ontology in text and locating their arguments is an especially challenging task because events vary widely in their textual realizations and their arguments are often spread across multiple clauses or sentences.', '1809.05576-1-1-1': 'Most event research has been in the context of the 2005 NIST Automatic Content Extraction (ace) sentence-level event mention task [CITATION], which also provides the standard corpus.', '1809.05576-1-1-2': 'Recently, tac kbp has introduced document-level event argument extraction shared tasks for 2014 and 2015 (kbp ea).', '1809.05576-1-2-0': 'Progress on events since ACE has been limited.', '1809.05576-1-2-1': 'Most subsequent work has tried improve performance through the use of more complex inference [CITATION], by transductively drawing on outside sources of information, or both [CITATION].', '1809.05576-1-2-2': 'Such approaches have produced modest reductions in error over a pipeline of simple classifiers trained on ACE.', '1809.05576-1-3-0': 'In our efforts to improve on the kbp ea 2014 systems, we were stymied by a lack of data, especially for rarer event types.', '1809.05576-1-3-1': 'Ten of the 33 event types have fewer than 25 training examples in ace, and even for more frequent events, many trigger words and classes of arguments occurred only once.', '1809.05576-1-3-2': 'Furthermore, the 2015 task would include new argument types.', '1809.05576-1-3-3': 'These problems motivated the following question: (a) are we at a plateau in the performance vs. annotation time curve?', '1809.05576-1-3-4': '(b) is there an viable alternative to full-document annotation, especially for rarer event types?', '1809.05576-1-3-5': '(c) for novel event types or languages, how quickly can a useful event model be trained?', '1809.05576-1-4-0': 'In traditional annotation, a static corpus selected to be rich in the target event types is annotated.', '1809.05576-1-4-1': 'Active learning augments existing training data by having a human oracle annotate system queries (or features [CITATION]).', '1809.05576-1-4-2': 'We explored a novel form of annotation, curated training (ct), in which teachers (annotators) actively seek out informative training examples.', '1809.05576-1-5-0': '# Curated Training', '1809.05576-1-6-0': "In ct the teacher created a prioritized indicator list of words and phrases which could indicate a target event's presence.", '1809.05576-1-6-1': 'Given a tool with a search box, a document list, and a document text pane, teachers searched for indicators in priority order and annotated ten documents each.', '1809.05576-1-6-2': "On loading a document, they used their browser's search to locate a single sentence containing the indicator.", '1809.05576-1-7-0': 'If the sentence mentioned multiple instances of the target event or was unclear, it was skipped.', '1809.05576-1-7-1': 'If it contained no mention of the event, they marked it negative.', '1809.05576-1-7-2': "Otherwise, they (a) marked the sentence as event-present; (b) applied the anchor annotation to the tokens whose presence makes the presence of the event likely; (c) marked each argument span within the selected sentence; and (d) marked any other spans they thought might be 'educational' as interesting.", '1809.05576-1-7-3': '(d) was also done for negative sentences.', '1809.05576-1-8-0': 'Teachers were permitted to annotate extra documents if an indicator seemed ambiguous.', '1809.05576-1-8-1': 'They looked very briefly (2-3 seconds) in the context of selected sentences to see if there were additional informative instances to annotate.', '1809.05576-1-8-2': 'If any non-indicator anchor was marked, it was added to the indicator list with high priority.', '1809.05576-1-8-3': 'The process was repeated for four hours or until the teacher felt additional ct would not be useful.', '1809.05576-1-9-0': '## Data Gathered', '1809.05576-1-10-0': 'We recruited three teachers without NLP backgrounds but with annotation experience.', '1809.05576-1-10-1': 'We consider here only the teacher (a) who completed all event types in time for assessment.', '1809.05576-1-10-2': 'Teacher a averaged seven minutes brainstorming indicators and produced 6,205 event presence, 5137 negative, and 13,459 argument annotations.', '1809.05576-1-10-3': 'Every teacher action was time-stamped.', '1809.05576-1-10-4': 'For analysis, we updated the timestamps to remove breaks longer than two minutes.', '1809.05576-1-11-0': 'Since the ct was stored as character offsets, we aligned it to parses to get ace-style event mentions.', '1809.05576-1-11-1': 'For training our argument attachment models, we omit any event mentions where any annotation failed to project.', '1809.05576-1-11-2': "Projecting Teacher a's data produced 5792 event mentions for trigger training, 5221 for argument training, and 4,954 negatives.", '1809.05576-1-12-0': '# Evaluation', '1809.05576-1-13-0': 'Our target evaluation task is kbp-ea [CITATION] which requires mapping a document to a set of [MATH] tuples indicating that an entity [MATH] plays the role [MATH] in an event of type [MATH] with realis [MATH].', '1809.05576-1-13-1': 'Scoring is F1 over these tuples.', '1809.05576-1-13-2': 'We evalute over the the 2014 newswire evaluation corpus [CITATION] using the scorer on the evaluation key augmented with assessments by Teacher a of responses from our system not found therein.', '1809.05576-1-13-3': 'To focus on event detection and argument attachment, we enabled the neutralizeRealis and attemptToNeutralizeCoref scorer options.', '1809.05576-1-14-0': '## Baseline', '1809.05576-1-15-0': 'The highest-performing system in kbp ea 2014, bbn1, ran a pipeline of four log-linear classifiers (trigger detection, argument attachment, genericity assignment, and a trigger-less argument model) in a high-recall mode which output all event mentions and arguments scoring above 10% probability.', '1809.05576-1-15-1': 'This output was fed into a series of inference rules and a score was computed based on the sub-model scores and the inference rules applied [CITATION].', '1809.05576-1-16-0': 'We used this evaluation system for the experiments in this paper with two changes.', '1809.05576-1-16-1': 'First, bbn1 used a multi-class model for trigger detection, while we use one binary model per event type because with ct each type has a different set of negative examples.', '1809.05576-1-16-2': "Second, we omitted the 'trigger-less' argument classifier for simplicity.", '1809.05576-1-16-3': "This version, baseline, lags bbn1's performance by 0.8 F1 but outperforms all other 2014 evaluation systems by a large margin.", '1809.05576-1-17-0': 'To compare against full document annotation, we needed to estimate how long the event-only portion of ace annotation took.', '1809.05576-1-17-1': 'The LDC ventured a rough estimate of 1500 words per hour (about twenty minutes per ace document).', '1809.05576-1-17-2': 'The LDC human annotator in kbp-ea 2014 was allocated thirty minutes per document [CITATION].', '1809.05576-1-17-3': 'We use the former estimate.', '1809.05576-1-17-4': 'To estimate performance with a fraction of ace, we used the first [MATH]% documents as needed.', '1809.05576-1-18-0': '# Analysis', '1809.05576-1-19-0': "In aggregate ct's performance closely tracked ace for small amounts of mean annotation time per event (Figure [REF]).", '1809.05576-1-19-1': 'However, the performance of ct plateaus more slowly than ACE, beginning to diverge around ninety minutes per event, and continuing to increase sharply at the end of our annotation, leaving unclear what the potential performance of the technique is.', '1809.05576-1-19-2': 'When added to ace, the ct improves performance somewhat, reducing error of P/R/F 1%/5%/6% at ninety minutes per event before plateauing.', '1809.05576-1-19-3': 'ct has a substantial advantage over ace for event types which are rare in ace, but lags significantly for event types abundant in ace (Figure [REF]).', '1809.05576-1-20-0': 'The annotation tool designer and two other NLP experts also did ct for conflict.demonstrate (Figure [REF]; Table [REF]).', '1809.05576-1-20-1': 'All experts significantly outperformed Teacher a and ace in terms of F1.', '1809.05576-1-20-2': 'In two cases this is because the experts sacrificed precision for recall.', '1809.05576-1-20-3': "The second expert matched Teacher a's precision with much higher recall.", '1809.05576-1-20-4': 'Annotators varied widely in the volume of their annotation and indicator searches, but this did not have a clear relationship to performance.', '1809.05576-1-21-0': '## Possible Confounding Factors', '1809.05576-1-22-0': 'Because Teacher a both provided ct and did the output assessment, improvements may reflect the system learning their biases.', '1809.05576-1-22-1': 'We controlled for this somewhat by having Teacher B dual-assess several hundred responses, resulting in encouraging agreement rates of 95% for event presence, 98% for role selection, and 98% for argument assessment.', '1809.05576-1-22-2': "For some events, the guidelines changed from ace to kbp ea 2014 by eliminating 'trumping' rules and expanding allowable inference, which could also account for some improvement.", '1809.05576-1-22-3': 'If either of these were significant factors, it would suggest that ct may be a useful tool for retargetting systems to new, related tasks.'}

{'1809.05576-2-0-0': '# Introduction', '1809.05576-2-1-0': 'Identifying events from a given ontology in text and locating their arguments is an especially challenging task because events vary widely in their textual realizations and their arguments are often spread across multiple clauses or sentences.', '1809.05576-2-1-1': 'Most event research has been in the context of the 2005 NIST Automatic Content Extraction (ace) sentence-level event mention task [CITATION], which also provides the standard corpus.', '1809.05576-2-1-2': 'Recently, tac kbp has introduced document-level event argument extraction shared tasks for 2014 and 2015 (kbp ea).', '1809.05576-2-2-0': 'Progress on events since ACE has been limited.', '1809.05576-2-2-1': 'Most subsequent work has tried improve performance through the use of more complex inference [CITATION], by transductively drawing on outside sources of information, or both [CITATION].', '1809.05576-2-2-2': 'Such approaches have produced modest reductions in error over a pipeline of simple classifiers trained on ACE.', '1809.05576-2-3-0': 'In our efforts to improve on the kbp ea 2014 systems, we were stymied by a lack of data, especially for rarer event types.', '1809.05576-2-3-1': 'Ten of the 33 event types have fewer than 25 training examples in ace, and even for more frequent events, many trigger words and classes of arguments occurred only once.', '1809.05576-2-3-2': 'Furthermore, the 2015 task would include new argument types.', '1809.05576-2-3-3': 'These problems motivated the following question: (a) are we at a plateau in the performance vs. annotation time curve?', '1809.05576-2-3-4': '(b) is there an viable alternative to full-document annotation, especially for rarer event types?', '1809.05576-2-3-5': '(c) for novel event types or languages, how quickly can a useful event model be trained?', '1809.05576-2-4-0': 'In traditional annotation, a static corpus selected to be rich in the target event types is annotated.', '1809.05576-2-4-1': 'Active learning augments existing training data by having a human oracle annotate system queries (or features [CITATION]).', '1809.05576-2-4-2': 'We explored a novel form of annotation, curated training (ct), in which teachers (annotators) actively seek out informative training examples.', '1809.05576-2-5-0': '# Curated Training', '1809.05576-2-6-0': "In ct the teacher created a prioritized indicator list of words and phrases which could indicate a target event's presence.", '1809.05576-2-6-1': 'Given a tool with a search box, a document list, and a document text pane, teachers searched for indicators in priority order and annotated ten documents each.', '1809.05576-2-6-2': "On loading a document, they used their browser's search to locate a single sentence containing the indicator.", '1809.05576-2-7-0': 'If the sentence mentioned multiple instances of the target event or was unclear, it was skipped.', '1809.05576-2-7-1': 'If it contained no mention of the event, they marked it negative.', '1809.05576-2-7-2': "Otherwise, they (a) marked the sentence as event-present; (b) applied the anchor annotation to the tokens whose presence makes the presence of the event likely; (c) marked each argument span within the selected sentence; and (d) marked any other spans they thought might be 'educational' as interesting.", '1809.05576-2-7-3': '(d) was also done for negative sentences.', '1809.05576-2-8-0': 'Teachers were permitted to annotate extra documents if an indicator seemed ambiguous.', '1809.05576-2-8-1': 'They looked very briefly (2-3 seconds) in the context of selected sentences to see if there were additional informative instances to annotate.', '1809.05576-2-8-2': 'If any non-indicator anchor was marked, it was added to the indicator list with high priority.', '1809.05576-2-8-3': 'The process was repeated for four hours or until the teacher felt additional ct would not be useful.', '1809.05576-2-9-0': '## Data Gathered', '1809.05576-2-10-0': 'We recruited three teachers without NLP backgrounds but with annotation experience.', '1809.05576-2-10-1': 'We consider here only the teacher (a) who completed all event types in time for assessment.', '1809.05576-2-10-2': 'Teacher a averaged seven minutes brainstorming indicators and produced 6,205 event presence, 5137 negative, and 13,459 argument annotations.', '1809.05576-2-10-3': 'Every teacher action was time-stamped.', '1809.05576-2-10-4': 'For analysis, we updated the timestamps to remove breaks longer than two minutes.', '1809.05576-2-11-0': 'Since the ct was stored as character offsets, we aligned it to parses to get ace-style event mentions.', '1809.05576-2-11-1': 'For training our argument attachment models, we omit any event mentions where any annotation failed to project.', '1809.05576-2-11-2': "Projecting Teacher a's data produced 5792 event mentions for trigger training, 5221 for argument training, and 4,954 negatives.", '1809.05576-2-12-0': '# Evaluation', '1809.05576-2-13-0': 'Our target evaluation task is kbp-ea [CITATION] which requires mapping a document to a set of [MATH] tuples indicating that an entity [MATH] plays the role [MATH] in an event of type [MATH] with realis [MATH].', '1809.05576-2-13-1': 'Scoring is F1 over these tuples.', '1809.05576-2-13-2': 'We evalute over the the 2014 newswire evaluation corpus [CITATION] using the scorer on the evaluation key augmented with assessments by Teacher a of responses from our system not found therein.', '1809.05576-2-13-3': 'To focus on event detection and argument attachment, we enabled the neutralizeRealis and attemptToNeutralizeCoref scorer options.', '1809.05576-2-14-0': '## Baseline', '1809.05576-2-15-0': 'The highest-performing system in kbp ea 2014, bbn1, ran a pipeline of four log-linear classifiers (trigger detection, argument attachment, genericity assignment, and a trigger-less argument model) in a high-recall mode which output all event mentions and arguments scoring above 10% probability.', '1809.05576-2-15-1': 'This output was fed into a series of inference rules and a score was computed based on the sub-model scores and the inference rules applied [CITATION].', '1809.05576-2-16-0': 'We used this evaluation system for the experiments in this paper with two changes.', '1809.05576-2-16-1': 'First, bbn1 used a multi-class model for trigger detection, while we use one binary model per event type because with ct each type has a different set of negative examples.', '1809.05576-2-16-2': "Second, we omitted the 'trigger-less' argument classifier for simplicity.", '1809.05576-2-16-3': "This version, baseline, lags bbn1's performance by 0.8 F1 but outperforms all other 2014 evaluation systems by a large margin.", '1809.05576-2-17-0': 'To compare against full document annotation, we needed to estimate how long the event-only portion of ace annotation took.', '1809.05576-2-17-1': 'The LDC ventured a rough estimate of 1500 words per hour (about twenty minutes per ace document).', '1809.05576-2-17-2': 'The LDC human annotator in kbp-ea 2014 was allocated thirty minutes per document [CITATION].', '1809.05576-2-17-3': 'We use the former estimate.', '1809.05576-2-17-4': 'To estimate performance with a fraction of ace, we used the first [MATH]% documents as needed.', '1809.05576-2-18-0': '# Analysis', '1809.05576-2-19-0': "In aggregate ct's performance closely tracked ace for small amounts of mean annotation time per event (Figure [REF]).", '1809.05576-2-19-1': 'However, the performance of ct plateaus more slowly than ACE, beginning to diverge around ninety minutes per event, and continuing to increase sharply at the end of our annotation, leaving unclear what the potential performance of the technique is.', '1809.05576-2-19-2': 'When added to ace, the ct improves performance somewhat, reducing error of P/R/F 1%/5%/6% at ninety minutes per event before plateauing.', '1809.05576-2-19-3': 'ct has a substantial advantage over ace for event types which are rare in ace, but lags significantly for event types abundant in ace (Figure [REF]).', '1809.05576-2-20-0': 'The annotation tool designer and two other NLP experts also did ct for conflict.demonstrate (Figure [REF]; Table [REF]).', '1809.05576-2-20-1': 'All experts significantly outperformed Teacher a and ace in terms of F1.', '1809.05576-2-20-2': 'In two cases this is because the experts sacrificed precision for recall.', '1809.05576-2-20-3': "The second expert matched Teacher a's precision with much higher recall.", '1809.05576-2-20-4': 'Annotators varied widely in the volume of their annotation and indicator searches, but this did not have a clear relationship to performance.', '1809.05576-2-21-0': '## Possible Confounding Factors', '1809.05576-2-22-0': 'Because Teacher a both provided ct and did the output assessment, improvements may reflect the system learning their biases.', '1809.05576-2-22-1': 'We controlled for this somewhat by having Teacher B dual-assess several hundred responses, resulting in encouraging agreement rates of 95% for event presence, 98% for role selection, and 98% for argument assessment.', '1809.05576-2-22-2': "For some events, the guidelines changed from ace to kbp ea 2014 by eliminating 'trumping' rules and expanding allowable inference, which could also account for some improvement.", '1809.05576-2-22-3': 'If either of these were significant factors, it would suggest that ct may be a useful tool for retargetting systems to new, related tasks.'}

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{'1': 'http://creativecommons.org/licenses/by/4.0/', '2': 'http://creativecommons.org/licenses/by/4.0/'}

https://arxiv.org/abs/1809.05576

1107.0881

{'1107.0881-1-0-0': 'We have shown that Wightman function of a free quantum field generates any complete set of solutions of relativistic wave equations.', '1107.0881-1-0-1': 'Using this approach we have constructed the complete set of solutions to 2d Dirac equation consisting of eigenfunctions of the generator of Lorentz rotations (boost operator).', '1107.0881-1-0-2': 'It is shown that at the surface of the light cone the boost modes for a fermion field contain [MATH]-function of a complex argument.', '1107.0881-1-0-3': 'Due to the presence of such singularity exclusion even of a single mode with an arbitrary value of the boost quantum number makes the set of boost modes incomplete.', '1107.0881-1-1-0': '# Introduction', '1107.0881-1-2-0': 'It is well known that quantization procedure in a quantum field theory implies expansion of the field operator in terms of a complete set of modes which are solutions of the corresponding (Dirac or Klein-Gordon) classical field equation.', '1107.0881-1-2-1': 'Therefore finding exact solutions for these equations is one of the central points of the quantum field theory in the presence of a classical background.', '1107.0881-1-3-0': 'One of the most powerful instruments for solving partial differential equations is based on symmetry properties of the physical system described by the equation.', '1107.0881-1-3-1': "According to the Noether's theorem [CITATION] any differentiable symmetry of the action of a physical system has a corresponding conservation law.", '1107.0881-1-3-2': 'One can always choose a set of variables in such a way that a change in one of them corresponds to the symmetry transformation.', '1107.0881-1-3-3': 'Since the generator of the symmetry transformation commutes with the differential operator of the equation, the variable corresponding to it can be separated.', '1107.0881-1-3-4': 'If there are enough symmetries for the equation, such that their generators commute with each other, the problem of finding the solution can be reduced to solving of a generally second order ordinary equation.', '1107.0881-1-3-5': 'The set of such solutions is labeled by the eigenvalues of the generators and is complete.', '1107.0881-1-4-0': 'All most important and widely used solutions for relativistic quantum equations in the presence of classical external fields were obtained using this method.', '1107.0881-1-4-1': 'We mean Coulomb field, see, e.g., [CITATION], constant magnetic [CITATION] and electric [CITATION] fields, the field of a plane electromagnetic wave [CITATION], and some other fields of more sophisticated configurations [CITATION].', '1107.0881-1-4-2': 'In all these cases the solutions were eigenfunctions of linear combinations of generators of time and space translations or space rotations, and were labeled by values of components of either 4-momentum, or angular momentum respectively.', '1107.0881-1-4-3': 'However, the Poincare group, the group of isometries of Minkowski spacetime (MS), includes also Lorentz rotations (boosts).', '1107.0881-1-4-4': 'The boost symmetry was almost never used for field quantization.', '1107.0881-1-4-5': 'Certainly, this is because the boost generator does not commute with the Hamiltonian, and thus the boost quantum number and energy cannot be parts of any complete set of observables simultaneously.', '1107.0881-1-4-6': 'Nevertheless, the boost modes can be very useful, especially for the quantum field theory in a curved space, or if the symmetry of MS with respect to time and/or space translations is broken in the presence of a classical background.', '1107.0881-1-4-7': 'In these cases the boost symmetry can appear to be the sole symmetry for the quantum field.', '1107.0881-1-4-8': 'Dilaton gravity in two dimensions [CITATION], as well as Schwarzschild geometry [CITATION], are the examples.', '1107.0881-1-4-9': 'Therefore analysis of the properties of boost modes is important.', '1107.0881-1-5-0': 'For the first time the boost modes for a free massive scalar field were discussed in Ref. [CITATION] by W. Unruh, though their explicit form was not specified.', '1107.0881-1-5-1': 'Then some properties of scalar boost modes were studied in Refs. [CITATION].', '1107.0881-1-5-2': 'In particular, the remarkable properties of boost modes were ascertained in Ref. [CITATION].', '1107.0881-1-5-3': 'It was shown that (i) the zero boost mode of a free massive scalar field coincides up to a trivial constant factor with the positive-frequency Whightman function, and (ii) the boost modes considered as functions of the boost quantum number possess a [MATH]-function singularity at the surface of the light cone.', '1107.0881-1-5-4': 'The properties of the boost modes for the case of a free massless fermion field appeared to be even more interesting.', '1107.0881-1-5-5': 'It was shown in Ref. [CITATION] that boost modes of two-dimensional massless fermions on a light cone are expressed in terms of the Dirac delta function of a complex argument.', '1107.0881-1-6-0': 'In this paper we consider boost modes for a free massive fermion field.', '1107.0881-1-6-1': 'To construct them we start from the Wightman functions.', '1107.0881-1-6-2': 'The positive and negative frequency Wightman functions for free fields are explicitly determined (with accuracy to a constant factor) by Lorentz and translational invariance of the theory [CITATION].', '1107.0881-1-6-3': 'And vice versa, be the Wightman functions known one can use those symmetries to construct any complete set of positive and negative solutions of Klein-Fock-Gordon (KFG) or Dirac equations.', '1107.0881-1-6-4': 'In fact, this is the direct consequence of the Wightman reconstruction theorem [CITATION].', '1107.0881-1-7-0': 'Since the Lorentz rotation singles out a two-dimensional plane in MS, we will discuss the specific properties of boost modes by examples of free two-dimensional KFG and Dirac equations.', '1107.0881-1-7-1': 'In the next section we will realize the above formulated approach by the example of a single-component massive neutral field.', '1107.0881-1-7-2': 'It is worth noting that, as opposed to the case of plane wave modes, analytic properties of boost modes change dramatically when one passes to multi-component fields which are discussed in Sec. III.', '1107.0881-1-7-3': 'The discussion of the results and conclusions are given in Sec. IV.', '1107.0881-1-8-0': '# Bosons', '1107.0881-1-9-0': 'In this section we will consider the case of a free neutral massive scalar field in two-dimensional MS. We will start from two-point Wightman function [CITATION] which for a free field theory coincides with the positive-frequency part of the commutator of two field operators (Pauli-Jordan function in 4d theory of a scalar field).', '1107.0881-1-9-1': 'Positive-frequency Wightman function [MATH] for a massive scalar field satisfies KFG equation [EQUATION] contains only positive frequencies and is invariant with respect to Lorentz rotations (boosts) [EQUATION]', '1107.0881-1-9-2': 'These conditions determine [MATH] accurate within a constant factor.', '1107.0881-1-9-3': 'Indeed, any positive-frequency solution of KFG equation can be written as follows [EQUATION] where [MATH] is a certain function of 2-vector [MATH].', '1107.0881-1-9-4': 'After the change of variables [EQUATION] where [MATH] is rapidity, and integration over [MATH] we obtain the equation [EQUATION]', '1107.0881-1-9-5': 'It can be easily seen that condition ([REF]) is satisfied only if [MATH].', '1107.0881-1-9-6': 'Choosing [MATH] we arrive to the standard representation for [MATH], compare, e.g., [CITATION] [EQUATION]', '1107.0881-1-9-7': 'It is assumed in ([REF]) that an infinitely small negative imaginary part is added to [MATH] [CITATION].', '1107.0881-1-9-8': 'Another representation for [MATH] reads, see, e.g., [CITATION], [EQUATION]', '1107.0881-1-9-9': 'According to the Wightman reconstruction theorem [CITATION] the two-point Wightman function uniquely determines the quantum theory of a free field.', '1107.0881-1-9-10': 'Particularly, it allows to reconstruct any orthonormalized and complete set of solutions for the field equation.', '1107.0881-1-9-11': 'Indeed, it follows from the translational invariance of the theory that Wightman function with an arbitrarily shifted argument [EQUATION] satisfies KFG equation.', '1107.0881-1-9-12': 'Functions ([REF]) constitute an overcomplete set of modes since they are labeled by two independent parameters [MATH].', '1107.0881-1-9-13': 'Anyway, any complete set of positive-frequency solutions [MATH] [EQUATION] orthonormalized by the condition [EQUATION] can be represented as [EQUATION]', '1107.0881-1-9-14': 'However, since the set of solutions ([REF]) is overcomplete, the coefficient functions [MATH] cannot be determined uniquely.', '1107.0881-1-9-15': 'To make the choice of [MATH] unique one should impose a restriction on [MATH] which can be chosen for reasons of symmetry.', '1107.0881-1-10-0': 'First, we will illustrate this procedure for the trivial case of plane waves [MATH] which are eigenfunctions of the generator of spatial translations [EQUATION]', '1107.0881-1-10-1': 'In this case it is convenient to limit the family of [MATH] only by functions spatially shifted with respect to each other.', '1107.0881-1-10-2': 'This means that coefficient functions [MATH] have the form [MATH], so that [EQUATION]', '1107.0881-1-10-3': 'Let us substitute this expansion into Eq. ([REF]).', '1107.0881-1-10-4': 'Taking into account that Wightman function [MATH] exponentially tends to zero when [MATH], we can integrate the left-hand side of the resulting equation by part and arrive to the following equation for [MATH] [EQUATION] so that [MATH].', '1107.0881-1-10-5': 'Substituting now [MATH] of that form into ([REF]) and using ([REF]) we finally obtain after integration over [MATH] [EQUATION]', '1107.0881-1-10-6': 'The normalization constant here was determined from the condition ([REF]).', '1107.0881-1-11-0': 'Functions [MATH] satisfy the relation [EQUATION] and thus the set ([REF]) is complete.', '1107.0881-1-12-0': 'We are interested in the set of boost modes [MATH] which are eigenfunctions of boost operator [MATH] [EQUATION]', '1107.0881-1-12-1': 'It is worth noting from the very beginning that Wightman function [MATH] is a zero mode [MATH] of this set as it follows from Eqs. ([REF]) and ([REF]).', '1107.0881-1-13-0': 'To obtain the boost modes we will fix [MATH] in ([REF]) at one of the orbits of the restricted Lorentz group [EQUATION] so that [MATH].', '1107.0881-1-13-1': 'Choosing the upper sign in ([REF]) for definiteness, we put [EQUATION] and rewrite Eq. ([REF]) in the form [EQUATION]', '1107.0881-1-13-2': 'Substituting ([REF]) into Eq. ([REF]) and using the relation [EQUATION] which can be straightforwardly obtained with the help of representation ([REF]), we get the following equation for [MATH] [EQUATION].', '1107.0881-1-13-3': 'Hence, [MATH], and we finally get from ([REF]) with due regard for condition ([REF]) the following representation for the positive frequency boost modes [EQUATION] which was earlier obtained in Refs. [CITATION] by another method.', '1107.0881-1-13-4': 'The negative frequency boost modes [MATH] are defined [CITATION] as [EQUATION]', '1107.0881-1-13-5': 'Boost modes ([REF]), ([REF]) are distributions with respect to both [MATH] and spectral parameter [MATH].', '1107.0881-1-13-6': 'They are defined on the class of smooth functions of rapid enough descent.', '1107.0881-1-14-0': 'The modes ([REF]) constitute a complete set since they satisfy the condition [EQUATION] and hence can be used as a basis for quantization of a neutral scalar field, see Ref. [CITATION].', '1107.0881-1-14-1': 'Since the sign of a particle energy is Lorentz invariant, the vacuum states in the boost and plane-wave quantization schemes are identical.', '1107.0881-1-14-2': 'Hence, these two quantization schemes are unitary equivalent, see Ref. [CITATION].', '1107.0881-1-15-0': 'The most remarkable property of boost modes is their behavior at the light cone.', '1107.0881-1-15-1': 'It is easily seen from ([REF]), ([REF]) that at the vertex of the light cone [MATH] possess a [MATH]-function singularity [EQUATION]', '1107.0881-1-15-2': 'It was shown in Ref. [CITATION] that [MATH] possess a [MATH]-function singularity at the lines [MATH] as well.', '1107.0881-1-15-3': 'Therefore contribution of the single spectral point [MATH] to physical quantities can be finite.', '1107.0881-1-15-4': 'We will illustrate this by the example of Wightman function.', '1107.0881-1-16-0': 'Taking into account the property of translational invariance we can rewrite Eq. ([REF]) in the form [EQUATION]', '1107.0881-1-16-1': 'Using now Eq. ([REF]) we obtain [EQUATION] i.e. the integral over [MATH] in ([REF]) is determined by the point [MATH] entirely.', '1107.0881-1-17-0': 'This result means that the point [MATH] cannot be deleted from the spectrum, or in other words the integral over [MATH] in ([REF]),([REF]) cannot be changed by its principal value [EQUATION]', '1107.0881-1-17-1': 'Thereby, the family of boost modes does not constitute a complete set in MS after excluding the zero mode.', '1107.0881-1-17-2': 'However, the equality in Eq. ([REF]) could be restored if we cut the light cone out of MS, compare [CITATION].', '1107.0881-1-17-3': 'This is because all the points where boost modes possess [MATH] singularity are located just at the light cone.', '1107.0881-1-18-0': 'Furthermore, the Wightman function Eq. ([REF]) cannot be represented in the form [EQUATION]', '1107.0881-1-18-1': 'If it were so, we could rewrite ([REF]), due to translational invariance of the Wightman function and the boost mode property ([REF]), as follows [EQUATION]', '1107.0881-1-18-2': 'But Eq. ([REF]) is evidently meaningless.', '1107.0881-1-18-3': 'Indeed, the distribution [MATH] is defined on the class of functions continuous at the interval including the point [MATH].', '1107.0881-1-18-4': 'Therefore Eq. ([REF]) should be understood as [EQUATION] where [MATH] is the Heaviside step function.', '1107.0881-1-18-5': 'However, the product of two distributions [MATH] is not defined.', '1107.0881-1-19-0': 'The authors of Ref. [CITATION] do not agree with this statement.', '1107.0881-1-19-1': 'Discussing their Eq. (2.129), which in our notations coincides with ([REF]), they admit that this expression is undefined if either of the two points [MATH] is located on the light cone.', '1107.0881-1-19-2': 'They think that smearing of the distribution [MATH] with compactly supported functions [MATH] and [MATH] improve the situation and the smeared "Wightman function" [MATH](2.131) is well defined on the whole MS. Our analysis shows however that this is not correct.', '1107.0881-1-19-3': 'Indeed, if we use Eq. ([REF]) in the form ([REF]), we see that this expression is undefined for arbitrary [MATH].', '1107.0881-1-19-4': 'Moreover, smearing cannot improve the situation since it does not influence the [MATH]-function in ([REF]).', '1107.0881-1-19-5': 'The correct expression for the smeared Wightman function can be easily obtained from Eq. ([REF]).', '1107.0881-1-19-6': 'It reads [EQUATION] where [EQUATION] is the smeared boost mode.', '1107.0881-1-19-7': 'We see that again only the single spectral point [MATH] contributes to the integral ([REF]) for the smeared Wightman function.', '1107.0881-1-20-0': 'Note, that we could try to use expression ([REF]) directly, not applying the property of translational invariance to it.', '1107.0881-1-20-1': 'In that case we should have attach exact mathematical meaning to ([REF]) first.', '1107.0881-1-20-2': 'Namely, we should write it in the form [EQUATION]', '1107.0881-1-20-3': 'But it can be easily seen that function [MATH] in ([REF]) is not translational invariant and thus has nothing to do with Wightman function for a free field in MS. Indeed, if it were translational invariant, it could be written down as [EQUATION] and would be equal to zero identically.', '1107.0881-1-21-0': 'Singular behavior of the boost mode at the point [MATH] can be understood in terms of classical trajectories of free particles with a given value of boost parameter [MATH] [EQUATION] where [MATH] is the momentum, [MATH] - the energy of a particle, [MATH].', '1107.0881-1-21-1': 'Let us rewrite Eq. ([REF]) in the the form [EQUATION]', '1107.0881-1-21-2': 'After differentiating both sides of Eq. ([REF]) with time one can easily find its regular [EQUATION] and singular [EQUATION] solutions.', '1107.0881-1-22-0': 'When [MATH] the singular solution is represented by two branches of hyperbola ([REF]).', '1107.0881-1-22-1': 'The branch with [MATH] is located in the right wedge of MS, while the branch with [MATH] in the left one, see Figs. 1a,b. Free boost particles with the given value of [MATH] propagate along the regular world lines which are different straight lines tangent to the corresponding branches of hyperbola ([REF]).', '1107.0881-1-22-2': 'It is worth noting that classical particles with positive values of boost parameter [MATH] cannot penetrate to the left wedge, as well as particles with [MATH] cannot penetrate to the right one.', '1107.0881-1-23-0': 'The world lines of boost particles with [MATH] are given by [EQUATION]', '1107.0881-1-23-1': 'All of them cross the vertex of the light cone [MATH], see Fig. 1c, and any world line crossing [MATH] belongs to the family of trajectories with [MATH].', '1107.0881-1-23-2': 'Hence, elimination of the point [MATH] from the spectrum is equivalent to prohibition for particles to cross the point [MATH], or to pricking it out of MS. Besides, deleting the point [MATH] from the spectrum, we delete a bunch of infinite number of trajectories, thus loosing a substantial part of degrees of freedom.', '1107.0881-1-23-3': 'This means that the point [MATH] is of nonzero measure at [MATH] and explains the [MATH]-function singularity of the boost mode there.', '1107.0881-1-24-0': 'The points lying at the light cone surface need a separate consideration.', '1107.0881-1-24-1': 'The lines [MATH] cannot be trajectories of massive particles.', '1107.0881-1-24-2': 'But the branches of hyperbola ([REF]) degenerate into lines constituting the surface of the light cone in our 2d problem when [MATH].', '1107.0881-1-24-3': 'As it can easily be seen from Figs. 1a,b, all regular trajectories except those which are tangent to the hyperbola branches at the point [MATH] also tend to the lines [MATH] in the limit [MATH].', '1107.0881-1-24-4': 'Hence, these lines [MATH] are the lines of condensation for regular trajectories with [MATH].', '1107.0881-1-24-5': 'Possibly this explains the existence of [MATH]-function singularity of boost modes at the light cone surface.', '1107.0881-1-25-0': 'It is worth noting that there exist another representation for the boost modes ([REF]), ([REF]).', '1107.0881-1-25-1': 'Keeping in mind that an infinitely small negative imaginary part [MATH] is added to [MATH] in ([REF]), after the change of the variable of integration [MATH] we obtain for [MATH] the following expression [EQUATION] with [MATH] and [MATH].', '1107.0881-1-25-2': 'Using now formula 3.471(9) of Ref. [CITATION] we get [EQUATION] where [MATH] is the MacDonald function, and distributions [MATH] should be understood as [EQUATION] see Ref. [CITATION].', '1107.0881-1-25-3': 'Using Eq. ([REF]) we obtain [EQUATION] see [CITATION], where [EQUATION] are the expressions for the boost modes in the past (P), right (R), future (F) and left (L) sectors of MS respectively.', '1107.0881-1-25-4': 'Here [MATH] are Hankel functions.', '1107.0881-1-26-0': 'Using equations ([REF]) and ([REF]), we can write down a new representation for the Wightman functions [MATH], [EQUATION]', '1107.0881-1-26-1': 'Particularly, for the positive frequency function we obtain [EQUATION] in agreement with the Wightman result [CITATION].', '1107.0881-1-27-0': '# Massive fermions', '1107.0881-1-28-0': 'A set of orthonormalized positive (negative) frequency solutions [MATH] of Dirac equation in 2d MS [EQUATION] ([MATH] - two-dimensional Dirac matrices) is complete if [EQUATION] where the positive [MATH] and negative [MATH] frequency Wightman functions for the fermion field are equal to [EQUATION] and [MATH] is defined in ([REF]).', '1107.0881-1-28-1': 'Choosing Dirac matrices in the form convenient for the limit processing [MATH], [MATH], [MATH]), [MATH] - Pauli matrices, we obtain for the Wightman functions the following expression [EQUATION]', '1107.0881-1-28-2': 'Again, we assume that an infinitely small negative (positive) imaginary part is added to [MATH] in the expression for positive (negative) frequency Wightman function.', '1107.0881-1-29-0': 'Any set [MATH] can be constructed from the set of matrices [MATH] in perfect analogy with the boson case.', '1107.0881-1-29-1': 'In particular, it is easy to check that functions [EQUATION] ([MATH] are defined in Eq. ([REF])) are the eigenfunctions of the fermion boost generator [EQUATION] if [EQUATION] where [MATH] is an arbitrary constant two-component spinor.', '1107.0881-1-30-0': 'It is easy to see that [EQUATION]', '1107.0881-1-30-1': 'Then for the boost modes we have [EQUATION]', '1107.0881-1-30-2': 'Using exactly the same procedure as for the scalar boost modes ([REF]) we can transform Eq. ([REF]) to the form [EQUATION] where again [MATH].', '1107.0881-1-31-0': 'Eq. ([REF]) could be also rewritten in the form ([REF]).', '1107.0881-1-31-1': 'However, we will not do it here since the corresponding formulas are too cumbersome.', '1107.0881-1-31-2': 'We will note only that an attempt to give a representation for fermion boost modes in the form ([REF]) was done for the first time in Ref. [CITATION].', '1107.0881-1-31-3': 'The authors of Ref. [CITATION] have chosen another set of Dirac [MATH]matrices, and therefore their result cannot be compared to our one straightforwardly.', '1107.0881-1-31-4': 'However, it is clear that they have omitted the required factor [MATH] in their formulas.', '1107.0881-1-31-5': 'The authors of Ref. [CITATION] have derived them by means of analytical extension of the solution for Dirac equation in the Rindler space (restriction of MS to the right wedge) to the whole MS but did not take into account that implementation of transition from Rindler to Cartesian coordinates, [MATH], [MATH], requires the following rule of transformation for fermion wave function [CITATION] [EQUATION]', '1107.0881-1-31-6': 'The boost modes ([REF]), ([REF]) are orthonormalized by the condition [EQUATION] constitute a complete set in the sense ([REF]), and thus can serve a basis for quantization of the fermion field in MS, [EQUATION] where [MATH] and [MATH] are annihilation operators for fermion particles and antiparticles respectively, which obey the standard commutation relations.', '1107.0881-1-31-7': 'By the same reasons as in the boson case, boost quantization of the fermion field is unitary equivalent to quantization in the plane wave basis [EQUATION] where [EQUATION]', '1107.0881-1-31-8': 'It is not difficult to ascertain that the boost modes ([REF]) are connected with the plane waves ([REF]) through the following integral transformation [EQUATION]', '1107.0881-1-31-9': 'The modes [MATH] and [MATH] are distributions with respect to variables [MATH] and [MATH].', '1107.0881-1-31-10': 'This means that they define linear functionals [EQUATION] on some sets of test functions [MATH], [MATH] respectively.', '1107.0881-1-31-11': 'Such functionals naturally appear when we calculate, e.g., matrix elements of the field operators ([REF]), ([REF]) between the vacuum state [MATH] and the states which are particle (or antiparticle) wave packets, [EQUATION]', '1107.0881-1-31-12': 'Physical one-particle states must be normalized, [EQUATION]', '1107.0881-1-31-13': 'Hence, [MATH], [MATH] must be square-integrable functions of [MATH] and [MATH] respectively.', '1107.0881-1-31-14': 'Then it is easy to see that matrix elements ([REF]), ([REF]) are square-integrable functions of [MATH].', '1107.0881-1-31-15': 'It is clear that functionals ([REF]), ([REF]) are the elements of one and the same functional space.', '1107.0881-1-31-16': 'Hence, there exists one to one correspondence between test functions [MATH] and [MATH]: [EQUATION] where [MATH] is rapidity.', '1107.0881-1-31-17': 'For the sake of convenience we will confine ourselves to [MATH] belonging to the class of continuous piecewise smooth functions descending faster than [MATH] at [MATH].', '1107.0881-1-31-18': 'This requirement guarantees finiteness of the mean value of the energy of the one-particle state [MATH] ([REF]), [EQUATION]', '1107.0881-1-31-19': 'To ascertain the properties of test functions [MATH] we will use the relation ([REF]).', '1107.0881-1-32-0': 'First, we will illustrate by a simple example that, unlike the case of the functional ([REF]), we need know the behavior of test functions [MATH] for the functional ([REF]) not only on the real axis but also in the complex plane.', '1107.0881-1-32-1': 'Consider [EQUATION] which evidently satisfies the above formulated requirements to functions [MATH].', '1107.0881-1-32-2': 'Using Eq. ([REF]) we obtain [EQUATION]', '1107.0881-1-32-3': 'Direct calculation of the integral ([REF]) at the point [MATH] then yields [EQUATION]', '1107.0881-1-32-4': 'So we see that the value of the functional ([REF]) at the vertex of the light cone is determined by the values of the test function [MATH] at imaginary points [MATH].', '1107.0881-1-33-0': 'Let us now revert to Eq. ([REF]).', '1107.0881-1-33-1': 'After the change of the variable of integration [MATH] it can be written as Fourier transform of function [MATH] [EQUATION]', '1107.0881-1-33-2': 'Under our assumptions [EQUATION] as [MATH].', '1107.0881-1-33-3': 'Then, according to the Paley-Wiener theorem [CITATION] test functions [MATH] ([REF]) are analytic in the strip [EQUATION]', '1107.0881-1-33-4': 'Let us calculate the integral in Eq. ([REF]) by parts.', '1107.0881-1-33-5': 'Since the test functions [MATH] are continuous and descend exponentially as [MATH], see Eq. ([REF]), the first integrated term is equal to zero.', '1107.0881-1-33-6': 'Hence, the functions [MATH] in their domain of analyticity ([REF]) descend at least as [MATH] when [MATH].', '1107.0881-1-33-7': 'This means that the path of integration in ([REF]) can be shifted in the range of the strip ([REF]).', '1107.0881-1-34-0': 'Consider the functional ([REF]) at the point [MATH].', '1107.0881-1-34-1': 'Shifting the path of integration for the upper component upward, and for the lower one downward, by [MATH], we obtain [EQUATION]', '1107.0881-1-34-2': 'Eq. ([REF]) generalizes formula ([REF]) to the class of functions [MATH] analytic in the strip ([REF]) and descending rapidly enough when [MATH].', '1107.0881-1-35-0': 'The obtained result makes it possible to write down the fermion boost mode in the vertex of the light cone in terms of [MATH]-functions of complex argument [EQUATION]', '1107.0881-1-35-1': 'The functionals [MATH] can be also written as integrals around the closed contours represented in Figs. [REF]a,b [EQUATION]', '1107.0881-1-35-2': 'Indeed, the explicit form for [MATH] can be easily derived from Eq. ([REF]).', '1107.0881-1-35-3': 'Putting [MATH] there and using the ascending series for [MATH] [CITATION], we get [EQUATION]', '1107.0881-1-35-4': "Then, taking into account that the integrand in ([REF]) has simple poles at [MATH] and using the Cauchy's residue theorem, we immediately reproduce the result ([REF]).", '1107.0881-1-35-5': 'Since the integrals taken along the vertical segments of the contours [MATH] are evidently equal to zero, this means, in particular, that the integrals along the upper (lower) piece of the contour [MATH]) is also equal to zero in the limit [MATH].', '1107.0881-1-35-6': 'The latter statement can be easily checked by direct calculation.', '1107.0881-1-36-0': "[MATH]-function of complex argument was first introduced by Gel'fand and Shilov in Ref. [CITATION] on the basis of Cauchy residue theorem, [EQUATION] where [MATH] is a contour enclosing the point [MATH].", '1107.0881-1-36-1': 'The distribution ([REF]) was defined in [CITATION] on some class of entire functions [MATH], the so called [MATH]-class.', '1107.0881-1-36-2': 'Our class of test functions analytic in the strip ([REF]) differs from [MATH].', '1107.0881-1-36-3': "Therefore some properties of the Gel'fand's [MATH]-function [CITATION] cannot be applied to our one.", '1107.0881-1-36-4': "However, the opposite statement is true: all properties of our [MATH]-function hold valid for the Gel'fand's one.", '1107.0881-1-37-0': 'So, on the class of test functions analytic in the strip ([REF]), as well as for the [MATH]-class, the following equality of distributions is valid [CITATION] [EQUATION]', '1107.0881-1-37-1': 'Let us show now that the modes ([REF]), ([REF]) possess [MATH]-function singularities not only at the vertex of the light cone but at lines [MATH] as well.', '1107.0881-1-37-2': 'Indeed, at the surface of the light cone the arguments of [MATH] functions in ([REF]) are small, [MATH].', '1107.0881-1-37-3': 'Therefore using the ascending series for Macdonald functions [CITATION] we get [EQUATION]', '1107.0881-1-37-4': 'So we see that at the surface of the light cone [MATH] contains [MATH] at [MATH] in the upper, and [MATH] at [MATH] in the lower component.', '1107.0881-1-38-0': 'It was shown in the preceding section that in the case of a scalar field the family of boost modes does not constitute a complete set in MS after excluding the zero mode.', '1107.0881-1-38-1': 'This is because the scalar boost modes possess a [MATH]-function singularity at the surface of the light cone, and hence the point [MATH] gives finite contribution to physical quantities.', '1107.0881-1-38-2': 'Some of them, e.g., Wightman function, are determined by the spectral point [MATH] entirely.', '1107.0881-1-38-3': 'Fermion boost modes ([REF]), ([REF]) do not possess singularities at the real axis of [MATH].', '1107.0881-1-38-4': 'There is a question whether it is possible to delete the point [MATH] from the spectrum in the fermion case.', '1107.0881-1-38-5': 'To clarify this we will consider the integral [EQUATION] where [MATH] is an arbitrary real number.', '1107.0881-1-38-6': 'To calculate its upper component we will use the contour [MATH] represented in Fig. [REF].', '1107.0881-1-38-7': 'The integrals along the upper piece of the contour, as well as along the vertical segments Re [MATH], are equal to zero as before.', '1107.0881-1-38-8': 'The integrals along the slanting segments compensate each other in the limit [MATH].', '1107.0881-1-38-9': 'Hence, the upper component of [MATH] is equal to the integral [EQUATION] around the unclosed circle [MATH] of radius [MATH] with the center at the point [MATH].', '1107.0881-1-38-10': 'After changing the variable of integration, [MATH], integral ([REF]) can be transformed to the form [EQUATION]', '1107.0881-1-38-11': 'The integral in ([REF]) is easily calculated in the limit [MATH], and finally we obtain [EQUATION]', '1107.0881-1-38-12': 'Similar result can be obtained for the lower component of ([REF]) as well.', '1107.0881-1-39-0': 'Eq. ([REF]) evidently shows that the result of calculation for [MATH] strongly depends on the sequence of limit processing [MATH] and [MATH].', '1107.0881-1-39-1': 'If we changed the sequence of limit processing in ([REF])-([REF]), we would return from [MATH] ([REF]) to [MATH] ([REF]).', '1107.0881-1-39-2': 'Not surprisingly, Eq. ([REF]) reproduces the result ([REF]) in this case.', '1107.0881-1-39-3': 'The accepted sequence which corresponds to calculation of the functional ([REF]) leads to a meaningless result.', '1107.0881-1-39-4': 'This means that pricking of any point out of the real axis [MATH] is inadmissible in the fermion case.', '1107.0881-1-39-5': 'In other words, the set of boost modes ([REF]),([REF]) does not constitute a complete set in MS after deleting a single arbitrary point from the spectrum.', '1107.0881-1-39-6': 'It can be complete only in the space which is MS without the line cone (four separate wedges of MS).', '1107.0881-1-39-7': 'In this sense our conclusion repeats the analogous result for boson field discussed in the preceding section.', '1107.0881-1-40-0': 'It is worth noting that the latter result was obtained under minimum possible restrictions as regards to test functions [MATH].', '1107.0881-1-40-1': 'If we restrict ourselves to [MATH] belonging to the class [MATH] of smooth functions with compact support, as it is often done in quantum field theory, functions [MATH] will belong to the class [MATH] [CITATION].', '1107.0881-1-40-2': 'In this case the proof of impossibility of deleting an arbitrary point [MATH] from the spectrum is especially simple.', '1107.0881-1-40-3': "Indeed, the Gel'fand's [MATH]-function is an analytic distribution in the whole complex plain [MATH].", '1107.0881-1-40-4': 'Thus it can be expanded in Taylor series around any point [MATH], [EQUATION] and the radius of convergence of the series ([REF]) is equal to infinity [CITATION].', '1107.0881-1-40-5': 'Here [MATH] denotes the [MATH]-th derivative of the [MATH]-function at the point [MATH].', '1107.0881-1-40-6': 'As a result, [EQUATION] and hence, on the class [MATH] of test functions, deleting of an arbitrary spectral point [MATH] from the spectrum leads to vanishing of functionals [MATH] ([REF]) at the vertex of the light cone, and its finite variation at other points of the cone surface.', '1107.0881-1-40-7': "However, since the Gel'fand's [MATH]-function is analytic on the whole plane of complex [MATH], it can be expanded in a Taylor series around another real point [MATH], [MATH].", '1107.0881-1-40-8': 'If we calculate integral ([REF]) using such representation of the [MATH]-function, we will see that deleting of the spectral point [MATH] from the spectrum does not influence the result of integration.', '1107.0881-1-40-9': 'Thus, since the value of the matrix element ([REF]) depends on the way of calculation, we conclude that the operation of exclusion of an arbitrary point from the spectrum is meaningless in agreement with the previous consideration, see discussion of Eq. ([REF]) in the preceding paragraph.', '1107.0881-1-41-0': 'As it was shown in the preceding section, Wightman functions [MATH] of a massive scalar field are determined by the zero boost modes [MATH], see Eq. ([REF]).', '1107.0881-1-41-1': 'This is a consequence of translational invariance of [MATH] and the presence of [MATH]-function singularity of [MATH] at the light cone.', '1107.0881-1-41-2': 'A similar result is valid also for the Wightman functions of a massive fermion field.', '1107.0881-1-41-3': 'Indeed, due to the property of translational invariance we can write, compare ([REF]), [EQUATION]', '1107.0881-1-41-4': 'Using now Eq. ([REF]) we get the following result for matrix elements [MATH] of Wightman function ([REF]), [EQUATION] where [MATH] are the [MATH]-components of the boost modes ([REF]), ([REF]).', '1107.0881-1-41-5': 'Thereby, we see that the matrix elements of [MATH] are determined by only two "spectral points" [MATH].', '1107.0881-1-41-6': 'Exactly as in the scalar case, it is easy to ascertain that this result holds valid for the smeared Wightman functions as well.', '1107.0881-1-42-0': 'Using Eq. ([REF]) one can express the Wightman functions ([REF]) in the form, compare ([REF]), [EQUATION]', '1107.0881-1-42-1': 'Consider now the case of a massless fermion field first studied in Ref. [CITATION].', '1107.0881-1-42-2': 'The Wightman function of the massless field [MATH] can be obtained easily by passage to the limit [MATH] in Eq. ([REF]).', '1107.0881-1-42-3': '[EQUATION]', '1107.0881-1-42-4': 'Just as in the cases considered earlier, one can use [MATH] to obtain any complete set of orthonormalized solutions of Dirac equation.', '1107.0881-1-42-5': 'In particular, for the boost modes we have, [EQUATION] where [MATH] is an arbitrary two-component column, compare ([REF]), ([REF]).', '1107.0881-1-42-6': 'We see that at [MATH] the Wightman function ([REF]) is diagonal.', '1107.0881-1-42-7': 'Owing to this property the upper and the lower components of the boost mode ([REF]) become independent: [EQUATION] where [MATH] are the eigenvectors of the Pauli matrix [MATH], [MATH].', '1107.0881-1-42-8': 'This is because there appears a new conservation law for the massless case, conservation of chirality, see, e.g., [CITATION].', '1107.0881-1-42-9': 'Thereby solutions of the Dirac equation ([REF]) are labeled by the additional quantum number [MATH], chirality.', '1107.0881-1-42-10': 'The coefficients [MATH] and [MATH] in ([REF]) are independent and can be determined by the normalization condition for solutions [MATH] and [MATH] separately.', '1107.0881-1-42-11': 'Therefore the sign [MATH] in front of the coefficient [MATH] is inessential and serves only a reminder about the representation for [MATH] in the form ([REF]).', '1107.0881-1-42-12': 'Note that in the massive case, since chirality is not conserved, the components of the column [MATH] ([REF]) cannot be determined separately.', '1107.0881-1-42-13': 'In that case their combination forms a monomial factor of the solution [MATH], see Eq. ([REF]).', '1107.0881-1-43-0': 'For the positive frequency functions [MATH] we have [EQUATION]', '1107.0881-1-43-1': 'Negative frequency functions [MATH] come out of ([REF]) after complex conjugation and the change [MATH].', '1107.0881-1-44-0': 'Plane wave modes are also labeled by the quantum number [MATH] in the massless case, [EQUATION]', '1107.0881-1-44-1': 'They are linked to the boost modes [MATH] through the Mellin transform [EQUATION]', '1107.0881-1-44-2': 'This means that, if we assume that distributions [MATH] are defined on the same class of test functions [MATH] as in the massive case, the distributions [MATH] will be defined on the class of test functions [MATH] analytic in the strip ([REF]) and descending at [MATH] in this strip.', '1107.0881-1-45-0': 'Then taking into account Eq. ([REF]) we conclude that boost modes of a massless fermion field at the surface of the light cone is a [MATH]-function of a complex argument, compare [CITATION], [EQUATION]', '1107.0881-1-45-1': 'Now we will show how the modes ([REF]) can be obtained from Eq. ([REF]) by limit processing [MATH].', '1107.0881-1-45-2': 'Using the ascending series [CITATION] for functions [MATH] we get [EQUATION] i.e., representation ([REF]) with coefficients [MATH] and [MATH] containing singular at [MATH] phase factors.', '1107.0881-1-45-3': 'However these factors do not influence the normalization constants, have no impact on any physical quantities and hence can be omitted.', '1107.0881-1-46-0': '# Concluding remarks', '1107.0881-1-47-0': 'We have shown that Wightman function of a free quantum field generates any complete set of solutions of relativistic wave equations.', '1107.0881-1-47-1': 'Using this approach we have constructed the complete sets of solutions to KFG and Dirac equations consisting of eigenfunctions of the generator of Lorentz rotations (boost operator).', '1107.0881-1-48-0': 'Boost modes are used as a basis for field quantization very rarely.', '1107.0881-1-48-1': 'Till now they were exploited only for analysis of the so-called "Unruh effect" [CITATION] and at attempts to quantize a charged massive scalar field in the presence of an external constant electric field [CITATION].', '1107.0881-1-48-2': 'However, there are many problems, especially in the quantum field theory in a curved space, where the boost symmetry may appear to be the only symmetry for the quantum field, and thus using it for separation of variables in a classical field equation is the only instrument to find solutions for such equations.', '1107.0881-1-49-0': 'The specific feature of the boost modes is that, taken at the surface of the light cone, they as functions of the boost quantum number [MATH] possess strong singularities.', '1107.0881-1-49-1': 'Certainly, the physical reason for these singularities is the singularity of Lorentz transformations at [MATH].', '1107.0881-1-50-0': 'For the case of a scalar field this is a [MATH]-function singularity [MATH] [CITATION].', '1107.0881-1-50-1': 'This leads to a special role of the zero boost mode: its exclusion from the set of the boost modes makes the latter incomplete.', '1107.0881-1-50-2': 'The exceptional role of the zero boost mode becomes quite clear if we recall that the zero value of [MATH] means that [MATH] is a Lorentz invariant positive frequency solution of KFG equation, i.e. the Wightman function for the quantum field coinciding with the positive frequency part of the commutator of two scalar field operators.', '1107.0881-1-50-3': 'Hence the exclusion of the zero boost mode results in a "quantum"[MATH] theory with commuting field operators.', '1107.0881-1-51-0': 'The singularities of the fermion boost modes are even stronger.', '1107.0881-1-51-1': 'It is shown in the present paper that at the surface of the light cone they possess [MATH]-function of a complex argument [MATH].', '1107.0881-1-51-2': "The [MATH]-function of a complex argument was first introduced by Gel'fand and Shilov in Ref. [CITATION], and was defined on the class of entire functions [MATH].", '1107.0881-1-51-3': 'Our [MATH]-function is defined on the class of test functions analytic in the strip ([REF]).', '1107.0881-1-51-4': 'Actually, the width of the strip is determined by physical requirements.', '1107.0881-1-51-5': 'Our choice provides square integrability of one-particle wave packets and finiteness of their energy.', '1107.0881-1-51-6': 'If we require finiteness of the squared energy we should narrow the class of test functions and extend the width of the strip to [EQUATION].', '1107.0881-1-51-7': 'Further toughening of requirements to physical states will lead to subsequent extension of the width of the strip of analyticity of the test functions.', '1107.0881-1-51-8': 'Therefore it is reasonable to generalize the concept of [MATH]-function of a complex argument and introduce the distribution [MATH] [EQUATION] defined on the class of test functions [MATH] analytic in the strip [EQUATION] and [MATH] belong to this strip.', '1107.0881-1-51-9': "In such notation the [MATH]-function introduced in Sec. III will look as [MATH], the Gel'fand [MATH]-function as [MATH] and the standard Dirac [MATH]-function as [MATH].", '1107.0881-1-52-0': 'The presence of a [MATH]-function of a complex argument in a boost mode at the surface of the light cone does not allow to exclude any point from the boost spectrum.', '1107.0881-1-52-1': 'We have shown this for the case [MATH], see Eqs. ([REF])-([REF]), and this statement proving does not change for an arbitrary value of [MATH].', '1107.0881-1-52-2': 'Another way to prove this statement was used for the distribution [MATH] based on its analyticity.', '1107.0881-1-52-3': 'The analogous proof could be given for the [MATH]-function as well.', '1107.0881-1-52-4': 'Let [MATH] be equal to [MATH] for definiteness.', '1107.0881-1-52-5': 'In this case, due to the finite value of radius of convergence [MATH], the expansion ([REF]) can be applied only for real [MATH].', '1107.0881-1-52-6': 'As a next step Dirac [MATH]-function [MATH] and every its derivative can be represented as a Taylor series of the type ([REF]) centered at the point [MATH] on the real axis, [MATH].', '1107.0881-1-52-7': 'So, [MATH] will be expanded in a [MATH]-multiple series of Dirac [MATH]-function and its derivatives.', '1107.0881-1-52-8': 'After a finite number of steps [MATH] we can reach an arbitrary spectral point [MATH] and hence obtain a representation of [MATH] in the form of a [MATH]-multiple series of Dirac [MATH]-function and its derivatives.', '1107.0881-1-52-9': 'It is clear that the point [MATH] cannot be excluded from the spectrum then.', '1107.0881-1-53-0': 'The latter reasoning clearly explains why the scalar case, when we have the only one distinguished point [MATH] which cannot be excluded from the spectrum, differs drastically from the fermion case.', '1107.0881-1-53-1': 'Indeed, the strip of analyticity ([REF]) for [MATH] degenerates into the real axis of the [MATH] complex plane at [MATH], so that the radius of convergence for the corresponding Taylor series becomes equal to zero, or in other words the Dirac [MATH]-function is not an analytic distribution.', '1107.0881-1-53-2': 'Thus the procedure discussed above cannot be realized.', '1107.0881-1-54-0': 'To conclude, it is worth emphasizing that we have shown explicitly that smearing of boost modes, or Wightman functions does not change our results, see also Ref. [CITATION].'}

{'1107.0881-2-0-0': 'We have shown that Wightman function of a free quantum field generates any complete set of solutions of relativistic wave equations.', '1107.0881-2-0-1': 'Using this approach we have constructed the complete set of solutions to 2d Dirac equation consisting of eigenfunctions of the generator of Lorentz rotations (boost operator).', '1107.0881-2-0-2': 'It is shown that at the surface of the light cone the boost modes for a fermion field contain [MATH]-function of a complex argument.', '1107.0881-2-0-3': 'Due to the presence of such singularity exclusion even of a single mode with an arbitrary value of the boost quantum number makes the set of boost modes incomplete.', '1107.0881-2-1-0': '# Introduction', '1107.0881-2-2-0': 'It is well known that quantization procedure in a quantum field theory implies expansion of the field operator in terms of a complete set of modes which are solutions of the corresponding (Dirac or Klein-Fock-Gordon) classical field equation.', '1107.0881-2-2-1': 'Therefore finding exact solutions for these equations is one of the central points of the quantum field theory in the presence of a classical background.', '1107.0881-2-3-0': 'One of the most powerful instruments for solving partial differential equations is based on symmetry properties of the physical system described by the equation.', '1107.0881-2-3-1': "According to the Noether's theorem [CITATION] any differentiable symmetry of the action of a physical system has a corresponding conservation law.", '1107.0881-2-3-2': 'One can always choose a set of variables in such a way that a change in one of them corresponds to the symmetry transformation.', '1107.0881-2-3-3': 'Since the generator of the symmetry transformation commutes with the differential operator of the equation, the variable corresponding to it can be separated.', '1107.0881-2-3-4': 'If there are enough symmetries for the equation, such that their generators commute with each other, the problem of finding the solution can be reduced to solving, generally speaking, of a second order ordinary equation.', '1107.0881-2-3-5': 'The set of such solutions is labeled by the eigenvalues of the generators and is complete.', '1107.0881-2-4-0': 'All most important and widely used solutions for relativistic quantum equations in the presence of classical external fields were obtained using this method.', '1107.0881-2-4-1': 'We mean the Coulomb field, see, e.g., [CITATION], the constant magnetic [CITATION] and electric [CITATION] fields, the field of a plane electromagnetic wave [CITATION], and some other fields of more sophisticated configurations [CITATION].', '1107.0881-2-4-2': 'In all these cases the solutions were eigenfunctions of linear combinations of generators of time and spatial translations or spatial rotations, and were labeled by values of components of either 4-momentum, or angular momentum respectively.', '1107.0881-2-4-3': 'However, the Poincare group, the group of isometries of Minkowski spacetime (MS), includes also Lorentz rotations (boosts).', '1107.0881-2-4-4': 'The boost symmetry was almost never used for field quantization.', '1107.0881-2-4-5': 'Certainly, this is because the boost generator does not commute with the Hamiltonian, and thus the boost quantum number and energy cannot be parts of any complete set of observables simultaneously.', '1107.0881-2-4-6': 'Nevertheless, the boost modes can be very useful, especially for the quantum field theory in a curved space, or if the symmetry of MS with respect to time and/or space translations is broken in the presence of a classical background.', '1107.0881-2-4-7': 'In these cases the boost symmetry can appear to be the sole symmetry for the quantum field.', '1107.0881-2-4-8': 'Dilaton gravity in two dimensions [CITATION], as well as Schwarzschild geometry [CITATION], are the examples.', '1107.0881-2-4-9': 'Therefore analysis of the properties of boost modes is important.', '1107.0881-2-5-0': 'For the first time the boost modes for a free massive scalar field were discussed in Ref. [CITATION] by W. Unruh, though their explicit form was not specified.', '1107.0881-2-5-1': 'Then some properties of scalar boost modes were studied in Refs. [CITATION].', '1107.0881-2-5-2': 'In particular, the remarkable properties of boost modes in 2d MS were ascertained in Ref. [CITATION].', '1107.0881-2-5-3': 'It was shown that (i) the zero boost mode of a free massive scalar field coincides up to a trivial constant factor with the positive-frequency Whightman function, and (ii) the boost modes considered as functions of the boost quantum number possess the Dirac [MATH]-function singularity at the surface of the light cone.', '1107.0881-2-5-4': 'The properties of the boost modes for the case of a free massless fermion field appeared to be even more interesting.', '1107.0881-2-5-5': 'It was shown in Ref. [CITATION] that boost modes of two-dimensional massless fermions on a light cone are expressed in terms of the delta function of a complex argument.', '1107.0881-2-5-6': 'Boost modes of a massive fermion field have been first considered in Refs. [CITATION], see also Ref. [CITATION].', '1107.0881-2-5-7': 'However the results of Refs. [CITATION] contain some discrepancies which we discuss in Sec. III of the present paper.', '1107.0881-2-6-0': 'In this paper we consider boost modes for a free massive fermion field.', '1107.0881-2-6-1': 'To construct them we start from the Wightman functions.', '1107.0881-2-6-2': 'The positive and negative frequency Wightman functions for free fields are explicitly determined (with accuracy to a constant factor) by Lorentz and translational invariance of the theory [CITATION].', '1107.0881-2-6-3': 'And vice versa, once the Wightman functions are known, one can use those symmetries to construct any complete set of positive and negative frequency solutions of Klein-Fock-Gordon (KFG) or Dirac equations.', '1107.0881-2-6-4': 'In fact, this is the direct consequence of the Wightman reconstruction theorem [CITATION].', '1107.0881-2-7-0': 'Since the Lorentz rotation singles out a two-dimensional plane in MS, we will discuss the specific properties of boost modes by examples of free 2d KFG and Dirac equations.', '1107.0881-2-7-1': 'In the next section we will realize the above formulated approach by the example of a single-component massive neutral field.', '1107.0881-2-7-2': 'It is worth noting that, as opposed to the case of plane wave modes, analytic properties of boost modes change dramatically when one passes to multi-component fields which are considered in Sec. III.', '1107.0881-2-7-3': 'The discussion of the results and conclusions are given in Sec. IV.', '1107.0881-2-8-0': '# Bosons', '1107.0881-2-9-0': 'In this section we will consider the case of a free neutral massive scalar field in two-dimensional MS. We will start from two-point Wightman function [CITATION] which for a free field theory coincides with the positive-frequency part of the commutator of two field operators (Pauli-Jordan function in 4d theory of a scalar field).', '1107.0881-2-9-1': 'Positive-frequency Wightman function [MATH] for a massive scalar field satisfies KFG equation [EQUATION] contains only positive frequencies and is invariant with respect to Lorentz rotations (boosts) [EQUATION]', '1107.0881-2-9-2': 'These conditions determine [MATH] accurate within a constant factor.', '1107.0881-2-9-3': 'Indeed, any positive-frequency solution of KFG equation can be written as follows [EQUATION] where [MATH] is a certain function of 2-vector [MATH], [MATH] is the Heaviside step function.', '1107.0881-2-9-4': 'After the change of variables [EQUATION] where [MATH] is rapidity, and integration over [MATH] we obtain the equation [EQUATION]', '1107.0881-2-9-5': 'It can be easily seen that condition ([REF]) is satisfied only if [MATH].', '1107.0881-2-9-6': 'Choosing [MATH] we arrive to the standard representation for [MATH], compare, e.g., [CITATION] [EQUATION]', '1107.0881-2-9-7': 'It is assumed in ([REF]) that an infinitely small negative imaginary part is added to [MATH] [CITATION].', '1107.0881-2-9-8': 'Another representation for [MATH] reads, see, e.g., [CITATION], [EQUATION] from now on we shall omit the index of the spatial component of the 2-vector [MATH].', '1107.0881-2-10-0': 'According to the Wightman reconstruction theorem [CITATION] the two-point Wightman function uniquely determines the quantum theory of a free field.', '1107.0881-2-10-1': 'Particularly, it allows to reconstruct any orthonormalized and complete set of solutions for the field equation.', '1107.0881-2-10-2': 'Indeed, it follows from the translational invariance of the theory that Wightman function with an arbitrarily shifted argument [EQUATION] satisfies KFG equation.', '1107.0881-2-10-3': 'Functions ([REF]) constitute an overcomplete set of modes since they are labeled by two independent parameters [MATH].', '1107.0881-2-10-4': 'Anyway, any complete set of positive-frequency solutions [MATH] [EQUATION] orthonormalized by the condition [EQUATION] can be represented as [EQUATION]', '1107.0881-2-10-5': 'However, since the set of solutions ([REF]) is overcomplete, the coefficient functions [MATH] cannot be determined uniquely.', '1107.0881-2-10-6': 'To make the choice of [MATH] unique one should impose a restriction on [MATH] which can be chosen for reasons of symmetry.', '1107.0881-2-11-0': 'First, we will illustrate this procedure for the trivial case of plane waves [MATH] which are eigenfunctions of the generator of spatial translations [EQUATION]', '1107.0881-2-11-1': 'In this case it is convenient to limit the family of [MATH] only by functions spatially shifted with respect to each other.', '1107.0881-2-11-2': 'This means that coefficient functions [MATH] have the form [MATH], so that [EQUATION]', '1107.0881-2-11-3': 'Let us substitute this expansion into Eq. ([REF]).', '1107.0881-2-11-4': 'Taking into account that Wightman function [MATH] exponentially tends to zero when [MATH], we can integrate the left-hand side of the resulting equation by part and arrive to the following equation for [MATH] [EQUATION] so that [MATH].', '1107.0881-2-11-5': 'Substituting now [MATH] of that form into ([REF]) and using ([REF]) we finally obtain after integration over [MATH] [EQUATION]', '1107.0881-2-11-6': 'The normalization constant here was determined from the condition ([REF]).', '1107.0881-2-12-0': 'Functions [MATH] satisfy the relation [EQUATION] and thus the set ([REF]) is complete.', '1107.0881-2-13-0': 'We are interested in the set of boost modes [MATH] which are eigenfunctions of boost operator [MATH] [EQUATION]', '1107.0881-2-13-1': 'It is worth noting from the very beginning that Wightman function [MATH] is a zero mode [MATH] of this set as it follows from Eqs. ([REF]) and ([REF]).', '1107.0881-2-14-0': 'To obtain the boost modes we will confine [MATH] in ([REF]) to one of the orbits of the restricted Lorentz group [EQUATION] so that [MATH].', '1107.0881-2-14-1': 'Choosing the upper sign in ([REF]) for definiteness, we put [EQUATION] and rewrite Eq. ([REF]) in the form [EQUATION]', '1107.0881-2-14-2': 'Substituting ([REF]) into Eq. ([REF]) and using the relation [EQUATION] which can be straightforwardly obtained with the help of representation ([REF]), we get the following equation for [MATH] [EQUATION].', '1107.0881-2-14-3': 'Hence, [MATH], and we finally get from ([REF]) with due regard for condition ([REF]) the following representation for the positive frequency boost modes [EQUATION] which was earlier obtained in Refs. [CITATION] by another method.', '1107.0881-2-14-4': 'The negative frequency boost modes [MATH] are defined [CITATION] as [EQUATION]', '1107.0881-2-14-5': 'Boost modes ([REF]), ([REF]) are distributions with respect to both [MATH] and spectral parameter [MATH].', '1107.0881-2-14-6': 'They are defined on the class of smooth functions of rapid enough descent.', '1107.0881-2-15-0': 'The modes ([REF]),([REF]) constitute a complete set since they satisfy the condition [EQUATION] and hence can be used as a basis for quantization of a neutral scalar field, see Ref. [CITATION].', '1107.0881-2-15-1': 'Since the sign of a particle energy is Lorentz invariant, the vacuum states in the boost and plane-wave quantization schemes are identical.', '1107.0881-2-15-2': 'Hence, these two quantization schemes are unitary equivalent, see Ref. [CITATION].', '1107.0881-2-16-0': 'The most remarkable property of boost modes is their behavior at the light cone.', '1107.0881-2-16-1': 'It is easily seen from ([REF]), ([REF]) that at the vertex of the light cone [MATH] possess the Dirac [MATH]-function singularity [EQUATION]', '1107.0881-2-16-2': 'It was shown in Ref. [CITATION] that [MATH] possess a [MATH]-function singularity at the lines [MATH] as well.', '1107.0881-2-16-3': 'Therefore contribution of the single spectral point [MATH] to physical quantities can be finite.', '1107.0881-2-16-4': 'We will illustrate this by the example of Wightman function.', '1107.0881-2-17-0': 'Taking into account the property of translational invariance we can rewrite Eq. ([REF]) in the form [EQUATION]', '1107.0881-2-17-1': 'Using now Eq. ([REF]) we obtain [EQUATION] i.e. the integral over [MATH] in ([REF]) is determined by the point [MATH] entirely.', '1107.0881-2-18-0': 'This result means that the point [MATH] cannot be deleted from the spectrum, or in other words the integral over [MATH] in ([REF]),([REF]) cannot be changed by its principal value [EQUATION]', '1107.0881-2-18-1': 'Thereby, the family of boost modes does not constitute a complete set in MS after excluding the zero mode.', '1107.0881-2-18-2': 'However, the equality in Eq. ([REF]) could be restored if we cut the light cone out of MS, compare [CITATION].', '1107.0881-2-18-3': 'This is because all the points where boost modes possess [MATH] singularity are located just at the light cone.', '1107.0881-2-19-0': 'Furthermore, the Wightman function Eq. ([REF]) cannot be represented in the form [EQUATION]', '1107.0881-2-19-1': 'If it were so, we could rewrite ([REF]), due to translational invariance of the Wightman function and the boost mode property ([REF]), as follows [EQUATION]', '1107.0881-2-19-2': 'But Eq. ([REF]) is evidently meaningless.', '1107.0881-2-19-3': 'Indeed, the distribution [MATH] is defined on the class of functions continuous at the interval including the point [MATH].', '1107.0881-2-19-4': 'Therefore Eq. ([REF]) should be understood as [EQUATION] where [MATH] is the Heaviside step function.', '1107.0881-2-19-5': 'However, the product of two distributions [MATH] is not defined.', '1107.0881-2-20-0': 'The authors of Ref. [CITATION] do not agree with this statement.', '1107.0881-2-20-1': 'Discussing their Eq. (2.129), which in our notations coincides with ([REF]), they admit that this expression is undefined if either of the two points [MATH] is located on the light cone.', '1107.0881-2-20-2': 'They think that smearing of the distribution [MATH] with compactly supported functions [MATH] and [MATH] improve the situation and the smeared "Wightman function" [MATH](2.131) is well defined on the whole MS. Our analysis shows however that this is not correct.', '1107.0881-2-20-3': 'Indeed, if we use Eq. ([REF]) in the form ([REF]), we see that this expression is undefined for arbitrary [MATH].', '1107.0881-2-20-4': 'Moreover, smearing cannot improve the situation since it does not influence the [MATH]-function in ([REF]).', '1107.0881-2-20-5': 'The correct expression for the smeared Wightman function can be easily obtained from Eq. ([REF]).', '1107.0881-2-20-6': 'It reads [EQUATION] where [EQUATION] is the smeared boost mode.', '1107.0881-2-20-7': 'We see that again only the single spectral point [MATH] contributes to the integral ([REF]) for the smeared Wightman function.', '1107.0881-2-21-0': 'Note, that we could try to use expression ([REF]) directly, not applying the property of translational invariance to it.', '1107.0881-2-21-1': 'In that case we should attach exact mathematical meaning to ([REF]) first.', '1107.0881-2-21-2': 'Namely, we should write it in the form [EQUATION]', '1107.0881-2-21-3': 'But it can be easily seen that function [MATH] in ([REF]) is not translationally invariant and thus has nothing to do with Wightman function for a free field in MS. Indeed, if it were translationally invariant, it could be written down as [EQUATION] and would be equal to zero identically.', '1107.0881-2-22-0': 'Singular behavior of the boost mode at the point [MATH] can be understood in terms of classical trajectories of free particles with a given value of the boost parameter [MATH] [EQUATION] where [MATH] is the momentum, [MATH] - the energy of a particle, [MATH].', '1107.0881-2-22-1': 'Let us rewrite Eq. ([REF]) in the the form [EQUATION]', '1107.0881-2-22-2': 'After differentiating both sides of Eq. ([REF]) with time one can easily find its regular [EQUATION] and singular [EQUATION] solutions.', '1107.0881-2-23-0': 'When [MATH] the singular solution is represented by two branches of hyperbola ([REF]).', '1107.0881-2-23-1': 'The branch with [MATH] is located in the right wedge of MS, while the branch with [MATH] in the left one, see Figs. 1a,b. Free boost particles with the given value of [MATH] propagate along the regular world lines which are different straight lines tangent to the corresponding branches of hyperbola ([REF]).', '1107.0881-2-23-2': 'It is worth noting that classical particles with positive values of the boost parameter [MATH] cannot penetrate to the left wedge, as well as particles with [MATH] cannot penetrate to the right one.', '1107.0881-2-24-0': 'The world lines of boost particles with [MATH] are given by [EQUATION]', '1107.0881-2-24-1': 'All of them cross the vertex of the light cone [MATH] and any world line crossing [MATH] belongs to the family of trajectories with [MATH] , see Fig. 1c.', '1107.0881-2-24-2': 'Hence, elimination of the point [MATH] from the spectrum is equivalent to prohibition for particles to cross the point [MATH], or to pricking it out of MS. Besides, deleting the point [MATH] from the spectrum, we delete a bunch of infinite number of trajectories, thus loosing a substantial part of degrees of freedom.', '1107.0881-2-24-3': 'This means that the point [MATH] is of nonzero measure at [MATH] and explains the [MATH]-function singularity of the boost mode there.', '1107.0881-2-25-0': 'The points lying at the light cone surface need a separate consideration.', '1107.0881-2-25-1': 'The lines [MATH] cannot be trajectories of massive particles.', '1107.0881-2-25-2': 'But the branches of hyperbola ([REF]) degenerate into lines constituting the surface of the light cone in our 2d problem when [MATH].', '1107.0881-2-25-3': 'As it can easily be seen from Figs. 1a,b, all regular trajectories except those which are tangent to the hyperbola branches at the point [MATH] also tend to the lines [MATH] in the limit [MATH].', '1107.0881-2-25-4': 'Hence, these lines [MATH] are the lines of condensation for regular trajectories with [MATH].', '1107.0881-2-25-5': 'Possibly this explains the existence of [MATH]-function singularity of boost modes at the light cone surface.', '1107.0881-2-26-0': 'It is worth noting that there exist another representation for the boost modes ([REF]), ([REF]).', '1107.0881-2-26-1': 'Keeping in mind that an infinitely small negative imaginary part [MATH] is added to [MATH] in ([REF]), after the change of the variable of integration [MATH] we obtain for [MATH] the following expression [EQUATION] with [MATH] and [MATH].', '1107.0881-2-26-2': 'Using now formula 3.471(9) of Ref. [CITATION] we get [EQUATION] where [MATH] is the Macdonald function, and distributions [MATH] should be understood as [EQUATION] see Ref. [CITATION].', '1107.0881-2-26-3': 'Using Eq. ([REF]) we obtain [EQUATION] see [CITATION], where [EQUATION] are the expressions for the boost modes in the past ([MATH]), right ([MATH]), future ([MATH]) and left ([MATH]) wedges of MS respectively.', '1107.0881-2-26-4': 'Here [MATH] are Hankel functions.', '1107.0881-2-27-0': 'Putting [MATH] in Eq. ([REF]) and taking into account Eq. ([REF]) one can easily reproduce the Wightman result [CITATION] for [MATH].', '1107.0881-2-27-1': 'Applying the same procedure to Eq. ([REF]) we get [EQUATION]', '1107.0881-2-27-2': 'This is a new compact representation for the Wightman function in 2d scalar theory.', '1107.0881-2-28-0': '# Fermions', '1107.0881-2-29-0': 'A set of orthonormalized positive (negative) frequency solutions [MATH] of Dirac equation in 2d MS [EQUATION] ([MATH] - two-dimensional Dirac matrices, [MATH], [MATH], [MATH]) is complete if [EQUATION] where the positive [MATH] and negative [MATH] frequency Wightman functions for the fermion field are equal to [EQUATION] and [MATH] is defined in ([REF]).', '1107.0881-2-30-0': 'Any set [MATH] can be constructed of the set of matrices [MATH] in the perfect analogy with the boson case.', '1107.0881-2-30-1': 'Taking into account that the Dirac operators [MATH] commute with the fermion boost operator [MATH], [EQUATION] it is easy to check that functions [EQUATION] ([MATH] are defined in ([REF])) are the eigenfunctions of this operator if [EQUATION] where [MATH] is an arbitrary constant column.', '1107.0881-2-30-2': 'Substituting ([REF]) into Eq. ([REF]) we easily get with the account of definition ([REF]) the following expression for the normalized boost modes [EQUATION]', '1107.0881-2-30-3': 'Here the arbitrary constant column [MATH] satisfies the relation [EQUATION]', '1107.0881-2-30-4': 'It is difficult to compare representation ([REF]) with the results of Refs. [CITATION] directly since the authors of these works by unknown reasons use in 2d spacetime [MATH] matrices which constitute a reducible representation of the Dirac matrices.', '1107.0881-2-30-5': 'The passage to their representation may be realized as follows [EQUATION] where [EQUATION]', '1107.0881-2-30-6': 'In particular, using the notations of Refs. [CITATION] [EQUATION] for the positive frequency solution in the Rindler wedge of 2d MS we obtain from Eq. ([REF]) with account of ([REF]),([REF]) [EQUATION]', '1107.0881-2-30-7': 'It is worth noting that the results of Refs. [CITATION], see Eqs. 21.64, 21.65, 21.77 and 21.78 in [CITATION], differ from ([REF]) by the absence of factors [MATH] in front of the functions [MATH] in ([REF]).', '1107.0881-2-30-8': 'In the framework of the method used by the authors of Refs. [CITATION], the latter should have appeared as a result of change of Rindler [MATH] by Cartesian coordinates [EQUATION] see, e.g., [CITATION].', '1107.0881-2-31-0': 'Besides, the modes in Refs. [CITATION] differ from ([REF]) by additional normalization factor [MATH].', '1107.0881-2-31-1': 'This discrepancy has arisen due to incorrect method of normalization used by the authors of Refs. [CITATION].', '1107.0881-2-31-2': 'Indeed, they have represented the normalization integral in 2d MS as a sum of four terms each equal to the normalization integral in one of four wedges of MS. However, the correct normalization procedure implies integration over some Cauchy surface in MS. The most convenient surface for this purpose is the surface [MATH], compare, e.g., [CITATION].', '1107.0881-2-31-3': 'Then only the solutions in [MATH] and [MATH] wedges of MS contribute to the normalization integral and the normalization factor coincides with that one in ([REF]).', '1107.0881-2-32-0': 'From now on, we will use a representation of Dirac matrices different from ([REF]), [EQUATION] where [MATH], [MATH] are Pauli matrices.', '1107.0881-2-32-1': 'Use of representation ([REF]) allows essentially simplify the form of solutions of Eq. ([REF]).', '1107.0881-2-32-2': 'We will see below that it also is very convenient for the procedure of transition to the limit [MATH].', '1107.0881-2-32-3': 'With this set of Dirac matrices the representation ([REF]) for the boost modes reduces to [EQUATION]', '1107.0881-2-32-4': 'Using exactly the same procedure as for the scalar boost modes ([REF]) we can transform Eq. ([REF]) to the form [EQUATION] where again [MATH].', '1107.0881-2-33-0': 'In perfect analogy with Eq. ([REF]) the positive frequency boost mode ([REF]) can be rewritten in the form [EQUATION] where in [MATH] wedge [EQUATION] in [MATH] wedge [EQUATION] in [MATH] wedge [EQUATION] and finally in [MATH] wedge [EQUATION]', '1107.0881-2-33-1': 'A similar representation is valid for the negative frequency modes [MATH] as well.', '1107.0881-2-34-0': 'The boost modes ([REF]), ([REF]) are orthonormalized by the condition [EQUATION] constitute a complete set in the sense ([REF]), and thus can serve a basis for quantization of the fermion field in MS, [EQUATION] where [MATH] and [MATH] are annihilation operators for boost fermion particles and antiparticles respectively, which obey the standard commutation relations.', '1107.0881-2-34-1': 'By the same reasons as in the boson case, boost quantization of the fermion field is unitary equivalent to quantization in the plane wave basis [EQUATION] where [EQUATION]', '1107.0881-2-34-2': 'It is not difficult to ascertain that the boost modes ([REF]) are connected with the plane waves ([REF]) through the following integral transformation [EQUATION]', '1107.0881-2-34-3': 'The modes [MATH] and [MATH] are distributions with respect to variables [MATH] and [MATH].', '1107.0881-2-34-4': 'This means that they define linear functionals [EQUATION] on some sets of test functions [MATH], [MATH] respectively.', '1107.0881-2-34-5': 'Such functionals naturally appear when we calculate, e.g., matrix elements of the field operators ([REF]), ([REF]) between the vacuum state [MATH] and the states which are particle (or antiparticle) wave packets, [EQUATION]', '1107.0881-2-34-6': 'Physical one-particle states must be normalized, [EQUATION]', '1107.0881-2-34-7': 'Hence, [MATH], [MATH] must be square-integrable functions of [MATH] and [MATH] respectively.', '1107.0881-2-34-8': 'Then it is easy to see that matrix elements ([REF]), ([REF]) are square-integrable functions of [MATH].', '1107.0881-2-34-9': 'It is clear that functionals ([REF]), ([REF]) are the elements of one and the same space of functions.', '1107.0881-2-34-10': 'Hence, there exists one to one correspondence between the test functions [MATH] and [MATH]: [EQUATION] where [MATH] is rapidity.', '1107.0881-2-34-11': 'For the sake of convenience we will confine ourselves to [MATH] belonging to the class of continuous piecewise smooth functions descending faster than [MATH] at [MATH].', '1107.0881-2-34-12': 'This requirement guarantees finiteness of the mean value of the energy of the one-particle state [MATH] ([REF]), [EQUATION]', '1107.0881-2-34-13': 'To ascertain the properties of test functions [MATH] we will use the relation ([REF]).', '1107.0881-2-35-0': 'First, we will illustrate by a simple example that, unlike the case of the functional ([REF]), we need know the behavior of test functions [MATH] for the functional ([REF]) not only on the real axis but also in the complex plane.', '1107.0881-2-35-1': 'Consider [EQUATION] which evidently satisfies the above formulated requirements to functions [MATH].', '1107.0881-2-35-2': 'Using Eq. ([REF]) we obtain [EQUATION]', '1107.0881-2-35-3': 'Direct calculation of the integral ([REF]) at the point [MATH] then yields [EQUATION]', '1107.0881-2-35-4': 'So we see that the value of the functional ([REF]) at the vertex of the light cone is determined by the values of the test function [MATH] at imaginary points [MATH].', '1107.0881-2-36-0': 'Let us now revert to Eq. ([REF]).', '1107.0881-2-36-1': 'After the change of the variable of integration [MATH] it can be written as a Fourier transform of function [MATH] [EQUATION]', '1107.0881-2-36-2': 'Under our assumptions [EQUATION] as [MATH].', '1107.0881-2-36-3': 'Then, according to the Paley-Wiener theorem [CITATION] the test functions [MATH] ([REF]) are analytic in the strip [EQUATION]', '1107.0881-2-36-4': 'Let us calculate the integral in Eq. ([REF]) by parts.', '1107.0881-2-36-5': 'Since the test functions [MATH] are continuous and descend exponentially as [MATH], see Eq. ([REF]), the first integrated term is equal to zero.', '1107.0881-2-36-6': 'Hence, the functions [MATH] in their domain of analyticity ([REF]) descend at least as [MATH] when [MATH].', '1107.0881-2-36-7': 'This means that the path of integration in ([REF]) can be shifted in the range of the strip ([REF]).', '1107.0881-2-37-0': 'Consider the functional ([REF]) at the point [MATH].', '1107.0881-2-37-1': 'Shifting the path of integration for the upper component upward, and for the lower one downward, by [MATH], we obtain [EQUATION]', '1107.0881-2-37-2': 'Eq. ([REF]) generalizes formula ([REF]) to the class of functions [MATH] analytic in the strip ([REF]) and descending rapidly enough when [MATH].', '1107.0881-2-38-0': 'The obtained result makes it possible to write down the fermion boost mode in the vertex of the light cone in terms of [MATH]-functions of complex argument [EQUATION]', '1107.0881-2-38-1': 'The functionals [MATH] can be also written as integrals around the closed contours represented in Figs. [REF]a,b [EQUATION]', '1107.0881-2-38-2': 'Indeed, the explicit form for [MATH] can be easily derived from Eq. ([REF]).', '1107.0881-2-38-3': 'Putting [MATH] there and using the ascending series for [MATH] [CITATION], we get [EQUATION]', '1107.0881-2-38-4': "Then, taking into account that the integrand in ([REF]) has simple poles at [MATH] and using the Cauchy's residue theorem, we immediately reproduce the result ([REF]).", '1107.0881-2-38-5': 'Since the integrals taken along the vertical segments of the contours [MATH] are evidently equal to zero, this means, in particular, that the integrals along the upper (lower) piece of the contour [MATH]) is also equal to zero in the limit [MATH].', '1107.0881-2-38-6': 'The latter statement can be easily checked by direct calculation.', '1107.0881-2-39-0': "[MATH]-function of complex argument was first introduced by Gel'fand and Shilov in Ref. [CITATION] on the basis of Cauchy residue theorem, [EQUATION] where [MATH] is a contour enclosing an arbitrary complex point [MATH].", '1107.0881-2-39-1': 'The distribution ([REF]) was defined in [CITATION] on some class of entire functions [MATH], the so called [MATH]-class.', '1107.0881-2-39-2': 'Our class of test functions analytic in the strip ([REF]) differs from [MATH].', '1107.0881-2-39-3': "Therefore some properties of the Gel'fand's [MATH]-function [CITATION] cannot be applied to our one.", '1107.0881-2-39-4': "However, the opposite statement is true: all properties of our [MATH]-function hold valid for the Gel'fand's one.", '1107.0881-2-40-0': 'So, on the class of test functions analytic in the strip ([REF]), as well as for the [MATH]-class, the following equality of distributions is valid [CITATION] [EQUATION]', '1107.0881-2-40-1': 'Let us show now that the modes ([REF]), ([REF]) possess [MATH]-function singularities not only at the vertex of the light cone but at lines [MATH] as well.', '1107.0881-2-40-2': 'Indeed, at the surface of the light cone the arguments of Macdonald functions in ([REF]) are small, [MATH].', '1107.0881-2-40-3': 'Therefore using the ascending series for Macdonald functions [CITATION] we get [EQUATION]', '1107.0881-2-40-4': 'So we see that at the surface of the light cone [MATH] contains [MATH] at [MATH] in the upper, and [MATH] at [MATH] in the lower component.', '1107.0881-2-41-0': 'It was shown in the preceding section that in the case of a scalar field the family of boost modes does not constitute a complete set in MS after excluding the zero mode.', '1107.0881-2-41-1': 'This is because the scalar boost modes possess a [MATH]-function singularity at the surface of the light cone, and hence the point [MATH] gives finite contribution to physical quantities.', '1107.0881-2-41-2': 'Some of them, e.g., Wightman function, are determined by the spectral point [MATH] entirely.', '1107.0881-2-41-3': 'There is a question whether it is possible to delete the point [MATH] from the spectrum in the fermion case.', '1107.0881-2-41-4': 'To clarify this issue we will consider the integral [EQUATION] where [MATH] is an arbitrary real number.', '1107.0881-2-42-0': 'Consider the upper component of the second term in the RHS of Eq. ([REF]) [EQUATION]', '1107.0881-2-42-1': 'The integral in ([REF]) can be easily calculated in the limit [MATH] and we obtain [EQUATION]', '1107.0881-2-42-2': 'Eq. ([REF]) evidently shows that the result of calculation for [MATH] strongly depends on the sequence of limit processing [MATH] and [MATH].', '1107.0881-2-42-3': 'If we changed the sequence of limit processing in ([REF]), ([REF]), we would return from [MATH] ([REF]) to [MATH] ([REF]).', '1107.0881-2-42-4': 'Not surprisingly, Eq. ([REF]) reproduces the result ([REF]) in this case.', '1107.0881-2-42-5': 'The accepted sequence which corresponds to calculation of the functional ([REF]) leads to a meaningless result.', '1107.0881-2-42-6': 'This means that pricking of any point out of the real axis [MATH] is inadmissible in the fermion case.', '1107.0881-2-42-7': 'In other words, the set of boost modes ([REF]),([REF]) does not constitute a complete set in MS after deleting a single arbitrary point from the spectrum.', '1107.0881-2-42-8': 'It can be complete only in the space which is MS without the line cone (four separate wedges of MS).', '1107.0881-2-42-9': 'In this sense our conclusion repeats the analogous result for boson field discussed in the preceding section.', '1107.0881-2-43-0': 'It is worth noting that the latter result was obtained under minimum possible restrictions as regards to the test functions [MATH].', '1107.0881-2-43-1': 'If we restrict ourselves to [MATH] belonging to the class [MATH] of smooth functions with compact support, as it is often done in quantum field theory, functions [MATH] would belong to the class [MATH] [CITATION].', '1107.0881-2-43-2': 'In this case the proof of impossibility of deleting an arbitrary point [MATH] from the spectrum is especially simple.', '1107.0881-2-43-3': "Indeed, the Gel'fand's [MATH]-function is an analytic distribution in the whole complex plain [MATH].", '1107.0881-2-43-4': 'Thus it can be expanded in a Taylor series around any point [MATH], [EQUATION] and the radius of convergence of the series ([REF]) is equal to infinity [CITATION].', '1107.0881-2-43-5': 'Here [MATH] denotes the [MATH]-th derivative of the [MATH]-function at the point [MATH].', '1107.0881-2-43-6': 'As a result, [EQUATION] and hence, on the class [MATH] of test functions, deleting of an arbitrary spectral point [MATH] from the spectrum leads to vanishing of functionals [MATH] ([REF]) at the vertex of the light cone, and its finite variation at other points of the cone surface.', '1107.0881-2-43-7': "However, since the Gel'fand's [MATH]-function is analytic on the whole plane of complex [MATH], it can be expanded in a Taylor series around another real point [MATH], [MATH].", '1107.0881-2-43-8': 'If we calculate integral ([REF]) using such representation of the [MATH]-function, we will see that deleting of the spectral point [MATH] from the spectrum does not influence the result of integration.', '1107.0881-2-43-9': 'Thus, since the value of the matrix element ([REF]) depends on the way of calculation, we conclude that the operation of exclusion of an arbitrary point from the spectrum is meaningless in agreement with the previous consideration, see discussion of Eq. ([REF]) in the preceding paragraph.', '1107.0881-2-44-0': 'As it was shown in the preceding section, Wightman functions [MATH] of a massive scalar field are determined by the zero boost modes [MATH], see Eq. ([REF]).', '1107.0881-2-44-1': 'This is a consequence of translational invariance of [MATH] and the presence of Dirac [MATH]-function singularity of [MATH] at the light cone.', '1107.0881-2-44-2': 'A similar result is valid also for the Wightman functions of a massive fermion field.', '1107.0881-2-44-3': 'Indeed, due to the property of translational invariance we can write, compare ([REF]), [EQUATION]', '1107.0881-2-44-4': 'Using now Eq. ([REF]) we get the following result for matrix elements [MATH] of Wightman function ([REF]), [EQUATION] where [MATH] are the [MATH]-components of the boost modes ([REF]), ([REF]).', '1107.0881-2-44-5': 'Thereby, we see that the matrix elements of [MATH] are determined by only two "spectral points" [MATH].', '1107.0881-2-44-6': 'Exactly as in the scalar case, it is easy to ascertain that this result holds valid for the smeared Wightman functions as well.', '1107.0881-2-45-0': 'Using Eq. ([REF]) one can express the Wightman functions ([REF]) in the form, compare ([REF]), [EQUATION] where [MATH] is the same as in Eq. ([REF]).', '1107.0881-2-46-0': 'Consider now the case of a massless fermion field first studied in Ref. [CITATION].', '1107.0881-2-46-1': 'The Wightman function of the massless field [MATH] can be obtained easily by passage to the limit [MATH] in Eq. ([REF]) [EQUATION]', '1107.0881-2-46-2': 'Just as in the cases considered earlier, one can use [MATH] to obtain any complete set of orthonormalized solutions of Dirac equation.', '1107.0881-2-46-3': 'In particular, for the boost modes we have, [EQUATION] where [MATH] is an arbitrary two-component column, compare ([REF]), ([REF]).', '1107.0881-2-46-4': 'We see that at [MATH] the Wightman function ([REF]) is antidiagonal.', '1107.0881-2-46-5': 'Owing to this property the upper and the lower components of the boost mode ([REF]) become independent: [EQUATION] where [MATH] are the eigenvectors of the Pauli matrix [MATH], [MATH].', '1107.0881-2-46-6': 'This is because there appears a new conservation law for the massless case, conservation of chirality, see, e.g., [CITATION].', '1107.0881-2-46-7': 'Thereby solutions of the Dirac equation ([REF]) are labeled by the additional quantum number [MATH], chirality.', '1107.0881-2-46-8': 'The coefficients [MATH] and [MATH] in ([REF]) are independent and can be determined by the normalization condition for solutions [MATH] and [MATH] separately.', '1107.0881-2-46-9': 'Note that in the massive case, since chirality is not conserved, the components of the column [MATH] ([REF]) cannot be determined separately.', '1107.0881-2-46-10': 'In that case their combination forms a monomial factor of the solution [MATH] ([REF]).', '1107.0881-2-47-0': 'For the normalized positive frequency functions [MATH] we have [EQUATION] where the phase factors were chosen for the sake of convenience.', '1107.0881-2-47-1': 'It is worth noting that the negative frequency functions [MATH] come out of ([REF]) by complex conjugation and the change [MATH].', '1107.0881-2-48-0': 'Plane waves are also labeled by the quantum number [MATH] in the massless case, [EQUATION]', '1107.0881-2-48-1': 'They are linked to the boost modes [MATH] through the Mellin transform [EQUATION]', '1107.0881-2-48-2': 'This means that, if we assume that distributions [MATH] are defined on the same class of test functions [MATH] as in the massive case, the distributions [MATH] will be defined on the class of test functions [MATH] analytic in the strip ([REF]) and descending at [MATH] in this strip.', '1107.0881-2-49-0': 'Then taking into account Eq. ([REF]) we conclude that boost modes of a massless fermion field at the surface of the light cone are [MATH]-functions of a complex argument, compare [CITATION], [EQUATION]', '1107.0881-2-49-1': 'Now we will show how the modes ([REF]) can be obtained from Eq. ([REF]) by limit processing [MATH].', '1107.0881-2-49-2': 'Using the ascending series [CITATION] for functions [MATH] we get [EQUATION] i.e., representation ([REF]) with coefficients [MATH] and [MATH] containing singular at [MATH] phase factors.', '1107.0881-2-49-3': 'However these factors do not influence the normalization constants, have no impact on any physical quantities and hence can be omitted.', '1107.0881-2-50-0': '# Concluding remarks', '1107.0881-2-51-0': 'We have shown that Wightman function of a free quantum field generates any complete set of solutions of relativistic wave equations.', '1107.0881-2-51-1': 'Using this approach we have constructed the complete sets of solutions to KFG and Dirac equations consisting of eigenfunctions of the generator of Lorentz rotations (boost operator).', '1107.0881-2-52-0': 'Boost modes are used as a basis for field quantization very rarely.', '1107.0881-2-52-1': 'Till now they were exploited only for analysis of the so-called "Unruh effect" [CITATION] and at attempts to quantize a charged massive scalar field in the presence of an external constant electric field [CITATION].', '1107.0881-2-52-2': 'However, there are many problems, especially in the quantum field theory in a curved space, where the boost symmetry may appear to be the only symmetry for the quantum field, and thus using it for separation of variables in a classical field equation is the only instrument to find solutions for such equations.', '1107.0881-2-53-0': 'The specific feature of the boost modes is that, taken at the surface of the light cone, they as functions of the boost quantum number [MATH] possess strong singularities.', '1107.0881-2-53-1': 'Certainly, the physical reason for these singularities is the singularity of Lorentz transformations at [MATH].', '1107.0881-2-54-0': 'For the case of a scalar field this is a Dirac [MATH]-function singularity [MATH] [CITATION].', '1107.0881-2-54-1': 'This leads to a special role of the zero boost mode: its exclusion from the set of the boost modes makes the latter incomplete.', '1107.0881-2-54-2': 'The exceptional role of the zero boost mode becomes quite clear if we recall that the zero value of [MATH] means that [MATH] is a Lorentz invariant positive frequency solution of KFG equation, i.e. the Wightman function for the quantum field coinciding with the positive frequency part of the commutator of two scalar field operators.', '1107.0881-2-54-3': 'Hence the exclusion of the zero boost mode results in a "quantum"[MATH] theory with commuting field operators.', '1107.0881-2-55-0': 'The singularities of the fermion boost modes are even stronger.', '1107.0881-2-55-1': 'It is shown in the present paper that at the surface of the light cone they possess [MATH]-function of a complex argument [MATH].', '1107.0881-2-55-2': "The [MATH]-function of a complex argument was first introduced by Gel'fand and Shilov in Ref. [CITATION], and was defined on the class [MATH] of entire functions.", '1107.0881-2-55-3': 'Our [MATH]-function is defined on the class of test functions analytic in the strip ([REF]).', '1107.0881-2-55-4': 'Actually, the width of the strip is determined by physical requirements.', '1107.0881-2-55-5': 'Our choice provides square integrability of one-particle wave packets and finiteness of their energy.', '1107.0881-2-55-6': 'If we require finiteness of the squared energy we should narrow the class of test functions and extend the width of the strip to [EQUATION].', '1107.0881-2-55-7': 'Further toughening of requirements to physical states will lead to subsequent extension of the width of the strip of analyticity of the test functions.', '1107.0881-2-55-8': 'Therefore it is reasonable to generalize the concept of [MATH]-function of a complex argument and introduce the distribution [MATH] [EQUATION] defined on the class [MATH] of test functions [MATH] analytic in the strip [EQUATION] and [MATH] belong to this strip.', '1107.0881-2-55-9': "In such notation the [MATH]-function introduced in Sec. III will look as [MATH], the Gel'fand [MATH]-function as [MATH] and the standard Dirac [MATH]-function as [MATH].", '1107.0881-2-56-0': 'It is worth noting that for [MATH]-function defined on the class [MATH] the following integral representation is valid [EQUATION] which at [MATH] is a straightforward generalization of the standard representation for the Dirac [MATH]-function.', '1107.0881-2-56-1': 'Besides, the following relations are valid, [EQUATION] compare Ref. [CITATION] for the case [MATH].', '1107.0881-2-57-0': 'The presence of a [MATH]-function of a complex argument in a boost mode at the surface of the light cone does not allow to exclude any point from the boost spectrum.', '1107.0881-2-57-1': 'We have shown this for the case [MATH], see Eqs. ([REF])-([REF]), and this statement proving does not change for an arbitrary value of [MATH].', '1107.0881-2-57-2': 'Another way to prove this statement was used for the distribution [MATH] based on its analyticity in the whole complex plane.', '1107.0881-2-57-3': 'The analogous proof could be given for the [MATH]-function as well.', '1107.0881-2-57-4': 'In this case, due to the finite value of radius of convergence [MATH], the expansion ([REF]) can be applied only for real [MATH].', '1107.0881-2-57-5': 'As a next step the Dirac [MATH]-function [MATH] and every its derivative can be represented as a Taylor series of the type ([REF]) centered at the point [MATH] on the real axis, [MATH].', '1107.0881-2-57-6': 'So, [MATH] will be expanded in a [MATH]-multiple series of the Dirac [MATH]-function and its derivatives.', '1107.0881-2-57-7': 'After a finite number of steps [MATH] we can reach an arbitrary spectral point [MATH] and hence obtain a representation of [MATH] in the form of a [MATH]-multiple series of the Dirac [MATH]-function and its derivatives.', '1107.0881-2-57-8': 'It is clear that the point [MATH] cannot be excluded from the spectrum then.', '1107.0881-2-58-0': 'The latter reasoning clearly explains why the scalar case, when we have the only one distinguished point [MATH] which cannot be excluded from the spectrum, differs drastically from the fermion case.', '1107.0881-2-58-1': 'Indeed, the strip of analyticity ([REF]) for [MATH] degenerates into the real axis of the [MATH] complex plane at [MATH], so that the radius of convergence for the corresponding Taylor series becomes equal to zero, or in other words the Dirac [MATH]-function is not an analytic distribution.', '1107.0881-2-58-2': 'Thus the procedure discussed above cannot be realized.', '1107.0881-2-59-0': 'To conclude, it is worth emphasizing that we have shown explicitly that smearing of boost modes, or Wightman functions does not change our results, see also Ref. [CITATION].'}

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'1107.0881-2-11-6'], ['1107.0881-1-41-0', '1107.0881-2-44-0'], ['1107.0881-1-41-2', '1107.0881-2-44-2'], ['1107.0881-1-41-3', '1107.0881-2-44-3'], ['1107.0881-1-41-4', '1107.0881-2-44-4'], ['1107.0881-1-41-5', '1107.0881-2-44-5'], ['1107.0881-1-41-6', '1107.0881-2-44-6'], ['1107.0881-1-12-0', '1107.0881-2-13-0'], ['1107.0881-1-12-1', '1107.0881-2-13-1'], ['1107.0881-1-2-1', '1107.0881-2-2-1'], ['1107.0881-1-45-1', '1107.0881-2-49-1'], ['1107.0881-1-45-2', '1107.0881-2-49-2'], ['1107.0881-1-45-3', '1107.0881-2-49-3'], ['1107.0881-1-36-1', '1107.0881-2-39-1'], ['1107.0881-1-36-2', '1107.0881-2-39-2'], ['1107.0881-1-36-3', '1107.0881-2-39-3'], ['1107.0881-1-36-4', '1107.0881-2-39-4'], ['1107.0881-1-40-2', '1107.0881-2-43-2'], ['1107.0881-1-40-3', '1107.0881-2-43-3'], ['1107.0881-1-40-5', '1107.0881-2-43-5'], ['1107.0881-1-40-6', '1107.0881-2-43-6'], ['1107.0881-1-40-7', '1107.0881-2-43-7'], ['1107.0881-1-40-8', '1107.0881-2-43-8'], ['1107.0881-1-40-9', '1107.0881-2-43-9'], ['1107.0881-1-13-1', '1107.0881-2-14-1'], ['1107.0881-1-13-2', '1107.0881-2-14-2'], ['1107.0881-1-13-3', '1107.0881-2-14-3'], ['1107.0881-1-13-4', '1107.0881-2-14-4'], ['1107.0881-1-13-5', '1107.0881-2-14-5'], ['1107.0881-1-13-6', '1107.0881-2-14-6'], ['1107.0881-1-17-0', '1107.0881-2-18-0'], ['1107.0881-1-17-1', '1107.0881-2-18-1'], ['1107.0881-1-17-2', '1107.0881-2-18-2'], ['1107.0881-1-17-3', '1107.0881-2-18-3'], ['1107.0881-1-50-1', '1107.0881-2-54-1'], ['1107.0881-1-50-2', '1107.0881-2-54-2'], ['1107.0881-1-50-3', '1107.0881-2-54-3'], ['1107.0881-1-3-0', '1107.0881-2-3-0'], ['1107.0881-1-3-1', '1107.0881-2-3-1'], ['1107.0881-1-3-2', '1107.0881-2-3-2'], ['1107.0881-1-3-3', '1107.0881-2-3-3'], ['1107.0881-1-3-5', '1107.0881-2-3-5'], ['1107.0881-1-47-0', '1107.0881-2-51-0'], ['1107.0881-1-47-1', '1107.0881-2-51-1'], ['1107.0881-1-54-0', '1107.0881-2-59-0'], ['1107.0881-1-48-0', '1107.0881-2-52-0'], ['1107.0881-1-48-1', '1107.0881-2-52-1'], ['1107.0881-1-48-2', '1107.0881-2-52-2'], ['1107.0881-1-49-0', '1107.0881-2-53-0'], ['1107.0881-1-49-1', '1107.0881-2-53-1'], ['1107.0881-1-32-0', '1107.0881-2-35-0'], ['1107.0881-1-32-1', '1107.0881-2-35-1'], ['1107.0881-1-32-2', '1107.0881-2-35-2'], ['1107.0881-1-32-3', '1107.0881-2-35-3'], ['1107.0881-1-32-4', '1107.0881-2-35-4'], ['1107.0881-1-23-0', '1107.0881-2-24-0'], ['1107.0881-1-23-2', '1107.0881-2-24-2'], ['1107.0881-1-23-3', '1107.0881-2-24-3'], ['1107.0881-1-0-0', '1107.0881-2-0-0'], ['1107.0881-1-0-1', '1107.0881-2-0-1'], ['1107.0881-1-0-2', '1107.0881-2-0-2'], ['1107.0881-1-0-3', '1107.0881-2-0-3'], ['1107.0881-1-15-0', '1107.0881-2-16-0'], ['1107.0881-1-15-2', '1107.0881-2-16-2'], ['1107.0881-1-15-3', '1107.0881-2-16-3'], ['1107.0881-1-15-4', '1107.0881-2-16-4'], ['1107.0881-1-42-1', '1107.0881-2-46-0'], ['1107.0881-1-42-4', '1107.0881-2-46-2'], ['1107.0881-1-42-5', '1107.0881-2-46-3'], ['1107.0881-1-42-7', '1107.0881-2-46-5'], ['1107.0881-1-42-8', '1107.0881-2-46-6'], ['1107.0881-1-42-9', '1107.0881-2-46-7'], ['1107.0881-1-42-10', '1107.0881-2-46-8'], ['1107.0881-1-42-12', '1107.0881-2-46-9'], ['1107.0881-1-9-0', '1107.0881-2-9-0'], ['1107.0881-1-9-1', '1107.0881-2-9-1'], ['1107.0881-1-9-2', '1107.0881-2-9-2'], ['1107.0881-1-9-4', '1107.0881-2-9-4'], ['1107.0881-1-9-5', '1107.0881-2-9-5'], ['1107.0881-1-9-6', '1107.0881-2-9-6'], ['1107.0881-1-9-7', '1107.0881-2-9-7'], ['1107.0881-1-9-9', '1107.0881-2-10-0'], ['1107.0881-1-9-10', '1107.0881-2-10-1'], ['1107.0881-1-9-11', '1107.0881-2-10-2'], ['1107.0881-1-9-12', '1107.0881-2-10-3'], ['1107.0881-1-9-13', '1107.0881-2-10-4'], ['1107.0881-1-9-14', '1107.0881-2-10-5'], ['1107.0881-1-9-15', '1107.0881-2-10-6']]

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[]

[['1107.0881-1-43-0', '1107.0881-2-47-0'], ['1107.0881-1-43-1', '1107.0881-2-47-1'], ['1107.0881-1-42-0', '1107.0881-2-45-0'], ['1107.0881-1-9-8', '1107.0881-2-9-8']]

[]

['1107.0881-1-42-3']

{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1107.0881

0903.0904

{'0903.0904-1-0-0': 'We construct a class of exact solutions of the noncommutative vacuum Einstein field equations, which are noncommutative analogues of the plane-fronted gravitational waves in classical gravity.', '0903.0904-1-1-0': '# Introduction', '0903.0904-1-2-0': 'Isometric embeddings [CITATION] of spacetime into pseudo-Euclidean spaces provide a mechanism for investigating geometric symmetries of the spacetime in classical gravity.', '0903.0904-1-2-1': 'Indeed symmetries of the pseudo-Euclidean space that preserves the embedding give rise to spacetime symmetries automatically.', '0903.0904-1-2-2': 'There was also a proposal (see, e.g. [CITATION]) in the late 1960s that the origin of gauge symmetries in particle physics might be understood from spacetime embeddings.', '0903.0904-1-3-0': 'Recently a formalism for spacetime quantisation is proposed [CITATION] by using isometric embeddings.', '0903.0904-1-3-1': 'In this formalism, one first finds a global embedding of a spacetime into some pseudo-Euclidean space, whose existence is guaranteed by theorems of Nash, Clarke and Greene [CITATION].', '0903.0904-1-3-2': 'Then one quantises the spacetime following the strategy of deformation quantisation [CITATION] by deforming [CITATION] the algebra of functions in the pseudo-Euclidean space to a noncommutative associate algebra known as the Moyal algebra.', '0903.0904-1-3-3': 'Through this mechanism, classical spacetime metrics will deform to "quantum" noncommutative metrics which acquire quantum fluctuations.', '0903.0904-1-3-4': 'In particular, certain anti-symmetric components arise in the deformed metrics, which involve the Planck constant and vanish in the classical limit.', '0903.0904-1-4-0': 'The theory of [CITATION] retains the notions of connections and curvatures in the noncommutative setting in a mathematically consistent manner.', '0903.0904-1-4-1': 'In particular, the quantum deformed noncommutative Ricci curvatures can be defined in a unique way, and noncommutative Einstein field equations make sense in this spacetime quantisation scheme [CITATION].', '0903.0904-1-4-2': 'An important problem, which has not been studied up to now, is to find exact solutions of the noncommutative Einstein field equations.', '0903.0904-1-5-0': 'In this letter we construct a class of exact solutions of the noncommutative vacuum Einstein field equations.', '0903.0904-1-5-1': 'These solutions are quantum deformations of the plane-fronted gravitational waves.', '0903.0904-1-6-0': 'The plane-fronted gravitational waves in classical general relativity are an important family of exact solutions of the Einstein field equations.', '0903.0904-1-6-1': 'They were first studied by Brinkmann in 1925 [CITATION] and have since been rediscovered several times (e.g. [CITATION]).', '0903.0904-1-6-2': "These solutions provide the gravitational field analogue of electromagnetic plane wave solutions in Maxwell's theory and model spacetimes moving at the speed of light and radiating energy.", '0903.0904-1-7-0': 'The quantum deformed plane-fronted gravitational waves discovered here should be interpreted similarly.', '0903.0904-1-7-1': 'We expect them to have a role to play in future investigations of quantum gravity.', '0903.0904-1-8-0': 'We should point out that in general the quantum deformed metrics of most classical spacetimes, e.g., the Schwarzschild spacetime, satisfy the field equations only up to some order of the Planck constant [CITATION].', '0903.0904-1-8-1': 'It is truly remarkable that the quantum deformed plane-fronted gravitational waves solve the noncommutative vacuum Einstein field equations exactly.', '0903.0904-1-9-0': '# Noncommutative differential geometry', '0903.0904-1-10-0': 'In this section, we give a brief introduction to the local noncommutative differential geometry developed in [CITATION].', '0903.0904-1-11-0': 'Let [MATH] be an [MATH]-dimensional pseudo-Riemannian manifold whose metric [MATH] has signature [MATH].', '0903.0904-1-11-1': "By results of [CITATION], which extends Nash's isometric embedding [CITATION] to pesudo-Riemannian manifolds, there exist positive integers [MATH], [MATH], [MATH] and a set of smooth function [MATH] on [MATH] such that [EQUATION]", '0903.0904-1-11-2': 'Let [MATH] be a diagonal [MATH] matrix with [MATH] of the diagonal entries being [MATH], and [MATH] of them being [MATH].', '0903.0904-1-11-3': 'Denote by [MATH] the embedding.', '0903.0904-1-12-0': 'Let [MATH] be a coordinate chart of [MATH] with natural coordinates [MATH].', '0903.0904-1-12-1': 'Let [MATH] be a real indeterminate, and denote by [MATH] the ring of formal power series in [MATH].', '0903.0904-1-12-2': 'Let [MATH] be the set of formal power series in [MATH] with coefficients being real smooth functions on [MATH].', '0903.0904-1-12-3': 'Namely, every element of [MATH] is of the form [MATH] where [MATH] are smooth functions on [MATH].', '0903.0904-1-12-4': 'Then [MATH] is an [MATH]-module.', '0903.0904-1-13-0': 'Given any two smooth functions [MATH] and [MATH] on [MATH], we denote by [MATH] the usual point-wise product of the two functions.', '0903.0904-1-13-1': 'We also define their star-product (or more precisely, Moyal product) [MATH] on [MATH] by [EQUATION] where [MATH], and [MATH] is a constant skew symmetric [MATH] matrix.', '0903.0904-1-13-2': 'It is well known that such a multiplication is associative.', '0903.0904-1-13-3': 'Thus [MATH] equipped with the Moyal product is a deformation of the algebra of functions on [MATH] in the sense of [CITATION].', '0903.0904-1-13-4': 'Since [MATH] is constant, the Leibniz rule remains valid in the present case: [EQUATION].', '0903.0904-1-14-0': 'In noncommutative geometry [CITATION], the associative algebra [MATH] is regarded as defining some quantum deformation of the region [MATH], and finitely generated projective modules over [MATH] are regarded as (spaces of sections of) noncommutative vector bundles on the quantum deformation of [MATH] (defined by the noncommutative algebra [MATH]).', '0903.0904-1-14-1': 'Given an integer [MATH], we let [MATH] (resp. [MATH]) be the set of [MATH]-tuples with entries in [MATH] written as rows (resp. columns).', '0903.0904-1-14-2': 'We shall regard [MATH] (resp. [MATH]) as a left (resp. right) [MATH]-module with the action defined by multiplication from the left (resp. right).', '0903.0904-1-14-3': 'More explicitly, for [MATH], and [MATH], we have [MATH].', '0903.0904-1-14-4': 'Similarly for [MATH], we have [MATH].', '0903.0904-1-14-5': 'Let [MATH] be the set of [MATH]-matrices with entries in [MATH].', '0903.0904-1-14-6': 'We define matrix multiplication in the usual way but by using the Moyal product for products of matrix entries, and still denote the corresponding matrix multiplication by [MATH].', '0903.0904-1-14-7': 'Now for [MATH] and [MATH], we have [MATH] with [MATH].', '0903.0904-1-14-8': 'Then [MATH] is an [MATH]-algebra, which has a natural left (resp. right) action on [MATH] (resp. [MATH]).', '0903.0904-1-15-0': 'A finitely generated projective left (reps. right) [MATH]-module is isomorphic to some direct summand of [MATH] (resp. [MATH]) for some [MATH].', '0903.0904-1-15-1': 'If [MATH] satisfies the condition [MATH], that is, it is an idempotent, then', '0903.0904-1-16-0': '[EQUATION] are respectively projective left and right [MATH]-modules.', '0903.0904-1-16-1': 'Furthermore, every projective left (right) [MATH]-module is isomorphic to an [MATH] (resp. [MATH]) constructed this way by using some idempotent [MATH].', '0903.0904-1-17-0': 'Given [MATH] in [MATH], we define an [MATH] matrix [MATH] with entries [EQUATION].', '0903.0904-1-17-1': 'We call [MATH] a noncommutative space embedded in [MATH] if the matrix [MATH] is invertible over [MATH].', '0903.0904-1-17-2': 'Denote its inverse matrix by [MATH].', '0903.0904-1-17-3': 'Let [EQUATION] for [MATH], where [MATH] denotes the transpose of [MATH].', '0903.0904-1-17-4': 'Then the matrix [MATH] given by [EQUATION] is an idempotent, which satisfies [MATH] for all [MATH].', '0903.0904-1-17-5': 'The projective module associated to [MATH] is the quantised tangent bundle.', '0903.0904-1-18-0': 'The matrix [MATH] yields a metric on the embedded noncommutative surface [MATH] over [MATH].', '0903.0904-1-18-1': 'We shall call [MATH] the local quantum deformation of spacetime metric [MATH] over [MATH].', '0903.0904-1-18-2': 'As the noncommutative geometries on the left module [MATH] and right module [MATH] are equivalent, we need only to investigate [MATH].', '0903.0904-1-18-3': 'Let [MATH] be the canonical connections on [MATH].', '0903.0904-1-18-4': 'The covariant derivative on the noncommutative bundle [MATH] is given by [EQUATION].', '0903.0904-1-18-5': 'The curvature of [MATH] associated with the connection [MATH] is given by [EQUATION] where [MATH] is the commutator.', '0903.0904-1-18-6': 'The component of the curvature [MATH] is defined as [EQUATION].', '0903.0904-1-18-7': 'Denote [MATH], [MATH], where [EQUATION]', '0903.0904-1-18-8': 'It can be shown by some lengthy calculations that [EQUATION]', '0903.0904-1-18-9': 'A new feature is that there are two consistent ways to contract [MATH], leading to two distinct noncommutative Ricci curvatures [MATH] and [MATH] respectively defined by [EQUATION]', '0903.0904-1-18-10': 'Both [MATH] and [MATH] reduce to the usual Ricci curvature in the commutative limit.', '0903.0904-1-18-11': 'Also, it was shown [CITATION] that the noncommutative scalar curvatures obtained from the two noncommutative Ricci curvatures are the same, that is, [MATH].', '0903.0904-1-19-0': 'The noncommutative vacuum Einstein field equations are given by [EQUATION]', '0903.0904-1-19-1': 'The main aim of this note is to find a class of exact solutions of the equations.', '0903.0904-1-20-0': '# Noncommutative plane-fronted gravitational waves', '0903.0904-1-21-0': 'In this section, we shall show that quantum deformed plane-fronted gravitational waves are exact solutions of the noncommutative vacuum Einstein field equations.', '0903.0904-1-22-0': '## Plane-fronted gravitational waves in classical gravity', '0903.0904-1-23-0': 'According to Brinkmann [CITATION], the plane-fronted gravitational waves have the following metric [EQUATION] where [MATH] is a smooth function.', '0903.0904-1-23-1': 'They satisfy the vacuum Einstein field equations if and only if [EQUATION]', '0903.0904-1-23-2': 'The plane-fronted gravitational waves have numerous striking properties.', '0903.0904-1-23-3': 'For instance, if two plane-fronted gravitational waves with functions [MATH], [MATH] respectively satisfy the vacuum Einstein field equations, then the metric ([REF]) with [MATH] is also a vacuum solution.', '0903.0904-1-23-4': 'This is a rather nontrivial fact since the Einstein field equations are highly nonlinear, and it is extremely rare to have this additivity property.', '0903.0904-1-23-5': 'Moreover, Penrose [CITATION] observed that near a null geodesic, every spacetime can be blown up so that the given null geodesic becomes the covariantly constant null geodesic congruence of a plane wave.', '0903.0904-1-24-0': 'The plane-fronted gravitational waves can be embedded into [MATH] via the embedding [CITATION] [EQUATION] and the metric ([REF]) can be induced from [EQUATION]', '0903.0904-1-25-0': '## Noncommutative plane-fronted gravitational waves', '0903.0904-1-26-0': 'Now we turn to quantum deformation of the plane-fronted gravitational waves.', '0903.0904-1-26-1': 'Let [MATH] be an arbitrary constant skew symmetric [MATH] matrix, and endow the space of functions of the variables [MATH] with the Moyal product defined with respect to [MATH].', '0903.0904-1-26-2': 'We denote the resulting noncommutative algebra by [MATH].', '0903.0904-1-27-0': 'Now we consider a noncommutative space [MATH] embedded in [MATH] by a map of the form [REF] but with the component functions interpreted as belonging to [MATH].', '0903.0904-1-28-0': 'A straightforward but very lengthy computation yields the following noncommutative metric for [MATH]: [EQUATION]', '0903.0904-1-28-1': 'Further tedious computation give us the following inverse metric: [EQUATION] with [EQUATION]', '0903.0904-1-28-2': 'Using these formulae we can compute [MATH] and [MATH], the nonzero components of which are given by [EQUATION]', '0903.0904-1-28-3': 'Remarkably, explicit formulae for curvatures can also be obtained, even though the noncommutativity of the [MATH]-product complicates the computations enormously.', '0903.0904-1-28-4': 'As an example, we give the computation of [MATH] here: [EQUATION]', '0903.0904-1-28-5': 'We find the other components of the curvature in the same way.', '0903.0904-1-28-6': 'We have [EQUATION]', '0903.0904-1-28-7': 'Thus the nonzero components of [MATH] are [EQUATION]', '0903.0904-1-28-8': 'We finally obtain the nonzero components of the Ricci curvature [EQUATION]', '0903.0904-1-28-9': 'Therefore the quantum deformed plane-fronted gravitational waves satisfy the noncommutative vacuum Einstein field equations ([REF]) if and only if ([REF]) holds.', '0903.0904-1-29-0': 'It is clear from [REF] that quantum deformed plane-fronted gravitational waves also satisfy the additivity property.', '0903.0904-1-30-0': '# Discussions', '0903.0904-1-31-0': 'Working within the framework of the noncommutative Riemannian geometry of [CITATION], we have obtained in this paper exact solutions of the quantum noncommutative vacuum Einstein field equations, which noncommutative analogues of the plane-fronted gravitational waves in classical general relativity [CITATION].', '0903.0904-1-31-1': 'In the classical setting, the plane-fronted gravitational waves model spacetimes moving at the speed of light and radiating energy.', '0903.0904-1-31-2': 'We expect the noncommutative plane-fronted waves to play a similar role in the noncommutative case.', '0903.0904-1-31-3': 'It will be very interesting to investigate the physical applications of these solutions.', '0903.0904-1-32-0': 'We should note that a large volume of work on noncommutative relativity has been carried out by many researchers in recent years.', '0903.0904-1-32-1': 'We mention in particular that several tentative proposals for a theory of noncommutative relativity exist in the literature [CITATION] beside that of [CITATION].', '0903.0904-1-32-2': 'In [CITATION] general relativity on a noncommutative spacetime is treated as a nonconmmutative gauge theory of a deformed Lorentzian algebra analogous to the classical picture of Utiyama and Kibble, while in [CITATION] noncommutativity was introduced into gravity by deforming the diffeomorphism algebra.', '0903.0904-1-32-3': 'Very recently, the authors of [CITATION] explored a possible moving frame formalism for a noncommutative geometry on the Moyal space as the first step toward setting up a framework for noncommutative general relativity.', '0903.0904-1-32-4': 'Much work has also been done to investigate implications of spacetime noncommutativity to black hole physics.', '0903.0904-1-33-0': 'However, no exact solutions of noncommutative Einstein field equations have been investigated in the literature as far as we know.', '0903.0904-1-33-1': 'Presumably this is partly due to the lack of mathematical rigour in many of the proposed theories of noncommutative relativity.', '0903.0904-1-33-2': 'Also the mathematical complexities introduced by spacetime noncommutativity makes it extremely difficult to study exact solutions of the field equations.', '0903.0904-1-34-0': 'It is quite striking that the quantum noncommutative Einstein field equations [CITATION], complicated as they are, actually admit classes of exact solutions.', '0903.0904-1-34-1': 'This may be a good indication that the noncommutative Riemannian geometry proposed in [CITATION] is a promising theory mathematically.', '0903.0904-1-34-2': 'We hope that it will develop into a coherent framework for studying the structure of spacetime at the Planck scale.'}

{'0903.0904-2-0-0': 'We construct a class of exact solutions of the noncommutative Einstein field equations in the vacuum, which are noncommutative analogues of the plane-fronted gravitational waves in classical gravity.', '0903.0904-2-1-0': '# Introduction', '0903.0904-2-2-0': 'There have been intensive research activities on noncommutative relativity in recent years.', '0903.0904-2-2-1': 'In particular, several tentative proposals [CITATION] for a theory of noncommutative relativity have been put forward.', '0903.0904-2-2-2': 'In [CITATION] noncommutativity was introduced into gravity by deforming the diffeomorphism algebra.', '0903.0904-2-2-3': 'In [CITATION] general relativity on a noncommutative spacetime is treated as a noncommutative gauge theory.', '0903.0904-2-2-4': 'Very recently, the authors of [CITATION] explored a possible moving frame formalism for a noncommutative geometry on the Moyal space as the first step toward setting up a framework for noncommutative general relativity.', '0903.0904-2-2-5': 'Much work has also been done to investigate implications of spacetime noncommutativity to black hole physics [CITATION].', '0903.0904-2-3-0': 'In the papers [CITATION], a formalism for spacetime quantisation was proposed, which made use of isometric embeddings [CITATION] of spacetime into pseudo-Euclidean spaces.', '0903.0904-2-3-1': 'In this formalism, one first finds a global embedding of a spacetime into some pseudo-Euclidean space, whose existence is guaranteed by theorems of Nash, Clarke and Greene [CITATION].', '0903.0904-2-3-2': 'Then one quantises the spacetime following the strategy of deformation quantisation [CITATION] by deforming [CITATION] the algebra of functions in the pseudo-Euclidean space to a noncommutative associate algebra known as the Moyal algebra.', '0903.0904-2-3-3': 'Through this mechanism, classical spacetime metrics will deform to "quantum" noncommutative metrics which acquire quantum fluctuations.', '0903.0904-2-3-4': 'In particular, certain anti-symmetric components arise in the deformed metrics, which involve the Planck constant and vanish in the classical limit.', '0903.0904-2-4-0': 'The theory of [CITATION] can be formulated in an intrinsic way, free of the use of emebeddings.', '0903.0904-2-4-1': 'This theory retains the notions of connections and curvatures in the noncommutative setting in a mathematically consistent manner.', '0903.0904-2-4-2': 'In particular, the quantum deformed noncommutative Ricci curvatures can be defined in a unique way.', '0903.0904-2-4-3': 'This enabled one to develop a noncommutative analogue of the Einstein field equations [CITATION].', '0903.0904-2-5-0': 'It is important to solve the noncommutative Einstein field equations to construct quantum noncommutative spacetimes.', '0903.0904-2-5-1': 'In [CITATION], we obtained noncommutative analogues of Schwarzschild spacetime and de-Sitter Schwarzschild spacetime, which are approximate solutions of the noncommutative Einstein field equations exact to the first order of the deformation parameter.', '0903.0904-2-5-2': 'Quantum corrections to the area law of black hole entropy was observed for these solutions.', '0903.0904-2-6-0': 'In this letter we construct a class of exact solutions of the noncommutative Einstein field equations in the vacuum.', '0903.0904-2-6-1': 'These solutions are quantum deformations of the plane-fronted gravitational waves first constructed by Brinkmann in 1925 [CITATION] and have since been rediscovered several times (e.g. [CITATION]).', '0903.0904-2-6-2': 'Our solutions are noncommutative gravitational analogues of electromagnetic plane waves.', '0903.0904-2-6-3': 'We expect them to have an important role to play in future investigations of quantum gravity.', '0903.0904-2-7-0': 'Fuzzy pp-waves were constructed by Madore, Maceda and Robinson in [CITATION].', '0903.0904-2-7-1': 'These authors start with a given classical solution of the Einstein field equation in the vacuum and construct a noncommutative algebra and a differential calculus which supported the metric.', '0903.0904-2-7-2': 'The corresponding noncommutative scalar curvature was however nonzero.', '0903.0904-2-7-3': 'In general the quantum deformed metrics of most classical spacetimes satisfy the field equations only up to some order of the Planck constant.', '0903.0904-2-8-0': 'Exact (that is, not approximate) solutions of noncommutative Einstein field equations do not seem to have been investigated much in the literature.', '0903.0904-2-8-1': 'Presumably this is partly due to the fact that many of the proposals of noncommutative relativity are based on intuition.', '0903.0904-2-8-2': 'Much work has been done to investigate corrections to physically relevant quantities to the first order in the deformation parameter within the frameworks of the various proposals.', '0903.0904-2-8-3': 'However, to go beyond the first order approximation, one will need a mathematically more rigorous theory.', '0903.0904-2-8-4': 'In particular, we can only investigate exact solutions when precisely formulated noncommutative Einstein field equations are given.', '0903.0904-2-9-0': 'We mention that even within the mathematically rigorous formulations like those of [CITATION] and [CITATION], the mathematical complexities introduced by spacetime noncommutativity makes it extremely difficult to study exact solutions of the noncommutative Einstein field equations.', '0903.0904-2-9-1': 'Thus it is quite remarkable that the quantum deformed plane-fronted gravitational waves constructed here solve the noncommutative vacuum Einstein field equations exactly.', '0903.0904-2-10-0': '# Noncommutative Einstein equations', '0903.0904-2-11-0': 'In order to set up the noncommutative Einstein equations, we need to have a noncommutative differential geometry which retains the notions of metric, connection and curvature.', '0903.0904-2-11-1': 'Such a theory was constructed in [CITATION].', '0903.0904-2-11-2': 'We describe the theory very briefly here; details can be found in [CITATION].', '0903.0904-2-12-0': '## A local noncommutative differential geometry', '0903.0904-2-13-0': 'Let [MATH] be a domain in [MATH] with natural coordinates [MATH].', '0903.0904-2-13-1': 'Let [MATH] be a real indeterminate, and denote by [MATH] the ring of formal power series in [MATH].', '0903.0904-2-13-2': 'Let [MATH] be the set of formal power series in [MATH] with coefficients being real smooth functions on [MATH].', '0903.0904-2-13-3': 'Namely, every element of [MATH] is of the form [MATH] where [MATH] are smooth functions on [MATH].', '0903.0904-2-13-4': 'Then [MATH] is an [MATH]-module.', '0903.0904-2-14-0': 'Given any two smooth functions [MATH] and [MATH] on [MATH], we denote by [MATH] the usual point-wise product of the two functions.', '0903.0904-2-14-1': 'We also define their star-product (or more precisely, Moyal product) [MATH] on [MATH] by [EQUATION] where [MATH], and [MATH] is a constant skew symmetric [MATH] matrix.', '0903.0904-2-14-2': 'It is well known that such a multiplication is associative.', '0903.0904-2-14-3': 'Thus [MATH] equipped with the Moyal product is a deformation of the algebra of functions on [MATH] in the sense of [CITATION].', '0903.0904-2-14-4': 'Since [MATH] is constant, the Leibniz rule remains valid in the present case: [EQUATION].', '0903.0904-2-15-0': 'In noncommutative geometry [CITATION], the associative algebra [MATH] is regarded as defining some quantum deformation of the region [MATH], and finitely generated projective modules over [MATH] are regarded as (spaces of sections of) noncommutative vector bundles on the quantum deformation of [MATH] (defined by the noncommutative algebra [MATH]).', '0903.0904-2-15-1': 'Given an integer [MATH], we let [MATH] (resp. [MATH]) be the set of [MATH]-tuples with entries in [MATH] written as rows (resp. columns).', '0903.0904-2-15-2': 'We shall regard [MATH] (resp. [MATH]) as a left (resp. right) [MATH]-module with the action defined by multiplication from the left (resp. right).', '0903.0904-2-15-3': 'More explicitly, for [MATH], and [MATH], we have [MATH].', '0903.0904-2-15-4': 'Similarly for [MATH], we have [MATH].', '0903.0904-2-15-5': 'Let [MATH] be the set of [MATH]-matrices with entries in [MATH].', '0903.0904-2-15-6': 'We define matrix multiplication in the usual way but by using the Moyal product for products of matrix entries, and still denote the corresponding matrix multiplication by [MATH].', '0903.0904-2-15-7': 'Now for [MATH] and [MATH], we have [MATH] with [MATH].', '0903.0904-2-15-8': 'Then [MATH] is an [MATH]-algebra, which has a natural left (resp. right) action on [MATH] (resp. [MATH]).', '0903.0904-2-16-0': 'A finitely generated projective left (reps. right) [MATH]-module is isomorphic to some direct summand of [MATH] (resp. [MATH]) for some [MATH].', '0903.0904-2-16-1': 'If [MATH] satisfies the condition [MATH], that is, it is an idempotent, then', '0903.0904-2-17-0': '[EQUATION] are respectively projective left and right [MATH]-modules.', '0903.0904-2-17-1': 'Furthermore, every projective left (right) [MATH]-module is isomorphic to an [MATH] (resp. [MATH]) constructed this way by using some idempotent [MATH].', '0903.0904-2-18-0': 'As the noncommutative geometries on the left module [MATH] and right module [MATH] are equivalent, we need only to investigate [MATH].', '0903.0904-2-18-1': 'Let [EQUATION] be the canonical connections on [MATH].', '0903.0904-2-18-2': 'The covariant derivative on the noncommutative bundle [MATH] is given by [EQUATION].', '0903.0904-2-18-3': 'The curvature of [MATH] associated with the connection [MATH] is given by [EQUATION] where [MATH] is the commutator.', '0903.0904-2-19-0': 'Let [MATH] be a diagonal [MATH] matrix with [MATH] of the diagonal entries being [MATH], and [MATH] of them being [MATH] for some [MATH].', '0903.0904-2-19-1': 'The fibre metric is the [MATH]-bilinear map [EQUATION] where for any [MATH] and [MATH], we have [MATH].', '0903.0904-2-19-2': 'The metric compatibility of the connection and also the Bianchi identities for the Riemannian curvature were discussed in [CITATION].', '0903.0904-2-20-0': 'In certain situations, we may regard [MATH] and [MATH] as noncommutative tangent bundles of some noncommutative space.', '0903.0904-2-20-1': 'Consider the case when there exists a finite set of [MATH]-generators [MATH]) of [MATH] with the following properties.', '0903.0904-2-20-2': 'The column vectors [MATH] (where [MATH] are the transposes of [MATH]) generate [MATH], and the [MATH] matrix [MATH] with [EQUATION] is invertible over [MATH].', '0903.0904-2-20-3': 'Then the idempotent [MATH] is given by [MATH].', '0903.0904-2-20-4': 'In this case, we call the matrix [MATH] the metric.', '0903.0904-2-21-0': 'We may consider the components of the curvature [MATH]: [EQUATION] where [MATH], for [MATH].', '0903.0904-2-22-0': 'A new feature is that there are two consistent ways to contract [MATH], leading to two distinct noncommutative Ricci curvatures [MATH] and [MATH] respectively defined by [EQUATION]', '0903.0904-2-22-1': 'Correspondingly there are two scalar curvatures [MATH] and [MATH].', '0903.0904-2-22-2': 'Both [MATH] and [MATH] reduce to the usual Ricci curvature in the commutative limit.', '0903.0904-2-23-0': 'We now state the noncommutative Einstein field equations in the vacuum (for unknowns [MATH]) in this theory, which are given by [EQUATION]', '0903.0904-2-23-1': 'The aim of this note is to construct exact solutions of the equations.', '0903.0904-2-24-0': '## Embedded noncommutative spaces', '0903.0904-2-25-0': 'Embedded noncommutative spaces are elementary and manifestly consistent realisations of the local differential geometry discussed above.', '0903.0904-2-25-1': 'Given [MATH] in [MATH], we define an [MATH] matrix [MATH] with entries [EQUATION] where [MATH] are the matrix elements of the diagonal matrix [MATH].', '0903.0904-2-26-0': 'The matrix [MATH] is invertible over [MATH] if and only if [MATH] is invertible.', '0903.0904-2-26-1': 'We denote the inverse matrix of [MATH] by [MATH].', '0903.0904-2-26-2': 'In this case, [MATH] reduces to an embedded space with metric [MATH] in the commutative limit with [MATH].', '0903.0904-2-26-3': 'Therefore, we call [MATH] a noncommutative space embedded in [MATH] in analogy to the classical case.', '0903.0904-2-27-0': 'Let [MATH] for [MATH].', '0903.0904-2-27-1': 'Then [MATH].', '0903.0904-2-27-2': 'The matrix [EQUATION] is an idempotent and satisfies [MATH] and [MATH] for all [MATH].', '0903.0904-2-27-3': 'The left (resp. right) projective [MATH]-module [MATH] (resp. [MATH]) associated to [MATH] is the quantised left (resp. right) tangent bundle of the embedded noncommutative space.', '0903.0904-2-27-4': 'It is easy to show that the metric defined by the [MATH]-bilinear map [REF] agrees with [REF] in the present case.', '0903.0904-2-28-0': 'We may cast the formulation of the embedded noncommutative space into a more familiar form.', '0903.0904-2-28-1': 'The connection is now given by [EQUATION] where [MATH] can be explicitly described in the following way.', '0903.0904-2-28-2': 'Let [MATH].', '0903.0904-2-28-3': 'We have [EQUATION] with [EQUATION] where the object [MATH] is called the noncommutative torsion in [CITATION].', '0903.0904-2-28-4': 'The curvatures are given by [EQUATION]', '0903.0904-2-28-5': 'It was shown in [CITATION] that the two noncommutative scalar curvatures [MATH] and [MATH] coincide in the present case.', '0903.0904-2-29-0': '# Exact solutions of noncommutative Einstein field equations', '0903.0904-2-30-0': 'We shall now construct a class of exact solutions of the noncommutative vacuum Einstein field equations.', '0903.0904-2-30-1': 'The solutions are quantum deformed analogues of plane-fronted gravitational waves [CITATION].', '0903.0904-2-31-0': 'Let [MATH] be an arbitrary constant skew symmetric [MATH] matrix, and endow the space of functions of the variables [MATH] with the Moyal product defined with respect to [MATH].', '0903.0904-2-31-1': 'We denote the resulting noncommutative algebra by [MATH].', '0903.0904-2-32-0': 'Now we consider a noncommutative space [MATH] embedded in [MATH] by a map of the form [EQUATION] where, needless to say, the component functions are elements of [MATH].', '0903.0904-2-33-0': 'Let us take [MATH], and construct the noncommutative metric [MATH] by using the formula [REF] for this embedded noncommutative space.', '0903.0904-2-33-1': 'A very lengthy calculation yields the following result: [EQUATION]', '0903.0904-2-33-2': 'It is useful to note that in the classical limit with all [MATH], the metric has Minkowski signature.', '0903.0904-2-33-3': 'In fact it reduces to the matrix [MATH], which diagonalises to [MATH].', '0903.0904-2-33-4': 'Further tedious computations produce the following inverse metric: [EQUATION] with [EQUATION]', '0903.0904-2-33-5': 'Using these formulae we can compute [MATH] and [MATH], the nonzero components of which are given below: [EQUATION]', '0903.0904-2-33-6': 'Remarkably, explicit formulae for curvatures can also be obtained, even though the noncommutativity of the [MATH]-product complicates the computations enormously.', '0903.0904-2-33-7': 'As an example, we give the computation of [MATH] here: [EQUATION]', '0903.0904-2-33-8': 'The other components of the curvature can be obtained in the same way.', '0903.0904-2-33-9': 'We have [EQUATION]', '0903.0904-2-33-10': 'Thus the nonzero components of [MATH] are [EQUATION]', '0903.0904-2-33-11': 'From these formulae, we obtain the nonzero components of the Ricci curvature: [EQUATION]', '0903.0904-2-33-12': 'Thus the noncommutative vacuum Einstein field equations ([REF]) are satisfied if and only if the following equation holds: [EQUATION]', '0903.0904-2-33-13': 'Solutions of this linear equation for [MATH] exist in abundance.', '0903.0904-2-33-14': 'Each solution leads to an exact solution of the noncommutative vacuum Einstein field equations.', '0903.0904-2-33-15': 'If we set [MATH] to zero, we recover from such a solution the plane-fronted gravitational wave [CITATION] in classical general relativity.', '0903.0904-2-33-16': 'Thus we shall call such a solution of ([REF]) a plane-fronted noncommutative gravitational wave.', '0903.0904-2-34-0': 'It is clear from [REF] that plane-fronted noncommutative gravitational waves satisfy the additivity property.', '0903.0904-2-34-1': 'Explicitly, if the noncommutative metrics of [EQUATION] are plane-fronted noncommutative gravitational waves, we let [MATH], and set [EQUATION]', '0903.0904-2-34-2': 'Then the noncommutative metric of [MATH] is also a plane-fronted noncommutative gravitational wave.', '0903.0904-2-34-3': 'This is a rather nontrivial fact since the noncommutative Einstein field equations are highly nonlinear in [MATH], and it is extremely rare to have this additivity property.', '0903.0904-2-35-0': 'At this point, it is appropriate to point out that the embedding [REF] is only used as a device for constructing the metric and the connection, from which the curvatures are derived.', '0903.0904-2-35-1': 'However, we should observe the power of embeddings in solving the noncommutative Einstein field equations.', '0903.0904-2-35-2': 'Without using the embedding [REF], it would be very difficult to come up with elegant solutions like what we have obtained here.', '0903.0904-2-36-0': '# Discussions', '0903.0904-2-37-0': 'Working within the framework of the noncommutative Riemannian geometry of [CITATION], we have obtained in this paper exact solutions of the quantum noncommutative vacuum Einstein field equations, which are noncommutative analogues of the plane-fronted gravitational waves in classical general relativity [CITATION].', '0903.0904-2-38-0': 'In the classical setting, the plane-fronted gravitational waves model spacetimes moving at the speed of light and radiating energy.', '0903.0904-2-38-1': 'Furthermore, Penrose [CITATION] observed that near a null geodesic, every spacetime can be blown up so that the given null geodesic becomes the covariantly constant null geodesic congruence of a plane wave.', '0903.0904-2-38-2': 'We expect the plane-fronted noncommutative gravitational waves to play a similar role.', '0903.0904-2-38-3': 'It will be very interesting to investigate the physical applications of these solutions.', '0903.0904-2-39-0': 'It is quite striking that the quantum noncommutative Einstein field equations [CITATION], complicated as they are, admit explicit exact solutions as simple as the ones constructed here.', '0903.0904-2-39-1': 'This indicates the promise of the theory of noncommutative Riemannian geometry proposed in [CITATION].', '0903.0904-2-39-2': 'We hope that the theory will develop into a coherent framework for studying the structure of spacetime at the Planck scale.'}

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[]

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{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/0903.0904

1209.4282

{'1209.4282-1-0-0': 'Nuclear-spin-dependent (NSD) parity violating (PV) effects can be strongly enhanced in diatomic molecules containing heavy atoms.', '1209.4282-1-0-1': 'Future measurements are anticipated to provide nuclear anapole moments and strength constants for PV nuclear forces.', '1209.4282-1-0-2': 'In light molecules, the NSD electroweak electron-nucleus interaction may also be detected.', '1209.4282-1-0-3': 'Here we calculate NSD PV effects for molecular ions.', '1209.4282-1-0-4': 'Our calculations are motivated by rapid developments in trapping techniques for such systems at low temperatures.', '1209.4282-1-1-0': 'It was previously shown that nuclear spin-dependent (NSD) parity violation (PV) effects are enhanced by a factor of [MATH] in diatomic molecules with [MATH] and [MATH] electronic states due to the mixing of close rotational states of opposite parity ([MATH]-doublet for [MATH]) [CITATION].', '1209.4282-1-1-1': 'DeMille and co-workers suggested measuring NSD PV effects by using neutral diatomic molecules in a Stark interference experiment to determine the mixing between opposite-parity rotational/hyperfine levels [CITATION].', '1209.4282-1-1-2': 'Another proposal was published in Ref. [CITATION], and corresponding experiments have already started.', '1209.4282-1-1-3': 'Recently, it was demonstrated that positive molecular ions may be easily trapped and studied at low temperatures [CITATION], which motivated us to perform calculations on NSD PV effects in such systems.', '1209.4282-1-2-0': 'The term in the Hamiltonian operator arising from the NSD parity violating electron-nucleus interaction is [EQUATION]', '1209.4282-1-2-1': 'Here, and throughout the text, we use atomic units.', '1209.4282-1-2-2': 'In Eq. ([REF]), [MATH] is the dimensionless strength constant, [MATH] a.u. is the Fermi constant, [MATH] is a vector comprised of the conventional Dirac matrices, [MATH] is the nuclear spin, [MATH] is the displacement of the valence electron from the nucleus, and [MATH] is the (normalized) nuclear density.', '1209.4282-1-2-3': 'There are three sources for this interaction: the first contribution arises from the electroweak neutral coupling between electron vector and nucleon axial-vector currents ([MATH]) [CITATION].', '1209.4282-1-2-4': 'The second contribution comes from the nuclear-spin-independent weak interaction combined with the hyperfine interaction [CITATION].', '1209.4282-1-2-5': 'Finally, the nuclear anapole moment contribution, which scales with the number of nucleons [MATH] as [MATH], becomes the dominant contribution in spin-dependent atomic PV effects for a sufficiently large nuclear charge [MATH] [CITATION].', '1209.4282-1-3-0': 'The anapole moment was first predicted by Zeldovich [CITATION] in 1958 as a new parity ([MATH]) violating and time ([MATH]) reversal conserving moment of an elementary particle.', '1209.4282-1-3-1': 'It appears in the second-order multipole expansion of the magnetic vector-potential simultaneously with the [MATH]- and [MATH]- violating magnetic quadrupole moment [CITATION].', '1209.4282-1-3-2': 'The nuclear anapole moment was experimentally discovered in the [MATH]Cs atom in 1997 [CITATION] following a proposal by Flambaum and Khriplovich [CITATION], who showed that the nuclear anapole provides the dominant contribution to the nuclear-spin-dependent parity violating effect in atoms and molecules.', '1209.4282-1-4-0': 'The nuclear anapole requires nuclear spin [MATH] and in a simple valence model has the following value [CITATION], [EQUATION]', '1209.4282-1-4-1': 'Here, [MATH], [MATH] is the orbital angular momentum of the external unpaired nucleon [MATH]; [MATH], [MATH].', '1209.4282-1-4-2': 'Theoretical estimates give the strength constant for nucleon-nucleus weak potential [MATH] for a proton and [MATH] for a neutron [CITATION].', '1209.4282-1-4-3': 'The aim of anapole measurements is to provide accurate values for these constants, thus obtaining important information about hadronic weak coupling.', '1209.4282-1-5-0': 'In a recent paper [CITATION] we presented Dirac Hartree-Fock and relativistic density-functional calculations of the electronic [MATH] factor of the diatomic group-2 and -12 fluorides and a number of other diatomic compounds.', '1209.4282-1-5-1': 'Here, we investigate the nuclear spin-dependent parity violation effects in a different type of system, i.e. positively charged dimers, as these systems have an experimental advantage of being easily trapped [CITATION].', '1209.4282-1-5-2': 'We use the combination of methods presented in Ref. [CITATION] to calculate the [MATH] factors of positively ionized group-13 and group-15 fluorides ([MATH] and [MATH] ground states, respectively), and a number of other positive diatomic ions having a [MATH] ground state.', '1209.4282-1-6-0': 'For [MATH] and [MATH] electronic states, the interaction ([REF]) can be replaced by the effective operator, which appears in the spin-rotational Hamiltonian [CITATION], [EQUATION] where [MATH] is the effective spin and [MATH] is the unit vector directed along the molecular axis from the heavier to the lighter nucleus.', '1209.4282-1-6-1': 'The electronic factor [MATH] is found from evaluating the matrix elements of the [MATH] operator in the molecular spinor basis [CITATION].', '1209.4282-1-6-2': 'The [MATH] and the [MATH] open-shell electronic states are two-fold degenerate, corresponding to the two possible projections of electronic angular momentum along [MATH], i.e. [MATH].', '1209.4282-1-6-3': 'When operating within this degenerate space, the operator [MATH] is equivalent to [MATH]) (Eq. ([REF])).', '1209.4282-1-6-4': 'Time-reversal symmetry ensures that only the matrix elements that are off-diagonal in [MATH] are non-vanishing.', '1209.4282-1-6-5': 'This symmetry rule is encapsulated within the effective operator [MATH] by the angular factor [MATH].', '1209.4282-1-6-6': 'Here the effective spin [MATH] generates rotations in the degenerate subspace analogously to usual spin operator [MATH] in a spin-1/2 system.', '1209.4282-1-7-0': 'The calculations were carried out within the open-shell single determinant average-of-configuration Dirac-Hartree-Fock approach (DHF) [CITATION] and within the relativistic density functional theory (DFT) [CITATION], employing quaternion symmetry [CITATION].', '1209.4282-1-7-1': 'A finite nucleus, modeled by the Gaussian charge distribution was used [CITATION].', '1209.4282-1-7-2': "All the calculations were performed using the developer's version of the DIRAC10 program package [CITATION].", '1209.4282-1-8-0': 'For the lighter elements (boron to phosphorus), uncontracted aug-cc-pVTZ basis sets were used [CITATION].', '1209.4282-1-8-1': "For the rest of the atoms, we employed Faegri's dual family basis sets [CITATION].", '1209.4282-1-8-2': 'As a good description of the electronic wave function in the nuclear region is essential for obtaining reliable results for parity violating properties [CITATION], we augmented the basis sets with high exponent [MATH] and [MATH] functions, which brings about an increase of around [MATH] in the calculated values of [MATH].', '1209.4282-1-8-3': 'The basis sets were increased, both in the core and in the valence regions, to convergence with respect to the calculated [MATH] constants.', '1209.4282-1-8-4': 'The final basis sets can be found in Table [REF].', '1209.4282-1-9-0': 'Where available, experimentally determined bond distances [MATH] were used.', '1209.4282-1-9-1': 'For molecules where [MATH] is not known experimentally, we optimized the bond distance using relativistic coupled cluster theory with single, double, and perturbative triple excitations, CCSD(T) [CITATION].', '1209.4282-1-9-2': 'To reduce the computational effort, we employed an infinite order two-component relativistic Hamiltonian obtained after the Barysz-Sadlej-Snijders (BSS) transformation of the Dirac Hamiltonian in a finite basis set [CITATION].', '1209.4282-1-9-3': 'Our calculated [MATH] are typically within 0.01 AA of the experimental values, where available.', '1209.4282-1-9-4': 'The experimental/calculated equilibrium distances can be found in Table [REF].', '1209.4282-1-10-0': 'In the DFT calculations we used the Coulomb-attenuated B3LYP functional (CAMB3LYP*), the parameters of which were adjusted by Thierfelder et al. [CITATION] to reproduce the PV energy shifts obtained using coupled cluster calculations (the newly adjusted parameters are [MATH], [MATH], and [MATH]).', '1209.4282-1-11-0': 'In our previous work [CITATION] we have examined and compared various schemes for adding electron correlation to the Dirac-Hartree-Fock [MATH] values, and core-polarization contributions to the DFT results.', '1209.4282-1-11-1': 'Here, we correct the calculated DHF and DFT [MATH] for core polarization using a scaling parameter, [MATH].', '1209.4282-1-11-2': 'This parameter is obtained from atomic calculations as described in the following.', '1209.4282-1-11-3': 'The main contribution to the matrix elements of the NSD interaction for the valence molecular electrons comes from short distances around the heavy nucleus, where the total molecular potential is spherically symmetric to very high precision, and the core of the heavy atom is practically unaffected by the presence of the second atom, justifying our use of the atomic model.', '1209.4282-1-11-4': 'The molecular orbitals of the valence electron can thus be expanded in this region, using spherical harmonics centered at the heavy nucleus, [EQUATION]', '1209.4282-1-11-5': 'Only [MATH] and [MATH] terms of this expansion give significant contribution to the matrix elements of the weak interaction.', '1209.4282-1-11-6': 'These functions can be considered as states of an atomic valence electron and are calculated using standard atomic techniques in two different approximations: one that includes electron correlation and another that does not.', '1209.4282-1-12-0': 'The single electron DHF Hamiltonian is given by [EQUATION] where [MATH] and [MATH] are the Dirac matrices and [MATH] is the self-consistent DHF potential due to atomic electrons.', '1209.4282-1-13-0': 'The self-consistent DHF procedure is first performed for the closed shell ion, from which the valence electron is removed.', '1209.4282-1-13-1': 'Then the core potential [MATH] is frozen and the valence [MATH] and [MATH] states are calculated by solving the DHF equation for the valence electron, [EQUATION] where [MATH] is given by ([REF]).', '1209.4282-1-14-0': 'The core polarization can be understood as the change of the self-consistent DHF potential due to the effect of the extra term (the weak interaction operator [MATH]) in the Hamiltonian.', '1209.4282-1-14-1': 'The inclusion of the core polarization in a self-consistent way is equivalent to the random-phase approximation (RPA, see, e.g. [CITATION]).', '1209.4282-1-14-2': 'The change in the DHF potential is found by solving the RPA-type equations self-consistently for all states in the atomic core, [EQUATION]', '1209.4282-1-14-3': 'Here, [MATH] is the DHF Hamiltonian ([REF]), index [MATH] enumerates the states in the core, [MATH] is the correction to the core state [MATH] due to weak interaction [MATH], and [MATH] is the correction to the self-consistent core potential due to the change of all core functions.', '1209.4282-1-14-4': 'Once [MATH] is found, the core polarization can be included into a matrix element for valence states [MATH] and [MATH] via the redefinition of the weak interaction Hamiltonian, [EQUATION]', '1209.4282-1-14-5': 'We then obtain the scaling parameter for core-polarization effects, [MATH], from [EQUATION]', '1209.4282-1-14-6': 'It should be noted that for the positively charged group 15 fluorides we have only calculated the correlations between the valence electrons and the core; the correlations between the valence [MATH] and [MATH] electrons are not included.', '1209.4282-1-15-0': 'We investigated two types of positively ionized diatomic molecules: those with a [MATH] ground state, including group 13 fluorides and a number of other systems, and molecular ions with a [MATH] ground state (represented here by group 15 fluorides).', '1209.4282-1-15-1': 'The values of [MATH] for all the systems under study are presented in Table [REF], together with the DHF and the DFT [MATH] constants.', '1209.4282-1-15-2': 'As the final recommended value for the [MATH] parameter we take an average of [MATH](DHF)[MATH] and [MATH](DFT)[MATH].', '1209.4282-1-15-3': 'The estimate of the accuracy in our previous work [CITATION] has shown that it is about 15% for molecules in the [MATH] electronic state and 20-30% for the [MATH] state.', '1209.4282-1-16-0': 'The magnitude of [MATH] in the [MATH] electronic state is expected to scale as [MATH] [CITATION], where [MATH] is the relativistic parameter, [EQUATION]', '1209.4282-1-16-1': 'In Eq. ([REF]), [MATH] is the Bohr radius, [MATH] m, and [MATH] is the fine-structure constant.', '1209.4282-1-16-2': 'The [MATH] parameters are shown in Table [REF] for each of the metal atoms.', '1209.4282-1-16-3': 'In Fig. [REF] we plot [MATH] as a function of [MATH] for both groups of dimers.', '1209.4282-1-16-4': 'For group-13 fluorides the scaling is, indeed, close to [MATH].', '1209.4282-1-16-5': 'In the case of group 15 fluorides, however, the ground state is [MATH], for which the [MATH] parameter vanishes in the non-relativistic limit, since in this limit it does not contain the [MATH]-wave electronic orbital and can not provide the matrix element [MATH].', '1209.4282-1-16-6': 'The effect appears due to the mixing of [MATH] and [MATH] electronic states by the spin-orbit interaction, and gives an extra factor of [MATH] in the [MATH]-dependence of [MATH], as seen in Fig [REF].', '1209.4282-1-17-0': 'To summarize, here we have performed calculations of the [MATH]-odd interaction constant [MATH] in singly-ionized group-13 and group-15 fluorides as well as other select singly-ionized diatomic systems.', '1209.4282-1-17-1': 'To the best of our knowledge, this is the first investigation of nuclear spin-dependent parity violation effects in molecular ions and it is motivated by progress in the cooling and trapping of such systems.', '1209.4282-1-18-0': 'This work was supported by the Marsden Fund (Royal Society of New Zealand), the Australian Research Council, the Alexander von Humboldt Foundation (Bonn), and the Slovak Research and Development Agency (grant number APVV-0059-10).'}

{'1209.4282-2-0-0': 'Nuclear-spin-dependent (NSD) parity violating (PV) effects can be strongly enhanced in diatomic molecules containing heavy atoms.', '1209.4282-2-0-1': 'Future measurements are anticipated to provide nuclear anapole moments and strength constants for PV nuclear forces.', '1209.4282-2-0-2': 'In light molecules, the NSD electroweak electron-nucleus interaction may also be detected.', '1209.4282-2-0-3': 'Here we calculate NSD PV effects for molecular ions.', '1209.4282-2-0-4': 'Our calculations are motivated by rapid developments in trapping techniques for such systems at low temperatures.', '1209.4282-2-1-0': 'It was previously shown that nuclear spin-dependent (NSD) parity violation (PV) effects are enhanced by a factor of [MATH] in diatomic molecules with [MATH] and [MATH] electronic states due to the mixing of close rotational states of opposite parity ([MATH]-doublet for [MATH]) [CITATION].', '1209.4282-2-1-1': 'DeMille and co-workers suggested measuring NSD PV effects by using neutral diatomic molecules in a Stark interference experiment to determine the mixing between opposite-parity rotational/hyperfine levels [CITATION].', '1209.4282-2-1-2': 'Another proposal was published in Ref. [CITATION], and corresponding experiments have already started.', '1209.4282-2-1-3': 'Recently, it was demonstrated that positive molecular ions may be easily trapped and studied at low temperatures [CITATION], which motivated us to perform calculations on NSD PV effects in such systems.', '1209.4282-2-2-0': 'The term in the Hamiltonian operator arising from the NSD parity violating electron-nucleus interaction is [EQUATION]', '1209.4282-2-2-1': 'Here, and throughout the text, we use atomic units.', '1209.4282-2-2-2': 'In Eq. ([REF]), [MATH] is the dimensionless strength constant, [MATH] a.u. is the Fermi constant, [MATH] is a vector comprised of the conventional Dirac matrices, [MATH] is the nuclear spin, [MATH] is the displacement of the valence electron from the nucleus, and [MATH] is the (normalized) nuclear density.', '1209.4282-2-2-3': 'There are three sources for this interaction: the first contribution arises from the electroweak neutral coupling between electron vector and nucleon axial-vector currents ([MATH]) [CITATION].', '1209.4282-2-2-4': 'The second contribution comes from the nuclear-spin-independent weak interaction combined with the hyperfine interaction [CITATION].', '1209.4282-2-2-5': 'Finally, the nuclear anapole moment contribution, which scales with the number of nucleons [MATH] as [MATH], becomes the dominant contribution in spin-dependent atomic PV effects for a sufficiently large nuclear charge [MATH] [CITATION].', '1209.4282-2-3-0': 'The anapole moment was first predicted by Zeldovich [CITATION] in 1958 as a new parity ([MATH]) violating and time ([MATH]) reversal conserving moment of an elementary particle.', '1209.4282-2-3-1': 'It appears in the second-order multipole expansion of the magnetic vector-potential simultaneously with the [MATH]- and [MATH]- violating magnetic quadrupole moment [CITATION].', '1209.4282-2-3-2': 'The nuclear anapole moment was experimentally discovered in the [MATH]Cs atom in 1997 [CITATION] following a proposal by Flambaum and Khriplovich [CITATION], who showed that the nuclear anapole provides the dominant contribution to the nuclear-spin-dependent parity violating effect in atoms and molecules.', '1209.4282-2-4-0': 'The nuclear anapole requires nuclear spin [MATH] and in a simple valence model has the following value [CITATION], [EQUATION]', '1209.4282-2-4-1': 'Here, [MATH], [MATH] is the orbital angular momentum of the external unpaired nucleon [MATH]; [MATH], [MATH].', '1209.4282-2-4-2': 'Theoretical estimates give the strength constant for nucleon-nucleus weak potential [MATH] for a proton and [MATH] for a neutron [CITATION].', '1209.4282-2-4-3': 'The aim of anapole measurements is to provide accurate values for these constants, thus obtaining important information about hadronic weak coupling.', '1209.4282-2-5-0': 'A number of theoretical investigations of the nuclear spin-dependent parity violation in diatomic molecules have been performed in recent years, using both semiempirical [CITATION] and ab initio methods [CITATION] .', '1209.4282-2-5-1': 'In a recent paper [CITATION] we presented Dirac Hartree-Fock and relativistic density-functional calculations of the electronic [MATH] factor of the diatomic group-2 and -12 fluorides and a number of other diatomic compounds.', '1209.4282-2-5-2': 'In this work we investigate the nuclear spin-dependent parity violation effects in a different type of system, i.e. positively charged dimers, as these systems have an experimental advantage of being easily trapped [CITATION].', '1209.4282-2-5-3': 'Diatomic ions have also been proposed for the search for the electron electric-dipole moment (eEDM) [CITATION], and the preliminary experiments are currently being conducted [CITATION].', '1209.4282-2-5-4': 'Here we use the combination of methods presented in Ref. [CITATION] to calculate the [MATH] factors of positively ionized group-13 and group-15 fluorides ([MATH] and [MATH] ground states, respectively), and a number of other positive diatomic ions having a [MATH] ground state.', '1209.4282-2-6-0': 'For [MATH] and [MATH] electronic states, the interaction ([REF]) can be replaced by the effective operator, which appears in the spin-rotational Hamiltonian [CITATION], [EQUATION] where [MATH] is the effective spin and [MATH] is the unit vector directed along the molecular axis from the heavier to the lighter nucleus.', '1209.4282-2-6-1': 'The electronic factor [MATH] is found from evaluating the matrix elements of the [MATH] operator in the molecular spinor basis [CITATION].', '1209.4282-2-6-2': 'The [MATH] and the [MATH] open-shell electronic states are two-fold degenerate, corresponding to the two possible projections of electronic angular momentum along [MATH], i.e. [MATH].', '1209.4282-2-6-3': 'When operating within this degenerate space, the operator [MATH] is equivalent to [MATH]) (Eq. ([REF])).', '1209.4282-2-6-4': 'Time-reversal symmetry ensures that only the matrix elements that are off-diagonal in [MATH] are non-vanishing.', '1209.4282-2-6-5': 'This symmetry rule is encapsulated within the effective operator [MATH] by the angular factor [MATH].', '1209.4282-2-6-6': 'Here the effective spin [MATH] generates rotations in the degenerate subspace analogously to usual spin operator [MATH] in a spin-1/2 system.', '1209.4282-2-7-0': 'The calculations were carried out within the open-shell single determinant average-of-configuration Dirac-Hartree-Fock approach (DHF) [CITATION] and within the relativistic density functional theory (DFT) [CITATION], employing quaternion symmetry [CITATION].', '1209.4282-2-7-1': 'A finite nucleus, modeled by the Gaussian charge distribution was used [CITATION].', '1209.4282-2-7-2': "All the calculations were performed using the developer's version of the DIRAC10 program package [CITATION].", '1209.4282-2-8-0': 'For the lighter elements (boron to phosphorus), uncontracted aug-cc-pVTZ basis sets were used [CITATION].', '1209.4282-2-8-1': "For the rest of the atoms, we employed Faegri's dual family basis sets [CITATION].", '1209.4282-2-8-2': 'As a good description of the electronic wave function in the nuclear region is essential for obtaining reliable results for parity violating properties [CITATION], we augmented the basis sets with high exponent [MATH] and [MATH] functions, which brings about an increase of around [MATH] in the calculated values of [MATH].', '1209.4282-2-8-3': 'The basis sets were increased, both in the core and in the valence regions, to convergence with respect to the calculated [MATH] constants.', '1209.4282-2-8-4': 'The final basis sets can be found in Table [REF].', '1209.4282-2-9-0': 'Where available, experimentally determined bond distances [MATH] were used.', '1209.4282-2-9-1': 'For molecules where [MATH] is not known experimentally, we optimized the bond distance using relativistic coupled cluster theory with single, double, and perturbative triple excitations, CCSD(T) [CITATION].', '1209.4282-2-9-2': 'To reduce the computational effort, we employed an infinite order two-component relativistic Hamiltonian obtained after the Barysz-Sadlej-Snijders (BSS) transformation of the Dirac Hamiltonian in a finite basis set [CITATION].', '1209.4282-2-9-3': 'Our calculated [MATH] are typically within 0.01 AA of the experimental values, where available.', '1209.4282-2-9-4': 'The experimental/calculated equilibrium distances can be found in Table [REF].', '1209.4282-2-10-0': 'In the DFT calculations we used the Coulomb-attenuated B3LYP functional (CAMB3LYP*), the parameters of which were adjusted by Thierfelder et al. [CITATION] to reproduce the PV energy shifts obtained using coupled cluster calculations (the newly adjusted parameters are [MATH], [MATH], and [MATH]).', '1209.4282-2-11-0': 'In our previous work [CITATION] we have examined and compared various schemes for adding electron correlation to the Dirac-Hartree-Fock [MATH] values, and core-polarization contributions to the DFT results.', '1209.4282-2-11-1': 'Here, we correct the calculated DHF and DFT [MATH] for core polarization using a scaling parameter, [MATH].', '1209.4282-2-11-2': 'This parameter is obtained from atomic calculations as described in the following.', '1209.4282-2-11-3': 'The main contribution to the matrix elements of the NSD interaction for the valence molecular electrons comes from short distances around the heavy nucleus, where the total molecular potential is spherically symmetric to very high precision, and the core of the heavy atom is practically unaffected by the presence of the second atom, justifying our use of the atomic model.', '1209.4282-2-11-4': 'The molecular orbitals of the valence electron can thus be expanded in this region, using spherical harmonics centered at the heavy nucleus, [EQUATION]', '1209.4282-2-11-5': 'Only [MATH] and [MATH] terms of this expansion give significant contribution to the matrix elements of the weak interaction.', '1209.4282-2-11-6': 'These functions can be considered as states of an atomic valence electron and are calculated using standard atomic techniques in two different approximations: one that includes electron correlation and another that does not.', '1209.4282-2-12-0': 'The single electron DHF Hamiltonian is given by [EQUATION] where [MATH] and [MATH] are the Dirac matrices and [MATH] is the self-consistent DHF potential due to atomic electrons.', '1209.4282-2-13-0': 'The self-consistent DHF procedure is first performed for the closed shell ion, from which the valence electron is removed.', '1209.4282-2-13-1': 'Then the core potential [MATH] is frozen and the valence [MATH] and [MATH] states are calculated by solving the DHF equation for the valence electron, [EQUATION] where [MATH] is given by ([REF]).', '1209.4282-2-14-0': 'The core polarization can be understood as the change of the self-consistent DHF potential due to the effect of the extra term (the weak interaction operator [MATH]) in the Hamiltonian.', '1209.4282-2-14-1': 'The inclusion of the core polarization in a self-consistent way is equivalent to the random-phase approximation (RPA, see, e.g. [CITATION]).', '1209.4282-2-14-2': 'The change in the DHF potential is found by solving the RPA-type equations self-consistently for all states in the atomic core, [EQUATION]', '1209.4282-2-14-3': 'Here, [MATH] is the DHF Hamiltonian ([REF]), index [MATH] enumerates the states in the core, [MATH] is the correction to the core state [MATH] due to weak interaction [MATH], and [MATH] is the correction to the self-consistent core potential due to the change of all core functions.', '1209.4282-2-14-4': 'Once [MATH] is found, the core polarization can be included into a matrix element for valence states [MATH] and [MATH] via the redefinition of the weak interaction Hamiltonian, [EQUATION]', '1209.4282-2-14-5': 'We then obtain the scaling parameter for core-polarization effects, [MATH], from [EQUATION]', '1209.4282-2-14-6': 'It should be noted that for the positively charged group 15 fluorides we have only calculated the correlations between the valence electrons and the core; the correlations between the valence [MATH] and [MATH] electrons are not included.', '1209.4282-2-15-0': 'We investigated two types of positively ionized diatomic molecules: those with a [MATH] ground state, including group 13 fluorides and a number of other systems, and molecular ions with a [MATH] ground state (represented here by group 15 fluorides).', '1209.4282-2-15-1': 'The values of [MATH] for all the systems under study are presented in Table [REF], together with the DHF and the DFT [MATH] constants.', '1209.4282-2-15-2': 'As the final recommended value for the [MATH] parameter we take an average of [MATH](DHF)[MATH] and [MATH](DFT)[MATH].', '1209.4282-2-15-3': 'The estimate of the accuracy in our previous work [CITATION] has shown that it is about 15% for molecules in the [MATH] electronic state and 20-30% for the [MATH] state.', '1209.4282-2-16-0': 'The magnitude of [MATH] in the [MATH] electronic state is expected to scale as [MATH] [CITATION], where [MATH] is the relativistic parameter, [EQUATION]', '1209.4282-2-16-1': 'In Eq. ([REF]), [MATH] is the Bohr radius, [MATH] m, and [MATH] is the fine-structure constant.', '1209.4282-2-16-2': 'The [MATH] parameters are shown in Table [REF] for each of the metal atoms.', '1209.4282-2-16-3': 'In Fig. [REF] we plot [MATH] as a function of [MATH] for both groups of dimers.', '1209.4282-2-16-4': 'For group-13 fluorides the scaling is, indeed, [MATH].', '1209.4282-2-16-5': 'In the case of group 15 fluorides, however, the ground state is [MATH], for which the [MATH] parameter vanishes in the non-relativistic limit, since in this limit it does not contain the [MATH]-wave electronic orbital and can not provide the matrix element [MATH].', '1209.4282-2-16-6': 'The effect appears due to the mixing of [MATH] and [MATH] electronic states by the spin-orbit interaction, and gives an extra factor of [MATH] in the [MATH]-dependence of [MATH], as seen in Fig [REF].', '1209.4282-2-17-0': 'To summarize, here we have performed calculations of the [MATH]-odd interaction constant [MATH] in singly-ionized group-13 and group-15 fluorides as well as other select singly-ionized diatomic systems.', '1209.4282-2-17-1': 'To the best of our knowledge, this is the first investigation of nuclear spin-dependent parity violation effects in molecular ions and it is motivated by progress in the cooling and trapping of such systems.', '1209.4282-2-18-0': 'This work was supported by the Marsden Fund (Royal Society of New Zealand), the Australian Research Council, the Alexander von Humboldt Foundation (Bonn), and the Slovak Research and Development Agency (grant number APVV-0059-10).', '1209.4282-2-18-1': 'The authors are grateful to R. Berger and T. Isaev for critical comments.'}

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{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1209.4282

1510.07611

{'1510.07611-1-0-0': 'Sampling is at the core of deep learning and more general machine learning applications; an increase in its efficiency would have a significant impact across several domains.', '1510.07611-1-0-1': 'With the advent of more mature quantum computing technologies, quantum annealers have been proposed as a potential candidate to speed up these tasks, but several limitations still bar these state-of-the-art technologies from being used effectively.', '1510.07611-1-0-2': "One of the main limitations, and the focus of this work, is that using the device's experimentally accessible temperature as a reference for sampling purposes leads to very poor correlation with the Boltzmann distribution it is programmed to sample from.", '1510.07611-1-0-3': 'Based on quantum dynamical arguments, one can expect that if the device indeed happens to be sampling from a Boltzmann-like distribution, it will correspond to one with an instance-dependent effective temperature.', '1510.07611-1-0-4': 'Unless this unknown temperature can be unveiled, it might not be possible to effectively use a quantum annealer for Boltzmann sampling processes.', '1510.07611-1-0-5': 'In this work, we propose a strategy to overcome this challenge with a simple effective-temperature estimation algorithm.', '1510.07611-1-0-6': 'To highlight the importance of this estimation, we provide a systematic study assessing the impact of the effective temperatures in the training of a kind of restricted Boltzmann machine on quantum hardware, which can serve as a building block for deep learning architectures.', '1510.07611-1-0-7': 'We also provide a comparison to [MATH]-step contrastive divergence(CD-[MATH]) with [MATH] up to 100 to have better assessment of the performance.', '1510.07611-1-0-8': 'Although our suggested quantum-assisted training algorithm as well as other simplifications that assume a fixed effective temperature clearly outperform conventional contrastive divergence (CD-1), only our algorithm is capable of matching CD-100 for the Boltzmann machine trained in this work.', '1510.07611-1-1-0': '# Introduction', '1510.07611-1-2-0': 'The use of quantum computing technologies for sampling and machine learning applications is attracting increasing interest from the research community in recent years [CITATION].', '1510.07611-1-2-1': 'Although the main focus of the quantum annealing computational paradigm [CITATION] has been in solving discrete optimization problems in a wide variety of application domains [CITATION], it has been also introduced as a potential candidate to speed up computations in sampling applications.', '1510.07611-1-2-2': 'Indeed, it is an important open research question whether or not quantum annealers can sample from Boltzmann distributions more efficiently than traditional techniques [CITATION].', '1510.07611-1-3-0': 'However, there are some challenges that need to be overcome before uncovering the potential of quantum annealing hardware for sampling problems.', '1510.07611-1-3-1': 'One of the main difficulties is that the device does not necessarily sample from the Boltzmann distribution that would correspond to the physical temperature and the user-specified control parameters (couplings and fields) of the device.', '1510.07611-1-3-2': 'Instead, there might be some instance-dependent corrections leading, in principle, to instance-dependent effective temperature [CITATION].', '1510.07611-1-3-3': 'Bian et al. [CITATION] has used the maximum likelihood method to estimate such an instance-dependent temperature and introduced some additional shifts in the control parameter of the quantum device, for several realizations of small eight-qubit instances on an early generation of quantum annealers produced by D-Wave Systems.', '1510.07611-1-3-4': 'The authors showed that, with these additional estimated shifts in place, the empirical probability distribution obtained from the D-Wave appears to correlate very well with the corresponding Boltzmann distribution.', '1510.07611-1-3-5': 'Further experimental evidence of this effective temperature can be found in Ref. [CITATION] where a proper determination of this effective temperature was needed to determine residual bias in the programmable parameters of the device.', '1510.07611-1-4-0': 'Recently some authors have explored the use of quantum annealing hardware for the training of Boltzmann machines and deep neural networks [CITATION].', '1510.07611-1-4-1': 'Training a general Boltzmann machine or a deep neural network is in general intractable due to long equilibration times of sampling techniques like Markov Chain Monte Carlo (MCMC).', '1510.07611-1-4-2': 'One of the strategies that have made possible the recent spectacular success [CITATION] of these techniques is to deal with less general architectures that allow for substantial speedups.', '1510.07611-1-4-3': 'Restricted Boltzmann Machines (RBMs) [CITATION] are an important example of this kind that moreover serve as a suitable building block for deeper architectures.', '1510.07611-1-4-4': 'Still, quantum annealers has the potential to allow for training more complex architectures.', '1510.07611-1-5-0': 'When applying quantum annealing hardware to the training of Boltzmann machines the interest is in finding the optimal control parameters that best represent the empirical distribution of a dataset.', '1510.07611-1-5-1': 'However, estimating additional shifts for the control parameters, as done by Bian et al. [CITATION], would not be practical since it is in a sense similar to the very kind of problem that a Boltzmann machine attempts to solve.', '1510.07611-1-5-2': 'One could then ask what is the meaning of using a quantum annealer for learning the parameters of a distribution, if to do it we need to use standard techniques to learn the corrections to the control parameters.', '1510.07611-1-6-0': 'Here we explore a different approach by taking into account only the possibility of an instance-dependent effective temperature without the need of considering further instance-dependent shifts in the control parameters.', '1510.07611-1-6-1': 'We devise a technique to estimate the effective temperature associated to a given instance by generating only two sets of samples from the machine and performing a linear regression.', '1510.07611-1-6-2': 'The samples used in our effective-temperature estimation algorithm are the same ones used towards achieving the final goal of the sampling application.', '1510.07611-1-6-3': 'This is in contrast with the approach taken in [CITATION] which needs in principle many evaluations of the gradient of the log-likelihood of a set of samples from the device, making it impractical for large problem instances.', '1510.07611-1-7-0': 'We test our ideas in the training of Boltzmann machines.', '1510.07611-1-7-1': 'In the next section we shall present a brief overview of the ideas related to Boltzmann machines and discuss how quantum annealing hardware can be used to assist their training.', '1510.07611-1-7-2': 'Afterwards we discuss related work.', '1510.07611-1-7-3': 'In the section that follows we introduce our technique to estimate the effective temperature associated to a given instance.', '1510.07611-1-7-4': 'We then show an implementation of these ideas for the quantum-assisted training (QuAT) of a Chimera-RBM on the Bars and Stripes dataset [CITATION], implemented in the Dwave 2X device (DW2X) located at NASA Ames Research Center.', '1510.07611-1-7-5': 'Finally, we present the conclusions of the work and some perspectives of the future work we will be exploring.', '1510.07611-1-8-0': '# General considerations', '1510.07611-1-9-0': '## General Boltzmann machines', '1510.07611-1-10-0': 'Consider a data set [MATH] whose empiric distribution is [MATH]; here each datapoint can be represented as an array of Ising variables, i.e. [MATH] with [MATH].', '1510.07611-1-10-1': 'A Boltzmann machine models the data via a model distribution [MATH], where [MATH] is a Boltzmann distribution on a possibly extended sample space [MATH], [MATH] is the corresponding energy function, and [MATH] is the normalization constant or partition function.', '1510.07611-1-10-2': 'Notice that in this case we do not need a temperature parameter, since it only amounts at a rescaling of the model parameters that we want to learn.', '1510.07611-1-10-3': "Here [MATH] are the 'unseen' or 'hidden' variables, that help capture higher level structure in the data, and [MATH] are the 'visible' variables, that correspond to the data themselves.", '1510.07611-1-10-4': 'Denoting these variables collectively by [MATH] we can write [EQUATION] where [MATH] and [MATH] are the model parameters that has to be adjusted to fit the data; here [MATH] and [MATH] are the set of vertices and edges, respectively, that make up the interaction graph [MATH].', '1510.07611-1-11-0': 'The task is then to find the model parameters that make the model distribution [MATH] as close as possible to the data distribution [MATH].', '1510.07611-1-11-1': 'This can be accomplished by minimizing the KL divergence between [MATH] and [MATH] or, equivalently, by maximizing the log-likelihood [CITATION] [EQUATION] with respect to the model parameters [MATH] and [MATH].', '1510.07611-1-12-0': 'Gradient ascent is a standard method to carry out this optimization via the rule [EQUATION] where [MATH] is the learning rate, and the gradient of the log-likelihood function is given by [CITATION] [EQUATION]', '1510.07611-1-12-1': 'Here [MATH] denotes the ensemble average with respect to the distribution [MATH] that involves the data.', '1510.07611-1-12-2': 'Similarly, [MATH] denotes the ensemble average with respect to the distribution [MATH] that involves exclusively the model.', '1510.07611-1-12-3': 'Such averages can be estimated by standard sampling techniques, such as MCMC.', '1510.07611-1-12-4': 'Another possibility, explored in this work, is to rely on a physical process that naturally generates samples from a Boltzmann distribution.', '1510.07611-1-13-0': '## Quantum annealing', '1510.07611-1-14-0': 'Quantum annealing is an algorithm that attempts to exploit quantum effects to find the configurations with the lowest cost of a function describing a problem of interest [CITATION].', '1510.07611-1-14-1': 'It relies on finding a mapping of such a function into the energy function of an equivalent physical system.', '1510.07611-1-14-2': 'The latter is suitably modified to incorporate quantum fluctuations whose purpose is to maintain the system in its lowest-energy solution space.', '1510.07611-1-15-0': 'In short, the algorithm consists in slowly transforming the ground state of an initial quantum system, that is relatively easy to prepare, into the ground state of a final Hamiltonian that encodes the problem to be solved.', '1510.07611-1-15-1': 'The device produced by D-Wave Systems [CITATION] is a realization of this idea for solving quadratic unconstrained optimization problems on binary variables.', '1510.07611-1-15-2': 'It implements the Hamiltonian [EQUATION] where [MATH] are Pauli matrices that operate on spin or qubit [MATH].', '1510.07611-1-15-3': 'The control parameters of the D-Wave machine are composed of a field [MATH] for each qubit [MATH] and a coupling [MATH] for each pair of interacting qubits [MATH] and [MATH].', '1510.07611-1-15-4': 'The topology of the interactions between qubits in the D-Wave is given by a so-called Chimera graph [MATH].', '1510.07611-1-15-5': 'This is made up of elementary cells of [MATH] complete bi-partite graphs that are coupled as shown in Fig. [REF] (a).', '1510.07611-1-15-6': 'The transformation from the simple Hamiltonian [MATH] to the problem Hamiltonian [MATH] is controlled by time-dependent monotonic functions [MATH] and [MATH], such that [MATH] and [MATH].', '1510.07611-1-15-7': 'Here [MATH], where [MATH] is the physical time and [MATH] is the anealing time, i.e. the time that it takes to transform Hamiltonian [MATH] into Hamiltonian [MATH].', '1510.07611-1-16-0': 'Although quantum annealers were designed with the purpose of reaching a ground state of the problem Hamiltonian [MATH], there are theoretical arguments [CITATION] and experimental evidence [CITATION] suggesting that under certain conditions the device samples from an approximately Boltzmann distribution at a given effective temperature.', '1510.07611-1-16-1': 'Such an effective temperature depends on the specific energy landscape that is realized, and so on the particular instance.', '1510.07611-1-16-2': 'How to exploit this sampling feature of the quantum device is the main focus of this work.', '1510.07611-1-17-0': '## Chimera Restricted Boltzmann machines', '1510.07611-1-18-0': 'Training a general Boltzmann machine is in general intractable due to long equilibration time of sampling techniques like MCMC.', '1510.07611-1-18-1': 'One way to escape this issue is to use less general architectures.', '1510.07611-1-18-2': 'One of the most investigated architectures is the Restricted Boltzmann Machine (RBM).', '1510.07611-1-18-3': 'The interaction graph [MATH] of an RBM is a fully bipartite graph in which visible and hidden units interact with each other but not among themselves.', '1510.07611-1-18-4': 'This implies that the conditional distributions [MATH] and [MATH] factorize in terms of single variable marginals, which substantially simplifies the problem of obtaining samples from them.', '1510.07611-1-18-5': 'This leads to the idea of [MATH]-step contrastive divergence learning (CD-[MATH]) where the model expectation values are approximated as follows: first we start with a datapoint [MATH]; then we sample [MATH] from [MATH], and subsequently sample [MATH] from [MATH] and so on for [MATH] steps.', '1510.07611-1-18-6': 'At the end of this process we obtain a sample [MATH] [CITATION].', '1510.07611-1-19-0': 'It is in principle possible to embed a restricted Boltzmann machine (RBM) in quantum annealing hardware [CITATION].', '1510.07611-1-19-1': 'However, due to limited connectivity of the device, the resulting physical representation would involve a larger number of physical variables (qubits) and of physical links (couplings) than the original RBM being represented.', '1510.07611-1-19-2': 'It would be preferable to use an alternative model that can be naturally represented in the device.', '1510.07611-1-19-3': 'For this reason we will focus on the kind of models that are obtained after removing from a given RBM all the links that are not present in the D-Wave machine [CITATION].', '1510.07611-1-19-4': 'We will call this type of models Chimera Restricted Boltzmann Machines (Chimera-RBM).', '1510.07611-1-19-5': 'Fig. [REF] shows an example of a Chimera-RBM (a) and a possible embedding of the pixels of an image into its visible units (b).', '1510.07611-1-20-0': '# Related work', '1510.07611-1-21-0': 'Dumoulin et al. [CITATION] have studied the impact of different limitations of quantum annealing hardware for training restricted Boltzmann machines.', '1510.07611-1-21-1': 'The authors have focused on three kinds of limitations: noisy parameters, limited parameter range, and restricted architecture.', '1510.07611-1-21-2': 'The training method used was persistent contrastive divergence where the model ensemble averages were estimated with samples from simulated quantum hardware while the data ensemble averages were estimated by exact mean field.', '1510.07611-1-22-0': 'For assessing the impact of limited connectivity, Dumoulin et al. investigated a Chimera-RBM.', '1510.07611-1-22-1': 'They found that limited connectivity is the most relevant limitation in this context.', '1510.07611-1-22-2': 'In a sense this is understandable as standard RBMs are based on complete bipartite graphs while the Chimera graph is sparse.', '1510.07611-1-22-3': 'Roughly speaking, this means that if the number of variables is of order [MATH], the number of parameters present in a Chimera graph is a vanishing fraction (of order [MATH]) of the number of parameters in the corresponding RBM.', '1510.07611-1-22-4': 'Furthermore, connections in the Chimera graph are rather localized.', '1510.07611-1-22-5': 'This feature may make more difficult to capture higher level correlations.', '1510.07611-1-23-0': 'The authors also found that noisy parameters in an RBM is the next relevant limitation and that the noise in the couplings is more relevant than the noise in the fields.', '1510.07611-1-23-1': 'This could simply be due to the fact that the number of fields is a vanishing fraction (of order [MATH]) of the number of couplings in an RBM.', '1510.07611-1-23-2': 'This argument is not valid anymore in a Chimera-RBM, though.', '1510.07611-1-23-3': "Now, the authors also mention that the noise in the couplings is 'quenched', i.e. it only changes every time the coupling values change, while the noise in the fields is 'annealed', i.e. it changes in every sample generated.", '1510.07611-1-23-4': 'If this is indeed the case, this could be another reason for the higher relevance of the noise in the couplings than the noise in the fields.', '1510.07611-1-24-0': 'Finally, an upper bound in the magnitude of the model parameters, similar to the one present in the D-Wave device, does not seem to have much impact.', '1510.07611-1-24-1': 'In this respect, we should notice that current D-Wave devices are designed with the aim of reaching only the ground state.', '1510.07611-1-24-2': 'In contrast, typical applications of Boltzmann machines deals with heterogeneous real data which contains a relatively large level of uncertainty, and are expected to exploit a wider range of configurations.', '1510.07611-1-24-3': 'This suggests that in sampling applications control parameters are typically smaller than those explored in applications to combinatorial optimization.', '1510.07611-1-24-4': 'This suggests that potential lower bounds in the magnitude of the control parameter can turn out to be more relevant for sampling applications.', '1510.07611-1-24-5': 'In this respect, it is important to notice that noise in the control parameters can lead to an effective lower bound.', '1510.07611-1-25-0': 'While Dumoulin et al. modeled the instance-dependent corrections as independent Gaussian noise around the user defined parameter values, Denil and Freitas [CITATION] devised a way to by-pass this problem altogether.', '1510.07611-1-25-1': 'For doing this, the authors have optimized the one-step reconstruction error as a black box function and approximate its gradient empirically.', '1510.07611-1-25-2': 'They do this by a technique called simultaneous perturbation stochastic approximation.', '1510.07611-1-25-3': 'However, with this approach it is not possible to decouple the model from the machine.', '1510.07611-1-25-4': 'Furthermore, it is not clear what is the efficiency of this technique nor how to extend it to deal with the more robust log-likelihood function rather than the reconstruction error.', '1510.07611-1-25-5': 'In their approach only the hidden layer is embedded in the D-Wave, and qubit interactions are exploited to build a semi-restricted Boltzmann machine.', '1510.07611-1-25-6': 'They report encouraging results, although they are still no conclusive since the log-likelihood function was not the function being optimized.', '1510.07611-1-26-0': '# Quantum-assisted training of Boltzmann machines', '1510.07611-1-27-0': "A promising idea for training more general Boltzmann machines is to rely on a physical device, like D-Wave's, that naturally generates samples from a Boltzmann distribution [CITATION].", '1510.07611-1-27-1': 'Due to various sources of noise, the D-Wave device produces samples from an approximately Boltzmann distribution, but at a temperature different from the physical temperature of the device.', '1510.07611-1-27-2': 'Indeed, there might be some instance-dependent corrections [CITATION]; we will only take into account corrections to the temperature and neglect any possible shifts in the control parameters.', '1510.07611-1-27-3': 'We will assume a Boltzmann distribution defined by an energy function as in [REF], with [MATH] and [MATH], where [MATH] can be instance-dependent.', '1510.07611-1-27-4': 'While the control parameters for the D-Wave are couplings and fields, i.e. [MATH] and [MATH], the learning takes place on the ratio of the control parameters to the temperature, i.e. [MATH] and [MATH].', '1510.07611-1-27-5': 'Inferring temperature is therefore a fundamental step in order to be able to use the samples from the D-Wave for learning.', '1510.07611-1-27-6': 'A learning algorithm for the quantum-assisted training (QuAT) of a Boltzmann machine needs access to a method that estimates the temperature at each iteration.', '1510.07611-1-27-7': 'The learning algorithm we propose is initialized as follows:', '1510.07611-1-28-0': 'Then it iterates as follows:', '1510.07611-1-29-0': 'A few comments are in order.', '1510.07611-1-29-1': 'First, the two sets of samples that we mention above are for estimating model and data ensemble averages.', '1510.07611-1-29-2': 'For the former we just need to run the device with the specified control parameters.', '1510.07611-1-29-3': 'For the latter we need to generate samples with the visible units clamped to the datapoints, which can be done by applying suitable fields to the corresponding qubits.', '1510.07611-1-29-4': 'However, in the case of restricted Boltzmann machines we can avoid this last step as it is possible to compute exactly the data ensemble averages.', '1510.07611-1-29-5': 'Second, notice that to compute the new control parameters at step [MATH] we should have used temperature [MATH] at the same step.', '1510.07611-1-29-6': 'However, to obtain such a temperature we would need to know which are the parameters at time [MATH].', '1510.07611-1-29-7': 'To escape this vicious cycle we have done [MATH].', '1510.07611-1-29-8': 'In the next section we discuss a method for estimating this instance-dependent temperature.', '1510.07611-1-30-0': '# Temperature estimation', '1510.07611-1-31-0': 'Consider the Boltzmann distribution [MATH] at a generic inverse temperature [MATH] as a function of the energy [MATH].', '1510.07611-1-31-1': 'Here [MATH] is the degeneracy of the energy level [MATH] and [MATH] is the partition function.', '1510.07611-1-31-2': 'We want to devise an efficient method for estimating the effective temperature associated to a given instance.', '1510.07611-1-31-3': 'To do this, consider the log-likelihood ratio of two different states or energy levels, [MATH] and [MATH], given by [EQUATION] where [MATH].', '1510.07611-1-31-4': 'We can estimate this log-likelihood ratio by estimating the frequencies of the two states involved.', '1510.07611-1-31-5': 'We could in principle do this for different values of the parameter [MATH] by rescaling the control parameters of the device.', '1510.07611-1-31-6': 'Indeed, rescaling the control parameters by a factor [MATH] is equivalent to setting a parameter [MATH], where [MATH] is the inverse of the effective temperature [MATH] associated to a given instance.', '1510.07611-1-31-7': 'Notice that this is only true under the assumption that [MATH], despite being instance-dependent, does not change appreciably under small rescalings of the control parameters.', '1510.07611-1-31-8': 'In this way, we can expect that by plotting the log-likelihood ratio [MATH] against the scaling parameter [MATH], we should obtain a straight line whose slope and intercept are given by [MATH] and [MATH], respectively.', '1510.07611-1-31-9': 'Since we know the energy levels we can in principle infer [MATH].', '1510.07611-1-31-10': 'However, the performance of this method was rather poor in all experiments we did (not shown).', '1510.07611-1-31-11': 'A reason for this could be that to perform the linear regression to extract the corresponding effective temperature, several values of [MATH] need to be explored in a relatively wide range.', '1510.07611-1-31-12': 'Next we present a proposal that mitigate this limitation.', '1510.07611-1-32-0': 'The previous approach relied on several values of the scaling parameter [MATH] but only two energy levels.', '1510.07611-1-32-1': 'We were not exploiting all the information available in the other states in the sample set from the quantum annealer.', '1510.07611-1-32-2': 'We can exploit such an information to obtain a more robust estimate of the temperature by sampling only for the original control parameters and a single rescaling of them.', '1510.07611-1-32-3': 'The idea here is to take the difference [MATH] of the log-likelihood ratios of pairs of energy levels, with [MATH] and [MATH], to eliminate the unknown degeneracies altogether, yielding [EQUATION] where [MATH].', '1510.07611-1-32-4': "In this way, by generating a second set of samples at a 'suitable' value of [MATH] and then taking the differences of, say, the most populated level and all the remaining relevant ones we can plot [MATH] Vs. [MATH].", '1510.07611-1-32-5': 'According to Eq. [REF] this is expected to be a straight line with slope given by [MATH].', '1510.07611-1-32-6': 'In this way we can in principle obtain [MATH] by generating only a new set of samples for parameters scaled by [MATH].', '1510.07611-1-32-7': 'The choice of [MATH] matters: if it is too small no informative changes, other than noise, would be detected; if it is too large some levels would become unpopulated and we would not be able to compare them at both the original and rescaled control parameters.', '1510.07611-1-33-0': 'Fig. [REF]a shows an instance of temperature estimation using 1000 samples from the DW2X and a scaling parameter [MATH].', '1510.07611-1-33-1': 'Despite the dispersion of the data we can observe a somewhat robust behavior and a clear linear trend.', '1510.07611-1-33-2': 'A least squares linear regression of the data yields a temperature of [MATH].', '1510.07611-1-33-3': 'Fig. [REF]b shows a comparison of the performance of the temperature estimation technique for different values of the scaling parameter [MATH], using [MATH] as a reference.', '1510.07611-1-33-4': 'Overall we can see that [MATH] typically displays a better performance.', '1510.07611-1-33-5': 'This value has also worked best for training a Chimera-RBM in the different experiments we have run.', '1510.07611-1-33-6': 'Notice also that although there is some dependency on the performance as a function of [MATH], and [MATH] seemed to be the best here, there is no strong dependence of this value so a rule of thumb of selecting [MATH] could be suggested.', '1510.07611-1-34-0': '# Training of a Boltzmann machine assisted by the D-Wave 2X', '1510.07611-1-35-0': 'Now that we have at our disposal a robust temperature estimation technique, we can use it for training Boltzmann machines.', '1510.07611-1-35-1': 'We will focus here on the training of a Chimera-RBM.', '1510.07611-1-35-2': 'We decided to restrict to Chimera-RBM for two reasons.', '1510.07611-1-35-3': 'On the one hand, although an RBM can be embedded into quantum hardware [CITATION], it requires to represent single variables with chains of qubits coupled via ferromagnetic interactions of a given strength.', '1510.07611-1-35-4': 'We consider that rather than forcing couplings to take a specific value in order to meet certain preconceived design, it should be better to give the device the freedom to find the parameters that works best for a particular application.', '1510.07611-1-35-5': 'On the other hand, the focus of our work is in better understanding the challenges that need to be overcome for using quantum annealers like DW2X for sampling applications, before moving forward to tackle problems of larger complexity.', '1510.07611-1-36-0': 'To the best of our knowledge, this is the first systematic study providing both, an assessment of the use of the DWave in training Boltzmann machines and studying the impact of the effective temperature in the learning performance.', '1510.07611-1-36-1': 'We consider that it is important to asses the performance of the different training methods by exactly computing the log-likelihood during the training process.', '1510.07611-1-36-2': 'Otherwise, we could not be sure whether a difference in performance is due to the new training method or due to errors in the approximation of the log-likelihood.', '1510.07611-1-36-3': 'For this reason we tested the method in a small dataset called Bars and Stripes (BAS) and computed exhaustively the corresponding log-likelihood for evaluation.', '1510.07611-1-36-4': 'The BAS dataset consists of pictures generated by setting all the pixels of each row (or column) to either black (−1) or white (+1), at random [CITATION].', '1510.07611-1-36-5': 'A further technical difficulty that we meet is that while generating (say) 1000 samples in the DW2X for a given instance can take about 20 ms, the waiting time for accessing the machine to generate a new set of samples for a different instance can vary widely depending on the amount of jobs that are scheduled.', '1510.07611-1-36-6': 'So, while training a Chimera-RBM could take say about one hour had we exclusive access to the device, the waiting times of the different jobs can increase this time by an order of magnitude or so.', '1510.07611-1-36-7': 'Currently, we are running experiments with much larger instances and evaluating the performance in more tractable ways.', '1510.07611-1-37-0': 'We modeled the BAS dataset with a Chimera-RBM of 16 visible and 16 hidden units with the topology shown in Fig. [REF] (a).', '1510.07611-1-37-1': 'The mapping of pixels to visible units is shown in Fig. [REF] (b) (cf. [CITATION]).', '1510.07611-1-37-2': 'Fig. [REF] (a) shows the evolution of the log-likelihood during the training of a Chimera-RBM on the BAS dataset under different methods, all of them with learning rate [MATH].', '1510.07611-1-37-3': 'We can observe that QuAT (blue diagonal crosses [MATH]) outperforms CD-[MATH] (blue solid squares) after about 300 iterations.', '1510.07611-1-37-4': 'However, within the 3000 iterations shown in the figure, QuAT has not yet been able to outperform CD-25 nor CD-100, although there is a clear trend in that direction.', '1510.07611-1-37-5': 'Interestingly, CD-[MATH] reaches their best log-likelihood values after a relatively small number of iterations while QuAT, in contrast, increases slowly and steadily.', '1510.07611-1-37-6': 'One may be inclined to think this is because CD-[MATH] estimates the model averages from samples that are generated by running a [MATH]-step Markov chain initialized at each data point.', '1510.07611-1-37-7': 'In this way CD-[MATH] is using information contained in the data from the very beginning for the estimation of the model ensemble averages, while QuAT ignores them altogether.', '1510.07611-1-37-8': 'However, if this were indeed the case one should expect such a trend to diminish for increasing values of [MATH], something that is not observed in the figure.', '1510.07611-1-37-9': 'A better understanding of this point has the potential to considerably improve the performance of QuAT.', '1510.07611-1-38-0': 'To asses the relevance of temperature estimation for QuAT, we also show in Fig. [REF] the evolution of the log-likelihood for quantum assisted training keeping the temperature to a fixed value, rather than evaluating it at each iteration.', '1510.07611-1-38-1': 'First, we can observe that using the physical temperature of the device (pink empty squares in Fig. [REF] (b)), which is estimated to be [MATH] mK), leads to a very poor performance.', '1510.07611-1-38-2': 'Fixing the temperature to the average [MATH] over all temperatures found during the QuAT discussed before leads to a better performance (red empty circles) but still well below that displayed by QuAT.', '1510.07611-1-38-3': 'Interestingly, fixing the temperature to [MATH] leads to a better performance than that displayed with [MATH], still below the performance of QuAT.', '1510.07611-1-38-4': 'Fixing the effective temperature to [MATH], on the other hand, leads to a decrease in performance with respect to that displayed with [MATH].', '1510.07611-1-39-0': 'Fig. [REF] (c) shows the effective temperatures estimated at each iteration of QuAT of the Chimera-RBM on the BAS dataset.', '1510.07611-1-39-1': 'We can observe that it starts with values closed to the temperature of the device [MATH] and starts increasing, initially at a steady pace and then with a slower trend.', '1510.07611-1-40-0': '# Conclusions and Future Work', '1510.07611-1-41-0': 'Training a general Boltzmann machine is in general intractable due to long equilibration times of sampling techniques like MCMC.', '1510.07611-1-41-1': 'In this work we propose a strategy to overcome one of the main limitations when intending to use a quantum annealer to sample from Boltzmann distributions: the determination of effective temperatures in quantum annealers.', '1510.07611-1-41-2': 'The simple technique proposed in this work uses the samples obtained from the quantum annealer (the DW2X at NASA is our experimental implementation) to estimate both, the effective temperature and the hard-to-compute term in the log-likelihood gradient, i.e., the averages over the model distribution, needed to determine the next step in the training process.', '1510.07611-1-41-3': 'We present a systematic study of the impact of the effective-temperature in the training of a Chimera-RBM model with 16 visible and 16 hidden units, and we compare the different settings with different effective temperatures to the performance of a CD-[MATH] implementation, with [MATH] up to 100.', '1510.07611-1-42-0': 'The Chimera-RBM model itself is much less powerful than the full RBM model.', '1510.07611-1-42-1': 'While the former is sparse with a number of parameters increasing linearly with the number of variables, the latter is dense with a number of parameters increasing quadratically instead.', '1510.07611-1-42-2': 'RBMs have the nice feature that sampling in one layer (conditioned to a configuration in the other layer) can be done in parallel and in one step; this is one of the key features of full RBMs and the main reason for their wide adoption.', '1510.07611-1-42-3': 'This nice feature does not hold true anymore once we have non-trivial lateral connections in one of the layers, which is the concept behind more powerful general BMs, and where we think it is one for the most promising direction to explore with QuAT algorithms.', '1510.07611-1-42-4': 'By restricting QuAT to study Chimera-RBM models (or, in general, RBM models), one is paying the price of using a device that is in principle more powerful, but we are not taking the gains of having a more general model.', '1510.07611-1-42-5': 'It is important to investigate how to take full advantage of the DW2X by designing more suitable models based on the Chimera graph.', '1510.07611-1-42-6': 'An interesting possibility is the one explored in Ref. [CITATION] where the Chimera graph of the DW2X is used as a hidden layer to build a semi-restricted Boltzmann machine, which therefore have lateral connections in the hidden layer.', '1510.07611-1-43-0': 'However, the goal of this first QuAT implementation on small Chimera-RBMs serves several purposes.', '1510.07611-1-43-1': 'When dealing with large datasets the log-likelihood cannot be exhaustively computed due to the intractability of computing the partition function.', '1510.07611-1-43-2': 'Log-likelihood is the gold standard metric but it becomes intractable for large systems.', '1510.07611-1-43-3': 'In these cases, other performance metrics have been used such as reconstruction error or cross entropy error, and although widely used these are not 100% reliable for our benchmarking purpose [CITATION].', '1510.07611-1-43-4': 'Since these other proxies are meant to approximate the log-likelihood, if we were to use these we could not be sure that we would be drawing the right conclusions.', '1510.07611-1-43-5': 'This justifies why we used a moderately small dataset with 16 visible and 16 hidden units, and even though computing the log-likelihood was computational intensive for the study performed here, having 32 units in total was still a manageable size.', '1510.07611-1-43-6': 'Through the computation of the exact likelihood we were able to examine in more detail some of the goals proposed here: being able to assess the best effective temperature fit to the desired Boltzmann distribution and to show that using a constant temperature different from the estimated one with our approach might lead to severe suboptimal setting in terms of the training performance.', '1510.07611-1-44-0': 'Another aspect we explore in this study was to go beyond the conventional CD-1, with the purpose of having a more fair comparison to the results that might be expected from the entirely classical algorithm counterpart.', '1510.07611-1-44-1': 'Previous results from our research group [CITATION], as well as others reported by other researchers [CITATION], are limited to comparing the performance of quantum annealers to the quick but suboptimal CD-1.', '1510.07611-1-44-2': 'As shown in those studies, even with a suboptimal constant temperature one might be drawn to conclude that QuAT is outperforming conventional CD.', '1510.07611-1-44-3': 'Similar conclusions might be drawn from the curves for constant but suboptimal [MATH] and [MATH] vs. CD-1 in Fig. [REF].', '1510.07611-1-44-4': 'As shown in Fig. [REF], this conclusion does not hold anymore for higher values of [MATH], while the method using the effective-temperature estimation proposed here is the only one showing a steady increase in performance, close to matching the largest value of [MATH] tried here, i.e. [MATH].', '1510.07611-1-44-5': 'A follow up study of this work will assess how close QuAT is following the exact log-likelihood gradient during the training and how different are the distributions learned with the classical algorithm and the ones obtained with the QuAT algorithm.', '1510.07611-1-45-0': 'Another important point that we are currently studying is whether the differences observed in performance remain for larger and more complex datasets.', '1510.07611-1-45-1': 'We would expect that the performance of CD-[MATH] degrades with larger number of variables as equilibration times are expected to grow fast with the number of variables once the probability distribution starts having non-trivial structure.', '1510.07611-1-45-2': 'From this perspective, it is important to notice that QuAT is expected to display a more uniform exploration of configuration space.', '1510.07611-1-45-3': 'This feature may render unecessary adding auxiliary terms to prevent distortions in the learning process.', '1510.07611-1-45-4': 'Indeed, CD-[MATH] is usually regularized by adding a term that penalizes the size of the weights and that, if properly tuned, can avoid the drop in likelihood [CITATION].', '1510.07611-1-45-5': 'We can observe this effect in Fig. [REF] after about 500 iterations for CD-1 and about 800 iterations for CD-25, for instance.', '1510.07611-1-45-6': 'QuAT, instead, increases the likelihood slowly but steadily without the need of an explicit regularization, at least within the 3000 iterations that we have investigated.', '1510.07611-1-46-0': 'Several other interesting studies follow from having a robust method for estimating [MATH], and that we will be exploring to speedup the first version of our first QuAT protocol proposed here.', '1510.07611-1-46-1': 'For example, it seems reasonable to estimate it only in every certain number of training iterations.', '1510.07611-1-46-2': 'From Fig. [REF] it can be seen that a good strategy would be to do it more frequently at the beginning and less frequent as it stabilizes or seem to converge around certain value.', '1510.07611-1-46-3': 'Still this has the drawback that if by any chance the temperature happens to be estimated badly, such an error would persist for several iterations.', '1510.07611-1-46-4': 'Another simple strategy not used in this study is to recycle also the samples taken at the value of [MATH] and that was used thus far only for the estimation of [MATH].', '1510.07611-1-46-5': 'Since the two distributions are expected to be close enough, importance sampling as used in Ref. [CITATION] can be used to connect the obtained samples and still use them in the model average computed along with the samples at [MATH].', '1510.07611-1-47-0': 'There are other ways in which the ideas explored here could be extended.', '1510.07611-1-47-1': 'For instance, we can go beyond restricted Boltzmann machines to build deep learning architectures or beyond unsupervised learning to build discriminative models.', '1510.07611-1-47-2': "In principle the speed of learning could be increased by adding a 'momentum' term to the gradient-ascent learning rule [CITATION].", '1510.07611-1-47-3': 'Indeed, Addachi and Henderson have started exploring these ideas in a contemporary work [CITATION].', '1510.07611-1-47-4': 'Instead, our approach has focused on first trying to better understand the basics before adding more (classical) complexity to the learning algorithms that we feel have the risk to obscure the actual contributions from the new approach.', '1510.07611-1-48-0': 'Note that since CD-[MATH] seems to be faster than the QuAT in the first iterations in Fig. 3, having the capabilities of efficiently estimating effective temperatures allows us for a simple hybrid classical-quantum learning approach.', '1510.07611-1-48-1': 'For example, one can think of using the classical algorithm to be used to generate a reasonably good starting Boltzmann model for the QuAT algorithm.', '1510.07611-1-48-2': 'More specifically, one could use the simple and cheap CD-1 until the learning progress plateaus, and then use the expensive QuAT to bootstrap and enhance the classically obtained model.', '1510.07611-1-48-3': 'This may be useful in helping to diminish the larger impact that the noise in the device might have when the control parameters are somewhat small.', '1510.07611-1-48-4': 'This work will be explored in detail in a following publication.', '1510.07611-1-49-0': '# Supplementary information'}

{'1510.07611-2-0-0': 'Sampling is at the core of deep learning and more general machine learning applications; an increase in its efficiency would have a significant impact across several domains.', '1510.07611-2-0-1': 'With the advent of more mature quantum computing technologies, quantum annealers have been proposed as a potential candidate to speed up these tasks, but several limitations still bar these state-of-the-art technologies from being used effectively.', '1510.07611-2-0-2': "One of the main limitations, and the focus of this work, is that using the device's experimentally accessible temperature as a reference for sampling purposes leads to very poor correlation with the Boltzmann distribution it is programmed to sample from.", '1510.07611-2-0-3': 'Based on quantum dynamical arguments, one can expect that if the device indeed happens to be sampling from a Boltzmann-like distribution, it will correspond to one with an instance-dependent effective temperature.', '1510.07611-2-0-4': 'Unless this unknown temperature can be unveiled, it might not be possible to effectively use a quantum annealer for Boltzmann sampling processes.', '1510.07611-2-0-5': 'In this work, we propose a strategy to overcome this challenge with a simple effective-temperature estimation algorithm.', '1510.07611-2-0-6': 'To highlight the importance of this estimation, we provide a systematic study assessing the impact of the effective temperatures in the training of a kind of restricted Boltzmann machine on quantum hardware, which can serve as a building block for deep learning architectures.', '1510.07611-2-0-7': 'We also provide a comparison to [MATH]-step contrastive divergence(CD-[MATH]) with [MATH] up to 100 to have better assessment of the performance.', '1510.07611-2-0-8': 'Although our suggested quantum-assisted training algorithm as well as other simplifications that assume a fixed effective temperature clearly outperform conventional contrastive divergence (CD-1), only our algorithm is capable of matching CD-100 for the Boltzmann machine trained in this work.', '1510.07611-2-1-0': '# Introduction', '1510.07611-2-2-0': 'The use of quantum computing technologies for sampling and machine learning applications is attracting increasing interest from the research community in recent years [CITATION].', '1510.07611-2-2-1': 'Although the main focus of the quantum annealing computational paradigm [CITATION] has been in solving discrete optimization problems in a wide variety of application domains [CITATION], it has been also introduced as a potential candidate to speed up computations in sampling applications.', '1510.07611-2-2-2': 'Indeed, it is an important open research question whether or not quantum annealers can sample from Boltzmann distributions more efficiently than traditional techniques [CITATION].', '1510.07611-2-3-0': 'However, there are some challenges that need to be overcome before uncovering the potential of quantum annealing hardware for sampling problems.', '1510.07611-2-3-1': 'One of the main difficulties is that the device does not necessarily sample from the Boltzmann distribution that would correspond to the physical temperature and the user-specified control parameters (couplings and fields) of the device.', '1510.07611-2-3-2': 'Instead, there might be some instance-dependent corrections leading, in principle, to instance-dependent effective temperature [CITATION].', '1510.07611-2-3-3': 'Bian et al. [CITATION] has used the maximum likelihood method to estimate such an instance-dependent temperature and introduced some additional shifts in the control parameter of the quantum device, for several realizations of small eight-qubit instances on an early generation of quantum annealers produced by D-Wave Systems.', '1510.07611-2-3-4': 'The authors showed that, with these additional estimated shifts in place, the empirical probability distribution obtained from the D-Wave appears to correlate very well with the corresponding Boltzmann distribution.', '1510.07611-2-3-5': 'Further experimental evidence of this effective temperature can be found in Ref. [CITATION] where a proper determination of this effective temperature was needed to determine residual bias in the programmable parameters of the device.', '1510.07611-2-4-0': 'Recently some authors have explored the use of quantum annealing hardware for the training of Boltzmann machines and deep neural networks [CITATION].', '1510.07611-2-4-1': 'Training a general Boltzmann machine or a deep neural network is in general intractable due to long equilibration times of sampling techniques like Markov Chain Monte Carlo (MCMC).', '1510.07611-2-4-2': 'One of the strategies that have made possible the recent spectacular success [CITATION] of these techniques is to deal with less general architectures that allow for substantial speedups.', '1510.07611-2-4-3': 'Restricted Boltzmann Machines (RBMs) [CITATION] are an important example of this kind that moreover serve as a suitable building block for deeper architectures.', '1510.07611-2-4-4': 'Still, quantum annealers has the potential to allow for training more complex architectures.', '1510.07611-2-5-0': 'When applying quantum annealing hardware to the training of Boltzmann machines the interest is in finding the optimal control parameters that best represent the empirical distribution of a dataset.', '1510.07611-2-5-1': 'However, estimating additional shifts for the control parameters, as done by Bian et al. [CITATION], would not be practical since it is in a sense similar to the very kind of problem that a Boltzmann machine attempts to solve.', '1510.07611-2-5-2': 'One could then ask what is the meaning of using a quantum annealer for learning the parameters of a distribution, if to do it we need to use standard techniques to learn the corrections to the control parameters.', '1510.07611-2-6-0': 'Here we explore a different approach by taking into account only the possibility of an instance-dependent effective temperature without the need of considering further instance-dependent shifts in the control parameters.', '1510.07611-2-6-1': 'We devise a technique to estimate the effective temperature associated to a given instance by generating only two sets of samples from the machine and performing a linear regression.', '1510.07611-2-6-2': 'The samples used in our effective-temperature estimation algorithm are the same ones used towards achieving the final goal of the sampling application.', '1510.07611-2-6-3': 'This is in contrast with the approach taken in [CITATION] which needs in principle many evaluations of the gradient of the log-likelihood of a set of samples from the device, making it impractical for large problem instances.', '1510.07611-2-7-0': 'We test our ideas in the training of Boltzmann machines.', '1510.07611-2-7-1': 'In the next section we shall present a brief overview of the ideas related to Boltzmann machines and discuss how quantum annealing hardware can be used to assist their training.', '1510.07611-2-7-2': 'Afterwards we discuss related work.', '1510.07611-2-7-3': 'In the section that follows we introduce our technique to estimate the effective temperature associated to a given instance.', '1510.07611-2-7-4': 'We then show an implementation of these ideas for the quantum-assisted training (QuAT) of a Chimera-RBM on the Bars and Stripes dataset [CITATION], implemented in the Dwave 2X device (DW2X) located at NASA Ames Research Center.', '1510.07611-2-7-5': 'Finally, we present the conclusions of the work and some perspectives of the future work we will be exploring.', '1510.07611-2-8-0': '# General considerations', '1510.07611-2-9-0': '## General Boltzmann machines', '1510.07611-2-10-0': 'Consider a data set [MATH] whose empiric distribution is [MATH]; here each datapoint can be represented as an array of Ising variables, i.e. [MATH] with [MATH].', '1510.07611-2-10-1': 'A Boltzmann machine models the data via a model distribution [MATH], where [MATH] is a Boltzmann distribution on a possibly extended sample space [MATH], [MATH] is the corresponding energy function, and [MATH] is the normalization constant or partition function.', '1510.07611-2-10-2': 'Notice that in this case we do not need a temperature parameter, since it only amounts at a rescaling of the model parameters that we want to learn.', '1510.07611-2-10-3': "Here [MATH] are the 'unseen' or 'hidden' variables, that help capture higher level structure in the data, and [MATH] are the 'visible' variables, that correspond to the data themselves.", '1510.07611-2-10-4': 'Denoting these variables collectively by [MATH] we can write [EQUATION] where [MATH] and [MATH] are the model parameters that has to be adjusted to fit the data; here [MATH] and [MATH] are the set of vertices and edges, respectively, that make up the interaction graph [MATH].', '1510.07611-2-11-0': 'The task is then to find the model parameters that make the model distribution [MATH] as close as possible to the data distribution [MATH].', '1510.07611-2-11-1': 'This can be accomplished by minimizing the KL divergence between [MATH] and [MATH] or, equivalently, by maximizing the log-likelihood [CITATION] [EQUATION] with respect to the model parameters [MATH] and [MATH].', '1510.07611-2-12-0': 'Gradient ascent is a standard method to carry out this optimization via the rule [EQUATION] where [MATH] is the learning rate, and the gradient of the log-likelihood function is given by [CITATION] [EQUATION]', '1510.07611-2-12-1': 'Here [MATH] denotes the ensemble average with respect to the distribution [MATH] that involves the data.', '1510.07611-2-12-2': 'Similarly, [MATH] denotes the ensemble average with respect to the distribution [MATH] that involves exclusively the model.', '1510.07611-2-12-3': 'Such averages can be estimated by standard sampling techniques, such as MCMC.', '1510.07611-2-12-4': 'Another possibility, explored in this work, is to rely on a physical process that naturally generates samples from a Boltzmann distribution.', '1510.07611-2-13-0': '## Quantum annealing', '1510.07611-2-14-0': 'Quantum annealing is an algorithm that attempts to exploit quantum effects to find the configurations with the lowest cost of a function describing a problem of interest [CITATION].', '1510.07611-2-14-1': 'It relies on finding a mapping of such a function into the energy function of an equivalent physical system.', '1510.07611-2-14-2': 'The latter is suitably modified to incorporate quantum fluctuations whose purpose is to maintain the system in its lowest-energy solution space.', '1510.07611-2-15-0': 'In short, the algorithm consists in slowly transforming the ground state of an initial quantum system, that is relatively easy to prepare, into the ground state of a final Hamiltonian that encodes the problem to be solved.', '1510.07611-2-15-1': 'The device produced by D-Wave Systems [CITATION] is a realization of this idea for solving quadratic unconstrained optimization problems on binary variables.', '1510.07611-2-15-2': 'It implements the Hamiltonian [EQUATION] where [MATH] are Pauli matrices that operate on spin or qubit [MATH].', '1510.07611-2-15-3': 'The control parameters of the D-Wave machine are composed of a field [MATH] for each qubit [MATH] and a coupling [MATH] for each pair of interacting qubits [MATH] and [MATH].', '1510.07611-2-15-4': 'The topology of the interactions between qubits in the D-Wave is given by a so-called Chimera graph [MATH].', '1510.07611-2-15-5': 'This is made up of elementary cells of [MATH] complete bi-partite graphs that are coupled as shown in Fig. [REF] (a).', '1510.07611-2-15-6': 'The transformation from the simple Hamiltonian [MATH] to the problem Hamiltonian [MATH] is controlled by time-dependent monotonic functions [MATH] and [MATH], such that [MATH] and [MATH].', '1510.07611-2-15-7': 'Here [MATH], where [MATH] is the physical time and [MATH] is the anealing time, i.e. the time that it takes to transform Hamiltonian [MATH] into Hamiltonian [MATH].', '1510.07611-2-16-0': 'Although quantum annealers were designed with the purpose of reaching a ground state of the problem Hamiltonian [MATH], there are theoretical arguments [CITATION] and experimental evidence [CITATION] suggesting that under certain conditions the device samples from an approximately Boltzmann distribution at a given effective temperature.', '1510.07611-2-16-1': 'Such an effective temperature depends on the specific energy landscape that is realized, and so on the particular instance.', '1510.07611-2-16-2': 'How to exploit this sampling feature of the quantum device is the main focus of this work.', '1510.07611-2-17-0': '## Chimera Restricted Boltzmann machines', '1510.07611-2-18-0': 'Training a general Boltzmann machine is in general intractable due to long equilibration time of sampling techniques like MCMC.', '1510.07611-2-18-1': 'One way to escape this issue is to use less general architectures.', '1510.07611-2-18-2': 'One of the most investigated architectures is the Restricted Boltzmann Machine (RBM).', '1510.07611-2-18-3': 'The interaction graph [MATH] of an RBM is a fully bipartite graph in which visible and hidden units interact with each other but not among themselves.', '1510.07611-2-18-4': 'This implies that the conditional distributions [MATH] and [MATH] factorize in terms of single variable marginals, which substantially simplifies the problem of obtaining samples from them.', '1510.07611-2-18-5': 'This leads to the idea of [MATH]-step contrastive divergence learning (CD-[MATH]) where the model expectation values are approximated as follows: first we start with a datapoint [MATH]; then we sample [MATH] from [MATH], and subsequently sample [MATH] from [MATH] and so on for [MATH] steps.', '1510.07611-2-18-6': 'At the end of this process we obtain a sample [MATH] [CITATION].', '1510.07611-2-19-0': 'It is in principle possible to embed a restricted Boltzmann machine (RBM) in quantum annealing hardware [CITATION].', '1510.07611-2-19-1': 'However, due to limited connectivity of the device, the resulting physical representation would involve a larger number of physical variables (qubits) and of physical links (couplings) than the original RBM being represented.', '1510.07611-2-19-2': 'It would be preferable to use an alternative model that can be naturally represented in the device.', '1510.07611-2-19-3': 'For this reason we will focus on the kind of models that are obtained after removing from a given RBM all the links that are not present in the D-Wave machine [CITATION].', '1510.07611-2-19-4': 'We will call this type of models Chimera Restricted Boltzmann Machines (Chimera-RBM).', '1510.07611-2-19-5': 'Fig. [REF] shows an example of a Chimera-RBM (a) and a possible embedding of the pixels of an image into its visible units (b).', '1510.07611-2-20-0': '# Related work', '1510.07611-2-21-0': 'Dumoulin et al. [CITATION] have studied the impact of different limitations of quantum annealing hardware for training restricted Boltzmann machines.', '1510.07611-2-21-1': 'The authors have focused on three kinds of limitations: noisy parameters, limited parameter range, and restricted architecture.', '1510.07611-2-21-2': 'The training method used was persistent contrastive divergence where the model ensemble averages were estimated with samples from simulated quantum hardware while the data ensemble averages were estimated by exact mean field.', '1510.07611-2-22-0': 'For assessing the impact of limited connectivity, Dumoulin et al. investigated a Chimera-RBM.', '1510.07611-2-22-1': 'They found that limited connectivity is the most relevant limitation in this context.', '1510.07611-2-22-2': 'In a sense this is understandable as standard RBMs are based on complete bipartite graphs while the Chimera graph is sparse.', '1510.07611-2-22-3': 'Roughly speaking, this means that if the number of variables is of order [MATH], the number of parameters present in a Chimera graph is a vanishing fraction (of order [MATH]) of the number of parameters in the corresponding RBM.', '1510.07611-2-22-4': 'Furthermore, connections in the Chimera graph are rather localized.', '1510.07611-2-22-5': 'This feature may make more difficult to capture higher level correlations.', '1510.07611-2-23-0': 'The authors also found that noisy parameters in an RBM is the next relevant limitation and that the noise in the couplings is more relevant than the noise in the fields.', '1510.07611-2-23-1': 'This could simply be due to the fact that the number of fields is a vanishing fraction (of order [MATH]) of the number of couplings in an RBM.', '1510.07611-2-23-2': 'This argument is not valid anymore in a Chimera-RBM, though.', '1510.07611-2-23-3': "Now, the authors also mention that the noise in the couplings is 'quenched', i.e. it only changes every time the coupling values change, while the noise in the fields is 'annealed', i.e. it changes in every sample generated.", '1510.07611-2-23-4': 'If this is indeed the case, this could be another reason for the higher relevance of the noise in the couplings than the noise in the fields.', '1510.07611-2-24-0': 'Finally, an upper bound in the magnitude of the model parameters, similar to the one present in the D-Wave device, does not seem to have much impact.', '1510.07611-2-24-1': 'In this respect, we should notice that current D-Wave devices are designed with the aim of reaching only the ground state.', '1510.07611-2-24-2': 'In contrast, typical applications of Boltzmann machines deals with heterogeneous real data which contains a relatively large level of uncertainty, and are expected to exploit a wider range of configurations.', '1510.07611-2-24-3': 'This suggests that in sampling applications control parameters are typically smaller than those explored in applications to combinatorial optimization.', '1510.07611-2-24-4': 'This suggests that potential lower bounds in the magnitude of the control parameter can turn out to be more relevant for sampling applications.', '1510.07611-2-24-5': 'In this respect, it is important to notice that noise in the control parameters can lead to an effective lower bound.', '1510.07611-2-25-0': 'While Dumoulin et al. modeled the instance-dependent corrections as independent Gaussian noise around the user defined parameter values, Denil and Freitas [CITATION] devised a way to by-pass this problem altogether.', '1510.07611-2-25-1': 'For doing this, the authors have optimized the one-step reconstruction error as a black box function and approximate its gradient empirically.', '1510.07611-2-25-2': 'They do this by a technique called simultaneous perturbation stochastic approximation.', '1510.07611-2-25-3': 'However, with this approach it is not possible to decouple the model from the machine.', '1510.07611-2-25-4': 'Furthermore, it is not clear what is the efficiency of this technique nor how to extend it to deal with the more robust log-likelihood function rather than the reconstruction error.', '1510.07611-2-25-5': 'In their approach only the hidden layer is embedded in the D-Wave, and qubit interactions are exploited to build a semi-restricted Boltzmann machine.', '1510.07611-2-25-6': 'They report encouraging results, although they are still no conclusive since the log-likelihood function was not the function being optimized.', '1510.07611-2-26-0': '# Quantum-assisted training of Boltzmann machines', '1510.07611-2-27-0': "A promising idea for training more general Boltzmann machines is to rely on a physical device, like D-Wave's, that naturally generates samples from a Boltzmann distribution [CITATION].", '1510.07611-2-27-1': 'Due to various sources of noise, the D-Wave device produces samples from an approximately Boltzmann distribution, but at a temperature different from the physical temperature of the device.', '1510.07611-2-27-2': 'Indeed, there might be some instance-dependent corrections [CITATION]; we will only take into account corrections to the temperature and neglect any possible shifts in the control parameters.', '1510.07611-2-27-3': 'We will assume a Boltzmann distribution defined by an energy function as in [REF], with [MATH] and [MATH], where [MATH] can be instance-dependent.', '1510.07611-2-27-4': 'While the control parameters for the D-Wave are couplings and fields, i.e. [MATH] and [MATH], the learning takes place on the ratio of the control parameters to the temperature, i.e. [MATH] and [MATH].', '1510.07611-2-27-5': 'Inferring temperature is therefore a fundamental step in order to be able to use the samples from the D-Wave for learning.', '1510.07611-2-27-6': 'A learning algorithm for the quantum-assisted training (QuAT) of a Boltzmann machine needs access to a method that estimates the temperature at each iteration.', '1510.07611-2-27-7': 'The learning algorithm we propose is initialized as follows:', '1510.07611-2-28-0': 'Then it iterates as follows:', '1510.07611-2-29-0': 'A few comments are in order.', '1510.07611-2-29-1': 'First, the two sets of samples that we mention above are for estimating model and data ensemble averages.', '1510.07611-2-29-2': 'For the former we just need to run the device with the specified control parameters.', '1510.07611-2-29-3': 'For the latter we need to generate samples with the visible units clamped to the datapoints, which can be done by applying suitable fields to the corresponding qubits.', '1510.07611-2-29-4': 'However, in the case of restricted Boltzmann machines we can avoid this last step as it is possible to compute exactly the data ensemble averages.', '1510.07611-2-29-5': 'Second, notice that to compute the new control parameters at step [MATH] we should have used temperature [MATH] at the same step.', '1510.07611-2-29-6': 'However, to obtain such a temperature we would need to know which are the parameters at time [MATH].', '1510.07611-2-29-7': 'To escape this vicious cycle we have done [MATH].', '1510.07611-2-29-8': 'In the next section we discuss a method for estimating this instance-dependent temperature.', '1510.07611-2-30-0': '# Temperature estimation', '1510.07611-2-31-0': 'Consider the Boltzmann distribution [MATH] at a generic inverse temperature [MATH] as a function of the energy [MATH].', '1510.07611-2-31-1': 'Here [MATH] is the degeneracy of the energy level [MATH] and [MATH] is the partition function.', '1510.07611-2-31-2': 'We want to devise an efficient method for estimating the effective temperature associated to a given instance.', '1510.07611-2-31-3': 'To do this, consider the log-likelihood ratio of two different states or energy levels, [MATH] and [MATH], given by [EQUATION] where [MATH].', '1510.07611-2-31-4': 'We can estimate this log-likelihood ratio by estimating the frequencies of the two states involved.', '1510.07611-2-31-5': 'We could in principle do this for different values of the parameter [MATH] by rescaling the control parameters of the device.', '1510.07611-2-31-6': 'Indeed, rescaling the control parameters by a factor [MATH] is equivalent to setting a parameter [MATH], where [MATH] is the inverse of the effective temperature [MATH] associated to a given instance.', '1510.07611-2-31-7': 'Notice that this is only true under the assumption that [MATH], despite being instance-dependent, does not change appreciably under small rescalings of the control parameters.', '1510.07611-2-31-8': 'In this way, we can expect that by plotting the log-likelihood ratio [MATH] against the scaling parameter [MATH], we should obtain a straight line whose slope and intercept are given by [MATH] and [MATH], respectively.', '1510.07611-2-31-9': 'Since we know the energy levels we can in principle infer [MATH].', '1510.07611-2-31-10': 'However, the performance of this method was rather poor in all experiments we did (not shown).', '1510.07611-2-31-11': 'A reason for this could be that to perform the linear regression to extract the corresponding effective temperature, several values of [MATH] need to be explored in a relatively wide range.', '1510.07611-2-31-12': 'Next we present a proposal that mitigate this limitation.', '1510.07611-2-32-0': 'The previous approach relied on several values of the scaling parameter [MATH] but only two energy levels.', '1510.07611-2-32-1': 'We were not exploiting all the information available in the other states in the sample set from the quantum annealer.', '1510.07611-2-32-2': 'We can exploit such an information to obtain a more robust estimate of the temperature by sampling only for the original control parameters and a single rescaling of them.', '1510.07611-2-32-3': 'The idea here is to take the difference [MATH] of the log-likelihood ratios of pairs of energy levels, with [MATH] and [MATH], to eliminate the unknown degeneracies altogether, yielding [EQUATION] where [MATH].', '1510.07611-2-32-4': "In this way, by generating a second set of samples at a 'suitable' value of [MATH] and then taking the differences of, say, the most populated level and all the remaining relevant ones we can plot [MATH] Vs. [MATH].", '1510.07611-2-32-5': 'According to Eq. [REF] this is expected to be a straight line with slope given by [MATH].', '1510.07611-2-32-6': 'In this way we can in principle obtain [MATH] by generating only a new set of samples for parameters scaled by [MATH].', '1510.07611-2-32-7': 'The choice of [MATH] matters: if it is too small no informative changes, other than noise, would be detected; if it is too large some levels would become unpopulated and we would not be able to compare them at both the original and rescaled control parameters.', '1510.07611-2-33-0': 'Fig. [REF]a shows an instance of temperature estimation using 1000 samples from the DW2X and a scaling parameter [MATH].', '1510.07611-2-33-1': 'Despite the dispersion of the data we can observe a somewhat robust behavior and a clear linear trend.', '1510.07611-2-33-2': 'A least squares linear regression of the data yields a temperature of [MATH].', '1510.07611-2-33-3': 'Fig. [REF]b shows a comparison of the performance of the temperature estimation technique for different values of the scaling parameter [MATH], using [MATH] as a reference.', '1510.07611-2-33-4': 'Overall we can see that [MATH] typically displays a better performance.', '1510.07611-2-33-5': 'This value has also worked best for training a Chimera-RBM in the different experiments we have run.', '1510.07611-2-33-6': 'Notice also that although there is some dependency on the performance as a function of [MATH], and [MATH] seemed to be the best here, there is no strong dependence of this value so a rule of thumb of selecting [MATH] could be suggested.', '1510.07611-2-34-0': '# Training of a Boltzmann machine assisted by the D-Wave 2X', '1510.07611-2-35-0': 'Now that we have at our disposal a robust temperature estimation technique, we can use it for training Boltzmann machines.', '1510.07611-2-35-1': 'We will focus here on the training of a Chimera-RBM.', '1510.07611-2-35-2': 'We decided to restrict to Chimera-RBM for two reasons.', '1510.07611-2-35-3': 'On the one hand, although an RBM can be embedded into quantum hardware [CITATION], it requires to represent single variables with chains of qubits coupled via ferromagnetic interactions of a given strength.', '1510.07611-2-35-4': 'We consider that instead of forcing couplings to take a specific value in order to meet a preconceived design, it might be better to allow the learning algorithm itself to find the parameter values that work best for a particular application.', '1510.07611-2-35-5': 'On the other hand, the focus of our work is in better understanding the challenges that need to be overcome for using quantum annealers like DW2X for sampling applications, before moving forward to tackle problems of larger complexity.', '1510.07611-2-36-0': 'To the best of our knowledge, this is the first systematic study providing both, an assessment of the use of the DWave in training Boltzmann machines and studying the impact of the effective temperature in the learning performance.', '1510.07611-2-36-1': 'We consider that it is important to asses the performance of the different training methods by exactly computing the log-likelihood during the training process.', '1510.07611-2-36-2': 'Otherwise, we could not be sure whether a difference in performance is due to the new training method or due to errors in the approximation of the log-likelihood.', '1510.07611-2-36-3': 'For this reason we tested the method in a small dataset called Bars and Stripes (BAS) and computed exhaustively the corresponding log-likelihood for evaluation.', '1510.07611-2-36-4': 'The BAS dataset consists of pictures generated by setting all the pixels of each row (or column) to either black (-1) or white (+1), at random [CITATION].', '1510.07611-2-36-5': 'A further technical difficulty that we meet is that while generating (say) 1000 samples in the DW2X for a given instance can take about 20 ms, the waiting time for accessing the machine to generate a new set of samples for a different instance can vary widely depending on the amount of jobs that are scheduled.', '1510.07611-2-36-6': 'So, while training a Chimera-RBM could take say about one hour had we exclusive access to the device, the waiting times of the different jobs can increase this time by an order of magnitude or so.', '1510.07611-2-36-7': 'Currently, we are running experiments with much larger instances and evaluating the performance in more tractable ways.', '1510.07611-2-37-0': 'We modeled the BAS dataset with a Chimera-RBM of 16 visible and 16 hidden units with the topology shown in Fig. [REF] (a).', '1510.07611-2-37-1': 'The mapping of pixels to visible units is shown in Fig. [REF] (b) (cf. [CITATION]).', '1510.07611-2-37-2': 'Fig. [REF] (a) shows the evolution of the log-likelihood during the training of a Chimera-RBM on the BAS dataset under different learning algorithms, all of them with learning rate [MATH].', '1510.07611-2-37-3': 'We can observe that the quantum assisted training algorithm with effective-temperature estimation at each iteration (QuAT@[MATH], blue diagonal crosses [MATH]) outperforms CD-[MATH] (blue solid squares) after about 300 iterations.', '1510.07611-2-37-4': 'However, within the 3000 iterations shown in the figure, QuAT@[MATH] has not yet been able to outperform CD-25 nor CD-100, although there is a clear trend in that direction.', '1510.07611-2-37-5': 'Interestingly, CD-[MATH] reaches their best log-likelihood values after a relatively small number of iterations while QuAT@[MATH], in contrast, increases slowly and steadily.', '1510.07611-2-37-6': 'One may be inclined to think this is because CD-[MATH] estimates the model averages from samples that are generated by running a [MATH]-step Markov chain initialized at each data point.', '1510.07611-2-37-7': 'In this way CD-[MATH] is using information contained in the data from the very beginning for the estimation of the model ensemble averages, while QuAT@[MATH] ignores them altogether.', '1510.07611-2-37-8': 'However, if this were indeed the case one should expect such a trend to diminish for increasing values of [MATH], something that is not observed in the figure.', '1510.07611-2-37-9': 'A better understanding of this point has the potential to considerably improve the performance of QuAT@[MATH].', '1510.07611-2-38-0': 'To asses the relevance of temperature estimation for QuAT@[MATH], we also show in Fig. [REF] the evolution of the log-likelihood under quantum assisted training at a fixed temperature [MATH] (QuAT@[MATH]).', '1510.07611-2-38-1': 'First, we can observe that using the physical temperature of the device (pink empty squares in Fig. [REF] (b)), which is estimated to be [MATH] mK), leads to a very poor performance.', '1510.07611-2-38-2': 'Fixing the temperature to the average [MATH] over all temperatures found during the run of QuAT@[MATH] discussed before leads to a better performance (red empty circles) but still well below that displayed by QuAT@[MATH] itself.', '1510.07611-2-38-3': 'Interestingly, fixing the temperature to [MATH] leads to a better performance than that displayed with [MATH], still below the performance of QuAT@[MATH].', '1510.07611-2-38-4': 'Fixing the effective temperature to [MATH], on the other hand, leads to a decrease in performance with respect to that displayed with [MATH].', '1510.07611-2-39-0': 'Fig. [REF] (c) shows the effective temperatures estimated at each iteration of QuAT@[MATH] of the Chimera-RBM on the BAS dataset.', '1510.07611-2-39-1': 'We can observe that it starts with values closed to the temperature of the device [MATH] and starts increasing, initially at a steady pace and then with a slower trend.', '1510.07611-2-40-0': '# Conclusions and Future Work', '1510.07611-2-41-0': 'Training a general Boltzmann machine is in general intractable due to long equilibration times of sampling techniques like MCMC.', '1510.07611-2-41-1': 'In this work we propose a strategy to overcome one of the main limitations when intending to use a quantum annealer to sample from Boltzmann distributions: the determination of effective temperatures in quantum annealers.', '1510.07611-2-41-2': 'The simple technique proposed in this work uses the samples obtained from the quantum annealer (the DW2X at NASA is our experimental implementation) to estimate both, the effective temperature and the hard-to-compute term in the log-likelihood gradient, i.e., the averages over the model distribution, needed to determine the next step in the training process.', '1510.07611-2-41-3': 'We present a systematic study of the impact of the effective-temperature in the training of a Chimera-RBM model with 16 visible and 16 hidden units, and we compare the different settings with different effective temperatures to the performance of a CD-[MATH] implementation, with [MATH] up to 100.', '1510.07611-2-42-0': 'The Chimera-RBM model itself is much less powerful than the full RBM model.', '1510.07611-2-42-1': 'While the former is sparse with a number of parameters increasing linearly with the number of variables, the latter is dense with a number of parameters increasing quadratically instead.', '1510.07611-2-42-2': 'RBMs have the nice feature that sampling in one layer (conditioned to a configuration in the other layer) can be done in parallel and in one step; this is one of the key features of full RBMs and the main reason for their wide adoption.', '1510.07611-2-42-3': 'This nice feature does not hold true anymore once we have non-trivial lateral connections in one of the layers, which is the concept behind more powerful general BMs, and where we think it is one for the most promising direction to explore with the quantum-assisted training (QuAT) algorithms.', '1510.07611-2-42-4': 'By restricting QuAT to study Chimera-RBM models (or, in general, RBM models), one is paying the price of using a device that is in principle more powerful, but we are not taking the gains of having a more general model.', '1510.07611-2-42-5': 'It is important to investigate how to take full advantage of the DW2X by designing more suitable models based on the Chimera graph.', '1510.07611-2-42-6': 'An interesting possibility is the one explored in Ref. [CITATION] where the Chimera graph of the DW2X is used as a hidden layer to build a semi-restricted Boltzmann machine, which therefore have lateral connections in the hidden layer.', '1510.07611-2-43-0': 'However, the goal of this first QuAT implementation on small Chimera-RBMs serves several purposes.', '1510.07611-2-43-1': 'When dealing with large datasets the log-likelihood cannot be exhaustively computed due to the intractability of computing the partition function.', '1510.07611-2-43-2': 'Log-likelihood is the gold standard metric but it becomes intractable for large systems.', '1510.07611-2-43-3': 'In these cases, other performance metrics have been used such as reconstruction error or cross entropy error, and although widely used these are not 100% reliable for our benchmarking purpose [CITATION].', '1510.07611-2-43-4': 'Since these other proxies are meant to approximate the log-likelihood, if we were to use these we could not be sure that we would be drawing the right conclusions.', '1510.07611-2-43-5': 'This justifies why we used a moderately small dataset with 16 visible and 16 hidden units, and even though computing the log-likelihood was computational intensive for the study performed here, having 32 units in total was still a manageable size.', '1510.07611-2-43-6': 'Through the computation of the exact likelihood we were able to examine in more detail some of the goals proposed here: being able to assess the best effective temperature fit to the desired Boltzmann distribution and to show that using a constant temperature different from the estimated one with our approach might lead to severe suboptimal setting in terms of the training performance.', '1510.07611-2-44-0': 'Another aspect we explore in this study was to go beyond the conventional CD-1, with the purpose of having a more fair comparison to the results that might be expected from the entirely classical algorithm counterpart.', '1510.07611-2-44-1': 'Previous results from our research group [CITATION], as well as others reported by other researchers [CITATION], are limited to comparing the performance of quantum annealers to the quick but suboptimal CD-1.', '1510.07611-2-44-2': 'As shown in those studies, even with a suboptimal constant temperature one might be drawn to conclude that QuAT is outperforming conventional CD.', '1510.07611-2-44-3': 'Similar conclusions might be drawn from the curves for constant but suboptimal [MATH] and [MATH] vs. CD-1 in Fig. [REF].', '1510.07611-2-44-4': 'As shown in Fig. [REF], this conclusion does not hold anymore for higher values of [MATH], while the method using the effective-temperature estimation proposed here is the only one showing a steady increase in performance, close to matching the largest value of [MATH] tried here, i.e. [MATH].', '1510.07611-2-44-5': 'A follow up study of this work will assess how close QuAT is following the exact log-likelihood gradient during the training and how different are the distributions learned with the classical algorithm and the ones obtained with the QuAT algorithm.', '1510.07611-2-45-0': 'Another important point that we are currently studying is whether the differences observed in performance remain for larger and more complex datasets.', '1510.07611-2-45-1': 'We would expect that the performance of CD-[MATH] degrades with larger number of variables as equilibration times are expected to grow fast with the number of variables once the probability distribution starts having non-trivial structure.', '1510.07611-2-45-2': 'From this perspective, it is important to notice that QuAT is expected to display a more uniform exploration of configuration space.', '1510.07611-2-45-3': 'This feature may render unecessary adding auxiliary terms to prevent distortions in the learning process.', '1510.07611-2-45-4': 'Indeed, CD-[MATH] is usually regularized by adding a term that penalizes the size of the weights and that, if properly tuned, can avoid the drop in likelihood [CITATION].', '1510.07611-2-45-5': 'We can observe this effect in Fig. [REF] after about 500 iterations for CD-1 and about 800 iterations for CD-25, for instance.', '1510.07611-2-45-6': 'QuAT, instead, increases the likelihood slowly but steadily without the need of an explicit regularization, at least within the 3000 iterations that we have investigated.', '1510.07611-2-46-0': 'Several other interesting studies follow from having a robust method for estimating [MATH], and that we will be exploring to speedup the first version of our first QuAT protocol proposed here.', '1510.07611-2-46-1': 'For example, it seems reasonable to estimate it only in every certain number of training iterations.', '1510.07611-2-46-2': 'From Fig. [REF] it can be seen that a good strategy would be to do it more frequently at the beginning and less frequent as it stabilizes or seem to converge around certain value.', '1510.07611-2-46-3': 'Still this has the drawback that if by any chance the temperature happens to be estimated badly, such an error would persist for several iterations.', '1510.07611-2-46-4': 'Another simple strategy not used in this study is to recycle also the samples taken at the value of [MATH] and that was used thus far only for the estimation of [MATH].', '1510.07611-2-46-5': 'Since the two distributions are expected to be close enough, importance sampling as used in Ref. [CITATION] can be used to connect the obtained samples and still use them in the model average computed along with the samples at [MATH].', '1510.07611-2-47-0': 'There are other ways in which the ideas explored here could be extended.', '1510.07611-2-47-1': 'For instance, we can go beyond restricted Boltzmann machines to build deep learning architectures or beyond unsupervised learning to build discriminative models.', '1510.07611-2-47-2': "In principle the speed of learning could be increased by adding a 'momentum' term to the gradient-ascent learning rule [CITATION].", '1510.07611-2-47-3': 'Indeed, Addachi and Henderson have started exploring these ideas in a contemporary work [CITATION].', '1510.07611-2-47-4': 'Instead, we have focused on first trying to better understand the basics before adding more (classical) complexity to the learning algorithms that we feel have the risk that it might obscure the actual contributions from the new approach.', '1510.07611-2-48-0': 'Note that since CD-[MATH] seems to be faster than the QuAT in the first iterations in Fig. 3, having the capabilities of efficiently estimating effective temperatures allows us for a simple hybrid classical-quantum learning approach.', '1510.07611-2-48-1': 'For example, one can think of using the classical algorithm to be used to generate a reasonably good starting Boltzmann model for the QuAT algorithm.', '1510.07611-2-48-2': 'More specifically, one could use the simple and cheap CD-1 until the learning progress plateaus, and then use the expensive QuAT to bootstrap and enhance the classically obtained model.', '1510.07611-2-48-3': 'This may be useful in helping to diminish the larger impact that the noise in the device might have when the control parameters are somewhat small.', '1510.07611-2-48-4': 'This work will be explored in detail in a following publication.'}

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[]

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[]

['1510.07611-1-27-7', '1510.07611-1-28-0', '1510.07611-2-27-7', '1510.07611-2-28-0', '1510.07611-3-35-4', '1510.07611-3-36-0', '1510.07611-4-35-4', '1510.07611-4-36-0']

{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '3': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '4': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1510.07611

{'1510.07611-3-0-0': 'An increase in the efficiency of sampling from Boltzmann distributions would have a significant impact in deep learning and other machine learning applications.', '1510.07611-3-0-1': 'Recently, quantum annealers have been proposed as a potential candidate to speed up this task, but several limitations still bar these state-of-the-art technologies from being used effectively.', '1510.07611-3-0-2': 'One of the main limitations is that, while the device may indeed sample from a Boltzmann-like distribution, quantum dynamical arguments suggests it will do so with an instance-dependent effective temperature, different from its physical temperature.', '1510.07611-3-0-3': 'Unless this unknown temperature can be unveiled, it might not be possible to effectively use a quantum annealer for Boltzmann sampling.', '1510.07611-3-0-4': 'In this work, we propose a strategy to overcome this challenge with a simple effective-temperature estimation algorithm.', '1510.07611-3-0-5': 'We provide a systematic study assessing the impact of the effective temperatures in the learning of a kind of restricted Boltzmann machine embedded on quantum hardware, which can serve as a building block for deep learning architectures.', '1510.07611-3-0-6': 'We also provide a comparison to [MATH]-step contrastive divergence (CD-[MATH]) with [MATH] up to 100.', '1510.07611-3-0-7': 'Although assuming a suitable fixed effective temperature also allows to outperform one step contrastive divergence (CD-1), only when using an instance-dependent effective temperature we find a performance close to that of CD-100 for the case studied here.', '1510.07611-3-1-0': '# Introduction', '1510.07611-3-2-0': 'The use of quantum computing technologies for sampling and machine learning applications is attracting increasing interest from the research community in recent years [CITATION].', '1510.07611-3-2-1': 'Although the main focus of the quantum annealing computational paradigm [CITATION] has been in solving discrete optimization problems in a wide variety of application domains [CITATION], it has been also introduced as a potential candidate to speed up computations in sampling applications.', '1510.07611-3-2-2': 'Indeed, it is an important open research question whether or not quantum annealers can sample from Boltzmann distributions more efficiently than traditional techniques [CITATION].', '1510.07611-3-3-0': 'There are some challenges that need to be overcome before uncovering the potential of quantum annealing hardware for sampling problems.', '1510.07611-3-3-1': 'One of the main difficulties is that the device does not necessarily sample from the Boltzmann distribution associated with the physical temperature and the user-specified control parameters of the device.', '1510.07611-3-3-2': 'Instead, there might be some instance-dependent corrections leading, in principle, to instance-dependent effective temperature [CITATION].', '1510.07611-3-3-3': 'Bian et al. [CITATION] has used the maximum likelihood method to estimate such an instance-dependent temperature and introduced some additional shifts in the control parameters of the quantum device; this was done for several realizations of small eight-qubit instances on an early generation of quantum annealers produced by D-Wave Systems.', '1510.07611-3-3-4': 'The authors showed that, with these additional estimated shifts in place, the empirical probability distribution obtained from the D-Wave appears to correlate very well with the corresponding Boltzmann distribution.', '1510.07611-3-3-5': 'Further experimental evidence of this effective temperature can be found in Ref. [CITATION] where its proper estimation is needed to determine residual bias in the programmable parameters of the device.', '1510.07611-3-4-0': 'Recently, some authors have explored the use of quantum annealing hardware for the learning of Boltzmann machines and deep neural networks [CITATION].', '1510.07611-3-4-1': 'Learning a Boltzmann machine or a deep neural network is in general intractable due to long equilibration times of sampling techniques like Markov Chain Monte Carlo (MCMC).', '1510.07611-3-4-2': 'One of the strategies that have made possible the recent spectacular success [CITATION] of these techniques is to deal with less general architectures that allow for substantial algorithmic speedups.', '1510.07611-3-4-3': 'Restricted Boltzmann Machines (RBMs) [CITATION] are an important example of this kind that moreover serve as a suitable building block for deeper architectures.', '1510.07611-3-4-4': 'Still, quantum annealers have the potential to allow for learning more complex architectures.', '1510.07611-3-5-0': 'When applying quantum annealing hardware to the learning of Boltzmann machines, the interest is in finding the optimal control parameters that best represent the empirical distribution of a dataset.', '1510.07611-3-5-1': 'However, estimating additional shifts for the control parameters, as done by Bian et al. [CITATION], would not be practical since it is in a sense similar to the very kind of problem that a Boltzmann machine attempts to solve.', '1510.07611-3-5-2': 'One could then ask what is the meaning of using a quantum annealer for learning the parameters of a distribution, if to do so we need to use standard techniques to learn the corrections to the control parameters.', '1510.07611-3-6-0': 'Here we explore a different approach by taking into account only the possibility of an instance-dependent effective temperature without the need of considering further instance-dependent shifts in the control parameters.', '1510.07611-3-6-1': 'We devise a technique to estimate the effective temperature associated to a given instance by generating only two sets of samples from the machine and performing a linear regression.', '1510.07611-3-6-2': 'The samples used in our effective-temperature estimation algorithm are the same ones used towards achieving the final goal of the sampling application.', '1510.07611-3-6-3': 'This is in contrast with the approach taken in Ref. [CITATION] which needs many evaluations of the gradient of the log-likelihood of a set of samples from the device, making it impractical for large problem instances.', '1510.07611-3-7-0': 'We test our ideas in the learning of a kind of restricted Boltzmann machines.', '1510.07611-3-7-1': 'In the next section we shall present a brief overview of Boltzmann machines and discuss how quantum annealing hardware can be used to assist their learning.', '1510.07611-3-7-2': 'Afterwards, we discuss related work.', '1510.07611-3-7-3': 'In the section that follows we introduce our technique to estimate the effective temperature associated to a given instance.', '1510.07611-3-7-4': 'We then show an implementation of these ideas for our Quantum-Assisted Learning (QuALe) of a Chimera-RBM on the Bars And Stripes (BAS) dataset [CITATION], implemented in the D-Wave 2X device (DW2X) located at NASA Ames Research Center.', '1510.07611-3-7-5': 'Finally, we present the conclusions of the work and some perspectives of the future work we shall be exploring.', '1510.07611-3-8-0': '# General considerations', '1510.07611-3-9-0': '## Boltzmann machines', '1510.07611-3-10-0': 'Consider a binary data set [MATH] whose empiric distribution is [MATH]; here each datapoint can be represented as an array of Ising variables, i.e. [MATH] with [MATH], for [MATH].', '1510.07611-3-10-1': 'A Boltzmann machine models the data via a probability distribution [MATH], where [MATH] is a Boltzmann distribution on a possibly extended sample space [MATH].', '1510.07611-3-10-2': "Here [MATH] are the 'unobservable' or 'hidden' variables, that help capture higher level structure in the data [CITATION], and [MATH] are the 'visible' variables, that correspond to the data themselves.", '1510.07611-3-10-3': 'More precisely, denoting these variables collectively by [MATH] we can write [EQUATION] where [EQUATION] is the corresponding energy function, and [MATH] is the normalization constant or partition function.', '1510.07611-3-10-4': 'Notice that in this case we do not need a temperature parameter, since it only amounts at a rescaling of the model parameters [MATH] and [MATH] that we want to learn.', '1510.07611-3-10-5': 'Here [MATH] and [MATH] are the set of vertices and edges, respectively, that make up the interaction graph [MATH].', '1510.07611-3-11-0': 'The task is then to find the model parameters that make the model distribution [MATH] as close as possible to the data distribution [MATH].', '1510.07611-3-11-1': 'This can be accomplished by minimizing the Kullback-Leibler (KL) divergence [CITATION] [EQUATION] between [MATH] and [MATH] or, equivalently, by maximizing the average log-likelihood [EQUATION] with respect to the model parameters [MATH] and [MATH].', '1510.07611-3-12-0': 'Gradient ascent is a standard method to carry out this optimization via the rule [EQUATION] where [MATH] is the learning rate, and the gradient of the average log-likelihood function is given by [CITATION] [EQUATION]', '1510.07611-3-12-1': 'Here [MATH] denotes the ensemble average with respect to the distribution [MATH] that involves the data.', '1510.07611-3-12-2': 'Similarly, [MATH] denotes the ensemble average with respect to the distribution [MATH] that involves exclusively the model.', '1510.07611-3-12-3': 'Such averages can be estimated by standard sampling techniques, such as MCMC.', '1510.07611-3-12-4': 'Another possibility, explored in this work, is to rely on a physical process that naturally generates samples from a Boltzmann distribution.', '1510.07611-3-13-0': '## Quantum annealing', '1510.07611-3-14-0': 'Quantum annealing is an algorithm that attempts to exploit quantum effects to find the configurations with the lowest cost of a function describing a problem of interest [CITATION].', '1510.07611-3-14-1': 'It relies on finding a mapping of such a function into the energy function of an equivalent physical system.', '1510.07611-3-14-2': 'The latter is suitably modified to incorporate quantum fluctuations whose purpose is to maintain the system in its lowest-energy solution space.', '1510.07611-3-15-0': 'In short, the algorithm consists in slowly transforming the ground state of an initial quantum system, that is relatively easy to prepare, into the ground state of a final Hamiltonian that encodes the problem to be solved.', '1510.07611-3-15-1': 'The device produced by D-Wave Systems [CITATION] is a realization of this idea for solving quadratic unconstrained optimization problems on binary variables.', '1510.07611-3-15-2': 'It implements the Hamiltonian [EQUATION] where [MATH] are Pauli matrices that operate on spin or qubit [MATH].', '1510.07611-3-15-3': 'The control parameters of the D-Wave machine are composed of a field [MATH] for each qubit [MATH] and a coupling [MATH] for each pair of interacting qubits [MATH] and [MATH].', '1510.07611-3-15-4': 'The topology of the interactions between qubits in the D-Wave is given by a so-called Chimera graph [MATH].', '1510.07611-3-15-5': 'This is made up of elementary cells of [MATH] complete bipartite graphs that are coupled as shown in Fig. [REF] (a).', '1510.07611-3-15-6': 'The transformation from the simple Hamiltonian [MATH] to the problem Hamiltonian [MATH] is controlled by time-dependent monotonic functions [MATH] and [MATH], such that [MATH] and [MATH].', '1510.07611-3-15-7': 'Here [MATH], where [MATH] is the physical time and [MATH] is the annealing time, i.e. the time that it takes to transform Hamiltonian [MATH] into Hamiltonian [MATH].', '1510.07611-3-16-0': 'Although quantum annealers were designed with the purpose of reaching a ground state of the problem Hamiltonian [MATH], there are theoretical arguments [CITATION] and experimental evidence [CITATION] suggesting that under certain conditions the device can sample from an approximately Boltzmann distribution at a given effective temperature, as described in more detail in the next section.', '1510.07611-3-17-0': '## Quantum annealing for sampling applications', '1510.07611-3-18-0': 'There are many classical computations that are intrinsically hard and that might benefit from quantum technologies.', '1510.07611-3-18-1': "Common tasks include the factoring of large numbers into its basic primes, as is the case with Shor's algorithm [CITATION] in the gate model of quantum computation.", '1510.07611-3-18-2': 'Another one described above consists of finding the global minimum of a hard-to-optimize cost function, where quantum annealing is the most natural paradigm.', '1510.07611-3-18-3': 'As described at the end of Sec. [REF], another computationally hard problem, key for the successful training of Boltzmann machines and related machine learning tasks, is for example the estimation of averages [MATH] over probability distribution functions [MATH].', '1510.07611-3-18-4': 'In the case of models with a slow mixing rate, the standard MCMC approaches would have a hard time obtaining reliable samples from the probability distribution [MATH] [CITATION].', '1510.07611-3-18-5': 'As long as the quantum annealer can sample more reliably or more efficiently from this Boltzmann distribution, then we can find value in using it to solve a problem where MCMC might become intractable.', '1510.07611-3-18-6': 'It has been pointed out in the literature [CITATION] by several experts in the field that to a large extent the key to success of unsupervised learning relies on breakthroughs towards efficient sampling algorithms.', '1510.07611-3-19-0': 'Several key questions arise when considering quantum annealers as potential technologies for providing an algorithmic speed up in sampling applications.', '1510.07611-3-19-1': 'Why is a quantum annealer expected to sample from a classical Boltzmann distribution [MATH], given that it is a quantum device?', '1510.07611-3-19-2': "Shouldn't we expect the quantum annealer to sample from a quantum distribution instead?", '1510.07611-3-19-3': 'When and why could we expect the quantum annealer to do better than classical MCMC approaches?', '1510.07611-3-20-0': 'There are several competing dynamical processes happening at different time scales, with the time per annealing cycle being one, while decoherence and relaxation processes having their intrinsic timescale as well.', '1510.07611-3-20-1': 'For example, if the annealing time is much larger that the thermal equilibration timescale, the system will remain in its thermal equilibrium until the end of the annealing schedule.', '1510.07611-3-20-2': 'On the contrary, if it is too short, diabatic transitions promoting undesirable population flux from the ground state to excited states, would become relevant, leading it to be in a non-equilibrium state.', '1510.07611-3-21-0': 'For quantum annealers that have a strong interaction with the environment leading to relatively fast thermalization and decoherence, theory suggests that the relevant quantum dynamics during an annealing essentially freezes somewhere between the critical point associated with the minimum gap and the end of the annealing schedule [CITATION].', '1510.07611-3-21-1': 'In a quasistatic regime [CITATION], the system happens to be close to a Boltzmann distribution but at a certain effective temperature that is in general different from the physical temperature of the device.', '1510.07611-3-21-2': 'Such a freezing point [MATH] tends to coincide with the coefficients in Eq. [REF] satisfying [MATH], which suggests that the system being quantum annealed might end up in a Boltzmann distribution of the classical cost function encoded in [MATH].', '1510.07611-3-22-0': 'The intuition behind this phenomenon is that the dominant coupling of the qubits to the environment/bath degrees of freedom is via the [MATH] operator (for details, see supplementary material of Refs. [CITATION]).', '1510.07611-3-22-1': 'Since at the freezing point we have [MATH], and the interaction with the bath lacks a strong [MATH] component capable of causing relaxation between the states of the computational basis (i.e. eigenstates of [MATH]), then the system cannot relax its population anymore; in other words, its population dynamics freezes.', '1510.07611-3-22-2': 'Since around [MATH] the full Hamiltonian driving the dynamics is [MATH], if a Boltzmann distribution is indeed reached, it would correspond to an effective temperature [MATH] different from the physical temperature of the device.', '1510.07611-3-22-3': 'Here we will follow the convention that the units of temperature are given in a dimensionless energy scale where 1.0 is the maximum programmable value for the [MATH] couplers.', '1510.07611-3-22-4': 'According to Eq. [REF] the total Hamiltonian at the end of the annealing ([MATH]) is given by [MATH], so [MATH] would correspond to an energy value given by [MATH].', '1510.07611-3-22-5': 'For the DW2X at NASA, [MATH] corresponds to [MATH] GHz.', '1510.07611-3-22-6': 'For example, the physical fridge temperature of this quantum annealer, [MATH] mK, corresponds to [MATH] in the dimensionless units we follow in this paper.', '1510.07611-3-22-7': 'The effective temperature would be [MATH]; since [MATH], then [MATH].', '1510.07611-3-22-8': 'Such an effective temperature is expected to depend on the specific instance being studied and on the details of its energy landscape.', '1510.07611-3-22-9': 'Some recent unpublished work in our research team indicates that the effective temperature could also be influenced by the noise in the programmable parameters and by its interplay with the specific instance studied, making an a priori estimation a daunting task.', '1510.07611-3-22-10': 'The approach we take in this work is to estimate this effective temperature from the same samples that would be eventually used for the subsequent training process.', '1510.07611-3-23-0': 'We could wonder why a quantum annealer is expected to help in this computational task?', '1510.07611-3-23-1': 'It has been shown that quantum tunneling [CITATION] might be a powerful computational resource for keeping the system close to the ground state and to the proper thermal distribution.', '1510.07611-3-23-2': 'It is these quantum resources, available during the quantum dynamics before the freezing point, that might assist and speed up the thermalization process, making sampling more efficient than other classical approaches, such as MCMC.', '1510.07611-3-23-3': 'In this work we focus in unveiling the effective temperature that properly defines the distribution we are sampling from.', '1510.07611-3-23-4': 'The question of whether or not quantum mechanics is playing any role in the specific dataset considered in this paper is beyond the scope of this work and will be addressed in future work.', '1510.07611-3-24-0': '## Chimera restricted Boltzmann machines', '1510.07611-3-25-0': 'Learning a Boltzmann machine is in general intractable due to long equilibration time of sampling techniques like MCMC.', '1510.07611-3-25-1': 'One way to escape this issue is to use less general architectures.', '1510.07611-3-25-2': 'One of the most investigated architectures is the Restricted Boltzmann Machine (RBM).', '1510.07611-3-25-3': 'The interaction graph [MATH] of an RBM is a complete bipartite graph in which visible and hidden units interact with each other, but not among themselves.', '1510.07611-3-25-4': 'This implies that the conditional distributions [MATH] and [MATH] factorize in terms of single variable marginals, which substantially simplifies the problem.', '1510.07611-3-25-5': 'One the one hand, data averages [MATH] can be computed exactly in one shot.', '1510.07611-3-25-6': 'On the other hand, model averages [MATH] can be approximated by [MATH]-step contrastive divergence (CD-[MATH]): first, we start with a datapoint [MATH]; then we sample [MATH] from [MATH], and subsequently sample [MATH] from [MATH] and so on for [MATH] steps.', '1510.07611-3-25-7': 'At the end of this process we obtain samples [MATH] and [MATH] from which it is possible to estimate model averages [CITATION].', '1510.07611-3-25-8': 'CD-[MATH] is not guarantee to give correct results [CITATION] nor does it actually follow the log-likelihood gradient nor the gradient of any function indeed.', '1510.07611-3-25-9': 'Better sampling methods can have therefore a positive impact in the kind of models learned.', '1510.07611-3-26-0': 'It is in principle possible to embed an RBM in quantum annealing hardware [CITATION].', '1510.07611-3-26-1': 'However, due to limited connectivity of the device, the resulting physical representation would involve a number of qubits and couplings between them much larger than the number of logical variables and weights in the original RBM being represented.', '1510.07611-3-26-2': 'It would be preferable to use an alternative model that can be naturally represented in the device.', '1510.07611-3-26-3': 'For this reason we will focus on the kind of models that are obtained after removing from a given RBM all the links that are not present in the D-Wave machine [CITATION].', '1510.07611-3-26-4': 'We will call this type of model a Chimera Restricted Boltzmann Machine (Chimera-RBM).', '1510.07611-3-26-5': 'Fig. [REF] (a) shows an example of a Chimera-RBM and Fig. [REF] (b) shows a possible embedding of the pixels of an image into its visible units (cf. Ref. [CITATION]).', '1510.07611-3-27-0': '# Related work', '1510.07611-3-28-0': 'Dumoulin et al. [CITATION] have studied the impact of different limitations of quantum annealing hardware for learning restricted Boltzmann machines.', '1510.07611-3-28-1': 'The authors have focused on three kinds of limitations: noisy parameters, limited parameter range, and restricted architecture.', '1510.07611-3-28-2': 'The learning method used was persistent contrastive divergence where the model ensemble averages were estimated with samples from simulated quantum hardware while the data ensemble averages were estimated by exact mean field.', '1510.07611-3-29-0': 'For assessing the impact of limited connectivity, Dumoulin et al. investigated a Chimera-RBM.', '1510.07611-3-29-1': 'They found that limited connectivity is the most relevant limitation in this context.', '1510.07611-3-29-2': 'In a sense this is understandable as RBMs are based on complete bipartite graphs while Chimera-RBMs are sparse.', '1510.07611-3-29-3': 'Roughly speaking, this means that if the number of variables is of order [MATH], the number of parameters present in a Chimera-RBM is a vanishing fraction (of order [MATH]) of the number of parameters in the corresponding RBM.', '1510.07611-3-29-4': 'Furthermore, connections in a Chimera-RBM are rather localized.', '1510.07611-3-29-5': 'This feature may make more difficult to capture higher level correlations.', '1510.07611-3-30-0': 'The authors also found that noise in the parameters of an RBM is the next relevant limitation and that noise in the weights [MATH] is more relevant than noise in the biases [MATH].', '1510.07611-3-30-1': 'This could happen because the number of biases is a vanishing fraction of the number of weights in an RBM.', '1510.07611-3-30-2': 'This argument is not valid anymore in a Chimera-RBM, though.', '1510.07611-3-30-3': 'Now, the authors also mention that noise in the weights changes only when these change, while noise in the biases changes in every sample generated.', '1510.07611-3-30-4': 'If this is indeed the case, this could be another reason for the higher relevance of noise in the weights than noise in the biases.', '1510.07611-3-31-0': 'Finally, an upper bound in the magnitude of the model parameters, similar to the one present in the D-Wave device, does not seem to have much impact.', '1510.07611-3-31-1': 'In this respect, we should notice that current D-Wave devices are designed with the sole aim of consistently reaching the ground state.', '1510.07611-3-31-2': 'In contrast, typical applications of Boltzmann machines deals with heterogeneous real data which contains a relatively high level of uncertainty, and are expected to exploit a wider range of configurations.', '1510.07611-3-31-3': 'This suggests that in sampling applications control parameters are typically smaller than those explored in combinatorial optimization applications.', '1510.07611-3-31-4': 'If this is indeed the case, potential lower bounds in the magnitude of the control parameters can turn out to be more relevant for sampling applications.', '1510.07611-3-31-5': 'In this respect, it is important to notice that noise in the control parameters can lead to an effective lower bound.', '1510.07611-3-32-0': 'While Dumoulin et al. modeled the instance-dependent corrections as independent Gaussian noise around the user defined parameter values, Denil and Freitas [CITATION] devised a way to by-pass this problem altogether.', '1510.07611-3-32-1': 'For doing this, the authors have optimized the one-step reconstruction error as a black-box function and approximate its gradient empirically.', '1510.07611-3-32-2': 'They do this by a technique called simultaneous perturbation stochastic approximation.', '1510.07611-3-32-3': 'However, with this approach, it is not possible to decouple the model from the machine.', '1510.07611-3-32-4': 'Furthermore, it is not clear what is the efficiency of this technique nor how to extend it to deal with the more robust log-likelihood function instead of the reconstruction error.', '1510.07611-3-32-5': 'In their approach only the hidden layer is embedded in the D-Wave, and qubit interactions are exploited to build a semi-restricted Boltzmann machine.', '1510.07611-3-32-6': 'Although they report encouraging results, the authors acknowledge that these are still not conclusive.', '1510.07611-3-33-0': 'More recently, Adachi and Henderson [CITATION] have devised a way to embed an RBM on a D-Wave chip with Chimera topology.', '1510.07611-3-33-1': 'They do this by representing each logical variable by a string of qubits with strong ferromagnetic interactions.', '1510.07611-3-33-2': 'Furthermore, they implement a simple strategy to average out the effects of the noise in the D-Wave control parameters.', '1510.07611-3-33-3': 'They use the quantum annealer to estimate model averages as in Ref. [CITATION] for pre-training a two-layer neural network.', '1510.07611-3-33-4': 'However, the authors do not evaluate the performance of the quantum device at this stage; they rather post-train the model with (classical) discriminative techniques for learning the labels of a coarse-grained version of the MNIST dataset and compute the classification error.', '1510.07611-3-33-5': 'They report that this approach outperforms the standard approach where CD-1, instead of quantum annealing, is used for pre-training the generative model.', '1510.07611-3-34-0': '# Quantum-assisted learning of Boltzmann machines', '1510.07611-3-35-0': 'In this work we assume that quantum annealers, like those produced by D-Wave Systems, sample from a Boltzmann distribution defined by an energy function as in Eq. [REF], with [MATH] and [MATH], where [MATH] can be instance-dependent.', '1510.07611-3-35-1': 'While the control parameters for the D-Wave are couplings and fields, i.e. [MATH] and [MATH], the learning takes place on the ratio of the control parameters to the temperature, i.e. [MATH] and [MATH].', '1510.07611-3-35-2': 'Inferring temperature is therefore a fundamental step to be able to use samples from a device like D-Wave for learning, since it provides a translation from [MATH] to the [MATH] and from the [MATH] to the [MATH].', '1510.07611-3-35-3': 'We propose a quantum-assisted learning (QuALe) technique that includes an efficient estimation of the effective temperature.', '1510.07611-3-35-4': 'It is initialized as follows:', '1510.07611-3-36-0': 'Then it iterates as follows:', '1510.07611-3-37-0': 'A few comments are in order.', '1510.07611-3-37-1': 'First, for each sample step we need to generate samples for estimating model and data ensemble averages.', '1510.07611-3-37-2': 'For the former we just need to run the device with the specified control parameters.', '1510.07611-3-37-3': 'For the latter we need to generate samples with the visible units clamped to the data points, which can be done by applying suitable fields to the corresponding qubits.', '1510.07611-3-37-4': 'However, in the case of restricted Boltzmann machines we can avoid this last step as it is possible to compute exactly the data ensemble averages.', '1510.07611-3-37-5': 'Second, notice that to compute the new control parameters at step [MATH] it would have been ideal to estimate the temperature [MATH] at the same step.', '1510.07611-3-37-6': 'However, to estimate such a temperature we would need to know which are the parameters at time [MATH].', '1510.07611-3-37-7': 'To escape this vicious cycle we have done [MATH].', '1510.07611-3-37-8': 'Finally, notice that if we think the learning process in terms of the control parameters [MATH] and [MATH], we may get the impression that the learning rate is temperature-dependent.', '1510.07611-3-37-9': 'We would like to emphasize that the learning operates on the model parameters [MATH] and [MATH], which are those that actually shape the Boltzmann distribution, through the update rules given by Eqs. [REF] and [REF].', '1510.07611-3-37-10': 'So, the actual learning rate is given by [MATH] in the update equations above; if we fix [MATH] to a constant, it would remain so.', '1510.07611-3-37-11': 'We need [MATH] only to estimate the required control parameters.', '1510.07611-3-37-12': 'Still, the approximation [MATH] and the error in their estimation can introduce some noise that may deviate the learning process from the actual update rules given by Eqs. [REF] and [REF].', '1510.07611-3-37-13': 'It would be interesting to investigate what is the impact of this noise in contrast to that due to the estimation of the log-likelihood gradient with a finite number of samples.', '1510.07611-3-37-14': 'In the next section we discuss a method for estimating this instance-dependent temperature.', '1510.07611-3-38-0': '# Temperature estimation', '1510.07611-3-39-0': '## Extracting temperature from two sample sets', '1510.07611-3-40-0': 'At a generic inverse temperature [MATH], the probability of observing a configuration of energy [MATH] is given by [MATH] .', '1510.07611-3-40-1': 'Here [MATH] is the degeneracy of the energy level [MATH] and the normalization factor, [MATH], is the partition function.', '1510.07611-3-40-2': 'We want to devise an efficient method for estimating the effective temperature associated with a given instance.', '1510.07611-3-40-3': 'To do this, consider the log-ratio of probabilities associated with two different energy levels, [MATH] and [MATH], given by [EQUATION] where [MATH].', '1510.07611-3-40-4': 'We can estimate this log-ratio by estimating the frequencies of the two energy levels involved; in practice, we may have to do a suitable binning to have more robust statistics.', '1510.07611-3-40-5': 'Although we cannot control the physical temperature, we could in principle do this for different values of the parameter [MATH] by rescaling the control parameters of the device.', '1510.07611-3-40-6': 'Indeed, rescaling the control parameters by a factor [MATH].', '1510.07611-3-40-7': 'This is equivalent to setting a parameter [MATH], where [MATH] is the inverse of the effective temperature [MATH] associated to the instance of interest.', '1510.07611-3-40-8': 'Notice that this is only true under the assumption that [MATH], despite being generally dependent on arbitrary variations of the control parameters, does not change appreciably under these small rescalings.', '1510.07611-3-40-9': 'By plotting the log-ratio [MATH] against the scaling parameter [MATH], we should obtain a straight line whose slope and intercept are given by [MATH] and [MATH], respectively.', '1510.07611-3-40-10': 'Since we know the energy levels we can in principle infer [MATH].', '1510.07611-3-40-11': 'However, the performance of this method was rather poor in all experiments we carried out (not shown).', '1510.07611-3-40-12': 'A reason could be that to perform the linear regression and extract the corresponding effective temperature, several values of [MATH] need to be explored in a relatively wide range.', '1510.07611-3-40-13': 'Next we present a proposal that mitigates this limitation, which also happens to be much more efficient.', '1510.07611-3-41-0': 'The previous approach relied on several values of the scaling parameter [MATH] but only two energy levels.', '1510.07611-3-41-1': 'We were not exploiting all the information available in the other energy levels sampled from the quantum annealer.', '1510.07611-3-41-2': 'We can exploit such an information to obtain a more robust estimate of the temperature by sampling only for the original control parameters and a single rescaling of them.', '1510.07611-3-41-3': 'The idea is to take the difference [MATH], with [MATH] and [MATH], to eliminate the unknown degeneracies altogether, yielding [EQUATION] where [MATH].', '1510.07611-3-41-4': 'In this way, by generating a second set of samples at a suitable value of [MATH] and then taking the differences of, say, the most populated level and all the remaining relevant ones we can plot [MATH] against [MATH].', '1510.07611-3-41-5': 'According to Eq. [REF] this is expected to be a straight line with slope given by [MATH].', '1510.07611-3-41-6': 'Notice that to estimate the energy histograms from the two sets of samples, we compute the energies corresponding to the sampled configurations using only the original values of the control parameters in both cases, not the rescaled ones.', '1510.07611-3-41-7': 'This is because we have already counted the effect of the rescaling in a different inverse effective temperature [MATH].', '1510.07611-3-42-0': 'The choice of [MATH] matters: if it is too small no informative changes would be detected, other than noise due to finite sampling and uncontrolled physical processes in the device.', '1510.07611-3-42-1': 'If it is too large, some levels would become unpopulated and we would not be able to compare them at both the original and rescaled control parameters; moreover, the assumption of the invariance of [MATH] under small perturbations around the original control parameters would be less likely to be valid.', '1510.07611-3-42-2': 'Next we discuss some ideas on how to choose the value of [MATH].', '1510.07611-3-43-0': '## A rule of thumb for the scaling factor', '1510.07611-3-44-0': 'We can rely on concepts of information theory to guide the choice of the scaling factor [MATH].', '1510.07611-3-44-1': 'The idea here is to choose the value of [MATH] as close as possible to one that still allows us to distinguish between the two sets of samples of a given size.', '1510.07611-3-44-2': "Via Sanov's theorem, the KL divergence provides a natural way to characterize the notion of distinguishability in this case [CITATION].", '1510.07611-3-44-3': 'Here we will briefly discuss the main ideas in a rather informal way; the interested reader can refer to Ref. [CITATION] for details.', '1510.07611-3-44-4': 'We want to know whether we can distinguish between two Boltzmann distributions at different inverse temperatures [MATH] and [MATH] from a set of [MATH] samples.', '1510.07611-3-44-5': 'For doing this, it is useful to imagine that we compute the maximum likelihood estimate of the inverse temperature [MATH] from the sample set corresponding to inverse temperature [MATH].', '1510.07611-3-44-6': 'We can imagine that we repeat this procedure many times so we can compute the probability distribution of [MATH].', '1510.07611-3-44-7': 'The two Boltzmann distributions are said to be distinguishable from a set of [MATH] samples if the probability of [MATH] being close to [MATH] is smaller than a given tolerance [MATH], i.e. if [EQUATION] where [MATH] is some suitably small constant.', '1510.07611-3-44-8': "From Sanov's theorem it follows that when [MATH] is large enough [EQUATION] where the factor [MATH] gathers sub-dominant terms in [MATH].", '1510.07611-3-44-9': 'So, if [MATH] the two Boltzmann distributions are distinguishable in the sense defined above.', '1510.07611-3-45-0': 'Assuming that [MATH] and [MATH] are close enough, the KL divergence can be expanded up to second order to yield [EQUATION] where [EQUATION] is known in information theory as the Fisher information, or generalized succeptibility; in this case, it is essentially the specific heat.', '1510.07611-3-45-1': 'When [MATH] is large enough, the right hand side in Eq. [REF] becomes appreciable only for [MATH] and [MATH] very close.', '1510.07611-3-45-2': 'So, for large [MATH] we can replace the KL divergence by the Fisher information in Eq. [REF].', '1510.07611-3-46-0': 'Example: Consider a simple model of an Ising variable [MATH] on a field [MATH], i.e. [MATH].', '1510.07611-3-46-1': 'The maximum likelihood estimator of the inverse temperature from [MATH] independent samples [MATH] is [EQUATION]', '1510.07611-3-46-2': 'Using the central limit theorem we obtain for sufficiently large [MATH] and small enough [MATH] [EQUATION] where [MATH] and [MATH] is the Fisher information of a single spin.', '1510.07611-3-46-3': 'The models with [MATH] and [MATH] are distinguishable if [EQUATION] neglecting a term [MATH].', '1510.07611-3-46-4': 'For example, let us take arbitrarily [MATH], [MATH], and [MATH]; then the maximum value of the right hand is about one.', '1510.07611-3-47-0': 'Following these ideas, we propose to choose the scaling factor [MATH] such that [MATH], where [MATH] is a given constant (cf. Ref. [CITATION]).', '1510.07611-3-47-1': 'Eqs. [REF] and [REF] yield [EQUATION]', '1510.07611-3-47-2': 'Some remarks are in order: (i) Eq. [REF] gives a rule of thumb to choose a suitable value of [MATH] for estimating [MATH]; however, the latter also appears in this expression.', '1510.07611-3-47-3': 'To deal with this we will guess an inital value of [MATH] to estimate the initial temperature and then iteratively use the previous estimated temperature to pick [MATH] for the next temperature estimation.', '1510.07611-3-47-4': '(ii) The sign in Eq. [REF] could be chosen positive during the first iterations to avoid the rescaled control parameters to be below the noise level of the device, and negative afterwards to avoid the rescaled control parameters to be above the allowed range.', '1510.07611-3-47-5': '(iii) Since [MATH], Eq. [REF] says that [MATH], i.e. the larger [MATH] or [MATH] the closer [MATH] and [MATH] are.', '1510.07611-3-47-6': 'The dependence with [MATH] is a consequence of the fact that the more samples we have the closer models we can distinguish.', '1510.07611-3-47-7': 'The dependence with [MATH] arises because the average energy and corresponding standard deviation are typically [MATH] and [MATH], respectively.', '1510.07611-3-47-8': 'Therefore, for fixed [MATH] and [MATH], the overlap between the two corresponding distributions decreases with increasing [MATH].', '1510.07611-3-47-9': '(iv) Eq. [REF] has been derived assuming that values of the KL divergence about [MATH] can be well approximated with the Fisher information.', '1510.07611-3-47-10': 'These assumption may fail in practice when [MATH] is relatively small or when [MATH] is far from the reference value at [MATH].', '1510.07611-3-48-0': '# A few gadgets to improve performance', '1510.07611-3-49-0': 'In this section we discuss three techniques that help improve the performance of our quantum-assisted learning algorithm.', '1510.07611-3-49-1': 'First of all, it is known that the performance of quantum annealers can be significantly impaired by the presence of both persistent and random biases between the actual values of the control parameters and the user-specified values.', '1510.07611-3-49-2': 'Perdomo-Ortiz et al. [CITATION] have developed a technique for determining and correcting the persistent biases and have shown evidence that this recalibration procedure can enhance the performance of the device for solving combinatorial optimization problems.', '1510.07611-3-49-3': 'In the next section we will show evidence that correcting for persistent biases can also enhance the performance of quantum annealers for sampling applications.', '1510.07611-3-50-0': 'Second, noise in the control parameters can hinder the initial stage of learning, when these are typically small.', '1510.07611-3-50-1': 'In order to avoid this situation we can run CD-[MATH] for a few iterations until we find some meaningful initial values for the control parameters that are above the noise level of the device and then restart with QuALe.', '1510.07611-3-50-2': 'This is exclusively due to the current state of quantum annealing technologies and it is expeted to be further mitigated in new generations of these devices.', '1510.07611-3-50-3': 'We emphasize that the number of iterations with CD-1 has to be small to keep the weights within the dynamical range of the device.', '1510.07611-3-51-0': 'Finally, for estimating the effective temperature associated to a given instance we need to generate two sets of samples: one corresponding to the actual values of the control parameters that we are interested in, and another corresponding to these values rescaled by a factor [MATH].', '1510.07611-3-51-1': 'According to the discussion in the previous section, the scaling factor is chosen in such a way that the two probability distributions are as close as possible, yet distinguishable.', '1510.07611-3-51-2': 'So, we expect that the samples obtained at [MATH] can also be used for the estimation of the log-likelihood gradient, given by Eqs. [REF] and [REF], at [MATH] via the technique of importance sampling [CITATION].', '1510.07611-3-51-3': 'In short, we can use a set of samples [MATH] extracted from a Boltzmann distribution at inverse temperature [MATH] to estimate ensamble averages of an arbitrary observable [MATH] with a Boltzmann distribution at inverse temperature [MATH] as [EQUATION] where [MATH] is the ratio between the unnormalized probabilities.', '1510.07611-3-51-4': 'The approximation is expected to be good as long as the two distributions are close enough [CITATION].', '1510.07611-3-51-5': 'In the next section we will show evidence that including the set of samples corresponding to the rescaled control parameters indeed improves the performance of QuALe.', '1510.07611-3-52-0': 'From now on, when refering to the QuALe algorithm we imply that these three gadgets are also included, unless otherwise specified.', '1510.07611-3-53-0': '# Learning a Boltzmann machine assisted by the D-Wave 2X', '1510.07611-3-54-0': 'Now that we have at our disposal a robust temperature estimation technique, we can use it for learning Boltzmann machines.', '1510.07611-3-54-1': 'We decided to focus on the learning of a Chimera-RBM for two reasons.', '1510.07611-3-54-2': 'On the one hand, although an RBM can be embedded into quantum hardware [CITATION], it requires to represent single variables with chains of qubits coupled via ferromagnetic interactions of a given strength.', '1510.07611-3-54-3': 'Instead of forcing couplings to take a specific value to meet a preconceived design, it might be better to allow the learning algorithm itself to find the parameter values that work best for a particular application.', '1510.07611-3-54-4': 'On the other hand, the focus of our work is in better understanding the challenges that need to be overcome for using quantum annealers for sampling applications, and taking the necessary steps towards an effective implementation of deep learning applications on quantum annealers.', '1510.07611-3-55-0': 'To the best of our knowledge, this is the first systematic study providing both, an assessment of the use of the D-Wave in learning Boltzmann machines and studying the impact of the effective temperature in the learning performance.', '1510.07611-3-55-1': 'We consider that it is important to assess the performance of the different methods by computing the exact log-likelihood during the learning process.', '1510.07611-3-55-2': 'Otherwise, we could not be sure whether a difference in performance is due to the new learning method or due to errors in the approximation of the log-likelihood.', '1510.07611-3-55-3': 'For this reason we tested the method on a small synthetic dataset called Bars and Stripes (BAS) and computed exhaustively the corresponding log-likelihood for evaluation.', '1510.07611-3-55-4': 'The BAS dataset consists of [MATH] pictures generated by setting the four pixels of each row (or column) to either black (-1) or white (+1), at random [CITATION].', '1510.07611-3-55-5': 'Another reason to focus on this small synthetic dataset is that while generating, e.g., 2000 samples in the DW2X for a given instance can take about 40 ms, the waiting time for accessing the machine to generate a new set of samples for a different instance can vary widely depending on the amount of jobs that are scheduled.', '1510.07611-3-55-6': 'So, while running QuALe with 2000 samples per iteration on the whole chip (1097 qubits) for [MATH] iterations could take in principle about 7 minutes had we exclusive access to the device, the waiting times of the different jobs can increase this time by several orders of magnitude.', '1510.07611-3-56-0': 'We modeled the BAS dataset with a Chimera-RBM of 16 visible and 16 hidden units with the topology shown in Fig. [REF] (a).', '1510.07611-3-56-1': 'The mapping of pixels to visible units is shown in Fig. [REF] (b) (cf. [CITATION]).', '1510.07611-3-56-2': 'We run all algorithms with learning rate [MATH], which is the best value we found among five values in the range [MATH].', '1510.07611-3-56-3': 'To begin with, Fig. [REF] shows an instance of temperature estimation using [MATH] samples from the DW2X and [MATH], for some generic control parameters found during the learning process (cf. Fig. [REF]).', '1510.07611-3-56-4': 'This value of [MATH] is the one that worked best out of a few trial values; it corresponds to a value per variable which is about fifteen times larger than the lower bound estimated for a single independent variable after Eq. [REF].', '1510.07611-3-56-5': 'Fig. [REF]a shows the histograms corresponding to samples obtained at the actual control parameters (blue, shifted to the left) and the rescaled ones (orange, shifted to the right).', '1510.07611-3-56-6': 'A suitable binning of the energy levels has been done to get more robust statistics.', '1510.07611-3-56-7': 'Fig. [REF]b shows a plot of [MATH] against [MATH] for all energy values that appear in the overlap of the two histograms.', '1510.07611-3-56-8': 'We can observe a rather clear linear trend as predicted by Eq. [REF], which is confirmed by a relatively high regression coefficient, [MATH].', '1510.07611-3-56-9': 'From the slope [MATH] of the regression line we can obtain the effective temperature by solving [MATH].', '1510.07611-3-57-0': 'Fig. [REF]a shows the impact of bias correction on the performance of the QuALe algorithm.', '1510.07611-3-57-1': 'The performance is measured in terms of the average log-likelihood [MATH], which has been evaluated exhaustively every fifty iterations.', '1510.07611-3-57-2': 'These results are obtained by implementing the Chimera-RBM on five different locations of the DW2X chip and running the QuALe algorithm three times on each location, for a total of fifteen runs.', '1510.07611-3-57-3': 'The points correspond to the average of [MATH] over those fifteen runs and the error bars to one standard deviation.', '1510.07611-3-57-4': 'We can see that QuALe with persistent bias correction (blue crosses) outperforms QuALe without it (pink triangles).', '1510.07611-3-57-5': 'Fig [REF]b, on the other hand, shows the QuALe algorithm with (blue crosses) and without (pink triangles) taking into account the samples obtained at [MATH] for the estimation of the log-likelihood gradient, via importance sampling.', '1510.07611-3-57-6': 'The points correspond to the average of [MATH] over five runs of QuALe on a single location of the DW2X chip.', '1510.07611-3-57-7': 'Finally, Fig. [REF]c shows the positive impact of carrying out a few iterations of CD-1 to generate suitable initial conditions for QuALe.', '1510.07611-3-58-0': 'Fig. [REF] shows the evolution of [MATH] during the learning of a Chimera-RBM on the BAS dataset under different learning algorithms, all of them with learning rate [MATH].', '1510.07611-3-58-1': 'We can observe that the quantum assisted learning algorithm with effective-temperature estimation at each iteration (QuALe@[MATH], blue diagonal crosses) outperforms CD-[MATH] (blue solid squares) after about 300 iterations and CD-10 (green solid circles) after about 1500 iterations.', '1510.07611-3-58-2': 'However, within the 5000 iterations shown in the figure, QuALe@[MATH] has not yet been able to outperform CD-100, although there is a clear trend in that direction.', '1510.07611-3-58-3': 'Interestingly, all CD-[MATH] reach their best average performance after a relatively small number of iterations while QuALe@[MATH], in contrast, increases slowly and steadily.', '1510.07611-3-58-4': 'One may be inclined to think this is because CD-[MATH] estimates the model averages from samples generated by a [MATH]-step Markov chain initialized at each data point.', '1510.07611-3-58-5': 'In this way CD-[MATH] is using information contained in the data from the very beginning for the estimation of the model ensemble averages, while QuALe@[MATH] ignores them altogether.', '1510.07611-3-58-6': 'However, if this were indeed the case one should expect such a trend to diminish for increasing values of [MATH], something that is not observed in the figure.', '1510.07611-3-58-7': 'A better understanding of this point has the potential to considerably improve the performance of QuALe@[MATH].', '1510.07611-3-59-0': 'To assess the relevance of temperature estimation for QuALe@[MATH], we also show in Fig. [REF] the average performance under quantum assisted learning at a fixed temperature.', '1510.07611-3-59-1': 'First, it is worth mentioning that using the physical temperature of the device, [MATH] (corresponding to [MATH] mK as explained in the caption of Fig. [REF]), leads to a very poor performance, reaching values [MATH] (not shown).', '1510.07611-3-59-2': 'Fixing the temperature to the average QuALe@[MATH] over all temperatures found during the run of QuALe@[MATH] leads to a better performance (red empty circles), but still well below that displayed by QuALe@[MATH] itself.', '1510.07611-3-59-3': 'Fixing the temperature to [MATH] (QuALe@[MATH]) and to [MATH] (QuALe@[MATH]) leads to a decrease in performance with respect to that displayed with [MATH].', '1510.07611-3-60-0': 'In Fig. [REF] we can observe the variation of the effective temperature estimated during a window of 80 iterations of QuALe@[MATH] (green line).', '1510.07611-3-60-1': 'To evaluate whether such a variation is within the finite sampling error, we estimated the effective temperature 15 times at each iteration.', '1510.07611-3-60-2': 'The (blue) circles show the median of [MATH] and the error bars represent the corresponding first and third quartiles.', '1510.07611-3-60-3': 'Clearly, this variation cannot be explained as due to finite sampling error.', '1510.07611-3-60-4': 'We emphasize that during the execution of QuALe the effective temperature is estimated only once.', '1510.07611-3-61-0': '# Conclusions and Future Work', '1510.07611-3-62-0': 'Applications that rely on sampling, such as learning Boltzmann machines, are in general intractable due to long equilibration times of sampling techniques like MCMC.', '1510.07611-3-62-1': 'Some authors have conjectured quantum anneling could have an advantage in sampling applications.', '1510.07611-3-62-2': 'In this work we proposed a strategy to overcome one of the main limitations when intending to use a quantum annealer to sample from Boltzmann distributions: the determination of effective temperatures.', '1510.07611-3-62-3': 'The simple technique proposed in this work uses samples obtained from a quantum annealer (the DW2X at NASA is our experimental implementation) to estimate both the effective temperature and the hard-to-compute term in the log-likelihood gradient, i.e., the averages over the model distribution; these are needed to determine the next step in the learning process.', '1510.07611-3-62-4': 'We present a systematic study of the impact of the effective-temperature in the learning of a Chimera-RBM model with 16 visible and 16 hidden units.', '1510.07611-3-62-5': 'For doing so, we compared the QuALe algorithm with both instance-dependent effective temperature and different constant effective temperatures to the performance of a CD-[MATH] implementation, with [MATH] equal to [MATH], [MATH], and [MATH].', '1510.07611-3-63-0': 'The Chimera-RBM model itself is much less powerful than the RBM model.', '1510.07611-3-63-1': 'While the former is sparse with a number of parameters increasing linearly with the number of variables, the latter is dense with a number of parameters increasing quadratically.', '1510.07611-3-63-2': 'For instance, the Chimera-RBM that we have studied here, with 16 hidden and 16 visible variables, has only about 31% of the weight parameters that a corresponding RBM of the same size has.', '1510.07611-3-63-3': 'This is reflected in that a Chimera-RBM, learned either with QuALe or with standard classical techniques, struggles to generate samples faithfully resembling the [MATH] BAS dataset on which it was trained (not shown).', '1510.07611-3-63-4': 'In this first study, we have decided to omit any regularization of the learning process.', '1510.07611-3-63-5': 'We have done this to keep the focus as clear as possible on the potential gains obtained by using QuALe and to avoid the search of optimal regularization parameters that could be very expensive due to the accessing time to the DW2X.', '1510.07611-3-63-6': 'While this may lead to drops in likelihood [CITATION], we expect that the substantial reduction in the number of parameters mentioned above may act as an implicit regularizing sparsity constraint.', '1510.07611-3-63-7': 'Since we have neglected regularization altogether in all the learning algorithms, we expect the comparison to be fair.', '1510.07611-3-63-8': 'Moreover, as the work by Dumoulin et al. [CITATION] suggests, the Chimera-RBM model we have investigated has a limited expressive power.', '1510.07611-3-63-9': 'So we have decided to delay the investigation of the role of regularization for when we deal with more expressive models that can be naturally represented in a Chimera topology.', '1510.07611-3-64-0': 'RBMs have the nice feature that sampling in one layer conditioned to a configuration in the other layer can be done in parallel and in one step; this is one of the main reasons for their wide adoption.', '1510.07611-3-64-1': 'This feature does not hold true anymore once we have non-trivial lateral connections in one of the layers, which is the concept behind more powerful Boltzmann machines.', '1510.07611-3-64-2': 'We think this is one of the most promising directions to explore with the quantum-assisted learning (QuALe) algorithm.', '1510.07611-3-64-3': 'By restricting QuALe to study RBM or Chimera-RBM models, we are paying the price of using a device that is in principle more powerful, but we are not taking advantage of having a more general model.', '1510.07611-3-64-4': 'It is important to investigate how to take full advantage of the DW2X by designing more suitable models based on the Chimera graph.', '1510.07611-3-64-5': 'An interesting possibility is the one explored in Ref. [CITATION] where the Chimera graph of the DW2X is used as a hidden layer to build a semi-restricted Boltzmann machine, which therefore has lateral connections in the hidden layer.', '1510.07611-3-64-6': 'When dealing with more general Boltzmann machines it would be interesting to compare the performance of QuALe against mean field methods.', '1510.07611-3-64-7': 'Recently, there has been some interest in applying mean field techniques for learning restricted Boltzmann machines too [CITATION].', '1510.07611-3-64-8': "Future work should explore how the performance of mean field techniques compares with QuALe's.", '1510.07611-3-65-0': 'However, the goal of this first QuALe implementation on small Chimera-RBMs serves several purposes.', '1510.07611-3-65-1': 'When dealing with large datasets the log-likelihood cannot be exhaustively computed due to the intractability of computing the partition function.', '1510.07611-3-65-2': 'Log-likelihood is the gold standard metric, but it becomes intractable for large systems.', '1510.07611-3-65-3': 'In these cases, other performance metrics such as reconstruction error or cross-entropy error turn out to be more convinient, but although widely used, they are rough approximations to the log-likelihood [CITATION].', '1510.07611-3-65-4': 'If we were to use these proxies we could not be sure that we would be drawing the right conclusions.', '1510.07611-3-65-5': 'This justifies why we used a moderately small dataset with 16 visible and 16 hidden units, and even though computing the log-likelihood was computationally expensive for the study performed here, having 32 units in total was still a manageable size.', '1510.07611-3-65-6': 'Through the computation of the exact likelihood we were able to examine in more detail some of the goals proposed here: being able to assess the best effective temperature fit to the desired Boltzmann distribution and to show that using a constant temperature different from the one estimated with our approach might lead to severe suboptimal performance.', '1510.07611-3-66-0': 'Another aspect we explored in this study was to go beyond the conventional CD-1, with the purpose of having a fairer comparison to the results that might be expected from the entirely classical algorithm counterpart.', '1510.07611-3-66-1': 'Previous results from our research group [CITATION], as well as others reported by other researchers [CITATION], are limited to comparing the performance of quantum annealers to the quick but suboptimal CD-1.', '1510.07611-3-66-2': 'As shown in those studies, even with a suboptimal constant temperature one might be drawn to conclude that QuALe is outperforming conventional CD.', '1510.07611-3-66-3': 'Similar conclusions might be drawn from the curves for constant but suboptimal [MATH] and [MATH] vs. CD-1 in Fig. [REF].', '1510.07611-3-66-4': 'As shown in Fig. [REF], this conclusion does not hold anymore for higher values of [MATH], while the method using the effective-temperature estimation proposed here is the only one showing a steady increase in performance, close to matching the largest value of [MATH] tried here, i.e. [MATH].', '1510.07611-3-67-0': 'Another important point to investigate in the future is whether the differences observed in performance remain for larger and more complex datasets.', '1510.07611-3-67-1': 'We would expect that the performance of CD-[MATH] degrades with larger instances as equilibration times are expected to grow fast with the number of variables once the probability distribution starts having non-trivial structure.', '1510.07611-3-67-2': 'From this perspective, it is important to notice that QuALe is expected to display a more uniform exploration of configuration space.', '1510.07611-3-68-0': 'There are other ways in which the ideas explored here could be extended.', '1510.07611-3-68-1': 'For instance, we can go beyond restricted Boltzmann machines to build deep learning architectures or beyond unsupervised learning to build discriminative models.', '1510.07611-3-68-2': "In principle the speed of learning could be increased by adding a 'momentum' term to the gradient-ascent learning rule [CITATION].", '1510.07611-3-68-3': 'Indeed, Adachi and Henderson have started exploring these ideas in a contemporary work [CITATION].', '1510.07611-3-68-4': 'Instead, we have focused on first trying to better understand the basics before adding more (classical) complexity to the learning algorithms that we feel have the risk to obscure the actual contributions from the new approach.'}

{'1510.07611-4-0-0': 'An increase in the efficiency of sampling from Boltzmann distributions would have a significant impact on deep learning and other machine-learning applications.', '1510.07611-4-0-1': 'Recently, quantum annealers have been proposed as a potential candidate to speed up this task, but several limitations still bar these state-of-the-art technologies from being used effectively.', '1510.07611-4-0-2': 'One of the main limitations is that, while the device may indeed sample from a Boltzmann-like distribution, quantum dynamical arguments suggest it will do so with an instance-dependent effective temperature, different from its physical temperature.', '1510.07611-4-0-3': 'Unless this unknown temperature can be unveiled, it might not be possible to effectively use a quantum annealer for Boltzmann sampling.', '1510.07611-4-0-4': 'In this work, we propose a strategy to overcome this challenge with a simple effective-temperature estimation algorithm.', '1510.07611-4-0-5': 'We provide a systematic study assessing the impact of the effective temperatures in the learning of a special class of a restricted Boltzmann machine embedded on quantum hardware, which can serve as a building block for deep-learning architectures.', '1510.07611-4-0-6': 'We also provide a comparison to [MATH]-step contrastive divergence (CD-[MATH]) with [MATH] up to 100.', '1510.07611-4-0-7': 'Although assuming a suitable fixed effective temperature also allows us to outperform one step contrastive divergence (CD-1), only when using an instance-dependent effective temperature do we find a performance close to that of CD-100 for the case studied here.', '1510.07611-4-1-0': '# Introduction', '1510.07611-4-2-0': 'The use of quantum computing technologies for sampling and machine learning applications has attracted increasing interest from the research community in recent years [CITATION].', '1510.07611-4-2-1': 'Although the main focus of the quantum annealing computational paradigm [CITATION] has been on solving discrete optimization problems in a wide variety of application domains [CITATION], it has been also introduced as a potential candidate to speed up computations in sampling applications.', '1510.07611-4-2-2': 'Indeed, it is an important open research question whether or not quantum annealers can sample from Boltzmann distributions more efficiently than traditional techniques [CITATION].', '1510.07611-4-3-0': 'There are challenges that need to be overcome before uncovering the potential of quantum annealing hardware for sampling problems.', '1510.07611-4-3-1': 'One of the main difficulties is that the device does not necessarily sample from the Boltzmann distribution associated with the physical temperature and the user-specified control parameters of the device.', '1510.07611-4-3-2': 'Instead, there might be instance-dependent corrections leading, in principle, to instance-dependent effective temperature [CITATION].', '1510.07611-4-3-3': 'Bian et al. [CITATION] used the maximum-likelihood method to estimate such an instance-dependent temperature and introduced additional shifts in the control parameters of the quantum device; this was done for several realizations of small eight-qubit instances on an early generation of quantum annealers produced by D-Wave Systems.', '1510.07611-4-3-4': 'The authors showed that, with these additional estimated shifts in place, the empirical probability distribution obtained from the D-Wave appears to correlate very well with the corresponding Boltzmann distribution.', '1510.07611-4-3-5': 'Further experimental evidence of this effective temperature can be found in Ref. [CITATION] where its proper estimation is needed to determine residual bias in the programmable parameters of the device.', '1510.07611-4-4-0': 'Recent works have explored the use of quantum annealing hardware for the learning of Boltzmann machines and deep neural networks [CITATION].', '1510.07611-4-4-1': 'Learning of a Boltzmann machine or a deep neural network is in general intractable due to long equilibration times of sampling techniques like Markov chain Monte Carlo (MCMC) [CITATION].', '1510.07611-4-4-2': 'One of the strategies that have made possible the recent spectacular success [CITATION] of these techniques is to deal with less general architectures that allow for substantial algorithmic speedups.', '1510.07611-4-4-3': 'Restricted Boltzmann machines (RBMs) [CITATION] are an important example of this kind that, moreover, serve as a suitable building block for deeper architectures.', '1510.07611-4-4-4': 'Still, quantum annealers have the potential to allow for learning more complex architectures.', '1510.07611-4-5-0': 'When applying quantum annealing hardware to the learning of Boltzmann machines, the interest is in finding the optimal control parameters that best represent the empirical distribution of a dataset.', '1510.07611-4-5-1': 'However, estimating additional shifts for the control parameters, as done by Bian et al. [CITATION], would not be practical since it is in a sense similar to the very kind of problem that a Boltzmann machine attempts to solve.', '1510.07611-4-5-2': 'One could then ask what is the meaning of using a quantum annealer for learning the parameters of a distribution, if to do so we need to use standard techniques to learn the corrections to the control parameters.', '1510.07611-4-6-0': 'Here we explore a different approach by taking into account only the possibility of an instance-dependent effective temperature without the need to consider further instance-dependent shifts in the control parameters.', '1510.07611-4-6-1': 'We devise a technique to estimate the effective temperature associated with a given instance by generating only two sets of samples from the machine and performing a linear regression.', '1510.07611-4-6-2': 'The samples used in our effective-temperature estimation algorithm are the same ones used towards achieving the final goal of the sampling application.', '1510.07611-4-6-3': 'This is in contrast to the approach taken in Ref. [CITATION], which needs many evaluations of the gradient of the log likelihood of a set of samples from the device, making it impractical for large problem instances.', '1510.07611-4-7-0': 'We test our ideas in the learning of a special class of restricted Boltzmann machines.', '1510.07611-4-7-1': 'In the next section we shall present a brief overview of Boltzmann machines and discuss how quantum annealing hardware can be used to assist their learning.', '1510.07611-4-7-2': 'Afterwards, we discuss related work.', '1510.07611-4-7-3': 'In the section that follows we introduce our technique to estimate the effective temperature associated with a given instance.', '1510.07611-4-7-4': 'We then show an implementation of these ideas for our quantum-assisted learning (QuALe) of a chimera-RBM on the Bars And Stripes (BAS) dataset [CITATION], implemented in the D-Wave 2X device (DW2X) located at the NASA Ames Research Center.', '1510.07611-4-7-5': 'Finally, we present the conclusions of the work and some perspectives of the future work we shall be exploring.', '1510.07611-4-8-0': '# General considerations', '1510.07611-4-9-0': '## Boltzmann machines', '1510.07611-4-10-0': 'Consider a binary data set [MATH] whose empiric distribution is [MATH]; here each datapoint can be represented as an array of Ising variables, i.e. [MATH] with [MATH], for [MATH].', '1510.07611-4-10-1': 'A Boltzmann machine models the data via a probability distribution [MATH], where [MATH] is a Boltzmann distribution on a possibly extended sample space [MATH].', '1510.07611-4-10-2': "Here [MATH] are the 'unobservable' or 'hidden' variables, that help capture higher level structure in the data [CITATION], and [MATH] are the 'visible' variables, that correspond to the data themselves.", '1510.07611-4-10-3': 'More precisely, denoting these variables collectively by [MATH], we can write [EQUATION] where [EQUATION] is the corresponding energy function, and [MATH] is the normalization constant or partition function.', '1510.07611-4-10-4': 'Notice that in this case we do not need a temperature parameter, since it only amounts to a rescaling of the model parameters [MATH] and [MATH] that we want to learn.', '1510.07611-4-10-5': 'Here [MATH] and [MATH] are the set of vertices and edges, respectively, that make up the interaction graph [MATH].', '1510.07611-4-11-0': 'The task is then to find the model parameters that make the model distribution [MATH] as close as possible to the data distribution [MATH].', '1510.07611-4-11-1': 'This can be accomplished by minimizing the Kullback-Leibler (KL) divergence [CITATION] [EQUATION] between [MATH] and [MATH] or, equivalently, by maximizing the average log likelihood [EQUATION] with respect to the model parameters [MATH] and [MATH].', '1510.07611-4-12-0': 'Gradient ascent is a standard method to carry out this optimization via the rule [EQUATION] where [MATH] is the learning rate, and the gradient of the average log-likelihood function is given by [CITATION] [EQUATION]', '1510.07611-4-12-1': 'Here [MATH] denotes the ensemble average with respect to the distribution [MATH] that involves the data.', '1510.07611-4-12-2': 'Similarly, [MATH] denotes the ensemble average with respect to the distribution [MATH] that involves exclusively the model.', '1510.07611-4-12-3': 'Such averages can be estimated by standard sampling techniques, such as MCMC.', '1510.07611-4-12-4': 'Another possibility, explored in this work, is to rely on a physical process that naturally generates samples from a Boltzmann distribution.', '1510.07611-4-13-0': '## Quantum annealing', '1510.07611-4-14-0': 'Quantum annealing is an algorithm that attempts to exploit quantum effects to find the configurations with the lowest cost of a function describing a problem of interest [CITATION].', '1510.07611-4-14-1': 'It relies on finding a mapping of such a function into the energy function of an equivalent physical system.', '1510.07611-4-14-2': 'The latter is suitably modified to incorporate quantum fluctuations whose purpose is to maintain the system in its lowest-energy solution space.', '1510.07611-4-15-0': 'In short, the algorithm consists of slowly transforming the ground state of an initial quantum system, which is relatively easy to prepare, into the ground state of a final Hamiltonian that encodes the problem to be solved.', '1510.07611-4-15-1': 'The device produced by D-Wave Systems [CITATION] is a realization of this idea for solving quadratic unconstrained optimization problems on binary variables.', '1510.07611-4-15-2': 'It implements the Hamiltonian [EQUATION] where [MATH] are Pauli matrices that operate on spin or qubit [MATH].', '1510.07611-4-15-3': 'The control parameters of the D-Wave machine are composed of a field [MATH] for each qubit [MATH] and a coupling [MATH] for each pair of interacting qubits [MATH] and [MATH].', '1510.07611-4-15-4': 'The topology of the interactions between qubits in the D-Wave is given by a so-called Chimera graph [MATH].', '1510.07611-4-15-5': 'This is made up of elementary cells of [MATH] complete bipartite graphs that are coupled as shown in Fig. [REF] (a).', '1510.07611-4-15-6': 'The transformation from the simple Hamiltonian [MATH] to the problem Hamiltonian [MATH] is controlled by time-dependent monotonic functions [MATH] and [MATH], such that [MATH] and [MATH].', '1510.07611-4-15-7': 'Here [MATH], where [MATH] is the physical time and [MATH] is the annealing time, i.e., the time that it takes to transform Hamiltonian [MATH] into Hamiltonian [MATH].', '1510.07611-4-16-0': 'Although quantum annealers were designed with the purpose of reaching a ground state of the problem Hamiltonian [MATH], there are theoretical arguments [CITATION] and experimental evidence [CITATION] suggesting that under certain conditions the device can sample from an approximately Boltzmann distribution at a given effective temperature, as described in more detail in the next section.', '1510.07611-4-17-0': '## Quantum annealing for sampling applications', '1510.07611-4-18-0': 'There are many classical computations that are intrinsically hard and that might benefit from quantum technologies.', '1510.07611-4-18-1': "Common tasks include the factoring of large numbers into its basic primes, as is the case with Shor's algorithm [CITATION] in the gate model of quantum computation.", '1510.07611-4-18-2': 'Another one described above consists of finding the global minimum of a hard-to-optimize cost function, where quantum annealing is the most natural paradigm.', '1510.07611-4-18-3': 'As described at the end of Sec. [REF], another computationally hard problem, key for the successful training of Boltzmann machines and related machine learning tasks, is for example the estimation of averages [MATH] over probability distribution functions [MATH].', '1510.07611-4-18-4': 'In the case of models with a slow mixing rate, the standard MCMC approaches would have a hard time obtaining reliable samples from the probability distribution [MATH] [CITATION].', '1510.07611-4-18-5': 'As long as the quantum annealer can sample more reliably or more efficiently from this Boltzmann distribution, then we can find value in using it to solve a problem where MCMC might become intractable.', '1510.07611-4-18-6': 'It has been pointed out in the literature [CITATION] by several experts in the field that to a large extent the key to success of unsupervised learning relies on breakthroughs in efficient sampling algorithms.', '1510.07611-4-19-0': 'Several key questions arise when considering quantum annealers as potential technologies for providing an algorithmic speed up in sampling applications.', '1510.07611-4-19-1': 'Why is a quantum annealer expected to sample from a classical Boltzmann distribution [MATH], given that it is a quantum device?', '1510.07611-4-19-2': "Shouldn't we expect the quantum annealer to sample from a quantum distribution instead?", '1510.07611-4-19-3': 'When and why should we expect the quantum annealer to do better than classical MCMC approaches?', '1510.07611-4-20-0': 'There are several competing dynamical processes happening at different time scales, with the time per annealing cycle being one, and decoherence and relaxation processes having their intrinsic timescale as well.', '1510.07611-4-20-1': 'For example, if the annealing time is much larger that the thermal equilibration timescale, the system will remain in its thermal equilibrium until the end of the annealing schedule.', '1510.07611-4-20-2': 'On the contrary, if it is too short, diabatic transitions promoting undesirable population flux from the ground state to excited states, would become relevant, leading it to be in a non-equilibrium state.', '1510.07611-4-21-0': 'For quantum annealers that have a strong interaction with the environment leading to relatively fast thermalization and decoherence, theory suggests that the relevant quantum dynamics during an annealing essentially freezes somewhere between the critical point associated with the minimum gap and the end of the annealing schedule [CITATION].', '1510.07611-4-21-1': 'In a quasistatic regime [CITATION], the system happens to be close to a Boltzmann distribution but at a certain effective temperature that is in general different from the physical temperature of the device.', '1510.07611-4-21-2': 'Such a freezing point [MATH] tends to coincide with the coefficients in Eq. [REF] satisfying [MATH], which suggests that the system being quantum annealed might end up in a Boltzmann distribution of the classical cost function encoded in [MATH].', '1510.07611-4-22-0': 'The intuition behind this phenomenon is that the dominant coupling of the qubits to the environment or bath degrees of freedom is via the [MATH] operator (for details, see the supplementary material of Refs. [CITATION]).', '1510.07611-4-22-1': 'Since at the freezing point we have [MATH], and the interaction with the bath lacks a strong [MATH] component capable of causing relaxation between the states of the computational basis (i.e., eigenstates of [MATH]), the system cannot relax its population anymore; in other words, its population dynamics freezes.', '1510.07611-4-22-2': 'Since around [MATH] the full Hamiltonian driving the dynamics is [MATH], if a Boltzmann distribution is indeed reached, it would correspond to an effective temperature [MATH] different from the physical temperature of the device.', '1510.07611-4-22-3': 'Here we will follow the convention that the units of temperature are given in a dimensionless energy scale where 1.0 is the maximum programmable value for the [MATH] couplers.', '1510.07611-4-22-4': 'According to Eq. [REF] the total Hamiltonian at the end of the annealing ([MATH]) is given by [MATH], so [MATH] would correspond to an energy value given by [MATH].', '1510.07611-4-22-5': 'For the DW2X at NASA, [MATH] corresponds to [MATH] GHz.', '1510.07611-4-22-6': 'For example, the physical fridge temperature of this quantum annealer, [MATH] mK, corresponds to [MATH] in the dimensionless units we follow in this paper.', '1510.07611-4-22-7': 'The effective temperature would be [MATH]; since [MATH], then [MATH].', '1510.07611-4-22-8': 'Such an effective temperature is expected to depend on the specific instance being studied and on the details of its energy landscape.', '1510.07611-4-22-9': 'Some recent unpublished work of our research team indicates that the effective temperature could also be influenced by the noise in the programmable parameters and by its interplay with the specific instance studied, making an a priori estimation a daunting task.', '1510.07611-4-22-10': 'The approach we take in this work is to estimate this effective temperature from the same samples that would be eventually used for the subsequent training process.', '1510.07611-4-23-0': 'We could wonder why a quantum annealer is expected to help in this computational task?', '1510.07611-4-23-1': 'It has been shown that quantum tunneling [CITATION] might be a powerful computational resource for keeping the system close to the ground state and to the proper thermal distribution.', '1510.07611-4-23-2': 'It is these quantum resources, available during the quantum dynamics before the freezing point, that might assist and speed up the thermalization process, making sampling more efficient than other classical approaches, such as MCMC.', '1510.07611-4-23-3': 'It is important to mention that such a quantum advantage is not expected for all energy landscapes; there will be instances that will be hard for both classical annealers and quantum annealers.', '1510.07611-4-23-4': 'The answer to this question will be highly dependent on the quantum resources available and on the complexity of the energy landscape itself.', '1510.07611-4-23-5': 'This is an important and interesting question in its own right that we will address in future work.', '1510.07611-4-23-6': 'In this work we focus on unveiling the effective temperature that properly defines the distribution we are sampling from and test our method in the context of a machine-learning problem related to the training of Boltzmann machines.', '1510.07611-4-24-0': '## Chimera restricted Boltzmann machines', '1510.07611-4-25-0': 'Learning of a Boltzmann machine is, in general, intractable due to the long equilibration time of sampling techniques like MCMC.', '1510.07611-4-25-1': 'One way to escape this issue is to use less general architectures.', '1510.07611-4-25-2': 'One of the most investigated architectures is the RBM.', '1510.07611-4-25-3': 'The interaction graph [MATH] of an RBM is a complete bipartite graph in which visible and hidden units interact with each other, but not among themselves.', '1510.07611-4-25-4': 'This implies that the conditional distributions [MATH] and [MATH] factorize in terms of single variable marginals, which substantially simplifies the problem.', '1510.07611-4-25-5': 'One the one hand, data averages [MATH] can be computed exactly in one shot.', '1510.07611-4-25-6': 'On the other hand, model averages [MATH] can be approximated by [MATH]-step contrastive divergence (CD-[MATH]): first, we start with a datapoint [MATH]; then we sample [MATH] from [MATH], and subsequently sample [MATH] from [MATH] and so on for [MATH] steps.', '1510.07611-4-25-7': 'At the end of this process we obtain samples [MATH] and [MATH] from which it is possible to estimate model averages [CITATION].', '1510.07611-4-25-8': 'CD-[MATH] is not guaranteed to give correct results [CITATION] nor does it actually follow the log-likelihood gradient or, indeed, the gradient of any function.', '1510.07611-4-25-9': 'Better sampling methods can have, therefore, a positive impact on the kind of models learned.', '1510.07611-4-26-0': 'It is, in principle, possible to embed an RBM in quantum annealing hardware [CITATION].', '1510.07611-4-26-1': 'However, due to limited connectivity of the device, the resulting physical representation would involve a number of qubits and couplings between them much larger than the number of logical variables and weights in the original RBM being represented.', '1510.07611-4-26-2': 'It would be preferable to use an alternative model that can be naturally represented in the device.', '1510.07611-4-26-3': 'For this reason we will focus on the kinds of models that are obtained after removing from a given RBM all the links that are not present in the D-Wave machine [CITATION].', '1510.07611-4-26-4': 'We will call this type of model a Chimera Restricted Boltzmann Machine (Chimera-RBM).', '1510.07611-4-26-5': 'Figure [REF] -(a) shows an example of a Chimera-RBM, and Fig. [REF] (b) shows a possible embedding of the pixels of an image into its visible units (cf. Ref. [CITATION]).', '1510.07611-4-27-0': '# Related work', '1510.07611-4-28-0': 'Dumoulin et al. [CITATION] have studied the impact of different limitations of quantum annealing hardware for the learning of restricted Boltzmann machines.', '1510.07611-4-28-1': 'The authors have focused on three kinds of limitations: noisy parameters, limited parameter range, and restricted architecture.', '1510.07611-4-28-2': 'The learning method used was persistent contrastive divergence where the model ensemble averages were estimated with samples from simulated quantum hardware while the data ensemble averages were estimated by exact mean field.', '1510.07611-4-29-0': 'To assess the impact of limited connectivity, Dumoulin et al. investigated a Chimera-RBM.', '1510.07611-4-29-1': 'They found that limited connectivity is the most relevant limitation in this context.', '1510.07611-4-29-2': 'In a sense this is understandable as RBMs are based on complete bipartite graphs while Chimera-RBMs are sparse.', '1510.07611-4-29-3': 'Roughly speaking, this means that if the number of variables is of order [MATH], the number of parameters present in a Chimera-RBM is a vanishing fraction (of order [MATH]) of the number of parameters in the corresponding RBM.', '1510.07611-4-29-4': 'Furthermore, connections in a Chimera-RBM are rather localized.', '1510.07611-4-29-5': 'This feature may make capturing higher-level correlations more difficult.', '1510.07611-4-30-0': 'The authors also found that noise in the parameters of an RBM is the next relevant limitation and that noise in the weights [MATH] is more relevant than noise in the biases [MATH].', '1510.07611-4-30-1': 'This could happen because the number of biases is a vanishing fraction of the number of weights in an RBM.', '1510.07611-4-30-2': 'This argument is no longer valid in a Chimera-RBM, however.', '1510.07611-4-30-3': 'The authors also mentioned that noise in the weights changes only when the instance changes, while noise in the biases changes in every sample generated.', '1510.07611-4-30-4': 'If this is indeed the case, this could be another reason for the higher relevance of noise in the weights than noise in the biases.', '1510.07611-4-31-0': 'Finally, an upper bound in the magnitude of the model parameters, similar to the one present in the D-Wave device, does not seem to have much impact.', '1510.07611-4-31-1': 'In this respect, we should notice that current D-Wave devices are designed with the sole aim of consistently reaching the ground state.', '1510.07611-4-31-2': 'In contrast, typical applications of Boltzmann machines deals with heterogeneous real data which contain a relatively high level of uncertainty, and are expected to exploit a wider range of configurations.', '1510.07611-4-31-3': 'This suggests that in sampling applications control parameters are typically smaller than those explored in combinatorial optimization applications.', '1510.07611-4-31-4': 'If this is indeed the case, potential lower bounds in the magnitude of the control parameters can turn out to be more relevant for sampling applications.', '1510.07611-4-31-5': 'In this respect, it is important to notice that noise in the control parameters can lead to an effective lower bound.', '1510.07611-4-32-0': 'While Dumoulin et al. modeled the instance-dependent corrections as independent Gaussian noise around the user defined parameter values, Denil and De Freitas [CITATION] devised a way to by-pass this problem altogether.', '1510.07611-4-32-1': 'To do this, the authors optimized the one-step reconstruction error as a black-box function and approximated its gradient empirically.', '1510.07611-4-32-2': 'They did this using a technique called simultaneous perturbation stochastic approximation.', '1510.07611-4-32-3': 'However, with this approach, it is not possible to decouple the model from the machine.', '1510.07611-4-32-4': 'Furthermore, it is not clear what the efficiency of this technique is or how to extend it to deal with the more robust log-likelihood function instead of the reconstruction error.', '1510.07611-4-32-5': 'In their approach only the hidden layer is embedded in the D-Wave, and qubit interactions are exploited to build a semi-restricted Boltzmann machine.', '1510.07611-4-32-6': 'Although they reported encouraging results, the authors acknowledged that they are still not conclusive.', '1510.07611-4-33-0': 'More recently, Adachi and Henderson [CITATION] devised a way to embed an RBM on a D-Wave chip with Chimera topology.', '1510.07611-4-33-1': 'They did this by representing each logical variable by a string of qubits with strong ferromagnetic interactions.', '1510.07611-4-33-2': 'Furthermore, they implemented a simple strategy to average out the effects of the noise in the D-Wave control parameters.', '1510.07611-4-33-3': 'They used the quantum annealer to estimate model averages as in Ref. [CITATION] for pre-training a two-layer neural network.', '1510.07611-4-33-4': 'However, the authors did not evaluate the performance of the quantum device at this stage; they rather post trained the model with (classical) discriminative techniques for learning the labels of a coarse-grained version of a well-known data set of handwritten digits called MNIST and computed the classification error.', '1510.07611-4-33-5': 'They reported that this approach outperforms the standard approach where CD-1, instead of quantum annealing, is used for pre training the generative model.', '1510.07611-4-34-0': '# Quantum-assisted learning of Boltzmann machines', '1510.07611-4-35-0': 'In this work we assume that quantum annealers, like those produced by D-Wave Systems, sample from a Boltzmann distribution defined by an energy function as in Eq. [REF], with [MATH] and [MATH], where [MATH] can be instance-dependent.', '1510.07611-4-35-1': 'While the control parameters for the D-Wave are couplings and fields, i.e. [MATH] and [MATH], the learning takes place on the ratio of the control parameters to the temperature, i.e. [MATH] and [MATH].', '1510.07611-4-35-2': 'Inferring temperature is therefore a fundamental step to be able to use samples from a device like D-Wave for learning, since it provides a translation from [MATH] to [MATH] and from [MATH] to [MATH].', '1510.07611-4-35-3': 'We propose a QuALe technique that includes an efficient estimation of the effective temperature.', '1510.07611-4-35-4': 'It is initialized as follows:', '1510.07611-4-36-0': 'Then the technique iterates as follows:', '1510.07611-4-37-0': 'A few comments are in order.', '1510.07611-4-37-1': 'First, for each sample step we need to generate samples for estimating model and data ensemble averages.', '1510.07611-4-37-2': 'For the former we just need to run the device with the specified control parameters.', '1510.07611-4-37-3': 'For the latter we need to generate samples with the visible units clamped to the data points, which can be done by applying suitable fields to the corresponding qubits.', '1510.07611-4-37-4': 'However, in the case of restricted Boltzmann machines we can avoid this last step as it is possible to compute exactly the data ensemble averages.', '1510.07611-4-37-5': 'Second, notice that to compute the new control parameters in step [MATH] it would have been ideal to estimate the temperature [MATH] in the same step.', '1510.07611-4-37-6': 'However, to estimate such a temperature we would need to know what the parameters are at time [MATH].', '1510.07611-4-37-7': 'To escape this vicious cycle we have set [MATH].', '1510.07611-4-37-8': 'Finally, notice that if we think of the learning process in terms of the control parameters [MATH] and [MATH], we may get the impression that the learning rate is temperature-dependent.', '1510.07611-4-37-9': 'We would like to emphasize that the learning operates on the model parameters [MATH] and [MATH], which are the one that actually shape the Boltzmann distribution through the update rules given by Eqs. [REF] and [REF].', '1510.07611-4-37-10': 'So, the actual learning rate is given by [MATH] in the update equations above; if we fix [MATH] to a constant, it would remain so.', '1510.07611-4-37-11': 'We need [MATH] only to estimate the required control parameters.', '1510.07611-4-37-12': 'Still, the approximation [MATH] and the error in their estimation can introduce noise that cause the learning process to deviate from the actual update rules given by Eqs. [REF] and [REF].', '1510.07611-4-37-13': 'It would be interesting to investigate what the impact of this noise is in contrast to that due to the estimation of the log-likelihood gradient with a finite number of samples.', '1510.07611-4-37-14': 'In the next section we discuss a method for estimating this instance-dependent temperature.', '1510.07611-4-38-0': '# Temperature estimation', '1510.07611-4-39-0': '## Extracting temperature from two sample sets', '1510.07611-4-40-0': 'At a generic inverse temperature [MATH], the probability of observing a configuration of energy [MATH] is given by [MATH] .', '1510.07611-4-40-1': 'Here [MATH] is the degeneracy of the energy level [MATH] and the normalization factor, [MATH], is the partition function.', '1510.07611-4-40-2': 'We want to devise an efficient method for estimating the effective temperature associated with a given instance.', '1510.07611-4-40-3': 'To do this, consider the log-ratio of probabilities associated with two different energy levels, [MATH] and [MATH], given by [EQUATION] where [MATH].', '1510.07611-4-40-4': 'We can estimate this log-ratio by estimating the frequencies of the two energy levels involved; in practice, we may have to do a suitable binning to have more robust statistics.', '1510.07611-4-40-5': 'Although we cannot control the physical temperature, we could in principle do this for different values of the parameter [MATH] by rescaling the control parameters of the device by a factor [MATH].', '1510.07611-4-40-6': 'This is equivalent to setting a parameter [MATH], where [MATH] is the inverse of the effective temperature [MATH] associated with the instance of interest.', '1510.07611-4-40-7': 'Notice that this is only true under the assumption that [MATH], despite being generally dependent on arbitrary variations of the control parameters, does not change appreciably under these small rescalings.', '1510.07611-4-40-8': 'By plotting the log-ratio [MATH] against the scaling parameter [MATH], we should obtain a straight line whose slope and intercept are given by [MATH] and [MATH], respectively.', '1510.07611-4-40-9': 'Since we know the energy levels we can in principle infer [MATH].', '1510.07611-4-40-10': 'However, the performance of this method was rather poor in all experiments we carried out (not shown).', '1510.07611-4-40-11': 'A reason could be that to perform the linear regression and extract the corresponding effective temperature, several values of [MATH] need to be explored in a relatively wide range.', '1510.07611-4-40-12': 'Next we present a proposal that mitigates this limitation, which also happens to be much more efficient.', '1510.07611-4-41-0': 'The previous approach relied on several values of the scaling parameter [MATH] but only two energy levels.', '1510.07611-4-41-1': 'We were not exploiting all the information available in the other energy levels sampled from the quantum annealer.', '1510.07611-4-41-2': 'We can exploit such information to obtain a more robust estimate of the temperature by sampling only for the original control parameters and a single rescaling of them.', '1510.07611-4-41-3': 'The idea is to take the difference [MATH], with [MATH] and [MATH], to eliminate the unknown degeneracies altogether, yielding [EQUATION] where [MATH].', '1510.07611-4-41-4': 'In this way, by generating a second set of samples at a suitable value of [MATH] and then taking the differences of all pairs of populated levels, we can plot [MATH] against [MATH].', '1510.07611-4-41-5': 'According to Eq. [REF] this is expected to be a straight line with slope given by [MATH].', '1510.07611-4-41-6': 'In practice, one has to choose a binning strategy and use the same bin intervals in both histograms so that the overlap makes sense.', '1510.07611-4-41-7': 'For example, by setting the number of bins to [MATH], where [MATH] is the number of samples per set, one obtains [MATH] data points for linear regression.', '1510.07611-4-41-8': 'Notice that the raw energies computed before binning refer to the original values of the control parameters in both cases, not the rescaled ones.', '1510.07611-4-41-9': 'This is because we have already counted the effect of the rescaling in a different inverse effective temperature [MATH].', '1510.07611-4-41-10': 'Finally, the energy levels obtained after binning correspond to the midpoint of each bin.', '1510.07611-4-42-0': 'The choice of [MATH] matters: if it is too small no informative changes would be detected, other than noise due to finite sampling and uncontrolled physical processes in the device.', '1510.07611-4-42-1': 'If it is too large, several levels would become unpopulated and we would not be able to compare them at both the original and rescaled control parameters; moreover, the assumption of the invariance of [MATH] under small perturbations around the original control parameters would be less likely to be valid.', '1510.07611-4-42-2': 'Next, we discuss how to choose the value of [MATH].', '1510.07611-4-43-0': '## A rule of thumb for the scaling factor', '1510.07611-4-44-0': 'We can rely on concepts of information theory to guide the choice of the scaling factor [MATH].', '1510.07611-4-44-1': 'The idea here is to choose the value of [MATH] as close as possible to one that still allows us to distinguish between the two sets of samples of a given size.', '1510.07611-4-44-2': "Via Sanov's theorem, the KL divergence provides a natural way to characterize the notion of distinguishability in this case [CITATION].", '1510.07611-4-44-3': 'Here we will briefly discuss the main ideas in a rather informal way; the interested reader can refer to Ref. [CITATION] for details.', '1510.07611-4-44-4': 'We want to know whether we can distinguish between two Boltzmann distributions at different inverse temperatures [MATH] and [MATH] from a set of [MATH] samples.', '1510.07611-4-44-5': 'To do this, it is useful to consider that we compute the maximum-likelihood estimate of the inverse temperature [MATH] from the sample set corresponding to inverse temperature [MATH].', '1510.07611-4-44-6': 'We can consider that we repeat this procedure many times so we can compute the probability distribution of [MATH].', '1510.07611-4-44-7': 'The two Boltzmann distributions are said to be distinguishable from a set of [MATH] samples if the probability of [MATH] being close to [MATH] is smaller than a given tolerance [MATH], i.e., if [EQUATION] where [MATH] is a suitably small constant.', '1510.07611-4-44-8': "From Sanov's theorem it follows that when [MATH] is large enough [EQUATION] where the factor [MATH] gathers sub-dominant terms in [MATH].", '1510.07611-4-44-9': 'So, if [MATH] the two Boltzmann distributions are distinguishable in the sense defined above.', '1510.07611-4-45-0': 'Assuming that [MATH] and [MATH] are close enough, the KL divergence can be expanded up to second order to yield [EQUATION] where [EQUATION] is known in information theory as the Fisher information, or generalized susceptibility; in this case, it is essentially the specific heat.', '1510.07611-4-45-1': 'When [MATH] is large enough, the right hand side in Eq. [REF] becomes appreciable only for very close [MATH] and [MATH].', '1510.07611-4-45-2': 'So for large [MATH] we can replace the KL divergence by the Fisher information in Eq. [REF].', '1510.07611-4-46-0': 'Following these ideas, we propose to choose the scaling factor [MATH] such that [MATH], where [MATH] is a given constant (cf. Ref. [CITATION]).', '1510.07611-4-46-1': 'Eqs. [REF] and [REF] yield [EQUATION]', '1510.07611-4-46-2': 'Some remarks are in order: (i) Eq. [REF] gives a rule of thumb to choose a suitable value of [MATH] for estimating [MATH]; however, the latter also appears in this expression.', '1510.07611-4-46-3': 'We can initiate [MATH] by either making a reasonable guess or using the pseudo-likelihood estimate (see the Appendix [REF]).', '1510.07611-4-46-4': '(ii) The sign in Eq. [REF] could be chosen to be positive during the first iterations to avoid the rescaled control parameters being below the noise level of the device, and negative afterwards to avoid the rescaled control parameters being above the allowed range.', '1510.07611-4-46-5': '(iii) Equation [REF] has been derived assuming that values of the KL divergence about [MATH] can be well approximated with the Fisher information.', '1510.07611-4-46-6': 'These assumption may fail in practice when [MATH] is relatively small or when [MATH] is far from the reference value at [MATH].', '1510.07611-4-46-7': '(iv) In principle, as long as the samples generated by the quantum annealer follow a Boltzmann distribution and the effective temperature remains constant under re-scalings of the control parameters, our temperature estimation technique is exact if there are enough samples.', '1510.07611-4-46-8': 'Still, the number of samples needed could grow exponentially with problem size due to the bias and variance associated with our estimator, whose study we leave for future work.', '1510.07611-4-46-9': '(v) Finally, the linear regression to compute our estimator may be affected by noise due to energy bands with very low frequency; in principle, this could be mitigated by relying on a weighted linear regression giving more weight to points associated with higher frequencies.', '1510.07611-4-47-0': '# A few gadgets to improve performance', '1510.07611-4-48-0': 'In this section we discuss three techniques that help improve the performance of our quantum-assisted learning algorithm.', '1510.07611-4-48-1': 'First of all, it is known that the performance of quantum annealers can be significantly impaired by the presence of both persistent and random biases between the actual values of the control parameters and the user-specified values.', '1510.07611-4-48-2': 'Perdomo-Ortiz et al. [CITATION] have developed a technique for determining and correcting the persistent biases and have shown evidence that this recalibration procedure can enhance the performance of the device for solving combinatorial optimization problems.', '1510.07611-4-48-3': 'In the next section we will show evidence that correcting for persistent biases can also enhance the performance of quantum annealers for sampling applications.', '1510.07611-4-49-0': 'Second, noise in the control parameters can hinder the initial stage of learning when they are typically small.', '1510.07611-4-49-1': 'In order to avoid this situation we can run CD-[MATH] for a few iterations until we find meaningful initial values for the control parameters that are above the noise level of the device and then restart with QuALe.', '1510.07611-4-49-2': 'This is exclusively due to the current state of quantum annealing technologies and it is expected to be further mitigated in new generations of these devices.', '1510.07611-4-49-3': 'We emphasize that the number of iterations with CD-1 has to be small to keep the weights within the dynamical range of the device.', '1510.07611-4-50-0': 'Finally, to estimate the effective temperature associated with a given instance we need to generate two sets of samples: one corresponding to the actual values of the control parameters that we are interested in, and another corresponding to these values rescaled by a factor [MATH].', '1510.07611-4-50-1': 'According to the discussion in the previous section, the scaling factor is chosen in such a way that the two probability distributions are as close as possible, yet distinguishable.', '1510.07611-4-50-2': 'So, we expect that the samples obtained at [MATH] can also be used for the estimation of the log-likelihood gradient, given by Eqs. [REF] and [REF], at [MATH] via the technique of importance sampling [CITATION].', '1510.07611-4-50-3': 'In short, we can use a set of samples [MATH] extracted from a Boltzmann distribution at inverse temperature [MATH] to estimate ensamble averages of an arbitrary observable [MATH] with a Boltzmann distribution at inverse temperature [MATH] as [EQUATION] where [MATH] is the ratio between the unnormalized probabilities.', '1510.07611-4-50-4': 'The approximation is expected to be good as long as the two distributions are close enough [CITATION].', '1510.07611-4-50-5': 'In the next section we will show evidence that including the set of samples corresponding to the rescaled control parameters indeed improves the performance of QuALe.', '1510.07611-4-51-0': 'From now on, when refering to the QuALe algorithm we imply that these three gadgets are also included, unless otherwise specified.', '1510.07611-4-52-0': '# Learning of a Boltzmann machine assisted by the D-Wave 2X', '1510.07611-4-53-0': 'Now that we have at our disposal a robust temperature estimation technique, we can use it for learning Boltzmann machines.', '1510.07611-4-53-1': 'We decided to focus on the learning of a Chimera-RBM for two reasons.', '1510.07611-4-53-2': 'On the one hand, although an RBM can be embedded into quantum hardware [CITATION], it requires us to represent single variables with chains of qubits coupled via ferromagnetic interactions of a given strength.', '1510.07611-4-53-3': 'Instead of forcing couplings to take a specific value to meet a preconceived design, it might be better to allow the learning algorithm itself to find the parameter values that work best for a particular application.', '1510.07611-4-53-4': 'On the other hand, the focus of our work is on better understanding the challenges that need to be overcome for using quantum annealers for sampling applications and taking the necessary steps towards an effective implementation of deep learning applications on quantum annealers.', '1510.07611-4-54-0': 'This systematic study provides both an assessment of the use of the D-Wave in learning Boltzmann machines and a study of the impact of the effective temperature in the learning performance.', '1510.07611-4-54-1': 'We consider it important to assess the performance of the different methods by computing the exact log-likelihood during the learning process.', '1510.07611-4-54-2': 'Otherwise, we could not be sure whether a difference in performance is due to the new learning method or due to errors in the approximation of the log-likelihood.', '1510.07611-4-54-3': 'For this reason we tested the method on a small synthetic data set called Bars and Stripes (BAS) and computed exhaustively the corresponding log-likelihood for evaluation.', '1510.07611-4-54-4': 'The BAS dataset consists of [MATH] pictures generated by setting the four pixels of each row (or column) to either black (-1) or white (+1), at random [CITATION].', '1510.07611-4-54-5': 'Another reason to focus on this small synthetic dataset is that while generating, e.g., 2000 samples in the DW2X for a given instance can take about 40 ms, the waiting time for accessing the machine to generate a new set of samples for a different instance can vary widely depending on the amount of jobs that are scheduled.', '1510.07611-4-54-6': 'So, while running QuALe with 2000 samples per iteration on the whole chip (1097 qubits) for [MATH] iterations could take in principle about 7 min if we had exclusive access to the device, the waiting times of the different jobs could increase this time by several orders of magnitude.', '1510.07611-4-55-0': 'We modeled the BAS dataset with a Chimera-RBM of 16 visible and 16 hidden units with the topology shown in Fig. [REF] (a).', '1510.07611-4-55-1': 'The mapping of pixels to visible units is shown in Fig. [REF] (b) (cf. [CITATION]).', '1510.07611-4-55-2': 'We run all algorithms with learning rate [MATH], which is the best value we found among five values in the range [MATH].', '1510.07611-4-55-3': 'To begin with, Fig. [REF] shows an instance of temperature estimation using [MATH] samples from the DW2X and [MATH], for generic control parameters found during the learning process (see Fig. [REF]).', '1510.07611-4-55-4': 'This value of [MATH] is the one that worked best out of a few trial values.', '1510.07611-4-55-5': 'Fig. [REF]a shows the histograms corresponding to [MATH] bins of samples obtained at the actual control parameters (blue histogram, shifted to the left) and the rescaled ones (pink histogram, shifted to the right).', '1510.07611-4-55-6': 'Fig. [REF]b shows a plot of [MATH] against [MATH] for all energy values that appear in the overlap of the two histograms.', '1510.07611-4-55-7': 'We can observe a rather clear linear trend as predicted by Eq. [REF], which is confirmed by a relatively high regression coefficient, [MATH].', '1510.07611-4-55-8': 'From the slope [MATH] of the regression line we can obtain the effective temperature by solving [MATH].', '1510.07611-4-56-0': 'Fig. [REF]a shows the impact of bias correction on the performance of the QuALe algorithm.', '1510.07611-4-56-1': 'The performance is measured in terms of the average log-likelihood [MATH], which has been evaluated exhaustively every 50 iterations.', '1510.07611-4-56-2': 'These results are obtained by implementing the Chimera-RBM on five different locations of the DW2X chip and running the QuALe algorithm three times on each location, for a total of 15 runs.', '1510.07611-4-56-3': 'The points correspond to the average of [MATH] over those 15 runs and the error bars correspond to one standard deviation.', '1510.07611-4-56-4': 'We can see that QuALe with persistent bias correction (blue crosses) outperforms QuALe without it (pink triangles).', '1510.07611-4-56-5': 'Fig [REF]b, on the other hand, shows the QuALe algorithm with (blue crosses) and without (pink triangles) taking into account the samples obtained at [MATH] for the estimation of the log-likelihood gradient, via importance sampling.', '1510.07611-4-56-6': 'The points correspond to the average of [MATH] over five runs of QuALe on a single location of the DW2X chip.', '1510.07611-4-56-7': 'Finally, Fig. [REF]c shows the positive impact of carrying out a few iterations of CD-1 to generate suitable initial conditions for QuALe.', '1510.07611-4-57-0': 'Fig. [REF] shows the evolution of [MATH] during the learning of a Chimera-RBM on the BAS dataset under different learning algorithms, all of them with learning rate [MATH].', '1510.07611-4-57-1': 'We can observe that the quantum assisted learning algorithm with effective-temperature estimation at each iteration (QuALe@[MATH], blue diagonal crosses) outperforms CD-[MATH] (blue solid squares) after about 300 iterations and CD-10 (green solid circles) after about 1500 iterations.', '1510.07611-4-57-2': 'However, within the 5000 iterations shown in the figure, QuALe@[MATH] has not yet been able to outperform CD-100, although there is a clear trend in that direction.', '1510.07611-4-57-3': 'As we did not observe any significant improvement when using larger values of [MATH], we expect that CD-[MATH] is close to an exact computation (cf. Theorem 5.1 in [CITATION]).', '1510.07611-4-57-4': 'Interestingly, all CD-[MATH] reach their best average performance after a relatively small number of iterations while QuALe@[MATH], in contrast, increases slowly and steadily.', '1510.07611-4-57-5': 'One may be inclined to think this is because CD-[MATH] estimates the model averages from samples generated by a [MATH]-step Markov chain initialized at each data point.', '1510.07611-4-57-6': 'In this way CD-[MATH] is using information contained in the data from the very beginning for the estimation of the model ensemble averages, while QuALe@[MATH] ignores them altogether.', '1510.07611-4-57-7': 'However, if this were indeed the case one should expect such a trend to diminish for increasing values of [MATH], something that is not observed in the figure.', '1510.07611-4-57-8': 'A better understanding of this point has the potential to considerably improve the performance of QuALe@[MATH].', '1510.07611-4-58-0': 'To assess the relevance of temperature estimation for QuALe@[MATH], we also show in Fig. [REF] the average performance under quantum assisted learning at a fixed temperature.', '1510.07611-4-58-1': 'First, it is worth mentioning that using the physical temperature of the device, [MATH] (corresponding to [MATH] mK as explained in the caption of Fig. [REF]), leads to a very poor performance, reaching values [MATH] (not shown).', '1510.07611-4-58-2': 'Fixing the temperature to the average QuALe@[MATH] over all temperatures found during the run of QuALe@[MATH] leads to a better performance (red empty circles), but still well below that displayed by QuALe@[MATH] itself.', '1510.07611-4-58-3': 'Fixing the temperature to [MATH] (QuALe@[MATH]) and to [MATH] (QuALe@[MATH]) leads to a decrease in performance with respect to that displayed with [MATH].', '1510.07611-4-59-0': 'In Fig. [REF] we can observe the variation of the effective temperature estimated during a window of 80 iterations of QuALe@[MATH] (green line).', '1510.07611-4-59-1': 'To evaluate whether such a variation is within the finite sampling error, we estimated the effective temperature 15 times at each iteration.', '1510.07611-4-59-2': 'The blue circles show the median of [MATH] and the error bars represent the corresponding first and third quartiles.', '1510.07611-4-59-3': 'Clearly, this variation cannot be explained as due to finite sampling error.', '1510.07611-4-59-4': 'We emphasize that during the execution of QuALe the effective temperature is estimated only once.', '1510.07611-4-60-0': '# Conclusions and Future Work', '1510.07611-4-61-0': 'Applications that rely on sampling, such as learning Boltzmann machines, are in general intractable due to long equilibration times of sampling techniques like MCMC [CITATION].', '1510.07611-4-61-1': 'Some authors have conjectured quantum annealing could have an advantage in sampling applications.', '1510.07611-4-61-2': 'In this work we proposed a strategy to overcome one of the main limitations when intending to use a quantum annealer to sample from Boltzmann distributions: the determination of effective temperatures.', '1510.07611-4-61-3': 'The simple technique proposed in this work uses samples obtained from a quantum annealer (the DW2X at NASA is our experimental implementation) to estimate both the effective temperature and the hard-to-compute term in the log-likelihood gradient, i.e., the averages over the model distribution; these are needed to determine the next step in the learning process.', '1510.07611-4-61-4': 'We present a systematic study of the impact of the effective-temperature in the learning of a Chimera-RBM model with 16 visible and 16 hidden units.', '1510.07611-4-61-5': 'To do so, we compared the QuALe algorithm with both instance-dependent effective temperature and different constant effective temperatures to the performance of a CD-[MATH] implementation, with [MATH] equal to [MATH], [MATH], and [MATH].', '1510.07611-4-62-0': 'The Chimera-RBM model itself is much less powerful than the RBM model.', '1510.07611-4-62-1': 'While the former is sparse with a number of parameters increasing linearly with the number of variables, the latter is dense with a number of parameters increasing quadratically.', '1510.07611-4-62-2': 'For instance, the Chimera-RBM that we have studied here, with 16 hidden and 16 visible variables, has only about 31% of the weight parameters that a corresponding RBM of the same size has.', '1510.07611-4-62-3': 'This is reflected by the fact that a Chimera-RBM, trained either with QuALe or with standard classical techniques, struggles to generate samples faithfully resembling the [MATH] BAS dataset on which it was trained (not shown).', '1510.07611-4-62-4': 'In this first study, we have decided to omit any regularization of the learning process.', '1510.07611-4-62-5': 'We have done this to keep the focus as clear as possible on the potential gains obtained by using QuALe and to avoid the search of optimal regularization parameters that could be very expensive due to the time to access the DW2X.', '1510.07611-4-62-6': 'While this may lead to drops in likelihood [CITATION], we expect that the substantial reduction in the number of parameters mentioned above may act as an implicit regularizing sparsity constraint.', '1510.07611-4-62-7': 'Since we have neglected regularization altogether in all the learning algorithms, we expect the comparison to be fair.', '1510.07611-4-62-8': 'Moreover, as the work by Dumoulin et al. [CITATION] suggests, the Chimera-RBM model we have investigated has a limited expressive power.', '1510.07611-4-62-9': 'So we have decided to delay the investigation of the role of regularization for when we deal with more expressive models that can be naturally represented in a Chimera topology.', '1510.07611-4-63-0': 'RBMs have the nice feature that sampling in one layer conditioned to a configuration in the other layer can be done in parallel and in one step; this is one of the main reasons for their wide adoption.', '1510.07611-4-63-1': 'This feature does not hold true anymore once we have non-trivial lateral connections in one of the layers, which is the concept behind more powerful Boltzmann machines [CITATION].', '1510.07611-4-63-2': 'We think this is one of the most promising directions to explore with the quantum-assisted learning (QuALe) algorithm.', '1510.07611-4-63-3': 'By restricting QuALe to study RBM or Chimera-RBM models, we are paying the price of using a device that is in principle more powerful, but we are not taking advantage of having a more general model.', '1510.07611-4-63-4': 'It is important to investigate how to take full advantage of the DW2X by designing more suitable models based on the Chimera graph.', '1510.07611-4-63-5': 'An interesting possibility is the one explored in Ref. [CITATION] where the Chimera graph of the DW2X is used as a hidden layer to build a semi-restricted Boltzmann machine, which therefore has lateral connections in the hidden layer.', '1510.07611-4-63-6': 'When dealing with more general Boltzmann machines it would be interesting to compare the performance of QuALe against mean field methods.', '1510.07611-4-63-7': 'Recently, there has been interest in applying mean field techniques for learning restricted Boltzmann machines too [CITATION].', '1510.07611-4-63-8': "Future work should explore how the performance of mean field techniques compares with QuALe's.", '1510.07611-4-64-0': 'However, the goal of this first QuALe implementation on small Chimera-RBMs serves several purposes.', '1510.07611-4-64-1': 'When dealing with large datasets the log-likelihood cannot be exhaustively computed due to the intractability of computing the partition function.', '1510.07611-4-64-2': 'Log-likelihood is the gold standard metric, but it becomes intractable for large systems.', '1510.07611-4-64-3': 'In these cases, other performance metrics such as reconstruction error or cross-entropy error turn out to be more convenient, but although widely used, they are rough approximations to the log-likelihood [CITATION].', '1510.07611-4-64-4': 'If we were to use these proxies we could not be sure that we would be drawing the right conclusions.', '1510.07611-4-64-5': 'This justifies why we used a moderately small dataset with 16 visible and 16 hidden units, and even though computing the log-likelihood was computationally expensive for the study performed here, having 32 units in total was still a manageable size.', '1510.07611-4-64-6': 'Through the computation of the exact likelihood we were able to examine in more detail some of the goals proposed here: being able to assess the best effective temperature fit to the desired Boltzmann distribution and to show that using a constant temperature different from the one estimated with our approach might lead to severe suboptimal performance.', '1510.07611-4-65-0': 'Another aspect we explored in this study was to go beyond the conventional CD-1, with the purpose of having a fairer comparison to the results that might be expected from the entirely classical algorithm counterpart.', '1510.07611-4-65-1': 'Previous results from our research group [CITATION], as well as others reported by other researchers [CITATION], are limited to comparing the performance of quantum annealers to the quick but suboptimal CD-1.', '1510.07611-4-65-2': 'As shown in those studies, even with a suboptimal constant temperature one might be drawn to conclude that QuALe is outperforming conventional CD.', '1510.07611-4-65-3': 'Similar conclusions might be drawn from the curves for constant but suboptimal [MATH] and [MATH] vs. CD-1 in Fig. [REF].', '1510.07611-4-65-4': 'As shown in Fig. [REF], this conclusion does not hold anymore for higher values of [MATH], while the method using the effective-temperature estimation proposed here is the only one showing a steady increase in performance, close to matching the largest value of [MATH] tried here, i.e. [MATH].', '1510.07611-4-66-0': 'Another important point to investigate in the future is whether the differences observed in performance remain for larger and more complex datasets.', '1510.07611-4-66-1': 'We would expect that the performance of CD-[MATH] degrades with larger instances as equilibration times are expected to grow fast with the number of variables once the probability distribution starts having non-trivial structure.', '1510.07611-4-66-2': 'From this perspective, it is important to notice that QuALe is expected to display a more uniform exploration of configuration space.', '1510.07611-4-67-0': 'A related important question has to do with the scalability of our temperature estimation technique, i.e. how should the number of samples grow with problem size?', '1510.07611-4-67-1': 'In principle, as long as the quantum annealer converges to an approximately Boltzmann distribution and the effective temperature remains constant under rescalings of its control parameters, our method is exact given enough samples.', '1510.07611-4-67-2': 'We have left this question for future work as we consider that there are more pressing issues, i.e. limited connectivity and noise, that need to be addressed before we can say something conclusive about scalability.', '1510.07611-4-67-3': 'Needless to say, the validity of the assumptions on which our work relies should also be investigated in more detail.', '1510.07611-4-67-4': 'It is also important to devise more controlled experiments that allow us to isolate the different phenomena involved.', '1510.07611-4-67-5': 'Two months after submission of this manuscript, we learned of ongoing work addressing some of these issues and putting forward other temperature estimation techniques [CITATION].', '1510.07611-4-67-6': 'Finally, an investigation on the bias and variance of our effective temperature estimator is an interesting theoretical question that we expect to address in future work.', '1510.07611-4-68-0': 'There are other ways in which the ideas explored here could be extended.', '1510.07611-4-68-1': 'For instance, we can go beyond restricted Boltzmann machines to build deep learning architectures or beyond unsupervised learning to build discriminative models.', '1510.07611-4-68-2': "In principle the speed of learning could be increased by adding a 'momentum' term to the gradient-ascent learning rule [CITATION].", '1510.07611-4-68-3': 'Indeed, Adachi and Henderson have started exploring these ideas in a recent work [CITATION].', '1510.07611-4-68-4': 'Instead, we have focused on first trying to better understand the basics before adding more (classical) complexity to the learning algorithms that we feel have the risk to obscure the actual contributions from our approach.', '1510.07611-4-69-0': '# Comparison to alternative temperature estimation techniques', '1510.07611-4-70-0': 'Here we discuss alternative techniques to approximately estimate the instance-dependent effective temperature [MATH], which are in principle efficient too, and show evidence that our method produces superior results.', '1510.07611-4-71-0': 'One of the mainstream approaches in statistical physics to estimate parameters of an Ising model goes under the name of inverse Ising model [CITATION].', '1510.07611-4-71-1': 'One of the most investigated techniques for solving the inverse Ising model relies on mean field approximations [CITATION], due to its relative simplicity.', '1510.07611-4-71-2': 'These techniques fail, though, for low temperatures where low-energy configurations are arranged in a non-trivial clustered phase [CITATION].', '1510.07611-4-71-3': 'On the other hand, the so-called pseudo-likelihood method [CITATION] is recognized as the state-of-the art in solving this problem.', '1510.07611-4-71-4': 'Recently, it has been suggested that by suitably introducing information about the clustered phase into mean field methods, these can yield comparable results to the pseudo-likelihood method [CITATION].', '1510.07611-4-72-0': 'We first devised a simple strategy to test the feasibility of a mean field approach before attempting to develop a technique specifically targeted to the estimation of [MATH] alone.', '1510.07611-4-72-1': 'Indeed, since we know the control parameters [MATH] and [MATH], we can in principle estimate [MATH] by first determining [MATH] and [MATH] using the Bethe approximation [CITATION], and then finding the value of [MATH] that minimizes some distance between the control parameters and the estimated ones.', '1510.07611-4-72-2': 'However, the estimation of [MATH] and [MATH] using the samples from the DW2X along the learning path only produces real values up to about the first hundred iterations (not shown).', '1510.07611-4-72-3': 'This suggests the Bethe approximation is not suitable for the parameter regime traversed when learning the BAS dataset studied here.', '1510.07611-4-73-0': 'Since, as we mentioned above, the pseudo-likelihood method [CITATION] is considered the state of the art technique for estimating the parameters of an Ising model we will focus from now on in such an approach.', '1510.07611-4-73-1': 'We will see that our method displays a much better performance on the BAS dataset studied here.', '1510.07611-4-74-0': 'Given a set of samples [MATH], where [MATH] with [MATH], generated by a quantum annealer with control parameters [MATH] and [MATH], we can estimate the effective temperature [MATH] by maximizing the average pseudo-likelihood [CITATION] [EQUATION] where [MATH] denotes the set of neighbors of [MATH].', '1510.07611-4-75-0': 'In contrast to the approach in Ref. [CITATION], here the only unknown is [MATH].', '1510.07611-4-75-1': "We can find a maximum average pseudo-likelihood estimator for the effective temperature [MATH] via second order Newton's method [CITATION].", '1510.07611-4-75-2': 'In our experiments, we start from [MATH] and iterate until the update is smaller than a tolerance level of [MATH].', '1510.07611-4-75-3': 'Fig. [REF]a shows a comparison of the performance of our quantum-assisted learning algorithm QuALe@[MATH] with [MATH] estimated with the pseudo-likelihood method (pink circles) as described here and estimated with the method introduced in Sec. [REF] (blue crosses) of the present work.', '1510.07611-4-75-4': 'We can observe that while QuALe@[MATH] with the pseudo-likelihood method performs better on the first about 1000 iterations, QuALe@[MATH] with linear regression performs better afterwards, reaching higher values for the likelihood function.', '1510.07611-4-75-5': 'Fig. [REF]b shows the values of effective temperatures estimated by the two techniques along the learning path; interestingly, the effective temperatures estimated by the pseudo-likelihood (pink points on the bottom) are consistently smaller and have less variability than those estimated with our linear regression technique (blue points on the top).'}

1204.5504

{'1204.5504-1-0-0': 'Using a network of seismometers and sets of optimal filters, we implemented a feed-forward control technique to minimize the seismic contribution to multiple interferometric degrees of freedom of the LIGO interferometers.', '1204.5504-1-0-1': 'The filters are constructed by using the Levinson-Durbin recursion relation to approximate the optimal Wiener filter.', '1204.5504-1-0-2': 'By reducing the RMS of the interferometer feedback signals below [MATH]10 Hz, we have improved the stability and duty cycle of the joint network of gravitational wave detectors.', '1204.5504-1-0-3': 'By suppressing the large control forces and mirror motions, we have dramatically reduced the rate of non-Gaussian transients in the gravitational wave signal stream.', '1204.5504-1-1-0': '# Introduction', '1204.5504-1-2-0': 'The Laser Interferometer Gravitational-wave Observatory (LIGO) [CITATION] consists of detectors located in Livingston, Louisiana (LLO) and Hanford, Washington (LHO).', '1204.5504-1-2-1': 'The goal of the experiment is to measure minute disturbances in the relative positions of test mass mirrors generated by gravitational waves, using laser interferometry.', '1204.5504-1-2-2': 'Passing gravitational waves are expected to produce a quadrupolar strain in the local space-time metric, resulting in a differential phase shift between the laser fields traveling in the two arms of the interferometer.', '1204.5504-1-2-3': 'Potential astrophysical sources of gravitational waves include supernovae and compact binary inspirals.', '1204.5504-1-2-4': 'The expected strain from such sources requires a detector sensitive to displacements on the order of [MATH] m/[MATH] over a few km baseline, at frequencies ranging from [MATH]50 Hz to 7 kHz.', '1204.5504-1-2-5': 'Figure [REF] shows a typical displacement sensitivity for the LIGO interferometers.', '1204.5504-1-3-0': "Using Fabry-Perot cavities for the interferometer arms as well as power recycling [CITATION] helps improve the detector's sensitivity.", '1204.5504-1-3-1': 'Figure [REF] shows a simplified schematic of a LIGO interferometer.', '1204.5504-1-3-2': 'The motion of the ground is many orders of magnitude above the required sensitivity, so the mirrors are seismically isolated, using both passive and active techniques.', '1204.5504-1-3-3': 'The cavities are held on optical resonance by suppressing external disturbances with a family of length and alignment feedback control loops.', '1204.5504-1-3-4': 'Suspending the mirrors as pendulums provides passive vibration isolation above the pendular resonance, which is arranged to be [MATH]1 Hz.', '1204.5504-1-3-5': 'The support point of the pendulum is attached to a passive in-vacuum seismic isolation stack, consisting of four mass and spring layers.', '1204.5504-1-3-6': 'This payload is attached to tubes which exit the vacuum and are supported by an external active seismic isolation system.', '1204.5504-1-3-7': 'Small magnets are glued to the backs of the mirrors to allow for actuation via magnetic fields generated by currents flowing in nearby wire coils.', '1204.5504-1-3-8': 'Above [MATH]50 Hz, where ground vibrations have been sufficiently suppressed, the detector is fundamentally limited by thermal noise in the mirrors and seismic isolation systems and by photon shot noise.', '1204.5504-1-4-0': 'There are four length degrees of freedom that need to be controlled.', '1204.5504-1-4-1': 'They are defined in terms of differential and common length changes, [EQUATION] where [MATH], [MATH], [MATH], and [MATH] are as shown in Figure [REF].', '1204.5504-1-4-2': 'An RF modulation scheme [CITATION] is used to generate control signals by relating position and angle fluctuations of the test mass mirrors to power fluctuations measured by photodetectors at various locations around the interferometer [CITATION].', '1204.5504-1-4-3': 'The detector is brought to and held at its operating point by sequentially bringing cavities into resonance using dynamically calculated error signals [CITATION], a process called lock acquisition.', '1204.5504-1-5-0': 'Precision measurement of the differential arm length degree of freedom, called [MATH] or DARM, enables the detector to find potential gravitational waves.', '1204.5504-1-5-1': 'The common (mean) arm length, [MATH] or CARM, is also sensitive to gravitational waves but is used as a reference to stabilize the frequency of the laser.', '1204.5504-1-5-2': 'When the arm lengths are set such that the antisymmetric port is near the dark fringe most of the input power is sent back towards the laser, such that the interferometer acts as a highly reflecting mirror.', '1204.5504-1-5-3': 'By controlling the common power recycling cavity length, [MATH] or PRC, another resonant cavity can be formed between the power recycling mirror and the Fabry-Perot Michelson, increasing the power on the beamsplitter and in the arms.', '1204.5504-1-5-4': 'In order to maintain the Michelson fringe offset required for this configuration the short Michelson cavity length, [MATH] or MICH, must also be controlled.', '1204.5504-1-6-0': "Of the external disturbances that the control systems must overcome, one of the most problematic is the persistent and large 0.1 - 0.3 Hz double-frequency 'microseismic' peak, generated by wave activity in the oceans [CITATION].", '1204.5504-1-6-1': 'At the microseismic peak the amplitude is on the order of several m/[MATH], but the level of seismicity present at the observatories varies widely (by a factor of [MATH]10) [CITATION].', '1204.5504-1-6-2': 'The relative displacement of the test masses in this band is of the same order as the horizontal displacements caused locally by the microseism.', '1204.5504-1-6-3': 'These levels of seismic noise present a host of problems, including:', '1204.5504-1-7-0': 'At frequencies where the motion of the ground and the length fluctuations in the cavity are coherent the seismic noise can be actively subtracted from the cavity motion using a feed-forward signal.', '1204.5504-1-7-1': 'Feed-forward cancellation of the microseism was demonstrated at LLO during the commissioning of the detector, using piezoelectric actuators and filters created via system-identification [CITATION].', '1204.5504-1-7-2': "Between LIGO's third and fourth science runs (S3 and S4) a new external seismic isolation system, HEPI, was installed at LLO, which used feed-forward signals to subtract local ground motion from the isolated platform motion with a technique often called 'sensor correction' [CITATION].", '1204.5504-1-7-3': 'Those techniques were used to the absolute motion of each individual platform.', '1204.5504-1-7-4': "This paper will describe an improved feed-forward scheme which uses combinations of seismometers located several km's apart and Wiener filtering to reduce the differential motion between the platforms, and thereby quiet interferometric control signals.", '1204.5504-1-7-5': "We show results achieved during LIGO's sixth science run (S6) which lasted from July 2009 until October 2010.", '1204.5504-1-7-6': 'In other work, Wiener filtering has been used for real-time cancellation of phase noise from vibrations in cavity stabilized lasers [CITATION], and a similar technique was used in a 40 meter long suspended interferometer [CITATION].', '1204.5504-1-8-0': '# Method', '1204.5504-1-9-0': 'In order to optimize the noise subtraction we want to find the Wiener filter [CITATION], [MATH], that processes the witness seismometer signals, [MATH], into the best possible estimate of the targeted cavity length control signal, [MATH].', '1204.5504-1-9-1': 'We find [MATH] by minimizing the mean square error between [MATH] and [MATH], [EQUATION] with respect to [MATH], where [MATH] is the filter order (number of taps).', '1204.5504-1-9-2': 'This is done by setting the derivative, [EQUATION] which yields the Wiener-Hopf equations, [EQUATION] where [MATH] is the autocorrelation matrix of the seismometer signal and [MATH] is the cross-correlation between the seismometer and the control signals.', '1204.5504-1-9-3': 'The FIR Wiener filters were estimated in MATLAB [CITATION] using the Levinson-Durbin algorithm [CITATION], and then fit to a set of IIR coefficients to reduce the computational time in the real-time control system.', '1204.5504-1-9-4': 'The conversion from FIR to IIR was done with the Vectfit [CITATION] software package.', '1204.5504-1-9-5': "The intermediate FIR filters were composed of 1000's of taps and processed data with a sample rate of 64 Hz, allowing for subtraction down to a few 10's of mHz.", '1204.5504-1-9-6': 'Approximately 1 hour of data was used to train each filter.', '1204.5504-1-9-7': 'Each test mass is in the vicinity of at least one seismometer (see Figure [REF]), which measures motion in three perpendicular directions, X, Y, and Z, where Z is vertical and the two horizontal directions are roughly aligned to the arms of the interferometer.', '1204.5504-1-9-8': 'The length control is coupled most strongly to motion along the axes of the respective cavities, but cross-couplings to the other directions are non-negligible.', '1204.5504-1-9-9': 'Regardless of the direction of the witness signal generating a feed-forward correction the actuation is applied in the direction in which the laser beam is propagating for that particular chamber.', '1204.5504-1-10-0': 'Offloading control signals to actuators located in the external seismic isolation systems mitigates several of the problems mentioned in Section [REF].', '1204.5504-1-10-1': 'In order to properly subtract the filtered witness signals, the transfer function from our point of actuation to the cavity control signal must be measured and divided out.', '1204.5504-1-10-2': 'A diagram showing the relevant pieces of the mechanical structure can be seen in Figure [REF].', '1204.5504-1-10-3': 'The number of mechanical components separating the mirrors from the ground creates a complicated transfer function with many resonant features.', '1204.5504-1-11-0': 'At LLO, the active seismic isolation system is HEPI (hydraulic external pre-isolator) [CITATION].', '1204.5504-1-11-1': 'A similar system installed at LHO, PEPI, used piezoelectric actuation.', '1204.5504-1-11-2': 'HEPI is an Advanced LIGO (aLIGO) system which will be in place at both LLO and LHO in the future.', '1204.5504-1-11-3': 'Figure [REF] shows a schematic of the HEPI system [CITATION].', '1204.5504-1-11-4': "The Wiener filter feed-forward path was added in parallel to the existing sensor correction path, which contained hand-tuned filters designed to match HEPI's position sensors to local seismometers on the ground.", '1204.5504-1-11-5': 'Typically this sensor correction reduced the differential motion sensed by the suspended cavities to [MATH] of the ground motion, in the microseism band.', '1204.5504-1-11-6': 'All results shown are improvements on top of this existing isolation.', '1204.5504-1-11-7': 'The HEPI actuators [CITATION] provide the ability to move its payload by [MATH] 700 m; the maximum range of PEPI was [MATH] m.', '1204.5504-1-11-8': 'We excited these actuators while the detector was locked and monitored the mirror motion to measure the mechanical response.', '1204.5504-1-11-9': 'We again used Vectfit to fit these measurements.', '1204.5504-1-11-10': 'An example of these transfer function measurements can be seen in Figure [REF].', '1204.5504-1-11-11': 'An example Wiener filter can be seen in Figure [REF].', '1204.5504-1-11-12': 'Both the Wiener filter and mechanical response are only fit accurately up to few Hz.', '1204.5504-1-11-13': 'Low-pass filtering was applied to the feed-forward signal to prevent noise injection at higher frequencies.', '1204.5504-1-12-0': '# Results and Limitations', '1204.5504-1-13-0': 'Before the laser beam enters interferometer, higher order spatial modes are removed by a suspended, triangular cavity called the code cleaner (MC).', '1204.5504-1-13-1': 'The length of the mode cleaner is controlled such that only the TEM00 mode is resonant.', '1204.5504-1-13-2': 'The cavity is [MATH]12 m long, with its mirrors split between isolated tables in two vacuum tanks, MC1 and MC2, as shown in Figure [REF].', '1204.5504-1-13-3': 'At LLO these chambers are equipped with HEPI and served as the initial testbed for the Wiener filter feed-forward implementation.', '1204.5504-1-13-4': 'Figure [REF] shows the control signal holding the mode cleaner on resonance with and without the Wiener feed-forward signal path enabled.', '1204.5504-1-13-5': 'A factor of up to [MATH]5 improvement was realized around the microseismic peak, and the RMS of the signal was reduced by a factor of [MATH]2.', '1204.5504-1-13-6': 'The pre-existing sensor correction filters are left in their nominal state for all results shown in this paper.', '1204.5504-1-14-0': 'Successful implementation of Wiener filter feed-forward on the mode cleaner length control prompted commissioning of feed-forward paths for the other interferometric control signals, starting with the power recycling cavity control at LLO.', '1204.5504-1-14-1': "This work was performed during a time of particularly high microseismic activity which impaired the Livingston detector's ability to remain in lock.", '1204.5504-1-14-2': 'As can be seen in Figure [REF](a), a significant reduction in control signal was achieved, with the RMS being reduced by more than a factor of 2.', '1204.5504-1-14-3': "As a result there was also a reduction in the power fluctuations in the recycling cavity, which tended to drive the interferometer control loops towards instability, and greatly improved the instrument's duty cycle.", '1204.5504-1-14-4': 'This cavity is nearly geometrically degenerate [CITATION] (g-factor [MATH]1) and therefore is especially sensitive to misalignments.', '1204.5504-1-15-0': 'Sending feed-forward signals developed to minimize one cavity control signal may inject noise into other length controls, since the cavities share mirrors.', '1204.5504-1-15-1': 'To avoid this problem, feed-forward paths were implemented in a serial fashion, such that extra motion injected into other degrees of freedom could be corrected by the Wiener filters constructed for subsequent feed-forward paths.', '1204.5504-1-15-2': 'For example, Figure [REF](b) shows the effect of filters designed to minimize the short Michelson control signal, calculated on top of the existing power recycling feed-forward path.', '1204.5504-1-15-3': 'Above [MATH]0.2 Hz the Michelson signal is reduced, with the improved isolation of the power recycling cavity preserved, albeit with some noise injection above [MATH]1 Hz.', '1204.5504-1-15-4': 'Since the overall RMS is still reduced this noise injection was considered an acceptable tradeoff.', '1204.5504-1-16-0': 'Global feed-forward for the differential arm length control was implemented at both observatories with similar results.', '1204.5504-1-16-1': 'Since the DARM degree of freedom is sensitive to the motions of mirrors in both the end buildings as well as the corner station seismometers in all three locations were incorporated into the calculation of the optimal feed-forward filter.', '1204.5504-1-16-2': 'Figures [REF] and [REF] show that the overall RMS of the control signal is reduced by a factor of [MATH]2.5 for each interferometer.', '1204.5504-1-16-3': 'For LLO this improvement in performance was balanced by a slight noise increase above 1 Hz.', '1204.5504-1-16-4': 'Some noise was also injected by Wiener filter feed-forward at LHO, below 0.1 Hz, and above 3 Hz.', '1204.5504-1-17-0': 'The couplings of ground motion to the feedback signal can potentially be time dependent.', '1204.5504-1-17-1': "Since the approach used here is not adaptive, this could lead to a degradation of the filter's subtraction efficacy over time.", '1204.5504-1-17-2': 'Figure [REF] shows the performance of DARM feed-forward at LLO when first implemented and 8 months later.', '1204.5504-1-17-3': 'While the overall reduction in RMS motion was originally a factor of [MATH]2.5 the same feed-forward filters provided [MATH]20% less isolation after 8 months, reducing the RMS motion by a factor of 2.', '1204.5504-1-17-4': 'There is no obstacle to retraining new Wiener filters to potentially recover subtraction performance, however if there is a change in the mechanical plant remeasuring the transfer function is a time consuming process, due to the high accuracy required.', '1204.5504-1-18-0': 'As mentioned in Section [REF], we have observed that the rate of non-Gaussian events in the gravitational wave band increases when large forces are required to maintain detector lock.', '1204.5504-1-18-1': 'To measure the effect of improved isolation on the rate of background events, two 30 minute segments of data were collected for L1, with the DARM Wiener feed-forward first enabled and then disabled.', '1204.5504-1-18-2': 'A templated search, using a sine-Gaussian waveform basis [CITATION], was performed and the number of triggers per second reported are shown in Figure [REF].', '1204.5504-1-18-3': 'Only triggers with frequencies between 40 and 150 Hz are included, and the threshold SNR was set to 5.', '1204.5504-1-18-4': 'While both distributions show an excess of non-Gaussian events, reducing the low frequency control signal clearly results in a lower rate of signal band transients.', '1204.5504-1-18-5': 'The stationary noise floor of the detector did not vary significantly during the time this data was taken.', '1204.5504-1-19-0': 'Fundamentally the ability to provide feed-forward subtraction is limited by the coherence between the witness signals and the cavity control signal.', '1204.5504-1-19-1': 'A drawback particular to the scheme detailed in this paper is the lack of witnesses measuring all signals which contribute to the cavity motion.', '1204.5504-1-19-2': 'Since the seismometers cannot distinguish between translation and tilt, additional sensors such as tilt meters could contribute additional isolation through more feed-forward paths [CITATION].', '1204.5504-1-19-3': 'There is the potential to be limited by the noise floors of motion sensors, however at the microseismic peak the signal to noise ratio is very large; see Figure [REF].', '1204.5504-1-19-4': 'Numerical errors can arise in the calculation of the FIR Wiener filter as well as in the fitting of both the IIR coefficients and the mechanical transfer function.', '1204.5504-1-19-5': 'These inaccuracies also have the potential to limit the feed-forward subtraction.', '1204.5504-1-20-0': '# Conclusions', '1204.5504-1-21-0': "We have demonstrated the ability to use the LIGO detector's active seismic isolation system to reduce the relative motion between optics on a 4 km scale through seismometer feed-forward control.", '1204.5504-1-21-1': 'By sending the filtered seismometer signals to the HEPI/PEPI actuators we were able to quiet the relative motion of the mirrors, offloading control signal from the test mass actuators to the seismic isolation system.', '1204.5504-1-21-2': 'During times of elevated microseismic activity this allowed for unprecedented interferometer stability.', '1204.5504-1-21-3': 'The rate of audio band transients is significantly reduced by having less motion and less applied force.', '1204.5504-1-21-4': 'This method will be directly applicable to the second generation gravitational-wave detectors.'}

{'1204.5504-2-0-0': 'Using a network of seismometers and sets of optimal filters, we implemented a feed-forward control technique to minimize the seismic contribution to multiple interferometric degrees of freedom of the LIGO interferometers.', '1204.5504-2-0-1': 'The filters are constructed by using the Levinson-Durbin recursion relation to approximate the optimal Wiener filter.', '1204.5504-2-0-2': 'By reducing the RMS of the interferometer feedback signals below [MATH]10 Hz, we have improved the stability and duty cycle of the joint network of gravitational wave detectors.', '1204.5504-2-0-3': 'By suppressing the large control forces and mirror motions, we have dramatically reduced the rate of non-Gaussian transients in the gravitational wave signal stream.', '1204.5504-2-1-0': '# Introduction', '1204.5504-2-2-0': 'The Laser Interferometer Gravitational-wave Observatory (LIGO) [CITATION] consists of detectors located in Livingston, Louisiana (LLO) and Hanford, Washington (LHO).', '1204.5504-2-2-1': 'The goal of the experiment is to measure minute disturbances in the relative positions of test mass mirrors generated by gravitational waves, using laser interferometry.', '1204.5504-2-2-2': 'Passing gravitational waves are expected to produce a quadrupolar strain in the local space-time metric, resulting in a differential phase shift between the laser fields traveling in the two arms of the interferometer.', '1204.5504-2-2-3': 'Potential astrophysical sources of gravitational waves include supernovae and compact binary inspirals.', '1204.5504-2-2-4': 'The expected strain from such sources requires a detector sensitive to displacements on the order of [MATH] m/[MATH] over a few km baseline, at frequencies ranging from [MATH]50 Hz to 7 kHz.', '1204.5504-2-2-5': 'Figure [REF] shows a typical displacement sensitivity for the LIGO interferometers.', '1204.5504-2-3-0': "Using Fabry-Perot cavities for the interferometer arms as well as power recycling [CITATION] helps improve the detector's sensitivity.", '1204.5504-2-3-1': 'Figure [REF] shows a simplified schematic of a LIGO interferometer.', '1204.5504-2-3-2': 'The motion of the ground is many orders of magnitude above the required sensitivity, so the mirrors are seismically isolated, using both passive and active techniques.', '1204.5504-2-3-3': 'The cavities are held on optical resonance by suppressing external disturbances with a family of length and alignment feedback control loops.', '1204.5504-2-3-4': 'Suspending the mirrors as pendulums provides passive vibration isolation above the pendular resonance, which is arranged to be [MATH]1 Hz.', '1204.5504-2-3-5': 'The support point of the pendulum is attached to a passive in-vacuum seismic isolation stack, consisting of four mass and spring layers.', '1204.5504-2-3-6': 'This payload is attached to tubes which exit the vacuum and are supported by an external active seismic isolation system.', '1204.5504-2-3-7': 'Small magnets are glued to the backs of the mirrors to allow for actuation via magnetic fields generated by currents flowing in nearby wire coils.', '1204.5504-2-3-8': 'Above [MATH]50 Hz, where ground vibrations have been sufficiently suppressed, the detector is fundamentally limited by thermal noise in the mirrors and seismic isolation systems and by photon shot noise.', '1204.5504-2-4-0': 'There are four length degrees of freedom that need to be controlled.', '1204.5504-2-4-1': 'They are defined in terms of differential and common length changes, [EQUATION] where [MATH], [MATH], [MATH], and [MATH] are as shown in Figure [REF].', '1204.5504-2-4-2': 'An RF modulation scheme [CITATION] is used to generate control signals by relating position and angle fluctuations of the test mass mirrors to power fluctuations measured by photodetectors at various locations around the interferometer [CITATION].', '1204.5504-2-4-3': 'The detector is brought to and held at its operating point by sequentially bringing cavities into resonance using dynamically calculated error signals [CITATION], a process called lock acquisition.', '1204.5504-2-5-0': 'Precision measurement of the differential arm length degree of freedom, called [MATH] or DARM, enables the detector to find potential gravitational waves.', '1204.5504-2-5-1': 'The common (mean) arm length, [MATH] or CARM, is also sensitive to gravitational waves but is used as a reference to stabilize the frequency of the laser.', '1204.5504-2-5-2': 'When the arm lengths are set such that the antisymmetric port is near the dark fringe most of the input power is sent back towards the laser, such that the interferometer acts as a highly reflecting mirror.', '1204.5504-2-5-3': 'By controlling the common power recycling cavity length, [MATH] or PRC, another resonant cavity can be formed between the power recycling mirror and the Fabry-Perot Michelson, increasing the power on the beamsplitter and in the arms.', '1204.5504-2-5-4': 'In order to maintain the Michelson fringe offset required for this configuration the short Michelson cavity length, [MATH] or MICH, must also be controlled.', '1204.5504-2-6-0': "Of the external disturbances that the control systems must overcome, one of the most problematic is the persistent and large 0.1 - 0.3 Hz double-frequency 'microseismic' peak, generated by wave activity in the oceans [CITATION].", '1204.5504-2-6-1': 'At the microseismic peak the amplitude is on the order of several m/[MATH], but the level of seismicity present at the observatories varies widely (by a factor of [MATH]10) [CITATION].', '1204.5504-2-6-2': 'The relative displacement of the test masses in this band is of the same order as the horizontal displacements caused locally by the microseism.', '1204.5504-2-6-3': 'These levels of seismic noise present a host of problems, including:', '1204.5504-2-7-0': 'At frequencies where the motion of the ground and the length fluctuations in the cavity are coherent the seismic noise can be actively subtracted from the cavity motion using a feed-forward signal.', '1204.5504-2-7-1': 'Feed-forward cancellation of the microseism was demonstrated at LLO during the commissioning of the detector, using piezoelectric actuators and filters created via system-identification [CITATION].', '1204.5504-2-7-2': "Between LIGO's third and fourth science runs (S3 and S4) a new external seismic isolation system, HEPI, was installed at LLO, which used feed-forward signals to subtract local ground motion from the isolated platform motion with a technique often called 'sensor correction' [CITATION].", '1204.5504-2-7-3': 'Those techniques were used to the absolute motion of each individual platform.', '1204.5504-2-7-4': "This paper will describe an improved feed-forward scheme which uses combinations of seismometers located several km's apart and Wiener filtering to reduce the differential motion between the platforms, and thereby quiet interferometric control signals.", '1204.5504-2-7-5': "We show results achieved during LIGO's sixth science run (S6) which lasted from July 2009 until October 2010.", '1204.5504-2-7-6': 'In other work, Wiener filtering has been used for real-time cancellation of phase noise from vibrations in cavity stabilized lasers [CITATION], and a similar technique was used in a 40 meter long suspended interferometer [CITATION].', '1204.5504-2-8-0': '# Method', '1204.5504-2-9-0': 'In order to optimize the noise subtraction we want to find the Wiener filter [CITATION], [MATH], that processes the witness seismometer signals, [MATH], into the best possible estimate of the targeted cavity length control signal, [MATH].', '1204.5504-2-9-1': 'We find [MATH] by minimizing the mean square error between [MATH] and [MATH], [EQUATION] with respect to [MATH], where [MATH] is the filter order (number of taps).', '1204.5504-2-9-2': 'This is done by setting the derivative, [EQUATION] which yields the Wiener-Hopf equations, [EQUATION] where [MATH] is the autocorrelation matrix of the seismometer signal and [MATH] is the cross-correlation between the seismometer and the control signals.', '1204.5504-2-9-3': 'The FIR Wiener filters were estimated in MATLAB [CITATION] using the Levinson-Durbin algorithm [CITATION], and then fit to a set of IIR coefficients to reduce the computational time in the real-time control system.', '1204.5504-2-9-4': 'The conversion from FIR to IIR was done with the Vectfit [CITATION] software package.', '1204.5504-2-9-5': "The intermediate FIR filters were composed of 1000's of taps and processed data with a sample rate of 64 Hz, allowing for subtraction down to a few 10's of mHz.", '1204.5504-2-9-6': 'Approximately 1 hour of data was used to train each filter.', '1204.5504-2-9-7': 'Each test mass is in the vicinity of at least one seismometer (see Figure [REF]), which measures motion in three perpendicular directions, X, Y, and Z, where Z is vertical and the two horizontal directions are roughly aligned to the arms of the interferometer.', '1204.5504-2-9-8': 'The length control is coupled most strongly to motion along the axes of the respective cavities, but cross-couplings to the other directions are non-negligible.', '1204.5504-2-9-9': 'Regardless of the direction of the witness signal generating a feed-forward correction the actuation is applied in the direction in which the laser beam is propagating for that particular chamber.', '1204.5504-2-10-0': 'Offloading control signals to actuators located in the external seismic isolation systems mitigates several of the problems mentioned in Section [REF].', '1204.5504-2-10-1': 'In order to properly subtract the filtered witness signals, the transfer function from our point of actuation to the cavity control signal must be measured and divided out.', '1204.5504-2-10-2': 'A diagram showing the relevant pieces of the mechanical structure can be seen in Figure [REF].', '1204.5504-2-10-3': 'The number of mechanical components separating the mirrors from the ground creates a complicated transfer function with many resonant features.', '1204.5504-2-11-0': 'At LLO, the active seismic isolation system is HEPI (hydraulic external pre-isolator) [CITATION].', '1204.5504-2-11-1': 'A similar system installed at LHO, PEPI, used piezoelectric actuation.', '1204.5504-2-11-2': 'HEPI is an Advanced LIGO (aLIGO) system which will be in place at both LLO and LHO in the future.', '1204.5504-2-11-3': 'Figure [REF] shows a schematic of the HEPI system [CITATION].', '1204.5504-2-11-4': "The Wiener filter feed-forward path was added in parallel to the existing sensor correction path, which contained hand-tuned filters designed to match HEPI's position sensors to local seismometers on the ground.", '1204.5504-2-11-5': 'Typically this sensor correction reduced the differential motion sensed by the suspended cavities to [MATH] of the ground motion, in the microseism band.', '1204.5504-2-11-6': 'All results shown are improvements on top of this existing isolation.', '1204.5504-2-11-7': 'The HEPI actuators [CITATION] provide the ability to move its payload by [MATH] 700 m; the maximum range of PEPI was [MATH] m.', '1204.5504-2-11-8': 'We excited these actuators while the detector was locked and monitored the mirror motion to measure the mechanical response.', '1204.5504-2-11-9': 'We again used Vectfit to fit these measurements.', '1204.5504-2-11-10': 'An example of these transfer function measurements can be seen in Figure [REF].', '1204.5504-2-11-11': 'An example Wiener filter can be seen in Figure [REF].', '1204.5504-2-11-12': 'Both the Wiener filter and mechanical response are only fit accurately up to few Hz.', '1204.5504-2-11-13': 'Low-pass filtering was applied to the feed-forward signal to prevent noise injection at higher frequencies.', '1204.5504-2-12-0': '# Results and Limitations', '1204.5504-2-13-0': 'Before the laser beam enters interferometer, higher order spatial modes are removed by a suspended, triangular cavity called the code cleaner (MC).', '1204.5504-2-13-1': 'The length of the mode cleaner is controlled such that only the TEM00 mode is resonant.', '1204.5504-2-13-2': 'The cavity is [MATH]12 m long, with its mirrors split between isolated tables in two vacuum tanks, MC1 and MC2, as shown in Figure [REF].', '1204.5504-2-13-3': 'At LLO these chambers are equipped with HEPI and served as the initial testbed for the Wiener filter feed-forward implementation.', '1204.5504-2-13-4': 'Figure [REF] shows the control signal holding the mode cleaner on resonance with and without the Wiener feed-forward signal path enabled.', '1204.5504-2-13-5': 'A factor of up to [MATH]5 improvement was realized around the microseismic peak, and the RMS of the signal was reduced by a factor of [MATH]2.', '1204.5504-2-13-6': 'The pre-existing sensor correction filters are left in their nominal state for all results shown in this paper.', '1204.5504-2-14-0': 'Successful implementation of Wiener filter feed-forward on the mode cleaner length control prompted commissioning of feed-forward paths for the other interferometric control signals, starting with the power recycling cavity control at LLO.', '1204.5504-2-14-1': "This work was performed during a time of particularly high microseismic activity which impaired the Livingston detector's ability to remain in lock.", '1204.5504-2-14-2': 'As can be seen in Figure [REF](a), a significant reduction in control signal was achieved, with the RMS being reduced by more than a factor of 2.', '1204.5504-2-14-3': "As a result there was also a reduction in the power fluctuations in the recycling cavity, which tended to drive the interferometer control loops towards instability, and greatly improved the instrument's duty cycle.", '1204.5504-2-14-4': 'This cavity is nearly geometrically degenerate [CITATION] (g-factor [MATH]1) and therefore is especially sensitive to misalignments.', '1204.5504-2-15-0': 'Sending feed-forward signals developed to minimize one cavity control signal may inject noise into other length controls, since the cavities share mirrors.', '1204.5504-2-15-1': 'To avoid this problem, feed-forward paths were implemented in a serial fashion, such that extra motion injected into other degrees of freedom could be corrected by the Wiener filters constructed for subsequent feed-forward paths.', '1204.5504-2-15-2': 'For example, Figure [REF](b) shows the effect of filters designed to minimize the short Michelson control signal, calculated on top of the existing power recycling feed-forward path.', '1204.5504-2-15-3': 'Above [MATH]0.2 Hz the Michelson signal is reduced, with the improved isolation of the power recycling cavity preserved, albeit with some noise injection above [MATH]1 Hz.', '1204.5504-2-15-4': 'Since the overall RMS is still reduced this noise injection was considered an acceptable tradeoff.', '1204.5504-2-16-0': 'Global feed-forward for the differential arm length control was implemented at both observatories with similar results.', '1204.5504-2-16-1': 'Since the DARM degree of freedom is sensitive to the motions of mirrors in both the end buildings as well as the corner station seismometers in all three locations were incorporated into the calculation of the optimal feed-forward filter.', '1204.5504-2-16-2': 'Figures [REF] and [REF] show that the overall RMS of the control signal is reduced by a factor of [MATH]2.5 for each interferometer.', '1204.5504-2-16-3': 'For LLO this improvement in performance was balanced by a slight noise increase above 1 Hz.', '1204.5504-2-16-4': 'Some noise was also injected by Wiener filter feed-forward at LHO, below 0.1 Hz, and above 3 Hz.', '1204.5504-2-17-0': 'The couplings of ground motion to the feedback signal can potentially be time dependent.', '1204.5504-2-17-1': "Since the approach used here is not adaptive, this could lead to a degradation of the filter's subtraction efficacy over time.", '1204.5504-2-17-2': 'Figure [REF] shows the performance of DARM feed-forward at LLO when first implemented and 8 months later.', '1204.5504-2-17-3': 'While the overall reduction in RMS motion was originally a factor of [MATH]2.5 the same feed-forward filters provided [MATH]20% less isolation after 8 months, reducing the RMS motion by a factor of 2.', '1204.5504-2-17-4': 'There is no obstacle to retraining new Wiener filters to potentially recover subtraction performance, however if there is a change in the mechanical plant remeasuring the transfer function is a time consuming process, due to the high accuracy required.', '1204.5504-2-18-0': 'As mentioned in Section [REF], we have observed that the rate of non-Gaussian events in the gravitational wave band increases when large forces are required to maintain detector lock.', '1204.5504-2-18-1': 'To measure the effect of improved isolation on the rate of background events, two 30 minute segments of data were collected for L1, with the DARM Wiener feed-forward first enabled and then disabled.', '1204.5504-2-18-2': 'A templated search, using a sine-Gaussian waveform basis [CITATION], was performed and the number of triggers per second reported are shown in Figure [REF].', '1204.5504-2-18-3': 'Only triggers with frequencies between 40 and 150 Hz are included, and the threshold SNR was set to 5.', '1204.5504-2-18-4': 'While both distributions show an excess of non-Gaussian events, reducing the low frequency control signal clearly results in a lower rate of signal band transients.', '1204.5504-2-18-5': 'The stationary noise floor of the detector did not vary significantly during the time this data was taken.', '1204.5504-2-19-0': 'Fundamentally the ability to provide feed-forward subtraction is limited by the coherence between the witness signals and the cavity control signal.', '1204.5504-2-19-1': 'A drawback particular to the scheme detailed in this paper is the lack of witnesses measuring all signals which contribute to the cavity motion.', '1204.5504-2-19-2': 'Since the seismometers cannot distinguish between translation and tilt, additional sensors such as tilt meters could contribute additional isolation through more feed-forward paths [CITATION].', '1204.5504-2-19-3': 'There is the potential to be limited by the noise floors of motion sensors, however at the microseismic peak the signal to noise ratio is very large; see Figure [REF].', '1204.5504-2-19-4': 'Numerical errors can arise in the calculation of the FIR Wiener filter as well as in the fitting of both the IIR coefficients and the mechanical transfer function.', '1204.5504-2-19-5': 'These inaccuracies also have the potential to limit the feed-forward subtraction.', '1204.5504-2-20-0': '# Conclusions', '1204.5504-2-21-0': "We have demonstrated the ability to use the LIGO detector's active seismic isolation system to reduce the relative motion between optics on a 4 km scale through seismometer feed-forward control.", '1204.5504-2-21-1': 'By sending the filtered seismometer signals to the HEPI/PEPI actuators we were able to quiet the relative motion of the mirrors, offloading control signal from the test mass actuators to the seismic isolation system.', '1204.5504-2-21-2': 'During times of elevated microseismic activity this allowed for unprecedented interferometer stability.', '1204.5504-2-21-3': 'The rate of audio band transients is significantly reduced by having less motion and less applied force.', '1204.5504-2-21-4': 'This method will be directly applicable to the second generation gravitational-wave detectors.'}

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[]

[]

[]

[]

['1204.5504-1-6-3', '1204.5504-2-6-3']

{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1204.5504

astro-ph-0208041

{'astro-ph-0208041-1-0-0': "We study the formation of the Milky Way's halo in a [MATH]CDM cosmology by scaling down a high resolution simulation of the formation of a cluster of galaxies.", 'astro-ph-0208041-1-0-1': 'We determine how much phase-space substructure is left over from the objects that merge to build up the present galaxy.', 'astro-ph-0208041-1-0-2': 'We study the debris streams originating from such objects and find that their evolution can be explained simply in terms of the conservation of phase-space density.', 'astro-ph-0208041-1-0-3': 'Analysing the mass growth history of the halo we find that its inner regions have been in place for more than 10 Gyr, but that the growth of the halo as a whole is more gradual.', 'astro-ph-0208041-1-0-4': 'Recent accretion contributes to the inner 10 kpc of the halo only at the 10[MATH] level.', 'astro-ph-0208041-1-0-5': 'Finally we determine the number of dark-matter streams as a function of distance from the centre of the halo.', 'astro-ph-0208041-1-0-6': 'In the equivalent of the "Solar vicinity", we find that the dark-matter is smoothly distributed in space, and that the velocity ellipsoid is formed by hundreds of thousands of streams, most of which have velocity dispersions of the order of 1 km s[MATH] or less.', 'astro-ph-0208041-1-1-0': '# Introduction', 'astro-ph-0208041-1-2-0': 'Over the last twenty years, the hierarchical paradigm has emerged as the standard model to describe the formation of structure in the Universe.', 'astro-ph-0208041-1-2-1': 'As embodied in the current "concordance" [MATH]CDM model it appears to be consistent with a very wide range of cosmological data ranging from fluctuations in the Cosmic Microwave Background through the structure of Ly[MATH] forest absorption in QSO spectra and the gravitational shear induced by dark-matter structures to the observed large scale structure in the galaxy distribution.', 'astro-ph-0208041-1-2-2': 'An important characteristic of such models is that they are based on a set of well-defined and testable assumptions.', 'astro-ph-0208041-1-2-3': 'This renders possible the detailed modelling of the formation and evolution of galactic systems, and a later comparison to observations of the properties of these systems as a function of environment or redshift (e.g. Diaferio et al. 2001; Benson et al. 2001; Somerville, Primack Faber 2001).', 'astro-ph-0208041-1-3-0': 'It is also possible to test the hierarchical paradigm on our Galaxy (e.g. Hernandez, Avila-Reese Firmani 2001).', 'astro-ph-0208041-1-3-1': 'Several groups (Moore et al. 1999; Klypin et al. 1999; Klypin, Zhao Somerville 2002) have focused on the properties of dark halos, hoping to constrain the nature of dark-matter.', 'astro-ph-0208041-1-3-2': 'These groups performed high resolution simulations of a galactic size halo in CDM cosmologies.', 'astro-ph-0208041-1-3-3': 'They confirmed earlier analytic claims (Kauffmann, White Guiderdoni 1993) that the predicted number of satellites exceeds the known population in the Local Group by a factor of ten.', 'astro-ph-0208041-1-3-4': 'Some attempts have been made to account for the disagreement, by changing the nature of the dark-matter (Spergel Steinhardt 2000; Bode, Ostriker Turok 2001), by modifying the initial power spectrum of density fluctuations (Kamionkowski Liddle 2000) or by taking into account the effects of a reionising background which may inhibit star formation in the smallest mass halos (Kauffmann et al. 1993; Bullock, Kravtsov Weinberg 2001; Benson et al. 2002).', 'astro-ph-0208041-1-3-5': 'The recent results by Kleyna et al. (2002) on the mass distribution in the Draco dSph (see also Mateo 1997 and Lokas 2001 for a similar study on Fornax) favour an astrophysical explanation since the actual circular velocities of the other satellite galaxies of the Milky Way are in fact several tens of km s[MATH] larger than previously thought, and agree with those expected for the most massive substructures in a [MATH]CDM universe (Stoehr et al. 2002).', 'astro-ph-0208041-1-4-0': 'Broadly speaking, the hierarchical paradigm predicts that the Milky Way formed through mergers of smaller systems (White Rees 1978).', 'astro-ph-0208041-1-4-1': 'These systems would contribute to the dark halo, the spheroid (the bulge and the stellar halo) and to the Galactic gas reservoir.', 'astro-ph-0208041-1-4-2': 'It may be very difficult to determine the relative gas contribution of these progenitor objects to the present Galaxy, since gas "easily forgets" its site of origin.', 'astro-ph-0208041-1-4-3': 'However for collisionless stars and dark-matter the situation can be quite different.', 'astro-ph-0208041-1-4-4': 'If the dynamical mixing timescales are sufficiently long (i.e. longer than the age of the Universe) it may be possible to "break-up" the Galactic spheroid (stars and may be even dark matter particles) into coherent structures in phase-space directly related to the systems that merged to form the Milky Way we observe today.', 'astro-ph-0208041-1-5-0': 'A first attempt at determining whether the merging history of the Milky Way may be imprinted in the phase-space structure of nearby halo stars, and thus be recovered, was made by Helmi White (1999; hereafter HW).', 'astro-ph-0208041-1-5-1': 'They studied the infall of satellites onto a fixed Galactic potential, and the evolution of the debris in phase-space.', 'astro-ph-0208041-1-5-2': 'They found that after 10 Gyr stars having a common origin are distributed smoothly in space, but appear very clumped in velocity space, where they define streams with very small velocity dispersions.', 'astro-ph-0208041-1-5-3': 'The expected number of such streams scales with the initial size [MATH], velocity dispersion [MATH] and orbital period [MATH], of the disrupted object: [EQUATION]', 'astro-ph-0208041-1-5-4': 'The total number of stars associated with the object is [MATH] (from the virial theorem) while the volume [MATH] over which they are spread scales with the cube of the size of the orbit, and so approximately as [MATH].', 'astro-ph-0208041-1-5-5': 'Hence the number of stars per stream in the Solar neighbourhood scales as [MATH]; objects with large initial velocity dispersion and small initial size should produce the most easily detectable streams with little dependence on initial period.', 'astro-ph-0208041-1-5-6': 'Such arguments suggest that the Solar neighbourhood velocity ellipsoid is composed of [MATH] kinematically coherent structures which originated in past merger and accretion events.', 'astro-ph-0208041-1-5-7': 'A pair of halo streams that can perhaps be directly linked to a disrupted satellite were detected in the Solar neighbourhood by Helmi et al.(1999).', 'astro-ph-0208041-1-5-8': 'The progenitor of these two streams was probably similar to the dwarf galaxy Fornax.', 'astro-ph-0208041-1-5-9': 'Substructure in the outer halo also appears to be ubiquitous, and has been found by several surveys over the last few years (e.g. Ivezic et al. (2000) and Yanny et al. (2000) for the SDSS; Dohm-Palmer et al. (2001) for the SPS; Vivas et al. (2001) for QUEST).', 'astro-ph-0208041-1-5-10': "Most of these recently discovered structures can be associated to just one of Milky Way's satellites: the Sagittarius dwarf galaxy which is in the process of being completely disrupted (Ibata et al. 2001; Martinez-Delgado et al. 2001; Helmi White 2001).", 'astro-ph-0208041-1-6-0': 'A weak point of the HW analysis and of similar studies (e.g. Johnston, Hernquist Bolte 1996; Johnston 1998), is the assumption of a fixed, smooth potential onto which galaxies are accreted.', 'astro-ph-0208041-1-6-1': 'In hierarchical clustering, galaxy potentials are constantly changing, and can vary very violently during mergers.', 'astro-ph-0208041-1-6-2': 'Large numbers of clumps orbit the centre of even a "virialised" halo.', 'astro-ph-0208041-1-6-3': 'These clumps may have substantial effects on the structure of debris streams (e.g Johnston, Spergel Haydn 2002; Ibata et al. 2002; Mayer et al. 2002).', 'astro-ph-0208041-1-7-0': 'The main goal of the present paper is to understand the phase-space structure of cold dark-matter halos.', 'astro-ph-0208041-1-7-1': 'In particular, we want to study the evolution of satellite debris, and to quantify the expected amount of substructure.', 'astro-ph-0208041-1-7-2': 'We also want to determine to what extent previous results are valid in the truly hierarchical regime of the build-up of a galaxy.', 'astro-ph-0208041-1-7-3': 'We tackle these problems by scaling down to galactic size a high-resolution simulation of the formation of a cluster in a [MATH]CDM cosmology (Springel et al. 2001).', 'astro-ph-0208041-1-8-0': 'The paper is organised as follows.', 'astro-ph-0208041-1-8-1': 'In Sec.2 we describe the simulations, in Sec.3 we follow the evolution in phase-space of debris streams, and compare to the analytic model of HW in Section 3.2.', 'astro-ph-0208041-1-8-2': 'Sec.4 describes the mass-growth history of the simulated dark-matter halo, and in Section5 we determine the number of streams and their internal properties as function of distance from the dark-matter halo centre.', 'astro-ph-0208041-1-8-3': 'We leave the summary and discussion of our results for Section 6.', 'astro-ph-0208041-1-9-0': '# Methodology', 'astro-ph-0208041-1-10-0': '## The simulations', 'astro-ph-0208041-1-11-0': 'The simulation we analyse here was carried out using a parallel tree-code (Springel, Yoshida White 2001) on the Cray T3E at the Garching Computing Centre of the Max Planck Society.', 'astro-ph-0208041-1-11-1': 'Its initial conditions were generated by zooming in and re-creating with higher resolution a particular galaxy cluster and its surroundings formed within a cosmological simulation (as in Tormen, Bouchet White 1997).', 'astro-ph-0208041-1-11-2': 'The original parent [MATH]CDM cosmological simulation (from Kauffmann et al. 1999) had parameters [MATH], [MATH], [MATH] and [MATH].', 'astro-ph-0208041-1-11-3': 'The cluster selected for re-simulation was the second most massive cluster in this simulation, having a virial mass of [MATH].', 'astro-ph-0208041-1-11-4': 'The particles that end up in the final cluster of the cosmological simulation and in its immediate surroundings (defined by a comoving sphere of [MATH] Mpc radius) were traced back to their Lagrangian region in the initial conditions for re-simulation.', 'astro-ph-0208041-1-11-5': 'The initial mass distribution between [MATH] and [MATH] Mpc was represented by [MATH] particles.', 'astro-ph-0208041-1-11-6': 'In the inner region, where the original simulation had [MATH] particles, new initial conditions were created using [MATH] particles.', 'astro-ph-0208041-1-11-7': 'Small scale power was added in accordance with the better k-space sampling allowed by the larger number of particles.', 'astro-ph-0208041-1-11-8': 'The original force softening was also decreased to obtain better spatial resolution.', 'astro-ph-0208041-1-11-9': 'The new simulation was run from high redshift until [MATH], and was analysed in considerable detail in Springel et al. (2001).', 'astro-ph-0208041-1-11-10': 'In this high resolution simulation there are about 20 million particles within the virialised region of the cluster halo.', 'astro-ph-0208041-1-12-0': '## Scaling to a Milky Way halo', 'astro-ph-0208041-1-13-0': 'In Figure [REF] we show the circular velocity profile of the cluster at [MATH].', 'astro-ph-0208041-1-13-1': 'We determine the cluster centre by successively refining a mesh located on the cluster, and determining the cell containing the largest number of particles.', 'astro-ph-0208041-1-13-2': 'This process is repeated until the largest number of particles in a given (now small size) cell is sufficiently small to determine by simple counts which particle has the largest number of neighbours.', 'astro-ph-0208041-1-13-3': 'Its position corresponds to the point of maximum density and it is defined as the cluster centre.', 'astro-ph-0208041-1-13-4': 'This determination is robust against changes in the mesh shape and size, and we estimate the error in the final position of the centre of the cluster to be of the order of 0.7 kpc, comparable to the scale of the gravitational softening used in the simulation.', 'astro-ph-0208041-1-13-5': 'The circular velocity is then determined after spherically averaging the mass distribution around the centre of the cluster and is derived from [MATH].', 'astro-ph-0208041-1-14-0': 'By fitting a Navarro, Frenk White (NFW; 1996) profile, we find that the concentration of the cluster halo is [MATH], the scale radius [MATH] kpc and the virial radius [MATH] kpc.', 'astro-ph-0208041-1-15-0': 'We can scale the cluster to a "Milky Way" halo, by requiring that its maximum circular velocity is equal to 220 km s[MATH].', 'astro-ph-0208041-1-15-1': 'The required scaling factor [MATH] is given by [EQUATION]', 'astro-ph-0208041-1-15-2': 'With this scaling, the virial radius of our simulated Milky Way dark-matter halo is 228 kpc.', 'astro-ph-0208041-1-15-3': 'Its virial mass is [MATH], the mass of an individual particle is [MATH], and the gravitational force in the final object has an equivalent Plummer softening of 0.11 kpc.', 'astro-ph-0208041-1-16-0': 'Our argument that with the simple scaling of Eq. ([REF]) this simulation is a fair representation of the growth of the Milky Way relies on both theoretical and numerical results (Lacey Cole 1993; Moore et al. 1999).', 'astro-ph-0208041-1-16-1': 'Numerical simulations by Moore et al. (1999) have shown that galaxy and cluster halos have similar properties in terms of final structure (density profile), number of satellites, etc., despite typically assembling at systematically different redshifts.', 'astro-ph-0208041-1-16-2': 'Jing Suto (2000) have also performed high resolution simulations of several galaxy, group and cluster size halos.', 'astro-ph-0208041-1-16-3': 'Although they find that galaxy halos have a steeper inner profile than larger mass halos, they also find that the scatter in the properties of objects belonging to the same class is as large as the systematic differences between classes.', 'astro-ph-0208041-1-16-4': 'We expect therefore, that our scaled simulation will represent reasonably well the formation process of a galactic halo, except that its assembly occurs at later times than the majority of such halos.', 'astro-ph-0208041-1-16-5': 'From now on we will use this scaling in the paper, and assume, unless otherwise stated, that the simulation represents a galaxy halo.', 'astro-ph-0208041-1-17-0': '# The phase-space evolution of debris', 'astro-ph-0208041-1-18-0': 'Here we study the phase-space evolution of debris from the disrupted halos that end up forming the "Galactic" dark halo at the present time.', 'astro-ph-0208041-1-18-1': 'With this goal in mind, we proceed by identifying halos at high-redshift which are directly accreted onto the main progenitor of the "Milky Way\'s" halo.', 'astro-ph-0208041-1-19-0': 'We identify halos at each output time using a Friends-of-Friends (FOF) algorithm, which links particles separated by less than 20% of the mean interparticle separation.', 'astro-ph-0208041-1-19-1': 'In this way we can construct a detailed merger history of the galaxy.', 'astro-ph-0208041-1-19-2': "As we step back in redshift we identify at each output the most massive halo which is part of the galaxy's main progenitor at the subsequent time.", 'astro-ph-0208041-1-19-3': 'We say that a halo identified at redshift [MATH] will be directly accreted onto the main progenitor at [MATH] (the redshift of the next simulation output) if at least half of its particles and the most-bound particle have become part of the main progenitor at [MATH].', 'astro-ph-0208041-1-19-4': 'Here the most-bound particle refers to the particle with the minimum potential energy in the (to be) accreted halo.', 'astro-ph-0208041-1-20-0': 'The centre of mass position [MATH] and velocity [MATH] of the main progenitor are computed as follows.', 'astro-ph-0208041-1-20-1': 'We first determine the location of the most bound particle [MATH].', 'astro-ph-0208041-1-20-2': 'We define concentric spheres of successively smaller radii (down to 10 kpc radius) around this particle, and compute the centre of mass [MATH] from the particles within these spheres.', 'astro-ph-0208041-1-20-3': 'We stop this iterative procedure when [MATH], where [MATH] kpc.', 'astro-ph-0208041-1-20-4': 'This is then defined as the location of centre of mass of the main progenitor.', 'astro-ph-0208041-1-20-5': 'The velocity of the centre of mass is then [MATH], measured by the velocities of the particles within the largest sphere for which the above condition is satisfied.', 'astro-ph-0208041-1-21-0': "This is like following the (thickest) trunk of the merger tree, which would correspond to the galaxy's main progenitor, and studying what happens to halos which join from other tree branches as time progresses.", 'astro-ph-0208041-1-21-1': 'The idea is illustrated in Figure [REF].', 'astro-ph-0208041-1-22-0': 'For our high resolution simulation we have identified 752 halos with at least 1000 particles (which corresponds to a minimum mass of [MATH]), which fall onto the main progenitor between redshift [MATH] and the present day.', 'astro-ph-0208041-1-22-1': 'The accreted halos have a large spread in mass as shown in the bottom panel of Figure [REF].', 'astro-ph-0208041-1-22-2': 'From the top panel of this Figure we note that in some cases the satellite-to-primary mass ratio is close to unity, corresponding to a major merger.', 'astro-ph-0208041-1-22-3': 'Such mergers, although few in number, contribute a substantial fraction of the total mass growth.', 'astro-ph-0208041-1-23-0': '## Spatial and kinematic evolution of halo debris', 'astro-ph-0208041-1-24-0': 'We now study in detail how the debris streams from one of the above 752 halos evolve in time.', 'astro-ph-0208041-1-24-1': 'We follow a halo that merges with the main progenitor at redshift 1.8, and has an initial mass of [MATH].', 'astro-ph-0208041-1-24-2': 'This halo is denoted with a diamond in Figure [REF].', 'astro-ph-0208041-1-24-3': 'Its mass at the time of accretion was about 25% that of the main progenitor.', 'astro-ph-0208041-1-24-4': 'This satellite decays through dynamical friction to the centre of the main progenitor where it is fully disrupted.', 'astro-ph-0208041-1-24-5': 'The mass stripped off at earlier times mainly populates the outer regions of the final object, while that lost at late times ends up closer to its centre.', 'astro-ph-0208041-1-25-0': '### Evolution of debris in the outer galaxy', 'astro-ph-0208041-1-26-0': 'We identify material from the accreted halo which is part of a tidal stream in the outer galaxy at the present time.', 'astro-ph-0208041-1-26-1': 'We select a reference particle in this structure, which can be traced back to [MATH], and then followed forwards in time.', 'astro-ph-0208041-1-26-2': 'In Figure [REF] we show a time sequence of the spatial distribution of particles lost at [MATH].', 'astro-ph-0208041-1-26-3': 'We say that a particle has been lost by its progenitor halo if its binding energy has become positive.', 'astro-ph-0208041-1-26-4': 'The binding energy [MATH] of a particle located at [MATH] and with velocity [MATH] with respect to the centre of mass of the satellite is defined as [MATH], where [MATH] is the potential energy at [MATH] due to all particles in the satellite.', 'astro-ph-0208041-1-26-5': 'From Fig. [REF] we see that the initial distribution of particles is relatively compact, and that as time passes by the material is strung out in a characteristic stream-like structure (e.g at [MATH] Gyr) over several tens to hundreds of kiloparsecs. At the final time the material appears to be more smoothly distributed over the whole box, which is 400 kpc (in physical units) on a side.', 'astro-ph-0208041-1-26-6': 'Our simulations are not well-suited to address the effects of other dark-matter lumps on tidal tails (cf Johnston et al. 2001), which could be related to some of the transient structures observed in this Figure.', 'astro-ph-0208041-1-26-7': 'Our main limitation is the large time interval between outputs which prevents us from determining the effect of close encounters and their relation to the features seen in Figure [REF].', 'astro-ph-0208041-1-27-0': 'In many of the snapshots tight small substructures can be observed.', 'astro-ph-0208041-1-27-1': 'These objects were already present as subhalos within the satellite halo before it was accreted by the galaxy.', 'astro-ph-0208041-1-27-2': 'After accretion, they were released from their parent satellite, becoming subhalos of the main galaxy.', 'astro-ph-0208041-1-27-3': 'It is worth mentioning that these subhalos only constitute a small fraction of the debris material lost by the satellite (less than 10%, e.g. Ghigna et al. 2000; Springel et al. 2001).', 'astro-ph-0208041-1-28-0': 'In Figure [REF] we show the velocities of debris particles that are relatively close (inside a cube of side 20 kpc) to the reference particle, at three different times.', 'astro-ph-0208041-1-28-1': 'For comparison, note that the virial radius of the satellite at the time of infall was approximately 37 kpc.', 'astro-ph-0208041-1-28-2': 'The solid grey circles correspond to particles which', 'astro-ph-0208041-1-29-0': 'were always neighbours of the reference particle: to be in this set, particles need to have [MATH], [MATH] and [MATH] coordinates within 10 kpc of those of the reference particle in all previous outputs;', 'astro-ph-0208041-1-30-0': 'are within 5 kpc of the reference particle in each coordinate in the current output.', 'astro-ph-0208041-1-31-0': 'We note that the initial velocity distribution of the halo is broad, and relatively clumpy.', 'astro-ph-0208041-1-31-1': 'As discussed before, this clumpiness reflects the internal structure of the object that fell in.', 'astro-ph-0208041-1-31-2': 'As time goes by, the motions of the neighbours of the reference particle become more similar, and the velocity-box is empty except for velocities close to that of the reference particle (middle row).', 'astro-ph-0208041-1-31-3': 'At later times (bottom row), other moving groups are visible, showing that the system has now produced multiple intersecting streams even in the outer galaxy.', 'astro-ph-0208041-1-32-0': 'The evolution of the velocity dispersion [MATH] in the neighbourhood of the reference particle is shown in the top panel of Figure [REF].', 'astro-ph-0208041-1-32-1': 'Here [MATH] is defined as [MATH], and is therefore independent of the choice of the coordinate system.', 'astro-ph-0208041-1-32-2': 'It is measured for the set of particles shown as grey solid circles in Figure [REF], that is for particles that satisfy both conditions (i) and (ii) for being neighbours of the reference particle.', 'astro-ph-0208041-1-32-3': 'We also measure [MATH] for a subset of even closer neighbouring particles (whose [MATH], [MATH], and [MATH] coordinates in all outputs before the one under study were within 5 kpc of those of the reference particle, and in the current output within 2.5 kpc).', 'astro-ph-0208041-1-32-4': 'The different values of the velocity dispersion in the stream obtained for these two cases are due to velocity gradients, which can be as large as the measured dispersions themselves.', 'astro-ph-0208041-1-32-5': 'Note as well the decrease by roughly a factor 10 in the velocity dispersions in only 2 Gyr.', 'astro-ph-0208041-1-33-0': "The stream's density is shown in the bottom panel of the same figure.", 'astro-ph-0208041-1-33-1': 'It is measured at each output by the number of neighbours in the stream in boxes of a given size (either of 5 or 10 kpc on a side) around the reference particle.', 'astro-ph-0208041-1-33-2': 'For the larger volume, this number evolves from being slightly larger than five thousand to only thirteen after less than 2 Gyr of evolution.', 'astro-ph-0208041-1-33-3': 'This implies that after this time, the number counts are dominated by Poisson noise.', 'astro-ph-0208041-1-34-0': '### Evolution of debris closer to the centre of the galaxy', 'astro-ph-0208041-1-35-0': 'To follow the evolution of streams closer to the centre of the galaxy we focus on material lost a few Gyr after the satellite halo shown as a diamond in Fig. [REF] was accreted.', 'astro-ph-0208041-1-35-1': 'We here focus on debris lost from this satellite halo at [MATH] Gyr, or 3.5 Gyr after infall.', 'astro-ph-0208041-1-35-2': 'This is deposited at an intermediate distance from the centre of the galaxy.', 'astro-ph-0208041-1-35-3': 'We decided against tracking material lost even later because it turns out to be very difficult to follow streams from such material with our numerical resolution and number of stored outputs.', 'astro-ph-0208041-1-35-4': 'The material lost at later times mixes on shorter timescales.', 'astro-ph-0208041-1-35-5': 'Streams more rapidly reach very small densities, and we are unable to determine their properties reliably.', 'astro-ph-0208041-1-36-0': 'Figure [REF] shows the evolution of the spatial distribution of the particles lost at [MATH] Gyr.', 'astro-ph-0208041-1-36-1': 'Note that the particles are more smoothly distributed than in the case of the outer halo material shown in Figure [REF].', 'astro-ph-0208041-1-36-2': 'This is because the inner regions of (satellite) halos are in general much less lumpy than the outskirts due to the shorter dynamical timescales.', 'astro-ph-0208041-1-37-0': 'As in the previous section we choose a reference particle (amongst all those lost at [MATH] Gyr) that orbits through the intermediate regions of the galaxy.', 'astro-ph-0208041-1-37-1': 'Proceeding as before, we follow the evolution of the velocities of particles located close to the reference particle at all times.', 'astro-ph-0208041-1-37-2': 'Figure [REF] shows the velocity distribution in the neighbourhood of the reference particle.', 'astro-ph-0208041-1-37-3': 'By comparing to the analogue for the outer halo stream (Fig. [REF]) we note that, not just the spatial distribution, but also the velocity distribution is much smoother initially.', 'astro-ph-0208041-1-37-4': 'This is also true at late times, and very rapidly a regime is reached where multiple streams can be observed.', 'astro-ph-0208041-1-37-5': 'Figure [REF] corresponds to the evolution of the velocity dispersion and density of a stream.', 'astro-ph-0208041-1-37-6': 'Again we are not able to follow the evolution of the velocity dispersion of the stream at late times, because we run out of neighbouring particles quite rapidly, particularly for the smaller box.', 'astro-ph-0208041-1-37-7': 'After [MATH] Gyr, or 2 Gyr of evolution, the number of neighbours is dominated by Poisson noise.', 'astro-ph-0208041-1-37-8': 'After this time, the values of the 1-d velocity dispersion, when measured, are based on the velocities of only two or three particles.', 'astro-ph-0208041-1-38-0': '## Mixing in phase-space: Comparison to analytic models of stream evolution', 'astro-ph-0208041-1-39-0': 'In the previous section we found a very rapid decline in the density and velocity dispersion of streams, especially in the first few Gyr of evolution.', 'astro-ph-0208041-1-39-1': 'After this initial period, the number of particles in a stream is so small in our simulation that it is dominated by Poisson noise.', 'astro-ph-0208041-1-39-2': 'In this regime, it is difficult to quantify the properties of the stream, and it is therefore hard to determine whether the lack of variation of the velocity dispersion at late times is due to numerical limitations or is a real effect.', 'astro-ph-0208041-1-40-0': 'To gain insight into these issues, we will analyse the expected behaviour of streams evolving in a smooth time-independent potential, which should resemble that of the galaxy halo.', 'astro-ph-0208041-1-40-1': 'Using the approach developed by HW, we can follow the evolution of streams produced in a spherical and static NFW potential.', 'astro-ph-0208041-1-40-2': 'The basic idea here is to map the initial system onto action-angle space, then follow the much simpler evolution in this space, and finally transform back locally onto observable coordinates (all these being linear transformations; for details see HW).', 'astro-ph-0208041-1-40-3': 'This method, which uses action-angle variables, is limited to applications in which the potential is separable (cf Goldstein 1953; Binney Tremaine 1987).', 'astro-ph-0208041-1-40-4': 'This includes all spherically symmetric potentials but only few axisymmetric and triaxial ones, such as the general class of Stackel potentials (e.g. Lynden-Bell 1962; de Zeeuw 1985; Dejonghe de Zeeuw 1988).', 'astro-ph-0208041-1-41-0': 'We also approximate the phase-space density around the reference particle by a multivariate Gaussian distribution.', 'astro-ph-0208041-1-41-1': 'This is possible because the multivariate Gaussian is determined from the properties of particles in a volume much smaller than the size of the halo.', 'astro-ph-0208041-1-42-0': 'As discussed in Section 2.2, the (galaxy) halo can be fit by an NFW profile: [EQUATION] where [MATH] and [MATH] is a function of the concentration of the halo [MATH]: [EQUATION]', 'astro-ph-0208041-1-42-1': 'Recall that [MATH].', 'astro-ph-0208041-1-42-2': "The potential associated with this density can be obtained by integrating Poisson's equation, and is found to be: [EQUATION]", 'astro-ph-0208041-1-42-3': 'Here [MATH].', 'astro-ph-0208041-1-42-4': 'The values of the parameters [MATH] and [MATH] given in Section 2.2 should be multiplied by the factor [MATH] for the scaled halo.', 'astro-ph-0208041-1-43-0': 'In this (spherically averaged) potential we integrate the orbit of the reference particle in time.', 'astro-ph-0208041-1-43-1': 'This integration is done in two complementary ways:', 'astro-ph-0208041-1-44-0': 'starting from the position and velocity of the reference particle at the "time of formation of the stream" (this is the time when the particles become unbound from their parent halo).', 'astro-ph-0208041-1-44-1': 'In this case, the integration is performed forwards in time, and the potential used is given by Eq. ([REF]), where [MATH] is the redshift of formation of the stream.', 'astro-ph-0208041-1-45-0': 'starting from the position and velocity of the reference particle at the present time.', 'astro-ph-0208041-1-45-1': 'The orbit is then integrated backwards in time, until the "time of formation of the stream", in the present-day potential ([MATH]).', 'astro-ph-0208041-1-46-0': 'In both cases, the orbits are integrated in the reference frame of the centre of mass of the main progenitor of the galaxy.', 'astro-ph-0208041-1-47-0': 'In the top panel of Figure [REF] we plot the radial oscillations of the orbit for the outer halo reference particle.', 'astro-ph-0208041-1-47-1': 'The solid curve corresponds to the NFW potential at redshift [MATH], while the dashed curve to that at [MATH].', 'astro-ph-0208041-1-47-2': 'We note that for the reference particle in the outer halo stream, the best agreement is obtained when the integration is performed backwards in time.', 'astro-ph-0208041-1-48-0': 'The analogous plots for the reference particle orbiting the intermediate halo are shown in the bottom panel.', 'astro-ph-0208041-1-48-1': 'Recall that here [MATH].', 'astro-ph-0208041-1-48-2': 'In this case, none of the proposed orbits fit very well the actual orbit of the particle, which may be considered to lie in between these two cases.', 'astro-ph-0208041-1-48-3': 'Thus, the characteristics and the evolution of the debris streams predicted for either of the two proposed orbits may be expected to encompass the actual behaviour of the streams in our simulations.', 'astro-ph-0208041-1-49-0': 'In Figures [REF] and [REF] we show the predicted evolution of the 1-d velocity dispersion and the behaviour of the density in the spherical static NFW potential for the outer and intermediate halo streams, respectively.', 'astro-ph-0208041-1-49-1': 'We may directly compare our predictions to the actual behaviour of the stream in the simulation, using Figs. [REF] and [REF] for the outer halo stream.', 'astro-ph-0208041-1-49-2': 'For the stream orbiting the intermediate regions of the halo, we use Figs. [REF] and [REF].', 'astro-ph-0208041-1-49-3': 'In both cases, we find that the decrease observed in the simulated streams is close to what is predicted for the evolution of streams moving in a static potential.', 'astro-ph-0208041-1-49-4': 'This is the case at least for the first few Gyrs of evolution.', 'astro-ph-0208041-1-49-5': 'Although after this period we expect some differences, due to the fact that the orbit evolves in shape, the agreement still appears to be quite good.', 'astro-ph-0208041-1-49-6': 'We predict a rapidly varying velocity dispersion (and density) on top of a secular evolution.', 'astro-ph-0208041-1-49-7': 'The spikes take place when particles in a stream go through a caustic surface defined by their orbital turning points.', 'astro-ph-0208041-1-49-8': 'To observe this behaviour in the simulation would require output times spaced by 1/2 of the radial period at most, or roughly 0.25 Gyr for the stream orbiting the intermediate regions of the halo.', 'astro-ph-0208041-1-49-9': 'Our outputs, on the other hand, are logarithmically spaced in the cosmological expansion factor [MATH], so that at [MATH] the time elapsed between two outputs is [MATH] Gyr, and by [MATH], [MATH] Gyr.', 'astro-ph-0208041-1-50-0': '# Mass growth history of the galaxy', 'astro-ph-0208041-1-51-0': 'Here we focus on determining the mass growth history of the galaxy as a function of distance from the galaxy centre.', 'astro-ph-0208041-1-51-1': 'This is relevant in two different ways.', 'astro-ph-0208041-1-51-2': 'First, debris from satellites accreted at late times will be generally less mixed, and could thus produce more massive streams.', 'astro-ph-0208041-1-51-3': 'Determining where to expect these streams will enable us to understand the properties of the dark halo of our Galaxy.', 'astro-ph-0208041-1-51-4': 'Secondly such satellites will have had more time to form stars in them (prior to their merging), thereby providing the galaxy with younger stars.', 'astro-ph-0208041-1-51-5': 'The time of merging could thus be used as an indicator of the expected age distribution of stars in different regions of the Galaxy.', 'astro-ph-0208041-1-52-0': 'We select all satellites that merged with the galaxy since redshift [MATH], and determine when these mergers took place and what is their final debris distribution.', 'astro-ph-0208041-1-52-1': 'We proceed by dividing the halo in six spherical shells around the galaxy centre.', 'astro-ph-0208041-1-52-2': 'These shells are located at: [MATH] kpc, [MATH] kpc, [MATH] kpc, [MATH] kpc, [MATH] kpc and [MATH] kpc.', 'astro-ph-0208041-1-52-3': 'For each particle in a given shell, we determine when it was accreted by the main progenitor of the galaxy.', 'astro-ph-0208041-1-52-4': 'In Figure [REF] we show the fraction of mass accreted normalised to the present mass for each shell as a function of redshift (and time).', 'astro-ph-0208041-1-52-5': 'We note that the formation time of the inner galaxy is strongly biased towards high redshifts, with more than 60 of the mass already present at [MATH] or 11 Gyr ago.', 'astro-ph-0208041-1-52-6': 'Conversely, we find that mergers in the last 3 Gyr, contributed a relatively small amount of mass, less than 0.1, to this region of the galaxy halo.', 'astro-ph-0208041-1-52-7': 'For the outer regions of the galaxy, we note that the growth is much more gradual in time, with accretion being almost equally important at all times.', 'astro-ph-0208041-1-53-0': 'Of course, the detailed shape of the mass growth history depends on the detailed merger history of the halo, since the peaks observed in the different histograms correspond to individual mergers taking place at those times.', 'astro-ph-0208041-1-53-1': 'However, since most of the mass in our simulation was already in place in the halo 10 Gyr ago, this history would be consistent with that of the Milky Way.', 'astro-ph-0208041-1-53-2': 'The Galactic thin disk contains 10 Gyr old stars, implying that no major merger could have taken place after this time.', 'astro-ph-0208041-1-54-0': 'Another way of understanding the age structure of the galaxy halo is by focusing on what fraction of the mass was in place by different redshifts as a function of distance from the galaxy centre.', 'astro-ph-0208041-1-54-1': 'To some extent this is the cumulative distribution of the plots shown in Fig. [REF] for each shell.', 'astro-ph-0208041-1-54-2': 'We focus on the fraction of the present-day mass in the shell that was in place at four different redshifts [MATH], 1.5, 0.84, and 0.35.', 'astro-ph-0208041-1-54-3': 'The results are shown in Figure [REF].', 'astro-ph-0208041-1-54-4': 'Again we notice that 60% of the mass in the inner galaxy was in place by [MATH], and more than 90% by [MATH], i.e. 9.5 Gyr ago.', 'astro-ph-0208041-1-54-5': 'Thus any accreted stellar populations in the inner regions of the galaxy are predicted to be old.', 'astro-ph-0208041-1-54-6': 'On the other hand, only about half of the particles in the intermediate regions of the galaxy were present 9.5 Gyr ago (90% by [MATH]).', 'astro-ph-0208041-1-54-7': 'The formation of the outer galaxy is more biased towards late times, with half of its particles coming into place in the last 7 Gyr.', 'astro-ph-0208041-1-55-0': 'As a cautionary remark, let us recall that although we discuss here the expected "stellar" populations, we are actually describing the growth in mass of the dark-matter halo, rather than of the stellar halo.', 'astro-ph-0208041-1-55-1': 'To describe properly the age distribution of the stellar halo we would need to take into account that the stars in the different satellites that merged to build up the halo probably were located primarily in their innermost regions.', 'astro-ph-0208041-1-55-2': 'This would mean, as shown in Section [REF], that they should end up closer in the galaxy than most of the dark-matter particles of the same satellite (see also White Springel 2000).', 'astro-ph-0208041-1-55-3': 'Moreover, we are also not taking into account the fact that satellites orbiting the intermediate and outer regions of the halo can survive until the present day.', 'astro-ph-0208041-1-55-4': 'In those cases it is possible that they continue to form stars, even while orbiting inside the dark halo of the galaxy (like the Magellanic Clouds).', 'astro-ph-0208041-1-55-5': 'This means that the material that is stripped off from these systems could also contain younger stars.', 'astro-ph-0208041-1-56-0': '# The number of streams', 'astro-ph-0208041-1-57-0': 'The results obtained in Section [REF] suggest that the number of streams in the inner galaxy, and in particular in the vicinity of the Sun should be quite large.', 'astro-ph-0208041-1-57-1': 'In this section we will estimate this quantity, as well as the characteristics of the streams, such as internal velocity dispersion and density.', 'astro-ph-0208041-1-57-2': 'We will compute the number of streams inside cubes located in the inner [MATH] kpc[MATH] in the scaled "Milky Way" halo.', 'astro-ph-0208041-1-57-3': 'We partition this space into boxes of 2 kpc on a side.', 'astro-ph-0208041-1-58-0': '## General definitions', 'astro-ph-0208041-1-59-0': 'Let there be [MATH] separate true streams in a given box, each having mass [MATH]).', 'astro-ph-0208041-1-59-1': 'Let us assume that the observed number of particles in a stream follows a Poissonian distribution, and that the actual count is [MATH] particles ([MATH]).', 'astro-ph-0208041-1-59-2': 'Then the expectation value of [MATH] is [MATH], and [MATH].', 'astro-ph-0208041-1-59-3': '(Here we measure mass in units of the particle mass of the simulation.)', 'astro-ph-0208041-1-60-0': 'The mass-weighted mean mass per stream [MATH] is [EQUATION]', 'astro-ph-0208041-1-60-1': 'Therefore our estimate for [MATH] corrected for Poisson noise will be [EQUATION]', 'astro-ph-0208041-1-60-2': 'In the limit of very massive streams, the Poisson correction will be negligible since [MATH].', 'astro-ph-0208041-1-60-3': 'In the limit of small number of particles per stream, the correction will be of the same order as the quantity we measure.', 'astro-ph-0208041-1-60-4': 'Note as well, that streams which in this realization do not have any particle or just have one, do not contribute to the numerator of Eq. ([REF]).', 'astro-ph-0208041-1-60-5': 'However single particle streams do contribute to the total number of particles in the box.', 'astro-ph-0208041-1-60-6': 'If in a box we find mostly one-particle streams, then [MATH] can (correctly) become very small.', 'astro-ph-0208041-1-61-0': 'We define the mass-weighted number of streams [MATH] in a box as the ratio of the total mass in the box to the mass-weighted mean mass [MATH] per stream in the box.', 'astro-ph-0208041-1-61-1': 'Therefore [MATH] is [EQUATION] and our Poisson corrected estimate of [MATH] is [EQUATION] or [EQUATION]', 'astro-ph-0208041-1-61-2': 'For example, if in our realization all of the streams have only two particles [MATH], our estimate of the mass-weighted filling factor becomes [MATH], where [MATH] is the total number of particles in the box.', 'astro-ph-0208041-1-61-3': 'In the regime where one massive stream dominates the distribution, [MATH] and will thus be close to unity.', 'astro-ph-0208041-1-61-4': 'Also note that [MATH] can be larger than [MATH] which will happen when most of the streams contain only 1 particle.', 'astro-ph-0208041-1-62-0': '## The minimum number of streams: the number of halos', 'astro-ph-0208041-1-63-0': 'We would like to obtain an estimate of the number of disrupted halos contributing to the density at each location in the galaxy.', 'astro-ph-0208041-1-63-1': 'This is a lower limit to the total number of streams present, since a halo can (and usually does) give rise to multiple and spatially overlapping structures, as shown in Sec. [REF].', 'astro-ph-0208041-1-63-2': 'This lower limit will be particularly unrepresentative of the true number of streams in the inner halo both because of the very short dynamical timescales there and because more than 60% of the mass in this region comes from just one object identified at [MATH], as shown in Sec. [REF].', 'astro-ph-0208041-1-64-0': 'To obtain an estimate of the number of halos contributing to any given position in the Galaxy, we determine which of the halos identified at [MATH] contribute to each given box and with how many particles.', 'astro-ph-0208041-1-64-1': 'In Figure [REF] we plot the (Poisson corrected) mass-weighted mean number of particles per halo [MATH] as a function of distance from the galaxy centre for each of the boxes considered.', 'astro-ph-0208041-1-64-2': 'The thick black line corresponds to the mean number of particles in a box, averaged over all boxes at the same location.', 'astro-ph-0208041-1-64-3': 'This Figure shows that most of the particles in the inner galaxy come only from a handful of disrupted halos.', 'astro-ph-0208041-1-64-4': 'This can also be seen from Figure [REF], where we plot the Poisson corrected estimate of the mass-weighted number of halos [MATH] as a function of distance from the galaxy centre and for each one of the boxes considered.', 'astro-ph-0208041-1-64-5': 'We note that the mean number of halos per box in the outer galaxy is large, but that each contributes only a handful of particles.', 'astro-ph-0208041-1-64-6': 'However, the inner galaxy is dominated by just a few halos making up most of the mass.', 'astro-ph-0208041-1-64-7': 'This trend (of increasing number of halos with distance) is due to a form of mass segregation: heavy halos can sink by dynamical friction in short timescales to the centre of the newly formed system, whereas lighter halos, unable to decay quickly, can only deposit their mass in the outskirts.', 'astro-ph-0208041-1-65-0': '## The observed number of streams in the simulations', 'astro-ph-0208041-1-66-0': 'In general, a stream may be defined by those particles having the same orbital phase and coming from the same halo at some initial (high) redshift.', 'astro-ph-0208041-1-66-1': 'The orbital phase of a particle may be determined, in principle, by counting how many times it has crossed the planes [MATH].', 'astro-ph-0208041-1-66-2': 'However, because of the limited number of outputs which are logarithmically spaced in the expansion factor, it is difficult to carry this through correctly.', 'astro-ph-0208041-1-66-3': 'As in section [REF], we therefore define a stream as a set of particles which', 'astro-ph-0208041-1-67-0': 'have been neighbours in all previous outputs; are relatively close in space at the present time.', 'astro-ph-0208041-1-68-0': 'In practice, we make a coarse partition of the 3-dimensional space, whose elements are boxes of 15 kpc on a side.', 'astro-ph-0208041-1-68-1': 'At each output, we check in which box of the partition any given particle is located.', 'astro-ph-0208041-1-68-2': 'We tag the particle by this box ID, and by the IDs of those nearest to it.', 'astro-ph-0208041-1-68-3': 'So for example, a given particle will generally not be located right at the centre of a box, but will be closer to one of its edges.', 'astro-ph-0208041-1-68-4': 'Thus, this particle is assigned six different numbers (corresponding to the IDs of eight neighbouring boxes) for every output, as shown in Figure [REF] for the two-dimensional analogue.', 'astro-ph-0208041-1-68-5': 'We repeat this procedure for each output of our simulation.', 'astro-ph-0208041-1-68-6': 'Particles which are on the same stream should have the same box tags or have been in neighbouring boxes at all previous outputs.', 'astro-ph-0208041-1-68-7': 'This is almost equivalent to defining a box of 15 kpc on a side around each particle at each output, and finding which particles fall within that region.', 'astro-ph-0208041-1-68-8': 'Our procedure is however, much more efficient computationally.', 'astro-ph-0208041-1-69-0': 'For the output corresponding to the present day, and for each particle in one of the 2 kpc boxes of the partition of the dark-matter halo, we find which other particles in that same box, satisfy condition (i) above (condition (ii) is automatically fulfilled).', 'astro-ph-0208041-1-69-1': 'Thus at the end we obtain a link list, where particles have the same tags if they belong to the same stream.', 'astro-ph-0208041-1-70-0': 'In Figure [REF] we plot the (Poisson corrected) mass-weighted mean mass per stream [MATH] as a function of distance from the galaxy centre for each one of the boxes considered.', 'astro-ph-0208041-1-70-1': 'This Figure shows that [MATH] is generally smaller than unity.', 'astro-ph-0208041-1-70-2': 'This happens when inside a box there are one or two streams with several particles, and the rest are one-particle structures.', 'astro-ph-0208041-1-70-3': 'Clearly this is the case for the vast majority of the boxes.', 'astro-ph-0208041-1-70-4': 'However, at larger distances from the Galactic centre, massive streams can be found.', 'astro-ph-0208041-1-71-0': 'In Figure [REF] we plot the Poisson corrected estimate of the mass-weighted number of streams [MATH] as a function of distance from the galaxy centre and for each one of the boxes considered.', 'astro-ph-0208041-1-71-1': 'The thick grey line shows the median value of [MATH], where we consider all boxes located at the same distance from the centre.', 'astro-ph-0208041-1-71-2': 'For the "Solar neighbourhood" the estimate of the mass-weighted number of streams is roughly [MATH] streams, showing that the local dark halo of the galaxy is extremely well-mixed.', 'astro-ph-0208041-1-71-3': 'The variations in the number of streams at fixed distance from the galaxy centre are due to variations in the number of particles themselves.', 'astro-ph-0208041-1-71-4': 'These result from the flattened shape of the galaxy halo.', 'astro-ph-0208041-1-72-0': '## An analytic estimate for the number of streams', 'astro-ph-0208041-1-73-0': 'It may at first seem counterintuitive to find a number of streams larger than the number of particles actually observed in a box.', 'astro-ph-0208041-1-73-1': 'This effect is due to the small number of particles populating each stream, which boosts up the Poisson correction.', 'astro-ph-0208041-1-73-2': 'It is fair to say that the number of streams we are determining is based on the actual presence of a few hundred streams, each detected with only two or three particles in the inner 30 kpc region of the halo.', 'astro-ph-0208041-1-74-0': 'To ensure that the determination is meaningful we estimate how many streams we expect to find using the analytic prescription for the evolution of streams developed by HW.', 'astro-ph-0208041-1-74-1': 'In Sec. [REF] we saw that this prescription does represent a reasonable approximation to the evolution of debris streams, even during the hierarchical build up of a CDM halo, such as that studied here.', 'astro-ph-0208041-1-75-0': 'To obtain an analytic estimate for the number of streams expected in a given box we proceed as follows:', 'astro-ph-0208041-1-76-0': 'We select 20 representative boxes at different distances from the halo centre.', 'astro-ph-0208041-1-76-1': 'We identify the progenitor halos of the individual particles in each of these boxes.', 'astro-ph-0208041-1-76-2': 'Among the particles belonging to the same progenitor halo, we select, when possible, 10 particles (for boxes out to [MATH] kpc).', 'astro-ph-0208041-1-76-3': 'In all other cases, we choose 2 particles, or just 1, if this is all the halo contributes.', 'astro-ph-0208041-1-76-4': 'For each of these particles we integrate the orbit:', 'astro-ph-0208041-1-77-0': 'Once the orbit is known we can compute the evolution (always forwards in time) of the density in the stream where the particle is located.', 'astro-ph-0208041-1-77-1': 'This is done using the action-angle formalism of HW.', 'astro-ph-0208041-1-77-2': 'From the median (in time) density of a stream we determine the number of streams from a given progenitor halo.', 'astro-ph-0208041-1-77-3': 'The number of streams is given by the ratio of the mean (coarse-grained) density of the debris to the actual density of the stream where the particle [MATH] is located, i.e. [EQUATION]', 'astro-ph-0208041-1-77-4': 'Here [MATH], [MATH], and [MATH], where we assume [MATH].', 'astro-ph-0208041-1-77-5': 'Therefore we can express the number of streams as [EQUATION]', 'astro-ph-0208041-1-77-6': 'Having obtained the number of streams for each particle, we derive the average number of streams for each halo from all [MATH] particles that fall within the same box and belonged to this same halo: [EQUATION]', 'astro-ph-0208041-1-77-7': 'The total number-weighted number of streams in a box is now obtained by adding over all halos contributing to the box, where the weights are given by the number of particles from each halo in the box [MATH]: [EQUATION]', 'astro-ph-0208041-1-77-8': 'In Figure [REF] we plot our estimate for the number of streams as a function of distance from the Galactic centre.', 'astro-ph-0208041-1-77-9': 'The solid curve corresponds to the estimate obtained by integrating the orbit backwards in time, from the present day conditions.', 'astro-ph-0208041-1-77-10': 'The dotted curve, on the other hand, corresponds to the orbit integration performed forwards in time from [MATH].', 'astro-ph-0208041-1-77-11': 'The geometric mean of these two estimates is the dashed curve in the same figure.', 'astro-ph-0208041-1-77-12': 'Notably, the predicted number of streams given by the geometric mean is in reasonable agreement with what was shown in Figure [REF].', 'astro-ph-0208041-1-78-0': '## Characteristics of the streams', 'astro-ph-0208041-1-79-0': 'Another interesting quantity which we can calculate for each box is the characteristic mass-weighted dispersion within a stream: [EQUATION] where the [MATH] sum runs over all particle pairs in the [MATH]-th stream in a given box, [MATH] is the number of particles in this stream in this box, and as before, [MATH] is the total number of particles in the box.', 'astro-ph-0208041-1-79-1': 'This mass-weighted velocity dispersion is expected to be relatively small in view of our results of the phase-space evolution of halo debris.', 'astro-ph-0208041-1-79-2': 'We plot this quantity in Figure [REF] as a function of distance from the galaxy centre for each one of the boxes considered.', 'astro-ph-0208041-1-79-3': 'For our boxes of 2 kpc on a side we find a typical velocity dispersion of 1 km s[MATH] at the position of the "Sun".', 'astro-ph-0208041-1-80-0': '# Discussion', 'astro-ph-0208041-1-81-0': 'We have studied the phase-space evolution of debris from the progenitors that merge to build up a dark-matter halo in a [MATH]CDM cosmology.', 'astro-ph-0208041-1-81-1': 'Our analysis has shown that the debris streams originating in progenitors of different sizes and orbital characteristics all behave in a similar way: with velocity dispersions decreasing in time and with local space density decreasing as [MATH].', 'astro-ph-0208041-1-81-2': 'The evolution of the debris streams that we were able to follow until the present time is consistent with phase mixing.', 'astro-ph-0208041-1-81-3': 'Even for halos that we could not follow for a very long time - because of their smaller initial number of particles or their shorter orbital timescales- we find the debris to show similar behaviour.', 'astro-ph-0208041-1-81-4': 'On the scales of the fine-grained distribution function, mixing is apparently not strongly chaotic.', 'astro-ph-0208041-1-81-5': 'On the contrary, the phase-space evolution appears to be quite organised and simple, very similar to the mixing of streams orbiting in an idealised static potential.', 'astro-ph-0208041-1-82-0': 'In principle, a dark-matter halo formed in a [MATH]CDM cosmology is not a smooth entity.', 'astro-ph-0208041-1-82-1': 'Not only do dark-matter halos contain a large number of dark satellites, they also have large amounts of substructure in the form of streams.', 'astro-ph-0208041-1-82-2': 'We predict, however, that dark matter in the Solar neighbourhood should be clumped in a few hundred thousand streams, producing a velocity ellipsoid which is close to a multivariate Gaussian (Helmi, White Springel 2002).', 'astro-ph-0208041-1-82-3': 'These streams have their origin in the different halos that merged to form the dark halo of the Galaxy.', 'astro-ph-0208041-1-82-4': 'Most of these halos give rise to a large number of intersecting streams in the inner Galaxy.', 'astro-ph-0208041-1-83-0': 'Determining the characteristics of these hundred thousand streams is difficult even with the high-resolution simulation used here.', 'astro-ph-0208041-1-83-1': 'We are mostly limited by the number of particles.', 'astro-ph-0208041-1-83-2': 'Although the simulation as a whole has 66 million particles inside a 4 kpc box centred on the "Sun", we find only a couple of thousand particles.', 'astro-ph-0208041-1-83-3': 'In such boxes we find on average two hundred streams with more than one particle, and typically each has only two particles!', 'astro-ph-0208041-1-83-4': 'The internal stream velocity dispersion that we measure at the present-day is extremely small, of the order of only 1 km s[MATH].', 'astro-ph-0208041-1-84-0': 'It is encouraging that we find reasonable agreement between the behaviour of debris streams in static potentials and that observed in this high resolution simulation.', 'astro-ph-0208041-1-84-1': 'This implies that our earlier estimates of the number of stellar streams in the vicinity of the Sun (Helmi White 1999) may indeed apply.', 'astro-ph-0208041-1-84-2': 'Since the initial stellar phase-space distribution in the progenitor objects was probably of lower dimensionality than we here assume - stars tend to form in disks so that the distribution in at least two of the six phase-space coordinates collapses - star streams may be colder than the dark matter streams we have analysed, and so may be more easily distinguishable.', 'astro-ph-0208041-1-84-3': 'It is also interesting to note that because the material that ends up populating the inner galaxy was already in place 10 Gyr ago, the oldest stars are predicted to be near the galactic centre (White Springel 2000).', 'astro-ph-0208041-1-84-4': 'Because this material comes from only a few objects, one might expect the stellar populations to be quite homogeneous, although this of course depends on whether the stars themselves formed in these few massive objects, or whether they were accreted into these objects in the first place.', 'astro-ph-0208041-1-85-0': 'Considerably more insight into the properties of the Galactic stellar halo would be obtained by combining semi-analytic techniques (e.g Kauffmann et al. 1993) with high-resolution simulations (see, for example, Springel et al. 2001).', 'astro-ph-0208041-1-85-1': 'This would enable one to predict trends in the chemical composition, age, spatial distribution and kinematics of halo stars as a function of position throughout the Galaxy.'}

{'astro-ph-0208041-2-0-0': "We study the formation of the Milky Way's halo in a [MATH]CDM cosmology by scaling down a high resolution simulation of the formation of a cluster of galaxies.", 'astro-ph-0208041-2-0-1': 'We determine how much phase-space substructure is left over from the objects that merge to build up the present galaxy.', 'astro-ph-0208041-2-0-2': 'We study the debris streams originating from such objects and find that their evolution can be explained simply in terms of the conservation of phase-space density.', 'astro-ph-0208041-2-0-3': 'Analysing the mass growth history of our halo we find that its inner regions have been in place for more than 10 Gyr, but that the growth of the halo as a whole is more gradual, in agreement with other high resolution simulations of dark-matter halos.', 'astro-ph-0208041-2-0-4': 'Recent accretion contributes to the inner 10 kpc of the halo only at the 10[MATH] level.', 'astro-ph-0208041-2-0-5': 'Finally we determine the number of dark-matter streams as a function of distance from the centre of the halo.', 'astro-ph-0208041-2-0-6': 'In the equivalent of the "Solar vicinity", we find that the dark-matter is smoothly distributed in space, and that the velocity ellipsoid is formed by hundreds of thousands of streams, most of which have velocity dispersions of the order of 1 km s[MATH] or less.', 'astro-ph-0208041-2-1-0': '# Introduction', 'astro-ph-0208041-2-2-0': 'Over the last twenty years, the hierarchical paradigm has emerged as the standard model to describe the formation of structure in the Universe.', 'astro-ph-0208041-2-2-1': 'As embodied in the current "concordance" [MATH]CDM model it appears to be consistent with a very wide range of cosmological data ranging from fluctuations in the Cosmic Microwave Background through the structure of Ly[MATH] forest absorption in QSO spectra and the gravitational shear induced by dark-matter structures to the observed large scale structure in the galaxy distribution.', 'astro-ph-0208041-2-2-2': 'An important characteristic of such models is that they are based on a set of well-defined and testable assumptions.', 'astro-ph-0208041-2-2-3': 'This renders possible the detailed modelling of the formation and evolution of galactic systems, and a later comparison to observations of the properties of these systems as a function of environment or redshift (e.g. Diaferio et al. 2001; Benson et al. 2001; Somerville, Primack Faber 2001).', 'astro-ph-0208041-2-3-0': 'It is also possible to test the hierarchical paradigm on our Galaxy (e.g. Hernandez, Avila-Reese Firmani 2001).', 'astro-ph-0208041-2-3-1': 'Several groups (Moore et al. 1999; Klypin et al. 1999; Klypin, Zhao Somerville 2002) have focused on the properties of dark halos, hoping to constrain the nature of dark-matter.', 'astro-ph-0208041-2-3-2': 'These groups performed high resolution simulations of a galactic size halo in CDM cosmologies.', 'astro-ph-0208041-2-3-3': 'They confirmed earlier analytic claims (Kauffmann, White Guiderdoni 1993) that the predicted number of satellites exceeds the known population in the Local Group by a factor of ten.', 'astro-ph-0208041-2-3-4': 'Some attempts have been made to account for the disagreement, by changing the nature of the dark-matter (Spergel Steinhardt 2000; Bode, Ostriker Turok 2001), by modifying the initial power spectrum of density fluctuations (Kamionkowski Liddle 2000) or by taking into account the effects of a reionising background which may inhibit star formation in the smallest mass halos (Kauffmann et al. 1993; Bullock, Kravtsov Weinberg 2001; Benson et al. 2002).', 'astro-ph-0208041-2-3-5': 'The recent results by Kleyna et al. (2002) on the mass distribution in the Draco dSph (see also Mateo 1997 and Lokas 2001 for a similar study on Fornax) favour an astrophysical explanation since the actual circular velocities of the other satellite galaxies of the Milky Way are in fact several tens of km s[MATH] larger than previously thought, and agree with those expected for the most massive substructures in a [MATH]CDM universe (Stoehr et al. 2002).', 'astro-ph-0208041-2-4-0': 'Broadly speaking, the hierarchical paradigm predicts that the Milky Way formed through mergers of smaller systems (White Rees 1978).', 'astro-ph-0208041-2-4-1': 'These systems would contribute to the dark halo, the spheroid (the bulge and the stellar halo) and to the Galactic gas reservoir.', 'astro-ph-0208041-2-4-2': 'It may be very difficult to determine the relative gas contribution of these progenitor objects to the present Galaxy, since gas "easily forgets" its site of origin.', 'astro-ph-0208041-2-4-3': 'However for collisionless stars and dark-matter the situation can be quite different.', 'astro-ph-0208041-2-4-4': 'If the dynamical mixing timescales are sufficiently long (i.e. longer than the age of the Universe) it may be possible to "break-up" the Galactic spheroid (stars and may be even dark matter particles) into coherent structures in phase-space directly related to the systems that merged to form the Milky Way we observe today.', 'astro-ph-0208041-2-5-0': 'A first attempt at determining whether the merging history of the Milky Way may be imprinted in the phase-space structure of nearby halo stars, and thus be recovered, was made by Helmi White (1999; hereafter HW).', 'astro-ph-0208041-2-5-1': 'They studied the infall of satellites onto a fixed Galactic potential, and the evolution of the debris in phase-space.', 'astro-ph-0208041-2-5-2': 'They found that after 10 Gyr stars having a common origin are distributed smoothly in space, but appear very clumped in velocity space, where they define streams with very small velocity dispersions.', 'astro-ph-0208041-2-5-3': 'The expected number of such streams scales with the initial size [MATH], velocity dispersion [MATH] and orbital period [MATH], of the disrupted object: [EQUATION]', 'astro-ph-0208041-2-5-4': 'The total number of stars associated with the object is [MATH] (from the virial theorem) while the volume [MATH] over which they are spread scales with the cube of the size of the orbit, and so approximately as [MATH].', 'astro-ph-0208041-2-5-5': 'Hence the number of stars per stream in the Solar neighbourhood scales as [MATH]; objects with large initial velocity dispersion and small initial size should produce the most easily detectable streams with little dependence on initial period.', 'astro-ph-0208041-2-5-6': 'Such arguments suggest that the Solar neighbourhood velocity ellipsoid is composed of [MATH] kinematically coherent structures which originated in past merger and accretion events.', 'astro-ph-0208041-2-5-7': 'A pair of halo streams that can perhaps be directly linked to a disrupted satellite were detected in the Solar neighbourhood by Helmi et al.(1999).', 'astro-ph-0208041-2-5-8': 'The progenitor of these two streams was probably similar to the dwarf galaxy Fornax.', 'astro-ph-0208041-2-5-9': 'Substructure in the outer halo also appears to be ubiquitous, and has been found by several surveys over the last few years (e.g. Ivezic et al. (2000) and Yanny et al. (2000) for the SDSS; Dohm-Palmer et al. (2001) for the SPS; Vivas et al. (2001) for QUEST).', 'astro-ph-0208041-2-5-10': "Most of these recently discovered structures can be associated to just one of Milky Way's satellites: the Sagittarius dwarf galaxy which is in the process of being completely disrupted (Ibata et al. 2001; Martinez-Delgado et al. 2001; Helmi White 2001).", 'astro-ph-0208041-2-6-0': 'A weak point of the HW analysis and of similar studies (e.g. Johnston, Hernquist Bolte 1996; Johnston 1998), is the assumption of a fixed, smooth potential onto which galaxies are accreted.', 'astro-ph-0208041-2-6-1': 'In hierarchical clustering, galaxy potentials are constantly changing, and can vary very violently during mergers.', 'astro-ph-0208041-2-6-2': 'Large numbers of clumps orbit the centre of even a "virialised" halo.', 'astro-ph-0208041-2-6-3': 'These clumps may have substantial effects on the structure of debris streams (e.g Johnston, Spergel Haydn 2002; Ibata et al. 2002; Mayer et al. 2002).', 'astro-ph-0208041-2-7-0': 'The main goal of the present paper is to understand the phase-space structure of cold dark-matter halos.', 'astro-ph-0208041-2-7-1': 'In particular, we want to study the evolution of satellite debris, and to quantify the expected amount of substructure.', 'astro-ph-0208041-2-7-2': 'We also want to determine to what extent previous results are valid in the truly hierarchical regime of the build-up of a galaxy.', 'astro-ph-0208041-2-7-3': 'We tackle these problems by scaling down to galactic size a high-resolution simulation of the formation of a cluster in a [MATH]CDM cosmology (Springel et al. 2001).', 'astro-ph-0208041-2-8-0': 'The paper is organised as follows.', 'astro-ph-0208041-2-8-1': 'In Sec.2 we describe the simulations, in Sec.3 we follow the evolution in phase-space of debris streams, and compare to the analytic model of HW in Section 3.2.', 'astro-ph-0208041-2-8-2': 'Sec.4 describes the mass-growth history of the simulated dark-matter halo, and in Section 5 we determine the number of streams and their internal properties as function of distance from the dark-matter halo centre.', 'astro-ph-0208041-2-8-3': 'We leave the summary and discussion of our results for Section 6.', 'astro-ph-0208041-2-9-0': '# Methodology', 'astro-ph-0208041-2-10-0': '## The simulations', 'astro-ph-0208041-2-11-0': 'The simulation we analyse here was carried out using a parallel tree-code (Springel, Yoshida White 2001) on the Cray T3E at the Garching Computing Centre of the Max Planck Society.', 'astro-ph-0208041-2-11-1': 'Its initial conditions were generated by zooming in and re-creating with higher resolution a particular galaxy cluster and its surroundings formed within a cosmological simulation (as in Tormen, Bouchet White 1997).', 'astro-ph-0208041-2-11-2': 'The original parent [MATH]CDM cosmological simulation (from Kauffmann et al. 1999) had parameters [MATH], [MATH], [MATH] and [MATH].', 'astro-ph-0208041-2-11-3': 'The cluster selected for re-simulation was the second most massive cluster in this simulation, having a virial mass of [MATH].', 'astro-ph-0208041-2-11-4': 'The particles that end up in the final cluster of the cosmological simulation and in its immediate surroundings (defined by a comoving sphere of [MATH] Mpc radius) were traced back to their Lagrangian region in the initial conditions for re-simulation.', 'astro-ph-0208041-2-11-5': 'The initial mass distribution between [MATH] and [MATH] Mpc was represented by [MATH] particles.', 'astro-ph-0208041-2-11-6': 'In the inner region, where the original simulation had [MATH] particles, new initial conditions were created using [MATH] particles.', 'astro-ph-0208041-2-11-7': 'Small scale power was added in accordance with the better k-space sampling allowed by the larger number of particles.', 'astro-ph-0208041-2-11-8': 'The original force softening was also decreased to obtain better spatial resolution.', 'astro-ph-0208041-2-11-9': 'The new simulation was run from high redshift until [MATH], and was analysed in considerable detail in Springel et al. (2001).', 'astro-ph-0208041-2-11-10': 'In this high resolution simulation there are about 20 million particles within the virialised region of the cluster halo.', 'astro-ph-0208041-2-12-0': '## Scaling to a Milky Way halo', 'astro-ph-0208041-2-13-0': 'In Figure [REF] we show the circular velocity profile of the cluster at [MATH].', 'astro-ph-0208041-2-13-1': 'We determine the cluster centre by successively refining a mesh located on the cluster, and determining the cell containing the largest number of particles.', 'astro-ph-0208041-2-13-2': 'This process is repeated until the largest number of particles in a given (now small size) cell is sufficiently small to determine by simple counts which particle has the largest number of neighbours.', 'astro-ph-0208041-2-13-3': 'Its position corresponds to the point of maximum density and it is defined as the cluster centre.', 'astro-ph-0208041-2-13-4': 'This determination is robust against changes in the mesh shape and size, and we estimate the error in the final position of the centre of the cluster to be of the order of 0.7 kpc, comparable to the scale of the gravitational softening used in the simulation.', 'astro-ph-0208041-2-13-5': 'The circular velocity is then determined after spherically averaging the mass distribution around the centre of the cluster and is derived from [MATH].', 'astro-ph-0208041-2-14-0': 'By fitting a Navarro, Frenk White (NFW; 1996) profile, we find that the concentration of the cluster halo is [MATH], the scale radius [MATH] kpc and the virial radius [MATH] kpc.', 'astro-ph-0208041-2-15-0': 'We can scale the cluster to a "Milky Way" halo, by requiring that its maximum circular velocity be equal to 220 km s[MATH].', 'astro-ph-0208041-2-15-1': 'The required scaling factor [MATH] is given by [EQUATION]', 'astro-ph-0208041-2-15-2': 'With this scaling, the virial radius of our simulated Milky Way dark-matter halo is 228 kpc.', 'astro-ph-0208041-2-15-3': 'Its virial mass is [MATH], the mass of an individual particle is [MATH], and the gravitational force in the final object has an equivalent Plummer softening of 0.11 kpc.', 'astro-ph-0208041-2-16-0': 'Our argument that with the simple scaling of Eq. ([REF]) this simulation is a fair representation of the growth of the Milky Way relies on both theoretical and numerical results (Lacey Cole 1993; Moore et al. 1999).', 'astro-ph-0208041-2-16-1': 'Numerical simulations by Moore et al. (1999) have shown that galaxy and cluster halos have similar properties in terms of final structure (density profile), number of satellites, etc., despite typically assembling at systematically different redshifts.', 'astro-ph-0208041-2-16-2': 'Jing Suto (2000) have also performed high resolution simulations of several galaxy, group and cluster size halos.', 'astro-ph-0208041-2-16-3': 'Although they find that galaxy halos have a steeper inner profile than larger mass halos, they also find that the scatter in the properties of objects belonging to the same class is as large as the systematic differences between classes.', 'astro-ph-0208041-2-16-4': 'We expect therefore, that our scaled simulation will represent reasonably well the formation process of a galactic halo, except that its assembly occurs at later times than the majority of such halos.', 'astro-ph-0208041-2-16-5': 'From now on we will use this scaling in the paper, and assume, unless otherwise stated, that the simulation represents a galaxy halo.', 'astro-ph-0208041-2-17-0': '# The phase-space evolution of debris', 'astro-ph-0208041-2-18-0': 'Here we study the phase-space evolution of debris from the disrupted halos that end up forming the "Galactic" dark halo at the present time.', 'astro-ph-0208041-2-18-1': 'With this goal in mind, we proceed by identifying halos at high-redshift which are directly accreted onto the main progenitor of the "Milky Way\'s" halo.', 'astro-ph-0208041-2-19-0': 'We identify halos at each output time using a Friends-of-Friends (FOF) algorithm, which links particles separated by less than 20% of the mean interparticle separation.', 'astro-ph-0208041-2-19-1': 'In this way we can construct a detailed merger history of the galaxy.', 'astro-ph-0208041-2-19-2': "As we step back in redshift we identify at each output the most massive halo which is part of the galaxy's main progenitor at the subsequent time.", 'astro-ph-0208041-2-19-3': 'We say that a halo identified at redshift [MATH] will be directly accreted onto the main progenitor at [MATH] (the redshift of the next simulation output) if at least half of its particles and the most-bound particle have become part of the main progenitor at [MATH].', 'astro-ph-0208041-2-19-4': 'Here the most-bound particle refers to the particle with the minimum potential energy in the (to be) accreted halo.', 'astro-ph-0208041-2-20-0': 'The centre of mass position [MATH] and velocity [MATH] of the main progenitor are computed as follows.', 'astro-ph-0208041-2-20-1': 'We first determine the location of the most bound particle [MATH].', 'astro-ph-0208041-2-20-2': 'We define concentric spheres of successively smaller radii (down to 10 kpc radius) around this particle, and compute the centre of mass [MATH] from the particles within these spheres.', 'astro-ph-0208041-2-20-3': 'We stop this iterative procedure when [MATH], where [MATH] kpc.', 'astro-ph-0208041-2-20-4': 'This is then defined as the location of centre of mass of the main progenitor.', 'astro-ph-0208041-2-20-5': 'The velocity of the centre of mass is then [MATH], measured by the velocities of the particles within the largest sphere for which the above condition is satisfied.', 'astro-ph-0208041-2-21-0': "This is like following the (thickest) trunk of the merger tree, which would correspond to the galaxy's main progenitor, and studying what happens to halos which join from other tree branches as time progresses.", 'astro-ph-0208041-2-21-1': 'The idea is illustrated in Figure [REF].', 'astro-ph-0208041-2-22-0': 'For our high resolution simulation we have identified 752 halos with at least 1000 particles (which corresponds to a minimum mass of [MATH]), which fall onto the main progenitor between redshift [MATH] and the present day.', 'astro-ph-0208041-2-22-1': 'The accreted halos have a large spread in mass as shown in the bottom panel of Figure [REF].', 'astro-ph-0208041-2-22-2': 'From the top panel of this Figure we note that in some cases the satellite-to-primary mass ratio is close to unity, corresponding to a major merger.', 'astro-ph-0208041-2-22-3': 'Such mergers, although few in number, contribute a substantial fraction of the total mass growth.', 'astro-ph-0208041-2-23-0': '## Spatial and kinematic evolution of halo debris', 'astro-ph-0208041-2-24-0': 'We now study in detail how the debris streams from one of the above 752 halos evolve in time.', 'astro-ph-0208041-2-24-1': 'We follow a halo that merges with the main progenitor at redshift 1.8, and has an initial mass of [MATH].', 'astro-ph-0208041-2-24-2': 'This halo is denoted with a diamond in Figure [REF].', 'astro-ph-0208041-2-24-3': 'Its mass at the time of accretion was about 25% that of the main progenitor.', 'astro-ph-0208041-2-24-4': 'This satellite decays through dynamical friction to the centre of the main progenitor where it is fully disrupted.', 'astro-ph-0208041-2-24-5': 'The mass stripped off at earlier times mainly populates the outer regions of the final object, while that lost at late times ends up closer to its centre.', 'astro-ph-0208041-2-25-0': '### Evolution of debris in the outer galaxy', 'astro-ph-0208041-2-26-0': 'We identify material from the accreted halo which is part of a tidal stream in the outer galaxy at the present time.', 'astro-ph-0208041-2-26-1': 'We select a reference particle in this structure, which can be traced back to [MATH], and then followed forwards in time.', 'astro-ph-0208041-2-26-2': 'In Figure [REF] we show a time sequence of the spatial distribution of particles lost at [MATH].', 'astro-ph-0208041-2-26-3': 'We say that a particle has been lost by its progenitor halo if its binding energy has become positive.', 'astro-ph-0208041-2-26-4': 'The binding energy [MATH] of a particle located at [MATH] and with velocity [MATH] with respect to the centre of mass of the satellite is defined as [MATH], where [MATH] is the potential energy at [MATH] due to all particles in the satellite.', 'astro-ph-0208041-2-26-5': 'From Fig. [REF] we see that the initial distribution of particles is relatively compact, and that as time passes by the material is strung out in a characteristic stream-like structure (e.g at [MATH] Gyr) over several tens to hundreds of kiloparsecs. At the final time the material appears to be more smoothly distributed over the whole box, which is 400 kpc (in physical units) on a side.', 'astro-ph-0208041-2-26-6': 'Our simulations are not well-suited to address the effects of other dark-matter lumps on tidal tails (cf Johnston et al. 2001), which could be related to some of the transient structures observed in this Figure.', 'astro-ph-0208041-2-26-7': 'Our main limitation is the large time interval between stored outputs which prevents us from determining the effect of close encounters and their relation to the features seen in Figure [REF].', 'astro-ph-0208041-2-27-0': 'In many of the snapshots tight small substructures can be observed.', 'astro-ph-0208041-2-27-1': 'These objects were already present as subhalos within the satellite halo before it was accreted by the galaxy.', 'astro-ph-0208041-2-27-2': 'After accretion, they were released from their parent satellite, becoming subhalos of the main galaxy.', 'astro-ph-0208041-2-27-3': 'It is worth mentioning that these subhalos only constitute a small fraction of the debris material lost by the satellite (less than 10%, e.g. Ghigna et al. 2000; Springel et al. 2001).', 'astro-ph-0208041-2-28-0': 'In Figure [REF] we show the velocities of debris particles that are relatively close (inside a cube of side 20 kpc) to the reference particle, at three different times.', 'astro-ph-0208041-2-28-1': 'For comparison, note that the virial radius of the satellite at the time of infall was approximately 37 kpc.', 'astro-ph-0208041-2-28-2': 'The solid grey circles correspond to particles which', 'astro-ph-0208041-2-29-0': 'were always neighbours of the reference particle: to be in this set, particles need to have [MATH], [MATH] and [MATH] coordinates within 10 kpc of those of the reference particle in all previous outputs;', 'astro-ph-0208041-2-30-0': 'are within 5 kpc of the reference particle in each coordinate in the current output.', 'astro-ph-0208041-2-31-0': 'We note that the initial velocity distribution of the halo is broad, and relatively clumpy.', 'astro-ph-0208041-2-31-1': 'As discussed before, this clumpiness reflects the internal structure of the object that fell in.', 'astro-ph-0208041-2-31-2': 'As time goes by, the motions of the neighbours of the reference particle become more similar, and the velocity-box is empty except for velocities close to that of the reference particle (middle row).', 'astro-ph-0208041-2-31-3': 'At later times (bottom row), other moving groups are visible, showing that the system has now produced multiple intersecting streams even in the outer galaxy.', 'astro-ph-0208041-2-32-0': 'The evolution of the velocity dispersion [MATH] in the neighbourhood of the reference particle is shown in the top panel of Figure [REF].', 'astro-ph-0208041-2-32-1': 'Here [MATH] is defined as [MATH], and is therefore independent of the choice of the coordinate system.', 'astro-ph-0208041-2-32-2': 'It is measured for the set of particles shown as grey solid circles in Figure [REF], that is for particles that satisfy both conditions (i) and (ii) for being neighbours of the reference particle.', 'astro-ph-0208041-2-32-3': 'We also measure [MATH] for a subset of even closer neighbouring particles (whose [MATH], [MATH], and [MATH] coordinates in all outputs before the one under study were within 5 kpc of those of the reference particle, and in the current output are within 2.5 kpc).', 'astro-ph-0208041-2-32-4': 'The different values of the velocity dispersion in the stream obtained for these two cases are due to velocity gradients, which can be as large as the measured dispersions themselves.', 'astro-ph-0208041-2-32-5': 'Note as well the decrease by roughly a factor 10 in the velocity dispersions in only 2 Gyr.', 'astro-ph-0208041-2-33-0': "The stream's density is shown in the bottom panel of the same figure.", 'astro-ph-0208041-2-33-1': 'It is measured at each output by the number of neighbours in the stream in boxes of a given size (either of 5 or 10 kpc on a side) around the reference particle.', 'astro-ph-0208041-2-33-2': 'For the larger volume, this number evolves from being slightly larger than five thousand to only thirteen after less than 2 Gyr of evolution.', 'astro-ph-0208041-2-33-3': 'This implies that after this time, the number counts are dominated by Poisson noise.', 'astro-ph-0208041-2-34-0': '### Evolution of debris closer to the centre of the galaxy', 'astro-ph-0208041-2-35-0': 'To follow the evolution of streams closer to the centre of the galaxy we focus on material lost a few Gyr after the satellite halo shown as a diamond in Fig. [REF] was accreted.', 'astro-ph-0208041-2-35-1': 'We here focus on debris lost from this satellite halo at [MATH] Gyr, or 3.5 Gyr after infall.', 'astro-ph-0208041-2-35-2': 'This is deposited at an intermediate distance from the centre of the galaxy.', 'astro-ph-0208041-2-35-3': 'We decided against tracking material lost even later because it turns out to be very difficult to follow streams from such material with our numerical resolution and number of stored outputs.', 'astro-ph-0208041-2-35-4': 'The material lost at later times mixes on shorter timescales.', 'astro-ph-0208041-2-35-5': 'Streams more rapidly reach very small densities, and we are unable to determine their properties reliably.', 'astro-ph-0208041-2-36-0': 'Figure [REF] shows the evolution of the spatial distribution of the particles lost at [MATH] Gyr.', 'astro-ph-0208041-2-36-1': 'Note that the particles are more smoothly distributed than in the case of the outer halo material shown in Figure [REF].', 'astro-ph-0208041-2-36-2': 'This is because the inner regions of (satellite) halos are in general much less lumpy than the outskirts due to the shorter dynamical timescales.', 'astro-ph-0208041-2-37-0': 'As in the previous section we choose a reference particle (amongst all those lost at [MATH] Gyr) that orbits through the intermediate regions of the galaxy.', 'astro-ph-0208041-2-37-1': 'Proceeding as before, we follow the evolution of the velocities of particles located close to the reference particle at all times.', 'astro-ph-0208041-2-37-2': 'Figure [REF] shows the velocity distribution in the neighbourhood of the reference particle.', 'astro-ph-0208041-2-37-3': 'By comparing to the analogue for the outer halo stream (Fig. [REF]) we note that, not just the spatial distribution, but also the velocity distribution is much smoother initially.', 'astro-ph-0208041-2-37-4': 'This is also true at late times, and very rapidly a regime is reached where multiple streams can be observed.', 'astro-ph-0208041-2-37-5': 'Figure [REF] corresponds to the evolution of the velocity dispersion and density of a stream.', 'astro-ph-0208041-2-37-6': 'Again we are not able to follow the evolution of the velocity dispersion of the stream at late times, because we run out of neighbouring particles quite rapidly, particularly for the smaller box.', 'astro-ph-0208041-2-37-7': 'After [MATH] Gyr, or 2 Gyr of evolution, the number of neighbours is dominated by Poisson noise.', 'astro-ph-0208041-2-37-8': 'After this time, the values of the 1-d velocity dispersion, when measured, are based on the velocities of only two or three particles.', 'astro-ph-0208041-2-38-0': 'The evolution of the intermediate and outer halo streams is characteristic of all streams originating in directly accreted halos.', 'astro-ph-0208041-2-38-1': 'Naturally the properties of streams and their location in the galaxy halo will depend on their progenitor, in particular on its initial mass.', 'astro-ph-0208041-2-38-2': 'In this respect, streams originating in smaller halos are narrower, more clearly defined, and, typically, they phase-mix on longer timescales.', 'astro-ph-0208041-2-39-0': '## Mixing in phase-space: Comparison to analytic models of stream evolution', 'astro-ph-0208041-2-40-0': 'In the previous section we found a very rapid decline in the density and velocity dispersion of streams, especially in the first few Gyr of evolution.', 'astro-ph-0208041-2-40-1': 'After this initial period, the number of particles in a stream is so small in our simulation that it is dominated by Poisson noise.', 'astro-ph-0208041-2-40-2': 'In this regime, it is difficult to quantify the properties of the stream, and it is therefore hard to determine whether the lack of variation of the velocity dispersion at late times is due to numerical limitations or is a real effect.', 'astro-ph-0208041-2-41-0': 'To gain insight into these issues, we will analyse the expected behaviour of streams evolving in a smooth time-independent potential, which should resemble that of the galaxy halo.', 'astro-ph-0208041-2-41-1': 'Using the approach developed by HW, we can follow the evolution of streams produced in a spherical and static NFW potential.', 'astro-ph-0208041-2-41-2': 'The basic idea here is to map the initial system onto action-angle space, then follow the much simpler evolution in this space, and finally transform back locally onto observable coordinates (all these being linear transformations; for details see HW).', 'astro-ph-0208041-2-41-3': 'This method, which uses action-angle variables, is limited to applications in which the potential is separable (cf Goldstein 1953; Binney Tremaine 1987).', 'astro-ph-0208041-2-41-4': 'This includes all spherically symmetric potentials but only few axisymmetric and triaxial ones, such as the general class of Stackel potentials (e.g. Lynden-Bell 1962; de Zeeuw 1985; Dejonghe de Zeeuw 1988).', 'astro-ph-0208041-2-42-0': 'We also approximate the phase-space density around the reference particle by a multivariate Gaussian distribution.', 'astro-ph-0208041-2-42-1': 'This is possible because the multivariate Gaussian is determined from the properties of particles in a volume much smaller than the size of the halo.', 'astro-ph-0208041-2-43-0': 'As discussed in Section 2.2, the (galaxy) halo can be fit by an NFW profile: [EQUATION] where [MATH] and [MATH] is a function of the concentration of the halo [MATH]: [EQUATION]', 'astro-ph-0208041-2-43-1': 'Recall that [MATH].', 'astro-ph-0208041-2-43-2': "The potential associated with this density can be obtained by integrating Poisson's equation, and is found to be: [EQUATION]", 'astro-ph-0208041-2-43-3': 'Here [MATH].', 'astro-ph-0208041-2-43-4': 'The values of the parameters [MATH] and [MATH] given in Section 2.2 should be multiplied by the factor [MATH] for the scaled halo.', 'astro-ph-0208041-2-44-0': 'In this (spherically averaged) potential we integrate the orbit of the reference particle in time.', 'astro-ph-0208041-2-44-1': 'This integration is done in two complementary ways:', 'astro-ph-0208041-2-45-0': 'starting from the position and velocity of the reference particle at the "time of formation of the stream" (this is the time when the particles become unbound from their parent halo).', 'astro-ph-0208041-2-45-1': 'In this case, the integration is performed forwards in time, and the potential used is given by Eq. ([REF]), where [MATH] is the redshift of formation of the stream.', 'astro-ph-0208041-2-46-0': 'starting from the position and velocity of the reference particle at the present time.', 'astro-ph-0208041-2-46-1': 'The orbit is then integrated backwards in time, until the "time of formation of the stream", in the present-day potential ([MATH]).', 'astro-ph-0208041-2-47-0': 'In both cases, the orbits are integrated in the reference frame of the centre of mass of the main progenitor of the galaxy.', 'astro-ph-0208041-2-48-0': 'In the top panel of Figure [REF] we plot the radial oscillations of the orbit for the outer halo reference particle.', 'astro-ph-0208041-2-48-1': 'The solid curve corresponds to the NFW potential at redshift [MATH], while the dashed curve to that at [MATH].', 'astro-ph-0208041-2-48-2': 'We note that for the reference particle in the outer halo stream, the best agreement is obtained when the integration is performed backwards in time.', 'astro-ph-0208041-2-48-3': 'We also note that the potential seems to have fluctuated dramatically until [MATH] Gyr, inducing strong changes in the radial oscillations of the reference particle.', 'astro-ph-0208041-2-48-4': 'As a whole, the orbit has evolved to a more bound state, due to the aggregation of mass during mergers.', 'astro-ph-0208041-2-49-0': 'The analogous plots for the reference particle orbiting the intermediate halo are shown in the bottom panel.', 'astro-ph-0208041-2-49-1': 'Recall that here [MATH].', 'astro-ph-0208041-2-49-2': 'In this case, none of the proposed orbits fit the actual orbit very well; it may be considered to lie in between these two cases.', 'astro-ph-0208041-2-49-3': 'Thus, the characteristics and the evolution of the debris streams predicted for the two proposed orbits may perhaps encompass the actual behaviour of the streams in our simulations.', 'astro-ph-0208041-2-49-4': 'In this "intermediate halo" case, we do not find a clear indication of evolution in the orbit.', 'astro-ph-0208041-2-49-5': 'This is likely due to the mild increase in the mass within the orbit since [MATH] (see Sec. 4)', 'astro-ph-0208041-2-50-0': 'In Figures [REF] and [REF] we show the predicted evolution of the 1-d velocity dispersion and the behaviour of the density in the spherical static NFW potential for the outer and intermediate halo streams, respectively.', 'astro-ph-0208041-2-50-1': 'The actual behaviour of the streams in the simulation is also shown for direct comparison.', 'astro-ph-0208041-2-50-2': 'In both cases, we find that the decrease observed in the simulated streams is close to what is predicted for the evolution of streams moving in a static potential.', 'astro-ph-0208041-2-50-3': 'This is the case at least for the first few Gyrs of evolution.', 'astro-ph-0208041-2-50-4': 'Although after this period we expect some differences, due to the fact that the orbit evolves in shape, the agreement still appears to be quite good.', 'astro-ph-0208041-2-50-5': 'We predict a rapidly varying velocity dispersion (and density) on top of a secular evolution.', 'astro-ph-0208041-2-50-6': 'The spikes take place when particles in a stream go through a caustic surface defined by their orbital turning points.', 'astro-ph-0208041-2-50-7': 'To observe this behaviour in the simulation would require output times spaced by 1/2 of the radial period at most, or roughly 0.25 Gyr for the stream orbiting the intermediate regions of the halo.', 'astro-ph-0208041-2-50-8': 'Our outputs, on the other hand, are logarithmically spaced in the cosmological expansion factor [MATH], so that at [MATH] the time elapsed between two outputs is [MATH] Gyr, and by [MATH], [MATH] Gyr.', 'astro-ph-0208041-2-51-0': '# Mass growth history of the galaxy', 'astro-ph-0208041-2-52-0': 'Here we focus on determining the mass growth history of the halo as a function of distance from the halo centre.', 'astro-ph-0208041-2-52-1': 'This is relevant in two different ways.', 'astro-ph-0208041-2-52-2': 'First, debris from satellites accreted at late times will be generally less mixed, and could thus produce more massive streams.', 'astro-ph-0208041-2-52-3': 'Determining where to expect these streams will enable us to understand the properties of the dark halo of our Galaxy.', 'astro-ph-0208041-2-52-4': 'Secondly such satellites will have had more time to form stars in them (prior to their merging), thereby providing the galaxy with younger stars.', 'astro-ph-0208041-2-52-5': 'The time of merging could thus be used as an indicator of the expected age distribution of stars in different regions of our Galaxy.', 'astro-ph-0208041-2-53-0': 'We select all satellites that merged with the galaxy since redshift [MATH], and determine when these mergers took place and what is their final debris distribution.', 'astro-ph-0208041-2-53-1': 'We proceed by dividing the halo in six spherical shells around the galaxy centre.', 'astro-ph-0208041-2-53-2': 'These shells are located at: [MATH] kpc, [MATH] kpc, [MATH] kpc, [MATH] kpc, [MATH] kpc and [MATH] kpc.', 'astro-ph-0208041-2-53-3': 'For each particle in a given shell, we determine when it was accreted by the main progenitor of the galaxy.', 'astro-ph-0208041-2-53-4': 'In Figure [REF] we show the fraction of mass accreted normalised to the present mass for each shell as a function of redshift (and time).', 'astro-ph-0208041-2-53-5': 'We note that the formation time of the inner galaxy is strongly biased towards high redshifts, with more than 60 of the mass already present at [MATH] or 11 Gyr ago.', 'astro-ph-0208041-2-53-6': 'Conversely, we find that mergers in the last 3 Gyr, contributed a relatively small amount of mass, less than 0.1, to this region of the galaxy halo.', 'astro-ph-0208041-2-53-7': 'For the outer regions of the galaxy, we note that the growth is much more gradual in time, with accretion being almost equally important at all times.', 'astro-ph-0208041-2-54-0': 'Of course, the detailed shape of the mass growth history depends on the detailed merger history of the halo, since the peaks observed in the different histograms correspond to individual mergers taking place at those times.', 'astro-ph-0208041-2-54-1': 'Nonetheless, several authors have found that there is an almost "universal" form of the mass accretion histories of galaxies and galaxy clusters (e.g. van den Bosch 2002; Wechsler et al. 2002), which would also represent reasonably well the growth of our simulated halo.', 'astro-ph-0208041-2-54-2': 'On the other hand, Zhao et al. (2002) have shown using high-resolution simulations that the mass in the inner regions of dark-matter halos (which they define as the mass within a scale radius, which in our case corresponds to approximately 30 kpc), is generally in place by [MATH].', 'astro-ph-0208041-2-54-3': 'These results suggest that the formation history of our simulated halo is rather typical.', 'astro-ph-0208041-2-55-0': 'Another way of understanding the age structure of the galaxy halo is by focusing on what fraction of the mass was in place by different redshifts as a function of distance from the galaxy centre.', 'astro-ph-0208041-2-55-1': 'To some extent this is the cumulative distribution of the plots shown in Fig. [REF] for each shell.', 'astro-ph-0208041-2-55-2': 'We focus on the fraction of the present-day mass in the shell that was in place at four different redshifts [MATH], 1.5, 0.84, and 0.35.', 'astro-ph-0208041-2-55-3': 'The results are shown in Figure [REF].', 'astro-ph-0208041-2-55-4': 'Again we notice that 60% of the mass in the inner galaxy was in place by [MATH], and more than 90% by [MATH], i.e. 9.5 Gyr ago.', 'astro-ph-0208041-2-55-5': 'Thus any accreted stellar populations in the inner regions of the galaxy are predicted to be old.', 'astro-ph-0208041-2-55-6': 'On the other hand, only about half of the particles in the intermediate regions of the galaxy were present 9.5 Gyr ago (90% by [MATH]).', 'astro-ph-0208041-2-55-7': 'The formation of the outer galaxy is more biased towards late times, with half of its particles coming into place in the last 7 Gyr.', 'astro-ph-0208041-2-56-0': 'As a cautionary remark, let us recall that although we discuss here the expected "stellar" populations, we are actually describing the growth in mass of the dark-matter halo, rather than of the stellar halo.', 'astro-ph-0208041-2-56-1': 'To describe properly the age distribution of the stellar halo we would need to take into account that the stars in the different satellites that merged to build up the halo probably were located primarily in their innermost regions.', 'astro-ph-0208041-2-56-2': 'This would mean, as shown in Section [REF], that they should end up closer to the galactic centre than most of the dark-matter particles of the same satellite (see also White Springel 2000).', 'astro-ph-0208041-2-56-3': 'Moreover, we are also ignoring the fact that satellites orbiting the intermediate and outer regions of the halo can survive until the present day.', 'astro-ph-0208041-2-56-4': 'In those cases it is possible that they continue to form stars, even while orbiting inside the dark halo of the galaxy (like the Magellanic Clouds).', 'astro-ph-0208041-2-56-5': 'This means that the material that is stripped off from these systems could also contain younger stars.', 'astro-ph-0208041-2-57-0': '# The number of streams', 'astro-ph-0208041-2-58-0': 'The results obtained in Section [REF] suggest that the number of streams in the inner galaxy, and in particular in the vicinity of the Sun should be quite large.', 'astro-ph-0208041-2-58-1': 'In this section we will estimate this quantity, as well as other characteristics of the streams, such as internal velocity dispersion and density.', 'astro-ph-0208041-2-58-2': 'We will compute the number of streams inside cubes located in the inner [MATH] kpc[MATH] in the scaled "Milky Way" halo.', 'astro-ph-0208041-2-58-3': 'We partition this space into boxes of 2 kpc on a side.', 'astro-ph-0208041-2-59-0': '## General definitions', 'astro-ph-0208041-2-60-0': 'Let there be [MATH] separate true streams in a given box, each having mass [MATH]).', 'astro-ph-0208041-2-60-1': 'Let us assume that the observed number of particles in a stream follows a Poissonian distribution, and that the actual count is [MATH] particles ([MATH]).', 'astro-ph-0208041-2-60-2': 'Then the expectation value of [MATH] is [MATH], and [MATH].', 'astro-ph-0208041-2-60-3': '(Here we measure mass in units of the particle mass of the simulation.)', 'astro-ph-0208041-2-61-0': 'The mass-weighted mean mass per stream [MATH] is [EQUATION]', 'astro-ph-0208041-2-61-1': 'Therefore our estimate for [MATH] corrected for Poisson noise will be [EQUATION]', 'astro-ph-0208041-2-61-2': 'In the limit of very massive streams, the Poisson correction will be negligible since [MATH].', 'astro-ph-0208041-2-61-3': 'In the limit of small number of particles per stream, the correction will be of the same order as the quantity we measure.', 'astro-ph-0208041-2-61-4': 'Note as well, that streams which in this realization do not have any particle or just have one, do not contribute to the numerator of Eq. ([REF]).', 'astro-ph-0208041-2-61-5': 'However single particle streams do contribute to the total number of particles in the box.', 'astro-ph-0208041-2-61-6': 'If in a box we find mostly one-particle streams, then [MATH] can (correctly) become very small.', 'astro-ph-0208041-2-62-0': 'We define the mass-weighted number of streams [MATH] in a box as the ratio of the total mass in the box to the mass-weighted mean mass [MATH] per stream in the box.', 'astro-ph-0208041-2-62-1': 'Therefore [MATH] is [EQUATION] and our Poisson corrected estimate of [MATH] is [EQUATION] or [EQUATION]', 'astro-ph-0208041-2-62-2': 'For example, if in our realization all of the streams have only two particles [MATH], our estimate of the mass-weighted filling factor becomes [MATH], where [MATH] is the total number of particles in the box.', 'astro-ph-0208041-2-62-3': 'In the regime where one massive stream dominates the distribution, [MATH] and will thus be close to unity.', 'astro-ph-0208041-2-62-4': 'Also note that [MATH] can be larger than [MATH] which will happen when most of the streams contain only 1 particle.', 'astro-ph-0208041-2-63-0': '## The minimum number of streams: the number of halos', 'astro-ph-0208041-2-64-0': 'We would like to obtain an estimate of the number of disrupted halos contributing to the density at each location in the galaxy.', 'astro-ph-0208041-2-64-1': 'This is a lower limit to the total number of streams present, since a halo can (and usually does) give rise to multiple and spatially overlapping structures, as shown in Sec. [REF].', 'astro-ph-0208041-2-64-2': 'This lower limit will be particularly unrepresentative of the true number of streams in the inner halo both because of the very short dynamical timescales there and because more than 60% of the mass in this region comes from just one object identified at [MATH], as shown in Sec. [REF].', 'astro-ph-0208041-2-65-0': 'To obtain an estimate of the number of halos contributing to any given position in the Galaxy, we determine which of the halos identified at [MATH] contribute to each given box and with how many particles.', 'astro-ph-0208041-2-65-1': 'In Figure [REF] we plot the (Poisson corrected) mass-weighted mean number of particles per halo [MATH] as a function of distance from the galaxy centre for each of the boxes considered.', 'astro-ph-0208041-2-65-2': 'The thick black line corresponds to the mean number of particles in a box, averaged over all boxes at the same location.', 'astro-ph-0208041-2-65-3': 'This Figure shows that most of the particles in the inner galaxy come only from a handful of disrupted halos.', 'astro-ph-0208041-2-65-4': 'This can also be seen from Figure [REF], where we plot the Poisson corrected estimate of the mass-weighted number of halos [MATH] as a function of distance from the galaxy centre and for each one of the boxes considered.', 'astro-ph-0208041-2-65-5': 'We note that the mean number of halos per box in the outer galaxy is large, but that each contributes only a handful of particles.', 'astro-ph-0208041-2-65-6': 'However, the inner galaxy is dominated by just a few halos making up most of the mass.', 'astro-ph-0208041-2-65-7': 'This trend (of increasing number of halos with distance) is due to a form of mass segregation: heavy halos can sink by dynamical friction in short timescales to the centre of the newly formed system, whereas lighter halos, unable to decay quickly, can only deposit their mass in the outskirts.', 'astro-ph-0208041-2-66-0': '## The observed number of streams in the simulations', 'astro-ph-0208041-2-67-0': 'In general, a stream may be defined by those particles having the same orbital phase and coming from the same halo at some initial (high) redshift.', 'astro-ph-0208041-2-67-1': 'The orbital phase of a particle may be determined, in principle, by counting how many times it has crossed the planes [MATH].', 'astro-ph-0208041-2-67-2': 'However, because of the limited number of outputs which are logarithmically spaced in the expansion factor, it is difficult to carry this through correctly.', 'astro-ph-0208041-2-67-3': 'As in section [REF], we therefore define a stream as a set of particles which', 'astro-ph-0208041-2-68-0': 'have been neighbours in all previous outputs; are relatively close in space at the present time.', 'astro-ph-0208041-2-69-0': 'In practice, we make a coarse partition of the 3-dimensional space, whose elements are boxes of 15 kpc on a side.', 'astro-ph-0208041-2-69-1': 'At each output, we check in which box of the partition any given particle is located.', 'astro-ph-0208041-2-69-2': 'We tag the particle by this box ID, and by the IDs of those nearest to it.', 'astro-ph-0208041-2-69-3': 'So for example, a given particle will generally not be located right at the centre of a box, but will be closer to one of its edges.', 'astro-ph-0208041-2-69-4': 'Thus, this particle is assigned six different numbers (corresponding to the IDs of eight neighbouring boxes) for every output, as shown in Figure [REF] for the two-dimensional analogue.', 'astro-ph-0208041-2-69-5': 'We repeat this procedure for each output of our simulation.', 'astro-ph-0208041-2-69-6': 'Particles which are on the same stream should have the same box tags or have been in neighbouring boxes at all previous outputs.', 'astro-ph-0208041-2-69-7': 'This is almost equivalent to defining a box of 15 kpc on a side around each particle at each output, and finding which particles fall within that region.', 'astro-ph-0208041-2-69-8': 'Our procedure is however, much more efficient computationally.', 'astro-ph-0208041-2-70-0': 'For the output corresponding to the present day, and for each particle in one of the 2 kpc boxes of the partition of the dark-matter halo, we find which other particles in that same box, satisfy condition (i) above (condition (ii) is automatically fulfilled).', 'astro-ph-0208041-2-70-1': 'Thus at the end we obtain a link list, where particles have the same tags if they belong to the same stream.', 'astro-ph-0208041-2-71-0': 'In Figure [REF] we plot the (Poisson corrected) mass-weighted mean mass per stream [MATH] as a function of distance from the galaxy centre for each one of the boxes considered.', 'astro-ph-0208041-2-71-1': 'This Figure shows that [MATH] is generally smaller than unity.', 'astro-ph-0208041-2-71-2': 'This happens when inside a box there are one or two streams with several particles, and the rest are one-particle structures.', 'astro-ph-0208041-2-71-3': 'Clearly this is the case for the vast majority of the boxes.', 'astro-ph-0208041-2-71-4': 'However, at larger distances from the Galactic centre, massive streams can be found.', 'astro-ph-0208041-2-72-0': 'In Figure [REF] we plot the Poisson corrected estimate of the mass-weighted number of streams [MATH] as a function of distance from the galaxy centre and for each one of the boxes considered.', 'astro-ph-0208041-2-72-1': 'The thick grey line shows the median value of [MATH], where we consider all boxes located at the same distance from the centre.', 'astro-ph-0208041-2-72-2': 'For the "Solar neighbourhood" the estimate of the mass-weighted number of streams is roughly [MATH] streams, showing that the local dark halo of the galaxy is extremely well-mixed.', 'astro-ph-0208041-2-72-3': 'The variations in the number of streams at fixed distance from the galaxy centre are due to variations in the number of particles themselves.', 'astro-ph-0208041-2-72-4': 'These result from the flattened shape of the galaxy halo.', 'astro-ph-0208041-2-73-0': '## An analytic estimate for the number of streams', 'astro-ph-0208041-2-74-0': 'It may at first seem counterintuitive to find a number of streams larger than the number of particles actually observed in a box.', 'astro-ph-0208041-2-74-1': 'This effect is due to the small number of particles populating each stream, which boosts up the Poisson correction.', 'astro-ph-0208041-2-74-2': 'It is fair to say that the number of streams we are determining is based on the actual presence of a few hundred streams, each detected with only two or three particles, in the inner 30 kpc region of the halo.', 'astro-ph-0208041-2-75-0': 'To ensure that the determination is meaningful we estimate how many streams we expect to find using the analytic prescription for the evolution of streams developed by HW.', 'astro-ph-0208041-2-75-1': 'In Sec. [REF] we saw that this prescription does represent a reasonable approximation to the evolution of debris streams, even during the hierarchical build up of a CDM halo, such as that studied here.', 'astro-ph-0208041-2-76-0': 'To obtain an analytic estimate for the number of streams expected in a given box we proceed as follows:', 'astro-ph-0208041-2-77-0': 'We select 20 representative boxes at different distances from the halo centre.', 'astro-ph-0208041-2-77-1': 'We identify the progenitor halos of the individual particles in each of these boxes.', 'astro-ph-0208041-2-77-2': 'Among the particles belonging to the same progenitor halo, we select, when possible, 10 particles (for boxes out to [MATH] kpc).', 'astro-ph-0208041-2-77-3': 'In all other cases, we choose 2 particles, or just 1, if this is all the halo contributes.', 'astro-ph-0208041-2-77-4': 'For each of these particles we integrate the orbit:', 'astro-ph-0208041-2-78-0': 'Once the orbit is known we can compute the evolution (always forwards in time) of the density in the stream where the particle is located.', 'astro-ph-0208041-2-78-1': 'This is done using the action-angle formalism of HW.', 'astro-ph-0208041-2-78-2': 'From the median (in time) density of a stream we determine the number of streams from a given progenitor halo.', 'astro-ph-0208041-2-78-3': 'The number of streams is given by the ratio of the mean (coarse-grained) density of the debris to the actual density of the stream where the particle [MATH] is located, i.e. [EQUATION]', 'astro-ph-0208041-2-78-4': 'Here [MATH], [MATH], and [MATH], where we assume [MATH].', 'astro-ph-0208041-2-78-5': 'Therefore we can express the number of streams as [EQUATION]', 'astro-ph-0208041-2-78-6': 'Having obtained the number of streams for each particle, we derive the average number of streams for each halo from all [MATH] particles that fall within the same box and belonged to this same halo: [EQUATION]', 'astro-ph-0208041-2-78-7': 'The total number-weighted number of streams in a box is now obtained by adding over all halos contributing to the box, where the weights are given by the number of particles from each halo in the box [MATH]: [EQUATION]', 'astro-ph-0208041-2-78-8': 'In Figure [REF] we plot our estimate for the number of streams as a function of distance from the Galactic centre.', 'astro-ph-0208041-2-78-9': 'The solid curve corresponds to the estimate obtained by integrating the orbit backwards in time, from the present day conditions.', 'astro-ph-0208041-2-78-10': 'The dotted curve, on the other hand, corresponds to the orbit integration performed forwards in time from [MATH].', 'astro-ph-0208041-2-78-11': 'The geometric mean of these two estimates is the dashed curve in the same figure.', 'astro-ph-0208041-2-78-12': 'The predicted number of streams given by the geometric mean is in reasonable agreement with what was shown in Figure [REF].', 'astro-ph-0208041-2-79-0': '## Characteristics of the streams', 'astro-ph-0208041-2-80-0': 'Another interesting quantity which we can calculate for each box is the characteristic mass-weighted dispersion within a stream: [EQUATION] where the [MATH] sum runs over all particle pairs in the [MATH]-th stream in a given box, [MATH] is the number of particles in this stream in this box, and as before, [MATH] is the total number of particles in the box.', 'astro-ph-0208041-2-80-1': 'This mass-weighted velocity dispersion is expected to be relatively small in view of our results of the phase-space evolution of halo debris.', 'astro-ph-0208041-2-80-2': 'We plot this quantity in Figure [REF] as a function of distance from the galaxy centre for each one of the boxes considered.', 'astro-ph-0208041-2-80-3': 'For our boxes of 2 kpc on a side we find a typical velocity dispersion of 1 km s[MATH] at the position of the "Sun".', 'astro-ph-0208041-2-80-4': 'The values observed are probably upper limits to the typical velocity dispersion in a stream, since they could only be measured for the densest streams (with two or more particles), and so are biased towards high velocity dispersions.', 'astro-ph-0208041-2-81-0': '# Discussion', 'astro-ph-0208041-2-82-0': 'We have studied the phase-space evolution of debris from the progenitors that merge to build up a dark-matter halo in a [MATH]CDM cosmology.', 'astro-ph-0208041-2-82-1': 'Our analysis has shown that the debris streams originating in progenitors of different sizes and orbital characteristics all behave in a similar way: with velocity dispersions and local space density decreasing in time.', 'astro-ph-0208041-2-82-2': 'The evolution of the debris streams that we were able to follow until the present time is consistent with phase mixing.', 'astro-ph-0208041-2-82-3': 'Even for halos that we could not follow for a very long time - because of their smaller initial number of particles or their shorter orbital timescales- we find the debris to show similar behaviour.', 'astro-ph-0208041-2-82-4': 'On the scales of the fine-grained distribution function, mixing is apparently not strongly chaotic.', 'astro-ph-0208041-2-82-5': 'On the contrary, the phase-space evolution appears to be quite organised and simple, very similar to the mixing of streams orbiting in an idealised static potential.', 'astro-ph-0208041-2-83-0': 'In principle, a dark-matter halo formed in a [MATH]CDM cosmology is not a smooth entity.', 'astro-ph-0208041-2-83-1': 'Not only do dark-matter halos contain a large number of dark satellites, they also have large amounts of substructure in the form of streams.', 'astro-ph-0208041-2-83-2': 'We predict, however, that dark matter in the Solar neighbourhood should be clumped in a few hundred thousand streams, producing a velocity ellipsoid which is close to a multivariate Gaussian (Helmi, White Springel 2002).', 'astro-ph-0208041-2-83-3': 'These streams have their origin in the different halos that merged to form the dark halo of the Galaxy.', 'astro-ph-0208041-2-83-4': 'Most of these halos give rise to a large number of intersecting streams in the inner Galaxy.', 'astro-ph-0208041-2-84-0': 'Determining the characteristics of these hundred thousand streams is difficult even with the high-resolution simulation used here.', 'astro-ph-0208041-2-84-1': 'We are mostly limited by the number of particles.', 'astro-ph-0208041-2-84-2': 'Although the simulation as a whole has 66 million particles, inside a 4 kpc box centred on the "Sun", we find only a couple of thousand particles.', 'astro-ph-0208041-2-84-3': 'In such boxes we find on average two hundred streams with more than one particle, and typically each has only two particles!', 'astro-ph-0208041-2-84-4': 'The internal stream velocity dispersion that we measure at the present-day is extremely small, of the order of only 1 km s[MATH].', 'astro-ph-0208041-2-85-0': 'It is encouraging that we find reasonable agreement between the behaviour of debris streams in static potentials and that observed in this high resolution simulation.', 'astro-ph-0208041-2-85-1': 'This implies that our earlier estimates of the number of stellar streams in the vicinity of the Sun (Helmi White 1999) may indeed apply.', 'astro-ph-0208041-2-85-2': 'Since the initial stellar phase-space distribution in the progenitor objects was probably of lower dimensionality than we here assume - stars tend to form in disks so that the distribution in at least two of the six phase-space coordinates collapses - star streams may be colder than the dark matter streams we have analysed, and so may be more easily distinguishable.', 'astro-ph-0208041-2-85-3': 'It is also interesting to note that because the material that ends up populating the inner galaxy was already in place 10 Gyr ago, the oldest stars are predicted to be near the galactic centre (White Springel 2000).', 'astro-ph-0208041-2-85-4': 'Because this material comes from only a few objects, one might expect the stellar populations to be quite homogeneous, although this of course depends on whether the stars themselves formed in these few massive objects, or whether they were accreted into these objects in the first place.', 'astro-ph-0208041-2-86-0': 'A crude estimate of the stellar content of a stream in the Solar neighbourhood can be obtained as follows.', 'astro-ph-0208041-2-86-1': 'Let us first estimate the "mass-to-light " ratio per particle [MATH] as the ratio of dark-matter mass to stellar halo light enclosed in a shell of thickness 2 kpc at the location of the Sun.', 'astro-ph-0208041-2-86-2': 'We use the NFW profile of Eq. ([REF]) for the dark-matter distribution.', 'astro-ph-0208041-2-86-3': 'For the stellar distribution we assume a density profile [MATH], where [MATH] kpc[MATH], and where [MATH] is the core radius, and is much smaller than [MATH].', 'astro-ph-0208041-2-86-4': 'We obtain for the Solar Neighbourhood [MATH], for [MATH].', 'astro-ph-0208041-2-86-5': 'For one of our most massive streams (with 3 particles), this would imply an average [MATH] in stars in a sphere of 100 pc radius centred on the Sun.', 'astro-ph-0208041-2-86-6': 'Assuming a Salpeter initial mass function, and down to an absolute magnitude of [MATH] (Bergbusch VandenBerg 1992), i.e. [MATH] at a distance of 100 pc, this corresponds to approximately 2 stars per stream.', 'astro-ph-0208041-2-86-7': 'On the other hand, streams originate from very localised regions of phase-space in the progenitor objects, so it seems more likely that only about one stream in 250 has any stars at all, that these streams have [MATH] characteristic of the "stellar" regions of their progenitors, and that all the other streams are dark.', 'astro-ph-0208041-2-86-8': 'In this case a massive, "luminous" stream might contain [MATH] in stars in a sphere of 100 pc radius centred on the Sun, implying approximately [MATH] stars down to [MATH] for a Salpeter IMF.', 'astro-ph-0208041-2-87-0': 'More reliable estimates of the stellar content and number of streams, as well as considerably more insight into the properties of the Galactic stellar halo would be obtained by combining semi-analytic techniques (e.g Kauffmann et al. 1993) with high-resolution simulations (see, for example, Springel et al. 2001).', 'astro-ph-0208041-2-87-1': 'This would enable one to predict trends in the chemical composition, age, spatial distribution and kinematics of halo stars as a function of position throughout the Galaxy.'}

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'astro-ph-0208041-2-16-3'], ['astro-ph-0208041-1-16-4', 'astro-ph-0208041-2-16-4'], ['astro-ph-0208041-1-16-5', 'astro-ph-0208041-2-16-5'], ['astro-ph-0208041-1-29-0', 'astro-ph-0208041-2-29-0'], ['astro-ph-0208041-1-41-0', 'astro-ph-0208041-2-42-0'], ['astro-ph-0208041-1-41-1', 'astro-ph-0208041-2-42-1'], ['astro-ph-0208041-1-66-0', 'astro-ph-0208041-2-67-0'], ['astro-ph-0208041-1-66-1', 'astro-ph-0208041-2-67-1'], ['astro-ph-0208041-1-66-2', 'astro-ph-0208041-2-67-2'], ['astro-ph-0208041-1-66-3', 'astro-ph-0208041-2-67-3'], ['astro-ph-0208041-1-14-0', 'astro-ph-0208041-2-14-0'], ['astro-ph-0208041-1-20-0', 'astro-ph-0208041-2-20-0'], ['astro-ph-0208041-1-20-1', 'astro-ph-0208041-2-20-1'], ['astro-ph-0208041-1-20-2', 'astro-ph-0208041-2-20-2'], ['astro-ph-0208041-1-20-3', 'astro-ph-0208041-2-20-3'], ['astro-ph-0208041-1-20-4', 'astro-ph-0208041-2-20-4'], ['astro-ph-0208041-1-20-5', 'astro-ph-0208041-2-20-5'], ['astro-ph-0208041-1-71-0', 'astro-ph-0208041-2-72-0'], 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['astro-ph-0208041-1-70-3', 'astro-ph-0208041-2-71-3'], ['astro-ph-0208041-1-70-4', 'astro-ph-0208041-2-71-4'], ['astro-ph-0208041-1-49-0', 'astro-ph-0208041-2-50-0'], ['astro-ph-0208041-1-49-3', 'astro-ph-0208041-2-50-2'], ['astro-ph-0208041-1-49-4', 'astro-ph-0208041-2-50-3'], ['astro-ph-0208041-1-49-5', 'astro-ph-0208041-2-50-4'], ['astro-ph-0208041-1-49-6', 'astro-ph-0208041-2-50-5'], ['astro-ph-0208041-1-49-7', 'astro-ph-0208041-2-50-6'], ['astro-ph-0208041-1-49-8', 'astro-ph-0208041-2-50-7'], ['astro-ph-0208041-1-49-9', 'astro-ph-0208041-2-50-8'], ['astro-ph-0208041-1-33-0', 'astro-ph-0208041-2-33-0'], ['astro-ph-0208041-1-33-1', 'astro-ph-0208041-2-33-1'], ['astro-ph-0208041-1-33-2', 'astro-ph-0208041-2-33-2'], ['astro-ph-0208041-1-33-3', 'astro-ph-0208041-2-33-3'], ['astro-ph-0208041-1-82-0', 'astro-ph-0208041-2-83-0'], ['astro-ph-0208041-1-82-1', 'astro-ph-0208041-2-83-1'], ['astro-ph-0208041-1-82-2', 'astro-ph-0208041-2-83-2'], ['astro-ph-0208041-1-82-3', 'astro-ph-0208041-2-83-3'], ['astro-ph-0208041-1-82-4', 'astro-ph-0208041-2-83-4'], ['astro-ph-0208041-1-3-0', 'astro-ph-0208041-2-3-0'], ['astro-ph-0208041-1-3-1', 'astro-ph-0208041-2-3-1'], ['astro-ph-0208041-1-3-2', 'astro-ph-0208041-2-3-2'], ['astro-ph-0208041-1-3-3', 'astro-ph-0208041-2-3-3'], ['astro-ph-0208041-1-3-4', 'astro-ph-0208041-2-3-4'], ['astro-ph-0208041-1-3-5', 'astro-ph-0208041-2-3-5'], ['astro-ph-0208041-1-6-0', 'astro-ph-0208041-2-6-0'], ['astro-ph-0208041-1-6-1', 'astro-ph-0208041-2-6-1'], ['astro-ph-0208041-1-6-2', 'astro-ph-0208041-2-6-2'], ['astro-ph-0208041-1-6-3', 'astro-ph-0208041-2-6-3'], ['astro-ph-0208041-1-22-0', 'astro-ph-0208041-2-22-0'], ['astro-ph-0208041-1-22-1', 'astro-ph-0208041-2-22-1'], ['astro-ph-0208041-1-22-2', 'astro-ph-0208041-2-22-2'], ['astro-ph-0208041-1-22-3', 'astro-ph-0208041-2-22-3'], ['astro-ph-0208041-1-39-0', 'astro-ph-0208041-2-40-0'], ['astro-ph-0208041-1-39-1', 'astro-ph-0208041-2-40-1'], ['astro-ph-0208041-1-39-2', 'astro-ph-0208041-2-40-2'], ['astro-ph-0208041-1-26-0', 'astro-ph-0208041-2-26-0'], ['astro-ph-0208041-1-26-1', 'astro-ph-0208041-2-26-1'], ['astro-ph-0208041-1-26-2', 'astro-ph-0208041-2-26-2'], ['astro-ph-0208041-1-26-3', 'astro-ph-0208041-2-26-3'], ['astro-ph-0208041-1-26-4', 'astro-ph-0208041-2-26-4'], ['astro-ph-0208041-1-26-5', 'astro-ph-0208041-2-26-5'], ['astro-ph-0208041-1-26-6', 'astro-ph-0208041-2-26-6'], ['astro-ph-0208041-1-52-0', 'astro-ph-0208041-2-53-0'], ['astro-ph-0208041-1-52-1', 'astro-ph-0208041-2-53-1'], ['astro-ph-0208041-1-52-2', 'astro-ph-0208041-2-53-2'], ['astro-ph-0208041-1-52-3', 'astro-ph-0208041-2-53-3'], ['astro-ph-0208041-1-52-4', 'astro-ph-0208041-2-53-4'], ['astro-ph-0208041-1-52-5', 'astro-ph-0208041-2-53-5'], ['astro-ph-0208041-1-52-6', 'astro-ph-0208041-2-53-6'], ['astro-ph-0208041-1-52-7', 'astro-ph-0208041-2-53-7'], ['astro-ph-0208041-1-30-0', 'astro-ph-0208041-2-30-0']]

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[]

[['astro-ph-0208041-1-0-3', 'astro-ph-0208041-2-0-3']]

[]

['astro-ph-0208041-1-42-1', 'astro-ph-0208041-1-42-3', 'astro-ph-0208041-1-43-1', 'astro-ph-0208041-1-75-0', 'astro-ph-0208041-1-76-4', 'astro-ph-0208041-2-43-1', 'astro-ph-0208041-2-43-3', 'astro-ph-0208041-2-44-1', 'astro-ph-0208041-2-76-0', 'astro-ph-0208041-2-77-4']

{'1': 'http://arxiv.org/licenses/assumed-1991-2003/', '2': 'http://arxiv.org/licenses/assumed-1991-2003/'}

https://arxiv.org/abs/astro-ph/0208041

1811.12330

{'1811.12330-1-0-0': 'Let [MATH] be a non-compact almost Kahler manifold.', '1811.12330-1-0-1': 'In this paper we provide various criteria that assure that [MATH] induces a non trivial class in the reduced [MATH] maximal/minimal cohomology of [MATH].', '1811.12330-1-0-2': 'Furthermore in the last part we explore some topological applications of our results.', '1811.12330-1-1-0': 'Keywords: Almost Kahler manifolds, symplectic manifolds, [MATH]-cohomology, [MATH]-parabolicity, Kahler spaces.', '1811.12330-1-2-0': 'Mathematics subject classification: 53D05, 58J10, 31C12, 32C18.', '1811.12330-1-3-0': '# Introduction', '1811.12330-1-4-0': 'Let [MATH] be a compact symplectic manifold of dimension [MATH].', '1811.12330-1-4-1': 'Among its basic properties there is the well known fact that [MATH] induces a non trivial class in the de Rham cohomology of [MATH], that is [EQUATION] for any [MATH].', '1811.12330-1-4-2': 'Consider now a non-compact symplectic manifold [MATH].', '1811.12330-1-4-3': 'Let [MATH] be an almost complex structure compatible with [MATH] and let [MATH] be the Riemannian metric induced by [MATH] and [MATH], that is [MATH] for any [MATH].', '1811.12330-1-4-4': 'Usually in the literature a manifold [MATH] equipped with three tensors [MATH], [MATH] and [MATH] as above is called an almost Kahler manifold, see e.g. [CITATION].', '1811.12330-1-4-5': 'For various reasons, besides the usual de Rham cohomology, in the non-compact setting it is also interesting to consider the [MATH]-de Rham cohomology which, roughly speaking, is defined as the quotient between [MATH]-closed forms modulo [MATH]-exact forms .', '1811.12330-1-4-6': 'Thus, looking at [REF], it is natural to wonder whether something similar holds true also for the [MATH]-cohomology of a non-compact almost Kahler manifold.', '1811.12330-1-4-7': 'Certainly we cannot expect that a generalization of [REF] holds true for an arbitrary non-compact almost Kahler manifold without any further assumption on [MATH] or [MATH].', '1811.12330-1-4-8': 'First, as [MATH] is parallel with respect to some Hermitian connection, it is necessary to assume that [MATH] in order to have [MATH] and [MATH].', '1811.12330-1-4-9': 'Secondly there are celebrated vanishing theorems for the [MATH]-cohomology of certain complete Kahler manifolds [MATH] based on the fact that the corresponding Kahler form [MATH] admits a primitive in [MATH], that is there exists [MATH] such that [MATH], see e.g. [CITATION] and [CITATION].', '1811.12330-1-4-10': 'Thus we can say that the aim of this paper is to provide an answer to the following question:', '1811.12330-1-5-0': 'We have already mentioned above that we are led to assume [MATH].', '1811.12330-1-5-1': 'In what follows we will see that another important property is the [MATH]-parabolicity (and other criteria inspired by the notion of [MATH]-parabolicity) of [MATH].', '1811.12330-1-5-2': 'Let us now provide some more details by explaining the structure of this paper.', '1811.12330-1-5-3': 'The first section is devoted to the background material about [MATH]-cohomology and almost Kahler manifolds.', '1811.12330-1-5-4': 'The second section is split in two parts.', '1811.12330-1-5-5': 'Its first subsection collects various technical propositions that will be needed later.', '1811.12330-1-5-6': 'The second subsection contains the main results of this paper.', '1811.12330-1-5-7': 'More precisely it is devoted to various criteria assuring the non-vanishing of [MATH] in [MATH], where [MATH] is the reduced [MATH]-maximal/minimal cohomology of [MATH] respectively, see [REF] and [REF].', '1811.12330-1-5-8': 'Concerning the non-vanishing of [MATH] in [MATH] we prove the following', '1811.12330-1-6-0': 'Let [MATH] be a possibly incomplete almost Kahler manifold of finite volume and dimension [MATH].', '1811.12330-1-6-1': 'Assume that [MATH] for some [MATH].', '1811.12330-1-6-2': 'Then [MATH] induces a non trivial class in [MATH] for any [MATH] and [MATH].', '1811.12330-1-7-0': 'Requiring [MATH] to be [MATH]-parabolic, see Def. [REF], we can also deduce the non-vanishing of [MATH] in [MATH] for certain [MATH].', '1811.12330-1-7-1': 'More precisely we have', '1811.12330-1-8-0': 'Let [MATH] be a possibly incomplete almost Kahler manifold of finite volume and dimension [MATH].', '1811.12330-1-8-1': 'Assume that [MATH] is [MATH]-parabolic with [MATH] and let [MATH].', '1811.12330-1-8-2': 'Then:', '1811.12330-1-9-0': 'For any [MATH], [MATH] induces a non trivial class in [MATH] for any [MATH].', '1811.12330-1-9-1': 'For any [MATH], [MATH] induces a non trivial class in [MATH] for any [MATH].', '1811.12330-1-10-0': 'Finally the last section contains various examples and applications.', '1811.12330-1-10-1': 'We exhibit, especially in the framework of Kahler manifolds, various examples of metrics satisfying the above theorems.', '1811.12330-1-10-2': 'Moreover we show some topological applications of our results.', '1811.12330-1-10-3': 'In particular we prove the following', '1811.12330-1-11-0': 'Let [MATH] be a compact and irreducible Kahler space of complex dimension [MATH].', '1811.12330-1-11-1': 'Assume that every point [MATH] has a local base of open neighborhoods whose regular parts are connected.', '1811.12330-1-11-2': 'Then [EQUATION] for each [MATH].', '1811.12330-1-11-3': 'In particular if [MATH] is a compact and irreducible normal Kahler space then [EQUATION] for each [MATH].'}

{'1811.12330-2-0-0': 'Let [MATH] be a non-compact almost Kahler manifold.', '1811.12330-2-0-1': 'In this paper we provide various criteria that assure that [MATH] induces a non trivial class in the reduced [MATH] maximal/minimal cohomology of [MATH].', '1811.12330-2-0-2': 'Furthermore in the last part we explore some topological applications of our results.', '1811.12330-2-1-0': 'Keywords: Almost Kahler manifolds, symplectic manifolds, [MATH]-cohomology, [MATH]-parabolicity, Kahler spaces.', '1811.12330-2-2-0': 'Mathematics subject classification: 53D05, 58J10, 31C12, 32C18.', '1811.12330-2-3-0': '# Introduction', '1811.12330-2-4-0': 'Let [MATH] be a compact symplectic manifold of dimension [MATH].', '1811.12330-2-4-1': 'Among its basic properties there is the well known fact that [MATH] induces a non trivial class in the de Rham cohomology of [MATH], that is [EQUATION] for any [MATH].', '1811.12330-2-4-2': 'Consider now a non-compact symplectic manifold [MATH].', '1811.12330-2-4-3': 'Let [MATH] be an almost complex structure compatible with [MATH] and let [MATH] be the Riemannian metric induced by [MATH] and [MATH], that is [MATH] for any [MATH].', '1811.12330-2-4-4': 'Usually in the literature a manifold [MATH] equipped with three tensors [MATH], [MATH] and [MATH] as above is called an almost Kahler manifold, see e.g. [CITATION].', '1811.12330-2-4-5': 'For various reasons, besides the usual de Rham cohomology, in the non-compact setting it is also interesting to consider the [MATH]-de Rham cohomology which, roughly speaking, is defined as the quotient between [MATH]-closed forms modulo [MATH]-exact forms .', '1811.12330-2-4-6': 'Thus, looking at [REF], it is natural to wonder whether something similar holds true also for the [MATH]-cohomology of a non-compact almost Kahler manifold.', '1811.12330-2-4-7': 'Certainly we cannot expect that a generalization of [REF] holds true for an arbitrary non-compact almost Kahler manifold without any further assumption on [MATH] or [MATH].', '1811.12330-2-4-8': 'First, as [MATH] is parallel with respect to some Hermitian connection, it is necessary to assume that [MATH] in order to have [MATH] and [MATH].', '1811.12330-2-4-9': 'Secondly there are celebrated vanishing theorems for the [MATH]-cohomology of certain complete Kahler manifolds [MATH] based on the fact that the corresponding Kahler form [MATH] admits a primitive in [MATH], that is there exists [MATH] such that [MATH], see e.g. [CITATION] and [CITATION].', '1811.12330-2-4-10': 'Thus we can say that the aim of this paper is to provide an answer to the following question:', '1811.12330-2-5-0': 'We have already mentioned above that we are led to assume [MATH].', '1811.12330-2-5-1': 'In what follows we will see that another important property is the [MATH]-parabolicity (and other criteria inspired by the notion of [MATH]-parabolicity) of [MATH].', '1811.12330-2-5-2': 'Let us now provide some more details by explaining the structure of this paper.', '1811.12330-2-5-3': 'The first section is devoted to the background material about [MATH]-cohomology and almost Kahler manifolds.', '1811.12330-2-5-4': 'The second section is split in two parts.', '1811.12330-2-5-5': 'Its first subsection collects various technical propositions that will be needed later.', '1811.12330-2-5-6': 'The second subsection contains the main results of this paper.', '1811.12330-2-5-7': 'More precisely it is devoted to various criteria assuring the non-vanishing of [MATH] in [MATH], where [MATH] is the reduced [MATH]-maximal/minimal cohomology of [MATH] respectively, see [REF] and [REF].', '1811.12330-2-5-8': 'Concerning the non-vanishing of [MATH] in [MATH] we prove the following', '1811.12330-2-6-0': 'Let [MATH] be a possibly incomplete almost Kahler manifold of finite volume and dimension [MATH].', '1811.12330-2-6-1': 'Assume that [MATH] for some [MATH].', '1811.12330-2-6-2': 'Then [MATH] induces a non trivial class in [MATH] for any [MATH] and [MATH].', '1811.12330-2-7-0': 'Requiring [MATH] to be [MATH]-parabolic, see Def. [REF], we can also deduce the non-vanishing of [MATH] in [MATH] for certain [MATH].', '1811.12330-2-7-1': 'More precisely we have', '1811.12330-2-8-0': 'Let [MATH] be a possibly incomplete almost Kahler manifold of finite volume and dimension [MATH].', '1811.12330-2-8-1': 'Assume that [MATH] is [MATH]-parabolic with [MATH] and let [MATH].', '1811.12330-2-8-2': 'Then:', '1811.12330-2-9-0': 'For any [MATH], [MATH] induces a non trivial class in [MATH] for any [MATH].', '1811.12330-2-9-1': 'For any [MATH], [MATH] induces a non trivial class in [MATH] for any [MATH].', '1811.12330-2-10-0': 'Finally the last section contains various examples and applications.', '1811.12330-2-10-1': 'We exhibit, especially in the framework of Kahler manifolds, various examples of metrics satisfying the above theorems.', '1811.12330-2-10-2': 'Moreover we show some topological applications of our results.', '1811.12330-2-10-3': 'In particular we prove the following', '1811.12330-2-11-0': 'Let [MATH] be a compact and irreducible Kahler space of complex dimension [MATH].', '1811.12330-2-11-1': 'Assume that every point [MATH] has a local base of open neighborhoods whose regular parts are connected.', '1811.12330-2-11-2': 'Then [EQUATION] for each [MATH].', '1811.12330-2-11-3': 'In particular if [MATH] is a compact and irreducible normal Kahler space then [EQUATION] for each [MATH].'}

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[['1811.12330-1-7-0', '1811.12330-2-7-0'], ['1811.12330-1-7-1', '1811.12330-2-7-1'], ['1811.12330-1-4-0', '1811.12330-2-4-0'], ['1811.12330-1-4-1', '1811.12330-2-4-1'], ['1811.12330-1-4-2', '1811.12330-2-4-2'], ['1811.12330-1-4-3', '1811.12330-2-4-3'], ['1811.12330-1-4-4', '1811.12330-2-4-4'], ['1811.12330-1-4-5', '1811.12330-2-4-5'], ['1811.12330-1-4-6', '1811.12330-2-4-6'], ['1811.12330-1-4-7', '1811.12330-2-4-7'], ['1811.12330-1-4-8', '1811.12330-2-4-8'], ['1811.12330-1-4-9', '1811.12330-2-4-9'], ['1811.12330-1-8-0', '1811.12330-2-8-0'], ['1811.12330-1-8-1', '1811.12330-2-8-1'], ['1811.12330-1-11-0', '1811.12330-2-11-0'], ['1811.12330-1-11-1', '1811.12330-2-11-1'], ['1811.12330-1-11-2', '1811.12330-2-11-2'], ['1811.12330-1-11-3', '1811.12330-2-11-3'], ['1811.12330-1-10-0', '1811.12330-2-10-0'], ['1811.12330-1-10-1', '1811.12330-2-10-1'], ['1811.12330-1-10-2', '1811.12330-2-10-2'], ['1811.12330-1-10-3', '1811.12330-2-10-3'], ['1811.12330-1-5-0', '1811.12330-2-5-0'], ['1811.12330-1-5-1', '1811.12330-2-5-1'], ['1811.12330-1-5-2', '1811.12330-2-5-2'], ['1811.12330-1-5-3', '1811.12330-2-5-3'], ['1811.12330-1-5-4', '1811.12330-2-5-4'], ['1811.12330-1-5-5', '1811.12330-2-5-5'], ['1811.12330-1-5-6', '1811.12330-2-5-6'], ['1811.12330-1-5-7', '1811.12330-2-5-7'], ['1811.12330-1-5-8', '1811.12330-2-5-8'], ['1811.12330-1-0-0', '1811.12330-2-0-0'], ['1811.12330-1-0-1', '1811.12330-2-0-1'], ['1811.12330-1-0-2', '1811.12330-2-0-2'], ['1811.12330-1-9-0', '1811.12330-2-9-0'], ['1811.12330-1-9-1', '1811.12330-2-9-1'], ['1811.12330-1-6-0', '1811.12330-2-6-0'], ['1811.12330-1-6-1', '1811.12330-2-6-1'], ['1811.12330-1-6-2', '1811.12330-2-6-2']]

[]

[]

[]

[]

['1811.12330-1-1-0', '1811.12330-1-2-0', '1811.12330-1-4-10', '1811.12330-1-8-2', '1811.12330-2-1-0', '1811.12330-2-2-0', '1811.12330-2-4-10', '1811.12330-2-8-2']

{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1811.12330

1610.01449

{'1610.01449-1-0-0': 'It is shown that if two hyperbolic polynomials have a particular factorization into quadratics, then their roots satisfy a power majorization relation whenever key coefficients in their factorizations satisfy a corresponding majorization relation.', '1610.01449-1-0-1': 'In particular, a numerical observation by Klemes [CITATION] is confirmed.', '1610.01449-1-1-0': '[2010]26B10, 26C25', '1610.01449-1-2-0': '# Introduction', '1610.01449-1-3-0': 'Let [MATH] and [MATH] be two [MATH]-tuples of real numbers.', '1610.01449-1-3-1': 'We recall ([CITATION]) that [MATH] is said to be majorized by [MATH] if the sum of [MATH] largest entries of [MATH] is less or equal to the sum of [MATH] largest entries of [MATH], where [MATH] ranges from [MATH] to [MATH], and equality holds at [MATH].', '1610.01449-1-3-2': 'There are several useful characterizations of majorization, perhaps the most famous one is due to Hardy, Littlewood and Polya [CITATION] which says that [MATH] is majorized by [MATH] if and only if [EQUATION] for every convex function [MATH].', '1610.01449-1-4-0': 'Mainly motivated by the study of [MATH]-means, for two [MATH]-tuples of positive real numbers, we say that [MATH] is power majorized by [MATH] provided that [EQUATION] whenever [MATH], with reversal of the inequality sign when [MATH].', '1610.01449-1-5-0': 'Clearly, majorization implies power majorization.', '1610.01449-1-5-1': 'But the converse is not true.', '1610.01449-1-5-2': 'The first example illustrating the difference between majorization and power majorization was given in [CITATION].', '1610.01449-1-5-3': 'Unlike the rich theory on the majorization relation, little is known about power majorization.', '1610.01449-1-5-4': 'Indeed, it is in general difficult to determine whether one vector is power majorized by another.', '1610.01449-1-5-5': 'Our investigation in this paper stems from the following example.', '1610.01449-1-6-0': 'Suppose that one is interested in comparing the [MATH] norms of the eigenvalues [MATH] and [MATH] respectively of the [MATH] matrices [MATH] and [MATH] defined by [MATH], [MATH], where [EQUATION]', '1610.01449-1-6-1': 'On numerical evidence, it was suggested in [CITATION] that [MATH] is power majorized by [MATH].', '1610.01449-1-6-2': 'The author of [CITATION] asked for an “enlightening” proof or disproof.', '1610.01449-1-6-3': 'We confirm this numerical guess.', '1610.01449-1-7-0': 'A simple calculation gives [EQUATION] and their characteristic polyonmials [EQUATION]', '1610.01449-1-8-0': 'It can be shown that in this example [MATH] is not majorized by [MATH] (as the sum of the two largest entries of [MATH] is larger than the sum of the two largest entries of [MATH]).', '1610.01449-1-8-1': 'We observe that in the above two factors of [MATH] and [MATH], the coefficients of [MATH] satisfy a majorization relation.', '1610.01449-1-8-2': 'More precisely, [MATH] is majorized by [MATH].', '1610.01449-1-8-3': 'This simple observation hints at the statement of our main result.', '1610.01449-1-9-0': '# Main Results', '1610.01449-1-10-0': 'A hyperbolic polynomial is a polynomial whose roots are all real.', '1610.01449-1-10-1': 'Our main result is the following theorem, which gives a condition for power majorization between the roots of two hyperbolic polynomials having a certain factorization.', '1610.01449-1-10-2': 'We remark that relevant studies on the majorization relation between the roots of hyperbolic polynomials (the so called spectral order) are given in [CITATION].', '1610.01449-1-11-0': 'Consider two polyonomials [EQUATION]', '1610.01449-1-11-1': 'Let [MATH] and [MATH] be the vectors of the roots of [MATH] and [MATH], respectively.', '1610.01449-1-11-2': 'If [MATH] is majorized by [MATH], then [MATH] is power majorized by [MATH].', '1610.01449-1-12-0': 'We need some basic facts about Schur-convex functions ([CITATION]).', '1610.01449-1-12-1': 'A real-valued function [MATH] defined on a set [MATH] is said to be Schur-convex on [MATH] if for every [MATH], [MATH] being majorized by [MATH] implies [MATH].', '1610.01449-1-12-2': 'We say [MATH] is Schur-concave if [MATH] is Schur-convex.', '1610.01449-1-13-0': 'Let [MATH] be an open interval and let [MATH] be continuously differentiable.', '1610.01449-1-13-1': 'The well-known Schur condition ([CITATION]) says that [MATH] is Schur-convex on [MATH] if and only if [MATH] is symmetric on [MATH] and [EQUATION]', '1610.01449-1-13-2': 'Proof of Theorem [REF].', '1610.01449-1-13-3': 'The roots of [MATH] are [MATH], [MATH] and the roots of [MATH] are [MATH], [MATH].', '1610.01449-1-13-4': 'We need to show that for any fixed [MATH], [EQUATION] and that the inequality reverses for [MATH].', '1610.01449-1-14-0': 'This would follow if one could show [EQUATION] where [MATH], is Schur convex for [MATH] and is Schur concave for [MATH].', '1610.01449-1-15-0': 'Assume first that [MATH].', '1610.01449-1-15-1': "Clearly, [MATH] is symmetric; by Schur's condition, it remains to show that [EQUATION].", '1610.01449-1-16-0': 'Without loss of generality, we assume [MATH].', '1610.01449-1-16-1': 'Then it suffices to show [EQUATION]', '1610.01449-1-16-2': 'Compute [EQUATION]', '1610.01449-1-16-3': 'Thus, ([REF]) would follow if we can show [EQUATION] is an increasing function for [MATH].', '1610.01449-1-17-0': 'A simple calculation gives [EQUATION]', '1610.01449-1-17-1': 'With [MATH], this becomes [EQUATION]', '1610.01449-1-17-2': 'To see that [MATH], it suffices to show that for [MATH], [EQUATION]', '1610.01449-1-17-3': 'When [MATH] this is clear.', '1610.01449-1-17-4': 'When [MATH] it is equivalent to showing [EQUATION]', '1610.01449-1-17-5': 'But [MATH] and [EQUATION] so [MATH] for [MATH].', '1610.01449-1-17-6': 'This completes the proof of the [MATH] case.', '1610.01449-1-18-0': 'If [MATH], the argument is similar to the proof of the [MATH] case.', '1610.01449-1-18-1': 'It suffices to show [MATH] defined above is decreasing for [MATH].', '1610.01449-1-18-2': 'To see that [MATH], we need to verify that for [MATH], [EQUATION]', '1610.01449-1-18-3': 'Equivalently, [MATH] defined above should be nonpositive.', '1610.01449-1-18-4': 'But in this case, [MATH] and [EQUATION]', '1610.01449-1-18-5': 'So the proof of Theorem [REF] is complete.'}

{'1610.01449-2-0-0': 'It is shown that if two hyperbolic polynomials have a particular factorization into quadratics, then their roots satisfy a power majorization relation whenever key coefficients in their factorizations satisfy a corresponding majorization relation.', '1610.01449-2-0-1': 'In particular, a numerical observation by Klemes [CITATION] is confirmed.', '1610.01449-2-1-0': '[2010]26B10, 26C25', '1610.01449-2-2-0': '# Introduction', '1610.01449-2-3-0': 'Let [MATH] and [MATH] be two [MATH]-tuples of real numbers.', '1610.01449-2-3-1': 'We recall ([CITATION]) that [MATH] is said to be majorized by [MATH] if the sum of [MATH] largest entries of [MATH] is less or equal to the sum of [MATH] largest entries of [MATH], where [MATH] ranges from [MATH] to [MATH], and equality holds at [MATH].', '1610.01449-2-3-2': 'There are several useful characterizations of majorization, perhaps the most famous one is due to Hardy, Littlewood and Polya [CITATION] which says that [MATH] is majorized by [MATH] if and only if [EQUATION] for every convex function [MATH].', '1610.01449-2-4-0': 'Mainly motivated by the study of [MATH]-means, for two [MATH]-tuples of positive real numbers, we say that [MATH] is power majorized by [MATH] provided that [EQUATION] whenever [MATH], with reversal of the inequality sign when [MATH].', '1610.01449-2-5-0': 'Clearly, majorization implies power majorization.', '1610.01449-2-5-1': 'But the converse is not true.', '1610.01449-2-5-2': 'The first example illustrating the difference between majorization and power majorization was given in [CITATION].', '1610.01449-2-5-3': 'Unlike the rich theory on the majorization relation, little is known about power majorization.', '1610.01449-2-5-4': 'Indeed, it is in general difficult to determine whether one vector is power majorized by another.', '1610.01449-2-5-5': 'Our investigation in this paper stems from the following example.', '1610.01449-2-6-0': 'Suppose that one is interested in comparing the [MATH] norms of the eigenvalues [MATH] and [MATH] respectively of the [MATH] matrices [MATH] and [MATH] defined by [MATH], [MATH], where [EQUATION]', '1610.01449-2-6-1': 'On numerical evidence, it was suggested in [CITATION] that [MATH] is power majorized by [MATH].', '1610.01449-2-6-2': 'The author of [CITATION] asked for an “enlightening” proof or disproof.', '1610.01449-2-6-3': 'We confirm this numerical guess.', '1610.01449-2-7-0': 'A simple calculation gives [EQUATION] and their characteristic polyonmials [EQUATION]', '1610.01449-2-8-0': 'It can be shown that in this example [MATH] is not majorized by [MATH] (as the sum of the two largest entries of [MATH] is larger than the sum of the two largest entries of [MATH]).', '1610.01449-2-8-1': 'We observe that in the above two factors of [MATH] and [MATH], the coefficients of [MATH] satisfy a majorization relation.', '1610.01449-2-8-2': 'More precisely, [MATH] is majorized by [MATH].', '1610.01449-2-8-3': 'This simple observation hints at the statement of our main result.', '1610.01449-2-9-0': '# Main Result', '1610.01449-2-10-0': 'A hyperbolic polynomial is a polynomial whose roots are all real.', '1610.01449-2-10-1': 'We refer the interested reader to the classical text [CITATION] on the roots of hyperbolic polynomials.', '1610.01449-2-10-2': 'Our main result is the following theorem, which gives a condition for power majorization between the roots of two hyperbolic polynomials having a certain factorization.', '1610.01449-2-10-3': 'We remark that relevant studies on the majorization relation between the roots of hyperbolic polynomials (the so called spectral order) are given in [CITATION].', '1610.01449-2-11-0': 'Consider two polynomials [EQUATION]', '1610.01449-2-11-1': 'Let [MATH] and [MATH] be the vectors of the roots of [MATH] and [MATH], respectively.', '1610.01449-2-11-2': 'If [MATH] is majorized by [MATH], then [MATH] is power majorized by [MATH].', '1610.01449-2-12-0': 'We need some basic facts about Schur-convex functions ([CITATION]).', '1610.01449-2-12-1': 'A real-valued function [MATH] defined on a set [MATH] is said to be Schur-convex on [MATH] if for every [MATH], [MATH] being majorized by [MATH] implies [MATH].', '1610.01449-2-12-2': 'We say [MATH] is Schur-concave if [MATH] is Schur-convex.', '1610.01449-2-13-0': 'Let [MATH] be an open interval and let [MATH] be continuously differentiable.', '1610.01449-2-13-1': 'The well-known Schur condition ([CITATION]) says that [MATH] is Schur-convex on [MATH] if and only if [MATH] is symmetric on [MATH] and [EQUATION]', '1610.01449-2-13-2': 'Proof of Theorem [REF].', '1610.01449-2-13-3': 'The roots of [MATH] are [MATH], [MATH] and the roots of [MATH] are [MATH], [MATH].', '1610.01449-2-13-4': 'We need to show that for any fixed [MATH], [EQUATION] and that the inequality reverses for [MATH].', '1610.01449-2-14-0': 'This would follow if one could show [EQUATION] where [MATH], is Schur convex for [MATH] and is Schur concave for [MATH].', '1610.01449-2-15-0': 'Assume first that [MATH].', '1610.01449-2-15-1': "Clearly, [MATH] is symmetric; by Schur's condition, it remains to show that [EQUATION].", '1610.01449-2-16-0': 'Without loss of generality, we assume [MATH].', '1610.01449-2-16-1': 'Then it suffices to show [EQUATION]', '1610.01449-2-16-2': 'Compute [EQUATION]', '1610.01449-2-16-3': 'Thus, ([REF]) would follow if we can show [EQUATION] is an increasing function for [MATH].', '1610.01449-2-17-0': 'A simple calculation gives [EQUATION]', '1610.01449-2-17-1': 'With [MATH], this becomes [EQUATION]', '1610.01449-2-17-2': 'To see that [MATH], it suffices to show that for [MATH], [EQUATION]', '1610.01449-2-17-3': 'When [MATH] this is clear.', '1610.01449-2-17-4': 'When [MATH] it is equivalent to showing [EQUATION]', '1610.01449-2-17-5': 'But [MATH] and [EQUATION] so [MATH] for [MATH].', '1610.01449-2-17-6': 'This completes the proof of the [MATH] case.', '1610.01449-2-18-0': 'If [MATH], the argument is similar to the proof of the [MATH] case.', '1610.01449-2-18-1': 'It suffices to show [MATH] defined above is decreasing for [MATH].', '1610.01449-2-18-2': 'To see that [MATH], we need to verify that for [MATH], [EQUATION]', '1610.01449-2-18-3': 'Equivalently, [MATH] defined above should be nonpositive.', '1610.01449-2-18-4': 'But in this case, [MATH] and [EQUATION]', '1610.01449-2-18-5': 'So the proof of Theorem [REF] is complete.'}

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[]

[]

[['1610.01449-1-11-0', '1610.01449-2-11-0']]

[]

['1610.01449-1-1-0', '1610.01449-1-7-0', '1610.01449-1-16-2', '1610.01449-1-17-5', '1610.01449-2-1-0', '1610.01449-2-7-0', '1610.01449-2-16-2', '1610.01449-2-17-5']

{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1610.01449

0801.3238

{'0801.3238-1-0-0': 'We analyze in detail, beyond the usual scaling hypothesis, the finite-size convergence of static quantities toward the thermodynamic limit.', '0801.3238-1-0-1': 'In this way we are able to obtain sequences of pseudo-critical points which display a faster convergence rate as compared to currently used methods.', '0801.3238-1-0-2': 'The approaches are valid in any spatial dimension and for any value of the dynamic exponent.', '0801.3238-1-0-3': 'We demonstrate the effectiveness of our methods both analytically on the basis of the one dimensional XY model, and numerically considering [MATH] transitions occurring in non integrable spin models.', '0801.3238-1-0-4': 'In particular, we show that these general methods are able to locate precisely the onset of the Berezinskii-Kosterlitz-Thouless transition making only use of ground-state properties on relatively small systems.', '0801.3238-1-1-0': '# Introduction', '0801.3238-1-2-0': 'In the study of physical properties of phase transitions a basic prerequisite is a reliable method to locate the critical point, whenever the latter is not known a priori from symmetry or duality arguments.', '0801.3238-1-2-1': 'Typically, in numerical or even experimental studies on finite samples one obtains a sequence of pseudocritical points (in the sense specified below) to be extrapolated to the true critical point in the thermodynamic limit (TL).', '0801.3238-1-2-2': 'The extrapolation may be done with some polynomial fit in the inverse size of the system or, better, exploiting some fitting function derived on the basis of a scaling ansatz or through the renormalization group (RG).', '0801.3238-1-2-3': 'The point is especially relevant in the context of quantum phase transitions (QPT) [CITATION] in lattice systems where the exponential growth of the dimension of the Hilbert space with the number of sites is a strong limitation on the accessible sizes with the current computational power and algorithms.', '0801.3238-1-2-4': 'One of the most used algorithms is still the Lanczos method for the virtually exact extraction of the low-lying energy levels; in the most favorable case of spin-1/2 models one cannot go beyond some tens of sites.', '0801.3238-1-2-5': 'This limit can be moved to maybe a few thousands of sites using the so-called density matrix renormalization group (DMRG) [CITATION] that has become the method of choice for 1D problems due to its high level of accuracy.', '0801.3238-1-2-6': 'Nonetheless, if one considers two or even three dimensional systems the situation is much worse: with the Lanczos algorithm the largest lattices have only a few sites of linear extension and the DMRG is not particularly efficient.', '0801.3238-1-2-7': 'At present, the only other choice is Quantum Monte Carlo (QMC) (see, for instance, [CITATION]) that, however, suffers from a sign problem in the case of fermionic or frustrated systems and does not reach the level of accuracy of the DMRG.', '0801.3238-1-2-8': 'Very recently there have been attempts to exploit both DMRG-like features and the QMC sampling tricks, to design hybrid methods [CITATION] that are however still under verification.', '0801.3238-1-3-0': 'It is generally believed that a sequence of pseudocritical points, for example the loci of maxima of finite-size susceptibilities, converges to the critical point as a power law [MATH], with a so-called shift exponent [MATH] given by the inverse of the correlation length exponent [MATH].', '0801.3238-1-3-1': 'Hence, generally speaking, the larger is [MATH] the slower is the convergence.', '0801.3238-1-3-2': 'This difficulty reaches its maximum for Berezinskii-Kosterlitz-Thouless (BKT) transitions, in which the correlation length diverges with an essential singularity or, loosely speaking, "[MATH]".', '0801.3238-1-3-3': 'However, already in the seminal paper by Fisher and Barber [CITATION], it was pointed out that the relation [MATH] is not always valid and [MATH] depends, among other factors, on the boundary conditions.', '0801.3238-1-4-0': 'The most used method to locate quantum critical points in [MATH] by means of finite-size data is the so-called Phenomenological Renormalization Group (PRG), reviewed for instance in [CITATION].', '0801.3238-1-4-1': 'Another convenient approach, the Finite-size Crossing Method (FSCM), was recently proposed in [CITATION].', '0801.3238-1-4-2': 'The aim of this paper is to improve both of them by means of criteria that produce sequences of pseudocritical points that converge more rapidly.', '0801.3238-1-4-3': 'We will show that in our sequences the shift exponent will have the form [MATH], where [MATH] and/or [MATH] are larger than the corresponding values in the usual methods and therefore allow for a better convergence.', '0801.3238-1-5-0': 'The paper is organized as follows.', '0801.3238-1-5-1': 'In Section [REF] we illustrate the general arguments leading to the enhanced sequences, both in the framework of the FSCM and of the PRG (Subsec. [REF]).', '0801.3238-1-5-2': 'Special cases as the BKT transition (Subsec. [REF]) and that of logarithmic divergences (Subsec. [REF]) are discussed separately.', '0801.3238-1-5-3': 'In Section [REF] we illustrate the usefulness of the methods on the hand of analytic and numerical tests.', '0801.3238-1-5-4': 'In Subsec. [REF] we treat the XY spin-1/2 chain, using a series of exact calculations reported in the Appendix.', '0801.3238-1-5-5': 'Then we move to two cases of spin chains for which no exact solution is available: in Subsec. [REF] we consider a spin-1 model with anisotropies in a parameter range that gives rise to a large value of [MATH] and in Subsec. [REF] we study the spin-1/2 model with next to nearest neighbor interactions that is known to undergo a BKT transition.', '0801.3238-1-5-6': 'In this case we find a value for the critical coupling in agreement with the accepted one, which was found using a model-specific investigation of the excited states [CITATION].', '0801.3238-1-5-7': 'Section [REF] is devoted to conclusions.', '0801.3238-1-6-0': '# Derivation of rapidly-converging sequences', '0801.3238-1-7-0': 'We consider systems in [MATH] spatial dimensions of linear size [MATH] and periodic boundary conditions (PBC).', '0801.3238-1-7-1': 'Let the transition be driven by a linear parameter [MATH] such that the Hamiltonian is', '0801.3238-1-8-0': '[EQUATION].', '0801.3238-1-9-0': 'Dealing with QPT we consider the case of strictly zero temperature, [MATH], even if the arguments presented below can be simply extended to the finite-temperature case, replacing the parameter [MATH] with [MATH] (and without using the dimensional crossover rule used below).', '0801.3238-1-9-1': 'The free energy density reduces to the ground-state (GS) energy density which, close to the critical point [MATH], shows a singularity in the second (or higher) derivatives with respect to [MATH]:', '0801.3238-1-10-0': '[EQUATION] where [MATH] is the correlation length.', '0801.3238-1-10-1': 'Note that, as a consequence of the scaling hypothesis, the singular part of the energy [MATH] is a universal quantity that depends only on [MATH], the relevant length scale close to the critical point.', '0801.3238-1-10-2': 'Hence, [MATH] may be considered quite in general an even function of [MATH] that vanishes at the critical point.', '0801.3238-1-11-0': 'On the other hand, the bulk energy density at the critical point behaves as (Privman-Fisher hypothesis) [EQUATION] where [MATH] is a sort of Casimir-like term that may depend on the actual geometry of the lattice.', '0801.3238-1-11-1': 'Note that this hypothesis has to be changed properly if one or more of the spatial dimensions are of infinite extent.', '0801.3238-1-11-2': 'Moreover, Eq. ([REF]) has been written in analogy with Eq. (11.29) of ref. [CITATION] using the dimensional crossover rule according to which the partition function and the thermodynamic (static) properties of a [MATH]-dimensional quantum system are equivalent to those of [MATH])-dimensional classical counterpart [CITATION], where [MATH] is the dynamic exponent [CITATION].', '0801.3238-1-11-3': 'Then, for the implementation of our methods we need to know by some other means the value of [MATH] relating the energy gap [MATH] and the correlation length [MATH]: [MATH].', '0801.3238-1-11-4': 'Typically, but not always, energy and momentum in the continuum limit at the critical point satisfy a linear dispersion relation, [MATH], for small [MATH] so that [MATH] and a relativistic effective field theory can be used to describe the universal features of the transition.', '0801.3238-1-11-5': 'In [MATH] the scale invariance at the critical point is often sufficient to imply also conformal invariance (see Ch. 2 of [CITATION]), thanks to which several exact results can be obtained using the powerful predictions of conformal field theories (CFT).', '0801.3238-1-11-6': 'For example, by mapping the space-time complex plane onto a cylinder whose circumference represents the finite chain of length [MATH] we can identify [MATH] in Eq. ([REF]) where [MATH] is the central charge of the theory.', '0801.3238-1-11-7': 'In the RG sense, moving away from criticality corresponds to perturbing the CFT with a relevant operator that destroys conformal invariance.', '0801.3238-1-11-8': 'However, this is not the only effect of varying the microscopic parameter [MATH] out of [MATH]: in general also the speed of elementary excitations gets renormalized in the unperturbed CFT part.', '0801.3238-1-11-9': 'For this reason we say that [MATH] (henceforth [MATH]) depends on [MATH], in the vicinity of [MATH].', '0801.3238-1-12-0': 'Scaling and dimensional arguments imply that, in the thermodynamic, off-critical regime [MATH], the singular part of the energy behaves as[EQUATION] with [MATH].', '0801.3238-1-12-1': 'For a second order phase transition [MATH].', '0801.3238-1-13-0': 'After introducing the scaling variable [MATH], the finite-size scaling (FSS) theory asserts [CITATION] that in a system of length [MATH], [EQUATION] where [MATH] is a universal function that, in the off-critical regime [MATH], must behave as [MATH] in order to recover Eq. ([REF]) .', '0801.3238-1-13-1': 'Instead, for [MATH] we are in the critical regime and [MATH] behaves as an analytic function that vanishes for [MATH].', '0801.3238-1-13-2': 'Here we assume that the leading term in [MATH] is quadratic in [MATH] (see below why it cannot be linear), but the following arguments are easily generalizable to higher integer powers.', '0801.3238-1-14-0': 'Differentiating [MATH] with respect to [MATH], gives the mean value [MATH], whose singular part [MATH] behaves as[EQUATION]', '0801.3238-1-14-1': 'Considering FSS for the combination of Eqs. ([REF]) and ([REF]) and then differentiating we find[EQUATION] where the subscript "[MATH], reg" hereafter means regular in the TL.', '0801.3238-1-14-2': 'In order to write down the expression above we used [MATH] and assumed that the powers neglected in the last term are just larger than [MATH].', '0801.3238-1-14-3': 'To illustrate this point we could consider the irrelevant operator with the smallest scaling dimension [MATH].', '0801.3238-1-14-4': 'At first order in perturbation theory with the renormalized coupling [MATH] the corrections to the GS energy density are of the form [MATH] so that [MATH].', '0801.3238-1-14-5': 'Note also that the amplitude [MATH] can vanish and [MATH] terms have to be included.', '0801.3238-1-14-6': 'For these and more details we leave the reader to Ref. [CITATION].', '0801.3238-1-14-7': 'Generically we admit corrections with [MATH] that may come either from irrelevant operator in the continuum theory or from lattice effects.', '0801.3238-1-14-8': 'The case [MATH] corresponds to marginal perturbations and typically leads to logarithmic corrections.', '0801.3238-1-14-9': 'Notice that now the leading term of [MATH] is linear in [MATH] in the critical region.', '0801.3238-1-14-10': 'If we would have admitted a linear term in [MATH] then [MATH] and a finite jump discontinuity at finite [MATH] would be present in [MATH].', '0801.3238-1-15-0': 'We can also calculate [MATH], yielding to a singular part that is similar to Eq. ([REF]) but with a changed sign.', '0801.3238-1-15-1': 'In fact, for [MATH] the leading singular parts of [MATH] and [MATH] must cancel in the sum that gives back the energy [MATH], which does not contain that singularity.', '0801.3238-1-15-2': 'In particular, the scaling is[EQUATION]', '0801.3238-1-15-3': 'The FSCM [CITATION] identifies the critical point with the limit of the sequence [MATH] of single crossing points [EQUATION] with [MATH].', '0801.3238-1-15-4': 'Applying this criterion to Eq. ([REF]), we obtain (for [MATH]) [EQUATION]', '0801.3238-1-15-5': 'This equation defines the shift exponent [MATH] and may converge very slowly when [MATH].', '0801.3238-1-15-6': 'The extremely difficult case is the BKT transition where formally [MATH], but this latter situation must be treated in a different way (see Subsec. [REF]).', '0801.3238-1-16-0': 'Now, consider the quantity [MATH] and suppose to be able to tune [MATH] exactly at [EQUATION]', '0801.3238-1-16-1': 'It easily seen that [MATH] does not contain the Casimir-like term responsible for the critical point shift.', '0801.3238-1-16-2': 'In fact, the scaling of [MATH] is When [MATH] is equal to [MATH] given in Eq. ([REF]), the critical point found by this crossing method is approached as [MATH], with a shift exponent [MATH] (again this holds true provided that [MATH]).', '0801.3238-1-16-3': 'The additional term [MATH] allows, in general, for a better convergence of the sequence [MATH].', '0801.3238-1-17-0': 'A possible algorithm for finding numerically the critical point in such a way is the following.', '0801.3238-1-17-1': 'If [MATH] is at a crossing point, of [MATH], then we have [EQUATION].', '0801.3238-1-17-2': 'Notice that putting [MATH] is equivalent to the FSCM applied to [MATH] and the denominator has definite sign about the critical point.', '0801.3238-1-17-3': 'Now we find [MATH] requiring that [MATH], i.e. [EQUATION] or in the continuum version [EQUATION]', '0801.3238-1-17-4': 'Calling [MATH] and rewriting only the essential terms in the scaling ansatze we have the simplified forms [EQUATION]', '0801.3238-1-17-5': 'Now, putting these two relations in ([REF]) we obtain [EQUATION] that gives a shift exponent [MATH] as anticipated.', '0801.3238-1-18-0': 'The main result of this section is the crossing criterion ([REF]) that identifies the rapidly converging sequence ([REF]) to the critical value.', '0801.3238-1-19-0': '## Homogeneity condition', '0801.3238-1-20-0': 'In the previous section we have shown how to obtain a sequence of pseudocritical points with an improved shift exponent [MATH].', '0801.3238-1-20-1': 'Here we provide another, yet simpler equation for the determination of the critical point.', '0801.3238-1-20-2': 'The resulting pseudocritical sequence is characterized by the same shift exponent [MATH] .', '0801.3238-1-20-3': 'However in this case we are able to prove convergence toward the critical point even in the extreme case of a BKT transition (see Subsec. [REF]).', '0801.3238-1-21-0': 'The idea is to require that at the critical point the [MATH]-dependent part of [MATH] is dominated by the Casimir-like term with power [MATH] (see Eq. ([REF])).', '0801.3238-1-21-1': 'This condition is translated into the requirement that the [MATH]-derivative of [MATH] is a homogeneous function of degree [MATH], i.e. [EQUATION]', '0801.3238-1-21-2': 'Consequently, the corresponding sequence of pseudocritical points [MATH] scales as[EQUATION]', '0801.3238-1-21-3': 'The equation ([REF]) represents the homogeneity condition method (HCM) that we are proposing for the efficient location of critical points.', '0801.3238-1-21-4': 'We stress here that, being [MATH] a GS property, this criterion does not require knowledge of excited states as in the case of the PRG method.', '0801.3238-1-21-5': 'This is a point in favor to the HCM since excited states are typically assessed with less numerical accuracy.', '0801.3238-1-21-6': 'In addition, the HCM is superior to the PRG in that it produces a faster converging sequence (see Subsec. [REF]).', '0801.3238-1-22-0': '## Case [MATH] with logarithmic divergences', '0801.3238-1-23-0': 'For completeness we consider the case [MATH] that was excluded in the previous treatment.', '0801.3238-1-23-1': 'In this situation, the ansatz requires the inclusion of logarithmic corrections [EQUATION].', '0801.3238-1-23-2': 'The calculation of the critical point with the FSCM gives in this case', '0801.3238-1-24-0': '[EQUATION] whereas the HCM Eq. ([REF]) yields [EQUATION]', '0801.3238-1-24-1': 'These results are compatible with the exact calculations of the XY model (see Subsec. [REF]).', '0801.3238-1-24-2': 'Note that in general, we should to perform a similar calculation for [EQUATION] namely when the [MATH]-th derivative of the free energy diverges logarithmically.', '0801.3238-1-25-0': '## A scaling ansatz for the BKT case', '0801.3238-1-26-0': 'For [MATH] at the BKT the correlation function in the TL behaves like [MATH].', '0801.3238-1-26-1': 'In the typical example of the classical two dimensional XY model it is known that [MATH].', '0801.3238-1-26-2': 'Instead, for the quantum Heisenberg model with frustration (which we will consider in Subsec. [REF]) Haldane suggested [MATH] [CITATION].', '0801.3238-1-26-3': 'We also set [MATH] because the (effective) dimensionality in the BKT scenario is two.', '0801.3238-1-26-4': 'The singular part of the finite-size energy density now is conveniently expressed in terms of [MATH] [CITATION] so that [EQUATION] where [MATH] is a universal function that, in the off-critical regime [MATH], must behave as [MATH].', '0801.3238-1-26-5': 'Again, in the quasicritical regime [MATH] at any finite [MATH] the energy density and its derivatives must be analytic in [MATH].', '0801.3238-1-26-6': 'The value [MATH] can be absorbed in [MATH] and it can be checked directly that the first contribution has to be at least quadratic in [MATH] because otherwise a finite-size discontinuity in [MATH] would be generated.', '0801.3238-1-26-7': 'For [MATH] we adopt the following ansatz (justified from perturbed conformal field theory [CITATION]): [EQUATION] with [MATH] a constant and [MATH] an integer larger than 1 ([MATH] from Eq (22) in [CITATION]).', '0801.3238-1-26-8': 'Hence Now we want to get rid of all the [MATH] contributions that "hinder" the location of finite-size pseudocritical points.', '0801.3238-1-26-9': 'Hence we first differentiate with respect to [MATH] to eliminate [MATH], then multiply by [MATH] to isolate the term in square brackets in Eq.([REF]) and finally set to zero a further difference in [MATH] in order to drop [MATH].', '0801.3238-1-26-10': 'Formally, in the region [MATH], we can write down the condition [EQUATION]', '0801.3238-1-26-11': 'It is worth noticing that the latter condition is equal to the HCM Eq. ([REF]) when [MATH].', '0801.3238-1-26-12': 'Treating [MATH] as a continuous variable one can read off the shift exponent for the sequence [MATH] that turns out to be [MATH]; if [MATH] then the corrections to scaling are also governed by (another) marginal operator and we expect [MATH] with [MATH] a positive integer ([MATH] from Eq. (22) in [CITATION]).', '0801.3238-1-27-0': 'Dealing with numerical simulations it is very important to specify how one implements the finite-size differences in [MATH].', '0801.3238-1-27-1': 'In fact, there are several finite-difference expressions used in the literature to express the derivatives and here the requirement is that they all reproduce Eq. ([REF]) in the limit [MATH].', '0801.3238-1-27-2': 'For example, if one takes a uniform step [MATH] then the following symmetric expression can be built[EQUATION] and then the precise condition to cancel the term [MATH] becomes[EQUATION]', '0801.3238-1-27-3': 'In the limit of large [MATH] we recover ([REF]) as required.', '0801.3238-1-27-4': 'Clearly, the correct discretization prescription must be identified not only for the BKT; when [MATH] and/or [MATH] are not one, the suitable variant of ([REF]) for finite [MATH] has to be adopted, with [MATH] weighted by [MATH] and [MATH] weighted by a polynomial in [MATH] of degree [MATH] whose coefficients depend on [MATH].', '0801.3238-1-28-0': '## PRG revisited', '0801.3238-1-29-0': 'The PRG method identifies the critical point with the limit of the sequence [MATH] of crossing points satisfying[EQUATION]', '0801.3238-1-29-1': 'Here [MATH] is the finite-size energy gap of the spectrum for which we may adopt the following form [EQUATION] with [MATH] and [MATH] a prefactor depending on the excited state [MATH] we are considering.', '0801.3238-1-29-2': 'The standard PRG approach in [MATH] relies upon Eq. ([REF]) with [MATH].', '0801.3238-1-29-3': 'More generally, a first test to identify [MATH] is done by plotting the usual scaled gaps [MATH] and see if at the critical point they settle to a constant or not.', '0801.3238-1-29-4': 'If they do not, one is led to search for a better value of [MATH] and solve ([REF]) with the correct value of the dynamic exponent.', '0801.3238-1-29-5': 'Note that the corrections come from the irrelevant/lattice contributions and the Casimir-like term for the gap is actually the constant term in [MATH].', '0801.3238-1-29-6': 'For quantum systems in [MATH], CFT ([MATH]) predicts [MATH], where [MATH] is the scaling dimension of the operator [MATH] that generates the excited state [MATH].', '0801.3238-1-29-7': 'In addition [MATH] can be computed in the framework of perturbed CFT, in the regime [MATH] when a relevant operator [MATH] with scaling dimension [MATH] is added to the critical field theory.', '0801.3238-1-29-8': 'For instance, from Eqs. (7) and (10) of Ref. [CITATION] we have [MATH] where [MATH] is the structure constant that appears as prefactor of the three points correlation function [MATH].', '0801.3238-1-30-0': 'Now, in general the PRG method gives [MATH] with [MATH] [CITATION].', '0801.3238-1-30-1': 'However, it may happen that [MATH] in the scaling function (examples are discussed in Subsec. [REF] and at the end of Subsec. [REF] for a specific excited state of the XY model).', '0801.3238-1-30-2': 'In this case, the shift exponent in such a case decreases to [MATH].', '0801.3238-1-30-3': 'Nonetheless there is a way to improve this behavior.', '0801.3238-1-30-4': 'In fact, the "extremum" (instead of the zeroes) of the quantity in left side of Eq. ([REF]) with the ansatz ([REF]) is located exactly at the critical point [MATH].', '0801.3238-1-30-5': 'In order to appreciate a shift from criticality we have to include higher orders of [MATH] in the irrelevant non-scaling term, i.e. [EQUATION].', '0801.3238-1-30-6': 'In this case, the convergence is [MATH] with [MATH], which is better than usual PRG not only because of the double exponent but also thanks to the coefficient in front: it is proportional to [MATH] which is usually a small quantity.', '0801.3238-1-30-7': 'It is also worth noticing that in this case [MATH].', '0801.3238-1-31-0': '# Testing the methods', '0801.3238-1-32-0': '## XY model in transverse field', '0801.3238-1-33-0': 'To check our methods analytically, we consider the 1D spin-1/2 XY model given by[EQUATION]', '0801.3238-1-33-1': 'Throughout the paper we will consider [MATH] even and PBC.', '0801.3238-1-33-2': 'This model can be solved exactly [CITATION] by means of a Jordan-Wigner transformation from spins to spinless fermions followed by a Bogoliubov transformation to arrive at a Hamiltonian of free quasiparticles.', '0801.3238-1-33-3': 'The number of original fermions [MATH], is not a conserved quantity, but its parity [MATH] corresponds to a [MATH]-rotation around the [MATH]-axis, and therefore is conserved.', '0801.3238-1-33-4': 'One should beware of a delicate issue concerning boundary conditions.', '0801.3238-1-33-5': 'Starting with PBC in Eq. ([REF]), the fermionic Hamiltonian turns to have PBC in the sector of odd parity [MATH].', '0801.3238-1-33-6': 'Instead in the even parity sector [MATH] - which comprises the ground state - antiperiodic boundary conditions must be used.', '0801.3238-1-33-7': 'In this sector the model becomes[EQUATION] where [MATH] are Bogoliubov quasiparticles, [MATH] ranges in the first Brillouin zone and the dispersion relation is[EQUATION]', '0801.3238-1-33-8': 'This model displays an Ising transition at [MATH] with exponents [MATH], which means that the scaling variable is [MATH].', '0801.3238-1-33-9': 'To test our ideas we need to calculate, for finite [MATH], the GS energy [MATH] and the average potential [MATH], in the quasi-critical region given by [MATH].', '0801.3238-1-33-10': 'According to Eq. [REF] the GS is given by [MATH] with [MATH].', '0801.3238-1-33-11': 'We then expand the argument of the sum up to the desired order in [MATH].', '0801.3238-1-33-12': 'For our purposes we need [MATH] up to [MATH] and [MATH] up to [MATH].', '0801.3238-1-33-13': 'The resulting sums are then evaluated with the aid of Euler-Maclaurin formula [CITATION].', '0801.3238-1-33-14': 'The results and the details are given in the Appendix [REF].', '0801.3238-1-33-15': 'From Eqs. [REF] and [REF], one sees explicitly that terms of order [MATH] are absent both from [MATH] and [MATH] so that according to our definition Eq. [REF] we find [MATH].', '0801.3238-1-33-16': 'In passing we notice that, from Eq. ([REF]), the Casimir-like term is [MATH] consistent with the CFT formula [MATH].', '0801.3238-1-33-17': 'In fact the central charge is [MATH] and, from the dispersion relation ([REF]) the spin velocity turns out to be [MATH].', '0801.3238-1-34-0': 'Now, using Eqs. [REF] and [REF], we have all the elements to derive the sequences of pseudocritical points analytically.', '0801.3238-1-34-1': 'As far as the FSCM is concerned, the pseudo critical points are obtained imposing [MATH], or formally [MATH].', '0801.3238-1-34-2': 'Up to leading order in [MATH] the solution is [EQUATION] as already obtained in Ref. [CITATION] for the Ising model [MATH].', '0801.3238-1-34-3': 'This shows explicitly that [MATH] consistent with our prediction [MATH] (remind that [MATH]).', '0801.3238-1-35-0': 'For what concerns the "balancing trick" discussed in Sec. [REF], we can show that the solution of Eq. ([REF]), is given by [MATH].', '0801.3238-1-35-1': 'The pseudocritical points are then given imposing [MATH].', '0801.3238-1-35-2': 'In this case, at leading order, the solution is [EQUATION] which means [MATH] once again consistent with our prediction [MATH] (remind that [MATH]).', '0801.3238-1-35-3': 'From a numerical point of view it is more profitable to use the HCM.', '0801.3238-1-35-4': 'From [MATH] we obtain [EQUATION] with the same exponent [MATH].', '0801.3238-1-36-0': 'We pass now to treat the PRG method for which knowledge of the lowest gap is required.', '0801.3238-1-36-1': 'The first excited state belongs to the sector with odd parity.', '0801.3238-1-36-2': 'Correspondingly, the finite size gap is given, besides a constant term, by the difference between two Riemann sums where the sampling is taken over odd and even (in units of [MATH]) wavenumbers respectively.', '0801.3238-1-36-3': 'In analogy with Ref. [CITATION] the lowest gap can be eventually written as [MATH], where [MATH].', '0801.3238-1-36-4': 'Again we refer to the Appendix for the details.', '0801.3238-1-36-5': 'Using the form of the gap Eq. [REF] we can evaluate the various terms of the scaling function in Eq. [REF][EQUATION]', '0801.3238-1-36-6': 'Hence [MATH], consistently with Ref. [CITATION] for [MATH] and with the known result [MATH] for the first excited state of the [MATH] minimal model [CITATION].', '0801.3238-1-36-7': 'Moreover, since [MATH], according to the general analysis in Subsec. [REF], there is no advantage in searching the minima or maxima in [MATH] of the left hand side of Eq. ([REF]).', '0801.3238-1-36-8': 'Looking for the zeros by imposing the PRG equation [MATH], yields [EQUATION].', '0801.3238-1-36-9': 'This result for generic [MATH], shows explicitly that the PRG shift exponent is [MATH] and proves a conjecture put forward by Hamer and Barber in Ref. [CITATION] for the Ising model.', '0801.3238-1-36-10': 'From Eq. ([REF]) together with Eq. ([REF]) we read [MATH] as previously, so that the calculated exponent is consistent with the prediction [MATH].', '0801.3238-1-37-0': 'The explicit calculation of the various shift exponents, confirms that the HCM given by Eq. ([REF]) is superior to the standard PRG method.', '0801.3238-1-37-1': 'However, the things are different if we considered the excitation obtained creating a well-defined quasiparticle with the smallest momentum [MATH], instead of the very first excitation gap.', '0801.3238-1-37-2': 'In that case, the energy gap is given by Eq. ([REF]).', '0801.3238-1-37-3': 'Using the method of the extrema of the PRG quantity in the left hand side of Eq. ([REF]), we would have obtained a convergence to the critical point as [MATH], while with the standard PRG only as [MATH].', '0801.3238-1-38-0': '## [MATH] transition (non BKT)', '0801.3238-1-39-0': 'We choose the spin-1 [MATH] model on a chain ([MATH]) with PBC[EQUATION] because the transition from the Haldane phase to the phase at large [MATH] is described by a [MATH] CFT ([MATH]) with continuously varying exponents.', '0801.3238-1-39-1': 'The Hamiltonian ([REF]) has been used to describe the magnetic properties of different quasi 1D compounds (see [CITATION] for a brief account).', '0801.3238-1-39-2': 'Here, by fixing [MATH] and [MATH], for which it has been already estimated [MATH] [CITATION], we wish to test the methods described above in a severe case in which a [MATH] scaling would give sublinear convergence in [MATH].', '0801.3238-1-39-3': 'Indeed for the FSCM we expect [MATH].', '0801.3238-1-39-4': 'The previous estimate using the log-log plots of the finite-size gaps was [MATH] [CITATION], while in Ref. [CITATION] it is found [MATH] using the method of level crossing with antiperiodic boundary conditions, which is however specific to this transition.', '0801.3238-1-39-5': 'The first irrelevant operator allowed by the lattice symmetries has scaling dimension [MATH] where [MATH] so that [MATH] where with [MATH] we denote the smallest exponent of corrections arising from lattice contributions at finite [MATH] that are not captured in the framework of the (relativistic) continuum theory.', '0801.3238-1-40-0': 'As discussed in Refs. [CITATION] this is a case in which the linear term [MATH] in the scaling function of the PRG vanishes because [MATH] for the sine-Gordon model, the effective field theory that describe the surroundings of the [MATH] line.', '0801.3238-1-40-1': 'So it is convenient to use also our improved version of the PRG as discussed in Subsec. [REF].', '0801.3238-1-40-2': 'The expected shift exponent is [MATH].', '0801.3238-1-41-0': 'In order to have an idea of the range of values of [MATH] to be used let us imagine that [MATH] so that [MATH].', '0801.3238-1-41-1': 'With [MATH] this exponent leads to variations in the pseudocritical points [MATH] smaller than [MATH].', '0801.3238-1-41-2': 'Hence we prefer to illustrate the method with virtually exact numerical data obtained with the Lanczos algorithm using [MATH].', '0801.3238-1-41-3': 'The reason is that the DMRG would give rather accurate values for the energies but the estimates for [MATH] could not be sufficiently precise to appreciate the variations in [MATH] obtained from the crossings.', '0801.3238-1-41-4': 'So we use the DMRG only to extend the data to [MATH] with [MATH] optimized states.', '0801.3238-1-41-5': 'In any case the GS belongs to the [MATH] sector.', '0801.3238-1-42-0': 'The FSCM with [MATH] yields [MATH] with [MATH] in reasonable agreement with the CFT expectations.', '0801.3238-1-42-1': 'The high-precision procedure based on Eq. ([REF]) yields [MATH] with [MATH].', '0801.3238-1-42-2': 'The value of the shift exponent is definitely larger than what expected, which could be due to the vanishing of the coefficient of the first irrelevant contribution with scaling dimensions [MATH].', '0801.3238-1-42-3': 'In any case, from Fig. [REF] one can clearly appreciate that the sequence [MATH] converges more slowly than [MATH].', '0801.3238-1-42-4': 'As far as the PRG is concerned, with the standard procedure of finding the zeroes of Eq. ([REF]) we get [MATH] with [MATH].', '0801.3238-1-42-5': 'Instead, using the improved method estimation by looking for the extremal value of the quantity on the left side of ([REF]) we find a rapidly converging sequence that however also oscillates between [MATH] and [MATH].', '0801.3238-1-42-6': 'Finally, a small oscillation in the fourth decimal place about [MATH] is seen also in the sequence [MATH] obtained through the homogeneity criterion ([REF]).', '0801.3238-1-42-7': 'As expected the latter sequence converges in a fashion similar to [MATH].', '0801.3238-1-42-8': 'All these results are summarized in Fig. [REF].', '0801.3238-1-43-0': 'The data show that subtracting out the terms that induce a slow convergence of the pseudocritical points, one remains with quantities that inherit a residual (possibly oscillating) [MATH]-dependence from the specific lattice model and that could be very difficult to account for.', '0801.3238-1-43-1': 'Other factors that affect the extrapolation of the critical points at this level of accuracy are the sampling [MATH] and the trade-off between computational accuracy and the maximum available size.', '0801.3238-1-44-0': 'To summarize we observe that our improved methods yield very fast convergence to the critical point that we estimate to be [MATH], consistently with Ref. [CITATION].', '0801.3238-1-45-0': '## [MATH] BKT transition', '0801.3238-1-46-0': 'We consider now the spin-1/2 Heisenberg model with frustration due to next-to-nearest neighbors interaction [EQUATION].', '0801.3238-1-46-1': 'The model is equivalent to a 2-legs zigzag ladder with [MATH] rungs.', '0801.3238-1-46-2': 'The best estimate of the critical point was made by Okamoto and Nomura using a model-specific crossing method [CITATION]; with exact diagonalizations up to [MATH] they determined [MATH].', '0801.3238-1-46-3': 'The model is gapless for [MATH] and has a doubly degenerate GS in the TL for [MATH].', '0801.3238-1-46-4': 'Again the GS has [MATH].', '0801.3238-1-46-5': 'Without exploiting a priori information about the BKT character of transition (if not the value of the dynamic exponent [MATH]), we tested the homogeneity criterion ([REF]) using ladders with PBC of up to [MATH] rungs, an effective [MATH] and [MATH] DMRG states that ensure an accuracy of [MATH] on the values of [MATH].', '0801.3238-1-46-6': 'The results reported in Fig. [REF] are encouraging: while with the FSCM we would get no crossings at all, the zeroes of Eq. ([REF]) yield a sequence of points converging to [MATH].', '0801.3238-1-46-7': 'The main problem comes from the left side of the transition where the truncation DMRG error for [MATH] and [MATH] induces some oscillations on the plotted quantity.', '0801.3238-1-46-8': 'We content ourselves with linear fits in [MATH].', '0801.3238-1-46-9': 'If we exclude the point with [MATH] the fit is better even if we find [MATH]; by selecting all the available points, instead, the fit is visibly worse but the extrapolated value is [MATH], in agreement with Ref. [CITATION].', '0801.3238-1-46-10': 'As above, apart from the details of the extrapolation procedure, we see that the homogeneity criterion provides a viable procedure to locate the critical point in a BKT transition, where almost all existing generic methods fail.', '0801.3238-1-46-11': 'We remark that this analysis is based solely on [MATH], namely an observable evaluated on the GS, without invoking further assumptions on the nature of the excitations.', '0801.3238-1-47-0': '# Conclusions', '0801.3238-1-48-0': 'Making only use of finite-size quantities related to the ground state, we show how to generate sequences of pseudocritical points that converge very fast to the infinite-size critical point.', '0801.3238-1-48-1': 'The convergence is of the form [MATH] with a shift exponent [MATH].', '0801.3238-1-48-2': 'In this article we propose a homogeneity condition method (HCM) which is faster than the standard phenomenological renormalization group (PRG) in locating the critical points.', '0801.3238-1-48-3': 'Moreover its validity is more general as it can be applied without modification to the difficult case of a Berezinskii-Kosterlitz-Thouless transition.', '0801.3238-1-48-4': 'The homogeneity method requires only the knowledge of [MATH], that is, the expectation value of the term that drives the transition.', '0801.3238-1-49-0': 'We also presented an improvement to the PRG method, which allows, under certain conditions, to obtain pseudocritical sequences characterized by the same shift exponent as for the HCM.', '0801.3238-1-49-1': 'However, this modification, relying on a particular form of the gap scaling function, is not valid in general.', '0801.3238-1-49-2': 'It holds true, for instance for the sine-Gordon model that underlies a variety of transitions in [MATH] dimension.', '0801.3238-1-50-0': 'The formulations of the approaches are sufficiently general to be applied in any spatial dimensionality.', '0801.3238-1-50-1': 'Even if we are primarily interested in quantum phase transitions ideally at zero temperature, in principle the methods can be extended to problems of finite-temperature statistical mechanics.', '0801.3238-1-50-2': 'At variance with other accelerating methods found in the literature, e.g. the van den Broeck-Schwartz or the Bulirsch-Stoer ones (reviewed in Ch. 9 of [CITATION]), the procedures presented here rely on the scaling behavior of thermodynamical quantities expected from physical and renormalization arguments.', '0801.3238-1-51-0': 'The validity of the methods has been tested with analytical calculations on the one dimensional XY model in transverse field and numerically on a nontrivial spin-1 chain with anisotropy.', '0801.3238-1-51-1': 'As extreme case, we have shown that the homogeneity condition method provides a satisfactory location of the critical point also in the case of Berezinskii-Kosterlitz-Thouless transitions.', '0801.3238-1-51-2': 'These confirmations motivate us to consider systems in higher spatial dimensionality.', '0801.3238-1-51-3': 'In these cases the numerical data are restricted to smaller system-size and the need for fast-converging pseudo-critical sequences is a prerequisite for the precise location of the critical points.', '0801.3238-1-52-0': 'This work was partially supported by the Italian MiUR through the PRIN grant n. 2005021773.', '0801.3238-1-52-1': 'Numerical calculations were performed on a cluster of machines made available by the Theoretical Group of the Bologna Section of the INFN.', '0801.3238-1-53-0': '#', '0801.3238-1-54-0': 'Here we indicate how to compute the mean energy per site [MATH], and the average potential [MATH], as require id in Subsec. [REF].', '0801.3238-1-54-1': 'Consider for example the energy sum [EQUATION] where [MATH] is given by Eq. ([REF]).', '0801.3238-1-54-2': 'We need to investigate the above sum in the quasi-critical region [MATH] so it is sufficient to expand [MATH] in powers of [MATH][EQUATION]', '0801.3238-1-54-3': 'The resulting sums can be computed using the Euler-Maclaurin formula (see e.g. [CITATION])[EQUATION] valid for a function [MATH] with at least [MATH] continuous derivatives in [MATH].', '0801.3238-1-54-4': 'Here [MATH] are the Bernoulli numbers and the remainder [MATH] depends on [MATH] on [MATH].', '0801.3238-1-54-5': 'Some care must be taken when the function [MATH] diverges at the border of the Brillouin zone, in this case one must keep [MATH] and [MATH] away from the borders.', '0801.3238-1-54-6': 'Moreover, sending [MATH] to infinity in Eq. [REF], some sums must be regularized using a Borel summation technique.', '0801.3238-1-54-7': 'The final result for the ground state energy is[EQUATION] where [MATH] is the Euler-Mascheroni constant and the thermodynamic values are given by [EQUATION]', '0801.3238-1-54-8': 'Using similar procedures we obtain the following expression for average potential [MATH]:[EQUATION]', '0801.3238-1-54-9': 'Finally, the sum [EQUATION] for the evaluation of the finite-size gap in the PRG method, can be treated along similar lines.', '0801.3238-1-54-10': 'The final result for the gap is[EQUATION]'}

{'0801.3238-2-0-0': 'We analyze in detail, beyond the usual scaling hypothesis, the finite-size convergence of static quantities toward the thermodynamic limit.', '0801.3238-2-0-1': 'In this way we are able to obtain sequences of pseudo-critical points which display a faster convergence rate as compared to currently used methods.', '0801.3238-2-0-2': 'The approaches are valid in any spatial dimension and for any value of the dynamic exponent.', '0801.3238-2-0-3': 'We demonstrate the effectiveness of our methods both analytically on the basis of the one dimensional XY model, and numerically considering [MATH] transitions occurring in non integrable spin models.', '0801.3238-2-0-4': 'In particular, we show that these general methods are able to locate precisely the onset of the Berezinskii-Kosterlitz-Thouless transition making only use of ground-state properties on relatively small systems.', '0801.3238-2-1-0': '# Introduction', '0801.3238-2-2-0': 'In the study of physical properties of phase transitions a basic prerequisite is a reliable method to locate the critical point, whenever the latter is not known a priori from symmetry or duality arguments.', '0801.3238-2-2-1': 'Typically, in numerical or even experimental studies on finite samples one obtains a sequence of pseudocritical points (in the sense specified below) to be extrapolated to the true critical point in the thermodynamic limit (TL).', '0801.3238-2-2-2': 'The extrapolation may be done with some polynomial fit in the inverse size of the system or, better, exploiting some fitting function derived on the basis of a scaling ansatz or through the renormalization group (RG).', '0801.3238-2-2-3': 'The point is especially relevant in the context of quantum phase transitions (QPT) [CITATION] in lattice systems where the exponential growth of the dimension of the Hilbert space with the number of sites is a strong limitation on the accessible sizes with the current computational power and algorithms.', '0801.3238-2-2-4': 'One of the most used algorithms is still the Lanczos method for the virtually exact extraction of the low-lying energy levels; in the most favorable case of spin-1/2 models one cannot go beyond some tens of sites.', '0801.3238-2-2-5': 'This limit can be moved to maybe a few thousands of sites using the so-called density matrix renormalization group (DMRG) [CITATION] that has become the method of choice for 1D problems due to its high level of accuracy.', '0801.3238-2-2-6': 'Nonetheless, if one considers two or even three dimensional systems the situation is much worse: with the Lanczos algorithm the largest lattices have only a few sites of linear extension and the DMRG is not particularly efficient.', '0801.3238-2-2-7': 'At present, the only other choice is Quantum Monte Carlo (QMC) (see, for instance, [CITATION]) that, however, suffers from a sign problem in the case of fermionic or frustrated systems and does not reach the level of accuracy of the DMRG.', '0801.3238-2-2-8': 'Very recently there have been attempts to exploit both DMRG-like features and the QMC sampling tricks, to design hybrid methods [CITATION] that are however still under verification.', '0801.3238-2-3-0': 'It is generally believed that a sequence of pseudocritical points, for example the loci of maxima of finite-size susceptibilities, converges to the critical point as a power law [MATH], with a so-called shift exponent [MATH] given by the inverse of the correlation length exponent [MATH].', '0801.3238-2-3-1': 'Hence, generally speaking, the larger is [MATH] the slower is the convergence.', '0801.3238-2-3-2': 'This difficulty reaches its maximum for Berezinskii-Kosterlitz-Thouless (BKT) transitions, in which the correlation length diverges with an essential singularity or, loosely speaking, "[MATH]".', '0801.3238-2-3-3': 'However, already in the seminal paper by Fisher and Barber [CITATION], it was pointed out that the relation [MATH] is not always valid and [MATH] depends, among other factors, on the boundary conditions.', '0801.3238-2-4-0': 'The most used method to locate quantum critical points in [MATH] by means of finite-size data is the so-called Phenomenological Renormalization Group (PRG), reviewed for instance in [CITATION].', '0801.3238-2-4-1': 'Another convenient approach, the Finite-size Crossing Method (FSCM), was recently proposed in [CITATION].', '0801.3238-2-4-2': 'The aim of this paper is to improve both of them by means of criteria that produce sequences of pseudocritical points that converge more rapidly.', '0801.3238-2-4-3': 'We will show that in our sequences the shift exponent will have the form [MATH], where [MATH] and/or [MATH] are larger than the corresponding values in the usual methods and therefore allow for a better convergence.', '0801.3238-2-5-0': 'The paper is organized as follows.', '0801.3238-2-5-1': 'In Section [REF] we illustrate the general arguments leading to the enhanced sequences, both in the framework of the FSCM and of the PRG (Subsec. [REF]).', '0801.3238-2-5-2': 'Special cases as the BKT transition (Subsec. [REF]) and that of logarithmic divergences (Subsec. [REF]) are discussed separately.', '0801.3238-2-5-3': 'In Section [REF] we illustrate the usefulness of the methods on the hand of analytic and numerical tests.', '0801.3238-2-5-4': 'In Subsec. [REF] we treat the XY spin-1/2 chain, using a series of exact calculations reported in the Appendix.', '0801.3238-2-5-5': 'Then we move to two cases of spin chains for which no exact solution is available: in Subsec. [REF] we consider a spin-1 model with anisotropies in a parameter range that gives rise to a large value of [MATH] and in Subsec. [REF] we study the spin-1/2 model with next to nearest neighbor interactions that is known to undergo a BKT transition.', '0801.3238-2-5-6': 'In this case we find a value for the critical coupling in agreement with the accepted one, which was found using a model-specific investigation of the excited states [CITATION].', '0801.3238-2-5-7': 'Section [REF] is devoted to conclusions.', '0801.3238-2-6-0': '# Derivation of rapidly-converging sequences', '0801.3238-2-7-0': 'We consider systems in [MATH] spatial dimensions of linear size [MATH] and periodic boundary conditions (PBC).', '0801.3238-2-7-1': 'Let the transition be driven by a linear parameter [MATH] such that the Hamiltonian is', '0801.3238-2-8-0': '[EQUATION].', '0801.3238-2-9-0': 'Dealing with QPT we consider the case of strictly zero temperature, [MATH], even if the arguments presented below can be simply extended to the finite-temperature case, replacing the parameter [MATH] with [MATH] (and without using the dimensional crossover rule used below).', '0801.3238-2-9-1': 'The free energy density reduces to the ground-state (GS) energy density which, close to the critical point [MATH], shows a singularity in the second (or higher) derivatives with respect to [MATH]:', '0801.3238-2-10-0': '[EQUATION] where [MATH] is the correlation length.', '0801.3238-2-10-1': 'Note that, as a consequence of the scaling hypothesis, the singular part of the energy [MATH] is a universal quantity that depends only on [MATH], the relevant length scale close to the critical point.', '0801.3238-2-10-2': 'Hence, [MATH] may be considered quite in general an even function of [MATH] that vanishes at the critical point.', '0801.3238-2-11-0': 'On the other hand, the bulk energy density at the critical point behaves as (Privman-Fisher hypothesis) [EQUATION] where [MATH] is a sort of Casimir-like term that may depend on the actual geometry of the lattice.', '0801.3238-2-11-1': 'Note that this hypothesis has to be changed properly if one or more of the spatial dimensions are of infinite extent.', '0801.3238-2-11-2': 'Moreover, Eq. ([REF]) has been written in analogy with Eq. (11.29) of ref. [CITATION] using the dimensional crossover rule according to which the partition function and the thermodynamic (static) properties of a [MATH]-dimensional quantum system are equivalent to those of a [MATH])-dimensional classical counterpart [CITATION], where [MATH] is the dynamic exponent [CITATION].', '0801.3238-2-11-3': 'Then, for the implementation of our methods we need to know by some other means the value of [MATH] relating the energy gap [MATH] and the correlation length [MATH]: [MATH].', '0801.3238-2-11-4': 'Typically, but not always, energy and momentum in the continuum limit at the critical point satisfy a linear dispersion relation, [MATH], for small [MATH] so that [MATH] and a relativistic effective field theory can be used to describe the universal features of the transition.', '0801.3238-2-11-5': 'In [MATH] the scale invariance at the critical point is often sufficient to imply also conformal invariance (see Ch. 2 of [CITATION]), thanks to which several exact results can be obtained using the powerful predictions of conformal field theories (CFT).', '0801.3238-2-11-6': 'For example, by mapping the space-time complex plane onto a cylinder whose circumference represents the finite chain of length [MATH] we can identify [MATH] in Eq. ([REF]) where [MATH] is the central charge of the theory.', '0801.3238-2-11-7': 'In the RG sense, moving away from criticality corresponds to perturbing the CFT with a relevant operator that destroys conformal invariance.', '0801.3238-2-11-8': 'However, this is not the only effect of varying the microscopic parameter [MATH] out of [MATH]: in general also the speed of elementary excitations gets renormalized in the unperturbed CFT part.', '0801.3238-2-11-9': 'For this reason we say that [MATH] (henceforth [MATH]) depends on [MATH], in the vicinity of [MATH].', '0801.3238-2-12-0': 'Scaling and dimensional arguments imply that, in the thermodynamic, off-critical regime [MATH], the singular part of the energy behaves as[EQUATION] with [MATH].', '0801.3238-2-12-1': 'For a second order phase transition [MATH].', '0801.3238-2-13-0': 'After introducing the scaling variable [MATH], the finite-size scaling (FSS) theory asserts [CITATION] that in a system of length [MATH], [EQUATION] where [MATH] is a universal function that, in the off-critical regime [MATH], must behave as [MATH] in order to recover Eq. ([REF]) .', '0801.3238-2-13-1': 'Instead, for [MATH] we are in the critical regime and [MATH] behaves as an analytic function that vanishes for [MATH].', '0801.3238-2-13-2': 'Here we assume that the leading term in [MATH] is quadratic in [MATH] (see below why it cannot be linear), but the following arguments are easily generalizable to higher integer powers.', '0801.3238-2-13-3': 'The constant term [MATH] is already adsorbed in the nonuniversal part of the energy density at [MATH] as shown in Eq. ([REF]).', '0801.3238-2-14-0': 'Differentiating [MATH] with respect to [MATH], gives the mean value [MATH], whose singular part [MATH] behaves as[EQUATION]', '0801.3238-2-14-1': 'Considering FSS for the combination of Eqs. ([REF]) and ([REF]) and then differentiating we find[EQUATION] where the subscript "[MATH], reg" hereafter means regular in the TL.', '0801.3238-2-14-2': 'In order to write down the expression above we used [MATH] and assumed that the powers neglected in the last term are just larger than [MATH].', '0801.3238-2-14-3': 'To illustrate this point we could consider the irrelevant operator with the smallest scaling dimension [MATH].', '0801.3238-2-14-4': 'At first order in perturbation theory with the renormalized coupling [MATH] the corrections to the GS energy density are of the form [MATH] so that [MATH].', '0801.3238-2-14-5': 'Note also that the amplitude [MATH] can vanish and [MATH] terms have to be included.', '0801.3238-2-14-6': 'For these and more details we leave the reader to Ref. [CITATION].', '0801.3238-2-14-7': 'Generically we admit corrections with [MATH] that may come either from irrelevant operator in the continuum theory or from lattice effects.', '0801.3238-2-14-8': 'The case [MATH] corresponds to marginal perturbations and typically leads to logarithmic corrections.', '0801.3238-2-14-9': 'Notice that now the leading term of [MATH] is linear in [MATH] in the critical region.', '0801.3238-2-14-10': 'If we would have admitted a linear term in [MATH] then [MATH] and a finite jump discontinuity at finite [MATH] would be present in [MATH].', '0801.3238-2-15-0': 'We can also calculate [MATH], yielding to a singular part that is similar to Eq. ([REF]) but with a changed sign.', '0801.3238-2-15-1': 'In fact, for [MATH] the leading singular parts of [MATH] and [MATH] must cancel in the sum that gives back the energy [MATH], which does not contain that singularity.', '0801.3238-2-15-2': 'In particular, the scaling is[EQUATION]', '0801.3238-2-15-3': 'The FSCM [CITATION] identifies the critical point with the limit of the sequence [MATH] of single crossing points [EQUATION] with [MATH].', '0801.3238-2-15-4': 'Applying this criterion to Eq. ([REF]), we obtain (for [MATH]) [EQUATION]', '0801.3238-2-15-5': 'This equation defines the shift exponent [MATH] and may converge very slowly when [MATH].', '0801.3238-2-15-6': 'The extremely difficult case is the BKT transition where formally [MATH], but this latter situation must be treated in a different way (see Subsec. [REF]).', '0801.3238-2-16-0': 'Now, consider the quantity [MATH] and suppose to be able to tune [MATH] exactly at [EQUATION]', '0801.3238-2-16-1': 'It easily seen that [MATH] does not contain the Casimir-like term responsible for the critical point shift.', '0801.3238-2-16-2': 'In fact, the scaling of [MATH] is When [MATH] is equal to [MATH] given in Eq. ([REF]), the critical point found by this crossing method is approached as [MATH], with a shift exponent [MATH] (again this holds true provided that [MATH]).', '0801.3238-2-16-3': 'The additional term [MATH] allows, in general, for a better convergence of the sequence [MATH].', '0801.3238-2-17-0': 'A possible algorithm for finding numerically the critical point in such a way is the following.', '0801.3238-2-17-1': 'If [MATH] is at a crossing point, of [MATH], then we have [EQUATION].', '0801.3238-2-17-2': 'Notice that putting [MATH] is equivalent to the FSCM applied to [MATH] and the denominator has definite sign about the critical point.', '0801.3238-2-17-3': 'Now we find [MATH] requiring that [MATH], i.e. [EQUATION] or in the continuum version [EQUATION]', '0801.3238-2-17-4': 'Calling [MATH] and rewriting only the essential terms in the scaling ansatze we have the simplified forms [EQUATION]', '0801.3238-2-17-5': 'Now, putting these two relations in ([REF]) we obtain [EQUATION] that gives a shift exponent [MATH] as anticipated.', '0801.3238-2-18-0': 'The main result of this section is the crossing criterion ([REF]) that identifies the rapidly converging sequence ([REF]) to the critical value.', '0801.3238-2-19-0': '## Homogeneity condition', '0801.3238-2-20-0': 'In the previous section we have shown how to obtain a sequence of pseudocritical points with an improved shift exponent [MATH].', '0801.3238-2-20-1': 'Here we provide another, yet simpler equation for the determination of the critical point.', '0801.3238-2-20-2': 'The resulting pseudocritical sequence is characterized by the same shift exponent [MATH] .', '0801.3238-2-20-3': 'However in this case we are able to prove convergence toward the critical point even in the extreme case of a BKT transition (see Subsec. [REF]).', '0801.3238-2-21-0': 'The idea is to require that at the critical point the [MATH]-dependent part of [MATH] is dominated by the Casimir-like term with power [MATH] (see Eq. ([REF])).', '0801.3238-2-21-1': 'This condition is translated into the requirement that the [MATH]-derivative of [MATH] is a homogeneous function of degree [MATH], i.e. [EQUATION]', '0801.3238-2-21-2': 'Consequently, the corresponding sequence of pseudocritical points [MATH] scales as[EQUATION]', '0801.3238-2-21-3': 'The equation ([REF]) represents the homogeneity condition method (HCM) that we are proposing for the efficient location of critical points.', '0801.3238-2-21-4': 'We stress here that, being [MATH] a GS property, this criterion does not require knowledge of excited states as in the case of the PRG method.', '0801.3238-2-21-5': 'This is a point in favor to the HCM since excited states are typically assessed with less numerical accuracy.', '0801.3238-2-21-6': 'In addition, the HCM is superior to the PRG in that it produces a faster converging sequence (see Subsec. [REF]).', '0801.3238-2-22-0': '## Case [MATH] with logarithmic divergences', '0801.3238-2-23-0': 'For completeness we consider the case [MATH] that was excluded in the previous treatment.', '0801.3238-2-23-1': 'In this situation, the ansatz requires the inclusion of logarithmic corrections [EQUATION].', '0801.3238-2-23-2': 'The calculation of the critical point with the FSCM gives in this case', '0801.3238-2-24-0': '[EQUATION] whereas the HCM Eq. ([REF]) yields [EQUATION]', '0801.3238-2-24-1': 'These results are compatible with the exact calculations of the XY model (see Subsec. [REF]).', '0801.3238-2-24-2': 'Note that in general, we should to perform a similar calculation for [EQUATION] namely when the [MATH]-th derivative of the free energy diverges logarithmically.', '0801.3238-2-25-0': '## A scaling ansatz for the BKT case', '0801.3238-2-26-0': 'For [MATH] at the BKT the correlation function in the TL behaves like [MATH].', '0801.3238-2-26-1': 'In the typical example of the classical two dimensional XY model it is known that [MATH].', '0801.3238-2-26-2': 'Instead, for the quantum Heisenberg model with frustration (which we will consider in Subsec. [REF]) Haldane suggested [MATH] [CITATION].', '0801.3238-2-26-3': 'We also set [MATH] because the (effective) dimensionality in the BKT scenario is two.', '0801.3238-2-26-4': 'The singular part of the finite-size energy density now is conveniently expressed in terms of [MATH] [CITATION] so that [EQUATION] where [MATH] is a universal function that, in the off-critical regime [MATH], must behave as [MATH].', '0801.3238-2-26-5': 'Again, in the quasicritical regime [MATH] at any finite [MATH] the energy density and its derivatives must be analytic in [MATH].', '0801.3238-2-26-6': 'The value [MATH] can be absorbed in [MATH] and it can be checked directly that the first contribution has to be at least quadratic in [MATH] because otherwise a finite-size discontinuity in [MATH] would be generated.', '0801.3238-2-26-7': 'For [MATH] we adopt the following ansatz (justified from perturbed conformal field theory [CITATION]): [EQUATION] with [MATH] a constant and [MATH] an integer larger than 1 ([MATH] from Eq (22) in [CITATION]).', '0801.3238-2-26-8': 'Hence Now we want to get rid of all the [MATH] contributions that "hinder" the location of finite-size pseudocritical points.', '0801.3238-2-26-9': 'Hence we first differentiate with respect to [MATH] to eliminate [MATH], then multiply by [MATH] to isolate the term in square brackets in Eq.([REF]) and finally set to zero a further difference in [MATH] in order to drop [MATH].', '0801.3238-2-26-10': 'Formally, in the region [MATH], we can write down the condition [EQUATION]', '0801.3238-2-26-11': 'It is worth noticing that the latter condition is equal to the HCM Eq. ([REF]) when [MATH].', '0801.3238-2-26-12': 'Treating [MATH] as a continuous variable one can read off the shift exponent for the sequence [MATH] that turns out to be [MATH]; if [MATH] then the corrections to scaling are also governed by (another) marginal operator and we expect [MATH] with [MATH] a positive integer ([MATH] from Eq. (22) in [CITATION]).', '0801.3238-2-27-0': 'Dealing with numerical simulations it is very important to specify how one implements the finite-size differences in [MATH].', '0801.3238-2-27-1': 'In fact, there are several finite-difference expressions used in the literature to express the derivatives and here the requirement is that they all reproduce Eq. ([REF]) in the limit [MATH].', '0801.3238-2-27-2': 'For example, if one takes a uniform step [MATH] then the following symmetric expression can be built[EQUATION] and then the precise condition to cancel the term [MATH] becomes[EQUATION]', '0801.3238-2-27-3': 'In the limit of large [MATH] we recover ([REF]) as required.', '0801.3238-2-27-4': 'Clearly, the correct discretization prescription must be identified not only for the BKT; when [MATH] and/or [MATH] are not one, the suitable variant of ([REF]) for finite [MATH] has to be adopted, with [MATH] weighted by [MATH] and [MATH] weighted by a polynomial in [MATH] of degree [MATH] whose coefficients depend on [MATH].', '0801.3238-2-28-0': '## PRG revisited', '0801.3238-2-29-0': 'The PRG method identifies the critical point with the limit of the sequence [MATH] of crossing points satisfying[EQUATION]', '0801.3238-2-29-1': 'Here [MATH] is the finite-size energy gap of the spectrum for which we may adopt the following form [EQUATION] with [MATH] and [MATH] a prefactor depending on the excited state [MATH] we are considering.', '0801.3238-2-29-2': 'The standard PRG approach in [MATH] relies upon Eq. ([REF]) with [MATH].', '0801.3238-2-29-3': 'More generally, a first test to identify [MATH] is done by plotting the usual scaled gaps [MATH] and see if at the critical point they settle to a constant or not.', '0801.3238-2-29-4': 'If they do not, one is led to search for a better value of [MATH] and solve ([REF]) with the correct value of the dynamic exponent.', '0801.3238-2-29-5': 'Note that the corrections come from the irrelevant/lattice contributions and the Casimir-like term for the gap is actually the constant term in [MATH].', '0801.3238-2-29-6': 'For quantum systems in [MATH], CFT ([MATH]) predicts [MATH], where [MATH] is the scaling dimension of the operator [MATH] that generates the excited state [MATH].', '0801.3238-2-29-7': 'In addition [MATH] can be computed in the framework of perturbed CFT, in the regime [MATH] when a relevant operator [MATH] with scaling dimension [MATH] is added to the critical field theory.', '0801.3238-2-29-8': 'For instance, from Eqs. (7) and (10) of Ref. [CITATION] we have [MATH] where [MATH] is the structure constant that appears as prefactor of the three points correlation function [MATH].', '0801.3238-2-30-0': 'Now, in general the PRG method gives [MATH] with [MATH] [CITATION].', '0801.3238-2-30-1': 'However, it may happen that [MATH] in the scaling function (examples are discussed in Subsec. [REF] and at the end of Subsec. [REF] for a specific excited state of the XY model).', '0801.3238-2-30-2': 'The shift exponent in such a case decreases to [MATH].', '0801.3238-2-30-3': 'Nonetheless there is a way to improve this behavior.', '0801.3238-2-30-4': 'In fact, the "extremum" (instead of the zeroes) of the quantity in left side of Eq. ([REF]) with the ansatz ([REF]) is located exactly at the critical point [MATH].', '0801.3238-2-30-5': 'In order to appreciate a shift from criticality we have to include higher orders of [MATH] in the irrelevant non-scaling term, i.e. [EQUATION].', '0801.3238-2-30-6': 'In this case, the convergence is [MATH] with [MATH], which is better than usual PRG not only because of the double exponent but also thanks to the coefficient in front: it is proportional to [MATH] which is usually a small quantity.', '0801.3238-2-30-7': 'It is also worth noticing that in this case [MATH].', '0801.3238-2-31-0': '# Testing the methods', '0801.3238-2-32-0': '## XY model in transverse field', '0801.3238-2-33-0': 'To check our methods analytically, we consider the 1D spin-1/2 XY model given by[EQUATION]', '0801.3238-2-33-1': 'Throughout the paper we will consider [MATH] even and PBC.', '0801.3238-2-33-2': 'This model can be solved exactly [CITATION] by means of a Jordan-Wigner transformation from spins to spinless fermions followed by a Bogoliubov transformation to arrive at a Hamiltonian of free quasiparticles.', '0801.3238-2-33-3': 'The number of original fermions [MATH], is not a conserved quantity, but its parity [MATH] corresponds to a [MATH]-rotation around the [MATH]-axis, and therefore is conserved.', '0801.3238-2-33-4': 'One should beware of a delicate issue concerning boundary conditions.', '0801.3238-2-33-5': 'Starting with PBC in Eq. ([REF]), the fermionic Hamiltonian turns to have PBC in the sector of odd parity [MATH].', '0801.3238-2-33-6': 'Instead in the even parity sector [MATH] - which comprises the ground state - antiperiodic boundary conditions must be used.', '0801.3238-2-33-7': 'In this sector the model becomes[EQUATION] where [MATH] are Bogoliubov quasiparticles, [MATH] ranges in the first Brillouin zone and the dispersion relation is[EQUATION]', '0801.3238-2-33-8': 'For [MATH] this model displays an Ising transition at [MATH] with exponents [MATH], which means that the scaling variable is [MATH].', '0801.3238-2-33-9': 'To test our ideas we need to calculate, for finite [MATH], the GS energy [MATH] and the average potential [MATH], in the quasi-critical region given by [MATH].', '0801.3238-2-33-10': 'According to Eq. [REF] the GS is given by [MATH] with [MATH].', '0801.3238-2-33-11': 'We then expand the argument of the sum up to the desired order in [MATH].', '0801.3238-2-33-12': 'For our purposes we need [MATH] up to [MATH] and [MATH] up to [MATH].', '0801.3238-2-33-13': 'The resulting sums are then evaluated with the aid of Euler-Maclaurin formula [CITATION].', '0801.3238-2-33-14': 'The results and the details are given in the Appendix [REF].', '0801.3238-2-33-15': 'From Eqs. [REF] and [REF], one sees explicitly that terms of order [MATH] are absent both from [MATH] and [MATH] so that according to our definition Eq. [REF] we find [MATH].', '0801.3238-2-33-16': 'In passing we notice that, from Eq. ([REF]), the Casimir-like term is [MATH] consistent with the CFT formula [MATH].', '0801.3238-2-33-17': 'In fact the central charge is [MATH] and, from the dispersion relation ([REF]) the spin velocity turns out to be [MATH].', '0801.3238-2-34-0': 'Now, using Eqs. [REF] and [REF], we have all the elements to derive the sequences of pseudocritical points analytically.', '0801.3238-2-34-1': 'As far as the FSCM is concerned, the pseudo critical points are obtained imposing [MATH], or formally [MATH].', '0801.3238-2-34-2': 'Up to leading order in [MATH] the solution is [EQUATION] as already obtained in Ref. [CITATION] for the Ising model [MATH].', '0801.3238-2-34-3': 'This shows explicitly that [MATH] consistent with our prediction [MATH] (remind that [MATH]).', '0801.3238-2-35-0': 'For what concerns the "balancing trick" discussed in Sec. [REF], we can show that the solution of Eq. ([REF]), is given by [MATH].', '0801.3238-2-35-1': 'The pseudocritical points are then given imposing [MATH].', '0801.3238-2-35-2': 'In this case, at leading order, the solution is [EQUATION] which means [MATH] once again consistent with our prediction [MATH] (remind that [MATH]).', '0801.3238-2-35-3': 'From a numerical point of view it is more profitable to use the HCM.', '0801.3238-2-35-4': 'From [MATH] we obtain [EQUATION] with the same exponent [MATH].', '0801.3238-2-36-0': 'Now we proceed to discuss the PRG method for which knowledge of the lowest gap is required.', '0801.3238-2-36-1': 'The first excited state belongs to the sector with odd parity.', '0801.3238-2-36-2': 'Correspondingly, the finite size gap is given, besides a constant term, by the difference between two Riemann sums where the sampling is taken over odd and even (in units of [MATH]) wavenumbers respectively.', '0801.3238-2-36-3': 'In analogy with Ref. [CITATION] the lowest gap can be eventually written as [MATH], where [MATH].', '0801.3238-2-36-4': 'Again we refer to the Appendix for the details.', '0801.3238-2-36-5': 'Using the form of the gap Eq. [REF] we can evaluate the various terms of the scaling function in Eq. [REF][EQUATION]', '0801.3238-2-36-6': 'Hence [MATH], consistently with Ref. [CITATION] for [MATH] and with the known result [MATH] for the first excited state of the [MATH] minimal model [CITATION].', '0801.3238-2-36-7': 'Moreover, since [MATH], according to the general analysis in Subsec. [REF], there is no advantage in searching the minima or maxima in [MATH] of the left hand side of Eq. ([REF]).', '0801.3238-2-36-8': 'Looking for the zeros by imposing the PRG equation [MATH], yields [EQUATION].', '0801.3238-2-36-9': 'This result for generic [MATH], shows explicitly that the PRG shift exponent is [MATH] and proves a conjecture put forward by Hamer and Barber in Ref. [CITATION] for the Ising model.', '0801.3238-2-36-10': 'From Eq. ([REF]) together with Eq. [REF] we read [MATH] as previously, so that the calculated exponent is consistent with the prediction [MATH].', '0801.3238-2-37-0': 'The explicit calculation of the various shift exponents, confirms that the HCM given by Eq. ([REF]) is superior to the standard PRG method.', '0801.3238-2-37-1': 'However, the things are different if we considered the excitation obtained creating a well-defined quasiparticle with the smallest momentum [MATH], instead of the very first excitation gap.', '0801.3238-2-37-2': 'In that case, the energy gap is given by Eq. ([REF]).', '0801.3238-2-37-3': 'Using the method of the extrema of the PRG quantity in the left hand side of Eq. ([REF]), we would have obtained a convergence to the critical point as [MATH], while with the standard PRG only as [MATH].', '0801.3238-2-38-0': '## [MATH] transition (non BKT)', '0801.3238-2-39-0': 'We choose the spin-1 [MATH] model on a chain ([MATH]) with PBC[EQUATION] because the transition from the Haldane phase to the phase at large [MATH] is described by a [MATH] CFT ([MATH]) with continuously varying exponents.', '0801.3238-2-39-1': 'The Hamiltonian ([REF]) has been used to describe the magnetic properties of different quasi 1D compounds (see [CITATION] for a brief account).', '0801.3238-2-39-2': 'Here, by fixing [MATH] and [MATH], for which it has been already estimated [MATH] [CITATION], we wish to test the methods described above in a severe case in which a [MATH] scaling would give sublinear convergence in [MATH].', '0801.3238-2-39-3': 'Indeed for the FSCM we expect [MATH].', '0801.3238-2-39-4': 'The previous estimate using the log-log plots of the finite-size gaps was [MATH] [CITATION], while in Ref. [CITATION] it is found [MATH] using the method of level crossing with antiperiodic boundary conditions, which is however specific to this transition.', '0801.3238-2-39-5': 'The first irrelevant operator allowed by the lattice symmetries has scaling dimension [MATH] where [MATH] so that [MATH] where with [MATH] we denote the smallest exponent of corrections arising from lattice contributions at finite [MATH] that are not captured in the framework of the (relativistic) continuum theory.', '0801.3238-2-40-0': 'As discussed in Refs. [CITATION] this is a case in which the linear term [MATH] in the scaling function of the PRG vanishes because [MATH] for the sine-Gordon model, the effective field theory that describe the surroundings of the [MATH] line.', '0801.3238-2-40-1': 'So it is convenient to use also our improved version of the PRG as discussed in Subsec. [REF].', '0801.3238-2-40-2': 'The expected shift exponent is [MATH].', '0801.3238-2-41-0': 'In order to have an idea of the range of values of [MATH] to be used let us imagine that [MATH] so that [MATH].', '0801.3238-2-41-1': 'With [MATH] this exponent leads to variations in the pseudocritical points [MATH] smaller than [MATH].', '0801.3238-2-41-2': 'Hence we prefer to illustrate the method with virtually exact numerical data obtained with the Lanczos algorithm using [MATH].', '0801.3238-2-41-3': 'The reason is that the DMRG would give rather accurate values for the energies but the estimates for [MATH] could not be sufficiently precise to appreciate the variations in [MATH] obtained from the crossings.', '0801.3238-2-41-4': 'So we use the DMRG only to extend the data to [MATH] with [MATH] optimized states.', '0801.3238-2-41-5': 'In any case the GS belongs to the [MATH] sector.', '0801.3238-2-42-0': 'The FSCM with [MATH] yields [MATH] with [MATH] in reasonable agreement with the CFT expectations.', '0801.3238-2-42-1': 'The high-precision procedure based on Eq. ([REF]) yields [MATH] with [MATH].', '0801.3238-2-42-2': 'The value of the shift exponent is definitely larger than what expected, which could be due to the vanishing of the coefficient of the first irrelevant contribution with scaling dimensions [MATH].', '0801.3238-2-42-3': 'In any case, from Fig. [REF] one can clearly appreciate that the sequence [MATH] converges more slowly than [MATH].', '0801.3238-2-42-4': 'As far as the PRG is concerned, with the standard procedure of finding the zeroes of Eq. ([REF]) we get [MATH] with [MATH].', '0801.3238-2-42-5': 'Instead, using the improved method estimation by looking for the extremal value of the quantity on the left side of ([REF]) we find a rapidly converging sequence that however also oscillates between [MATH] and [MATH].', '0801.3238-2-42-6': 'Finally, a small oscillation in the fourth decimal place about [MATH] is seen also in the sequence [MATH] obtained through the homogeneity criterion ([REF]).', '0801.3238-2-42-7': 'As expected the latter sequence converges in a fashion similar to [MATH].', '0801.3238-2-42-8': 'All these results are summarized in Fig. [REF].', '0801.3238-2-43-0': 'The data show that subtracting out the terms that induce a slow convergence of the pseudocritical points, one remains with quantities that inherit a residual (possibly oscillating) [MATH]-dependence from the specific lattice model and that could be very difficult to account for.', '0801.3238-2-43-1': 'Other factors that affect the extrapolation of the critical points at this level of accuracy are the sampling [MATH] and the trade-off between computational accuracy and the maximum available size.', '0801.3238-2-44-0': 'To summarize we observe that our improved methods yield very fast convergence to the critical point that we estimate to be [MATH], consistently with Ref. [CITATION].', '0801.3238-2-45-0': '## [MATH] BKT transition', '0801.3238-2-46-0': 'We consider now the spin-1/2 Heisenberg model with frustration due to next-to-nearest neighbors interaction [EQUATION].', '0801.3238-2-46-1': 'The model is equivalent to a 2-legs zigzag ladder with [MATH] rungs.', '0801.3238-2-46-2': 'The best estimate of the critical point was made by Okamoto and Nomura using a model-specific crossing method [CITATION]; with exact diagonalizations up to [MATH] they determined [MATH].', '0801.3238-2-46-3': 'The model is gapless for [MATH] and has a doubly degenerate GS in the TL for [MATH].', '0801.3238-2-46-4': 'Again the GS has [MATH].', '0801.3238-2-46-5': 'Without exploiting a priori information about the BKT character of transition (if not the value of the dynamic exponent [MATH]), we tested the homogeneity criterion ([REF]) using ladders with PBC of up to [MATH] rungs, an effective [MATH] and [MATH] DMRG states that ensure an accuracy of [MATH] on the values of [MATH].', '0801.3238-2-46-6': 'The results reported in Fig. [REF] are encouraging: while with the FSCM we would get no crossings at all, the zeroes of Eq. ([REF]) yield a sequence of points converging to [MATH].', '0801.3238-2-46-7': 'The main problem comes from the left side of the transition where the truncation DMRG error for [MATH] and [MATH] induces some oscillations on the plotted quantity.', '0801.3238-2-46-8': 'We content ourselves with linear fits in [MATH].', '0801.3238-2-46-9': 'If we exclude the point with [MATH] the fit is better even if we find [MATH]; by selecting all the available points, instead, the fit is visibly worse but the extrapolated value is [MATH], in agreement with Ref. [CITATION].', '0801.3238-2-46-10': 'As above, apart from the details of the extrapolation procedure, we see that the homogeneity criterion provides a viable procedure to locate the critical point in a BKT transition, where almost all existing generic methods fail.', '0801.3238-2-46-11': 'We remark that this analysis is based solely on [MATH], namely an observable evaluated on the GS, without invoking further assumptions on the nature of the excitations.', '0801.3238-2-47-0': '# Conclusions', '0801.3238-2-48-0': 'Making only use of finite-size quantities related to the ground state, we show how to generate sequences of pseudocritical points that converge very fast to the infinite-size critical point.', '0801.3238-2-48-1': 'The convergence is of the form [MATH] with a shift exponent [MATH].', '0801.3238-2-48-2': 'In this article we propose a homogeneity condition method (HCM) which is faster than the standard phenomenological renormalization group (PRG) in locating the critical points.', '0801.3238-2-48-3': 'Moreover its validity is more general as it can be applied without modification to the difficult case of a Berezinskii-Kosterlitz-Thouless transition.', '0801.3238-2-48-4': 'The homogeneity method requires only the knowledge of [MATH], that is, the expectation value of the term that drives the transition.', '0801.3238-2-49-0': 'We also presented an improvement to the PRG method, which allows, under certain conditions, to obtain pseudocritical sequences characterized by the same shift exponent as for the HCM.', '0801.3238-2-49-1': 'However, this modification, relying on a particular form of the gap scaling function, is not valid in general.', '0801.3238-2-49-2': 'It holds true, for instance for the sine-Gordon model that underlies a variety of transitions in [MATH] dimension.', '0801.3238-2-50-0': 'The formulations of the approaches are sufficiently general to be applied in any spatial dimensionality.', '0801.3238-2-50-1': 'Even if we are primarily interested in quantum phase transitions ideally at zero temperature, in principle the methods can be extended to problems of finite-temperature statistical mechanics.', '0801.3238-2-50-2': 'At variance with other accelerating methods found in the literature, e.g. the van den Broeck-Schwartz or the Bulirsch-Stoer ones (reviewed in Ch. 9 of [CITATION]), the procedures presented here rely on the scaling behavior of thermodynamical quantities expected from physical and renormalization arguments.', '0801.3238-2-51-0': 'The validity of the methods has been tested with analytical calculations on the one dimensional XY model in transverse field and numerically on a nontrivial spin-1 chain with anisotropy.', '0801.3238-2-51-1': 'As extreme case, we have shown that the homogeneity condition method provides a satisfactory location of the critical point also in the case of Berezinskii-Kosterlitz-Thouless transitions.', '0801.3238-2-51-2': 'These confirmations motivate us to consider systems in higher spatial dimensionality.', '0801.3238-2-51-3': 'In these cases the numerical data are restricted to smaller system-size and the need for fast-converging pseudo-critical sequences is a prerequisite for the precise location of the critical points.', '0801.3238-2-52-0': 'This work was partially supported by the Italian MiUR through the PRIN grant n. 2005021773.', '0801.3238-2-52-1': 'M.R. acknowledges support from the EU (SCALA).', '0801.3238-2-52-2': 'Numerical calculations were performed on a cluster of machines made available by the Theoretical Group of the Bologna Section of the INFN.', '0801.3238-2-53-0': '#', '0801.3238-2-54-0': 'Here we indicate how to compute the mean energy per site [MATH], and the average potential [MATH], as require id in Subsec. [REF].', '0801.3238-2-54-1': 'Consider for example the energy sum [EQUATION] where [MATH] is given by Eq. ([REF]).', '0801.3238-2-54-2': 'We need to investigate the above sum in the quasi-critical region [MATH] so it is sufficient to expand [MATH] in powers of [MATH][EQUATION]', '0801.3238-2-54-3': 'The resulting sums can be computed using the Euler-Maclaurin formula (see e.g. [CITATION])[EQUATION] valid for a function [MATH] with at least [MATH] continuous derivatives in [MATH].', '0801.3238-2-54-4': 'Here [MATH] are the Bernoulli numbers and the remainder [MATH] depends on [MATH] on [MATH].', '0801.3238-2-54-5': 'Some care must be taken when the function [MATH] diverges at the border of the Brillouin zone, in this case one must keep [MATH] and [MATH] away from the borders.', '0801.3238-2-54-6': 'Moreover, sending [MATH] to infinity in Eq. [REF], some sums must be regularized using a Borel summation technique.', '0801.3238-2-54-7': 'The final result for the ground state energy is[EQUATION] where [MATH] is the Euler-Mascheroni constant and the thermodynamic values are given by [EQUATION]', '0801.3238-2-54-8': 'Using similar procedures we obtain the following expression for average potential [MATH]:[EQUATION]', '0801.3238-2-54-9': 'Finally, the sum [EQUATION] for the evaluation of the finite-size gap in the PRG method, can be treated along similar lines.', '0801.3238-2-54-10': 'The final result for the gap is[EQUATION]'}

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[]

[]

[]

['0801.3238-1-8-0', '0801.3238-1-9-1', '0801.3238-1-53-0', '0801.3238-2-8-0', '0801.3238-2-9-1', '0801.3238-2-53-0']

{'1': 'http://arxiv.org/licenses/assumed-1991-2003/', '2': 'http://arxiv.org/licenses/assumed-1991-2003/'}

https://arxiv.org/abs/0801.3238

gr-qc-0507109

{'gr-qc-0507109-1-0-0': 'We determine the expression of the tangential velocity of test objects following a stable circular equatorial orbit in the spacetime of a dilatonic current-carrying string.', 'gr-qc-0507109-1-0-1': 'We impose the condition, established by observations in large samples of disks galaxies, that the magnitude of such velocity is radii independent.', 'gr-qc-0507109-1-0-2': 'In doing this, we find a constraint equation for the metric coefficients of the dilatonic cosmic string.', 'gr-qc-0507109-1-0-3': 'This solution, stemming from the string theory, is a useful constraint for the underlying microscopic field theory.', 'gr-qc-0507109-1-1-0': '# Introduction:', 'gr-qc-0507109-1-2-0': 'Most extensions of the standard model of particle physics predict that extra scalar fields, in addition to the ordinary Higgs field, whose experimental detection is still to be done, should exist in Nature.', 'gr-qc-0507109-1-2-1': 'At low energies (compared to the Planck scale), they appear to be classifiable into essentially two main categories, namely those which couple in a straightforward way to the other particle fields (as, e.g. grand unification breaking Higgs fields, supersymmetric partners of ordinary fermions or extra bosonic degrees of freedom coming from the Neveu-Schwartz sector of superstring theory [CITATION]), and those whose most important coupling is to gravity, such as the dilaton, whose origin can be traced to the Ramond sector in the superstring context.', 'gr-qc-0507109-1-2-2': 'Both kinds, coupled or decoupled, have been studied from different (and often disjoint) perspectives, and both have various cosmological and astrophysically observable consequences; these terms permit us, for instance, to obtain fully nonsingular cosmologies [CITATION].', 'gr-qc-0507109-1-2-3': 'In particular, scalar-tensor theories of gravity [CITATION] may provide a natural solution to the problem of terminating inflation [CITATION], whilst grand unified theory (GUT) scalars, being symmetry breakers, may lead to the formation of topological defects [CITATION], of which only cosmic strings are viable candidates from the point of view of cosmology.', 'gr-qc-0507109-1-3-0': 'Among these theories, some predict both kinds of fields.', 'gr-qc-0507109-1-3-1': 'As a result, one expects that cosmic strings could exist whose coupling to gravity would be altered by inclusion of dilaton effects.', 'gr-qc-0507109-1-3-2': 'In Ref. [CITATION], a local cosmic string solution was considered in the framework of low energy string theory which is reminiscent of the scalar-tensor theories of gravity [CITATION].', 'gr-qc-0507109-1-3-3': 'Indeed, a massless dilaton is shown to obey a least coupling principle [CITATION], e.g. to decouple from matter by cosmological attraction in much the same way as the generic attractor mechanism of the scalar-tensor theories of gravity [CITATION].', 'gr-qc-0507109-1-3-4': 'It was found [CITATION] then that the metric around a cosmic string in the framework of scalar-tensor gravity is of the Taub-Kasner type [CITATION] so that the particle and light propagation resembles that around a wiggly cosmic string in ordinary general relativity [CITATION], although the effect was expected to be one order of magnitude stronger.', 'gr-qc-0507109-1-4-0': 'In a number of papers [CITATION], a current has been included in the internal structure of the cosmic string.', 'gr-qc-0507109-1-4-1': 'The most noticeable consequence of a current-like effect is to modify the internal dynamics of cosmic strings in such a way that new states are reachable.', 'gr-qc-0507109-1-4-2': 'Indeed, the breaking of the Lorentz boost invariance along the worldsheet allows rotating equilibrium configurations, called vortons, which, if they are stable, can overclose the universe, thereby leading to a catastrophe for the theory that predicts them [CITATION].', 'gr-qc-0507109-1-4-3': 'Finally, inclusion of such an internal structure could drastically change the predictions of a cosmic string model [CITATION] in the microwave background anisotropies [CITATION].', 'gr-qc-0507109-1-4-4': 'In [CITATION], it is shown that the long-range effect on a cosmologically relevant network of strings is vanishing on average, but that vorton-like states can be reached by microscopically small loops.', 'gr-qc-0507109-1-5-0': 'Here, we would like to point out another effect which might come out from the inclusion of a current in the internal structure of a dilatonic string.', 'gr-qc-0507109-1-5-1': 'Namely, we will address ourselves to a well-posed problem at the galactic scale: The measurements of rotation curves in galaxies show that the coplanar orbital motion of gas in the outer parts of the galaxies keeps a constant velocity up to several luminous radii [CITATION].', 'gr-qc-0507109-1-5-2': 'The most accepted explanation for this effect is that there exists a spherical halo of dark matter which surrounds the galaxy and account for the missing mass needed to produce the flat behaviour of the rotational curves.', 'gr-qc-0507109-1-6-0': 'In what follows, after having set the relevant gravitational theory and notations, we derive the geometry of an electrically charged cosmic string in the dilatonic theory of gravity in Sec. 2.', 'gr-qc-0507109-1-6-1': 'Then, in the Sec. 3, we compute the tangential velocity for test particles imposing that its magnitude is independent of the radius.', 'gr-qc-0507109-1-6-2': 'In doing this, we find a constraint equation for the metric coefficients of the dilatonic cosmic string.', 'gr-qc-0507109-1-6-3': 'Section 4 summarizes our findings and discusses the relevant conclusions.', 'gr-qc-0507109-1-7-0': '# Exterior Solution for a Timelike Current-Carrying String:', 'gr-qc-0507109-1-8-0': '## The Model:', 'gr-qc-0507109-1-9-0': 'In this section we will mainly review the obtention of the gravitational field generated by a string carrying a current of timelike-type as presented in the Refs. [CITATION].', 'gr-qc-0507109-1-9-1': 'We start with the action in the Jordan-Fierz frame: [EQUATION] [MATH] is the physical metric which contains both scalar and tensor degrees of freedom, [MATH] is the curvature scalar associated to it and [MATH] is the action for general matter fields which, at this point, is left arbitrary.', 'gr-qc-0507109-1-9-2': 'The metric signature is assumed to be [MATH].', 'gr-qc-0507109-1-10-0': 'In what follows, we will concentrate our attention to superconducting vortex configurations which arise from the spontaneous breaking of the symmetry [MATH].', 'gr-qc-0507109-1-10-1': 'Therefore, the action for the matter fields will be composed by two pairs of coupled complex scalar and gauge fields [MATH] and [MATH].', 'gr-qc-0507109-1-10-2': 'Also, for technical purposes, it is preferable to work in the so-called Einstein (or conformal) frame, in which the scalar and tensor degrees of freedom do not mix: [EQUATION] where [MATH] , [MATH] and the potential is suitably chosen in order that the pair [MATH] breaks one symmetry [MATH] in vacuum (giving rise to the vortex configuration) and the second pair [MATH] breaks the symmetry [MATH] in the core of the vortex (giving rise to the superconducting properties): [EQUATION]', 'gr-qc-0507109-1-10-3': 'We restrict, then, ourselves to the configurations corresponding to an isolated, static current-carrying vortex lying in the [MATH].', 'gr-qc-0507109-1-10-4': 'In a cylindrical coordinate system [MATH] such that [MATH] and [MATH], we make the following ansatz: [EQUATION]', 'gr-qc-0507109-1-10-5': 'The pair [MATH], which is responsible for the superconducting properties of the vortex, is set in the form: [EQUATION] where [MATH] corresponds to the electric field which leads to a timelike current in the vortex.', 'gr-qc-0507109-1-10-6': 'We also require that the functions [MATH] must be regular everywhere and must satisfy the usual boundary conditions of vortex [CITATION] and superconducting configurations [CITATION].', 'gr-qc-0507109-1-11-0': 'The action (2) is obtained from ([REF]) by a conformal transformation [EQUATION] and by the redefinition of the quantity [EQUATION] which makes evident the feature that any gravitational phenomena will be affected by the variation of the gravitation constant [MATH] in the scalar-tensor gravity, and by introducing a new parameter [EQUATION]', 'gr-qc-0507109-1-11-1': "Variation of the action (2) with respect to the metric [MATH] and to the dilaton field [MATH] gives the modified Einstein's equations and a wave equation for the dilaton, respectively.", 'gr-qc-0507109-1-11-2': 'Namely, [EQUATION]', 'gr-qc-0507109-1-11-3': 'Where [MATH] is the energy-momentum tensor which is obtained by [EQUATION]', 'gr-qc-0507109-1-11-4': 'We note, in passing, that, in the conformal frame, this tensor is not conserved providing us with an additional equation [EQUATION].', 'gr-qc-0507109-1-12-0': 'In what follows, we will write the general static metric with cylindrical symmetry corresponding to the electric case in the form: [EQUATION] where [MATH] are functions of [MATH] only.', 'gr-qc-0507109-1-13-0': 'The non-vanishing components of the energy-momentum tensor using ([REF]) are [EQUATION]', 'gr-qc-0507109-1-13-1': 'Therefore, for the electric case, eqs. (6) are written as [EQUATION]', 'gr-qc-0507109-1-13-2': 'In order to solve the above equations we will divide the space in two regions: the exterior region, [MATH], in which only the electric component of the Maxwell tensor contributes to the energy-momentum tensor and the internal region, [MATH], where all matter fields survive.', 'gr-qc-0507109-1-13-3': '[MATH] is the string thickness.', 'gr-qc-0507109-1-14-0': '## Form of the Line Element for the Exterior Region:', 'gr-qc-0507109-1-15-0': "Due to the specific properties of the Maxwell tensor [EQUATION] the Einstein's equations ([REF]) may be transformed into some algebraic relations called Rainich algebra [CITATION]: [EQUATION] which, for the electric case, have the form: [EQUATION].", 'gr-qc-0507109-1-15-1': 'Then we have [EQUATION] the solution is: [EQUATION]', 'gr-qc-0507109-1-15-2': 'Therefore, the exterior metric for a timelike current-carrying string is: [EQUATION] where [EQUATION]', 'gr-qc-0507109-1-15-3': 'The constants [MATH] will be determined after the inclusion of the matter fields.', 'gr-qc-0507109-1-16-0': '# Stable Circular Geodesics Around the Cosmic String:', 'gr-qc-0507109-1-17-0': 'In this section we will derive the geodesic equations in the equatorial plane ([MATH]), where dot stands for "derivative with respect to the proper time [MATH]".', 'gr-qc-0507109-1-17-1': 'First of all, let us re-write metric ([REF]) in a more compact way: [EQUATION] with [EQUATION]', 'gr-qc-0507109-1-17-2': 'The Lagrangian for a test particle moving on this spacetime is given by: [EQUATION] and the associated canonical momenta, [MATH], are: [EQUATION]', 'gr-qc-0507109-1-17-3': 'Because of the symmetries of this particular spacetime, the quantities [MATH] and [MATH] are constants for each geodesic and, because this spacetime is static, the Hamiltonian, [MATH], is also a constant.', 'gr-qc-0507109-1-17-4': 'Combining this information with the restriction of a motion in an equatorial plane, we arrive to the following equation for the radial geodesic: [EQUATION]', 'gr-qc-0507109-1-17-5': 'In this work, we will concentrate on stable circular motion.', 'gr-qc-0507109-1-17-6': 'Therefore, we have to satisfy three conditions simultaneously.', 'gr-qc-0507109-1-17-7': 'Namely:', 'gr-qc-0507109-1-18-0': 'Consequently, we have: [EQUATION]', 'gr-qc-0507109-1-18-1': 'Expressing [MATH] and [MATH] in terms of the constant quantities [MATH] and [MATH], respectively, we can re-write the above equations as: [EQUATION] where prime means "derivative with respect to the coordinate [MATH]", which finally gives us the expressions for [MATH] and [MATH]: [EQUATION]', 'gr-qc-0507109-1-18-2': 'Recalling that the angular velocity of a test particle moving in a circular motion in an orbital plane is [MATH], we have: [EQUATION]', 'gr-qc-0507109-1-18-3': 'We are now in a position to compute the tangential velocity of the moving particles in a circular orbit in the equatorial plane.', 'gr-qc-0507109-1-18-4': 'From now on, we will follow the prescription established by Chandrasekhar in [CITATION].', 'gr-qc-0507109-1-18-5': 'Let us re-express the metric ([REF]) in terms of the proper time [MATH], as [MATH]: [EQUATION] and comparing with the expression [EQUATION] where [MATH], we can easily obtain the spatial velocity [MATH]: [EQUATION] whose components are, respectively: [EQUATION]', 'gr-qc-0507109-1-18-6': 'In order to have stable circular orbits, the tangential velocity [MATH] must be constant at different radii at the equatorial plane.', 'gr-qc-0507109-1-18-7': 'Therefore, we can impose: [EQUATION]', 'gr-qc-0507109-1-18-8': 'At this point we would like to notice the following theorem [CITATION]:', 'gr-qc-0507109-1-19-0': 'Theorem: The tangential velocity of circular stable equatorial orbits is constant iff the coefficient metrics are related as [EQUATION] provided [MATH].', 'gr-qc-0507109-1-20-0': 'This theorem implies that the line element in the equatorial plane must be: [EQUATION]', 'gr-qc-0507109-1-20-1': 'This form clearly is not asymptotically flat and neither describes a spacetime corresponding to a central black hole.', 'gr-qc-0507109-1-20-2': 'Therefore, we can infer that it describes solely the region where the tangential velocity of the test particles is constant, being probably joined in the interior and exterior regions with other metrics, suitably chosen in order to ensure regularity in the asymptotic limits.', 'gr-qc-0507109-1-21-0': 'Let us notice, however, that this metric has the form which has been found in [CITATION], after identifying [MATH] with the appropriate constant parameters which depend on the microscopic details of the model.', 'gr-qc-0507109-1-21-1': 'The calculations are straightforward but lengthy.', 'gr-qc-0507109-1-21-2': 'We will skip here the details and provide directly the results.', 'gr-qc-0507109-1-21-3': 'For the details of these calculations, we refer the reader to Refs. [CITATION].', 'gr-qc-0507109-1-21-4': 'For this particular configuration, consisting of an electrically charged dilatonic string, we have: [EQUATION] where [MATH] and [MATH] are the energy per unit length, the tension per unit length and the current of the string, respectively.', 'gr-qc-0507109-1-21-5': '[MATH] measures the coupling of the dilaton to the matter fields.', 'gr-qc-0507109-1-21-6': 'For cosmic strings formed at GUT scales, [MATH], and for a coupling [MATH] which is compatible with present experimental data [CITATION], [MATH], the parameter [MATH] (and, thus, the tangential velocity [MATH]) seems to be too small.', 'gr-qc-0507109-1-21-7': 'The observed magnitude of the tangential velocity [CITATION] being [MATH] cannot be explained by a single dilatonic current-carrying string in this case.', 'gr-qc-0507109-1-21-8': 'As argued by Lee in the Ref. [CITATION], if a bundle of [MATH] cosmic strings formed at GUT scales seeded one galaxy, then the total magnitude of the tangential velocity would be [MATH].', 'gr-qc-0507109-1-21-9': 'In our case, to be compatible with the astronomical observations, one must have a bundle of [MATH] strings seeding a galaxy.', 'gr-qc-0507109-1-22-0': '# Conclusions:', 'gr-qc-0507109-1-23-0': 'The measurements of rotation curves in galaxies show that the coplanar orbital motion of gas in the outer parts of the galaxies keeps a constant velocity up to several luminous radii.', 'gr-qc-0507109-1-23-1': 'The most accepted explanation for this effect is that there exists a spherical halo of dark matter which surrounds the galaxy and account for the missing mass needed to produce the flat behaviour of the rotational curves.', 'gr-qc-0507109-1-24-0': 'In the Ref. [CITATION], it has been shown that in a static, axially symmetric spacetime, a sufficient and necessary condition in order to have a flat behaviour for the rotational curves in galaxies is that the metric assumes the form ([REF]).', 'gr-qc-0507109-1-24-1': 'This form clearly is not asymptotically flat and neither describes a spacetime corresponding to a central black hole.', 'gr-qc-0507109-1-24-2': 'Therefore, we can infer that it describes solely the region where the tangential velocity of the test particles is constant, being probably joined in the interior and exterior regions with other metrics, suitably chosen in order to ensure regularity in the asymptotic limits.', 'gr-qc-0507109-1-25-0': 'In previous papers [CITATION], we have found a metric corresponding to the spacetime generated by an electrically charged dilatonic string which possesses the form ([REF]), with appropriate parameters which are related to the microscopic details of the model ([REF]).', 'gr-qc-0507109-1-25-1': 'The observed magnitude of the tangential velocity cannot be explained by a single dilatonic current-carrying string in this case.', 'gr-qc-0507109-1-25-2': 'However, if we consider that a bundle of [MATH] cosmic strings formed at GUT scales seeded one galaxy, then the total magnitude of the tangential velocity would be [MATH].', 'gr-qc-0507109-1-25-3': 'In our case, to be compatible with the astronomical observations, one must have a bundle of [MATH] strings seeding one galaxy.'}

{'gr-qc-0507109-2-0-0': 'We analyse the flat behaviour of the rotational curves in some galaxies in the framework of a dilatonic, current-carrying string.', 'gr-qc-0507109-2-0-1': 'We determine the expression of the tangential velocity of test objects following a stable circular equatorial orbit in this spacetime.', 'gr-qc-0507109-2-1-0': '# Introduction:', 'gr-qc-0507109-2-2-0': 'The measurements of rotation curves in some galaxies show that the coplanar orbital motion of gas in the outer parts of the galaxies keeps a constant velocity up to several luminous radii [CITATION].', 'gr-qc-0507109-2-2-1': 'The most accepted explanation for this effect is that there exists a spherical halo of dark matter which surrounds the galaxy and account for the missing mass needed to produce the flat behavior of the rotational curves.', 'gr-qc-0507109-2-3-0': 'In this work, we would like to analyse this effect in the framework of a dilatonic, current-carrying cosmic string.', 'gr-qc-0507109-2-3-1': 'In a number of papers [5-11], a current has been included in the internal structure of the cosmic string.', 'gr-qc-0507109-2-3-2': 'The most noticeable consequence of a current-like effect is to modify the internal dynamics of cosmic strings in such a way that new states are reachable.', 'gr-qc-0507109-2-3-3': 'Indeed, the breaking of the Lorentz boost invariance along the worldsheet allows rotating equilibrium configurations, called vortons, which, if they are stable, can overclose the universe, thereby leading to a catastrophe for the theory that predicts them [CITATION].', 'gr-qc-0507109-2-3-4': 'Finally, inclusion of such an internal structure could drastically change the predictions of a cosmic string model [CITATION] in the microwave background anisotropies [CITATION].', 'gr-qc-0507109-2-3-5': 'In [CITATION], it is shown that the long-range effect on a cosmologically relevant network of strings is vanishing on average, but that vorton-like states can be reached by microscopically small loops.', 'gr-qc-0507109-2-3-6': 'Here, we would like address ourselves to a well-posed problem at the galactic scale.', 'gr-qc-0507109-2-4-0': 'In what follows, after having set the relevant gravitational theory and notations, we derive the geometry of an electrically charged cosmic string in the dilatonic theory of gravity in Sec. 2.', 'gr-qc-0507109-2-4-1': 'Then, in the Sec. 3, we compute the tangential velocity for test particles imposing that its magnitude is independent of the radius.', 'gr-qc-0507109-2-4-2': 'In doing this, we find a constraint equation for the metric coefficients of the dilatonic cosmic string.', 'gr-qc-0507109-2-4-3': 'Unfortunately, as we will see later, we also find that the tangential velocity cannot be explained by a single string of the kind proposed in our model.', 'gr-qc-0507109-2-4-4': 'Instead, in order to be compatible with the observed magnitude, one must have a bundle of [MATH] strings seeding a galaxy.', 'gr-qc-0507109-2-4-5': 'With such a density, a cosmic string network would be dominating the universe, and its dynamics would be completely different.', 'gr-qc-0507109-2-4-6': 'Section 4 summarizes our findings and discusses the relevant conclusions.', 'gr-qc-0507109-2-5-0': '# Exterior Solution for a Timelike Current-Carrying String:', 'gr-qc-0507109-2-6-0': '## The Model:', 'gr-qc-0507109-2-7-0': 'In this section we will mainly review the obtention of the gravitational field generated by a string carrying a current of timelike-type as presented in the Refs. [CITATION].', 'gr-qc-0507109-2-8-0': 'We will concentrate our attention to superconducting vortex configurations which arise from the spontaneous breaking of the symmetry [MATH].', 'gr-qc-0507109-2-8-1': 'Therefore, the action for the matter fields will be composed by two pairs of coupled complex scalar and gauge fields [MATH] and [MATH].', 'gr-qc-0507109-2-8-2': 'Also, for technical purposes, it is preferable to work in the so-called Einstein (or conformal) frame, in which the scalar and tensor degrees of freedom do not mix: [EQUATION] where [MATH] , [MATH] and the potential is suitably chosen in order that the pair [MATH] breaks one symmetry [MATH] in vacuum (giving rise to the vortex configuration) and the second pair [MATH] breaks the symmetry [MATH] in the core of the vortex (giving rise to the superconducting properties): [MATH].', 'gr-qc-0507109-2-9-0': 'In what follows, we will write the general static metric with cylindrical symmetry corresponding to the electric case in the form [EQUATION] where [MATH] are functions of [MATH] only.', 'gr-qc-0507109-2-10-0': 'In order to solve the equations we will divide the space in two regions: the exterior region, [MATH], in which only the electric component of the Maxwell tensor contributes to the energy-momentum tensor and the internal region, [MATH], where all matter fields survive.', 'gr-qc-0507109-2-10-1': '[MATH] is the string thickness.', 'gr-qc-0507109-2-11-0': '## Form of the Line Element for the Exterior Region:', 'gr-qc-0507109-2-12-0': 'Due to the specific properties of the Maxwell tensor [EQUATION] we can find the metric through some algebraic relations called Rainich algebra [CITATION], which, for the electric case, have the form: [EQUATION].', 'gr-qc-0507109-2-13-0': 'Therefore, the exterior metric for a timelike current-carrying string is: [EQUATION] where [EQUATION]', 'gr-qc-0507109-2-13-1': 'The constants [MATH] will be determined after the inclusion of the matter fields.', 'gr-qc-0507109-2-14-0': '# Stable Circular Geodesics Around the Cosmic String:', 'gr-qc-0507109-2-15-0': 'In this section we will derive the geodesic equations in the equatorial plane ([MATH]), where dot stands for "derivative with respect to the proper time [MATH]".', 'gr-qc-0507109-2-15-1': 'First of all, let us re-write metric (4) in a more compact way: [EQUATION] with [EQUATION]', 'gr-qc-0507109-2-15-2': 'The Lagrangian for a test particle moving on this spacetime is given by: [EQUATION] and the associated canonical momenta, [MATH], are: [EQUATION]', 'gr-qc-0507109-2-15-3': 'Because of the symmetries of this particular spacetime, the quantities [MATH] and [MATH] are constants for each geodesic and, because this spacetime is static, the Hamiltonian, [MATH], is also a constant.', 'gr-qc-0507109-2-15-4': 'Combining this information with the restriction of a motion in an equatorial plane, we arrive to the following equation for the radial geodesic: [EQUATION]', 'gr-qc-0507109-2-15-5': 'In this work, we will concentrate on stable circular motion.', 'gr-qc-0507109-2-15-6': 'Therefore, we have to satisfy three conditions simultaneously.', 'gr-qc-0507109-2-15-7': 'Namely:', 'gr-qc-0507109-2-16-0': 'Consequently, we have: [EQUATION]', 'gr-qc-0507109-2-16-1': 'Expressing [MATH] and [MATH] in terms of the constant quantities [MATH] and [MATH], respectively, we can get their expressions : [EQUATION] where prime means "derivative with respect to the coordinate [MATH]".', 'gr-qc-0507109-2-17-0': 'Recalling that the angular velocity of a test particle moving in a circular motion in an orbital plane is [MATH], we have: [EQUATION]', 'gr-qc-0507109-2-17-1': 'We are now in a position to compute the tangential velocity of the moving particles in a circular orbit in the equatorial plane.', 'gr-qc-0507109-2-17-2': 'From now on, we will follow the prescription established by Chandrasekhar in [CITATION].', 'gr-qc-0507109-2-17-3': 'Let us re-express the metric (6) in terms of the proper time [MATH], as [MATH]: [EQUATION] and comparing with the expression [MATH] where [MATH], we can easily obtain the spatial velocity [MATH]: [EQUATION] from which we can obtain all the components of the spatial velocities.', 'gr-qc-0507109-2-17-4': 'However, we are particularly interested in the tangential component [MATH]: [EQUATION]', 'gr-qc-0507109-2-17-5': 'In order to have stable circular orbits, the tangential velocity [MATH] must be constant at different radii at the equatorial plane.', 'gr-qc-0507109-2-17-6': 'Therefore, we can impose: [EQUATION]', 'gr-qc-0507109-2-17-7': 'At this point we would like to notice the following theorem [CITATION]:', 'gr-qc-0507109-2-18-0': 'Theorem: The tangential velocity of circular stable equatorial orbits is constant iff the coefficient metrics are related as [EQUATION] provided [MATH].', 'gr-qc-0507109-2-19-0': 'This theorem implies that the line element in the equatorial plane must be: [EQUATION]', 'gr-qc-0507109-2-19-1': 'This form clearly is not asymptotically flat and neither describes a spacetime corresponding to a central black hole.', 'gr-qc-0507109-2-19-2': 'Therefore, we can infer that it describes solely the region where the tangential velocity of the test particles is constant, being probably joined in the interior and exterior regions with other metrics, suitably chosen in order to ensure regularity in the asymptotic limits.', 'gr-qc-0507109-2-20-0': 'Let us notice, however, that this metric has the form which has been found in [CITATION], after identifying [MATH] with the appropriate constant parameters which depend on the microscopic details of the model.', 'gr-qc-0507109-2-20-1': 'The calculations are straightforward but lengthy.', 'gr-qc-0507109-2-20-2': 'We will skip here the details and provide directly the results.', 'gr-qc-0507109-2-20-3': 'For the details of these calculations, we refer the reader to Refs. [CITATION].', 'gr-qc-0507109-2-20-4': 'For this particular configuration, consisting of an electrically charged dilatonic string, we have: [EQUATION] where [MATH] and [MATH] are the energy per unit length, the tension per unit length and the current of the string, respectively.', 'gr-qc-0507109-2-20-5': '[MATH] measures the coupling of the dilaton to the matter fields.', 'gr-qc-0507109-2-20-6': 'For cosmic strings formed at GUT scales, [MATH], and for a coupling [MATH] which is compatible with present experimental data [CITATION], [MATH], the parameter [MATH] (and, thus, the tangential velocity [MATH]) seems to be too small.', 'gr-qc-0507109-2-20-7': 'The observed magnitude of the tangential velocity [CITATION] being [MATH] cannot be explained by a single dilatonic current-carrying string in this case.', 'gr-qc-0507109-2-20-8': 'As argued by Lee in the Ref. [CITATION], if a bundle of [MATH] cosmic strings formed at GUT scales seeded one galaxy, then the total magnitude of the tangential velocity would be [MATH].', 'gr-qc-0507109-2-20-9': 'In our case, to be compatible with the astronomical observations, one must have a bundle of [MATH] strings seeding a galaxy.', 'gr-qc-0507109-2-20-10': 'With such a density, a cosmic string network would be dominating the universe, and its dynamics would be completely different.', 'gr-qc-0507109-2-20-11': 'The only situation where such a high number of strings could be possible is at much lower energy scales (electroweak scale, say) but then of course the energy scale is far too low to have any relevance for structure formation.', 'gr-qc-0507109-2-21-0': 'If, on the other hand, one supposes that one single string could explain the observed values of the tangential velocity, one should therefore impose that such a string is formed at Planck scales, which at the moment seems not quite realistic conclusion as well.', 'gr-qc-0507109-2-22-0': '# Conclusions:', 'gr-qc-0507109-2-23-0': 'The measurements of rotation curves in galaxies show that the coplanar orbital motion of gas in the outer parts of the galaxies keeps a constant velocity up to several luminous radii.', 'gr-qc-0507109-2-23-1': 'The most accepted explanation for this effect is that there exists a spherical halo of dark matter which surrounds the galaxy and account for the missing mass needed to produce the flat behavior of the rotational curves.', 'gr-qc-0507109-2-24-0': 'In the Ref. [CITATION], it has been shown that in a static, axially symmetric spacetime, a sufficient and necessary condition in order to have a flat behavior for the rotational curves in galaxies is that the metric assumes the form ([REF]).', 'gr-qc-0507109-2-24-1': 'This form clearly is not asymptotically flat and neither describes a spacetime corresponding to a central black hole.', 'gr-qc-0507109-2-24-2': 'Therefore, we can infer that it describes solely the region where the tangential velocity of the test particles is constant, being probably joined in the interior and exterior regions with other metrics, suitably chosen in order to ensure regularity in the asymptotic limits.', 'gr-qc-0507109-2-25-0': 'In previous papers [CITATION], we have found a metric corresponding to the spacetime generated by an electrically charged dilatonic string which possesses the form ([REF]), with appropriate parameters which are related to the microscopic details of the model ([REF]).', 'gr-qc-0507109-2-25-1': 'The observed magnitude of the tangential velocity cannot be explained by a single dilatonic current-carrying string in this case.', 'gr-qc-0507109-2-25-2': 'However, if we consider that a bundle of [MATH] cosmic strings formed at GUT scales seeded one galaxy, then the total magnitude of the tangential velocity would be [MATH].', 'gr-qc-0507109-2-25-3': 'In our case, to be compatible with the astronomical observations, one must have a bundle of [MATH] strings seeding one galaxy!', 'gr-qc-0507109-2-25-4': 'The only situation where such a high number of strings could be possible is at much lower energy scales (electroweak scale, say) but then the energy scale is far too low to have any relevance for structure formation.'}

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[]

[['gr-qc-0507109-1-18-1', 'gr-qc-0507109-2-16-1'], ['gr-qc-0507109-1-0-0', 'gr-qc-0507109-2-0-1'], ['gr-qc-0507109-1-5-1', 'gr-qc-0507109-2-2-0']]

[['gr-qc-0507109-1-15-0', 'gr-qc-0507109-2-12-0']]

['gr-qc-0507109-1-11-2', 'gr-qc-0507109-1-17-7', 'gr-qc-0507109-1-18-8', 'gr-qc-0507109-2-15-7', 'gr-qc-0507109-2-17-7']

{'1': 'http://arxiv.org/licenses/assumed-1991-2003/', '2': 'http://arxiv.org/licenses/assumed-1991-2003/'}

https://arxiv.org/abs/gr-qc/0507109

1003.2691

{'1003.2691-1-0-0': 'We discuss the non-thermal leptogenesis in the scheme of 5D orbifold SO(10) GUT with the smooth hybrid inflation.', '1003.2691-1-0-1': 'With an unambiguously determined Dirac Yukawa couplings and an assumption for the neutrino mixing matrix of the tri-bimaximal from, we analyze baryon asymmetry of the universe via non-thermal leptogenesis in two typical cases for the light neutrino mass spectrum, the normal and inverted hierarchical cases.', '1003.2691-1-0-2': 'The resultant baryon asymmetry is obtained as a function of the lightest mass eigenvalue of the light neutrinos, and we find that a suitable amount of baryon asymmetry of the universe can be produced in the normal hierarchical case, while in the inverted hierarchical case the baryon asymmetry is too small to be consistent with the observation.', '1003.2691-1-1-0': 'The so-called renormalizable minimal SO(10) GUT model has been paid a particular attention, where two Higgs multiplets [MATH] are utilized for the Yukawa couplings with matters [MATH] is the generation index) [CITATION].', '1003.2691-1-1-1': 'A remarkable feature of the model is its high predictive power for the neutrino oscillation parameters, in reproducing charged fermion masses and mixing angles.', '1003.2691-1-1-2': 'The unambiguously determined Yukawa couplings play a crucial role for predictions of the model in other phenomena, such as the lepton flavor violation [CITATION] and proton decay [CITATION].', '1003.2691-1-1-3': 'The Higgs superpotential of the model has been investigated and the explicit pattern of the SO(10) gauge symmetry to the standard model one has been shown [CITATION].', '1003.2691-1-1-4': 'On the other hand, the explicit representation of intermediate energy scales revealed in these papers gives rise to the deviation of gauge coupling unification [CITATION].', '1003.2691-1-1-5': 'Also the minimal SO(10) model likely predict too short proton lifetime and has some deviation from the precise measurements of the neutrino oscillation data [CITATION].', '1003.2691-1-2-0': 'In order to remedy these problems, we have argued SO(10) GUT in the context of the orbifold GUT [CITATION] and proposed a simple supersymmetric (SUSY) SO(10) model in 5D [CITATION].', '1003.2691-1-2-1': 'In this model, the SO(10) symmetry in 5D is broken by orbifold boundary conditions to the Pati-Salam (PS) symmetry SU(4)[MATH] SU(2)[MATH]SU(2)[MATH].', '1003.2691-1-2-2': 'All matter and Higgs multiplets reside only on a brane (PS brane) where the PS gauge symmetry is manifest, so that low energy effective description of this model is nothing but the PS model in 4D with a special set of matter and Higgs multiplets.', '1003.2691-1-2-3': 'At energies higher than the compactification scale, the Kaluza-Klein (KK) modes of the bulk SO(10) gauge multiplet are involved in the particle contents and in fact, the gauge coupling unification was shown to be successfully realized by incorporating the KK mode threshold corrections into the gauge coupling running [CITATION].', '1003.2691-1-2-4': 'The unification scale ([MATH]) and the compactification scale ([MATH]) which was set to be the same as the PS symmetry breaking scale ([MATH]) for simplicity were found to be [MATH] GeV and [MATH] GeV, respectively.', '1003.2691-1-3-0': 'More recently, it has been shown [CITATION] that this orbifold GUT model is applicable to the smooth hybrid inflation [CITATION].', '1003.2691-1-3-1': 'Interestingly, this inflation model can fit the WMAP data [CITATION] very well by utilizing the PS breaking scale ([MATH]) and the gauge coupling unification scale predicted independently of cosmological considerations.', '1003.2691-1-3-2': 'Another cosmological issue, the dark matter candidate of the model has been investigated in [CITATION].', '1003.2691-1-3-3': 'In the paper, the sparticle mass spectrum is calculated in the context of the gaugino mediated supersymmetry breaking [CITATION] which can be naturally incorporated in the model and it has shown that the neutralino LSP as the dark matter candidate can be realized when the compactification scale is taken to be slightly bigger than the PS symmetry breaking scale, while keeping the successful gauge coupling unification.', '1003.2691-1-4-0': 'In the present paper, we apply our model to the leptogenesis scenario for creating the baryon asymmetry of the universe.', '1003.2691-1-4-1': 'In order to produce a suitable amount of the baryon asymmetry of the universe in the thermal leptogenesis scenario [CITATION], the scale of right-handed (scalar) neutrino masses should be grater than [MATH] GeV [CITATION] and hence the reheating temperature after inflation should also be beyond this scale.', '1003.2691-1-4-2': 'However, in supersymmetric models, the reheating temperature is severely constrained by Big Bang Nucleosynthesis (BBN) to be [MATH] GeV [CITATION] (gravitino problem), and the conventional thermal leptogenesis scenario cannot work.', '1003.2691-1-5-0': 'In this case, we consider the so-called non-thermal leptogenesis [CITATION] in which the right-handed (scalar) neutrinos are non-thermally produced by the decay of inflaton and their decays can produce a suitable amount of baryon asymmetry of the universe even if the reheating temperature is low.', '1003.2691-1-5-1': 'We adopt the non-thermal leptogenesis to our hybrid inflation scenario [CITATION] and show that the non-thermal leptogenesis is successful with a suitable choice of the model parameters which are consistent with the results in the previous works [CITATION].', '1003.2691-1-6-0': 'Let us begin with a brief review of the orbifold SO(10) GUT proposed in Ref. [CITATION].', '1003.2691-1-6-1': 'The model is described in 5D and the 5th dimension is compactified on the orbifold [MATH] [CITATION].', '1003.2691-1-6-2': 'A circle [MATH] with radius [MATH] is divided by a [MATH] orbifold transformation [MATH] is the fifth dimensional coordinate [MATH]) and this segment is further divided by a [MATH] transformation [MATH] with [MATH].', '1003.2691-1-6-3': 'There are two inequivalent orbifold fixed points at [MATH] and [MATH].', '1003.2691-1-6-4': 'Under this orbifold compactification, a general bulk wave function is classified with respect to its parities, [MATH] and [MATH], under [MATH] and [MATH], respectively.', '1003.2691-1-7-0': 'Assigning suitable parities ([MATH]) to the bulk SO(10) gauge multiplet [CITATION], only the PS gauge multiplet has zero-mode and the bulk 5D N=1 SUSY SO(10) gauge symmetry is broken to 4D N=1 supersymmetric PS gauge symmetry.', '1003.2691-1-7-1': 'All vector multiplets has wave functions on the brane at [MATH], SO(10) gauge symmetry is respected there, while only the PS symmetry is on the brane at [MATH] (PS brane).', '1003.2691-1-8-0': 'We place all the matter and Higgs multiplets on the PS brane, where only the PS symmetry is manifest, so that the particle contents are in the representation under the PS gauge symmetry, not necessary to be in SO(10) representation.', '1003.2691-1-8-1': 'For a different setup, see [CITATION].', '1003.2691-1-8-2': 'The matter and Higgs in our model is listed in Table 1.', '1003.2691-1-8-3': 'For later conveniences, let us introduce the following notations: [EQUATION]', '1003.2691-1-8-4': 'Superpotential relevant for fermion masses is given by [EQUATION] where [MATH] is the 5D Planck mass.', '1003.2691-1-8-5': 'The product, [MATH], effectively works as [MATH], while [MATH] effectively works as [MATH], and is responsible for the right-handed Majorana neutrino masses.', '1003.2691-1-8-6': 'Assuming appropriate VEVs for Higgs multiplets, fermion mass matrices are obtained, which we parameterize as the following form [CITATION]: [EQUATION]', '1003.2691-1-8-7': 'Here, [MATH] and [MATH] are the mass matrices of up and down type quarks, Dirac neutrino and charged lepton, respectively, while [MATH] is right-handed Majorana neutrino mass matrix.', '1003.2691-1-9-0': 'The following two points should be remarked:', '1003.2691-1-10-0': '1.', '1003.2691-1-10-1': 'The combination of two mass matrices of [MATH] and [MATH] among [MATH], and [MATH] in the PS symmetry is the same as that of [MATH] and [MATH] in the minimal SO(10) model (see [CITATION] for notation) and, therefore, the procedure for fitting the realistic Dirac fermion mass matrices is the same as in the minimal SO(10) model.', '1003.2691-1-11-0': 'On the other hand,', '1003.2691-1-12-0': '2.', '1003.2691-1-12-1': '[MATH] is fully independent on the above four Dirac Fermion mass matrices in the PS group, whereas in the minimal SO(10) model it is described by [MATH] and not independent.', '1003.2691-1-12-2': 'This fact enables us to improve the precise data fitting on the neutrino oscillation parameters.', '1003.2691-1-13-0': 'Now we discuss the smooth hybrid inflation model [CITATION] in the context of the orbifold SO(10) GUT model.', '1003.2691-1-13-1': 'Introducing a singlet chiral superfield [MATH], we consider the superpotential , [EQUATION] where [MATH] is a dimensionless coefficient, [MATH] is a dimensionful parameter, and [MATH] is the 5D Planck mass.', '1003.2691-1-13-2': 'SUSY vacuum conditions lead to non-zero VEVs for [MATH], by which the PS symmetry is broken down to the SM one, and thus [EQUATION]', '1003.2691-1-13-3': 'It is theoretically natural to identify [MATH] as the GUT scale, [MATH].', '1003.2691-1-13-4': 'From the analysis of the gauge coupling unification in the context of the 5D orbifold GUT [CITATION], we found that [MATH] GeV and [MATH] GeV.', '1003.2691-1-13-5': 'Independently of the analysis of the gauge coupling unification, it has shown in [CITATION] that this smooth hybrid inflation model, where the inflation trajectory is approximately parameterized by the scalar component of [MATH], can reproduce the WMAP data by [MATH] GeV and [MATH] being the the same order of magnitude as [MATH].', '1003.2691-1-14-0': 'Now we discuss the main topic of this paper: the non-thermal leptogenesis.', '1003.2691-1-14-1': 'The relevant part of the superpotential is [EQUATION] where without loss of generality, we work on the mass diagonal basis of the right-handed neutrinos.', '1003.2691-1-14-2': 'The inflaton which is the scalar component of [MATH] couples with the scalar right-handed neutrinos in the scalar potential, [EQUATION]', '1003.2691-1-14-3': 'Parameterizing the inflaton field [MATH], the inflaton mass is found to be [EQUATION] and the interaction between the inflaton and the scalar right-handed neutrinos [EQUATION] where [MATH] is mass of the (scalar) right-handed neutrino of the [MATH]-th generation, and we set [MATH] without loss of generality.', '1003.2691-1-14-4': 'The partial decay width of the inflaton into the [MATH]-th generation scalar right-handed neutrino, if kinematically allowed, given by [EQUATION]', '1003.2691-1-14-5': 'Here [MATH] denotes the scalar right-handed neutrino in the [MATH]-th generation.', '1003.2691-1-14-6': 'Since the inflaton and the superfields, [MATH] and [MATH], have the same mass, the inflaton cannot decay into the superfields.', '1003.2691-1-15-0': 'In non-thermal leptogenesis, the inflaton decays into (scalar) right-handed neutrinos and then, the CP-violating decay of the neutrinos generates lepton asymmetry of the universe, which is finally converted into baryon asymmetry via the sphaleron processes.', '1003.2691-1-15-1': 'The resultant baryon asymmetry of the universe is evaluated as [EQUATION] where the sum is taken to be scalar right-handed neutrinos kinematically allowed, and the CP-violating parameter is given by [CITATION] [EQUATION] with the Dirac neutrino Yukawa coupling [MATH] and [EQUATION]', '1003.2691-1-15-2': 'Here we have assumed that masses of all scalar right-handed neutrinos are greater than the reheating temperature after inflation.', '1003.2691-1-15-3': 'This assumption is crucial because if a scalar right-handed neutrino is lighter than the reheating temperature, the scenario becomes thermal leptogenesis and the baryon asymmetry produced is not enough for a low reheating temperature.', '1003.2691-1-16-0': 'For the prediction of the resultant baryon asymmetry, we need the information of the Dirac Yukawa coupling, the mass spectra of the scalar right-handed neutrinos and light neutrinos, and the neutrino mixing matrix.', '1003.2691-1-16-1': 'Through the seesaw mechanism [CITATION], the light neutrino mass matrix is given by [EQUATION] in the basis where the mass matrix of charged lepton is diagonal.', '1003.2691-1-16-2': 'Here [MATH] is the VEV of the up-type Higgs doublet, [MATH] is the mass matrix of the right-handed neutrinos, and [MATH] is the diagonal mass matrix of light neutrinos.', '1003.2691-1-16-3': 'In this paper, we consider two typical cases for the light neutrino mass spectrum and describe [MATH] in terms of the lightest mass eigenvalue [MATH] and the mass squared differences: [EQUATION] for the normal hierarchical case, and [EQUATION] for the inverted hierarchical case.', '1003.2691-1-16-4': 'Here we adopted the neutrino oscillation data [CITATION]: [EQUATION]', '1003.2691-1-16-5': 'In addition, we assume the mixing matrix of the so-called tri-bimaximal form [CITATION] [EQUATION] which is in very good agreement with the current best fit values of the neutrino oscillation data [CITATION].', '1003.2691-1-16-6': 'As we mentioned above, the data fit for the realistic Dirac mass matrices of the present model is the same as in the minimal SO(10) model, and as an example, we here use the numerical value [MATH] obtained in [CITATION] at the GUT scale usual in 4D models [MATH] GeV for [MATH]: [EQUATION]', '1003.2691-1-16-7': 'In this way, we can obtain the (scalar) right-handed neutrino mass matrix as a function of [MATH], [EQUATION] with [MATH] assumed to be the tri-bimaximal mixing matrix.', '1003.2691-1-17-0': 'Now we are ready to analyze the baryon asymmetry produced in the model.', '1003.2691-1-17-1': 'Once the parameters in the smooth hybrid inflation, [MATH], [MATH] and [MATH], are fixed, the baryon asymmetry of the universe of Eq. ([REF]) is calculated as a function of only [MATH], providing the reheating temperature evaluated as [EQUATION] which is also a function of [MATH].', '1003.2691-1-17-2': 'Here [MATH] is the total decay width of the inflaton, [MATH] GeV, and [MATH].', '1003.2691-1-18-0': 'In our analysis, we take [MATH] GeV and [MATH] GeV according to the values found in [CITATION] which realize the gauge coupling unification.', '1003.2691-1-18-1': 'As an example, we fix [MATH].', '1003.2691-1-18-2': 'For these parameters fixed, Fig. 1 shows the mass spectrum of the scalar right-handed neutrinos as a function of [MATH], together with [MATH], in the normal hierarchical case.', '1003.2691-1-18-3': 'For [MATH] eV, the inflaton decays into only a pairs of the scalar right-handed neutrinos in the first generation.', '1003.2691-1-19-0': 'The reheating temperature as a function of [MATH] is depicted in Fig. 2.', '1003.2691-1-19-1': 'The jump around [MATH] eV is because the decay channel of inflaton into the scalar right-handed neutrino in the second generation is opened up there and the reheating temperature becomes higher than [MATH], [MATH], so that we exclude the region [MATH] eV in our analysis.', '1003.2691-1-20-0': 'In Fig. 2, the reheating temperature exceeds its BBN bound [MATH].', '1003.2691-1-20-1': 'However, this bound is not applicable if the gravitino is heavy, [MATH] TeV, in which case gravitino in the early universe decays before BBN takes place.', '1003.2691-1-20-2': 'As has been investigated in [CITATION], the gaugino mediated supersymmetry breaking is naturally incorporated in our 5D SO(10) GUT, where SUSY breaking is assumed to occur on the brane at [MATH] and the SO(10) gaugino residing in the bulk directly couples with the SUSY breaking sector, [EQUATION] where [MATH] is a singlet chiral superfield which breaks SUSY by its F-component VEV ([MATH]), and [MATH] is a dimensionless constant.', '1003.2691-1-20-3': 'Then, the gaugino obtains the SUSY breaking soft mass, [EQUATION] where the compactification scale [MATH] comes from the wave function normalization of the bulk gaugino, we have used the relation between the 4D and 5D Planck masses, [MATH] with the reduced Planck mass [MATH] GeV, and [MATH] is gravitino mass.', '1003.2691-1-20-4': 'In this paper we adopt [MATH], so that [MATH] TeV for [MATH] GeV.', '1003.2691-1-21-0': 'Finally, in the normal hierarchical case for the light neutrino mass spectrum, we show the resultant baryon asymmetry of the universe generated via the non-thermal leptogenesis as a function of [MATH] in Fig. 3, together with the currently observed value [CITATION] [EQUATION]', '1003.2691-1-21-1': 'We find the observed value is reproduced for [MATH] eV.', '1003.2691-1-22-0': 'We repeat the same analysis for the inverted hierarchical case.', '1003.2691-1-22-1': 'Fig. 4 shows the mass spectrum of the scalar right-handed neutrinos, and the reheating temperature is depicted in Fig. 5.', '1003.2691-1-22-2': 'The resultant baryon asymmetry is shown in Fig. 6 as a function of [MATH].', '1003.2691-1-22-3': 'We find that in the inverted hierarchical case the baryon asymmetry produced in non-thermal leptogenesis is too small to be consistent with the observation.', '1003.2691-1-23-0': 'In summary, we have studied the non-thermal leptogenesis in the scheme of 5D orbifold SO(10) GUT with the smooth hybrid inflation.', '1003.2691-1-23-1': 'With an unambiguously determined Dirac Yukawa couplings and an assumption for the neutrino mixing matrix of the tri-bimaximal from, we have analyzed the baryon asymmetry of the universe via non-thermal leptogenesis in two typical cases for the light neutrino mass spectrum, the normal and inverted hierarchical cases.', '1003.2691-1-23-2': 'The resultant baryon asymmetry is given as a function of the lightest mass eigenvalue of the light neutrinos [MATH].', '1003.2691-1-23-3': 'In the normal hierarchical case, for [MATH] eV, the model predicts a suitable amount of the baryon asymmetry through non-thermal leptogenesis, while in the inverted hierarchical case, the predicted asymmetry is too small to be consistent with the observations.', '1003.2691-1-23-4': 'As can be seen from Eqs. ([REF]) and ([REF]), a mildly small [MATH] guarantees [MATH] and this is crucial for the realization of non-thermal leptogenesis where we can neglect wash-out processes.', '1003.2691-1-24-0': 'Our 5D orbifold SO(10) GUT was originally constructed in order to remedy problems of the minimal SO(10) GUT in particle physics.', '1003.2691-1-24-1': 'It is very interesting that the parameters determined from particle physics give the consistent observational values of WMAP coming from quite different origins of cosmology.', '1003.2691-1-24-2': 'Leptogenesis may be placed just in the midst of particle physics and cosmology among others and is very sensitive to the parameter of particle physics.', '1003.2691-1-24-3': 'Our theory is consistent with it, giving additional constraints on the lightest neutrino mass.'}

{'1003.2691-2-0-0': 'We discuss the non-thermal leptogenesis in the scheme of 5D orbifold SO(10) GUT with the smooth hybrid inflation.', '1003.2691-2-0-1': 'With an unambiguously determined Dirac Yukawa couplings and an assumption for the neutrino mixing matrix of the tri-bimaximal from, we analyze baryon asymmetry of the universe via non-thermal leptogenesis in two typical cases for the light neutrino mass spectrum, the normal and inverted hierarchical cases.', '1003.2691-2-0-2': 'The resultant baryon asymmetry is obtained as a function of the lightest mass eigenvalue of the light neutrinos, and we find that a suitable amount of baryon asymmetry of the universe can be produced in the normal hierarchical case, while in the inverted hierarchical case the baryon asymmetry is too small to be consistent with the observation.', '1003.2691-2-1-0': 'The so-called renormalizable minimal SO(10) GUT model has been paid a particular attention, where two Higgs multiplets [MATH] are utilized for the Yukawa couplings with matters [MATH] is the generation index) [CITATION].', '1003.2691-2-1-1': 'A remarkable feature of the model is its high predictive power for the neutrino oscillation parameters, in reproducing charged fermion masses and mixing angles.', '1003.2691-2-1-2': 'The unambiguously determined Yukawa couplings play a crucial role for predictions of the model in other phenomena, such as the lepton flavor violation [CITATION] and proton decay [CITATION].', '1003.2691-2-1-3': 'The Higgs superpotential of the model has been investigated and the explicit pattern of the SO(10) gauge symmetry to the standard model one has been shown [CITATION].', '1003.2691-2-1-4': 'On the other hand, the explicit representation of intermediate energy scales revealed in these papers gives rise to the deviation of gauge coupling unification [CITATION].', '1003.2691-2-1-5': 'Also the minimal SO(10) model likely predict too short proton lifetime and has some deviation from the precise measurements of the neutrino oscillation data [CITATION] (see however [CITATION]).', '1003.2691-2-2-0': 'In order to remedy these problems, we have argued SO(10) GUT in the context of the orbifold GUT [CITATION] and proposed a simple supersymmetric (SUSY) SO(10) model in 5D [CITATION].', '1003.2691-2-2-1': 'In this model, the SO(10) symmetry in 5D is broken by orbifold boundary conditions to the Pati-Salam (PS) symmetry SU(4)[MATH] SU(2)[MATH]SU(2)[MATH].', '1003.2691-2-2-2': 'All matter and Higgs multiplets reside only on a brane (PS brane) where the PS gauge symmetry is manifest, so that low energy effective description of this model is nothing but the PS model in 4D with a special set of matter and Higgs multiplets.', '1003.2691-2-2-3': 'At energies higher than the compactification scale, the Kaluza-Klein (KK) modes of the bulk SO(10) gauge multiplet are involved in the particle contents and in fact, the gauge coupling unification was shown to be successfully realized by incorporating the KK mode threshold corrections into the gauge coupling running [CITATION].', '1003.2691-2-2-4': 'The unification scale ([MATH]) and the compactification scale ([MATH]) which was set to be the same as the PS symmetry breaking scale ([MATH]) for simplicity were found to be [MATH] GeV and [MATH] GeV, respectively.', '1003.2691-2-3-0': 'More recently, it has been shown [CITATION] that this orbifold GUT model is applicable to the smooth hybrid inflation [CITATION].', '1003.2691-2-3-1': 'Interestingly, this inflation model can fit the WMAP data [CITATION] very well by utilizing the PS breaking scale ([MATH]) and the gauge coupling unification scale predicted independently of cosmological considerations.', '1003.2691-2-3-2': 'Another cosmological issue, the dark matter candidate of the model has been investigated in [CITATION].', '1003.2691-2-3-3': 'In the paper, the sparticle mass spectrum is calculated in the context of the gaugino mediated supersymmetry breaking [CITATION] which can be naturally incorporated in the model and it has shown that the neutralino LSP as the dark matter candidate can be realized when the compactification scale is taken to be slightly bigger than the PS symmetry breaking scale, while keeping the successful gauge coupling unification.', '1003.2691-2-4-0': 'In the present paper, we apply our model to the leptogenesis scenario for creating the baryon asymmetry of the universe.', '1003.2691-2-4-1': 'In order to produce a suitable amount of the baryon asymmetry of the universe in the thermal leptogenesis scenario [CITATION], the scale of right-handed (scalar) neutrino masses should be grater than [MATH] GeV [CITATION] and hence the reheating temperature after inflation should also be beyond this scale.', '1003.2691-2-4-2': 'However, in supersymmetric models, the reheating temperature is severely constrained by Big Bang Nucleosynthesis (BBN) to be [MATH] GeV [CITATION] (gravitino problem [CITATION]), and the conventional thermal leptogenesis scenario cannot work.', '1003.2691-2-5-0': 'In this case, we consider the so-called non-thermal leptogenesis [CITATION] in which the right-handed (scalar) neutrinos are non-thermally produced by the decay of inflaton and their decays can produce a suitable amount of baryon asymmetry of the universe even if the reheating temperature is low.', '1003.2691-2-5-1': 'We adopt the non-thermal leptogenesis to our hybrid inflation scenario [CITATION] and show that the non-thermal leptogenesis is successful with a suitable choice of the model parameters which are consistent with the results in the previous works [CITATION].', '1003.2691-2-6-0': 'Let us begin with a brief review of the orbifold SO(10) GUT proposed in Ref. [CITATION].', '1003.2691-2-6-1': 'The model is described in 5D and the 5th dimension is compactified on the orbifold [MATH] [CITATION].', '1003.2691-2-6-2': 'A circle [MATH] with radius [MATH] is divided by a [MATH] orbifold transformation [MATH] is the fifth dimensional coordinate [MATH]) and this segment is further divided by a [MATH] transformation [MATH] with [MATH].', '1003.2691-2-6-3': 'There are two inequivalent orbifold fixed points at [MATH] and [MATH].', '1003.2691-2-6-4': 'Under this orbifold compactification, a general bulk wave function is classified with respect to its parities, [MATH] and [MATH], under [MATH] and [MATH], respectively.', '1003.2691-2-7-0': 'Assigning suitable parities ([MATH]) to the bulk SO(10) gauge multiplet [CITATION], only the PS gauge multiplet has zero-mode and the bulk 5D N=1 SUSY SO(10) gauge symmetry is broken to 4D N=1 supersymmetric PS gauge symmetry.', '1003.2691-2-7-1': 'All vector multiplets has wave functions on the brane at [MATH], SO(10) gauge symmetry is respected there, while only the PS symmetry is on the brane at [MATH] (PS brane).', '1003.2691-2-8-0': 'We place all the matter and Higgs multiplets on the PS brane, where only the PS symmetry is manifest, so that the particle contents are in the representation under the PS gauge symmetry, not necessary to be in SO(10) representation.', '1003.2691-2-8-1': 'For a different setup, see [CITATION].', '1003.2691-2-8-2': 'The matter and Higgs in our model is listed in Table 1.', '1003.2691-2-8-3': 'For later conveniences, let us introduce the following notations: [EQUATION]', '1003.2691-2-8-4': 'Superpotential relevant for fermion masses is given by [EQUATION] where [MATH] is the 5D Planck mass.', '1003.2691-2-8-5': 'The product, [MATH], effectively works as [MATH], while [MATH] effectively works as [MATH], and is responsible for the right-handed Majorana neutrino masses.', '1003.2691-2-8-6': 'Assuming appropriate VEVs for Higgs multiplets, fermion mass matrices are obtained, which we parameterize as the following form [CITATION]: [EQUATION]', '1003.2691-2-8-7': 'Here, [MATH] and [MATH] are the mass matrices of up and down type quarks, Dirac neutrino and charged lepton, respectively, while [MATH] is right-handed Majorana neutrino mass matrix.', '1003.2691-2-9-0': 'The following two points should be remarked:', '1003.2691-2-10-0': '1.', '1003.2691-2-10-1': 'The combination of two mass matrices of [MATH] and [MATH] among [MATH], and [MATH] in the PS symmetry is the same as that of [MATH] and [MATH] in the minimal SO(10) model (see [CITATION] for notation) and, therefore, the procedure for fitting the realistic Dirac fermion mass matrices is the same as in the minimal SO(10) model.', '1003.2691-2-11-0': 'On the other hand,', '1003.2691-2-12-0': '2.', '1003.2691-2-12-1': '[MATH] is fully independent on the above four Dirac Fermion mass matrices in the PS group, whereas in the minimal SO(10) model it is described by [MATH] and not independent.', '1003.2691-2-12-2': 'This fact enables us to improve the precise data fitting on the neutrino oscillation parameters.', '1003.2691-2-13-0': 'Now we discuss the smooth hybrid inflation model [CITATION] in the context of the orbifold SO(10) GUT model.', '1003.2691-2-13-1': 'Introducing a singlet chiral superfield [MATH], we consider the superpotential , [EQUATION] where [MATH] is a dimensionless coefficient, [MATH] is a dimensionful parameter, and [MATH] is the 5D Planck mass.', '1003.2691-2-13-2': 'SUSY vacuum conditions lead to non-zero VEVs for [MATH], by which the PS symmetry is broken down to the SM one, and thus [EQUATION]', '1003.2691-2-13-3': 'It is theoretically natural to identify [MATH] as the GUT scale, [MATH].', '1003.2691-2-13-4': 'From the analysis of the gauge coupling unification in the context of the 5D orbifold GUT [CITATION], we found that [MATH] GeV and [MATH] GeV.', '1003.2691-2-13-5': 'Independently of the analysis of the gauge coupling unification, it has shown in [CITATION] that this smooth hybrid inflation model, where the inflation trajectory is approximately parameterized by the scalar component of [MATH], can reproduce the WMAP data by [MATH] GeV and [MATH] being the the same order of magnitude as [MATH].', '1003.2691-2-14-0': 'Now we discuss the main topic of this paper: the non-thermal leptogenesis.', '1003.2691-2-14-1': 'The relevant part of the superpotential is [EQUATION] where without loss of generality, we work on the mass diagonal basis of the right-handed neutrinos.', '1003.2691-2-14-2': 'The inflaton which is the scalar component of [MATH] couples with the scalar right-handed neutrinos in the scalar potential, [EQUATION]', '1003.2691-2-14-3': 'Parameterizing the inflaton field [MATH], the inflaton mass is found to be [EQUATION] and the interaction between the inflaton and the scalar right-handed neutrinos [EQUATION] where [MATH] is mass of the (scalar) right-handed neutrino of the [MATH]-th generation, and we set [MATH] without loss of generality.', '1003.2691-2-14-4': 'The partial decay width of the inflaton into the [MATH]-th generation scalar right-handed neutrino, if kinematically allowed, given by [EQUATION]', '1003.2691-2-14-5': 'Here [MATH] denotes the scalar right-handed neutrino in the [MATH]-th generation.', '1003.2691-2-14-6': 'Since the inflaton and the superfields, [MATH] and [MATH], have the same mass, the inflaton cannot decay into the superfields.', '1003.2691-2-15-0': 'In non-thermal leptogenesis, the inflaton decays into (scalar) right-handed neutrinos and then, the CP-violating decay of the neutrinos generates lepton asymmetry of the universe, which is finally converted into baryon asymmetry via the sphaleron processes.', '1003.2691-2-15-1': 'The resultant baryon asymmetry of the universe is evaluated as [EQUATION] where the sum is taken to be scalar right-handed neutrinos kinematically allowed, and the CP-violating parameter is given by [CITATION] [EQUATION] with the Dirac neutrino Yukawa coupling [MATH] and [EQUATION]', '1003.2691-2-15-2': 'Here we have assumed that masses of all scalar right-handed neutrinos are greater than the reheating temperature after inflation.', '1003.2691-2-15-3': 'This assumption is crucial because if a scalar right-handed neutrino is lighter than the reheating temperature, the scenario becomes thermal leptogenesis and the baryon asymmetry produced is not enough for a low reheating temperature.', '1003.2691-2-16-0': 'For the prediction of the resultant baryon asymmetry, we need the information of the Dirac Yukawa coupling, the mass spectra of the scalar right-handed neutrinos and light neutrinos, and the neutrino mixing matrix.', '1003.2691-2-16-1': 'Through the seesaw mechanism [CITATION], the light neutrino mass matrix is given by [EQUATION] in the basis where the mass matrix of charged lepton is diagonal.', '1003.2691-2-16-2': 'Here [MATH] is the VEV of the up-type Higgs doublet, [MATH] is the mass matrix of the right-handed neutrinos, and [MATH] is the diagonal mass matrix of light neutrinos.', '1003.2691-2-16-3': 'In this paper, we consider two typical cases for the light neutrino mass spectrum and describe [MATH] in terms of the lightest mass eigenvalue [MATH] and the mass squared differences: [EQUATION] for the normal hierarchical case, and [EQUATION] for the inverted hierarchical case.', '1003.2691-2-16-4': 'Here we adopted the neutrino oscillation data [CITATION]: [EQUATION]', '1003.2691-2-16-5': 'In addition, we assume the mixing matrix of the so-called tri-bimaximal form [CITATION] [EQUATION] which is in very good agreement with the current best fit values of the neutrino oscillation data [CITATION].', '1003.2691-2-16-6': 'As we mentioned above, the data fit for the realistic Dirac mass matrices of the present model is the same as in the minimal SO(10) model, and as an example, we here use the numerical value [MATH] obtained in [CITATION] at the GUT scale usual in 4D models [MATH] GeV for [MATH]: [EQUATION]', '1003.2691-2-16-7': 'In this way, we can obtain the (scalar) right-handed neutrino mass matrix as a function of [MATH], [EQUATION] with [MATH] assumed to be the tri-bimaximal mixing matrix.', '1003.2691-2-17-0': 'Now we are ready to analyze the baryon asymmetry produced in the model.', '1003.2691-2-17-1': 'Once the parameters in the smooth hybrid inflation, [MATH], [MATH] and [MATH], are fixed, the baryon asymmetry of the universe of Eq. ([REF]) is calculated as a function of only [MATH], providing the reheating temperature evaluated as [EQUATION] which is also a function of [MATH].', '1003.2691-2-17-2': 'Here [MATH] is the total decay width of the inflaton, [MATH] GeV, and [MATH].', '1003.2691-2-18-0': 'In our analysis, we take [MATH] GeV and [MATH] GeV according to the values found in [CITATION] which realize the gauge coupling unification.', '1003.2691-2-18-1': 'As an example, we fix [MATH].', '1003.2691-2-18-2': 'For these parameters fixed, Fig. 1 shows the mass spectrum of the scalar right-handed neutrinos as a function of [MATH], together with [MATH], in the normal hierarchical case.', '1003.2691-2-18-3': 'For [MATH] eV, the inflaton decays into only a pairs of the scalar right-handed neutrinos in the first generation.', '1003.2691-2-19-0': 'The reheating temperature as a function of [MATH] is depicted in Fig. 2.', '1003.2691-2-19-1': 'The jump around [MATH] eV is because the decay channel of inflaton into the scalar right-handed neutrino in the second generation is opened up there and the reheating temperature becomes higher than [MATH], [MATH], so that we exclude the region [MATH] eV in our analysis.', '1003.2691-2-20-0': 'In Fig. 2, the reheating temperature exceeds its BBN bound [MATH].', '1003.2691-2-20-1': 'However, this bound is not applicable if the gravitino is heavy, [MATH] TeV, in which case gravitino in the early universe decays before BBN takes place.', '1003.2691-2-20-2': 'As has been investigated in [CITATION], the gaugino mediated supersymmetry breaking is naturally incorporated in our 5D SO(10) GUT, where SUSY breaking is assumed to occur on the brane at [MATH] and the SO(10) gaugino residing in the bulk directly couples with the SUSY breaking sector, [EQUATION] where [MATH] is a singlet chiral superfield which breaks SUSY by its F-component VEV ([MATH]), and [MATH] is a dimensionless constant.', '1003.2691-2-20-3': 'Then, the gaugino obtains the SUSY breaking soft mass, [EQUATION] where the compactification scale [MATH] comes from the wave function normalization of the bulk gaugino, we have used the relation between the 4D and 5D Planck masses, [MATH] with the reduced Planck mass [MATH] GeV, and [MATH] is gravitino mass.', '1003.2691-2-20-4': 'In this paper we adopt [MATH], so that [MATH] TeV for [MATH] GeV.', '1003.2691-2-21-0': 'Finally, in the normal hierarchical case for the light neutrino mass spectrum, we show the resultant baryon asymmetry of the universe generated via the non-thermal leptogenesis as a function of [MATH] in Fig. 3, together with the currently observed value [CITATION] [EQUATION]', '1003.2691-2-21-1': 'We find the observed value is reproduced for [MATH] eV.', '1003.2691-2-22-0': 'We repeat the same analysis for the inverted hierarchical case.', '1003.2691-2-22-1': 'Fig. 4 shows the mass spectrum of the scalar right-handed neutrinos, and the reheating temperature is depicted in Fig. 5.', '1003.2691-2-22-2': 'The resultant baryon asymmetry is shown in Fig. 6 as a function of [MATH].', '1003.2691-2-22-3': 'We find that in the inverted hierarchical case the baryon asymmetry produced in non-thermal leptogenesis is too small to be consistent with the observation.', '1003.2691-2-23-0': 'In summary, we have studied the non-thermal leptogenesis in the scheme of 5D orbifold SO(10) GUT with the smooth hybrid inflation.', '1003.2691-2-23-1': 'With an unambiguously determined Dirac Yukawa couplings and an assumption for the neutrino mixing matrix of the tri-bimaximal from, we have analyzed the baryon asymmetry of the universe via non-thermal leptogenesis in two typical cases for the light neutrino mass spectrum, the normal and inverted hierarchical cases.', '1003.2691-2-23-2': 'The resultant baryon asymmetry is given as a function of the lightest mass eigenvalue of the light neutrinos [MATH].', '1003.2691-2-23-3': 'In the normal hierarchical case, for [MATH] eV, the model predicts a suitable amount of the baryon asymmetry through non-thermal leptogenesis, while in the inverted hierarchical case, the predicted asymmetry is too small to be consistent with the observations.', '1003.2691-2-23-4': 'As can be seen from Eqs. ([REF]) and ([REF]), a mildly small [MATH] guarantees [MATH] and this is crucial for the realization of non-thermal leptogenesis where we can neglect wash-out processes.', '1003.2691-2-24-0': 'Our 5D orbifold SO(10) GUT was originally constructed in order to remedy problems of the minimal SO(10) GUT in particle physics.', '1003.2691-2-24-1': 'It is very interesting that the parameters determined from particle physics give the consistent observational values of WMAP coming from quite different origins of cosmology.', '1003.2691-2-24-2': 'Leptogenesis may be placed just in the midst of particle physics and cosmology among others and is very sensitive to the parameter of particle physics.', '1003.2691-2-24-3': 'Our theory is consistent with it, giving additional constraints on the lightest neutrino mass.'}

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[]

[]

[]

['1003.2691-1-9-0', '1003.2691-1-10-0', '1003.2691-1-11-0', '1003.2691-1-12-0', '1003.2691-2-9-0', '1003.2691-2-10-0', '1003.2691-2-11-0', '1003.2691-2-12-0']

{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1003.2691

0905.2902

{'0905.2902-1-0-0': "In a previous paper [CITATION] we proposed a purely mathematical way to quantum mechanics based on Cartan's simple spinors in their most elementary form of 2 components spinors.", '0905.2902-1-0-1': 'Here we proceed along that path proposing, this time, a symmetric tensor, quadrilinear in simple spinors, as a candidate for the symmetric tensor of general relativity.', '0905.2902-1-1-0': "The procedure resembles closely that in which one builds bilinearly from simple spinors an asymmetric electromagnetic tensor, from which easily descend Maxwell's equations and the photon can be seen as a bilinear combination of neutrinos.", '0905.2902-1-1-1': 'Here Lorentzian spaces result compact, building up spheres, where hopefully the problem of the Standard Model could be solved.', '0905.2902-1-2-0': 'std', '0905.2902-1-3-0': 'for_Paolo', '0905.2902-1-4-0': '# Introduction', '0905.2902-1-5-0': 'In a previous paper [CITATION] it was shown how, from the simplest, non trivial, two component spinors, which are pure, we may bilinearly and quite naturally obtain null vectors which are building up Minkowski momentum space.', '0905.2902-1-5-1': 'With these, in fact, one may formulate the well-known Weyl equations of motion for massless neutrinos as follows: [EQUATION] where [MATH] is a spinor (Dirac) associated with the Clifford algebra [MATH] with [MATH] its generators, and [MATH] its volume element.', '0905.2902-1-6-0': 'In [CITATION] [MATH] are bilinearly obtained precisely from those two component pure spinors from which we started.', '0905.2902-1-7-0': 'This was a first elementary result of Ref. [CITATION] which was aimed at searching a purely mathematical formulation of quantum mechanics.', '0905.2902-1-7-1': 'A similar way was followed by Einstein, Poincare, and other outstanding mathematicians in the early part of last century, when they arrived at special and general relativity.', '0905.2902-1-7-2': 'In this way one could avoid the plague of paradoxes which instead, up to our time, ruined the development of quantum mechanics.', '0905.2902-1-7-3': 'In Ref. [CITATION] two more main results were obtained:', '0905.2902-1-8-0': 'I.', '0905.2902-1-8-1': "While the classical dynamics of macroscopic bodies has to be formulated and dealt with in ordinary (Minkowski possibly curved) space-time, as we learned from Newton, Lagrange, Hamilton, Jacobi, Einstein, Poincare and others, so here the basic geometry is an Euclid's one, including the concept of point-event, which is necessary to represent the center of mass of the macroscopic bodies while running along their trajectories or orbits (like the Kepler orbits).", '0905.2902-1-9-0': 'For atomic physics instead, the appropriate space for the formulation of dynamical quantum equations and for their solution is momentum-space bilinearly constructed from pure spinors, as it appears in (1) the first elementary example of the Weyl equations.', '0905.2902-1-10-0': 'II.', '0905.2902-1-10-1': 'Pure Spinors.', '0905.2902-1-10-2': 'E. Cartan, the discoverer of spinors, underlined the great elegance of the spinors he named simple [CITATION] (renamed pure by Chevalley [CITATION]).', '0905.2902-1-11-0': 'He started by considering the correlations between spinors and totally null planes as follows: let [MATH] represent a complex Euclidean, [MATH] dimensional space, with well-defined quadratic forms.', '0905.2902-1-11-1': 'Then we may define the corresponding Clifford algebra [MATH] which, as known, may be considered as the endomorphism algebra in a [MATH] dimensional space [MATH] of spinors, and we write: [EQUATION]', '0905.2902-1-11-2': 'A spinor [MATH] may be defined by the equation [EQUATION] where [MATH] is a vector of [MATH] (referred to a Cartesian orthonormal coordinate system) and [MATH], called the generators of [MATH], represent the univectors in the direction of the [MATH] orthonormal coordinates of [MATH].', '0905.2902-1-12-0': 'The generators [MATH] obey the anticommutation relations.', '0905.2902-1-12-1': '[EQUATION]', '0905.2902-1-12-2': 'Let us now multiply equation (3) from the left by [MATH] and we obtain (because of equation (4)): [EQUATION] and, if [MATH]: [EQUATION] which means that the non zero spinor [MATH], satisfying equation (3) (sometimes called Dirac spinor) defines a subspace of [MATH] whose vectors are all null and/or mutually orthogonal, it is called the totally null plane associated to [MATH] and indicated with [MATH] where [MATH] is its dimension.', '0905.2902-1-12-3': 'It is known, and easy to prove, that the maximal possible value of [MATH] is [MATH], that is, one half of the dimension [MATH] of the space [MATH].', '0905.2902-1-13-0': 'In order to arrive at the Cartan definition of simple or pure spinors, let us define the "volume element" of [MATH]: [EQUATION]', '0905.2902-1-13-1': 'It is easy to show that [MATH] anticommutes with all the [MATH] and that [MATH] generate the Clifford algebra [MATH] whose even subalgebra [MATH] is isomorphic to the simple algebra [MATH]: [EQUATION]', '0905.2902-1-13-2': 'With [MATH] one may define what are called the Weyl spinors [MATH]: [EQUATION] where [MATH] is the Dirac spinor defined by equation (3).', '0905.2902-1-14-0': 'The Weyl spinors may be defined by the Weyl equations [EQUATION]', '0905.2902-1-14-1': 'Obviously also [MATH] will define totally null planes in [MATH].', '0905.2902-1-14-2': 'Well E. Cartan showed how a simple or pure spinor is isomorphic, up to a sign, to the maximal totally null plane of the Weyl spinor associated with a given Clifford algebra, and this property renders pure spinors complicated geometrical objects to deal with.', '0905.2902-1-14-3': 'In fact, while the dimension of the maximal totally null planes increase linearly with [MATH], that of the spinors, increase with [MATH] like [MATH] and then high dimensional spinors which are used to explain some phenomena of elementary particle physics (up to [MATH] component spinors), need a lot of (ten for 32 components) constraint equations in their components to render them pure.', '0905.2902-1-14-4': 'In short, all spinors up to [MATH] components are pure (or equivalent to pure).', '0905.2902-1-15-0': 'At present several people have tried to use pure spinors to explain both the high energy phenomena of the elementary particles and the unsolved problem of quantization of the gravitational field (through superstring theory) but with no acceptable results.', '0905.2902-1-16-0': 'In fact, the so-called Standard model, now represented by the symbols [MATH], which represents the symmetries presented in high-energy elementary particle phenomena, contains more than 20 constants representing charges, masses and so on, which may not be computed deriving them from acceptable theory.', '0905.2902-1-17-0': 'But now with the possible prescription found in [CITATION] that quantum dynamical problems have to be dealt with in momentum-space rather than in space-time, and because of a theorem we recently discovered, the situation could drastically change.', '0905.2902-1-18-0': 'III.', '0905.2902-1-18-1': 'The theorem.', '0905.2902-1-18-2': 'Consider two Weyl spinors: [MATH] and [MATH] associated with a Clifford algebra [MATH] and define the vector of the space [MATH], with components: [EQUATION] where [MATH].', '0905.2902-1-18-3': 'The vector [MATH] is null: [MATH], if and only if one of the two spinors, either [MATH] or [MATH], is pure.', '0905.2902-1-18-4': 'The proof is in Ref. [CITATION].', '0905.2902-1-19-0': 'It is known that in [MATH] dimensional spaces with Lorentzian signatures like [MATH] or [MATH] the vector components will be real if of the form [MATH], where [MATH] and [MATH] means [MATH] hermitian conjugate while [MATH] is the time-like generator.', '0905.2902-1-20-0': 'Suppose now [MATH] to be pure then we will have that: [EQUATION] with real components, (12) may be written in the form: [EQUATION] where [MATH]; which, might arrive at [MATH] (exploiting the isomorphism between Dirac and Weyl spinors doublets and conformal covariance [CITATION]).', '0905.2902-1-21-0': 'It is interesting to observe that in the momentum space, where according to the conclusion of Ref. [CITATION], we have to formulate the dynamical problems of quantum physics, pure spinors may bilinearly define in Lorentzian momentum spaces compact manifolds consisting of spheres (with Poincare invariant radii).', '0905.2902-1-21-1': 'This might obviously encourage the hope that not only the mathematical origin of quantum jumps might finally find a simple geometrical explanation, (the auto-vibrations of spheres are discrete) but also that, hopefully, the transitory sickness of the otherwise beautiful Standard Model, manifested by the symptom of its more than 20 unexplained constants, may finally be cured.', '0905.2902-1-22-0': 'IV.', '0905.2902-1-22-1': 'First applications and results.', '0905.2902-1-22-2': 'It might sound strange, but it is true: the first convincing confirmation of the above predictions was found by V. Fock more than 70 years ago: in 1935 [CITATION].', '0905.2902-1-22-3': 'He dealt with the historical problem (the first one to deal with atomic quantization) of the H-atom stationary states.', '0905.2902-1-23-0': 'At that time the problem was solved through the Schrodinger equation for the electron possible orbits in the field of the proton.', '0905.2902-1-23-1': 'Fock, anticipating the conclusions of [CITATION], formulated it in momentum space on a sphere [MATH] conceived as the one point compactification of ordinary [MATH] dimensional momentum space [MATH] with the following integral equation: [EQUATION] where [MATH] is the volume of the [MATH] unit sphere; [MATH] is the fine structure constant, [MATH] a unit of momentum, [MATH] the (reduced) electron mass and [MATH] is a unit vector indicating a point on the unit sphere [MATH], equation (14) is the Fock equation in adimensional form.', '0905.2902-1-23-2': 'Fock solved it through harmonic analysis and found that setting [MATH] he obtained for [MATH], representing the eigen vibrations of the sphere [MATH], the energy levels [MATH] of the H-atom stationary states, which in turn explain the Balmer spectrum of the H-atom.', '0905.2902-1-23-3': 'But it was V. Fock himself to draw from equation (14) the fundamental discovery that an integral equation on the sphere [MATH], like equation (14), has to forsee the symmetry [MATH] of its solutions (including obviously their classical counterparts - the solar planets - which is true, as later underlined by the great W. Pauli, Nobel prize winner).', '0905.2902-1-24-0': 'Now we may go one step further representing the H-atom with a quadruplet of spinors say (Proton, Neutron, Neutrino and Electron) and adopt equation (13) with [MATH] [CITATION]: since it is well known that spinors up to 4 components are pure (or equivalent to pure, and this equivalence is exploited here) [CITATION] [EQUATION] from which we obtain [MATH] with Poincare invariant radius [MATH] therefore we may expect that the spinorial [MATH] may reflect the Poincare invariance; and in the energy levels found by Fock we have the obviously relativistic form [CITATION]: [EQUATION]', '0905.2902-1-24-1': 'If we take this first result as an indication of the validity of the hypothesis formulated above [CITATION] we may affirm that the first example of atomic quantization confirms that the H-atom discrete energy levels (generating its discrete optical spectrum - Balmer series) are generated by the computed manifolds determined by pure spinors in momentum space.', '0905.2902-1-24-2': 'In particular by the eigen vibrations of those spheres we may also geometrically explain the quantum-jumps.', '0905.2902-1-25-0': 'Should this result have confirmations by further examples then the historical way to arrive at the solution of the problem of the H-atom stationary states would not only be more complicated but also wrong.', '0905.2902-1-25-1': 'It would be wrong to consider the H-atom as a proton plus an orbitating point electron to be then substituted arbitrarily by wave functions.', '0905.2902-1-26-0': 'As stated in [CITATION] we have to formulate the quantum problems in momentum space where we have the mathematical possibility to integrate null vectors to generate strings [CITATION], which are non local objects and might be at the origin of the concept of wave function without giving origin to paradoxes, as will be further discussed, elsewhere.', '0905.2902-1-27-0': 'V.', '0905.2902-1-27-1': 'The possibility to cure the Standard Model.', '0905.2902-1-27-2': 'V. Fock solved equation (14) computing the adimensional factors [MATH] through harmonic analysis from the ball [MATH] to [MATH].', '0905.2902-1-27-3': 'We tried (with P. Nurowski) to compute, through harmonic analysis, also the other adimensional factor [MATH] in equation (14) obtaining all the factors but one.', '0905.2902-1-27-4': 'In fact, [MATH] was computed for the first time by M.A. Wyler in 1969 [CITATION] with the result: [EQUATION] differing less than [MATH] from the experimental value.', '0905.2902-1-28-0': 'He used group theoretical methods, however declared (private communication) to be not familiar with spinor theory.', '0905.2902-1-28-1': 'There are more authors who with other methods computed other quantum constants (including [MATH]) relevant for particle physics [CITATION].', '0905.2902-1-28-2': 'Now the point is if their computations may be correlated or derived from the compact manifolds in momentum space derived from pure spinors.', '0905.2902-1-29-0': 'VI.', '0905.2902-1-29-1': 'The wave function.', '0905.2902-1-29-2': 'If we follow the suggestion of Ref. [CITATION] that quantum dynamics has to be formulated and dealt with in momentum space, then we have to abandon the concept of wave function representing the electron say, in the Schrodinger equation (at the source of several paradoxes).', '0905.2902-1-29-3': 'As we have seen in momentum space there is no possibility of defining the point event.', '0905.2902-1-29-4': 'We may instead define integrals of null vectors, bilinear in pure spinors which uniquely define strings, which are non local but relativistically covariant; furthermore, for their existence in nature, there is experimental evidence since they were discovered from the interpretation of some experiments performed at CERN (Geneve).', '0905.2902-1-29-5': 'It should not be difficult to obtain from relativistic non local objects also non relativistic ones like in the Fock integral equation, which, as seen above, presents relativistic properties.', '0905.2902-1-30-0': 'VII.', '0905.2902-1-30-1': 'From the quantum to the classical field equations.', '0905.2902-1-30-2': "It is well known how, from the Weyl equation for massless neutrinos, equation (1) one may easily obtain the classical Maxwell's equations which, when quantized give the equations of motion for the massless and chargeless photon.", '0905.2902-1-30-3': 'It also results from [CITATION] that for [MATH] dimensional momentum space [MATH]; that is [MATH].', '0905.2902-1-31-0': 'In this way the massless photon may be conceived as bilinearly generated by massless neutrinos representing elementary pure spinors (but notoriously not their bound states).', '0905.2902-1-32-0': 'Already E. Cartan named electromagnetic tensor the one contained in the Clifford algebra [MATH]: [EQUATION]', '0905.2902-1-32-1': "In fact, if we apply the Weyl operator of equation (1) we obtain: [EQUATION] which are the well known Maxwell's equations in vacuum.", '0905.2902-1-33-0': 'At this point a natural question presents itself - why not try to get from pure spinors also the gravitational field equations?', '0905.2902-1-33-1': 'These equations present enormous difficulties that for a long time blocked the progress of theoretical physics and the efforts of the best present physicists in Trieste (and no longer present, like my dear friend Dennis Shama).', '0905.2902-1-33-2': 'The best answer for a long time has been: because it is too difficult to get symmetric tensors from Clifford Algebras.', '0905.2902-1-34-0': 'But now, there is a way to generate symmetric tensors from Clifford algebras: they are entered through the [MATH] of the electroweak model (represented by [MATH] in the standard model).', '0905.2902-1-34-1': 'And then let us try.', '0905.2902-1-35-0': 'Let us then consider an element of the electroweak model of the form: [EQUATION] it is obviously symmetric like should be the metric tensor (fundamental for general relativity!)', '0905.2902-1-35-1': 'and it is quadrilinear in pure spinors.', '0905.2902-1-36-0': "The equation for the massless and chargeless photon has been derived from Maxwell's equations.", '0905.2902-1-36-1': "A similar result should also be obtained for the massless and chargeless graviton, which however will have to result as quadrilinear of massless neutrinos since its spin is [MATH] instead of [MATH]; and as we will see, this can be done in the following way, precisely starting from the standard model in its form: [MATH] where [MATH] represents the origin of the Electro Weak mode and contains products of the left-handed currents which, through equation (1) for the massless neutrinos, may produce the gravitational equation in flat space in a similar way as they produce Maxwell's equations for the massless photons.", '0905.2902-1-37-0': 'In fact, suppose that in equation (18) [MATH] represents the existing weak decay, Neutron [MATH] Proton [MATH] lepton [MATH] neutrinos (or the ones in which the fermions are substituted by one of the 3 twins of the same family), and we would get the graviton if we could consider chargeless the [MATH] proton and [MATH] lepton which appear on the right-hand side that should be both chargeless and massless like the two neutrinos.', '0905.2902-1-37-1': 'This possibility should certainly be discussed and analyzed further.', '0905.2902-1-38-0': "The situation appears difficult but not desperate since from the Weyl equation for the neutrino (1) we obtain the Maxwell's equations (17), notoriously extendable also to charges.", '0905.2902-1-38-1': 'From these in turn, we may derive the equations for the chargeless and massless photon.', '0905.2902-1-38-2': 'A similar procedure could be possible also for gravity since, as well known, its renormalizability was not solved through sophisticated supergravity and it is quite possible that it could be attacked going back to the first steps as suggested above.', '0905.2902-1-39-0': 'A possible way could be: let us start from the obviously symmetric tensor [MATH] in (18) and postulate the obviously equivalent equations [EQUATION] which are similar to the first of the equations (17) from which one derives the zero mass of the photon.', '0905.2902-1-39-1': 'It remains to be discussed if and which phenomena are represented by the equations with the emisymmetric tensor [MATH].', '0905.2902-1-40-0': 'VIII.', '0905.2902-1-40-1': 'Conclusions and outlook.', '0905.2902-1-40-2': 'In this paper we followed the suggestions of one of our great predecessors: Elie Cartan, the discoverer of spinors.', '0905.2902-1-40-3': "The main Cartan's merits are to have underlined the importance of simple spinors and of being the first to see in Clifford algebras an emisymmetric tensor, that he named electromagnetic tensor.", '0905.2902-1-40-4': "From this tensor one may derive the Maxwell's equations (17) from which only the first one, the simplest, is necessary to derive the zero mass and charge of the photon which, then, may be interpreted as bilinear in massless neutrinos (but not their bound state).", '0905.2902-1-41-0': 'Another great physicist was P.A.M. Dirac who, in 1938 discovered the "deep connection in Nature between cosmology and atomic physics" [CITATION] and which we used in [CITATION] in order to search (and perhaps to find) a mathematical way to quantum mechanics.', '0905.2902-1-42-0': 'Also for Dirac we could guess some hidden suggestion of a great greek philosopher, Parmenides that defined "to be" or "existence" as that what neither was nor will be but always is ("nor was once, nor will be, since is, now, all together" I apologize if my memory is not perfect).', '0905.2902-1-42-1': 'In this way, for us, Parmenides connected very distant concepts.', '0905.2902-1-43-0': 'But for us it would be wonderful if from equations (19) one might derive the zero mass and charge of the graviton represented by the superposition of four neutrinos (but not their bound state).', '0905.2902-1-44-0': 'for_Paolo', '0905.2902-1-45-0': 'std'}

{'0905.2902-2-0-0': "In a previous paper [CITATION] we proposed a purely mathematical way to quantum mechanics based on Cartan's simple spinors in their most elementary form of 2 components spinors.", '0905.2902-2-0-1': 'Here we proceed along that path proposing, this time, a symmetric tensor, quadrilinear in simple spinors, as a candidate for the symmetric tensor of general relativity.', '0905.2902-2-1-0': "The procedure resembles closely that in which one builds bilinearly from simple spinors an asymmetric electromagnetic tensor, from which easily descend Maxwell's equations and the photon can be seen as a bilinear combination of neutrinos.", '0905.2902-2-1-1': 'Here Lorentzian spaces result compact, building up spheres, where hopefully the problems of the Standard Model could be solved.', '0905.2902-2-2-0': 'std', '0905.2902-2-3-0': 'for_Paolo', '0905.2902-2-4-0': '# Introduction', '0905.2902-2-5-0': 'In a previous paper [CITATION] it was shown how, from the simplest, non trivial, two component spinors, which are pure, we may bilinearly and quite naturally obtain null vectors which are building up Minkowski momentum space.', '0905.2902-2-5-1': 'With these, in fact, one may formulate the well-known Weyl equations of motion for massless neutrinos as follows: [EQUATION] where [MATH] is a spinor (Dirac) associated with the Clifford algebra [MATH] with [MATH] its generators, and [MATH] its volume element.', '0905.2902-2-6-0': 'In [CITATION] [MATH] are bilinearly obtained precisely from those two component pure spinors from which we started.', '0905.2902-2-7-0': 'This was a first elementary result of Ref. [CITATION] which was aimed at searching a purely mathematical formulation of quantum mechanics.', '0905.2902-2-7-1': 'A similar way was followed by Einstein, Poincare, and other outstanding mathematicians in the early part of last century, when they arrived at special and general relativity.', '0905.2902-2-7-2': 'In this way one could avoid the plague of paradoxes which instead, up to our time, ruined the development of quantum mechanics.', '0905.2902-2-7-3': 'In Ref. [CITATION] two more main results were obtained:', '0905.2902-2-8-0': 'I.', '0905.2902-2-8-1': "While the classical dynamics of macroscopic bodies has to be formulated and dealt with in ordinary (Minkowski possibly curved) space-time, as we learned from Newton, Lagrange, Hamilton, Jacobi, Einstein, Poincare and others, so here the basic geometry is an Euclid's one, including the concept of point-event, which is necessary to represent the center of mass of the macroscopic bodies while running along their trajectories or orbits (like the Kepler orbits).", '0905.2902-2-9-0': 'For atomic physics instead, the appropriate space for the formulation of dynamical quantum equations and for their solution is momentum-space bilinearly constructed from pure spinors, as it appears in (1) the first elementary example of the Weyl equations.', '0905.2902-2-10-0': 'II.', '0905.2902-2-10-1': 'Pure Spinors.', '0905.2902-2-10-2': 'E. Cartan, the discoverer of spinors, underlined the great elegance of the spinors he named simple [CITATION] (renamed pure by Chevalley [CITATION]).', '0905.2902-2-11-0': 'He started by considering the correlations between spinors and totally null planes as follows: let [MATH] represent a complex Euclidean, [MATH] dimensional space, with well-defined quadratic forms.', '0905.2902-2-11-1': 'Then we may define the corresponding Clifford algebra [MATH] which, as known, may be considered as the endomorphism algebra in a [MATH] dimensional space [MATH] of spinors, and we write: [EQUATION]', '0905.2902-2-11-2': 'A spinor [MATH] may be defined by the equation [EQUATION] where [MATH] is a vector of [MATH] (referred to a Cartesian orthonormal coordinate system) and [MATH], called the generators of [MATH], represent the univectors in the direction of the [MATH] orthonormal coordinates of [MATH].', '0905.2902-2-12-0': 'The generators [MATH] obey the anticommutation relations.', '0905.2902-2-12-1': '[EQUATION]', '0905.2902-2-12-2': 'Let us now multiply equation (3) from the left by [MATH] and we obtain (because of equation (4)): [EQUATION] and, if [MATH]: [EQUATION] which means that the non zero spinor [MATH], satisfying equation (3) (sometimes called Dirac spinor) defines a subspace of [MATH] whose vectors are all null and/or mutually orthogonal, it is called the totally null plane associated to [MATH] and indicated with [MATH] where [MATH] is its dimension.', '0905.2902-2-12-3': 'It is known, and easy to prove, that the maximal possible value of [MATH] is [MATH], that is, one half of the dimension [MATH] of the space [MATH].', '0905.2902-2-13-0': 'In order to arrive at the Cartan definition of simple or pure spinors, let us define the "volume element" of [MATH]: [EQUATION]', '0905.2902-2-13-1': 'It is easy to show that [MATH] anticommutes with all the [MATH] and that [MATH] generate the Clifford algebra [MATH] whose even subalgebra [MATH] is isomorphic to the simple algebra [MATH]: [EQUATION]', '0905.2902-2-13-2': 'With [MATH] one may define what are called the Weyl spinors [MATH]: [EQUATION] where [MATH] is the Dirac spinor defined by equation (3).', '0905.2902-2-14-0': 'The Weyl spinors may be defined by the Weyl equations [EQUATION]', '0905.2902-2-14-1': 'Obviously also [MATH] will define totally null planes in [MATH].', '0905.2902-2-14-2': 'Well E. Cartan showed how a simple or pure spinor is isomorphic, up to a sign, to the maximal totally null plane of the Weyl spinor associated with a given Clifford algebra, and this property renders pure spinors complicated geometrical objects to deal with.', '0905.2902-2-14-3': 'In fact, while the dimension of the maximal totally null planes increase linearly with [MATH], that of the spinors, increase with [MATH] like [MATH] and then high dimensional spinors which are used to explain some phenomena of elementary particle physics (up to [MATH] component spinors), need a lot of (ten for 32 components) constraint equations in their components to render them pure.', '0905.2902-2-14-4': 'In short, all spinors up to [MATH] components are pure (or equivalent to pure).', '0905.2902-2-15-0': 'At present several people have tried to use pure spinors to explain both the high energy phenomena of the elementary particles and the unsolved problem of quantization of the gravitational field (through superstring theory) but with no acceptable results.', '0905.2902-2-16-0': 'In fact, the so-called Standard model, now represented by the symbols [MATH], which represents the symmetries presented in high-energy elementary particle phenomena, contains more than 20 constants representing charges, masses and so on, which may not be computed deriving them from acceptable theory.', '0905.2902-2-17-0': 'But now with the possible prescription found in [CITATION] that quantum dynamical problems have to be dealt with in momentum-space rather than in space-time, and because of a theorem we recently discovered, the situation could drastically change.', '0905.2902-2-18-0': 'III.', '0905.2902-2-18-1': 'The theorem.', '0905.2902-2-18-2': 'Consider two Weyl spinors: [MATH] and [MATH] associated with a Clifford algebra [MATH] and define the vector of the space [MATH], with components: [EQUATION] where [MATH].', '0905.2902-2-18-3': 'The vector [MATH] is null: [MATH], if and only if one of the two spinors, either [MATH] or [MATH], is pure.', '0905.2902-2-18-4': 'The proof is in Ref. [CITATION].', '0905.2902-2-19-0': 'It is known that in [MATH] dimensional spaces with Lorentzian signatures like [MATH] or [MATH] the vector components will be real if of the form [MATH], where [MATH] and [MATH] means [MATH] hermitian conjugate while [MATH] is the time-like generator.', '0905.2902-2-20-0': 'Suppose now [MATH] to be pure then we will have that: [EQUATION] with real components, (12) may be written in the form: [EQUATION] where [MATH]; which, might arrive at [MATH] (exploiting the isomorphism between Dirac and Weyl spinors doublets and conformal covariance [CITATION]).', '0905.2902-2-21-0': 'It is interesting to observe that in the momentum space, where according to the conclusion of Ref. [CITATION], we have to formulate the dynamical problems of quantum physics, pure spinors may bilinearly define in Lorentzian momentum spaces compact manifolds consisting of spheres (with Poincare invariant radii).', '0905.2902-2-21-1': 'This might obviously encourage the hope that not only the mathematical origin of quantum jumps might finally find a simple geometrical explanation, (the auto-vibrations of spheres are discrete) but also that, hopefully, the transitory sickness of the otherwise beautiful Standard Model, manifested by the symptom of its more than 20 unexplained constants, may finally be cured.', '0905.2902-2-22-0': 'IV.', '0905.2902-2-22-1': 'First applications and results.', '0905.2902-2-22-2': 'It might sound strange, but it is true: the first convincing confirmation of the above predictions was found by V. Fock more than 70 years ago: in 1935 [CITATION].', '0905.2902-2-22-3': 'He dealt with the historical problem (the first one to deal with atomic quantization) of the H-atom stationary states.', '0905.2902-2-23-0': 'At that time the problem was solved through the Schrodinger equation for the electron possible orbits in the field of the proton.', '0905.2902-2-23-1': 'Fock, anticipating the conclusions of [CITATION], formulated it in momentum space on a sphere [MATH] conceived as the one point compactification of ordinary [MATH] dimensional momentum space [MATH] with the following integral equation: [EQUATION] where [MATH] is the volume of the [MATH] unit sphere; [MATH] is the fine structure constant, [MATH] a unit of momentum, [MATH] the (reduced) electron mass and [MATH] is a unit vector indicating a point on the unit sphere [MATH], equation (14) is the Fock equation in adimensional form.', '0905.2902-2-23-2': 'Fock solved it through harmonic analysis and found that setting [MATH] he obtained for [MATH], representing the eigen vibrations of the sphere [MATH], the energy levels [MATH] of the H-atom stationary states, which in turn explain the Balmer spectrum of the H-atom.', '0905.2902-2-23-3': 'But it was V. Fock himself to draw from equation (14) the fundamental discovery that an integral equation on the sphere [MATH], like equation (14), has to forsee the symmetry [MATH] of its solutions (including obviously their classical counterparts - the solar planets - which is true, as later underlined by the great W. Pauli, Nobel prize winner).', '0905.2902-2-24-0': 'Now we may go one step further representing the H-atom with a quadruplet of spinors say (Proton, Neutron, Neutrino and Electron) and adopt equation (13) with [MATH] [CITATION]: since it is well known that spinors up to 4 components are pure (or equivalent to pure, and this equivalence is exploited here) [CITATION] [EQUATION] from which we obtain [MATH] with Poincare invariant radius [MATH] therefore we may expect that the spinorial [MATH] may reflect the Poincare invariance; and in the energy levels found by Fock we have the obviously relativistic form [CITATION]: [EQUATION]', '0905.2902-2-24-1': 'If we take this first result as an indication of the validity of the hypothesis formulated above [CITATION] we may affirm that the first example of atomic quantization confirms that the H-atom discrete energy levels (generating its discrete optical spectrum - Balmer series) are generated by the computed manifolds determined by pure spinors in momentum space.', '0905.2902-2-24-2': 'In particular by the eigen vibrations of those spheres we may also geometrically explain the quantum-jumps.', '0905.2902-2-25-0': 'Should this result have confirmations by further examples then the historical way to arrive at the solution of the problem of the H-atom stationary states would not only be more complicated but also wrong.', '0905.2902-2-25-1': 'It would be wrong to consider the H-atom as a proton plus an orbitating point electron to be then substituted arbitrarily by wave functions.', '0905.2902-2-26-0': 'As stated in [CITATION] we have to formulate the quantum problems in momentum space where we have the mathematical possibility to integrate null vectors to generate strings [CITATION], which are non local objects and might be at the origin of the concept of wave function without giving origin to paradoxes, as will be further discussed, elsewhere.', '0905.2902-2-27-0': 'V.', '0905.2902-2-27-1': 'The possibility to cure the Standard Model.', '0905.2902-2-27-2': 'V. Fock solved equation (14) computing the adimensional factors [MATH] through harmonic analysis from the ball [MATH] to [MATH].', '0905.2902-2-27-3': 'We tried (with P. Nurowski) to compute, through harmonic analysis, also the other adimensional factor [MATH] in equation (14) obtaining all the factors but one.', '0905.2902-2-27-4': 'In fact, [MATH] was computed for the first time by M.A. Wyler in 1969 [CITATION] with the result: [EQUATION] differing less than [MATH] from the experimental value.', '0905.2902-2-28-0': 'He used group theoretical methods, however declared (private communication) to be not familiar with spinor theory.', '0905.2902-2-28-1': 'There are more authors who with other methods computed other quantum constants (including [MATH]) relevant for particle physics [CITATION].', '0905.2902-2-28-2': 'Now the point is if their computations may be correlated or derived from the compact manifolds in momentum space derived from pure spinors.', '0905.2902-2-29-0': 'VI.', '0905.2902-2-29-1': 'The wave function.', '0905.2902-2-29-2': 'If we follow the suggestion of Ref. [CITATION] that quantum dynamics has to be formulated and dealt with in momentum space, then we have to abandon the concept of wave function representing the electron say, in the Schrodinger equation (at the source of several paradoxes).', '0905.2902-2-29-3': 'As we have seen in momentum space there is no possibility of defining the point event.', '0905.2902-2-29-4': 'We may instead define integrals of null vectors, bilinear in pure spinors which uniquely define strings, which are non local but relativistically covariant; furthermore, for their existence in nature, there is experimental evidence since they were discovered from the interpretation of some experiments performed at CERN (Geneve).', '0905.2902-2-29-5': 'It should not be difficult to obtain from relativistic non local objects also non relativistic ones like in the Fock integral equation, which, as seen above, presents relativistic properties.', '0905.2902-2-30-0': 'VII.', '0905.2902-2-30-1': 'From the quantum to the classical field equations.', '0905.2902-2-30-2': "It is well known how, from the Weyl equation for massless neutrinos, equation (1) one may easily obtain the classical Maxwell's equations which, when quantized give the equations of motion for the massless and chargeless photon.", '0905.2902-2-30-3': 'It also results from [CITATION] that for [MATH] dimensional momentum space [MATH]; that is [MATH].', '0905.2902-2-31-0': 'In this way the massless photon may be conceived as bilinearly generated by massless neutrinos representing elementary pure spinors (but notoriously not their bound states).', '0905.2902-2-32-0': 'Already E. Cartan named electromagnetic tensor the one contained in the Clifford algebra [MATH]: [EQUATION]', '0905.2902-2-32-1': "In fact, if we apply the Weyl operator of equation (1) we obtain: [EQUATION] which are the well known Maxwell's equations in vacuum.", '0905.2902-2-33-0': 'At this point a natural question presents itself - why not try to get from pure spinors also the gravitational field equations?', '0905.2902-2-33-1': 'These equations present enormous difficulties that for a long time blocked the progress of theoretical physics and the efforts of the best present physicists in Trieste (and no longer present, like my dear friend Dennis Shama).', '0905.2902-2-33-2': 'The best answer for a long time has been: because it is too difficult to get symmetric tensors from Clifford Algebras.', '0905.2902-2-34-0': 'But now, there is a way to generate symmetric tensors from Clifford algebras: they are entered through the [MATH] of the electroweak model (represented by [MATH] in the standard model).', '0905.2902-2-34-1': 'And then let us try.', '0905.2902-2-35-0': 'Let us then consider an element of the electroweak model of the form: [EQUATION] it is obviously symmetric like should be the metric tensor (fundamental for general relativity!)', '0905.2902-2-35-1': 'and it is quadrilinear in pure spinors.', '0905.2902-2-36-0': "The equation for the massless and chargeless photon has been derived from Maxwell's equations.", '0905.2902-2-36-1': "A similar result should also be obtained for the massless and chargeless graviton, which however will have to result as quadrilinear of massless neutrinos since its spin is [MATH] instead of [MATH]; and as we will see, this can be done in the following way, precisely starting from the standard model in its form: [MATH] where [MATH] represents the origin of the Electro Weak mode and contains products of the left-handed currents which, through equation (1) for the massless neutrinos, may produce the gravitational equation in flat space in a similar way as they produce Maxwell's equations for the massless photons.", '0905.2902-2-37-0': 'In fact, suppose that in equation (18) [MATH] represents the existing weak decay, Neutron [MATH] Proton [MATH] lepton [MATH] neutrinos (or the ones in which the fermions are substituted by one of the 3 twins of the same family), and we would get the graviton if we could consider chargeless the [MATH] proton and [MATH] lepton which appear on the right-hand side that should be both chargeless and massless like the two neutrinos.', '0905.2902-2-37-1': 'This possibility should certainly be discussed and analyzed further.', '0905.2902-2-38-0': "The situation appears difficult but not desperate since from the Weyl equation for the neutrino (1) we obtain the Maxwell's equations (17), notoriously extendable also to charges.", '0905.2902-2-38-1': 'From these in turn, we may derive the equations for the chargeless and massless photon.', '0905.2902-2-38-2': 'A similar procedure could be possible also for gravity since, as well known, its renormalizability was not solved through sophisticated supergravity and it is quite possible that it could be attacked going back to the first steps as suggested above.', '0905.2902-2-39-0': 'A possible way could be: let us start from the obviously symmetric tensor [MATH] in (18) and postulate the obviously equivalent equations [EQUATION] which are similar to the first of the equations (17) from which one derives the zero mass of the photon.', '0905.2902-2-39-1': 'It remains to be discussed if and which phenomena are represented by the equations with the emisymmetric tensor [MATH].', '0905.2902-2-40-0': 'VIII.', '0905.2902-2-40-1': 'Conclusions and outlook.', '0905.2902-2-40-2': 'In this paper we followed the suggestions of one of our great predecessors: Elie Cartan, the discoverer of spinors.', '0905.2902-2-40-3': "The main Cartan's merits are to have underlined the importance of simple spinors and of being the first to see in Clifford algebras an emisymmetric tensor, that he named electromagnetic tensor.", '0905.2902-2-40-4': "From this tensor one may derive the Maxwell's equations (17) from which only the first one, the simplest, is necessary to derive the zero mass and charge of the photon which, then, may be interpreted as bilinear in massless neutrinos (but not their bound state).", '0905.2902-2-41-0': 'Another great physicist was P.A.M. Dirac who, in 1938 discovered the "deep connection in Nature between cosmology and atomic physics" [CITATION] and which we used in [CITATION] in order to search (and perhaps to find) a mathematical way to quantum mechanics.', '0905.2902-2-42-0': 'Also for Dirac we could guess some hidden suggestion of a great greek philosopher, Parmenides that defined "to be" or "existence" as that what neither was nor will be but always is ("nor was once, nor will be, since is, now, all together" I apologize if my memory is not perfect).', '0905.2902-2-42-1': 'In this way, for us, Parmenides connected very distant concepts.', '0905.2902-2-43-0': 'But for us it would be wonderful if from equations (19) one might derive the zero mass and charge of the graviton represented by the superposition of four neutrinos (but not their bound state).', '0905.2902-2-44-0': 'for_Paolo', '0905.2902-2-45-0': 'std'}

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{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '3': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '4': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/0905.2902

{'0905.2902-3-0-0': "In a previous paper [CITATION] we proposed a purely mathematical way to quantum mechanics based on Cartan's simple spinors in their most elementary form of 2 components spinors.", '0905.2902-3-0-1': 'Here we proceed along that path proposing, this time, a symmetric tensor, quadrilinear in simple spinors, as a candidate for the symmetric tensor of general relativity.', '0905.2902-3-1-0': "The procedure resembles closely that in which one builds bilinearly from simple spinors an antisymmetric electromagnetic tensor, from which easily descend Maxwell's equations and the photon can be seen as a bilinear combination of neutrinos.", '0905.2902-3-1-1': 'Here Lorentzian spaces result compact, building up spheres, where hopefully the problems of the Standard Model could be solved, but, to obtain the graviton, one has to use a symmetric tensor quadrilinear in Weyl spinors.', '0905.2902-3-2-0': 'std', '0905.2902-3-3-0': 'for_Paolo', '0905.2902-3-4-0': '# Introduction', '0905.2902-3-5-0': 'In a previous paper [CITATION] it was shown how, from the simplest, non trivial, two component spinors, which are pure, we may bilinearly and quite naturally obtain null vectors which are building up Minkowski momentum space.', '0905.2902-3-5-1': 'With these, in fact, one may formulate the well-known Weyl equations of motion for massless neutrinos as follows: [EQUATION] where [MATH] is a spinor (Dirac) associated with the Clifford algebra [MATH] with [MATH] its generators, and [MATH] its volume element.', '0905.2902-3-6-0': 'In [CITATION] [MATH] are bilinearly obtained precisely from those two component pure spinors from which we started.', '0905.2902-3-7-0': 'This was a first elementary result of Ref. [CITATION] which was aimed at searching a purely mathematical formulation of quantum mechanics.', '0905.2902-3-7-1': 'A similar way was followed by Einstein, Poincare, and other outstanding mathematicians in the early part of last century, when they arrived at special and general relativity.', '0905.2902-3-7-2': 'In this way one could avoid the plague of paradoxes which instead, up to our time, ruined the development of quantum mechanics.', '0905.2902-3-7-3': 'In Ref. [CITATION] two more main results were obtained:', '0905.2902-3-8-0': "I. while the classical dynamics of macroscopic bodies has to be formulated and dealt with in ordinary (Minkowski possibly curved) space-time, as we learned from Newton, Lagrange, Hamilton, Jacobi, Einstein, Poincare and others, so here the basic geometry is an Euclid's one, including the concept of point-event, which is necessary to represent the center of mass of the macroscopic bodies while running along their trajectories or orbits (like the Kepler orbits).", '0905.2902-3-9-0': 'II.', '0905.2902-3-9-1': 'For atomic physics instead, the appropriate space for the formulation of dynamical quantum equations and for their solution is momentum-space bilinearly constructed from pure spinors, as it appears in (1) the first elementary example of the Weyl equations.', '0905.2902-3-10-0': '# Pure Spinors', '0905.2902-3-11-0': 'E. Cartan, the discoverer of spinors, underlined the great elegance of the spinors he named simple [CITATION] (renamed pure by Chevalley [CITATION]).', '0905.2902-3-12-0': 'He started by considering the correlations between spinors and totally null planes as follows: let [MATH] represent a complex Euclidean, [MATH] dimensional space, with well-defined quadratic forms.', '0905.2902-3-12-1': 'Then we may define the corresponding Clifford algebra [MATH] which, as known, may be considered as the endomorphism algebra in a [MATH] dimensional space [MATH] of spinors, and we write: [EQUATION]', '0905.2902-3-12-2': 'A spinor [MATH] may be defined by the equation [EQUATION] where [MATH] is a vector of [MATH] (referred to a Cartesian orthonormal coordinate system) and [MATH], called the generators of [MATH], represent the univectors in the direction of the [MATH] orthonormal coordinates of [MATH].', '0905.2902-3-13-0': 'The generators [MATH] obey the anticommutation relations.', '0905.2902-3-13-1': '[EQUATION]', '0905.2902-3-13-2': 'Let us now multiply equation (3) from the left by [MATH] and we obtain (because of equation (4)): [EQUATION] and, if [MATH]: [EQUATION] which means that the non zero spinor [MATH], satisfying equation (3) (sometimes called Dirac spinor) defines a subspace of [MATH] whose vectors are all null and/or mutually orthogonal, it is called the totally null plane associated to [MATH] and indicated with [MATH] where [MATH] is its dimension.', '0905.2902-3-13-3': 'It is known, and easy to prove, that the maximal possible value of [MATH] is [MATH], that is, one half of the dimension [MATH] of the space [MATH].', '0905.2902-3-14-0': 'In order to arrive at the Cartan definition of simple or pure spinors, let us define the "volume element" of [MATH]: [EQUATION]', '0905.2902-3-14-1': 'It is easy to show that [MATH] anticommutes with all the [MATH] and that [MATH] generate the Clifford algebra [MATH] whose even subalgebra [MATH] is isomorphic to the simple algebra [MATH]: [EQUATION]', '0905.2902-3-14-2': 'With [MATH] one may define what are called the Weyl spinors [MATH]: [EQUATION] where [MATH] is the Dirac spinor defined by equation (3).', '0905.2902-3-15-0': 'The Weyl spinors may be defined by the Weyl equations [EQUATION]', '0905.2902-3-15-1': 'Obviously also [MATH] will define totally null planes in [MATH].', '0905.2902-3-15-2': 'Well E. Cartan showed how a simple or pure spinor is isomorphic, up to a sign, to the maximal totally null plane of the Weyl spinor associated with a given Clifford algebra, and this property renders pure spinors complicated geometrical objects to deal with.', '0905.2902-3-15-3': 'In fact, while the dimension of the maximal totally null planes increase linearly with [MATH], that of the spinors, increase with [MATH] like [MATH] and then high dimensional spinors which are used to explain some phenomena of elementary particle physics (up to [MATH] component spinors), need a lot of (ten for 32 components) constraint equations in their components to render them pure.', '0905.2902-3-15-4': 'In short, all spinors up to [MATH] components are pure (or equivalent to pure).', '0905.2902-3-16-0': 'At present several people have tried to use pure spinors to explain both the high energy phenomena of the elementary particles and the unsolved problem of quantization of the gravitational field (through superstring theory) but with no acceptable results.', '0905.2902-3-17-0': 'In fact, the so-called Standard model, now represented by the symbols [MATH], which represents the symmetries presented in high-energy elementary particle phenomena, contains more than 20 constants representing charges, masses and so on, which may not be computed deriving them from acceptable theory.', '0905.2902-3-18-0': 'But now with the possible prescription found in [CITATION] that quantum dynamical problems have to be dealt with in momentum-space rather than in space-time, and because of a theorem we recently discovered, the situation could drastically change.', '0905.2902-3-19-0': '# The theorem', '0905.2902-3-20-0': 'Consider two Weyl spinors: [MATH] and [MATH] associated with a Clifford algebra [MATH] and define the vector of the space [MATH], with components: [EQUATION] where [MATH].', '0905.2902-3-20-1': 'The vector [MATH] is null: [MATH], if and only if one of the two spinors, either [MATH] or [MATH], is pure.', '0905.2902-3-20-2': 'The proof is in Ref. [CITATION].', '0905.2902-3-21-0': 'It is known that in [MATH] dimensional spaces with Lorentzian signatures like [MATH] or [MATH] the vector components will be real if of the form [MATH], where [MATH] and [MATH] means [MATH] hermitian conjugate while [MATH] is the time-like generator.', '0905.2902-3-22-0': 'Suppose now [MATH] to be pure then we will have that: [EQUATION] with real components, (12) may be written in the form: [EQUATION] where [MATH]; which, might arrive (in the r.h.s.) at [MATH] (exploiting the isomorphism between Dirac and Weyl spinors doublets and conformal covariance [CITATION]).', '0905.2902-3-23-0': 'It is interesting to observe that in the momentum space, where according to the conclusion of Ref. [CITATION], we have to formulate the dynamical problems of quantum physics, pure spinors may bilinearly define in Lorentzian momentum spaces compact manifolds consisting of spheres (with Poincare invariant radii).', '0905.2902-3-23-1': 'This might obviously encourage the hope that not only the mathematical origin of quantum jumps might finally find a simple geometrical explanation, (the auto-vibrations of spheres are discrete) but also that, hopefully, the transitory sickness of the otherwise beautiful Standard Model, manifested by the symptom of its more than 20 unexplained constants, may finally be cured.', '0905.2902-3-24-0': '# First applications and results', '0905.2902-3-25-0': 'It might sound strange, but it is true: the first convincing confirmation of the above predictions was found by V. Fock more than 70 years ago: in 1935 [CITATION].', '0905.2902-3-25-1': 'He dealt with the historical problem (the first one to deal with atomic quantization) of the H-atom stationary states.', '0905.2902-3-26-0': 'At that time the problem was solved through the Schrodinger equation for the electron possible orbits in the field of the proton.', '0905.2902-3-26-1': 'Fock, anticipating the conclusions of [CITATION], formulated it in momentum space on a sphere [MATH] conceived as the one point compactification of ordinary [MATH] dimensional momentum space [MATH] with the following integral equation: [EQUATION] where [MATH] is the volume of the [MATH] unit sphere; [MATH] is the fine structure constant, [MATH] a unit of momentum, [MATH] the (reduced) electron mass and [MATH] is a unit vector indicating a point on the unit sphere [MATH], equation (14) is the Fock equation in adimensional form.', '0905.2902-3-26-2': 'Fock solved it through harmonic analysis and found that setting [MATH] he obtained for [MATH], representing the eigen vibrations of the sphere [MATH], the energy levels [MATH] of the H-atom stationary states, which in turn explain the Balmer spectrum of the H-atom.', '0905.2902-3-26-3': 'But it was V. Fock himself to draw from equation (14) the fundamental discovery that an integral equation on the sphere [MATH], like equation (14), has to forsee the symmetry [MATH] of its solutions (including obviously their classical counterparts - the solar planets - which is true, as later underlined by the great W. Pauli, Nobel prize winner).', '0905.2902-3-27-0': 'Now we may go one step further representing the H-atom with a quadruplet of spinors say (Proton, Neutron, Neutrino and Electron) and adopt equation (13) with [MATH] [CITATION]: since it is well known that spinors up to 4 components are pure (or equivalent to pure, and this equivalence is exploited here) [CITATION] [EQUATION] from which we obtain [MATH] with Poincare invariant radius [MATH] therefore we may expect that the spinorial [MATH] may reflect the Poincare invariance; and in the energy levels found by Fock we have the obviously relativistic form [CITATION]: [EQUATION]', '0905.2902-3-27-1': 'If we take this first result as an indication of the validity of the hypothesis formulated above [CITATION] we may affirm that the first example of atomic quantization confirms that the H-atom discrete energy levels (generating its discrete optical spectrum - Balmer series) are generated by the computed manifolds determined by pure spinors in momentum space.', '0905.2902-3-27-2': 'In particular by the eigen vibrations of those spheres we may also geometrically explain the quantum-jumps.', '0905.2902-3-28-0': 'Should this result have confirmations by further examples then the historical way to arrive at the solution of the problem of the H-atom stationary states would not only be more complicated but also wrong.', '0905.2902-3-28-1': 'It would be wrong to consider the H-atom as a proton plus an orbitating point electron to be then substituted arbitrarily by wave functions.', '0905.2902-3-29-0': 'As stated in [CITATION] we have to formulate the quantum problems in momentum space where we have the mathematical possibility to integrate null vectors to generate strings [CITATION], which are non local objects and might be at the origin of the concept of wave function without giving origin to paradoxes, as will be further discussed, elsewhere.', '0905.2902-3-30-0': '# The possibility to cure the Standard Model', '0905.2902-3-31-0': 'V. Fock solved equation (14) computing the adimensional factors [MATH] through harmonic analysis from the ball [MATH] to [MATH].', '0905.2902-3-31-1': 'We tried (with P. Nurowski) to compute, through harmonic analysis, also the other adimensional factor [MATH] in equation (14) obtaining all the factors but one.', '0905.2902-3-31-2': 'In fact, [MATH] was computed for the first time by M.A. Wyler in 1969 [CITATION] with the result: [EQUATION] differing less than [MATH] from the experimental value.', '0905.2902-3-32-0': 'He used group theoretical methods, however declared (private communication) to be not familiar with spinor theory.', '0905.2902-3-32-1': 'There are more authors who with other methods computed other quantum constants (including [MATH]) relevant for particle physics [CITATION].', '0905.2902-3-32-2': 'Now the point is if their computations may be correlated or derived from the compact manifolds in momentum space derived from pure spinors.', '0905.2902-3-33-0': '# The wave function', '0905.2902-3-34-0': 'If we follow the suggestion of Ref. [CITATION] that quantum dynamics has to be formulated and dealt with in momentum space, then we have to abandon the concept of wave function representing the electron say, in the Schrodinger equation (at the source of several paradoxes).', '0905.2902-3-34-1': 'As we have seen in momentum space there is no possibility of defining the point event.', '0905.2902-3-34-2': 'We may instead define integrals of null vectors, bilinear in pure spinors which uniquely define strings, which are non local but relativistically covariant; furthermore, for their existence in nature, there is experimental evidence since they were discovered from the interpretation of some experiments performed at CERN (Geneve).', '0905.2902-3-34-3': 'It should not be difficult to obtain from relativistic non local objects also non relativistic ones like in the Fock integral equation, which, as seen above, presents relativistic properties.', '0905.2902-3-35-0': '# From the quantum to the classical field equations', '0905.2902-3-36-0': "It is well known how, from the Weyl equation for massless neutrinos, equation (1) one may easily obtain the classical Maxwell's equations which, when quantized give the equations of motion for the massless and chargeless photon.", '0905.2902-3-36-1': 'It also results from [CITATION] that for [MATH] dimensional momentum space [MATH]; that is [MATH].', '0905.2902-3-37-0': 'In this way the massless photon may be conceived as bilinearly generated by massless neutrinos representing elementary pure spinors (but notoriously not their bound states).', '0905.2902-3-38-0': 'Already E. Cartan named electromagnetic tensor the one contained in the Clifford algebra [MATH]: [EQUATION]', '0905.2902-3-38-1': "In fact, if we apply the Weyl operator of equation (1) we obtain: [EQUATION] which are the well known Maxwell's equations in vacuum.", '0905.2902-3-39-0': 'At this point a natural question presents itself - why not try to get from pure spinors also the gravitational field equations?', '0905.2902-3-39-1': 'These equations present enormous difficulties that for a long time blocked the progress of theoretical physics and the efforts of the best present physicists in Trieste (and no longer present, like my dear friend Dennis Shama).', '0905.2902-3-39-2': 'The best answer for a long time has been: because it is too difficult to get symmetric tensors from Clifford Algebras.', '0905.2902-3-40-0': 'But now, there is a way to generate symmetric tensors from Clifford algebras: they are entered through the [MATH] of the electroweak model (represented by [MATH] in the standard model).', '0905.2902-3-40-1': 'And then let us try.', '0905.2902-3-41-0': 'Let us then consider an element of the electroweak model of the form: [EQUATION] it is obviously symmetric like should be the metric tensor (fundamental for general relativity!)', '0905.2902-3-41-1': 'and it is quadrilinear in pure spinors.', '0905.2902-3-42-0': "The equation for the massless and chargeless photon has been derived from Maxwell's equations.", '0905.2902-3-42-1': "A similar result should also be obtained for the massless and chargeless graviton, which however will have to result as quadrilinear of massless neutrinos since its spin is [MATH] instead of [MATH]; and as we will see, this can be done in the following way, precisely starting from the standard model in its form: [MATH] where [MATH] represents the origin of the Electro Weak mode and contains products of the left-handed currents which, through equation (1) for the massless neutrinos, may produce the gravitational equation in flat space in a similar way as they produce Maxwell's equations for the massless photons.", '0905.2902-3-43-0': 'In fact, suppose that in equation (18) [MATH] represents the existing weak decay, Neutron [MATH] Proton [MATH] lepton [MATH] neutrinos (or the ones in which the fermions are substituted by one of the 3 twins of the same family), and we would get the graviton if we could consider chargeless the [MATH] proton and [MATH] lepton which appear on the right-hand side that should be both chargeless and massless like the two neutrinos.', '0905.2902-3-43-1': 'This possibility should certainly be discussed and analyzed further.', '0905.2902-3-44-0': "The situation appears difficult but not desperate since from the Weyl equation for the neutrino (1) we obtain the Maxwell's equations (17), notoriously extendable also to charges.", '0905.2902-3-44-1': 'From these in turn, we may derive the equations for the chargeless and massless photon.', '0905.2902-3-44-2': 'A similar procedure could be possible also for gravity since, as well known, its renormalizability was not solved through sophisticated supergravity and it is quite possible that it could be attacked going back to the first steps as suggested above.', '0905.2902-3-45-0': 'A possible way could be: let us start from the obviously symmetric tensor [MATH] in (18) and postulate the obviously equivalent equations [EQUATION] which are similar to the first of the equations (17) from which one derives the zero mass of the photon.', '0905.2902-3-45-1': 'It remains to be discussed if and which phenomena are represented by the equations with the emisymmetric tensor [MATH].', '0905.2902-3-46-0': '# Conclusions and outlook', '0905.2902-3-47-0': 'In this paper we followed the suggestions of one of our great predecessors: Elie Cartan, the discoverer of spinors.', '0905.2902-3-47-1': "One of the main Cartan's merits is to have underlined the importance of simple spinors and of being the first to see in Clifford algebras an emisymmetric tensor, that he named electromagnetic tensor.", '0905.2902-3-47-2': "From this tensor one may derive the Maxwell's equations (17) from which only the first one, the simplest, is necessary to derive the zero mass and charge of the photon which, then, may be interpreted as bilinear in massless neutrinos (but not their bound state).", '0905.2902-3-48-0': 'Another great physicist was P.A.M. Dirac who, in 1938 discovered the "deep connection in Nature between cosmology and atomic physics" [CITATION] and which we used in [CITATION] in order to search (and perhaps to find) a mathematical way to quantum mechanics.', '0905.2902-3-49-0': 'Also for Dirac we could guess some hidden suggestion of a great greek philosopher, Parmenides that defined "to be" or "existence" as that what neither was nor will be but always is ("nor was once, nor will be, since is, now, all together" I apologize if my memory is not perfect).', '0905.2902-3-49-1': 'In this way, for us, Parmenides connected very distant concepts.', '0905.2902-3-50-0': 'But for us it would be wonderful if from equations (19) one might derive the zero mass and charge of the graviton represented by the superposition of four neutrinos (but not their bound state).', '0905.2902-3-51-0': 'for_Paolo', '0905.2902-3-52-0': 'std'}

{'0905.2902-4-0-0': "In a previous paper [CITATION] we proposed a purely mathematical way to quantum mechanics based on Cartan's simple spinors in their most elementary form of 2 component spinors.", '0905.2902-4-0-1': 'Here we proceed along that path proposing, this time, a symmetric tensor, quadrilinear in simple spinors, as a candidate for the symmetric tensor of general relativity.', '0905.2902-4-0-2': 'This is allowed now, after the discovery of the electro-weak model and its introduction in the Standard Model with [MATH].', '0905.2902-4-1-0': 'The procedure resembles closely that in which one builds bilinearly from simple spinors an antisymmetric "electromagnetic tensor", from which easily descend Maxwell\'s equations and the photon can be seen as a bilinear combination of neutrinos.', '0905.2902-4-1-1': 'Here Lorentzian spaces result compact, building up spheres, where hopefully some of the problems of the Standard Model could be solved as pointed out in the conclusions.', '0905.2902-4-2-0': 'std', '0905.2902-4-3-0': 'for_Paolo', '0905.2902-4-4-0': '# Introduction', '0905.2902-4-5-0': 'In a previous paper [CITATION] it was shown how, from the simplest, non trivial, two component spinors, which are pure, we may bilinearly and quite naturally obtain null vectors which are building up Minkowski momentum space.', '0905.2902-4-5-1': 'With these, in fact, one may formulate the well-known Weyl equations of motion for massless neutrinos as follows: [EQUATION] where [MATH] is a spinor (Dirac) associated with the Clifford algebra [MATH] with [MATH] its generators, and [MATH] its volume element.', '0905.2902-4-6-0': 'In [CITATION] [MATH] are bilinearly obtained precisely from those two component pure spinors from which we started.', '0905.2902-4-7-0': 'This was a first elementary result of Ref. [CITATION] which was aimed at searching a purely mathematical formulation of quantum mechanics.', '0905.2902-4-7-1': 'A similar way was followed by Einstein, Poincare, and other outstanding mathematicians in the early part of last century, when they arrived at special and general relativity.', '0905.2902-4-7-2': "In this way one could avoid the plague of paradoxes which instead, up to our time, ruined the development of quantum mechanics mostly deriving from the naive postulate that the Schrodinger (or Dirac) wave function [MATH] has to collapse in the point [MATH] if the particle it represents is observed and that is at the origin of many paradoxes (like the famous Einstein's bed and Schrodinger's cat).", '0905.2902-4-7-3': 'In Ref. [CITATION] two more main results were obtained:', '0905.2902-4-8-0': "I. while the classical dynamics of macroscopic bodies has to be formulated and dealt with in ordinary (Minkowski possibly curved) space-time, as we learned from Newton, Lagrange, Hamilton, Jacobi, Einstein, Poincare and others, so here the basic geometry is an Euclid's one, including the concept of point-event, which is necessary to represent the center of mass of the macroscopic bodies while running along their trajectories or orbits (like the Kepler orbits).", '0905.2902-4-9-0': 'II.', '0905.2902-4-9-1': 'For atomic physics instead, the appropriate space for the formulation of dynamical quantum equations and for their solution is momentum-space bilinearly constructed from pure spinors, as it appears in (1) the first elementary example of the Weyl equations.', '0905.2902-4-10-0': '# Pure Spinors', '0905.2902-4-11-0': 'E. Cartan, the discoverer of spinors, underlined the great elegance of the spinors he named simple [CITATION] (renamed pure by Chevalley [CITATION]).', '0905.2902-4-12-0': 'He started by considering the correlations between spinors and totally null planes as follows: let [MATH] represent a complex Euclidean, [MATH] dimensional space, with well-defined quadratic forms.', '0905.2902-4-12-1': 'Then we may define the corresponding Clifford algebra [MATH] which, as known, may be considered as the endomorphism algebra in a [MATH] dimensional space [MATH] of spinors, and we write: [EQUATION]', '0905.2902-4-12-2': 'A spinor [MATH] may be defined by the equation [EQUATION] where [MATH] is a vector of [MATH] (referred to a Cartesian orthonormal coordinate system) and [MATH], called the generators of [MATH], represent the univectors in the direction of the [MATH] orthonormal coordinates of [MATH].', '0905.2902-4-13-0': 'The generators [MATH] obey the anticommutation relations.', '0905.2902-4-13-1': '[EQUATION]', '0905.2902-4-13-2': 'Let us now multiply equation (3) from the left by [MATH] and we obtain (because of equation (4)): [EQUATION] and, if [MATH]: [EQUATION] which means that the non zero spinor [MATH], satisfying equation (3) (sometimes called Dirac spinor) defines a subspace of [MATH] whose vectors are all null and/or mutually orthogonal, it is called the totally null plane associated to [MATH] and indicated with [MATH] where [MATH] is its dimension.', '0905.2902-4-13-3': 'It is known, and easy to prove, that the maximal possible value of [MATH] is [MATH], that is, one half of the dimension [MATH] of the space [MATH].', '0905.2902-4-14-0': 'In order to arrive at the Cartan definition of simple or pure spinors, let us define the "volume element" of [MATH]: [EQUATION]', '0905.2902-4-14-1': 'It is easy to show that [MATH] anticommutes with all the [MATH] and that [MATH] generate the Clifford algebra [MATH] whose even subalgebra [MATH] is isomorphic to the simple algebra [MATH]: [EQUATION]', '0905.2902-4-14-2': 'With [MATH] one may define what are called the Weyl spinors [MATH]: [EQUATION] where [MATH] is the Dirac spinor defined by equation (3).', '0905.2902-4-15-0': 'The Weyl spinors may be defined by the Weyl equations [EQUATION]', '0905.2902-4-15-1': 'Obviously also [MATH] will define totally null planes in [MATH].', '0905.2902-4-15-2': 'Well E. Cartan showed how a simple or pure spinor is isomorphic, up to a sign, to the maximal totally null plane of the Weyl spinor associated with a given Clifford algebra, and this property renders pure spinors complicated geometrical objects to deal with.', '0905.2902-4-15-3': 'In fact, while the dimension of the maximal totally null planes increase linearly with [MATH], that of the spinors, increase with [MATH] like [MATH] and then high dimensional spinors which are used to explain some phenomena of elementary particle physics (up to [MATH] component spinors), need a lot of (ten for 32 components) constraint equations in their components to render them pure.', '0905.2902-4-15-4': 'In short, all spinors up to [MATH] components are pure (or equivalent to pure).', '0905.2902-4-16-0': 'At present several people have tried to use pure spinors to explain both the high energy phenomena of the elementary particles and the unsolved problem of quantization of the gravitational field (through superstring theory) but with no acceptable results.', '0905.2902-4-17-0': 'In fact, the so-called Standard model, now represented by the symbols [MATH], which represents the symmetries presented in high-energy elementary particle phenomena, contains more than 20 constants representing charges, masses and so on, which may not be computed deriving them from acceptable theory.', '0905.2902-4-18-0': 'But now with the possible prescription found in [CITATION] that quantum dynamical problems have to be dealt with in momentum-space rather than in space-time, and because of a theorem we recently discovered, the situation could drastically change.', '0905.2902-4-19-0': '# The theorem', '0905.2902-4-20-0': 'Consider two Weyl spinors: [MATH] and [MATH] associated with a Clifford algebra [MATH] and define the vector of the space [MATH], with components: [EQUATION] where [MATH].', '0905.2902-4-20-1': 'The vector [MATH] is null: [MATH], if and only if one of the two spinors, either [MATH] or [MATH], is pure.', '0905.2902-4-20-2': 'The proof is in Ref. [CITATION].', '0905.2902-4-21-0': 'It is known that in [MATH] dimensional spaces with Lorentzian signatures like [MATH] or [MATH] the vector components will be real if of the form [MATH], where [MATH] and [MATH] means [MATH] hermitian conjugate while [MATH] is the time-like generator.', '0905.2902-4-22-0': 'Suppose now [MATH] to be pure then we will have that: [EQUATION] with real components, (12) may be written in the form: [EQUATION] where [MATH]; which, might arrive (in the r.h.s.) at [MATH] (exploiting the isomorphism between Dirac and Weyl spinors doublets and conformal covariance [CITATION]).', '0905.2902-4-23-0': 'It is interesting to observe that in the momentum space, where according to the conclusion of Ref. [CITATION], we have to formulate the dynamical problems of quantum physics, pure spinors may bilinearly define in Lorentzian momentum spaces compact manifolds consisting of spheres (with Poincare invariant radii).', '0905.2902-4-23-1': 'This might obviously encourage the hope that not only the mathematical origin of quantum jumps might finally find a simple geometrical explanation, (the auto-vibrations of spheres are discrete) but also that, hopefully, the transitory sickness of the otherwise beautiful Standard Model, manifested by the symptom of its more than 20 unexplained constants, may finally be cured.', '0905.2902-4-24-0': '# First applications and results', '0905.2902-4-25-0': 'It might sound strange, but it is true: the first convincing confirmation of the above predictions was found by V. Fock more than 70 years ago: in 1935 [CITATION].', '0905.2902-4-25-1': 'He dealt with the historical problem (the first one to deal with atomic quantization) of the H-atom stationary states.', '0905.2902-4-26-0': 'At that time the problem was solved through the Schrodinger equation for the electron possible orbits in the field of the proton.', '0905.2902-4-26-1': 'Fock, anticipating the conclusions of [CITATION], formulated it in momentum space on a sphere [MATH] conceived as the one point compactification of ordinary [MATH] dimensional momentum space [MATH] with the following integral equation: [EQUATION] where [MATH] is the volume of the [MATH] unit sphere; [MATH] is the fine structure constant, [MATH] a unit of momentum, [MATH] the (reduced) electron mass and [MATH] is a unit vector indicating a point on the unit sphere [MATH], equation (14) is the Fock equation in adimensional form.', '0905.2902-4-26-2': 'Fock solved it through harmonic analysis and found that setting [MATH] he obtained for [MATH], representing the eigen vibrations of the sphere [MATH], the energy levels [MATH] of the H-atom stationary states, which in turn explain the Balmer spectrum of the H-atom.', '0905.2902-4-26-3': 'But it was V. Fock himself to draw from equation (14) the fundamental discovery that an integral equation on the sphere [MATH], like equation (14), has to forsee the symmetry [MATH] of its solutions (including obviously their classical counterparts - the solar planets - which is true, as later underlined by the great W. Pauli, Nobel prize winner).', '0905.2902-4-27-0': 'Now we may go one step further representing the H-atom with a quadruplet of spinors say (Proton, Neutron, Neutrino and Electron) and adopt equation (13) with [MATH] [CITATION]: since it is well known that spinors up to 4 components are pure (or equivalent to pure, and this equivalence is exploited here) [CITATION] [EQUATION] from which we obtain [MATH] with Poincare invariant radius [MATH] therefore we may expect that the spinorial [MATH] may reflect the Poincare invariance; and in the energy levels found by Fock we have the obviously relativistic form [CITATION]: [EQUATION]', '0905.2902-4-27-1': 'If we take this first result as an indication of the validity of the hypothesis formulated above [CITATION] we may affirm that the first example of atomic quantization confirms that the H-atom discrete energy levels (generating its discrete optical spectrum - Balmer series) are generated by the computed manifolds determined by pure spinors in momentum space.', '0905.2902-4-27-2': 'In particular by the eigen vibrations of those spheres we may also geometrically explain the quantum-jumps.', '0905.2902-4-28-0': 'Should this result have confirmations by further examples then the historical way to arrive at the solution of the problem of the H-atom stationary states would not only be more complicated but also wrong.', '0905.2902-4-28-1': 'It would be wrong to consider the H-atom as a proton plus an orbitating point electron to be then substituted arbitrarily by wave functions.', '0905.2902-4-29-0': 'As stated in [CITATION] we have to formulate the quantum problems in momentum space where we have the mathematical possibility to integrate null vectors to generate strings [CITATION], which are non local objects and might be at the origin of the concept of wave function without giving origin to paradoxes, as will be further discussed, elsewhere.', '0905.2902-4-30-0': '# The possibility to cure the Standard Model', '0905.2902-4-31-0': 'V. Fock solved equation (14) computing the adimensional factors [MATH] through harmonic analysis from the ball [MATH] to [MATH].', '0905.2902-4-31-1': 'We tried (with P. Nurowski) to compute, through harmonic analysis, also the other adimensional factor [MATH] in equation (14) obtaining all the factors but one.', '0905.2902-4-31-2': 'In fact, [MATH] was computed for the first time by M.A. Wyler in 1969 [CITATION] with the result: [EQUATION] differing less than [MATH] from the experimental value.', '0905.2902-4-32-0': 'He used group theoretical methods, however declared (private communication) to be not familiar with spinor theory.', '0905.2902-4-32-1': 'There are more authors who with other methods computed other quantum constants (including [MATH]) relevant for particle physics [CITATION].', '0905.2902-4-32-2': 'Now the point is if their computations may be correlated or derived from the compact manifolds in momentum space derived from pure spinors.', '0905.2902-4-33-0': '# The wave function', '0905.2902-4-34-0': 'If we follow the suggestion of Ref. [CITATION] that quantum dynamics has to be formulated and dealt with in momentum space, then we have to abandon the concept of wave function representing the electron say, in the Schrodinger equation (at the source of several paradoxes).', '0905.2902-4-34-1': 'As we have seen in momentum space there is no possibility of defining the point event.', '0905.2902-4-34-2': 'We may instead define integrals of null vectors, bilinear in pure spinors which uniquely define strings, which are non local but relativistically covariant; furthermore, for their existence in nature, there is experimental evidence since they were discovered from the interpretation of some experiments performed at CERN (Geneve).', '0905.2902-4-34-3': 'It should not be difficult to obtain from relativistic non local objects also non relativistic ones like in the Fock integral equation, which, as seen above, presents relativistic properties.', '0905.2902-4-35-0': '# From the quantum to the classical field equations', '0905.2902-4-36-0': "It is well known how, from the Weyl equation for massless neutrinos, equation (1) one may easily obtain the classical Maxwell's equations which, when quantized give the equations of motion for the massless and chargeless photon.", '0905.2902-4-36-1': 'It also results from [CITATION] that for [MATH] dimensional momentum space [MATH]; that is [MATH].', '0905.2902-4-37-0': 'In this way the massless photon may be conceived as bilinearly generated by massless neutrinos representing elementary pure spinors (but notoriously not their bound states).', '0905.2902-4-38-0': 'Already E. Cartan named electromagnetic tensor the one contained in the Clifford algebra [MATH]: [EQUATION]', '0905.2902-4-38-1': "In fact, if we apply the Weyl operator of equation (1) we obtain: [EQUATION] which are the well known Maxwell's equations in vacuum.", '0905.2902-4-39-0': 'At this point a natural question presents itself - why not try to get from pure spinors also the gravitational field equations?', '0905.2902-4-39-1': 'These equations present enormous difficulties that for a long time blocked the progress of theoretical physics and the efforts of the best present physicists in Trieste (and no longer present, like my dear friend Dennis Shama).', '0905.2902-4-39-2': 'The best answer for a long time has been: because it is too difficult to get symmetric tensors from Clifford Algebras.', '0905.2902-4-40-0': 'But now, there is a way to generate symmetric tensors from Clifford algebras: they are entered through the [MATH] of the electroweak model (represented by [MATH] in the standard model).', '0905.2902-4-40-1': 'And then let us try.', '0905.2902-4-41-0': 'Let us then consider an element of the electroweak model of the form: [EQUATION] it is obviously symmetric like should be the metric tensor (fundamental for general relativity!)', '0905.2902-4-41-1': 'and it is quadrilinear in pure spinors.', '0905.2902-4-42-0': "The equation for the massless and chargeless photon has been derived from Maxwell's equations.", '0905.2902-4-42-1': "A similar result should also be obtained for the massless and chargeless graviton, which however will have to result as quadrilinear of massless neutrinos since its spin is [MATH] instead of [MATH]; and as we will see, this can be done in the following way, precisely starting from the standard model in its form: [MATH] where [MATH] represents the origin of the Electro Weak mode and contains products of the left-handed currents which, through equation (1) for the massless neutrinos, may produce the gravitational equation in flat space in a similar way as they produce Maxwell's equations for the massless photons.", '0905.2902-4-43-0': 'In fact, suppose that in equation (18) [MATH] represents the existing weak decay, Neutron [MATH] Proton [MATH] lepton [MATH] neutrinos (or the ones in which the fermions are substituted by one of the 3 twins of the same family), and we would get the graviton if we could consider chargeless the [MATH] proton and [MATH] lepton which appear on the right-hand side that should be both chargeless and massless like the two neutrinos.', '0905.2902-4-43-1': 'This possibility should certainly be discussed and analyzed further.', '0905.2902-4-44-0': "The situation appears difficult but not desperate since from the Weyl equation for the neutrino (1) we obtain the Maxwell's equations (17), notoriously extendable also to charges.", '0905.2902-4-44-1': 'From these in turn, we may derive the equations for the chargeless and massless photon.', '0905.2902-4-44-2': 'A similar procedure could be possible also for gravity since, as well known, its renormalizability was not solved through sophisticated supergravity and it is quite possible that it could be attacked going back to the first steps as suggested above.', '0905.2902-4-45-0': 'A possible way could be: let us start from the obviously symmetric tensor [MATH] in (18) and postulate the obviously equivalent equations [EQUATION] which are similar to the first of the equations (17) from which one derives the zero mass of the photon.', '0905.2902-4-45-1': 'It remains to be discussed if and which phenomena are represented by the equations with the emisymmetric tensor [MATH].', '0905.2902-4-46-0': '# Conclusions and possible follow-up', '0905.2902-4-47-0': 'In this paper we followed the suggestions of one of our great predecessors: Elie Cartan, the discoverer of spinors.', '0905.2902-4-47-1': "One of the main Cartan's merits is to have underlined the importance of simple spinors and of being the first to see in Clifford algebras an emisymmetric tensor, that he named electromagnetic tensor.", '0905.2902-4-47-2': "From this tensor one may derive the Maxwell's equations (17) from which only the first one, the simplest, is necessary to derive the zero mass and charge of the photon which, then, may be interpreted as bilinear in massless neutrinos (but not their bound state).", '0905.2902-4-48-0': 'Another great physicist was P.A.M. Dirac who, in 1938 discovered the "deep connection in Nature between cosmology and atomic physics" [CITATION] used in [CITATION].', '0905.2902-4-49-0': 'To get the graviton, one needs a symmetric tensor that can be built quadrilinearly in Weyl spinors.', '0905.2902-4-49-1': 'This construction could hopefully also provide suggestions on the quantization of the gravitational field which failed up to now despite several efforts (and whose history still waits to be written).', '0905.2902-4-50-0': 'Being in the frame of general relativity the appropriate field where to look for suggestions is probably Cosmology, as suggested in the paper, where certainly the space-time curvature must play a central role.', '0905.2902-4-51-0': 'std, for_Paolo'}

1806.02096

{'1806.02096-1-0-0': 'The reactor antineutrino anomaly might be explained by the oscillation of reactor antineutrinos towards a sterile neutrino of eV mass.', '1806.02096-1-0-1': 'In order to explore this hypothesis, the Stereo experiment measures the antineutrino energy spectrum in six different detector cells covering baselines between 9 and 11 meters from the compact core of the ILL research reactor.', '1806.02096-1-0-2': 'In this article, results from 66 days of reactor-on and 138 days of reactor-off are reported.', '1806.02096-1-0-3': 'A novel method to extract the antineutrino rates has been developed based on the distribution of the pulse shape discrimination parameter.', '1806.02096-1-0-4': 'The test of a new oscillation toward a sterile neutrino is performed by comparing ratios of cells, independent of absolute normalization and of the prediction of the reactor spectrum.', '1806.02096-1-0-5': 'The results are found compatible with the null oscillation hypothesis and the best fit of the reactor antineutrino anomaly is excluded at 97.5% C.L.', '1806.02096-1-1-0': 'Neutrino oscillation experiments of the last two decades have measured all mixing angles and mass splittings in a three flavor framework [CITATION].', '1806.02096-1-1-1': 'In the three-neutrino model, no significant disappearance of neutrinos of few MeV energy is expected at baselines of less than 100 m. Nevertheless, many experiments in the vicinity of nuclear reactors have observed a lower electron antineutrino flux than predicted at such distances.', '1806.02096-1-1-2': 'There are basically two possible explanations for this observation known as the Reactor Antineutrino Anomaly (RAA) [CITATION].', '1806.02096-1-1-3': 'One is a deficient prediction of the antineutrino flux and spectrum from reactors, due to underestimated systematics of the measurements of beta spectra emitted after fission [CITATION] or of the conversion method [CITATION], see [CITATION] for recent reviews.', '1806.02096-1-1-4': 'The other one proposes new physics beyond the Standard Model of particle physics considering an oscillation from active towards a sterile neutrino state [CITATION].', '1806.02096-1-1-5': 'This sterile neutrino option could also explain the deficits observed by the solar neutrino experiments GALLEX and SAGE in their calibrations with intense [MATH]Cr and [MATH]Ar neutrino sources [CITATION].', '1806.02096-1-1-6': 'The original contours of allowed regions in the ([MATH], [MATH]) plane given in [CITATION] and their best fit values ([MATH], [MATH]=2.4 eV[MATH]) are used as benchmark in this paper.', '1806.02096-1-1-7': 'A recent review of light sterile neutrinos in the context of these anomalies and fits in different scenarios can be found in [CITATION].', '1806.02096-1-1-8': 'In contrast, other experimental results strongly constrain oscillations to sterile neutrinos in different channels, putting tension on global fits [CITATION].', '1806.02096-1-1-9': 'In particular appearance and disappearance data appear incompatible.', '1806.02096-1-2-0': 'Both solutions to the RAA, deficient predictions of reactor antineutrino spectra and intensities as well as new particle physics, can be studied with the data of the Stereo experiment.', '1806.02096-1-2-1': 'Stereo is installed at the high flux reactor of the Institut Laue Langevin operating with highly enriched [MATH]U (93%).', '1806.02096-1-2-2': 'Therefore, contributions from fission of other isotopes is negligible and Stereo will provide a pure [MATH]U antineutrino spectrum measured at 10 m baseline.', '1806.02096-1-2-3': 'However, in this paper we concentrate on the sterile neutrino hypothesis as solution of the RAA which has triggered a series of reactor antineutrino experiments at very short baselines [CITATION].', '1806.02096-1-2-4': 'Results of the first two of these experiments, DANSS [CITATION] and NEOS [CITATION], exclude significant parts of allowed regions from Ref. [CITATION].', '1806.02096-1-2-5': 'At the same time, a combined analysis of all reactor antineutrino disappearance experiments still favors oscillations involving a fourth neutrino state at the 3[MATH] level [CITATION].', '1806.02096-1-2-6': 'The best fit parameters driven by the new DANSS and NEOS results suggest a mass splitting of [MATH] eV[MATH] and a mixing angle of [MATH], which is slightly outside the favored regions of Ref. [CITATION] towards a lower mixing angle.', '1806.02096-1-2-7': 'This result is independent from flux predictions, since it is based on the comparison of purely spectral information.', '1806.02096-1-2-8': 'The analysis of DANSS compares the antineutrino energy spectrum of the moveable detector for two baselines.', '1806.02096-1-2-9': 'However, it awaits calculation of the final systematic uncertainties [CITATION].', '1806.02096-1-2-10': 'NEOS relies on a non-trivial comparison of their data to the measured Daya Bay spectrum [CITATION] obtained at different reactors with different detectors where the correction of the spectra requires inputs from predictions.', '1806.02096-1-3-0': 'In Stereo, the antineutrino spectrum with energies up to about 10 MeV is measured in a segmented detector using six identical target cells, each having a slightly different baseline.', '1806.02096-1-3-1': 'The detector center is located at about 10 m distance from the ILL core.', '1806.02096-1-3-2': 'The sterile neutrino hypothesis can be tested by comparing the measured antineutrino energy spectra of the different cells.', '1806.02096-1-3-3': 'A neutrino oscillation with a mass splitting in the eV region would manifest in a clear spectral pattern of a distance-dependent distortion of the energy spectrum along the detector axis.', '1806.02096-1-3-4': 'In this first sterile neutrino analysis, spectra ratios using one cell as a reference are used.', '1806.02096-1-3-5': 'In this way, no reactor spectrum prediction is needed and the analysis is independent from the absolute flux normalization, minimizing systematic uncertainties.', '1806.02096-1-4-0': 'The Stereo detector system [CITATION] (see Figure [REF]) consists of an antineutrino detector, a muon veto on top and several calibration devices.', '1806.02096-1-4-1': 'The antineutrinos are detected via the inverse beta decay reaction (IBD) on hydrogen nuclei in an organic liquid scintillator: [MATH].', '1806.02096-1-4-2': 'The six optically separated cells of the Target (TG) volume are filled with a gadolinium (Gd) loaded liquid scintillator for a total of almost 2 m[MATH].', '1806.02096-1-4-3': 'They are read out from the top by 4 photomultiplier tubes (PMT) per cell.', '1806.02096-1-4-4': 'The IBD signature is a delayed coincidence of a prompt positron and a delayed neutron capture event.', '1806.02096-1-4-5': 'The antineutrino energy is directly inferred from the prompt event.', '1806.02096-1-4-6': 'The neutron from the IBD reaction is moderated and then mainly captured by Gd isotopes.', '1806.02096-1-4-7': 'This capture creates a characteristic delayed signal of a gamma cascade with about 8 MeV total energy.', '1806.02096-1-4-8': 'These gammas can interact in the TG and in an outer volume, called Gamma-Catcher (GC), which surrounds the TG.', '1806.02096-1-4-9': 'It is filled with liquid scintillator without Gd and equipped with a total of 24 PMTs.', '1806.02096-1-4-10': 'In some cases, the GC serves also for the total positron energy, detecting annihilation gammas escaping the TG.', '1806.02096-1-4-11': 'The mean capture time of the coincidence signal is about 16 [MATH]s allowing for efficient discrimination of accidental background.', '1806.02096-1-4-12': 'Moreover, background events are strongly reduced by a thorough passive shielding design of various materials with a total mass of about 65 tons.', '1806.02096-1-4-13': 'On the ILL site, Stereo is installed underneath a water channel providing, together the reactor building, an overburden of 15 m w.e. against cosmic radiation.', '1806.02096-1-4-14': 'Remaining background can be measured during phases with the reactor turned off.', '1806.02096-1-4-15': 'A method has been developed to convert the PMT signals into a reconstructed energy, taking into account light cross-talk between cells.', '1806.02096-1-4-16': 'The reconstructed energy resolution ([MATH]) for [MATH]Mn [MATH]-rays (0.835MeV) is about 9%.', '1806.02096-1-4-17': 'Energy non-linearity, due to the quenching effects, is measured precisely and reproduced in the Monte Carlo (MC) at the percent level.', '1806.02096-1-4-18': 'More information on the detector and its performances can be found in ref. [CITATION].', '1806.02096-1-4-19': 'The analysis presented in this article concerns the phase-I of the experiment with about 66 days of reactor-on and 138 days of reactor-off [CITATION].', '1806.02096-1-5-0': 'Table [REF] lists the set of IBD selection cuts corresponding to the best compromise between detection efficiency and background rejection, although the results remain quite stable around the chosen values.', '1806.02096-1-5-1': 'Beyond the basic cuts on energy and capture time (cuts 1-3 in table [REF]), advantage is taken of the segmentation of the detector to better tag the topology of energy deposits of IBD events: a compact prompt event with the only potentially escaping particles being the 511 keV annihilation [MATH]-rays (cuts 4 and 5), the expanded energy deposition pattern of the n-Gd capture ensuring a minimal deposit in the TG (cut 6) and a maximum distance between the reconstructed vertices of prompt and delayed signals (cut 7).', '1806.02096-1-5-2': 'Large part of the cosmic-rays induced background is rejected by applying a 100[MATH]s muon veto (cut 8) and an isolation cut to get rid of multi-neutron cascades (cut 9).', '1806.02096-1-5-3': 'Non-tagged muons that stop and decay in the very top layer of the detector, without depositing more than 7.1 MeV energy, may be mistaken as IBD candidates.', '1806.02096-1-5-4': 'For these events, the light distribution between PMTs of the vertex cell is more asymmetric than for events in the detector bulk.', '1806.02096-1-5-5': 'Therefore, the asymmetry, defined as the maximum of single PMT charge in the cell divided by the total PMT charge, allows to remove the majority of them (cut 10).', '1806.02096-1-5-6': 'The main contributions to the dead time are from the muon veto and isolation cuts.', '1806.02096-1-5-7': 'The total correction ranges from 10 to 15 depending on the single rates induced by the activities of the neighboring experiments.', '1806.02096-1-5-8': 'It is accurately computed using two independent methods and leads to a relative uncertainty of 0.3 over the data taking time.', '1806.02096-1-6-0': 'Inside the above selection cuts, an average of [MATH]/day is detected with a signal to background ratio of about 0.9, averaged over the prompt energy window.', '1806.02096-1-6-1': 'This signal has been separated from the remaining background using a Pulse Shape Discrimination (PSD) parameter, defined as the ratio of the pulse tail charge and the pulse total charge.', '1806.02096-1-6-2': 'The PSD distribution of the prompt event of all pair candidates is shown in figure [REF] for one of the 11 reconstructed energy bins defined in the analysis.', '1806.02096-1-6-3': 'Two classes of events clearly appear, the proton recoils induced by fast neutrons, showing up at high PSD, and the electron recoils at low PSD, where the IBD positrons are expected.', '1806.02096-1-7-0': 'The reactor-off data collected by Stereo are used to parameterize accurately the PSD distribution of the correlated background induced by cosmic rays.', '1806.02096-1-7-1': 'Data are split into time bins of 1 week and energy bins of 500 keV width.', '1806.02096-1-7-2': 'For each bin, the PSD distribution is modeled with a multi-Gaussian function: one Gaussian for electronic recoils, one for proton recoils and one for accidental coincidences.', '1806.02096-1-7-3': 'The accidental coincidences component is determined by fitting, in a combined way, its PSD distribution obtained very accurately by opening many (typically 100) delayed windows for each prompt candidate and rescaling the obtained distributions by the number of windows.', '1806.02096-1-7-4': 'The PSD distribution of the random coincidences is almost purely electronic since gammas dominate the single rates.', '1806.02096-1-7-5': 'In the fitting procedure, the position and the width of the electronic recoil Gaussian are constrained using their very accurate determination on the PSD distribution of single events.', '1806.02096-1-7-6': 'The positions of the Gaussians drift with time due to temperature changes and to the evolution of the light collection efficiency and cross-talk of the cells during phase-I.', '1806.02096-1-7-7': 'However, the ratio of the areas, [MATH], was found to be compatible with a constant and in particular, independent on the atmospheric pressure.', '1806.02096-1-7-8': 'The average over time of this ratio for each energy bin is the only parameter of the reactor-off data transposed to the analysis of the reactor-on data.', '1806.02096-1-7-9': 'The reactor-on PSD distribution for each cell and time-energy bin is then fit using this background model with an additional Gaussian to reproduce the antineutrino signal.', '1806.02096-1-7-10': 'The area of the additional Gaussian gives the numlber of neutrino for the time-energy bin.', '1806.02096-1-7-11': 'The mean and sigma values of the antineutrino Gaussian are let free to vary by about 15% (1[MATH] value of a pull term) around these values.', '1806.02096-1-7-12': 'This range was set to include all observed fluctuations between the electronic recoil peak and the neutrino peak.', '1806.02096-1-7-13': 'In contrast to a fixed cut on the PSD value, this novel method permits a full separation of electronic and proton recoils in spite of the overlapping distributions and accounts for slow drifts in the PSD distribution.', '1806.02096-1-7-14': 'The method is insensitive to dead time differences between reactor-on and reactor-off runs since rates entering in the ratios are measured simultaneously and only ratios are transferred between reactor-on and reactor-off measurements.', '1806.02096-1-7-15': 'The remaining systematics due to the deviation of the PSD shape model from the true shape is controlled by the high goodness of fit for all energy bins of reactor-off PSD distributions.', '1806.02096-1-7-16': 'Moreover, since this model is applied to all cells, potential deviation from the model will be further suppressed in the ratio of spectra used in the oscillation search.', '1806.02096-1-8-0': 'To search for a possible oscillation toward a sterile neutrino in the data, a ratio method is used.', '1806.02096-1-8-1': 'It consists in dividing bin by bin the spectrum of cells 2 to 6 by the spectrum of cell 1, which serves as a reference, and comparing these ratios between data and MC.', '1806.02096-1-8-2': 'This formalism is insensitive to the model of the reactor spectrum and relies only on the relative difference between cells.', '1806.02096-1-8-3': 'However, the variance of the ratio cannot be properly computed when the denominator approaches zero within few sigma units.', '1806.02096-1-8-4': 'Therefore, this analysis has been limited to [MATH]7.125MeV.', '1806.02096-1-8-5': 'A profile [MATH] method is used with: [EQUATION] [MATH] and [MATH] are five-dimensional vectors (cell 2 to cell 6) corresponding to the measured and the MC ratios, respectively, for the [MATH] energy bin.', '1806.02096-1-8-6': 'Since all cell ratios are computed using the cell 1 spectrum, the covariance matrix of the [MATH] energy bin ratios, [MATH], is not diagonal.', '1806.02096-1-8-7': 'It has been estimated by MC considering Gaussian uncertainty for the neutrino rate of each bin.', '1806.02096-1-8-8': 'Nuisance parameters [MATH] are added to take into account systematic uncertainties: [MATH] are the relative normalizations of the cells due to the uncertainties on the volume and detection efficiencies ([MATH] and [MATH] because of reduced optical coupling for cell 4, see [CITATION]), [MATH] are the uncorrelated energy scale uncertainty driven by the cell-wise residual discrepancies between the energy response of data and MC ([MATH]) and [MATH] corresponds to the energy scale bias common to all cells due to the time-wise evolution of the energy response ([MATH]).', '1806.02096-1-9-0': 'Nuisance parameters [MATH] are taken into account in the model as following: [EQUATION] where [MATH] are the predicted spectra including oscillation and detector response.', '1806.02096-1-10-0': 'First, the null oscillation hypothesis has been tested.', '1806.02096-1-10-1': 'Figure [REF] compares the measured ratios and the simulated ratios without oscillation after minimization letting free the nuisance parameters.', '1806.02096-1-10-2': 'The simulated ratios are not perfectly flat because the energy response can slightly vary from one cell to another.', '1806.02096-1-10-3': 'The [MATH] value is 87.8 to be compared with 78.7, the value obtained with free nuisance parameters and oscillation parameters.', '1806.02096-1-10-4': 'From the probability density function (PDF) obtained by MC, the [MATH] of 9.1 corresponds to a p-value of 0.34.', '1806.02096-1-10-5': 'Hence, the null oscillation hypothesis cannot be rejected.', '1806.02096-1-11-0': 'To infer an exclusion contour in the oscillation parameter space, a raster scan method [CITATION] has been used.', '1806.02096-1-11-1': 'It consists in dividing the 2D parameter space into slices with one slice per [MATH] bin and computing for each slice the [MATH] as a function of [MATH] with free nuisance parameters.', '1806.02096-1-11-2': 'Then, the [MATH] values are computed using the minimum value of each slice and not the global minimum.', '1806.02096-1-11-3': 'The 90% C.L. exclusion contour corresponds to the parameter space where the [MATH] is higher than the value giving a one sided p-value of 0.1 in the PDF obtained by MC for each bin of the parameter space.', '1806.02096-1-11-4': 'The result is shown in Figure [REF].', '1806.02096-1-11-5': 'The obtained exclusion contour is centered around the sensitivity contour with oscillations due to the statistical fluctuations.', '1806.02096-1-11-6': 'The original RAA best fit is excluded at 97.5 C.L. For the first time, an experiment has been able to measure, at the same time, the energy spectrum at different distances from a compact core of a research reactor and to exclude the RAA best fit values without any assumption on the emitted spectrum.', '1806.02096-1-12-0': 'These first results demonstrate the ability of the Stereo experiment to detect antineutrinos above the residual background, dominated by cosmic-ray induced events.', '1806.02096-1-12-1': 'With the novel method presented in this paper, the proton recoil component of this background is measured in the temperature and pressure conditions of the reactor-on data taking while the associated relative contamination of electronic recoils is well-constrained from the reactor-off data.', '1806.02096-1-12-2': 'The accuracy of the background subtraction is thus driven by the statistics, which naturally improves as more reactor-off data are acquired between the reactor-on periods.', '1806.02096-1-12-3': 'The Stereo data taking is still in progress and should reach the nominal statistics, 300 days at nominal reactor power, before the end of 2019.', '1806.02096-1-13-0': 'We would like to thank G. Mention for the many discussions on the implementation of the ratio method.', '1806.02096-1-13-1': "This work is funded by the French National Research Agency (ANR) within the project ANR-13-BS05-0007 and the 'Investments for the future' programs P2IO LabEx (ANR-10-LABX-0038) and ENIGMASS LabEx (ANR-11-LABX-0012)."}

{'1806.02096-2-0-0': 'The reactor antineutrino anomaly might be explained by the oscillation of reactor antineutrinos toward a sterile neutrino of eV mass.', '1806.02096-2-0-1': 'In order to explore this hypothesis, the STEREO experiment measures the antineutrino energy spectrum in six different detector cells covering baselines between 9 and 11 m from the compact core of the ILL research reactor.', '1806.02096-2-0-2': 'In this Letter, results from 66 days of reactor turned on and 138 days of reactor turned off are reported.', '1806.02096-2-0-3': 'A novel method to extract the antineutrino rates has been developed based on the distribution of the pulse shape discrimination parameter.', '1806.02096-2-0-4': 'The test of a new oscillation toward a sterile neutrino is performed by comparing ratios of cells, independent of absolute normalization and of the prediction of the reactor spectrum.', '1806.02096-2-0-5': 'The results are found to be compatible with the null oscillation hypothesis and the best fit of the reactor antineutrino anomaly is excluded at 97.5% C.L.', '1806.02096-2-1-0': 'Neutrino oscillation experiments of the last two decades have measured all mixing angles and mass splittings in a three flavor framework [CITATION].', '1806.02096-2-1-1': 'Within this framework, no significant disappearance of neutrinos of few MeV energy is expected at baselines of less than 100 m. Nevertheless, many experiments at such baselines from nuclear reactors have observed a lower electron antineutrino flux than predicted.', '1806.02096-2-1-2': 'There are basically two possible explanations for this observation known as the reactor antineutrino anomaly (RAA) [CITATION].', '1806.02096-2-1-3': 'One is a deficient prediction of the antineutrino flux and spectrum from reactors, due to underestimated systematics of the measurements of beta spectra emitted after fission [CITATION] or of the conversion method [CITATION], see [CITATION] for recent reviews.', '1806.02096-2-1-4': 'The other one proposes new physics beyond the standard model of particle physics considering an oscillation from active toward a sterile neutrino state [CITATION].', '1806.02096-2-1-5': 'The resulting disappearance probability for a neutrino of energy [MATH] at distance [MATH] from the source can be written as [MATH] where [MATH] is the mixing angle and [MATH] the difference of the mass squares of the mass eigenstates.', '1806.02096-2-1-6': 'This sterile neutrino option could also explain the deficits observed by the solar neutrino experiments GALLEX and SAGE in their calibrations with intense [MATH]Cr and [MATH]Ar neutrino sources [CITATION].', '1806.02096-2-1-7': 'The original contours of allowed regions given in [CITATION] and their best fit values [[MATH], [MATH]=2.4 eV[MATH]] are used as a benchmark in this Letter.', '1806.02096-2-1-8': 'A recent review of light sterile neutrinos in this context and fits in different scenarios can be found in [CITATION].', '1806.02096-2-1-9': 'In contrast, other experimental results strongly constrain oscillations to sterile neutrinos in different channels, putting tension on global fits [CITATION].', '1806.02096-2-1-10': 'In particular appearance and disappearance data appear incompatible.', '1806.02096-2-2-0': 'Both explanations of the RAA can be studied with the data of the STEREO experiment.', '1806.02096-2-2-1': 'STEREO is installed at the High Flux Reactor of the Institut Laue-Langevin whose compact core (80 cm high, 40 cm diameter) operates with highly enriched [MATH]U (93%).', '1806.02096-2-2-2': 'Therefore, contributions from fission of other isotopes are negligible and STEREO will provide a pure [MATH]U antineutrino spectrum measured at a 10 m baseline.', '1806.02096-2-2-3': 'However, in this Letter we concentrate on the sterile neutrino hypothesis which has triggered a series of reactor antineutrino experiments at very short baselines [CITATION].', '1806.02096-2-2-4': 'Results of the first two experiments, DANSS [CITATION] and NEOS [CITATION], exclude significant parts of the allowed region from Ref. [CITATION], but a combined analysis of all reactor antineutrino disappearance experiments still favors oscillations involving a fourth neutrino state at the 3[MATH] level [CITATION].', '1806.02096-2-2-5': 'The best fit parameters driven by the new DANSS and NEOS results suggest a mass splitting of [MATH] eV[MATH] and a mixing angle of [MATH], which is slightly outside the favored regions of Ref. [CITATION] toward a lower mixing angle.', '1806.02096-2-2-6': 'This result is based on the comparison of purely spectral information.', '1806.02096-2-2-7': 'The analysis of DANSS compares the antineutrino spectrum of the movable detector for two baselines.', '1806.02096-2-2-8': 'However, it awaits calculation of the final systematic uncertainties [CITATION].', '1806.02096-2-2-9': 'NEOS relies on a nontrivial comparison of their data to the measured Daya Bay spectrum [CITATION] obtained at different reactors with different detectors where the correction of the spectra requires inputs from predictions.', '1806.02096-2-2-10': 'Recently PROSPECT [CITATION] and NEUTRINO-4 [CITATION] have presented first results.', '1806.02096-2-3-0': 'In STEREO, the antineutrino spectrum with energies up to about 10 MeV is measured in a segmented detector using six identical target cells of 37 cm length, whose centers are placed from 9.4 to 11.1 m from the reactor core.', '1806.02096-2-3-1': 'The sterile neutrino hypothesis can be tested by comparing the measured antineutrino energy spectra of the different cells.', '1806.02096-2-3-2': 'A neutrino oscillation with a mass splitting in the electron Volt region would manifest in a clear spectral pattern of a distance-dependent distortion of the energy spectrum.', '1806.02096-2-3-3': 'The analysis presented here uses spectra ratios with one cell as reference.', '1806.02096-2-3-4': 'It does not require a reactor spectrum prediction and is independent from the absolute flux normalization, minimizing systematic uncertainties.', '1806.02096-2-4-0': 'The STEREO detector system [CITATION] (see Fig. [REF]) consists of an antineutrino detector, a muon veto on top and several calibration devices.', '1806.02096-2-4-1': 'The antineutrinos are detected via the inverse beta decay reaction (IBD) on hydrogen nuclei in an organic liquid scintillator: [MATH].', '1806.02096-2-4-2': 'The six optically separated cells of the target volume are filled with a gadolinium (Gd) loaded liquid scintillator for a total of almost 2 m[MATH].', '1806.02096-2-4-3': 'They are read out from the top by four photomultiplier tubes (PMT) per cell.', '1806.02096-2-4-4': 'The IBD signature is a coincidence of a prompt positron and a delayed neutron capture event.', '1806.02096-2-4-5': 'The antineutrino energy is directly inferred from the prompt event as [MATH] MeV.', '1806.02096-2-4-6': 'The neutron from the IBD reaction is moderated and then mainly captured by Gd isotopes.', '1806.02096-2-4-7': 'This capture creates a gamma cascade with about 8 MeV total energy.', '1806.02096-2-4-8': 'These gammas can interact in the target and in the gamma catcher, a segmented volume surrounding the target, filled with liquid scintillator without Gd and equipped with 24 PMTs.', '1806.02096-2-4-9': 'In some cases, the gamma catcher serves also for the total positron energy, detecting annihilation gammas escaping the target.', '1806.02096-2-4-10': 'The mean capture time of the coincidence signal is about 16 [MATH]s allowing for efficient discrimination of accidental background.', '1806.02096-2-4-11': 'Moreover, background events are strongly reduced by a thorough passive shielding design of various materials with a total mass of about 65 tons.', '1806.02096-2-4-12': 'STEREO is installed underneath a water channel providing, together with the reactor building, an overburden of 15 m water equivalent against cosmic radiation.', '1806.02096-2-4-13': 'The remaining background can be measured during phases with the reactor turned off.', '1806.02096-2-4-14': 'A Geant4 [CITATION] (version 10.1) Monte Carlo model (MC) based on DCGLG4sim [CITATION] describes detector geometry, shielding, position to the reactor core, particle interactions including neutron moderation and capture; light production, transport including cross talks between cells and detection, and signal conversion in the electronics.', '1806.02096-2-4-15': 'A method has been developed to convert the measured (or simulated) PMT signals into a reconstructed energy, taking into account light cross-talk between cells which ranges up to 15 .', '1806.02096-2-4-16': 'The reconstructed energy resolution ([MATH]) for [MATH]Mn [MATH] rays (0.835MeV) is about 9.', '1806.02096-2-4-17': 'Energy nonlinearity, due to quenching effects, is measured precisely and reproduced by the MC at the percent level.', '1806.02096-2-4-18': 'Drifts of the reconstructed energy are at the subpercent level.', '1806.02096-2-4-19': 'More information on the detector and its performances can be found in [CITATION].', '1806.02096-2-4-20': 'The analysis presented in this Letter concerns phase I of the experiment with 66 days of reactor turned on and 138 days of reactor turned off [CITATION].', '1806.02096-2-5-0': 'Table [REF] lists the set of IBD selection cuts corresponding to the best compromise between detection efficiency and background rejection, where the results remain quite stable around the chosen values.', '1806.02096-2-5-1': 'Beyond the basic cuts on energy and capture time (cuts 1-3), the detector segmentation is exploited to tag the topology of energy deposition of IBD events: compact prompt event only allowing for escaping 511 keV annihilation [MATH] rays (cuts 4 and 5), lower energy deposition threshold in the target for the expanded deposition pattern of the n-Gd capture (cut 6), and upper distance threshold between the reconstructed vertices of prompt and delayed signals (cut 7).', '1806.02096-2-5-2': 'A 100[MATH]s muon veto (cut 8) and an isolation cut against multineutron cascades (cut 9) reject a large part of the cosmic-ray induced background.', '1806.02096-2-5-3': 'Untagged muons stopping and decaying in the top layer of the detector, without depositing more than 7.125 MeV energy, may be mistaken as IBD candidates.', '1806.02096-2-5-4': 'They are removed by the asymmetry of their light distribution between the PMTs of the vertex cell (ratio of maximum charge in a single PMT to total charge), which is larger than for events in the detector bulk (cut 10).', '1806.02096-2-5-5': 'The effects of these cuts on spectra and cell efficiencies are well described by the MC which was studied using measurements with sources as well as antineutrino runs.', '1806.02096-2-5-6': 'For example, measurements with an AmBe neutron source at various positions in the detector demonstrated that cell-to-cell differences in the data-to-MC ratio of the cut efficiencies were less than 1%.', '1806.02096-2-5-7': 'These differences are included in the systematic uncertainties.', '1806.02096-2-5-8': 'The main contributions to the dead time are from the muon veto and isolation cuts.', '1806.02096-2-5-9': 'The total correction ranges from 10 to 15 depending on the single rates induced by the activities of the neighboring experiments.', '1806.02096-2-5-10': 'It is accurately computed using two independent methods and leads to a relative uncertainty of 0.3 over the data taking time.', '1806.02096-2-6-0': 'The antineutrino signal is separated from the remaining background using a pulse shape discrimination (PSD) parameter defined as the ratio of the pulse tail to total charge.', '1806.02096-2-6-1': 'The PSD distribution of the prompt event of all pair candidates passing the IBD selection cuts is shown in Fig. [REF] for one of the eleven reconstructed energy bins defined in the analysis.', '1806.02096-2-6-2': 'Two classes of events clearly appear, the proton recoils due to muon-induced fast neutrons at high PSD and the electronic recoils at low PSD.', '1806.02096-2-6-3': 'The electronic recoil class comprises IBD events, correlated electronic background induced by cosmic rays, and accidental coincidences (the single rates being dominated by gammas).', '1806.02096-2-7-0': 'After splitting the data into time bins of 1 week and energy bins of 500 keV width for each cell, the PSD distribution of each bin is modeled as sum of a proton recoil, an electron recoil and - for reactor-on data - an IBD Gaussian.', '1806.02096-2-7-1': 'Area [MATH], position and width of the proton recoil Gaussian are determined directly from the fit.', '1806.02096-2-7-2': 'Position and width of the electronic recoil Gaussian, comprising accidentals and correlated events, are fixed to the values [MATH] and [MATH] of the PSD distribution of singles obtained with negligible statistical uncertainty.', '1806.02096-2-7-3': 'The area can be separated into [MATH].', '1806.02096-2-7-4': '[MATH] is determined by a simultaneous fit to the PSD distribution of accidental events, extracted with high statistics by looking for random delayed events in many (typically 100) delayed windows for each prompt candidate and rescaling by the number of windows.', '1806.02096-2-7-5': 'This accounts correctly for changing uncorrelated background.', '1806.02096-2-7-6': '[MATH] parametrizes the ratio of correlated electronic recoils to proton recoils.', '1806.02096-2-7-7': 'Whereas the rates of both event types depend on atmospheric pressure, their ratio was found to be compatible with a constant.', '1806.02096-2-7-8': 'This can be understood since, within the applied cuts, electronic recoils in the prompt event are dominantly created by primary (multi)neutron spallation, e.g., via [MATH] or gammas of double neutron capture events.', '1806.02096-2-7-9': '[MATH] is determined from reactor-off data for each energy bin and its time average and statistical uncertainty are the only parameters transferred to the analysis of the reactor-on data, as a pull term in the PSD fits.', '1806.02096-2-7-10': 'The PSD distribution of IBD prompt events is slightly different from that of singles because of the positron annihilation gammas.', '1806.02096-2-7-11': 'This is accounted for by constraining position and width of the IBD Gaussian only moderately in the fit, to [MATH] and [MATH], respectively.', '1806.02096-2-7-12': 'These constraints have been estimated from the difference of the reactor-on and reactor-off PSD distributions.', '1806.02096-2-7-13': 'Finally, the area of the IBD Gaussian [MATH] yields the number of antineutrinos for the respective time-energy bin.', '1806.02096-2-8-0': 'In contrast to a fixed cut on the PSD value, this novel method permits a full separation of the different contributions to electronic and proton recoils, in spite of the overlapping distributions, and accounts for slow drifts in the PSD distribution.', '1806.02096-2-8-1': 'The method is insensitive to dead time differences between reactor-on and reactor-off runs since rates entering in the ratios are measured simultaneously and only ratios are transferred between reactor-on and reactor-off measurements.', '1806.02096-2-8-2': 'The remaining systematics due to deviations of the model from the true PSD distribution is controlled by the high goodness of fit and the stability with respect to the fit ranges for all energy bins of reactor-off PSD distributions.', '1806.02096-2-8-3': 'Moreover, since this model is applied to all cells, potential deviations from the model will be further suppressed in the ratio of spectra used in the oscillation search.', '1806.02096-2-9-0': 'The resulting total antineutrino rate is [MATH]/day with a signal to background ratio of about 0.9, determined from integrating over the region of interest in the PSD parameter.', '1806.02096-2-9-1': 'To search for a possible oscillation toward a sterile neutrino in the data, a ratio method is used.', '1806.02096-2-9-2': 'It consists of dividing bin by bin the spectrum of cells 2 to 6 by the spectrum of cell 1, which serves as reference, and comparing these ratios between data and MC.', '1806.02096-2-9-3': 'This formalism is insensitive to the model of the reactor spectrum and relies only on the relative difference between cells.', '1806.02096-2-9-4': 'However, the variance of the ratio is not well defined when the denominator approaches zero within few [MATH] units.', '1806.02096-2-9-5': 'Therefore, this analysis has been limited to [MATH]7.125MeV.', '1806.02096-2-9-6': 'In this range, the smallest denominator value is 4.7 [MATH] away from zero.', '1806.02096-2-9-7': 'A profile [MATH] method is used with [EQUATION] [MATH] and [MATH] are five-dimensional vectors (cell 2 to cell 6) corresponding to the measured and the MC ratios, respectively, for the ith energy bin.', '1806.02096-2-9-8': 'The MC takes into account the spatial distribution of IBD events for antineutrinos from the reactor core, the energy nonlinearities and the applied cuts in order to simulate the expected energy spectra.', '1806.02096-2-9-9': 'Since the energy spectrum of cell 1 is used as a common denominator for all ratios, the ith energy bins of all ratios are highly correlated.', '1806.02096-2-9-10': 'This effect is coded in the covariance matrices [MATH], whose off-diagonal elements have been determined by random sampling considering Gaussian uncertainties for the antineutrino rates of each bin.', '1806.02096-2-9-11': 'Nuisance parameters [MATH] are added to take into account systematic uncertainties: [MATH] are the relative normalizations of the cells due to the uncertainties on the volume and detection efficiencies ([MATH] and [MATH] because of reduced optical coupling for cell 4, see [CITATION]), [MATH] are the uncorrelated energy scale uncertainties driven by the cellwise residual discrepancies between the energy response of data and MC ([MATH]) and [MATH] corresponds to the energy scale bias common to all cells due to the timewise evolution of the energy response ([MATH]).', '1806.02096-2-9-12': 'They enter into [MATH] as follows: [EQUATION] where [MATH] are the predicted spectra including oscillation and detector response and [MATH] describe the changes obtained from neighbor energy bins depending on the energy scale parameters.', '1806.02096-2-10-0': 'First, the null oscillation hypothesis has been tested.', '1806.02096-2-10-1': 'Fig. [REF] compares the measured and the simulated ratios without oscillation (and with oscillations with the RAA best fit values from [CITATION]) after minimization with free nuisance parameters.', '1806.02096-2-10-2': 'The decrease of the mean value of the ratios with increasing distance reflects the [MATH] flux dependence, where the cell detection efficiencies have to be taken into account.', '1806.02096-2-10-3': 'This dependence is confirmed quantitatively since the fitted cell normalization parameters [MATH] were found within the expected uncertainties.', '1806.02096-2-10-4': 'The simulated ratios are not perfectly flat because the energy response can slightly vary from one cell to another.', '1806.02096-2-10-5': 'From the probability density function of [MATH] obtained by generating a large number of pseudoexperiments, the [MATH] of 9.1 with respect to the minimum in the [MATH] plane corresponds to a p value of 0.34.', '1806.02096-2-10-6': 'Hence, the null oscillation hypothesis cannot be rejected.', '1806.02096-2-11-0': 'To infer an exclusion contour in the oscillation parameter space, a raster scan method [CITATION] has been used.', '1806.02096-2-11-1': 'It consists of dividing the 2D parameter space into slices, with one slice per [MATH] bin, and computing for each slice the [MATH] as a function of [MATH] with free nuisance parameters.', '1806.02096-2-11-2': 'Then, the [MATH] values are computed using the minimum value of each slice and not the global minimum.', '1806.02096-2-11-3': 'The 90% C.L. exclusion contour corresponds to the parameter space where the [MATH] is higher than the value giving a one sided p value of 0.1 in the probability density function obtained from pseudoexperiments for each bin of the parameter space.', '1806.02096-2-11-4': 'The result is shown in Fig. [REF].', '1806.02096-2-11-5': 'The exclusion contour is centered around the sensitivity contour, also computed with a raster scan, with oscillations due to the statistical fluctuations.', '1806.02096-2-11-6': 'The original RAA best fit is excluded at 97.5% C.L.', '1806.02096-2-12-0': 'These first results demonstrate the ability of the STEREO experiment to detect antineutrinos above the residual background, dominated by cosmic-ray induced events.', '1806.02096-2-12-1': 'With the novel method presented in this Letter, the proton recoil component of this background is measured in the temperature and pressure conditions of the reactor-on data taking while the associated relative contamination of electronic recoils is well constrained from the reactor-off data.', '1806.02096-2-12-2': 'The accuracy of the background subtraction is thus driven by the statistics and improves with more reactor-off data acquired.', '1806.02096-2-12-3': 'The STEREO data taking is in progress and should reach the envisaged statistics, 300 days at nominal reactor power, before the end of 2019.', '1806.02096-2-13-0': 'We would like to thank G. Mention for discussions on the implementation of the ratio method.', '1806.02096-2-13-1': 'This work is supported by the French National Research Agency (ANR) within the project ANR-13-BS05-0007 and the programs P2IO LabEx (ANR-10-LABX-0038) and ENIGMASS LabEx (ANR-11-LABX-0012).', '1806.02096-2-13-2': 'We acknowledge the support of the CEA, CNRS/IN2P3, the ILL and the Max Planck Gesellschaft.'}

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[]

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[]

[]

{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1806.02096

1511.03380

{'1511.03380-1-0-0': 'Oscillations between photons and axion-like particles (ALP) travelling in intergalactic magnetic fields have been invoked to explain a number of astrophysical phenomena, or used to constrain ALP properties using observations.', '1511.03380-1-0-1': 'One example is the anomalous transparency of the universe to TeV gamma-rays.', '1511.03380-1-0-2': 'The intergalactic magnetic field is usually modeled as patches of coherent domains, each with a uniform magnetic field, but the field orientation changes randomly from one domain to the next ("discrete-[MATH] model").', '1511.03380-1-0-3': 'We show in this paper that in more realistic situations, when the magnetic field direction varies continuously along the propagation path, the photon-to-ALP conversion probability [MATH] can be significantly different from the discrete-[MATH] model.', '1511.03380-1-0-4': 'In particular, [MATH] has a distinct dependence on the photon energy and ALP mass, and can be as large as [MATH].', '1511.03380-1-0-5': 'This result may affect previous constraints on ALP properties based on ALP-photon propagation in intergalactic magnetic fields.', '1511.03380-1-1-0': 'Introduction.', '1511.03380-1-1-1': 'Axion is particle first introduced to solve the strong CP problem [CITATION].', '1511.03380-1-1-2': 'Axion-like particles (ALPs) also appear in many theoretically well-motivated extensions of the standard model of particle physics [CITATION].', '1511.03380-1-1-3': 'A general property of ALPs (represented by the field [MATH]) is that they can couple to photons (represented by [MATH]) in the presence of an external magnetic field [MATH] through the interaction Lagrangian [MATH].', '1511.03380-1-1-4': 'While for axions there exists a relation between the coupling constant [MATH] and the axion mass [MATH], in general [MATH] and [MATH] are unrelated for ALPs.', '1511.03380-1-2-0': 'As a result of the photon-ALP coupling, a photon can oscillate into an ALP and vise versa in an external magnetic field.', '1511.03380-1-2-1': 'Such ALP-photon oscillations have been invoked to explain a variety of astrophysical phenomena, or conversely used to constrain the properties of ALPs using observations [CITATION].', '1511.03380-1-2-2': 'Examples include the apparent dimming of distant supernovae [CITATION], spectral distortions of the cosmic microwave background [CITATION], and the dispersion of QSO spectra [CITATION].', '1511.03380-1-2-3': 'Recently, ALP-photon oscillation has been used to explain anomalous lack of opacity of the Universe to gamma-rays [CITATION]: High-energy Gamma-ray photons from Active Galactic Nuclei (AGNs) at cosmological distances should not reach Earth due to electron-positron pair production on the extragalactic background infrared radiation [CITATION].', '1511.03380-1-2-4': 'However, the Universe appears more transparent to gamma-rays according to the observations of HESS and Fermi [CITATION].', '1511.03380-1-2-5': 'A possible explanation to the transparency problem is that because of the ALP-photon mixing, radiation from AGNs travels in the form of ALPs on a significant fraction of distance (without producing pairs) and converts back to photons before their detections [CITATION]', '1511.03380-1-3-0': 'ALP-photon propagation over cosmological distance is strongly affected by the magnetic field structure.', '1511.03380-1-3-1': 'The extragalactic magnetic field is most likely random and could described as patches of coherent domains with a typical scale length of order 1 Mpc.', '1511.03380-1-3-2': 'Previous studies have adopted a simple model, in which the magnetic field is uniform in each domain, but the field orientation (characterized by the angle [MATH]) changes in a random fashion from one domain to the next.', '1511.03380-1-3-3': 'Note that in this "discrete-[MATH]" model, the photon-to-ALP conversion probability in each domain [MATH] can be easily derived [see Eq. ([REF])] (since [MATH] is constant in each domain).', '1511.03380-1-3-4': "Assuming that [MATH]'s for different domains are random, Grossman et al. [CITATION] then derived an expression for the photon-to-ALP conversion probability through a large number of domains [see Eq. ([REF])], and this expression has been widely used in many previous studies.", '1511.03380-1-4-0': 'In realistic situations, the magnetic field and its orientation angle [MATH] should vary continuously across neighboring domains.', '1511.03380-1-4-1': 'In fact, for a wide range of interesting ALP/magnetic field parameter space, the variation of [MATH] with distance is sufficiently rapid that it cannot be neglected in almost all regions along the path of propagation.', '1511.03380-1-4-2': 'We show in this paper that a proper treatment of the random variation of the intergalactic magnetic field gives a qualitatively different result for the photon-to-ALP conversion probability compared to that obtained in the "discrete-[MATH]" model.', '1511.03380-1-5-0': 'For concreteness, we will focus on TeV photon-ALP propagating through intergalactic medium over cosmological distances, but our analysis and method can be easily rescaled to other situations.', '1511.03380-1-6-0': 'Equations.', '1511.03380-1-6-1': 'The evolution equation of the photon electric field [MATH] and ALP field [MATH] of a given angular frequency [MATH] or energy [MATH] (so that [MATH]), expressed in a fixed Cartesian coordinates [MATH] (with the [MATH]-axis along the direction of propagation), takes the form [EQUATION]', '1511.03380-1-6-2': 'Here the superscript [MATH] stands for [MATH], and [MATH] is the azimuthal angle of the magnetic field [MATH] (more precisely, [MATH] is the angle between [MATH], the projection of [MATH] in the [MATH]-plane, and the [MATH]-axis).', '1511.03380-1-6-3': 'The ALP-mass-related parameter [MATH] and the ALP-photon coupling parameter [MATH] are given by [EQUATION] where [MATH] is the ALP mass, [MATH] is the photon energy, [MATH] is the axion-photon interaction parameter.', '1511.03380-1-6-4': 'We adopt units such that [MATH], and define dimensionless quantities [MATH], [MATH], [MATH], and [MATH].', '1511.03380-1-6-5': 'The plasma parameter [MATH] (where [MATH] is the electron plasma frequency and [MATH] is the electron density) is unimportant for the parameter regime considered in this paper and will be neglected.', '1511.03380-1-6-6': 'Also, Eq. ([REF]) does not include the QED effect, which is negligible for typical nG intergalactic magnetic fields [CITATION].', '1511.03380-1-7-0': 'Analytical Results.', '1511.03380-1-7-1': 'For a given magnetic field structure, the ALP-photon evolution can be obtained by integrating Eq. ([REF]) along the ray.', '1511.03380-1-7-2': 'Before studying complex random fields, we first consider two simple "single-domain" cases: (i) [MATH] independent of [MATH]; (ii) [MATH] increases linearly with [MATH], with [MATH].', '1511.03380-1-7-3': 'Some numerical results are plotted in Fig. [REF]), showing that the photon-to-ALP conversion probabilities are quite different in the two cases.', '1511.03380-1-8-0': 'To understand the difference analytically, we consider the evolution of [MATH] and [MATH], the components of [MATH] parallel and perpendicular to [MATH], respectively.', '1511.03380-1-8-1': 'Since [MATH], [MATH], Eq. ([REF]) can be rewritten as [EQUATION] where we have dropped the non-essential term [MATH] in the diagonal elements.', '1511.03380-1-9-0': 'If [MATH], Eq. ([REF]) can be simplified to the evolution equation of [MATH] and [MATH]: [EQUATION]', '1511.03380-1-9-1': 'This equation has been widely discussed in previous works.', '1511.03380-1-9-2': 'If the magnetic field strength varies slowly (we assume [MATH] is constant along the ray for simplicity), the mode evolution is said to be "adiabatic", and the photon-to-ALP conversion probability is given by the well-known formula [CITATION]: [EQUATION] with [MATH].', '1511.03380-1-9-3': 'In the limit of [MATH], Eq. ([REF]) simplifies to [MATH].', '1511.03380-1-10-0': 'Intergalactic magnetic fields can often have [MATH], much larger than [MATH] and [MATH] (see Eqs. 2-3).', '1511.03380-1-10-1': 'If [MATH], the electric field can be solved as [MATH], [MATH] assuming [MATH], [MATH] at [MATH], i.e., [MATH].', '1511.03380-1-10-2': 'Substitute this electric field into Eq. ([REF]), we find the evolution equation for the ALP field, [MATH], with the solution [EQUATION]', '1511.03380-1-10-3': 'For [MATH] with constant [MATH], we obtain the photon-to-ALP conversion probability', '1511.03380-1-11-0': 'P=a(z)^2_M^2_a^2(1-[(', '1511.03380-1-12-0': '+(/_a-_az)^2].', '1511.03380-1-13-0': 'This equation accurately describes the numerical result of Fig. [REF] for various values of [MATH].', '1511.03380-1-13-1': 'For example, in the limit of [MATH], Eq. ([REF]) simplifies to [MATH], which has an oscillation length [MATH] and is independent of the ALP mass and energy.', '1511.03380-1-14-0': 'Results for Random Magnetic Fields.', '1511.03380-1-14-1': 'The magnetic field in the intergalactic medium is randomly distributed, with the expected coherent length of order 0.1-1 Mpc (about the size of galaxy clusters).', '1511.03380-1-14-2': 'In general, numerical integrations are necessary to obtain the photon-to-axion conversion probability for a given realization of the random magnetic field distributions in addition to the relevant ALP parameters.', '1511.03380-1-14-3': 'A "discrete-[MATH]" model has been widely used in previous studies: The path of propagation is divided into many domains, each has the same size [MATH] and a uniform magnetic field, with the magnetic orientation angle [MATH] changing randomly but discretely from one domain to the next.', '1511.03380-1-14-4': 'Based on this model, Ref. [CITATION] derived an analytic expression for the mean value of the photo-to-ALP conversion probability after propagating through [MATH] domains (over distance [MATH]): [EQUATION] where on the right-hand side, [MATH] is given by Eq. ([REF]) evaluated at [MATH].', '1511.03380-1-14-5': 'Note that for [MATH], we have [MATH], an upper limit for the conversion probability.', '1511.03380-1-14-6': 'In Fig. [REF] we depict an example of the discrete-[MATH] model (with [MATH] Mpc) and the numerical results for the conversion probabilities at different values of [MATH] (black lines).', '1511.03380-1-14-7': 'These numerical results are in agreement with Eq. ([REF]) in the statistical sense.', '1511.03380-1-15-0': 'As discussed above, we expect that the discrete-[MATH] model may be problematic since in most regions of the intergalactic medium [MATH] can be much larger than [MATH].', '1511.03380-1-15-1': 'In Fig. [REF] (see the red lines) we consider a "linearly-continuous-[MATH]" model: The path of propagation is again divided into many equal-sized domains; in each domain, [MATH] varies linearly from one random value to another (thus, [MATH] is always continuous, [MATH] is constant inside each domain but changes across the domain boundary).', '1511.03380-1-15-2': 'Our numerical results show that this continuous-[MATH] model can yield completely different conversion probabilities compared to the discrete-[MATH] model.', '1511.03380-1-15-3': 'In particular, [MATH] exhibits quasi-periodicity along the ray (with the period dependent on [MATH]) and can be close to unity for large values of [MATH].', '1511.03380-1-16-0': 'To understand these numerical results, we apply Eq. ([REF]) to the linearly-continuous-[MATH] model.', '1511.03380-1-16-1': 'The ALP amplitude after traversing [MATH] domains is given by [EQUATION]', '1511.03380-1-16-2': 'In the [MATH]-th domain, [MATH], with [MATH].', '1511.03380-1-16-3': 'For [MATH] and [MATH] (these two conditions are similar since [MATH]), Eq. ([REF]) can be simplified, giving [EQUATION] where', '1511.03380-1-17-0': 'A_j(_j-_j-1)^2-_a^2l^2 +i_al(_j-_j-1)^2-_a^2l^2,', '1511.03380-1-18-0': 'with [MATH].', '1511.03380-1-18-1': 'For random [MATH] (varying between [MATH] and [MATH]), [MATH] can be characterized by the mean [MATH] and variance [MATH].', '1511.03380-1-18-2': 'The mean photo-to-ALP conversion probability [MATH] is then [EQUATION] where we have used [MATH].', '1511.03380-1-18-3': 'The variance [MATH] can be calculated using Monte-Carlo method, and we find [MATH] and [MATH] for [MATH] and [MATH]>[MATH][MATH] (corresponding to [MATH] and [MATH]<[MATH][MATH]).', '1511.03380-1-18-4': 'Note that the validity of Eq. ([REF]) requires [MATH], [MATH] and [MATH] (or [MATH]).', '1511.03380-1-18-5': 'Under the same condition, Eq. ([REF]) reduces to [MATH], similar to the second term in Eq. ([REF]).', '1511.03380-1-19-0': 'Equation ([REF]) indicates that the photon-to-ALP conversion probability has a cosine function dependence, with the oscillation length [MATH] Mpc.', '1511.03380-1-19-1': 'This is in agreement with numerical results presented in Fig. [REF], especially for [MATH]<[MATH][MATH]<[MATH][MATH].', '1511.03380-1-19-2': 'For [MATH]<[MATH][MATH], the inequality [MATH] is not well satisfied; for [MATH]>[MATH][MATH], the ALP amplitude can be comparable to [MATH], making Eq. ([REF]) inaccurate.', '1511.03380-1-20-0': 'Distribution of Conversion Probability.', '1511.03380-1-20-1': 'Because the intergalactic magnetic field has random orientations along the propagation path, the conversion probability [MATH] has a distribution with finite spread.', '1511.03380-1-20-2': 'To obtain the [MATH] distribution, we carry out Monte-Carlo calculations of the photon-ALP propagations for [MATH] times, each time with the same set of ALP and magnetic field parameters, but for different random values of [MATH] in each domain.', '1511.03380-1-20-3': 'We consider both the discrete [MATH] model and the linearly-continuous [MATH] model as discussed above.', '1511.03380-1-20-4': 'The results are shown in Fig. [REF].', '1511.03380-1-21-0': 'For the discrete-[MATH] model, the [MATH]-distribution function, [MATH], is a skewed Gaussian (see the upper panel of Fig. [REF]).', '1511.03380-1-21-1': 'The peak of the distribution is accurately predicted by Eq. ([REF]).', '1511.03380-1-21-2': 'We find that [MATH] is almost the same for different values of [MATH], except that for [MATH]>[MATH][MATH] the distribution becomes a broader.', '1511.03380-1-22-0': 'The lower panel of Fig. [REF] shows the [MATH]-distribution function [MATH] for the linearly-continuous [MATH] model.', '1511.03380-1-22-1': 'For [MATH]<[MATH][MATH] the distribution function is similar to that of the discrete-[MATH] model.', '1511.03380-1-22-2': 'However, as [MATH] increases, the peak of the distribution shifts to larger values and the width becomes narrower - these features are in marked contrast to the discrete-[MATH] model.', '1511.03380-1-23-0': 'To characterize how the [MATH]-distribution function varies for different parameters, we show in Fig. [REF] [MATH] and [MATH], the peak and half-peak width of the distribution for two different values of [MATH], as a function of [MATH] (the other parameters are the same as in Fig. [REF]).', '1511.03380-1-23-1': 'For the discrete-[MATH] model, [MATH] is almost independent of [MATH], except when the oscillation length is smaller than domain size, i.e., when [MATH]>[MATH][MATH] or [MATH]<[MATH][MATH].', '1511.03380-1-23-2': 'The value of [MATH] can be accurately predicted by Eq. ([REF]).', '1511.03380-1-23-3': 'The width of the distribution is almost constant except for [MATH]>[MATH][MATH].', '1511.03380-1-23-4': 'Note that for [MATH], the conversion probability is close to the upper limit 1/3, and the [MATH] distribution is not a Gaussian.', '1511.03380-1-24-0': 'For the linearly-continuous [MATH] model, larger [MATH] generally leads to larger [MATH] and smaller [MATH].', '1511.03380-1-24-1': 'Interestingly, both [MATH] and [MATH] are not a monotonous function of [MATH], but have oscillations.', '1511.03380-1-24-2': 'This oscillation can be described by Eq. ([REF]), as shown by the dotted lines in Fig. [REF] [Note that Eq. ([REF]) is valid only for [MATH], so we choose the dotted lines start from [MATH]].', '1511.03380-1-24-3': 'For [MATH]<[MATH][MATH], both [MATH] and [MATH] are almost the same as in the discrete-[MATH] model.', '1511.03380-1-24-4': 'In the case of [MATH] (blue lines), Eq. ([REF]) agrees very well with the numerical result, since the assumption [MATH] always tenable.', '1511.03380-1-24-5': 'For [MATH] [or [MATH]], the conversion probability reaches its maximum [MATH].', '1511.03380-1-24-6': 'In the case of [MATH], the peak conversion probability [MATH] approaches unity for [MATH]>[MATH][MATH], implying a nearly [MATH] photon-to-ALP conversion [Of course, the analytical expression ([REF]) is less accurate when [MATH] since [MATH] is invalid].', '1511.03380-1-25-0': 'In the above, we have focused on the linearly-continuous-[MATH] model, since in this case we can derive analytical equations [see Eqs. ([REF]) and ([REF])] to help understand our numerical results.', '1511.03380-1-25-1': 'We have performed calculations for other continuous-[MATH] models, e.g., using the spline function or sine function to link the random [MATH] values in multiple domains.', '1511.03380-1-25-2': 'We find that the results for the [MATH] distribution are similar to the linearly-continuous [MATH] model, although the conversion probabilities are slightly lower because in the spline and sine [MATH] models there always exist some regions with [MATH].', '1511.03380-1-25-3': 'We have also considered the case of varying magnetic field strength in different domains; we find that the [MATH]-distribution in this case is similar to the other continuous-[MATH] models.', '1511.03380-1-26-0': 'Discussion.', '1511.03380-1-26-1': 'We have shown that a proper treatment of the inhomogeneity of intergalactic magnetic fields can lead to very different photon-to-ALP conversion probabilities compared to the "discrete-[MATH]" model widely used in previous studies.', '1511.03380-1-26-2': 'The difference is particularly striking when [MATH]>[MATH][MATH] [the first term of Eq. ([REF]) larger than the second term; here [MATH] is the distance and [MATH] is the coherence length of the magnetic field].', '1511.03380-1-26-3': 'In the discrete-[MATH] model, the conversion probability is determined by [MATH] and almost does not depend on [MATH] (assuming [MATH] and [MATH]), and never exceeds [MATH] [see Eq. ([REF])].', '1511.03380-1-26-4': 'By contrast, in the continuous-[MATH] model, the photon-to-axion conversion probability has a distinct dependence on [MATH]; it becomes significant when [MATH]<[MATH][MATH] [see Eqs. (2)-(3)] and can be as large as 100% (see Figs. [REF]-[REF]).', '1511.03380-1-26-5': 'Our analytic expression ([REF]) (valid for [MATH], [MATH] and [MATH]<[MATH][MATH]) approximately captures these features.', '1511.03380-1-27-0': 'A number of recent works have highlighted the possible roles of ALP-photon oscillations on the observations of TeV gamma-ray sources.', '1511.03380-1-27-1': 'For example, Wouters Brun [CITATION] proposed to search irregularities in gammy ray source spectra as a signature of ALPs; Horns et al. [CITATION] explored the hardening of the TeV photon spectrum of Blazars due to the [MATH] conversions in galactic and extragalatic magnetic fields; Meyer et al. [CITATION] discussed ALP-photon oscillations in turbulent and coherent magnetic fields and their impacts on gamma-ray detection.', '1511.03380-1-27-2': 'In all these works, the discrete-[MATH] model and Eq. ([REF]) are used to estimate the photon-to-ALP conversion probabilities.', '1511.03380-1-27-3': 'In light of the significant difference between the discrete and continuous-[MATH] models of magnetic fields, a re-evaluation of the previous results is warranted.', '1511.03380-1-28-0': 'This work has been supported in part by the National Natural Science Foundation of China (11273029), and by NSF grant AST-1211061 and NASA grant NNX14AG94G.'}

{'1511.03380-2-0-0': '# Introduction.', '1511.03380-2-1-0': 'Axion is particle first introduced to solve the strong CP problem [CITATION].', '1511.03380-2-1-1': 'Axion-like particles (ALPs) also appear in many theoretically well-motivated extensions of the standard model of particle physics [CITATION].', '1511.03380-2-1-2': 'A general property of ALPs (represented by the field [MATH]) is that they can couple to photons (represented by [MATH]) in the presence of an external magnetic field [MATH] through the interaction Lagrangian [MATH].', '1511.03380-2-1-3': 'While for axions there exists a relation between the coupling constant [MATH] and the axion mass [MATH], in general [MATH] and [MATH] are unrelated for ALPs.', '1511.03380-2-2-0': 'As a result of the photon-ALP coupling, a photon can oscillate into an ALP and vise versa in an external magnetic field.', '1511.03380-2-2-1': 'Such ALP-photon oscillations have been invoked to explain a variety of astrophysical phenomena, or conversely used to constrain the properties of ALPs using observations [CITATION].', '1511.03380-2-2-2': 'Examples include the apparent dimming of distant supernovae [CITATION], spectral distortions of the cosmic microwave background [CITATION], and the dispersion of QSO spectra [CITATION], et al. Recently, ALP-photon oscillation has been used to explain anomalous lack of opacity of the Universe to gamma rays: high energy gamma ray photons from Active Galactic Nuclei (AGNs) at cosmological distances have been detected by HESS, MAGIC and Fermi [CITATION].', '1511.03380-2-2-3': 'These photons can suffer significant attenuation before reaching Earth due to electron-positron pair production on the extragalactic background infrared radiation.', '1511.03380-2-2-4': 'Several analysis suggest that the Universe appears more transparent than expected based on recent extra-galactic background light models( [CITATION]; however see [CITATION]).', '1511.03380-2-2-5': 'A possible explanation to the transparency problem is that because of the ALP-photon mixing, radiation from AGNs travels in the form of ALPs on a significant fraction of distance (without producing pairs) and converts back to photons before their detections [CITATION].', '1511.03380-2-2-6': 'Another example concerns the possibilities that the recent observed 3.55keV photon line [CITATION] may arise from dark matter decay to ALPs and then convert to photons due to oscillations in the magnetic field of M31 and the Milky Way [CITATION].', '1511.03380-2-3-0': 'ALP-photon propagation over cosmological distance is strongly affected by the magnetic field structure.', '1511.03380-2-3-1': 'The primordial extragalactic magnetic field is most likely random and could described as patches of coherent domains with a typical magnitude upper limit of a few nG [CITATION] and scale length of order a few Mpc [CITATION].', '1511.03380-2-3-2': 'Previous studies have adopted a simple model, in which the magnetic field is uniform in each domain, but the field orientation (characterized by the angle [MATH]) changes in a random fashion from one domain to the next.', '1511.03380-2-3-3': 'Note that in this "discrete-[MATH]" model, the photon-to-ALP conversion probability in each domain [MATH] can be easily derived [see Eq. ([REF])] (since [MATH] is constant in each domain).', '1511.03380-2-3-4': "Assuming that [MATH]'s for different domains are random, Grossman et al. [CITATION] then derived an expression for the photon-to-ALP conversion probability through a large number of domains [see Eq. ([REF])], and this expression has been widely used in many previous studies.", '1511.03380-2-4-0': 'In realistic situations, the magnetic field and its orientation angle [MATH] should vary continuously across neighboring domains.', '1511.03380-2-4-1': 'In fact, for a wide range of interesting ALP/magnetic field parameter space, the variation of [MATH] with distance is sufficiently rapid that it cannot be neglected in almost all regions along the path of propagation.', '1511.03380-2-4-2': 'We show in this paper that a proper treatment of the random variation of the intergalactic magnetic field gives a qualitatively different result for the photon-to-ALP conversion probability compared to that obtained in the "discrete-[MATH]" model.', '1511.03380-2-5-0': 'For concreteness, we will focus on TeV photon-ALP propagating through intergalactic medium over cosmological distances, but our analysis and method can be easily re-scaled to other situations such as the Milky way or galaxy clusters, as well as for different photon energies.', '1511.03380-2-6-0': '# Equations', '1511.03380-2-7-0': 'The evolution equation of the photon electric field [MATH] and ALP field [MATH] of a given angular frequency [MATH] or energy [MATH] (so that [MATH]), expressed in a fixed Cartesian coordinates [MATH] (with the [MATH]-axis along the direction of propagation), takes the form [EQUATION]', '1511.03380-2-7-1': 'Here the superscript [MATH] stands for [MATH], and [MATH] is the azimuthal angle of the magnetic field [MATH] (more precisely, [MATH] is the angle between [MATH], the projection of [MATH] in the [MATH]-plane, and the [MATH]-axis).', '1511.03380-2-7-2': 'The ALP-mass-related parameter [MATH] and the ALP-photon coupling parameter [MATH] are given by [EQUATION] where [MATH] is the ALP mass, [MATH] is the photon energy, [MATH] is the axion-photon interaction parameter.', '1511.03380-2-7-3': 'We adopt units such that [MATH], and define dimensionless quantities [EQUATION]', '1511.03380-2-7-4': 'The plasma parameter [MATH] (where [MATH] is the electron plasma frequency and [MATH] is the electron density) is unimportant for the parameter regime considered in this paper and will be neglected.', '1511.03380-2-7-5': 'Also, Eq. ([REF]) does not include the QED effect, which is negligible for typical nG intergalactic magnetic fields [CITATION].', '1511.03380-2-7-6': 'All the numerical results presented in this paper are based on Eq. ([REF]).', '1511.03380-2-8-0': '# Analytical Results', '1511.03380-2-9-0': 'For a given magnetic field structure, the ALP-photon evolution can be obtained by integrating Eq. ([REF]) along the ray.', '1511.03380-2-9-1': 'Before studying complex random fields, we first consider two simple "single-domain" cases: (i) [MATH] independent of [MATH]; (ii) [MATH] increases linearly with [MATH], with [MATH].', '1511.03380-2-9-2': 'Typical intergalactic magnetic fields have a coherence length of order [MATH]Mpc, we define [EQUATION]', '1511.03380-2-9-3': 'Note that, as a function of [MATH], the photon-to-ALP conversion probability [MATH] depends only on the dimensionless quantities [MATH] and [MATH], and thus on [MATH] and [MATH].', '1511.03380-2-9-4': 'Some numerical results are plotted in Fig. [REF], showing that the photon-to-ALP conversion probabilities are quite different in the two cases.', '1511.03380-2-10-0': 'To understand the difference analytically, we consider the evolution of [MATH] and [MATH], the components of [MATH] parallel and perpendicular to [MATH], respectively.', '1511.03380-2-10-1': 'Since [MATH], [MATH], Eq. ([REF]) can be rewritten as [EQUATION] where we have dropped the non-essential term [MATH] in the diagonal elements.', '1511.03380-2-11-0': 'If [MATH], Eq. ([REF]) can be simplified to the evolution equation of [MATH] and [MATH]: [EQUATION]', '1511.03380-2-11-1': 'This equation has been widely discussed in previous works.', '1511.03380-2-11-2': 'If the magnetic field strength varies slowly (we assume [MATH] is constant along the ray for simplicity), the mode evolution is said to be "adiabatic", and the photon-to-ALP conversion probability is given by the well-known formula [CITATION]: [EQUATION] with [MATH].', '1511.03380-2-11-3': 'In the limit of [MATH], Eq. ([REF]) simplifies to [MATH].', '1511.03380-2-12-0': 'Intergalactic magnetic fields can often have [MATH], much larger than [MATH] and [MATH] (see Eqs. 2-3).', '1511.03380-2-12-1': 'If [MATH], the electric field can be solved as [MATH], [MATH] assuming [MATH], [MATH] at [MATH], i.e., [MATH].', '1511.03380-2-12-2': 'Substitute this electric field into Eq. ([REF]), we find the evolution equation for the ALP field, [MATH], with the solution [EQUATION]', '1511.03380-2-12-3': 'For [MATH] with constant [MATH], we obtain the photon-to-ALP conversion probability', '1511.03380-2-13-0': 'P=a(z)^2_M^2_a^2(1-[(', '1511.03380-2-14-0': '+(/_a-_az)^2].', '1511.03380-2-15-0': 'This equation accurately describes the numerical result of Fig. [REF] for various values of [MATH].', '1511.03380-2-15-1': 'For example, in the limit of [MATH], Eq. ([REF]) simplifies to [MATH], which has an oscillation length [MATH] and is independent of the ALP mass and energy.', '1511.03380-2-16-0': '# Results for Random Magnetic Fields', '1511.03380-2-17-0': 'The magnetic field in the intergalactic medium is randomly distributed, with the expected coherent length of order 0.1-1 Mpc (about the size of galaxy clusters).', '1511.03380-2-17-1': 'In general, numerical integrations are necessary to obtain the photon-to-axion conversion probability for a given realization of the random magnetic field distributions in addition to the relevant ALP parameters.', '1511.03380-2-17-2': 'A "discrete-[MATH]" model has been widely used in previous studies: The path of propagation is divided into many domains, each has the same size [MATH] and a uniform magnetic field, with the magnetic orientation angle [MATH] changing randomly but discretely from one domain to the next.', '1511.03380-2-17-3': 'Based on this model, Ref. [CITATION] derived an analytic expression for the mean value of the photo-to-ALP conversion probability after propagating through [MATH] domains (over distance [MATH]): [EQUATION] where on the right-hand side, [MATH] is given by Eq. ([REF]) evaluated at [MATH].', '1511.03380-2-17-4': 'Note that for [MATH], we have [MATH], an upper limit for the conversion probability.', '1511.03380-2-17-5': 'In Fig. [REF] we depict an example of the discrete-[MATH] model and the numerical results for the conversion probabilities at different values of [MATH] (black lines).', '1511.03380-2-17-6': 'These numerical results are in agreement with Eq. ([REF]) in the statistical sense.', '1511.03380-2-18-0': 'As discussed above, we expect that the discrete-[MATH] model may be problematic since in most regions of the intergalactic medium [MATH] can be much larger than [MATH].', '1511.03380-2-18-1': 'In Fig. [REF] (see the red lines) we consider a "linearly-continuous-[MATH]" model: The path of propagation is again divided into many equal-sized domains; in each domain, [MATH] varies linearly from one random value to another (thus, [MATH] is always continuous, [MATH] is constant inside each domain but changes across the domain boundary).', '1511.03380-2-18-2': 'Our numerical results show that this continuous-[MATH] model can yield completely different conversion probabilities compared to the discrete-[MATH] model.', '1511.03380-2-18-3': 'In particular, [MATH] exhibits quasi-periodicity along the ray (with the period dependent on [MATH]) and can be close to unity for large values of [MATH].', '1511.03380-2-19-0': 'To understand these numerical results, we apply Eq. ([REF]) to the linearly-continuous-[MATH] model.', '1511.03380-2-19-1': 'The ALP amplitude after traversing [MATH] domains is given by [EQUATION]', '1511.03380-2-19-2': 'In the [MATH]-th domain, [MATH], with [MATH].', '1511.03380-2-19-3': 'For [MATH] and [MATH] (these two conditions are similar since [MATH]), Eq. ([REF]) can be simplified, giving [EQUATION] where', '1511.03380-2-20-0': 'A_j(_j-_j-1)^2-_a^2l^2 +i_al(_j-_j-1)^2-_a^2l^2,', '1511.03380-2-21-0': 'with [MATH].', '1511.03380-2-21-1': 'For random [MATH] (varying between [MATH] and [MATH]), [MATH] can be characterized by the mean [MATH] and variance [MATH].', '1511.03380-2-21-2': 'The mean photo-to-ALP conversion probability [MATH] is then [EQUATION] where we have used [MATH].', '1511.03380-2-21-3': 'The variance [MATH] can be calculated using Monte-Carlo method, and we find [MATH] and [MATH] for [MATH] and [MATH]>[MATH][MATH] (corresponding to [MATH] and [MATH]<[MATH][MATH]).', '1511.03380-2-21-4': 'Note that the validity of Eq. ([REF]) requires [MATH], [MATH] and [MATH] (or [MATH]).', '1511.03380-2-21-5': 'Under the same condition, Eq. ([REF]) reduces to [MATH], similar to the second term in Eq. ([REF]).', '1511.03380-2-22-0': 'Equation ([REF]) indicates that the photon-to-ALP conversion probability has a cosine function dependence, with the oscillation length (in units of [MATH].', '1511.03380-2-22-1': 'This is in agreement with numerical results presented in Fig. [REF], especially for [MATH]<[MATH][MATH]<[MATH][MATH].', '1511.03380-2-22-2': 'For [MATH]<[MATH][MATH], the inequality [MATH] is not well satisfied; for [MATH]>[MATH][MATH], the ALP amplitude can be comparable to [MATH], making Eq. ([REF]) inaccurate.', '1511.03380-2-23-0': '# Distribution of Conversion Probability', '1511.03380-2-24-0': 'Because the intergalactic magnetic field has random orientations along the propagation path, the conversion probability [MATH] has a distribution with finite spread.', '1511.03380-2-24-1': 'To obtain the [MATH] distribution, we carry out Monte-Carlo calculations of the photon-ALP propagations for [MATH] times, each time with the same set of ALP and magnetic field parameters, but for different random values of [MATH] in each domain.', '1511.03380-2-24-2': 'We consider both the discrete [MATH] model and the linearly-continuous [MATH] model as discussed above.', '1511.03380-2-24-3': 'The results are shown in Fig. [REF].', '1511.03380-2-25-0': 'For the discrete-[MATH] model, the [MATH]-distribution function, [MATH], is a skewed Gaussian (see the upper panel of Fig. [REF]).', '1511.03380-2-25-1': 'The peak of the distribution is accurately predicted by Eq. ([REF]).', '1511.03380-2-25-2': 'We find that [MATH] is almost the same for different values of [MATH], except that for [MATH]>[MATH][MATH] the distribution becomes a broader.', '1511.03380-2-26-0': 'The lower panel of Fig. [REF] shows the [MATH]-distribution function [MATH] for the linearly-continuous [MATH] model.', '1511.03380-2-26-1': 'For [MATH]<[MATH][MATH] the distribution function is similar to that of the discrete-[MATH] model.', '1511.03380-2-26-2': 'However, as [MATH] increases, the peak of the distribution shifts to larger values and the width becomes narrower - these features are in marked contrast to the discrete-[MATH] model.', '1511.03380-2-27-0': 'To characterize how the [MATH]-distribution function varies for different parameters, we show in Fig. [REF] [MATH] and [MATH], the peak and half-peak width of the distribution for two different values of [MATH], as a function of [MATH] (the other parameters are the same as in Fig. [REF]).', '1511.03380-2-27-1': 'For the discrete-[MATH] model (blue lines in Fig. [REF]), [MATH] is almost independent of [MATH], except when the oscillation length is smaller than domain size, i.e., when [MATH]>[MATH][MATH] or [MATH]<[MATH][MATH].', '1511.03380-2-27-2': 'The value of [MATH] can be accurately predicted by Eq. ([REF]).', '1511.03380-2-27-3': 'The width of the distribution is almost constant except for [MATH]>[MATH][MATH].', '1511.03380-2-27-4': 'Note that for [MATH], the conversion probability is close to the upper limit 1/3, and the [MATH] distribution is not a Gaussian.', '1511.03380-2-28-0': 'For the linearly-continuous [MATH] model (red lines in Fig. [REF]), larger [MATH] generally leads to larger [MATH] and smaller [MATH].', '1511.03380-2-28-1': 'Interestingly, both [MATH] and [MATH] are not a monotonous function of [MATH], but have oscillations.', '1511.03380-2-28-2': 'This oscillation can be described by Eq. ([REF]), as shown by the black lines in Fig. [REF] [Note that Eq. ([REF]) is valid only for [MATH], so we choose the dotted lines start from [MATH]].', '1511.03380-2-28-3': 'For [MATH]<[MATH][MATH], both [MATH] and [MATH] are almost the same as in the discrete-[MATH] model.', '1511.03380-2-28-4': 'In the case of [MATH] (blue lines), Eq. ([REF]) agrees very well with the numerical result, since the assumption [MATH] always tenable.', '1511.03380-2-28-5': 'For [MATH] [or [MATH]], the conversion probability reaches its maximum [MATH].', '1511.03380-2-28-6': 'In the case of [MATH], the peak conversion probability [MATH] approaches unity for [MATH]>[MATH][MATH], implying a nearly [MATH] photon-to-ALP conversion [Of course, the analytical expression ([REF]) is less accurate when [MATH] since [MATH] is invalid].', '1511.03380-2-29-0': 'In the above, we have focused on the linearly-continuous-[MATH] model, since in this case we can derive analytical equations [see Eqs. ([REF]) and ([REF])] to help understand our numerical results.', '1511.03380-2-29-1': 'We have performed calculations for other continuous-[MATH] models, e.g., using the spline function or sine function to link the random [MATH] values in multiple domains (see Fig. [REF]).', '1511.03380-2-29-2': 'We find that the results for the [MATH] distribution are similar to the linearly-continuous-[MATH] model, although the conversion probabilities are slightly lower because in the spline and sine [MATH] models there always exist some regions with [MATH].', '1511.03380-2-30-0': 'So far in this paper we have assumed that the magnetic field has the same strength [MATH] in different domains but with varying orientations.', '1511.03380-2-30-1': 'What happens when [MATH] also varies?', '1511.03380-2-30-2': 'For concreteness, we consider a simple model where the values of [MATH] in different domains are randomly distributed in the range between [MATH] and [MATH].', '1511.03380-2-30-3': 'Our numerical results for the final [MATH] distributions (for both discrete-[MATH] and continuous-[MATH] models) for various values of [MATH], [MATH] and [MATH] are shown in Figure [REF].', '1511.03380-2-30-4': 'The results are very similar to the constant [MATH] case shown in Fig. [REF].', '1511.03380-2-30-5': 'For the discrete-[MATH] model (Fig. [REF]a), we can derive an analytical expression of the final conversion probability using Eq. ([REF]) (with [MATH]constant in each domain, but [MATH] and [MATH] have different values in different domains).', '1511.03380-2-30-6': 'For [MATH], we find [EQUATION] where [MATH].', '1511.03380-2-30-7': 'The above equation is the same as the case with constant [MATH] and discrete [MATH]), if we replace [MATH] by [MATH].', '1511.03380-2-30-8': 'Equation ([REF]) agrees well with the numerical results shown in Fig. [REF]a.', '1511.03380-2-30-9': 'The same applies for the linearly-continuous-[MATH] model: the [MATH] curves shown in Fig. [REF]b are almost the same as the constant [MATH] case if we replace [MATH] by [MATH].', '1511.03380-2-31-0': '# Discussion', '1511.03380-2-32-0': 'We have shown that a proper treatment of the inhomogeneity of intergalactic magnetic fields can lead to very different photon-to-ALP conversion probabilities compared to the "discrete-[MATH]" model widely used in previous studies.', '1511.03380-2-32-1': 'The difference is particularly striking when [MATH]>[MATH][MATH] [the first term of Eq. ([REF]) larger than the second term; here [MATH] is the coherence length of the magnetic field domain and [MATH] the domain numbers across distance [MATH]].', '1511.03380-2-32-2': 'In the discrete-[MATH] model, the conversion probability is determined by [MATH] and almost does not depend on [MATH] or [MATH] (assuming [MATH] and [MATH]), and never exceeds [MATH] [see Eq. ([REF])].', '1511.03380-2-32-3': 'By contrast, in the continuous-[MATH] model, the photon-to-axion conversion probability has a distinct dependence on [MATH]; it becomes significant when [MATH]<[MATH][MATH] [see Eqs. (2)-(3)] and can be as large as 100% (see Figs. [REF]-[REF]).', '1511.03380-2-32-4': 'Our analytic expression ([REF]) (valid for [MATH], [MATH] and [MATH]<[MATH][MATH]) approximately captures these features.', '1511.03380-2-33-0': 'Note that although we have considered TeV photon-ALP propagation in intergalactic magnetic fields (with [MATH], [MATH]), our results can be easily re-scaled to different situations.', '1511.03380-2-33-1': 'For example, Ref. [CITATION] explored the hardening of the TeV photon spectrum of Blazars due to the [MATH] conversions in the magnetic fields of galaxy clusters and Milky Way.', '1511.03380-2-33-2': 'In galaxy clusters, [MATH], [MATH], we have [EQUATION]', '1511.03380-2-33-3': 'Thus with [MATH]eV, [MATH]GeV (see [CITATION]), the conversion of TeV photon-to-ALP in the galaxy clusters can be significantly affected by our results.', '1511.03380-2-33-4': 'ALPs can convert back to be TeV photons in the magnetic field of the Milky Way.', '1511.03380-2-33-5': 'Similar ALP-to-photon conversion in M31 and the Milky Way are used to explain the recent observations of the 3.55 keV photon line [CITATION].', '1511.03380-2-33-6': 'Our results can be easily adapted to the typical galactic magnetic field ([MATH], [MATH]pc-[MATH]kpc) with different [MATH], [MATH] and [MATH].', '1511.03380-2-34-0': 'In summary, many previous works use the discrete-[MATH] model and Eq. ([REF]) to estimate the photon-to-ALP conversion probabilities.', '1511.03380-2-34-1': 'In light of the significant difference between the discrete and continuous-[MATH] models of magnetic fields, a re-evaluation of the previous results is warranted.'}

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[]

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[]

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{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1511.03380

1904.03047

{'1904.03047-1-0-0': 'We explore the dynamics and self-affinity of magnetic domain wall driven by electrical current, in a ferromagnetic (Ga,Mn)(As,P) thin film with perpendicular magnetic anisotropy.', '1904.03047-1-0-1': 'We show that common universal behaviors corresponding to the so-called quenched Edwards-Wilkinson universality class are shared by current and magnetic field driven domain wall motion.', '1904.03047-1-0-2': 'The universal behaviors of the creep and depinning dynamical regimes can be described by a unique self-consistent model.', '1904.03047-1-0-3': 'However, the transverse orientation between current and domain wall is found to be unstable, leading to the formation of faceted structures typical of the quenched Kardar-Parisi-Zhang universality class.', '1904.03047-1-1-0': 'The displacement of small spin texture as magnetic domain walls (DWs) thanks to spin torque effects is at the basis of potential applications to magnetic memory storage [CITATION].', '1904.03047-1-1-1': 'An important effort is dedicated to search for magnetic materials [CITATION] with large and well controlled DW velocities.', '1904.03047-1-1-2': 'However, DWs are very sensitive to weak pinning defects [CITATION], which strongly reduce their mobility and produce roughening and stochastic avalanche-like motion [CITATION].', '1904.03047-1-1-3': 'Therefore, it is particularly interesting to better understand the contribution of pinning to current induced DW dynamics.', '1904.03047-1-2-0': 'Magnetic domain walls [CITATION] present surprising universal critical behaviors, encountered in a wide variety of moving interfaces such as the reaction front propagation in disordered flows [CITATION], growing bacterial colonies [CITATION], wetting [CITATION], motion of ferroelectric domain walls [CITATION], to name a few.', '1904.03047-1-2-1': 'The displacement interfaces presents self-affine scaling variations: [MATH], where [MATH] is the position of interface, [MATH] a distance between two points of the interface and [MATH] the roughness exponent.', '1904.03047-1-2-2': 'Moreover, a depinning driving force [MATH] separates the so-called creep ([MATH]) and depinning ([MATH]) regimes.', '1904.03047-1-2-3': 'In the creep regime, the velocity varies [CITATION] as an Arrhenius law [MATH], where [MATH] is the thermal fluctuation energy.', '1904.03047-1-2-4': '[MATH] is the effective pinning energy barriers height, which follows a universal power law variation with the drive [MATH], where [MATH] is the creep exponent.', '1904.03047-1-2-5': 'In the depinning regime [CITATION], the effective pinning barriers are collapsed.', '1904.03047-1-2-6': 'The velocity presents power law variations with drive [MATH] and temperature [MATH]: [MATH] and [MATH], where [MATH] and [MATH] are the depinning and thermal rounding exponents, respectively.', '1904.03047-1-3-0': 'Universal behaviors have been extensively investigated for DWs driven by magnetic field ([MATH]) in ferromagnetic ultrathin films.', '1904.03047-1-3-1': 'For a large variety of materials, the measured values of the creep ([MATH]) [CITATION] and roughness ([MATH] [CITATION] and [MATH] [CITATION] ) exponents, are compatible with the prediction for the motion of an elastic 1D line in a short-range weak pinning disorder, described by the so-called quenched Edwards Wilkinson (qEW) universality class with [CITATION], and without [CITATION] anharmonic contributions, respectively.', '1904.03047-1-3-2': 'For the depinning transition, the compatibility predictions for the qEW universalty class ([MATH] [CITATION], and [MATH] [CITATION]) was evidenced recently [CITATION].', '1904.03047-1-4-0': 'In contrast, the universal behaviors of DW motion induced by electric current are more contentious.', '1904.03047-1-4-1': 'To the best of our knowledge, the universality of depinning transition have not yet been explored.', '1904.03047-1-4-2': 'For the creep motion, a compatibility with [MATH] is suggested for DW driven by the conventional spin transfer torque (STT) in Pt/Co/Pt nanowires [CITATION] and by spin orbit torque (SOT), in ferrimagnets [CITATION].', '1904.03047-1-4-3': 'However, different values were reported for other materials ([MATH] for (Ga,Mn)As [CITATION] and [MATH] for Ta/CoFeB/MgO [CITATION]), which are difficult to interpret.', '1904.03047-1-4-4': 'Moreover, an important difference between magnetic field and current driven motion is reported in the literature.', '1904.03047-1-4-5': 'The current produces tilting and faceting [CITATION] of DWs.', '1904.03047-1-4-6': 'Those features could suggest a compatibility with the so-called quenched Kardar-Parisi-Zhang (qKPZ) universality class [CITATION].', '1904.03047-1-4-7': 'However, in the direction perpendicular to DW, the roughness is characterized by an exponent ([MATH]) independent of DW tilting angle and compatible the measurement obtained for field driven DW motion ([MATH]), while a different value ([MATH]) is obtained in the direction of current.', '1904.03047-1-4-8': 'Therefore, whether distinct or common universality classes describe the motion of DWs driven by current and magnetic field remains an open question.', '1904.03047-1-5-0': 'In this letter, we report a study of DW tilting and faceting process complemented by extensive investigations on the DW dynamics and self-affinity.', '1904.03047-1-5-1': 'Independent measurements of creep and roughness critical exponents reveal common universal behaviors of current and magnetic field driven DW motion, which are found to extend up to the depinning transition.', '1904.03047-1-6-0': 'Experimental technique.', '1904.03047-1-6-1': 'The experiments were performed with rectangles of a 4 nm thick (Ga,Mn)(As,P)/(Ga,Mn)As bilayer film patterned by lithography.', '1904.03047-1-6-2': 'The film was grown on a (001) GaAs/AlAs buffer [CITATION].', '1904.03047-1-6-3': 'It has an effective perpendicular anisotropy and a Curie temperature ([MATH]) of 65 K.', '1904.03047-1-6-4': 'The sizes of rectangles were 133 [MATH] 210, 228 [MATH] 302, and 323 [MATH]m[MATH] (see supplemental material [CITATION] for the details).', '1904.03047-1-6-5': 'Two 40 [MATH]m wide gold electrodes (separated by 110, 204, and 300 [MATH]m, respectively) were deposited by evaporation parallel to the narrow sides of rectangles [CITATION], and were used to generate an homogeneous current density.', '1904.03047-1-6-6': 'The pulse amplitude varied between 0 and 11 [MATH].', '1904.03047-1-6-7': 'We verified that the Joule effect had a negligible contribution on DW dynamics [CITATION].', '1904.03047-1-6-8': 'Perpendicular magnetic field pulses of adjustable amplitude (0-65 mT) were produced by a 75 turns small coil (diameter [MATH] 1 mm) mounted on the sample.', '1904.03047-1-6-9': 'The set-up was placed in an optical He-flow cryostat allowing to stabilize the temperature between 5.7 K and [MATH].', '1904.03047-1-6-10': 'The DW displacement is observed using a magneto-optical Kerr microscope (resolution [MATH]).', '1904.03047-1-6-11': 'The DW velocity is defined as the ratio between the average displacement [MATH] and the pulse duration [MATH] [CITATION], which varies between 1 [MATH]s and 120 s.', '1904.03047-1-7-0': 'Evolution of DW shape.', '1904.03047-1-7-1': 'The time-evolution of an initially almost flat DW when driven by magnetic field and electrical current is compared in Fig. [REF].', '1904.03047-1-7-2': 'For field driven DW motion, the average successive displacements are relatively similar.', '1904.03047-1-7-3': 'The initial DW shape is almost conserved during the motion.', '1904.03047-1-7-4': 'The DWs become sometimes strongly pinned and curved (see Fig. [REF] (A)) but flatted again when depinned, due to the combined effects of DW elasticity and driving force, which acts as a pressure ([MATH]).', '1904.03047-1-7-5': 'In contrast, the initial DW shape is significantly altered by the current (see Fig. [REF] (B)).', '1904.03047-1-7-6': 'DWs form faceted structures.', '1904.03047-1-7-7': 'There is a clear reduction of DW displacements with increasing tilting angle [MATH] until DWs stop (on the experiment time scale) at a critical angle ([MATH]).', '1904.03047-1-7-8': 'The formation of faceted DWs suggests different universal behaviors for current- and field-driven DW motion [CITATION].', '1904.03047-1-8-0': 'Origin of domain wall faceting.', '1904.03047-1-8-1': 'In order to investigate the origin of DW faceting in details, we first analyze the evolution of an initially almost rectangular domain placed in a uniform current (see Fig. [REF]).', '1904.03047-1-8-2': 'As it can be observed, the side DWs, which are aligned along the current (i.e., [MATH] , where [MATH] is the direction normal to DW) remain almost motionless.', '1904.03047-1-8-3': 'In contrast, the back and front DWs perpendicular to the current (i.e., [MATH]) are significantly displaced.', '1904.03047-1-8-4': 'Surprisingly, the back DW moves faster than the front DW, which causes the collapse of the domain (see Fig. [REF] (E-F)).', '1904.03047-1-8-5': 'Another interesting feature is the increasingly pointed shape of the front DW (not observed for the back DW).', '1904.03047-1-8-6': 'Here, the faceting of the front DW develops without any contribution of "strong" pinning sites observed in Fig. [REF] (B), which suggests that the transverse orientation between DW and current is unstable.', '1904.03047-1-9-0': 'To explore more quantitatively the contribution of different forces on the shape evolution of DWs, we measured systematically DW displacements [MATH] along the direction [MATH] as a function of the angle [MATH] between [MATH] and [MATH], for a fixed magnitude of current density and pulse duration [MATH] (see Fig. [REF] (G and H)).', '1904.03047-1-9-1': 'As shown in Fig. [REF] (I), [MATH] varies linearly with [MATH], and there is a critical angle [MATH] for which [MATH].', '1904.03047-1-9-2': 'This suggests a simple model describing DW displacement as a balance between driving [MATH] and pinning [MATH] forces: [MATH].', '1904.03047-1-9-3': 'Assuming [MATH] and a constant value for [MATH]) leads to the relation [MATH], where the critical angle is defined by [MATH].', '1904.03047-1-9-4': 'The rather good agreement between the model and experimental results shown in Fig. [REF] (I) suggests that the DW faceting originates from the directionality of driving force ([MATH]), which tends to destabilize the transverse orientation of DWs with current and to reduce DW velocity, and a critical tilting angle [MATH] determined by the ratio between magnitude of pinning and drive.', '1904.03047-1-9-5': 'Taking this into account, the different shape evolutions of the front and back DWs observed in Fig. [REF] (A-F) can be interpreted as a result of opposite contributions of DW elasticity.', '1904.03047-1-9-6': 'The two side DWs pull the extremities of back (front) DWs in the [MATH]) direction, which tends to stabilize (destabilize) the transverse DW orientation.', '1904.03047-1-10-0': 'DW dynamics and creep exponent.', '1904.03047-1-10-1': 'Let us now discuss of DW universal behaviors starting with investigations on current driven DW dynamics.', '1904.03047-1-10-2': 'In order to circumvent the variation of driving force with DW tilt, the velocity measurements presented later are all performed from almost flat DWs transverse to current ([MATH]) as the initial states shown in Fig. [REF].', '1904.03047-1-10-3': 'The velocity curves are reported in Fig. [REF] and show similar features to those usually encountered in the literature for magnetic field driven DW dynamics [CITATION].', '1904.03047-1-10-4': 'At low drive ([MATH]), the velocity follows a strong non-linear variation with drive and temperature, which characterizes the thermally activated creep regime (see the inset in Fig. [REF]).', '1904.03047-1-10-5': 'The curves present a change of curvature below and above the depinning threshold ([MATH]).', '1904.03047-1-10-6': 'The linear variation observed well above threshold corresponds to the flow regime.', '1904.03047-1-10-7': 'Here the exploration of DW dynamics was limited to the temperature range [MATH] 49-59 [MATH], due to large thermal fluctuations impeding accurate displacement measurements above 59 [MATH] and to an increase of the depinning threshold beyond experimental access below 49 [MATH].', '1904.03047-1-11-0': 'For more quantitative insights on universal behaviors of DW dynamics, we follow the analysis developed in Ref. [CITATION] for magnetic field driven DW motion.', '1904.03047-1-11-1': 'The data were compared to the creep and depinning laws: [EQUATION] where [MATH] is the velocity at depinning threshold, and [MATH] the velocity that DW would reach at [MATH] without pinning, i.e. within the flow regime.', '1904.03047-1-11-2': 'For the creep regime ([MATH]), the energy barrier height is given by [MATH], where [MATH] is the characteristic height of effective pinning barrier.', '1904.03047-1-11-3': 'For the depinning regime ([MATH]), Eq. [REF] is only valid within the range: [MATH], with [MATH] (see Ref. [CITATION] for details).', '1904.03047-1-11-4': 'The depinning exponents are [MATH], and [MATH], and the critical parameter [MATH] [CITATION].', '1904.03047-1-12-0': 'We performed a global fit of Eqs. [REF] for all the temperatures.', '1904.03047-1-12-1': 'The exponent [MATH] was set as a free shared parameter while the material and temperature dependent pinning parameters ([MATH], [MATH], and [MATH]) were taken as independent.', '1904.03047-1-12-2': 'As it can be observed in Fig. [REF], the fits present a good agreement with the data.', '1904.03047-1-12-3': 'The same analysis was performed for magnetic field driven DW motion [CITATION].', '1904.03047-1-12-4': 'A comparison of the pinning parameters obtained in both cases is shown in Table [REF].', '1904.03047-1-12-5': 'The pretty close heights of effective pinning energy barrier ([MATH]) reported for each temperature indicates that a similar weak pinning disorder control both dynamics.', '1904.03047-1-12-6': 'Moreover, the values of creep exponent obtained with field ([MATH]) [CITATION] and current ([MATH]) match within experimental error bars [CITATION].', '1904.03047-1-12-7': 'A good agreement is also found with the self-description of the creep and depinning regimes, already proposed for magnetic field driven DW dynamic [CITATION].', '1904.03047-1-12-8': 'Therefore, DW motion induced by transverse current and magnetic field presents common universal dynamical behaviors.', '1904.03047-1-13-0': 'Roughness exponent.', '1904.03047-1-13-1': 'In order to discuss this issue from an independent measurement, we have investigated the DW roughness in the creep regime.', '1904.03047-1-13-2': 'The DW self affinity was tested using the displacement-displacement correlation function [CITATION]: [EQUATION] where [MATH] is the DW displacement measured parallel to current and [MATH] the length of DW segment along the axis [MATH] transverse to current (see the inset of Fig. [REF] (A)).', '1904.03047-1-13-3': 'For a self-affine interface ([MATH]), the function [MATH] is expected to follow a power law variation [MATH], where [MATH] is the roughness exponent.', '1904.03047-1-13-4': 'Typical variations of [MATH] versus [MATH] obtained for field and current induced motion are compared in Fig. [REF] (B) in log-log scale.', '1904.03047-1-13-5': 'As it can be observed, above the microscope resolution ([MATH]), both curves present a linear variation over the range [MATH] 1-10 [MATH] with similar slopes ([MATH]).', '1904.03047-1-13-6': 'In order to get a more quantitative comparison, the slope of linear fits were systematically measured for successions of DW positions (as shown in Fig. [REF]) and a temperature varying over one decade ([MATH]4.5-59 [MATH]).', '1904.03047-1-13-7': 'No significant variation of [MATH] was observed with DW tilting (see Ref. [CITATION] for details) and [MATH] remained always smaller than the value ([MATH]) reported in Ref. [CITATION].', '1904.03047-1-13-8': 'This is most probably associated to a much lower DW tilting observed for (Ga,Mn)(As,P) than for Pt/Co/Pt.', '1904.03047-1-13-9': 'The mean and standard deviation of the roughness exponent for current ([MATH]) and field ([MATH]) driven DW motion is reported in Fig. [REF] as a function of temperature.', '1904.03047-1-13-10': 'As expected for universal critical exponents, the values of [MATH] and [MATH] do not vary significantly.', '1904.03047-1-13-11': 'Their mean values ([MATH] and [MATH]), calculated from all measurements, agree well within experimental error.', '1904.03047-1-13-12': 'Both are compatible with the value ([MATH] [CITATION]) obtained by Moon et al. [CITATION] for a DW roughness analyzed in the direction ([MATH]) normal to DW and with usual experimental results reported in the literature for field driven DW motion [CITATION].', '1904.03047-1-13-13': 'A good agreement is also obtained with theoretical predictions ([MATH]) [CITATION] for the universality class of the qEW model with short range disorder and elasticity including an-harmonic correction.', '1904.03047-1-14-0': 'The common universal behaviors observed both for the roughness and dynamics of field and current driven DW motion seems a priori surprising since no faceting of DW is expected for the qEW universality class.', '1904.03047-1-14-1': 'This apparent paradox can be explained from the variation of driving force with DW tilting discussed at the beginning of the letter.', '1904.03047-1-14-2': 'Along [MATH], the magnitude of driving force is proportional to the product [MATH].', '1904.03047-1-14-3': 'The second order expansion of [MATH]), introduces a term in the equation of motion similar to the so-called KPZ term producing interface tilting: [MATH], with [MATH] [CITATION].', '1904.03047-1-14-4': 'However, in the direction [MATH] perpendicular to DW, the driving force is simply proportional to the current density ([MATH]) and acts as a magnetic field ([MATH]), which explains a compatibility with the qEW universality class.', '1904.03047-1-15-0': 'In conclusion, the common universal behaviors with field driven DW motion and in particular the compatibility with the self-consistent description of creep and depinning regimes could help to better understand current induced DW motion experiments [CITATION] since it allows a clear identification of dynamical regimes and to discriminate universal from material dependent behaviors [CITATION].', '1904.03047-1-15-1': 'It would be also very interesting to study the universal behaviors of DWs driven by spin-orbit-torque [CITATION] and of DWs in antiferromagnets [CITATION].', '1904.03047-1-15-2': 'Moreover, the instability of transverse alignment between DW and current should have direct implications for potential applications based on the controlled motion of DWs [CITATION] in nanowires.', '1904.03047-1-16-0': 'We wish to thank S. Bustingorry, A. Kolton, and K. Wiese for fruitful discussions.', '1904.03047-1-16-1': 'This work was partly supported by the projects DIM CNano IdF (Region Ile-de-France) and the Labex NanoSaclay (ANR-10-LABX-0035).', '1904.03047-1-16-2': 'R.D.P. thanks the Mexican council CONACyT for the PhD fellowship n0: 449563.'}

{'1904.03047-2-0-0': 'We explore universal behaviors of magnetic domain wall driven by the spin-transfer of an electrical current, in a ferromagnetic (Ga,Mn)(As,P) thin film with perpendicular magnetic anisotropy.', '1904.03047-2-0-1': 'For a current direction transverse to domain wall, the dynamics of the thermally activated creep regime and the depinning transition are found to be compatible with a self-consistent universal description of magnetic field induced domain wall dynamics.', '1904.03047-2-0-2': 'This common universal behavior, characteristic of the so-called quenched Edwards-Wilkinson universality class, is confirmed by a complementary and independent analysis of domain wall roughness.', '1904.03047-2-0-3': 'However, the tilting of domain walls and the formation of facets is produced by the directionality of interaction with the current, which acts as a magnetic field only in the direction transverse to domain wall.', '1904.03047-2-1-0': 'The displacement of small spin texture as magnetic domain walls (DWs) thanks to spin torque effects is at the basis of potential applications to magnetic memory storage [CITATION].', '1904.03047-2-1-1': 'An important effort is dedicated to search for magnetic materials [CITATION] with large and well controlled DW velocities.', '1904.03047-2-1-2': 'However, DWs are very sensitive to weak pinning defects [CITATION], which strongly reduce their mobility and produce roughening and stochastic avalanche-like motion [CITATION].', '1904.03047-2-1-3': 'Therefore, it is particularly interesting to better understand the contribution of pinning to current induced DW dynamics.', '1904.03047-2-2-0': 'Magnetic domain walls [CITATION] present surprising universal critical behaviors, encountered in a wide variety of moving interfaces such as the reaction front propagation in disordered flows [CITATION], growing bacterial colonies [CITATION], wetting [CITATION], motion of ferroelectric domain walls [CITATION], to name a few.', '1904.03047-2-2-1': 'The interfaces are rough with self-affine width growing as [MATH], where [MATH] the distance between two points of the interface and [MATH] the roughness exponent.', '1904.03047-2-2-2': 'Moreover, a depinning driving force [MATH] separates the so-called creep ([MATH]) and depinning ([MATH]) regimes.', '1904.03047-2-2-3': 'In the creep regime, the velocity varies as an Arrhenius law [MATH] [CITATION], where [MATH] is the thermal fluctuation energy.', '1904.03047-2-2-4': '[MATH] is the effective pinning energy barrier height, which follows a universal power law variation with the driving force [MATH], where [MATH] is the creep exponent.', '1904.03047-2-2-5': 'In the depinning regime [CITATION], the effective pinning barriers are collapsed.', '1904.03047-2-2-6': 'The velocity presents power law variations with drive [MATH] and temperature [MATH]: [MATH] and [MATH], where [MATH] and [MATH] are the depinning and thermal rounding exponents, respectively.', '1904.03047-2-3-0': 'Universal behaviors have been extensively investigated for DWs driven by magnetic field ([MATH]) in ferromagnetic ultrathin films.', '1904.03047-2-3-1': 'For a large variety of materials, the measured values of the creep ([MATH]) [CITATION] and roughness ([MATH] [CITATION] and [MATH] [CITATION] ) exponents, are compatible with the prediction for the motion of an elastic 1D line in a short-range weak pinning disorder, described by the so-called quenched Edwards Wilkinson (qEW) universality class with [CITATION], and without [CITATION] anharmonic contributions, respectively.', '1904.03047-2-3-2': 'Moreover, it was recently shown [CITATION] that the depinning transition is compatible with the predictions for the qEW universality class ([MATH] [CITATION], and [MATH] [CITATION]).', '1904.03047-2-4-0': 'In contrast, the universal behaviors of DW motion induced by spin-polarized electric current are more contentious.', '1904.03047-2-4-1': 'To the best of our knowledge, the universality of the depinning transition has not yet been explored.', '1904.03047-2-4-2': 'For the creep motion, a compatibility with [MATH] is suggested for DW driven by the conventional spin transfer torque (STT) in Pt/Co/Pt nanowires [CITATION] and by spin orbit torque (SOT), in ferrimagnets [CITATION].', '1904.03047-2-4-3': 'However, rather intriguing differences between current and magnetic field driven motion are also reported in the literature.', '1904.03047-2-4-4': 'Different values of the creep exponent were reported for other materials ([MATH] for (Ga,Mn)As [CITATION] and [MATH] for Ta/CoFeB/MgO [CITATION]), which are difficult to interpret.', '1904.03047-2-4-5': 'The tilting and faceting [CITATION] of DWs, produced by the current could suggest a compatibility with the so-called quenched Kardar-Parisi-Zhang (qKPZ) universality class [CITATION].', '1904.03047-2-4-6': 'However, in the direction perpendicular to DW, the roughness is characterized by an exponent ([MATH]) independent of DW tilting angle and compatible with the measurement obtained for field driven DW motion ([MATH]), while a different value ([MATH]) is obtained in the direction of current.', '1904.03047-2-4-7': 'Therefore, whether distinct or common universality classes describe the motion of DWs produced by current and magnetic field remains an open question.', '1904.03047-2-4-8': 'In this letter, we show that the roughness and universal dynamics of spin-transfer-torque (STT) driven domain wall are particularly well described within the qEW universality class.', '1904.03047-2-5-0': 'Experimental techniques.', '1904.03047-2-5-1': 'The experiments were performed with rectangles of a 4 nm thick (Ga,Mn)(As,P)/(Ga,Mn)As bilayer film patterned by lithography.', '1904.03047-2-5-2': 'The film was grown on a (001) GaAs/AlAs buffer [CITATION].', '1904.03047-2-5-3': 'It has an effective perpendicular anisotropy and a Curie temperature ([MATH]) of 65 K.', '1904.03047-2-5-4': 'The sizes of rectangles were 133 [MATH] 210, 228 [MATH] 302, and 323 [MATH]m[MATH] (see supplemental material [CITATION] for the details).', '1904.03047-2-5-5': 'Two 40 [MATH]m wide gold electrodes (separated by 110, 204, and 300 [MATH]m, respectively) were deposited by evaporation parallel to the narrow sides of rectangles [CITATION].', '1904.03047-2-5-6': 'They were used to generate an homogeneous current density producing DW motion by STT.', '1904.03047-2-5-7': 'The pulse amplitude varied between 0 and 11 [MATH].', '1904.03047-2-5-8': 'We verified that the Joule effect had a negligible contribution on DW dynamics [CITATION].', '1904.03047-2-5-9': 'Perpendicular magnetic field pulses of adjustable amplitude (0-65 mT) were produced by a 75 turns small coil (diameter [MATH] 1 mm) mounted on the sample.', '1904.03047-2-5-10': 'The set-up was placed in an optical He-flow cryostat allowing to stabilize the temperature between 5.7 K and [MATH].', '1904.03047-2-5-11': 'The DW displacement is observed using a magneto-optical Kerr microscope (resolution [MATH]).', '1904.03047-2-5-12': 'The DW velocity is defined as the ratio between the average displacement [MATH] and the pulse duration [MATH] [CITATION], which varies between 1 [MATH]s and 120 s.', '1904.03047-2-6-0': 'DW dynamics and creep exponent.', '1904.03047-2-6-1': 'Let us start with investigations of current driven DW dynamics.', '1904.03047-2-6-2': 'In order to circumvent the variation of driving force with DW tilt, the velocity measurements were all performed from almost flat DWs transverse to current ([MATH]) (see [CITATION]).', '1904.03047-2-6-3': 'The velocity curves are reported in Fig. [REF] and show similar features to those usually encountered in the literature for magnetic field driven DW dynamics [CITATION].', '1904.03047-2-6-4': 'At low drive ([MATH]), the velocity follows a strong non-linear variation with drive and temperature, which characterizes the thermally activated creep regime (see the inset in Fig. [REF]).', '1904.03047-2-6-5': 'The curves present a change of curvature below and above the depinning threshold ([MATH]).', '1904.03047-2-6-6': 'The linear variation observed well above threshold corresponds to the flow regime.', '1904.03047-2-6-7': 'Here the exploration of DW dynamics was limited to the temperature range [MATH] 49-59 [MATH], due to large thermal fluctuations impeding accurate displacement measurements above 59 [MATH] and to an increase of the depinning threshold beyond experimental access below 49 [MATH].', '1904.03047-2-7-0': 'For more quantitative insights on dynamical universal behaviors, the velocity curves were compared to the self-consistent description of creep, depinning and flow regimes developed for magnetic field driven DW motion [CITATION]: [EQUATION] where [MATH] is the velocity at depinning threshold, and [MATH] the velocity that DW would reach at [MATH] without pinning, i.e. within the flow regime.', '1904.03047-2-7-1': 'For the creep regime, the energy barrier height is given by [MATH], where [MATH] is the characteristic height of effective pinning barrier.', '1904.03047-2-7-2': 'For the depinning regime, Eq. [REF] is only valid over a limited range: it does not account for the effect of thermal fluctuations occurring just above the depinning threshold nor for the crossover to linear flow regime [CITATION].', '1904.03047-2-7-3': 'The depinning exponents are [MATH], and [MATH], and the critical parameter [MATH] [CITATION].', '1904.03047-2-7-4': 'Finally, the linear flow regime in Eq. [REF] is only reached at the end of crossover.', '1904.03047-2-8-0': 'For each temperature, the fit of three regimes described by Eqs. [REF] (see Fig. [REF]) relies only on three independent adjustable parameters ([MATH], [MATH], and [MATH]) and was performed simultaneously.', '1904.03047-2-8-1': 'As the depinning threshold [MATH] is not known a priori, the following procedure was used.', '1904.03047-2-8-2': 'The fit of creep regime was performed between the lowest measured velocity and increasing values of [MATH] on the velocity curve.', '1904.03047-2-8-3': '[MATH] was taken as a free parameter, and [MATH].', '1904.03047-2-8-4': 'The threshold velocity [MATH] was assumed to correspond to the best simultaneous fit of Eqs. [REF].', '1904.03047-2-8-5': 'As it can be observed in Fig. [REF], a good agreement is obtained with the data.', '1904.03047-2-8-6': 'The same analysis was performed for magnetic field driven DW motion [CITATION].', '1904.03047-2-8-7': 'A comparison of the pinning parameters obtained in both cases is shown in Table [REF].', '1904.03047-2-8-8': 'The pretty close heights of effective pinning energy barrier ([MATH]) reported for each temperature indicates that a similar weak pinning disorder controls both dynamics.', '1904.03047-2-8-9': 'In order to estimate precisely the value of creep exponent [MATH], it was set as a free shared parameter and we perform a global fit of the creep power law [MATH], for [MATH], and all the temperatures.', '1904.03047-2-8-10': 'The obtained values for field ([MATH]) [CITATION] and current ([MATH]) driven DW motion match within experimental error bars [CITATION].', '1904.03047-2-8-11': 'Therefore, a good agreement is obtained both for the creep exponent and self-consistent description of the creep and depinning universal dynamics.', '1904.03047-2-8-12': 'This is a strong evidence that DW motion induced by transverse current through the STT mechanism shares common universal creep and depinning behaviors of DW driven by magnetic field observed in thin magnetic films made of different materials [CITATION].', '1904.03047-2-9-0': 'Roughness exponent.', '1904.03047-2-10-0': 'As this finding do not confirm the analysis reported in Refs. [CITATION], we have investigated the DW roughness in the creep regime, in order to obtain a complementary and independent test of universal behavior.', '1904.03047-2-10-1': 'The DW self affinity was studied using the displacement-displacement correlation function [CITATION]: [EQUATION] where [MATH] is the DW displacement measured parallel to current and [MATH] the length of DW segment along the axis [MATH] transverse to current (see the inset of Fig. [REF] (A)).', '1904.03047-2-10-2': 'For a self-affine interface, the function [MATH] is expected to follow a power law variation [MATH], where [MATH] is the roughness exponent.', '1904.03047-2-10-3': 'Typical variations of [MATH] versus [MATH] obtained for field and current induced motion are compared in Fig. [REF] (B) in log-log scale.', '1904.03047-2-10-4': 'As it can be observed, both curves present a linear variation with similar slopes ([MATH]), between the microscope resolution ([MATH]) and [MATH].', '1904.03047-2-10-5': 'In order to get more statistics, the slopes were systematically determined for successions of DW positions (see Ref. [CITATION]) and a temperature varying over one decade ([MATH]4.5-59 [MATH]).', '1904.03047-2-10-6': 'The mean and standard deviation of the roughness exponent for current ([MATH]) and field ([MATH]) driven DW motion is reported in Fig. [REF] as a function of temperature.', '1904.03047-2-10-7': 'As expected for universal critical exponents, the values of [MATH] and [MATH] do not vary significantly.', '1904.03047-2-10-8': 'Their mean values ([MATH] and [MATH]), calculated from all measurements, agree well within experimental error.', '1904.03047-2-10-9': 'Here, it is important to notice some differences and similarities with the results reported for Pt/Co/Pt in Ref. [CITATION].', '1904.03047-2-10-10': 'For Ga/Mn/As, [MATH] presents no significant variation with DW tilting (see [CITATION] for details) and remains always significantly smaller than the value ([MATH]) reported for DW displacements measured in the direction of current [CITATION].', '1904.03047-2-10-11': 'In contrast, the value [MATH] is compatible with the results ([MATH]) of Ref. [CITATION] for a DW roughness analyzed in the direction ([MATH]) normal to DW.', '1904.03047-2-10-12': 'The good agreement is also obtained with usual experimental results reported in the literature for field driven DW motion [CITATION], and with theoretical predictions ([MATH]) [CITATION] for the universality class of the qEW model with short range disorder and elasticity including an-harmonic correction.', '1904.03047-2-10-13': 'Therefore, both the analysis of DW dynamics and roughness lead to the same conclusion of common universal behaviors for current and field driven DW motion.', '1904.03047-2-11-0': 'Origin of Domain wall faceting.', '1904.03047-2-11-1': 'This common behavior, which is a priori difficult to reconcile with the faceting of DW produced by the current calls for investigation on the origin of DW faceting.', '1904.03047-2-11-2': 'First, we analyze the evolution of an initially almost rectangular domain subjected to a large current density ([MATH]).', '1904.03047-2-11-3': 'As it can be observed in Fig. [REF], the edges of the domain aligned along the current (i.e., [MATH] , where [MATH] is the direction normal to DW) remain almost motionless.', '1904.03047-2-11-4': 'In contrast, the back and front DWs perpendicular to the current (i.e., [MATH]) are significantly displaced.', '1904.03047-2-11-5': 'Surprisingly, the back DW moves faster than the front DW, which causes the collapse of the domain (see Fig. [REF] (E-F)).', '1904.03047-2-11-6': 'Another interesting feature is the increasingly pointed shape of the front DW (not observed for the back DW).', '1904.03047-2-11-7': 'Here, the faceting of the front DW develops without any contribution of "strong" pinning sites (see [CITATION]), which suggests that the transverse orientation between DW and current is unstable.', '1904.03047-2-11-8': 'Consequently, the different shape evolutions of the front and back DWs observed in Fig. [REF] (A-F) can be interpreted as a result of opposite contributions of DW elasticity.', '1904.03047-2-11-9': 'The two side DWs pull the extremities of back (front) DWs in the [MATH]) direction, which tends to stabilize (destabilize) the transverse DW orientation.', '1904.03047-2-12-0': 'To explore more quantitatively the directionality of interaction between DW and current, we have measured the DW displacements [MATH] along the direction [MATH] as a function of the angle [MATH] between [MATH] and [MATH], for a fixed magnitude of current density and pulse duration [MATH] (see Fig. [REF] (G and H)).', '1904.03047-2-12-1': 'We have then deduced the variation of velocity [MATH] as a function of [MATH], which are reported Fig. [REF] (I).', '1904.03047-2-12-2': 'As expected the DW velocity decreases as the tilting angle increases (i.e.,[MATH] decreases).', '1904.03047-2-12-3': 'For [MATH], the displacement [MATH] becomes lower than the spatial resolution so that the estimation for [MATH] is zero.', '1904.03047-2-12-4': 'The dispersion of data most probably results from the contributions of DW elasticity, which tends to reduce DW velocity.', '1904.03047-2-12-5': 'Therefore, the velocity DW without contribution of elasticity should correspond to the upper measured values.', '1904.03047-2-12-6': 'Interestingly, those values present a rather good overlap with the velocity curves of Fig. [REF], which were obtained as a function of the current density and for [MATH].', '1904.03047-2-12-7': 'This indicates that DW tilting reduces the driving force in the direction normal [MATH] to the DW and suggests that the DW faceting originates from the directionality of driving force [MATH].', '1904.03047-2-13-0': 'Let us now discuss the implication of the variation of the driving force with DW tilting ([MATH]) on universal behaviors.', '1904.03047-2-13-1': 'The second order expansion of [MATH]) introduces a so-called KPZ term ([MATH] with [MATH]) in the equation of motion [CITATION], which is usually used to describe interface tilting.', '1904.03047-2-13-2': 'However, at large tilting angle, the relation [MATH] should not be compatible with the KPZ minimal model.', '1904.03047-2-13-3': 'In contrast, in the direction [MATH] perpendicular to DW, for a fixed DW tilting, the driving force due to spin-transfer-torque is simply proportional to [MATH] and acts as a magnetic field ([MATH]).', '1904.03047-2-13-4': 'This could explain the compatibility of current induced DW motion with the qEW universality class, which is observed experimentally.', '1904.03047-2-14-0': 'In conclusion, the common universal behaviors with field driven DW motion and in particular the agreement with the self-consistent description of creep and depinning regimes could help to better understand spin-transfer-torque induced DW motion experiments [CITATION] since it allows a clear identification of dynamical regimes and to discriminate universal from material dependent behaviors [CITATION].', '1904.03047-2-14-1': 'It would be also very interesting to study the universal behaviors of DWs driven by spin-orbit-torque [CITATION] and of DWs in antiferromagnets [CITATION].', '1904.03047-2-14-2': 'Moreover, the instability of transverse alignment between DW and current should have direct implications for potential applications based on the controlled motion of DWs [CITATION] in nanowires.', '1904.03047-2-15-0': 'We wish to thank S. Bustingorry, A. Kolton, and K. Wiese for fruitful discussions.', '1904.03047-2-15-1': 'This work was partly supported by the projects DIM CNano IdF (Region Ile-de-France) and the Labex NanoSaclay (ANR-10-LABX-0035).', '1904.03047-2-15-2': 'R.D.P. thanks the Mexican council CONACyT for the PhD fellowship n0: 449563.'}

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[]

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{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1904.03047

0709.4377

{'0709.4377-1-0-0': 'The transport coefficients of sulfonated polystyrene in the oligomer range are determined by coarse-grained molecular dynamics simulation and compared to data obtained by capillary electrophoresis and electrophoresis NMR.', '0709.4377-1-0-1': 'We find excellent agreement of experiments and simulations when hydrodynamic interactions are properly included in the simulations.', '0709.4377-1-0-2': 'We show that the electrophoretic mobility exhibits a maximum in the oligomer range and demonstrate that this maximum is due to the hydrodynamical interactions between the chain monomers and the solvent.', '0709.4377-1-0-3': 'Our findings demonstrate convincingly that it is possible to model transport properties of short polyelectrolytes correctly using a coarse grained model for the polyelectrolyte and the solvent, if hydrodynamic interactions are included.', '0709.4377-1-1-0': '# Introduction', '0709.4377-1-2-0': 'Electrophoresis methods are widely used to separate (macro)biomolecules [CITATION] such as peptides, proteins, and DNA, as well as synthetic polymers [CITATION].', '0709.4377-1-2-1': 'Short polyelectrolytes can conveniently be separated in free solution without the aid of a gel by capillary electrophoresis (CE).', '0709.4377-1-2-2': 'Additionally, CE is employed to characterize the hydrodynamic properties of charged biomolecules, in particular the hydrodynamic radius, the diffusion coefficient, and the electrophoretic mobility [CITATION].', '0709.4377-1-3-0': 'Alternatively, these properties can be determined by pulsed field gradient (PFG) NMR [CITATION].', '0709.4377-1-3-1': 'With a combination of diffusion NMR and electrophoresis NMR the charge of macromolecules [CITATION], the influence of ionic strength [CITATION] and the dielectric constant [CITATION] of the solution have been investigated.', '0709.4377-1-4-0': 'Several studies of polyelectrolytes of well defined length in the short chain regime have shown that the free-solution electrophoretic mobility [MATH] exhibits a characteristic behaviour [CITATION]: from the monomer mobility [MATH] onwards, the mobility increases towards a maximum [MATH] that occurs for chains of a specific degree of polymerization [MATH].', '0709.4377-1-4-1': 'After this maximum, the mobility decreases slightly to reach a constant value [MATH], the so-called free draining mobility.', '0709.4377-1-5-0': 'Whether or not a mobility maximum appears and the precise value of [MATH] depend on the flexibility of the chain.', '0709.4377-1-5-1': 'While a maximum has been observed for flexible polyelectrolytes such as single stranded DNA or sulfonated polystyrene (PSS) at values around 10 repeat units for [MATH][CITATION], semi-flexible double-stranded DNA does not exhibit it.', '0709.4377-1-5-2': 'The small number for [MATH] and the strong influence of the persistence length shows that a short chain behavior must be responsible for this.', '0709.4377-1-5-3': 'Furthermore, it has been shown that the [MATH] is diminished under elevated salt conditions.', '0709.4377-1-6-0': 'Existing theories describing the electrophoretic properties of polyelectrolytes [CITATION] have been successful in describing the qualitative behaviour of an initially rising mobility as well as the constant long-chain limit, but they have not been able to reproduce this maximum or explain its origin.', '0709.4377-1-7-0': 'The focus of this article is to shed light on the origin of the maximum in the case of PSS.', '0709.4377-1-7-1': 'We do this by coarse-grained Molecular Dynamics (MD) simulations that take all charged particles in the system (i.e. polyelectrolyte repeat units, counter ions, and additional salt) explicitly and hydrodynamics implicitly into account.', '0709.4377-1-7-2': 'The simulation results are compared to data sets obtained by two different experimental techniques.', '0709.4377-1-7-3': 'The combination of individual findings is then used to describe the size-dependence of the electrophoretic mobility for short polyelectrolyte chains.', '0709.4377-1-8-0': '# Simulations', '0709.4377-1-9-0': 'Model.', '0709.4377-1-9-1': 'We simulate a flexible polyelectrolyte using a bead-spring model employing the ESPResSo package ([CITATION]).', '0709.4377-1-9-2': 'All parameter values are given in reduced units with [MATH] and [MATH]A being the energy and the relevant length scale.', '0709.4377-1-9-3': 'The beads (chain monomers) are connected by FENE bonds with stiffness [MATH], and maximum extension [MATH].', '0709.4377-1-9-4': 'Additionally a truncated Lennard-Jones or WCA potential with depth [MATH] and width [MATH], is used for excluded volume interactions.', '0709.4377-1-9-5': 'Each monomer has a charge of [MATH] in units of [MATH].', '0709.4377-1-9-6': 'Monovalent counter ions, [MATH], and monovalent positive and negative salt ions are subjected to the same LJ potential giving all particles the same size.', '0709.4377-1-9-7': 'The simulations are carried out under periodic boundary conditions in a rectangular simulation box of size [MATH] (for [MATH]) to [MATH] for ([MATH]) resulting in constant monomer concentration.', '0709.4377-1-9-8': 'All electrostatic interactions are calculated with the P3M algorithm.', '0709.4377-1-9-9': 'The Bjerrrum length [MATH] in simulation units corresponds to 7.1 A (the Bjerrum length for water at room temperature).', '0709.4377-1-9-10': "Together with the model's average bond length of [MATH], we compute a Manning factor of [MATH].", '0709.4377-1-9-11': 'The inclusion of hydrodynamic interactions is done via frictional coupling of the beads to a Lattice Boltzmann (LB) fluid as detailed in [CITATION].', '0709.4377-1-9-12': 'The modeled fluid has a kinematic viscosity [MATH], a fluid density [MATH], and is discretized by a grid with spacing [MATH].', '0709.4377-1-9-13': 'The coupling parameter is [MATH].', '0709.4377-1-9-14': 'The simulation time step is [MATH].', '0709.4377-1-10-0': 'To determine the impact of hydrodynamic interactions, we compare the results to simulations with a simple Langevin thermostat that does not recover long-range hydrodynamic interactions between the monomers, but only offers local interaction with the solvent.', '0709.4377-1-11-0': 'Transport coefficients.', '0709.4377-1-11-1': 'We determine two different transport coefficients for the model polyelectrolyte that are likewise determined in the associated experiments.', '0709.4377-1-11-2': 'The single chain diffusion coefficient is obtained from the center of mass velocity auto correlation function: [MATH].', '0709.4377-1-12-0': 'In CE experiments, the electrophoretic mobility [MATH] of the solute is determined by [MATH], where [MATH] is the velocity, [MATH] is the electric field, [MATH] is the applied voltage, [MATH] is the total length of the capillary, [MATH] is the migration (or effective) length up to the detector and [MATH] is the detection time of the solute.', '0709.4377-1-13-0': 'In the simulations, we use a Green-Kubo relation to obtain the electrophoretic mobility at zero electric field.', '0709.4377-1-13-1': 'This approach has been successfully applied in simulations to determine the electrophoretic mobility of charged colloids ([CITATION]).', '0709.4377-1-13-2': 'The chain mobility is calculated from the correlation function between the center of mass velocity of the polyelectrolyte chain and the velocities of all charged particles in the system: [MATH].', '0709.4377-1-13-3': 'This method guarantees that no conformational changes of the chain structure or the ion distribution are induced by an artificial high external field, which is sometimes used in other simulations to separate the directed electrophoretic motion from Brownian motion within reasonable computing time.', '0709.4377-1-13-4': 'Another beneficial side effect of this method is that both transport properties can be obtained from the same simulation trajectories without additional computational effort.', '0709.4377-1-14-0': '# Experiments', '0709.4377-1-15-0': 'Capillary electrophoresis (CE) is an analytical separation technique based on the differential migration of ionic species under electric field [CITATION].', '0709.4377-1-16-0': 'The CE experiments were performed using an Agilent technologies capillary electrophoresis system (Agilent, Waldbronn, Germany).', '0709.4377-1-16-1': 'Capillaries of 33.5 cm (25 cm to the detector) length, and 50 [MATH]m diameter were prepared from bare silica tubing purchased from Supelco (Bellefonte, PA, USA).', '0709.4377-1-16-2': 'New capillaries were conditioned by performing the following flushes: 1M NaOH for 30 min, 0.1 M NaOH for 10 min, and water for 5 min.', '0709.4377-1-16-3': 'Samples were introduced hydrodynamically ([MATH] nL) at 0.5 g/L concentration ([MATH]mMol/l monomer concentration).', '0709.4377-1-16-4': 'The electrolyte was pure water.', '0709.4377-1-16-5': 'Solutes were detected at 225 nm.', '0709.4377-1-16-6': 'The electric field was kept constant at 224 V/cm (V=+7.5 kV).', '0709.4377-1-16-7': 'The polarity of the applied voltage on the inlet side of the capillaries was positive.', '0709.4377-1-16-8': 'All the experiments were performed at 27 [MATH]C. Electro osmotic mobilities were determined from the migration time of a neutral marker (mesityl oxide, [MATH]% (v/v) in the electrolyte).', '0709.4377-1-16-9': 'The sodium polystyrene sulfonate standards ([MATH]; [MATH] ca 1.1) were purchased from American Polymer Standards Corp. (Mentor, OH, USA).', '0709.4377-1-16-10': 'Borax (disodium tetraborate decahydrate) was from Prolabo (Paris, France).', '0709.4377-1-16-11': 'Mesityl oxide used as neutral marker was obtained from Aldrich.', '0709.4377-1-16-12': 'Purified water delivered by an Alpha-Q system (Millipore, Molsheim, France) was used to prepare all electrolytes and sample solutions.', '0709.4377-1-17-0': 'In electrophoresis NMR, diffusion and electrophoretic motion are separated by the design of the experiment [CITATION].', '0709.4377-1-17-1': 'No gel has been used, so that self diffusion and free electrophoresis are measured.', '0709.4377-1-17-2': 'The PSS samples have been obtained from Fluka.', '0709.4377-1-17-3': 'To minimize the effects of variations of the ionic strength [CITATION], samples have been dialyzed against water (cut off volume 0.5kDA), and subsequently dried under vacuum.', '0709.4377-1-17-4': 'For all experiments a concentration of 5mMol/l (monomer) in deuterated water has been used.', '0709.4377-1-17-5': 'The diffusion experiments have been performed on a Bruker Avance 500 NMR spectrometer operating at a Larmor frequency of 500MHz for protons equipped with a DIFF30 probe head generating a maximum pulsed field gradient strength of 12T/m.', '0709.4377-1-17-6': 'The gradient pulse duration [MATH] and diffusion times [MATH] have been adjusted between 0.8 to 2ms for [MATH] and 8 to 20ms for [MATH] respectively for optimal resolution for each molecular weight resulting in different diffusion coefficients.', '0709.4377-1-17-7': 'Because of their narrow molecular weight distribution, diffusion coefficients have been determined by a linear fit to the Stejskal-Tanner equation [CITATION].', '0709.4377-1-17-8': 'Electrophoresis NMR experiments have been performed on a Bruker Avance 300 NMR spectrometer operating at a Larmor frequency of 300MHz for protons with an in-house-built electrophoresis probe head utilizing a Bruker micro2.5 imaging gradient system generating magnetic field gradient strength of up to 1T/m.', '0709.4377-1-17-9': 'The flow times [MATH] varied between 15 and 50 ms and the gradient pulse duration [MATH] between 3 and 6 ms respectively with gradient amplitudes between 0.3 and 0.6 T/m.', '0709.4377-1-17-10': 'The electric field has been linearly incremented between -140 V/cm and +140 V/cm.', '0709.4377-1-17-11': 'The electrophoretic mobility is a model-free read out from the two-dimensional electrophoresis NMR spectrum correlating the chemical shift, that identifies the moving species, with electrophoretic mobility [CITATION].', '0709.4377-1-18-0': '# Discussion', '0709.4377-1-19-0': 'Diffusion.', '0709.4377-1-19-1': 'Recently, Stellwagen et.', '0709.4377-1-19-2': 'al. [CITATION] observed that the diffusion coefficient [MATH] of PSS can be approximated by a power law scaling [MATH], where [MATH] is the monomer diffusion coefficient, and [MATH] is the scaling exponent.', '0709.4377-1-19-3': 'In Figure [REF], we compare the diffusion coefficient obtained from simulations to the results from the NMR study.', '0709.4377-1-19-4': 'The simulated data is normalized by [MATH] as obtained from the power law fit, and the experimental data by the monomer diffusion coefficient of [MATH] m[MATH]/s.', '0709.4377-1-20-0': 'The simulations with hydrodynamic interactions result in a scaling exponent of [MATH], which is in good agreement with value obtained from experiments, [MATH] [CITATION], and a previously reported result, [MATH], by [CITATION].', '0709.4377-1-20-1': 'Only for the very short chains ([MATH]), a deviation from the prediction is observable and a higher diffusion coefficient is found in the simulations.', '0709.4377-1-20-2': 'For intermediate chain length, the coarse-grained simulation model with hydrodynamic interactions, reproduces the experimentally observed behaviour.', '0709.4377-1-21-0': 'Without hydrodynamic interactions, the chains show the expected Rouse diffusion with an exponent of [MATH].', '0709.4377-1-21-1': 'This simple model is clearly not applicable to mimic the experimental behaviour of short polyelectrolyte chains.', '0709.4377-1-22-0': 'Electrophoretic mobility ([MATH]).', '0709.4377-1-22-1': 'The results for measurements of [MATH] in pure water without additional salt are shown in Fig. [REF].', '0709.4377-1-22-2': 'To account for the different viscosity of the solvents, we rescale the mobilities by the free-draining mobility [MATH] as obtained for long chains.', '0709.4377-1-23-0': 'The experimental data sets agree within the accuracies of the individual methods and show the characteristic behaviour of the mobility in dependence to chain length.', '0709.4377-1-23-1': 'A mobility maximum for [MATH] is observed with capillary electrophoresis.', '0709.4377-1-23-2': 'This maximum for intermediate chains as well as the long chain behaviour is successfully reproduced in simulations with hydrodynamic interactions.', '0709.4377-1-23-3': 'For the first few oligomers, we observe a small difference which is in line with the deviation for the diffusion.', '0709.4377-1-24-0': 'On the other hand, as illustrated in the inset of Fig. [REF], the simulation without hydrodynamic interactions fails completely to describe the short chain behaviour and can only be mapped to the experimental data in the long chain limit.', '0709.4377-1-24-1': 'Therefore, we infer that the mobility maximum can only be explained when taking into account hydrodynamic interactions between the polyelectrolyte and the surrounding solvent .', '0709.4377-1-25-0': 'To further illustrate this, we estimate the effective hydrodynamic friction [MATH] and the effective charge [MATH] in dependence of its length.', '0709.4377-1-25-1': 'During electrophoresis, the electrical driving force [MATH] is balanced by the frictional force with the solvent [MATH], where [MATH] is the migration velocity induced by the electric field [MATH].', '0709.4377-1-25-2': 'For the mobility, we then obtain: [MATH].', '0709.4377-1-25-3': 'The PSS chain is surrounded by oppositely charged counter ions, some of which moving with the chain and thus reducing its effective charge.', '0709.4377-1-25-4': 'We estimate this charge reduction by subtracting the number of counter ions that are found within 2 units of the chain from the bare charge [MATH].', '0709.4377-1-25-5': 'This estimate is used together with the obtained mobility to calculate the effective friction as shown in Fig. [REF].', '0709.4377-1-25-6': 'The effective charge for long chains is in agreement with the Manning prediction at infinite dilution, yielding [MATH].', '0709.4377-1-25-7': 'From Fig. [REF], we see the impact of hydrodynamic interactions that results in the mobility maximum.', '0709.4377-1-25-8': 'Additional monomers are partly shielded from the flow by the other monomers decreasing the effective friction per monomer with chain length.', '0709.4377-1-25-9': 'This shielding is effective on short length scales and leads to a stronger initial decrease of the friction than the counter ion condensation reduces the effective charge.', '0709.4377-1-25-10': 'This leads to an increasing mobility for intermediate chains.', '0709.4377-1-25-11': 'While the effective friction levels of quickly, the effective charge per monomer decreases more slowly, reducing the mobility and causing the maximum.', '0709.4377-1-25-12': 'For long chains, effective charge and effective friction per monomer become constant, leading to the well-known free-draining behaviour.', '0709.4377-1-26-0': '# Conclusion', '0709.4377-1-27-0': 'We performed a detailed study investigating the transport properties of short PSS chains via MD simulations of a coarse grained model and via two different experimental approaches.', '0709.4377-1-27-1': 'The results of experiments and simulations can be quantitatively matched and agree with the existing theory and predictions, as long as the simulation model correctly includes long-range hydrodynamic interactions.', '0709.4377-1-27-2': 'A simulation model that neglects hydrodynamic interactions fails to reproduce the short-length scale behaviour of the PSS diffusion coefficient and of the electrophoretic mobility.', '0709.4377-1-28-0': 'To our knowledge we demonstrated for the first time, that the transport coefficient of short polyelectrolytes can be quantitatively modeled by coarse grained simulation techniques.', '0709.4377-1-28-1': 'No chemical details are needed to explain the experimental results.', '0709.4377-1-28-2': 'Our model allows to simulate time scales otherwise out of reach for atomistic simulations.', '0709.4377-1-28-3': 'The short chain behaviour as observed in experiments is accurately reproduced when hydrodynamic interactions are included.', '0709.4377-1-28-4': 'From our observation we conclude, that hydrodynamic interactions between the chain monomers are the major reason for the existence of the maximum at intermediate length.', '0709.4377-1-29-0': 'Having a simulation model at hand, that confirms the experimental data, opens new possibilities of investigating the electrophoretic behaviour of short polyelectrolytes, which so far has not been fully explained by the existing theories.'}

{'0709.4377-2-0-0': 'The transport coefficients of sulfonated polystyrene in the oligomer range are determined by coarse-grained molecular dynamics simulation and compared to data obtained by capillary electrophoresis and electrophoresis NMR.', '0709.4377-2-0-1': 'We find excellent agreement of experiments and simulations when hydrodynamic interactions are properly included in the simulations.', '0709.4377-2-0-2': 'We show that the electrophoretic mobility exhibits a maximum in the oligomer range and demonstrate that this maximum is due to the hydrodynamical interactions between the chain monomers and the solvent.', '0709.4377-2-0-3': 'Our findings demonstrate convincingly that it is possible to model transport properties of short polyelectrolytes correctly using a coarse grained model for the polyelectrolyte and the solvent, if hydrodynamic interactions are included.', '0709.4377-2-1-0': '# Introduction', '0709.4377-2-2-0': 'Electrophoresis methods are widely used to separate (macro)biomolecules [CITATION] such as peptides, proteins, and DNA, as well as synthetic polymers [CITATION].', '0709.4377-2-2-1': 'Short polyelectrolytes can conveniently be separated in free solution without the aid of a gel by capillary electrophoresis (CE).', '0709.4377-2-2-2': 'Additionally, CE is employed to characterize the hydrodynamic properties of charged biomolecules, in particular the hydrodynamic radius, the diffusion coefficient, and the electrophoretic mobility [CITATION].', '0709.4377-2-3-0': 'Alternatively, these properties can be determined by pulsed field gradient (PFG) NMR [CITATION].', '0709.4377-2-3-1': 'With a combination of diffusion NMR and electrophoresis NMR the charge of macromolecules [CITATION], the influence of ionic strength [CITATION] and the dielectric constant [CITATION] of the solution have been investigated.', '0709.4377-2-4-0': 'Several studies of polyelectrolytes of well defined length in the short chain regime have shown that the free-solution electrophoretic mobility [MATH] exhibits a characteristic behaviour [CITATION]: from the monomer mobility [MATH] onwards, the mobility increases towards a maximum [MATH] that occurs for chains of a specific degree of polymerization [MATH].', '0709.4377-2-4-1': 'After this maximum, the mobility decreases slightly to reach a constant value [MATH], the so-called free draining mobility.', '0709.4377-2-5-0': 'Whether or not a mobility maximum appears and the precise value of [MATH] depend on the flexibility of the chain.', '0709.4377-2-5-1': 'While a maximum has been observed for flexible polyelectrolytes such as single stranded DNA or sulfonated polystyrene (PSS) at values around 10 repeat units for [MATH][CITATION], semi-flexible double-stranded DNA does not exhibit it.', '0709.4377-2-5-2': 'The small number for [MATH] and the strong influence of the persistence length shows that a short chain behavior must be responsible for this.', '0709.4377-2-5-3': 'Furthermore, it has been shown that the [MATH] is diminished under elevated salt conditions.', '0709.4377-2-6-0': 'Existing theories describing the electrophoretic properties of polyelectrolytes [CITATION] have been successful in describing the qualitative behaviour of an initially rising mobility as well as the constant long-chain limit, but they have not been able to reproduce this maximum or explain its origin.', '0709.4377-2-7-0': 'The focus of this article is to shed light on the origin of the maximum in the case of PSS.', '0709.4377-2-7-1': 'We do this by coarse-grained Molecular Dynamics (MD) simulations that take all charged particles in the system (i.e. polyelectrolyte repeat units, counter ions, and additional salt) explicitly and hydrodynamics implicitly into account.', '0709.4377-2-7-2': 'The simulation results are compared to data sets obtained by two different experimental techniques.', '0709.4377-2-7-3': 'The combination of individual findings is then used to describe the size-dependence of the electrophoretic mobility for short polyelectrolyte chains.', '0709.4377-2-8-0': '# Simulations', '0709.4377-2-9-0': 'Model.', '0709.4377-2-9-1': 'We simulate a flexible polyelectrolyte using a bead-spring model employing the ESPResSo package ([CITATION]).', '0709.4377-2-9-2': 'All parameter values are given in reduced units with [MATH] and [MATH]A being the energy and the relevant length scale.', '0709.4377-2-9-3': 'The beads (chain monomers) are connected by FENE bonds with stiffness [MATH], and maximum extension [MATH].', '0709.4377-2-9-4': 'Additionally a truncated Lennard-Jones or WCA potential with depth [MATH] and width [MATH], is used for excluded volume interactions.', '0709.4377-2-9-5': 'Each monomer has a charge of [MATH] in units of [MATH].', '0709.4377-2-9-6': 'Monovalent counter ions, [MATH], and monovalent positive and negative salt ions are subjected to the same LJ potential giving all particles the same size.', '0709.4377-2-9-7': 'The simulations are carried out under periodic boundary conditions in a rectangular simulation box of size [MATH] (for [MATH]) to [MATH] for ([MATH]) resulting in constant monomer concentration.', '0709.4377-2-9-8': 'All electrostatic interactions are calculated with the P3M algorithm.', '0709.4377-2-9-9': 'The Bjerrrum length [MATH] in simulation units corresponds to 7.1 A (the Bjerrum length for water at room temperature).', '0709.4377-2-9-10': "Together with the model's average bond length of [MATH], we compute a Manning factor of [MATH].", '0709.4377-2-9-11': 'The inclusion of hydrodynamic interactions is done via frictional coupling of the beads to a Lattice Boltzmann (LB) fluid as detailed in [CITATION].', '0709.4377-2-9-12': 'The modeled fluid has a kinematic viscosity [MATH], a fluid density [MATH], and is discretized by a grid with spacing [MATH].', '0709.4377-2-9-13': 'The coupling parameter is [MATH].', '0709.4377-2-9-14': 'The simulation timestep is [MATH].', '0709.4377-2-10-0': 'To determine the impact of hydrodynamic interactions, we compare the results to simulations with a simple Langevin thermostat that does not recover long-range hydrodynamic interactions between the monomers, but only offers local interaction with the solvent.', '0709.4377-2-11-0': 'Transport coefficients.', '0709.4377-2-11-1': 'We determine two different transport coefficients for the model polyelectrolyte that are likewise determined in the associated experiments.', '0709.4377-2-11-2': 'The single chain diffusion coefficient is obtained from the center of mass velocity auto correlation function: [MATH].', '0709.4377-2-12-0': 'In CE experiments, the electrophoretic mobility [MATH] of the solute is determined by [MATH], where [MATH] is the velocity, [MATH] is the electric field, [MATH] is the applied voltage, [MATH] is the total length of the capillary, [MATH] is the migration (or effective) length up to the detector and [MATH] is the detection time of the solute.', '0709.4377-2-13-0': 'In the simulations, we use a Green-Kubo relation to obtain the electrophoretic mobility at zero electric field.', '0709.4377-2-13-1': 'This approach has been successfully applied in simulations to determine the electrophoretic mobility of charged colloids ([CITATION]).', '0709.4377-2-13-2': 'The chain mobility is calculated from the correlation function between the center of mass velocity of the polyelectrolyte chain and the velocities of all charged particles in the system: [MATH].', '0709.4377-2-13-3': 'This method guarantees that no conformational changes of the chain structure or the ion distribution are induced by an artificial high external field, which is sometimes used in other simulations to separate the directed electrophoretic motion from Brownian motion within reasonable computing time.', '0709.4377-2-13-4': 'Another benefitial side effect of this method is that both transport properties can be obtained from the same simulation trajectories without additional computational effort.', '0709.4377-2-14-0': '# Experiments', '0709.4377-2-15-0': 'Capillary electrophoresis (CE) is an analytical separation technique based on the differential migration of ionic species under electric field [CITATION].', '0709.4377-2-16-0': 'The CE experiments were performed using an Agilent technologies capillary electrophoresis system (Agilent, Waldbronn, Germany).', '0709.4377-2-16-1': 'Capillaries of 33.5 cm (25 cm to the detector) length, and 50 [MATH]m diameter were prepared from bare silica tubing purchased from Supelco (Bellefonte, PA, USA).', '0709.4377-2-16-2': 'New capillaries were conditioned by performing the following flushes: 1M NaOH for 30 min, 0.1 M NaOH for 10 min, and water for 5 min.', '0709.4377-2-16-3': 'Samples were introduced hydrodynamically ([MATH] nL) at 0.5 g/L concentration ([MATH]mMol/l monomer concentration).', '0709.4377-2-16-4': 'The electrolyte was pure water.', '0709.4377-2-16-5': 'Solutes were detected at 225 nm.', '0709.4377-2-16-6': 'The electric field was kept constant at 224 V/cm (V=+7.5 kV).', '0709.4377-2-16-7': 'The polarity of the applied voltage on the inlet side of the capillaries was positive.', '0709.4377-2-16-8': 'All the experiments were performed at 27 [MATH]C. Electroosmotic mobilities were determined from the migration time of a neutral marker (mesityl oxide, [MATH]% (v/v) in the electrolyte).', '0709.4377-2-16-9': 'The sodium polystyrenesulfonate standards ([MATH]; [MATH] ca 1.1) were purchased from American Polymer Standards Corp. (Mentor, OH, USA).', '0709.4377-2-16-10': 'Borax (disodium tetraborate decahydrate) was from Prolabo (Paris, France).', '0709.4377-2-16-11': 'Mesityl oxide used as neutral marker was obtained from Aldrich.', '0709.4377-2-16-12': 'Purified water delivered by an Alpha-Q system (Millipore, Molsheim, France) was used to prepare all electrolytes and sample solutions.', '0709.4377-2-17-0': 'In electrophoresis NMR, diffusion and electrophoretic motion are separated by the design of the experiment [CITATION].', '0709.4377-2-17-1': 'No gel has been used, so that self diffusion and free electrophoresis are measured.', '0709.4377-2-17-2': 'The PSS samples have been obtained from Fluka.', '0709.4377-2-17-3': 'To minimize the effects of variations of the ionic strength [CITATION], samples have been dialyzed against water (cut off volume 0.5kDA), and subsequently dried under vacuum.', '0709.4377-2-17-4': 'For all experiments a concentration of 5mMol/l (monomer) in deuterated water has been used.', '0709.4377-2-17-5': 'The diffusion experiments have been performed on a Bruker Avance 500 NMR spectrometer operating at a Larmor frequency of 500MHz for protons equipped with a DIFF30 probehead generating a maximum pulsed field gradient strength of 12T/m.', '0709.4377-2-17-6': 'The gradient pulse duration [MATH] and diffusion times [MATH] have been adjusted between 0.8 to 2ms for [MATH] and 8 to 20ms for [MATH] respectively for optimal resolution for each molecular weight resulting in different diffusion coefficients.', '0709.4377-2-17-7': 'Because of their narrow molecular weight distribution, diffusion coefficients have been determined by a linear fit to the Stejskal-Tanner equation [CITATION].', '0709.4377-2-17-8': 'Electrophoresis NMR experiments have been performed on a Bruker Avance 300 NMR spectrometer operating at a Larmor frequency of 300MHz for protons with an in-house-built electrophoresis probehead utilizing a Bruker micro2.5 imaging gradient system generating magnetic field gradient strength of up to 1T/m.', '0709.4377-2-17-9': 'The flow times [MATH] varied between 15 and 50 ms and the gradient pulse duration [MATH] between 3 and 6 ms respectively with gradient amplitudes between 0.3 and 0.6 T/m.', '0709.4377-2-17-10': 'The electric field has been linearly incremented between -140 V/cm and +140 V/cm.', '0709.4377-2-17-11': 'The electrophoretic mobility is a model-free read out from the two-dimensional electrophoresis NMR spectrum correllating the chemical shift, that identifies the moving species, with electrophoretic mobility [CITATION].', '0709.4377-2-18-0': '# Discussion', '0709.4377-2-19-0': 'Diffusion.', '0709.4377-2-19-1': 'Recently, Stellwagen et.', '0709.4377-2-19-2': 'al. [CITATION] observed that the diffusion coefficient [MATH] of PSS can be approximated by a power law scaling [MATH], where [MATH] is the monomer diffusion coefficient, and [MATH] is the scaling exponent.', '0709.4377-2-19-3': 'In Figure [REF], we compare the diffusion coefficient obtained from simulations to the results from the NMR study.', '0709.4377-2-19-4': 'The simulated data is normalized by [MATH] as obtained from the power law fit, and the experimental data by the monomer diffusion coefficient of [MATH] m[MATH]/s.', '0709.4377-2-20-0': 'The simulations with hydrodynamic interactions result in a scaling exponent of [MATH], which is in good agreement with value obtained from experiments, [MATH] [CITATION], and a previously reported result, [MATH], by [CITATION].', '0709.4377-2-20-1': 'Only for the very short chains ([MATH]), a deviation from the prediction is observable and a higher diffusion coefficient is found in the simulations.', '0709.4377-2-20-2': 'For intermediate chain length, the coarse-grained simulation model with hydrodynamic interactions, reproduces the experimentally observed behaviour.', '0709.4377-2-21-0': 'Without hydrodynamic interactions, the chains show the expected Rouse diffusion with an exponent of [MATH].', '0709.4377-2-21-1': 'This simple model is clearly not applicable to mimic the experimental behaviour of short polyelectrolyte chains.', '0709.4377-2-22-0': 'Electrophoretic mobility ([MATH]).', '0709.4377-2-22-1': 'The results for measurements of [MATH] in pure water without additional salt are shown in Fig. [REF].', '0709.4377-2-22-2': 'To account for the different viscosity of the solvents, we rescale the mobilities by the free-draining mobility [MATH] as obtained for long chains.', '0709.4377-2-23-0': 'The experimental data sets agree within the accuracies of the individual methods and show the characteristic behaviour of the mobility in dependence to chain length.', '0709.4377-2-23-1': 'A mobility maximum for [MATH] is observed with capillary electrophoresis.', '0709.4377-2-23-2': 'This maximum for intermediate chains as well as the long chain behaviour is successfully reproduced in simulations with hydrodynamic interactions.', '0709.4377-2-23-3': 'For the first few oligomers, we observe a small difference which is inline with the deviation for the diffusion.', '0709.4377-2-24-0': 'On the other hand, as illustrated in the inset of Fig. [REF], the simulation without hydrodynamic interactions fails completly to describe the short chain behaviour and can only be mapped to the experimental data in the long chain limit.', '0709.4377-2-24-1': 'Therefore, we infer that the mobility maximum can only be explained when taking into account hydrodynamic interactions between the polyelectrolyte and the surrounding solvent.', '0709.4377-2-25-0': 'To further illustrate this, we estimate the effective hydrodynamic friction [MATH] and the effective charge [MATH] in dependence of its length.', '0709.4377-2-25-1': 'During electrophoresis, the electrical driving force [MATH] is balanced by the frictional force with the solvent [MATH], where [MATH] is the migration velocity induced by the electric field [MATH].', '0709.4377-2-25-2': 'For the mobility, we then obtain: [MATH].', '0709.4377-2-25-3': 'The PSS chain is surrounded by oppositely charged counterions, some of which moving with the chain and thus reducing its effective charge.', '0709.4377-2-25-4': 'We estimate this charge reduction by substracting the number of counterions that are found within 2 units of the chain from the bare charge [MATH].', '0709.4377-2-25-5': 'This estimate is used together with the obtained mobility to calculate the effective friction as shown in Fig. [REF].', '0709.4377-2-25-6': 'The effective charge for long chains is in agreement with the Manning prediction at infinite dilution, yielding [MATH].', '0709.4377-2-25-7': 'From Fig. [REF], we see the impact of hydrodynamic interactions that results in the mobility maximum.', '0709.4377-2-25-8': 'Additional monomers are partly shielded from the flow by the other monomers decreasing the effective friction per monomer with chain length.', '0709.4377-2-25-9': 'This shielding is effective on short length scales and leads to a stronger initial decrease of the friction than the counterion condensation reduces the effective charge.', '0709.4377-2-25-10': 'This leads to an increasing mobility for intermediate chains.', '0709.4377-2-25-11': 'While the effective friction levels of quickly, the effective charge per monomer decreases more slowly, reducing the mobility and causing the maximum.', '0709.4377-2-25-12': 'For long chains, effective charge and effective friction per monomer become constant, leading to the well-known free-draining behaviour.', '0709.4377-2-26-0': '# Conclusion', '0709.4377-2-27-0': 'We performed a detailed study investigating the transport properties of short PSS chains via MD simulations of a coarse grained model and via two different experimental approaches.', '0709.4377-2-27-1': 'The results of experiments and simulations can be quantitatively matched and agree with the existing theory and predictions, as long as the simulation model correctly includes long-range hydrodynamic interactions.', '0709.4377-2-27-2': 'A simulation model that neglects hydrodynamic interactions fails to reproduce the short-length scale behaviour of the PSS diffusion coefficient and of the electrophoretic mobility.', '0709.4377-2-28-0': 'To our knowledge we demonstrated for the first time, that the transport coefficient of short polyelectrolytes can be quantitatively modeled by coarse grained simulation techniques.', '0709.4377-2-28-1': 'No chemical details are needed to explain the experimental results.', '0709.4377-2-28-2': 'Our model allows to simulate time scales otherwise out of reach for atomistic simulations.', '0709.4377-2-28-3': 'The short chain behaviour as observed in experiments is accurately reproduced when hydrodynamic interactions are included.', '0709.4377-2-28-4': 'From our observation we conclude, that hydrodynamic interactions between the chain monomers are the major reason for the existence of the maximum at intermediate length.', '0709.4377-2-29-0': 'Having a simulation model at hand, that confirms the experimental data, opens new possibilities of investigating the electrophoretic behaviour of short polyelectrolytes, which so far has not been fully explained by the existing theories.'}

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[]

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[]

['0709.4377-1-9-0', '0709.4377-1-11-0', '0709.4377-1-19-0', '0709.4377-1-19-1', '0709.4377-1-22-0', '0709.4377-2-9-0', '0709.4377-2-11-0', '0709.4377-2-19-0', '0709.4377-2-19-1', '0709.4377-2-22-0', '0709.4377-3-9-0', '0709.4377-3-11-0', '0709.4377-3-19-0', '0709.4377-3-21-0']

{'1': 'http://arxiv.org/licenses/assumed-1991-2003/', '2': 'http://arxiv.org/licenses/assumed-1991-2003/', '3': 'http://arxiv.org/licenses/assumed-1991-2003/'}

https://arxiv.org/abs/0709.4377

{'0709.4377-3-0-0': 'The dynamic behavior of polyelectrolyte chains in the oligomer range is investigated with coarse-grained molecular dynamics simulation and compared to data obtained by two different experimental methods, namely capillary electrophoresis and electrophoresis NMR.', '0709.4377-3-0-1': 'We find excellent agreement of experiments and simulations when hydrodynamic interactions are accounted for in the simulations.', '0709.4377-3-0-2': 'We show that the electrophoretic mobility exhibits a maximum in the oligomer range and for the first time illustrate that this maximum is due to the hydrodynamical shielding between the chain monomers.', '0709.4377-3-0-3': 'Our findings demonstrate convincingly that it is possible to model dynamic behavior of polyelectrolytes using coarse grained models for both, the polyelectrolyte chains and the solvent induced hydrodynamic interactions.', '0709.4377-3-1-0': '# Introduction', '0709.4377-3-2-0': 'Electrophoresis methods are widely used to separate (macro)biomolecules [CITATION] such as peptides, proteins, and DNA, as well as synthetic polymers [CITATION].', '0709.4377-3-2-1': 'Short polyelectrolytes (PEs) can conveniently be separated in free solution without the aid of a gel by capillary electrophoresis (CE).', '0709.4377-3-2-2': 'Additionally, CE is employed to characterize the hydrodynamic properties of charged biomolecules, in particular the electrophoretic mobility, [MATH], the diffusion coefficient, and the hydrodynamic radius [CITATION].', '0709.4377-3-3-0': 'Alternatively, these properties can be determined by pulsed field gradient (PFG) NMR [CITATION].', '0709.4377-3-3-1': 'With a combination of diffusion NMR and electrophoretic NMR, the charge of macromolecules [CITATION], the influence of ionic strength and the dielectric constant of the solution have been investigated [CITATION].', '0709.4377-3-4-0': 'Several studies of PEs of well defined length in the short chain regime have shown that the free-solution mobility [MATH] exhibits a characteristic behavior [CITATION]: from the monomer mobility, [MATH], onwards, [MATH] increases towards a maximum, [MATH], that occurs for chains of a specific degree of polymerization, [MATH].', '0709.4377-3-4-1': 'After this maximum, [MATH] decreases slightly to reach a constant value, [MATH], the so-called free draining mobility.', '0709.4377-3-4-2': 'Very little is known about the origin of the maximum, yet the knowledge of the precise dynamical behavior of a PE is a prerequisite for designing possible applications in microfluidic devices, such as electrophoretic separation or gene sequencing.', '0709.4377-3-5-0': 'Whether or not a mobility maximum appears and the precise value of [MATH] seems to depend on the flexibility of the chain.', '0709.4377-3-5-1': 'While a maximum has been observed for flexible PEs such as single stranded DNA or sulfonated polystyrene (PSS) at values around 10 repeat units for [MATH][CITATION], semi-flexible double-stranded DNA does not exhibit it.', '0709.4377-3-5-2': 'The small number for [MATH] and the strong influence of the persistence length shows that a short chain behavior must be responsible for this.', '0709.4377-3-5-3': 'Furthermore, it has been shown that the [MATH] is diminished under elevated salt conditions.', '0709.4377-3-6-0': 'Existing theories [CITATION] have been successful in describing the qualitative behavior of an initially rising mobility as well as the constant long-chain limit, but they have not been able to reproduce this maximum or explain its origin.', '0709.4377-3-7-0': 'To provide a fundamental understanding of the involved dynamics of PEs, we propose a coarse-grained Molecular Dynamics model that takes all charged particles (i.e. PE repeat units, counter ions, and additional salt) and electrostatic and hydrodynamic interactions (HI) between them into account.', '0709.4377-3-7-1': 'In the following we demonstrate that this model quantitatively reproduces experimental results obtained by two completely different experimental techniques.', '0709.4377-3-7-2': 'In addition we suggest a microscopic interpretation of the size-dependence of [MATH] for short PE chains, based on hydrodynamic shielding, which gives fundamental insight into the interplay of hydrodynamic friction and charge correlations for charged macromolecules.', '0709.4377-3-8-0': '# Simulations', '0709.4377-3-9-0': 'Model.', '0709.4377-3-9-1': 'We simulate a flexible PE using a bead-spring model employing the ESPResSo package ([CITATION]).', '0709.4377-3-9-2': 'All parameter values are chosen to match the properties of PSS used in the experiments and given in reduced units with [MATH] and [MATH]A being the energy and the relevant length scale.', '0709.4377-3-9-3': 'The beads (chain monomers) are connected by FENE bonds with stiffness [MATH], and maximum extension [MATH].', '0709.4377-3-9-4': 'Additionally a truncated Lennard-Jones or WCA potential with depth [MATH] and width [MATH], is used for excluded volume interactions.', '0709.4377-3-9-5': 'Each monomer has a charge of [MATH] in units of [MATH].', '0709.4377-3-9-6': 'Monovalent counter ions, [MATH], and monovalent positive and negative salt ions are subjected to the same LJ potential giving all particles the same size.', '0709.4377-3-9-7': 'The simulations are carried out under periodic boundary conditions in a rectangular simulation box of size [MATH] (for [MATH]) to [MATH] for ([MATH]) resulting in constant monomer concentration of approximately 1 g/l or 5 mM.', '0709.4377-3-9-8': 'All electrostatic interactions are calculated with the P3M algorithm.', '0709.4377-3-9-9': 'The Bjerrum length [MATH] in simulation units corresponds to 7.1 A (the Bjerrum length for water at room temperature).', '0709.4377-3-9-10': "Together with the model's average bond length of [MATH], we compute a Manning factor of [MATH].", '0709.4377-3-9-11': 'The inclusion of HI is done via frictional coupling of the beads to a Lattice Boltzmann (LB) fluid as detailed in [CITATION].', '0709.4377-3-9-12': 'The modeled fluid has a kinematic viscosity [MATH], a fluid density [MATH], and is discretized by a grid with spacing [MATH].', '0709.4377-3-9-13': 'The coupling parameter is [MATH].', '0709.4377-3-9-14': 'The simulation time step is [MATH].', '0709.4377-3-10-0': 'To determine the impact of HI, we compare the results to simulations with a simple Langevin thermostat that does not recover long-range hydrodynamic interactions between the monomers, but only offers local interaction with the solvent.', '0709.4377-3-11-0': 'Transport coefficients.', '0709.4377-3-11-1': 'We determine two different transport coefficients for the model PE that are likewise determined in the associated experiments.', '0709.4377-3-11-2': 'The single chain diffusion coefficient is obtained from the center of mass velocity auto correlation function: [MATH].', '0709.4377-3-12-0': 'In CE experiments, the electrophoretic mobility [MATH] of the solute is determined by [MATH], where [MATH] is the velocity, [MATH] is the electric field, [MATH] is the applied voltage, [MATH] is the total length of the capillary, [MATH] is the migration (or effective) length up to the detector and [MATH] is the detection time of the solute.', '0709.4377-3-13-0': 'In the simulations, we use a Green-Kubo relation to obtain [MATH] at zero electric field.', '0709.4377-3-13-1': 'This approach has been successfully applied in simulations to determine the electrophoretic mobility of charged colloids ([CITATION]).', '0709.4377-3-13-2': 'The chain mobility is calculated from the correlation function between the center of mass velocity of the polyelectrolyte chain and the velocities of all charged particles in the system: [MATH].', '0709.4377-3-13-3': 'This method guarantees that no conformational changes of the chain structure or the ion distribution are induced by an artificial high external field, which is sometimes used in other simulations to separate the directed electrophoretic motion from Brownian motion within reasonable computing time.', '0709.4377-3-13-4': 'Our method enables us furthermore to obtain both transport properties from the same simulation trajectories without additional computational effort.', '0709.4377-3-14-0': '# Experiments', '0709.4377-3-15-0': 'Capillary electrophoresis (CE) is an analytical separation technique based on the differential migration of ionic species under electric field [CITATION].', '0709.4377-3-16-0': 'The CE experiments were performed using an Agilent technologies capillary electrophoresis system (Agilent, Waldbronn, Germany).', '0709.4377-3-16-1': 'Capillaries of 33.5 cm (25 cm to the detector) length, and 50 [MATH]m diameter were prepared from bare silica tubing purchased from Supelco (Bellefonte, PA, USA).', '0709.4377-3-16-2': 'New capillaries were conditioned by performing the following flushes: 1M NaOH for 30 min, 0.1 M NaOH for 10 min, and water for 5 min.', '0709.4377-3-16-3': 'Samples were introduced hydrodynamically ([MATH] nL) at 0.5 g/L concentration ([MATH] mM monomer concentration).', '0709.4377-3-16-4': 'The electrolyte was pure water.', '0709.4377-3-16-5': 'Solutes were detected at 225 nm.', '0709.4377-3-16-6': 'The electric field was kept constant at 224 V/cm (V=+7.5 kV).', '0709.4377-3-16-7': 'The polarity of the applied voltage on the inlet side of the capillaries was positive.', '0709.4377-3-16-8': 'All the experiments were performed at 27 [MATH]C. Electro osmotic mobilities were determined from the migration time of a neutral marker (mesityl oxide, [MATH]% (v/v) in the electrolyte).', '0709.4377-3-16-9': 'The sodium polystyrene sulfonate standards ([MATH]; [MATH] ca 1.1) were purchased from American Polymer Standards Corp. (Mentor, OH, USA).', '0709.4377-3-16-10': 'These standards have verified sulfonation rates larger than 88% and are almost fully charged.', '0709.4377-3-16-11': 'Purified water delivered by an Alpha-Q system (Millipore, Molsheim, France) was used to prepare all electrolytes and sample solutions.', '0709.4377-3-17-0': 'In electrophoretic NMR, diffusion and electrophoretic motion are separated by the design of the experiment [CITATION].', '0709.4377-3-17-1': 'No gel has been used, so that self diffusion and free electrophoresis are measured.', '0709.4377-3-17-2': 'The PSS samples have been obtained from Fluka.', '0709.4377-3-17-3': 'To minimize the effects of variations of the ionic strength [CITATION], samples have been dialyzed against water (cut off volume 0.5kDA), and subsequently dried under vacuum.', '0709.4377-3-17-4': 'For all experiments a monomer concentration of 5 mM in deuterated water has been used.', '0709.4377-3-17-5': 'The diffusion experiments have been performed on a Bruker Avance 500 NMR spectrometer operating at a Larmor frequency of 500MHz for protons equipped with a DIFF30 probe head generating a maximum pulsed field gradient strength of 12T/m.', '0709.4377-3-17-6': 'The gradient pulse duration and diffusion times have been adjusted between 0.8 to 2ms and 8 to 20ms respectively for optimal resolution for each molecular weight resulting in different diffusion coefficients.', '0709.4377-3-17-7': 'Because of their narrow molecular weight distribution, diffusion coefficients have been determined by a linear fit to the Stejskal-Tanner equation [CITATION].', '0709.4377-3-17-8': 'Electrophoresis NMR experiments have been performed on a Bruker Avance 300 NMR spectrometer operating at a Larmor frequency of 300MHz for protons with an in-house-built electrophoresis probe head utilizing a Bruker micro2.5 imaging gradient system generating magnetic field gradient strength of up to 1T/m.', '0709.4377-3-17-9': 'The flow times varied between 15 and 50 ms and the gradient pulse duration between 3 and 6 ms respectively with gradient amplitudes between 0.3 and 0.6 T/m.', '0709.4377-3-17-10': 'The electric field has been linearly incremented between -140 V/cm and +140 V/cm.', '0709.4377-3-17-11': 'The electrophoretic mobility is a model-free read out from the two-dimensional electrophoresis NMR spectrum correlating the chemical shift, that identifies the moving species, with electrophoretic mobility [CITATION].', '0709.4377-3-18-0': '# Discussion', '0709.4377-3-19-0': 'Diffusion.', '0709.4377-3-19-1': 'Recently, Stellwagen et al. [CITATION] observed that the diffusion coefficient, [MATH], of PSS can be approximated by a power law scaling [MATH], where [MATH] is the monomer diffusion coefficient, and [MATH] is the scaling exponent.', '0709.4377-3-19-2': 'In Figure [REF], we compare the diffusion coefficient obtained from simulations to the results from the NMR study.', '0709.4377-3-19-3': 'The simulated data is normalized by [MATH] as obtained from the power law fit, and the experimental data by the monomer diffusion coefficient of [MATH] m[MATH]/s.', '0709.4377-3-19-4': 'The simulations with HI result in a scaling exponent of [MATH], which is in good agreement with value obtained from experiments, [MATH] [CITATION], and a previously reported result, [MATH], by [CITATION].', '0709.4377-3-19-5': 'Only for the very short chains, ([MATH]), is a deviation from the prediction observed and a higher diffusion coefficient found in the simulations.', '0709.4377-3-19-6': 'For intermediate chain length, the coarse-grained simulation model with HI, reproduces the experimentally observed behavior.', '0709.4377-3-20-0': 'Without HI, the chains show the expected Rouse diffusion with an exponent of [MATH].', '0709.4377-3-20-1': 'This simple model is clearly not applicable to mimic the experimental behavior of short PE chains.', '0709.4377-3-21-0': 'Electrophoretic mobility ([MATH]).', '0709.4377-3-21-1': 'The results for measurements of [MATH] in pure water without additional salt are shown in Fig. [REF].', '0709.4377-3-21-2': 'To account for the different viscosity of the solvents, we rescale the mobilities by the free-draining mobility [MATH] as obtained for long chains.', '0709.4377-3-22-0': 'The experimental data sets agree within the accuracies of the individual methods and show the characteristic behavior of the mobility in dependence to chain length.', '0709.4377-3-22-1': 'A mobility maximum for [MATH] is observed with capillary electrophoresis.', '0709.4377-3-22-2': 'This maximum for intermediate chains as well as the long chain behavior is successfully reproduced in simulations with HI.', '0709.4377-3-22-3': 'For the first few oligomers, we observe a small difference which is in line with the deviation for the diffusion.', '0709.4377-3-23-0': 'On the other hand, as illustrated in the inset of Fig. [REF], the simulation without hydrodynamic interactions fails completely to describe the short chain behavior and can only be mapped to the experimental data in the long chain limit.', '0709.4377-3-23-1': 'Therefore, we infer that the mobility maximum can only be explained when taking into account HI between the PE and the surrounding solvent .', '0709.4377-3-24-0': 'To further illustrate this, we estimate the effective hydrodynamic friction, [MATH], and the effective charge, [MATH], in dependence of its length.', '0709.4377-3-24-1': 'During electrophoresis, the electrical driving force [MATH] is balanced by the frictional force with the solvent [MATH], where [MATH] is the migration velocity induced by the electric field [MATH].', '0709.4377-3-24-2': 'For the mobility, we then obtain: [MATH].', '0709.4377-3-24-3': 'The PE is surrounded by oppositely charged counter ions, some of which moving with the chain and thus reducing its effective charge.', '0709.4377-3-24-4': 'We estimate this charge reduction by subtracting the number of counter ions that are found within 2 units of the chain from the bare charge [MATH].', '0709.4377-3-24-5': 'This estimate is used together with the obtained mobility to calculate the effective friction as shown in Fig. [REF].', '0709.4377-3-24-6': 'The effective charge for long chains is in agreement with the Manning prediction at infinite dilution, yielding [MATH].', '0709.4377-3-24-7': 'From Fig. [REF], we see the impact of HI that results in the mobility maximum.', '0709.4377-3-24-8': 'Additional monomers are partly shielded from the flow by the other monomers decreasing the effective friction per monomer with chain length.', '0709.4377-3-24-9': 'This shielding is effective on short length scales and leads to a stronger initial decrease of the friction than the counter ion condensation reduces the effective charge.', '0709.4377-3-24-10': 'This leads to an increasing mobility for intermediate chains.', '0709.4377-3-24-11': 'While the effective friction levels off quickly, the effective charge per monomer keeps decreasing slowly, reducing the mobility and causing the maximum.', '0709.4377-3-24-12': 'For long chains, effective charge and effective friction per monomer become constant, leading to the well-known free-draining behavior.', '0709.4377-3-25-0': '# Conclusion', '0709.4377-3-26-0': 'We performed a detailed study investigating the dynamic behavior of short PE chains via MD simulations of a coarse grained model and via two different experimental approaches.', '0709.4377-3-26-1': 'The results of experiments and simulations can be quantitatively matched and agree with the existing theory and predictions, as long as the simulation model correctly includes long-range HI.', '0709.4377-3-26-2': 'A simulation model that neglects HI fails to reproduce the short-length scale behavior of the diffusion coefficient and of the electrophoretic mobility.', '0709.4377-3-27-0': 'To our knowledge we demonstrated for the first time, that the transport coefficient of short PEs can be quantitatively modeled by coarse grained simulation techniques.', '0709.4377-3-27-1': 'No chemical details are needed to explain the experimental results.', '0709.4377-3-27-2': 'Our model allows to simulate time scales otherwise out of reach for atomistic simulations and provides a microscopic understanding of the observed maximum in [MATH].', '0709.4377-3-27-3': 'From our observation we conclude, that hydrodynamic shielding between the chain monomers is the major cause for this previously little understood behavior, and that it is vitally important to include HI to model such systems.', '0709.4377-3-28-0': 'Having a simulation model at hand, that confirms the experimental data, opens new possibilities of investigating the electrophoretic behavior of short PEs, which so far has not been fully explained by the existing theories.', '0709.4377-3-29-0': 'We thank B. Dunweg, U. Schiller, and G. Slater for helpful remarks.', '0709.4377-3-29-1': 'Funds from the the Volkswagen foundation, the DAAD, and DFG under the TR6 are gratefully acknowledged.'}

0811.4095

{'0811.4095-1-0-0': 'This article introduces Grapham, an open-source implementation covering several previously proposed adaptive random walk Metropolis algorithms.', '0811.4095-1-0-1': 'The target distribution, or "model", is specified in Grapham by determining a set of conditional probabilities-as a "graphical model".', '0811.4095-1-0-2': 'Grapham implements the Adaptive Metropolis (AM) algorithm of [CITATION] ([CITATION]) based on using the estimated covariance matrix in the proposal distribution.', '0811.4095-1-0-3': 'The implemented algorithms include also AM with delayed rejection, and an algorithm optimising the expected acceptance probability.', '0811.4095-1-0-4': 'All the algorithms can be used block-wise, with arbitrary blocking of the variables in the model.', '0811.4095-1-0-5': 'The software is written in C, and the model is specified using a simple high-level language Lua.', '0811.4095-1-1-0': '# Introduction', '0811.4095-1-2-0': 'Markov chain Monte Carlo (MCMC) is a general framework for computing expectations over complicated distributions in general state spaces.', '0811.4095-1-2-1': 'The methods are based on constructing a Markov chain [MATH] so that the ergodic averages [MATH] converge to [MATH] as [MATH], where [MATH] is the "target" distribution of interest.', '0811.4095-1-2-2': 'Such a chain is often easy to construct using the Metropolis-Hastings algorithm; see, e.g, [CITATION].', '0811.4095-1-2-3': 'Depending on [MATH], however, it may be difficult to design a practical algorithm so that [MATH] would approximate [MATH] well with moderate number of samples [MATH].', '0811.4095-1-3-0': 'Recently proposed adaptive MCMC algorithms adjust the parameters of the algorithm (the proposal distribution) on-the-fly, aiming to efficient simulation.', '0811.4095-1-3-1': 'They have gained increasing interest in the last few years, after the seminal paper [CITATION] presented the Adaptive Metropolis (AM) algorithm.', '0811.4095-1-3-2': 'It was shown in [CITATION] that AM provides valid simulation of a target distribution with a bounded and compactly supported density.', '0811.4095-1-3-3': 'After that, several authors have described more general conditions under which AM is valid, and introduced alternative adaptive MCMC algorithms; see, e.g., .', '0811.4095-1-4-0': 'This paper introduces Grapham [CITATION], a free open-source implementation covering several of the previously proposed adaptive MCMC algorithms.', '0811.4095-1-4-1': 'The purpose of Grapham is to provide an experimental tool for evaluating the performance of the adaptive MCMC algorithms with practical problems, especially in Bayesian statistics.', '0811.4095-1-4-2': 'The target distribution, the "model", is specified in Grapham by a set of conditional distributions-as a "graphical model" (see, e.g., [CITATION] and references therein).', '0811.4095-1-4-3': 'The model specification philosophy of Grapham thus reminds that of BUGS [CITATION].', '0811.4095-1-4-4': 'However, Grapham is an experimental project very different to BUGS in many aspects, and not intended to be an alternative to it.', '0811.4095-1-4-5': 'The paper [CITATION] gives a brief review of other software for graphical models.', '0811.4095-1-5-0': 'Rosenthal describes another adaptive MCMC implementation: AMCMC [CITATION].', '0811.4095-1-5-1': 'It is an R interface to one adaptive MCMC algorithm.', '0811.4095-1-5-2': 'Grapham differs from AMCMC in that it relies on a hierarchical model specification, and provides more alternative algorithms.', '0811.4095-1-5-3': 'Unlike AMCMC, Grapham also provides a set of ready-made standard distribution functions the user can employ as a part of their model specification.', '0811.4095-1-5-4': 'This is intended to facilitate faster development, while permitting the user to define arbitrary distributions easily.', '0811.4095-1-6-0': '# The Implementation', '0811.4095-1-7-0': 'The emphasis of the implementation is on the Adaptive Metropolis (AM) algorithm introduced by [CITATION].', '0811.4095-1-7-1': 'The variant of the AM algorithm Grapham implements can be summarised as follows.', '0811.4095-1-7-2': 'Begin with some initial [MATH] and let [MATH] be an initial positive definite matrix.', '0811.4095-1-7-3': 'Then, recursively for [MATH], set [EQUATION] where [MATH] is a Metropolis kernel using a Gaussian proposal distribution with covariance matrix [MATH], the constant [MATH] is a scaling parameter, and [MATH] denotes the average of [MATH].', '0811.4095-1-8-0': 'The implementation allows one to use block-wise sampling, by arbitrary blocks of variables, in the spirit of single component AM .', '0811.4095-1-8-1': 'There is a possibility to use delayed rejection with AM, as suggested by .', '0811.4095-1-8-2': 'In addition, one can use an algorithm that adaptively scales the proposal distribution according to the observed acceptance probability, as presented in .', '0811.4095-1-8-3': 'This adaptive scaling framework can be used with AM, as suggested in , in which case the algorithm would only add a changing scaling parameter [MATH] in [REF].', '0811.4095-1-8-4': 'The various configuration options enable to design and experiment with new variants of the algorithms.', '0811.4095-1-9-0': 'Grapham includes some numerical subroutines from NetlibfootnoteIn particular, the Cholesky factor update function DCHUD of Netlib allows [MATH] update of the Cholesky factor of dimension [MATH] in each step [REF] of the AM algorithm.', '0811.4095-1-9-1': 'Straightforward implementation with full Cholesky factorisation would take [MATH] operations.', '0811.4095-1-9-2': 'Notice that one needs [MATH] operations to generate a sample from the multivariate Gaussian proposal distribution.', '0811.4095-1-9-3': '[CITATION] and can optionally be compiled with the dSFMT random number generator [CITATION] instead of using the random number generators provided by the C standard libraries.', '0811.4095-1-9-4': 'The Numeric Lua package [CITATION] can be compiled with Grapham, in particular to allow easy working with vector-valued variables.', '0811.4095-1-10-0': 'The model is specified in Grapham using the Lua programming language, mainly due to the following three main reasons:', '0811.4095-1-11-0': 'It is also convenient that Lua is a full-featured programming language.', '0811.4095-1-11-1': 'This allows, for example, including data to the model from some other file.', '0811.4095-1-11-2': 'In fact, Grapham includes a function written in Lua to load data from a CSV (comma separated values) file into the model.', '0811.4095-1-11-3': 'The conditional density associated to a variable in the model can be defined in Lua, or alternatively (to increase simulation speed) in a separate C library.', '0811.4095-1-12-0': 'The samples simulated by Grapham can be used directly to estimate the expectation of some functional, specified in either Lua or C. Alternatively, the samples can be saved into a file in CSV or in a simple binary format.', '0811.4095-1-12-1': 'The former allows the samples to be imported easily to many other environments.', '0811.4095-1-12-2': 'In addition, there are ready-made functions for loading the binary data files into R [CITATION] and Matlab [CITATION] environments.', '0811.4095-1-13-0': '# An Example', '0811.4095-1-14-0': 'Consider the baseball example that is used in the AMCMC package .', '0811.4095-1-14-1': 'It consists of a hierarchical model having 20 unknown real-valued variables.', '0811.4095-1-14-2': 'The model is depicted in Figure [REF]', '0811.4095-1-15-0': 'and the model specification in Grapham is given below.', '0811.4095-1-16-0': 'const = v = 0.00434', '0811.4095-1-17-0': 'model = mu = density = "duniform" , t = parents = "mu","a", init_val = 1, density = "dnorm" , y = parents = "t", "v", density = "dnorm" , a = init_val = 1, density = function(a_) return dexp(1/a_, 1/2) end ,', '0811.4095-1-18-0': 'tmp, y = read_csv("models/baseball.data") repeat_block("y","t", y[1]) function functional() return t1, mu, a end para = niter = 30e3, nburn = 10e3, algorithm="ascm",', '0811.4095-1-19-0': 'In Grapham, the model is defined in the Lua table model, in which the density of each of the variables conditional to the "parent variables" is described.', '0811.4095-1-19-1': 'The other variables in the example have standard distributions, excluding a whose distribution is defined as a Lua function (calling also a standard distribution function dexp).', '0811.4095-1-19-2': 'The model has, in fact, variables [MATH] and [MATH], whose values are loaded with the read_csv function.', '0811.4095-1-19-3': 'The block of variables [MATH] in the model is then replicated using function repeat_block, which also fixes the values of [MATH] to the values read from the file baseball.data.', '0811.4095-1-20-0': 'Grapham was run with the above model specification having 20 unknown variables.', '0811.4095-1-20-1': 'The run consisted of 40000 iterations of an adaptively scaling Metropolis within Gibbs algorithm similar to the one implemented in AMCMC.', '0811.4095-1-20-2': 'The running time of Grapham was approximately 1.1 seconds with Intel Pentium 4 at 2.80GHz.', '0811.4095-1-20-3': 'As a comparison, the same run with AMCMC [CITATION] took approximately 3.8 seconds.', '0811.4095-1-20-4': 'The faster simulation speed of Grapham is explained by the hierarchical model setup, which Grapham can take advantage on.', '0811.4095-1-21-0': '# Conclusions and Future Work', '0811.4095-1-22-0': 'Grapham provides a free open-source test bed for evaluating the performance of different adaptive MCMC algorithms with practical distributions.', '0811.4095-1-22-1': 'It provides a relatively simple way of determining models, distributions, and incorporating data into the model using the high-level language Lua.', '0811.4095-1-22-2': 'The speed of the simulation with Grapham is relatively good.', '0811.4095-1-22-3': 'There are number of ready-made density functions and tools for importing the generated samples into R and Matlab.', '0811.4095-1-23-0': 'Grapham is at its present state very experimental.', '0811.4095-1-23-1': 'The current implementation covers a number of proposed adaptive MCMC methods, but there are several variants or new algorithms that are not currently implemented.', '0811.4095-1-23-2': 'In addition to new algorithms, the interface with other computing environments, such as R, could be improved.'}

{'0811.4095-2-0-0': 'This article introduces Grapham, an open-source implementation of several previously proposed adaptive random walk Metropolis algorithms, with emphasis on graphical (hierarchical) models.', '0811.4095-2-0-1': 'The implemented algorithms include the seminal Adaptive Metropolis algorithm adjusting the proposal covariance according to the history of the chain, and an adaptive scaling Metropolis algorithm optimising the expected acceptance probability.', '0811.4095-2-0-2': 'Different variants of the algorithms allow one, e.g., to use these two algorithms together, employ delayed rejection, and adjust several parameters of the algorithms.', '0811.4095-2-0-3': 'The implemented Metropolis-within-Gibbs update allows arbitrary sampling blocks.', '0811.4095-2-0-4': 'The software is written in C, and uses a simple extension language Lua in configuration.', '0811.4095-2-1-0': '# Introduction', '0811.4095-2-2-0': 'Markov chain Monte Carlo (MCMC) is a general framework for computing expectations over complicated distributions in general state spaces.', '0811.4095-2-2-1': 'The methods are based on constructing a Markov chain [MATH] so that the ergodic averages [MATH] converge to [MATH] as [MATH], where [MATH] is the target distribution of interest.', '0811.4095-2-2-2': 'Such a chain is often easy to construct using the Metropolis-Hastings algorithm; see, e.g, [CITATION].', '0811.4095-2-2-3': 'Depending on [MATH], however, it may be difficult to design a practical algorithm so that [MATH] would approximate [MATH] well with moderate number of samples [MATH].', '0811.4095-2-3-0': 'Recently proposed adaptive MCMC algorithms adjust the parameters of the algorithm (the proposal distribution) on-the-fly, aiming to allow efficient simulation.', '0811.4095-2-3-1': 'They have gained increasing interest in the last few years, after [CITATION] presented the seminal Adaptive Metropolis (AM) algorithm, and [CITATION] related adaptive MCMC to the general context of Robbins-Monro stochastic approximation.', '0811.4095-2-3-2': 'After that, several authors have proposed new algorithms and variations, and provided theoretical validation of the methods ; see also the recent review and references therein.', '0811.4095-2-4-0': 'This paper introduces Grapham, a publicly available open-source implementation of several adaptive MCMC algorithms based on the random walk Metropolis sampler.', '0811.4095-2-4-1': 'The purpose of Grapham is to provide an experimental tool for evaluating the performance of such algorithms with practical problems, especially in Bayesian statistics.', '0811.4095-2-4-2': 'The target distribution (model) is specified in Grapham in a graphical (hierarchical) manner, by a set of conditional distributions; see, e.g., [CITATION] and references therein.', '0811.4095-2-4-3': 'The underlying philosophy of Grapham thus reminds that of BUGS ; see also the review of other software for graphical models.', '0811.4095-2-4-4': 'Grapham is, however, an experimental project not offering the versatility and maturity of BUGS, hence not to be considered an "alternative" to it.', '0811.4095-2-5-0': '[CITATION] describes another adaptive MCMC implementation: AMCMC.', '0811.4095-2-5-1': 'It is an R interface to one (acceptance rate optimising) adaptive MCMC algorithm.', '0811.4095-2-5-2': 'Grapham differs from AMCMC in that it relies on a hierarchical model specification, and provides more alternative algorithms.', '0811.4095-2-5-3': 'Unlike AMCMC, Grapham also provides a set of ready-made standard distribution functions the user can employ as a part of their model specification.', '0811.4095-2-5-4': 'This is intended to facilitate faster development, while permitting the user to define arbitrary distributions easily.', '0811.4095-2-6-0': '# Algorithms', '0811.4095-2-7-0': 'The general form of the algorithms Grapham implements can be described as follows.', '0811.4095-2-7-1': 'Let [MATH] be a given starting point for the state chain, and [MATH] and [MATH] stand for the initial scaling parameter and the (lower-diagonal with non-zero diagonal) shape matrix, respectively.', '0811.4095-2-7-2': 'Then, for [MATH], follow the recursion [(S1)]', '0811.4095-2-8-0': 'form a proposal [MATH], where [MATH] is an independent sample from a symmetric proposal distribution [MATH],', '0811.4095-2-9-0': 'with probability [MATH] the proposal accepted and [MATH]; otherwise the proposal is rejected and [MATH], and', '0811.4095-2-10-0': 'update the scaling parameter [MATH] and the shape [MATH] according to the selected adaptive algorithm.', '0811.4095-2-11-0': 'The steps (S[REF]) and (S[REF]) implement an iteration of the random-walk Metropolis algorithm with the proposal distribution [MATH] scaled by the factor [MATH].', '0811.4095-2-11-1': 'Step (S[REF]) implements the adaptation, changing the scaling parameters [MATH] and [MATH] based on the history of the chain.', '0811.4095-2-12-0': 'Instead of applying the recursion (S[REF])-(S[REF]) at once to all the elements of the vector [MATH], one may use Metropolis-within-Gibbs and apply the recursion sequentially to subsets of the elements of [MATH], in the spirit of [CITATION].', '0811.4095-2-12-1': 'These sampling blocks can be selected freely in Grapham.', '0811.4095-2-12-2': 'The proposal distribution [MATH] in (S[REF]) can also be chosen.', '0811.4095-2-12-3': 'Grapham currently implements (multivariate) Gaussian, student, uniform (in a cube), and a ([MATH]-fold product of) Laplace proposal distributions.', '0811.4095-2-13-0': 'The adaptation of (S[REF]) depends on the selected algorithm.', '0811.4095-2-13-1': 'The Adaptive Metropolis (AM) algorithm of [CITATION] implies constant scaling [MATH] for all [MATH].', '0811.4095-2-13-2': 'The shape matrix [MATH] is the Cholesky factor of a covariance estimate of the chain.', '0811.4095-2-13-3': 'In particular, [MATH] with a positive definite [MATH] defined through [EQUATION] with [MATH].', '0811.4095-2-13-4': 'The weight sequence [MATH] can be selected arbitrarily.', '0811.4095-2-13-5': 'In the original setting [MATH], in which case [MATH] and [MATH] coincide with the average, and (asymptotically) the sample covariance of [MATH], respectively.', '0811.4095-2-13-6': 'The updated Cholesky factor [MATH] of [MATH] is computed efficiently from [MATH] by a well-known rank one update, requiring [MATH] operations.', '0811.4095-2-13-7': 'Observe that the same order of operations is needed when forming the proposal [MATH] in (S[REF]).', '0811.4095-2-14-0': 'The adaptive scaling Metropolis (ASCM) algorithm as proposed in', '0811.4095-2-15-0': 'leaves the shape matrix constant [MATH] for all [MATH].', '0811.4095-2-15-1': 'The scaling parameter [MATH] is updated according to the observed acceptance probability.', '0811.4095-2-15-2': 'The default update in Grapham is [EQUATION] where [MATH] is the desired acceptance probability.', '0811.4095-2-15-3': 'The user can, however, supply an alternative, arbitrary update function easily.', '0811.4095-2-16-0': 'These two algorithms, AM and ASCM, can be used simultaneously, as suggested in .', '0811.4095-2-16-1': 'Additional flavors to the algorithms include a Rao-Blackwellised version of AM modifying the update formulae [REF] and [REF] to [EQUATION]', '0811.4095-2-16-2': 'There is a possibility to use (two-stage) delayed rejection (DR) with AM .', '0811.4095-2-16-3': 'DR can also be applied when using ASCM, so that only the first-stage acceptance probability [MATH] is employed in [REF].', '0811.4095-2-17-0': 'Grapham implements three different burn-in strategies for adaptation.', '0811.4095-2-17-1': 'The straightforward "greedy" strategy performs continuous adaptation during the whole MCMC run.', '0811.4095-2-17-2': 'The "traditional" strategy as proposed in uses a fixed proposal for the burn-in, and then performs continuous adaptation during the rest of the simulation.', '0811.4095-2-17-3': 'One may also apply a safe "freeze" strategy adapting only during burn-in, and keeping the obtained parameters fixed during the estimation run.', '0811.4095-2-18-0': 'It is possible to employ a mixture of two proposal density components, a fixed and an adaptive one .', '0811.4095-2-18-1': 'This is implemented in Grapham so that with probability [MATH], the initial parameters [MATH] and [MATH] are used in (S[REF]) instead of the adapted values [MATH] and [MATH].', '0811.4095-2-18-2': 'The user may define also a non-constant mixing probability [MATH].', '0811.4095-2-18-3': 'This feature can be used, e.g., to introduce a "gradual burn-in," by defining a decaying sequence [MATH].', '0811.4095-2-19-0': '# Implementation', '0811.4095-2-20-0': 'Grapham does not have an interactive "user interface."', '0811.4095-2-20-1': 'It is simply executed from the command prompt (shell) with input file names as parameters.', '0811.4095-2-20-2': 'The input files contain the model specification and the simulation parameters.', '0811.4095-2-20-3': 'It is also possible to define the functional of interest in the input files.', '0811.4095-2-20-4': 'For more complicated functionals, however, it may be convenient to store (a subset of) the samples simulated by Grapham, and process them in another environment.', '0811.4095-2-20-5': 'The samples can be saved into a file in the CSV (comma separated values) format or in a simple binary format.', '0811.4095-2-20-6': 'The former allows the samples to be imported easily to many other environments.', '0811.4095-2-20-7': 'There are ready-made functions for loading the binary data files into R , Matlabtextsuperscript (The MathWorks, Natick, Massachusetts), and Octave environments.', '0811.4095-2-21-0': 'The core of Grapham is implemented in C.', '0811.4095-2-21-1': 'It includes some numerical Fortran subroutines from the Netlib repository and can optionally be compiled with the dSFMT random number generator of [CITATION] instead of using the random number generators provided by the C standard libraries.', '0811.4095-2-21-2': 'The configuration of Grapham is done using the small and publicly available extension language Lua .', '0811.4095-2-21-3': 'While minimalistic, Lua is in fact a full-featured programming language offering a great flexibility.', '0811.4095-2-21-4': 'For example, the user can supply functions as configuration parameters, e.g. an alternative to [REF], and to apply data from external files in the model.', '0811.4095-2-21-5': 'In fact, Grapham includes functions written in Lua for reading data files in the CSV format.', '0811.4095-2-21-6': 'The Numeric Lua package can also be compiled with Grapham to allow easy working with vector-valued variables.', '0811.4095-2-22-0': 'There are numerous ready-made distribution functions available for defining the conditional densities associated with the variables.', '0811.4095-2-22-1': 'The densities can also be defined arbitrarily as Lua functions.', '0811.4095-2-22-2': 'Likewise, the functional of interest may be written in Lua.', '0811.4095-2-22-3': 'However, to allow optimal performance, Grapham allows the user to supply densities and functionals in a separate C library with ease.', '0811.4095-2-23-0': '# An Example Session', '0811.4095-2-24-0': 'Consider the baseball example of [CITATION] considered also with AMCMC in .', '0811.4095-2-24-1': 'It consists of a hierarchical model having 20 unknown real-valued variables.', '0811.4095-2-24-2': 'The model is depicted in Figure [REF]', '0811.4095-2-25-0': 'and the model specification in Grapham is given below.', '0811.4095-2-26-0': 'const = v = 0.00434', '0811.4095-2-27-0': 'model = mu = density = "duniform" , t = parents = "mu","a", init_val = 1, density = "dnorm" , y = parents = "t", "v", density = "dnorm" , a = init_val = 1, density = function(a_) return dexp(1/a_, 1/2) end ,', '0811.4095-2-28-0': '_, y = read_csv("models/baseball.data") repeat_block("y","t", y[1]) function functional() return t1, mu, a end para = niter = 30000, nburn = 10000, algorithm = "ascm",', '0811.4095-2-29-0': 'In Grapham, the model is defined in the Lua table model, containing an entry corresponding each variable.', '0811.4095-2-29-1': 'Each entry contains a logarithmic density, conditional on the "parent variables."', '0811.4095-2-29-2': 'The variables [MATH], [MATH], and [MATH] in the example have standard distributions: [MATH] has an (improper) uniform distribution over [MATH] while [MATH] and [MATH] are conditionally Gaussian with means [MATH] and [MATH] and variances [MATH] and [MATH], respectively.', '0811.4095-2-29-3': 'The reciprocal of the variable [MATH] is exponentially distributed; this is defined through a Lua function calling dexp, the exponential distribution function.', '0811.4095-2-29-4': 'The model is, in fact, then modified by the function repeat_block.', '0811.4095-2-29-5': 'The block of variables [MATH] in the model is replicated 18 times to obtain blocks [MATH].', '0811.4095-2-29-6': 'At the same time, the function repeat_block sets the values of [MATH] to the 18 values read from the CSV file baseball.data using the function read_csv.', '0811.4095-2-30-0': 'The following shows an example run of Grapham with the above model specification stored in file models/baseball.lua.', '0811.4095-2-31-0': '[MATH][t_1,,a][MATH]t_1[MATH]^*[MATH] .', '0811.4095-2-31-1': '/grapham models/baseball.lua -e "para.outfile=\'bb.bin\'"', '0811.4095-2-32-0': "This command includes the the chunk of Lua code para.outfile='bb.bin' after reading baseball.lua.", '0811.4095-2-32-1': 'As a consequence, the simulated samples are written in the file bb.bin.', '0811.4095-2-32-2': 'In R, one could write for example', '0811.4095-2-33-0': '> source("tools/grapham_read.', '0811.4095-2-33-1': 'r") > data <- grapham_read("bb.bin", nthin=10) > plot(data[MATH]t1)', '0811.4095-2-34-0': 'which would plot every tenth of the 30000 simulated samples of [MATH] in the same figure.', '0811.4095-2-35-0': '# Conclusions', '0811.4095-2-36-0': 'Grapham provides a flexible open-source test bed for evaluating the performance of different adaptive random walk Metropolis algorithms, especially with hierarchical models often encountered in Bayesian statistics.', '0811.4095-2-36-1': 'It provides a relatively simple and general way of determining models and functionals, and for incorporating data into the model.', '0811.4095-2-36-2': 'The simulation speed of Grapham is good, even in a relatively high-dimensional setting, as the implemented algorithms involve at most a quadratic number of operations with respect to the dimension.', '0811.4095-2-36-3': 'The user has extensive control over the various parameters of the algorithms, facilitating thorough testing of different adaptation strategies.', '0811.4095-2-36-4': 'Moreover, new adaptive algorithms of the similar random walk type can be added to Grapham relatively easily.'}

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['0811.4095-2-20-2', '0811.4095-3-20-2'], ['0811.4095-2-20-3', '0811.4095-3-20-3'], ['0811.4095-2-20-5', '0811.4095-3-20-5'], ['0811.4095-2-29-4', '0811.4095-3-26-4'], ['0811.4095-2-29-5', '0811.4095-3-26-5'], ['0811.4095-2-29-6', '0811.4095-3-26-6'], ['0811.4095-2-2-0', '0811.4095-3-2-0'], ['0811.4095-2-2-1', '0811.4095-3-2-1'], ['0811.4095-2-5-0', '0811.4095-3-5-0'], ['0811.4095-2-5-3', '0811.4095-3-5-3'], ['0811.4095-2-7-1', '0811.4095-3-8-1'], ['0811.4095-2-13-0', '0811.4095-3-14-0'], ['0811.4095-2-13-1', '0811.4095-3-14-1'], ['0811.4095-2-13-2', '0811.4095-3-14-2'], ['0811.4095-2-13-7', '0811.4095-3-14-9'], ['0811.4095-2-11-0', '0811.4095-3-12-0'], ['0811.4095-2-11-1', '0811.4095-3-12-1'], ['0811.4095-2-21-0', '0811.4095-3-21-0'], ['0811.4095-2-21-1', '0811.4095-3-21-1'], ['0811.4095-2-21-2', '0811.4095-3-21-2'], ['0811.4095-2-21-3', '0811.4095-3-21-3'], ['0811.4095-2-21-6', '0811.4095-3-21-6'], ['0811.4095-2-16-2', '0811.4095-3-16-2'], ['0811.4095-2-16-3', '0811.4095-3-16-3'], 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'0811.4095-3-6-0'], ['0811.4095-2-4-4', '0811.4095-3-6-4'], ['0811.4095-2-30-0', '0811.4095-3-25-0'], ['0811.4095-2-30-0', '0811.4095-3-27-0'], ['0811.4095-1-11-2', '0811.4095-2-21-5']]

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[]

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[['0811.4095-1-11-2', '0811.4095-2-21-5']]

['0811.4095-1-10-0', '0811.4095-1-15-0', '0811.4095-1-16-0', '0811.4095-1-17-0', '0811.4095-1-18-0', '0811.4095-2-8-0', '0811.4095-2-14-0', '0811.4095-2-25-0', '0811.4095-2-26-0', '0811.4095-2-27-0', '0811.4095-2-28-0', '0811.4095-2-31-0', '0811.4095-2-31-1', '0811.4095-2-33-0', '0811.4095-2-33-1', '0811.4095-3-9-0', '0811.4095-3-28-0', '0811.4095-3-28-1', '0811.4095-3-28-2', '0811.4095-3-31-0']

{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '3': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/0811.4095

{'0811.4095-3-0-0': 'Recently developed adaptive Markov chain Monte Carlo (MCMC) methods have been applied successfully to many problems in Bayesian statistics.', '0811.4095-3-0-1': 'Grapham is a new open source implementation covering several such methods, with emphasis on graphical models for directed acyclic graphs.', '0811.4095-3-0-2': 'The implemented algorithms include the seminal Adaptive Metropolis algorithm adjusting the proposal covariance according to the history of the chain and a Metropolis algorithm adjusting the proposal scale based on the observed acceptance probability.', '0811.4095-3-0-3': 'Different variants of the algorithms allow one, for example, to use these two algorithms together, employ delayed rejection and adjust several parameters of the algorithms.', '0811.4095-3-0-4': 'The implemented Metropolis-within-Gibbs update allows arbitrary sampling blocks.', '0811.4095-3-0-5': 'The software is written in C and uses a simple extension language Lua in configuration.', '0811.4095-3-1-0': '# Introduction', '0811.4095-3-2-0': 'Markov chain Monte Carlo (MCMC) is a general framework for computing expectations over complicated distributions in general state spaces.', '0811.4095-3-2-1': 'The methods are based on constructing a Markov chain [MATH] so that the ergodic averages [MATH] converge to [MATH] as [MATH], where [MATH] is the target distribution of interest.', '0811.4095-3-2-2': 'Such a chain is often easy to construct using the Metropolis-Hastings algorithm; see, for example, [CITATION].', '0811.4095-3-2-3': 'Depending on [MATH], however, it may be difficult to design a practical algorithm so that [MATH] would approximate [MATH] well with a moderate number of samples [MATH].', '0811.4095-3-3-0': 'Recently proposed adaptive MCMC algorithms adjust the parameters of the algorithm (the proposal distribution) on-the-fly, aiming to allow efficient simulation.', '0811.4095-3-3-1': 'They have attracted increasing attention in the last few years, after [CITATION] presented the seminal Adaptive Metropolis (AM) algorithm, and [CITATION] related adaptive MCMC to the general context of the Robbins-Monro stochastic approximation.', '0811.4095-3-3-2': 'After that, several authors have proposed new algorithms and variations, and provided theoretical validation of the methods ; see also the recent review by [CITATION] and references therein.', '0811.4095-3-4-0': 'Grapham is an open source implementation of several adaptive MCMC algorithms based on the random walk Metropolis sampler.', '0811.4095-3-4-1': 'The purpose of Grapham is to provide an experimental tool for evaluating the performance of such algorithms with practical problems, especially in Bayesian statistics.', '0811.4095-3-4-2': 'The source code of the software and additional documentation are available for downloading in http://iki.fi/mvihola/grapham/.', '0811.4095-3-5-0': '[CITATION] describes another adaptive MCMC implementation: AMCMC.', '0811.4095-3-5-1': "It is an R interface to one adaptive MCMC algorithm (referred to as 'ASCM' in Section [REF] below).", '0811.4095-3-5-2': 'Grapham differs from AMCMC in that it relies on a hierarchical model specification and provides more alternative algorithms.', '0811.4095-3-5-3': 'Unlike AMCMC, Grapham also provides a set of ready-made standard distribution functions the user can employ as a part of their model specification.', '0811.4095-3-5-4': 'This is intended to allow faster development while permitting the user to define arbitrary distributions easily.', '0811.4095-3-6-0': 'The models are specified in Grapham by defining a set of variables with their conditional distributions.', '0811.4095-3-6-1': "Such models are often referred to as 'graphical models'; see, for example, [CITATION] and references therein.", '0811.4095-3-6-2': 'This underlying philosophy of Grapham reminds that of BUGS ; see also the review [CITATION] of other software for graphical models.', '0811.4095-3-6-3': 'The advantage of Grapham over BUGS is that the adaptive MCMC algorithms can be much more efficient than the non-adaptive (Metropolis-within-)Gibbs algorithms of BUGS.', '0811.4095-3-6-4': 'One should, however, notice that Grapham is an experimental project not offering the versatility and maturity of BUGS.', '0811.4095-3-6-5': 'It is also likely that BUGS performs better than Grapham with many simpler models.', '0811.4095-3-7-0': '# Algorithms', '0811.4095-3-8-0': 'The general form of the algorithms implemented in Grapham can be described as follows.', '0811.4095-3-8-1': 'Let [MATH] be a given starting point for the state chain, and [MATH] and [MATH] stand for the initial scaling parameter and the (lower-diagonal with non-zero diagonal) shape matrix, respectively.', '0811.4095-3-8-2': 'For [MATH], the recursion follows: [(S1)]', '0811.4095-3-9-0': 'form a proposal [MATH], where [MATH] is an independent sample from a symmetric proposal distribution [MATH],', '0811.4095-3-10-0': 'with probability [MATH], the proposal is accepted and [MATH]; otherwise, the proposal is rejected and [MATH], and', '0811.4095-3-11-0': 'update the scaling parameter [MATH] and the shape [MATH] according to the selected adaptive algorithm.', '0811.4095-3-12-0': 'The steps (S[REF]) and (S[REF]) implement an iteration of the random-walk Metropolis algorithm with the proposal distribution [MATH] scaled by the factor [MATH].', '0811.4095-3-12-1': 'Step (S[REF]) implements the adaptation, changing the scaling parameters [MATH] and [MATH] based on the history of the chain.', '0811.4095-3-12-2': 'Examples of such updates are given below.', '0811.4095-3-13-0': 'Instead of applying the iteration (S[REF])-(S[REF]) at once to all the elements of the vector [MATH], one may use Metropolis-within-Gibbs and apply the iteration sequentially to subsets of the elements of [MATH], as in the single component AM algorithm suggested by [CITATION].', '0811.4095-3-13-1': 'These sampling blocks can be selected freely in Grapham.', '0811.4095-3-13-2': 'The proposal distribution [MATH] in (S[REF]) can also be chosen.', '0811.4095-3-13-3': 'Grapham currently implements (multivariate) Gaussian, student, uniform (in a cube) and (a [MATH]-fold product of) Laplace proposal distributions.', '0811.4095-3-14-0': 'The adaptation of (S[REF]) depends on the selected algorithm.', '0811.4095-3-14-1': 'The Adaptive Metropolis (AM) algorithm of [CITATION] implies constant scaling [MATH] for all [MATH].', '0811.4095-3-14-2': 'The shape matrix [MATH] is the Cholesky factor of a covariance estimate of the chain.', '0811.4095-3-14-3': 'In particular, [MATH] with a positive definite [MATH] and defined through [EQUATION] with [MATH].', '0811.4095-3-14-4': 'The weight sequence [MATH] can be selected arbitrarily, but it is recommended to choose [MATH] decaying to zero.', '0811.4095-3-14-5': 'For example, setting [MATH] for all [MATH] results in an algorithm similar to the Adaptive Proposal (AP) algorithm .', '0811.4095-3-14-6': 'This algorithm does not, in general, provide valid simulation; see the example in [CITATION].', '0811.4095-3-14-7': 'The original AM algorithm employs the default value [MATH], in which case [MATH] and [MATH] coincide with the average and (asymptotically) the sample covariance of [MATH], respectively.', '0811.4095-3-14-8': 'The updated Cholesky factor [MATH] of [MATH] is computed efficiently from [MATH] by a rank one update requiring [MATH] operations .', '0811.4095-3-14-9': 'Observe that the same order of operations is needed when forming the proposal [MATH] in (S[REF]).', '0811.4095-3-15-0': 'The adaptive scaling Metropolis (ASCM) algorithm as proposed by [CITATION] and [CITATION] leaves the shape matrix constant [MATH] for all [MATH].', '0811.4095-3-15-1': 'The scaling parameter [MATH] is updated according to the observed acceptance probability.', '0811.4095-3-15-2': 'The default update in Grapham is [EQUATION] where [MATH] is the desired acceptance probability.', '0811.4095-3-15-3': 'The default values for [MATH] are [MATH] in dimension one and [MATH] otherwise following [CITATION].', '0811.4095-3-15-4': 'The user can also supply an alternative, arbitrary update function easily, as exemplified in Section [REF].', '0811.4095-3-16-0': 'These two algorithms, AM and ASCM, can be used simultaneously, as suggested in [CITATION] and [CITATION].', '0811.4095-3-16-1': 'Additional flavours to the algorithms include a Rao-Blackwellised version of AM modifying the update formulae [REF] and [REF] to [EQUATION]', '0811.4095-3-16-2': 'There is a possibility to use (two-stage) delayed rejection (DR) with AM .', '0811.4095-3-16-3': 'DR can also be applied when using ASCM, so that only the first-stage acceptance probability [MATH] is employed in [REF].', '0811.4095-3-17-0': 'Grapham implements three different burn-in strategies for adaptation.', '0811.4095-3-17-1': "The default 'greedy' strategy performs continuous adaptation during the whole MCMC run.", '0811.4095-3-17-2': "The 'traditional' strategy as proposed in [CITATION] uses a fixed proposal for the burn-in and then performs continuous adaptation during the rest of the simulation.", '0811.4095-3-17-3': "One may also apply a 'freeze' strategy adapting only during the burn-in and keeping the obtained parameters fixed during the estimation run.", '0811.4095-3-18-0': 'It is possible to employ a mixture of two proposal density components, a fixed and an adaptive one .', '0811.4095-3-18-1': 'This is implemented in Grapham so that, with probability [MATH], the initial parameters [MATH] and [MATH] are used in (S[REF]) instead of the adapted values [MATH] and [MATH].', '0811.4095-3-18-2': 'The user may define also a non-constant mixing probability [MATH].', '0811.4095-3-18-3': "This feature can be used, for example, to introduce a 'gradual burn-in,' by defining a decaying sequence [MATH].", '0811.4095-3-19-0': '# Implementation', '0811.4095-3-20-0': "Grapham does not have an interactive 'user interface.'", '0811.4095-3-20-1': 'It is simply executed from the command prompt (shell) with input file names as parameters.', '0811.4095-3-20-2': 'The input files contain the model specification and the simulation parameters.', '0811.4095-3-20-3': 'It is also possible to define the functional of interest in the input files.', '0811.4095-3-20-4': 'For more complicated functionals, however, it may be convenient to store (a subset of) the samples simulated by Grapham and process them in another environment.', '0811.4095-3-20-5': 'The samples can be saved into a file in the CSV (comma separated values) format or in a simple binary format.', '0811.4095-3-20-6': 'The former allows the samples to be easily imported to many other environments.', '0811.4095-3-20-7': 'There are ready-made functions for loading the binary data files into R , Matlabtextsuperscript (The MathWorks, Natick, Massachusetts) and Octave environments.', '0811.4095-3-21-0': 'The core of Grapham is implemented in C.', '0811.4095-3-21-1': 'It includes some numerical Fortran subroutines from the Netlib repository and can optionally be compiled with the dSFMT random number generator of [CITATION] instead of using the random number generators provided by the C standard libraries.', '0811.4095-3-21-2': 'The configuration of Grapham is done using the small and publicly available extension language Lua .', '0811.4095-3-21-3': 'While minimalistic, Lua is in fact a full-featured programming language offering a great flexibility.', '0811.4095-3-21-4': 'For example, the user can supply functions as configuration parameters and apply data from external files in the model.', '0811.4095-3-21-5': 'In fact, Grapham includes some functions written in Lua, for example for reading data files in the CSV format.', '0811.4095-3-21-6': 'The Numeric Lua package can also be compiled with Grapham to allow easy working with vector-valued variables.', '0811.4095-3-22-0': 'There are numerous ready-made distribution functions available for defining the conditional densities associated with the variables.', '0811.4095-3-22-1': 'The densities can also be defined arbitrarily as Lua functions.', '0811.4095-3-22-2': 'Likewise, the functional of interest may be written in Lua.', '0811.4095-3-22-3': 'However, to allow optimal performance, Grapham allows the user to supply densities and functionals in a separate C library with ease.', '0811.4095-3-23-0': '# An Example Session', '0811.4095-3-24-0': 'Consider the baseball model of [CITATION] used as an example also with AMCMC .', '0811.4095-3-24-1': 'It consists of 38 real-valued variables, defined hierarchically as depicted in Fig. [REF].', '0811.4095-3-25-0': 'The file specifying this model in Grapham is shown in Fig. [REF].', '0811.4095-3-26-0': 'The model is defined in the Lua table model, defined in lines 4-20.', '0811.4095-3-26-1': 'Each variable is defined by an entry containing a logarithmic density, conditional on the parent variables.', '0811.4095-3-26-2': 'The variables [MATH], [MATH] and [MATH] in the example have standard distributions: [MATH] has (an improper) uniform distribution over [MATH], while [MATH] and [MATH] are conditionally Gaussian with means [MATH] and [MATH] and variances [MATH] and [MATH], respectively.', '0811.4095-3-26-3': 'The reciprocal of the variable [MATH] is exponentially distributed; this is defined through a Lua function defined in lines 16-18, calling dexp, the exponential distribution function.', '0811.4095-3-26-4': 'The model is, in fact, then modified by the function repeat_block.', '0811.4095-3-26-5': 'The block of variables [MATH] in the model is replicated 18 times to obtain blocks [MATH].', '0811.4095-3-26-6': 'At the same time, the function repeat_block sets the values of [MATH] to the 18 values read from the CSV file baseball.data using the function read_csv.', '0811.4095-3-27-0': 'The following shows an example run of Grapham with the model specification of Fig. [REF].', '0811.4095-3-28-0': '[MATH][t_1,,a][MATH]t_1[MATH]^*[MATH]t[MATH] .', '0811.4095-3-28-1': '/grapham models/baseball.lua models/amcmc_dr.', '0811.4095-3-28-2': 'lua Functional average = [ 0.392465 0.266204 0.321466 ] Acceptance rates: ( a ): 70.26 (47.55', '0811.4095-3-29-0': 'In this case, the total acceptance rate of each block is around 70%, of which roughly two thirds are accepted in the first stage and one third in the second, delayed rejection stage.', '0811.4095-3-29-1': 'The estimates obtained for [MATH], [MATH] and [MATH] appear similar to the first run.', '0811.4095-3-30-0': 'Finally, to exemplify how the data simulated by Grapham can be used in other environments, let us run Grapham with the command line', '0811.4095-3-31-0': '[MATH]a, data[MATH](a,t_1)[MATH]'}

astro-ph-0509656

{'astro-ph-0509656-1-0-0': 'The presence of a second planet in a known, transiting-planet system will cause the time between transits to vary.', 'astro-ph-0509656-1-0-1': 'These variations can be used to constrain the orbital elements and mass of the perturbing planet.', 'astro-ph-0509656-1-0-2': 'We analyse the set of transit times of the TrES-1 system given in [CITATION].', 'astro-ph-0509656-1-0-3': 'We find no convincing evidence for a second planet in the TrES-1 system from that data.', 'astro-ph-0509656-1-0-4': 'By further analysis, we constrain the mass that a perturbing planet could have as a function of the semi-major axis ratio of the two planets and the eccentricity of the perturbing planet.', 'astro-ph-0509656-1-0-5': 'Near low-order, mean-motion resonances (within [MATH] fractional deviation), we find that a secondary planet must generally have a mass comparable to or less than the mass of the Earth-showing that this data is the first to have sensitivity to sub Earth-mass planets.', 'astro-ph-0509656-1-0-6': 'We compare the sensitivity of this technique to the mass of the perturbing planet with future, high-precision radial velocity measurements.', 'astro-ph-0509656-1-1-0': '# Introduction', 'astro-ph-0509656-1-2-0': 'Since the first discovery, a decade ago, of a planet orbiting a main-sequence star other than the Sun , several search techniques have been employed to identify additional extrasolar planets.', 'astro-ph-0509656-1-2-1': 'These techniques include radial velocity , astrometry , planetary microlensing , and planetary transits .', 'astro-ph-0509656-1-2-2': 'A recent addition to the repertoire of planet search techniques consists of looking for additional planets in a known, transiting system by analyzing the variation in the time between planetary transits.', 'astro-ph-0509656-1-2-3': 'These transit timing variations (TTV) can be used to constrain the orbital elements of an unseen, perturbing planet, even if its mass is comparable to the mass of the Earth .', 'astro-ph-0509656-1-2-4': 'Thus, for the near term, TTV can detect planets around main-sequence stars that are too small to detect by any other means.', 'astro-ph-0509656-1-2-5': 'This sensitivity is particularly manifest near mean-motion resonances, which recent works by [CITATION], [CITATION], and [CITATION] suggest might be very common.', 'astro-ph-0509656-1-3-0': 'The study of extrasolar planetary systems allows us to address fundamental questions about mechanisms of planet formation, the prevalence of small, rocky planets like our Earth, and the evolution of multiple planetary systems.', 'astro-ph-0509656-1-3-1': 'Multiple planet systems are an important subset of extrasolar planets because they give insight into our own solar system.', 'astro-ph-0509656-1-3-2': 'TTV should prove to be an important tool in sampling the population of such planets since it has the ability to discern the presence of bodies with very little mass ([MATH]).', 'astro-ph-0509656-1-4-0': 'The application of TTV depends upon the discovery and monitoring of transiting planetary systems.', 'astro-ph-0509656-1-4-1': 'The first detection of a planetary transit was for the previously known planet, HD209458b .', 'astro-ph-0509656-1-4-2': 'The first extrasolar planet to be discovered from transit data was the OGLE-TR-56b system reported by [CITATION].', 'astro-ph-0509656-1-4-3': 'Existing and future searches for planetary transits, such as the COROT , XO , and Kepler missions, are expected to provide many candidate systems where the timing of the transits can be analysed.', 'astro-ph-0509656-1-5-0': 'The Tr-ES1 planetary system, reported by [CITATION], was also discovered via planetary transits.', 'astro-ph-0509656-1-5-1': 'In a recent paper by [CITATION] (hereafter C05), the detection of thermal emission from the surface of TrES-1b was announced.', 'astro-ph-0509656-1-5-2': 'Table 1 of C05 gives the timing of 12 planetary transits.', 'astro-ph-0509656-1-5-3': 'We analyse the deviations from periodicity in those data in order to identify and constrain potential companion planets in that system.', 'astro-ph-0509656-1-6-0': '# The Data and Software', 'astro-ph-0509656-1-7-0': 'The timing data reported in C05 were derived from the 11 transits reported by [CITATION] with an additional transit that was observed at the IAC 80cm telescope after [CITATION] went to press.', 'astro-ph-0509656-1-7-1': 'One transit was excluded from their analysis because it constituted a 6-[MATH] departure from a constant period and because of anomalous features in the ingress and egress.', 'astro-ph-0509656-1-7-2': 'That point, if it is valid, is the most interesting point for our purposes because the TTV signal is defined by such deviations.', 'astro-ph-0509656-1-7-3': 'Consequently, we analyse two different sets of data from C05; the "12-point" set which includes this point, and the "11-point" set which does not.', 'astro-ph-0509656-1-7-4': 'This study is the first analysis of TTV as presented by [CITATION] and [CITATION].', 'astro-ph-0509656-1-7-5': 'And, it may be the first search for planets around main-sequence stars that can probe masses smaller than the mass of the Earth.', 'astro-ph-0509656-1-8-0': 'To calculate the time of transit, we integrate the equations of motion using the Burlisch-Stoer integration algorithm.', 'astro-ph-0509656-1-8-1': 'When the transiting planet crosses the line of sight, the transit time is identified and tabulated.', 'astro-ph-0509656-1-8-2': 'These simulated transit times are compared to the observed transit times and the quality of the fit of the orbital elements of the simulated system with those of the actual system is characterized by the [MATH] statistic.', 'astro-ph-0509656-1-8-3': 'For this work we restrict our analysis to planets whose orbits are coplanar and edge-on.', 'astro-ph-0509656-1-9-0': 'To locate the set of orbital elements that gives the minimum [MATH], we sequentially employ a random and then a direct minimization algorithm.', 'astro-ph-0509656-1-9-1': 'For the random portion we generate many sets of orbital elements and record the [MATH] for each.', 'astro-ph-0509656-1-9-2': 'All parameters are drawn from a uniform distribution, except for the eccentricity and mass which are logarithmically uniform; between [MATH] and [MATH] for the eccentricity and between [MATH] and [MATH] for the ratio of the planet and stellar masses.', 'astro-ph-0509656-1-9-3': 'We chose the logarithmic distribution for the mass because planetary masses generally span several orders of magnitude and one would expect more small objects orbiting the star than large objects.', 'astro-ph-0509656-1-9-4': 'The logarithmic distribution was chosen for eccentricity because tidal circularization and orbital stability constraints favor small eccentricities ([MATH]).', 'astro-ph-0509656-1-9-5': 'For completeness, we also conducted a study where the eccentricity of the perturbing planet began at zero.', 'astro-ph-0509656-1-10-0': 'We tabulate the two systems with the lowest [MATH] as starting points for the direct minimization algorithm.', 'astro-ph-0509656-1-10-1': 'Experiments with simulated data showed no significant improvement in the final [MATH] when more than two solutions were retained.', 'astro-ph-0509656-1-10-2': 'We assume, following the random minimization, that at least one set of orbital elements is near the global minimum of the [MATH].', 'astro-ph-0509656-1-10-3': 'That is, the nearest local minimum is the global minimum and that the local gradient points toward that minimum.', 'astro-ph-0509656-1-10-4': 'Tests of this approach with simulated data confirm this assumption.', 'astro-ph-0509656-1-10-5': 'From these points we use a direct minimization algorithm to locate the bottom of the global minimum.', 'astro-ph-0509656-1-11-0': '# Results', 'astro-ph-0509656-1-12-0': '## Search for Secondary Planets', 'astro-ph-0509656-1-13-0': 'We conducted a variety of searches for the best-fitting perturbing planet in the TrES-1 system.', 'astro-ph-0509656-1-13-1': 'These searches included different combinations of orbital elements for TrES-1b.', 'astro-ph-0509656-1-13-2': 'We found that any reduction in the overall [MATH] obtained by including the parameters [MATH] and [MATH] for TrES-1b was offset by the loss of a degree of freedom.', 'astro-ph-0509656-1-13-3': 'Therefore, we report results from the search where the eccentricity of TrES-1b was fixed at zero.', 'astro-ph-0509656-1-14-0': 'We stepped through the semi-major axis ratio of the putative secondary planet and TrES-1b.', 'astro-ph-0509656-1-14-1': 'At each point we minimized over six parameters: the eccentricity, longitude of pericentre, time of pericentre passage, and mass of the secondary planet and the period and the initial longitude of TrES-1b.', 'astro-ph-0509656-1-14-2': 'The inclination and ascending node of the perturbing planet were identical to the values for Tr-ES1b.', 'astro-ph-0509656-1-14-3': 'We analysed the data for both interior and exterior perturbers.', 'astro-ph-0509656-1-15-0': 'Our analysis did not produce any promising solution; though we present one interesting case for an interior perturber found from the 12-point analysis.', 'astro-ph-0509656-1-15-1': 'This solution is not near a low-order mean-motion resonance.', 'astro-ph-0509656-1-15-2': 'Indeed, the [MATH] was generally much higher near the [MATH]:[MATH] and the [MATH]:1 resonances than in the regions between them.', 'astro-ph-0509656-1-15-3': 'Figure [REF] compares the timing residuals for this solution with the data.', 'astro-ph-0509656-1-15-4': 'The reduced [MATH] for this system is 2.8 on [MATH] degrees of freedom (where [MATH] is the number of data) compared with 6.3 for no perturber ([MATH]).', 'astro-ph-0509656-1-16-0': 'This solution, while it would have been detected from RV measurements, is interesting because the average size of the timing deviations is larger than that of the data-making the variations easier to detect.', 'astro-ph-0509656-1-16-1': 'However, we suspect that while this solution is numerically valid, it is merely an artifact of the gap between the two primary epochs of observation.', 'astro-ph-0509656-1-16-2': 'For this solution, and others like it, the simulated timing residuals consist of small, short-term variations superposed on a large-amplitude, long-period variation with a period that is a multiple of the difference between the two epochs.', 'astro-ph-0509656-1-16-3': 'Several candiate systems for both the 11-point and the 12-point analyses showed similar behaviour.', 'astro-ph-0509656-1-16-4': 'Additional data, taken at a time that is not commensurate with this period would remove false solutions of this type.', 'astro-ph-0509656-1-17-0': 'From the above analysis, we conclude that there is not sufficient information in the data to uniquely and satisfactorally determine the characteristics of a secondary planet in the TrES-1 system.', 'astro-ph-0509656-1-17-1': 'This is in part because the number of model parameters is not much larger than the number of data and because the typical timing precision, [MATH], is not a sufficiently small fraction of the orbital period of the transiting planet (about [MATH]) to distinguish between different solutions.', 'astro-ph-0509656-1-17-2': 'In addition, the gap in coverage appears to affect the minimization dramatically.', 'astro-ph-0509656-1-17-3': 'We believe that it is primarily responsible for the observed fact that two nearby sets of orbital elements will have very different values of [MATH] -a slight change in the long-term variation will cause the simulated transit times to miss the second epoch of observations.', 'astro-ph-0509656-1-17-4': 'Additional timing data, with precision comparable to the most precise of the given data, [MATH], and at an epoch that is not commensurate with the existing gap in coverage will allow for a more complete investigation of the system.', 'astro-ph-0509656-1-18-0': '## Constraints on Secondary Planets', 'astro-ph-0509656-1-19-0': 'That the results of the planet search were inconclusive is not particularly surprising since the transit timings show little variation from a constant period and the point that deviates the most is suspect.', 'astro-ph-0509656-1-19-1': 'If a satisfactory solution cannot be determined from these data, we can still use them to place constraints on the orbital elements of a two-planet system.', 'astro-ph-0509656-1-19-2': 'Of particular interest is a constraint on the mass that the secondary planet could have as a function of various orbital elements; similar to the constraint that resulted from analyzing the OGLE-1998-BUL-14 microlensing event .', 'astro-ph-0509656-1-19-3': 'As the mass of the hypothetical perturbing planet increases, the [MATH] of the model should grow significantly regardless of the values of the remaining orbital elements.', 'astro-ph-0509656-1-19-4': 'Therefore, we systematically studied a grid of values on the semi-major axis/eccentricity plane of the perturbing planet-assuming that the orbit of TrES-1b is initially circular.', 'astro-ph-0509656-1-19-5': 'At each point we identified the mass that a perturber needs in order to produce a large deviation from the data.', 'astro-ph-0509656-1-20-0': 'At each location in the [MATH]/[MATH] plane we set the mass of the companion to be very small ([MATH]) and, for a random value for the longitude of pericenter and the time of pericenter passage, we calculated the [MATH] of the timing residuals.', 'astro-ph-0509656-1-20-1': 'We increased the mass of the secondary planet until the [MATH] grew by some fiducial amount.', 'astro-ph-0509656-1-20-2': 'At that point we minimized the [MATH] over the longitude of pericentre and the time of pericentre passage of the secondary planet and the initial longitude of TrES-1b.', 'astro-ph-0509656-1-20-3': 'Following the minimization the [MATH] typically fell below the fiducial amount and the mass of the perturber was again increased.', 'astro-ph-0509656-1-20-4': 'However, if after the minimization the [MATH] remained above the threshold, the orbital elements of the system were recorded, the [MATH] threshold increased, and the process repeated until either a maximum mass of the perturbing planet or a maximum [MATH] was achieved.', 'astro-ph-0509656-1-20-5': 'This procedure gives the minimum [MATH] as a function of perturbing planet mass for each point in the [MATH]/[MATH] plane.', 'astro-ph-0509656-1-21-0': 'The mass that corresponds to a 3-[MATH] increase in the [MATH] constitutes our estimate for the maximum allowed mass of a secondary planet.', 'astro-ph-0509656-1-21-1': 'We use the approach outlined by [CITATION] where we locate the mass that causes the difference between the [MATH] of the null hypothesis and the [MATH] obtained with a secondary planet to equal [MATH].', 'astro-ph-0509656-1-21-2': 'This gives the 3-[MATH] limit because we allow only one parameter to vary-the remaining parameters are either fixed or marginalized at each point.', 'astro-ph-0509656-1-21-3': 'This maximum mass is shown as a function of [MATH] and [MATH] for the 11-point set in Figure [REF] for both an interior and an exterior perturber.', 'astro-ph-0509656-1-21-4': 'The contours correspond to 100, 10, and 1 [MATH].', 'astro-ph-0509656-1-21-5': 'The dark portion in the upper-left corners are where the orbits overlap and we assign a mass of [MATH] to those locations.', 'astro-ph-0509656-1-21-6': 'We see from these figures many regions where the mass of a companion must be comparable to or less than the mass of the Earth regardless of its orbital eccentricity.', 'astro-ph-0509656-1-21-7': 'The most stringent constraints are near the [MATH]:[MATH] mean-motion resonances.', 'astro-ph-0509656-1-21-8': 'Of particular interest are the very tight constraints for low-eccentricity perturbers (which constraints continue to an eccentricity of zero) because tidal circularization drives the eccentricity down.', 'astro-ph-0509656-1-21-9': 'The 12-point analysis gave similar results which are not shown.', 'astro-ph-0509656-1-22-0': '## Comparison with Radial Velocity', 'astro-ph-0509656-1-23-0': 'Much of the region where a secondary planet is not tightly constrained by our analysis could be limited more efficiently by radial velocity measurements.', 'astro-ph-0509656-1-23-1': 'Figure [REF] shows the constraint achieved from this TTV analysis and the constraint from a hypothetical RV analysis, with the same number of data, as a function of the period ratio of the two planets.', 'astro-ph-0509656-1-23-2': 'We assume that the RV measurements have a precision of both 5m/s and 1m/s and that the orbit of the perturber can be treated as circular.', 'astro-ph-0509656-1-23-3': 'Figures [REF] and [REF] are the same comparison focussed on the region surrounding the 2:1 and 3:2 mean-motion resonances respectively.', 'astro-ph-0509656-1-24-0': 'We derived the RV limit in these figures by assuming that the velocity residuals that remain after removing the effects of Tr-ES1b randomly sample the phase of the putative second planet and that the noise is uncorrelated with the RV measurements.', 'astro-ph-0509656-1-24-1': 'If no secondary planet exists, then the residuals should surround zero with a variance equal to [MATH] where [MATH] is the precision of the measurements.', 'astro-ph-0509656-1-24-2': 'The expected [MATH] is then equal to [MATH] where [MATH] is the number of model parameters.', 'astro-ph-0509656-1-24-3': "On the other hand, if a secondary planet exists, the expected [MATH] is [EQUATION] where [MATH] is the planet to star mass ratio, [MATH] is the mass of the star, [MATH] is Newton's constant, and [MATH] is the number of model parameters.", 'astro-ph-0509656-1-24-4': 'By subtracting [MATH] from [MATH] and solving for the mass ratio we obtain the mass of a secondary planet that is detectable with a given [MATH] threshold as a function of the period of the planet and the precision of the radial velocity measurements [EQUATION]', 'astro-ph-0509656-1-24-5': 'We see from Figure [REF] that for low-order, mean-motion resonances the TTV analysis is able to place constraints on the mass that are nearly an order-of-magnitude smaller than the RV technique with 5m/s precision and that there are regions where it is more sensitive than RV measurements with 1m/s precision.', 'astro-ph-0509656-1-24-6': 'Meanwhile, in nonresonant regimes the RV method remains superior for a large portion of the parameter space.', 'astro-ph-0509656-1-24-7': 'Additional transit timing data, particularly with precision that is comparable to the most precise of the given data, would lower the entire limit from TTV, provided that no secondary planet exists.', 'astro-ph-0509656-1-24-8': 'Such data would render the TTV approach superior over a larger range of periods.', 'astro-ph-0509656-1-25-0': '# Discussion', 'astro-ph-0509656-1-26-0': 'Ultimately a combined analysis of all available data, including studies of the stability of candidate systems, will provide the most robust and sensitive determination of the parameters of each planetary system.', 'astro-ph-0509656-1-26-1': 'None of the planetary systems that compose the limits in Figures [REF] and [REF] below [MATH] are stable for more than [MATH] orbits; though stable orbits with comparable masses, periods, and eccentricities exist.', 'astro-ph-0509656-1-26-2': 'An overall stability analysis to accompany these TTV analyses was prohibitively expensive.', 'astro-ph-0509656-1-26-3': 'We estimate that for Figure [REF] there were [MATH] potential systems that were analysed and [MATH] systems were analysed for each curve in Figures [REF] and [REF]-requiring a total of [MATH] hours of processor time.', 'astro-ph-0509656-1-26-4': 'General stability analyses are often better suited to confirm or study a particular candidate system (e.g. [CITATION]) than to constrain candidate systems in the manner that we have done in this section.', 'astro-ph-0509656-1-26-5': 'We did conduct a stability analysis for the system shown in Figure [REF] and found it to be unstable within [MATH] orbits; no neighboring stable systems of comparable [MATH] were found.', 'astro-ph-0509656-1-27-0': 'It is unclear what fraction of probable planetary companions are excluded by our results.', 'astro-ph-0509656-1-27-1': 'Aside from the fact that many of the known, multi-planetary systems are in mean-motion orbital resonance (e.g. GJ 876), recent works by [CITATION], [CITATION], and [CITATION] show that, under fairly general initial conditions, small planets are readily trapped in low-order, mean-motion resonances with a gas-giant as it interacts with a protoplanetary disk and migrates inward.', 'astro-ph-0509656-1-27-2': 'These results may imply that resonant systems are common among multiple-planet systems.', 'astro-ph-0509656-1-27-3': 'If this is true, then results like Figure [REF] actually exclude a much larger fraction of the probable orbits than one might infer from the projection of the excluded regions onto the [MATH]/[MATH] plane.', 'astro-ph-0509656-1-27-4': 'In addition, [CITATION] show that a large fraction of existing, terrestrial planets can survive the migration of a Jupiter-mass planet, though only a fraction of the survivors will be in resonance.', 'astro-ph-0509656-1-28-0': 'The sensitivity of TTV to the mass of a perturbing planet renders it ideal for discovering and constraining the presence of additional planets in transiting systems like TrES-1.', 'astro-ph-0509656-1-28-1': 'These studies can help determine the ubiquity of multiple planet systems and resonant systems-including the distribution of mass in those systems.', 'astro-ph-0509656-1-28-2': 'Moreover, TTV analyses of several systems can play a role in identifying the importance of various planet-formation mechanisms.', 'astro-ph-0509656-1-28-3': 'For example, the presence of close-in terrestrial planets favors a sequential-accretion model of planet formation over a gravitational instability model .', 'astro-ph-0509656-1-29-0': 'For the case of Tr-ES1, while a companion planet with a mass larger than the Earth is ruled out in a portion of the available parameter space, there remain large regions where additional planets could reside; this includes mutually inclined orbits which were not studied in this work (we believe that these results are valid provided that the angle of mutual inclination [MATH] satisfies [MATH]).', 'astro-ph-0509656-1-29-1': 'The 11 timings analysed in this work, excepting the questionable "12[MATH]" point, deviate from a constant period by an amount that is difficult to interpret; having a [MATH] of about 2 per degree of freedom.', 'astro-ph-0509656-1-29-2': 'Additional observations, with higher precision, will allow us to more fully constrain the system and to interpret the existing transit timing variations.'}

{'astro-ph-0509656-2-0-0': 'The presence of a second planet in a known, transiting-planet system will cause the time between transits to vary.', 'astro-ph-0509656-2-0-1': 'These variations can be used to constrain the orbital elements and mass of the perturbing planet.', 'astro-ph-0509656-2-0-2': 'We analyse the set of transit times of the TrES-1 system given in [CITATION].', 'astro-ph-0509656-2-0-3': 'We find no convincing evidence for a second planet in the TrES-1 system from that data.', 'astro-ph-0509656-2-0-4': 'By further analysis, we constrain the mass that a perturbing planet could have as a function of the semi-major axis ratio of the two planets and the eccentricity of the perturbing planet.', 'astro-ph-0509656-2-0-5': 'Near low-order, mean-motion resonances (within [MATH] fractional deviation), we find that a secondary planet must generally have a mass comparable to or less than the mass of the Earth-showing that this data is the first to have sensitivity to sub Earth-mass planets.', 'astro-ph-0509656-2-0-6': 'We compare the sensitivity of this technique to the mass of the perturbing planet with future, high-precision radial velocity measurements.', 'astro-ph-0509656-2-1-0': '# Introduction', 'astro-ph-0509656-2-2-0': 'Since the first discovery, a decade ago, of a planet orbiting a distant main-sequence star , several search techniques have been employed to identify additional extrasolar planets.', 'astro-ph-0509656-2-2-1': 'These techniques include radial velocity , astrometry , planetary microlensing , and planetary transits .', 'astro-ph-0509656-2-2-2': 'A recent addition to the repertoire of planet search techniques consists of looking for additional planets in a known, transiting system by analyzing the variation in the time between planetary transits.', 'astro-ph-0509656-2-2-3': 'These transit timing variations (TTV) can be used to constrain the orbital elements of an unseen, perturbing planet, even if its mass is comparable to the mass of the Earth .', 'astro-ph-0509656-2-2-4': 'Thus, for the near term, TTV can detect planets around main-sequence stars that are too small to detect by any other means.', 'astro-ph-0509656-2-2-5': 'This sensitivity is particularly manifest near mean-motion resonances, which recent works by [CITATION], [CITATION], and [CITATION] suggest might be very common.', 'astro-ph-0509656-2-3-0': 'The study of extrasolar planetary systems allows us to address fundamental questions about mechanisms of planet formation, the prevalence of small, rocky planets like our Earth, and the evolution of multiple planetary systems.', 'astro-ph-0509656-2-3-1': 'Multiple planet systems are an important subset of extrasolar planets because they give insight into our own solar system.', 'astro-ph-0509656-2-3-2': 'TTV should prove to be an important tool in sampling the population of such planets since it has the ability to discern the presence of bodies with very little mass ([MATH]).', 'astro-ph-0509656-2-4-0': 'The application of TTV depends upon the discovery and monitoring of transiting planetary systems.', 'astro-ph-0509656-2-4-1': 'The first successful detection of a planetary transit was for a planet identified spectroscopically, HD209458b .', 'astro-ph-0509656-2-4-2': 'The first extrasolar planet to be discovered from transit data was the OGLE-TR-56b system reported by [CITATION].', 'astro-ph-0509656-2-4-3': 'Existing and future searches for planetary transits, such as the COROT , XO , and Kepler missions, are expected to provide many candidate systems where the timing of the transits can be analysed.', 'astro-ph-0509656-2-5-0': 'The TrES-1 planetary system, reported by [CITATION], was also discovered via planetary transits.', 'astro-ph-0509656-2-5-1': 'In a recent paper by [CITATION] (hereafter C05), the detection of thermal emission from the surface of TrES-1b was announced.', 'astro-ph-0509656-2-5-2': 'Table 1 of C05 gives the timing of 12 planetary transits.', 'astro-ph-0509656-2-5-3': 'We analyse the deviations from periodicity in those data in order to identify and constrain potential companion planets in that system.', 'astro-ph-0509656-2-6-0': '# The Data and Software', 'astro-ph-0509656-2-7-0': 'The timing data reported in C05 were derived from the 11 transits reported by [CITATION] with an additional transit that was observed at the IAC 80cm telescope after [CITATION] went to press.', 'astro-ph-0509656-2-7-1': 'One transit was excluded from their analysis because it constituted a 6-[MATH] departure from a constant period and because of anomalous features in the ingress and egress.', 'astro-ph-0509656-2-7-2': 'That point, if it is valid, is the most interesting point for our purposes because the TTV signal is defined by such deviations.', 'astro-ph-0509656-2-7-3': 'Consequently, we analyse two different sets of data from C05; the "12-point" set which includes this point, and the "11-point" set which does not.', 'astro-ph-0509656-2-7-4': 'This study is the first analysis of TTV as presented by [CITATION] and [CITATION].', 'astro-ph-0509656-2-7-5': 'And, it may be the first search for planets around main-sequence stars that can probe masses smaller than the mass of the Earth.', 'astro-ph-0509656-2-8-0': 'To calculate the time of transit, we integrate the equations of motion using the Burlisch-Stoer integration algorithm.', 'astro-ph-0509656-2-8-1': 'When the transiting planet crosses the line of sight, the transit time is identified and tabulated.', 'astro-ph-0509656-2-8-2': 'These simulated transit times are compared to the observed transit times and the quality of the fit of the orbital elements of the simulated system with those of the actual system is characterized by the [MATH] statistic.', 'astro-ph-0509656-2-8-3': 'For this work we restrict our analysis to planets whose orbits are coplanar and edge-on.', 'astro-ph-0509656-2-9-0': 'To locate the set of orbital elements that gives the minimum [MATH], we sequentially employ a random and then a direct minimization algorithm.', 'astro-ph-0509656-2-9-1': 'For the random portion we generate many sets of orbital elements and record the [MATH] for each.', 'astro-ph-0509656-2-9-2': 'All parameters are drawn from a uniform distribution, except for the eccentricity and mass which are logarithmically uniform; between [MATH] and [MATH] for the eccentricity and between [MATH] and [MATH] for the ratio of the planet to stellar mass.', 'astro-ph-0509656-2-9-3': 'We chose the logarithmic distribution so as to not assume a mass scale a priori-our limits corresponding approximately to masses too small to be seen and too large to have not been seen.', 'astro-ph-0509656-2-9-4': 'This approach agrees with the empirical distribution of masses determined by [CITATION].', 'astro-ph-0509656-2-9-5': 'The logarithmic distribution was chosen for eccentricity because tidal circularization and orbital stability constraints favor small eccentricities ([MATH]).', 'astro-ph-0509656-2-9-6': 'For completeness, we also conducted a study where the eccentricity of the perturbing planet began at zero.', 'astro-ph-0509656-2-10-0': 'We tabulate the two systems with the lowest [MATH] as starting points for the direct minimization algorithm.', 'astro-ph-0509656-2-10-1': 'Experiments with simulated data showed no significant improvement in the final [MATH] when more than two solutions were retained.', 'astro-ph-0509656-2-10-2': 'We assume, following the random minimization, that at least one set of orbital elements is near the global minimum of the [MATH].', 'astro-ph-0509656-2-10-3': 'That is, the nearest local minimum is the global minimum and that the local gradient points toward that minimum.', 'astro-ph-0509656-2-10-4': 'Tests of this approach with simulated data confirm this assumption.', 'astro-ph-0509656-2-10-5': 'From these points we use a direct minimization algorithm to locate the bottom of the global minimum.', 'astro-ph-0509656-2-11-0': '# Results', 'astro-ph-0509656-2-12-0': '## Search for Secondary Planets', 'astro-ph-0509656-2-13-0': 'We conducted a variety of searches for the best-fitting perturbing planet in the TrES-1 system.', 'astro-ph-0509656-2-13-1': 'These searches included different combinations of orbital elements for TrES-1b.', 'astro-ph-0509656-2-13-2': 'We found that any reduction in the overall [MATH] obtained by including the parameters [MATH] and [MATH] for TrES-1b was offset by the loss of a degree of freedom.', 'astro-ph-0509656-2-13-3': 'Therefore, we report results from the search where the eccentricity of TrES-1b was fixed at zero.', 'astro-ph-0509656-2-14-0': 'We stepped through the semi-major axis ratio of the putative secondary planet and TrES-1b.', 'astro-ph-0509656-2-14-1': 'At each point we minimized over six parameters: the eccentricity, longitude of pericentre, time of pericentre passage, and mass of the secondary planet and the period and the initial longitude of TrES-1b.', 'astro-ph-0509656-2-14-2': 'The inclination and ascending node of the perturbing planet were identical to the values for TrES-1b.', 'astro-ph-0509656-2-14-3': 'We analysed the data for both interior and exterior perturbers.', 'astro-ph-0509656-2-15-0': 'Our analysis did not produce any promising solution; though we present one interesting case for an interior perturber found from the 12-point analysis.', 'astro-ph-0509656-2-15-1': 'This solution is not near a low-order mean-motion resonance.', 'astro-ph-0509656-2-15-2': 'Indeed, the [MATH] was generally much higher near the [MATH]:[MATH] and the [MATH]:1 resonances than in the regions between them.', 'astro-ph-0509656-2-15-3': 'Figure [REF] compares the timing residuals for this solution with the data.', 'astro-ph-0509656-2-15-4': 'The reduced [MATH] for this system is 2.8 on [MATH] degrees of freedom (where [MATH] is the number of data) compared with 6.3 for no perturber ([MATH]).', 'astro-ph-0509656-2-16-0': 'This solution, while it would have been detected from RV measurements, is interesting because the average size of the timing deviations is larger than that of the data-making the variations easier to detect.', 'astro-ph-0509656-2-16-1': 'However, we suspect that while this solution is numerically valid, it is merely an artifact of the gap between the two primary epochs of observation.', 'astro-ph-0509656-2-16-2': 'For this solution, and others like it, the simulated timing residuals consist of small, short-term variations superposed on a large-amplitude, long-period variation with a period that is a multiple of the difference between the two epochs.', 'astro-ph-0509656-2-16-3': 'Several candiate systems for both the 11-point and the 12-point analyses showed similar behaviour.', 'astro-ph-0509656-2-16-4': 'Additional data, taken at a time that is not commensurate with this period would remove false solutions of this type.', 'astro-ph-0509656-2-17-0': 'From the above analysis, we conclude that there is not sufficient information in the data to uniquely and satisfactorally determine the characteristics of a secondary planet in the TrES-1 system.', 'astro-ph-0509656-2-17-1': 'This is in part because the number of model parameters is not much larger than the number of data and because the typical timing precision, [MATH], is not a sufficiently small fraction of the orbital period of the transiting planet (about [MATH]) to distinguish between different solutions.', 'astro-ph-0509656-2-17-2': 'In addition, the gap in coverage appears to affect the minimization dramatically.', 'astro-ph-0509656-2-17-3': 'We believe that it is primarily responsible for the observed fact that two nearby sets of orbital elements will have very different values of [MATH] -a slight change in the long-term variation will cause the simulated transit times to miss the second epoch of observations.', 'astro-ph-0509656-2-17-4': 'Additional timing data, with precision comparable to the most precise of the given data, [MATH], and at an epoch that is not commensurate with the existing gap in coverage will allow for a more complete investigation of the system.', 'astro-ph-0509656-2-18-0': '## Constraints on Secondary Planets', 'astro-ph-0509656-2-19-0': 'That the results of the planet search were inconclusive is not particularly surprising since the transit timings show little variation from a constant period and the point that deviates the most is suspect.', 'astro-ph-0509656-2-19-1': 'If a satisfactory solution cannot be determined from these data, we can still use them to place constraints on the orbital elements of a two-planet system.', 'astro-ph-0509656-2-19-2': 'Of particular interest is a constraint on the mass that the secondary planet could have as a function of various orbital elements; similar to the constraint that resulted from analyzing the OGLE-1998-BUL-14 microlensing event .', 'astro-ph-0509656-2-19-3': 'As the mass of the hypothetical perturbing planet increases, the [MATH] of the model should grow significantly regardless of the values of the remaining orbital elements.', 'astro-ph-0509656-2-19-4': 'Therefore, we systematically studied a grid of values on the semi-major axis/eccentricity plane of the perturbing planet-assuming that the orbit of TrES-1b is initially circular.', 'astro-ph-0509656-2-19-5': 'At each point we identified the mass that a perturber needs in order to produce a large deviation from the data.', 'astro-ph-0509656-2-20-0': 'At each location in the [MATH]/[MATH] plane we set the mass of the companion to be very small ([MATH]) and, for a random value for the longitude of pericenter and the time of pericenter passage, we calculated the [MATH] of the timing residuals.', 'astro-ph-0509656-2-20-1': 'We increased the mass of the secondary planet until the [MATH] grew by some fiducial amount.', 'astro-ph-0509656-2-20-2': 'At that point we minimized the [MATH] over the longitude of pericentre and the time of pericentre passage of the secondary planet and the initial longitude of TrES-1b.', 'astro-ph-0509656-2-20-3': 'Following the minimization the [MATH] typically fell below the fiducial amount and the mass of the perturber was again increased.', 'astro-ph-0509656-2-20-4': 'However, if after the minimization the [MATH] remained above the threshold, the orbital elements of the system were recorded, the [MATH] threshold increased, and the process repeated until either a maximum mass of the perturbing planet or a maximum [MATH] was achieved.', 'astro-ph-0509656-2-20-5': 'This procedure gives the minimum [MATH] as a function of perturbing planet mass for each point in the [MATH]/[MATH] plane.', 'astro-ph-0509656-2-21-0': 'The mass that corresponds to a 3-[MATH] increase in the [MATH] constitutes our estimate for the maximum allowed mass of a secondary planet.', 'astro-ph-0509656-2-21-1': 'We use the approach outlined by [CITATION] where we locate the mass that causes the difference between the [MATH] of the null hypothesis and the [MATH] obtained with a secondary planet to equal [MATH].', 'astro-ph-0509656-2-21-2': 'This gives the 3-[MATH] limit because we allow only one parameter to vary-the remaining parameters are either fixed or marginalized at each point.', 'astro-ph-0509656-2-21-3': 'This maximum mass is shown as a function of [MATH] and [MATH] for the 11-point set in Figure [REF] for both an interior and an exterior perturber.', 'astro-ph-0509656-2-21-4': 'The contours correspond to 100, 10, and 1 [MATH].', 'astro-ph-0509656-2-21-5': 'The dark portion in the upper-left corners are where the orbits overlap and we assign a mass of [MATH] to those locations.', 'astro-ph-0509656-2-21-6': 'We see from these figures many regions where the mass of a companion must be comparable to or less than the mass of the Earth regardless of its orbital eccentricity.', 'astro-ph-0509656-2-21-7': 'The most stringent constraints are near the [MATH]:[MATH] mean-motion resonances.', 'astro-ph-0509656-2-21-8': 'Of particular interest are the very tight constraints for low-eccentricity perturbers (which constraints continue to an eccentricity of zero) because tidal circularization drives the eccentricity down.', 'astro-ph-0509656-2-21-9': 'The 12-point analysis gave similar results which are not shown.', 'astro-ph-0509656-2-22-0': '## Comparison with Radial Velocity', 'astro-ph-0509656-2-23-0': 'Much of the region where a secondary planet is not tightly constrained by our analysis could be limited more efficiently by radial velocity measurements.', 'astro-ph-0509656-2-23-1': 'Figure [REF] shows the constraint achieved from this TTV analysis and the constraint from a hypothetical RV analysis, with the same number of data, as a function of the period ratio of the two planets.', 'astro-ph-0509656-2-23-2': 'We assume that the RV measurements have a precision of both 5 and 1 m s[MATH] and that the orbit of the perturber can be treated as circular.', 'astro-ph-0509656-2-23-3': 'Figures [REF] and [REF] are the same comparison focussed on the region surrounding the 2:1 and 3:2 mean-motion resonances respectively.', 'astro-ph-0509656-2-24-0': 'We derived the RV limit in these figures by assuming that the velocity residuals that remain after removing the effects of TrES-1b randomly sample the phase of the putative second planet and that the noise is uncorrelated with the RV measurements.', 'astro-ph-0509656-2-24-1': 'If no secondary planet exists, then the residuals should surround zero with a variance equal to [MATH] where [MATH] is the precision of the measurements.', 'astro-ph-0509656-2-24-2': 'The expected [MATH] is then equal to [MATH] where [MATH] is the number of model parameters.', 'astro-ph-0509656-2-24-3': "On the other hand, if a secondary planet exists, the expected [MATH] is [EQUATION] where [MATH] is the planet to star mass ratio, [MATH] is the mass of the star, [MATH] is Newton's constant, and [MATH] is the number of model parameters.", 'astro-ph-0509656-2-24-4': 'By subtracting [MATH] from [MATH] and solving for the mass ratio we obtain the mass of a secondary planet that is detectable with a given [MATH] threshold as a function of the period of the planet and the precision of the radial velocity measurements [EQUATION]', 'astro-ph-0509656-2-24-5': 'We see from Figure [REF] that for low-order, mean-motion resonances the TTV analysis is able to place constraints on the mass that are nearly an order-of-magnitude smaller than the RV technique with 5 m s[MATH] precision and that there are regions where it is more sensitive than RV measurements with 1 m s[MATH] precision.', 'astro-ph-0509656-2-24-6': 'Meanwhile, in nonresonant regimes the RV method remains superior for a large portion of the parameter space.', 'astro-ph-0509656-2-24-7': 'Additional transit timing data, particularly with precision that is comparable to the most precise of the given data, would lower the entire limit from TTV, provided that no secondary planet exists.', 'astro-ph-0509656-2-24-8': 'Such data would render the TTV approach superior over a larger range of periods.', 'astro-ph-0509656-2-25-0': '# Discussion', 'astro-ph-0509656-2-26-0': 'Ultimately a combined analysis of all available data, including studies of the stability of candidate systems, will provide the most robust and sensitive determination of the parameters of each planetary system.', 'astro-ph-0509656-2-26-1': 'None of the planetary systems that compose the limits in Figures [REF] and [REF] below [MATH] are stable for more than [MATH] orbits; though stable orbits with comparable masses, periods, and eccentricities exist.', 'astro-ph-0509656-2-26-2': 'An overall stability analysis to accompany these TTV analyses was prohibitively expensive.', 'astro-ph-0509656-2-26-3': 'We estimate that for Figure [REF] there were [MATH] potential systems that were analysed and [MATH] systems were analysed for each curve in Figures [REF] and [REF]-requiring a total of [MATH] hours of processor time.', 'astro-ph-0509656-2-26-4': 'General stability analyses are often better suited to confirm or study a particular candidate system (e.g. [CITATION]) than to constrain candidate systems in the manner that we have done in this section.', 'astro-ph-0509656-2-26-5': 'We did conduct a stability analysis for the system shown in Figure [REF] and found it to be unstable within [MATH] orbits; no neighboring stable systems of comparable [MATH] were found.', 'astro-ph-0509656-2-27-0': 'It is unclear what fraction of probable planetary companions are excluded by our results.', 'astro-ph-0509656-2-27-1': 'Aside from the fact that many of the known, multi-planetary systems are in mean-motion orbital resonance (e.g. GJ 876), recent works by [CITATION], [CITATION], and [CITATION] show that, under fairly general initial conditions, small planets are readily trapped in low-order, mean-motion resonances with a gas-giant as it interacts with a protoplanetary disk and migrates inward.', 'astro-ph-0509656-2-27-2': 'These results may imply that resonant systems are common among multiple-planet systems.', 'astro-ph-0509656-2-27-3': 'If this is true, then results like Figure [REF] actually exclude a much larger fraction of the probable orbits than one might infer from the projection of the excluded regions onto the [MATH]/[MATH] plane.', 'astro-ph-0509656-2-27-4': 'In addition, [CITATION] show that a large fraction of existing, terrestrial planets can survive the migration of a Jupiter-mass planet, though only a fraction of the survivors will be in resonance.', 'astro-ph-0509656-2-28-0': 'The sensitivity of TTV to the mass of a perturbing planet renders it ideal for discovering and constraining the presence of additional planets in transiting systems like TrES-1.', 'astro-ph-0509656-2-28-1': 'These studies can help determine the ubiquity of multiple planet systems and resonant systems-including the distribution of mass in those systems.', 'astro-ph-0509656-2-28-2': 'Moreover, TTV analyses of several systems can play a role in identifying the importance of various planet-formation mechanisms.', 'astro-ph-0509656-2-28-3': 'For example, the presence of close-in terrestrial planets favors a sequential-accretion model of planet formation over a gravitational instability model .', 'astro-ph-0509656-2-29-0': 'For the case of TrES-1, while a companion planet with a mass larger than the Earth is ruled out in a portion of the available parameter space, there remain large regions where additional planets could reside; this includes mutually inclined orbits which were not studied in this work (we believe that these results are valid provided that the angle of mutual inclination [MATH] satisfies [MATH]).', 'astro-ph-0509656-2-29-1': 'The 11 timings analysed in this work, excepting the questionable "12[MATH]" point, deviate from a constant period by an amount that is difficult to interpret; having a [MATH] of about 2 per degree of freedom.', 'astro-ph-0509656-2-29-2': 'Additional observations, with higher precision, will allow us to more fully constrain the system and to interpret the existing transit timing variations.'}

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[]

[['astro-ph-0509656-1-4-1', 'astro-ph-0509656-2-4-1'], ['astro-ph-0509656-1-9-3', 'astro-ph-0509656-2-9-3']]

[]

[]

{'1': 'http://arxiv.org/licenses/assumed-1991-2003/', '2': 'http://arxiv.org/licenses/assumed-1991-2003/'}

https://arxiv.org/abs/astro-ph/0509656

1811.04339

{'1811.04339-1-0-0': 'Very large-scale motions (VLSM) and large-scale motions (LSM) coexist at moderate Reynolds numbers (e.g. [MATH]) in a very long open channel flow.', '1811.04339-1-0-1': 'Direct numerical simulations two-way coupled with inertial particles are analysed using spectral information to investigate the modulation of VLSMs.', '1811.04339-1-0-2': 'Particle distributions show two layers corresponding to LSMs and VLSMs in the wall-normal direction.', '1811.04339-1-0-3': "This results in particle inertia's non-monotonic effects on the VLSMs: low and high inertia both strengthen the VLSM whereas moderate and very high inertia have very little influence.", '1811.04339-1-0-4': 'Through conditional tests, low and high inertia particles enhance VLSMs following two distinct paths.', '1811.04339-1-0-5': 'Low inertial particles promote VLSMs indirectly through the enhancement of the regeneration cycle (LSMs) in the inner region whereas high inertial particles enhance the VLSM directly through contribution to the Reynolds shear stress (upwelling/downwelling fluid motion) at similar temporal scales in the outer region.', '1811.04339-1-0-6': 'This understanding also provides more general insight into inner-outer interaction in high Reynolds number, wall-bounded flows.', '1811.04339-1-1-0': '# Introduction', '1811.04339-1-2-0': 'Very large-scale motions (VLSMs) extending to over [MATH] (where [MATH] is the boundary layer thickness) are found in very high Reynolds number, wall-bounded turbulent flows and are distinct from the wall-understood large-scale motions (LSMs) which form canonical streaks and hairpin vortices .', '1811.04339-1-2-1': 'These long, meandering features are observed to be energetic, carrying [MATH] of the kinetic energy and [MATH] Reynolds shear stress in pipe flow , which is contradictory to the notion of "inactive" motion proposed by [CITATION].', '1811.04339-1-2-2': 'In environmental flows, these anisotropic structures also have significant influence on the dispersion of pollutants, sand, and other constituents.', '1811.04339-1-2-3': 'At the same time, understanding the modulation of turbulence by inertial particles is itself a formidable challenge , and nearly all numerical studies of two-way coupling in particle-laden wall turbulence have been restricted to low Reynolds numbers.', '1811.04339-1-2-4': 'It is therefore the aim of this investigation to study the effects of particles on VLSMs, in particular focusing on the question of whether particles act directly or indirectly on these large motions.', '1811.04339-1-3-0': 'The importance of LSMs to the flow dynamics in the near-wall region has been demonstrated in many different contexts.', '1811.04339-1-3-1': 'The LSMs are found to follow a self-sustaining process (i.e. a regeneration cycle) characterized by three key structures shown in the lower left of figure [REF]: large-scale streaks (LSSs), large-scale vortices (LSVs), and meandering streaks.', '1811.04339-1-3-2': 'Associated with these structures are three regeneration processes: streak formation, stream breakdown, and vortex regeneration .', '1811.04339-1-3-3': 'The typical scales of VLSMs, meanwhile, are far larger than LSMs, with their spanwise wavelength [MATH] and their streamwise wavelength [MATH] .', '1811.04339-1-3-4': 'These structures co-exist with LSMs, and the interaction between them is still an open question.', '1811.04339-1-3-5': 'It is generally accepted that the regeneration cycle of LSMs does not require the existence of VLSMs , and [CITATION] propose that the formation of LSMs and VLSMs results from different mechanisms.', '1811.04339-1-3-6': '[CITATION] argue that VLSMs are self-sustained and do not draw energy from LSMs in buffer layer, however, [CITATION] and [CITATION] suggest that VLSMs are not a new type of turbulent structure but merely the consequence of the alignment of coherent LSMs.', '1811.04339-1-3-7': '[CITATION] show numerically that LSMs and VLSMs interact in a co-supporting cycle.', '1811.04339-1-3-8': 'In the current work, we find that the enhancement of VLSMs can be caused by the promotion of LSMs via inertial particles.', '1811.04339-1-4-0': 'As indicated in figure [REF], particles can directly impact velocity fluctuations and turbulent kinetic energy through momentum coupling .', '1811.04339-1-4-1': 'This yields the possibility that particle feedback on structures near the wall (e.g. LSMs, see [CITATION]) can have upscale, indirect influences on large turbulent structures (e.g. VLSMs, see [CITATION]) via nonlinear energy transfer.', '1811.04339-1-4-2': 'For inertial particles, [CITATION] and [CITATION] use numerical simulations to demonstrate that this upscale influence is a strong function of particle inertia, and as a result, particles can influence turbulence scales far removed from their own response time scale.', '1811.04339-1-4-3': 'To better understand this indirect modulation, [CITATION] further investigated small particles and their ability to enhance the LSM regeneration cycle (depending non-monotonically on particle inertia, see also [CITATION]), with the assumption that this was a route through which particles could modify even larger scales in high Reynolds number flow.', '1811.04339-1-4-4': 'Therefore as a follow-up, and since to date computational costs have precluded particle-laden direct numerical simulations at sufficiently high Reynolds number, we for the first time examine the effects of a wide range of particle inertia on VLSMs in open channel flow at Reynolds numbers of [MATH] and [MATH].', '1811.04339-1-5-0': '# Numerical parameters', '1811.04339-1-6-0': 'Direct numerical simulations of the Eulerian flow are performed for an incompressible Newtonian fluid using the same numerical implementation as [CITATION] and [CITATION].', '1811.04339-1-6-1': 'A pseudospectral method is employed in the periodic directions (streamwise [MATH] and spanwise [MATH]), and second-order finite differences are used for spatial discretization in the wall-normal ([MATH]) direction.', '1811.04339-1-6-2': 'We simulate pressure-driven open channel flow because it is characterized by features similar to closed channel flow, while also exhibiting the signatures of VLSMs at a lower, more computationally accessible Reynolds numbers .', '1811.04339-1-6-3': 'A no-slip condition is imposed on the bottom wall and a shear-free condition is imposed on the upper surface.', '1811.04339-1-6-4': 'The solution is advanced in time by a third-order Runge-Kutta scheme.', '1811.04339-1-6-5': 'A single-sided stretched grid (fine grid close to the wall, coarse grid close to the free surface) is used, and a comparison with [CITATION] at [MATH] produces nearly identical velocity fluctuations and 1D energy spectra.', '1811.04339-1-7-0': 'Particle trajectories and suspension flow dynamics are based on the Lagrangian point-particle approximation where the particle-to-fluid density ratio [MATH] and the particle size is smaller than the smallest viscous dissipation scales of the turbulence.', '1811.04339-1-7-1': 'Only the Stokes drag force and two-way coupling have been incorporated since we restrict our study to low volume concentration [MATH] [CITATION].', '1811.04339-1-7-2': 'Gravitational settling is not considered in order to highlight the effect of the particle response time.', '1811.04339-1-7-3': 'Particles experience a purely elastic collision with the lower wall and upper rigid free surface.', '1811.04339-1-7-4': 'Two-way coupling is implemented via a particle-in-cell scheme, and has been validated against [CITATION] and [CITATION] in turbulent channel flow.', '1811.04339-1-7-5': 'Grid convergence of both the flow and of the two-way coupling scheme have been verified as well .', '1811.04339-1-8-0': 'Particle modulation of turbulence is associated with the relative time scales between particles and local turbulent structures.', '1811.04339-1-8-1': 'The multiple turbulent structures spanning a wide spatial and temporal range (e.g. LSMs and VLSMs) result in a wide parameter space of the particle inertia to be investigated.', '1811.04339-1-8-2': 'As shown in table [REF], we choose [MATH] based on the viscous scale in the range of [MATH], corresponding to [MATH] based on the Kolmogorov scale in the range of [MATH] at [MATH] and [MATH] at [MATH].', '1811.04339-1-8-3': 'For [MATH] the streamwise domain extent [MATH] was determined by gradually increasing it until after the appearance of a bimodal energy spectra in the spanwise direction.', '1811.04339-1-8-4': 'Cases [MATH] are then designed to investigate the effects of particle inertia by systematically increasing the particle Stokes number.', '1811.04339-1-8-5': 'In single-phase flow it is well-known that VLSMs are very long in the streamwise direction and fully capturing their extent is computationally expensive .', '1811.04339-1-8-6': 'Therefore as a test, [MATH] doubles the streamwise extent for single-phase flow in order to check any effects of streamwise confinement on VLSMs by comparing to [MATH] (negligible differences were observed).', '1811.04339-1-8-7': 'In order to further examine particle direct modulation of VLSMs, [MATH] and [MATH] are performed at a higher [MATH] for single-phase and particle-laden flow - these ultimately yield identical conclusions.', '1811.04339-1-9-0': '# Results', '1811.04339-1-10-0': '## Particle distribution in two distinct layers', '1811.04339-1-11-0': 'It is well-established that for wall-bounded turbulent flow, low-inertia particles tend to distribute homogeneously in wall-normal planes, while intermediate Stokes numbers exhibit particle clustering in near-wall streaks and high-inertia particles behave with ballistic trajectories (thus eliminating much of the clustering).', '1811.04339-1-11-1': 'This qualitative transition with [MATH] is observed in the inner region of the simulated open channel flow (i.e. near the wall).', '1811.04339-1-11-2': 'Figures [REF] (b-f) present isosurfaces of particle concentration in two layers (inner region: [MATH] times the bulk concentration and outer region: [MATH] times the bulk concentration) for increasing Stokes numbers [MATH] to [MATH], where we find particles accumulating in the inner-flow streaks at intermediate [MATH]; the same as observed in [CITATION].', '1811.04339-1-11-3': 'At the same time, new type of organized, particle-clustering structure in the outer flow region is formed in low-vorticity regions ejected from the inner region, especially at high [MATH].', '1811.04339-1-11-4': 'However, with a very high [MATH], particles behave ballistically in the outer flow region, tending to distribute more as tracers similar to low [MATH].', '1811.04339-1-11-5': 'This non-monotonic particles accumulation behaviour has an influence on the non-monotonic modulation of the VLSMs in the outer region.', '1811.04339-1-12-0': '## Premultiplied 1D u-spectra', '1811.04339-1-13-0': 'Figures [REF](a,d) display as a function of [MATH] the premultiplied, one-dimensional [MATH]spectra [MATH], where [MATH], as functions of the normalized streamwise and spanwise wavelengths [MATH] and [MATH] (where [MATH] is the Fourier coefficient of [MATH]).', '1811.04339-1-13-1': 'As a reference, we compare with the results of wall-bounded channel flow from [CITATION] at the same [MATH], who find that the turbulence in the outer flow is approaching isotropic if VLSMs are artificially removed (i.e. VLSMs introduce anisotropy).', '1811.04339-1-13-2': 'Comparing between [CITATION] (who filter out high wavelengths, so only [MATH] and [MATH] are plotted) and the present (unfiltered) simulation in figures [REF](a,d), we observe that for unladen flow, VLSMs hardly affect the spectral signature of LSMs , whereas the contribution from VLSMs forms a bimodal spanwise spectra at [MATH]) at heights above the inner layer.', '1811.04339-1-13-3': 'The large-scale energetic structures (larger than LSMs) within the inner layer are also attributed to VLSMs, which are referred to as deep [MATH]modes or "footprints" of VLSMs .', '1811.04339-1-13-4': 'Figure [REF](b,c) and (e,f) show the streamwise and spanwise u-spectra at [MATH] and [MATH] normalized by the local streamwise velocity RMS, respectively.', '1811.04339-1-13-5': 'As noted previously, any effect of a limited streamwise domain extent [MATH] is minimal, since figure [REF](c) shows that the energy contained in VLSMs in a short domain ([MATH]) is nearly identical to the long domain ([MATH]).', '1811.04339-1-13-6': 'Thus overall our unladen simulations are consistent with the current understanding of VLSMs.', '1811.04339-1-14-0': 'Particle inertia has a non-monotonic effect on LSMs in the inner region , which can be seen in figure [REF](b): the energy contained in LSMs first decreases then increases with an increased [MATH].', '1811.04339-1-14-1': 'At the same time, the energetic LSMs in [MATH] are elongated compared to single-phase flow (i.e. the peak shifts to larger wavelengths).', '1811.04339-1-14-2': 'Focusing on VLSM modulation by inertial particles, figures [REF](c,f) show that low inertia ([MATH]) and high inertia ([MATH]) particles significantly enhance the VLSM energy content in the outer region whereas intermediate inertia ([MATH] and [MATH]) and very high inertia ([MATH]) cases have very little impact on the VLSMs.', '1811.04339-1-14-3': 'The enhancement of the VLSM signature at high inertia is also found at [MATH], as shown in figure [REF](e,f).', '1811.04339-1-14-4': 'The penetration of VLSMs from the outer region to the inner region can be observed in figure [REF](e), since the energy contained in turbulent scales at [MATH] is simultaneously enhanced with the same spanwise wavelength as in figure [REF](f) for [MATH] and [MATH].', '1811.04339-1-14-5': 'Additional evidence can be seen in figure [REF], which provides a snapshot of the streamwise velocity fluctuation in the [MATH] plane at [MATH].', '1811.04339-1-14-6': 'Comparing between figures [REF](b,c) and (a), it is evident that VLSMs are better-organized when [MATH]) or [MATH]) compared to single-phase flow ([MATH]).', '1811.04339-1-14-7': 'We now turn our investigation to understanding why the VLSM modulation appears to have two distinct peaks in Stokes number.', '1811.04339-1-15-0': '## Conditional tests of the particle coupling effect', '1811.04339-1-16-0': 'In spectral space, the modulation of LSMs and VLSMs by a dispersed phase is at least partially related to the direct influence on particle fluctuations, which in turn can modify the production of turbulent kinetic energy (TKE) and/or Reynolds shear stress.', '1811.04339-1-16-1': 'This is demonstrated more quantitatively in figures [REF](a,d), where we present the [MATH] spectral production term [MATH] as a function of [MATH] and the wall-normal direction for [MATH] and [MATH] in comparison with [MATH].', '1811.04339-1-16-2': 'Comparing figures [REF](a,d) with figure [REF](d), we find that they have a similar overall shape, and the bimodal spectrum appears both in the premultipled [MATH]spectra as well as the production term [MATH], which is enhanced in [MATH] and [MATH] in comparison with the single-phase flow.', '1811.04339-1-16-3': 'In addition to the modifications to streamwise TKE production, particles can also act as a direct source/sink in the spectral TKE and Reynolds stress budgets.', '1811.04339-1-16-4': 'See for example the schematic in figure [REF]: particle feedback contributes to [MATH], [MATH] and [MATH] budgets in the inner/outer region, corresponding to indirect/direct modulation of VLSMs, respectively.', '1811.04339-1-16-5': 'In the spectral energy budget particle sources are denoted as [MATH] to the [MATH] budget (figures [REF](b,e)) and [MATH] to [MATH] budget (figures [REF](c,f)), where [MATH] stands for the real part and [MATH] is the Fourier transform of the particle coupling force.', '1811.04339-1-16-6': 'We find that [MATH], which is positive in [MATH] whereas negative in [MATH] and confined largely to the inner region, and [MATH], which is negative in [MATH] whereas positive in [MATH] and extended throughout the entire wall-normal extent, play opposite roles in the turbulent energy and Reynolds stress budgets depending on whether one considers the inner or outer regions.', '1811.04339-1-16-7': 'This nearly opposite behavior indicates that there might be two underpinning mechanisms of VLSM enhancement induced by low and high Stokes numbers particles.', '1811.04339-1-17-0': '[CITATION] finds that this Stokes-number-dependent source/sink varies with wavelength, and is possibly associated with the particle clusters themselves even though it always responsible for the underlying changes to TKE .', '1811.04339-1-17-1': 'In the inner flow, [CITATION] find low inertial particles ([MATH]) enhance LSMs whereas high inertial particles ([MATH]) attenuate LSMs, corresponding to a positive particle feedback [MATH] in [MATH] (figure [REF](b)) but negative in [MATH] (figure [REF](e)).', '1811.04339-1-17-2': 'In addition, as shown in figure [REF](c), particle feedback [MATH] in [MATH] always attenuates the generation of [MATH].', '1811.04339-1-17-3': 'Therefore in [MATH], the positive feedback [MATH] in the inner flow is the most likely responsible for the enhancement of VLSMs in the outer flow.', '1811.04339-1-17-4': 'This process of particles inducing upscale energy transfer (or a reverse cascade), tending to build up the energy level at high wavelengths due to the modulation of small-scale turbulent motions, is also observed in homogeneous turbulence (see [CITATION] and [CITATION]) and in turbulent Couette flow (see [CITATION]).', '1811.04339-1-18-0': 'In contrast to [MATH], it is more straightforward to understand the VLSM modulation in [MATH].', '1811.04339-1-18-1': 'As shown in figure [REF](f), throughout the whole domain, we observe that [MATH] in [MATH] always has a positive contribution to the [MATH] budget and is at the same spatial locations as the production [MATH] of the [MATH] budget (seen also at [MATH] in [MATH], figure is not shown), whereas [MATH] in [MATH] (figure [REF](c)) tends to suppress the generation of [MATH].', '1811.04339-1-18-2': 'However, as shown in figure [REF](e), the negative particle feedback [MATH] in the inner region tends to attenuate [MATH] in [MATH].', '1811.04339-1-18-3': 'This ultimately results in [MATH] exerted in the outer region as the most likely explanation for the enhancement of VLSMs.', '1811.04339-1-18-4': 'As the source of Reynolds shear stress [MATH], the ejection/sweep cycles in very large scales are directly enhanced by the presence of high inertial particles in the outer flow, and these very large upwelling/downwelling structures further extract energy from the mean flow by working with local mean shear, see [CITATION].', '1811.04339-1-19-0': "In order to verify the above hypothesis that the VLSM enhancement in [MATH] is due to particles' modulation of LSMs in the inner flow (we refer to this as indirect modulation of VLSMs) whereas in [MATH] it is due to the particles' direct modulation on the VLSMs in the outer flow, we perform a conditional numerical test to identify the particles' effective region of influence regarding VLSM enhancement by artificially applying the particle feedback force only in one of three locations: (1) the viscous sublayer ([MATH]), (2) the regeneration cycle region ([MATH]), or (3) the outer flow ([MATH]), separately.", '1811.04339-1-19-1': 'The premultiplied two-dimensional energy spectrum of streamwise velocity [MATH], where [MATH], is shown in figure [REF] as a function of wall-normal direction.', '1811.04339-1-19-2': 'The figure exhibits a "boot-shaped" isosurface of [MATH].', '1811.04339-1-19-3': 'The "forefoot" illustrates the LSMs in the near-wall region whereas the "bootleg" is formed by the VLSMs.', '1811.04339-1-19-4': 'It can be seen that the VLSMs only experience enhancement in [MATH] when particle coupling is included in the regeneration cycle region.', '1811.04339-1-19-5': 'For [MATH], on the other hand, the opposite is true: VLSM enhancement is found only when particle coupling effects are included in the outer region.', '1811.04339-1-19-6': 'Both of these effects are observed throughout the entire range of [MATH].', '1811.04339-1-19-7': 'The tests of two-way coupling applied for [MATH] are not shown, but we find the spectrum in the range [MATH] and [MATH] is stronger than in the single-phase flow, but not in the range of VLSMs.', '1811.04339-1-20-0': '# Summary and discussion', '1811.04339-1-21-0': 'In this paper, we investigate the effect of inertial particles on VLSMs in moderate Reynolds number open channel flow.', '1811.04339-1-21-1': 'The particles are characterized by well-known clustering structures in the inner layer, and additional structures are formed in the outer flow as well but at different Stokes numbers.', '1811.04339-1-21-2': 'With very high particle inertia, at timescales slower than even the VLSMs, the particles approach a nearly uniform distribution.', '1811.04339-1-21-3': 'In terms of two-way coupling, we find that inertial particles have a non-monotonic effect on the VLSMs, where low and high inertia particles strengthen the VLSMs and the intermediate inertia particles hardly affect their structure and energy.', '1811.04339-1-22-0': 'The most direct route of particle modulation of turbulent motions comes from the particles feedback source in the turbulent energy budget.', '1811.04339-1-22-1': 'We find low inertial (high inertial) particles have a positive (negative) [MATH] in the inner flow and a negative (positive) [MATH] in the outer flow.', '1811.04339-1-22-2': 'By utilizing a conditional numerical test, we demonstrate that low inertia particles strengthen the VLSMs due to the enhancement of the LSMs in the inner flow.', '1811.04339-1-22-3': 'While the relationship between near-wall LSMs and the outer-scale VLSMs remains a subject of investigation, this suggests that there can exist an upscale transport of energy possible from LSMs to VLSMs.', '1811.04339-1-22-4': 'This is consistent with [CITATION] who show numerically that LSMs and VLSMs interact in a co-supporting cycle and [CITATION] who observe experimentally the high degree of velocity fluctuation correlation between the outer flow with the low-frequency content of the inner flow.', '1811.04339-1-22-5': 'In contrast, high inertia particles modulate the VLSM directly.', '1811.04339-1-22-6': 'They cluster in the low-vortical region of the outer flow and directly modulate the VLSMs, indicated by the particle feedback effect on the Reynolds shear stress budget with the same scale of VLSMs at the same spatial locations.', '1811.04339-1-22-7': 'The high inertia particle clustering signatures and the mechanism of the VLSM enhancement in the outer flow region still require investigation.'}

{'1811.04339-2-0-0': 'Very large-scale motions (VLSMs) and large-scale motions (LSMs) coexist at moderate Reynolds numbers in a very long open channel flow.', '1811.04339-2-0-1': 'Direct numerical simulations two-way coupled with inertial particles are analysed using spectral information to investigate the modulation of VLSMs.', '1811.04339-2-0-2': 'In the wall-normal direction, particle distributions (mean/preferential concentration) exhibit two distinct behaviors in the inner flow and outer flow, corresponding to two highly anisotropic turbulent structures, LSMs and VLSMs.', '1811.04339-2-0-3': "This results in particle inertia's non-monotonic effects on the VLSMs: low inertia (based on the inner scale) and high inertia (based on the outer scale) both strengthen the VLSMs whereas moderate and very high inertia have little influence.", '1811.04339-2-0-4': 'Through conditional tests, low and high inertia particles enhance VLSMs following two distinct routes.', '1811.04339-2-0-5': 'Low inertia particles promote VLSMs indirectly through the enhancement of the regeneration cycle (the self-sustaining mechanism of LSMs) in the inner region whereas high inertia particles enhance the VLSM directly through contribution to the Reynolds shear stress at similar temporal scales in the outer region.', '1811.04339-2-0-6': 'This understanding also provides more general insight into inner-outer interaction in high Reynolds number, wall-bounded flows.', '1811.04339-2-1-0': '# Introduction', '1811.04339-2-2-0': 'Very large-scale motions (VLSMs) extending to over [MATH] (where [MATH] is the boundary layer thickness) are found in very high Reynolds number, wall-bounded turbulent flows and are distinct from the well-understood large-scale motions (LSMs) which form canonical streaks and hairpin vortices .', '1811.04339-2-2-1': 'These long, meandering features are observed to be energetic, carrying [MATH] of the kinetic energy and [MATH] of the Reynolds shear stress in pipe flow or in turbulent boundary layers , which is contradictory to the notion of "inactive" motion proposed by [CITATION].', '1811.04339-2-2-2': 'In environmental flows, these anisotropic structures also have significant influence on the dispersion of pollutants, sand, and other constituents.', '1811.04339-2-2-3': 'At the same time, understanding the modulation of turbulence by inertial particles is itself a formidable challenge , and nearly all numerical studies of two-way coupling in particle-laden wall turbulence have been restricted to low Reynolds numbers.', '1811.04339-2-2-4': 'It is therefore the aim of this investigation to study the effects of particles on VLSMs, in particular focusing on the question of whether particles act directly or indirectly on these very large motions.', '1811.04339-2-3-0': 'In contrast with with VLSMs, the importance of LSMs on the flow dynamics in the near-wall region has been demonstrated in many different contexts.', '1811.04339-2-3-1': 'The LSMs are found to follow a self-sustaining process (i.e. a regeneration cycle) characterized by three key structures shown in the lower left of figure [REF]: large-scale streaks (LSSs), large-scale vortices (LSVs), and meandering streaks.', '1811.04339-2-3-2': 'Associated with these structures are three regeneration processes: streak formation, streak breakdown, and vortex regeneration .', '1811.04339-2-3-3': 'The particle modulation of the regeneration cycle has been investigated by [CITATION], [CITATION] and [CITATION] to explain the non-monotonic effect of the onset of transition with mass loading and particle size.', '1811.04339-2-3-4': 'The typical scales of VLSMs, meanwhile, are far larger than LSMs, with their spanwise wavelength [MATH] and their streamwise wavelength [MATH] .', '1811.04339-2-3-5': 'These structures co-exist with LSMs, and the interaction between them is still an open question.', '1811.04339-2-3-6': 'It is generally accepted that the regeneration cycle of LSMs does not require the existence of VLSMs , and [CITATION] propose that the formation of LSMs and VLSMs results from different mechanisms.', '1811.04339-2-3-7': '[CITATION] argue that VLSMs are self-sustained and do not draw energy from LSMs in the buffer layer, however, [CITATION] and [CITATION] suggest that VLSMs are not a new type of turbulent structure but merely the consequence of the alignment of coherent LSMs.', '1811.04339-2-3-8': '[CITATION] show numerically that LSMs and VLSMs interact in a co-supporting cycle.', '1811.04339-2-3-9': 'Recent work by [CITATION] reveals that very near the wall ([MATH]) there is a true inverse scale transfer from the dominant LSMs to VLSMs, which appears to be driven by interaction of the streaks with large-scale outer-layer structures.', '1811.04339-2-3-10': 'In the current work, we find that the enhancement of VLSMs can be caused by the promotion of LSMs via inertial particles.', '1811.04339-2-4-0': 'A wide ranging parameter space including flow type, mass fraction, and particle-to-fluid length and time scale ratios can be expected to complicate the picture of turbulence modulation, resulting in poorly mapped out mechanisms of particle two-way coupling .', '1811.04339-2-4-1': 'In isotropic turbulence, [CITATION] reviewed particle influences on the turbulent kinetic energy (TKE) spectrum of the carrier fluid, indicating that low wavenumbers are suppressed, while energy is gained at higher wave numbers.', '1811.04339-2-4-2': 'The physical explanations are still not well understood, however.', '1811.04339-2-4-3': 'The situation is even more complex for wall-bounded turbulence, especially in the logarithmic layer which contains a multiscale momentum cascade in three-dimensional space , while the particle-to-fluid length and time scale ratios also vary as a function of wall-normal height.', '1811.04339-2-4-4': 'In addition, the wall-normal TKE transport is also modulated by inertial particles, for example, [CITATION] found that inertial particles transported within streaky motions act as a carrier transferring TKE from the core region of the channel to the fluid close to the wall.', '1811.04339-2-4-5': 'There has been substantial progress in understanding inertial particle dynamics in the inner layer (i.e. related to LSMs), for instance the phenomena of particle clustering and segregation , drag reduction , high particle loading and cluster dynamics , particle inducing upscale energy transfer and transportation , and regeneration cycle modulation .', '1811.04339-2-4-6': 'However to the best of our knowledge, very little attention has been paid to particle clustering and modulation of VLSMs in the outer layer.', '1811.04339-2-5-0': 'More generally, it is inherently difficult to describe turbulence modulation by particles.', '1811.04339-2-5-1': 'The intensity of TKE is often used to indicate turbulence modulation (e.g. [CITATION], [CITATION] and [CITATION]), however, this can lead to somewhat contradictory descriptions.', '1811.04339-2-5-2': 'For instance, recent observations have shown that even though TKE is nearly unchanged, the onset of laminar-to-turbulent transition can be significantly advanced by particles with a low Stokes number ([MATH]; based on viscous time scale); see for example [CITATION], [CITATION], [CITATION], and [CITATION].', '1811.04339-2-5-3': 'In fact, not all scales of turbulence are enhanced during turbulence augmentation, so using the bulk TKE might misrepresent the modulation at certain length and time scales.', '1811.04339-2-5-4': 'This can be observed with spectral analysis, which is a natural means to study particles modulation of turbulence.', '1811.04339-2-5-5': 'For example [CITATION] found there is a possible so-called reverse cascade which tends to build up energy in large scale structures in homogeneous turbulence, and [CITATION] demonstrates that this upscale influence is a strong function of particle inertia.', '1811.04339-2-5-6': 'As a result, particles can influence turbulence scales far removed from their own response time scale in wall-bounded turbulence.', '1811.04339-2-6-0': 'In this study, as indicated in figure [REF], particles can directly impact TKE transport through momentum coupling , thereby modulating specific scales of turbulent structures directly, i.e. VLSMs in the outer layer (through route 1 in figure [REF]) or LSMs in the inner layer.', '1811.04339-2-6-1': 'As discussed above, particle modulation of LSMs in the inner layer also has the possible effect that particle feedback on LSMs near the wall can have upscale, indirect influences on VLSMs via nonlinear energy transfer (through route 2 in figure [REF]); see for example [CITATION] and [CITATION].', '1811.04339-2-6-2': 'To better understand particle modulation of LSMs in the inner layer, [CITATION] further investigated small particles and their ability to enhance the LSM regeneration cycle (depending non-monotonically on particle inertia, see also [CITATION]), with the assumption that this was a route through which particles could modify even larger scales in high Reynolds number flow.', '1811.04339-2-6-3': 'Therefore as a follow-up, and since to date computational costs have precluded particle-laden direct numerical simulations at sufficiently high Reynolds number, we for the first time examine the effects of a wide range of particle inertia on VLSMs in open channel flow at Reynolds numbers of [MATH] and [MATH].', '1811.04339-2-7-0': '# Numerical parameters', '1811.04339-2-8-0': 'Direct numerical simulations of the Eulerian flow are performed for an incompressible Newtonian fluid using the same numerical implementation as [CITATION] and [CITATION].', '1811.04339-2-8-1': 'A pseudospectral method is employed in the periodic directions (streamwise [MATH] and spanwise [MATH]), and second-order finite differences are used for spatial discretization in the wall-normal ([MATH]) direction.', '1811.04339-2-8-2': 'We simulate pressure-driven open channel flow because it is characterized by features of both closed channel flow and boundary layer , while also exhibiting the signatures of VLSMs at a more computationally accessible domain size and Reynolds number.', '1811.04339-2-8-3': 'A no-slip condition is imposed on the bottom wall and a shear-free condition is imposed on the upper surface, and such boundary conditions have been proven capable of capturing many of the phenomena (e.g. VLSMs) seen in experiments with shear-free upper boundaries; see [CITATION].', '1811.04339-2-8-4': 'We remark here that the inertial particles do not collect at the free surface in this study, which is also observed experimentally by [CITATION] and numerically by [CITATION].', '1811.04339-2-8-5': 'The solution is advanced in time by a third-order Runge-Kutta scheme.', '1811.04339-2-8-6': 'A single-sided stretched grid (fine grid close to the wall, coarse grid close to the free surface) is used in this study.', '1811.04339-2-8-7': 'Comparisons with a double-sided stretched grid (fine grid close to the wall and the free surface) and the simulations of [CITATION] at [MATH] produce nearly identical mean velocity profiles, shown in figure [REF](a), and turbulent intensity profiles, shown in figure [REF](b).', '1811.04339-2-8-8': 'In addition, the one-dimensional [MATH]spectra [MATH] for the single-sided stretched and double-sided stretched grids is shown in figure [REF] (c-d).', '1811.04339-2-8-9': 'Both close to the wall ([MATH]; figure [REF](c)) and near free-surface region ([MATH]; figure [REF](d)), single-sided and double-sided stretched grids agree with each other.', '1811.04339-2-9-0': 'Particle trajectories and suspension flow dynamics are based on the Lagrangian point-particle approximation where the particle-to-fluid density ratio [MATH] and the particle size is smaller than the smallest viscous dissipation scales of the turbulence.', '1811.04339-2-9-1': 'Only the Stokes drag force and two-way coupling have been incorporated since we restrict our study to low volume concentration [MATH] [CITATION].', '1811.04339-2-9-2': 'Gravitational settling is not considered in order to highlight the effect of the particle response time.', '1811.04339-2-9-3': 'Particles experience a purely elastic collision with the lower wall and upper rigid free surface.', '1811.04339-2-9-4': 'Two-way coupling is implemented via a particle-in-cell scheme, and has been validated against [CITATION] and [CITATION] in turbulent channel flow.', '1811.04339-2-9-5': 'Grid convergence of both the flow and of the two-way coupling scheme have been verified as well .', '1811.04339-2-10-0': 'Particle modulation of turbulence is often characterized by the relative time scales between particles and local turbulent structures.', '1811.04339-2-10-1': 'The multiple turbulent structures spanning a wide spatial and temporal range (e.g. LSMs and VLSMs) result in a wide parameter space of the particle inertia to be investigated.', '1811.04339-2-10-2': 'As shown in table [REF], we choose [MATH] in the range of [MATH] based on the inner viscous time scale, which corresponds to [MATH] ranging from [MATH], where [MATH] is based on the outer bulk flow time scale [MATH].', '1811.04339-2-10-3': 'This also corresponds to [MATH] in the range [MATH] based on the average Kolmogorov scale in the inner layer and [MATH] based on the average Kolmogorov scale in the outer layer.', '1811.04339-2-10-4': 'In single-phase channel flow, [CITATION] use [MATH] at [MATH] and [CITATION] choose [MATH] at [MATH] to study VLSMs.', '1811.04339-2-10-5': 'In current particle-laden flow, the domain size [MATH] is used, slightly shorter than [CITATION] whereas larger than [CITATION].', '1811.04339-2-10-6': 'With this domain size, we observe the appearance of a bimodal energy spectra in the spanwise direction and compare well with [CITATION] (see figure [REF]).', '1811.04339-2-10-7': 'In single-phase flow it is well-known that VLSMs are very long in the streamwise direction and fully capturing their extent is computationally expensive .', '1811.04339-2-10-8': 'Therefore as a test, [MATH] doubles the streamwise extent for single-phase flow in order to check any effects of streamwise confinement on VLSMs by comparing to [MATH] (streamwise velocity spectrum in spanwise direction is shown later in figure [REF](a); negligible differences were observed).', '1811.04339-2-10-9': '[MATH] are then designed to investigate the effects of particle inertia by systematically increasing the particle Stokes number.', '1811.04339-2-10-10': 'In order to further examine particle direct modulation of VLSMs, [MATH] are performed at a higher [MATH] for single-phase and particle-laden flow - these ultimately yield identical conclusions.', '1811.04339-2-11-0': '# Results', '1811.04339-2-12-0': '## Particle distribution in two distinct layers', '1811.04339-2-13-0': 'Mean particle volume concentrations in the inner layer and outer layer are shown in figure [REF](a), exhibiting a non-monotonic behaviour with Stokes number but with an opposite trend.', '1811.04339-2-13-1': 'In the inner layer it is maximized for [MATH], which corresponds to [MATH].', '1811.04339-2-13-2': 'At the same time, [MATH] also exhibits the minimum in the outer layer, where the more relevant Stokes number is [MATH].', '1811.04339-2-13-3': 'Stokes numbers lower or higher than [MATH] result in fewer particles in the inner layer whereas more particles in the outer layer.', '1811.04339-2-13-4': 'This often-observed behavior is due to turbophoresis , which induces a net particle flux towards the wall resulting in higher particle volume concentration in the inner layer than in the outer layer.', '1811.04339-2-14-0': 'It is also commonly accepted that inertial particles preferentially accumulate in low-speed streaks , which is also observed in this study.', '1811.04339-2-14-1': "Particle numbers in 'upwelling' and 'downwelling' regions can be straightforwardly counted by testing whether [MATH] or [MATH], where [MATH] is the fluid fluctuating velocity seen by the particle.", '1811.04339-2-14-2': "Then, the ratio of the number of particles in 'upwelling' and 'downwelling' regions can be used to compare across different wall-normal locations and Stokes numbers.", '1811.04339-2-14-3': 'This ratio, cast in terms of the effective volume concentration corresponding to these particle counts, is shown in figure [REF](b).', '1811.04339-2-14-4': 'Here, there is a clear non-monotonic trend with the Stokes number and with [MATH].', '1811.04339-2-14-5': "In the inner layer, there are more particles in the 'upwelling' fluid motions than in the 'downwelling' fluid motions, which is opposite compared to the outer layer.", '1811.04339-2-14-6': 'The lowest ratio appears for [MATH] at [MATH] in the inner layer, while the highest ratio appears for [MATH] at [MATH] in the outer layer.', '1811.04339-2-14-7': 'Thus in the inner layer, relatively low-inertia particles collect in the low-speed streaks, while in the outer layer, higher-inertia particles collect in the high-speed regions.', '1811.04339-2-14-8': 'The relevant Stokes numbers are different for each, since the respective fluid timescales are different.', '1811.04339-2-15-0': 'The relationship between particle distributions and the different turbulent structures (i.e. VLSMs and LSMs) is still poorly understood, however.', '1811.04339-2-15-1': 'To explore this, we introduce a mean fluid streamwise velocity fluctuation at the particle positions, but filtered according to the wavelengths associated with LSMs and VLSMs: [EQUATION]', '1811.04339-2-15-2': 'Here the interpolated velocity [MATH] is projected onto the Eulerian grid, allowing for the [MATH] field to be transferred to Fourier space ([MATH]).', '1811.04339-2-15-3': 'The goal is to artificially filter out targeted turbulent structures (e.g. removing wavelengths with [MATH], [MATH] to isolate VLSMs) in order to obtain [MATH] contributed by specific turbulent structures.', '1811.04339-2-15-4': 'For single-phase flow, [MATH] is set to one at all Eulerian grid points and the same procedure is followed.', '1811.04339-2-15-5': 'These quantities are shown in figure [REF].', '1811.04339-2-15-6': 'In the inner layer, [MATH] is negative in both VLSMs and LSMs, indicating that particles are more likely to reside in the large and very large scale low-speed streaks compared to the high-speed streaks.', '1811.04339-2-15-7': 'The minimum values of [MATH] for LSMs ([MATH] with [MATH] in figure [REF](a)) and VLSMs ([MATH] with [MATH] in figure [REF](b)) appear at different Stokes numbers, again since the flow timescales associated with LSMs and VLSMs are different.', '1811.04339-2-16-0': "In the outer layer, [MATH] is positive for both VLSMs and LSMs, indicating that particles tend to reside in the 'downwelling' regions at the scales of both the LSMs and the VLSMs.", '1811.04339-2-16-1': 'The maximum value of [MATH] for VLSMs (figure [REF](b)) appears for [MATH] with [MATH].', '1811.04339-2-16-2': 'Comparing the inner layer with the outer layer, [MATH] for LSMs is considerably stronger in the inner layer than in the outer layer as shown in figure [REF](a).', '1811.04339-2-16-3': 'Two possible reasons may explain this.', '1811.04339-2-16-4': 'One is that the intensity of LSMs is much weaker in the outer layer than in the inner layer (will be shown later in figure [REF](a)); the other is that particle preferential concentration is not as strongly correlated with streaky motions in the outer layer as compared to the inner layer (will be shown in figure [REF](a)).', '1811.04339-2-16-5': 'However for [MATH] in VLSMs (figure [REF](b)), the magnitude is comparable between the inner layer and the outer layer, albeit with opposite preferred signs.', '1811.04339-2-17-0': 'In terms of particle clustering behaviour, it is well-established that for wall-bounded turbulent flow in the inner layer, low-inertia particles ([MATH]) tend to distribute homogeneously in wall-normal planes , while intermediate Stokes numbers ([MATH]) exhibit particle clustering in near-wall streaks and high-inertia particles ([MATH]) behave with ballistic trajectories (thus eliminating much of the clustering).', '1811.04339-2-17-1': 'This qualitative transition with [MATH] is observed within the inner region of the simulated open channel flow.', '1811.04339-2-17-2': 'Figures [REF](a-e) present isosurfaces of particle concentration ([MATH] times the bulk [MATH]) for [MATH] in the inner and outer regions for increasing Stokes numbers.', '1811.04339-2-17-3': 'The advantage of showing concentration isosurfaces as opposed to individual Lagrangian points is that this method better visualizes the high-concentration particle clusters.', '1811.04339-2-17-4': 'The three panels across the horizontal represent three slabs at progressively increasing wall-normal distances (at the same snapshot in time): layer [MATH]: [MATH], layer [MATH]: [MATH], layer [MATH]: [MATH].', '1811.04339-2-18-0': 'Here we find particles accumulating in the inner-flow low-speed streaks at intermediate [MATH] in layer [MATH] of figures [REF](b-c)); this is similar to many other studies as noted above.', '1811.04339-2-18-1': 'At the same time, a new type of organized structure in the outer flow region is observed.', '1811.04339-2-18-2': 'These are especially clear at higher Stokes numbers, e.g. [MATH]) for [MATH] shown in layers [MATH] of figure [REF](d).', '1811.04339-2-18-3': 'However, with a very high particle inertia, [MATH]), particles behave ballistically in the outer flow region as shown in layers [MATH] of figure [REF](e), tending to distribute more homogeneously.', '1811.04339-2-18-4': 'These two distinct, non-monotonic particle accumulation behaviours in the inner and outer layers peak at different Stokes numbers ([MATH] in the inner layer and [MATH] in the outer layer) and have a strong influence on the non-monotonic modulation of the VLSMs in the outer region via the two routes indicated in figure [REF].', '1811.04339-2-18-5': 'This will be discussed further in section [REF].', '1811.04339-2-19-0': 'In order to further quantify the particle clustering behaviour, we employ a Voronoi diagram analysis, which compares the distribution of the tessellation areas in the particle-laden cases with the expected Poisson distribution if the particles were randomly distributed (see for example [CITATION]).', '1811.04339-2-19-1': 'A maximum clustering effect is typically observed for [MATH] around unity in isotropic turbulence and buoyancy-driven wall-bounded turbulence .', '1811.04339-2-19-2': 'Figures [REF](a,b) show the standard deviation ([MATH]) of the distribution of the normalized Voronoi area [MATH], where the inverse of the average Voronoi area [MATH] indicates the mean particle concentration.', '1811.04339-2-19-3': '[MATH] is scaled by the standard deviation of a random Poisson process (RPP; [MATH]).', '1811.04339-2-19-4': 'The ratio [MATH] exceeding unity indicates that particles are accumulating in clusters as compared to truly randomly distributed particles.', '1811.04339-2-20-0': 'Figure [REF](a) shows ratio [MATH] for multiple heights across all Stokes numbers, while figure [REF](b) shows the ratio as a function of [MATH] for two different representative heights ([MATH] and [MATH] are plotted based on [MATH] of the inner layer and outer layer, respectively, as provided in table [REF]).', '1811.04339-2-20-1': 'In the inner layer ([MATH]), from [MATH] to [MATH] or [MATH]), the clustering effect experiences a non-monotonic evolution as a function of Stokes number.', '1811.04339-2-20-2': 'The largest value appears at [MATH] and it gradually decreases to unity with a higher Stokes number.', '1811.04339-2-20-3': 'This is similar with the investigation of [CITATION] that particles with [MATH] preferentially accumulate in the streaks whereas higher inertial particles tend to spread throughout the inner layer.', '1811.04339-2-20-4': 'With increasing wall-normal distance, the ratio [MATH] decreases at very low Stokes number ([MATH]) whereas it increases in higher Stokes numbers ([MATH]).', '1811.04339-2-20-5': 'At still higher Stokes numbers ([MATH]), the ratio [MATH] again approaches unity.', '1811.04339-2-20-6': 'These very heavy particles cannot follow the streamlines, resulting in a nearly particle random distribution.', '1811.04339-2-20-7': 'In the outer layer ([MATH]), the largest value appears at [MATH]), similar in magnitude with previous investigations of clustering in isotropic turbulence; peaks were found for [MATH] and [MATH] by [CITATION] and [CITATION], respectively.', '1811.04339-2-21-0': '## Particle modulation of TKE in LSMs and VLSMs', '1811.04339-2-22-0': 'The premultiplied, two-dimensional energy spectrum of streamwise velocity, [MATH] where [MATH], is shown in figure [REF] for [MATH] as a function of wall-normal distance ([MATH] is the Fourier coefficient of [MATH]).', '1811.04339-2-22-1': 'The figure exhibits a "boot-shaped" structure, particularly well-defined for [MATH] (figure [REF](b)) and [MATH] (figure [REF](e)).', '1811.04339-2-22-2': 'The "forefoot" corresponds to the LSMs in the near-wall region whereas the "bootleg" corresponds to the VLSMs.', '1811.04339-2-22-3': 'The signature of VLSMs indeed appears at the upper-left corner (long and wide wavelengths in the streamwise and spanwise directions) for single-phase flow in figure [REF](a).', '1811.04339-2-22-4': 'This VLSM signature is nearly unchanged for [MATH] and [MATH] whereas it is slightly weakened in [MATH].', '1811.04339-2-22-5': 'It is clear, however, that for low Stokes number ([MATH] of [MATH] in figure [REF](b)) and high Stokes number ([MATH] of [MATH] in figure [REF](e)), energy contained by the VLSMs is enhanced by the presence of particles.', '1811.04339-2-22-6': 'In addition to the enhanced VLSMs observed in the outer layer, the large-scale energetic structures (e.g. the "bootleg" within [MATH] in figure [REF](b,e)) extend into the inner layer, which are referred to as deep [MATH]modes or VLSM "footprints" , and are a possible path for the inverse scale transfer from LSMs to VLSMs found by [CITATION].', '1811.04339-2-23-0': 'Figure [REF] displays as a function of [MATH] the premultiplied, one-dimensional [MATH]spectra [MATH], where [MATH], as a function of the normalized spanwise wavelength [MATH].', '1811.04339-2-23-1': 'As a reference, we compare with the results of [CITATION] for wall-bounded channel flow at the same [MATH], who find that the turbulence in the outer flow behaves roughly isotropic if VLSMs are artificially removed (i.e. VLSMs introduce anisotropy).', '1811.04339-2-23-2': 'Comparing between [CITATION] (who filter out high wavelengths, so [MATH] with only [MATH] are plotted) and the present (unfiltered) simulation in figures [REF](a), we observe that for unladen flow, with or without the contribution from turbulent structures of [MATH], the spectral signature of LSMs is hardly affected (see the spectrum below [MATH] with [MATH]) .', '1811.04339-2-23-3': 'As noted previously, any effect of a limited streamwise domain extent [MATH] is minimal, since figure [REF](a) shows that the energy contained in VLSMs in a short domain (contour represents for [MATH]) is nearly identical to the long domain ([MATH] is shown by lines).', '1811.04339-2-23-4': 'Thus overall our unladen simulations are consistent with the current understanding of VLSMs.', '1811.04339-2-23-5': 'We point out that in single-phase flow, only [MATH] times the maximum value based on the single-phase flow are displayed in figure [REF](a); therefore the second peak in the outer layer is not as readily observed as in figure [REF](a).', '1811.04339-2-23-6': 'For [MATH] (figure [REF](b)) and [MATH] (figure [REF](e)), the contribution from VLSMs forms a bimodal spanwise spectra at [MATH]) at heights above the inner layer.', '1811.04339-2-23-7': 'The enhancement of the VLSM signature is also found at [MATH], [MATH] as shown in figures [REF](g-i) for [MATH].', '1811.04339-2-23-8': 'The Stokes number based on the outer time scale for [MATH] is similar to that of [MATH] at [MATH].', '1811.04339-2-23-9': 'By investigating inertial particle modulation of the regeneration cycle of LSMs, [CITATION] found that particle inertia has a non-monotonic effect on LSMs in the inner region: low inertia ([MATH], e.g. [MATH] of [MATH]) promote the regeneration cycle whereas high inertia ([MATH], i.e. [MATH] of [MATH]) attenuate the regeneration cycle.', '1811.04339-2-23-10': 'However we see from figure [REF] that VLSM enhancement occurs at both low and high Stoke numbers (i.e. [MATH] and [MATH]).', '1811.04339-2-24-0': 'In order to quantify the TKE of VLSMs and LSMs modulated by particles and demonstrate their effect as function of Stokes number, we show [MATH] for LSMs (represented by [MATH]) and VLSMs (represented by [MATH]) in figures [REF](a) and (b), respectively.', '1811.04339-2-24-1': 'The TKE of LSMs is [MATH] times larger than the TKE of VLSMs.', '1811.04339-2-24-2': 'The [MATH] contributed by LSMs and VLSMs is shown as a function of Stokes number for the representative inner layer height [MATH] in figure [REF](c) and the representative outer layer height [MATH] in figure [REF](d).', '1811.04339-2-24-3': 'Figure [REF](c) shows that in the inner layer, the minimum TKE of LSMs appears for [MATH]), corresponding to the strongest turbulence attenuation observed by [CITATION].', '1811.04339-2-24-4': 'In the outer layer, the minimum TKE of LSMs appears at a higher Stokes number, i.e. [MATH]).', '1811.04339-2-24-5': 'However, the TKE modulation of VLSMs is distinct, and even opposite in behaviour, from that of LSMs.', '1811.04339-2-24-6': 'As shown in figure [REF](c), in the inner layer, the intensity of VLSMs reaches its maximum for [MATH]) whereas the TKE of VLSMs has two peaks in the outer layer: [MATH]) and [MATH]).', '1811.04339-2-24-7': 'As argued below, these two peaks correspond to indirect and direct modulation mechanisms, respectively, and will be discussed in section [REF].', '1811.04339-2-25-0': 'Additional qualitative evidence of VLSM enhancement can be seen in figure [REF], which provides a representative snapshot of the streamwise velocity fluctuation in the [MATH] plane at [MATH].', '1811.04339-2-25-1': 'Comparing the snapshots in figures [REF](c,e) and (a), it may not easy to detect VLSM modulation.', '1811.04339-2-25-2': 'However when the flow field is filtered by a threshold [MATH] and [MATH], as displayed in figures [REF](b,d,f), it is evident that VLSMs are stronger and more coherent in [MATH] (figure [REF](d)) and [MATH] (figure [REF](f)) as compared to single-phase flow.', '1811.04339-2-25-3': 'This is a more qualitative confirmation of the results shown in figures [REF], [REF], and [REF].', '1811.04339-2-25-4': 'We now turn our investigation to understanding why the VLSM modulation appears to have two distinct peaks in Stokes number - in particular demonstrated by [MATH] and [MATH].', '1811.04339-2-26-0': '## Two mechanisms of VLSMs enhancement by particles', '1811.04339-2-27-0': "In order to verify the above hypothesis that the VLSM enhancement in [MATH] is due to particles' modulation of LSMs in the inner flow (we refer to this as indirect modulation of VLSMs) whereas in [MATH] it is due to the particles' direct modulation on the VLSMs in the outer flow, we perform a conditional numerical test to identify the particles' effective region of influence regarding VLSM enhancement by artificially applying the particle feedback force only in one of three locations: (1) the viscous sublayer ([MATH]), (2) the regeneration cycle region ([MATH]), or (3) the outer flow ([MATH]), separately.", '1811.04339-2-27-1': 'The premultiplied two-dimensional energy spectrum of streamwise velocity [MATH] is shown in figure [REF] as a function of [MATH].', '1811.04339-2-27-2': 'Compared with figure [REF](a), it can be seen that the VLSMs only experience enhancement in [MATH] when particle coupling is included in the regeneration cycle region.', '1811.04339-2-27-3': 'For [MATH], on the other hand, the opposite is true: VLSM enhancement is found only when particle coupling effects are included in the outer region.', '1811.04339-2-27-4': 'Both of these effects are observed throughout the entire range of [MATH].', '1811.04339-2-27-5': 'The tests of two-way coupling applied for [MATH] are not shown, but we find the spectrum in the range [MATH]) and [MATH]) is stronger than in the single-phase flow, but shorter and narrower than the streamwise and spanwise scale of the second peak of the TKE spectrum.', '1811.04339-2-28-0': 'In spectral space, the modulation of LSMs and VLSMs by a dispersed phase is at least partially related to the direct influence on velocity fluctuations, which in turn can modify the production of TKE and/or Reynolds shear stress.', '1811.04339-2-28-1': 'This is demonstrated in figure [REF], where we present the [MATH] spectral production term [MATH] as well as the [MATH] spectral production term [MATH] as a function of [MATH] and the wall-normal direction [MATH] for two different Stokes numbers both previously seen to enhance VLSMs: low Stokes number [MATH] and high Stokes number [MATH].', '1811.04339-2-28-2': 'These are shown in comparison with single-phase flow [MATH].', '1811.04339-2-28-3': 'Throughout the wall-normal direction, [MATH] is positive whereas [MATH] is negative.', '1811.04339-2-28-4': 'Comparing figures [REF](a,b,c) with figures [REF](a,b,e) respectively, we find that they have a similar overall shape, and the bimodal spectrum appears both in the premultipled [MATH]spectra as well as the production term [MATH], which is enhanced in [MATH] and [MATH] in comparison with the single-phase flow.', '1811.04339-2-28-5': 'In regards to [MATH], a similar overall shape as compared to premultipled [MATH]-cospectra (figure is not shown) is observed even though the bimodal spectrum is not as obviously established as that for [MATH].', '1811.04339-2-28-6': 'The intensity of [MATH] is negative in the domain) is weakened in [MATH] and [MATH] in comparison with the single-phase flow by the presence of particles.', '1811.04339-2-28-7': 'Thus figure [REF] indicates that production of [MATH] is enhanced at the heights and wavenumbers associated with VLSMs, while at the same time, particularly for [MATH], the production of Reynolds shear stress is diminished at the same wavelengths and heights.', '1811.04339-2-29-0': 'In addition to the modifications to streamwise TKE and Reynolds shear stress production, particles can also act as a direct source/sink in the spectral TKE and Reynolds stress budgets.', '1811.04339-2-29-1': 'In the spectral energy budget, particle sources to the [MATH] budget are denoted as [MATH] and to the [MATH] budget as [MATH], where [MATH] stands for the real part and [MATH] is the Fourier transform of the particle coupling force.', '1811.04339-2-29-2': 'The mean value of [MATH] and [MATH] (where [MATH], [MATH] is the fluctuation of the particle feedback force on the carrier phase) in the inner layer and outer layer of all modes normalized by the bulk mass fraction [MATH], is shown in figure [REF](a) and figure [REF](a), respectively.', '1811.04339-2-29-3': 'In both the inner layer and outer layer, the sign of [MATH] is positive for [MATH] whereas it becomes negative for [MATH].', '1811.04339-2-29-4': 'This is opposite when compared to [MATH], indicating that [MATH] and [MATH] play opposite roles in the streamwise TKE budget and Reynolds stress budgets for the same Stokes number.', '1811.04339-2-29-5': 'In the spectral energy budget, the particle source to the [MATH] budget is denoted by [MATH] and to the [MATH] budget is [MATH] (both normalized by local particle mass fraction [MATH] in figures [REF] and [REF]).', '1811.04339-2-29-6': 'With increasing Stokes number as shown in figures [REF](b-f) and figures [REF](b-f), the regions of the highest magnitudes of [MATH] and [MATH] shift from the inner layer to the outer layer, and at the same time from low to high wavelength.', '1811.04339-2-29-7': 'In particular, it is found that for [MATH], a region of large, positive [MATH] (figure [REF](b)) appears in the inner layer with wavelengths associated with LSMs, whereas in [MATH], a region of large, positive [MATH] region instead develops at wavelengths associated with VLSMs (figure [REF](e)).', '1811.04339-2-29-8': 'In contrast, [MATH] exhibits a negative source of [MATH] at these wavelengths (figure [REF](b)), but for [MATH] there is at the same time a noticeable change in the sign of the contribution and location of [MATH] (figure [REF](e)).', '1811.04339-2-29-9': 'This picture is consistent with the conditional tests above that [MATH] works with LSMs in the inner layer whereas [MATH] works with VLSMs in the outer layer.', '1811.04339-2-29-10': 'This nearly opposite behavior indicates that there might be two underpinning mechanisms of VLSM enhancement induced by low and high Stokes numbers particles.', '1811.04339-2-30-0': 'In particle-laden flow, [CITATION] finds that the particle-induced, Stokes-number-dependent source/sink of TKE varies with wavelength, and is possibly associated with the particle clusters themselves .', '1811.04339-2-30-1': 'Focusing exclusively on the inner region, [CITATION] find that low inertia particles enhance LSMs whereas high inertia particles attenuate LSMs, which can help explain the positive particle feedback [MATH] for [MATH] (figure [REF](b)) but negative for [MATH] (figure [REF](e)).', '1811.04339-2-30-2': 'In addition, as shown in figure [REF](b), the particle source [MATH] in [MATH] always attenuates the generation of [MATH].', '1811.04339-2-30-3': 'Therefore in [MATH], the positive feedback [MATH] in the inner region is the most likely mechanism responsible for the enhancement of VLSMs in the outer region.', '1811.04339-2-30-4': 'This process of particles inducing upscale energy transfer (or a reverse cascade), tending to build up the energy level at high wavelengths due to the modulation of small-scale turbulent motions, is also observed in homogeneous turbulence (see [CITATION] and [CITATION]) and in turbulent Couette flow (see [CITATION]).', '1811.04339-2-31-0': 'At low Stokes number ([MATH]), [CITATION] found that the disturbance energy needed to induce turbulence is low at small mass fractions (i.e. [MATH]), whereas is high at large mass fraction (i.e. [MATH]), indicating a subtle dependence of two-way coupling effects on mass fraction at low Stokes number.', '1811.04339-2-31-1': 'In the current context, this understanding can be used to better interpret and understand the indirect modulation of VLSMs by low Stokes number particles.', '1811.04339-2-31-2': 'Figure [REF] shows one-dimensional [MATH]spectra [MATH] with increasing mass fraction ranging from [MATH] to [MATH], all compared with the unladen flow.', '1811.04339-2-31-3': 'We observe that the modulation of VLSMs as a function of mass fraction is consistent with the behaviour shown by [CITATION]: VLSMs are slightly enhanced by a small mass fraction of low Stokes number particles ([MATH] in figure [REF](b)) and significantly promoted by increasing the mass fraction to [MATH] (figure [REF](c)).', '1811.04339-2-31-4': 'With further increases in mass fraction, however ([MATH] in figure [REF](d)), this enhancement begins to diminish.', '1811.04339-2-31-5': 'Again, the non-monotonic response of LSM regeneration found in [CITATION] and [CITATION] appears to be linked to the underlying non-monotonic response of VLSM enhancement to low Stokes number particles.', '1811.04339-2-32-0': 'Finally, in contrast to [MATH], it is more straightforward to understand the VLSM modulation in [MATH].', '1811.04339-2-32-1': 'As shown in figure [REF](e), throughout the whole domain we observe that [MATH] for [MATH] always has a positive contribution to the [MATH] budget and is at the same spatial locations as the VLSMs of the [MATH]-spectra (seen also at [MATH] in [MATH], figure is not shown).', '1811.04339-2-32-2': 'On the other hand, [MATH] for [MATH] in figure [REF](e) tends to suppress the generation of [MATH].', '1811.04339-2-32-3': 'This ultimately results in [MATH] exerted in the outer region as being the most likely explanation for the enhancement of VLSMs.', '1811.04339-2-32-4': "As the source of Reynolds shear stress [MATH], the 'upwelling' and 'downwelling' cycles at very large scales are directly enhanced by the presence of high inertia particles ([MATH] of [MATH] at [MATH] and [MATH] of [MATH] at [MATH]) in the outer flow.", '1811.04339-2-32-5': 'These very large upwelling/downwelling structures further extract energy from the mean flow by working with local mean shear, as the production of streamwise turbulent kinetic energy budget ([MATH]); see for example the physical explanation from [CITATION].', '1811.04339-2-33-0': '# Conclusions', '1811.04339-2-34-0': 'In this paper, we have studied the effect of inertial particles on VLSMs in moderate Reynolds number in open channel flow.', '1811.04339-2-34-1': 'Higher particle concentrations are observed in the inner layer than the outer layer due to the towards-wall particle flux induced by turbophoresis.', '1811.04339-2-34-2': 'The particle concentration has a non-monotonic dependence on Stokes number whereas the trend is opposite between the inner layer and outer layer.', '1811.04339-2-34-3': 'In the inner layer, the particles are characterized by well-known preferential accumulation patterns in the anisotropic LSMs, especially in low-speed streaks, and this behaviour scales as [MATH] based on inner units.', '1811.04339-2-34-4': 'However with increasing wall-normal distance, additional clustering structures are formed in the outer flow.', '1811.04339-2-34-5': 'The clustering behaviour is found to be dependent on [MATH] based on the local Kolmogorov scale, similar to the traditional picture described in isotropic turbulence.', '1811.04339-2-34-6': "In addition, while particles preferentially accumulate in 'upwelling' LSMs within the inner layer (especially at [MATH] in [MATH]), in the outer layer they cluster both in 'upwelling' and 'downwelling' VLSMs (especially at [MATH] in [MATH]).", '1811.04339-2-34-7': 'The distinct bulk concentration and clustering behavior in the two layers are non-monotonically dependent on Stokes number, thereby influencing two-way coupling.', '1811.04339-2-34-8': 'This is observed primarily in spectral analysis, where we observe that inertial particles have a non-monotonic effect on the VLSM modulation: low and high inertia particles both strengthen the VLSMs but the intermediate inertia particles hardly affect their structure and energy.', '1811.04339-2-35-0': 'By utilizing a conditional numerical test, we demonstrate there are two distinct routes through which inertial particles enhance the VLSMs in the outer layer: low inertia ([MATH] based on the inner scale) particles strengthen the VLSMs due to the enhancement of the LSMs in the inner flow.', '1811.04339-2-35-1': 'On the contrary, high inertia ([MATH] based on the outer scale) particles strengthen the VLSMs due to direct interaction in the outer flow.', '1811.04339-2-35-2': 'The most direct route of particle modulation of turbulent motions comes from the particle feedback source in the turbulent energy budget.', '1811.04339-2-35-3': 'Correspondingly, we find that low inertia particles have a positive [MATH] in the inner flow and a negative [MATH] in the outer flow, which is opposite to high inertia particles.', '1811.04339-2-35-4': 'While the relationship between near-wall LSMs and the outer-scale VLSMs remains a subject of investigation, this suggests that there can exist an upscale transport of energy possible from LSMs to VLSMs.', '1811.04339-2-35-5': 'Inspired by previously observed, non-monotonic modulation of turbulence of low inertia particles in the inner layer with varying mass fraction , we observe a VLSM modulation pattern with respect to mass fraction in the outer layer which coincides with the turbulence instability response in the inner layer.', '1811.04339-2-35-6': 'This is consistent with [CITATION], who show numerically that LSMs and VLSMs interact in a co-supporting cycle, [CITATION] who observe experimentally the high degree of velocity fluctuation correlation between the outer flow with the low-frequency content of the inner flow, and [CITATION] who describe an inverse scale transfer from LSMs to VLSMs close to the wall.', '1811.04339-2-35-7': 'In contrast, high inertia particles modulate the VLSMs directly, indicated by the particle feedback effect on the Reynolds shear stress budget with the same scale of VLSMs at the same spatial locations.'}

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[]

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[]

[]

{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1811.04339

1807.06749

{'1807.06749-1-0-0': 'Most real-world systems are complex and hard to model accurately.', '1807.06749-1-0-1': 'Machine learning has been used to model complex dynamical systems (e.g. articulated robot structures, cable stretch) or coupled with reinforcement learning to learn new tasks based on vision and position sensors (e.g. grasping, reaching).', '1807.06749-1-0-2': 'To solve complex tasks using machine learning techniques, availability of a suitable dataset is an important factor.', '1807.06749-1-0-3': 'The robotic community still lacks public datasets, especially for problems that are complex to model like contact tasks, where it is difficult to obtain a precise model of the physical interaction between two objects.', '1807.06749-1-0-4': 'In this paper, we provide a public dataset for insertion of convex-shaped pegs in holes and analyze the nature of the task.', '1807.06749-1-0-5': 'We demonstrate using the data how a robot learns to insert polyhedral pegs into holes using only a 6-axis force/torque sensor.', '1807.06749-1-0-6': 'This dataset can also be used to learn other contact tasks such as shape recognition.', '1807.06749-1-1-0': '# Introduction', '1807.06749-1-2-0': 'Robot manufacturers are focused on making robots simpler to program to speed up the configuration of new assembly lines.', '1807.06749-1-2-1': 'Owing to recent advances in deep learning and machine learning, robots are becoming more flexible.', '1807.06749-1-2-2': 'Instead of manual programming, modern artificial intelligence allows robots to learn new tasks by looking at demonstrations or actively learning without explicit teaching.', '1807.06749-1-2-3': 'Recent works have already shown the potential to learn the robot dynamics [CITATION] or learn contact dynamics during a peg-in-the-hole task [CITATION].', '1807.06749-1-3-0': 'Data is a key to the success of machine learning for solving complex tasks.', '1807.06749-1-3-1': 'The emergence of large datasets has played a prominent role in different research communities where deep learning has provided state-of-the-art results, e.g. natural language processing [CITATION], image and score understanding [CITATION].', '1807.06749-1-3-2': 'The robotic community still lacks public datasets, especially for problems that are complex to model like contact tasks, where it is still difficult to obtain a precise model of the physical interaction between two objects [CITATION].', '1807.06749-1-3-3': 'Therefore, we believe that availability of more datasets collected using real robots is crucial.', '1807.06749-1-3-4': 'Towards this ambitious goal, Yu et al. [CITATION] is one of the first works to provide a large dataset on a robot contact task, with force information during pushing task.', '1807.06749-1-4-0': 'In this paper, we choose one of the most common industrial tasks: the peg-in-hole task.', '1807.06749-1-4-1': 'We provide a dataset of a force/torque (F/T) data of peg-in-hole operations with polyhedral pegs and holes.', '1807.06749-1-4-2': 'If the robot has a precise position control and the hole pose is estimated with enough accuracy, we can solve this problem using position commands.', '1807.06749-1-4-3': 'However, usually due to uncertainty of robotic assembly, the task becomes unsolvable by positioning alone; the sources of the uncertainty include object positioning errors, hole pose estimation inaccuracy and grasping inaccuracy.', '1807.06749-1-4-4': 'Hence, in this paper, we put emphasis on the F/T data of the task.', '1807.06749-1-4-5': 'The F/T dataset presented in this paper allow to assess the feasibility of novel techniques before further effort to realize them physically or to help pre-train neural networks for insertion tasks or shape recognition.', '1807.06749-1-5-0': '# Force-Based Insertion Dataset', '1807.06749-1-6-0': 'In this paper, we choose a strategy to solve the peg-in-hole [CITATION]: i) position the peg at a predefined height from the hole, ii) push the peg with a downwards force, iii) place the peg center within the clearance region of the hole center applying force/torque movements (search phase) and iv) push the peg with a downwards force (insertion phase).', '1807.06749-1-7-0': '## Data collection', '1807.06749-1-8-0': 'The dataset records object positions and interaction forces for a set of polyhedral pegs in contact with holes (see Fig. [REF]).', '1807.06749-1-8-1': 'The face of the polyhedron in contact with the environment is n-gon regular convex polygons with [MATH] in Fig. [REF].', '1807.06749-1-9-0': 'The data was collected by sending a sequence of robot commands: i) pick the peg, ii) rotate it in the given direction, iii) move to the center of hole with a predefined offset in [MATH] and [MATH] directions, iv) pushes the peg against the hole plate with a downwards force of 30 for 10.', '1807.06749-1-9-1': 'Pushing the plate for a long time help the controller passing form the transient to the steady situation.', '1807.06749-1-9-2': 'The force control is executed and recorded at 100.', '1807.06749-1-9-3': 'The value of the force torque sensor and the end effector position for each point is recorded and stored as vectors of [MATH].', '1807.06749-1-9-4': '[MATH], [MATH], [MATH], are the peg positions with respect to the hole; [MATH] is the peg angle with the respect to the hole angle; [MATH], [MATH], [MATH], [MATH], [MATH], [MATH] are the forces and moments in the force sensor frame; [MATH] is the time and [MATH] is a counter of the datapoints in Fig. [REF].', '1807.06749-1-10-0': 'As we are using a precisely calibrated table with a grid of screw holes, we can know the exact position of the hole.', '1807.06749-1-10-1': 'To ensure that the relative position of peg and the hole are known accurately, we start each data collection by manually inserting the peg in the hole at correct orientation, making both position and angle of the peg aligned with the hole.', '1807.06749-1-10-2': 'In this way, grasp errors do not have any effect on the experiments.', '1807.06749-1-10-3': 'The position of the force sensor with respect to the peg position is shown in Fig. [REF].', '1807.06749-1-10-4': 'The peg is moved by increments of 1.5 in [MATH] and [MATH] direction within the range of [MATH]8.5 from the center of the hole (see Fig. [REF]).', '1807.06749-1-11-0': 'Along with the dataset we also release the 3D models of all the objects used for collecting data.', '1807.06749-1-11-1': 'Details about the structure of the dataset are given inside the folders.', '1807.06749-1-12-0': 'The dataset is available at', '1807.06749-1-13-0': 'http://ibm.biz/multishapeinsertion', '1807.06749-1-14-0': '## Hardware', '1807.06749-1-15-0': 'Fig. [REF] shows the setup for collecting the data.', '1807.06749-1-16-0': '## Software', '1807.06749-1-17-0': 'The UR5 robot has many components available in the Robot Operating System (ROS) framework.', '1807.06749-1-17-1': 'We use such ROS nodes to collect our data.', '1807.06749-1-17-2': 'The F/T captured data are published as ROS topics and recorded at 100 Hz.', '1807.06749-1-17-3': 'The object position with respect to the hole is only given before the peg goes in contact with the hole.', '1807.06749-1-18-0': 'As we are in contact with the environment during the search phase and alignment, we adopt a common admittance control to stabilize the interaction betweeen robot and environment.', '1807.06749-1-18-1': 'This controller is common for many industrial manipulators controlled by a position controller [CITATION].', '1807.06749-1-19-0': '# A Method for Labeling the Data for Multi-Shape Insertions', '1807.06749-1-20-0': 'Here we illustrate a method to label each entry of the dataset for performing the peg-in-hole task.', '1807.06749-1-20-1': 'In the next section, we show how we can label each entry of the dataset for the shape recognition task.', '1807.06749-1-21-0': 'During the execution of the peg-in-hole task, the position and orientation of the hole are inaccurate due to different uncertainties.', '1807.06749-1-21-1': 'Using the accurate position and orientation of our dataset, we can calculate for each entry the best action that should be performed.', '1807.06749-1-21-2': 'The action is the label and the input is the force and moments.', '1807.06749-1-21-3': 'The position and orientation are then only used for labeling the data.', '1807.06749-1-22-0': 'Using only the peg positions [MATH], we label each entry of the dataset in 4 actions: move left, move down, move right and move up (see Fig. [REF]).', '1807.06749-1-22-1': 'These actions allows to reduce the position error of the peg respect to the hole during the search phase.', '1807.06749-1-23-0': 'To reduce the orientation error of the peg respect to the hole, we use the following labels:', '1807.06749-1-24-0': 'where [MATH] is the peg orientation respect to the hole and [MATH] is manually defined as a function of the clearance between the peg and hole.', '1807.06749-1-25-0': '# Analysis and Experiments', '1807.06749-1-26-0': '## Analysis of Force Data', '1807.06749-1-27-0': 'Fig. [REF] shows the forces and moments of the dataset for a peg with [MATH] at the top left position in Fig. [REF].', '1807.06749-1-28-0': 'In Fig. [REF], we can clearly distinguish three phases:', '1807.06749-1-29-0': '[label=*.]', '1807.06749-1-29-1': 'Non-contact: the peg is not in contact with the environment.', '1807.06749-1-29-2': 'This situation is not interesting for analyzing the contact and the insertion.', '1807.06749-1-29-3': 'Transient: the forces and moments keep changing with time.', '1807.06749-1-29-4': 'In this period, we can analyze the response of the robot to the interaction with the environment.', '1807.06749-1-29-5': 'Steady: the forces and moments of interactions remains almost unchanged in time.', '1807.06749-1-30-0': 'In the next sections, we will start to analyze situations B and C separately.', '1807.06749-1-30-1': 'Lastly, we also analyze the combination of the data from both situations.', '1807.06749-1-31-0': '## Comparaison of Classifiers', '1807.06749-1-32-0': 'We train multiple classifiers using different methods with a cross entropy cost function.', '1807.06749-1-32-1': 'The classifier aims to find the best action for the given input (forces and moments).', '1807.06749-1-32-2': 'Note that the accuracy of the classifier is the accuracy to provide a correct action given the forces and moments as inputs and is not the probability of entering the hole.', '1807.06749-1-33-0': 'To compare the classifiers using different machine learning (ML) methods, we prepare the training data for each method by the following procedure.', '1807.06749-1-33-1': 'First, we sample raw data from a time window starting at the index [MATH] and ending at the index [MATH].', '1807.06749-1-33-2': 'We separate the data to frames each of which has the length of [MATH].', '1807.06749-1-33-3': 'That results in [MATH] frames.', '1807.06749-1-33-4': 'In each frame, we calculate the average of the raw data.', '1807.06749-1-33-5': 'We use the obtained [MATH] average data as a training data.', '1807.06749-1-34-0': 'We compare the accuracy of the ML methods of two tasks using the label explained in Sec. [REF]; (T1) reduce the position error between peg and hole position using 4 force actions, and (T2) reduce the position error between peg and hole position using 3 moment actions.', '1807.06749-1-35-0': 'In Table [REF], we compare the classifiers using only shape [MATH] and the following 4 inputs: [EQUATION] with [MATH], [MATH], [MATH].', '1807.06749-1-35-1': 'We made the comparison using the following techniques: SVM is support vector machine classifier with linear kernel, DT is the Decision Tree, RNDF is the Random Forest method, ADA is Ada Boost classifier, GAUS is the Gaussian Naive Bayes method, LDA is a Linear Discriminant Analysis, QDA is a Quadratic Discriminant Analysis and MLP is the Multi Layer Perceptron.', '1807.06749-1-36-0': 'In Sec. [REF], we also compare the results of adding the remaining 2 F/T inputs.', '1807.06749-1-37-0': 'From Table [REF], MLP is the best choice in both tests.', '1807.06749-1-37-1': 'The MLP network is composed of 2 hidden layers of size [100, 50], the optimizer is lbfgs and activation function is rectified linear unit.', '1807.06749-1-37-2': 'In the next sections, we will only use this MLP network.', '1807.06749-1-38-0': '## Study of the different contact situations', '1807.06749-1-39-0': 'In Table [REF], we compare the accuracy for the transient and steady state.', '1807.06749-1-40-0': 'In Table [REF], we noticed that, for the task T2, taking only the average of the data points during the transient situation, the accuracy improved from c) [MATH] to a) [MATH].', '1807.06749-1-40-1': 'On the other hand, by taking the average of only steady situation the results decrease to b) [MATH].', '1807.06749-1-40-2': 'From this result, we can suppose that the information about the task T2 is mostly during transient situation.', '1807.06749-1-41-0': 'Another important result for T2 coming from the analysis of the dataset during the transient situation is that using the following parameters ([MATH], [MATH], [MATH]), the accuracy increases to d) [MATH].', '1807.06749-1-41-1': 'The input is a sequence of 8 data points, i.e. [(600-200)/50=8].', '1807.06749-1-42-0': 'For T1, the accuracy considering only the transient situation increase to a) [MATH] and using only the steady situation the accuracy decreases to b) [MATH].', '1807.06749-1-42-1': 'Using a sequence of 8 points as input during the transient situation the accuracy increase to d) [MATH].', '1807.06749-1-42-2': 'As shown in Table [REF] for ([MATH], [MATH], [MATH]), the steady contact situation is the most important phase for T1 and the accuracy increases to f) [MATH]).', '1807.06749-1-43-0': 'During the steady situation using the following parameters ([MATH], [MATH], [MATH]), the accuracy decrease to e) [MATH] for T1 and to e) [MATH] for T2.', '1807.06749-1-44-0': 'Analyzing these results, we can affirm that the dynamic during the impact between the peg and environment is very important to understand the insertion task for the search and alignment phases.', '1807.06749-1-45-0': 'We can conclude that for our analysis using MLP the main information for T2 is the transient, while for T1 is the whole contact phase.', '1807.06749-1-45-1': 'The parameters ([MATH], [MATH], [MATH]) in Table [REF] are a good compromise and we choose these parameters to analyze the results for different shapes.', '1807.06749-1-46-0': '## Study of different inputs', '1807.06749-1-47-0': 'Another important analysis of the dataset is to understand which inputs are the most important.', '1807.06749-1-47-1': 'Adding [MATH] as input the accuracy decrease from [MATH] to [MATH], adding [MATH] the accuracy is [MATH].', '1807.06749-1-47-2': 'Therefore, the main information is in [MATH], [MATH], [MATH], [MATH].', '1807.06749-1-48-0': '## Study of different shapes', '1807.06749-1-49-0': 'Table [REF] shows the results for the different shapes.', '1807.06749-1-49-1': 'From the table, we can clearly understand that T1 is easier than T2.', '1807.06749-1-49-2': 'In particular, we can notice that while the accuracy for the T1 increase with the number of sides, the accuracy for T2 is similar for all shapes.', '1807.06749-1-50-0': '## Robot Experiments', '1807.06749-1-51-0': 'The model learned using the dataset is used to perform the task on UR5 robot.', '1807.06749-1-51-1': 'As input we use the input of Eq. [REF] with [MATH], [MATH], [MATH].', '1807.06749-1-51-2': 'We use the following 4 actions during T1: [MATH], [MATH], [MATH], [MATH].', '1807.06749-1-51-3': 'And 3 actions during T2 [MATH], [MATH], [MATH].', '1807.06749-1-51-4': 'For the experiment, we fix [MATH], [MATH], [MATH].', '1807.06749-1-51-5': 'Using this parameters the robot perform the insertion task in average after [MATH] actions starting from a distance of the hole of 2 with 100% success rate.', '1807.06749-1-51-6': 'These results depend on the amplitude of the force and moment commands.', '1807.06749-1-52-0': 'The video is available at', '1807.06749-1-53-0': 'https://youtu.be/6rLc9fAtzAQ', '1807.06749-1-54-0': 'In the video, the robot used the learned model to perform the insertion for all shapes.', '1807.06749-1-55-0': '## Shape Recognition', '1807.06749-1-56-0': 'We use the force and moment during contact to recognize the shape of the peg and the hole.', '1807.06749-1-56-1': 'In our dataset, peg and the hole have the same shape.', '1807.06749-1-56-2': 'We label the data using 5 classes (one per shape).', '1807.06749-1-56-3': 'Using MLP and [MATH], [MATH], [MATH], we obtain an accuracy of [MATH].', '1807.06749-1-57-0': 'The result shows that using our dataset, the robot can also recognize the shape of the peg.', '1807.06749-1-57-1': 'If the robot has low confidence that it is holding the correct peg, it can generate an error with the reason of failure.', '1807.06749-1-58-0': '# Conclusions', '1807.06749-1-59-0': 'In this paper, we presented a dataset for multishape peg-in-hole.', '1807.06749-1-59-1': 'Using this dataset, we conducted several analysis and we trained a MLP network able to select the right action based on forces and moments.', '1807.06749-1-59-2': 'The learned motion was tested on the UR5 robot.', '1807.06749-1-60-0': 'In a near future, we would like to work with deeper hole.', '1807.06749-1-60-1': 'Moreover, the current data set does not consider angular alignment errors except for the rotation about the peg axis.', '1807.06749-1-60-2': 'We will investigate more in this direction.', '1807.06749-1-60-3': 'Another interesting area for future works would be transfer learning where the models are learned in simulation and fine-tuned on the real robot or where the insertion is learned from plastic pegs-holes and used with metal objects.'}

{'1807.06749-2-0-0': 'The accurate modeling of real-world systems and physical interactions is a common challenge towards the resolution of robotics tasks.', '1807.06749-2-0-1': 'Machine learning approaches have demonstrated significant results in the modeling of complex systems (e.g., articulated robot structures, cable stretch, fluid dynamics), or to learn robotics tasks (e.g., grasping, reaching) from raw sensor measurements without explicit programming, using reinforcement learning.', '1807.06749-2-0-2': 'However, a common bottleneck in machine learning techniques resides in the availability of suitable data.', '1807.06749-2-0-3': 'While many vision-based datasets have been released in the recent years, ones involving physical interactions, of particular interest for the robotic community, have been scarcer.', '1807.06749-2-0-4': 'In this paper, we present a public dataset on peg-in-hole insertion tasks containing force-torque and pose information for multiple variations of convex-shaped pegs.', '1807.06749-2-0-5': 'We demonstrate how this dataset can be used to train a robot to insert polyhedral pegs into holes using only 6-axis force/torque sensor measurements as inputs, as well as other tasks involving contact such as shape recognition.', '1807.06749-2-1-0': '# Introduction', '1807.06749-2-2-0': 'Robot manufacturers are focused on making robots simpler to program to speed up the configuration of new assembly lines.', '1807.06749-2-2-1': 'Owing to recent advances in deep learning and machine learning, robots are becoming more flexible.', '1807.06749-2-2-2': 'Instead of manual programming, modern artificial intelligence allows robots to learn new tasks by looking at demonstrations or actively learning without explicit teaching.', '1807.06749-2-2-3': 'Recent works have already shown the potential to learn the robot dynamics [CITATION] or learn contact dynamics during a peg-in-the-hole task [CITATION].', '1807.06749-2-3-0': 'Data is a key to the success of machine learning for solving complex tasks.', '1807.06749-2-3-1': 'The emergence of large datasets has played a prominent role in different research communities where deep learning has provided state-of-the-art results, e.g. natural language processing [CITATION], image and score understanding [CITATION].', '1807.06749-2-3-2': 'The robotic community still lacks public datasets, especially for problems that are complex to model like contact tasks, where it is still difficult to obtain a precise model of the physical interaction between two objects [CITATION].', '1807.06749-2-3-3': 'Therefore, we believe that availability of more datasets collected using real robots is crucial.', '1807.06749-2-3-4': 'Towards this ambitious goal, Yu et al. [CITATION] is one of the first works to provide a large dataset on a robot contact task, with force information during pushing task.', '1807.06749-2-4-0': 'In this paper, we choose one of the most common industrial tasks: the peg-in-hole task.', '1807.06749-2-4-1': 'We provide a dataset of a force/torque (F/T) data of peg-in-hole operations with polyhedral pegs and holes.', '1807.06749-2-4-2': 'If the robot has a precise position control and the hole pose is estimated with enough accuracy, we can solve this problem using position commands.', '1807.06749-2-4-3': 'However, usually due to uncertainty of robotic assembly, the task becomes unsolvable by positioning alone; the sources of the uncertainty include object positioning errors, hole pose estimation inaccuracy and grasping inaccuracy.', '1807.06749-2-4-4': 'Hence, in this paper, we put emphasis on the F/T data of the task.', '1807.06749-2-4-5': 'The F/T dataset presented in this paper allow to assess the feasibility of novel techniques before further effort to realize them physically or to help pre-train neural networks for insertion tasks or shape recognition.', '1807.06749-2-5-0': '# Force-Based Insertion Dataset', '1807.06749-2-6-0': 'In this paper, we choose a strategy to solve the peg-in-hole [CITATION]: i) position the peg at a predefined height from the hole, ii) push the peg with a downwards force, iii) place the peg center within the clearance region of the hole center applying force/torque movements (search phase) and iv) push the peg with a downwards force (insertion phase).', '1807.06749-2-7-0': '## Data collection', '1807.06749-2-8-0': 'The dataset records object positions and interaction forces for a set of polyhedral pegs in contact with holes (see Fig. [REF]).', '1807.06749-2-8-1': 'The face of the polyhedron in contact with the environment is n-gon regular convex polygons with [MATH] in Fig. [REF].', '1807.06749-2-9-0': 'The data was collected by sending a sequence of robot commands: i) pick the peg, ii) rotate it in the given direction, iii) move to the center of hole with a predefined offset in [MATH] and [MATH] directions, iv) pushes the peg against the hole plate with a downwards force of 30 for 10.', '1807.06749-2-9-1': 'Pushing the plate for a long time help the controller passing form the transient to the steady situation.', '1807.06749-2-9-2': 'The force control is executed and recorded at 100.', '1807.06749-2-9-3': 'The value of the force torque sensor and the end effector position for each point is recorded and stored as vectors of [MATH].', '1807.06749-2-9-4': '[MATH], [MATH], [MATH], are the peg positions with respect to the hole; [MATH] is the peg angle with the respect to the hole angle; [MATH], [MATH], [MATH], [MATH], [MATH], [MATH] are the forces and moments in the force sensor frame; [MATH] is the time and [MATH] is a counter of the datapoints in Fig. [REF].', '1807.06749-2-10-0': 'As we are using a precisely calibrated table with a grid of screw holes, we can know the exact position of the hole.', '1807.06749-2-10-1': 'To ensure that the relative position of peg and the hole are known accurately, we start each data collection by manually inserting the peg in the hole at correct orientation, making both position and angle of the peg aligned with the hole.', '1807.06749-2-10-2': 'In this way, grasp errors do not have any effect on the experiments.', '1807.06749-2-10-3': 'The position of the force sensor with respect to the peg position is shown in Fig. [REF].', '1807.06749-2-10-4': 'The peg is moved by increments of 1.5 in [MATH] and [MATH] direction within the range of [MATH]8.5 from the center of the hole (see Fig. [REF]).', '1807.06749-2-11-0': 'Along with the dataset we also release the 3D models of all the objects used for collecting data.', '1807.06749-2-11-1': 'Details about the structure of the dataset are given inside the folders.', '1807.06749-2-12-0': 'The dataset is available at', '1807.06749-2-13-0': 'http://ibm.biz/multishapeinsertion', '1807.06749-2-14-0': '## Hardware', '1807.06749-2-15-0': 'Fig. [REF] shows the setup for collecting the data.', '1807.06749-2-16-0': '## Software', '1807.06749-2-17-0': 'The UR5 robot has many components available in the Robot Operating System (ROS) framework.', '1807.06749-2-17-1': 'We use such ROS nodes to collect our data.', '1807.06749-2-17-2': 'The F/T captured data are published as ROS topics and recorded at 100 Hz.', '1807.06749-2-17-3': 'The object position with respect to the hole is only given before the peg goes in contact with the hole.', '1807.06749-2-18-0': 'As we are in contact with the environment during the search phase and alignment, we adopt a common admittance control to stabilize the interaction betweeen robot and environment.', '1807.06749-2-18-1': 'This controller is common for many industrial manipulators controlled by a position controller [CITATION].', '1807.06749-2-19-0': '# A Method for Labeling the Data for Multi-Shape Insertions', '1807.06749-2-20-0': 'Here we illustrate a method to label each entry of the dataset for performing the peg-in-hole task.', '1807.06749-2-20-1': 'In the next section, we show how we can label each entry of the dataset for the shape recognition task.', '1807.06749-2-21-0': 'During the execution of the peg-in-hole task, the position and orientation of the hole are inaccurate due to different uncertainties.', '1807.06749-2-21-1': 'Using the accurate position and orientation of our dataset, we can calculate for each entry the best action that should be performed.', '1807.06749-2-21-2': 'The action is the label and the input is the force and moments.', '1807.06749-2-21-3': 'The position and orientation are then only used for labeling the data.', '1807.06749-2-22-0': 'Using only the peg positions [MATH], we label each entry of the dataset in 4 actions: move left, move down, move right and move up (see Fig. [REF]).', '1807.06749-2-22-1': 'These actions allows to reduce the position error of the peg respect to the hole during the search phase.', '1807.06749-2-23-0': 'To reduce the orientation error of the peg respect to the hole, we use the following labels:', '1807.06749-2-24-0': 'where [MATH] is the peg orientation respect to the hole and [MATH] is manually defined as a function of the clearance between the peg and hole.', '1807.06749-2-25-0': '# Analysis and Experiments', '1807.06749-2-26-0': '## Analysis of Force Data', '1807.06749-2-27-0': 'Fig. [REF] shows the forces and moments of the dataset for a peg with [MATH] at the top left position in Fig. [REF].', '1807.06749-2-28-0': 'In Fig. [REF], we can clearly distinguish three phases:', '1807.06749-2-29-0': '[label=*.]', '1807.06749-2-29-1': 'Non-contact: the peg is not in contact with the environment.', '1807.06749-2-29-2': 'This situation is not interesting for analyzing the contact and the insertion.', '1807.06749-2-29-3': 'Transient: the forces and moments keep changing with time.', '1807.06749-2-29-4': 'In this period, we can analyze the response of the robot to the interaction with the environment.', '1807.06749-2-29-5': 'Steady: the forces and moments of interactions remains almost unchanged in time.', '1807.06749-2-30-0': 'In the next sections, we will start to analyze situations B and C separately.', '1807.06749-2-30-1': 'Lastly, we also analyze the combination of the data from both situations.', '1807.06749-2-31-0': '## Comparaison of Classifiers', '1807.06749-2-32-0': 'We train multiple classifiers using different methods with a cross entropy cost function.', '1807.06749-2-32-1': 'The classifier aims to find the best action for the given input (forces and moments).', '1807.06749-2-32-2': 'Note that the accuracy of the classifier is the accuracy to provide a correct action given the forces and moments as inputs and is not the probability of entering the hole.', '1807.06749-2-33-0': 'To compare the classifiers using different machine learning (ML) methods, we prepare the training data for each method by the following procedure.', '1807.06749-2-33-1': 'First, we sample raw data from a time window starting at the index [MATH] and ending at the index [MATH].', '1807.06749-2-33-2': 'We separate the data to frames each of which has the length of [MATH].', '1807.06749-2-33-3': 'That results in [MATH] frames.', '1807.06749-2-33-4': 'In each frame, we calculate the average of the raw data.', '1807.06749-2-33-5': 'We use the obtained [MATH] average data as a training data.', '1807.06749-2-34-0': 'We compare the accuracy of the ML methods of two tasks using the label explained in Sec. [REF]; (T1) reduce the position error between peg and hole position using 4 force actions, and (T2) reduce the position error between peg and hole position using 3 moment actions.', '1807.06749-2-35-0': 'In Table [REF], we compare the classifiers using only shape [MATH] and the following 4 inputs: [EQUATION] with [MATH], [MATH], [MATH].', '1807.06749-2-35-1': 'We made the comparison using the following techniques: SVM is support vector machine classifier with linear kernel, DT is the Decision Tree, RNDF is the Random Forest method, ADA is Ada Boost classifier, GAUS is the Gaussian Naive Bayes method, LDA is a Linear Discriminant Analysis, QDA is a Quadratic Discriminant Analysis and MLP is the Multi Layer Perceptron.', '1807.06749-2-36-0': 'In Sec. [REF], we also compare the results of adding the remaining 2 F/T inputs.', '1807.06749-2-37-0': 'From Table [REF], MLP is the best choice in both tests.', '1807.06749-2-37-1': 'The MLP network is composed of 2 hidden layers of size [100, 50], the optimizer is lbfgs and activation function is rectified linear unit.', '1807.06749-2-37-2': 'In the next sections, we will only use this MLP network.', '1807.06749-2-38-0': '## Study of the different contact situations', '1807.06749-2-39-0': 'In Table [REF], we compare the accuracy for the transient and steady state.', '1807.06749-2-40-0': 'In Table [REF], we noticed that, for the task T2, taking only the average of the data points during the transient situation, the accuracy improved from c) [MATH] to a) [MATH].', '1807.06749-2-40-1': 'On the other hand, by taking the average of only steady situation the results decrease to b) [MATH].', '1807.06749-2-40-2': 'From this result, we can suppose that the information about the task T2 is mostly during transient situation.', '1807.06749-2-41-0': 'Another important result for T2 coming from the analysis of the dataset during the transient situation is that using the following parameters ([MATH], [MATH], [MATH]), the accuracy increases to d) [MATH].', '1807.06749-2-41-1': 'The input is a sequence of 8 data points, i.e. [(600-200)/50=8].', '1807.06749-2-42-0': 'For T1, the accuracy considering only the transient situation increase to a) [MATH] and using only the steady situation the accuracy decreases to b) [MATH].', '1807.06749-2-42-1': 'Using a sequence of 8 points as input during the transient situation the accuracy increase to d) [MATH].', '1807.06749-2-42-2': 'As shown in Table [REF] for ([MATH], [MATH], [MATH]), the steady contact situation is the most important phase for T1 and the accuracy increases to f) [MATH]).', '1807.06749-2-43-0': 'During the steady situation using the following parameters ([MATH], [MATH], [MATH]), the accuracy decrease to e) [MATH] for T1 and to e) [MATH] for T2.', '1807.06749-2-44-0': 'Analyzing these results, we can affirm that the dynamic during the impact between the peg and environment is very important to understand the insertion task for the search and alignment phases.', '1807.06749-2-45-0': 'We can conclude that for our analysis using MLP the main information for T2 is the transient, while for T1 is the whole contact phase.', '1807.06749-2-45-1': 'The parameters ([MATH], [MATH], [MATH]) in Table [REF] are a good compromise and we choose these parameters to analyze the results for different shapes.', '1807.06749-2-46-0': '## Study of different inputs', '1807.06749-2-47-0': 'Another important analysis of the dataset is to understand which inputs are the most important.', '1807.06749-2-47-1': 'Adding [MATH] as input the accuracy decrease from [MATH] to [MATH], adding [MATH] the accuracy is [MATH].', '1807.06749-2-47-2': 'Therefore, the main information is in [MATH], [MATH], [MATH], [MATH].', '1807.06749-2-48-0': '## Study of different shapes', '1807.06749-2-49-0': 'Table [REF] shows the results for the different shapes.', '1807.06749-2-49-1': 'From the table, we can clearly understand that T1 is easier than T2.', '1807.06749-2-49-2': 'In particular, we can notice that while the accuracy for the T1 increase with the number of sides, the accuracy for T2 is similar for all shapes.', '1807.06749-2-50-0': '## Robot Experiments', '1807.06749-2-51-0': 'The model learned using the dataset is used to perform the task on UR5 robot.', '1807.06749-2-51-1': 'As input we use the input of Eq. [REF] with [MATH], [MATH], [MATH].', '1807.06749-2-51-2': 'We use the following 4 actions during T1: [MATH], [MATH], [MATH], [MATH].', '1807.06749-2-51-3': 'And 3 actions during T2 [MATH], [MATH], [MATH].', '1807.06749-2-51-4': 'For the experiment, we fix [MATH], [MATH], [MATH].', '1807.06749-2-51-5': 'Using this parameters the robot perform the insertion task in average after [MATH] actions starting from a distance of the hole of 2 with 100% success rate.', '1807.06749-2-51-6': 'These results depend on the amplitude of the force and moment commands.', '1807.06749-2-52-0': 'The video is available at', '1807.06749-2-53-0': 'https://youtu.be/6rLc9fAtzAQ', '1807.06749-2-54-0': 'In the video, the robot used the learned model to perform the insertion for all shapes.', '1807.06749-2-55-0': '## Shape Recognition', '1807.06749-2-56-0': 'We use the force and moment during contact to recognize the shape of the peg and the hole.', '1807.06749-2-56-1': 'In our dataset, peg and the hole have the same shape.', '1807.06749-2-56-2': 'We label the data using 5 classes (one per shape).', '1807.06749-2-56-3': 'Using MLP and [MATH], [MATH], [MATH], we obtain an accuracy of [MATH].', '1807.06749-2-57-0': 'The result shows that using our dataset, the robot can also recognize the shape of the peg.', '1807.06749-2-57-1': 'If the robot has low confidence that it is holding the correct peg, it can generate an error with the reason of failure.', '1807.06749-2-58-0': '# Conclusions', '1807.06749-2-59-0': 'In this paper, we presented a dataset for multishape peg-in-hole.', '1807.06749-2-59-1': 'Using this dataset, we conducted several analysis and we trained a MLP network able to select the right action based on forces and moments.', '1807.06749-2-59-2': 'The learned motion was tested on the UR5 robot.', '1807.06749-2-60-0': 'In a near future, we would like to work with deeper hole.', '1807.06749-2-60-1': 'Moreover, the current data set does not consider angular alignment errors except for the rotation about the peg axis.', '1807.06749-2-60-2': 'We will investigate more in this direction.', '1807.06749-2-60-3': 'Another interesting area for future works would be transfer learning where the models are learned in simulation and fine-tuned on the real robot or where the insertion is learned from plastic pegs-holes and used with metal objects.'}

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[]

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[['1807.06749-1-0-1', '1807.06749-2-0-1'], ['1807.06749-1-0-2', '1807.06749-2-0-2'], ['1807.06749-1-0-4', '1807.06749-2-0-4'], ['1807.06749-1-0-5', '1807.06749-2-0-5'], ['1807.06749-1-0-6', '1807.06749-2-0-5']]

[]

['1807.06749-1-12-0', '1807.06749-1-13-0', '1807.06749-1-15-0', '1807.06749-1-23-0', '1807.06749-1-28-0', '1807.06749-1-29-0', '1807.06749-1-41-1', '1807.06749-1-51-3', '1807.06749-1-52-0', '1807.06749-1-53-0', '1807.06749-2-12-0', '1807.06749-2-13-0', '1807.06749-2-15-0', '1807.06749-2-23-0', '1807.06749-2-28-0', '1807.06749-2-29-0', '1807.06749-2-41-1', '1807.06749-2-51-3', '1807.06749-2-52-0', '1807.06749-2-53-0']

{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1807.06749

hep-ph-0011088

{'hep-ph-0011088-1-0-0': 'Single particle strength restoration and nuclear transparency in high [MATH] exclusive (e,ep) reactions', 'hep-ph-0011088-1-1-0': 'ABSTRACT', 'hep-ph-0011088-1-2-0': "Quenching of the single particle strength which previously precluded unambiguous measurement of nuclear transparency in quasifree (e,e'p) reactions at [MATH] is evaluated from the cross sections of measured processes.", 'hep-ph-0011088-1-2-1': "We have demonstrated evidence of the single particle strength restoration with increase of [MATH] in the exclusive (e,e'p) reactions and argue that in practically interesting cases of study the nuclear transparency in these processes quenching weakly depends on the probability of short range nucleon correlations in nuclei.", 'hep-ph-0011088-1-2-2': "It is shown that use of Glauber approximation well describes E94-139 and NE18 (e,e'p) data at [MATH].", 'hep-ph-0011088-1-2-3': 'This gives a further support to our observation that the quenching of nuclear levels strongly depends on the resolution ([MATH]) and practically disappears [MATH].', 'hep-ph-0011088-1-3-0': '# Introduction', 'hep-ph-0011088-1-4-0': "The quasi-free knockout (e,e'p) reaction was used for a long time to study nuclear structure at the energy transfer [MATH], for review see [CITATION].", 'hep-ph-0011088-1-4-1': "During the last few years the (e,e'p) experiments at high energies and momentum transfers had been performed[CITATION] to study the nuclear transparency [MATH] as a function of the momentum transfer at [MATH] on several nuclei.", 'hep-ph-0011088-1-4-2': 'The main goal of these experiments was to search for the color transparency(CT) effects [CITATION].', 'hep-ph-0011088-1-5-0': 'The delicate point in extracting the nuclear transparency [MATH] defined as the ratio of the observed cross section to the cross section calculated in the impulse approximation is how large is cross section in the impulse approximation for the kinematics of the particular experiment.', 'hep-ph-0011088-1-5-1': 'Really, it was pointed out in [CITATION] that interpreting transparency measurements at high energies performed in the transverse kinematics with cuts on the momentum of the struck nucleon and the energy of the produced system requires a discussion of the quenching which was observed in the low energy domain.', 'hep-ph-0011088-1-5-2': 'In particular if one would use the same quenching for excitation of [MATH] hole states in the carbon as the one observed in the low [MATH] one would find that [MATH].', 'hep-ph-0011088-1-5-3': 'This number is much higher than the Glauber theory prediction which should be a very good approximation for [MATH] GeV, corresponding to [MATH].', 'hep-ph-0011088-1-5-4': 'At the same time we demonstrated that the carbon data at [MATH] for the differential [MATH] cross section appeared to be consistent with the Glauber calculation provided one assumes a strong reduction of the quenching effect at large [MATH].', 'hep-ph-0011088-1-5-5': 'We further argued that a [MATH] dependence of quenching should be a natural phenomenon reflecting transition from low [MATH] interactions where photon interacts with quasiparticles to the interaction with nucleons at larger [MATH].', 'hep-ph-0011088-1-6-0': 'In this paper we extend the analysis of [CITATION] in several directions: (i) We analyze the transparency measured recently in [CITATION] for a range of nuclei, focusing at [MATH] for which both integrated cross sections and differential cross sections are available.', 'hep-ph-0011088-1-6-1': 'Our choice of [MATH] is motivated by a very good understanding of [MATH] interactions for the corresponding energy of the ejected nucleon [MATH] - the Glauber theory is known to describe numerous data on elastic and quasielastic [MATH] interactions at this energy with a typical accuracy of few percent, see review in [CITATION].', 'hep-ph-0011088-1-6-2': 'Also, due to a weak energy dependence of [MATH] between [MATH] and [MATH] and smallness of the color transparency effects for [MATH] range one expects a very weak dependence of transparency on [MATH] for [MATH].', 'hep-ph-0011088-1-6-3': 'This is certainly consistent with the data.', 'hep-ph-0011088-1-6-4': 'Hence adding higher [MATH] data would not add much to the main trust of our analysis.', 'hep-ph-0011088-1-6-5': '(ii) We observe that measurements of [MATH] in the transverse kinematics of [MATH] are not sensitive to the high momentum component of the nuclear wave function since the cross section is proportional to [MATH] rather than to [MATH], where [MATH] is the nuclear spectral function.', 'hep-ph-0011088-1-6-6': 'Further reduction in the uncertainties is reached by using information on the cross sections of [MATH] scattering at [MATH] and [MATH] measured at Jlab [CITATION] which allows to determine independently [MATH] with accuracy of few %.', 'hep-ph-0011088-1-7-0': 'We check our conclusions about reduction of quenching by comparing the results of our calculations with the [MATH] data from the Jlab experiment [CITATION] for the differential cross sections.', 'hep-ph-0011088-1-7-1': 'Excellent agreement is observed without any adjusted parameters for the region [MATH] where contribution of the short-range correlations is small.', 'hep-ph-0011088-1-7-2': 'This provides a very strong new evidence for practical disappearance of the quenching at large [MATH].', 'hep-ph-0011088-1-8-0': 'In the end of the paper we consider implications for optimizing searches for the color transparency in high [MATH] processes.', 'hep-ph-0011088-1-8-1': 'Numerical predictions for the kinematics where onset of the color transparency is expected will be presented elsewhere.', 'hep-ph-0011088-1-9-0': '# Definition of transparency - fixing denominator', 'hep-ph-0011088-1-10-0': 'Current experiments which study nuclear transparency perform measurements in a restricted region of recoil nuclear momenta and excitation energies.', 'hep-ph-0011088-1-10-1': 'Hence to convert the measured cross section to the value of transparency [MATH] it is necessary to consider the ratio: [EQUATION] where [MATH] is the experimental spectral function which is equal to the total cross section divided by the kinematic factor [MATH] and the off-shell elastic electron-proton scattering cross section [MATH] [CITATION].', 'hep-ph-0011088-1-10-2': 'Since we are dealing with small nucleon momenta there are practically no difference between different models for off-shell extrapolation of the elastic [MATH] cross section.', 'hep-ph-0011088-1-10-3': 'Quantities [MATH] and [MATH] in ([REF]) define ranges of the missing momentum [MATH] and missing energy [MATH].', 'hep-ph-0011088-1-10-4': 'The value of transparency [MATH] is known to depend appreciably on the emission angle, excitation energy and the missing momentum.', 'hep-ph-0011088-1-11-0': 'In kinematics of the NE18 and E94-139 experiments [MATH] and [MATH] were restricted by 300 MeV/c and 80 MeV.', 'hep-ph-0011088-1-11-1': 'Besides, the kinematics was adjusted to satisfy condition [MATH].', 'hep-ph-0011088-1-11-2': 'Account for the kinematics of the quasielastic processes leads to the relationship between [MATH] and Bjorken scaling variable [MATH] valid at sufficiently large [MATH] [EQUATION]', 'hep-ph-0011088-1-11-3': 'Thus [MATH] condition implies in the kinematics of the NE18 and E94-139 experiments [MATH] and that the main contribution to the cross section is given by the region [EQUATION]', 'hep-ph-0011088-1-11-4': 'Obviously, if no restrictions other than [MATH] were imposed we would obtain the quasielastic contribution to the total cross section of the [MATH] cross section at [MATH] for the same [MATH].', 'hep-ph-0011088-1-11-5': 'At sufficiently high [MATH] this cross section is proportional to [MATH] which coincides with the integrated spectral function [MATH] in the y-scaling models for [MATH].', 'hep-ph-0011088-1-11-6': 'Important feature of this integral is that it has a much smaller contribution from the high momentum component of the spectral function than the normalization integral [MATH] since [EQUATION] leading to a strong enhancement of the small [MATH] region.', 'hep-ph-0011088-1-11-7': 'This, in turn, implies that for given kinematics contribution of the large excitation energies ([MATH]) which is predominantly due to the short range correlations is also insignificant.', 'hep-ph-0011088-1-11-8': 'Therefore, we can use the mean field models to calculate the value of [MATH] as measured in the [MATH] processes for [MATH] where inelastic contribution is still very small.', 'hep-ph-0011088-1-11-9': 'Note that the account of the inelastic contributions allows to describe well the [MATH] dependence of the ratio [MATH] at [MATH] in a wide range of [MATH] [CITATION].', 'hep-ph-0011088-1-11-10': 'In our calculations we used the Hartree-Fock-Skyrme model which describes well many global properties of nuclei namely energy binding, the spectrum of excited states, the shape of matter distribution [CITATION].', 'hep-ph-0011088-1-11-11': 'In this model the spectral function is given by [EQUATION]', 'hep-ph-0011088-1-11-12': 'The integral which determines the cross section of the quasielastic scattering at [MATH] is [EQUATION]', 'hep-ph-0011088-1-11-13': 'The results of the calculation are presented as the solid curve.', 'hep-ph-0011088-1-11-14': 'We have checked that calculations with different spectral functions, like for example those of [CITATION] coincide with our results within 5% hence confirming weak sensitivity of the integral ([REF]) to the choice of the wave function.', 'hep-ph-0011088-1-11-15': 'Our result(solid curve) is compared in Fig. [REF] to the values of [MATH] extracted from the data [CITATION] in vicinity of [MATH].', 'hep-ph-0011088-1-11-16': 'These experimental values were corrected for a small contribution of the inelastic processes using the analysis [CITATION] which described well the onset of the dominance of the inelastic contribution with increase of [MATH].', 'hep-ph-0011088-1-11-17': 'Correction is about 3% (6%) for [MATH]', 'hep-ph-0011088-1-12-0': 'One can see that we describe the data very well without any adjusted parameters.', 'hep-ph-0011088-1-12-1': 'Hence we conclude that the wave functions we use are sufficiently realistic and can be used for the calculation of the transparency in the [MATH] reactions.', 'hep-ph-0011088-1-13-0': "It is worth noting that already this comparison gives a new confirmation of our result for the value of the quenching factor [MATH] found in Ref. [CITATION] from the comparison of the calculated momentum distribution in the C(e,e'p) process to that measured in the NE18 experiment at [MATH].", 'hep-ph-0011088-1-14-0': '# Inclusive transparency in [MATH] reaction.', 'hep-ph-0011088-1-15-0': 'Recently the new data on inclusive transparency in [MATH] reactions were obtained in the E94-139 experiment at Jlab [CITATION] which are consistent with the NE18 data but have somewhat better accuracy.', 'hep-ph-0011088-1-15-1': 'In the previous section we have demonstrated that the denominator in the definition of the transparency in eq. [REF] in the kinematics of the NE18 and E94-139 is weakly sensitive to assumptions about the spectral function and is strongly constrained by the [MATH] data at [MATH] and [MATH].', 'hep-ph-0011088-1-15-2': 'This allows us to treat [MATH] with much smaller uncertainties than before and use it to verify evidence of the single particle strength restoration at high momentum transfer.', 'hep-ph-0011088-1-16-0': 'As a first step we calculate the distorted momentum distributions [MATH] for kinematics of E94-139 at [MATH] using the Glauber type model of the FSI for [MATH] reactions[CITATION] [EQUATION]', 'hep-ph-0011088-1-16-1': 'Here [EQUATION] where the [MATH] amplitude is given for the high energy protons by expression [EQUATION]', 'hep-ph-0011088-1-16-2': "The values of the total proton-nucleon cross section [MATH], slope parameter [MATH] and real-to-imaginary ratio [MATH] of the pN amplitude are well known[CITATION] for the 970 MeV protons outgoing in (e,e'p) at [MATH].", 'hep-ph-0011088-1-16-3': 'Integrating eq.([REF]) in the acceptance interval of [CITATION] we obtain values of [MATH] presented in Fig.[REF].', 'hep-ph-0011088-1-16-4': 'It follows from our analysis in section [REF] that there seem to be no reason for introducing any renormalization of the impulse approximation result as assumed in the procedure adopted in [CITATION] which introduced a 10(18,22) correction for [MATH] targets and would require a strong [MATH]-dependent modification of [MATH].', 'hep-ph-0011088-1-16-5': 'Removing these corrections results in the points shown in Fig.[REF].', 'hep-ph-0011088-1-17-0': 'We see that there is a good agreement between the calculation and the data.', 'hep-ph-0011088-1-18-0': '# Exclusive [MATH] cross section', 'hep-ph-0011088-1-19-0': 'A much more stringent test of the interpretation of the data as being due to unquenched scattering of low-momentum nucleons in nuclei can be reached using differential data from E94-139 [CITATION].', 'hep-ph-0011088-1-19-1': 'This allows us to avoid a certain inconsistency in our comparison with inclusive data for [MATH] due to the use in the experimental analysis of a shell-model spectral function which is somewhat different from our spectral function.', 'hep-ph-0011088-1-19-2': 'Results of our calculation for the distorted momentum distributions are compared to the data [CITATION] in Fig. [REF].', 'hep-ph-0011088-1-20-0': 'Taking into account that our calculations do not comprise any free parameters one can recognize a pretty fair agreement with experimental data at momenta of bound proton [MATH] 200 MeV/c.', 'hep-ph-0011088-1-20-1': 'A discrepancy at momenta above 200 MeV/c which increases with A can be considered as an evidence of the elastic incoherent rescattering processes for the outgoing nucleon (this effect will be considered elsewhere).', 'hep-ph-0011088-1-21-0': '# The [MATH] dependence of quenching', 'hep-ph-0011088-1-22-0': 'At first glance, the comparison with the data performed in previous sections leaves no room for the presence of the single particle strength quenching at [MATH].', 'hep-ph-0011088-1-22-1': 'However to make the final conclusion one should carefully take into account experimental errors and uncertainties of the calculations.', 'hep-ph-0011088-1-22-2': "Generally, the accuracy of the Glauber approach in description of the proton-nucleus interaction in high energy kinematics of (e,e'p) reaction at [MATH] is about few as long as no new physics like color transparency is present.", 'hep-ph-0011088-1-22-3': 'There exists also a few uncertainty due to the use in the calculation of the definite set of the Hartree-Fock wave functions.', 'hep-ph-0011088-1-22-4': 'Hence, a possibility of the quenching in range of [MATH] cannot be excluded.', 'hep-ph-0011088-1-22-5': 'However this is definitely much smaller than necessant to be agreed with [MATH] determined from the low [MATH] data [CITATION].', 'hep-ph-0011088-1-23-0': "It should be noted that the analysis of (e,e'p) data is evidently more definitive at high energy and high momentum transfer than in the low energy kinematics.", 'hep-ph-0011088-1-23-1': 'The kinematical off-shell effect in the [MATH] vertex due to the Fermi motion of nucleon studied by De Forest[CITATION] is minimized in the high energy limit.', 'hep-ph-0011088-1-23-2': 'Also, the renormalization of the [MATH] vertex due to inability for a low [MATH] photon of resolving the short-range and the long-range correlations of interacting proton with the rest of the nucleons is evidently more essential in the [MATH] kinematics.', 'hep-ph-0011088-1-23-3': 'Within the quasiparticle approach such a renormalization can be taken into account by using the form factor of quasiparticle which is softer than for a free nucleon because at low [MATH] resolution a low momentum bound nucleon in the nuclear medium is dressed by a cloud of virtual nuclear excitations.', 'hep-ph-0011088-1-23-4': 'With increase of the momentum transfer above the Fermi-momentum of bound nucleon [MATH] this renormalization of the electron-proton vertex disappears and we deal with the form factor of a free nucleon.', 'hep-ph-0011088-1-24-0': 'Besides, taking into account the FSI at low energies is more uncertain because one needs to deal with the optical potentials which are determined from the fits to the proton-nucleus elastic scattering data.', 'hep-ph-0011088-1-24-1': 'Such a treatment ignores a difference in the space geometry of the proton elastic scattering which is dominated by the interaction with the nuclear surface and the proton propagation in the nucleon knockout from the nucleus interior.', 'hep-ph-0011088-1-25-0': "To summarize, we have demonstrated based on the joint analysis of the exclusive A(e,e'p) and A(e,e')X data at [MATH] that the actual quenching factor which enters into cross sections of the exclusive quasielastic processes differs from the one used in [CITATION] and is practically insensitive to the probability of the short-range nucleon correlations in nuclei.", 'hep-ph-0011088-1-25-1': "We found further evidence for the dependence of the single particle strength quenching in the exclusive (e,e'p) reactions on the momentum transfer.", 'hep-ph-0011088-1-25-2': 'The strong effect (about 40 [MATH]) observed in the low energy phenomena practically disappears with increase of [MATH] when the quasiparticle structure of the nucleon due to the long range correlations inside the nuclear medium became resolved.', 'hep-ph-0011088-1-25-3': "Only a very modest quenching (less than 10[MATH]) in the exclusive (e,e'p) reaction at high [MATH] and [MATH] can survive due to the missed contribution of the short-range correlations which are strongly suppressed in the discussed kinematics.", 'hep-ph-0011088-1-26-0': 'A strong [MATH] dependence of quenching comes very naturally in the Fermi liquid theory [CITATION] and really represents the generic property of fermionic systems where the interaction between fermions is described by a renormalizable theory[CITATION], like QED or QCD since in this case the wave functions of constituents depend strongly on the resolution scale.', 'hep-ph-0011088-1-27-0': "A high precise measurements of the A(e,e') scattering and the differential cross sections of the exclusive A(e,e'p) reactions at [MATH] in the range [MATH] would be very useful for the accurate estimate of the quenching effect and determining the experimental values of the transparency.", 'hep-ph-0011088-1-27-1': 'The kinematics [MATH] appears to be optimal for searches of the color transparency at moderate and high momentum transfers in the [MATH] reactions, in order to understand the phenomenon of expanding of the small size quark configurations in hard processes.', 'hep-ph-0011088-1-28-0': 'We thank R.Ent for stimulating questions, M.Sargian for discussion of inelastic contributions.', 'hep-ph-0011088-1-28-1': "We also thank B.Birbrair, L. Lapikas and G. van der Steenhoven for numerous discussions of the relation between low [MATH] and high [MATH] (e,e'p) processes.", 'hep-ph-0011088-1-28-2': 'Research of M.S. and M.Z. was supported in part by the U.S. Department of Energy, research of L.F. was supported by the Israeli Academy of Science.'}

{'hep-ph-0011088-2-0-0': 'Single particle strength restoration and nuclear transparency in high [MATH] exclusive [MATH] reactions', 'hep-ph-0011088-2-1-0': 'ABSTRACT', 'hep-ph-0011088-2-2-0': 'Quasifree [MATH] reactions at [MATH] observed a strong quenching of the single-particle valence strength.', 'hep-ph-0011088-2-2-1': "This precluded unambiguous measurement of nuclear transparency in quasifree (e,e'p) reactions at [MATH].", 'hep-ph-0011088-2-2-2': 'We argue that the high-energy nuclear transparency in the transverse kinematics weakly depends on the probability of the short-range correlations and can be accurately determined using information on the cross section of the [MATH] reaction at [MATH] and [MATH].', 'hep-ph-0011088-2-2-3': "We find that the Glauber approximation well describes E91-013 and NE18 (e,e'p) data at [MATH] without any significant quenching.", 'hep-ph-0011088-2-2-4': 'This gives further support to our observation that the quenching of nuclear levels strongly depends on the resolution ([MATH]) and practically disappears at [MATH].', 'hep-ph-0011088-2-3-0': '# Introduction', 'hep-ph-0011088-2-4-0': "Quasi-free knockout (e,e'p) reactions were used for a long time to study nuclear structure at the energy transfer [MATH], for a review see [CITATION].", 'hep-ph-0011088-2-4-1': 'One of the important findings of these studies was an observation of suppression of the single-particle valence strength as compared to the calculations using the nuclear shell model wave functions.', 'hep-ph-0011088-2-4-2': 'This phenomenon of quenching in the low energy physics is naturally explained in the nuclear quasiparticle theory[CITATION] as a result of the single particle strength fragmentation over wide excitation energy range due to the long and short range nucleon-nucleon correlations(for the recent review see e.g. [CITATION]).', 'hep-ph-0011088-2-4-3': 'Recently a thorough comparison of all recent low energy data for [MATH] was performed in [CITATION].', 'hep-ph-0011088-2-4-4': 'It is found that the data from different experiments are consistent with each other and require a very substantial reduction of the s- and p- shell strength in [MATH] at [MATH] by the factor [EQUATION]', 'hep-ph-0011088-2-4-5': "If the quenching did not depend on [MATH], this finding would strongly affect the interpretation of the recent (e,e'p) experiments at high energies and momentum transfers [CITATION].", 'hep-ph-0011088-2-4-6': 'These experiments were performed to study the nuclear transparency [MATH] as a function of the momentum transfer at [MATH] on several nuclei with the main goal to search for the color transparency effects [CITATION].', 'hep-ph-0011088-2-4-7': 'According to the theoretical predictions[CITATION] the color transparency could be expected at these momentum transfers as a 10% increase of the nuclear transparency with increase of [MATH].', 'hep-ph-0011088-2-4-8': 'Hence, a high precision determination of all the nuclear characteristics influencing evaluation of [MATH] is very important.', 'hep-ph-0011088-2-5-0': 'Experimentally, the nuclear transparency [MATH] is defined as the ratio of the observed cross section to the cross section calculated in the plane-wave impulse approximation(PWIA).', 'hep-ph-0011088-2-5-1': 'The delicate point here is to estimate precisely how large the latter is for the kinematics of the particular experiment.', 'hep-ph-0011088-2-5-2': 'The PWIA cross section is [EQUATION] where [MATH] is the kinematic factor and [MATH] [CITATION] is the off-shell extrapolation of the elastic [MATH] cross section.', 'hep-ph-0011088-2-5-3': "If the momenta of bound nucleons, [MATH], and excitation energies of the residual nuclei, [MATH], in (e,e'p) are not too large, there are practically no differences between different models for off-shell extrapolation of [MATH].", 'hep-ph-0011088-2-5-4': 'Hence, the main problem is to determine accurately enough the integral over the nuclear spectral function [MATH].', 'hep-ph-0011088-2-5-5': "In high [MATH] (e,e'p) experiments [CITATION] the [MATH] was calculated in the independent particle shell model.", 'hep-ph-0011088-2-5-6': 'To correct this calculation for the missing single particle strength in the kinematics of the experiment an additional correction factor [MATH] was introduced, leading to [EQUATION]', 'hep-ph-0011088-2-5-7': 'In particular, the values [EQUATION] were used in [CITATION] for extracting the nuclear transparency from the data.', 'hep-ph-0011088-2-6-0': 'It was recently pointed out in [CITATION] that interpreting transparency measurements at high energies performed in the transverse kinematics, with the cuts on the momentum of the struck nucleon and the energy of the produced system, requires a re-evaluation of the quenching which was observed in the low energy domain.', 'hep-ph-0011088-2-6-1': 'In particular, if one would use the same quenching for excitation of [MATH] hole states in the carbon as the one observed at the low [MATH] (eq. [REF]), the transparency [MATH] for the [MATH] s- and p-valence state region is about 0.8 for [MATH].', 'hep-ph-0011088-2-6-2': 'This number is much higher than the predictions of Glauber theory which should be a reasonable approximation for [MATH] and should be a very good approximation for [MATH] GeV, corresponding to [MATH].', 'hep-ph-0011088-2-6-3': 'At the same time we demonstrated that the carbon data at [MATH] for the differential [MATH] cross section appeared to be consistent with the Glauber calculation provided one assumes a strong reduction of the quenching effect at large [MATH].', 'hep-ph-0011088-2-6-4': "In particular we used the NE-18 differential (e,e'p) cross sections for carbon to determine the quenching factor for [MATH] to be [MATH].", 'hep-ph-0011088-2-7-0': 'We further argued that a [MATH] dependence of quenching should be a natural phenomenon reflecting transition from low [MATH] interactions where photon interacts with quasiparticles to the interaction with nucleons at larger [MATH].', 'hep-ph-0011088-2-7-1': 'However, we see no reasons for a noticeable A-dependence of the correlation correction at high [MATH] and [MATH].', 'hep-ph-0011088-2-7-2': 'Indeed the main source of the [MATH] dependence at high [MATH] would be the [MATH]-dependence of the short-range correlation contribution.', 'hep-ph-0011088-2-7-3': 'According to the the analysis of high [MATH], [MATH] data(see e.g. [CITATION]) the effect of the short range correlations changes by [MATH] between A=12 and A=208.', 'hep-ph-0011088-2-7-4': 'However, as we show below,this contribution itself in the considered integral is just a few %([MATH] for carbon).', 'hep-ph-0011088-2-8-0': 'In this paper we extend the analysis of [CITATION] in several directions.', 'hep-ph-0011088-2-8-1': 'We analyze the transparency measured recently in [CITATION] for a range of nuclei, focusing at [MATH] for which both integrated cross sections and differential cross sections are available.', 'hep-ph-0011088-2-8-2': 'Our choice of [MATH] is motivated by a very good understanding of [MATH] interactions for the corresponding energy of the ejected nucleon [MATH] - the Glauber theory is known to describe numerous data on elastic and quasielastic [MATH] interactions at this energy with a typical accuracy of few percents, see review in [CITATION].', 'hep-ph-0011088-2-8-3': 'Also, due to a weak energy dependence of [MATH] between [MATH] and [MATH], and the smallness of the color transparency effects for the [MATH] range, one expects a very weak dependence of transparency on [MATH] for [MATH].', 'hep-ph-0011088-2-8-4': 'This is certainly consistent with the data.', 'hep-ph-0011088-2-8-5': 'Hence adding higher [MATH] data would not add much to the main thrust of our analysis.', 'hep-ph-0011088-2-8-6': 'We observe that measurements of [MATH] in the transverse kinematics of [MATH] are not sensitive to the high momentum component of the nuclear wave function since the cross section is proportional to [MATH] rather than to [MATH].', 'hep-ph-0011088-2-8-7': 'Further reduction in the uncertainties is reached by using information on the cross sections of [MATH] scattering at [MATH] and [MATH] measured at Jlab [CITATION] which allows to determine independently [MATH] with an accuracy of few %.', 'hep-ph-0011088-2-8-8': 'Using this information we calculated the transparency for the kinematics of the E91-013 experiment for carbon, iron and gold and find that with an appropriate normalization of the impulse approximation cross section we obtain a very good description of the data.', 'hep-ph-0011088-2-9-0': 'We also check our conclusions about the noted reduction of quenching by comparing the results of our calculations with the [MATH] data from the Jlab experiment [CITATION] for the differential cross sections.', 'hep-ph-0011088-2-9-1': 'We will show that excellent agreement is observed, without any adjusted parameters, for the region [MATH], where contribution of the short-range correlations is small.', 'hep-ph-0011088-2-9-2': 'This provides a very strong new evidence for the practical disappearance of the quenching at large [MATH].', 'hep-ph-0011088-2-10-0': 'In the end of the paper we consider implications for optimizing searches for color transparency in high [MATH] processes.', 'hep-ph-0011088-2-10-1': 'Numerical predictions for kinematics where the onset of the color transparency is expected will be presented elsewhere.', 'hep-ph-0011088-2-11-0': '# Definition of transparency - how large is the impulse approximation', 'hep-ph-0011088-2-12-0': 'Current experiments which study nuclear transparency perform measurements in a restricted region of recoil nuclear momenta and excitation energies.', 'hep-ph-0011088-2-12-1': 'Hence to convert the measured cross section to the value of transparency [MATH] it is necessary to consider the ratio: [EQUATION]', 'hep-ph-0011088-2-12-2': 'The quantities [MATH] and [MATH] in ([REF]) define the ranges in missing momentum [MATH] and missing energy [MATH].', 'hep-ph-0011088-2-12-3': 'The value of transparency [MATH] is known to depend appreciably on the excitation energy, the missing momentum and angle between [MATH] and [MATH][CITATION].', 'hep-ph-0011088-2-13-0': 'In the kinematics of the NE18 and E91-013 experiments [MATH] and [MATH] were restricted by 300 MeV/c and 80 MeV.', 'hep-ph-0011088-2-13-1': 'Besides, the transverse kinematics of the experiments corresponded approximately to [MATH].', 'hep-ph-0011088-2-13-2': 'Account for the kinematics of the quasielastic process leads to the following relationship between [MATH] and the Bjorken scaling variable [MATH] valid at sufficiently large [MATH] [EQUATION]', 'hep-ph-0011088-2-13-3': 'Thus, the [MATH] condition implies in the kinematics of the NE18 and E91-013 experiments [MATH], and that the main contribution to the cross section is given by the region [EQUATION]', 'hep-ph-0011088-2-13-4': 'Obviously, if no restrictions other than [MATH] were imposed we would obtain the quasielastic contribution to the total cross section of the [MATH] cross section at [MATH] for the same [MATH].', 'hep-ph-0011088-2-13-5': 'At sufficiently high [MATH] this cross section is proportional to [EQUATION] which coincides with the integrated spectral function [MATH] in the y-scaling models for [MATH].', 'hep-ph-0011088-2-13-6': 'An important feature of this integral is that it has a much smaller contribution from the high momentum component of the spectral function than the normalization integral [MATH] since [EQUATION] leading to a strong enhancement of the small [MATH] region.', 'hep-ph-0011088-2-13-7': 'This, in turn, implies that for a given kinematics the contribution of the large excitation energies ([MATH]), which is predominantly due to the short range correlations, is also insignificant.', 'hep-ph-0011088-2-13-8': 'Therefore, we can use mean field models to calculate the value of [MATH] as measured in the [MATH] processes for [MATH] where inelastic contributions are still very small.', 'hep-ph-0011088-2-13-9': 'Note that the account of the inelastic contributions allows for a good description of the [MATH] dependence of the ratio [MATH] at [MATH], over a wide range of [MATH] [CITATION].', 'hep-ph-0011088-2-13-10': 'In our calculations we used the Hartree-Fock-Skyrme model, which describes well many global properties of nuclei e.g. the energy binding, the spectra of the single particle states, root mean square radii and the shape of the proton and neutron matter distributions [CITATION].', 'hep-ph-0011088-2-13-11': 'In this model, the spectral function is given by [EQUATION]', 'hep-ph-0011088-2-13-12': 'The occupation probabilities for the filled nuclear levels are [MATH].', 'hep-ph-0011088-2-13-13': 'The integral which determines cross section of the quasielastic scattering at [MATH] and [MATH] is [EQUATION]', 'hep-ph-0011088-2-13-14': 'The results of the calculation are presented as the solid curve in Fig. [REF].', 'hep-ph-0011088-2-13-15': 'They are compared to the values of [MATH] extracted from the inclusive [MATH] data of [CITATION] in the vicinity of [MATH] at [MATH].', 'hep-ph-0011088-2-13-16': 'These experimental values were corrected for a small contribution of inelastic processes using the analysis of [CITATION], which described well the onset of the dominance of the inelastic contribution with increase of [MATH].', 'hep-ph-0011088-2-13-17': 'The correction is about 3% (6%) for [MATH].', 'hep-ph-0011088-2-14-0': 'One can see that the data are described very well without any adjusted parameters.', 'hep-ph-0011088-2-14-1': 'Note that the IPSM spectral function which was used in the experimental analysis of NE18 and E91-013 [CITATION] reproduces reasonably (dot-dashed line in Fig. [REF]) the weak [MATH]-dependence of the data though it somewhat overestimates the absolute value of the integral ([REF]).', 'hep-ph-0011088-2-14-2': 'At the same time an additional renormalization factor [MATH], Eq.( [REF]), introduced in [CITATION], does a good job for [MATH] but leads to much stronger A-dependence of the integral(dashed line in Fig. [REF]) than suggested by our calculations and by the data.', 'hep-ph-0011088-2-14-3': 'We have also checked that the estimate of the integral ([REF]) with the phenomenological models of the spectral functions, like, for example, those of [CITATION], which includes the 15 contribution of the short-range correlations from region [MATH] in the normalization integral for the spectral function, coincides with our results for ([REF]) within 5.', 'hep-ph-0011088-2-14-4': 'Also, in this model only 5 of the integral originates from the region of [MATH].', 'hep-ph-0011088-2-14-5': 'This confirms our conclusion about the strong suppression of the short range correlation contribution in the transverse kinematics of [CITATION] due to the specific properties of the integral ([REF]).', 'hep-ph-0011088-2-14-6': "It is worth noting that already this comparison gives a new confirmation of our result for the value of the quenching factor [MATH], extracted in Ref. [CITATION] from the comparison of the calculated momentum distribution in the C(e,e'p) process to that measured in the NE18 experiment at [MATH].", 'hep-ph-0011088-2-15-0': 'Hence we conclude that the wave functions we use are sufficiently realistic to estimate the integral quantity entering the denominator of ([REF]) in the calculation of the transparency in the [MATH] reactions.', 'hep-ph-0011088-2-16-0': '# Exclusive [MATH] cross section', 'hep-ph-0011088-2-17-0': 'A more stringent test of the wave functions and interpretation of the data can be reached using differential data from E91-013 [CITATION].', 'hep-ph-0011088-2-17-1': 'The differential cross section of the [MATH] were calculated using the Hartree-Fock-Skyrme wave functions and the Glauber type model of the FSI for [MATH] reactions[CITATION] .', 'hep-ph-0011088-2-17-2': 'The cross section can be represented in the same form as the impulse approximation with a substitution of the spectral function by the sum of the distorted shell momentum distributions given by [EQUATION]', 'hep-ph-0011088-2-17-3': 'Here [EQUATION] and the [MATH] amplitude for high energy protons is given by the expression [EQUATION].', 'hep-ph-0011088-2-17-4': 'The values of the total proton-nucleon cross section [MATH], the slope parameter [MATH] and the real-to-imaginary ratio [MATH] of the pN amplitude are well known[CITATION] for the 970 MeV protons corresponding to the [MATH] reaction at [MATH].', 'hep-ph-0011088-2-17-5': 'Results of our calculation for the distorted momentum distributions using eq. [REF] and the HF spectral function are compared to the data of [CITATION] in Figs. [REF], [REF], [REF].', 'hep-ph-0011088-2-18-0': 'Taking into account that our calculations do not comprise any free parameters, one observes a fair agreement with experimental data, at least for momenta of the bound proton [MATH] 200 MeV/c.', 'hep-ph-0011088-2-18-1': 'A discrepancy at momenta above 200 MeV/c, which increases with A, can be considered as an evidence for the elastic incoherent rescattering processes for the outgoing nucleon (this effect will be considered elsewhere).', 'hep-ph-0011088-2-19-0': '# Inclusive transparency in [MATH] reaction.', 'hep-ph-0011088-2-20-0': 'The E91-013 experiment at Jlab[CITATION] which we discussed in the previous section also reported new values for the nuclear transparency which are consistent with the NE18 data but has somewhat better accuracy.', 'hep-ph-0011088-2-20-1': 'In the previous sections we have demonstrated that both the numerator and the denominator in the definition of the transparency in Eq. ([REF]) in the kinematics of the NE18 and E91-013 are sensitive to assumptions about the spectral function, but are strongly constrained by the [MATH] and [MATH] data at [MATH] and [MATH].', 'hep-ph-0011088-2-20-2': 'This allows us to treat [MATH] with much smaller uncertainties than before and use it to obtain a suplementary evidence of the single particle strength restoration at high momentum transfer.', 'hep-ph-0011088-2-21-0': 'We obtain theoretical value of [MATH] from Eq. [REF] by using the spectral functions [MATH] (Eq. [REF]) and by integrating Eq.([REF]) to account for the FSI in the kinematics of [CITATION].', 'hep-ph-0011088-2-21-1': 'The result of calculation is presented by solid curve in Fig.[REF].', 'hep-ph-0011088-2-21-2': 'To compare these results with the data in a way consistent with our finding in section [REF] we need to use the impulse approximation cross section consistent with the results presented in Fig. [REF] and with the theoretical calculation of [MATH].', 'hep-ph-0011088-2-21-3': 'The simplest procedure is to correct the transparency values presented in [CITATION] using in the denominator of Eq.([REF]) the value of the cross section in the impulse approximation given by our spectral function which differ from the value used in [CITATION] by the factor [MATH] (the factors are 1.02 for carbon, 0.896 for iron and 0.83 for gold).', 'hep-ph-0011088-2-21-4': 'This leads to the corrected experimental values of the transparency shown in Fig.[REF] which are in a good agreement with results of calculations.', 'hep-ph-0011088-2-21-5': "We want to emphasize that this modification of the values of [MATH], presented in [CITATION], arises solely due to the change of the theoretical estimate of the impulse approximation cross section used in [CITATION] to ensure consistency with results of the (e,e') measurements.", 'hep-ph-0011088-2-22-0': '# The [MATH] dependence of quenching', 'hep-ph-0011088-2-23-0': 'At first glance, the comparison with the data performed in previous Sections leaves no room for the presence of single particle strength quenching at [MATH].', 'hep-ph-0011088-2-23-1': 'However to make the final conclusion one should carefully take into account experimental errors and uncertainties of the calculations.', 'hep-ph-0011088-2-23-2': "Generally, the accuracy of the Glauber approach in the description of the proton-nucleus interaction in high energy kinematics of the (e,e'p) reaction at [MATH] is estimated to be small (a few ) as long as no new physics like color transparency is present.", 'hep-ph-0011088-2-23-3': 'There exists also a few uncertainty due to the use in the calculation of a definite set of the Hartree-Fock wave functions and neglecting effects of the short-range correlations in the calculation of the normalization integral ([REF]).', 'hep-ph-0011088-2-23-4': 'Hence, a possibility of the quenching of order [MATH] cannot be excluded.', 'hep-ph-0011088-2-23-5': 'However, this is definitely much smaller than [MATH] determined from the low [MATH] data [CITATION].', 'hep-ph-0011088-2-24-0': "It should be noted that the analysis of (e,e'p) data is more definitive at high energy and high momentum transfer than in the low energy kinematics.", 'hep-ph-0011088-2-24-1': 'The kinematical off-shell effect in the [MATH] vertex due to the Fermi motion of nucleon, studied e.g. by De Forest[CITATION], is minimized in the high energy limit.', 'hep-ph-0011088-2-24-2': 'Also, the renormalization of the [MATH] vertex due to the inability of a low [MATH] photon to resolve the short-range and long-range correlations of interacting proton with the rest of the nucleons is evidently more essential in the low [MATH] kinematics.', 'hep-ph-0011088-2-24-3': 'Within the quasiparticle approach, such a renormalization can be performed by using the form factor of a quasiparticle which is softer than for a free nucleon, because at low resolution a low momentum bound nucleon in the nuclear medium is dressed by a cloud of virtual nuclear excitations.', 'hep-ph-0011088-2-24-4': 'With increase of the momentum transfer above the Fermi-momentum of the bound nucleon, [MATH], this renormalization of the electron-proton vertex disappears, and we deal with the form factor of a free nucleon.', 'hep-ph-0011088-2-25-0': 'Besides, taking into account the FSI at low energies is more cumbersome because one needs to deal with the optical potentials which are determined from fits to the proton-nucleus elastic scattering data.', 'hep-ph-0011088-2-25-1': 'Such a treatment ignores the difference in the space geometry of proton-nucleus elastic scattering, dominated by the interaction with the nuclear surface, and the proton propagation from the nucleus interior in the electron induced nucleon knockout reaction.', 'hep-ph-0011088-2-26-0': "To summarize, we have demonstrated, based on the joint analysis of the exclusive A(e,e'p) and A(e,e')X data at [MATH], that the actual quenching factor which enters into cross sections of the exclusive quasielastic processes differs from the one used in [CITATION] and is practically insensitive to the probability of the short-range nucleon correlations in nuclei.", 'hep-ph-0011088-2-26-1': "We found further evidence for the dependence of the single particle strength quenching in the exclusive (e,e'p) reactions on the momentum transfer.", 'hep-ph-0011088-2-26-2': 'The strong effect (about 40 [MATH]), observed in the low energy phenomena, practically disappears with increase of [MATH], when a probe resolves the quasiparticle structure of the nucleon due to the long-range correlations inside the nuclear medium.', 'hep-ph-0011088-2-26-3': "In the discussed transverse kinematics only a very modest quenching (less than 10[MATH]) may survive in the exclusive (e,e'p) reaction at high [MATH] and [MATH].", 'hep-ph-0011088-2-27-0': 'A strong [MATH] dependence of quenching comes very naturally in the Fermi liquid theory [CITATION] and really represents the generic property of fermionic systems where the interaction between fermions is described by a renormalizable theory[CITATION], like QED or QCD, since in this case the wave functions of constituents depend strongly on the resolution scale.', 'hep-ph-0011088-2-28-0': "A high precision measurements of A(e,e') scattering and the differential cross sections of the exclusive A(e,e'p) reactions at [MATH] in the range [MATH] would be very useful for an accurate estimate of the quenching effect, and to determine the experimental values of nuclear transparency.", 'hep-ph-0011088-2-28-1': 'The kinematics [MATH] appear to be optimal for searches of color transparency at moderate and high momentum transfers in the [MATH] reactions, in order to understand the phenomenon of expanding of small size quark configurations in hard processes.', 'hep-ph-0011088-2-29-0': 'We thank R.Ent for stimulating questions and comments, D.Dutta for the information about the spectral function used in the E91-013 experiment, M.Sargian for discussion of inelastic contributions.', 'hep-ph-0011088-2-29-1': "We also thank B.Birbrair, L. Lapikas and G. van der Steenhoven for numerous discussions of the relation between low [MATH] and high [MATH] (e,e'p) processes.", 'hep-ph-0011088-2-29-2': 'Research of M.S. and M.Z. was supported in part by the U.S. Department of Energy, research of L.F. was supported by the Israeli Academy of Science.'}

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[]

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[]

['hep-ph-0011088-1-1-0', 'hep-ph-0011088-2-1-0']

{'1': 'http://arxiv.org/licenses/assumed-1991-2003/', '2': 'http://arxiv.org/licenses/assumed-1991-2003/'}

https://arxiv.org/abs/hep-ph/0011088

hep-ph-0306213

{'hep-ph-0306213-1-0-0': 'The chiral symmetry is applied to an extended constituent quark potential model.', 'hep-ph-0306213-1-0-1': 'With random phase approximation (RPA), the model comprises sea quark excitations.', 'hep-ph-0306213-1-0-2': 'To obtain the pseudoscalar [MATH] meson as a Goldstone boson, the quark effective potentials are modified in the model to account for the dynamical breaking of chiral symmetry.', 'hep-ph-0306213-1-0-3': 'Also the vector [MATH] meson is calculated and the KSRF relation about [MATH] and [MATH] meson decay constants is derived in the model.', 'hep-ph-0306213-1-1-0': '# Introduction', 'hep-ph-0306213-1-2-0': 'Due to the complication of a non-Abelian [MATH] gauge theory, quantum chromodynamics (QCD), which describes the strong interaction, has many non-perturbative features such as the dynamical breaking of chiral symmetry and quark confinement.', 'hep-ph-0306213-1-2-1': 'In the study of hadron structure at low energy scale, two kinds of models of QCD are often used, each incorporating some important QCD features.', 'hep-ph-0306213-1-2-2': 'The constituent quark potential model which incorporates the QCD quark confinement has been impressively successful in hadron spectroscopy and decays [CITATION], except for the pseudo-scalar [MATH] meson which is a Goldstone boson and has very low mass.', 'hep-ph-0306213-1-2-3': 'It was shown recently that the important sea quark contribution must be considered in the treatment of pions [CITATION].', 'hep-ph-0306213-1-2-4': 'On the other hand, the Nambu-Jona-Lasinio (NJL) model [CITATION] and several of its extensions [CITATION] which incorporate the chiral symmetry can describe the [MATH] meson very well as a massless Goldstone boson of dynamical breaking of chiral symmetry.', 'hep-ph-0306213-1-2-5': 'However, the NJL model lacks the important QCD property of quark confinement.', 'hep-ph-0306213-1-3-0': 'In Ref. [CITATION], we proposed an extension to the constituent quark model to comprise the sea quark excitations.', 'hep-ph-0306213-1-3-1': 'As in NJL model, we start from an effective quark Hamiltonian.', 'hep-ph-0306213-1-3-2': 'The sea quark contribution is considered with the RPA approximation.', 'hep-ph-0306213-1-3-3': 'For meson structure, this leads to a Hamiltonian with two channels - the ordinary valence quark channel and a new sea quark channel, which is an extension of the quark potential model.', 'hep-ph-0306213-1-3-4': 'This extended quark potential model has a new coupling potential connecting couple the valence quark and the sea quark channel.', 'hep-ph-0306213-1-3-5': 'The coupling potential can be related to the ordinary valence quark potential in the model.', 'hep-ph-0306213-1-4-0': 'In this paper, we implement the chiral symmetry in the extended constituent quark potential model.', 'hep-ph-0306213-1-4-1': 'First we introduce the extended quark potential model.', 'hep-ph-0306213-1-4-2': 'In Sec. 3, we analyze [MATH] and [MATH] mesons using this framework.', 'hep-ph-0306213-1-4-3': 'The restriction from chiral symmetry is considered.', 'hep-ph-0306213-1-4-4': 'With the meson wave functions containing sea quark excitations, KSRF relation [CITATION] about [MATH] and [MATH] meson decay constants is derived.', 'hep-ph-0306213-1-4-5': 'Finally, we summarize our discussion.', 'hep-ph-0306213-1-5-0': '# Extending Quark Potential Model with RPA', 'hep-ph-0306213-1-6-0': 'QCD perturbation theory is very successful when applied in the high energy processes.', 'hep-ph-0306213-1-6-1': 'In the low energy regime, Hamiltonian approach can provide an alternative method to understand the non-perturbative features of QCD, such as the quark confinement.', 'hep-ph-0306213-1-6-2': 'Many efforts were made to obtain an effective Hamiltonian from the exact QCD Lagrangian [CITATION].', 'hep-ph-0306213-1-6-3': 'Here we start from the effective quark hamiltonian, assuming the quark fields have been separated approximately from the gluon fields [EQUATION] where [MATH] is the free current quark Hamiltonian and [MATH] is the current quark mass.', 'hep-ph-0306213-1-6-4': '[MATH] includes all effective quark interactions.', 'hep-ph-0306213-1-6-5': '[MATH] is the kernel function for coupling vertex [MATH].', 'hep-ph-0306213-1-7-0': 'Apart from a constant, the Hamiltonian ([REF]) can be written in normal order as [EQUATION] where [MATH] represents the free particle energy of constituent quark and its general form is [EQUATION] where the dependence on the three-vector momentum arises from the instantaneous approximation of quark interactions.', 'hep-ph-0306213-1-8-0': 'The two functions [MATH] and [MATH] should be obtained from a self-consistent mean field calculation.', 'hep-ph-0306213-1-8-1': 'However in a constituent quark potential model, the quark interactions contain the linear quark confinement which has severe infrared divergence in momentum space.', 'hep-ph-0306213-1-8-2': 'On the other hand, the constituent potential quark models always assume that the constituent quarks have fixed masses.', 'hep-ph-0306213-1-8-3': 'As an extension of the constituent potential quark model, here we assume that the [MATH] and [MATH] functions can be approximated as constants.', 'hep-ph-0306213-1-8-4': 'So we treat the constituent quark mass as a parameter, [MATH] (with [MATH]).', 'hep-ph-0306213-1-8-5': 'The free particle Hamiltonian becomes [EQUATION]', 'hep-ph-0306213-1-8-6': 'To account for the sea quark contribution, we consider the meson state as a superposition of creation and annihilation of [MATH] pairs on the vacuum, i.e., the excitation modes of the vacuum of the RPA type: [EQUATION]', 'hep-ph-0306213-1-8-7': 'For a meson in the rest frame, [MATH] excitation operator can be expressed with quark and antiquark creation and annihilation operators [EQUATION] where [MATH] and [MATH] are the RPA amplitudes and the time reversals are defined as [EQUATION]', 'hep-ph-0306213-1-8-8': 'From the well-known RPA equation of motion[CITATION] [EQUATION] we obtain the equations for RPA amplitudes [EQUATION]', 'hep-ph-0306213-1-8-9': 'Here we have introduced two effective potentials [MATH] and [MATH].', 'hep-ph-0306213-1-8-10': 'Their matrix elements are [EQUATION]', 'hep-ph-0306213-1-8-11': 'Thus, with the RPA approximation, the quark potential model is extended to a coupling system with two channels.', 'hep-ph-0306213-1-8-12': 'The contribution of valence quark excitations is described by x-amplitude [MATH] which we will call the x-channel wave function.', 'hep-ph-0306213-1-8-13': 'The contribution of sea quark excitations is given by the y-amplitude [MATH] which is the y-channel wave function.', 'hep-ph-0306213-1-8-14': 'The potential [MATH] interacts only within each individual channel while the new coupling potential [MATH] couples the two channels together.', 'hep-ph-0306213-1-8-15': 'If the coupling between the two channels [MATH] is small, the system decouples.', 'hep-ph-0306213-1-8-16': 'The extended potential model reduces to the ordinary potential model and the potential [MATH] is just the ordinary quark potential.', 'hep-ph-0306213-1-8-17': 'Here we will show that the coupling potential [MATH] is important due to chiral symmetry .', 'hep-ph-0306213-1-9-0': 'One can use the Dirac kets and bras to simplify notations.', 'hep-ph-0306213-1-9-1': '[MATH] and [MATH] are represented by two kets [MATH] and [MATH] respectively, [EQUATION]', 'hep-ph-0306213-1-9-2': 'Eqs. ([REF]) and ([REF]) can be written concisely as [EQUATION] or the familiar matrix form [EQUATION] where [MATH].', 'hep-ph-0306213-1-10-0': 'By definitions ([REF]) and ([REF]), the potentials [MATH] and [MATH] will be evaluated from the same kernels in the quark interaction [MATH] and thus they can be related together.', 'hep-ph-0306213-1-10-1': "In the non-relativistic limit [MATH], we have [EQUATION] where on the right hand side of equations, the kernels are divided into five types as in ref. [CITATION], i.e., scalar ([MATH]), pseudo-scalar ([MATH]), vector ([MATH]), axial-vector ([MATH]), and tensor ([MATH]); [MATH]'s are the pauli spin matrices.", 'hep-ph-0306213-1-11-0': 'To normalize the model wave functions, we apply the standard boson commutation relation [EQUATION] to rest mesons.', 'hep-ph-0306213-1-11-1': 'We obtain [EQUATION]', 'hep-ph-0306213-1-11-2': 'Then we find the normalization relation of RPA type [EQUATION]', 'hep-ph-0306213-1-11-3': 'Again with Dirac kets and bras, its form is [EQUATION]', 'hep-ph-0306213-1-11-4': 'With the normalized meson wave functions, one can calculate the static properties of mesons.', 'hep-ph-0306213-1-11-5': 'The [MATH] weak decay constant is defined as [EQUATION]', 'hep-ph-0306213-1-11-6': 'For pions at rest, the above equation becomes [EQUATION]', 'hep-ph-0306213-1-11-7': 'In the non-relativistic limit, we obtain [EQUATION] where [MATH] and [MATH] are the coordinate wave functions in x- and y- channel of the [MATH] meson with its total spin [MATH].', 'hep-ph-0306213-1-11-8': 'The [MATH] electro-magnetic decay constant is defined as [EQUATION] where [MATH] are the isospin indices, and [MATH] is the [MATH] meson polarization index.', 'hep-ph-0306213-1-11-9': 'For static [MATH] mesons [EQUATION]', 'hep-ph-0306213-1-11-10': 'In the non-relativistic limit, one obtains [EQUATION]', 'hep-ph-0306213-1-12-0': '# [MATH] and [MATH] meson Properties', 'hep-ph-0306213-1-13-0': 'Following the quark potential model [CITATION], we choose the effective quark interaction [MATH] as a vector interaction plus a scalar confinement interaction [EQUATION]', 'hep-ph-0306213-1-13-1': 'According to lattice calculation, the scalar confinement kernel is linear in [MATH] [EQUATION]', 'hep-ph-0306213-1-13-2': 'The vector kernel is taken from the one-gluon exchange plus a constant [EQUATION] where [MATH] is the QCD running coupling constant.', 'hep-ph-0306213-1-14-0': 'In our analysis, we use the simple harmonic oscillator wave functions for pions and rho mesons as a first approximation.', 'hep-ph-0306213-1-14-1': 'In this way the underlying physics is emphasized.', 'hep-ph-0306213-1-14-2': '[EQUATION] where [MATH], [MATH] are the spin quantum numbers of the meson, and [MATH] is the ground state wave function of harmonic oscillator in momentum space [EQUATION]', 'hep-ph-0306213-1-14-3': 'In the non-relativistic limit, the two potentials [MATH] and [MATH] are [EQUATION] where [EQUATION]', 'hep-ph-0306213-1-14-4': 'Here, one must be careful with the one-gluon exchange interaction [MATH].', 'hep-ph-0306213-1-14-5': 'It arises from the covariant form [MATH] in momentum space, where [MATH] is the 4-momentum of the exchanged gluon.', 'hep-ph-0306213-1-14-6': 'For potential [MATH], [MATH] and [MATH] are the 4-momenta of incoming and outgoing quarks respectively.', 'hep-ph-0306213-1-14-7': 'Thus in non-relativistic limit, [MATH] which is a Coulomb potential in coordinate space.', 'hep-ph-0306213-1-14-8': 'However for coupling potential [MATH], [MATH] and [MATH] are the 4-momenta of [MATH] quark pair which were created or annihilated on the vacuum.', 'hep-ph-0306213-1-14-9': 'So in non-relativistic limit, [MATH].', 'hep-ph-0306213-1-14-10': 'The term [MATH] is suppressed in [MATH].', 'hep-ph-0306213-1-15-0': 'It is well known in NJL model, the chiral symmetry can be broken dynamically with a nonzero vacuum condensate coming from quark interaction.', 'hep-ph-0306213-1-15-1': 'The current quarks gain dynamical masses and become constituent quarks which are the valence quarks in potential model.', 'hep-ph-0306213-1-15-2': 'The Goldstone boson pions are still massless because the same quark interaction is attractive and will produce large sea quark excitations in pseudo-scalar channel.', 'hep-ph-0306213-1-15-3': 'To account for this dynamical chiral symmetry breaking in our model, the coupling potential [MATH] must be large.', 'hep-ph-0306213-1-15-4': 'This is why we introduce a constant term into the vector kernel.', 'hep-ph-0306213-1-15-5': 'In this way, our model differs from the ordinary potential model (see ref. [CITATION]) which had the constant term in the scalar interaction.', 'hep-ph-0306213-1-15-6': 'In an ordinary potential model, this change will cause no difference (the potential [MATH] is not changed).', 'hep-ph-0306213-1-15-7': 'But in this extended potential model, the coupling potential [MATH] can be made strong enough to give [MATH] mesons very low masses.', 'hep-ph-0306213-1-16-0': 'Inserting the simple model wave functions ([REF]) and ([REF]) into Eq. ([REF]) with the potentials given by ([REF]) and ([REF]), we obtain an eigen equation of a [MATH] matrix [EQUATION] where [EQUATION]', 'hep-ph-0306213-1-16-1': 'The energy [MATH] can be easily obtained [EQUATION]', 'hep-ph-0306213-1-16-2': 'The wave function parameter [MATH] is determined by the minimum of the energy [EQUATION]', 'hep-ph-0306213-1-16-3': 'Since [MATH] is a constant, we get [EQUATION]', 'hep-ph-0306213-1-16-4': 'We approximate the average quark kinetic energy by the following expression.', 'hep-ph-0306213-1-16-5': '[EQUATION]', 'hep-ph-0306213-1-16-6': 'Numerically the deviation is very small and less than 5% (See Fig. [REF]).', 'hep-ph-0306213-1-17-0': 'The current quark mass [MATH] can be obtained from the partial conservation of axial-vector current (PCAC).', 'hep-ph-0306213-1-17-1': 'Consider the axial-vector current [EQUATION]', 'hep-ph-0306213-1-17-2': 'According to PCAC, its divergence is [EQUATION] where [MATH] and [MATH] are current masses of up and down quarks.', 'hep-ph-0306213-1-17-3': 'Let us calculate the matrix element [MATH] for [MATH] meson.', 'hep-ph-0306213-1-17-4': 'One obtains [EQUATION]', 'hep-ph-0306213-1-17-5': 'From right hand side of PCAC relation ([REF]), one also has [EQUATION]', 'hep-ph-0306213-1-17-6': 'Both of the matrix elements can be easily calculated using the model wave functions for a static [MATH] meson.', 'hep-ph-0306213-1-17-7': 'We obtain [EQUATION]', 'hep-ph-0306213-1-17-8': 'Before the numerical calculation, one can make some qualitative analysis of the properties of [MATH] and [MATH] mesons in the model.', 'hep-ph-0306213-1-17-9': 'Let [EQUATION]', 'hep-ph-0306213-1-17-10': 'The matrix equation ([REF]) can be written for [MATH], [MATH] [EQUATION]', 'hep-ph-0306213-1-17-11': 'Also the normalization relation ([REF]) becomes [EQUATION]', 'hep-ph-0306213-1-17-12': 'First, let us consider the [MATH] meson which is a Goldstone boson.', 'hep-ph-0306213-1-17-13': 'We have [EQUATION] where [EQUATION]', 'hep-ph-0306213-1-17-14': 'If the current quark mass [MATH], chiral symmetry is a strict QCD symmetry.', 'hep-ph-0306213-1-17-15': 'From the Goldstone theorem, [MATH] meson should be massless.', 'hep-ph-0306213-1-17-16': 'As an effective theory of QCD, the [MATH] meson mass in our model should also be zero.', 'hep-ph-0306213-1-17-17': 'Assuming [MATH] is the constituent quark mass in the limit of [MATH], we have [EQUATION]', 'hep-ph-0306213-1-17-18': 'In the real world, current quarks have small nonzero masses.', 'hep-ph-0306213-1-17-19': 'So does the [MATH] meson.', 'hep-ph-0306213-1-17-20': 'As the average mass of current quarks (u and d) slightly changes to [MATH], the constituent quark mass will also change a little bit to [MATH] with [MATH].', 'hep-ph-0306213-1-17-21': 'So we have [EQUATION] and [EQUATION]', 'hep-ph-0306213-1-17-22': 'Eqs. ([REF]) and ([REF]) become [EQUATION]', 'hep-ph-0306213-1-17-23': 'We obtain [EQUATION]', 'hep-ph-0306213-1-17-24': 'With the normalization [MATH], we obtain [EQUATION]', 'hep-ph-0306213-1-17-25': 'From PCAC (eq. ([REF])), the mass of current quark is related to [MATH] as [EQUATION]', 'hep-ph-0306213-1-17-26': 'Next we can calculate the weak decay constant of [MATH] meson to obtain [EQUATION]', 'hep-ph-0306213-1-17-27': 'Eq. ([REF]) means [MATH] and [MATH], which shows the importance of the contribution to the [MATH] meson from sea quark y-channel .', 'hep-ph-0306213-1-18-0': 'For the [MATH] meson, with the help of Eq. ([REF]), the equations are [EQUATION]', 'hep-ph-0306213-1-18-1': 'One simply obtains [EQUATION]', 'hep-ph-0306213-1-18-2': 'With the normalization [MATH], [EQUATION]', 'hep-ph-0306213-1-18-3': 'Next the electromagnetic decay constant of [MATH] meson can be calculated as [EQUATION]', 'hep-ph-0306213-1-18-4': 'A relation among masses of [MATH], [MATH] mesons and current quark mass [MATH] can be easily obtained [EQUATION]', 'hep-ph-0306213-1-18-5': 'From the experimental [MATH] meson mass [MATH] MeV, one can estimate that [MATH] MeV.', 'hep-ph-0306213-1-18-6': 'Then we can estimate the current quark mass to be [MATH] MeV from the physical pion mass [MATH] MeV.', 'hep-ph-0306213-1-19-0': 'Another relation about [MATH] and [MATH] decay constants can also be easily obtained from Eqs. ([REF]) and ([REF]), [EQUATION]', 'hep-ph-0306213-1-19-1': 'The KSRF relation [CITATION] gives [EQUATION] while the experiment value is [MATH].', 'hep-ph-0306213-1-19-2': 'In our approach [MATH] from Eq. ([REF]).', 'hep-ph-0306213-1-19-3': 'This means the contribution to the rho meson from sea quark y-channel is rather small.', 'hep-ph-0306213-1-19-4': 'If the sea quark excitations are completely neglected, i.e., [MATH], we may obtain [MATH] and [MATH].', 'hep-ph-0306213-1-19-5': 'Then [EQUATION] thus [EQUATION] which is exactly the KSRF relation.', 'hep-ph-0306213-1-19-6': 'The experimental value is between our result and KSRF relation.', 'hep-ph-0306213-1-20-0': 'In numerical calculation we parametrize the running coupling constant as [EQUATION] as in Ref. [CITATION] (with [MATH] in GeV).', 'hep-ph-0306213-1-20-1': 'Also the constituent quark mass [MATH] is fixed to be [MATH] MeV.', 'hep-ph-0306213-1-20-2': 'The results are listed in Table [REF].', 'hep-ph-0306213-1-20-3': 'First (Set 1), we choose the confinement parameter [MATH]GeV[MATH] according to lattice calculation.', 'hep-ph-0306213-1-20-4': 'The [MATH] meson mass is somehow larger.', 'hep-ph-0306213-1-20-5': 'Next (Set 2), we adjust the confinement parameter slightly to [MATH]GeV[MATH].', 'hep-ph-0306213-1-21-0': 'Both of pion and rho meson masses agree with the experimental values well.', 'hep-ph-0306213-1-21-1': 'The current and constituent quark masses are also reasonable in the ranges of theoretical and experimental estimations.', 'hep-ph-0306213-1-22-0': 'The decay constants [MATH] and [MATH] deviate from the experimental data by a factor 2.', 'hep-ph-0306213-1-22-1': 'This is not very surprising.', 'hep-ph-0306213-1-22-2': 'In the derivation of the effective Hamiltonian (which we skip in this work), the current quark field [MATH] is renormalized into constituent quark field.', 'hep-ph-0306213-1-22-3': 'This means that the constituent quark may include the contributions from configurations other than the single quark, like [MATH], [MATH] etc as indicated through the deep inelastic scattering experiments.', 'hep-ph-0306213-1-22-4': 'For example, we may assume the normalized constituent quark field [MATH] [EQUATION]', 'hep-ph-0306213-1-22-5': 'Here we have introduced the factor [MATH].', 'hep-ph-0306213-1-22-6': 'Roughly speaking, [MATH] is the probability of finding the "pure" current quark field in a complicated constituent quark.', 'hep-ph-0306213-1-22-7': 'However, from the electro-weak Lagrangian we know that what really participates in the electro-weak interactions are the current quarks.', 'hep-ph-0306213-1-22-8': 'In other words, only the "pure" current quark will contribute to the pion weak decay constant and rho meson electromagnetic decay constant.', 'hep-ph-0306213-1-22-9': 'In our calculation we need two real current quarks to annihilate.', 'hep-ph-0306213-1-22-10': 'So there will be a difference of factor [MATH].', 'hep-ph-0306213-1-22-11': 'Decay constants [MATH] and [MATH] can then be fit to the experimental values if we set the additional normalization factor [MATH] (In Table [REF] the results are listed in parentheses).', 'hep-ph-0306213-1-23-0': '# Summary', 'hep-ph-0306213-1-24-0': 'The chiral symmetry is studied in an extended constituent quark potential model which includes the sea quark contributions with RPA approximation.', 'hep-ph-0306213-1-24-1': 'With some modification of quark interaction in potential model, [MATH] meson is still a Goldstone boson.', 'hep-ph-0306213-1-24-2': 'The small mass of [MATH] meson can be connected to nonzero current quark mass from PCAC.', 'hep-ph-0306213-1-24-3': 'All other mesons such as the vector [MATH] meson can also be studied in this model like in the ordinary constituent quark models.', 'hep-ph-0306213-1-24-4': 'With the sea quark contribution, the well-known KSRF relation of the [MATH] and [MATH] meson decay constants is reestablished in the model.', 'hep-ph-0306213-1-25-0': 'This extension to constituent quark potential model aims at improving the constituent quark model to comprise sea quark excitations which are essential to [MATH] (and K) mesons.', 'hep-ph-0306213-1-25-1': 'To thoroughly investigate the effect of this extension, we need perform a full calculation of meson spectroscopy.'}

{'hep-ph-0306213-2-0-0': 'The chiral symmetry is applied to an extended constituent quark potential model.', 'hep-ph-0306213-2-0-1': 'With random phase approximation (RPA), the model comprises non-valence quark excitations.', 'hep-ph-0306213-2-0-2': 'To obtain the pseudoscalar [MATH] meson as a Goldstone boson, the quark effective potentials are modified in the model to account for the dynamical breaking of chiral symmetry.', 'hep-ph-0306213-2-0-3': 'Also the vector [MATH] meson is calculated and the KSRF relation about [MATH] and [MATH] meson decay constants is derived in the model.', 'hep-ph-0306213-2-1-0': '# Introduction', 'hep-ph-0306213-2-2-0': 'Due to the complication of a non-Abelian [MATH] gauge theory, quantum chromodynamics (QCD), which describes the strong interaction, has many non-perturbative features such as the dynamical breaking of chiral symmetry and quark confinement.', 'hep-ph-0306213-2-2-1': 'In the study of hadron structure at low energy scale, two kinds of models of QCD are often used, each incorporating some important QCD features.', 'hep-ph-0306213-2-2-2': 'The constituent quark potential model which incorporates the QCD quark confinement has been impressively successful in hadron spectroscopy and decays [CITATION], except for the pseudo-scalar [MATH] meson which is a Goldstone boson and has very low mass.', 'hep-ph-0306213-2-2-3': 'It was shown recently that the important non-valence quark contribution must be considered in the treatment of pions [CITATION].', 'hep-ph-0306213-2-2-4': 'On the other hand, the Nambu-Jona-Lasinio (NJL) model [CITATION] and several of its extensions [CITATION] which incorporate the chiral symmetry can describe the [MATH] meson very well as a massless Goldstone boson of dynamical breaking of chiral symmetry.', 'hep-ph-0306213-2-2-5': 'However, the NJL model lacks the important QCD property of quark confinement.', 'hep-ph-0306213-2-3-0': 'In Ref. [CITATION], we proposed an extension to the constituent quark model to comprise the non-valence quark excitations.', 'hep-ph-0306213-2-3-1': 'As in NJL model, we start from an effective quark Hamiltonian.', 'hep-ph-0306213-2-3-2': 'The non-valence quark contribution is considered with the RPA approximation.', 'hep-ph-0306213-2-3-3': 'For meson structure, this leads to a Hamiltonian with two channels - the ordinary valence quark channel and a new non-valence quark channel, which is an extension of the quark potential model.', 'hep-ph-0306213-2-3-4': 'This extended quark potential model has a new coupling potential connecting couple the valence quark and the non-valence quark channel.', 'hep-ph-0306213-2-3-5': 'The coupling potential can be related to the ordinary valence quark potential in the model.', 'hep-ph-0306213-2-4-0': 'In this paper, we implement the chiral symmetry in the extended constituent quark potential model.', 'hep-ph-0306213-2-4-1': 'First we introduce the extended quark potential model.', 'hep-ph-0306213-2-4-2': 'In Sec. 3, we analyze [MATH] and [MATH] mesons using this framework.', 'hep-ph-0306213-2-4-3': 'The restriction from chiral symmetry is considered.', 'hep-ph-0306213-2-4-4': 'With the meson wave functions containing non-valence quark excitations, KSRF relation [CITATION] about [MATH] and [MATH] meson decay constants is derived.', 'hep-ph-0306213-2-4-5': 'Finally, we summarize our discussion.', 'hep-ph-0306213-2-5-0': '# Extending Quark Potential Model with RPA', 'hep-ph-0306213-2-6-0': 'QCD perturbation theory is very successful when applied in the high energy processes.', 'hep-ph-0306213-2-6-1': 'In the low energy regime, Hamiltonian approach can provide an alternative method to understand the non-perturbative features of QCD, such as the quark confinement.', 'hep-ph-0306213-2-6-2': 'Many efforts were made to obtain an effective Hamiltonian from the exact QCD Lagrangian [CITATION].', 'hep-ph-0306213-2-6-3': 'Here we start from the effective quark hamiltonian, assuming the quark fields have been separated approximately from the gluon fields [EQUATION] where [MATH] is the free current quark Hamiltonian and [MATH] is the current quark mass.', 'hep-ph-0306213-2-6-4': '[MATH] includes all effective quark interactions.', 'hep-ph-0306213-2-6-5': '[MATH] is the kernel function for coupling vertex [MATH].', 'hep-ph-0306213-2-7-0': 'Apart from a constant, the Hamiltonian ([REF]) can be written in normal order as [EQUATION] where [MATH] represents the free particle energy of constituent quark and its general form is [EQUATION] where the dependence on the three-vector momentum arises from the instantaneous approximation of quark interactions.', 'hep-ph-0306213-2-8-0': 'The two functions [MATH] and [MATH] should be obtained from a self-consistent mean field calculation.', 'hep-ph-0306213-2-8-1': 'However in a constituent quark potential model, the quark interactions contain the linear quark confinement which has severe infrared divergence in momentum space.', 'hep-ph-0306213-2-8-2': 'On the other hand, the constituent potential quark models always assume that the constituent quarks have fixed masses.', 'hep-ph-0306213-2-8-3': 'As an extension of the constituent potential quark model, here we assume that the [MATH] and [MATH] functions can be approximated as constants.', 'hep-ph-0306213-2-8-4': 'So we treat the constituent quark mass as a parameter, [MATH] (with [MATH]).', 'hep-ph-0306213-2-8-5': 'The free particle Hamiltonian becomes [EQUATION]', 'hep-ph-0306213-2-8-6': 'In this paper, we will only consider meson structures.', 'hep-ph-0306213-2-8-7': 'In naive quark model, meson fields are approximately expressed in terms of local quark fields as [EQUATION] where [MATH] for iso-vector pseudo-scalar [MATH] mesons or [MATH] for iso-vector vector [MATH] mesons.', 'hep-ph-0306213-2-8-8': 'According to [MATH] expansion, the above meson structures are exact in the limit [MATH] [CITATION].', 'hep-ph-0306213-2-9-0': 'In the non-relativistic limit, [MATH] for quarks, one has[CITATION] [EQUATION] for [MATH] and [MATH] mesons.', 'hep-ph-0306213-2-9-1': 'On the other hand, the field operators of [MATH] and [MATH] can also be expressed in the [MATH] and [MATH] of meson creation and annihilation operators [EQUATION]', 'hep-ph-0306213-2-9-2': 'If the mesons are composed of valence quark pairs as in the potential models, one deduces from the equivalence of eqs. ([REF]) and ([REF]) [EQUATION]', 'hep-ph-0306213-2-9-3': 'However, generally one can only obtain [EQUATION]', 'hep-ph-0306213-2-9-4': 'This is just the RPA excitation operator - the meson state is a superposition of creation and annihilation of [MATH] pairs on the vacuum.', 'hep-ph-0306213-2-9-5': 'The second part is the non-valence quark contribution.', 'hep-ph-0306213-2-10-0': 'Thus, as an extension to the potential model, we take the mesons as excitation modes of the vacuum of the RPA type: [EQUATION]', 'hep-ph-0306213-2-10-1': 'For a meson in the rest frame, after we take account of the quark interactions, [MATH] excitation operators are [EQUATION] where the time reversals are defined as [EQUATION] and [MATH] and [MATH] are the RPA amplitudes determined from the well-known RPA equation of motion[CITATION] [EQUATION]', 'hep-ph-0306213-2-10-2': 'We obtain the equations for RPA amplitudes [EQUATION]', 'hep-ph-0306213-2-10-3': 'Here we have introduced two effective potentials [MATH] and [MATH].', 'hep-ph-0306213-2-10-4': 'Their matrix elements are [EQUATION]', 'hep-ph-0306213-2-10-5': 'Thus, with the RPA approximation, the quark potential model is extended to a coupling system with two channels.', 'hep-ph-0306213-2-10-6': 'The contribution of valence quark excitations is described by x-amplitude [MATH] which we will call the x-channel wave function.', 'hep-ph-0306213-2-10-7': 'The contribution of non-valence quark excitations is given by the y-amplitude [MATH] which is the y-channel wave function.', 'hep-ph-0306213-2-10-8': 'The potential [MATH] interacts only within each individual channel while the new coupling potential [MATH] couples the two channels together.', 'hep-ph-0306213-2-10-9': 'If the coupling between the two channels [MATH] is small, the system decouples.', 'hep-ph-0306213-2-10-10': 'The extended potential model reduces to the ordinary potential model and the potential [MATH] is just the ordinary quark potential.', 'hep-ph-0306213-2-10-11': 'Here we will show that the coupling potential [MATH] is important due to chiral symmetry .', 'hep-ph-0306213-2-11-0': 'One can use the Dirac kets and bras to simplify notations.', 'hep-ph-0306213-2-11-1': '[MATH] and [MATH] are represented by two kets [MATH] and [MATH] respectively, [EQUATION]', 'hep-ph-0306213-2-11-2': 'Eqs. ([REF]) and ([REF]) can be written concisely as [EQUATION] or the familiar matrix form [EQUATION] where [MATH].', 'hep-ph-0306213-2-12-0': 'By definitions ([REF]) and ([REF]), the potentials [MATH] and [MATH] will be evaluated from the same kernels in the quark interaction [MATH] and thus they can be related together.', 'hep-ph-0306213-2-12-1': "In the non-relativistic limit [MATH], we have [EQUATION] where on the right hand side of equations, the kernels are divided into five types as in ref. [CITATION], i.e., scalar ([MATH]), pseudo-scalar ([MATH]), vector ([MATH]), axial-vector ([MATH]), and tensor ([MATH]); [MATH]'s are the pauli spin matrices.", 'hep-ph-0306213-2-13-0': 'To normalize the model wave functions, we apply the standard boson commutation relation [EQUATION] to rest mesons.', 'hep-ph-0306213-2-13-1': 'We obtain [EQUATION]', 'hep-ph-0306213-2-13-2': 'Then we find the normalization relation of RPA type [EQUATION]', 'hep-ph-0306213-2-13-3': 'Again with Dirac kets and bras, its form is [EQUATION]', 'hep-ph-0306213-2-13-4': 'With the normalized meson wave functions, one can calculate the static properties of mesons.', 'hep-ph-0306213-2-13-5': 'The [MATH] weak decay constant is defined as [EQUATION]', 'hep-ph-0306213-2-13-6': 'For pions at rest, the above equation becomes [EQUATION]', 'hep-ph-0306213-2-13-7': 'In the non-relativistic limit, we obtain [EQUATION] where [MATH] and [MATH] are the coordinate wave functions in x- and y- channel of the [MATH] meson with its total spin [MATH].', 'hep-ph-0306213-2-13-8': 'The [MATH] electro-magnetic decay constant is defined as [EQUATION] where [MATH] are the isospin indices, and [MATH] is the [MATH] meson polarization index.', 'hep-ph-0306213-2-13-9': 'For static [MATH] mesons [EQUATION]', 'hep-ph-0306213-2-13-10': 'In the non-relativistic limit, one obtains [EQUATION]', 'hep-ph-0306213-2-14-0': '# [MATH] and [MATH] meson Properties', 'hep-ph-0306213-2-15-0': 'Following the quark potential model [CITATION], we choose the effective quark interaction [MATH] as a vector interaction plus a scalar confinement interaction [EQUATION]', 'hep-ph-0306213-2-15-1': 'According to lattice calculation, the scalar confinement kernel is linear in [MATH] [EQUATION]', 'hep-ph-0306213-2-15-2': 'The vector kernel is taken from the one-gluon exchange plus a constant [EQUATION] where [MATH] is the QCD running coupling constant.', 'hep-ph-0306213-2-16-0': 'In our analysis, we use the simple harmonic oscillator wave functions for pions and rho mesons as a first approximation.', 'hep-ph-0306213-2-16-1': 'In this way the underlying physics is emphasized.', 'hep-ph-0306213-2-16-2': '[EQUATION] where [MATH], [MATH] are the spin quantum numbers of the meson, and [MATH] is the ground state wave function of harmonic oscillator in momentum space [EQUATION]', 'hep-ph-0306213-2-16-3': 'In the non-relativistic limit, the two potentials [MATH] and [MATH] are [EQUATION] where [EQUATION]', 'hep-ph-0306213-2-16-4': 'Here, one must be careful with the one-gluon exchange interaction [MATH].', 'hep-ph-0306213-2-16-5': 'It arises from the covariant form [MATH] in momentum space, where [MATH] is the 4-momentum of the exchanged gluon.', 'hep-ph-0306213-2-16-6': 'For potential [MATH], [MATH] and [MATH] are the 4-momenta of incoming and outgoing quarks respectively.', 'hep-ph-0306213-2-16-7': 'Thus in non-relativistic limit, [MATH] which is a Coulomb potential in coordinate space.', 'hep-ph-0306213-2-16-8': 'However for coupling potential [MATH], [MATH] and [MATH] are the 4-momenta of [MATH] quark pair which were created or annihilated on the vacuum.', 'hep-ph-0306213-2-16-9': 'So in non-relativistic limit, [MATH].', 'hep-ph-0306213-2-16-10': 'The term [MATH] is suppressed in [MATH].', 'hep-ph-0306213-2-17-0': 'It is well known in NJL model, the chiral symmetry can be broken dynamically with a nonzero vacuum condensate coming from quark interaction.', 'hep-ph-0306213-2-17-1': 'The current quarks gain dynamical masses and become constituent quarks which are the valence quarks in potential model.', 'hep-ph-0306213-2-17-2': 'The Goldstone boson pions are still massless because the same quark interaction is attractive and will produce large non-valence quark excitations in pseudo-scalar channel.', 'hep-ph-0306213-2-17-3': 'To account for this dynamical chiral symmetry breaking in our model, the coupling potential [MATH] must be large.', 'hep-ph-0306213-2-17-4': 'This is why we introduce a constant term into the vector kernel.', 'hep-ph-0306213-2-17-5': 'In this way, our model differs from the ordinary potential model (see ref. [CITATION]) which had the constant term in the scalar interaction.', 'hep-ph-0306213-2-17-6': 'In an ordinary potential model, this change will cause no difference (the potential [MATH] is not changed).', 'hep-ph-0306213-2-17-7': 'But in this extended potential model, the coupling potential [MATH] can be made strong enough to give [MATH] mesons very low masses.', 'hep-ph-0306213-2-18-0': 'Inserting the simple model wave functions ([REF]) and ([REF]) into Eq. ([REF]) with the potentials given by ([REF]) and ([REF]), we obtain an eigen equation of a [MATH] matrix [EQUATION] where [EQUATION]', 'hep-ph-0306213-2-18-1': 'The energy [MATH] can be easily obtained [EQUATION]', 'hep-ph-0306213-2-18-2': 'The wave function parameter [MATH] is determined by the minimum of the energy [EQUATION]', 'hep-ph-0306213-2-18-3': 'Since [MATH] is a constant, we get [EQUATION]', 'hep-ph-0306213-2-18-4': 'We approximate the average quark kinetic energy by the following expression.', 'hep-ph-0306213-2-18-5': '[EQUATION]', 'hep-ph-0306213-2-18-6': 'Numerically the deviation is very small and less than 5% (See Fig. [REF]).', 'hep-ph-0306213-2-19-0': 'The current quark mass [MATH] can be obtained from the partial conservation of axial-vector current (PCAC).', 'hep-ph-0306213-2-19-1': 'Consider the axial-vector current [EQUATION]', 'hep-ph-0306213-2-19-2': 'According to PCAC, its divergence is [EQUATION] where [MATH] and [MATH] are current masses of up and down quarks.', 'hep-ph-0306213-2-19-3': 'Let us calculate the matrix element [MATH] for [MATH] meson with momentum [MATH].', 'hep-ph-0306213-2-19-4': 'One obtains [EQUATION]', 'hep-ph-0306213-2-19-5': 'From right hand side of PCAC relation ([REF]), one also has [EQUATION]', 'hep-ph-0306213-2-19-6': 'Both of the matrix elements can be easily calculated using the model wave functions for a static [MATH] meson.', 'hep-ph-0306213-2-19-7': 'We obtain [EQUATION]', 'hep-ph-0306213-2-19-8': 'Before the numerical calculation, one can make some qualitative analysis of the properties of [MATH] and [MATH] mesons in the model.', 'hep-ph-0306213-2-19-9': 'Let [EQUATION]', 'hep-ph-0306213-2-19-10': 'The matrix equation ([REF]) can be written for [MATH], [MATH] [EQUATION]', 'hep-ph-0306213-2-19-11': 'Also the normalization relation ([REF]) becomes [EQUATION]', 'hep-ph-0306213-2-19-12': 'First, let us consider the [MATH] meson which is a Goldstone boson.', 'hep-ph-0306213-2-19-13': 'We have [EQUATION] where [EQUATION]', 'hep-ph-0306213-2-19-14': 'If the current quark mass [MATH], chiral symmetry is a strict QCD symmetry.', 'hep-ph-0306213-2-19-15': 'From the Goldstone theorem, [MATH] meson should be massless.', 'hep-ph-0306213-2-19-16': 'As an effective theory of QCD, the [MATH] meson mass in our model should also be zero.', 'hep-ph-0306213-2-19-17': 'Assuming [MATH] is the constituent quark mass in the limit of [MATH], we have [EQUATION]', 'hep-ph-0306213-2-19-18': 'In the real world, current quarks have small nonzero masses.', 'hep-ph-0306213-2-19-19': 'So does the [MATH] meson.', 'hep-ph-0306213-2-19-20': 'As the average mass of current quarks (u and d) slightly changes to [MATH], the constituent quark mass will also change a little bit to [MATH] with [MATH].', 'hep-ph-0306213-2-19-21': 'So we have [EQUATION] and [EQUATION]', 'hep-ph-0306213-2-19-22': 'Eqs. ([REF]) and ([REF]) become [EQUATION]', 'hep-ph-0306213-2-19-23': 'We obtain [EQUATION]', 'hep-ph-0306213-2-19-24': 'With the normalization [MATH], we obtain [EQUATION]', 'hep-ph-0306213-2-19-25': 'From PCAC (eq. ([REF])), the mass of current quark is related to [MATH] as [EQUATION]', 'hep-ph-0306213-2-19-26': 'Next we can calculate the weak decay constant of [MATH] meson to obtain [EQUATION]', 'hep-ph-0306213-2-19-27': 'Eq. ([REF]) means [MATH] and [MATH], which shows the importance of the contribution to the [MATH] meson from non-valence quark y-channel .', 'hep-ph-0306213-2-20-0': 'For the [MATH] meson, with the help of Eq. ([REF]), the equations are [EQUATION]', 'hep-ph-0306213-2-20-1': 'One simply obtains [EQUATION]', 'hep-ph-0306213-2-20-2': 'With the normalization [MATH], [EQUATION]', 'hep-ph-0306213-2-20-3': 'Next the electromagnetic decay constant of [MATH] meson can be calculated as [EQUATION]', 'hep-ph-0306213-2-20-4': 'A relation among masses of [MATH], [MATH] mesons and current quark mass [MATH] can be easily obtained [EQUATION]', 'hep-ph-0306213-2-20-5': 'From the experimental [MATH] meson mass [MATH] MeV, one can estimate that [MATH] MeV.', 'hep-ph-0306213-2-20-6': 'Then we can estimate the current quark mass to be [MATH] MeV from the physical pion mass [MATH] MeV.', 'hep-ph-0306213-2-21-0': 'Another relation about [MATH] and [MATH] decay constants can also be easily obtained from Eqs. ([REF]) and ([REF]), [EQUATION]', 'hep-ph-0306213-2-21-1': 'The KSRF relation [CITATION] gives [EQUATION] while the experiment value is [MATH].', 'hep-ph-0306213-2-21-2': 'In our approach [MATH] from Eq. ([REF]).', 'hep-ph-0306213-2-21-3': 'This means the contribution to the rho meson from non-valence quark y-channel is rather small.', 'hep-ph-0306213-2-21-4': 'If the non-valence quark excitations are completely neglected, i.e., [MATH], we may obtain [MATH] and [MATH].', 'hep-ph-0306213-2-21-5': 'Then [EQUATION] thus [EQUATION] which is exactly the KSRF relation.', 'hep-ph-0306213-2-21-6': 'The experimental value is between our result and KSRF relation.', 'hep-ph-0306213-2-22-0': 'In numerical calculation we parametrize the running coupling constant as [EQUATION] as in Ref. [CITATION] (with [MATH] in GeV).', 'hep-ph-0306213-2-22-1': 'Also the constituent quark mass [MATH] is fixed to be [MATH] MeV.', 'hep-ph-0306213-2-22-2': 'The results are listed in Table [REF].', 'hep-ph-0306213-2-22-3': 'First (Set 1), we choose the confinement parameter [MATH]GeV[MATH] according to lattice calculation.', 'hep-ph-0306213-2-22-4': 'The [MATH] meson mass is somehow larger.', 'hep-ph-0306213-2-22-5': 'Next (Set 2), we adjust the confinement parameter slightly to [MATH]GeV[MATH].', 'hep-ph-0306213-2-23-0': 'Both of pion and rho meson masses agree with the experimental values well.', 'hep-ph-0306213-2-23-1': 'The current and constituent quark masses are also reasonable in the ranges of theoretical and experimental estimations.', 'hep-ph-0306213-2-24-0': 'The decay constants [MATH] and [MATH] deviate from the experimental data by a factor 2.', 'hep-ph-0306213-2-24-1': 'This is not very surprising.', 'hep-ph-0306213-2-24-2': 'In the derivation of the effective Hamiltonian (which we skip in this work), the current quark field [MATH] is renormalized into constituent quark field.', 'hep-ph-0306213-2-24-3': 'This means that the constituent quark may include the contributions from configurations other than the single quark, like [MATH], [MATH] etc as indicated through the deep inelastic scattering experiments.', 'hep-ph-0306213-2-24-4': 'For example, we may assume the normalized constituent quark field [MATH] [EQUATION]', 'hep-ph-0306213-2-24-5': 'Here we have introduced the factor [MATH].', 'hep-ph-0306213-2-24-6': 'Roughly speaking, [MATH] is the probability of finding the "pure" current quark field in a complicated constituent quark.', 'hep-ph-0306213-2-24-7': 'However, from the electro-weak Lagrangian we know that what really participates in the electro-weak interactions are the current quarks.', 'hep-ph-0306213-2-24-8': 'In other words, only the "pure" current quark will contribute to the pion weak decay constant and rho meson electromagnetic decay constant.', 'hep-ph-0306213-2-24-9': 'In our calculation we need two real current quarks to annihilate.', 'hep-ph-0306213-2-24-10': 'So there will be a difference of factor [MATH].', 'hep-ph-0306213-2-24-11': 'Decay constants [MATH] and [MATH] can then be fit to the experimental values if we set the additional normalization factor [MATH] (In Table [REF] the results are listed in parentheses).', 'hep-ph-0306213-2-25-0': '# Summary', 'hep-ph-0306213-2-26-0': 'The chiral symmetry is studied in an extended constituent quark potential model which includes the non-valence quark contributions with RPA approximation.', 'hep-ph-0306213-2-26-1': 'With some modification of quark interaction in potential model, [MATH] meson is still a Goldstone boson.', 'hep-ph-0306213-2-26-2': 'The small mass of [MATH] meson can be connected to nonzero current quark mass from PCAC.', 'hep-ph-0306213-2-26-3': 'All other mesons such as the vector [MATH] meson can also be studied in this model like in the ordinary constituent quark models.', 'hep-ph-0306213-2-26-4': 'With the non-valence quark contribution, the well-known KSRF relation of the [MATH] and [MATH] meson decay constants is reestablished in the model.', 'hep-ph-0306213-2-27-0': 'This extension to constituent quark potential model aims at improving the constituent quark model to comprise non-valence quark excitations which are essential to [MATH] (and K) mesons.', 'hep-ph-0306213-2-27-1': 'To thoroughly investigate the effect of this extension, we need perform a full calculation of meson spectroscopy.'}

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['hep-ph-0306213-2-15-2', 'hep-ph-0306213-3-15-2'], ['hep-ph-0306213-2-22-0', 'hep-ph-0306213-3-22-0'], ['hep-ph-0306213-2-22-1', 'hep-ph-0306213-3-22-1'], ['hep-ph-0306213-2-22-2', 'hep-ph-0306213-3-22-2'], ['hep-ph-0306213-2-22-3', 'hep-ph-0306213-3-22-3'], ['hep-ph-0306213-2-22-4', 'hep-ph-0306213-3-22-4'], ['hep-ph-0306213-2-22-5', 'hep-ph-0306213-3-22-5'], ['hep-ph-0306213-2-4-0', 'hep-ph-0306213-3-4-0'], ['hep-ph-0306213-2-4-1', 'hep-ph-0306213-3-4-1'], ['hep-ph-0306213-2-4-2', 'hep-ph-0306213-3-4-2'], ['hep-ph-0306213-2-4-3', 'hep-ph-0306213-3-4-3'], ['hep-ph-0306213-2-4-5', 'hep-ph-0306213-3-4-5'], ['hep-ph-0306213-1-8-0', 'hep-ph-0306213-2-8-0'], ['hep-ph-0306213-1-8-1', 'hep-ph-0306213-2-8-1'], ['hep-ph-0306213-1-8-2', 'hep-ph-0306213-2-8-2'], ['hep-ph-0306213-1-8-3', 'hep-ph-0306213-2-8-3'], ['hep-ph-0306213-1-8-4', 'hep-ph-0306213-2-8-4'], ['hep-ph-0306213-1-8-5', 'hep-ph-0306213-2-8-5'], ['hep-ph-0306213-1-8-9', 'hep-ph-0306213-2-10-3'], ['hep-ph-0306213-1-8-10', 'hep-ph-0306213-2-10-4'], ['hep-ph-0306213-1-8-11', 'hep-ph-0306213-2-10-5'], ['hep-ph-0306213-1-8-12', 'hep-ph-0306213-2-10-6'], ['hep-ph-0306213-1-8-14', 'hep-ph-0306213-2-10-8'], ['hep-ph-0306213-1-8-15', 'hep-ph-0306213-2-10-9'], ['hep-ph-0306213-1-8-16', 'hep-ph-0306213-2-10-10'], ['hep-ph-0306213-1-8-17', 'hep-ph-0306213-2-10-11']]

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[]

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[]

['hep-ph-0306213-1-11-1', 'hep-ph-0306213-1-16-5', 'hep-ph-0306213-1-17-4', 'hep-ph-0306213-1-17-7', 'hep-ph-0306213-1-17-9', 'hep-ph-0306213-1-17-22', 'hep-ph-0306213-1-17-23', 'hep-ph-0306213-2-13-1', 'hep-ph-0306213-2-18-5', 'hep-ph-0306213-2-19-4', 'hep-ph-0306213-2-19-7', 'hep-ph-0306213-2-19-9', 'hep-ph-0306213-2-19-22', 'hep-ph-0306213-2-19-23', 'hep-ph-0306213-3-13-1', 'hep-ph-0306213-3-18-5', 'hep-ph-0306213-3-19-6', 'hep-ph-0306213-3-19-8', 'hep-ph-0306213-3-19-21', 'hep-ph-0306213-3-19-22']

{'1': 'http://arxiv.org/licenses/assumed-1991-2003/', '2': 'http://arxiv.org/licenses/assumed-1991-2003/', '3': 'http://arxiv.org/licenses/assumed-1991-2003/'}

https://arxiv.org/abs/hep-ph/0306213

{'hep-ph-0306213-3-0-0': 'The chiral symmetry is applied to an extended constituent quark potential model.', 'hep-ph-0306213-3-0-1': 'With random phase approximation (RPA), a small component effect is added to the constituent quark model.', 'hep-ph-0306213-3-0-2': 'To obtain the pseudoscalar [MATH] meson as a Goldstone boson, the quark effective potentials are modified in the model to account for the dynamical breaking of chiral symmetry.', 'hep-ph-0306213-3-0-3': 'Also the vector [MATH] meson is calculated and the KSRF relation about [MATH] and [MATH] meson decay constants is derived in the model.', 'hep-ph-0306213-3-1-0': '# Introduction', 'hep-ph-0306213-3-2-0': 'Due to the complication of a non-Abelian [MATH] gauge theory, quantum chromodynamics (QCD), which describes the strong interaction, has many non-perturbative features such as the dynamical breaking of chiral symmetry and quark confinement.', 'hep-ph-0306213-3-2-1': 'In the study of hadron structure at low energy scale, two kinds of models of QCD are often used, each incorporating some important QCD features.', 'hep-ph-0306213-3-2-2': 'The constituent quark potential model which incorporates the QCD quark confinement has been impressively successful in hadron spectroscopy and decays [CITATION], except for the pseudo-scalar [MATH] meson which is a Goldstone boson and has very low mass.', 'hep-ph-0306213-3-2-3': 'It was shown recently that the RPA [MATH]-component must be considered in the treatment of pions [CITATION].', 'hep-ph-0306213-3-2-4': 'On the other hand, the Nambu-Jona-Lasinio (NJL) model [CITATION] and several of its extensions [CITATION] which incorporate the chiral symmetry can describe the [MATH] meson very well as a massless Goldstone boson of dynamical breaking of chiral symmetry.', 'hep-ph-0306213-3-2-5': 'However, the NJL model lacks the important QCD property of quark confinement.', 'hep-ph-0306213-3-3-0': 'In Ref. [CITATION], we proposed an extension to the constituent quark model to comprise the RPA [MATH]-component quark excitations.', 'hep-ph-0306213-3-3-1': 'As in NJL model, we start from an effective quark Hamiltonian.', 'hep-ph-0306213-3-3-2': 'For meson structure, this leads to a Hamiltonian with two channels - the ordinary valence quark [MATH]-channel and a new [MATH]-channel, which is an extension of the quark potential model.', 'hep-ph-0306213-3-3-3': 'This extended quark potential model has a new coupling potential connecting the ordinary valence quark [MATH]-channel and the small component [MATH]-channel.', 'hep-ph-0306213-3-3-4': 'The coupling potential can be related to the ordinary valence quark potential in the model.', 'hep-ph-0306213-3-4-0': 'In this paper, we implement the chiral symmetry in the extended constituent quark potential model.', 'hep-ph-0306213-3-4-1': 'First we introduce the extended quark potential model.', 'hep-ph-0306213-3-4-2': 'In Sec. 3, we analyze [MATH] and [MATH] mesons using this framework.', 'hep-ph-0306213-3-4-3': 'The restriction from chiral symmetry is considered.', 'hep-ph-0306213-3-4-4': 'With the meson wave functions containing [MATH]-channel quark excitations, KSRF relation [CITATION] about [MATH] and [MATH] meson decay constants is derived.', 'hep-ph-0306213-3-4-5': 'Finally, we summarize our discussion.', 'hep-ph-0306213-3-5-0': '# Extending Quark Potential Model with RPA', 'hep-ph-0306213-3-6-0': 'QCD perturbation theory is very successful when applied in the high energy processes.', 'hep-ph-0306213-3-6-1': 'In the low energy regime, Hamiltonian approach can provide an alternative method to understand the non-perturbative features of QCD, such as the quark confinement.', 'hep-ph-0306213-3-6-2': 'Many efforts were made to obtain an effective Hamiltonian from the exact QCD Lagrangian [CITATION].', 'hep-ph-0306213-3-6-3': 'Here we start from the effective quark hamiltonian, assuming the quark fields have been separated approximately from the gluon fields [EQUATION] where [MATH] is the free current quark Hamiltonian and [MATH] is the current quark mass.', 'hep-ph-0306213-3-6-4': '[MATH] includes all effective quark interactions.', 'hep-ph-0306213-3-6-5': '[MATH] is the kernel function for coupling vertex [MATH].', 'hep-ph-0306213-3-7-0': 'Apart from a constant, the Hamiltonian ([REF]) can be written in normal order as [EQUATION] where [MATH] represents the free particle energy of constituent quark and its general form is [EQUATION] where the dependence on the three-vector momentum arises from the instantaneous approximation of quark interactions.', 'hep-ph-0306213-3-8-0': 'The two functions [MATH] and [MATH] should be obtained from a self-consistent mean field calculation.', 'hep-ph-0306213-3-8-1': 'However in a constituent quark potential model, the quark interactions contain the linear quark confinement which has severe infrared divergence in momentum space.', 'hep-ph-0306213-3-8-2': 'On the other hand, the constituent potential quark models always assume that the constituent quarks have fixed masses.', 'hep-ph-0306213-3-8-3': 'As an extension of the constituent potential quark model, here we assume that the [MATH] and [MATH] functions can be approximated as constants.', 'hep-ph-0306213-3-8-4': 'So we treat the constituent quark mass as a parameter, [MATH] (with [MATH]).', 'hep-ph-0306213-3-8-5': 'The free particle Hamiltonian becomes [EQUATION]', 'hep-ph-0306213-3-8-6': 'In this paper, we will only consider meson structures.', 'hep-ph-0306213-3-8-7': 'In naive quark model, meson fields are approximately expressed in terms of local quark fields as [EQUATION] where [MATH] for iso-vector pseudo-scalar [MATH] mesons or [MATH] for iso-vector vector [MATH] mesons.', 'hep-ph-0306213-3-8-8': 'According to [MATH] expansion, the above meson structures are exact in the limit [MATH] [CITATION].', 'hep-ph-0306213-3-9-0': 'In the non-relativistic limit, as quark momentum [MATH], one has[CITATION] [EQUATION] for [MATH] and [MATH] mesons.', 'hep-ph-0306213-3-9-1': 'On the other hand, the field operators of [MATH] and [MATH] can also be expressed in the [MATH] and [MATH] of meson creation and annihilation operators [EQUATION]', 'hep-ph-0306213-3-9-2': 'If the mesons are composed of creation of valence quark pairs as in the potential models, one deduces from the equivalence of eqs. ([REF]) and ([REF]) [EQUATION]', 'hep-ph-0306213-3-9-3': 'However, generally one can only obtain [EQUATION]', 'hep-ph-0306213-3-9-4': 'This is just the RPA excitation operator - the meson state is a superposition of creation and annihilation of [MATH] pairs on the vacuum.', 'hep-ph-0306213-3-9-5': 'The second part, i.e. the [MATH]-component is simply discarded in the ordinary quark potential model as a small component.', 'hep-ph-0306213-3-10-0': 'Thus, as an extension to the potential model, we take the mesons as excitation modes of the vacuum of the RPA type: [EQUATION]', 'hep-ph-0306213-3-10-1': 'For a meson in the rest frame, after we take account of the quark interactions, [MATH] excitation operators are [EQUATION] where the time reversals are defined as [EQUATION] and [MATH] and [MATH] are the RPA amplitudes determined from the well-known RPA equation of motion[CITATION] [EQUATION]', 'hep-ph-0306213-3-10-2': 'We obtain the equations for RPA amplitudes [EQUATION]', 'hep-ph-0306213-3-10-3': 'Here we have introduced two effective potentials [MATH] and [MATH].', 'hep-ph-0306213-3-10-4': 'Their matrix elements are [EQUATION]', 'hep-ph-0306213-3-10-5': 'Thus, with the RPA approximation, the quark potential model is extended to a coupling system with two channels.', 'hep-ph-0306213-3-10-6': 'The first part, i.e. x-amplitude [MATH] which we will call the x-channel wave function, is the large component which is the sole valence quark contribution in the ordinary quark potential model.', 'hep-ph-0306213-3-10-7': 'The second part, i.e. the y-amplitude [MATH] which we will call the y-channel wave function, is the small component which is discarded in the ordinary quark potential model.', 'hep-ph-0306213-3-10-8': 'The potential [MATH] interacts only within each individual channel while the new coupling potential [MATH] couples the two channels together.', 'hep-ph-0306213-3-10-9': 'If the coupling between the two channels [MATH] is small, the system decouples.', 'hep-ph-0306213-3-10-10': 'The extended potential model reduces to the ordinary potential model and the potential [MATH] is just the ordinary quark potential.', 'hep-ph-0306213-3-10-11': 'Here we will show that the coupling potential [MATH] is important due to chiral symmetry .', 'hep-ph-0306213-3-11-0': 'One can use the Dirac kets and bras to simplify notations.', 'hep-ph-0306213-3-11-1': '[MATH] and [MATH] are represented by two kets [MATH] and [MATH] respectively, [EQUATION]', 'hep-ph-0306213-3-11-2': 'Eqs. ([REF]) and ([REF]) can be written concisely as [EQUATION] or the familiar matrix form [EQUATION] where [MATH].', 'hep-ph-0306213-3-12-0': 'By definitions ([REF]) and ([REF]), the potentials [MATH] and [MATH] will be evaluated from the same kernels in the quark interaction [MATH], thus they can be related to each other.', 'hep-ph-0306213-3-12-1': "In the non-relativistic limit [MATH], we have [EQUATION] where on the right hand side of equations, the kernels are divided into five types as in ref. [CITATION], i.e., scalar ([MATH]), pseudo-scalar ([MATH]), vector ([MATH]), axial-vector ([MATH]), and tensor ([MATH]); [MATH]'s are the pauli spin matrices.", 'hep-ph-0306213-3-13-0': 'To normalize the model wave functions, we apply the standard boson commutation relation [EQUATION] to rest mesons.', 'hep-ph-0306213-3-13-1': 'We obtain [EQUATION]', 'hep-ph-0306213-3-13-2': 'Then we find the normalization relation of RPA type [EQUATION]', 'hep-ph-0306213-3-13-3': 'Again with Dirac kets and bras, its form is [EQUATION]', 'hep-ph-0306213-3-13-4': 'With the normalized meson wave functions, one can calculate the static properties of mesons.', 'hep-ph-0306213-3-13-5': 'The [MATH] weak decay constant is defined as [EQUATION]', 'hep-ph-0306213-3-13-6': 'For pions at rest, the above equation becomes [EQUATION]', 'hep-ph-0306213-3-13-7': 'In the non-relativistic limit, we obtain [EQUATION] where [MATH] and [MATH] are the coordinate wave functions in x- and y- channel of the [MATH] meson with its total spin [MATH].', 'hep-ph-0306213-3-13-8': 'The [MATH] electro-magnetic decay constant is defined as [EQUATION] where [MATH] are the isospin indices, and [MATH] is the [MATH] meson polarization index.', 'hep-ph-0306213-3-13-9': 'For static [MATH] mesons [EQUATION]', 'hep-ph-0306213-3-13-10': 'In the non-relativistic limit, one obtains [EQUATION]', 'hep-ph-0306213-3-14-0': '# [MATH] and [MATH] meson Properties', 'hep-ph-0306213-3-15-0': 'Following the quark potential model [CITATION], we choose the effective quark interaction [MATH] as a vector interaction plus a scalar confinement interaction [EQUATION]', 'hep-ph-0306213-3-15-1': 'According to lattice calculation, the scalar confinement kernel is linear in [MATH] [EQUATION]', 'hep-ph-0306213-3-15-2': 'The vector kernel is taken from the one-gluon exchange plus a constant [EQUATION] where [MATH] is the QCD running coupling constant.', 'hep-ph-0306213-3-16-0': 'In our analysis, we use the simple harmonic oscillator wave functions for pions and rho mesons as a first approximation.', 'hep-ph-0306213-3-16-1': 'In this way the underlying physics is emphasized.', 'hep-ph-0306213-3-16-2': '[EQUATION] where [MATH], [MATH] are the spin quantum numbers of the meson, and [MATH] is the ground state wave function of harmonic oscillator in momentum space [EQUATION]', 'hep-ph-0306213-3-16-3': 'In the non-relativistic limit, the two potentials [MATH] and [MATH] are [EQUATION] where [EQUATION]', 'hep-ph-0306213-3-16-4': 'Here, one must be careful with the one-gluon exchange interaction [MATH].', 'hep-ph-0306213-3-16-5': 'It arises from the covariant form [MATH] in momentum space, where [MATH] is the 4-momentum of the exchanged gluon.', 'hep-ph-0306213-3-16-6': 'For potential [MATH], [MATH] and [MATH] are the 4-momenta of incoming and outgoing quarks respectively.', 'hep-ph-0306213-3-16-7': 'Thus in non-relativistic limit, [MATH] which is a Coulomb potential in coordinate space.', 'hep-ph-0306213-3-16-8': 'However for coupling potential [MATH], [MATH] and [MATH] are the 4-momenta of [MATH] quark pair which were created or annihilated on the vacuum.', 'hep-ph-0306213-3-16-9': 'So in non-relativistic limit, [MATH].', 'hep-ph-0306213-3-16-10': 'The term [MATH] is suppressed in [MATH].', 'hep-ph-0306213-3-17-0': 'It is well known in NJL model, the chiral symmetry can be broken dynamically with a nonzero vacuum condensate coming from quark interaction.', 'hep-ph-0306213-3-17-1': 'The current quarks gain dynamical masses and become constituent quarks which are the valence quarks in potential model.', 'hep-ph-0306213-3-17-2': 'The Goldstone boson pions are still massless.', 'hep-ph-0306213-3-17-3': 'To account for this dynamical chiral symmetry breaking in our model, the coupling potential [MATH] must be large.', 'hep-ph-0306213-3-17-4': 'This is why we introduce a constant term into the vector kernel.', 'hep-ph-0306213-3-17-5': 'In this way, our model differs from the ordinary potential model (see ref. [CITATION]) which had the constant term in the scalar interaction.', 'hep-ph-0306213-3-17-6': 'In an ordinary potential model, this change will cause no difference (the potential [MATH] is not changed).', 'hep-ph-0306213-3-17-7': 'But in this extended potential model, the coupling potential [MATH] can be made strong enough to give [MATH] mesons very low masses.', 'hep-ph-0306213-3-18-0': 'Inserting the simple model wave functions ([REF]) and ([REF]) into Eq. ([REF]) with the potentials given by ([REF]) and ([REF]), we obtain an eigen equation of a [MATH] matrix [EQUATION] where [EQUATION]', 'hep-ph-0306213-3-18-1': 'The energy [MATH] can be easily obtained [EQUATION]', 'hep-ph-0306213-3-18-2': 'The wave function parameter [MATH] is determined by the minimum of the energy [EQUATION]', 'hep-ph-0306213-3-18-3': 'Since [MATH] is a constant, we get [EQUATION]', 'hep-ph-0306213-3-18-4': 'We approximate the average quark kinetic energy by the following expression.', 'hep-ph-0306213-3-18-5': '[EQUATION]', 'hep-ph-0306213-3-18-6': 'Numerically the deviation is very small and less than 5% (See Fig. [REF]).', 'hep-ph-0306213-3-19-0': 'The current quark mass [MATH] can be obtained from the partial conservation of axial-vector current (PCAC).', 'hep-ph-0306213-3-19-1': 'Consider the axial-vector current [EQUATION]', 'hep-ph-0306213-3-19-2': 'According to PCAC, its divergence is [EQUATION] where [MATH] and [MATH] are current masses of up and down quarks.', 'hep-ph-0306213-3-19-3': 'Let us calculate the matrix element [MATH] for [MATH] meson with momentum [MATH].', 'hep-ph-0306213-3-19-4': 'One obtains [EQUATION] >From right hand side of PCAC relation ([REF]), one also has [EQUATION]', 'hep-ph-0306213-3-19-5': 'Both of the matrix elements can be easily calculated using the model wave functions for a static [MATH] meson.', 'hep-ph-0306213-3-19-6': 'We obtain [EQUATION]', 'hep-ph-0306213-3-19-7': 'Before the numerical calculation, one can make some qualitative analysis of the properties of [MATH] and [MATH] mesons in the model.', 'hep-ph-0306213-3-19-8': 'Let [EQUATION]', 'hep-ph-0306213-3-19-9': 'The matrix equation ([REF]) can be written for [MATH], [MATH] [EQUATION]', 'hep-ph-0306213-3-19-10': 'Also the normalization relation ([REF]) becomes [EQUATION]', 'hep-ph-0306213-3-19-11': 'First, let us consider the [MATH] meson which is a Goldstone boson.', 'hep-ph-0306213-3-19-12': 'We have [EQUATION] where [EQUATION]', 'hep-ph-0306213-3-19-13': 'If the current quark mass [MATH], chiral symmetry is a strict QCD symmetry.', 'hep-ph-0306213-3-19-14': 'From the Goldstone theorem, [MATH] meson should be massless.', 'hep-ph-0306213-3-19-15': 'As an effective theory of QCD, the [MATH] meson mass in our model should also be zero.', 'hep-ph-0306213-3-19-16': 'Assuming [MATH] is the constituent quark mass in the limit of [MATH], we have [EQUATION]', 'hep-ph-0306213-3-19-17': 'In the real world, current quarks have small nonzero masses.', 'hep-ph-0306213-3-19-18': 'So does the [MATH] meson.', 'hep-ph-0306213-3-19-19': 'As the average mass of current quarks (u and d) slightly changes to [MATH], the constituent quark mass will also change a little bit to [MATH] with [MATH].', 'hep-ph-0306213-3-19-20': 'So we have [EQUATION] and [EQUATION]', 'hep-ph-0306213-3-19-21': 'Eqs. ([REF]) and ([REF]) become [EQUATION]', 'hep-ph-0306213-3-19-22': 'We obtain [EQUATION]', 'hep-ph-0306213-3-19-23': 'With the normalization [MATH], we obtain [EQUATION] >From PCAC (eq. ([REF])), the mass of current quark is related to [MATH] as [EQUATION]', 'hep-ph-0306213-3-19-24': 'Next we can calculate the weak decay constant of [MATH] meson to obtain [EQUATION]', 'hep-ph-0306213-3-19-25': 'Eq. ([REF]) means [MATH] and [MATH], which shows the importance of the contribution to the [MATH] meson from the small component of y-channel .', 'hep-ph-0306213-3-20-0': 'For the [MATH] meson, with the help of Eq. ([REF]), the equations are [EQUATION]', 'hep-ph-0306213-3-20-1': 'One simply obtains [EQUATION]', 'hep-ph-0306213-3-20-2': 'With the normalization [MATH], [EQUATION]', 'hep-ph-0306213-3-20-3': 'Next the electromagnetic decay constant of [MATH] meson can be calculated as [EQUATION]', 'hep-ph-0306213-3-20-4': 'A relation among masses of [MATH], [MATH] mesons and current quark mass [MATH] can be easily obtained [EQUATION] >From the experimental [MATH] meson mass [MATH] MeV, one can estimate that [MATH] MeV.', 'hep-ph-0306213-3-20-5': 'Then we can estimate the current quark mass to be [MATH] MeV from the physical pion mass [MATH] MeV.', 'hep-ph-0306213-3-21-0': 'Another relation about [MATH] and [MATH] decay constants can also be easily obtained from Eqs. ([REF]) and ([REF]), [EQUATION]', 'hep-ph-0306213-3-21-1': 'The KSRF relation [CITATION] gives [EQUATION] while the experiment value is [MATH].', 'hep-ph-0306213-3-21-2': 'In our approach [MATH] from Eq. ([REF]).', 'hep-ph-0306213-3-21-3': 'This means the contribution to the rho meson from y-channel is rather small.', 'hep-ph-0306213-3-21-4': 'If the small component is completely neglected, i.e., [MATH], we may obtain [MATH] and [MATH].', 'hep-ph-0306213-3-21-5': 'Then [EQUATION] thus [EQUATION] which is exactly the KSRF relation.', 'hep-ph-0306213-3-21-6': 'The experimental value is between our result and KSRF relation.', 'hep-ph-0306213-3-22-0': 'In numerical calculation we parametrize the running coupling constant as [EQUATION] as in Ref. [CITATION] (with [MATH] in GeV).', 'hep-ph-0306213-3-22-1': 'Also the constituent quark mass [MATH] is fixed to be [MATH] MeV.', 'hep-ph-0306213-3-22-2': 'The results are listed in Table [REF].', 'hep-ph-0306213-3-22-3': 'First (Set 1), we choose the confinement parameter [MATH]GeV[MATH] according to lattice calculation.', 'hep-ph-0306213-3-22-4': 'The [MATH] meson mass is somehow larger.', 'hep-ph-0306213-3-22-5': 'Next (Set 2), we adjust the confinement parameter slightly to [MATH]GeV[MATH].', 'hep-ph-0306213-3-23-0': 'Both of pion and rho meson masses agree with the experimental values well.', 'hep-ph-0306213-3-23-1': 'The current and constituent quark masses are also reasonable in the ranges of theoretical and experimental estimations.', 'hep-ph-0306213-3-24-0': 'The decay constants [MATH] and [MATH] deviate from the experimental data by a factor 2.', 'hep-ph-0306213-3-24-1': 'This is not very surprising.', 'hep-ph-0306213-3-24-2': 'In the derivation of the effective Hamiltonian (which we skip in this work), the current quark field [MATH] is renormalized into constituent quark field.', 'hep-ph-0306213-3-24-3': 'This means that the constituent quark may include the contributions from configurations other than the single quark, like [MATH], [MATH] etc as indicated through the deep inelastic scattering experiments.', 'hep-ph-0306213-3-24-4': 'For example, we may assume the normalized constituent quark field [MATH] [EQUATION]', 'hep-ph-0306213-3-24-5': 'Here we have introduced the factor [MATH].', 'hep-ph-0306213-3-24-6': 'Roughly speaking, [MATH] is the probability of finding the "pure" current quark field in a complicated constituent quark.', 'hep-ph-0306213-3-24-7': 'However, from the electro-weak Lagrangian we know that what really participates in the electro-weak interactions are the current quarks.', 'hep-ph-0306213-3-24-8': 'In other words, only the "pure" current quark will contribute to the pion weak decay constant and rho meson electromagnetic decay constant.', 'hep-ph-0306213-3-24-9': 'In our calculation we need two real current quarks to annihilate.', 'hep-ph-0306213-3-24-10': 'So there will be a difference of factor [MATH].', 'hep-ph-0306213-3-24-11': 'Decay constants [MATH] and [MATH] can then be fit to the experimental values if we set the additional normalization factor [MATH] (In Table [REF] the results are listed in parentheses).', 'hep-ph-0306213-3-25-0': '# Summary', 'hep-ph-0306213-3-26-0': 'The chiral symmetry is studied in an extended constituent quark potential model which includes the small component effect with RPA approximation.', 'hep-ph-0306213-3-26-1': 'With some modification of quark interaction in potential model, [MATH] meson is still a Goldstone boson.', 'hep-ph-0306213-3-26-2': 'The small mass of [MATH] meson can be connected to nonzero current quark mass from PCAC.', 'hep-ph-0306213-3-26-3': 'All other mesons such as the vector [MATH] meson can also be studied in this model like in the ordinary constituent quark models.', 'hep-ph-0306213-3-26-4': 'With the small component effect of y-channel contribution, the well-known KSRF relation of the [MATH] and [MATH] meson decay constants is reestablished in the model.', 'hep-ph-0306213-3-27-0': 'This extension to constituent quark potential model aims at improving the constituent quark model to comprise the small component, i.e., the [MATH] channel quark excitations which are essential to [MATH] (and K) mesons.', 'hep-ph-0306213-3-27-1': 'To thoroughly investigate the effect of this extension, we need perform a full calculation of meson spectroscopy.'}

1802.03053

{'1802.03053-1-0-0': 'We study the asymptotics as [MATH] of stationary [MATH]-harmonic maps [MATH] from a compact manifold [MATH] to [MATH], satisfying the natural energy growth condition [EQUATION].', '1802.03053-1-0-1': 'Along a subsequence [MATH], we show that the singular sets [MATH] converge to the support of a stationary, rectifiable [MATH]-varifold [MATH] of density [MATH], given by the concentrated part of the measure [EQUATION].', '1802.03053-1-0-2': 'When [MATH], we show moreover that the density of [MATH] takes values in [MATH].', '1802.03053-1-0-3': 'Finally, on every compact manifold of dimension [MATH] we produce examples of nontrivial families [MATH] of such maps via natural min-max constructions.', '1802.03053-1-1-0': '# Introduction', '1802.03053-1-2-0': 'In their 1995 paper [CITATION], Hardt and Lin consider the following question: given a simply connected domain [MATH] and a map [MATH] of nonzero degree, what can be said about the limiting behavior of maps [EQUATION] minimizing the [MATH]-energy [EQUATION] as [MATH] approaches [MATH] from below?', '1802.03053-1-2-1': 'They succeed in showing-among other things-that away from a collection [MATH] of [MATH] singularities, a subsequence [MATH] converges strongly to a harmonic map [MATH], and the measures [EQUATION] converge to the sum [MATH] of Dirac masses on [MATH] [CITATION].', '1802.03053-1-2-2': 'Moreover, the singular set [MATH] minimizes a certain "renormalized energy" function [MATH] associated to [MATH], providing a strong constraint on the location of the singularities.', '1802.03053-1-2-3': 'In particular, though the homotopically nontrivial boundary map [MATH] admits no extension to an [MATH]-valued map of finite Dirichlet energy-i.e, [MATH]-the limit of the [MATH]-energy minimizers as [MATH] provides us with a natural candidate for the optimal harmonic extension of [MATH] to an [MATH]-valued map on [MATH].', '1802.03053-1-3-0': 'The results of [CITATION] were inspired in large part by the similar results of Bethuel, Brezis, and Helein-contained in the influential monograph [CITATION]-concerning the asymptotics for minimizers [MATH] of the Ginzburg-Landau functionals [EQUATION] as [MATH], with the measures [EQUATION] taking on the role played by the measures [MATH] in the setting of [CITATION].', '1802.03053-1-3-1': 'In recent decades, the asymptotics for critical points of the Ginzburg-Landau functionals [MATH] in higher dimensions have also been studied by a number of authors, often with an emphasis on the relationship betweeen concentration phenomena for the measures [MATH] and minimal submanifolds of codimension two (see, for instance, [CITATION],[CITATION],[CITATION],[CITATION],[CITATION],[CITATION], among many others).', '1802.03053-1-3-2': 'A typical result says roughly that if the measures [MATH] have uniformly bounded mass, then a subsequence [MATH]-converges as [MATH] to a limiting measure that decomposes into two pieces: a concentrated component given by a stationary, rectifiable varifold [MATH] of codimension two, and a diffuse measure of the form [MATH] for some harmonic one-form [MATH] (which vanishes under mild compactness assumptions) (see, e.g., [CITATION],[CITATION],[CITATION],[CITATION]).', '1802.03053-1-4-0': 'Results of this type point to the possibility of employing variational methods for the Ginzburg-Landau functionals to produce minimal submanifolds of codimension two, but for the complete success of such efforts, we need an improved understanding of the concentration of [MATH].', '1802.03053-1-4-1': 'In particular, the question of integrality (up to a factor of [MATH]) of the limiting varifold [MATH] has been resolved only in dimension two [CITATION] and for local minimizers in higher dimensions [CITATION].', '1802.03053-1-4-2': "If, on the other hand, one could establish integrality of [MATH] for general families of critical points (or families with bounded index), then the min-max methods of [CITATION] and [CITATION] would provide a new proof of the existence of nontrivial stationary, integral varifolds of codimension two, in the spirit of Guaraco's work for the Allen-Cahn equation in codimension one [CITATION].", '1802.03053-1-5-0': 'In this paper, motivated by analogy with the Ginzburg-Landau setting, we investigate the limiting behavior as [MATH] of stationary [MATH]-harmonic maps [MATH] from an arbitrary compact, oriented manifold [MATH] to the circle.', '1802.03053-1-5-1': 'At the global level, we find that the limiting behavior of the maps [MATH] and their energy measures strongly resembles the asymptotics described above for solutions to the Ginzburg-Landau equations.', '1802.03053-1-5-2': 'At the smallest scales, however, the comparatively straightforward blow-up analysis for [MATH]-harmonic maps leads us to some simpler arguments and sharper estimates than are currently available in the Ginzburg-Landau setting.', '1802.03053-1-6-0': 'Let [MATH] be a sequence with [MATH], and let [MATH] be a sequence of stationary [MATH]-harmonic maps from a compact, oriented Riemannian manifold [MATH] to the circle, satisfying [EQUATION]', '1802.03053-1-6-1': 'Then (a subsequence of) the energy measures [MATH] converge weakly in [MATH] to a limiting measure [MATH] of the form [EQUATION] where [MATH] is a harmonic one-form, and [MATH] is a stationary, rectifiable [MATH] varifold.', '1802.03053-1-6-2': 'Furthermore, the support of [MATH] is given by the Hausdorff limit [EQUATION] of the singular sets [MATH], and the density [MATH] satisfies [EQUATION]', '1802.03053-1-7-0': 'Suppose that, in addition to the hypotheses of Theorem [REF], either [MATH] or [EQUATION].', '1802.03053-1-7-1': 'Then (a subsequence of) the maps [MATH] converge weakly in [MATH] for all [MATH], and strongly in [MATH], to a limiting map [MATH] that is harmonic away from [MATH].', '1802.03053-1-8-0': 'In the situation of Theorem [REF], if [MATH], then the density of the concentration varifold [MATH] has the form [EQUATION] for some [MATH].', '1802.03053-1-9-0': 'As in the Ginzburg-Landau setting, the question of the integrality of [MATH] remains open in higher dimensions, but we suspect that the answer will be affirmative.', '1802.03053-1-10-0': 'On every compact Riemannian manifold [MATH] of dimension [MATH], there exists a family [MATH] of stationary [MATH]-harmonic maps to [MATH] for which [EQUATION]', '1802.03053-1-11-0': '## Outline of the Paper: In Section [REF], we review important facts about the structure of maps in [MATH], [MATH]-harmonic functions and weakly [MATH]-harmonic maps to [MATH], and stationary [MATH]-harmonic maps.', '1802.03053-1-12-0': 'In Section [REF], we record a sharp lower bound for the [MATH]-energy density of a stationary [MATH]-harmonic map [MATH] on its singular set [MATH]-a simpler and sharper analog of the [MATH]-ellipticity result (see [CITATION],[CITATION]) for solutions of the Ginzburg-Landau equations.', '1802.03053-1-12-1': 'We then use this to obtain [MATH]-independent estimates for the [MATH]-current [MATH] encoding the topological singularities of [MATH], in the dual Sobolev norms [MATH] for [MATH].', '1802.03053-1-13-0': 'In Section [REF], we employ the results of the preceding sections to estimate separately the components of the Hodge decomposition of the one-form [MATH], first globally in [MATH] for [MATH], then in stronger norms away from [MATH].', '1802.03053-1-14-0': 'In Section [REF], we use these estimates, together with some standard techniques from the study of energy concentration phenomena, to complete the proofs of Theorems [REF] and [REF].', '1802.03053-1-15-0': 'In Section [REF], we prove Theorem [REF], first under some compactness assumptions, using Theorem [REF] and a Pohozaev-type identity, and then for the general case, by showing that the compactness assumptions hold at scales outside of which the normalized energy measures vanish.', '1802.03053-1-16-0': "In Section [REF], we employ min-max arguments like those in [CITATION] together with Wang's results for generalized Ginzburg-Landau functionals [CITATION] to prove Theorem [REF].", '1802.03053-1-16-1': 'We also include a short appendix, containing the proofs of some estimates which are of use to us, but do not play a central role in the paper.', '1802.03053-1-17-0': '# Preliminaries: The Structure of [MATH] and Circle-Valued [MATH]-Harmonic Maps', '1802.03053-1-18-0': '## Topological Singularities and Lifting in [MATH]', '1802.03053-1-19-0': 'Let [MATH] be a compact, oriented Riemannian manifold, and consider the space [MATH] of circle-valued Sobolev maps, realized as the collection of complex-valued maps [MATH] satisfying [MATH] almost everywhere in [MATH].', '1802.03053-1-19-1': 'For each [MATH], we denote by [MATH] the one-form [EQUATION]', '1802.03053-1-19-2': 'Observe that [MATH] almost everywhere on [MATH], so that [MATH] belongs to [MATH].', '1802.03053-1-19-3': 'When [MATH] is smooth, the form [MATH] is obviously closed, and it is a straightforward consequence of the Poincare Lemma that [MATH] has a local lifting of the form [MATH] for some smooth, real-valued [MATH].', '1802.03053-1-20-0': 'For general [MATH], the exterior derivative [MATH] is no longer well-defined pointwise, but since [MATH] belongs to [MATH], we can still make sense of [MATH] as a distribution in [MATH].', '1802.03053-1-20-1': 'Namely, one defines the distributional Jacobian [MATH] of [MATH] to be the [MATH]-current acting on smooth [MATH]-forms [MATH] by [EQUATION]', '1802.03053-1-20-2': 'The analytic and measure-theoretic properties of distributional Jacobians for [MATH]-valued maps (and their analog for sphere-valued maps more generally) have been studied by a number of authors; we make no attempt to survey the many contributions here, but refer the reader to the papers [CITATION], [CITATION], [CITATION], and the references therein for a sample.', '1802.03053-1-20-3': 'Note that for smooth, complex-valued maps, we have the pointwise relation [EQUATION] and since [MATH] defines a continuous map from [MATH] to the space of [MATH] two-forms, it follows that [EQUATION] holds for all [MATH].', '1802.03053-1-20-4': 'In particular, since [MATH] almost everywhere, one deduces that [MATH] for all [MATH].', '1802.03053-1-20-5': 'On the other hand, for [MATH], and [MATH], the maps [MATH] given by [EQUATION] evidently lie in [MATH], with nontrivial distributional Jacobian [EQUATION]', '1802.03053-1-20-6': 'Observe now that if [MATH] has the form [MATH] for some real-valued [MATH], then [MATH] is given by [EQUATION] and since [MATH] can be approximated in [MATH] by smooth functions, it follows that [MATH].', '1802.03053-1-20-7': 'The following result of Demengel provides a useful converse-if the topological singularity vanishes, then [MATH] lifts locally to a real-valued function in the same Sobolev space:', '1802.03053-1-21-0': 'The significance of the lifting result for variational problems on [MATH] is clear: away from the support of the [MATH]-current [MATH], an [MATH]-valued solution [MATH] of some geometric p.d.e. lifts locally to a function [MATH] solving an associated scalar problem, for which a stronger regularity theory is often available.', '1802.03053-1-22-0': '## Weakly [MATH]-Harmonic Maps to [MATH]', '1802.03053-1-23-0': 'A map [MATH] for [MATH] is called weakly [MATH]-harmonic if it satisfies [EQUATION] for all [MATH].', '1802.03053-1-23-1': "Writing [MATH] in ([REF]), it's easy to see that ([REF]) holds if and only if [EQUATION] for all [MATH]-i.e., when [MATH] satisfies [EQUATION] distributionally on [MATH].", '1802.03053-1-23-2': 'From ([REF]), it is not hard to see that [MATH] is weakly [MATH]-harmonic precisely when [MATH] minimizes the [MATH]-energy [MATH] among all competitors of the form [MATH].', '1802.03053-1-24-0': 'In view of ([REF]), wherever [MATH] admits a local lifting [MATH] for some real-valued [MATH], we see that [MATH] is weakly [MATH]-harmonic if and only if [MATH] is a [MATH]-harmonic function-i.e., a weak solution of [EQUATION].', '1802.03053-1-24-1': 'For [MATH], the [MATH] regularity of [MATH]-harmonic functions was established by DiBenedetto [CITATION] and Lewis [CITATION].', '1802.03053-1-24-2': 'It is, moreover, possible to check that the Holder exponent [MATH] and other relevant constants in the central estimates of [CITATION] and [CITATION] can be taken independent of [MATH] for [MATH] bounded away from [MATH] and [MATH].', '1802.03053-1-24-3': 'Rather than using the full strength of the [MATH] regularity, we will employ in this paper the following simpler estimates, whose proof we sketch in the appendix:', '1802.03053-1-25-0': 'Combining this with the lifting criterion of Proposition [REF], one obtains the following partial regularity result for weakly [MATH]-harmonic maps to the circle:', '1802.03053-1-26-0': 'Let [MATH], and let [MATH] be a geodesic ball on a manifold [MATH] with [MATH].', '1802.03053-1-26-1': 'If [MATH] is a weakly [MATH]-harmonic map with vanishing distributional Jacobian [EQUATION] then [EQUATION] and [EQUATION]', '1802.03053-1-26-2': 'Though Corollary [REF] shows that weakly [MATH]-harmonic maps [MATH] are reasonably smooth (with effective estimates) away from the support of [MATH], observe that the weak [MATH]-harmonic condition alone gives no constraint on [MATH] itself.', '1802.03053-1-26-3': 'Indeed, given any [MATH], we can minimize [MATH] among all maps of the form [MATH] to find a weakly [MATH]-harmonic [MATH] with topological singularity [EQUATION] equal to that of [MATH].', '1802.03053-1-26-4': 'The problem of minimizing [MATH]-energy among [MATH]-valued maps with prescribed singularities in [MATH]-and, more generally, among [MATH]-valued maps with prescribed singularities in [MATH]-is studied in detail in [CITATION].', '1802.03053-1-27-0': '## [MATH]-Stationarity and Consequences', '1802.03053-1-28-0': 'A map [MATH] is said to be [MATH]-stationary, or simply stationary, if it is critical for the energy [MATH] with respect to perturbations of the form [MATH] for smooth families [MATH] of diffeomorphisms on [MATH].', '1802.03053-1-28-1': 'Equivalently, [MATH] is [MATH]-stationary if it satisfies the inner-variation equation [EQUATION] for every smooth, compactly supported vector field [MATH] on [MATH].', '1802.03053-1-29-0': 'The most-studied class of stationary [MATH]-harmonic maps (for arbitrary target manifolds) are the [MATH]-energy minimizers, whose regularity theory for [MATH] was first investigated in [CITATION] and [CITATION].', '1802.03053-1-29-1': 'On the other hand, as we discuss in Section [REF], one can also combine the results of [CITATION] with various min-max constructions to produce many examples non-minimizing stationary [MATH]-harmonic maps for certain non-integer values of [MATH].', '1802.03053-1-30-0': 'Given a stationary [MATH]-harmonic map [MATH], for each geodesic ball [MATH], we define the [MATH]-energy density [EQUATION]', '1802.03053-1-30-1': 'By standard arguments, it follows from the stationary equation ([REF]) that the density [MATH] is nearly monotonic in [MATH]: Namely, taking [MATH] in ([REF]) of the form [EQUATION] for some functions [MATH] approximating the characteristic function [MATH], and employing the Hessian comparison theorem to estimate the difference [MATH] in [MATH], one obtains the following well-known estimate (see, e.g., [CITATION], sections 4 and 7):', '1802.03053-1-31-0': 'Let [MATH] be a stationary [MATH]-harmonic map on a manifold [MATH] with [MATH].', '1802.03053-1-31-1': 'Then there is a constant [MATH] such that for any [MATH] and almost every [MATH], we have the inequality [EQUATION]', '1802.03053-1-31-2': 'In particular, [MATH] is monotone increasing in [MATH].', '1802.03053-1-32-0': 'In light of the monotonicity result, it makes sense to define the pointwise energy density [EQUATION]', '1802.03053-1-32-1': 'Perhaps the most significant consequence of Lemma [REF] is the boundedness of blow-up sequences: Given a sequence of radii [MATH], observe that the maps [MATH] defined by [EQUATION] are stationary [MATH]-harmonic with respect to the blown-up metrics [EQUATION] on [MATH], with [MATH]-energy given by [EQUATION] so it follows from Lemma [REF] that the [MATH]-energies [MATH] are uniformly bounded from above as [MATH].', '1802.03053-1-33-0': 'For a local minimizer [MATH] of the [MATH]-energy, one could then appeal to the compactness results of ([CITATION], Section 4) to conclude immediately that a subsequence [MATH] of such a blow-up sequence converges strongly to a minimizing tangent map [MATH].', '1802.03053-1-33-1': 'For [MATH]-the range of interest to us-it turns out that we can still obtain a strong convergence result without the minimizing assumption, but this relies on the following subtler result of [CITATION]:', '1802.03053-1-34-0': 'The significance of the condition [MATH] is that the [MATH]-energy has no conformally invariant dimension in this case, so that no bubbling can occur, and the proposition follows from arguments generalizing those of [CITATION] to the case [MATH] (see [CITATION]).', '1802.03053-1-34-1': 'The requirement that [MATH] be a homogeneous space is a technical one, arising from the fact that, at present, the most general [MATH]-regularity theorem available for stationary [MATH]-harmonic maps (when [MATH]) is that of [CITATION] for homogeneous targets.', '1802.03053-1-34-2': 'It may be of interest to note that [MATH]-regularity (and consequently Proposition [REF]) holds for arbitrary compact targets [MATH] for those stationary [MATH]-harmonic maps [MATH] constructed from critical points of generalized Ginzburg-Landau functionals, by virtue of Lemma 2.3 of [CITATION].', '1802.03053-1-35-0': 'Since [MATH] is in any case a compact homogeneous space, the result of Proposition [REF] applies to stationary [MATH]-harmonic maps to [MATH] for any [MATH], the range of interest.', '1802.03053-1-35-1': 'In particular, it follows that for any blow-up sequence [MATH], [MATH], we can extract a subsequence [MATH] such that the maps [MATH] converge strongly in [MATH] to a map [MATH] which is stationary [MATH]-harmonic with respect to the flat metric, and satisfies [EQUATION] for every [MATH].', '1802.03053-1-35-2': 'Following standard arguments (see, e.g., [CITATION]), we can then apply the Euclidean case of the monotonicity formula [REF] (in which [MATH]) to conclude that [MATH] must satisfy the [MATH]-homogeneity condition [EQUATION].', '1802.03053-1-35-3': 'In the next section, we will appeal to this strong convergence to tangent maps to obtain a sharp lower bound for the density [MATH] at singular points of [MATH], which will form the foundation for many of the estimates that follow.', '1802.03053-1-36-0': '# Sharp [MATH]-Regularity and Estimates for [MATH]', '1802.03053-1-37-0': 'A Sharp Lower Bound for Energy Density on [MATH] The analysis leading to Theorem [REF] rests largely on the following proposition-the comparatively simple counterpart in our setting to the "[MATH]-compactness"/"[MATH]-ellipticity" results of [CITATION],[CITATION]:', '1802.03053-1-38-0': 'Let [MATH]; by the small energy regularity theorem of [CITATION], this is equivalent to the positivity of the density [MATH].', '1802.03053-1-38-1': 'Taking a sequence of radii [MATH] and considering the blow-up sequence [MATH], we know from the discussion in Section [REF] that some subsequence [MATH] converges strongly in [MATH] to a nontrivial stationary [MATH]-harmonic map [EQUATION] satisfying [EQUATION]', '1802.03053-1-38-2': 'Since the tangent map [MATH] is radially homogeneous, it follows that its restriction [MATH] to the unit sphere defines a weakly [MATH]-harmonic map on [MATH].', '1802.03053-1-39-0': 'Next, we observe that if [MATH], the restriction [MATH] must again have a nontrivial singular set.', '1802.03053-1-39-1': 'Indeed, if [MATH] were [MATH], then since [MATH], we would have a lifting [MATH] for some [MATH]-harmonic function [MATH].', '1802.03053-1-39-2': 'The only [MATH]-harmonic functions on closed manifolds are the constants, so this would contradict the nontriviality of [MATH].', '1802.03053-1-39-3': 'Thus, [MATH] must have nonempty singular set on [MATH], and in particular, [MATH] must contain at least one ray in [MATH].', '1802.03053-1-40-0': 'We proceed now by a simple dimension reduction-type argument.', '1802.03053-1-40-1': 'Fixing some singular point [MATH] of [MATH] away from the origin, a standard application of ([REF]) and the monotonicity formula gives the density inequality [EQUATION]', '1802.03053-1-40-2': 'Thus, we can take a blow-up sequence for [MATH] at [MATH] to obtain a new tangent map [MATH] satisfying [EQUATION]', '1802.03053-1-40-3': 'This map [MATH] will again be radially homogeneous (by the monotonicity formula), and from the radial homogeneity [MATH] of [MATH], [MATH] inherits the additional translation symmetry [MATH] in the [MATH] direction.', '1802.03053-1-40-4': 'In particular, [MATH] is determined by its restriction to an [MATH]-sphere in the hyperplane perpendicular to [MATH], which defines a weakly [MATH]-harmonic map from [MATH] to [MATH].', '1802.03053-1-41-0': 'If [MATH], we can argue as before to see that [MATH] must have singularities on this [MATH]-sphere, and blow up again at some point [MATH].', '1802.03053-1-41-1': 'Carrying on in this way, we obtain finally a nontrivial stationary [MATH]-harmonic map [MATH] which is radially homogeneous, invariant under translation by some [MATH]-plane [MATH], and satisfies [EQUATION]', '1802.03053-1-41-2': "Now, it's easy to see that the only weakly [MATH]-harmonic maps from [MATH] to [MATH] are given by the identity [MATH] and its powers [MATH] for [MATH].", '1802.03053-1-41-3': 'In particular, letting [MATH] denote the projection of [MATH] onto [MATH], it follows that [EQUATION] for some [MATH].', '1802.03053-1-41-4': 'We can therefore compute [EQUATION]', '1802.03053-1-41-5': 'It then follows from ([REF]) that [EQUATION] as desired.', '1802.03053-1-42-0': 'Consequences of Proposition [REF] and Estimates for [MATH]', '1802.03053-1-43-0': 'Throughout this section, let [MATH] be an [MATH]-dimensional manifold satisfying the sectional curvature and injectivity radius bounds [EQUATION] and let [MATH].', '1802.03053-1-43-1': 'As a first consequence of Proposition [REF], we employ a simple Vitali covering argument (compare, e.g., Theorem 3.5 of [CITATION]) to obtain [MATH]-independent estimates for the [MATH]-content of the singular set [MATH] of a stationary [MATH]-harmonic map [MATH] to [MATH].', '1802.03053-1-44-0': 'Let [MATH] be a stationary [MATH]-harmonic map on a geodesic ball [MATH] of radius [MATH], satisfying the [MATH]-energy bound [EQUATION]', '1802.03053-1-44-1': 'For [MATH], the [MATH]-tubular neighborhood [MATH] about the singular set [MATH] then satisfies a volume bound of the form [EQUATION]', '1802.03053-1-44-2': 'Applying the Vitali covering lemma to the covering [EQUATION] of [MATH], we obtain a finite subcollection [MATH] for which [EQUATION] and [EQUATION]', '1802.03053-1-44-3': 'Now, by virtue of Proposition [REF] and Lemma [REF], we have for each [MATH] the lower energy bound [EQUATION] and from the disjointness ([REF]) of [MATH], it follows that [EQUATION].', '1802.03053-1-44-4': 'Since [MATH], this gives us an estimate of the form [EQUATION].', '1802.03053-1-44-5': 'By virtue of ([REF]), it then follows that [EQUATION] as claimed.', '1802.03053-1-45-0': 'For analysis purposes, this volume estimate is one of the most important consequences of Proposition [REF], and we will use it repeatedly throughout the remainder of the paper.', '1802.03053-1-45-1': 'The first application is a series of improved estimates for the distributional Jacobian [MATH] of [MATH].', '1802.03053-1-45-2': 'A priori, without further knowledge of the map [MATH], we know only that [EQUATION] since [EQUATION] for every [MATH] (with [MATH]).', '1802.03053-1-45-3': 'With Lemma [REF] in hand, however, we are able to show that, for stationary [MATH]-harmonic [MATH], if [MATH], then [MATH] is in fact uniformly bounded in various norms as [MATH].', '1802.03053-1-45-4': 'The first step in this direction is the following estimate:', '1802.03053-1-46-0': 'Fix [MATH] and set [MATH].', '1802.03053-1-46-1': 'Setting [MATH], we begin with the simple estimate [EQUATION] from which it follows that [EQUATION] for some [MATH].', '1802.03053-1-46-2': 'In particular, there is some scale [MATH] for which [EQUATION]', '1802.03053-1-46-3': 'Now, let [MATH], where [MATH] is given by [EQUATION] so that [MATH] on [MATH] and [MATH] outside [MATH].', '1802.03053-1-46-4': 'For any [MATH], we then have [EQUATION] where [MATH].', '1802.03053-1-46-5': 'Next, we note that [MATH], while [EQUATION] and, by ([REF]), [EQUATION] using all of this in the preceding estimate, we then obtain [EQUATION]', '1802.03053-1-46-6': 'Now, by Lemma [REF], we see that [EQUATION] and [EQUATION] which we use in ([REF]) to obtain [EQUATION]', '1802.03053-1-46-7': 'Recalling that [MATH] lies in the interval [MATH], it then follows that [EQUATION]', '1802.03053-1-46-8': 'Finally, we observe that [MATH] was arbitrary, so we can choose, for instance [EQUATION].', '1802.03053-1-46-9': 'If [MATH] then [EQUATION] and the desired estimate holds trivially.', '1802.03053-1-46-10': 'Otherwise, we have [MATH], so we can plug [MATH] into ([REF]), and using the fact that [MATH] is uniformly bounded for [MATH], we arrive an estimate of the desired form [EQUATION]', '1802.03053-1-46-11': 'By rescaling the result of Proposition [REF], we obtain the following statement at arbitrary small scales [MATH]:', '1802.03053-1-47-0': 'For [MATH], let [MATH] be a stationary [MATH]-harmonic map with [EQUATION].', '1802.03053-1-47-1': 'Then for every [MATH], we have [EQUATION]', '1802.03053-1-47-2': 'In particular, by virtue of energy monotonicity (Lemma [REF]), if [MATH] satisfies [MATH], then ([REF]) holds for all [MATH], for every ball [MATH].', '1802.03053-1-47-3': 'In the appendix, we establish the following general lemma, which will imply that these estimates, together with the volume bounds of Lemma [REF], yield [MATH]-independent bounds for [MATH] for any [MATH]:', '1802.03053-1-48-0': 'Let [MATH] be an [MATH]-current in [MATH] satisfying [EQUATION] for every ball [MATH].', '1802.03053-1-48-1': 'Suppose also that the [MATH]-tubular neighborhoods [MATH] about the support of [MATH] satisfy [EQUATION]', '1802.03053-1-48-2': 'Then there is a constant [MATH] such that for every [MATH], we have [EQUATION]', '1802.03053-1-48-3': 'In particular, combining the results of Lemma [REF] with Lemma [REF] and Corollary [REF], we find that', '1802.03053-1-49-0': 'Let [MATH] be a stationary [MATH]-harmonic map with [EQUATION].', '1802.03053-1-49-1': 'Then for every [MATH], we have [EQUATION]', '1802.03053-1-49-2': 'By the preceding discussion, we see that [EQUATION].', '1802.03053-1-49-3': 'To put this in the form ([REF]), we simply separate into two cases: if [MATH], then [MATH], and ([REF]) follows from the trivial estimate [EQUATION].', '1802.03053-1-49-4': 'On the other hand, if [MATH], then [MATH], and so [EQUATION] as claimed.', '1802.03053-1-50-0': 'Given the existence of estimates-like those of [CITATION]-which bound in certain weak norms the Jacobians of arbitrary maps in [MATH] by the normalized Ginzburg-Landau energies [MATH], it is natural to ask whether results along the lines of Proposition [REF] or Corollary [REF] can be obtained for arbitrary maps [MATH].', '1802.03053-1-50-1': 'If so, such estimates could be of use for the study of variational problems in [MATH], just as the results of [CITATION] have been in the Ginzburg-Landau setting (as in, e.g., [CITATION], [CITATION]).', '1802.03053-1-51-0': '# Estimates For the Hodge Decomposition of [MATH]', '1802.03053-1-52-0': 'Now, let [MATH] again be an arbitrary compact, oriented Riemannian manifold, and for [MATH], let [MATH] be a stationary [MATH]-harmonic map with [EQUATION]', '1802.03053-1-52-1': 'In our analysis of the global behavior of [MATH], just as in the Ginzburg-Landau setting, the Hodge decomposition [EQUATION] of [MATH] plays a central role.', '1802.03053-1-52-2': '(For more on Hodge decomposition in the space of [MATH] differential forms, we refer the reader to [CITATION].)', '1802.03053-1-52-3': 'Here, [MATH] is the function given by [EQUATION] [MATH] is the [MATH] two-form [EQUATION] and [MATH] is the remaining harmonic one-form, which we can write as [EQUATION] with respect to an [MATH]-orthonormal basis [MATH] for the space [MATH] of harmonic one-forms.', '1802.03053-1-52-4': 'We remark that, in our notation, [MATH] denotes the negative spectrum scalar Laplacian, but [MATH] is the usual positive spectrum Hodge Laplacian.', '1802.03053-1-53-0': 'Our first goal in this section is to establish estimates which show that, for each [MATH], and any sequence [MATH] as in Theorem [REF], the coexact component [MATH] remains bounded and the exact component [MATH] vanishes in [MATH] as [MATH].', '1802.03053-1-53-1': 'The second is to show that the same behavior holds in stronger norms away from the singular sets.', '1802.03053-1-54-0': 'For the harmonic form [MATH], we need only the trivial [MATH] estimate [EQUATION]', '1802.03053-1-54-1': 'For the exact and co-exact terms [MATH] and [MATH], we begin by establishing the following global estimates:', '1802.03053-1-55-0': 'First, note that we can apply Corollary [REF] (after some fixed rescaling) to a finite covering of [MATH] by geodesic balls, to obtain the [MATH] estimate [EQUATION] for the distributional Jacobian [MATH].', '1802.03053-1-55-1': 'Since [MATH] by definition, it follows from the [MATH] regularity of [MATH] that [EQUATION] as desired.', '1802.03053-1-56-0': 'To estimate [MATH], we begin by observing that since [MATH] is weakly [MATH]-harmonic, the distributional divergence [EQUATION] vanishes, and [MATH] can therefore be recast as [EQUATION].', '1802.03053-1-56-1': 'The [MATH] regularity of the Laplacian then gives [EQUATION] so it is enough produce an [MATH] estimate of the desired form for [MATH].', '1802.03053-1-57-0': 'To this end, we write [EQUATION]', '1802.03053-1-57-1': "It's easy to check that [EQUATION] so the [MATH] portion of the integral satisfies [EQUATION]", '1802.03053-1-57-2': 'To estimate the [MATH] portion of the integral, observe that [EQUATION] when [MATH], so fixing some [MATH], we split the integral again to see that [EQUATION] and since [MATH], this yields [EQUATION].', '1802.03053-1-57-3': 'Taking [MATH] we observe that [EQUATION] and [EQUATION] so putting this together with the preceding inequalities, we arrive at the estimate [EQUATION]', '1802.03053-1-57-4': 'Considering separately the cases [MATH] and [MATH] as in the proof of Corollary [REF], and recalling that [EQUATION] we arrive at an estimate of the desired form ([REF]).', '1802.03053-1-58-0': 'Next, we establish estimates resembling ([REF]) and ([REF]) in [MATH] norms away from the singular set [MATH].', '1802.03053-1-58-1': 'The simple estimates of Lemma [REF] below by no means represent the optimal bounds of this kind, but they will suffice for the purposes of this paper.', '1802.03053-1-59-0': 'Suppose now that [MATH], where [MATH] (so that [MATH] by Sobolev embedding).', '1802.03053-1-59-1': 'Letting [MATH], we have the [MATH] estimates [EQUATION] and [EQUATION]', '1802.03053-1-59-2': 'For [MATH], let [MATH] be given by [EQUATION] so that [MATH] on a neighborhood of [MATH], and [MATH].', '1802.03053-1-59-3': 'Then [MATH] is closed on the support of [MATH], and it follows that [EQUATION]', '1802.03053-1-59-4': 'Now, since [MATH], we have by Sobolev embedding and Proposition [REF] the estimate [EQUATION] applying this in the preceding inequality, it follows that [EQUATION].', '1802.03053-1-59-5': 'Taking [MATH] and appealing to the monotone convergence theorem, we arrive at ([REF]).', '1802.03053-1-60-0': 'For ([REF]), we proceed similarly: with [MATH] defined as above, we use the equation [EQUATION] (and the fact that [MATH]) to estimate [EQUATION]', '1802.03053-1-60-1': "With a few applications of Young's inequality, it then follows that [EQUATION]", '1802.03053-1-60-2': 'Now, by Sobolev embedding and Proposition [REF], we know that [EQUATION] so all that remains is to estimate [MATH].', '1802.03053-1-61-0': 'To this end, observe that the gradient estimate of Corollary [REF] implies [EQUATION] which together with the monotonicity of [MATH] yields the pointwise gradient estimate [EQUATION] in particular, it follows that [EQUATION]', '1802.03053-1-61-1': 'As in the proof of Proposition [REF], we note that [MATH] when [MATH], so that [EQUATION].', '1802.03053-1-61-2': 'Where [MATH], we can make repeated use of the pointwise estimate ([REF]), together with the fact that [EQUATION] to find [EQUATION]', '1802.03053-1-61-3': 'Splitting up the logarithm [EQUATION] we then see that [EQUATION]', '1802.03053-1-61-4': 'Finally, by the volume estimates of Lemma [REF], we know that [EQUATION] so that [EQUATION].', '1802.03053-1-61-5': 'In particular, it then follows that [EQUATION] which together with ([REF]) and ([REF]) gives [EQUATION]', '1802.03053-1-61-6': 'As before, we now take [MATH] and appeal to the Monotone Convergence Theorem to arrive at the desired estimate ([REF]).', '1802.03053-1-62-0': '# Limiting Behavior of the [MATH]-Energy Measure', '1802.03053-1-63-0': '## Generalized Varifolds', '1802.03053-1-64-0': 'Let [MATH] be a compact Riemannian manifold.', '1802.03053-1-64-1': 'For [MATH], denote by [MATH] the compact subbundle [EQUATION] of [MATH] consisting of symmetric endomorphisms with trace [MATH] and eigenvalues lying in [MATH].', '1802.03053-1-64-2': 'In [CITATION], Ambrosio and Soner define the space [MATH] of generalized [MATH]-varifolds to be the space of nonnegative Radon measures on [MATH].', '1802.03053-1-64-3': 'Note that [MATH] contains the standard [MATH]-varifolds-Radon measures on the Grassmannian bundle [MATH]-since identifying subspaces with the associated orthogonal projections gives a natural inclusion [MATH].', '1802.03053-1-65-0': 'As with standard varifolds (see [CITATION],[CITATION] for an introduction), for any [MATH], we define the weight measure [MATH] to be the pushforward [MATH] of [MATH] under the projection [MATH], and the first variation [MATH] to be the functional on [MATH] vector fields given by [EQUATION]', '1802.03053-1-65-1': 'A classical result of Allard (see [CITATION], Section 5) states that any (standard) [MATH]-varifold [MATH] whose first variation [MATH] is bounded in the [MATH] sense [EQUATION] restricts to an [MATH]-rectifiable varifold on the set [MATH] where its (upper-)[MATH]-dimensional density [EQUATION] is positive.', '1802.03053-1-65-2': 'In [CITATION], this result is extended to the setting of generalized varifolds as follows:', '1802.03053-1-66-0': 'In [CITATION], this result was originally used to study concentration of energy for solutions of the parabolic Ginzburg-Landau equations, as a means for constructing codimension-two Brakke flows.', '1802.03053-1-66-1': 'In the proof of Theorem [REF], we will use it similarly, to show that the concentrated part of [MATH] is given by the weight measure of a stationary, rectifiable [MATH]-varifold.', '1802.03053-1-67-0': '## Proof of Theorems [REF] and [REF]', '1802.03053-1-68-0': 'As in Theorem [REF], let [MATH] be a compact, oriented Riemannian manifold, let [MATH] with [MATH], and let [MATH] be a sequence of stationary [MATH]-harmonic maps satisfying [EQUATION]', '1802.03053-1-68-1': 'Passing to a subsequence, we can assume also that the [MATH]-energy measures [EQUATION] converge in [MATH] to a limiting measure [MATH], and that the singular sets [MATH] converge in the Hausdorff metric to a limiting set [EQUATION]', '1802.03053-1-68-2': 'Now, for each [MATH], consider as in Section [REF] the Hodge decomposition [EQUATION] of [MATH], and set [MATH].', '1802.03053-1-68-3': 'We associate to [MATH], [MATH] and [MATH], the following [MATH] sections of [MATH]: [EQUATION] and [EQUATION].', '1802.03053-1-68-4': 'As we shall see, the proof of Theorem [REF] rests largely on the following simple claim:', '1802.03053-1-69-0': '[EQUATION]', '1802.03053-1-69-1': 'Denoting by [MATH] the function [EQUATION] for [MATH], it is easy to check that [EQUATION] as an immediate consequence, we then have [EQUATION] for any [MATH].', '1802.03053-1-69-2': 'In particular, since [MATH], [MATH], and [MATH] (with [MATH]) by definition, it follows that [EQUATION] and [EQUATION]', '1802.03053-1-69-3': 'With this in mind, we estimate the [MATH] norm of [MATH] by splitting [MATH] into [MATH] and [MATH] for [MATH] small, writing [EQUATION]', '1802.03053-1-69-4': 'Now, since [EQUATION] and, by Lemma [REF], [EQUATION] we have the simple estimate [EQUATION]', '1802.03053-1-69-5': 'On the other hand, we know from Lemma [REF] that [EQUATION]', '1802.03053-1-69-6': 'Returning to our estimate for [MATH], it then follows that [EQUATION]', '1802.03053-1-69-7': 'Multiplying by [MATH] and taking [MATH], we arrive at the bound [EQUATION] since [MATH] was arbitrary, ([REF]) follows.', '1802.03053-1-70-0': "With this claim established, we next observe that measures [MATH] can be written as [EQUATION] and as a consequence of ([REF]), we see that [EQUATION] where in the last line we've set [EQUATION].", '1802.03053-1-70-1': 'Now, since [MATH] forms a bounded sequence in the space [MATH] of harmonic one-forms, by passing to a further subsequence, we can assume that it converges to some limit [EQUATION].', '1802.03053-1-70-2': "It's then clear that [EQUATION] pointwise, and we can therefore write [EQUATION]", '1802.03053-1-70-3': 'To complete the proof of Theorem [REF], it remains to realize the measure [EQUATION] as the weight measure of a stationary, rectifiable [MATH]-varifold satisfying the stated properties.', '1802.03053-1-70-4': 'To this end, we begin by remarking that, where [MATH], the tensor [EQUATION] belongs to [MATH], so we can define a sequence of generalized [MATH]-varifolds [MATH] by [EQUATION]', '1802.03053-1-70-5': 'The associated weight measures [MATH] are then given by [EQUATION] and since we have a uniform mass bound [EQUATION] we can pass to a further subsequence to obtain a weak limit [EQUATION] with weight measure [EQUATION]', '1802.03053-1-70-6': 'We claim next that [MATH].', '1802.03053-1-70-7': 'To see this, let [MATH] be a [MATH] vector field, so that [EQUATION]', '1802.03053-1-70-8': 'Appealing once more to ([REF]), we then see that [EQUATION]', '1802.03053-1-70-9': "Now, it's clear that [EQUATION] since [MATH] for harmonic [MATH].", '1802.03053-1-70-10': 'On the other hand, we know from the [MATH]-stationarity of [MATH] that [EQUATION] and consequently [EQUATION] as claimed.', '1802.03053-1-71-0': 'Since [MATH] for a generalized [MATH]-varifold [MATH] with [MATH], it will follow from Proposition [REF] that [MATH] is indeed the weight measure of a stationary, rectifiable [MATH]-varifold, once we show that [MATH] satisfies [EQUATION].', '1802.03053-1-71-1': 'In particular, to complete the proof of Theorem [REF], it now suffices to establish the following lemma:', '1802.03053-1-72-0': 'The support [MATH] of [MATH] is given by [EQUATION] and for [MATH], the density of [MATH] satisfies the lower bound [EQUATION]', '1802.03053-1-72-1': 'To establish ([REF]), first consider [MATH], and set [MATH].', '1802.03053-1-72-2': 'By definition of Hausdorff convergence, it follows that [EQUATION] and consequently [EQUATION] for [MATH] sufficiently large.', '1802.03053-1-72-3': 'Appealing once more to the estimates of Lemma [REF], we then see that [EQUATION] so that [MATH]; and since [MATH] was arbitrary, we therefore have [EQUATION]', '1802.03053-1-72-4': "Next, for [MATH], we'll show that [EQUATION]", '1802.03053-1-72-5': 'Indeed, if [MATH], then by definition there is a sequence [MATH] for which [EQUATION].', '1802.03053-1-72-6': 'By Lemma [REF], at each [MATH], we have [EQUATION] where [MATH] as [MATH].', '1802.03053-1-72-7': 'In particular, fixing [MATH] and appealing to the monotonicity of the [MATH]-energy (Lemma [REF]), we conclude that [EQUATION]', '1802.03053-1-72-8': 'Dividing through by [MATH] and letting [MATH], we arrive at the desired lower bound ([REF]).', '1802.03053-1-73-0': 'Finally, since the difference [EQUATION] clearly satisfies [EQUATION] we see that ([REF]) yields directly the desired density bound ([REF]) for [MATH] on [MATH].', '1802.03053-1-73-1': "Moreover, it follows immediately from ([REF]) that [MATH], and since we've already shown that [MATH], this completes the proof of ([REF]) as well.", '1802.03053-1-74-0': 'With the proof of Theorem [REF] completed, we turn our attention now to the proof of Theorem [REF], concerning compactness of the maps.', '1802.03053-1-74-1': 'Suppose that our sequence [MATH] either satisfies the additional bound [EQUATION] or that the first Betti number [MATH].', '1802.03053-1-74-2': 'In either case, it follows that the harmonic component [MATH] of [MATH] is uniformly bounded [EQUATION] as [MATH].', '1802.03053-1-74-3': 'Together with the [MATH] estimates of Proposition [REF], this implies immediately that [EQUATION] for any [MATH], so some subsequence of [MATH] must converge weakly in [MATH] to some limiting map [MATH].', '1802.03053-1-74-4': 'Moreover, since (by Proposition [REF]) the exact component [MATH] of [MATH] vanishes in [MATH] as [MATH], it follows that [EQUATION] weakly as [MATH], so the map [MATH] must satisfy [MATH] distributionally.', '1802.03053-1-75-0': 'Moreover, combining ([REF]) with Lemma [REF], it follows that, away from any [MATH]-neighborhood [MATH] of [MATH], we have [EQUATION] and putting this together with the local [MATH] estimate of Corollary [REF], we see that [EQUATION].', '1802.03053-1-75-1': "Of course, for [MATH], Rellich's theorem gives us compactness of the embedding [MATH]; hence, since [MATH] for [MATH] sufficiently large, there is indeed some subsequence of [MATH] which converges strongly in [MATH] to the limiting map [MATH] identified above.", '1802.03053-1-75-2': 'And since [MATH] and [MATH], it follows that [MATH] is indeed a strongly harmonic map in [MATH].', '1802.03053-1-76-0': '# Integrality of the Concentration Measure in Dimension 2', '1802.03053-1-77-0': 'By a simple blow-up argument, to establish the quantization result of Theorem [REF], it is enough to show the following: Let [MATH] be a sequence of metrics converging (in [MATH], say) to the Euclidean one on the disk [MATH] of radius [MATH], and let [MATH] be a sequence of stationary [MATH]-harmonic maps (with respect to [MATH]) for which [MATH] as [MATH].', '1802.03053-1-77-1': 'If the normalized energy measures [MATH] converge to a multiple [MATH] of the Dirac mass at [MATH], then this multiple [MATH] for some [MATH].', '1802.03053-1-78-0': 'Before diving in to the proof, we first note that one can easily follow the arguments of the preceding sections to establish local versions of Theorems [REF] and [REF].', '1802.03053-1-78-1': '(The estimates of Section [REF] are already stated in local form, and the estimates of Section [REF] are easily adapted to suitable local variants of the Hodge decomposition, like that which we employ later in this section in the proof of Theorem [REF].)', '1802.03053-1-78-2': 'In particular, for families of stationary [MATH]-harmonic maps [MATH] on the disk [MATH], we have the following:', '1802.03053-1-79-0': 'The first and most important step in the proof of Theorem [REF] is contained in the following result, which describes the limiting measure explicitly when the maps [MATH] converge to a limiting map [MATH]:', '1802.03053-1-80-0': 'For [MATH]-energy minimizers with respect to a fixed boundary condition, this result follows from the analysis of [CITATION], in which case all of the degrees [MATH] are either [MATH] or [MATH].', '1802.03053-1-80-1': 'It is also the immediate analog of the quantization result for 2-dimensional solutions of the Ginzburg-Landau equations in [CITATION], though the proof in our setting is much simpler.', '1802.03053-1-81-0': 'The reason for the relative simplicity in our setting is the form of the Pohozaev identity.', '1802.03053-1-81-1': 'In [CITATION], on their way to demonstrating the quantization of the energy measures [MATH], Comte and Mironescu appeal to the quantization results of [CITATION] and [CITATION] for the potential measures [MATH].', '1802.03053-1-81-2': 'These quantization results-though by no means trivial-can be derived in a relatively straightforward way from a Pohozaev identity that relates the integral of [MATH] on a disk to the behavior of [MATH] on its boundary.', '1802.03053-1-81-3': 'It is then observed in [CITATION] that the quantization of the potential measure gives strong constraints on the way that the degrees of the maps [MATH] can vary around clusters of zeroes (or "vortices") at different scales, which ultimately give rise to the quantization of the energy measures [MATH].', '1802.03053-1-82-0': 'In our setting, the path is much simpler, because the normalized [MATH]-energy [MATH] simultaneously plays the roles occupied by the energy and potential measures in the Ginzburg-Landau setting.', '1802.03053-1-82-1': 'In particular, we have the following nice Pohozaev-type identity:', '1802.03053-1-83-0': 'Let [MATH] be stationary [MATH]-harmonic on [MATH].', '1802.03053-1-83-1': 'On any annulus [EQUATION] we then have [EQUATION].', '1802.03053-1-84-0': 'The identity is simply a repackaging of the monotonicity formula in dimension two: By testing the inner variation equation [EQUATION] against vector fields of the form [MATH] for test functions [MATH] approximating the characteristic function [MATH], we find that [EQUATION] for almost every [MATH].', '1802.03053-1-84-1': 'Integrating over [MATH] then gives the desired equation.', '1802.03053-1-85-0': 'With this identity in hand, we can now argue in the spirit of [CITATION],[CITATION] to prove Proposition [REF]:', '1802.03053-1-86-0': '(Proof of Proposition [REF])', '1802.03053-1-87-0': 'Let [EQUATION] so that the limiting map [MATH] satisfies [EQUATION] where [MATH] denotes the degree of [MATH] about [MATH].', '1802.03053-1-87-1': 'Letting [MATH] be the map given by [EQUATION] we observe that [EQUATION] so the difference [MATH] is strongly harmonic.', '1802.03053-1-87-2': 'In particular, it follows that [EQUATION] for some harmonic function [MATH].', '1802.03053-1-88-0': 'Now, set [EQUATION] so that the density [MATH] of [MATH] at [MATH] is given by [EQUATION] for every [MATH].', '1802.03053-1-88-1': 'For any [MATH], it then follows from Lemma [REF] that [EQUATION]', '1802.03053-1-88-2': 'On the other hand, we also know that [MATH] strongly in [MATH] and [MATH] is uniformly bounded as [MATH], so it follows that [EQUATION]', '1802.03053-1-88-3': 'Since [MATH], we can expand [MATH] as [EQUATION] where for [MATH] we denote by [MATH] projection onto the line perpendicular to [MATH].', '1802.03053-1-88-4': 'In particular, if [MATH] for [MATH], then [MATH] for every [MATH], and it follows that [EQUATION].', '1802.03053-1-88-5': 'Combining this with the obvious estimate [EQUATION] (where [MATH] of course depends on [MATH]), we see that, on [MATH], [EQUATION] and [EQUATION]', '1802.03053-1-88-6': 'In particular, on the annulus [MATH], since [EQUATION] we can apply the preceding estimates to our computation of [MATH] to conclude that [EQUATION]', '1802.03053-1-88-7': 'Since [MATH] was arbitrary, it follows finally that [EQUATION] which is precisely what we wanted to show.', '1802.03053-1-89-0': 'Combining Proposition [REF] with a simple contradiction argument, and scaling, we can formulate the following lemma:', '1802.03053-1-90-0': 'For any [MATH] and [MATH], there exists [MATH] such that if [MATH], and [MATH] is stationary [MATH]-harmonic with [EQUATION] and [MATH], then [EQUATION]', '1802.03053-1-90-1': 'We now remove the requirement of a uniform [MATH] bound by arguing that, in the general setting of Theorem [REF], the normalized energy measures [MATH] are negligible on the complement of a collection of disks satisfying the conditions of Corollary [REF].', '1802.03053-1-91-0': '(Proof of Theorem [REF])', '1802.03053-1-92-0': 'Let [MATH] be a sequence of stationary [MATH]-harmonic maps as given, with [EQUATION] and [EQUATION].', '1802.03053-1-92-1': "We consider now a local version of the Hodge decomposition of Section [REF]; choosing some cutoff function [MATH] such that [MATH] on [MATH], we define [EQUATION] and [EQUATION] where [MATH] is the two-dimensional Green's function.", '1802.03053-1-93-0': 'Writing [EQUATION] we see that (distributionally) [EQUATION] and [EQUATION].', '1802.03053-1-93-1': 'In particular, [MATH] is harmonic on the disk [MATH] (where [MATH]), and it follows that [EQUATION]', '1802.03053-1-93-2': 'Next, consider the [MATH]-neighborhoods [EQUATION] about the singular sets [MATH].', '1802.03053-1-93-3': 'Our goal now is to show that [EQUATION] and that [MATH] is contained in a finite union of disks satisfying the hypotheses of Corollary [REF].', '1802.03053-1-93-4': 'To this end, we first observe that, by Corollary [REF] and Lemma [REF] of the appendix, [MATH] satisfies the pointwise bound [EQUATION] for [MATH].', '1802.03053-1-93-5': 'Putting this together with the volume estimate of Lemma [REF], we find that [EQUATION] and therefore [EQUATION]', '1802.03053-1-93-6': 'For [MATH], the arguments in the proofs of Proposition [REF] and Lemma [REF] again yield the local estimates [EQUATION] for [MATH] and [EQUATION] respectively.', '1802.03053-1-93-7': 'In particular, rearranging ([REF]) and recalling that [MATH] is uniformly bounded as [MATH], we see that [EQUATION]', '1802.03053-1-93-8': 'Now, to estimate [MATH], we observe that [EQUATION]', '1802.03053-1-93-9': 'Since [MATH] vanishes on compact subsets of [MATH] by assumption, it then follows that [EQUATION] and by ([REF]), we therefore have [EQUATION] where [MATH].', '1802.03053-1-94-0': 'Putting together the estimates ([REF]), ([REF]), and ([REF]), we see finally that [EQUATION] confirming ([REF]).', '1802.03053-1-95-0': 'Next, as in the proof of Lemma [REF], we know from a simple Vitali covering argument that [EQUATION] for some [MATH] such that [EQUATION] and it follows from Proposition [REF] that [EQUATION].', '1802.03053-1-95-1': 'In particular, [MATH] is uniformly bounded independent of [MATH], so passing to a subseqence, we can take [MATH] to be constant.', '1802.03053-1-95-2': 'Moreover, setting [EQUATION] we can pass to a further subsequence for which the (possibly infinite) limits [EQUATION] exist.', '1802.03053-1-95-3': 'Relabeling indices if necessary, there is then some [MATH] such that [EQUATION] and for every [MATH], there is some [MATH] for which [MATH].', '1802.03053-1-96-0': 'Now, let [EQUATION] and for [MATH], define the disks [EQUATION].', '1802.03053-1-96-1': 'For [MATH] sufficiently large, we then see that [EQUATION] and [EQUATION].', '1802.03053-1-96-2': 'In particular, since [EQUATION] by the monotonicity formula, our disks will satisfy the conditions of Corollary [REF] for some [MATH], once we show that [EQUATION]', '1802.03053-1-96-3': 'To establish ([REF]), we consider separately the components [MATH], [MATH], and [MATH] of the local Hodge decomposition.', '1802.03053-1-96-4': 'For [MATH], we have seen already that [EQUATION] where [MATH], so that [EQUATION] for [MATH] sufficiently large.', '1802.03053-1-96-5': 'For [MATH], recall that, for [MATH], [EQUATION] so that [EQUATION]', '1802.03053-1-96-6': 'For [MATH] sufficiently large, we can take [MATH] in the estimate above, to obtain [EQUATION]', '1802.03053-1-96-7': 'Next, employing the pointwise gradient estimate ([REF]) for [MATH] with Lemma [REF], we see that [EQUATION]', '1802.03053-1-96-8': 'And since [EQUATION] and [EQUATION] as [MATH], it follows that [EQUATION]', '1802.03053-1-96-9': 'Combining this with the preceding estimates for [MATH] and [MATH], we see that ([REF]) indeed holds.', '1802.03053-1-97-0': 'Finally, letting [EQUATION] and choosing an arbitrary [MATH], it follows from Corollary [REF] that for [MATH] sufficiently large, [EQUATION] in particular, we deduce that [EQUATION]', '1802.03053-1-97-1': 'Now, by the disjointness of the disks [MATH], we know that [EQUATION] and since [MATH], it follows that [EQUATION].', '1802.03053-1-97-2': 'On the other hand, since the disks [MATH] cover [MATH], we know from ([REF]) that [EQUATION]', '1802.03053-1-97-3': 'By ([REF]), it then follows that [EQUATION] as desired.', '1802.03053-1-98-0': 'In higher dimensions, one would like to show, analogously, that for a sequence [MATH] of stationary [MATH]-harmonic maps with energy concentrating along an [MATH]-plane [MATH], the limiting measure [MATH] must have the form [EQUATION] for some [MATH].', '1802.03053-1-98-1': 'As in the proof of Theorem [REF], it is possible to reduce the problem to the case where the maps [MATH] converge away from [MATH], but for the moment we have no higher-dimensional analog of Proposition [REF].', '1802.03053-1-99-0': 'In particular, if one naively attempts to generalize the Pohozaev identity to this setting (for instance, by testing the component of the position vector field perpendicular to [MATH] in the inner variation equation), the [MATH] components [MATH] of the derivatives of [MATH] invariably get in the way.', '1802.03053-1-99-1': 'And while it is easy to see that [MATH] under these assumptions, one would need the much stronger vanishing [MATH] for the arguments of this section to work.', '1802.03053-1-99-2': 'We are nonetheless optimistic about the prospect of extending the integrality result to higher dimensions, but a proof will undoubtedly require some interesting new ideas.', '1802.03053-1-100-0': '# Natural Min-Max Constructions', '1802.03053-1-101-0': '## Generalized Ginzburg-Landau Functionals', '1802.03053-1-102-0': 'Let [MATH] and [MATH] be compact Riemannian manifolds, with [MATH] isometrically embedded in some Euclidean space [MATH].', '1802.03053-1-102-1': 'For [MATH] and [MATH], Wang studies in [CITATION] the generalized Ginzburg-Landau functionals [EQUATION] given by [EQUATION] where the function [MATH] has the form [MATH] for a function [MATH] satisfying [MATH], [MATH].', '1802.03053-1-102-2': '[EQUATION] (Here, [MATH] is chosen such that nearest-point projection to [MATH] is well-defined and smooth on the [MATH]-neighborhood of [MATH] [CITATION].)', '1802.03053-1-102-3': 'Thus, as [MATH], the potential term in the energies [MATH] penalizes deviation from the target manifold [MATH], while for [MATH]-valued maps [MATH], one simply recovers the [MATH]-energy [MATH].', '1802.03053-1-103-0': 'For [MATH], the [MATH] asymptotics of bounded-energy critical points and negative gradient flows of these functionals had previously been studied in [CITATION], [CITATION], [CITATION] as regularizations of harmonic maps and harmonic map heat flows.', '1802.03053-1-103-1': 'While in the [MATH] setting one encounters the familiar bubbling phenomena that arise in the study of harmonic maps, for [MATH], Wang demonstrates that (much like Proposition [REF] for [MATH]-harmonic maps), bounded-energy sequences of critical points enjoy a strong compactness property: [EQUATION]', '1802.03053-1-103-2': 'As a consequence, for [MATH], the functionals [MATH] are naturally suited to the construction of stationary [MATH]-harmonic maps via min-max methods, in light of the following elementary fact:', '1802.03053-1-104-0': 'The generalized Ginzburg-Landau energy [MATH] is a [MATH] functional on [MATH], with derivative [EQUATION] and satisfies the following Palais-Smale condition: if [MATH] is a sequence satisfying [EQUATION] and [EQUATION] then [MATH] has a subsequence that converges strongly in [MATH].', '1802.03053-1-105-0': 'The first statement is trivial.', '1802.03053-1-105-1': 'The proof of the Palais-Smale condition is also quite standard, but we include it for completeness:', '1802.03053-1-106-0': "For a sequence [MATH] satisfying ([REF]), we know from Rellich's theorem that a subsequence (which we continue to denote by [MATH]) converges weakly in [MATH] and strongly in [MATH] to a limiting function [MATH].", '1802.03053-1-106-1': 'To confirm that the convergence is also strong in [MATH], it is enough to show that [EQUATION]', '1802.03053-1-106-2': 'And indeed, if the [MATH] also satisfies ([REF]), then we see that [EQUATION] from which ([REF]) follows, completing the proof.', '1802.03053-1-107-0': 'In the remainder of this section, we employ a simple min-max constructions for the energies [MATH] for [MATH] and [MATH], with arguments very similar to those of [CITATION], to prove Theorem [REF] of the introduction.', '1802.03053-1-107-1': 'Though we focus here on [MATH]-harmonic maps to [MATH] with [MATH], we remark that the following construction (though not the specific energy bounds) can be generalized to produce nontrivial stationary [MATH]-harmonic maps from manifolds of dimension [MATH] into arbitrary targets [MATH] with [MATH], for [MATH].', '1802.03053-1-107-2': 'A detailed study of this general construction is beyond the scope of this paper, but may be an interesting direction for further investigation.', '1802.03053-1-108-0': '## The Saddle Point Construction and an Upper Bound for the Energies', '1802.03053-1-109-0': 'We restrict ourselves now to the case where our target manifold [MATH] is [MATH], embedded in [MATH] as the boundary of the unit disk [MATH], and consider the collection [MATH] of two-parameter families [MATH] given by [EQUATION]', '1802.03053-1-109-1': 'For [MATH] and [MATH], we define the min-max energy levels [MATH] by [EQUATION] and the limiting energy levels [EQUATION]', '1802.03053-1-109-2': "Now, we've observed that [MATH] is a [MATH] functional on [MATH], which evidently vanishes on the circle of constant maps to [MATH].", '1802.03053-1-109-3': 'Thus, if we can show that [MATH] then we can apply the Saddle Point Theorem of Rabinowitz (see, e.g., Chapter 3 of [CITATION], Chapter 4 of [CITATION]) to conclude that for any minimizing sequence of families [EQUATION] there exists a sequence [MATH] for which [EQUATION] and [EQUATION]', '1802.03053-1-109-4': 'Moreover, since we can deform any minimizing sequence of families [MATH] to one whose maps take values in the unit disk by applying a retraction, we can obtain in this way a sequence [MATH] satisfying ([REF]) and a uniform bound [EQUATION].', '1802.03053-1-109-5': 'In particular, it will then follow from Lemma [REF] that there is indeed a critical point [MATH] of [MATH] with energy [EQUATION]', '1802.03053-1-109-6': 'To check that [MATH], we follow the same standard arguments as in [CITATION].', '1802.03053-1-109-7': 'Namely, for any [MATH], we note that the averaging map [EQUATION] defines a continuous map from [MATH] which restricts to the identity on [MATH], so that by elementary degree theory, there must be some [MATH] for which [MATH].', '1802.03053-1-109-8': "Now, the [MATH] Poincare inequality gives us a constant [MATH] such that [EQUATION] whenever [MATH], so by the preceding observation, for any [MATH], there is some [MATH] for which [MATH] satisfies [EQUATION] where we've set [MATH].", '1802.03053-1-109-9': 'In particular, since [EQUATION] it follows that, for [MATH], [EQUATION] for any family [MATH].', '1802.03053-1-109-10': 'Taking the infimum over [MATH], we confirm that [EQUATION]', '1802.03053-1-109-11': "In summary, we've so far established that For [MATH] and [MATH], there exists a critical point [MATH] of [MATH] satisfying [EQUATION]", '1802.03053-1-109-12': 'Next, we will establish an upper bound for the limiting energy levels [MATH]: namely, we show that', '1802.03053-1-110-0': 'There exists [MATH] independent of [MATH] such that [EQUATION]', '1802.03053-1-110-1': 'The finiteness of [MATH] will then allow us to apply Theorem [REF] to deduce the existence of a corresponding stationary [MATH]-harmonic map, while the boundedness of [MATH] will provide the upper bound in Theorem [REF].', '1802.03053-1-111-0': 'Again, the proof is very close to that of the analogous statement in Section 4 of [CITATION], albeit somewhat simpler, since in this case we can produce a single family [MATH] lying in [MATH] for every [MATH] and satisfying a bound [EQUATION] of the desired form.', '1802.03053-1-112-0': 'For [MATH], define [MATH] by [EQUATION] and set [MATH] for [MATH].', '1802.03053-1-112-1': 'Fix also a triangulation of [MATH]-that is, choose a bi-Lipschitz map [MATH] from [MATH] to the underlying space of a simplicial complex [MATH] in some Euclidean space [MATH].', '1802.03053-1-112-2': 'Applying a generic rotation, we can arrange that the projection map [MATH] from [MATH] to the plane [MATH] has full rank on the [MATH]-dimensional subspace parallel to each [MATH]-dimensional simplex [MATH].', '1802.03053-1-112-3': 'Denoting by [MATH] the composition [EQUATION] of the projection [MATH] with [MATH], we define the family [EQUATION]', '1802.03053-1-112-4': 'Our task now is to show that [MATH] defines a continuous map [MATH], satisfying ([REF]).', '1802.03053-1-112-5': 'First, since [MATH] is bi-Lipschitz and [MATH] is finite, we observe that it is enough to establish this for the family [EQUATION] on each [MATH]-dimensional face [MATH].', '1802.03053-1-112-6': "And since we've also chosen [MATH] such that the restriction [MATH] of the projection map to [MATH] has full rank, we can write [EQUATION] where [MATH] is an invertible affine-linear map, and [MATH] is simply the projection [EQUATION] onto the first two coordinates.", '1802.03053-1-112-7': 'In particular, it is enough to show that on a bounded domain [MATH], the family [EQUATION] is continuous in [MATH] for each [MATH], and satisfies [EQUATION]', '1802.03053-1-112-8': 'This is straightforward.', '1802.03053-1-112-9': 'By direct computation, the energy [MATH] of [MATH] on [MATH] satisfies [EQUATION]', '1802.03053-1-112-10': 'Thus, ([REF]) holds, and we see moreover that the energy [MATH] varies continuously in [MATH].', '1802.03053-1-112-11': 'Since the family [MATH] is obviously weakly continuous in [MATH], it follows that [MATH] is strongly continuous as well.', '1802.03053-1-113-0': 'We conclude finally that the families [MATH] defined by ([REF]) indeed belong to [MATH], and satisfy [EQUATION] where the constant [MATH] is determined by our choice of triangulation [MATH].', '1802.03053-1-113-1': 'In particular, it follows that [EQUATION] for every [MATH], and taking the supremum over [MATH], we therefore have [EQUATION] as desired.', '1802.03053-1-114-0': 'Since [MATH], we can apply Theorem [REF] to the min-max critical points [MATH] of Lemma [REF], to conclude that', '1802.03053-1-115-0': '## Lower Bounds for [MATH]', '1802.03053-1-116-0': 'To complete the proof of Theorem [REF], it remains to show that the min-max energies [MATH] satisfy a lower bound of the form [EQUATION]', '1802.03053-1-116-1': 'To achieve this, we argue as in Section 3 of [CITATION], with Proposition [REF] taking on the role played by the [MATH]-ellipticity theorem in the Ginzburg-Landau setting.', '1802.03053-1-117-0': 'We begin by observing that ([REF]) holds for the round sphere.', '1802.03053-1-117-1': 'As discussed in the proof of Proposition [REF], since [MATH], every nontrivial weakly [MATH]-harmonic map [MATH] must have singularities.', '1802.03053-1-117-2': 'In particular, the stationary [MATH]-harmonic maps [MATH] of energy [MATH] constructed above must have nontrivial singular set, and from Proposition [REF] and monotonicity, it indeed follows that [EQUATION]', '1802.03053-1-117-3': 'The estimate for arbitrary [MATH] is then an easy consequence of the following claim: There is a constant [MATH] such that [EQUATION] for every [MATH] and [MATH].', '1802.03053-1-118-0': 'We will construct a bounded linear map [MATH] that fixes the constant maps and satisfies [EQUATION] for all [MATH] and [MATH].', '1802.03053-1-118-1': 'For any family [MATH], we then see that [MATH] defines a family in [MATH], so that [EQUATION] and taking the infimum over [MATH] gives ([REF]).', '1802.03053-1-119-0': 'We construct this map [MATH] as follows.', '1802.03053-1-119-1': 'First, denote by [MATH] the northern hemisphere [MATH], and consider the reflection map [EQUATION] given by [EQUATION] [MATH] is clearly a bounded linear map which fixes the constants, and has the effect of doubling [MATH]-i.e., [EQUATION] for every [MATH].', '1802.03053-1-120-0': 'Next, since [MATH] is a topological ball, we can choose some smooth [MATH] which is a diffeomorphism onto its image.', '1802.03053-1-120-1': 'Fixing such an [MATH], we see that the pullback map [EQUATION] given by [EQUATION] is another bounded linear map that fixes the constant maps, and satisifies [EQUATION].', '1802.03053-1-120-2': 'In particular, taking [MATH] gives a map [MATH] satisfying the desired properties, confirming the claim.', '1802.03053-1-121-0': 'Finally, taking the supremum over [MATH] in ([REF]), we see that [EQUATION].', '1802.03053-1-121-1': 'Combining this with ([REF]), it follows finally that [EQUATION] as desired.', '1802.03053-1-121-2': 'In particular, putting this together with the conclusion of Proposition [REF], we arrive finally at the result of Theorem [REF].', '1802.03053-1-122-0': '# Appendix', '1802.03053-1-123-0': '## Proof of Proposition [REF]', '1802.03053-1-124-0': 'In this short section, we demonstrate the independence from the parameter [MATH] of some standard estimates for [MATH]-harmonic functions (namely, the [MATH] and [MATH] estimates discussed in Proposition [REF]).', '1802.03053-1-124-1': 'This is simply a matter of keeping track of [MATH] in the estimates of [CITATION] and [CITATION], but we give some details in the interest of completeness.', '1802.03053-1-125-0': 'Let [MATH] be a geodesic ball in a manifold [MATH] satisfying the sectional curvature bound [EQUATION] and let [MATH] be a [MATH]-harmonic function on [MATH] for [MATH].', '1802.03053-1-125-1': 'Recall that, by the convexity of the [MATH]-energy functional, [MATH] must be the unique minimizer for the [MATH]-energy with respect to its Dirichlet data.', '1802.03053-1-126-0': 'For [MATH], we consider as in [CITATION] the perturbed [MATH]-energy functionals [EQUATION] and let [MATH] minimize [MATH] with respect to the condition [MATH].', '1802.03053-1-126-1': 'Setting [EQUATION] we then have that [EQUATION] and by standard results on quasilinear equations of this form (see, e.g., Chapter 4 of [CITATION]), it follows that [MATH] is a smooth, classical solution of ([REF]).', '1802.03053-1-126-2': 'Moreover, since [MATH] is the unique [MATH]-energy minimizer with respect to its Dirichlet data, we know that [MATH] strongly in [MATH] as [MATH].', '1802.03053-1-126-3': 'The task now (as in [CITATION], [CITATION]) is to establish estimates of the form given in ([REF]) for the perturbed solutions [MATH], and pass them to the limit [MATH].', '1802.03053-1-127-0': 'As in [CITATION], we observe now that, for [MATH] solving ([REF]), the energy density [MATH] satisfies the divergence-form equation [EQUATION] where [MATH] denotes the composition [EQUATION] and [EQUATION]', '1802.03053-1-127-1': 'In particular, it follows that [EQUATION]', '1802.03053-1-127-2': "Now, since [MATH] when we integrate ([REF]) against a test function [MATH] with [MATH] on [MATH] and [MATH], we find that [EQUATION] and an application of Young's inequality yields [EQUATION]", '1802.03053-1-127-3': "In particular, since Holder's inequality gives [EQUATION] it follows that [EQUATION] and since [MATH], we can rewrite this as [EQUATION]", '1802.03053-1-127-4': 'To obtain [MATH] estimates for [MATH], we can apply Moser iteration to ([REF]).', '1802.03053-1-127-5': 'Since the eigenvalues of [EQUATION] are bounded between [MATH] and [MATH], and we are working with [MATH], it is easy to see that the resulting estimate has the desired form [EQUATION]', '1802.03053-1-127-6': 'Finally, since [MATH] strongly in [MATH], we have that [EQUATION] and it follows from ([REF]) and ([REF]) that [EQUATION] and [EQUATION]', '1802.03053-1-127-7': 'Proposition [REF] then follows by scaling.', '1802.03053-1-128-0': '## Proof of Lemma [REF]', '1802.03053-1-129-0': 'In this section, we prove Lemma [REF], which we employed in the proof of Corollary [REF].', '1802.03053-1-129-1': 'For convenience, we restate the lemma here:', '1802.03053-1-130-0': 'Let [MATH] be a geodesic ball in a manifold [MATH] of sectional curvature [MATH] and injectivity radius [MATH].', '1802.03053-1-130-1': 'Let [MATH] be an [MATH]-current in [MATH] satisfying, for some constant [MATH], [EQUATION] for every ball [MATH].', '1802.03053-1-130-2': 'Suppose also that the [MATH]-tubular neighborhoods [MATH] about the support of [MATH] satisfy [EQUATION]', '1802.03053-1-130-3': 'Then there is a constant [MATH] such that for every [MATH], we have [EQUATION]', '1802.03053-1-130-4': "Let's begin now by making some simple reductions.", '1802.03053-1-130-5': 'First, since the given metric [MATH] on [MATH] is uniformly equivalent to the flat one [MATH] with [EQUATION] for some constant [MATH], it will suffice to establish the lemma in the flat case.', '1802.03053-1-130-6': 'Next, we note that every [MATH]-current [MATH] in [MATH] is described by a finite collection of scalar distributions [MATH], where [EQUATION].', '1802.03053-1-130-7': 'Thus, it is enough to show that Lemma [REF] holds with a scalar distribution [MATH] in place of the [MATH]-current [MATH].', '1802.03053-1-130-8': 'Our first step in proving this is then the following observation:', '1802.03053-1-131-0': 'For [MATH], let [MATH] be a distribution on [MATH] satisfying the estimate [EQUATION] for every ball [MATH].', '1802.03053-1-131-1': "Fixing a cutoff function [MATH] such that [MATH] on [MATH], set [EQUATION] where [MATH] is the [MATH]-dimensional Euclidean Green's function.", '1802.03053-1-131-2': 'We then have for [MATH] a pointwise gradient estimate of the form [EQUATION]', '1802.03053-1-131-3': 'For [MATH], we observe that the pointwise derivatives [MATH] are well-defined, and given by [EQUATION] where [MATH] is a dimensional constant.', '1802.03053-1-132-0': 'To establish ([REF]), first choose a function [MATH] satisfying [EQUATION] and for [MATH], set [EQUATION].', '1802.03053-1-132-1': "Defining [EQUATION] it's easy to see that the functions [MATH] satisfy [EQUATION] and [EQUATION]", '1802.03053-1-132-2': 'Given [MATH], let [MATH], so that [EQUATION].', '1802.03053-1-132-3': 'Writing [EQUATION] and observing that [EQUATION] when [MATH] it follows that [EQUATION]', '1802.03053-1-132-4': 'Setting [EQUATION] we can then use ([REF])-([REF]) to see that [EQUATION] and [EQUATION].', '1802.03053-1-132-5': 'By ([REF]), it therefore follows that [EQUATION]', '1802.03053-1-132-6': 'Summing from [MATH] to [MATH], we obtain finally [EQUATION] giving the desired estimate ([REF]).', '1802.03053-1-133-0': 'Let [MATH] be as in Lemma [REF], satisfying [EQUATION] for every ball [MATH].', '1802.03053-1-133-1': 'In addition, suppose that the tubular neighborhoods [MATH] about the support of [MATH] satisfy the volume bound [EQUATION]', '1802.03053-1-133-2': 'Then there is a constant [MATH] depending only on [MATH] and [MATH] such that for every [MATH], we have the estimate [EQUATION]', '1802.03053-1-133-3': 'By Lemma [REF], there exists a function [MATH] satisfying [EQUATION] and [EQUATION] for [MATH].', '1802.03053-1-133-4': 'For any [MATH] and [MATH], we then have [EQUATION] while, by ([REF]) and ([REF]), we see that [EQUATION]', '1802.03053-1-133-5': 'Thus, we indeed have [EQUATION] the desired [MATH] estimate.', '1802.03053-1-134-0': 'As remarked previously, Lemma [REF] now follows by applying Corollary [REF] to the scalar component distributions of the [MATH]-current [MATH].'}

{'1802.03053-2-0-0': 'We study the asymptotics as [MATH] of stationary [MATH]-harmonic maps [MATH] from a compact manifold [MATH] to [MATH], satisfying the natural energy growth condition [EQUATION].', '1802.03053-2-0-1': 'Along a subsequence [MATH], we show that the singular sets [MATH] converge to the support of a stationary, rectifiable [MATH]-varifold [MATH] of density [MATH], given by the concentrated part of the measure [EQUATION].', '1802.03053-2-0-2': 'When [MATH], we show moreover that the density of [MATH] takes values in [MATH].', '1802.03053-2-0-3': 'Finally, on every compact manifold of dimension [MATH] we produce examples of nontrivial families [MATH] of such maps via natural min-max constructions.', '1802.03053-2-1-0': '# Introduction', '1802.03053-2-2-0': 'In their 1995 paper [CITATION], Hardt and Lin consider the following question: given a simply connected domain [MATH] and a map [MATH] of nonzero degree, what can be said about the limiting behavior of maps [EQUATION] minimizing the [MATH]-energy [EQUATION] as [MATH] approaches [MATH] from below?', '1802.03053-2-2-1': 'They succeed in showing-among other things-that away from a collection [MATH] of [MATH] singularities, a subsequence [MATH] converges strongly to a harmonic map [MATH], and the measures [EQUATION] converge to the sum [MATH] of Dirac masses on [MATH] [CITATION].', '1802.03053-2-2-2': 'Moreover, the singular set [MATH] minimizes a certain "renormalized energy" function [MATH] associated to [MATH], providing a strong constraint on the location of the singularities.', '1802.03053-2-2-3': 'In particular, though the homotopically nontrivial boundary map [MATH] admits no extension to an [MATH]-valued map of finite Dirichlet energy-i.e, [MATH]-the limit of the [MATH]-energy minimizers as [MATH] provides us with a natural candidate for the optimal harmonic extension of [MATH] to an [MATH]-valued map on [MATH].', '1802.03053-2-3-0': 'The results of [CITATION] were inspired in large part by the similar results of Bethuel, Brezis, and Helein-contained in the influential monograph [CITATION]-concerning the asymptotics for minimizers [MATH] of the Ginzburg-Landau functionals [EQUATION] as [MATH], with the measures [EQUATION] taking on the role played by the measures [MATH] in the setting of [CITATION].', '1802.03053-2-3-1': 'In recent decades, the asymptotics for critical points of the Ginzburg-Landau functionals [MATH] in higher dimensions have also been studied by a number of authors, often with an emphasis on the relationship betweeen concentration phenomena for the measures [MATH] and minimal submanifolds of codimension two (see, for instance, [CITATION],[CITATION],[CITATION],[CITATION],[CITATION],[CITATION], among many others).', '1802.03053-2-3-2': 'A typical result says roughly that if the measures [MATH] have uniformly bounded mass, then a subsequence [MATH]-converges as [MATH] to a limiting measure that decomposes into two pieces: a concentrated component given by a stationary, rectifiable varifold [MATH] of codimension two, and a diffuse measure of the form [MATH] for some harmonic one-form [MATH] (which vanishes under mild compactness assumptions) (see, e.g., [CITATION],[CITATION],[CITATION],[CITATION]).', '1802.03053-2-4-0': 'Results of this type point to the possibility of employing variational methods for the Ginzburg-Landau functionals to produce minimal submanifolds of codimension two, but for the complete success of such efforts, we need an improved understanding of the concentration of [MATH].', '1802.03053-2-4-1': 'In particular, the question of integrality (up to a factor of [MATH]) of the limiting varifold [MATH] has been resolved only in dimension two [CITATION] and for local minimizers in higher dimensions [CITATION].', '1802.03053-2-4-2': "If, on the other hand, one could establish integrality of [MATH] for general families of critical points (or families with bounded index), then the min-max methods of [CITATION] and [CITATION] would provide a new proof of the existence of nontrivial stationary, integral varifolds of codimension two, in the spirit of Guaraco's work for the Allen-Cahn equation in codimension one [CITATION].", '1802.03053-2-5-0': 'In this paper, motivated by analogy with the Ginzburg-Landau setting, we investigate the limiting behavior as [MATH] of stationary [MATH]-harmonic maps [MATH] from an arbitrary compact, oriented manifold [MATH] to the circle.', '1802.03053-2-5-1': 'At the global level, we find that the limiting behavior of the maps [MATH] and their energy measures strongly resembles the asymptotics described above for solutions to the Ginzburg-Landau equations.', '1802.03053-2-5-2': 'At the smallest scales, however, the comparatively straightforward blow-up analysis for [MATH]-harmonic maps leads us to some simpler arguments and sharper estimates than are currently available in the Ginzburg-Landau setting.', '1802.03053-2-6-0': 'Let [MATH] be a sequence with [MATH], and let [MATH] be a sequence of stationary [MATH]-harmonic maps from a compact, oriented Riemannian manifold [MATH] to the circle, satisfying [EQUATION]', '1802.03053-2-6-1': 'Then (a subsequence of) the energy measures [MATH] converge weakly in [MATH] to a limiting measure [MATH] of the form [EQUATION] where [MATH] is a harmonic one-form, and [MATH] is a stationary, rectifiable [MATH] varifold.', '1802.03053-2-6-2': 'Furthermore, the support of [MATH] is given by the Hausdorff limit [EQUATION] of the singular sets [MATH], and the density [MATH] satisfies [EQUATION]', '1802.03053-2-7-0': 'Suppose that, in addition to the hypotheses of Theorem [REF], either [MATH] or [EQUATION].', '1802.03053-2-7-1': 'Then (a subsequence of) the maps [MATH] converge weakly in [MATH] for all [MATH], and strongly in [MATH], to a limiting map [MATH] that is harmonic away from [MATH].', '1802.03053-2-8-0': 'In the situation of Theorem [REF], if [MATH], then the density of the concentration varifold [MATH] has the form [EQUATION] for some [MATH].', '1802.03053-2-9-0': 'As in the Ginzburg-Landau setting, the question of the integrality of [MATH] remains open in higher dimensions, but we suspect that the answer will be affirmative.', '1802.03053-2-10-0': 'On every compact Riemannian manifold [MATH] of dimension [MATH], there exists a family [MATH] of stationary [MATH]-harmonic maps to [MATH] for which [EQUATION]', '1802.03053-2-11-0': '## Outline of the Paper: In Section [REF], we review important facts about the structure of maps in [MATH], [MATH]-harmonic functions and weakly [MATH]-harmonic maps to [MATH], and stationary [MATH]-harmonic maps.', '1802.03053-2-12-0': 'In Section [REF], we record a sharp lower bound for the [MATH]-energy density of a stationary [MATH]-harmonic map [MATH] on its singular set [MATH]-a simpler and sharper analog of the [MATH]-ellipticity result (see [CITATION],[CITATION]) for solutions of the Ginzburg-Landau equations.', '1802.03053-2-12-1': 'We then use this to obtain [MATH]-independent estimates for the [MATH]-current [MATH] encoding the topological singularities of [MATH], in the dual Sobolev norms [MATH] for [MATH].', '1802.03053-2-13-0': 'In Section [REF], we employ the results of the preceding sections to estimate separately the components of the Hodge decomposition of the one-form [MATH], first globally in [MATH] for [MATH], then in stronger norms away from [MATH].', '1802.03053-2-14-0': 'In Section [REF], we use these estimates, together with some standard techniques from the study of energy concentration phenomena, to complete the proofs of Theorems [REF] and [REF].', '1802.03053-2-15-0': 'In Section [REF], we prove Theorem [REF], first under some compactness assumptions, using Theorem [REF] and a Pohozaev-type identity, and then for the general case, by showing that the compactness assumptions hold at scales outside of which the normalized energy measures vanish.', '1802.03053-2-16-0': "In Section [REF], we employ min-max arguments like those in [CITATION] together with Wang's results for generalized Ginzburg-Landau functionals [CITATION] to prove Theorem [REF].", '1802.03053-2-16-1': 'We also include a short appendix, containing the proofs of some estimates which are of use to us, but do not play a central role in the paper.'}

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[['1802.03053-1-3-0', '1802.03053-2-3-0'], ['1802.03053-1-3-1', '1802.03053-2-3-1'], ['1802.03053-1-3-2', '1802.03053-2-3-2'], ['1802.03053-1-4-0', '1802.03053-2-4-0'], ['1802.03053-1-4-1', '1802.03053-2-4-1'], ['1802.03053-1-4-2', '1802.03053-2-4-2'], ['1802.03053-1-8-0', '1802.03053-2-8-0'], ['1802.03053-1-9-0', '1802.03053-2-9-0'], ['1802.03053-1-12-0', '1802.03053-2-12-0'], ['1802.03053-1-12-1', '1802.03053-2-12-1'], ['1802.03053-1-15-0', '1802.03053-2-15-0'], ['1802.03053-1-14-0', '1802.03053-2-14-0'], ['1802.03053-1-5-0', '1802.03053-2-5-0'], ['1802.03053-1-5-1', '1802.03053-2-5-1'], ['1802.03053-1-5-2', '1802.03053-2-5-2'], ['1802.03053-1-0-0', '1802.03053-2-0-0'], ['1802.03053-1-0-1', '1802.03053-2-0-1'], ['1802.03053-1-0-2', '1802.03053-2-0-2'], ['1802.03053-1-0-3', '1802.03053-2-0-3'], ['1802.03053-1-7-0', '1802.03053-2-7-0'], ['1802.03053-1-7-1', '1802.03053-2-7-1'], ['1802.03053-1-16-0', '1802.03053-2-16-0'], ['1802.03053-1-16-1', '1802.03053-2-16-1'], ['1802.03053-1-2-0', '1802.03053-2-2-0'], ['1802.03053-1-2-1', '1802.03053-2-2-1'], ['1802.03053-1-2-2', '1802.03053-2-2-2'], ['1802.03053-1-2-3', '1802.03053-2-2-3'], ['1802.03053-1-10-0', '1802.03053-2-10-0'], ['1802.03053-1-13-0', '1802.03053-2-13-0'], ['1802.03053-1-6-0', '1802.03053-2-6-0'], ['1802.03053-1-6-1', '1802.03053-2-6-1'], ['1802.03053-1-6-2', '1802.03053-2-6-2']]

[]

[]

[]

[]

['1802.03053-1-20-7', '1802.03053-1-24-3', '1802.03053-1-25-0', '1802.03053-1-30-1', '1802.03053-1-33-1', '1802.03053-1-37-0', '1802.03053-1-42-0', '1802.03053-1-45-4', '1802.03053-1-46-11', '1802.03053-1-47-3', '1802.03053-1-54-1', '1802.03053-1-65-2', '1802.03053-1-68-4', '1802.03053-1-69-0', '1802.03053-1-71-1', '1802.03053-1-78-2', '1802.03053-1-79-0', '1802.03053-1-82-1', '1802.03053-1-83-0', '1802.03053-1-83-1', '1802.03053-1-85-0', '1802.03053-1-86-0', '1802.03053-1-89-0', '1802.03053-1-91-0', '1802.03053-1-94-0', '1802.03053-1-103-2', '1802.03053-1-105-0', '1802.03053-1-105-1', '1802.03053-1-112-8', '1802.03053-1-129-1', '1802.03053-1-130-8', '1802.03053-1-132-0', '1802.03053-1-132-1', '1802.03053-1-132-2', '1802.03053-1-132-3', '1802.03053-1-132-4', '1802.03053-1-132-5', '1802.03053-1-132-6']

{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1802.03053

1403.8002

{'1403.8002-1-0-0': 'We are interested in the following problem: given an open, bounded domain [MATH], what is the largest constant [MATH] such that there exist an infinite sequence of disks [MATH] and a sequence [MATH] with [MATH] such that [EQUATION] where [MATH] denotes the characteristic function?', '1403.8002-1-0-1': 'We prove that certain (somewhat peculiar) domains [MATH] satisfy the property with [MATH].', '1403.8002-1-0-2': 'For these domains there exists a sequence of points [MATH] in [MATH] with weights [MATH] such that for all harmonic functions [MATH] [EQUATION] where [MATH] depends only on [MATH].', '1403.8002-1-0-3': 'This gives a Quasi-Monte-Carlo method for harmonic functions which improves on the probabilistic Monte-Carlo bound [MATH] without introducing a dependence on the total variation.', '1403.8002-1-0-4': 'We do not know which decay rates are optimal.', '1403.8002-1-1-0': '# Introduction', '1403.8002-1-2-0': '## Harmonic functions.', '1403.8002-1-2-1': "This paper aims to describe some progress in a problem that arose at the Oberwolfach Workshop 1340 'Uniform Distribution Theory and Applications', where it was motivated by a talk of the author on a related problem [CITATION].", '1403.8002-1-2-2': 'We describe our question in its simplest possible setting: let [MATH] be some bounded domain and let [MATH] be a harmonic function, i.e. assume it satisfies [EQUATION] is the Laplacian.', '1403.8002-1-2-3': 'Is there a Quasi Monte Carlo method able to exploit this information effectively to compute an approximation (including an error estimate) of [EQUATION]', '1403.8002-1-2-4': 'The key ingredient suggesting that this might indeed be the case is the mean-value property: let [MATH] denote the disk with radius [MATH] centered at [MATH].', '1403.8002-1-2-5': 'If [MATH] is harmonic in a neighbourhood of [MATH], then [EQUATION].', '1403.8002-1-2-6': 'This means that exact integration over disks can be done with one function evaluation.', '1403.8002-1-2-7': 'In particular, if one had a sequence of disks [MATH] such that [EQUATION] for some [MATH], then this gives a Quasi Monte Carlo method for harmonic functions [EQUATION] where [MATH] is the center of [MATH] and [MATH].', '1403.8002-1-2-8': 'Conversely, since the constant function 1 is harmonic any such Quasi Monte Carlo method gives a sequence of disks [MATH] centered at [MATH] with radius [MATH] given via [MATH] and [MATH] such that [EQUATION]', '1403.8002-1-3-0': '## Main Result.', '1403.8002-1-4-0': "We will prove a result for the following (quite restricted but nontrivial) type of domains: we say [MATH] is finitely disk-covered if there exists a finite number of closed disks [MATH] such that any two disks meet at most in a single point that 'span' [MATH] in the following way: every point in [MATH] is either contained in one of the disks or lies in a region surrounded by three disks such that any two out of these three disks touch in a point.", '1403.8002-1-5-0': 'These sets are quite peculiar, however, at least any bounded, simply connected domain with a smooth boundary can be approximated in the Gromov-Hausdorff metric by a sequence of finitely disk-covered sets: it suffices to consider disks in a lattice arrangement (either hexagonal or rectangular in which case on has to add a final disk in the middle of every area) and approximate the desired domain using this lattice (rescaled to the desired level of accuracy of the approximation).', '1403.8002-1-6-0': 'Our consideration of this particular class of sets is twofold: it effectively cuts off the possibility of harmonic functions with large growth at the boundary of the domain since numerical integration close to the boundary of a finitely disk-covered domain can be done with finitely many function evaluations; secondly, the arising structure allows us to exploit recent advances in the study of Apollonian packings.', '1403.8002-1-6-1': 'Let [MATH] be finitely disk-covered.', '1403.8002-1-6-2': 'Then there exists a sequence of disks [MATH] such that [EQUATION] where [MATH] depends only on [MATH].', '1403.8002-1-7-0': 'We believe that the statement is not optimal and that one should be able to construct sequences with a larger exponent in [MATH] if one were to exploit the fact that some of the disks may have negative coefficients (something that is not used in the statement here).', '1403.8002-1-7-1': 'It should certainly be possible to prove some bounds for, say, the class of convex domains (numerical experiments suggest that the randomized greedy algorithm - picking a random point and, if it is not contained in one the existing disks, add the largest disk possible without introducing intersections - corresponds to [MATH] for convex domains).', '1403.8002-1-7-2': 'Since we never actually use the possibility of subtracting characteristic functions (i.e. [MATH]), we can immediately deduce that for any [MATH] [EQUATION] but there is no reason to assume that this should be in any way optimal.', '1403.8002-1-8-0': '## Quasi-Monte Carlo.', '1403.8002-1-8-1': 'As outlined above, the statement immediately implies a Quasi Monte Carlo method.', '1403.8002-1-8-2': 'The same considerations as above suggest that there is no reason to assume this might be optimal.', '1403.8002-1-8-3': 'Let [MATH] be finitely disk-covered.', '1403.8002-1-8-4': 'Then there exists a universal sequence [MATH] of points in [MATH] and a sequence [MATH] of nonnegative reals with the following property: if [EQUATION] then [EQUATION] where [MATH] depends only on [MATH].', '1403.8002-1-9-0': 'Let us emphasize the difference to classical QMC methods: a Quasi-Monte-Carlo method is based on the simple approximation [EQUATION] for a set of points [MATH].', '1403.8002-1-9-1': 'The well-known Koksma-Hlawka inequality gives [EQUATION] where [MATH] denotes the discrepancy of the point set and [MATH] the total variation in the sense of Hardy-Krause.', '1403.8002-1-9-2': 'We refer to the classical monographs of Kuipers Niederreiter [CITATION], Drmota Tichy [CITATION] and Dick Pillichshammer [CITATION] for further information.', '1403.8002-1-9-3': 'We emphasize that there exist point sets such that [MATH].', '1403.8002-1-9-4': 'However, and this is crucial, our bound is independent of the total variation of the function.', '1403.8002-1-9-5': 'Indeed, for the harmonic function (given in polar coordinates) [EQUATION] on some domain, we easily see that [MATH], which can be made arbitrarily large; in contrast, our bound is independent of [MATH].', '1403.8002-1-10-0': '## Possible extensions.', '1403.8002-1-10-1': 'If we were to modify the approximation scheme using a suitably rescaling, then for suitable points and weights the approximation [EQUATION] should yield even better results: what decay properties can be proven?', '1403.8002-1-10-2': 'Another natural conjecture is that, at least for finitely disk-covered domains, even [EQUATION] might be true.', '1403.8002-1-11-0': '# The Proof', '1403.8002-1-12-0': '[Proof of the Theorem.]', '1403.8002-1-12-1': 'The proof is constructive: since [MATH] is finitely disk-covered, we are initially given a finite set of disks [MATH] associated to [MATH] with centers [MATH].', '1403.8002-1-12-2': 'The mean-value theorem implies that for any harmonic [MATH] [EQUATION].', '1403.8002-1-12-3': 'This is already precise on some part of the domain.', '1403.8002-1-12-4': 'The idea is to cover the rest of the domain with smaller and smaller disks (on each of which exact integration can again be performed).', '1403.8002-1-12-5': 'Let us consider a connected component of [EQUATION].', '1403.8002-1-12-6': 'By assumption, it is bounded by three disks any two of which mutually touch in a point.', '1403.8002-1-12-7': 'Then there exists precisely one circle contained within the connected domain that is tangent to all three boundary circles: the statement dates back to Apollonius.', '1403.8002-1-12-8': 'Such a configuration of 4 circles is known as a Descartes configuration: given a Descartes configuration, it is possible to construct three additional circles within the three gaps.', '1403.8002-1-12-9': 'Iterating this process yields an Apollonian packing.', '1403.8002-1-13-0': 'For each connected component of [MATH] (of which there are only finitely many) we construct the associated Apollonian packing and then define an infinite sequence of disks [MATH] by ordering the union of the disks created by the Apollonian packings and the finitely many disks [MATH] by size.', '1403.8002-1-13-1': 'Let [MATH] denote the center of [MATH].', '1403.8002-1-13-2': 'Using the mean-value property, we get that [EQUATION].', '1403.8002-1-13-3': 'It remains to control the speed with which the disks exhaust the set.', '1403.8002-1-13-4': 'Here we use a recent result of Kontorovich Oh [CITATION]: generalizing an earlier result of Boyd [CITATION], they show the cardinality of disks with curvature [MATH] bounded from above by [MATH] behaves as [EQUATION] for a universal constant [MATH] (this approximation is due to McMullen [CITATION]) and constants [MATH] depending on the particular Apollonian packing.', '1403.8002-1-13-5': 'We consider merely finite number of Apollonian packings at the same time and may thus fix the constants [MATH] in what follows.', '1403.8002-1-13-6': 'This implies that [EQUATION].', '1403.8002-1-13-7': 'This estimate controls the number of disks with curvature in a certain interval and shows that for on average there are [MATH] disks with curvature [MATH].', '1403.8002-1-13-8': 'We have that [EQUATION] and therefore [EQUATION].', '1403.8002-1-13-9': 'A disk with curvature [MATH] has measure [MATH] and therefore using [MATH] [EQUATION] for some constant [MATH].', '1403.8002-1-13-10': 'If we define [MATH] to be the number of circes with curvature bounded from above by [MATH], then [EQUATION].', '1403.8002-1-13-11': 'Since [MATH], we have that [EQUATION] and this yields the result.', '1403.8002-1-14-0': '# Open problems', '1403.8002-1-15-0': '## Optimal decay rates.', '1403.8002-1-15-1': 'The natural question is which decay rates are optimal.', '1403.8002-1-15-2': 'Our proof may be regarded as a greedy algorithm: the big open question is the following: given a domain [MATH], is it true that the best approximation of [MATH] is always given by [EQUATION] for a sequence of balls [MATH] or whether there exist more interesting configurations for which [EQUATION] yields a better result for a suitable choice of signs.', '1403.8002-1-16-0': '## Harmonic functions on fractal sets.', '1403.8002-1-16-1': 'We conjecture that on finitely disk-covered domains for the sequence of disks constructed in the argument and an arbitrary harmonic function [MATH] actually the following stronger inequality should be true [EQUATION].', '1403.8002-1-16-2': 'We emphasize that this is not a geometric statement about the constructed packing of disks and that the statement is trivially false for arbitrary functions [MATH].', '1403.8002-1-16-3': 'Our reasoning behind conjecturing such an inequality is the fact that the set [EQUATION] while harmonic functions have strong ridigity properties.', '1403.8002-1-16-4': 'It seems extremely natural to assume that harmonic functions cannot differ too much on fractal sets from their average behavior.'}

{'1403.8002-2-0-0': 'We are interested in the following problem: given an open, bounded domain [MATH], what is the largest constant [MATH] such that there exist an infinite sequence of disks [MATH] and a sequence [MATH] with [MATH] such that [EQUATION] where [MATH] denotes the characteristic function?', '1403.8002-2-0-1': 'We prove that certain (somewhat peculiar) domains [MATH] satisfy the property with [MATH].', '1403.8002-2-0-2': 'For these domains there exists a sequence of points [MATH] in [MATH] with weights [MATH] such that for all harmonic functions [MATH] [EQUATION] where [MATH] depends only on [MATH].', '1403.8002-2-0-3': 'This gives a Quasi-Monte-Carlo method for harmonic functions which improves on the probabilistic Monte-Carlo bound [MATH] without introducing a dependence on the total variation.', '1403.8002-2-0-4': 'We do not know which decay rates are optimal.', '1403.8002-2-1-0': '# Introduction', '1403.8002-2-2-0': '## Harmonic functions.', '1403.8002-2-2-1': "This paper aims to describe some progress in a problem that arose at the Oberwolfach Workshop 1340 'Uniform Distribution Theory and Applications', where it was motivated by a talk of the author on a related problem [CITATION].", '1403.8002-2-2-2': 'We describe our question in its simplest possible setting: let [MATH] be some bounded domain and let [MATH] be a harmonic function, i.e. assume it satisfies [EQUATION] is the Laplacian.', '1403.8002-2-2-3': 'Is there a Quasi Monte Carlo method able to exploit this information effectively to compute an approximation (including an error estimate) of [EQUATION]', '1403.8002-2-2-4': 'The key ingredient suggesting that this might indeed be the case is the mean-value property: let [MATH] denote the disk with radius [MATH] centered at [MATH].', '1403.8002-2-2-5': 'If [MATH] is harmonic in a neighbourhood of [MATH], then [EQUATION].', '1403.8002-2-2-6': 'This means that exact integration over disks can be done with one function evaluation.', '1403.8002-2-2-7': 'In particular, if one had a sequence of disks [MATH] such that [EQUATION] for some [MATH], then this gives a Quasi Monte Carlo method for harmonic functions [EQUATION] where [MATH] is the center of [MATH] and [MATH].', '1403.8002-2-2-8': 'Conversely, since the constant function 1 is harmonic any such Quasi Monte Carlo method gives a sequence of disks [MATH] centered at [MATH] with radius [MATH] given via [MATH] and [MATH] such that [EQUATION]', '1403.8002-2-3-0': '## Main Result.', '1403.8002-2-4-0': "We will prove a result for the following (quite restricted but nontrivial) type of domains: we say [MATH] is finitely disk-covered if there exists a finite number of closed disks [MATH] such that any two disks meet at most in a single point that 'span' [MATH] in the following way: every point in [MATH] is either contained in one of the disks or lies in a region surrounded by three disks such that any two out of these three disks touch in a point.", '1403.8002-2-5-0': 'These sets are quite peculiar, however, at least any bounded, simply connected domain with a smooth boundary can be approximated in the Gromov-Hausdorff metric by a sequence of finitely disk-covered sets: it suffices to consider disks in a lattice arrangement (either hexagonal or rectangular in which case on has to add a final disk in the middle of every area) and approximate the desired domain using this lattice (rescaled to the desired level of accuracy of the approximation).', '1403.8002-2-6-0': 'Our consideration of this particular class of sets is twofold: it effectively cuts off the possibility of harmonic functions with large growth at the boundary of the domain since numerical integration close to the boundary of a finitely disk-covered domain can be done with finitely many function evaluations; secondly, the arising structure allows us to exploit recent advances in the study of Apollonian packings.', '1403.8002-2-6-1': 'Let [MATH] be finitely disk-covered.', '1403.8002-2-6-2': 'Then there exists a sequence of disks [MATH] such that [EQUATION] where [MATH] depends only on [MATH].', '1403.8002-2-7-0': 'We believe that the statement is not optimal and that one should be able to construct sequences with a larger exponent in [MATH] if one were to exploit the fact that some of the disks may have negative coefficients (something that is not used in the statement here).', '1403.8002-2-7-1': 'It should certainly be possible to prove some bounds for, say, the class of convex domains (numerical experiments suggest that the randomized greedy algorithm - picking a random point and, if it is not contained in one the existing disks, add the largest disk possible without introducing intersections - corresponds to [MATH] for convex domains).', '1403.8002-2-7-2': 'Since we never actually use the possibility of subtracting characteristic functions (i.e. [MATH]), we can immediately deduce that for any [MATH] [EQUATION] but there is no reason to assume that this should be in any way optimal.', '1403.8002-2-8-0': '## Quasi-Monte Carlo.', '1403.8002-2-8-1': 'As outlined above, the statement immediately implies a Quasi Monte Carlo method.', '1403.8002-2-8-2': 'The same considerations as above suggest that there is no reason to assume this might be optimal.', '1403.8002-2-8-3': 'Let [MATH] be finitely disk-covered.', '1403.8002-2-8-4': 'Then there exists a universal sequence [MATH] of points in [MATH] and a sequence [MATH] of nonnegative reals with the following property: if [EQUATION] then [EQUATION] where [MATH] depends only on [MATH].', '1403.8002-2-9-0': 'Let us emphasize the difference to classical QMC methods: a Quasi-Monte-Carlo method is based on the simple approximation [EQUATION] for a set of points [MATH].', '1403.8002-2-9-1': 'The well-known Koksma-Hlawka inequality gives [EQUATION] where [MATH] denotes the discrepancy of the point set and [MATH] the total variation in the sense of Hardy-Krause.', '1403.8002-2-9-2': 'We refer to the classical monographs of Kuipers Niederreiter [CITATION], Drmota Tichy [CITATION] and Dick Pillichshammer [CITATION] for further information.', '1403.8002-2-9-3': 'We emphasize that there exist point sets such that [MATH].', '1403.8002-2-9-4': 'However, and this is crucial, our bound is independent of the total variation of the function.', '1403.8002-2-9-5': 'Indeed, for the harmonic function (given in polar coordinates) [EQUATION] on some domain, we easily see that [MATH], which can be made arbitrarily large; in contrast, our bound is independent of [MATH].', '1403.8002-2-10-0': '## Possible extensions.', '1403.8002-2-10-1': 'If we were to modify the approximation scheme using a suitably rescaling, then for suitable points and weights the approximation [EQUATION] should yield even better results: what decay properties can be proven?', '1403.8002-2-10-2': 'Another natural conjecture is that, at least for finitely disk-covered domains, even [EQUATION] might be true.', '1403.8002-2-11-0': '# The Proof', '1403.8002-2-12-0': '[Proof of the Theorem.]', '1403.8002-2-12-1': 'The proof is constructive: since [MATH] is finitely disk-covered, we are initially given a finite set of disks [MATH] associated to [MATH] with centers [MATH].', '1403.8002-2-12-2': 'The mean-value theorem implies that for any harmonic [MATH] [EQUATION].', '1403.8002-2-12-3': 'This is already precise on some part of the domain.', '1403.8002-2-12-4': 'The idea is to cover the rest of the domain with smaller and smaller disks (on each of which exact integration can again be performed).', '1403.8002-2-12-5': 'Let us consider a connected component of [EQUATION].', '1403.8002-2-12-6': 'By assumption, it is bounded by three disks any two of which mutually touch in a point.', '1403.8002-2-12-7': 'Then there exists precisely one circle contained within the connected domain that is tangent to all three boundary circles: the statement dates back to Apollonius.', '1403.8002-2-12-8': 'Such a configuration of 4 circles is known as a Descartes configuration: given a Descartes configuration, it is possible to construct three additional circles within the three gaps.', '1403.8002-2-12-9': 'Iterating this process yields an Apollonian packing.', '1403.8002-2-13-0': 'For each connected component of [MATH] (of which there are only finitely many) we construct the associated Apollonian packing and then define an infinite sequence of disks [MATH] by ordering the union of the disks created by the Apollonian packings and the finitely many disks [MATH] by size.', '1403.8002-2-13-1': 'Let [MATH] denote the center of [MATH].', '1403.8002-2-13-2': 'Using the mean-value property, we get that [EQUATION].', '1403.8002-2-13-3': 'It remains to control the speed with which the disks exhaust the set.', '1403.8002-2-13-4': 'Here we use a recent result of Kontorovich Oh [CITATION]: generalizing an earlier result of Boyd [CITATION], they show the cardinality of disks with curvature [MATH] bounded from above by [MATH] behaves as [EQUATION] for a universal constant [MATH] (this approximation is due to McMullen [CITATION]) and constants [MATH] depending on the particular Apollonian packing.', '1403.8002-2-13-5': 'We consider merely finite number of Apollonian packings at the same time and may thus fix the constants [MATH] in what follows.', '1403.8002-2-13-6': 'This implies that [EQUATION].', '1403.8002-2-13-7': 'This estimate controls the number of disks with curvature in a certain interval and shows that for on average there are [MATH] disks with curvature [MATH].', '1403.8002-2-13-8': 'We have that [EQUATION] and therefore [EQUATION].', '1403.8002-2-13-9': 'A disk with curvature [MATH] has measure [MATH] and therefore using [MATH] [EQUATION] for some constant [MATH].', '1403.8002-2-13-10': 'If we define [MATH] to be the number of circes with curvature bounded from above by [MATH], then [EQUATION].', '1403.8002-2-13-11': 'Since [MATH], we have that [EQUATION] and this yields the result.', '1403.8002-2-14-0': '# Open problems', '1403.8002-2-15-0': '## Optimal decay rates.', '1403.8002-2-15-1': 'The natural question is which decay rates are optimal.', '1403.8002-2-15-2': 'Our proof may be regarded as a greedy algorithm: the big open question is the following: given a domain [MATH], is it true that the best approximation of [MATH] is always given by [EQUATION] for a sequence of balls [MATH] or whether there exist more interesting configurations for which [EQUATION] yields a better result for a suitable choice of signs.', '1403.8002-2-16-0': '## Harmonic functions on fractal sets.', '1403.8002-2-16-1': 'We conjecture that on finitely disk-covered domains for the sequence of disks constructed in the argument and an arbitrary harmonic function [MATH] actually the following stronger inequality should be true [EQUATION].', '1403.8002-2-16-2': 'We emphasize that this is not a geometric statement about the constructed packing of disks and that the statement is trivially false for arbitrary functions [MATH].', '1403.8002-2-16-3': 'Our reasoning behind conjecturing such an inequality is the fact that the set [EQUATION] while harmonic functions have strong ridigity properties.', '1403.8002-2-16-4': 'It seems extremely natural to assume that harmonic functions cannot differ too much on fractal sets from their average behavior.'}

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{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1403.8002

1605.04486

{'1605.04486-1-0-0': 'Alice and Bob are connected via a two-way binary channel.', '1605.04486-1-0-1': 'This paper describes an algorithm to enable Alice to send a message to Bob when 1) an oblivious adversary flips an unknown number of bits, [MATH], on the channel; and 2) the message length [MATH], and a desired error probability, [MATH] are public knowledge.', '1605.04486-1-1-0': 'With probability at least [MATH], our algorithm ensures that Bob receives the correct message, and that Alice and Bob terminate after sending a total of [MATH] bits.', '1605.04486-1-2-0': 'When [MATH] and [MATH] is large, the number of bits sent is [MATH], which is asymptotically optimal, assuming a conjecture from [CITATION].', '1605.04486-1-3-0': '# Introduction', '1605.04486-1-4-0': 'What if we want to tune the cost of sending a message over a noisy two-way channel to depend on the amount of noise, even when the noise rate is unknown?', '1605.04486-1-4-1': 'In particular, consider the case where Alice wants to send a message of length [MATH] to Bob, over a two-way channel, even in the presence of an oblivious adversary that can flip an unknown number of bits, [MATH], on the channel.', '1605.04486-1-4-2': 'We assume that [MATH], along with a desired error probability [MATH] are public knowledge.', '1605.04486-1-4-3': 'We assume that the oblivious adversary knows our algorithm, but does not know the private random bits of Alice and Bob, or the bits sent on the channel, except when these bits do not depend on the private random bits of the algorithm.', '1605.04486-1-4-4': 'Our main result is summarized in the following theorem.', '1605.04486-1-5-0': 'Our algorithm tolerates an unknown number of adversarial errors, [MATH], and for a given [MATH], succeeds with probability at least [MATH], and sends [MATH] bits.', '1605.04486-1-6-0': 'For the case where [MATH] and [MATH], the algorithm sends [MATH] bits.', '1605.04486-1-6-1': 'This is asymptotically optimal for any algorithm dealing with the case, assuming a conjecture from [CITATION].', '1605.04486-1-7-0': '## Related Work', '1605.04486-1-8-0': 'Interactive Communication The problem of interactive communication asks how two parties can run a protocol [MATH] over a noisy channel.', '1605.04486-1-8-1': 'This problem was first posed by Schulman [CITATION], who describes a deterministic method for simulating interactive protocols on noisy channels with only a constant-factor increase in the total communication complexity.', '1605.04486-1-8-2': 'This work spurred vigorous interest in the area (see [CITATION] for an excellent survey).', '1605.04486-1-9-0': "Schulman's scheme tolerates an adversarial noise rate of [MATH], even if the adversary is not oblivious.", '1605.04486-1-9-1': 'It critically depends on the notion of a tree code for which an exponential-time construction was originally provided.', '1605.04486-1-9-2': 'This exponential construction time motivated work on more efficient constructions [CITATION].', '1605.04486-1-9-3': 'There were also efforts to create alternative codes [CITATION].', '1605.04486-1-9-4': 'Recently, elegant computationally-efficient schemes that tolerate a constant adversarial noise rate have been demonstrated [CITATION].', '1605.04486-1-9-5': 'Additionally, a large number of results have improved the tolerable adversarial noise rate [CITATION], as well as tuning the communication costs to a known, but not necessarily constant, adversarial noise rate [CITATION].', '1605.04486-1-10-0': 'Our paper builds on a result on interactive communication by Dani et al [CITATION], which in contrast to previous work, assumes an oblivious adversary, but tolerates an unknown number of bit flips by such an adversary.', '1605.04486-1-10-1': 'Their algorithm sends [MATH] bits in expectation.', '1605.04486-1-10-2': 'They show that the adversary must be oblivious in order to tolerate unknown [MATH].', '1605.04486-1-11-0': 'We note that the problem in this paper can be viewed as a special case of interactive communication in the model from [CITATION], where the protocol [MATH] just sends [MATH] bits from Alice to Bob.', '1605.04486-1-11-1': 'However, since the problem in this paper is simpler, we can achieve better communication costs.', '1605.04486-1-11-2': 'In particular, when [MATH], our algorithm requires [MATH] bits.', '1605.04486-1-11-3': 'This is never worse than [CITATION], and can be significantly better.', '1605.04486-1-11-4': 'For example, when [MATH], our cost is [MATH] versus [MATH] from [CITATION].', '1605.04486-1-11-5': 'In general if [MATH] our cost is asymptotically better than [CITATION].', '1605.04486-1-12-0': 'Rateless Codes Rateless error correcting codes enable generation of potentially an infinite number of encoding symbols from a given set of source symbols with the property that given any subset of a sufficient number of encoded symbols (typically of the order of the number of source symbols), the original source symbols can ideally be recovered.', '1605.04486-1-12-1': 'Two examples of such codes are the Fountain codes [CITATION] and the LT codes [CITATION].', '1605.04486-1-12-2': 'Both of these belong to the class of erasure codes, enabling the recovery of the original message from only a subset of the encoded symbols.', '1605.04486-1-13-0': 'Erasure codes do not usually employ any feedback mechanisms other than for stopping transmission [CITATION] and for error detection [CITATION] at the receiver.', '1605.04486-1-13-1': 'Additionally, the feedback channel is typically assumed to be noise free.', '1605.04486-1-13-2': 'We differ from this model in that we allow the adversary to flip bits on the feedback channel.', '1605.04486-1-13-3': 'Additionally, our model of channel noise in both directions is more challenging than is typically assumed for rateless codes.', '1605.04486-1-13-4': 'In particular, adversarial bit flips are more difficult to tolerate than erasures.', '1605.04486-1-14-0': '## Formal Model', '1605.04486-1-15-0': 'Our algorithm takes as input the message [MATH], which is a sequence of [MATH] bits to be sent from Alice to Bob, and an error tolerance [MATH].', '1605.04486-1-15-1': 'We assume that both Alice and Bob know [MATH] as well as [MATH].', '1605.04486-1-16-0': 'Channel steps We assume that the communication over the channel is synchronous and individual computation is instantaneous.', '1605.04486-1-16-1': 'A channel step is defined as the amount of time that it takes to send one bit over the channel.', '1605.04486-1-17-0': 'Silence on the channel Similar to [CITATION], when neither Alice nor Bob sends in a channel step, we say that the channel is silent.', '1605.04486-1-17-1': 'In any contiguous sequence of silent channel steps, the bit received on the channel in the first step is set by the adversary for free.', '1605.04486-1-17-2': 'By default, the bit received in the subsequent steps of the sequence remains the same, unless the adversary pays for one bit flip each time it wants to change the value of the bit received in any contiguous sequence of silent steps.', '1605.04486-1-18-0': '## Paper organization', '1605.04486-1-19-0': 'This paper is organized as follows.', '1605.04486-1-19-1': 'Section [MATH] presents our main algorithm.', '1605.04486-1-19-2': 'Section [MATH] presents the analysis for failure probability, correctness, termination and number of bits sent by the algorithm.', '1605.04486-1-19-3': 'Finally, Section [MATH] concludes the paper by stating the main result and some open problems.', '1605.04486-1-20-0': '# Algorithm', '1605.04486-1-21-0': 'Our algorithm makes critical use of Reed-Solomon codes from [CITATION].', '1605.04486-1-21-1': 'Alice begins by encoding her message using a polynomial of degree [MATH] over [MATH], where [MATH].', '1605.04486-1-21-2': 'She sends the values of this polynomial computed at certain elements of the field as message symbols to Bob.', '1605.04486-1-21-3': 'Upon receiving an appropriate number of these points, Bob computes the polynomial using the Berlekamp-Welch algorithm [CITATION] and sends a fingerprint of his guess to Alice.', '1605.04486-1-21-4': 'Upon hearing this fingerprint, if Alice finds no errors, she echoes the fingerprint back to Bob, upon receiving a correct copy of which, Bob terminates the algorithm.', '1605.04486-1-21-5': 'Unless the adversary corrupts many bits, Alice terminates soon after.', '1605.04486-1-22-0': 'However, in the case where Alice does not receive a correct fingerprint of the polynomial from Bob, she sends two more evaluations of the polynomial to Bob.', '1605.04486-1-22-1': 'Bob keeps receiving extra evaluations and recomputing the polynomial until he receives the correct fingerprint echo from Alice.', '1605.04486-1-23-0': '## Notation', '1605.04486-1-24-0': 'Some helper functions and notation used in our algorithm are described in this section.', '1605.04486-1-24-1': 'We denote by [MATH] the fact that [MATH] is sampled uniformly at random from the set [MATH].', '1605.04486-1-24-2': 'Fingerprinting For fingerprinting, we use a well known theorem by Naor and Naor [CITATION], slightly reworded as follows:', '1605.04486-1-25-0': '[CITATION] Fix integer [MATH] and real [MATH].', '1605.04486-1-25-1': 'Then there exist constants [MATH] and algorithm h such that the following hold for a given string [MATH].', '1605.04486-1-26-0': 'For a string [MATH] of length at most [MATH], we have [MATH], where [MATH] is a string of length [MATH].', '1605.04486-1-26-1': 'For any bit strings [MATH] and [MATH] of length at most [MATH], if [MATH], then [MATH], else [MATH].', '1605.04486-1-27-0': 'We refer to [MATH] as the fingerprint of the message [MATH].', '1605.04486-1-28-0': 'GetPolynomial Let [MATH] be a multiset of tuples of the form [MATH].', '1605.04486-1-28-1': 'For each [MATH], we define [MATH] to be the tuple [MATH] that has the highest number of occurrences in [MATH], breaking ties arbitrarily.', '1605.04486-1-28-2': 'We define [MATH].', '1605.04486-1-28-3': 'Given the set [MATH], we define the function [MATH] which returns the degree-[MATH] polynomial over [MATH] that is supported by the largest number of points in [MATH], breaking ties arbitrarily.', '1605.04486-1-29-0': 'The following theorem from [CITATION] [CITATION] provides conditions under which [MATH] reconstructs the required polynomial.', '1605.04486-1-30-0': '[CITATION] [CITATION] Let [MATH] be a polynomial of degree [MATH] over some field [MATH], and [MATH].', '1605.04486-1-30-1': 'Let [MATH] be the number of elements [MATH] such that [MATH], and let [MATH].', '1605.04486-1-30-2': 'Then, if [MATH], we have [MATH].', '1605.04486-1-31-0': 'Algebraic Manipulation Detection Codes Our algorithm also makes use of Algebraic Manipulation Detection (AMD) codes from [CITATION].', '1605.04486-1-31-1': 'For a given [MATH], called the strength of AMD encoding, these codes provide three functions: [MATH], [MATH] and [MATH].', '1605.04486-1-31-2': 'The function [MATH] creates an AMD encoding of a message [MATH].', '1605.04486-1-31-3': 'The function [MATH] takes a message [MATH] and returns true if and only if there exists some message [MATH] such that [MATH].', '1605.04486-1-31-4': 'The function [MATH] takes a message [MATH] such that [MATH] and returns a message [MATH] such that [MATH].', '1605.04486-1-31-5': 'These functions enable detection of bit corruption in an encoded message with high probability.', '1605.04486-1-31-6': 'The following (slightly reworded) theorem from [CITATION] helps establish this:', '1605.04486-1-32-0': '[CITATION] For any [MATH], there exist functions [MATH], [MATH] and [MATH], such that for any bit string [MATH] of length [MATH]:', '1605.04486-1-33-0': '[MATH] is a string of length [MATH], for some constant [MATH] and [MATH] For any bit string [MATH] of length [MATH], we have [MATH]', '1605.04486-1-34-0': 'With the use of Naor-Naor hash functions along with AMD codes, we are able to provide the required security for messages with Alice and Bob.', '1605.04486-1-34-1': 'Assume that the Bob generates the fingerprint [MATH], which upon tampering by the adversary, is converted to [MATH] for some strings [MATH] of appropriate lengths.', '1605.04486-1-34-2': 'Upon receiving this, Alice compares it against the fingerprint of her message [MATH] by computing [MATH], for appropriately chosen [MATH].', '1605.04486-1-34-3': 'Then, we require that there exist a [MATH] such that for any choice of [MATH], [EQUATION] for any string [MATH].', '1605.04486-1-34-4': 'Theorem [REF] provides us with this guarantee directly.', '1605.04486-1-35-0': 'Error-correction Codes These codes enable us to encode a message so that it can be recovered even if the adversary corrupts a third of the bits.', '1605.04486-1-35-1': 'We will denote the encoding and decoding functions by ecEnc and ecDec, respectively.', '1605.04486-1-35-2': 'The following theorem, a slight restatement from [CITATION], gives the properties of these functions.', '1605.04486-1-36-0': '[CITATION] There is a constant [MATH] such that for any message [MATH], we have [MATH].', '1605.04486-1-36-1': 'Moreover, if [MATH] differs from [MATH] in at most one-third of its bits, then [MATH].', '1605.04486-1-37-0': 'Finally, we observe that the linearity of ecEnc and ecDec ensure that when the error correction is composed with the AMD code, the resulting code has the following properties:', '1605.04486-1-38-0': 'If at most a third of the bits of the message are flipped, then the original message can be uniquely reconstructed by rounding to the nearest codeword in the range of ecEnc.', '1605.04486-1-38-1': 'Even if an arbitrary set of bits is flipped, the probability of the change not being recognized is at most [MATH], i.e. the same guarantee as the AMD codes.', '1605.04486-1-39-0': 'This is because ecDec is linear, so when noise [MATH] is added by the adversary to the codeword [MATH], effectively what happens is the decoding function [MATH], where [MATH] is the AMD-encoded message.', '1605.04486-1-39-1': 'But now [MATH] is an obliviously selected string added to the AMD-encoded codeword.', '1605.04486-1-40-0': 'Silence In our algorithm, silence on the channel has a very specific meaning.', '1605.04486-1-40-1': 'We define the function [MATH] to return true iff the string [MATH] has fewer than [MATH] bit alternations.', '1605.04486-1-41-0': 'Other notation We use [MATH] to denote the [MATH]-bit string of all zeros, [MATH] for string concatenation and [MATH] to denote the function that returns the bits on the channel over the next [MATH] time steps.', '1605.04486-1-41-1': 'For the sake of convenience, we will use [MATH] to mean [MATH], unless specified otherwise.', '1605.04486-1-41-2': 'Let [MATH].', '1605.04486-1-42-0': '## Algorithm overview', '1605.04486-1-43-0': "[p] Alice's algorithm", '1605.04486-1-44-0': '[1] Alice[MATH] is a message of length [MATH] Field size [MATH] Degree of polynomial [MATH]degree-[MATH] polynomial encoding of [MATH] over [MATH] Send [MATH] Rounds for the algorithm [MATH] Message size in this round [MATH] Fingerprint from Bob [MATH] Send [MATH] Echo the fingerprint Send [MATH] if the fingerprint was not echoed.', '1605.04486-1-44-1': '[MATH] Bob has likely left [MATH]polynomial evaluation tuples of [MATH] at next two points of the field (cyclically) Send [MATH]', '1605.04486-1-45-0': "[p] Bob's algorithm", '1605.04486-1-46-0': "[1] Bob[MATH] Field size [MATH] Degree of polynomial [MATH] Listen to first [MATH] evaluations from Alice Add the corresponding polynomial evaluation tuples to [MATH] Message size in this round [MATH] Sample a string [MATH] Send [MATH] Send Alice the fingerprint of the polynomial [MATH] Listen to Alice's echo [MATH] Send a string [MATH] Receive polynomial evaluation tuples for the next two field elements and add to [MATH]", '1605.04486-1-47-0': 'We now present our main algorithm: Algorithm [REF] is what Alice follows and Algorithm [REF] is what Bob follows.', '1605.04486-1-47-1': 'Both algorithms share the knowledge of message length [MATH] and the error tolerance [MATH].', '1605.04486-1-48-0': '# Analysis', '1605.04486-1-49-0': 'We now prove that our algorithm is correct with probability at least [MATH], and compute the number of bits sent.', '1605.04486-1-49-1': 'Before proceeding to the proof, we define three bad events:', '1605.04486-1-50-0': 'Unintentional Silence.', '1605.04486-1-50-1': 'When Bob executes step 18 of his algorithm, the string received by Alice is interpreted as silence.', '1605.04486-1-50-2': 'Fingerprint Error.', '1605.04486-1-50-3': 'Fingerprint hash collision as per Theorem [REF].', '1605.04486-1-50-4': 'AMD Error.', '1605.04486-1-50-5': 'The adversary corrupts an AMD encoded message into an encoding of a different message.', '1605.04486-1-51-0': 'Rounds For both Alice and Bob, we define a round as one iteration of the for loop in our algorithm.', '1605.04486-1-51-1': 'We refer to the part of the algorithm before the for loop begins as round [MATH].', '1605.04486-1-51-2': 'The AMD encoding strength [MATH] is equal to [MATH] initially and decreases by a factor of [MATH] every [MATH] rounds.', '1605.04486-1-51-3': 'This way, the number of bits added to the messages increases linearly every [MATH] rounds, which enhances security against corruption.', '1605.04486-1-52-0': '## Correctness and Termination', '1605.04486-1-53-0': "We now prove that with probability at least [MATH], Bob terminates the algorithm with the correct guess of Alice's message.", '1605.04486-1-54-0': '### Unintentional Silence The following lemmas show that Alice terminates before Bob with probability at most [MATH].', '1605.04486-1-55-0': 'For [MATH], the probability that a [MATH]-bit string sampled uniformly at random from [MATH] has fewer than [MATH] bit alternations is at most [MATH].', '1605.04486-1-56-0': 'Let [MATH] be a string sampled uniformly at random from [MATH], where [MATH].', '1605.04486-1-56-1': 'Denote by [MATH] the [MATH] bit of [MATH].', '1605.04486-1-56-2': 'Let [MATH] be the indicator random variable for the event that [MATH], for [MATH].', '1605.04486-1-56-3': "Note that all [MATH]'s are mutually independent.", '1605.04486-1-56-4': 'Let [MATH] be the number of bit alternations in [MATH].', '1605.04486-1-56-5': 'Clearly, [MATH], which gives [MATH], using the linearity of expectation.', '1605.04486-1-56-6': 'Since [MATH] for all [MATH], we get [MATH].', '1605.04486-1-56-7': 'Using the multiplicative version of Chernoff bounds [CITATION] for [MATH], [EQUATION]', '1605.04486-1-56-8': 'To obtain [MATH], set [MATH] to get, [EQUATION]', '1605.04486-1-57-0': 'Alice terminates the algorithm before Bob with probability at most [MATH].', '1605.04486-1-58-0': 'Let [MATH] be the event that Alice terminates before Bob.', '1605.04486-1-58-1': 'This happens when the string sent by Bob in step 18 after possible adversarial corruptions is interpreted as silence by Alice.', '1605.04486-1-58-2': 'Let [MATH] be the event that Alice terminates before Bob in round [MATH] of the algorithm.', '1605.04486-1-58-3': 'Then, using a union bound over the rounds, the fact that [MATH] and Lemma [REF], we get [EQUATION]', '1605.04486-1-58-4': 'Note that Lemma [REF] is applicable here because for each [MATH], we have [MATH].', '1605.04486-1-58-5': 'To see this, use the fact that [MATH] and [MATH] to obtain the condition [MATH], which is always true because [MATH].', '1605.04486-1-59-0': '### Fingerprint Failure The following lemma proves that the fingerprint error happens with probability at most [MATH], ensuring the correctness of the algorithm.', '1605.04486-1-60-0': "Upon termination, Bob does not have the correct guess of Alice's message with probability at most [MATH].", '1605.04486-1-61-0': "Let [MATH] be the event that Bob does not have the correct guess of Alice's message upon termination.", '1605.04486-1-61-1': 'Note that in round [MATH], from Theorem [REF], the fingerprints fail with probability at most [MATH].', '1605.04486-1-61-2': 'Using a union bound over these rounds, we get [EQUATION]', '1605.04486-1-62-0': '### AMD Failure', '1605.04486-1-63-0': 'The probability of AMD failure is at most [MATH].', '1605.04486-1-64-0': 'Note that in round [MATH], from Theorem [REF], AMD failure occurs with probability at most [MATH].', '1605.04486-1-64-1': 'Hence, using a union bound over the rounds, the AMD failure occurs with probability [MATH].', '1605.04486-1-65-0': '## Probability of Failure', '1605.04486-1-66-0': 'Our algorithm succeeds with probability at least [MATH].', '1605.04486-1-67-0': 'Lemmas [REF], [REF] and [REF] ensure that the three bad events, as defined previously, each happen with probability at most [MATH].', '1605.04486-1-67-1': 'Hence, using a union bound over the occurrence of these three events, the total probability of failure of the algorithm is at most [MATH].', '1605.04486-1-67-2': 'If the three bad events do not occur, then Alice will continue to send evaluations of the polynomial until Bob has the correct message.', '1605.04486-1-67-3': 'Since [MATH] is finite, Bob will eventually have the correct message and terminate.', '1605.04486-1-68-0': '## Cost to the algorithm', '1605.04486-1-69-0': 'Recall that Alice and Bob compute their polynomials [MATH] and [MATH], respectively, over [MATH].', '1605.04486-1-69-1': 'We refer to every [MATH] that Bob stores after receiving the evaluation [MATH], that has potentially been tampered with, of the polynomial [MATH] at [MATH] from Alice as a polynomial evaluation tuple.', '1605.04486-1-69-2': "We call a polynomial evaluation tuple [MATH] in Bob's set [MATH] good if [MATH] and bad otherwise.", '1605.04486-1-70-0': 'We begin by stating two important lemmas that relate the number of bits flipped by the adversary to make [MATH] polynomial evaluation tuples bad to the number of bits required to send them.', '1605.04486-1-71-0': 'Let [MATH] be the number of bits flipped by the adversary to make [MATH] polynomial evaluation tuples bad.', '1605.04486-1-71-1': 'Then, [EQUATION]', '1605.04486-1-72-0': 'Let [MATH], where [MATH] is the number of polynomial evaluation tuples that were not encoded and [MATH] is the number of AMD and error-encoded polynomial evaluation tuples.', '1605.04486-1-72-1': 'Clearly, [MATH].', '1605.04486-1-72-2': 'Each of the remaining [MATH] polynomial evaluation tuples are sent in pairs, one pair per round.', '1605.04486-1-72-3': 'Since the adversary needs to flip at least a third of the number of bits for each encoded polynomial evaluation tuple to make it bad, we have [EQUATION]', '1605.04486-1-72-4': 'Since the number of bits per polynomial evaluation tuple increases monotonically, the expression above becomes: [EQUATION]', '1605.04486-1-73-0': 'Let [MATH] be the number of bits required to send [MATH] polynomial evaluation tuples, where [MATH].', '1605.04486-1-73-1': 'Then, [EQUATION]', '1605.04486-1-74-0': 'If [MATH], then we have [MATH], since each of these [MATH] polynomial evaluation tuples is of length [MATH].', '1605.04486-1-74-1': 'For [MATH], taking into account the fact that each round involves exchange of at most [MATH] messages between Alice and Bob, we get [EQUATION]', '1605.04486-1-75-0': 'Let [MATH], and [MATH] be any round at the end of which [MATH].', '1605.04486-1-75-1': 'Then the number of bad polynomial evaluation tuples through round [MATH] is at least [MATH].', '1605.04486-1-76-0': 'We call a field element [MATH] good if [MATH], and bad otherwise.', '1605.04486-1-76-1': 'Let [MATH] be the number of good field elements and [MATH] be the number of bad field elements up to round [MATH].', '1605.04486-1-76-2': 'Similarly, let [MATH] be the number of good polynomial evaluation tuples and [MATH] be the number of bad polynomial evaluation tuples up to round [MATH].', '1605.04486-1-76-3': 'Then, from Theorem [REF], we must have [MATH].', '1605.04486-1-76-4': 'Note that the total number of field elements for which Bob has received polynomial evaluation tuples from Alice through round [MATH] is [MATH].', '1605.04486-1-76-5': 'Adding this equality to the previous inequality, we have [EQUATION]', '1605.04486-1-76-6': 'The total number of polynomial evaluation tuples received by Bob up to round [MATH] is given by [EQUATION]', '1605.04486-1-76-7': 'Note that every bad field element is associated with at least [MATH] polynomial evaluation tuples.', '1605.04486-1-76-8': 'This gives [MATH].', '1605.04486-1-76-9': 'Using this inequality with Eqs. [REF] and [REF], we have [EQUATION]', '1605.04486-1-76-10': 'Case I: [MATH] For this case, we have [EQUATION]', '1605.04486-1-76-11': 'Case II: [MATH] For this case, we have [EQUATION] where the last inequality holds since [MATH] for [MATH].', '1605.04486-1-77-0': 'Combining Eqs. [REF] and [REF], we get [MATH].', '1605.04486-1-78-0': 'We now state a lemma that is crucial to the proof of Theorem [REF].', '1605.04486-1-79-0': 'If Bob terminates before Alice, the total number of bits sent by our algorithm is [EQUATION].', '1605.04486-1-80-0': 'Let [MATH] be the last round at the end of which [MATH], or [MATH] if [MATH] at the end of round [MATH] and for all subsequent rounds.', '1605.04486-1-80-1': 'Let [MATH] be the number of bits corrupted by the adversary through round [MATH].', '1605.04486-1-80-2': 'Let [MATH] represent the total cost through round [MATH] and [MATH] be the cost of the algorithm after round [MATH].', '1605.04486-1-80-3': "Note that after round [MATH], the adversary must corrupt one of either (1) the fingerprint, or (2) its echo, or (3) silence on the channel in Step 15 of Alice's algorithm, in every round to delay termination.", '1605.04486-1-80-4': 'Also, after round [MATH], Alice and Bob must exchange at least a fingerprint and an echo even if [MATH].', '1605.04486-1-80-5': 'Thus, we have, [EQUATION]', '1605.04486-1-81-0': 'Recall that the number of polynomial evaluation tuples sent up to round [MATH] is [MATH].', '1605.04486-1-81-1': 'Then, from Lemma [REF], we have [EQUATION]', '1605.04486-1-81-2': 'From Lemma [REF], we have that the number of bad polynomial evaluation tuples is at least [MATH].', '1605.04486-1-81-3': 'Thus, from Lemma [REF], we have [EQUATION]', '1605.04486-1-82-0': 'Case I : [MATH] Since [MATH] is at least the number of bad polynomial evaluation tuples, from Lemma [REF], we have [MATH], which gives [MATH].', '1605.04486-1-82-1': 'Hence, using Eq [REF], we get, [EQUATION] where the last equality holds because [MATH].', '1605.04486-1-83-0': 'Case II : [MATH] From Eq. [REF], we have [EQUATION]', '1605.04486-1-83-1': 'Since each summand in the inequality above is positive and [MATH], we get [MATH], which gives [EQUATION]', '1605.04486-1-83-2': 'Since [MATH], we have [MATH].', '1605.04486-1-83-3': 'Building on this, we get, [EQUATION]', '1605.04486-1-83-4': 'Hence, from Eqs. [REF], [REF] and [REF] , we get [EQUATION] where the last equality holds because [MATH] and [MATH] from inequality [REF].', '1605.04486-1-84-0': 'Combining Eqs. [REF], [REF] and [REF], the total number of bits sent by the algorithm becomes [MATH].', '1605.04486-1-85-0': 'Putting it all together, we are now ready to state our main theorem.', '1605.04486-1-85-1': 'Theorem 1.1.', '1605.04486-1-85-2': 'Our algorithm tolerates an unknown number of adversarial errors, [MATH], and for a given [MATH], succeeds with probability at least [MATH], and sends [MATH] bits.', '1605.04486-1-86-0': 'By Lemmas [REF], with probability at least [MATH], Bob terminates before Alice with the correct message.', '1605.04486-1-86-1': 'If this happens, then by Lemma [REF], the total number of bits sent is [MATH].', '1605.04486-1-87-0': '# Conclusion', '1605.04486-1-88-0': 'We have described an algorithm for one-way interactive communication that tolerates an unknown but finite amount of noise and provides error tolerance guarantee.', '1605.04486-1-88-1': 'Against an adversary that flips [MATH] bits, our algorithm succeeds with probability at least [MATH] and sends [MATH] bits, where [MATH] is the length of the message with Alice, and [MATH] is the error tolerance.', '1605.04486-1-88-2': 'When [MATH] and [MATH], the number of bits sent is [MATH], which is asymptotically optimal, assuming a conjecture from [CITATION].', '1605.04486-1-89-0': 'Several open problems remain including the following.', '1605.04486-1-89-1': 'First, can we adapt the results to interactive communication that involves more than two parties?', '1605.04486-1-89-2': 'Second, can we handle the case where [MATH] is unknown to Bob?', '1605.04486-1-89-3': "Finally, can we provide a proof to Haeupler's conjecture, establishing the lower bounds in this setting?"}

{'1605.04486-2-0-0': 'Alice and Bob are connected via a two-way channel, and Alice wants to send a message of [MATH] bits to Bob.', '1605.04486-2-0-1': 'An adversary flips an arbitrary but finite number of bits, [MATH], on the channel.', '1605.04486-2-0-2': "This adversary knows our algorithm and Alice's message, but does not know any private random bits generated by Alice or Bob, nor the bits sent over the channel, except when these bits can be predicted by knowledge of Alice's message or our algorithm.", '1605.04486-2-0-3': "We want Bob to receive Alice's message and for both players to terminate, with error probability at most [MATH], where [MATH] is a parameter known to both Alice and Bob.", '1605.04486-2-0-4': 'Unfortunately, the value [MATH] is unknown in advance to either Alice or Bob, and the value [MATH] is unknown in advance to Bob.', '1605.04486-2-1-0': 'We describe an algorithm to solve the above problem while sending an expected [MATH] bits.', '1605.04486-2-1-1': 'A special case is when [MATH], for some constant [MATH].', '1605.04486-2-1-2': 'Then when [MATH], the expected number of bits sent is [MATH], and when [MATH], the expected number of bits sent is [MATH], which is asymptotically optimal.', '1605.04486-2-2-0': '<ccs2012> <concept> <concept> <concept_id>10002978.10003014.10003015</concept_id> <concept_desc>Security and privacy Security protocols</concept_desc> <concept_significance>300</concept_significance> </concept> <concept_id>10002950.10003712</concept_id> <concept_desc>Mathematics of computing Information theory</concept_desc> <concept_significance>500</concept_significance> </concept> <concept> <concept_id>10010147.10010919.10010172</concept_id> <concept_desc>Computing methodologies Distributed algorithms</concept_desc> <concept_significance>500</concept_significance> </concept> </ccs2012>', '1605.04486-2-3-0': '[500]Mathematics of computing Information theory [500]Computing methodologies Distributed algorithms [300]Security and privacy Security protocols', '1605.04486-2-4-0': '# Introduction', '1605.04486-2-5-0': 'What if we want to send a message over a noisy two-way channel, and little is known in advance?', '1605.04486-2-5-1': 'In particular, imagine that Alice wants to send a message to Bob, but the number of bits flipped on the channel is unknown to either Alice or Bob in advance.', '1605.04486-2-5-2': "Further, the length of Alice's message is also unknown to Bob in advance.", '1605.04486-2-5-3': 'While this scenario seems like it would occur quite frequently, surprisingly little is known about it.', '1605.04486-2-6-0': 'In this paper, we describe an algorithm to efficiently address this problem.', '1605.04486-2-6-1': 'To do so, we make a critical assumption on the type of noise on the channel.', '1605.04486-2-6-2': 'We assume that an adversary flips bits on the channel, but this adversary is not completely omniscient.', '1605.04486-2-6-3': "The adversary knows our algorithm and Alice's message, but it does not know the private random bits of Alice and Bob, nor the bits that are sent over the channel, except when these bits do not depend on the random bits of Alice and Bob.", '1605.04486-2-6-4': 'Some assumption like this is necessary : if the adversary knows all bits sent on the channel and the number of bits it flips is unknown in advance, then no algorithm can succeed with better than constant probability (see Theorem 6.1 from [CITATION] for details).', '1605.04486-2-7-0': "Our algorithm assumes that a desired error probability, [MATH] is known to both Alice and Bob, that the adversary flips some number [MATH] bits that is finite but unknown in advance, and that the length of Alice's message, [MATH] is unknown to Bob in advance.", '1605.04486-2-7-1': 'Our main result is then summarized in the following theorem.', '1605.04486-2-8-0': 'Our algorithm tolerates an unknown number of adversarial errors, [MATH], and for any [MATH], succeeds in sending a message of length [MATH] with probability at least [MATH], and sends an expected [MATH] bits.', '1605.04486-2-9-0': 'An interesting case to consider is when the error probability is polynomially small in [MATH], i.e. when [MATH], for some constant [MATH].', '1605.04486-2-9-1': 'Then when [MATH], our algorithm sends [MATH] expected bits.', '1605.04486-2-9-2': 'When [MATH], the number of bits sent is [MATH], which is asymptotically optimal.', '1605.04486-2-10-0': '## Related Work', '1605.04486-2-11-0': 'Interactive Communication Our work is related to the area of interactive communication.', '1605.04486-2-11-1': 'The problem of interactive communication asks how two parties can run a protocol [MATH] over a noisy channel.', '1605.04486-2-11-2': 'This problem was first posed by Schulman [CITATION], who describes a deterministic method for simulating interactive protocols on noisy channels with only a constant-factor increase in the total communication complexity.', '1605.04486-2-11-3': 'This initial work spurred vigorous interest in the area (see [CITATION] for an excellent survey).', '1605.04486-2-12-0': "Schulman's scheme tolerates an adversarial noise rate of [MATH], even if the adversary is not oblivious.", '1605.04486-2-12-1': 'It critically depends on the notion of a tree code for which an exponential-time construction was originally provided.', '1605.04486-2-12-2': 'This exponential construction time motivated work on more efficient constructions [CITATION].', '1605.04486-2-12-3': 'There were also efforts to create alternative codes [CITATION].', '1605.04486-2-12-4': 'Recently, elegant computationally-efficient schemes that tolerate a constant adversarial noise rate have been demonstrated [CITATION].', '1605.04486-2-12-5': 'Additionally, a large number of results have improved the tolerable adversarial noise rate [CITATION], as well as tuning the communication costs to a known, but not necessarily constant, adversarial noise rate [CITATION].', '1605.04486-2-13-0': 'Interactive Communication with Private Channels Our paper builds on a recent result on interactive communication by Dani et al [CITATION].', '1605.04486-2-13-1': 'The model in [CITATION] is equivalent to the one in this paper except that 1) they assume that Alice and Bob are running an arbitrary protocol [MATH]; and 2) they assume that both Alice and Bob know the number of bits sent in [MATH].', '1605.04486-2-13-2': 'In particular, similar to this paper, they assume that the adversary flips an unknown number of bits [MATH], and that the adversary does not know the private random bits of Alice and Bob, or the bits sent over the channel.', '1605.04486-2-14-0': 'If the protocol [MATH] just sends [MATH] bits from Alice to Bob, then the algorithm from [CITATION] can solve the problem we consider here.', '1605.04486-2-14-1': 'In that case, the algorithm of [CITATION] will send an expected [MATH] bits, with a probability of error that is [MATH] for any fixed constant [MATH].', '1605.04486-2-15-0': 'For the same probability of error, the algorithm in this paper sends an expected [MATH] bits.', '1605.04486-2-15-1': 'This is never worse than [CITATION], and can be significantly better.', '1605.04486-2-15-2': 'For example, when [MATH], our cost is [MATH] versus [MATH] from [CITATION].', '1605.04486-2-15-3': 'In general if [MATH] our cost is asymptotically better than [CITATION].', '1605.04486-2-15-4': 'Additionally, unlike [CITATION], the algorithm in this paper does not assume that [MATH] is known in advance by Bob.', '1605.04486-2-16-0': 'An additional results of [CITATION] is a theorem showing that private channels are necessary in order to tolerate unknown [MATH] with better than constant probability of error.', '1605.04486-2-17-0': 'Rateless Codes Rateless error correcting codes enable generation of potentially an infinite number of encoding symbols from a given set of source symbols with the property that given any subset of a sufficient number of encoding symbols, the original source symbols can be recovered.', '1605.04486-2-17-1': 'Fountain codes [CITATION] and LT codes [CITATION] are two classic examples of rateless codes.', '1605.04486-2-17-2': 'Erasure codes employ feedback for stopping transmission [CITATION] and for error detection [CITATION] at the receiver.', '1605.04486-2-18-0': 'Critically, the feedback channel, i.e. the channel from Bob to Alice, is typically assumed to be noise free.', '1605.04486-2-18-1': 'We differ from this model in that we allow noise on the feedback channel, and additionally, we tolerate bit flips, while most rateless codes tolerate only bit erasures.', '1605.04486-2-19-0': '## Formal Model', '1605.04486-2-20-0': 'Initial State We assume that Alice initially knows some message [MATH] of length [MATH] bits that she wants to communicate to Bob, and that both Alice and Bob know an error tolerance parameter [MATH].', '1605.04486-2-20-1': 'However, Bob does not know [MATH] or any other information about [MATH] initially.', '1605.04486-2-20-2': 'Alice and Bob are connected by a two-way binary communication channel.', '1605.04486-2-21-0': 'The Adversary We assume an adversary can flip some a priori unknown, but finite number of bits [MATH] on the channel from Alice to Bob or from Bob to Alice.', '1605.04486-2-21-1': 'This adversary knows [MATH], and all of our algorithms.', '1605.04486-2-21-2': 'However, it does not know any random bits generated by Alice or Bob, or the bits sent over the channel, except when these can be determined from other known information.', '1605.04486-2-22-0': 'Channel steps We assume that communication over the channel is synchronous.', '1605.04486-2-22-1': 'A channel step is defined as the amount of time that it takes to send one bit over the channel.', '1605.04486-2-22-2': 'As is standard in distributed computing, we assume that all local computation is instantaneous.', '1605.04486-2-23-0': 'Silence on the channel Similar to [CITATION], when neither Alice nor Bob sends in a channel step, we say that the channel is silent.', '1605.04486-2-23-1': 'In any contiguous sequence of silent channel steps, the bit received on the channel in the first step is set by the adversary for free.', '1605.04486-2-23-2': 'By default, the bit received in the subsequent steps of the sequence remains the same, unless the adversary pays for one bit flip each time it wants to change the value of the bit received.', '1605.04486-2-24-0': '## Paper organization', '1605.04486-2-25-0': 'The rest of the paper is organized as follows.', '1605.04486-2-25-1': 'We first discuss an algorithm for the case when both Alice and Bob share the knowledge of [MATH] in Section [REF].', '1605.04486-2-25-2': 'We present the analysis for failure probability, correctness, termination and number of bits sent by this algorithm in Section [REF].', '1605.04486-2-25-3': 'Then, we remove the assumption of knowledge of [MATH] and provide an algorithm for the unknown [MATH] case in Section [REF], along with its analysis.', '1605.04486-2-25-4': 'Finally, in Section [REF], we conclude the paper by stating the main result and discuss some open problems.', '1605.04486-2-26-0': '# Known [MATH]', '1605.04486-2-27-0': 'We first discuss the case when Bob knows [MATH].', '1605.04486-2-27-1': 'We remove this assumption later in Section [REF].', '1605.04486-2-28-0': 'Our algorithm makes critical use of Reed-Solomon codes from [CITATION].', '1605.04486-2-28-1': 'Alice begins by encoding her message using a polynomial of degree [MATH] over [MATH], where [MATH].', '1605.04486-2-28-2': 'She sends the values of this polynomial computed at certain elements of the field as message symbols to Bob.', '1605.04486-2-28-3': 'Upon receiving an appropriate number of these points, Bob computes the polynomial using the Berlekamp-Welch algorithm [CITATION] and sends a fingerprint of his guess to Alice.', '1605.04486-2-28-4': 'Upon hearing this fingerprint, if Alice finds no errors, she echoes the fingerprint back to Bob, upon receiving a correct copy of which, Bob terminates the algorithm.', '1605.04486-2-28-5': 'Unless the adversary corrupts many bits, Alice terminates soon after.', '1605.04486-2-29-0': 'However, in the case where Alice does not receive a correct fingerprint of the polynomial from Bob, she sends two more evaluations of the polynomial to Bob.', '1605.04486-2-29-1': 'Bob keeps receiving extra evaluations and recomputing the polynomial until he receives the correct fingerprint echo from Alice.', '1605.04486-2-30-0': '## Notation', '1605.04486-2-31-0': 'Some helper functions and notation used in our algorithm are described in this section.', '1605.04486-2-31-1': 'We denote by [MATH] the fact that [MATH] is sampled uniformly at random from the set [MATH].', '1605.04486-2-31-2': 'Fingerprinting For fingerprinting, we use a well known theorem by Naor and Naor [CITATION], slightly reworded as follows:', '1605.04486-2-32-0': '[CITATION] Fix integer [MATH] and real [MATH].', '1605.04486-2-32-1': 'Then there exist constants [MATH] and algorithm h such that the following hold for a given string [MATH].', '1605.04486-2-33-0': 'For a string [MATH] of length at most [MATH], we have [MATH], where [MATH] is a string of length [MATH].', '1605.04486-2-33-1': 'For any bit strings [MATH] and [MATH] of length at most [MATH], if [MATH], then [MATH], else [MATH].', '1605.04486-2-34-0': 'We refer to [MATH] as the fingerprint of the message [MATH].', '1605.04486-2-35-0': 'GetPolynomial Let [MATH] be a multiset of tuples of the form [MATH].', '1605.04486-2-35-1': 'For each [MATH], we define [MATH] to be the tuple [MATH] that has the highest number of occurrences in [MATH], breaking ties arbitrarily.', '1605.04486-2-35-2': 'We define [MATH].', '1605.04486-2-35-3': 'Given the set [MATH], we define [MATH] as a function that returns the degree-[MATH] polynomial over [MATH] that is supported by the largest number of points in [MATH], breaking ties arbitrarily.', '1605.04486-2-36-0': 'The following theorem from [CITATION] [CITATION] provides conditions under which [MATH] reconstructs the required polynomial.', '1605.04486-2-37-0': '[CITATION] [CITATION] Let [MATH] be a polynomial of degree [MATH] over some field [MATH], and [MATH].', '1605.04486-2-37-1': 'Let [MATH] be the number of elements [MATH] such that [MATH], and let [MATH].', '1605.04486-2-37-2': 'Then, if [MATH], we have [MATH].', '1605.04486-2-38-0': 'Algebraic Manipulation Detection Codes Our algorithm also makes use of Algebraic Manipulation Detection (AMD) codes from [CITATION].', '1605.04486-2-38-1': 'For a given [MATH], called the strength of AMD encoding, these codes provide three functions: [MATH], [MATH] and [MATH].', '1605.04486-2-38-2': 'The function [MATH] creates an AMD encoding of a message [MATH].', '1605.04486-2-38-3': 'The function [MATH] takes a message [MATH] and returns true if and only if there exists some message [MATH] such that [MATH].', '1605.04486-2-38-4': 'The function [MATH] takes a message [MATH] such that [MATH] and returns a message [MATH] such that [MATH].', '1605.04486-2-38-5': 'These functions enable detection of bit corruption in an encoded message with high probability.', '1605.04486-2-38-6': 'The following (slightly reworded) theorem from [CITATION] helps establish this:', '1605.04486-2-39-0': '[CITATION] For any [MATH], there exist functions [MATH], [MATH] and [MATH], such that for any bit string [MATH] of length [MATH]:', '1605.04486-2-40-0': '[MATH] is a string of length [MATH], for some constant [MATH] and [MATH] For any bit string [MATH] of length [MATH], we have [EQUATION]', '1605.04486-2-41-0': 'With the use of Naor-Naor hash functions along with AMD codes, we are able to provide the required security for messages with Alice and Bob.', '1605.04486-2-41-1': 'Assume that the Bob generates the fingerprint [MATH], which upon tampering by the adversary, is converted to [MATH] for some strings [MATH] of appropriate lengths.', '1605.04486-2-41-2': 'Upon receiving this, Alice compares it against the fingerprint of her message [MATH] by computing [MATH], for appropriately chosen [MATH].', '1605.04486-2-41-3': 'Then, we require that there exist a [MATH] such that for any choice of [MATH], [EQUATION] for any string [MATH].', '1605.04486-2-41-4': 'Theorem [REF] provides us with this guarantee directly.', '1605.04486-2-42-0': 'Error-correcting Codes These codes enable us to encode a message so that it can be recovered even if the adversary corrupts a third of the bits.', '1605.04486-2-42-1': 'We will denote the encoding and decoding functions by ecEnc and ecDec, respectively.', '1605.04486-2-42-2': 'The following theorem, a slight restatement from [CITATION], gives the properties of these functions.', '1605.04486-2-43-0': '[CITATION] There is a constant [MATH] such that for any message [MATH], we have [MATH].', '1605.04486-2-43-1': 'Moreover, if [MATH] differs from [MATH] in at most one-third of its bits, then [MATH].', '1605.04486-2-44-0': 'Finally, we observe that the linearity of ecEnc and ecDec ensure that when the error correction is composed with the AMD code, the resulting code has the following properties:', '1605.04486-2-45-0': 'If at most a third of the bits of the message are flipped, then the original message can be uniquely reconstructed by rounding to the nearest codeword in the range of ecEnc.', '1605.04486-2-45-1': 'Even if an arbitrary set of bits is flipped, the probability of the change not being recognized is at most [MATH], i.e. the same guarantee as the AMD codes.', '1605.04486-2-46-0': 'This is because ecDec is linear, so when noise [MATH] is added by the adversary to the codeword [MATH], effectively what happens is the decoding function [MATH], where [MATH] is the AMD-encoded message.', '1605.04486-2-46-1': 'But now [MATH] is an random string that is added to the AMD-encoded codeword.', '1605.04486-2-47-0': 'Silence In our algorithm, silence on the channel has a very specific meaning.', '1605.04486-2-47-1': 'We define the function [MATH] to return true iff the string [MATH] has fewer than [MATH] bit alternations.', '1605.04486-2-48-0': 'Other notation We use [MATH] to denote the [MATH]-bit string of all zeros, [MATH] for string concatenation and [MATH] to denote the function that returns the bits on the channel over the next [MATH] time steps.', '1605.04486-2-48-1': 'For the sake of convenience, we will use [MATH] to mean [MATH], unless specified otherwise.', '1605.04486-2-48-2': 'Let [MATH].', '1605.04486-2-49-0': '## Algorithm overview', '1605.04486-2-50-0': "[t] Alice's algorithm", '1605.04486-2-51-0': '[1] Alice[MATH] is a message of length [MATH] Field size [MATH] Degree of polynomial [MATH]degree-[MATH] polynomial encoding of [MATH] over [MATH] Send [MATH] Rounds for the algorithm [MATH] Message size in this round [MATH] Fingerprint from Bob [MATH] Send [MATH] Echo the fingerprint Send [MATH] if the fingerprint was not echoed.', '1605.04486-2-51-1': '[MATH] Bob has likely left [MATH]polynomial evaluation tuples of [MATH] at next two points of the field (cyclically) Send [MATH]', '1605.04486-2-52-0': "[t] Bob's algorithm", '1605.04486-2-53-0': "[1] Bob[MATH] Field size [MATH] Degree of polynomial [MATH] Listen to first [MATH] evaluations from Alice Add the corresponding polynomial evaluation tuples to [MATH] Message size in this round [MATH] Sample a string [MATH] Send [MATH] Send Alice the fingerprint of the polynomial [MATH] Listen to Alice's echo [MATH] Send a string [MATH] Receive polynomial evaluation tuples for the next two field elements and add to [MATH]", '1605.04486-2-54-0': 'Our algorithm for the case when [MATH] is known is given in two parts: Algorithm [REF] is what Alice follows and Algorithm [REF] is what Bob follows.', '1605.04486-2-54-1': 'Both algorithms assume knowledge of the message length [MATH] and the error tolerance [MATH].', '1605.04486-2-54-2': 'The idea is for Alice to compute a degree-[MATH] polynomial encoding of [MATH] over a field of size [MATH].', '1605.04486-2-54-3': 'Here [MATH] and [MATH].', '1605.04486-2-54-4': 'She begins by sending evaluations of this polynomial over the first [MATH] field elements to Bob in plaintext, which Bob uses to reconstruct the polynomial and retrieve the message.', '1605.04486-2-54-5': 'He also computes a fingerprint of this polynomial and sends it back to Alice.', '1605.04486-2-54-6': 'He encodes this fingerprint with AMD encoding and then ECC encoding, so that any successful tampering will require at least a third of the bits in the encoded fingerprint to be flipped and will be detected with high probability.', '1605.04486-2-54-7': 'If Alice receives a correct fingerprint, she echoes it back to Bob.', '1605.04486-2-54-8': 'Upon listening to this echo, Bob terminates.', '1605.04486-2-54-9': 'The channel from Bob to Alice is now silent, after incepting which Alice terminates the protocol as well.', '1605.04486-2-55-0': "If the adversary flips bits on the channel so that Bob's fingerprint mismatches, Alice recognizes this mismatch with high probability and exchanges more evaluations of her polynomial with Bob, proceeding in rounds.", '1605.04486-2-55-1': 'In each round, Alice sends two more evaluations of the polynomial on the next two field elements and sends them to Bob.', '1605.04486-2-55-2': 'Bob uses these to reconstruct his polynomial and sends a fingerprint back to Alice.', '1605.04486-2-55-3': 'The next round only begins if Alice did not terminate in this round, which will require this fingerprint to match and for Alice to intercept silence after Bob has terminated.', '1605.04486-2-55-4': 'We will bound the number of rounds and the failure probability for our algorithm in the next section.', '1605.04486-2-56-0': '## Example Run', '1605.04486-2-57-0': 'We now discuss an example of a run of our protocol to make the different steps in the algorithm more clear.', '1605.04486-2-57-1': 'We illustrate this example in Fig. [REF] and provide a step-by-step explanation below.', '1605.04486-2-58-0': 'Alice begins by computing a polynomial [MATH] corresponding to the message and sends its evaluation on the first [MATH] field elements to Bob, in plaintext.', '1605.04486-2-58-1': 'The adversary now corrupts one of the evaluation tuples so that the polynomial [MATH] that Bob reconstructs is different than [MATH].', '1605.04486-2-58-2': 'Bob computes the fingerprint of this polynomial, depicted [MATH] for brevity, and sends it to Alice.', '1605.04486-2-58-3': 'Alice compares this fingerprint against the hash of her own polynomial, [MATH], and notices a mismatch.', '1605.04486-2-58-4': 'In response, Alice remains silent.', '1605.04486-2-58-5': 'Bob is now convinced that his version of the polynomial is incorrect, so he sends noise to Alice to ask her for a resend.', '1605.04486-2-58-6': 'Alice encodes two more evaluations of [MATH] at the next two field elements and sends them to Bob.', '1605.04486-2-58-7': 'The adversary tries to tamper with these evaluations by flipping some bits.', '1605.04486-2-58-8': 'For this example, we assume that he flips fewer than a third of the total number of bits in the encoded evaluations.', '1605.04486-2-58-9': 'Upon decoding, Bob is able to successfully recover both the evaluations and uses the [MATH] subroutine to recompute [MATH], which in this case matches [MATH].', '1605.04486-2-58-10': 'Bob computes [MATH] and sends it to Alice.', '1605.04486-2-58-11': 'Upon seeing this hash and verifying that it matches [MATH], Alice is now convinced that Bob has the correct copy of the polynomial, and hence, the original message.', '1605.04486-2-58-12': 'Alice echoes the hash back to Bob, upon hearing which Bob extracts the message from the polynomial (using its coefficients) and terminates the protocol.', '1605.04486-2-58-13': 'Silence follows on the channel from Bob to Alice.', '1605.04486-2-58-14': 'Alice intercepts silence and terminates the protocol as well.', '1605.04486-2-59-0': 'The message has now successfully been transmitted from Alice to Bob.', '1605.04486-2-60-0': '# Analysis', '1605.04486-2-61-0': 'We now prove that our algorithm is correct with probability at least [MATH], and compute the number of bits sent.', '1605.04486-2-61-1': 'Before proceeding to the proof, we define three bad events:', '1605.04486-2-62-0': 'Unintentional Silence.', '1605.04486-2-62-1': 'When Bob executes step 18 of his algorithm, the string received by Alice is interpreted as silence.', '1605.04486-2-62-2': 'Fingerprint Error.', '1605.04486-2-62-3': 'Fingerprint hash collision as per Theorem [REF].', '1605.04486-2-62-4': 'AMD Error.', '1605.04486-2-62-5': 'The adversary corrupts an AMD encoded message into an encoding of a different message.', '1605.04486-2-63-0': 'Rounds For both Alice and Bob, we define a round as one iteration of the for loop in our algorithm.', '1605.04486-2-63-1': 'We refer to the part of the algorithm before the for loop begins as round [MATH].', '1605.04486-2-63-2': 'The AMD encoding strength [MATH] is equal to [MATH] initially and decreases by a factor of [MATH] every [MATH] rounds.', '1605.04486-2-63-3': 'This way, the number of bits added to the messages increases linearly every [MATH] rounds, which enhances security against corruption.', '1605.04486-2-64-0': '## Correctness and Termination', '1605.04486-2-65-0': "We now prove that with probability at least [MATH], Bob terminates the algorithm with the correct guess of Alice's message.", '1605.04486-2-66-0': '### Unintentional Silence The following lemmas show that Alice terminates before Bob with probability at most [MATH].', '1605.04486-2-67-0': 'For [MATH], the probability that a [MATH]-bit string sampled uniformly at random from [MATH] has fewer than [MATH] bit alternations is at most [MATH].', '1605.04486-2-68-0': 'Let [MATH] be a string sampled uniformly at random from [MATH], where [MATH].', '1605.04486-2-68-1': 'Denote by [MATH] the [MATH] bit of [MATH].', '1605.04486-2-68-2': 'Let [MATH] be the indicator random variable for the event that [MATH], for [MATH].', '1605.04486-2-68-3': "Note that all [MATH]'s are mutually independent.", '1605.04486-2-68-4': 'Let [MATH] be the number of bit alternations in [MATH].', '1605.04486-2-68-5': 'Clearly, [MATH], which gives [MATH], using the linearity of expectation.', '1605.04486-2-68-6': 'Since [MATH] for all [MATH], we get [MATH].', '1605.04486-2-68-7': 'Using the multiplicative version of Chernoff bounds [CITATION] for [MATH], [EQUATION]', '1605.04486-2-68-8': 'To obtain [MATH], set [MATH] to get, [EQUATION]', '1605.04486-2-69-0': 'Alice terminates the algorithm before Bob with probability at most [MATH].', '1605.04486-2-70-0': 'Let [MATH] be the event that Alice terminates before Bob.', '1605.04486-2-70-1': 'This happens when the string sent by Bob in step 18 after possible adversarial corruptions is interpreted as silence by Alice.', '1605.04486-2-70-2': 'Let [MATH] be the event that Alice terminates before Bob in round [MATH] of the algorithm.', '1605.04486-2-70-3': 'Then, using a union bound over the rounds, the fact that [MATH] and Lemma [REF], we get [EQUATION]', '1605.04486-2-70-4': 'Note that Lemma [REF] is applicable here because for each [MATH], we have [MATH].', '1605.04486-2-70-5': 'To see this, use the fact that [MATH] and [MATH] to obtain the condition [MATH], which is always true because [MATH].', '1605.04486-2-71-0': '### Fingerprint Failure The following lemma proves that the fingerprint error happens with probability at most [MATH], ensuring the correctness of the algorithm.', '1605.04486-2-72-0': "Upon termination, Bob does not have the correct guess of Alice's message with probability at most [MATH].", '1605.04486-2-73-0': "Let [MATH] be the event that Bob does not have the correct guess of Alice's message upon termination.", '1605.04486-2-73-1': 'Note that in round [MATH], from Theorem [REF], the fingerprints fail with probability at most [MATH].', '1605.04486-2-73-2': 'Using a union bound over these rounds, we get [EQUATION]', '1605.04486-2-74-0': '### AMD Failure', '1605.04486-2-75-0': 'The probability of AMD failure is at most [MATH].', '1605.04486-2-76-0': 'Note that in round [MATH], from Theorem [REF], AMD failure occurs with probability at most [MATH].', '1605.04486-2-76-1': 'Hence, using a union bound over the rounds, the AMD failure occurs with probability [MATH].', '1605.04486-2-77-0': '## Probability of Failure', '1605.04486-2-78-0': 'Our algorithm succeeds with probability at least [MATH].', '1605.04486-2-79-0': 'Lemmas [REF], [REF] and [REF] ensure that the three bad events, as defined previously, each happen with probability at most [MATH].', '1605.04486-2-79-1': 'Hence, using a union bound over the occurrence of these three events, the total probability of failure of the algorithm is at most [MATH].', '1605.04486-2-79-2': 'If the three bad events do not occur, then Alice will continue to send evaluations of the polynomial until Bob has the correct message.', '1605.04486-2-79-3': 'Since [MATH] is finite, Bob will eventually have the correct message and terminate.', '1605.04486-2-80-0': '## Cost to the algorithm', '1605.04486-2-81-0': 'Recall that Alice and Bob compute their polynomials [MATH] and [MATH], respectively, over [MATH].', '1605.04486-2-81-1': 'We refer to every [MATH] that Bob stores after receiving the evaluation [MATH], that has potentially been tampered with, of the polynomial [MATH] at [MATH] from Alice as a polynomial evaluation tuple.', '1605.04486-2-81-2': "We call a polynomial evaluation tuple [MATH] in Bob's set [MATH] good if [MATH] and bad otherwise.", '1605.04486-2-82-0': 'We begin by stating two important lemmas that relate the number of bits flipped by the adversary to make [MATH] polynomial evaluation tuples bad to the number of bits required to send them.', '1605.04486-2-83-0': 'Let [MATH] be the number of bits flipped by the adversary to make [MATH] polynomial evaluation tuples bad.', '1605.04486-2-83-1': 'Then, [MATH] if [MATH], and [EQUATION] otherwise.', '1605.04486-2-84-0': 'Let [MATH], where [MATH] is the number of polynomial evaluation tuples that were not encoded and [MATH] is the number of AMD and error-encoded polynomial evaluation tuples.', '1605.04486-2-84-1': 'Clearly, [MATH].', '1605.04486-2-84-2': 'Each of the remaining [MATH] polynomial evaluation tuples are sent in pairs, one pair per round.', '1605.04486-2-84-3': 'Since the adversary needs to flip at least a third of the number of bits for each encoded polynomial evaluation tuple to make it bad, we have [EQUATION]', '1605.04486-2-84-4': 'Since the number of bits per polynomial evaluation tuple increases monotonically, the expression above becomes [MATH] if [MATH], and [EQUATION] otherwise.', '1605.04486-2-85-0': 'Let [MATH] be the number of bits required to send [MATH] polynomial evaluation tuples, where [MATH].', '1605.04486-2-85-1': 'Then, [EQUATION]', '1605.04486-2-86-0': 'If [MATH], then we have [MATH], since each of these [MATH] polynomial evaluation tuples is of length [MATH].', '1605.04486-2-86-1': 'For [MATH], taking into account the fact that each round involves exchange of at most [MATH] messages between Alice and Bob, we get [EQUATION]', '1605.04486-2-87-0': 'Let [MATH], and [MATH] be any round at the end of which [MATH].', '1605.04486-2-87-1': 'Then the number of bad polynomial evaluation tuples through round [MATH] is at least [MATH].', '1605.04486-2-88-0': 'We call a field element [MATH] good if [MATH], and bad otherwise.', '1605.04486-2-88-1': 'Let [MATH] be the number of good field elements and [MATH] be the number of bad field elements up to round [MATH].', '1605.04486-2-88-2': 'Similarly, let [MATH] be the number of good polynomial evaluation tuples and [MATH] be the number of bad polynomial evaluation tuples up to round [MATH].', '1605.04486-2-88-3': 'Then, from Theorem [REF], we must have [MATH].', '1605.04486-2-88-4': 'Note that the total number of field elements for which Bob has received polynomial evaluation tuples from Alice through round [MATH] is [MATH].', '1605.04486-2-88-5': 'Adding this equality to the previous inequality, we have [EQUATION]', '1605.04486-2-88-6': 'The total number of polynomial evaluation tuples received by Bob up to round [MATH] is given by [EQUATION]', '1605.04486-2-88-7': 'Note that every bad field element is associated with at least [MATH] polynomial evaluation tuples.', '1605.04486-2-88-8': 'This gives [MATH].', '1605.04486-2-88-9': 'Using this inequality with Eqs. [REF] and [REF], we have [EQUATION]', '1605.04486-2-88-10': 'Case I: [MATH] For this case, we have [EQUATION]', '1605.04486-2-88-11': 'Case II: [MATH] For this case, we have [EQUATION] where the last inequality holds since [MATH] for [MATH].', '1605.04486-2-89-0': 'Combining Eqs. [REF] and [REF], we get [MATH].', '1605.04486-2-90-0': 'We now state a lemma that is crucial to the proof of Theorem [REF].', '1605.04486-2-91-0': 'If Bob terminates before Alice, the total number of bits sent by our algorithm is [EQUATION].', '1605.04486-2-92-0': 'Let [MATH] be the last round at the end of which [MATH], or [MATH] if [MATH] at the end of round [MATH] and for all subsequent rounds.', '1605.04486-2-92-1': 'Let [MATH] be the number of bits corrupted by the adversary through round [MATH].', '1605.04486-2-92-2': 'Let [MATH] represent the total cost through round [MATH] and [MATH] be the cost of the algorithm after round [MATH].', '1605.04486-2-92-3': "Note that after round [MATH], the adversary must corrupt one of either (1) the fingerprint, or (2) its echo, or (3) silence on the channel in Step 15 of Alice's algorithm, in every round to delay termination.", '1605.04486-2-92-4': 'Also, after round [MATH], Alice and Bob must exchange at least a fingerprint and an echo even if [MATH].', '1605.04486-2-92-5': 'Thus, we have, [EQUATION]', '1605.04486-2-93-0': 'Recall that the number of polynomial evaluation tuples sent up to round [MATH] is [MATH].', '1605.04486-2-93-1': 'Then, from Lemma [REF], we have [EQUATION]', '1605.04486-2-94-0': 'From Lemma [REF], we have that the number of bad polynomial evaluation tuples is at least [MATH].', '1605.04486-2-94-1': 'Thus, from Lemma [REF], we have [MATH], which implies [MATH] if [MATH].', '1605.04486-2-94-2': 'Otherwise, we have [EQUATION]', '1605.04486-2-95-0': 'Case I : [MATH] Since [MATH] is at least the number of bad polynomial evaluation tuples, from Lemma [REF], we have [MATH], which gives [MATH].', '1605.04486-2-95-1': 'Hence, using Eq [REF], we get, [EQUATION] where the last equality holds because [MATH].', '1605.04486-2-96-0': 'Case II : [MATH] From Eq. [REF], we have [EQUATION]', '1605.04486-2-96-1': 'Since each summand in the inequality above is positive and [MATH], we get [MATH], which gives [EQUATION]', '1605.04486-2-96-2': 'Since [MATH], we have [MATH].', '1605.04486-2-96-3': 'Building on this, we get, [EQUATION]', '1605.04486-2-96-4': 'Hence, from Eqs. [REF], [REF] and [REF] , we get [EQUATION] where the last equality holds because [MATH] and [MATH] from inequality [REF].', '1605.04486-2-97-0': 'Combining Eqs. [REF], [REF] and [REF], the total number of bits sent by the algorithm becomes [EQUATION]', '1605.04486-2-98-0': 'Putting it all together, we are now ready to state our main theorem.', '1605.04486-2-99-0': 'Our algorithm tolerates an unknown number of adversarial errors, [MATH], and for a given [MATH], succeeds with probability at least [MATH], and sends [MATH] bits.', '1605.04486-2-100-0': 'By Lemmas [REF], with probability at least [MATH], Bob terminates before Alice with the correct message.', '1605.04486-2-100-1': 'If this happens, then by Lemma [REF], the total number of bits sent is [EQUATION]', '1605.04486-2-101-0': '# Unknown [MATH]', '1605.04486-2-102-0': 'We now discuss an algorithm for the case when the message length [MATH] is unknown to Bob.', '1605.04486-2-102-1': 'The only parameter now known to both Alice and Bob is [MATH].', '1605.04486-2-103-0': 'Our main idea is to make use of an algorithm from [CITATION], which enables Alice to send a message of unknown length to Bob in our model, but is inefficient.', '1605.04486-2-103-1': 'We thus use a two phase approach.', '1605.04486-2-103-2': 'First, we send the length of the message [MATH] (i.e. a total of [MATH] bits) from Alice to Bob using the algorithms of [CITATION].', '1605.04486-2-103-3': 'Second, once Bob learns the value [MATH], we use the algorithm from Section [REF] to communicate the message [MATH].', '1605.04486-2-103-4': 'We will show that the total number of bits sent by this two phase algorithm is asymptotically similar to the case when the message length is known by Bob in advance.', '1605.04486-2-104-0': '## Algorithm Overview', '1605.04486-2-105-0': 'Let [MATH] be a noise-free protocol in which Alice sends [MATH] to Bob, who is unaware of the length ([MATH] in this case) of the message.', '1605.04486-2-105-1': 'Let [MATH] be a noise-free protocol in which Alice sends [MATH] to Bob, who knows the length [MATH] a priori.', '1605.04486-2-105-2': 'W can write the noise-free protocol [MATH] to communicate [MATH] from Alice to Bob, who does not know [MATH], as a composition of [MATH] and [MATH] in this order.', '1605.04486-2-105-3': 'Let [MATH] and [MATH] be the simulations of [MATH] and [MATH], respectively, that are robust to adversarial bit flipping.', '1605.04486-2-106-0': 'To simulate [MATH] with desired error probability [MATH], we proceed in two steps.', '1605.04486-2-106-1': 'We first make [MATH] robust with [MATH] error tolerance using Algorithm [MATH] from [CITATION], setting [MATH].', '1605.04486-2-106-2': 'Then, we make [MATH] robust with [MATH] error tolerance using Algorithms [REF] and [REF].', '1605.04486-2-106-3': 'This way, when we compose the robust versions of [MATH] and [MATH], we get [MATH] with error probability at most [MATH] (by union bound).', '1605.04486-2-106-4': 'The correctness of [MATH] immediately follows from the correctness of [MATH] and [MATH], by construction.', '1605.04486-2-107-0': '## Probability of Failure', '1605.04486-2-108-0': 'The failure events for [MATH] are exactly the failure events for [MATH] and [MATH].', '1605.04486-2-108-1': 'In other words, we say [MATH] fails when one or both of [MATH] and [MATH] fail.', '1605.04486-2-108-2': 'Thus, the failure probability of [MATH] is at most [MATH], by a simple union bound over the two sub-protocols.', '1605.04486-2-109-0': '## Number of bits sent', '1605.04486-2-110-0': 'To analyze the number of bits sent, let [MATH] be the number of bits flipped by the adversary in [MATH] and [MATH] be the number of bits flipped by the adversary in [MATH].', '1605.04486-2-110-1': 'Recall that the length of the message from Alice to Bob in [MATH] is [MATH] and that in [MATH] is [MATH].', '1605.04486-2-110-2': 'Let [MATH] be the number of bits sent in [MATH] and [MATH] be the number of bits sent in [MATH].', '1605.04486-2-110-3': 'Thus, using Theorem [MATH] from [CITATION] (with [MATH] and [MATH]), we get [EQUATION]', '1605.04486-2-110-4': 'Similarly, using Theorem [REF] from this paper (with [MATH]), we get [EQUATION]', '1605.04486-2-110-5': 'Using [MATH], the total number of bits sent by [MATH] is then [MATH].', '1605.04486-2-110-6': 'The proof of Theorem [REF] now follows directly from the above analysis.', '1605.04486-2-111-0': 'Note that another approach to sending a message of unknown length from Alice to Bob would have been to directly use the algorithm in [CITATION] with [MATH].', '1605.04486-2-111-1': 'However, this would have incurred a higher blowup than the approach that we take in this paper.', '1605.04486-2-111-2': 'More specifically, when [MATH] is small, the direct use of the multiparty algorithm gives a multiplicative logarithmic blowup in the number of bits, while our current approach maintains the constant overall blowup in the number of bits by using the heavy weight protocol for the length of the message instead (which is exponentially smaller than the message).', '1605.04486-2-112-0': '# Conclusion', '1605.04486-2-113-0': 'We have described an algorithm for sending a message over a two-way noisy channel.', '1605.04486-2-113-1': 'Our algorithm is robust to an adversary that can flip an unknown but finite number of bits on the channel.', '1605.04486-2-113-2': 'The adversary knows our algorithm and the message to be sent, but does not know the random bits of the sender and receiver, nor the bits sent over the channel.', '1605.04486-2-113-3': 'The receiver of the message does not know the message length in advance.', '1605.04486-2-114-0': 'Assume the message length is [MATH], the number of bits flipped by the adversary is [MATH], and [MATH] is an error parameter known to both players.', '1605.04486-2-114-1': 'Then our algorithm sends an expected number of bits that is [MATH], and succeeds with probability at least [MATH].', '1605.04486-2-114-2': 'When [MATH] and [MATH] is polynomially small in [MATH], the number of bits sent is [MATH], which is asymptotically optimal; and when [MATH], the number of bits sent is [MATH].', '1605.04486-2-115-0': 'Many open problems remain including the following.', '1605.04486-2-115-1': 'First, Can we determine asymptotically matching upper and lower bounds on the number of bits required for our problem?', '1605.04486-2-115-2': 'Our current algorithm is optimal for [MATH], and seems close to optimal for [MATH], but is it optimal for intermediate values of [MATH]?', '1605.04486-2-115-3': 'Second, Can we tolerate a more powerful adversary or different types of adversaries?', '1605.04486-2-115-4': 'For example, it seems like our current algorithm can tolerate a completely omniscient adversary, if that adversary can only flip a chosen bit with some probability that is [MATH] for some fixed [MATH].', '1605.04486-2-115-5': 'Finally, can we extend our result to the problem of sending our message from a source to a target in an arbitrary network where nodes are connected via noisy two-way channels?', '1605.04486-2-115-6': 'This final problem seems closely related to the problem of network coding [CITATION], for the case where the amount of noise and the message size is not known in advance.', '1605.04486-2-115-7': 'In this final problem, since there are multiple nodes, we would likely also need to address problems of asynchronous communication.'}

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[]

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{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1605.04486

hep-th-0512221

{'hep-th-0512221-1-0-0': 'A pedagogical introduction to the heat kernel technique, [MATH]-function and Casimir effect is presented.', 'hep-th-0512221-1-0-1': 'Several applications are considered.', 'hep-th-0512221-1-0-2': 'First we derive the high temperature asymptotics of the free energy for boson fields in terms of the heat kernel expansion and [MATH]-function.', 'hep-th-0512221-1-0-3': 'Another application is chiral anomaly for local (MIT bag) boundary conditions.', 'hep-th-0512221-1-0-4': 'Then we rederive the Casimir energies for perfectly conducting cavities using a convenient new technique, main results are presented in a novel form.', 'hep-th-0512221-1-0-5': 'A geometric interpretation of the results for the cavities is suggested.', 'hep-th-0512221-1-1-0': '# Introduction', 'hep-th-0512221-1-2-0': 'The main problem of the quantum field theory with boundaries is its renormalization and physical meaning of the results obtained.', 'hep-th-0512221-1-2-1': 'Divergences that appear in quantum field theory make the problems on manifolds with boundaries more complicated than in infinite space.', 'hep-th-0512221-1-3-0': 'In the presence of boundaries or singularities the heat kernel technique is an effective tool for the analysis of the one loop effects (see reviews [CITATION], [CITATION]).', 'hep-th-0512221-1-3-1': 'Different applications of the heat kernel expansion exist.', 'hep-th-0512221-1-3-2': 'The heat kernel technique seems to be the easiest way for the calculation of quantum anomalies, calculation of effective actions based on finite-mode regularization and analysis of divergences in quantum field theory.', 'hep-th-0512221-1-4-0': 'Chiral anomaly, which was discovered more than 35 years ago [CITATION], still plays an important role in physics.', 'hep-th-0512221-1-4-1': 'On smooth manifolds without boundaries many successful approaches to the anomalies exist [CITATION].', 'hep-th-0512221-1-4-2': 'The heat kernel approach to the anomalies is essentially equivalent to the Fujikawa approach [CITATION] and to the calculations based on the finite-mode regularization [CITATION], but it can be more easily extended to complicated geometries.', 'hep-th-0512221-1-4-3': 'The local chiral anomaly in the case of non-trivial boundary conditions (MIT bag boundary conditions) has been calculated only recently [CITATION].', 'hep-th-0512221-1-5-0': 'Casimir effect [CITATION] is a macroscopic quantum effect.', 'hep-th-0512221-1-5-1': 'Briefly speaking, if we impose classical boundary conditions on a quantum field on some boundary surface than we get the Casimir effect.', 'hep-th-0512221-1-5-2': 'There are several different physical situations that should be distinguished in the Casimir effect.', 'hep-th-0512221-1-6-0': 'Suppose there are two spatially separated dielectrics, then in a dilute limit ([MATH]) the Casimir energy of this system is equal to the energy of pairwise interactions between dipoles of these two dielectrics via a Casimir-Polder retarded potential [CITATION].', 'hep-th-0512221-1-6-1': 'For a general case of separated dielectrics the Casimir energy can be calculated as in [CITATION] or [CITATION] (for a recent discussion of these issues see [CITATION] and a review [CITATION]).', 'hep-th-0512221-1-7-0': 'A different situation takes place when there is a dilute dielectric ball or any other simply connected dielectric under study (see a review [CITATION] for a discussion of related subjects and methods used).', 'hep-th-0512221-1-7-1': 'As it was pointed out in [CITATION] and then discussed in detail in [CITATION], microscopic interatomic distances should be taken into account to calculate the Casimir energy of a dilute dielectric ball.', 'hep-th-0512221-1-7-2': 'The average interatomic distance [MATH] serves as an effective physical cut off for simply connected dielectrics.', 'hep-th-0512221-1-8-0': 'The limit of perfect conductivity ([MATH]) is opposite to a dilute case.', 'hep-th-0512221-1-8-1': 'This is the strong coupling limit of the theory.', 'hep-th-0512221-1-8-2': 'Any results obtained in this limit are nonperturbative ones.', 'hep-th-0512221-1-9-0': 'The Casimir energy of a perfectly conducting rectangular cavity was first calculated in [CITATION] using exponential regularization.', 'hep-th-0512221-1-9-1': 'Later it was derived by some other methods (see references and numerical analysis in [CITATION], also a review [CITATION]).', 'hep-th-0512221-1-9-2': 'The Casimir energy changes its sign for different side lengths of the cavity.', 'hep-th-0512221-1-9-3': 'The Casimir energy of a rectangular cavity is an experimentally measurable quantity (we hope that this will be done in future experiments, see e.g. [CITATION]).', 'hep-th-0512221-1-10-0': 'In the present paper we derive the Casimir energy for rectangular cavities at zero temperature by a new method described in Sec. [MATH].', 'hep-th-0512221-1-10-1': 'What is interesting, by use of this method we could rewrite the Casimir energy for rectangular cavities in the form that makes transparent its geometric interpretation.', 'hep-th-0512221-1-11-0': 'The paper is organized as follows.', 'hep-th-0512221-1-11-1': 'In Sec. [MATH] we give an introduction to the formalism of the heat kernel and heat kernel expansion.', 'hep-th-0512221-1-11-2': 'Also we introduce a [MATH]-function and calculate the one loop effective action in terms of [MATH]- function.', 'hep-th-0512221-1-11-3': 'Then we consider two examples.', 'hep-th-0512221-1-11-4': 'First we derive the high temperature expansion of the free energy for boson fields [CITATION] in terms of the heat kernel expansion and [MATH]-function.', 'hep-th-0512221-1-11-5': 'Then we derive a chiral anomaly in four dimensions for an euclidean version of the MIT bag boundary conditions [CITATION].', 'hep-th-0512221-1-11-6': 'Sec. [MATH] is devoted to the Casimir energies of rectangular cavities.', 'hep-th-0512221-1-11-7': 'We introduce a regularization and a convenient new method of calculations using an example of two prefectly conducting parallel plates, then apply it to more complicated rectangular geometries.', 'hep-th-0512221-1-11-8': 'We present the Casimir energy of the cavity in the form ([REF]).', 'hep-th-0512221-1-11-9': 'Then we discuss an argument principle and [MATH]-functional regularization for the cavity.', 'hep-th-0512221-1-11-10': 'In Sec. [MATH] we rewrite the Casimir energy of the cavity in terms of geometric optics ([REF]).', 'hep-th-0512221-1-11-11': 'Finally we describe a possible experiment to verify the Casimir energies for perfectly conducting cavities and give the physical meaning of the force that can be measured.', 'hep-th-0512221-1-12-0': '# Spectral techniques', 'hep-th-0512221-1-13-0': '## Heat kernel', 'hep-th-0512221-1-14-0': 'Consider a second order elliptic partial differential operator [MATH] of Laplace type on an n-dimensional Riemannian manifold.', 'hep-th-0512221-1-14-1': 'Any operator of this type can be expanded locally as [EQUATION] where [MATH] and [MATH] are some matrix valued functions and [MATH] is the inverse metric tensor on the manifold.', 'hep-th-0512221-1-14-2': 'For a flat space [MATH].', 'hep-th-0512221-1-15-0': 'The heat kernel can be defined as follows: [EQUATION] where [MATH] is an eigenfunction of the operator [MATH] with the eigenvalue [MATH].', 'hep-th-0512221-1-16-0': 'It satisfies the heat equation [EQUATION] with an initial condition [EQUATION]', 'hep-th-0512221-1-16-1': 'If we consider the fields in a finite volume then it is necessary to specify boundary conditions.', 'hep-th-0512221-1-16-2': 'Different choices are possible.', 'hep-th-0512221-1-16-3': 'In section [MATH] we will consider the case of periodic boundary conditions on imaginary time coordinate, which are specific for boson fields.', 'hep-th-0512221-1-16-4': 'In section [MATH] we will study bag boundary conditions imposed on fermion fields.', 'hep-th-0512221-1-16-5': 'If the normal to the boundary component of the fermion current [MATH] vanishes at the boundary, one can impose bag boundary conditions, a particular case of mixed boundary conditions.', 'hep-th-0512221-1-16-6': 'We assume given two complementary projectors [MATH], [MATH] acting on a multi component field (the eigenfunction of the operator [MATH]) at each point of the boundary and define mixed boundary conditions by the relations [EQUATION] where [MATH] is a matrix valued function on the boundary.', 'hep-th-0512221-1-16-7': 'In other words, the components [MATH] satisfy Dirichlet boundary conditions, and [MATH] satisfy Robin (modified Neumann) ones.', 'hep-th-0512221-1-17-0': 'It is convenient to define [EQUATION]', 'hep-th-0512221-1-17-1': 'Let [MATH], [MATH] be a local orthonormal frame for the tangent space to the manifold and let on the boundary [MATH] be an inward pointing normal vector.', 'hep-th-0512221-1-18-0': 'The extrinsic curvature is defined by the equation [EQUATION] where [MATH] is the Christoffel symbol.', 'hep-th-0512221-1-18-1': 'For example, on the unit sphere [MATH] which bounds the unit ball in [MATH] the extrinsic curvature is [MATH].', 'hep-th-0512221-1-19-0': 'Curved space offers no complications in our approach compared to the flat case.', 'hep-th-0512221-1-19-1': 'Let [MATH] be the Riemann tensor, and let [MATH] be the Ricci tensor.', 'hep-th-0512221-1-19-2': 'With our sign convention the scalar curvature [MATH] is [MATH] on the unit sphere [MATH].', 'hep-th-0512221-1-19-3': 'In flat space the Riemann and Ricci tensors are equal to zero.', 'hep-th-0512221-1-20-0': 'One can always introduce a connection [MATH] and another matrix valued function [MATH] so that [MATH] takes the form: [EQUATION]', 'hep-th-0512221-1-20-1': 'Here [MATH] is a sum of covariant Riemannian derivative with respect to metric [MATH] and connection [MATH].', 'hep-th-0512221-1-20-2': 'One can, of course, express [MATH] and [MATH] in terms of [MATH], [MATH] and [MATH]: [EQUATION]', 'hep-th-0512221-1-20-3': 'For the future use we introduce also the field strength for [MATH]: [EQUATION]', 'hep-th-0512221-1-20-4': 'The connection [MATH] will be used to construct covariant derivatives.', 'hep-th-0512221-1-20-5': 'The subscript [MATH] will be used to denote repeated covariant derivatives with the connection [MATH] and the Christoffel connection on [MATH].', 'hep-th-0512221-1-20-6': 'The subscript [MATH] will denote repeated covariant derivatives containing [MATH] and the Christoffel connection on the boundary.', 'hep-th-0512221-1-20-7': 'Difference between these two covariant derivatives is measured by the extrinsic curvature ([REF]).', 'hep-th-0512221-1-20-8': 'For example, [MATH].', 'hep-th-0512221-1-21-0': 'Let us define an integrated heat kernel for a hermitian operator [MATH] by the equation: [EQUATION] where [MATH] is an hermitian matrix valued function, [MATH] here is over matrix indices.', 'hep-th-0512221-1-21-1': 'For the boundary conditions we consider in this paper there exists an asymptotic expansion [CITATION] as [MATH]: [EQUATION]', 'hep-th-0512221-1-21-2': 'According to the general theory [CITATION] the coefficients [MATH] are locally computable.', 'hep-th-0512221-1-21-3': 'This means that each [MATH] can be represented as a sum of volume and boundary integrals of local invariants constructed from [MATH], [MATH], [MATH], the curvature tensor, and their derivatives.', 'hep-th-0512221-1-21-4': 'Boundary invariants may also include [MATH], [MATH] and [MATH].', 'hep-th-0512221-1-21-5': 'Total mass dimension of such invariants should be [MATH] for the volume terms and [MATH] for the boundary ones.', 'hep-th-0512221-1-22-0': 'At the moment several coefficients of the expansion ([REF]) are known for the case of mixed boundary conditions ([REF]) and matrix valued function [MATH] (see [CITATION] for details of derivation; the formula ([REF]) for [MATH] was derived in [CITATION] with additional restrictions [MATH] and [MATH]) : [EQUATION]', 'hep-th-0512221-1-22-1': 'For a scalar function [MATH] and mixed boundary conditions the coefficients [MATH] and [MATH] were already derived [CITATION].', 'hep-th-0512221-1-23-0': '## [MATH]-function', 'hep-th-0512221-1-24-0': 'Zeta function of a positive operator [MATH] is defined by [EQUATION] where the sum is over all eigenvalues of the operator [MATH].', 'hep-th-0512221-1-24-1': 'The zeta function is related to the heat kernel by the transformation [EQUATION]', 'hep-th-0512221-1-24-2': 'Residues at the poles of the zeta function are related to the coefficients of the heat kernel expansion: [EQUATION]', 'hep-th-0512221-1-24-3': 'Here [MATH] is a unit matrix with a dimension of the matrix functions [MATH] in ([REF]).', 'hep-th-0512221-1-24-4': 'From ([REF]) it follows that [EQUATION]', 'hep-th-0512221-1-24-5': 'In Euclidean four dimensional space the zero temperature one-loop path integral over the boson fields [MATH] can be evaluated as follows (up to a normalization factor): [EQUATION]', 'hep-th-0512221-1-24-6': 'Here we introduced the constant [MATH] with a dimension of mass in order to keep a proper dimension of the measure in the functional integral.', 'hep-th-0512221-1-24-7': '[MATH] can be thought of as a number of eigenvalues of the operator [MATH].', 'hep-th-0512221-1-24-8': 'For the operator [MATH] in the form ([REF]) the number of eigenvalues is infinite, so [MATH] yields a regularized value for this number.', 'hep-th-0512221-1-25-0': 'The zero temperature one-loop effective action is defined then by [EQUATION]', 'hep-th-0512221-1-25-1': 'The term [MATH] in the effective action [MATH] determines the one-loop beta function, this term describes renormalization of the one-loop logarithmic divergences appearing in the theory.', 'hep-th-0512221-1-26-0': '## Free energy for boson fields', 'hep-th-0512221-1-27-0': 'A finite temperature field theory is defined in Euclidean space, since for boson fields one has to impose periodic boundary conditions on imaginary time coordinate (antiperiodic boundary conditions for fermion fields respectively).', 'hep-th-0512221-1-27-1': 'A partition function is defined by [EQUATION] where [MATH] is a hamiltonian of the problem and [MATH].', 'hep-th-0512221-1-27-2': 'Let us choose the lagrangian density [MATH] in the form [EQUATION] where [MATH] is an imaginary time coordinate and [MATH] is a three dimensional spatial part of the density in the form ([REF]).', 'hep-th-0512221-1-27-3': 'The free energy of the system is defined by [EQUATION] the integration is over all periodic fields satisfying [MATH] is a normalization coefficient).', 'hep-th-0512221-1-27-4': 'As a result the eigenfunctions of [MATH] have the form [MATH], where [MATH] and [MATH].', 'hep-th-0512221-1-27-5': 'The free energy is thus equal to [CITATION] [EQUATION] where we introduced [MATH]-function [EQUATION] and the parameter [MATH] with a mass dimensionality in order to make the argument of the logarithm dimensionless (also see a previous section).', 'hep-th-0512221-1-28-0': 'Then it is convenient to use the formula [EQUATION] and separate [MATH] and other terms in the sum.', 'hep-th-0512221-1-28-1': 'For [MATH] terms we substitute the heat kernel expansion for the operator [MATH] at small [MATH] [EQUATION] and perform [MATH] integration, then we arrive at the high temperature expansion ([MATH]) for the free energy [MATH]: [EQUATION]', 'hep-th-0512221-1-28-2': 'Here [MATH], [MATH] is a Riemann zeta function, [MATH] is a zeta function of an operator [MATH], [MATH] is the Euler constant.', 'hep-th-0512221-1-28-3': 'The first two terms on the r.h.s. of ([REF]) follow from the [MATH] term.', 'hep-th-0512221-1-29-0': 'The term [EQUATION] is the leading high temperature contribution to the free energy.', 'hep-th-0512221-1-30-0': 'The classical limit terms due to the equality [MATH] can be rewritten as follows: [EQUATION]', 'hep-th-0512221-1-30-1': 'The terms on the l.h.s. of ([REF]) yield a renormalized value of the terms on the r.h.s. of ([REF]), since the sum on the righthandsight is generally divergent when the number of modes is infinite.', 'hep-th-0512221-1-31-0': 'The term with [MATH] determines the part of the free energy that appears due to one-loop logarithmic divergences and thus it depends on the dimensional parameter [MATH] as in the zero temperature case.', 'hep-th-0512221-1-32-0': '## Chiral anomaly in four dimensions', 'hep-th-0512221-1-33-0': 'for MIT bag boundary conditions', 'hep-th-0512221-1-34-0': 'Consider the Dirac operator on an [MATH]-dimensional Riemannian manifold [EQUATION] in external vector [MATH] and axial vector [MATH] fields.', 'hep-th-0512221-1-34-1': 'We suppose that [MATH] and [MATH] are anti-hermitian matrices in the space of some representation of the gauge group.', 'hep-th-0512221-1-34-2': '[MATH] is the spin-connection.', 'hep-th-0512221-1-35-0': 'The Dirac operator transforms covariantly under infinitesimal local gauge transformations (the local gauge transformation is [MATH]): [EQUATION] and under infinitesimal local chiral transformations (the local chiral transformation is [MATH]): [EQUATION]', 'hep-th-0512221-1-35-1': 'The parameters [MATH] and [MATH] are anti-hermitian matrices.', 'hep-th-0512221-1-36-0': 'First we adopt the zeta-function regularization and write the one-loop effective action for Dirac fermions at zero temperature as [EQUATION] where [EQUATION] prime denotes differentiation with respect to [MATH], and [MATH] is the functional trace.', 'hep-th-0512221-1-37-0': 'The following identity holds: [EQUATION]', 'hep-th-0512221-1-37-1': 'Due to the identity ([REF]) [EQUATION] so the effective action ([REF]) is gauge invariant, [MATH].', 'hep-th-0512221-1-38-0': 'The chiral anomaly is by definition equal to the variation of [MATH] under an infinitesimal chiral transformation.', 'hep-th-0512221-1-38-1': 'Using ([REF]) we obtain: [EQUATION] and the anomaly reads [EQUATION]', 'hep-th-0512221-1-38-2': 'The heat kernel is related to the zeta function by the Mellin transformation: [EQUATION]', 'hep-th-0512221-1-38-3': 'In particular, after the substitution of the heat kernel expansion ([REF]) into the formula ([REF]) we obtain [EQUATION]', 'hep-th-0512221-1-38-4': 'The same expression for the anomaly follows also from the Fujikawa approach [CITATION].', 'hep-th-0512221-1-39-0': "One can also derive the expression for the anomaly ([REF]) from Schwinger's effective action.", 'hep-th-0512221-1-39-1': 'One should start from an identity: [EQUATION]', 'hep-th-0512221-1-39-2': 'Then the change in the effective action due to chiral transformations can be written: [EQUATION]', 'hep-th-0512221-1-39-3': 'We impose local boundary conditions: [EQUATION] which are nothing else than a Euclidean version of the MIT bag boundary conditions [CITATION].', 'hep-th-0512221-1-39-4': 'For these boundary conditions [MATH], and the normal component of the fermion current [MATH] vanishes on the boundary.', 'hep-th-0512221-1-39-5': 'Spectral properties of the Dirac operator for bag boundary conditions are intensively studied [CITATION].', 'hep-th-0512221-1-40-0': 'Since [MATH] is a first order differential operator it was enough to fix the boundary conditions ([REF]) on a half of the components.', 'hep-th-0512221-1-40-1': 'To proceed with a second order operator [MATH] we need boundary conditions on the remaining components as well.', 'hep-th-0512221-1-40-2': 'They are defined by the consistency condition [CITATION]: [EQUATION] which is equivalent to the Robin boundary condition [EQUATION] with [EQUATION]', 'hep-th-0512221-1-40-3': 'In the paper [CITATION] the following expression for a coefficient [MATH] with an hermitian matrix valued function [MATH] and conditions ([REF]), [MATH] (flat boundaries), [MATH] was obtained: [EQUATION]', 'hep-th-0512221-1-40-4': 'To obtain the chiral anomaly in four dimensions with MIT bag boundary conditions one has to calculate the coefficient [MATH] ([REF]) with [MATH], [MATH] and substitute it into ([REF]).', 'hep-th-0512221-1-40-5': 'We define [MATH], [MATH], [MATH].', 'hep-th-0512221-1-40-6': 'The anomaly contains two contributions: [EQUATION]', 'hep-th-0512221-1-40-7': 'In the volume part [EQUATION] only the [MATH] terms seem to be new [CITATION] (for flat space it can be found e.g. in [CITATION]).', 'hep-th-0512221-1-41-0': 'The boundary part [EQUATION] is new [CITATION].', 'hep-th-0512221-1-41-1': 'It has been derived under the two restrictions: [MATH] and [MATH].', 'hep-th-0512221-1-41-2': 'Note, that in the present context, the first condition ([MATH]) actually follows from the second one ([MATH]) due to ([REF]).', 'hep-th-0512221-1-42-0': '# Casimir energy for rectangular cavities', 'hep-th-0512221-1-43-0': '## Casimir energy of two perfectly conducting parallel plates', 'hep-th-0512221-1-44-0': 'The Casimir energy is usually defined as [EQUATION] where the sum is over all eigenfrequencies of the system.', 'hep-th-0512221-1-44-1': 'In what following we put [MATH].', 'hep-th-0512221-1-44-2': 'We start from the well known case of two perfectly conducting plates separated by a distance [MATH] from each other.', 'hep-th-0512221-1-44-3': 'In this case the eigenfrequencies [MATH] are defined as follows: [EQUATION] so that the Casimir energy can be written as [EQUATION] [MATH] is the surface of each plate.', 'hep-th-0512221-1-44-4': 'The first sum is equivalent to the sum over eigenfrequencies of the scalar field satisfying Dirichlet boundary conditions, the second sum is equivalent to the sum over eigenfrequencies of the scalar field satisfying Neumann boundary conditions.', 'hep-th-0512221-1-45-0': 'The expression for the Casimir energy written in this form is divergent.', 'hep-th-0512221-1-45-1': 'One has to regularize it somehow to obtain a finite answer for the energy.', 'hep-th-0512221-1-45-2': 'Different methods were used for this purpose.', 'hep-th-0512221-1-45-3': 'In the present paper we suggest a method which makes calculations of determinants in the rectangular geometry straightforward and easy to perform.', 'hep-th-0512221-1-46-0': 'By making use of an identity [EQUATION] we can see that up to an irrelevant constant the Casimir energy can be written in the form (we introduce a dimensional parameter [MATH] by the same reasoning as in ([REF]) or ([REF])): [EQUATION]', 'hep-th-0512221-1-46-1': 'Now the expression for the Casimir energy is written in the standard [MATH] form, which is usual for one-loop effective actions in quantum field theory.', 'hep-th-0512221-1-46-2': 'The coefficient [MATH] is equal to zero for our current choice of the operator [MATH] and boundary geometry.', 'hep-th-0512221-1-47-0': 'At this point we introduce a regularization - we restrict integrations over momenta by some cut off [MATH] in the momentum space.', 'hep-th-0512221-1-47-1': 'The sums over [MATH] are also restricted as follows: [EQUATION]', 'hep-th-0512221-1-47-2': 'It is convenient to perform a summation over [MATH] first.', 'hep-th-0512221-1-47-3': 'One can show that the following identity holds: [EQUATION] where [MATH] is a function that satisfies the condition [EQUATION]', 'hep-th-0512221-1-47-4': 'It is possible to add any finite number that does not depend on [MATH] to the regularized Casimir energy [MATH] ( the force between the plates is measured in experiments).', 'hep-th-0512221-1-47-5': 'We add the surface term (it does not depend on [MATH]) [EQUATION] to the regularized Casimir energy and after that perform the limit [MATH].', 'hep-th-0512221-1-47-6': 'Doing so we obtain [EQUATION]', 'hep-th-0512221-1-47-7': 'The first term in ([REF]) is twice the regularized Casimir energy of the scalar field in an infinite space since it can be rewritten as [EQUATION]', 'hep-th-0512221-1-47-8': 'This term should be subtracted because we are interested in the change of the ground state energy when the plates are inserted into the free space.', 'hep-th-0512221-1-48-0': 'Next we perform the limit [MATH].', 'hep-th-0512221-1-48-1': 'The Casimir energy is thus [EQUATION] which is the well known result by Casimir [CITATION].', 'hep-th-0512221-1-49-0': "It is often argued that only the volume term ([REF]) should be subtracted to obtain the finite answer for the Casimir energy in two plates' geometry.", 'hep-th-0512221-1-49-1': 'In our derivation it is shown that generally the surface term (in the present regularization we added ([REF])) should also be subtracted even for this simple geometry.', 'hep-th-0512221-1-50-0': 'After elaborations we summarize the main idea of the method.', 'hep-th-0512221-1-50-1': 'Suppose that we want to calculate [MATH] of the second order operator [MATH], where we denote dimensionalities of the operators by numbers.', 'hep-th-0512221-1-50-2': 'We denote the eigenmodes of these operators by [MATH] and [MATH] respectively.', 'hep-th-0512221-1-50-3': 'The following expression is finite (as can be seen from the heat kernel expansion) : [EQUATION]', 'hep-th-0512221-1-50-4': 'Here [MATH] is a dimensional parameter arising from [MATH].', 'hep-th-0512221-1-50-5': 'By ([REF]) we define the physically motivated subtraction from the trace [MATH] on a one-dimensional subspace.', 'hep-th-0512221-1-50-6': 'The key question that can be asked - what is the reason that the subtraction scheme can be defined by formula ([REF]) ?', 'hep-th-0512221-1-50-7': 'We tried to answer this question by a detailed derivation of the Casimir energy for two parallel plates (see a commentary before ([REF])).', 'hep-th-0512221-1-51-0': 'In rectangular cavities the function [MATH] has the following structure: [EQUATION] where [EQUATION] ([REF]) gives a contribution to the final finite answer for the Casimir energy of a rectangular cavity.', 'hep-th-0512221-1-51-1': 'The first term in the righthandsight of ([REF]) can be transformed to [MATH] in the same manner as in the beginning of this section (see a transition from ([REF]) to ([REF]) ).', 'hep-th-0512221-1-51-2': 'For the operator [MATH] we repeat the step ([REF]) and continue this cycle until the first term in the righthandsight of ([REF]) gets the form of the vacuum energy in an infinite space, i.e. the form ([REF]).', 'hep-th-0512221-1-51-3': '(The [MATH] term in ([REF]) yields a contribution to the Casimir energy proportional to [MATH], for two parallel plates it is equal to zero.)', 'hep-th-0512221-1-52-0': 'To implement ([REF]) for two parallel plates and rectangular cavities we used the following equality: [EQUATION] which is the central equality used in our method.', 'hep-th-0512221-1-53-0': '## Casimir energy of a perfectly conducting rectangular waveguide', 'hep-th-0512221-1-54-0': 'For a perfectly conducting rectangular waveguide technical issues can be done in analogy with two parallel plates.', 'hep-th-0512221-1-54-1': 'We tacitly assume that the reader understood how the regularization is introduced in our method, so we will write only main steps without bothering too much on divergent form of some expressions.', 'hep-th-0512221-1-54-2': 'The Casimir energy for unit length is: [EQUATION]', 'hep-th-0512221-1-54-3': 'For TM modes [MATH] and [MATH] take positive integer values from [MATH] to [MATH], for TE modes [MATH] and [MATH] take positive integer values and one of them can be equal to zero ([MATH] corresponds to the main wave case).', 'hep-th-0512221-1-55-0': 'So the energy can be rewritten as: [EQUATION] where [MATH] or [MATH].', 'hep-th-0512221-1-55-1': 'Using formula ([REF]) for multiplication over [MATH] we obtain for the energy: [EQUATION]', 'hep-th-0512221-1-55-2': 'A contribution from the term [MATH] in ([REF]) (this is just the [MATH] term in ([REF])) is proportional to the Seeley coefficient [MATH] as we already discussed, this coefficient is equal to zero in the present case.', 'hep-th-0512221-1-55-3': 'The part ([REF]) with logarithm is finite, it contributes to the finite final answer for the Casimir energy.', 'hep-th-0512221-1-56-0': 'For the first term in ([REF]) we get: [EQUATION] because up to a numerical coefficient the expression ([REF]) is just the same as the formula ([REF]).', 'hep-th-0512221-1-57-0': 'So the Casimir energy for unit length of a rectangular waveguide can be written as the sum of ([REF]) and ([REF]) : [EQUATION]', 'hep-th-0512221-1-58-0': '## Casimir energy of a perfectly conducting rectangular cavity', 'hep-th-0512221-1-59-0': 'The Casimir energy in this case can be written as: [EQUATION]', 'hep-th-0512221-1-59-1': 'Using formula ([REF]) and technique described in previous subsections we obtain: [EQUATION]', 'hep-th-0512221-1-59-2': 'The remaining terms should be calculated (using formula ([REF]) again) as follows: [EQUATION]', 'hep-th-0512221-1-59-3': 'As a result for the Casimir energy of the cavity we obtain: [EQUATION]', 'hep-th-0512221-1-60-0': '## Relation to the argument principle', 'hep-th-0512221-1-61-0': 'An argument principle is a convenient method of summation over the eigenmodes of the system (see [CITATION] and [CITATION] for its applications).', 'hep-th-0512221-1-61-1': 'The argument principle states: [EQUATION] where [MATH] are zeroes and [MATH] are poles of the function [MATH] inside the contour of integration.', 'hep-th-0512221-1-61-2': 'For the Casimir energy [MATH].', 'hep-th-0512221-1-61-3': 'We choose [EQUATION] in case of a scalar field satisfying Dirichlet boundary conditions on the plates.', 'hep-th-0512221-1-61-4': 'The contour lies on an imaginary axis, a contribution from the right semicircle with a large radius is negligible.', 'hep-th-0512221-1-61-5': 'A denominator is chosen in this form to remove [MATH] from the roots of the equation [MATH].', 'hep-th-0512221-1-61-6': 'In this case we proceed as follows: [EQUATION]', 'hep-th-0512221-1-61-7': 'Here [MATH].', 'hep-th-0512221-1-61-8': 'We see that the argument principle is in agreement with ([REF]).', 'hep-th-0512221-1-61-9': 'An important point: the surface contribution is divergent for the case of a scalar field with Dirichlet boundary conditions.', 'hep-th-0512221-1-61-10': 'Only the sum of Dirichlet modes and Neumann modes yields a finite surface contribution for the scalar field, which is just the same as the Casimir energy of electromagnetic field for two perfectly conducting plates.', 'hep-th-0512221-1-62-0': '## Zeta function regularization for the cavity', 'hep-th-0512221-1-63-0': '[MATH]-function has already been discussed in this paper, so it is natural to describe regularization of the Casimir energy for the cavity in terms of [MATH]-function.', 'hep-th-0512221-1-63-1': 'Usually the Casimir energy is regularized as follows: [EQUATION] where [MATH] is large enough to make ([REF]) convergent.', 'hep-th-0512221-1-63-2': 'Then we should continue analytically ([REF]) to the value [MATH] , this procedure yields the renormalized finite Casimir energy.', 'hep-th-0512221-1-63-3': 'In our case eigenfrequencies [MATH] should be taken from ([REF]).', 'hep-th-0512221-1-63-4': 'So the regularized Casimir energy of the cavity [MATH] can be written in terms of Epstein [MATH] and Riemann [MATH] zeta functions: [EQUATION]', 'hep-th-0512221-1-63-5': 'The prime means that the term with all [MATH] should be excluded from the sum.', 'hep-th-0512221-1-63-6': 'After analytical continuation by use of reflection formulas for zeta functions it can be shown that [EQUATION]', 'hep-th-0512221-1-63-7': 'The renormalized Casimir energy can therefore be written as: [EQUATION]', 'hep-th-0512221-1-63-8': 'The formulas ([REF]) and ([REF]) yield the Casimir energy for a perfectly conducting cavity.', 'hep-th-0512221-1-64-0': '## Geometric interpretation', 'hep-th-0512221-1-65-0': 'In this section we suggest a geometric interpretation of the main formulas in terms of geometric optics.', 'hep-th-0512221-1-65-1': 'This interpretation clarifies the physical meaning of the results ([REF]) , ([REF]) obtained, which is always important for further generalizations in more complicated cases.', 'hep-th-0512221-1-66-0': 'Two geometric interpretations - a semiclassical method [CITATION] and a method of geometric optics [CITATION] have been introduced recently for the evaluation of the Casimir energies.', 'hep-th-0512221-1-66-1': 'They were applied for the case of two parallel plates and some other geometries.', 'hep-th-0512221-1-66-2': 'For two parallel plates they give the same answer.', 'hep-th-0512221-1-67-0': 'At the moment no geometric interpretation for the Casimir energy of the cavities exists.', 'hep-th-0512221-1-67-1': 'Our formulas ([REF]), ([REF]) yield a simple geometric interpretation for the Casimir energy of the rectangular cavities.', 'hep-th-0512221-1-68-0': "Optical contributions to the Green's function of the scalar field with Dirichlet and Neumann boundary conditions have the form: [EQUATION]", 'hep-th-0512221-1-68-1': 'Here [MATH] is the length of the optical path that starts from [MATH] and arrives at [MATH] after n reflections from the boundary.', 'hep-th-0512221-1-68-2': '[MATH] is the enlargement factor of classical ray optics.', 'hep-th-0512221-1-68-3': 'For planar boundaries it is given by [MATH].', 'hep-th-0512221-1-68-4': 'From ([REF]) it follows that for two parallel plates the Casimir energy of the electromagnetic field can be expanded as: [EQUATION]', 'hep-th-0512221-1-68-5': 'Here [MATH] is a sum over all photon states (with frequencies [MATH]) in an infinite space.', 'hep-th-0512221-1-68-6': "The righthandsight of ([REF]) can be written in terms of optical Green's functions: [EQUATION]", 'hep-th-0512221-1-68-7': "Note that terms with odd reflections from Dirichlet and Neumann Green's functions cancel each other due to the factor [MATH] present in optical Dirichlet Green's function.", 'hep-th-0512221-1-68-8': 'This is why only periodic paths with even number of reflections from the boundary [MATH] enter into the expression for the Casimir energy.', 'hep-th-0512221-1-69-0': "Now consider the formula for the cavity ([REF]) (to obtain the Casimir energy of a waveguide ([REF]) in terms of optical Green's functions the arguments are the same, just start from two parallel plates).", 'hep-th-0512221-1-69-1': 'Imagine that there is a waveguide with side lengths [MATH] and [MATH].', 'hep-th-0512221-1-69-2': 'In order to obtain the rectangular cavity we have to insert two perfectly conducting plates with side lengths [MATH] and [MATH] (and a distance [MATH] apart) inside the waveguide.', 'hep-th-0512221-1-69-3': 'The eigenfrequencies that existed in a waveguide were equal to [MATH].', 'hep-th-0512221-1-69-4': 'Only the photons with frequencies [MATH] existed in a waveguide, and these photons start interacting with the plates inserted inside a waveguide.', 'hep-th-0512221-1-69-5': 'The optical contribution to the Casimir energy arising from the interaction of these [MATH] photons with inserted plates is equal to', 'hep-th-0512221-1-70-0': 'Here we sum over all eigenfrequencies of the electromagnetic field in [MATH] cases: when there is an infinite space ([MATH]), two parallel plates ([MATH]) and an infinite waveguide', 'hep-th-0512221-1-71-0': '([MATH]).', 'hep-th-0512221-1-72-0': 'The first two terms in ([REF]) may have the following geometric interpretation: from ([REF]) it follows that [EQUATION] where [MATH] is an eigenfrequency of a main wave in a waveguide.', 'hep-th-0512221-1-72-1': "So it is possible to express Casimir energies of perfectly conducting rectangular cavities in terms of optical Green's functions only.", 'hep-th-0512221-1-73-0': 'It is interesting that the Casimir energy of a perfectly conducting cavity can be written in terms of eigenfrequencies of the electromagnetic field in a free space, between two perfectly conducting plates and inside a perfectly conducting waveguide.', 'hep-th-0512221-1-74-0': '## The force', 'hep-th-0512221-1-75-0': 'For the experimental check of the Casimir energy for the rectangular cavity one should measure the force somehow.', 'hep-th-0512221-1-75-1': 'We think about the following possibility: one should insert two parallel perfectly conducting plates inside a perfectly conducting waveguide and measure the force acting on one of the plates as it is being moved through the waveguide.', 'hep-th-0512221-1-75-2': 'The distance between the inserted plates is [MATH] (see ([REF])).', 'hep-th-0512221-1-75-3': 'There is a permanent repulsive contribution to the force between the plates which is determined by: [EQUATION] because [MATH], which is equal to the Casimir energy on a unit length of an infinite perfectly conducting waveguide, is negative as one can check.', 'hep-th-0512221-1-76-0': 'The last term in ([REF]) yields an attractive contribution to the force between the plates.', 'hep-th-0512221-1-76-1': "So up to a positive constant ([MATH] doesn't depend on [MATH]) the contribution to the force between the plates is given by the last term in ([REF]) (The term [MATH] is canceled by the term with [MATH] in ([REF])).", 'hep-th-0512221-1-76-2': 'It can be said that the Casimir energy of a waveguide yields repulsive contribution to the force between the inserted plates, and the exchange of photons with the eigenfrequencies of a waveguide between the inserted plates yields attractive contribution to the force between the plates .'}

{'hep-th-0512221-2-0-0': 'A pedagogical introduction to the heat kernel technique, [MATH]-function and Casimir effect is presented.', 'hep-th-0512221-2-0-1': 'Several applications are considered.', 'hep-th-0512221-2-0-2': 'First we derive the high temperature asymptotics of the free energy for boson fields in terms of the heat kernel expansion and [MATH]-function.', 'hep-th-0512221-2-0-3': 'Another application is chiral anomaly for local (MIT bag) boundary conditions.', 'hep-th-0512221-2-0-4': 'Then we rederive the Casimir energies for perfectly conducting cavities using a convenient new technique, main results are presented in a novel form.', 'hep-th-0512221-2-0-5': 'A geometric interpretation of the results for the cavities is suggested.', 'hep-th-0512221-2-1-0': '# Introduction', 'hep-th-0512221-2-2-0': 'The main problem of the quantum field theory with boundaries is its renormalization and physical meaning of the results obtained.', 'hep-th-0512221-2-2-1': 'Divergences that appear in quantum field theory make the problems on manifolds with boundaries more complicated than in infinite space.', 'hep-th-0512221-2-3-0': 'In the presence of boundaries or singularities the heat kernel technique is an effective tool for the analysis of the one loop effects (see reviews [CITATION], [CITATION]).', 'hep-th-0512221-2-3-1': 'Different applications of the heat kernel expansion exist.', 'hep-th-0512221-2-3-2': 'The heat kernel technique seems to be the easiest way for the calculation of quantum anomalies, calculation of effective actions based on finite-mode regularization and analysis of divergences in quantum field theory.', 'hep-th-0512221-2-4-0': 'Chiral anomaly, which was discovered more than 35 years ago [CITATION], still plays an important role in physics.', 'hep-th-0512221-2-4-1': 'On smooth manifolds without boundaries many successful approaches to the anomalies exist [CITATION],[CITATION], [CITATION].', 'hep-th-0512221-2-4-2': 'The heat kernel approach to the anomalies is essentially equivalent to the Fujikawa approach [CITATION] and to the calculations based on the finite-mode regularization [CITATION], but it can be more easily extended to complicated geometries.', 'hep-th-0512221-2-4-3': 'The local chiral anomaly in the case of non-trivial boundary conditions (MIT bag boundary conditions) has been calculated only recently [CITATION].', 'hep-th-0512221-2-5-0': 'Casimir effect [CITATION] is a macroscopic quantum effect.', 'hep-th-0512221-2-5-1': 'Briefly speaking, if we impose classical boundary conditions on a quantum field on some boundary surface than we get the Casimir effect.', 'hep-th-0512221-2-5-2': 'There are several different physical situations that should be distinguished in the Casimir effect.', 'hep-th-0512221-2-6-0': 'Suppose there are two spatially separated dielectrics, then in a dilute limit ([MATH]) the Casimir energy of this system is equal to the energy of pairwise interactions between dipoles of these two dielectrics via a Casimir-Polder retarded potential [CITATION].', 'hep-th-0512221-2-6-1': 'For a general case of separated dielectrics the Casimir energy can be calculated as in [CITATION] or [CITATION] (for a recent discussion of these issues see [CITATION] and a review [CITATION]).', 'hep-th-0512221-2-7-0': 'A different situation takes place when there is a dilute dielectric ball or any other simply connected dielectric under study (see a review [CITATION] for a discussion of related subjects and methods used).', 'hep-th-0512221-2-7-1': 'As it was pointed out in [CITATION] and then discussed in detail in [CITATION], microscopic interatomic distances should be taken into account to calculate the Casimir energy of a dilute dielectric ball.', 'hep-th-0512221-2-7-2': 'The average interatomic distance [MATH] serves as an effective physical cut off for simply connected dielectrics.', 'hep-th-0512221-2-8-0': 'The limit of perfect conductivity ([MATH]) is opposite to a dilute case.', 'hep-th-0512221-2-8-1': 'This is the strong coupling limit of the theory.', 'hep-th-0512221-2-8-2': 'Any results obtained in this limit are nonperturbative ones.', 'hep-th-0512221-2-9-0': 'The Casimir energy of a perfectly conducting rectangular cavity was first calculated in [CITATION] using exponential regularization.', 'hep-th-0512221-2-9-1': 'Later it was derived by some other methods (see references and numerical analysis in [CITATION], also a review [CITATION]).', 'hep-th-0512221-2-9-2': 'The Casimir energy changes its sign for different side lengths of the cavity.', 'hep-th-0512221-2-9-3': 'The Casimir energy of a rectangular cavity is an experimentally measurable quantity (we hope that this will be done in future experiments, see e.g. [CITATION]).', 'hep-th-0512221-2-10-0': 'In the present paper we derive the Casimir energy for rectangular cavities at zero temperature by a new method described in Sec. [MATH].', 'hep-th-0512221-2-10-1': 'What is interesting, by use of this method we could rewrite the Casimir energy for rectangular cavities in the form that makes transparent its geometric interpretation.', 'hep-th-0512221-2-11-0': 'The paper is organized as follows.', 'hep-th-0512221-2-11-1': 'In Sec. [MATH] we give an introduction to the formalism of the heat kernel and heat kernel expansion.', 'hep-th-0512221-2-11-2': 'Also we introduce a [MATH]-function [CITATION] and calculate the one loop effective action in terms of [MATH]- function.', 'hep-th-0512221-2-11-3': 'Then we consider two examples.', 'hep-th-0512221-2-11-4': 'First we derive the high temperature expansion of the free energy for boson fields [CITATION] in terms of the heat kernel expansion and [MATH]-function.', 'hep-th-0512221-2-11-5': 'Then we derive a chiral anomaly in four dimensions for an euclidean version of the MIT bag boundary conditions [CITATION].', 'hep-th-0512221-2-11-6': 'Sec. [MATH] is devoted to the Casimir energies of rectangular cavities.', 'hep-th-0512221-2-11-7': 'We introduce a regularization and a convenient new method of calculations using an example of two prefectly conducting parallel plates, then apply it to more complicated rectangular geometries.', 'hep-th-0512221-2-11-8': 'We present the Casimir energy of the cavity in the form ([REF]).', 'hep-th-0512221-2-11-9': 'Then we discuss an argument principle and [MATH]-functional regularization for the cavity.', 'hep-th-0512221-2-11-10': 'In Sec. [MATH] we rewrite the Casimir energy of the cavity in terms of geometric optics ([REF]).', 'hep-th-0512221-2-11-11': 'Finally we describe a possible experiment to verify the Casimir energies for perfectly conducting cavities and give the physical meaning of the force that can be measured.', 'hep-th-0512221-2-12-0': '# Spectral techniques', 'hep-th-0512221-2-13-0': '## Heat kernel', 'hep-th-0512221-2-14-0': 'Consider a second order elliptic partial differential operator [MATH] of Laplace type on an n-dimensional Riemannian manifold.', 'hep-th-0512221-2-14-1': 'Any operator of this type can be expanded locally as [EQUATION] where [MATH] and [MATH] are some matrix valued functions and [MATH] is the inverse metric tensor on the manifold.', 'hep-th-0512221-2-14-2': 'For a flat space [MATH].', 'hep-th-0512221-2-15-0': 'The heat kernel can be defined as follows: [EQUATION] where [MATH] is an eigenfunction of the operator [MATH] with the eigenvalue [MATH].', 'hep-th-0512221-2-16-0': 'It satisfies the heat equation [EQUATION] with an initial condition [EQUATION]', 'hep-th-0512221-2-16-1': 'If we consider the fields in a finite volume then it is necessary to specify boundary conditions.', 'hep-th-0512221-2-16-2': 'Different choices are possible.', 'hep-th-0512221-2-16-3': 'In section [MATH] we will consider the case of periodic boundary conditions on imaginary time coordinate, which are specific for boson fields.', 'hep-th-0512221-2-16-4': 'In section [MATH] we will study bag boundary conditions imposed on fermion fields.', 'hep-th-0512221-2-16-5': 'If the normal to the boundary component of the fermion current [MATH] vanishes at the boundary, one can impose bag boundary conditions, a particular case of mixed boundary conditions.', 'hep-th-0512221-2-16-6': 'We assume given two complementary projectors [MATH], [MATH] acting on a multi component field (the eigenfunction of the operator [MATH]) at each point of the boundary and define mixed boundary conditions by the relations [EQUATION] where [MATH] is a matrix valued function on the boundary.', 'hep-th-0512221-2-16-7': 'In other words, the components [MATH] satisfy Dirichlet boundary conditions, and [MATH] satisfy Robin (modified Neumann) ones.', 'hep-th-0512221-2-17-0': 'It is convenient to define [EQUATION]', 'hep-th-0512221-2-17-1': 'Let [MATH], [MATH] be a local orthonormal frame for the tangent space to the manifold and let on the boundary [MATH] be an inward pointing normal vector.', 'hep-th-0512221-2-18-0': 'The extrinsic curvature is defined by the equation [EQUATION] where [MATH] is the Christoffel symbol.', 'hep-th-0512221-2-18-1': 'For example, on the unit sphere [MATH] which bounds the unit ball in [MATH] the extrinsic curvature is [MATH].', 'hep-th-0512221-2-19-0': 'Curved space offers no complications in our approach compared to the flat case.', 'hep-th-0512221-2-19-1': 'Let [MATH] be the Riemann tensor, and let [MATH] be the Ricci tensor.', 'hep-th-0512221-2-19-2': 'With our sign convention the scalar curvature [MATH] is [MATH] on the unit sphere [MATH].', 'hep-th-0512221-2-19-3': 'In flat space the Riemann and Ricci tensors are equal to zero.', 'hep-th-0512221-2-20-0': 'One can always introduce a connection [MATH] and another matrix valued function [MATH] so that [MATH] takes the form: [EQUATION]', 'hep-th-0512221-2-20-1': 'Here [MATH] is a sum of covariant Riemannian derivative with respect to metric [MATH] and connection [MATH].', 'hep-th-0512221-2-20-2': 'One can, of course, express [MATH] and [MATH] in terms of [MATH], [MATH] and [MATH]: [EQUATION]', 'hep-th-0512221-2-20-3': 'For the future use we introduce also the field strength for [MATH]: [EQUATION]', 'hep-th-0512221-2-20-4': 'The connection [MATH] will be used to construct covariant derivatives.', 'hep-th-0512221-2-20-5': 'The subscript [MATH] will be used to denote repeated covariant derivatives with the connection [MATH] and the Christoffel connection on [MATH].', 'hep-th-0512221-2-20-6': 'The subscript [MATH] will denote repeated covariant derivatives containing [MATH] and the Christoffel connection on the boundary.', 'hep-th-0512221-2-20-7': 'Difference between these two covariant derivatives is measured by the extrinsic curvature ([REF]).', 'hep-th-0512221-2-20-8': 'For example, [MATH].', 'hep-th-0512221-2-21-0': 'Let us define an integrated heat kernel for a hermitian operator [MATH] by the equation: [EQUATION] where [MATH] is an hermitian matrix valued function, [MATH] here is over matrix indices.', 'hep-th-0512221-2-21-1': 'For the boundary conditions we consider in this paper there exists an asymptotic expansion [CITATION] as [MATH]: [EQUATION]', 'hep-th-0512221-2-21-2': 'According to the general theory [CITATION] the coefficients [MATH] are locally computable.', 'hep-th-0512221-2-21-3': 'This means that each [MATH] can be represented as a sum of volume and boundary integrals of local invariants constructed from [MATH], [MATH], [MATH], the curvature tensor, and their derivatives.', 'hep-th-0512221-2-21-4': 'Boundary invariants may also include [MATH], [MATH] and [MATH].', 'hep-th-0512221-2-21-5': 'Total mass dimension of such invariants should be [MATH] for the volume terms and [MATH] for the boundary ones.', 'hep-th-0512221-2-22-0': 'At the moment several coefficients of the expansion ([REF]) are known for the case of mixed boundary conditions ([REF]) and matrix valued function [MATH] (see [CITATION] for details of derivation; the formula ([REF]) for [MATH] was derived in [CITATION] with additional restrictions [MATH] and [MATH]) : [EQUATION]', 'hep-th-0512221-2-22-1': 'For a scalar function [MATH] and mixed boundary conditions the coefficients [MATH] and [MATH] were already derived [CITATION].', 'hep-th-0512221-2-23-0': '## [MATH]-function', 'hep-th-0512221-2-24-0': 'Zeta function of a positive operator [MATH] is defined by [EQUATION] where the sum is over all eigenvalues of the operator [MATH].', 'hep-th-0512221-2-24-1': 'The zeta function is related to the heat kernel by the transformation [EQUATION]', 'hep-th-0512221-2-24-2': 'Residues at the poles of the zeta function are related to the coefficients of the heat kernel expansion: [EQUATION]', 'hep-th-0512221-2-24-3': 'Here [MATH] is a unit matrix with a dimension of the matrix functions [MATH] in ([REF]).', 'hep-th-0512221-2-24-4': 'From ([REF]) it follows that [EQUATION]', 'hep-th-0512221-2-24-5': 'In Euclidean four dimensional space the zero temperature one-loop path integral over the boson fields [MATH] can be evaluated as follows (up to a normalization factor): [EQUATION]', 'hep-th-0512221-2-24-6': 'Here we introduced the constant [MATH] with a dimension of mass in order to keep a proper dimension of the measure in the functional integral.', 'hep-th-0512221-2-24-7': '[MATH] can be thought of as a number of eigenvalues of the operator [MATH].', 'hep-th-0512221-2-24-8': 'For the operator [MATH] in the form ([REF]) the number of eigenvalues is infinite, so [MATH] yields a regularized value for this number.', 'hep-th-0512221-2-25-0': 'The zero temperature one-loop effective action is defined then by [EQUATION]', 'hep-th-0512221-2-25-1': 'The term [MATH] in the effective action [MATH] determines the one-loop beta function, this term describes renormalization of the one-loop logarithmic divergences appearing in the theory.', 'hep-th-0512221-2-26-0': '## Free energy for boson fields', 'hep-th-0512221-2-27-0': 'A finite temperature field theory is defined in Euclidean space, since for boson fields one has to impose periodic boundary conditions on imaginary time coordinate (antiperiodic boundary conditions for fermion fields respectively).', 'hep-th-0512221-2-27-1': 'A partition function is defined by [EQUATION] where [MATH] is a hamiltonian of the problem and [MATH].', 'hep-th-0512221-2-27-2': 'Let us choose the lagrangian density [MATH] in the form [EQUATION] where [MATH] is an imaginary time coordinate and [MATH] is a three dimensional spatial part of the density in the form ([REF]).', 'hep-th-0512221-2-27-3': 'The free energy of the system is defined by [EQUATION] the integration is over all periodic fields satisfying [MATH] is a normalization coefficient).', 'hep-th-0512221-2-27-4': 'As a result the eigenfunctions of [MATH] have the form [MATH], where [MATH] and [MATH].', 'hep-th-0512221-2-27-5': 'The free energy is thus equal to [CITATION] [EQUATION] where we introduced [MATH]-function [EQUATION] and the parameter [MATH] with a mass dimensionality in order to make the argument of the logarithm dimensionless (also see a previous section).', 'hep-th-0512221-2-28-0': 'Then it is convenient to use the formula [EQUATION] and separate [MATH] and other terms in the sum.', 'hep-th-0512221-2-28-1': 'For [MATH] terms we substitute the heat kernel expansion for the operator [MATH] at small [MATH] [EQUATION] and perform [MATH] integration, then we arrive at the high temperature expansion ([MATH]) for the free energy [MATH]: [EQUATION]', 'hep-th-0512221-2-28-2': 'Here [MATH], [MATH] is a Riemann zeta function, [MATH] is a zeta function of an operator [MATH], [MATH] is the Euler constant.', 'hep-th-0512221-2-28-3': 'The first two terms on the r.h.s. of ([REF]) follow from the [MATH] term.', 'hep-th-0512221-2-29-0': 'The term [EQUATION] is the leading high temperature contribution to the free energy.', 'hep-th-0512221-2-30-0': 'The classical limit terms due to the equality [MATH] can be rewritten as follows: [EQUATION]', 'hep-th-0512221-2-30-1': 'The terms on the l.h.s. of ([REF]) yield a renormalized value of the terms on the r.h.s. of ([REF]), since the sum on the righthandsight is generally divergent when the number of modes is infinite.', 'hep-th-0512221-2-31-0': 'The term with [MATH] determines the part of the free energy that appears due to one-loop logarithmic divergences and thus it depends on the dimensional parameter [MATH] as in the zero temperature case.', 'hep-th-0512221-2-32-0': '## Chiral anomaly in four dimensions', 'hep-th-0512221-2-33-0': 'for MIT bag boundary conditions', 'hep-th-0512221-2-34-0': 'Consider the Dirac operator on an [MATH]-dimensional Riemannian manifold [EQUATION] in external vector [MATH] and axial vector [MATH] fields.', 'hep-th-0512221-2-34-1': 'We suppose that [MATH] and [MATH] are anti-hermitian matrices in the space of some representation of the gauge group.', 'hep-th-0512221-2-34-2': '[MATH] is the spin-connection.', 'hep-th-0512221-2-35-0': 'The Dirac operator transforms covariantly under infinitesimal local gauge transformations (the local gauge transformation is [MATH]): [EQUATION] and under infinitesimal local chiral transformations (the local chiral transformation is [MATH]): [EQUATION]', 'hep-th-0512221-2-35-1': 'The parameters [MATH] and [MATH] are anti-hermitian matrices.', 'hep-th-0512221-2-36-0': 'First we adopt the zeta-function regularization and write the one-loop effective action for Dirac fermions at zero temperature as [EQUATION] where [EQUATION] prime denotes differentiation with respect to [MATH], and [MATH] is the functional trace.', 'hep-th-0512221-2-37-0': 'The following identity holds: [EQUATION]', 'hep-th-0512221-2-37-1': 'Due to the identity ([REF]) [EQUATION] so the effective action ([REF]) is gauge invariant, [MATH].', 'hep-th-0512221-2-38-0': 'The chiral anomaly is by definition equal to the variation of [MATH] under an infinitesimal chiral transformation.', 'hep-th-0512221-2-38-1': 'Using ([REF]) we obtain: [EQUATION] and the anomaly reads [EQUATION]', 'hep-th-0512221-2-38-2': 'The heat kernel is related to the zeta function by the Mellin transformation: [EQUATION]', 'hep-th-0512221-2-38-3': 'In particular, after the substitution of the heat kernel expansion ([REF]) into the formula ([REF]) we obtain [EQUATION]', 'hep-th-0512221-2-38-4': 'The same expression for the anomaly follows also from the Fujikawa approach [CITATION].', 'hep-th-0512221-2-39-0': "One can also derive the expression for the anomaly ([REF]) from Schwinger's effective action.", 'hep-th-0512221-2-39-1': 'One should start from an identity: [EQUATION]', 'hep-th-0512221-2-39-2': 'Then the change in the effective action due to chiral transformations can be written: [EQUATION]', 'hep-th-0512221-2-39-3': 'We impose local boundary conditions: [EQUATION] which are nothing else than a Euclidean version of the MIT bag boundary conditions [CITATION].', 'hep-th-0512221-2-39-4': 'For these boundary conditions [MATH], and the normal component of the fermion current [MATH] vanishes on the boundary.', 'hep-th-0512221-2-39-5': 'Spectral properties of the Dirac operator for bag boundary conditions are intensively studied [CITATION].', 'hep-th-0512221-2-40-0': 'Since [MATH] is a first order differential operator it was enough to fix the boundary conditions ([REF]) on a half of the components.', 'hep-th-0512221-2-40-1': 'To proceed with a second order operator [MATH] we need boundary conditions on the remaining components as well.', 'hep-th-0512221-2-40-2': 'They are defined by the consistency condition [CITATION]: [EQUATION] which is equivalent to the Robin boundary condition [EQUATION] with [EQUATION]', 'hep-th-0512221-2-40-3': 'In the paper [CITATION] the following expression for a coefficient [MATH] with an hermitian matrix valued function [MATH] and conditions ([REF]), [MATH] (flat boundaries), [MATH] was obtained: [EQUATION]', 'hep-th-0512221-2-40-4': 'To obtain the chiral anomaly in four dimensions with MIT bag boundary conditions one has to calculate the coefficient [MATH] ([REF]) with [MATH], [MATH] and substitute it into ([REF]).', 'hep-th-0512221-2-40-5': 'We define [MATH], [MATH], [MATH].', 'hep-th-0512221-2-40-6': 'The anomaly contains two contributions: [EQUATION]', 'hep-th-0512221-2-40-7': 'In the volume part [EQUATION] only the [MATH] terms seem to be new [CITATION] (for flat space it can be found e.g. in [CITATION]).', 'hep-th-0512221-2-41-0': 'The boundary part [EQUATION] is new [CITATION].', 'hep-th-0512221-2-41-1': 'It has been derived under the two restrictions: [MATH] and [MATH].', 'hep-th-0512221-2-41-2': 'Note, that in the present context, the first condition ([MATH]) actually follows from the second one ([MATH]) due to ([REF]).', 'hep-th-0512221-2-42-0': '# Casimir energy for rectangular cavities', 'hep-th-0512221-2-43-0': '## Casimir energy of two perfectly conducting parallel plates', 'hep-th-0512221-2-44-0': 'The Casimir energy is usually defined as [EQUATION] where the sum is over all eigenfrequencies of the system.', 'hep-th-0512221-2-44-1': 'In what following we put [MATH].', 'hep-th-0512221-2-44-2': 'We start from the well known case of two perfectly conducting plates separated by a distance [MATH] from each other.', 'hep-th-0512221-2-44-3': 'In this case the eigenfrequencies [MATH] are defined as follows: [EQUATION] so that the Casimir energy can be written as [EQUATION] [MATH] is the surface of each plate.', 'hep-th-0512221-2-44-4': 'The first sum is equivalent to the sum over eigenfrequencies of the scalar field satisfying Dirichlet boundary conditions, the second sum is equivalent to the sum over eigenfrequencies of the scalar field satisfying Neumann boundary conditions.', 'hep-th-0512221-2-45-0': 'The expression for the Casimir energy written in this form is divergent.', 'hep-th-0512221-2-45-1': 'One has to regularize it somehow to obtain a finite answer for the energy.', 'hep-th-0512221-2-45-2': 'Different methods were used for this purpose.', 'hep-th-0512221-2-45-3': 'In the present paper we suggest a method which makes calculations of determinants in the rectangular geometry straightforward and easy to perform.', 'hep-th-0512221-2-46-0': 'By making use of an identity [EQUATION] we can see that up to an irrelevant constant the Casimir energy can be written in the form (we introduce a dimensional parameter [MATH] by the same reasoning as in ([REF]) or ([REF])): [EQUATION]', 'hep-th-0512221-2-46-1': 'Now the expression for the Casimir energy is written in the standard [MATH] form, which is usual for one-loop effective actions in quantum field theory.', 'hep-th-0512221-2-46-2': 'The coefficient [MATH] is equal to zero for our current choice of the operator [MATH] and boundary geometry.', 'hep-th-0512221-2-47-0': 'At this point we introduce a regularization - we restrict integrations over momenta by some cut off [MATH] in the momentum space.', 'hep-th-0512221-2-47-1': 'The sums over [MATH] are also restricted as follows: [EQUATION]', 'hep-th-0512221-2-47-2': 'It is convenient to perform a summation over [MATH] first.', 'hep-th-0512221-2-47-3': 'One can show that the following identity holds: [EQUATION] where [MATH] is a function that satisfies the condition [EQUATION]', 'hep-th-0512221-2-47-4': 'It is possible to add any finite number that does not depend on [MATH] to the regularized Casimir energy [MATH] ( the force between the plates is measured in experiments).', 'hep-th-0512221-2-47-5': 'We add the surface term (it does not depend on [MATH]) [EQUATION] to the regularized Casimir energy and after that perform the limit [MATH].', 'hep-th-0512221-2-47-6': 'Doing so we obtain [EQUATION]', 'hep-th-0512221-2-47-7': 'The first term in ([REF]) is twice the regularized Casimir energy of the scalar field in an infinite space since it can be rewritten as [EQUATION]', 'hep-th-0512221-2-47-8': 'This term should be subtracted because we are interested in the change of the ground state energy when the plates are inserted into the free space.', 'hep-th-0512221-2-48-0': 'Next we perform the limit [MATH].', 'hep-th-0512221-2-48-1': 'The Casimir energy is thus [EQUATION] which is the well known result by Casimir [CITATION].', 'hep-th-0512221-2-49-0': "It is often argued that only the volume term ([REF]) should be subtracted to obtain the finite answer for the Casimir energy in two plates' geometry.", 'hep-th-0512221-2-49-1': 'In our derivation it is shown that generally the surface term (in the present regularization we added ([REF])) should also be subtracted even for this simple geometry.', 'hep-th-0512221-2-50-0': 'After elaborations we summarize the main idea of the method.', 'hep-th-0512221-2-50-1': 'Suppose that we want to calculate [MATH] of the second order operator [MATH], where we denote dimensionalities of the operators by numbers.', 'hep-th-0512221-2-50-2': 'We denote the eigenmodes of these operators by [MATH] and [MATH] respectively.', 'hep-th-0512221-2-50-3': 'The following expression is finite (as can be seen from the heat kernel expansion) : [EQUATION]', 'hep-th-0512221-2-50-4': 'Here [MATH] is a dimensional parameter arising from [MATH].', 'hep-th-0512221-2-50-5': 'By ([REF]) we define the physically motivated subtraction from the trace [MATH] on a one-dimensional subspace.', 'hep-th-0512221-2-50-6': 'The key question that can be asked - what is the reason that the subtraction scheme can be defined by formula ([REF]) ?', 'hep-th-0512221-2-50-7': 'We tried to answer this question by a detailed derivation of the Casimir energy for two parallel plates (see a commentary before ([REF])).', 'hep-th-0512221-2-51-0': 'In rectangular cavities the function [MATH] has the following structure: [EQUATION] where [EQUATION] ([REF]) gives a contribution to the final finite answer for the Casimir energy of a rectangular cavity.', 'hep-th-0512221-2-51-1': 'The first term in the righthandsight of ([REF]) can be transformed to [MATH] in the same manner as in the beginning of this section (see a transition from ([REF]) to ([REF]) ).', 'hep-th-0512221-2-51-2': 'For the operator [MATH] we repeat the step ([REF]) and continue this cycle until the first term in the righthandsight of ([REF]) gets the form of the vacuum energy in an infinite space, i.e. the form ([REF]).', 'hep-th-0512221-2-51-3': '(The [MATH] term in ([REF]) yields a contribution to the Casimir energy proportional to [MATH], for two parallel plates it is equal to zero.)', 'hep-th-0512221-2-52-0': 'To implement ([REF]) for two parallel plates and rectangular cavities we used the following equality: [EQUATION] which is the central equality used in our method.', 'hep-th-0512221-2-53-0': '## Casimir energy of a perfectly conducting rectangular waveguide', 'hep-th-0512221-2-54-0': 'For a perfectly conducting rectangular waveguide technical issues can be done in analogy with two parallel plates.', 'hep-th-0512221-2-54-1': 'We tacitly assume that the reader understood how the regularization is introduced in our method, so we will write only main steps without bothering too much on divergent form of some expressions.', 'hep-th-0512221-2-54-2': 'The Casimir energy for unit length is: [EQUATION]', 'hep-th-0512221-2-54-3': 'For TM modes [MATH] and [MATH] take positive integer values from [MATH] to [MATH], for TE modes [MATH] and [MATH] take positive integer values and one of them can be equal to zero ([MATH] corresponds to the main wave case).', 'hep-th-0512221-2-55-0': 'So the energy can be rewritten as: [EQUATION] where [MATH] or [MATH].', 'hep-th-0512221-2-55-1': 'Using formula ([REF]) for multiplication over [MATH] we obtain for the energy: [EQUATION]', 'hep-th-0512221-2-55-2': 'A contribution from the term [MATH] in ([REF]) (this is just the [MATH] term in ([REF])) is proportional to the Seeley coefficient [MATH] as we already discussed, this coefficient is equal to zero in the present case.', 'hep-th-0512221-2-55-3': 'The part ([REF]) with logarithm is finite, it contributes to the finite final answer for the Casimir energy.', 'hep-th-0512221-2-56-0': 'For the first term in ([REF]) we get: [EQUATION] because up to a numerical coefficient the expression ([REF]) is just the same as the formula ([REF]).', 'hep-th-0512221-2-57-0': 'So the Casimir energy for unit length of a rectangular waveguide can be written as the sum of ([REF]) and ([REF]) : [EQUATION]', 'hep-th-0512221-2-58-0': '## Casimir energy of a perfectly conducting rectangular cavity', 'hep-th-0512221-2-59-0': 'The Casimir energy in this case can be written as: [EQUATION]', 'hep-th-0512221-2-59-1': 'Using formula ([REF]) and technique described in previous subsections we obtain: [EQUATION]', 'hep-th-0512221-2-59-2': 'The remaining terms should be calculated (using formula ([REF]) again) as follows: [EQUATION]', 'hep-th-0512221-2-59-3': 'As a result for the Casimir energy of the cavity we obtain: [EQUATION]', 'hep-th-0512221-2-60-0': '## Relation to the argument principle', 'hep-th-0512221-2-61-0': 'An argument principle is a convenient method of summation over the eigenmodes of the system (see [CITATION] and [CITATION] for its applications).', 'hep-th-0512221-2-61-1': 'The argument principle states: [EQUATION] where [MATH] are zeroes and [MATH] are poles of the function [MATH] inside the contour of integration.', 'hep-th-0512221-2-61-2': 'For the Casimir energy [MATH].', 'hep-th-0512221-2-61-3': 'We choose [EQUATION] in case of a scalar field satisfying Dirichlet boundary conditions on the plates.', 'hep-th-0512221-2-61-4': 'The contour lies on an imaginary axis, a contribution from the right semicircle with a large radius is negligible.', 'hep-th-0512221-2-61-5': 'A denominator is chosen in this form to remove [MATH] from the roots of the equation [MATH].', 'hep-th-0512221-2-61-6': 'In this case we proceed as follows: [EQUATION]', 'hep-th-0512221-2-61-7': 'Here [MATH].', 'hep-th-0512221-2-61-8': 'We see that the argument principle is in agreement with ([REF]).', 'hep-th-0512221-2-61-9': 'An important point: the surface contribution is divergent for the case of a scalar field with Dirichlet boundary conditions.', 'hep-th-0512221-2-61-10': 'Only the sum of Dirichlet modes and Neumann modes yields a finite surface contribution for the scalar field, which is just the same as the Casimir energy of electromagnetic field for two perfectly conducting plates.', 'hep-th-0512221-2-62-0': '## Zeta function regularization for the cavity', 'hep-th-0512221-2-63-0': '[MATH]-function has already been discussed in this paper, so it is natural to describe regularization of the Casimir energy for the cavity in terms of [MATH]-function.', 'hep-th-0512221-2-63-1': 'Usually the Casimir energy is regularized as follows: [EQUATION] where [MATH] is large enough to make ([REF]) convergent.', 'hep-th-0512221-2-63-2': 'Then we should continue analytically ([REF]) to the value [MATH] , this procedure yields the renormalized finite Casimir energy.', 'hep-th-0512221-2-63-3': 'In our case eigenfrequencies [MATH] should be taken from ([REF]).', 'hep-th-0512221-2-63-4': 'So the regularized Casimir energy of the cavity [MATH] can be written in terms of Epstein [MATH] and Riemann [MATH] zeta functions: [EQUATION]', 'hep-th-0512221-2-63-5': 'The prime means that the term with all [MATH] should be excluded from the sum.', 'hep-th-0512221-2-63-6': 'After analytical continuation by use of reflection formulas for zeta functions it can be shown that [EQUATION]', 'hep-th-0512221-2-63-7': 'The renormalized Casimir energy can therefore be written as: [EQUATION]', 'hep-th-0512221-2-63-8': 'The formulas ([REF]) and ([REF]) yield the Casimir energy for a perfectly conducting cavity.', 'hep-th-0512221-2-64-0': '## Geometric interpretation', 'hep-th-0512221-2-65-0': 'In this section we suggest a geometric interpretation of the main formulas in terms of geometric optics.', 'hep-th-0512221-2-65-1': 'This interpretation clarifies the physical meaning of the results ([REF]) , ([REF]) obtained, which is always important for further generalizations in more complicated cases.', 'hep-th-0512221-2-66-0': 'Two geometric interpretations - a semiclassical method [CITATION] and a method of geometric optics [CITATION] have been introduced recently for the evaluation of the Casimir energies.', 'hep-th-0512221-2-66-1': 'They were applied for the case of two parallel plates and some other geometries.', 'hep-th-0512221-2-66-2': 'For two parallel plates they give the same answer.', 'hep-th-0512221-2-67-0': 'At the moment no geometric interpretation for the Casimir energy of the cavities exists.', 'hep-th-0512221-2-67-1': 'Our formulas ([REF]), ([REF]) yield a simple geometric interpretation for the Casimir energy of the rectangular cavities.', 'hep-th-0512221-2-68-0': "Optical contributions to the Green's function of the scalar field with Dirichlet and Neumann boundary conditions have the form: [EQUATION]", 'hep-th-0512221-2-68-1': 'Here [MATH] is the length of the optical path that starts from [MATH] and arrives at [MATH] after n reflections from the boundary.', 'hep-th-0512221-2-68-2': '[MATH] is the enlargement factor of classical ray optics.', 'hep-th-0512221-2-68-3': 'For planar boundaries it is given by [MATH].', 'hep-th-0512221-2-68-4': 'From ([REF]) it follows that for two parallel plates the Casimir energy of the electromagnetic field can be expanded as: [EQUATION]', 'hep-th-0512221-2-68-5': 'Here [MATH] is a sum over all photon states (with frequencies [MATH]) in an infinite space.', 'hep-th-0512221-2-68-6': "The righthandsight of ([REF]) can be written in terms of optical Green's functions: [EQUATION]", 'hep-th-0512221-2-68-7': "Note that terms with odd reflections from Dirichlet and Neumann Green's functions cancel each other due to the factor [MATH] present in optical Dirichlet Green's function.", 'hep-th-0512221-2-68-8': 'This is why only periodic paths with even number of reflections from the boundary [MATH] enter into the expression for the Casimir energy.', 'hep-th-0512221-2-69-0': "Now consider the formula for the cavity ([REF]) (to obtain the Casimir energy of a waveguide ([REF]) in terms of optical Green's functions the arguments are the same, just start from two parallel plates).", 'hep-th-0512221-2-69-1': 'Imagine that there is a waveguide with side lengths [MATH] and [MATH].', 'hep-th-0512221-2-69-2': 'In order to obtain the rectangular cavity we have to insert two perfectly conducting plates with side lengths [MATH] and [MATH] (and a distance [MATH] apart) inside the waveguide.', 'hep-th-0512221-2-69-3': 'The eigenfrequencies that existed in a waveguide were equal to [MATH].', 'hep-th-0512221-2-69-4': 'Only the photons with frequencies [MATH] existed in a waveguide, and these photons start interacting with the plates inserted inside a waveguide.', 'hep-th-0512221-2-69-5': 'The optical contribution to the Casimir energy arising from the interaction of these [MATH] photons with inserted plates is equal to', 'hep-th-0512221-2-70-0': 'Here we sum over all eigenfrequencies of the electromagnetic field in [MATH] cases: when there is an infinite space ([MATH]), two parallel plates ([MATH]) and an infinite waveguide', 'hep-th-0512221-2-71-0': '([MATH]).', 'hep-th-0512221-2-72-0': 'The first two terms in ([REF]) may have the following geometric interpretation: from ([REF]) it follows that [EQUATION] where [MATH] is an eigenfrequency of a main wave in a waveguide.', 'hep-th-0512221-2-72-1': "So it is possible to express Casimir energies of perfectly conducting rectangular cavities in terms of optical Green's functions only.", 'hep-th-0512221-2-73-0': 'It is interesting that the Casimir energy of a perfectly conducting cavity can be written in terms of eigenfrequencies of the electromagnetic field in a free space, between two perfectly conducting plates and inside a perfectly conducting waveguide.', 'hep-th-0512221-2-74-0': '## The force', 'hep-th-0512221-2-75-0': 'For the experimental check of the Casimir energy for the rectangular cavity one should measure the force somehow.', 'hep-th-0512221-2-75-1': 'We think about the following possibility: one should insert two parallel perfectly conducting plates inside a perfectly conducting waveguide and measure the force acting on one of the plates as it is being moved through the waveguide.', 'hep-th-0512221-2-75-2': 'The distance between the inserted plates is [MATH] (see ([REF])).', 'hep-th-0512221-2-75-3': 'There is a permanent repulsive contribution to the force between the plates which is determined by: [EQUATION] because [MATH], which is equal to the Casimir energy on a unit length of an infinite perfectly conducting waveguide, is negative as one can check.', 'hep-th-0512221-2-76-0': 'The last term in ([REF]) yields an attractive contribution to the force between the plates.', 'hep-th-0512221-2-76-1': "So up to a positive constant ([MATH] doesn't depend on [MATH]) the contribution to the force between the plates is given by the last term in ([REF]) (The term [MATH] is canceled by the term with [MATH] in ([REF])).", 'hep-th-0512221-2-76-2': 'It can be said that the Casimir energy of a waveguide yields repulsive contribution to the force between the inserted plates, and the exchange of photons with the eigenfrequencies of a waveguide between the inserted plates yields attractive contribution to the force between the plates .'}

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['hep-th-0512221-1-21-2', 'hep-th-0512221-2-21-2'], ['hep-th-0512221-1-21-3', 'hep-th-0512221-2-21-3'], ['hep-th-0512221-1-21-4', 'hep-th-0512221-2-21-4'], ['hep-th-0512221-1-21-5', 'hep-th-0512221-2-21-5'], ['hep-th-0512221-1-16-0', 'hep-th-0512221-2-16-0'], ['hep-th-0512221-1-16-1', 'hep-th-0512221-2-16-1'], ['hep-th-0512221-1-16-2', 'hep-th-0512221-2-16-2'], ['hep-th-0512221-1-16-3', 'hep-th-0512221-2-16-3'], ['hep-th-0512221-1-16-4', 'hep-th-0512221-2-16-4'], ['hep-th-0512221-1-16-5', 'hep-th-0512221-2-16-5'], ['hep-th-0512221-1-16-6', 'hep-th-0512221-2-16-6'], ['hep-th-0512221-1-16-7', 'hep-th-0512221-2-16-7'], ['hep-th-0512221-1-29-0', 'hep-th-0512221-2-29-0'], ['hep-th-0512221-1-38-0', 'hep-th-0512221-2-38-0'], ['hep-th-0512221-1-38-1', 'hep-th-0512221-2-38-1'], ['hep-th-0512221-1-38-2', 'hep-th-0512221-2-38-2'], ['hep-th-0512221-1-38-3', 'hep-th-0512221-2-38-3'], ['hep-th-0512221-1-38-4', 'hep-th-0512221-2-38-4'], ['hep-th-0512221-1-17-0', 'hep-th-0512221-2-17-0'], ['hep-th-0512221-1-17-1', 'hep-th-0512221-2-17-1'], ['hep-th-0512221-1-56-0', 'hep-th-0512221-2-56-0'], ['hep-th-0512221-1-59-0', 'hep-th-0512221-2-59-0'], ['hep-th-0512221-1-59-1', 'hep-th-0512221-2-59-1'], ['hep-th-0512221-1-59-2', 'hep-th-0512221-2-59-2'], ['hep-th-0512221-1-67-0', 'hep-th-0512221-2-67-0'], ['hep-th-0512221-1-67-1', 'hep-th-0512221-2-67-1'], ['hep-th-0512221-1-47-0', 'hep-th-0512221-2-47-0'], ['hep-th-0512221-1-47-1', 'hep-th-0512221-2-47-1'], ['hep-th-0512221-1-47-2', 'hep-th-0512221-2-47-2'], ['hep-th-0512221-1-47-3', 'hep-th-0512221-2-47-3'], ['hep-th-0512221-1-47-4', 'hep-th-0512221-2-47-4'], ['hep-th-0512221-1-47-5', 'hep-th-0512221-2-47-5'], ['hep-th-0512221-1-47-6', 'hep-th-0512221-2-47-6'], ['hep-th-0512221-1-47-7', 'hep-th-0512221-2-47-7'], ['hep-th-0512221-1-47-8', 'hep-th-0512221-2-47-8'], ['hep-th-0512221-1-25-0', 'hep-th-0512221-2-25-0'], ['hep-th-0512221-1-25-1', 'hep-th-0512221-2-25-1'], ['hep-th-0512221-1-30-0', 'hep-th-0512221-2-30-0'], ['hep-th-0512221-1-30-1', 'hep-th-0512221-2-30-1'], ['hep-th-0512221-1-46-0', 'hep-th-0512221-2-46-0'], ['hep-th-0512221-1-46-1', 'hep-th-0512221-2-46-1'], ['hep-th-0512221-1-46-2', 'hep-th-0512221-2-46-2'], ['hep-th-0512221-1-8-0', 'hep-th-0512221-2-8-0'], ['hep-th-0512221-1-8-1', 'hep-th-0512221-2-8-1'], ['hep-th-0512221-1-8-2', 'hep-th-0512221-2-8-2'], ['hep-th-0512221-1-68-0', 'hep-th-0512221-2-68-0'], ['hep-th-0512221-1-68-1', 'hep-th-0512221-2-68-1'], ['hep-th-0512221-1-68-2', 'hep-th-0512221-2-68-2'], ['hep-th-0512221-1-68-3', 'hep-th-0512221-2-68-3'], ['hep-th-0512221-1-68-4', 'hep-th-0512221-2-68-4'], ['hep-th-0512221-1-68-5', 'hep-th-0512221-2-68-5'], ['hep-th-0512221-1-68-6', 'hep-th-0512221-2-68-6'], ['hep-th-0512221-1-68-7', 'hep-th-0512221-2-68-7'], ['hep-th-0512221-1-68-8', 'hep-th-0512221-2-68-8'], ['hep-th-0512221-1-70-0', 'hep-th-0512221-2-70-0'], ['hep-th-0512221-1-28-0', 'hep-th-0512221-2-28-0'], ['hep-th-0512221-1-28-1', 'hep-th-0512221-2-28-1'], ['hep-th-0512221-1-28-2', 'hep-th-0512221-2-28-2'], ['hep-th-0512221-1-28-3', 'hep-th-0512221-2-28-3'], ['hep-th-0512221-1-66-0', 'hep-th-0512221-2-66-0'], ['hep-th-0512221-1-66-1', 'hep-th-0512221-2-66-1'], ['hep-th-0512221-1-66-2', 'hep-th-0512221-2-66-2'], ['hep-th-0512221-1-10-0', 'hep-th-0512221-2-10-0'], ['hep-th-0512221-1-10-1', 'hep-th-0512221-2-10-1'], ['hep-th-0512221-1-15-0', 'hep-th-0512221-2-15-0'], ['hep-th-0512221-1-22-0', 'hep-th-0512221-2-22-0'], ['hep-th-0512221-1-22-1', 'hep-th-0512221-2-22-1'], ['hep-th-0512221-1-35-0', 'hep-th-0512221-2-35-0'], ['hep-th-0512221-1-35-1', 'hep-th-0512221-2-35-1'], ['hep-th-0512221-1-4-0', 'hep-th-0512221-2-4-0'], ['hep-th-0512221-1-4-2', 'hep-th-0512221-2-4-2'], ['hep-th-0512221-1-4-3', 'hep-th-0512221-2-4-3'], ['hep-th-0512221-1-27-0', 'hep-th-0512221-2-27-0'], ['hep-th-0512221-1-27-1', 'hep-th-0512221-2-27-1'], ['hep-th-0512221-1-27-2', 'hep-th-0512221-2-27-2'], ['hep-th-0512221-1-27-3', 'hep-th-0512221-2-27-3'], ['hep-th-0512221-1-27-4', 'hep-th-0512221-2-27-4'], ['hep-th-0512221-1-27-5', 'hep-th-0512221-2-27-5'], ['hep-th-0512221-1-54-0', 'hep-th-0512221-2-54-0'], ['hep-th-0512221-1-54-1', 'hep-th-0512221-2-54-1'], ['hep-th-0512221-1-54-2', 'hep-th-0512221-2-54-2'], ['hep-th-0512221-1-54-3', 'hep-th-0512221-2-54-3'], ['hep-th-0512221-1-48-0', 'hep-th-0512221-2-48-0'], ['hep-th-0512221-1-48-1', 'hep-th-0512221-2-48-1'], ['hep-th-0512221-1-51-0', 'hep-th-0512221-2-51-0'], ['hep-th-0512221-1-51-1', 'hep-th-0512221-2-51-1'], ['hep-th-0512221-1-51-2', 'hep-th-0512221-2-51-2'], ['hep-th-0512221-1-51-3', 'hep-th-0512221-2-51-3'], ['hep-th-0512221-1-50-0', 'hep-th-0512221-2-50-0'], ['hep-th-0512221-1-50-1', 'hep-th-0512221-2-50-1'], ['hep-th-0512221-1-50-2', 'hep-th-0512221-2-50-2'], ['hep-th-0512221-1-50-3', 'hep-th-0512221-2-50-3'], ['hep-th-0512221-1-50-4', 'hep-th-0512221-2-50-4'], ['hep-th-0512221-1-50-5', 'hep-th-0512221-2-50-5'], ['hep-th-0512221-1-50-6', 'hep-th-0512221-2-50-6'], ['hep-th-0512221-1-50-7', 'hep-th-0512221-2-50-7'], ['hep-th-0512221-1-76-0', 'hep-th-0512221-2-76-0'], ['hep-th-0512221-1-76-1', 'hep-th-0512221-2-76-1'], ['hep-th-0512221-1-76-2', 'hep-th-0512221-2-76-2'], ['hep-th-0512221-1-37-0', 'hep-th-0512221-2-37-0'], ['hep-th-0512221-1-37-1', 'hep-th-0512221-2-37-1'], ['hep-th-0512221-1-3-0', 'hep-th-0512221-2-3-0'], ['hep-th-0512221-1-3-1', 'hep-th-0512221-2-3-1'], ['hep-th-0512221-1-3-2', 'hep-th-0512221-2-3-2'], ['hep-th-0512221-1-31-0', 'hep-th-0512221-2-31-0'], ['hep-th-0512221-1-73-0', 'hep-th-0512221-2-73-0'], ['hep-th-0512221-1-9-0', 'hep-th-0512221-2-9-0'], ['hep-th-0512221-1-9-1', 'hep-th-0512221-2-9-1'], ['hep-th-0512221-1-9-2', 'hep-th-0512221-2-9-2'], ['hep-th-0512221-1-9-3', 'hep-th-0512221-2-9-3'], ['hep-th-0512221-1-44-0', 'hep-th-0512221-2-44-0'], ['hep-th-0512221-1-44-1', 'hep-th-0512221-2-44-1'], ['hep-th-0512221-1-44-2', 'hep-th-0512221-2-44-2'], ['hep-th-0512221-1-44-3', 'hep-th-0512221-2-44-3'], ['hep-th-0512221-1-44-4', 'hep-th-0512221-2-44-4'], ['hep-th-0512221-1-41-0', 'hep-th-0512221-2-41-0'], ['hep-th-0512221-1-41-1', 'hep-th-0512221-2-41-1'], ['hep-th-0512221-1-41-2', 'hep-th-0512221-2-41-2'], ['hep-th-0512221-1-2-0', 'hep-th-0512221-2-2-0'], ['hep-th-0512221-1-2-1', 'hep-th-0512221-2-2-1'], ['hep-th-0512221-1-18-0', 'hep-th-0512221-2-18-0'], ['hep-th-0512221-1-18-1', 'hep-th-0512221-2-18-1'], ['hep-th-0512221-1-75-0', 'hep-th-0512221-2-75-0'], ['hep-th-0512221-1-75-1', 'hep-th-0512221-2-75-1'], ['hep-th-0512221-1-75-2', 'hep-th-0512221-2-75-2'], ['hep-th-0512221-1-75-3', 'hep-th-0512221-2-75-3'], ['hep-th-0512221-1-11-0', 'hep-th-0512221-2-11-0'], ['hep-th-0512221-1-11-1', 'hep-th-0512221-2-11-1'], ['hep-th-0512221-1-11-3', 'hep-th-0512221-2-11-3'], ['hep-th-0512221-1-11-4', 'hep-th-0512221-2-11-4'], ['hep-th-0512221-1-11-5', 'hep-th-0512221-2-11-5'], ['hep-th-0512221-1-11-6', 'hep-th-0512221-2-11-6'], ['hep-th-0512221-1-11-7', 'hep-th-0512221-2-11-7'], ['hep-th-0512221-1-11-8', 'hep-th-0512221-2-11-8'], ['hep-th-0512221-1-11-9', 'hep-th-0512221-2-11-9'], ['hep-th-0512221-1-11-10', 'hep-th-0512221-2-11-10'], ['hep-th-0512221-1-11-11', 'hep-th-0512221-2-11-11'], ['hep-th-0512221-1-24-0', 'hep-th-0512221-2-24-0'], ['hep-th-0512221-1-24-1', 'hep-th-0512221-2-24-1'], ['hep-th-0512221-1-24-2', 'hep-th-0512221-2-24-2'], ['hep-th-0512221-1-24-3', 'hep-th-0512221-2-24-3'], ['hep-th-0512221-1-24-4', 'hep-th-0512221-2-24-4'], ['hep-th-0512221-1-24-5', 'hep-th-0512221-2-24-5'], ['hep-th-0512221-1-24-6', 'hep-th-0512221-2-24-6'], ['hep-th-0512221-1-24-7', 'hep-th-0512221-2-24-7'], ['hep-th-0512221-1-24-8', 'hep-th-0512221-2-24-8'], ['hep-th-0512221-1-0-0', 'hep-th-0512221-2-0-0'], ['hep-th-0512221-1-0-1', 'hep-th-0512221-2-0-1'], ['hep-th-0512221-1-0-2', 'hep-th-0512221-2-0-2'], ['hep-th-0512221-1-0-3', 'hep-th-0512221-2-0-3'], ['hep-th-0512221-1-0-4', 'hep-th-0512221-2-0-4'], ['hep-th-0512221-1-0-5', 'hep-th-0512221-2-0-5'], ['hep-th-0512221-1-72-0', 'hep-th-0512221-2-72-0'], ['hep-th-0512221-1-72-1', 'hep-th-0512221-2-72-1'], ['hep-th-0512221-1-69-0', 'hep-th-0512221-2-69-0'], ['hep-th-0512221-1-69-1', 'hep-th-0512221-2-69-1'], ['hep-th-0512221-1-69-2', 'hep-th-0512221-2-69-2'], ['hep-th-0512221-1-69-3', 'hep-th-0512221-2-69-3'], ['hep-th-0512221-1-69-4', 'hep-th-0512221-2-69-4'], ['hep-th-0512221-1-69-5', 'hep-th-0512221-2-69-5'], ['hep-th-0512221-1-5-0', 'hep-th-0512221-2-5-0'], ['hep-th-0512221-1-5-1', 'hep-th-0512221-2-5-1'], ['hep-th-0512221-1-5-2', 'hep-th-0512221-2-5-2'], ['hep-th-0512221-1-49-0', 'hep-th-0512221-2-49-0'], ['hep-th-0512221-1-49-1', 'hep-th-0512221-2-49-1'], ['hep-th-0512221-1-57-0', 'hep-th-0512221-2-57-0'], ['hep-th-0512221-1-39-0', 'hep-th-0512221-2-39-0'], ['hep-th-0512221-1-39-1', 'hep-th-0512221-2-39-1'], ['hep-th-0512221-1-39-2', 'hep-th-0512221-2-39-2'], ['hep-th-0512221-1-39-3', 'hep-th-0512221-2-39-3'], ['hep-th-0512221-1-39-4', 'hep-th-0512221-2-39-4'], ['hep-th-0512221-1-39-5', 'hep-th-0512221-2-39-5'], ['hep-th-0512221-1-20-0', 'hep-th-0512221-2-20-0'], ['hep-th-0512221-1-20-1', 'hep-th-0512221-2-20-1'], ['hep-th-0512221-1-20-2', 'hep-th-0512221-2-20-2'], ['hep-th-0512221-1-20-3', 'hep-th-0512221-2-20-3'], ['hep-th-0512221-1-20-4', 'hep-th-0512221-2-20-4'], ['hep-th-0512221-1-20-5', 'hep-th-0512221-2-20-5'], ['hep-th-0512221-1-20-6', 'hep-th-0512221-2-20-6'], ['hep-th-0512221-1-20-7', 'hep-th-0512221-2-20-7'], ['hep-th-0512221-1-45-0', 'hep-th-0512221-2-45-0'], ['hep-th-0512221-1-45-1', 'hep-th-0512221-2-45-1'], ['hep-th-0512221-1-45-2', 'hep-th-0512221-2-45-2'], ['hep-th-0512221-1-45-3', 'hep-th-0512221-2-45-3'], ['hep-th-0512221-1-55-0', 'hep-th-0512221-2-55-0'], ['hep-th-0512221-1-55-1', 'hep-th-0512221-2-55-1'], ['hep-th-0512221-1-55-2', 'hep-th-0512221-2-55-2'], ['hep-th-0512221-1-55-3', 'hep-th-0512221-2-55-3'], ['hep-th-0512221-1-63-0', 'hep-th-0512221-2-63-0'], ['hep-th-0512221-1-63-1', 'hep-th-0512221-2-63-1'], ['hep-th-0512221-1-63-2', 'hep-th-0512221-2-63-2'], ['hep-th-0512221-1-63-3', 'hep-th-0512221-2-63-3'], ['hep-th-0512221-1-63-4', 'hep-th-0512221-2-63-4'], ['hep-th-0512221-1-63-5', 'hep-th-0512221-2-63-5'], ['hep-th-0512221-1-63-6', 'hep-th-0512221-2-63-6'], ['hep-th-0512221-1-63-7', 'hep-th-0512221-2-63-7'], ['hep-th-0512221-1-63-8', 'hep-th-0512221-2-63-8'], ['hep-th-0512221-1-19-0', 'hep-th-0512221-2-19-0'], ['hep-th-0512221-1-19-1', 'hep-th-0512221-2-19-1'], ['hep-th-0512221-1-19-2', 'hep-th-0512221-2-19-2'], ['hep-th-0512221-1-19-3', 'hep-th-0512221-2-19-3'], ['hep-th-0512221-1-34-0', 'hep-th-0512221-2-34-0'], ['hep-th-0512221-1-34-1', 'hep-th-0512221-2-34-1'], ['hep-th-0512221-1-34-2', 'hep-th-0512221-2-34-2'], ['hep-th-0512221-1-40-0', 'hep-th-0512221-2-40-0'], ['hep-th-0512221-1-40-1', 'hep-th-0512221-2-40-1'], ['hep-th-0512221-1-40-2', 'hep-th-0512221-2-40-2'], ['hep-th-0512221-1-40-3', 'hep-th-0512221-2-40-3'], ['hep-th-0512221-1-40-4', 'hep-th-0512221-2-40-4'], ['hep-th-0512221-1-40-6', 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['hep-th-0512221-1-21-2', 'hep-th-0512221-2-21-2'], ['hep-th-0512221-1-21-3', 'hep-th-0512221-2-21-3'], ['hep-th-0512221-1-21-4', 'hep-th-0512221-2-21-4'], ['hep-th-0512221-1-21-5', 'hep-th-0512221-2-21-5'], ['hep-th-0512221-1-16-0', 'hep-th-0512221-2-16-0'], ['hep-th-0512221-1-16-1', 'hep-th-0512221-2-16-1'], ['hep-th-0512221-1-16-2', 'hep-th-0512221-2-16-2'], ['hep-th-0512221-1-16-3', 'hep-th-0512221-2-16-3'], ['hep-th-0512221-1-16-4', 'hep-th-0512221-2-16-4'], ['hep-th-0512221-1-16-5', 'hep-th-0512221-2-16-5'], ['hep-th-0512221-1-16-6', 'hep-th-0512221-2-16-6'], ['hep-th-0512221-1-16-7', 'hep-th-0512221-2-16-7'], ['hep-th-0512221-1-29-0', 'hep-th-0512221-2-29-0'], ['hep-th-0512221-1-38-0', 'hep-th-0512221-2-38-0'], ['hep-th-0512221-1-38-1', 'hep-th-0512221-2-38-1'], ['hep-th-0512221-1-38-2', 'hep-th-0512221-2-38-2'], ['hep-th-0512221-1-38-3', 'hep-th-0512221-2-38-3'], ['hep-th-0512221-1-38-4', 'hep-th-0512221-2-38-4'], ['hep-th-0512221-1-17-0', 'hep-th-0512221-2-17-0'], 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['hep-th-0512221-1-28-2', 'hep-th-0512221-2-28-2'], ['hep-th-0512221-1-28-3', 'hep-th-0512221-2-28-3'], ['hep-th-0512221-1-66-0', 'hep-th-0512221-2-66-0'], ['hep-th-0512221-1-66-1', 'hep-th-0512221-2-66-1'], ['hep-th-0512221-1-66-2', 'hep-th-0512221-2-66-2'], ['hep-th-0512221-1-10-0', 'hep-th-0512221-2-10-0'], ['hep-th-0512221-1-10-1', 'hep-th-0512221-2-10-1'], ['hep-th-0512221-1-15-0', 'hep-th-0512221-2-15-0'], ['hep-th-0512221-1-22-0', 'hep-th-0512221-2-22-0'], ['hep-th-0512221-1-22-1', 'hep-th-0512221-2-22-1'], ['hep-th-0512221-1-35-0', 'hep-th-0512221-2-35-0'], ['hep-th-0512221-1-35-1', 'hep-th-0512221-2-35-1'], ['hep-th-0512221-1-4-0', 'hep-th-0512221-2-4-0'], ['hep-th-0512221-1-4-2', 'hep-th-0512221-2-4-2'], ['hep-th-0512221-1-4-3', 'hep-th-0512221-2-4-3'], ['hep-th-0512221-1-27-0', 'hep-th-0512221-2-27-0'], ['hep-th-0512221-1-27-1', 'hep-th-0512221-2-27-1'], ['hep-th-0512221-1-27-2', 'hep-th-0512221-2-27-2'], ['hep-th-0512221-1-27-3', 'hep-th-0512221-2-27-3'], ['hep-th-0512221-1-27-4', 'hep-th-0512221-2-27-4'], ['hep-th-0512221-1-27-5', 'hep-th-0512221-2-27-5'], ['hep-th-0512221-1-54-0', 'hep-th-0512221-2-54-0'], ['hep-th-0512221-1-54-1', 'hep-th-0512221-2-54-1'], ['hep-th-0512221-1-54-2', 'hep-th-0512221-2-54-2'], ['hep-th-0512221-1-54-3', 'hep-th-0512221-2-54-3'], ['hep-th-0512221-1-48-0', 'hep-th-0512221-2-48-0'], ['hep-th-0512221-1-48-1', 'hep-th-0512221-2-48-1'], ['hep-th-0512221-1-51-0', 'hep-th-0512221-2-51-0'], ['hep-th-0512221-1-51-1', 'hep-th-0512221-2-51-1'], ['hep-th-0512221-1-51-2', 'hep-th-0512221-2-51-2'], ['hep-th-0512221-1-51-3', 'hep-th-0512221-2-51-3'], ['hep-th-0512221-1-50-0', 'hep-th-0512221-2-50-0'], ['hep-th-0512221-1-50-1', 'hep-th-0512221-2-50-1'], ['hep-th-0512221-1-50-2', 'hep-th-0512221-2-50-2'], ['hep-th-0512221-1-50-3', 'hep-th-0512221-2-50-3'], ['hep-th-0512221-1-50-4', 'hep-th-0512221-2-50-4'], ['hep-th-0512221-1-50-5', 'hep-th-0512221-2-50-5'], ['hep-th-0512221-1-50-6', 'hep-th-0512221-2-50-6'], ['hep-th-0512221-1-50-7', 'hep-th-0512221-2-50-7'], ['hep-th-0512221-1-76-0', 'hep-th-0512221-2-76-0'], ['hep-th-0512221-1-76-1', 'hep-th-0512221-2-76-1'], ['hep-th-0512221-1-76-2', 'hep-th-0512221-2-76-2'], ['hep-th-0512221-1-37-0', 'hep-th-0512221-2-37-0'], ['hep-th-0512221-1-37-1', 'hep-th-0512221-2-37-1'], ['hep-th-0512221-1-3-0', 'hep-th-0512221-2-3-0'], ['hep-th-0512221-1-3-1', 'hep-th-0512221-2-3-1'], ['hep-th-0512221-1-3-2', 'hep-th-0512221-2-3-2'], ['hep-th-0512221-1-31-0', 'hep-th-0512221-2-31-0'], ['hep-th-0512221-1-73-0', 'hep-th-0512221-2-73-0'], ['hep-th-0512221-1-9-0', 'hep-th-0512221-2-9-0'], ['hep-th-0512221-1-9-1', 'hep-th-0512221-2-9-1'], ['hep-th-0512221-1-9-2', 'hep-th-0512221-2-9-2'], ['hep-th-0512221-1-9-3', 'hep-th-0512221-2-9-3'], ['hep-th-0512221-1-44-0', 'hep-th-0512221-2-44-0'], ['hep-th-0512221-1-44-1', 'hep-th-0512221-2-44-1'], ['hep-th-0512221-1-44-2', 'hep-th-0512221-2-44-2'], ['hep-th-0512221-1-44-3', 'hep-th-0512221-2-44-3'], ['hep-th-0512221-1-44-4', 'hep-th-0512221-2-44-4'], ['hep-th-0512221-1-41-0', 'hep-th-0512221-2-41-0'], ['hep-th-0512221-1-41-1', 'hep-th-0512221-2-41-1'], ['hep-th-0512221-1-41-2', 'hep-th-0512221-2-41-2'], ['hep-th-0512221-1-2-0', 'hep-th-0512221-2-2-0'], ['hep-th-0512221-1-2-1', 'hep-th-0512221-2-2-1'], ['hep-th-0512221-1-18-0', 'hep-th-0512221-2-18-0'], ['hep-th-0512221-1-18-1', 'hep-th-0512221-2-18-1'], ['hep-th-0512221-1-75-0', 'hep-th-0512221-2-75-0'], ['hep-th-0512221-1-75-1', 'hep-th-0512221-2-75-1'], ['hep-th-0512221-1-75-2', 'hep-th-0512221-2-75-2'], ['hep-th-0512221-1-75-3', 'hep-th-0512221-2-75-3'], ['hep-th-0512221-1-11-0', 'hep-th-0512221-2-11-0'], ['hep-th-0512221-1-11-1', 'hep-th-0512221-2-11-1'], ['hep-th-0512221-1-11-3', 'hep-th-0512221-2-11-3'], ['hep-th-0512221-1-11-4', 'hep-th-0512221-2-11-4'], ['hep-th-0512221-1-11-5', 'hep-th-0512221-2-11-5'], ['hep-th-0512221-1-11-6', 'hep-th-0512221-2-11-6'], ['hep-th-0512221-1-11-7', 'hep-th-0512221-2-11-7'], ['hep-th-0512221-1-11-8', 'hep-th-0512221-2-11-8'], ['hep-th-0512221-1-11-9', 'hep-th-0512221-2-11-9'], ['hep-th-0512221-1-11-10', 'hep-th-0512221-2-11-10'], ['hep-th-0512221-1-11-11', 'hep-th-0512221-2-11-11'], ['hep-th-0512221-1-24-0', 'hep-th-0512221-2-24-0'], ['hep-th-0512221-1-24-1', 'hep-th-0512221-2-24-1'], ['hep-th-0512221-1-24-2', 'hep-th-0512221-2-24-2'], ['hep-th-0512221-1-24-3', 'hep-th-0512221-2-24-3'], ['hep-th-0512221-1-24-4', 'hep-th-0512221-2-24-4'], ['hep-th-0512221-1-24-5', 'hep-th-0512221-2-24-5'], ['hep-th-0512221-1-24-6', 'hep-th-0512221-2-24-6'], ['hep-th-0512221-1-24-7', 'hep-th-0512221-2-24-7'], ['hep-th-0512221-1-24-8', 'hep-th-0512221-2-24-8'], ['hep-th-0512221-1-0-0', 'hep-th-0512221-2-0-0'], ['hep-th-0512221-1-0-1', 'hep-th-0512221-2-0-1'], ['hep-th-0512221-1-0-2', 'hep-th-0512221-2-0-2'], ['hep-th-0512221-1-0-3', 'hep-th-0512221-2-0-3'], ['hep-th-0512221-1-0-4', 'hep-th-0512221-2-0-4'], ['hep-th-0512221-1-0-5', 'hep-th-0512221-2-0-5'], ['hep-th-0512221-1-72-0', 'hep-th-0512221-2-72-0'], ['hep-th-0512221-1-72-1', 'hep-th-0512221-2-72-1'], ['hep-th-0512221-1-69-0', 'hep-th-0512221-2-69-0'], ['hep-th-0512221-1-69-1', 'hep-th-0512221-2-69-1'], ['hep-th-0512221-1-69-2', 'hep-th-0512221-2-69-2'], ['hep-th-0512221-1-69-3', 'hep-th-0512221-2-69-3'], ['hep-th-0512221-1-69-4', 'hep-th-0512221-2-69-4'], ['hep-th-0512221-1-69-5', 'hep-th-0512221-2-69-5'], ['hep-th-0512221-1-5-0', 'hep-th-0512221-2-5-0'], ['hep-th-0512221-1-5-1', 'hep-th-0512221-2-5-1'], ['hep-th-0512221-1-5-2', 'hep-th-0512221-2-5-2'], ['hep-th-0512221-1-49-0', 'hep-th-0512221-2-49-0'], ['hep-th-0512221-1-49-1', 'hep-th-0512221-2-49-1'], ['hep-th-0512221-1-57-0', 'hep-th-0512221-2-57-0'], ['hep-th-0512221-1-39-0', 'hep-th-0512221-2-39-0'], ['hep-th-0512221-1-39-1', 'hep-th-0512221-2-39-1'], ['hep-th-0512221-1-39-2', 'hep-th-0512221-2-39-2'], ['hep-th-0512221-1-39-3', 'hep-th-0512221-2-39-3'], ['hep-th-0512221-1-39-4', 'hep-th-0512221-2-39-4'], ['hep-th-0512221-1-39-5', 'hep-th-0512221-2-39-5'], ['hep-th-0512221-1-20-0', 'hep-th-0512221-2-20-0'], ['hep-th-0512221-1-20-1', 'hep-th-0512221-2-20-1'], ['hep-th-0512221-1-20-2', 'hep-th-0512221-2-20-2'], ['hep-th-0512221-1-20-3', 'hep-th-0512221-2-20-3'], ['hep-th-0512221-1-20-4', 'hep-th-0512221-2-20-4'], ['hep-th-0512221-1-20-5', 'hep-th-0512221-2-20-5'], ['hep-th-0512221-1-20-6', 'hep-th-0512221-2-20-6'], ['hep-th-0512221-1-20-7', 'hep-th-0512221-2-20-7'], ['hep-th-0512221-1-45-0', 'hep-th-0512221-2-45-0'], ['hep-th-0512221-1-45-1', 'hep-th-0512221-2-45-1'], ['hep-th-0512221-1-45-2', 'hep-th-0512221-2-45-2'], ['hep-th-0512221-1-45-3', 'hep-th-0512221-2-45-3'], ['hep-th-0512221-1-55-0', 'hep-th-0512221-2-55-0'], ['hep-th-0512221-1-55-1', 'hep-th-0512221-2-55-1'], ['hep-th-0512221-1-55-2', 'hep-th-0512221-2-55-2'], ['hep-th-0512221-1-55-3', 'hep-th-0512221-2-55-3'], ['hep-th-0512221-1-63-0', 'hep-th-0512221-2-63-0'], ['hep-th-0512221-1-63-1', 'hep-th-0512221-2-63-1'], ['hep-th-0512221-1-63-2', 'hep-th-0512221-2-63-2'], ['hep-th-0512221-1-63-3', 'hep-th-0512221-2-63-3'], ['hep-th-0512221-1-63-4', 'hep-th-0512221-2-63-4'], ['hep-th-0512221-1-63-5', 'hep-th-0512221-2-63-5'], ['hep-th-0512221-1-63-6', 'hep-th-0512221-2-63-6'], ['hep-th-0512221-1-63-7', 'hep-th-0512221-2-63-7'], ['hep-th-0512221-1-63-8', 'hep-th-0512221-2-63-8'], ['hep-th-0512221-1-19-0', 'hep-th-0512221-2-19-0'], ['hep-th-0512221-1-19-1', 'hep-th-0512221-2-19-1'], ['hep-th-0512221-1-19-2', 'hep-th-0512221-2-19-2'], ['hep-th-0512221-1-19-3', 'hep-th-0512221-2-19-3'], ['hep-th-0512221-1-34-0', 'hep-th-0512221-2-34-0'], ['hep-th-0512221-1-34-1', 'hep-th-0512221-2-34-1'], ['hep-th-0512221-1-34-2', 'hep-th-0512221-2-34-2'], ['hep-th-0512221-1-40-0', 'hep-th-0512221-2-40-0'], ['hep-th-0512221-1-40-1', 'hep-th-0512221-2-40-1'], ['hep-th-0512221-1-40-2', 'hep-th-0512221-2-40-2'], ['hep-th-0512221-1-40-3', 'hep-th-0512221-2-40-3'], ['hep-th-0512221-1-40-4', 'hep-th-0512221-2-40-4'], ['hep-th-0512221-1-40-6', 'hep-th-0512221-2-40-6'], ['hep-th-0512221-1-40-7', 'hep-th-0512221-2-40-7'], ['hep-th-0512221-1-14-0', 'hep-th-0512221-2-14-0'], ['hep-th-0512221-1-14-1', 'hep-th-0512221-2-14-1'], ['hep-th-0512221-1-14-2', 'hep-th-0512221-2-14-2'], ['hep-th-0512221-1-6-0', 'hep-th-0512221-2-6-0'], ['hep-th-0512221-1-6-1', 'hep-th-0512221-2-6-1'], ['hep-th-0512221-2-68-0', 'hep-th-0512221-3-66-0'], ['hep-th-0512221-2-68-1', 'hep-th-0512221-3-66-1'], ['hep-th-0512221-2-68-2', 'hep-th-0512221-3-66-2'], ['hep-th-0512221-2-68-3', 'hep-th-0512221-3-66-3'], ['hep-th-0512221-2-68-4', 'hep-th-0512221-3-66-4'], ['hep-th-0512221-2-68-5', 'hep-th-0512221-3-66-5'], ['hep-th-0512221-2-68-6', 'hep-th-0512221-3-66-6'], ['hep-th-0512221-2-68-7', 'hep-th-0512221-3-66-7'], ['hep-th-0512221-2-68-8', 'hep-th-0512221-3-66-8'], ['hep-th-0512221-2-59-0', 'hep-th-0512221-3-58-0'], ['hep-th-0512221-2-59-1', 'hep-th-0512221-3-58-1'], ['hep-th-0512221-2-59-2', 'hep-th-0512221-3-58-2'], ['hep-th-0512221-2-2-0', 'hep-th-0512221-3-2-0'], ['hep-th-0512221-2-2-1', 'hep-th-0512221-3-2-1'], ['hep-th-0512221-2-30-0', 'hep-th-0512221-3-29-0'], ['hep-th-0512221-2-30-1', 'hep-th-0512221-3-29-1'], ['hep-th-0512221-2-21-0', 'hep-th-0512221-3-20-0'], ['hep-th-0512221-2-21-1', 'hep-th-0512221-3-20-1'], ['hep-th-0512221-2-21-2', 'hep-th-0512221-3-20-2'], ['hep-th-0512221-2-21-3', 'hep-th-0512221-3-20-3'], ['hep-th-0512221-2-21-4', 'hep-th-0512221-3-20-4'], ['hep-th-0512221-2-21-5', 'hep-th-0512221-3-20-5'], ['hep-th-0512221-2-73-0', 'hep-th-0512221-3-71-0'], ['hep-th-0512221-2-65-0', 'hep-th-0512221-3-64-0'], ['hep-th-0512221-2-65-1', 'hep-th-0512221-3-64-1'], ['hep-th-0512221-2-44-0', 'hep-th-0512221-3-43-0'], ['hep-th-0512221-2-44-1', 'hep-th-0512221-3-43-1'], ['hep-th-0512221-2-44-2', 'hep-th-0512221-3-43-2'], ['hep-th-0512221-2-44-3', 'hep-th-0512221-3-43-3'], ['hep-th-0512221-2-44-4', 'hep-th-0512221-3-43-4'], ['hep-th-0512221-2-25-0', 'hep-th-0512221-3-24-0'], ['hep-th-0512221-2-25-1', 'hep-th-0512221-3-24-1'], ['hep-th-0512221-2-17-0', 'hep-th-0512221-3-16-0'], ['hep-th-0512221-2-17-1', 'hep-th-0512221-3-16-1'], ['hep-th-0512221-2-11-0', 'hep-th-0512221-3-10-0'], ['hep-th-0512221-2-11-1', 'hep-th-0512221-3-10-1'], ['hep-th-0512221-2-11-2', 'hep-th-0512221-3-10-2'], ['hep-th-0512221-2-11-3', 'hep-th-0512221-3-10-3'], ['hep-th-0512221-2-11-4', 'hep-th-0512221-3-10-4'], ['hep-th-0512221-2-11-5', 'hep-th-0512221-3-10-5'], ['hep-th-0512221-2-11-8', 'hep-th-0512221-3-10-8'], ['hep-th-0512221-2-11-9', 'hep-th-0512221-3-10-9'], ['hep-th-0512221-2-6-0', 'hep-th-0512221-3-6-0'], ['hep-th-0512221-2-45-0', 'hep-th-0512221-3-44-0'], ['hep-th-0512221-2-45-1', 'hep-th-0512221-3-44-1'], ['hep-th-0512221-2-45-2', 'hep-th-0512221-3-44-2'], ['hep-th-0512221-2-75-0', 'hep-th-0512221-3-73-0'], ['hep-th-0512221-2-31-0', 'hep-th-0512221-3-30-0'], ['hep-th-0512221-2-20-0', 'hep-th-0512221-3-19-0'], ['hep-th-0512221-2-20-1', 'hep-th-0512221-3-19-1'], ['hep-th-0512221-2-20-2', 'hep-th-0512221-3-19-2'], ['hep-th-0512221-2-20-3', 'hep-th-0512221-3-19-3'], ['hep-th-0512221-2-20-4', 'hep-th-0512221-3-19-4'], ['hep-th-0512221-2-20-5', 'hep-th-0512221-3-19-5'], ['hep-th-0512221-2-20-6', 'hep-th-0512221-3-19-6'], ['hep-th-0512221-2-20-7', 'hep-th-0512221-3-19-7'], ['hep-th-0512221-2-29-0', 'hep-th-0512221-3-28-0'], ['hep-th-0512221-2-69-0', 'hep-th-0512221-3-67-0'], ['hep-th-0512221-2-69-1', 'hep-th-0512221-3-67-1'], ['hep-th-0512221-2-69-2', 'hep-th-0512221-3-67-2'], ['hep-th-0512221-2-69-3', 'hep-th-0512221-3-67-3'], ['hep-th-0512221-2-69-4', 'hep-th-0512221-3-67-4'], ['hep-th-0512221-2-69-5', 'hep-th-0512221-3-67-5'], ['hep-th-0512221-2-15-0', 'hep-th-0512221-3-14-0'], ['hep-th-0512221-2-34-0', 'hep-th-0512221-3-33-0'], ['hep-th-0512221-2-34-1', 'hep-th-0512221-3-33-1'], ['hep-th-0512221-2-34-2', 'hep-th-0512221-3-33-2'], ['hep-th-0512221-2-70-0', 'hep-th-0512221-3-68-0'], ['hep-th-0512221-2-0-1', 'hep-th-0512221-3-0-1'], ['hep-th-0512221-2-0-3', 'hep-th-0512221-3-0-3'], ['hep-th-0512221-2-4-0', 'hep-th-0512221-3-4-0'], ['hep-th-0512221-2-4-1', 'hep-th-0512221-3-4-1'], ['hep-th-0512221-2-4-2', 'hep-th-0512221-3-4-2'], ['hep-th-0512221-2-4-3', 'hep-th-0512221-3-4-3'], ['hep-th-0512221-2-39-0', 'hep-th-0512221-3-38-0'], ['hep-th-0512221-2-39-1', 'hep-th-0512221-3-38-1'], ['hep-th-0512221-2-39-2', 'hep-th-0512221-3-38-2'], ['hep-th-0512221-2-39-3', 'hep-th-0512221-3-38-3'], ['hep-th-0512221-2-39-4', 'hep-th-0512221-3-38-4'], ['hep-th-0512221-2-39-5', 'hep-th-0512221-3-38-5'], ['hep-th-0512221-2-24-0', 'hep-th-0512221-3-23-0'], ['hep-th-0512221-2-24-1', 'hep-th-0512221-3-23-1'], ['hep-th-0512221-2-24-2', 'hep-th-0512221-3-23-2'], ['hep-th-0512221-2-24-3', 'hep-th-0512221-3-23-3'], ['hep-th-0512221-2-24-4', 'hep-th-0512221-3-23-4'], ['hep-th-0512221-2-24-5', 'hep-th-0512221-3-23-5'], ['hep-th-0512221-2-24-6', 'hep-th-0512221-3-23-6'], ['hep-th-0512221-2-24-7', 'hep-th-0512221-3-23-7'], ['hep-th-0512221-2-24-8', 'hep-th-0512221-3-23-8'], ['hep-th-0512221-2-55-0', 'hep-th-0512221-3-54-0'], ['hep-th-0512221-2-55-1', 'hep-th-0512221-3-54-1'], ['hep-th-0512221-2-5-0', 'hep-th-0512221-3-5-0'], ['hep-th-0512221-2-5-1', 'hep-th-0512221-3-5-1'], ['hep-th-0512221-2-5-2', 'hep-th-0512221-3-5-2'], ['hep-th-0512221-2-16-0', 'hep-th-0512221-3-15-0'], ['hep-th-0512221-2-16-1', 'hep-th-0512221-3-15-1'], ['hep-th-0512221-2-16-2', 'hep-th-0512221-3-15-2'], ['hep-th-0512221-2-16-3', 'hep-th-0512221-3-15-3'], ['hep-th-0512221-2-16-4', 'hep-th-0512221-3-15-4'], ['hep-th-0512221-2-16-5', 'hep-th-0512221-3-15-5'], ['hep-th-0512221-2-16-6', 'hep-th-0512221-3-15-6'], ['hep-th-0512221-2-16-7', 'hep-th-0512221-3-15-7'], ['hep-th-0512221-2-35-0', 'hep-th-0512221-3-34-0'], ['hep-th-0512221-2-35-1', 'hep-th-0512221-3-34-1'], ['hep-th-0512221-2-37-0', 'hep-th-0512221-3-36-0'], ['hep-th-0512221-2-37-1', 'hep-th-0512221-3-36-1'], ['hep-th-0512221-2-63-0', 'hep-th-0512221-3-62-0'], ['hep-th-0512221-2-63-1', 'hep-th-0512221-3-62-1'], ['hep-th-0512221-2-63-2', 'hep-th-0512221-3-62-2'], ['hep-th-0512221-2-63-3', 'hep-th-0512221-3-62-3'], ['hep-th-0512221-2-63-4', 'hep-th-0512221-3-62-4'], ['hep-th-0512221-2-63-5', 'hep-th-0512221-3-62-5'], ['hep-th-0512221-2-63-7', 'hep-th-0512221-3-62-8'], ['hep-th-0512221-2-18-0', 'hep-th-0512221-3-17-0'], ['hep-th-0512221-2-18-1', 'hep-th-0512221-3-17-1'], ['hep-th-0512221-2-72-0', 'hep-th-0512221-3-70-0'], ['hep-th-0512221-2-72-1', 'hep-th-0512221-3-70-1'], ['hep-th-0512221-2-36-0', 'hep-th-0512221-3-35-0'], ['hep-th-0512221-2-14-0', 'hep-th-0512221-3-13-0'], ['hep-th-0512221-2-14-1', 'hep-th-0512221-3-13-1'], ['hep-th-0512221-2-14-2', 'hep-th-0512221-3-13-2'], ['hep-th-0512221-2-27-0', 'hep-th-0512221-3-26-0'], ['hep-th-0512221-2-27-1', 'hep-th-0512221-3-26-1'], ['hep-th-0512221-2-27-2', 'hep-th-0512221-3-26-2'], ['hep-th-0512221-2-27-3', 'hep-th-0512221-3-26-3'], ['hep-th-0512221-2-27-4', 'hep-th-0512221-3-26-4'], ['hep-th-0512221-2-27-5', 'hep-th-0512221-3-26-5'], ['hep-th-0512221-2-19-0', 'hep-th-0512221-3-18-0'], ['hep-th-0512221-2-19-1', 'hep-th-0512221-3-18-1'], ['hep-th-0512221-2-19-2', 'hep-th-0512221-3-18-2'], ['hep-th-0512221-2-19-3', 'hep-th-0512221-3-18-3'], ['hep-th-0512221-2-38-0', 'hep-th-0512221-3-37-0'], ['hep-th-0512221-2-38-1', 'hep-th-0512221-3-37-1'], ['hep-th-0512221-2-38-2', 'hep-th-0512221-3-37-2'], ['hep-th-0512221-2-38-3', 'hep-th-0512221-3-37-3'], ['hep-th-0512221-2-38-4', 'hep-th-0512221-3-37-4'], ['hep-th-0512221-2-47-0', 'hep-th-0512221-3-46-0'], ['hep-th-0512221-2-47-1', 'hep-th-0512221-3-46-1'], ['hep-th-0512221-2-47-2', 'hep-th-0512221-3-46-3'], ['hep-th-0512221-2-47-6', 'hep-th-0512221-3-46-11'], ['hep-th-0512221-2-47-8', 'hep-th-0512221-3-46-13'], ['hep-th-0512221-2-28-0', 'hep-th-0512221-3-27-0'], ['hep-th-0512221-2-28-1', 'hep-th-0512221-3-27-1'], ['hep-th-0512221-2-28-2', 'hep-th-0512221-3-27-2'], ['hep-th-0512221-2-28-3', 'hep-th-0512221-3-27-3'], ['hep-th-0512221-2-46-0', 'hep-th-0512221-3-45-0'], ['hep-th-0512221-2-46-1', 'hep-th-0512221-3-45-1'], ['hep-th-0512221-2-46-2', 'hep-th-0512221-3-45-2'], ['hep-th-0512221-2-61-0', 'hep-th-0512221-3-60-0'], ['hep-th-0512221-2-61-1', 'hep-th-0512221-3-60-1'], ['hep-th-0512221-2-61-2', 'hep-th-0512221-3-60-2'], ['hep-th-0512221-2-61-3', 'hep-th-0512221-3-60-3'], ['hep-th-0512221-2-61-4', 'hep-th-0512221-3-60-4'], ['hep-th-0512221-2-61-5', 'hep-th-0512221-3-60-5'], ['hep-th-0512221-2-61-6', 'hep-th-0512221-3-60-6'], ['hep-th-0512221-2-61-8', 'hep-th-0512221-3-60-8'], ['hep-th-0512221-2-48-1', 'hep-th-0512221-3-47-1'], ['hep-th-0512221-2-8-1', 'hep-th-0512221-3-8-1'], ['hep-th-0512221-2-8-2', 'hep-th-0512221-3-8-2'], ['hep-th-0512221-2-7-0', 'hep-th-0512221-3-7-0'], ['hep-th-0512221-2-7-1', 'hep-th-0512221-3-7-1'], ['hep-th-0512221-2-7-2', 'hep-th-0512221-3-7-2'], ['hep-th-0512221-2-3-0', 'hep-th-0512221-3-3-0'], ['hep-th-0512221-2-3-1', 'hep-th-0512221-3-3-1'], ['hep-th-0512221-2-3-2', 'hep-th-0512221-3-3-2'], ['hep-th-0512221-2-54-1', 'hep-th-0512221-3-53-1'], ['hep-th-0512221-2-54-2', 'hep-th-0512221-3-53-2'], ['hep-th-0512221-2-54-3', 'hep-th-0512221-3-53-3'], ['hep-th-0512221-2-56-0', 'hep-th-0512221-3-55-0'], ['hep-th-0512221-2-40-0', 'hep-th-0512221-3-39-0'], ['hep-th-0512221-2-40-1', 'hep-th-0512221-3-39-1'], ['hep-th-0512221-2-40-2', 'hep-th-0512221-3-39-2'], ['hep-th-0512221-2-40-3', 'hep-th-0512221-3-39-3'], ['hep-th-0512221-2-40-4', 'hep-th-0512221-3-39-4'], ['hep-th-0512221-2-40-6', 'hep-th-0512221-3-39-6'], ['hep-th-0512221-2-40-7', 'hep-th-0512221-3-39-7'], ['hep-th-0512221-2-57-0', 'hep-th-0512221-3-56-0'], ['hep-th-0512221-2-22-0', 'hep-th-0512221-3-21-0'], ['hep-th-0512221-2-22-1', 'hep-th-0512221-3-21-1'], ['hep-th-0512221-2-41-0', 'hep-th-0512221-3-40-0'], ['hep-th-0512221-2-41-1', 'hep-th-0512221-3-40-1'], ['hep-th-0512221-2-41-2', 'hep-th-0512221-3-40-2'], ['hep-th-0512221-2-51-2', 'hep-th-0512221-3-49-2'], ['hep-th-0512221-2-9-1', 'hep-th-0512221-3-9-1'], ['hep-th-0512221-2-10-0', 'hep-th-0512221-3-9-2']]

[['hep-th-0512221-2-11-10', 'hep-th-0512221-3-10-10'], ['hep-th-0512221-1-4-1', 'hep-th-0512221-2-4-1'], ['hep-th-0512221-1-11-2', 'hep-th-0512221-2-11-2'], ['hep-th-0512221-2-11-7', 'hep-th-0512221-3-10-7'], ['hep-th-0512221-2-6-1', 'hep-th-0512221-3-6-1'], ['hep-th-0512221-2-45-3', 'hep-th-0512221-3-44-3'], ['hep-th-0512221-2-75-1', 'hep-th-0512221-3-73-1'], ['hep-th-0512221-2-0-0', 'hep-th-0512221-3-0-0'], ['hep-th-0512221-2-0-2', 'hep-th-0512221-3-0-2'], ['hep-th-0512221-2-55-3', 'hep-th-0512221-3-54-3'], ['hep-th-0512221-2-47-7', 'hep-th-0512221-3-46-12'], ['hep-th-0512221-2-48-0', 'hep-th-0512221-3-47-0'], ['hep-th-0512221-2-8-0', 'hep-th-0512221-3-8-0'], ['hep-th-0512221-2-50-1', 'hep-th-0512221-3-48-1'], ['hep-th-0512221-2-50-2', 'hep-th-0512221-3-48-3'], ['hep-th-0512221-2-50-3', 'hep-th-0512221-3-48-4'], ['hep-th-0512221-2-51-1', 'hep-th-0512221-3-49-1'], ['hep-th-0512221-2-9-0', 'hep-th-0512221-3-9-0'], ['hep-th-0512221-2-10-1', 'hep-th-0512221-3-9-3'], ['hep-th-0512221-2-66-0', 'hep-th-0512221-3-65-0'], ['hep-th-0512221-2-67-1', 'hep-th-0512221-3-65-1']]

[]

[['hep-th-0512221-2-11-6', 'hep-th-0512221-3-10-6'], ['hep-th-0512221-2-11-11', 'hep-th-0512221-3-10-6'], ['hep-th-0512221-2-75-2', 'hep-th-0512221-3-73-2'], ['hep-th-0512221-2-0-4', 'hep-th-0512221-3-0-4'], ['hep-th-0512221-2-63-8', 'hep-th-0512221-3-62-9'], ['hep-th-0512221-2-47-4', 'hep-th-0512221-3-46-9'], ['hep-th-0512221-2-47-4', 'hep-th-0512221-3-46-10'], ['hep-th-0512221-2-47-5', 'hep-th-0512221-3-46-10'], ['hep-th-0512221-2-76-2', 'hep-th-0512221-3-78-2'], ['hep-th-0512221-2-50-0', 'hep-th-0512221-3-48-0']]

[]

['hep-th-0512221-1-20-8', 'hep-th-0512221-1-33-0', 'hep-th-0512221-1-40-5', 'hep-th-0512221-1-59-3', 'hep-th-0512221-1-61-7', 'hep-th-0512221-1-71-0', 'hep-th-0512221-2-20-8', 'hep-th-0512221-2-33-0', 'hep-th-0512221-2-40-5', 'hep-th-0512221-2-59-3', 'hep-th-0512221-2-61-7', 'hep-th-0512221-2-71-0', 'hep-th-0512221-3-19-8', 'hep-th-0512221-3-32-0', 'hep-th-0512221-3-39-5', 'hep-th-0512221-3-52-0', 'hep-th-0512221-3-58-3', 'hep-th-0512221-3-60-7', 'hep-th-0512221-3-69-0', 'hep-th-0512221-3-79-3', 'hep-th-0512221-3-80-0']

{'1': 'http://arxiv.org/licenses/assumed-1991-2003/', '2': 'http://arxiv.org/licenses/assumed-1991-2003/', '3': 'http://arxiv.org/licenses/assumed-1991-2003/'}

https://arxiv.org/abs/hep-th/0512221

{'hep-th-0512221-3-0-0': 'A pedagogical introduction to the heat kernel technique, zeta function and Casimir effect is presented.', 'hep-th-0512221-3-0-1': 'Several applications are considered.', 'hep-th-0512221-3-0-2': 'First we derive the high temperature asymptotics of the free energy for boson fields in terms of the heat kernel expansion and zeta function.', 'hep-th-0512221-3-0-3': 'Another application is chiral anomaly for local (MIT bag) boundary conditions.', 'hep-th-0512221-3-0-4': 'Then we rederive the Casimir energies for perfectly conducting rectangular cavities using a new technique.', 'hep-th-0512221-3-0-5': 'The new results for the attractive Casimir force acting on each of the two perfectly conducting plates inside an infinite perfectly conducting waveguide of the same cross section as the plates are presented at zero and finite temperatures.', 'hep-th-0512221-3-1-0': '# Introduction', 'hep-th-0512221-3-2-0': 'The main problem of the quantum field theory with boundaries is its renormalization and physical meaning of the results obtained.', 'hep-th-0512221-3-2-1': 'Divergences that appear in quantum field theory make the problems on manifolds with boundaries more complicated than in infinite space.', 'hep-th-0512221-3-3-0': 'In the presence of boundaries or singularities the heat kernel technique is an effective tool for the analysis of the one loop effects (see reviews [CITATION], [CITATION]).', 'hep-th-0512221-3-3-1': 'Different applications of the heat kernel expansion exist.', 'hep-th-0512221-3-3-2': 'The heat kernel technique seems to be the easiest way for the calculation of quantum anomalies, calculation of effective actions based on finite-mode regularization and analysis of divergences in quantum field theory.', 'hep-th-0512221-3-4-0': 'Chiral anomaly, which was discovered more than 35 years ago [CITATION], still plays an important role in physics.', 'hep-th-0512221-3-4-1': 'On smooth manifolds without boundaries many successful approaches to the anomalies exist [CITATION],[CITATION], [CITATION].', 'hep-th-0512221-3-4-2': 'The heat kernel approach to the anomalies is essentially equivalent to the Fujikawa approach [CITATION] and to the calculations based on the finite-mode regularization [CITATION], but it can be more easily extended to complicated geometries.', 'hep-th-0512221-3-4-3': 'The local chiral anomaly in the case of non-trivial boundary conditions (MIT bag boundary conditions) has been calculated only recently [CITATION].', 'hep-th-0512221-3-5-0': 'Casimir effect [CITATION] is a macroscopic quantum effect.', 'hep-th-0512221-3-5-1': 'Briefly speaking, if we impose classical boundary conditions on a quantum field on some boundary surface than we get the Casimir effect.', 'hep-th-0512221-3-5-2': 'There are several different physical situations that should be distinguished in the Casimir effect.', 'hep-th-0512221-3-6-0': 'Suppose there are two spatially separated dielectrics, then in a dilute limit ([MATH]) the Casimir energy of this system is equal to the energy of pairwise interactions between dipoles of these two dielectrics via a Casimir-Polder retarded potential [CITATION].', 'hep-th-0512221-3-6-1': 'For a general case of separated dielectrics the Casimir energy can be calculated as in [CITATION] or [CITATION] (for a recent discussion of these issues see [CITATION] and a review [CITATION], new possible experiments in [CITATION] ).', 'hep-th-0512221-3-7-0': 'A different situation takes place when there is a dilute dielectric ball or any other simply connected dielectric under study (see a review [CITATION] for a discussion of related subjects and methods used).', 'hep-th-0512221-3-7-1': 'As it was pointed out in [CITATION] and then discussed in detail in [CITATION], microscopic interatomic distances should be taken into account to calculate the Casimir energy of a dilute dielectric ball.', 'hep-th-0512221-3-7-2': 'The average interatomic distance [MATH] serves as an effective physical cut off for simply connected dielectrics.', 'hep-th-0512221-3-8-0': 'The limit of a perfect conductivity ([MATH]) is opposite to a dilute case.', 'hep-th-0512221-3-8-1': 'This is the strong coupling limit of the theory.', 'hep-th-0512221-3-8-2': 'Any results obtained in this limit are nonperturbative ones.', 'hep-th-0512221-3-9-0': 'The Casimir energy for a perfectly conducting rectangular cavity was first calculated in [CITATION] using exponential regularization.', 'hep-th-0512221-3-9-1': 'Later it was derived by some other methods (see references and numerical analysis in [CITATION], also a review [CITATION]).', 'hep-th-0512221-3-9-2': 'In the present paper we derive the Casimir energy for rectangular cavities at zero temperature by a new method described in Sec. [MATH].', 'hep-th-0512221-3-9-3': 'By use of this method we could rewrite the Casimir energy for rectangular cavities in the form that makes transparent its geometric interpretation.', 'hep-th-0512221-3-9-4': 'Also this method yields new exact results for the Casimir force acting on two or more perfectly conducting plates of an arbitrary cross section inside an infinite perfectly conducting waveguide of the same cross section.', 'hep-th-0512221-3-10-0': 'The paper is organized as follows.', 'hep-th-0512221-3-10-1': 'In Sec. [MATH] we give an introduction to the formalism of the heat kernel and heat kernel expansion.', 'hep-th-0512221-3-10-2': 'Also we introduce a [MATH]-function [CITATION] and calculate the one loop effective action in terms of [MATH]- function.', 'hep-th-0512221-3-10-3': 'Then we consider two examples.', 'hep-th-0512221-3-10-4': 'First we derive the high temperature expansion of the free energy for boson fields [CITATION] in terms of the heat kernel expansion and [MATH]-function.', 'hep-th-0512221-3-10-5': 'Then we derive a chiral anomaly in four dimensions for an euclidean version of the MIT bag boundary conditions [CITATION].', 'hep-th-0512221-3-10-6': 'Sec.[MATH] is devoted to the Casimir effect for perfectly conducting cavities.', 'hep-th-0512221-3-10-7': 'In Sec.[MATH] we introduce a regularization and a convenient new method of calculations using an example of two perfectly conducting parallel plates, then apply it to more complicated rectangular geometries.', 'hep-th-0512221-3-10-8': 'We present the Casimir energy of the cavity in the form ([REF]).', 'hep-th-0512221-3-10-9': 'Then we discuss an argument principle and [MATH]-functional regularization for the cavity.', 'hep-th-0512221-3-10-10': 'In Sec.[MATH] we rewrite the Casimir energy of the cavity in terms of geometric optics ([REF]).', 'hep-th-0512221-3-10-11': 'In Sec.[MATH] we describe a possible experiment and derive the formula for the attractive force acting on each of the two parallel plates inside an infinite rectangular waveguide with the same cross section.', 'hep-th-0512221-3-10-12': 'Also we present a generalization of this result for the case of the two parallel perfectly conducting plates of an arbitrary cross section inside an infinite perfectly conducting waveguide with the same cross section at zero and finite temperatures.', 'hep-th-0512221-3-11-0': '# Spectral techniques', 'hep-th-0512221-3-12-0': '## Heat kernel', 'hep-th-0512221-3-13-0': 'Consider a second order elliptic partial differential operator [MATH] of Laplace type on an n-dimensional Riemannian manifold.', 'hep-th-0512221-3-13-1': 'Any operator of this type can be expanded locally as [EQUATION] where [MATH] and [MATH] are some matrix valued functions and [MATH] is the inverse metric tensor on the manifold.', 'hep-th-0512221-3-13-2': 'For a flat space [MATH].', 'hep-th-0512221-3-14-0': 'The heat kernel can be defined as follows: [EQUATION] where [MATH] is an eigenfunction of the operator [MATH] with the eigenvalue [MATH].', 'hep-th-0512221-3-15-0': 'It satisfies the heat equation [EQUATION] with an initial condition [EQUATION]', 'hep-th-0512221-3-15-1': 'If we consider the fields in a finite volume then it is necessary to specify boundary conditions.', 'hep-th-0512221-3-15-2': 'Different choices are possible.', 'hep-th-0512221-3-15-3': 'In section [MATH] we will consider the case of periodic boundary conditions on imaginary time coordinate, which are specific for boson fields.', 'hep-th-0512221-3-15-4': 'In section [MATH] we will study bag boundary conditions imposed on fermion fields.', 'hep-th-0512221-3-15-5': 'If the normal to the boundary component of the fermion current [MATH] vanishes at the boundary, one can impose bag boundary conditions, a particular case of mixed boundary conditions.', 'hep-th-0512221-3-15-6': 'We assume given two complementary projectors [MATH], [MATH] acting on a multi component field (the eigenfunction of the operator [MATH]) at each point of the boundary and define mixed boundary conditions by the relations [EQUATION] where [MATH] is a matrix valued function on the boundary.', 'hep-th-0512221-3-15-7': 'In other words, the components [MATH] satisfy Dirichlet boundary conditions, and [MATH] satisfy Robin (modified Neumann) ones.', 'hep-th-0512221-3-16-0': 'It is convenient to define [EQUATION]', 'hep-th-0512221-3-16-1': 'Let [MATH], [MATH] be a local orthonormal frame for the tangent space to the manifold and let on the boundary [MATH] be an inward pointing normal vector.', 'hep-th-0512221-3-17-0': 'The extrinsic curvature is defined by the equation [EQUATION] where [MATH] is the Christoffel symbol.', 'hep-th-0512221-3-17-1': 'For example, on the unit sphere [MATH] which bounds the unit ball in [MATH] the extrinsic curvature is [MATH].', 'hep-th-0512221-3-18-0': 'Curved space offers no complications in our approach compared to the flat case.', 'hep-th-0512221-3-18-1': 'Let [MATH] be the Riemann tensor, and let [MATH] be the Ricci tensor.', 'hep-th-0512221-3-18-2': 'With our sign convention the scalar curvature [MATH] is [MATH] on the unit sphere [MATH].', 'hep-th-0512221-3-18-3': 'In flat space the Riemann and Ricci tensors are equal to zero.', 'hep-th-0512221-3-19-0': 'One can always introduce a connection [MATH] and another matrix valued function [MATH] so that [MATH] takes the form: [EQUATION]', 'hep-th-0512221-3-19-1': 'Here [MATH] is a sum of covariant Riemannian derivative with respect to metric [MATH] and connection [MATH].', 'hep-th-0512221-3-19-2': 'One can, of course, express [MATH] and [MATH] in terms of [MATH], [MATH] and [MATH]: [EQUATION]', 'hep-th-0512221-3-19-3': 'For the future use we introduce also the field strength for [MATH]: [EQUATION]', 'hep-th-0512221-3-19-4': 'The connection [MATH] will be used to construct covariant derivatives.', 'hep-th-0512221-3-19-5': 'The subscript [MATH] will be used to denote repeated covariant derivatives with the connection [MATH] and the Christoffel connection on [MATH].', 'hep-th-0512221-3-19-6': 'The subscript [MATH] will denote repeated covariant derivatives containing [MATH] and the Christoffel connection on the boundary.', 'hep-th-0512221-3-19-7': 'Difference between these two covariant derivatives is measured by the extrinsic curvature ([REF]).', 'hep-th-0512221-3-19-8': 'For example, [MATH].', 'hep-th-0512221-3-20-0': 'Let us define an integrated heat kernel for a hermitian operator [MATH] by the equation: [EQUATION] where [MATH] is an hermitian matrix valued function, [MATH] here is over matrix indices.', 'hep-th-0512221-3-20-1': 'For the boundary conditions we consider in this paper there exists an asymptotic expansion [CITATION] as [MATH]: [EQUATION]', 'hep-th-0512221-3-20-2': 'According to the general theory [CITATION] the coefficients [MATH] are locally computable.', 'hep-th-0512221-3-20-3': 'This means that each [MATH] can be represented as a sum of volume and boundary integrals of local invariants constructed from [MATH], [MATH], [MATH], the curvature tensor, and their derivatives.', 'hep-th-0512221-3-20-4': 'Boundary invariants may also include [MATH], [MATH] and [MATH].', 'hep-th-0512221-3-20-5': 'Total mass dimension of such invariants should be [MATH] for the volume terms and [MATH] for the boundary ones.', 'hep-th-0512221-3-21-0': 'At the moment several coefficients of the expansion ([REF]) are known for the case of mixed boundary conditions ([REF]) and matrix valued function [MATH] (see [CITATION] for details of derivation; the formula ([REF]) for [MATH] was derived in [CITATION] with additional restrictions [MATH] and [MATH]) : [EQUATION]', 'hep-th-0512221-3-21-1': 'For a scalar function [MATH] and mixed boundary conditions the coefficients [MATH] and [MATH] were already derived [CITATION].', 'hep-th-0512221-3-22-0': '## [MATH]-function', 'hep-th-0512221-3-23-0': 'Zeta function of a positive operator [MATH] is defined by [EQUATION] where the sum is over all eigenvalues of the operator [MATH].', 'hep-th-0512221-3-23-1': 'The zeta function is related to the heat kernel by the transformation [EQUATION]', 'hep-th-0512221-3-23-2': 'Residues at the poles of the zeta function are related to the coefficients of the heat kernel expansion: [EQUATION]', 'hep-th-0512221-3-23-3': 'Here [MATH] is a unit matrix with a dimension of the matrix functions [MATH] in ([REF]).', 'hep-th-0512221-3-23-4': 'From ([REF]) it follows that [EQUATION]', 'hep-th-0512221-3-23-5': 'In Euclidean four dimensional space the zero temperature one-loop path integral over the boson fields [MATH] can be evaluated as follows (up to a normalization factor): [EQUATION]', 'hep-th-0512221-3-23-6': 'Here we introduced the constant [MATH] with a dimension of mass in order to keep a proper dimension of the measure in the functional integral.', 'hep-th-0512221-3-23-7': '[MATH] can be thought of as a number of eigenvalues of the operator [MATH].', 'hep-th-0512221-3-23-8': 'For the operator [MATH] in the form ([REF]) the number of eigenvalues is infinite, so [MATH] yields a regularized value for this number.', 'hep-th-0512221-3-24-0': 'The zero temperature one-loop effective action is defined then by [EQUATION]', 'hep-th-0512221-3-24-1': 'The term [MATH] in the effective action [MATH] determines the one-loop beta function, this term describes renormalization of the one-loop logarithmic divergences appearing in the theory.', 'hep-th-0512221-3-25-0': '## Free energy for boson fields', 'hep-th-0512221-3-26-0': 'A finite temperature field theory is defined in Euclidean space, since for boson fields one has to impose periodic boundary conditions on imaginary time coordinate (antiperiodic boundary conditions for fermion fields respectively).', 'hep-th-0512221-3-26-1': 'A partition function is defined by [EQUATION] where [MATH] is a hamiltonian of the problem and [MATH].', 'hep-th-0512221-3-26-2': 'Let us choose the lagrangian density [MATH] in the form [EQUATION] where [MATH] is an imaginary time coordinate and [MATH] is a three dimensional spatial part of the density in the form ([REF]).', 'hep-th-0512221-3-26-3': 'The free energy of the system is defined by [EQUATION] the integration is over all periodic fields satisfying [MATH] is a normalization coefficient).', 'hep-th-0512221-3-26-4': 'As a result the eigenfunctions of [MATH] have the form [MATH], where [MATH] and [MATH].', 'hep-th-0512221-3-26-5': 'The free energy is thus equal to [CITATION] [EQUATION] where we introduced [MATH]-function [EQUATION] and the parameter [MATH] with a mass dimensionality in order to make the argument of the logarithm dimensionless (also see a previous section).', 'hep-th-0512221-3-27-0': 'Then it is convenient to use the formula [EQUATION] and separate [MATH] and other terms in the sum.', 'hep-th-0512221-3-27-1': 'For [MATH] terms we substitute the heat kernel expansion for the operator [MATH] at small [MATH] [EQUATION] and perform [MATH] integration, then we arrive at the high temperature expansion ([MATH]) for the free energy [MATH]: [EQUATION]', 'hep-th-0512221-3-27-2': 'Here [MATH], [MATH] is a Riemann zeta function, [MATH] is a zeta function of an operator [MATH], [MATH] is the Euler constant.', 'hep-th-0512221-3-27-3': 'The first two terms on the r.h.s. of ([REF]) follow from the [MATH] term.', 'hep-th-0512221-3-28-0': 'The term [EQUATION] is the leading high temperature contribution to the free energy.', 'hep-th-0512221-3-29-0': 'The classical limit terms due to the equality [MATH] can be rewritten as follows: [EQUATION]', 'hep-th-0512221-3-29-1': 'The terms on the l.h.s. of ([REF]) yield a renormalized value of the terms on the r.h.s. of ([REF]), since the sum on the righthandsight is generally divergent when the number of modes is infinite.', 'hep-th-0512221-3-30-0': 'The term with [MATH] determines the part of the free energy that appears due to one-loop logarithmic divergences and thus it depends on the dimensional parameter [MATH] as in the zero temperature case.', 'hep-th-0512221-3-31-0': '## Chiral anomaly in four dimensions', 'hep-th-0512221-3-32-0': 'for MIT bag boundary conditions', 'hep-th-0512221-3-33-0': 'Consider the Dirac operator on an [MATH]-dimensional Riemannian manifold [EQUATION] in external vector [MATH] and axial vector [MATH] fields.', 'hep-th-0512221-3-33-1': 'We suppose that [MATH] and [MATH] are anti-hermitian matrices in the space of some representation of the gauge group.', 'hep-th-0512221-3-33-2': '[MATH] is the spin-connection.', 'hep-th-0512221-3-34-0': 'The Dirac operator transforms covariantly under infinitesimal local gauge transformations (the local gauge transformation is [MATH]): [EQUATION] and under infinitesimal local chiral transformations (the local chiral transformation is [MATH]): [EQUATION]', 'hep-th-0512221-3-34-1': 'The parameters [MATH] and [MATH] are anti-hermitian matrices.', 'hep-th-0512221-3-35-0': 'First we adopt the zeta-function regularization and write the one-loop effective action for Dirac fermions at zero temperature as [EQUATION] where [EQUATION] prime denotes differentiation with respect to [MATH], and [MATH] is the functional trace.', 'hep-th-0512221-3-36-0': 'The following identity holds: [EQUATION]', 'hep-th-0512221-3-36-1': 'Due to the identity ([REF]) [EQUATION] so the effective action ([REF]) is gauge invariant, [MATH].', 'hep-th-0512221-3-37-0': 'The chiral anomaly is by definition equal to the variation of [MATH] under an infinitesimal chiral transformation.', 'hep-th-0512221-3-37-1': 'Using ([REF]) we obtain: [EQUATION] and the anomaly reads [EQUATION]', 'hep-th-0512221-3-37-2': 'The heat kernel is related to the zeta function by the Mellin transformation: [EQUATION]', 'hep-th-0512221-3-37-3': 'In particular, after the substitution of the heat kernel expansion ([REF]) into the formula ([REF]) we obtain [EQUATION]', 'hep-th-0512221-3-37-4': 'The same expression for the anomaly follows also from the Fujikawa approach [CITATION].', 'hep-th-0512221-3-38-0': "One can also derive the expression for the anomaly ([REF]) from Schwinger's effective action.", 'hep-th-0512221-3-38-1': 'One should start from an identity: [EQUATION]', 'hep-th-0512221-3-38-2': 'Then the change in the effective action due to chiral transformations can be written: [EQUATION]', 'hep-th-0512221-3-38-3': 'We impose local boundary conditions: [EQUATION] which are nothing else than a Euclidean version of the MIT bag boundary conditions [CITATION].', 'hep-th-0512221-3-38-4': 'For these boundary conditions [MATH], and the normal component of the fermion current [MATH] vanishes on the boundary.', 'hep-th-0512221-3-38-5': 'Spectral properties of the Dirac operator for bag boundary conditions are intensively studied [CITATION].', 'hep-th-0512221-3-39-0': 'Since [MATH] is a first order differential operator it was enough to fix the boundary conditions ([REF]) on a half of the components.', 'hep-th-0512221-3-39-1': 'To proceed with a second order operator [MATH] we need boundary conditions on the remaining components as well.', 'hep-th-0512221-3-39-2': 'They are defined by the consistency condition [CITATION]: [EQUATION] which is equivalent to the Robin boundary condition [EQUATION] with [EQUATION]', 'hep-th-0512221-3-39-3': 'In the paper [CITATION] the following expression for a coefficient [MATH] with an hermitian matrix valued function [MATH] and conditions ([REF]), [MATH] (flat boundaries), [MATH] was obtained: [EQUATION]', 'hep-th-0512221-3-39-4': 'To obtain the chiral anomaly in four dimensions with MIT bag boundary conditions one has to calculate the coefficient [MATH] ([REF]) with [MATH], [MATH] and substitute it into ([REF]).', 'hep-th-0512221-3-39-5': 'We define [MATH], [MATH], [MATH].', 'hep-th-0512221-3-39-6': 'The anomaly contains two contributions: [EQUATION]', 'hep-th-0512221-3-39-7': 'In the volume part [EQUATION] only the [MATH] terms seem to be new [CITATION] (for flat space it can be found e.g. in [CITATION]).', 'hep-th-0512221-3-40-0': 'The boundary part [EQUATION] is new [CITATION].', 'hep-th-0512221-3-40-1': 'It has been derived under the two restrictions: [MATH] and [MATH].', 'hep-th-0512221-3-40-2': 'Note, that in the present context, the first condition ([MATH]) actually follows from the second one ([MATH]) due to ([REF]).', 'hep-th-0512221-3-41-0': '# Casimir effect for rectangular cavities', 'hep-th-0512221-3-42-0': '## Casimir energy of two perfectly conducting parallel plates', 'hep-th-0512221-3-43-0': 'The Casimir energy is usually defined as [EQUATION] where the sum is over all eigenfrequencies of the system.', 'hep-th-0512221-3-43-1': 'In what following we put [MATH].', 'hep-th-0512221-3-43-2': 'We start from the well known case of two perfectly conducting plates separated by a distance [MATH] from each other.', 'hep-th-0512221-3-43-3': 'In this case the eigenfrequencies [MATH] are defined as follows: [EQUATION] so that the Casimir energy can be written as [EQUATION] [MATH] is the surface of each plate.', 'hep-th-0512221-3-43-4': 'The first sum is equivalent to the sum over eigenfrequencies of the scalar field satisfying Dirichlet boundary conditions, the second sum is equivalent to the sum over eigenfrequencies of the scalar field satisfying Neumann boundary conditions.', 'hep-th-0512221-3-44-0': 'The expression for the Casimir energy written in this form is divergent.', 'hep-th-0512221-3-44-1': 'One has to regularize it somehow to obtain a finite answer for the energy.', 'hep-th-0512221-3-44-2': 'Different methods were used for this purpose.', 'hep-th-0512221-3-44-3': 'In the present paper we suggest a method which makes calculations of determinants straightforward and easy to perform.', 'hep-th-0512221-3-45-0': 'By making use of an identity [EQUATION] we can see that up to an irrelevant constant the Casimir energy can be written in the form (we introduce a dimensional parameter [MATH] by the same reasoning as in ([REF]) or ([REF])): [EQUATION]', 'hep-th-0512221-3-45-1': 'Now the expression for the Casimir energy is written in the standard [MATH] form, which is usual for one-loop effective actions in quantum field theory.', 'hep-th-0512221-3-45-2': 'The coefficient [MATH] is equal to zero for our current choice of the operator [MATH] and boundary geometry.', 'hep-th-0512221-3-46-0': 'At this point we introduce a regularization - we restrict integrations over momenta by some cut off [MATH] in the momentum space.', 'hep-th-0512221-3-46-1': 'The sums over [MATH] are also restricted as follows: [EQUATION]', 'hep-th-0512221-3-46-2': 'The regularized Casimir energy is defined by: [EQUATION]', 'hep-th-0512221-3-46-3': 'It is convenient to perform a summation over [MATH] first.', 'hep-th-0512221-3-46-4': 'The following identity holds: [EQUATION]', 'hep-th-0512221-3-46-5': 'The first sum in ([REF]) can be calculated in the [MATH] limit by use of an identity: [EQUATION]', 'hep-th-0512221-3-46-6': 'The second sum in ([REF]) can be derived by use of a Stirling formula (which is exact in the large [MATH] limit): [EQUATION] so in the large [MATH] limit it is possible to write: [EQUATION]', 'hep-th-0512221-3-46-7': 'In the large [MATH] limit the Dirichlet sum ([REF]) can be rewritten as: [EQUATION]', 'hep-th-0512221-3-46-8': 'The sum over Neumann modes can be rewritten as follows: [EQUATION]', 'hep-th-0512221-3-46-9': 'It is possible to add any finite number that does not depend on [MATH] to the regularized Casimir energy [MATH] ([REF]) (the force between the plates is being measured in experiments, so the energy can be defined up to a constant).', 'hep-th-0512221-3-46-10': 'We add the surface term [EQUATION] to the regularized Casimir energy [MATH] ([REF]).', 'hep-th-0512221-3-46-11': 'Doing so we obtain [EQUATION]', 'hep-th-0512221-3-46-12': 'The first term in ([REF]) is twice the regularized Casimir energy of the free scalar field since it can be rewritten as [EQUATION]', 'hep-th-0512221-3-46-13': 'This term should be subtracted because we are interested in the change of the ground state energy when the plates are inserted into the free space.', 'hep-th-0512221-3-47-0': 'Next we perform the limits [MATH].', 'hep-th-0512221-3-47-1': 'The Casimir energy is thus [EQUATION] which is the well known result by Casimir [CITATION].', 'hep-th-0512221-3-48-0': 'After elaborations we summarize the key points of the method, which is valid for the calculations in cylindrical cavities with arbitrary cross sections.', 'hep-th-0512221-3-48-1': 'Suppose we want to calculate [MATH] of the second order operator [MATH], where the dimensionalities of the operators are denoted by numbers.', 'hep-th-0512221-3-48-2': 'At zero temperature in our case of interest the operator [MATH] describes a scalar field inside an infinite waveguide of an arbitrary cross section with Dirichlet or Neumann boundary conditions imposed.', 'hep-th-0512221-3-48-3': 'The eigenmodes of the operators [MATH] and [MATH] are denoted by [MATH] and [MATH] respectively.', 'hep-th-0512221-3-48-4': 'The following expression is finite (as can be seen from the heat kernel expansion): [EQUATION]', 'hep-th-0512221-3-48-5': 'To obtain the initial determinant one should add to ([REF]) the term [EQUATION]', 'hep-th-0512221-3-48-6': 'The sum of ([REF]) and ([REF]) generally has the following structure (to obtain the total Casimir energy the sum over indices [MATH] and [MATH] has to be performed): [EQUATION] where [EQUATION] the term ([REF]) yields the energy of interaction for two flat parallel plates separated by a distance [MATH] inside an infinite waveguide of the same cross section as these parallel plates (the walls of a perfectly conducting waveguide are perpendicular to two flat parallel perfectly conducting plates inside it).', 'hep-th-0512221-3-48-7': 'The term ([REF]) yields an experimentally measurable contribution to the Casimir energy of the cavity (see Sec. [MATH] for details).', 'hep-th-0512221-3-49-0': 'The term [MATH] is equal to the self-energy of an infinite waveguide when [MATH].', 'hep-th-0512221-3-49-1': 'For rectangular cavities the term [MATH] can be transformed to [MATH] in the same manner as in the beginning of this section (see a transition from ([REF]) to ([REF]) ).', 'hep-th-0512221-3-49-2': 'For the operator [MATH] we repeat the step ([REF]) and continue this cycle until the first term in the righthandsight of ([REF]) gets the form of the vacuum energy in an infinite space, i.e. the form ([REF]).', 'hep-th-0512221-3-50-0': 'The [MATH] terms describe the self-energies of two parallel plates inside the waveguide due to Dirichlet and Neumann modes, these self-energies do not depend on [MATH].', 'hep-th-0512221-3-50-1': 'For flat boundaries Dirichlet and Neumann boundary contributions to the Casimir energy cancel each other identically as can be seen from the Seeley coefficient [MATH] ([REF]), for two parallel plates it can be seen from the expressions ([REF]) and ([REF]) .', 'hep-th-0512221-3-51-0': 'A contribution from the last term [MATH] is proportional to [MATH] (at zero temperature it is just the effective number of modes inside an infinite perfectly conducting waveguide, and thus it is not relevant to the energy of interaction between the two plates inside the waveguide) and [MATH] (this term is also not relevant when the interaction of the two plates inside an infinite waveguide is studied).', 'hep-th-0512221-3-52-0': 'To implement ([REF]) we used the following equality: [EQUATION]', 'hep-th-0512221-3-53-0': '## Casimir energy of a perfectly conducting rectangular waveguide For a perfectly conducting rectangular waveguide the technical issues can be done in analogy with two parallel plates.', 'hep-th-0512221-3-53-1': 'We tacitly assume that the reader understood how the regularization is introduced in our method, so we will write only main steps without bothering too much on divergent form of some expressions.', 'hep-th-0512221-3-53-2': 'The Casimir energy for unit length is: [EQUATION]', 'hep-th-0512221-3-53-3': 'For TM modes [MATH] and [MATH] take positive integer values from [MATH] to [MATH], for TE modes [MATH] and [MATH] take positive integer values and one of them can be equal to zero ([MATH] corresponds to the main wave case).', 'hep-th-0512221-3-54-0': 'So the energy can be rewritten as: [EQUATION] where [MATH] or [MATH].', 'hep-th-0512221-3-54-1': 'Using formula ([REF]) for multiplication over [MATH] we obtain for the energy: [EQUATION]', 'hep-th-0512221-3-54-2': 'A contribution from the term [MATH] in ([REF]) should be subtracted following the analysis of Section [MATH].', 'hep-th-0512221-3-54-3': 'The part ([REF]) with the logarithm is finite, it contributes to the finite final answer for the Casimir energy.', 'hep-th-0512221-3-55-0': 'For the first term in ([REF]) we get: [EQUATION] because up to a numerical coefficient the expression ([REF]) is just the same as the formula ([REF]).', 'hep-th-0512221-3-56-0': 'So the Casimir energy for unit length of a rectangular waveguide can be written as the sum of ([REF]) and ([REF]) : [EQUATION]', 'hep-th-0512221-3-57-0': '## Casimir energy of a perfectly conducting rectangular cavity', 'hep-th-0512221-3-58-0': 'The Casimir energy in this case can be written as: [EQUATION]', 'hep-th-0512221-3-58-1': 'Using formula ([REF]) and technique described in previous subsections we obtain: [EQUATION]', 'hep-th-0512221-3-58-2': 'The remaining terms should be calculated (using formula ([REF]) again) as follows: [EQUATION]', 'hep-th-0512221-3-58-3': 'As a result for the Casimir energy of the cavity we obtain: [EQUATION]', 'hep-th-0512221-3-59-0': '## Relation to the argument principle', 'hep-th-0512221-3-60-0': 'An argument principle is a convenient method of summation over the eigenmodes of the system (see [CITATION] and [CITATION] for its applications).', 'hep-th-0512221-3-60-1': 'The argument principle states: [EQUATION] where [MATH] are zeroes and [MATH] are poles of the function [MATH] inside the contour of integration.', 'hep-th-0512221-3-60-2': 'For the Casimir energy [MATH].', 'hep-th-0512221-3-60-3': 'We choose [EQUATION] in case of a scalar field satisfying Dirichlet boundary conditions on the plates.', 'hep-th-0512221-3-60-4': 'The contour lies on an imaginary axis, a contribution from the right semicircle with a large radius is negligible.', 'hep-th-0512221-3-60-5': 'A denominator is chosen in this form to remove [MATH] from the roots of the equation [MATH].', 'hep-th-0512221-3-60-6': 'In this case we proceed as follows: [EQUATION]', 'hep-th-0512221-3-60-7': 'Here [MATH].', 'hep-th-0512221-3-60-8': 'We see that the argument principle is in agreement with ([REF]).', 'hep-th-0512221-3-61-0': '## Zeta function regularization for the cavity', 'hep-th-0512221-3-62-0': '[MATH]-function has already been discussed in this paper, so it is natural to describe regularization of the Casimir energy for the cavity in terms of [MATH]-function.', 'hep-th-0512221-3-62-1': 'Usually the Casimir energy is regularized as follows: [EQUATION] where [MATH] is large enough to make ([REF]) convergent.', 'hep-th-0512221-3-62-2': 'Then we should continue analytically ([REF]) to the value [MATH] , this procedure yields the renormalized finite Casimir energy.', 'hep-th-0512221-3-62-3': 'In our case eigenfrequencies [MATH] should be taken from ([REF]).', 'hep-th-0512221-3-62-4': 'So the regularized Casimir energy of the cavity [MATH] can be written in terms of Epstein [MATH] and Riemann [MATH] zeta functions: [EQUATION]', 'hep-th-0512221-3-62-5': 'The prime means that the term with all [MATH] should be excluded from the sum.', 'hep-th-0512221-3-62-6': 'The reflection formulas for an analytical continuation of zeta functions exist: [EQUATION]', 'hep-th-0512221-3-62-7': 'By use of reflection formulas ([REF]), ([REF]) one gets: [EQUATION]', 'hep-th-0512221-3-62-8': 'The renormalized Casimir energy can therefore be written as: [EQUATION]', 'hep-th-0512221-3-62-9': 'One can check that the formulas ([REF]) and ([REF]) coincide identically and yield the Casimir energy for a perfectly conducting cavity.', 'hep-th-0512221-3-63-0': '## Geometric interpretation', 'hep-th-0512221-3-64-0': 'In this section we suggest a geometric interpretation of the main formulas in terms of geometric optics.', 'hep-th-0512221-3-64-1': 'This interpretation clarifies the physical meaning of the results ([REF]) , ([REF]) obtained, which is always important for further generalizations in more complicated cases.', 'hep-th-0512221-3-65-0': 'Several geometric approaches - a semiclassical method [CITATION], a worldline approach [CITATION] and a method of geometric optics [CITATION] have been introduced recently for the evaluation of the Casimir energies.', 'hep-th-0512221-3-65-1': 'Our formulas ([REF]), ([REF]) yield a simple geometric interpretation for the Casimir energy of the rectangular cavities in terms of geometric optics.', 'hep-th-0512221-3-66-0': "Optical contributions to the Green's function of the scalar field with Dirichlet and Neumann boundary conditions have the form: [EQUATION]", 'hep-th-0512221-3-66-1': 'Here [MATH] is the length of the optical path that starts from [MATH] and arrives at [MATH] after n reflections from the boundary.', 'hep-th-0512221-3-66-2': '[MATH] is the enlargement factor of classical ray optics.', 'hep-th-0512221-3-66-3': 'For planar boundaries it is given by [MATH].', 'hep-th-0512221-3-66-4': 'From ([REF]) it follows that for two parallel plates the Casimir energy of the electromagnetic field can be expanded as: [EQUATION]', 'hep-th-0512221-3-66-5': 'Here [MATH] is a sum over all photon states (with frequencies [MATH]) in an infinite space.', 'hep-th-0512221-3-66-6': "The righthandsight of ([REF]) can be written in terms of optical Green's functions: [EQUATION]", 'hep-th-0512221-3-66-7': "Note that terms with odd reflections from Dirichlet and Neumann Green's functions cancel each other due to the factor [MATH] present in optical Dirichlet Green's function.", 'hep-th-0512221-3-66-8': 'This is why only periodic paths with even number of reflections from the boundary [MATH] enter into the expression for the Casimir energy.', 'hep-th-0512221-3-67-0': "Now consider the formula for the cavity ([REF]) (to obtain the Casimir energy of a waveguide ([REF]) in terms of optical Green's functions the arguments are the same, just start from two parallel plates).", 'hep-th-0512221-3-67-1': 'Imagine that there is a waveguide with side lengths [MATH] and [MATH].', 'hep-th-0512221-3-67-2': 'In order to obtain the rectangular cavity we have to insert two perfectly conducting plates with side lengths [MATH] and [MATH] (and a distance [MATH] apart) inside the waveguide.', 'hep-th-0512221-3-67-3': 'The eigenfrequencies that existed in a waveguide were equal to [MATH].', 'hep-th-0512221-3-67-4': 'Only the photons with frequencies [MATH] existed in a waveguide, and these photons start interacting with the plates inserted inside a waveguide.', 'hep-th-0512221-3-67-5': 'The optical contribution to the Casimir energy arising from the interaction of these [MATH] photons with inserted plates is equal to', 'hep-th-0512221-3-68-0': 'Here we sum over all eigenfrequencies of the electromagnetic field in [MATH] cases: when there is an infinite space ([MATH]), two parallel plates ([MATH]) and an infinite waveguide', 'hep-th-0512221-3-69-0': '([MATH]).', 'hep-th-0512221-3-70-0': 'The first two terms in ([REF]) may have the following geometric interpretation: from ([REF]) it follows that [EQUATION] where [MATH] is an eigenfrequency of a main wave in a waveguide.', 'hep-th-0512221-3-70-1': "So it is possible to express Casimir energies of perfectly conducting rectangular cavities in terms of optical Green's functions only.", 'hep-th-0512221-3-71-0': 'It is interesting that the Casimir energy of a perfectly conducting cavity can be written in terms of eigenfrequencies of the electromagnetic field in a free space, between two perfectly conducting plates and inside a perfectly conducting waveguide.', 'hep-th-0512221-3-72-0': '## The experiment', 'hep-th-0512221-3-73-0': 'For the experimental check of the Casimir energy for the rectangular cavity one should measure the force somehow.', 'hep-th-0512221-3-73-1': 'We think about the following possibility: one should insert two parallel perfectly conducting plates inside an infinite perfectly conducting waveguide and measure the force acting on one of the plates as it is being moved through the waveguide.', 'hep-th-0512221-3-73-2': 'The distance between the inserted plates is [MATH].', 'hep-th-0512221-3-74-0': 'To calculate the force on each plate the following gedanken experiment is useful.', 'hep-th-0512221-3-74-1': 'Imagine that [MATH] parallel plates are inserted inside an infinite waveguide and then [MATH] exterior plates are moved to spatial infinities.', 'hep-th-0512221-3-74-2': 'This situation is exactly equivalent to [MATH] perfectly conducting cavities touching each other.', 'hep-th-0512221-3-74-3': 'From the energy of this system one has to subtract the Casimir energy of an infinite waveguide, only then do we obtain the energy of interaction between the interior parallel plates, the one that can be measured in the proposed experiment (the subtraction of the term ([REF]) is just the same subtraction for two parallel plates).', 'hep-th-0512221-3-74-4': 'Doing so we obtain the attractive force on each interior plate inside the waveguide: [EQUATION] where [EQUATION] coincides with ([REF]).', 'hep-th-0512221-3-75-0': 'We note that our formula ([REF]) for the special case [MATH] coincides with the formula (6) in reference [CITATION], there it was obtained using a different method and presented in a different mathematical form.', 'hep-th-0512221-3-76-0': 'To obtain the energy of interaction between the opposite sides of a single cavity one should subtract from the expression ([REF]) the Casimir energy of the same box without these two sides, i.e. one has to subtract from ([REF]) the expression for the Casimir energy of a waveguide of a finite length.', 'hep-th-0512221-3-76-1': 'To our knowledge the expression for the Casimir energy of a finite length waveguide is not known up to now.', 'hep-th-0512221-3-77-0': 'It was often argued that the constant repulsive force (for a fixed cross section) derived from ([REF]) can be measured in experiment.', 'hep-th-0512221-3-77-1': 'However, without the subtraction just mentioned it is not possible to measure the forces in any realistic experiment, this is why it is not possible to use the expression ([REF]) directly to calculate the force in the experiments.', 'hep-th-0512221-3-77-2': 'However, it can be used to derive a measurable in experiments expression for the force between the parallel plates inserted inside an infinite waveguide of the same cross section as the plates.', 'hep-th-0512221-3-78-0': 'Using the same technique as before it is possible to generalize our formulas ([REF]), ([REF]) for the case of an infinite waveguide with an arbitrary cross section.', 'hep-th-0512221-3-78-1': 'The force between the two plates inside this waveguide can be immediately written : [EQUATION] the sum here is over all TE and TM eigenfrequencies [MATH] for the waveguide with an arbitrary cross section and an infinite length.', 'hep-th-0512221-3-78-2': 'Thus it can be said that the exchange of photons with the eigenfrequencies of a waveguide between the inserted plates always yields the attractive force between the plates.', 'hep-th-0512221-3-79-0': 'To get the free energy [MATH] for bosons at nonzero temperatures [MATH] one has to make the substitutions (see Sec. [MATH], the formula ([REF])): [EQUATION]', 'hep-th-0512221-3-79-1': 'Thus the free energy describing the interaction of the two parallel perfectly conducting plates inside an infinite perfectly conducting waveguide of an arbitrary cross section has the form: [EQUATION] where [MATH] and [MATH] are eigenvalues of the two-dimensional Dirichlet and Neumann problems (a boundary here coincides with the boundary of each plate inside the waveguide): [EQUATION]', 'hep-th-0512221-3-79-2': 'The attractive force between the plates inside an infinite waveguide of the same cross section at nonzero temperatures is given by: [EQUATION]', 'hep-th-0512221-3-79-3': 'Here [MATH] and [MATH].', 'hep-th-0512221-3-80-0': 'The proof of these results will be presented elsewhere.'}

0904.1160

{'0904.1160-1-0-0': 'We study the dynamics of a trapped spin-1 condensate in a magnetic field.', '0904.1160-1-0-1': 'First, we analyze the homogeneous system, for which the dynamics can be understood in terms of orbits in phase space.', '0904.1160-1-0-2': 'We solve analytically for the dynamical evolution of the populations of the various Zeeman components of the homogeneous system.', '0904.1160-1-0-3': 'This result is then applied via a Local Density Approximation to trapped quasi-1D condensates.', '0904.1160-1-0-4': 'Our analysis of the trapped system in a magnetic field shows that both mean-field and Zeeman regimes are simultaneously present, and argue that the border between these two regions is where spin domains and phase defects are generated.', '0904.1160-1-0-5': 'We finally propose a method to experimentally tune the position of this border.', '0904.1160-1-1-0': '# Introduction', '0904.1160-1-2-0': 'Bose-Einstein condensates with a spin degree of freedom are an interesting field of research in many-body physics as they realize both superfluidity and magnetism in a well-controlled environment.', '0904.1160-1-2-1': 'First realized experimentally with [MATH]Na ten years ago [CITATION], their study has matured remarkably the last few years, with several groups studying their dynamics [CITATION] and thermodynamics [CITATION].', '0904.1160-1-2-2': 'Of particular interest is the study of the process by which spin domains are formed during time evolution, a phenomenon observed experimentally [CITATION] and in numerical simulations based on a mean-field approach [CITATION].', '0904.1160-1-3-0': 'The complicated dynamics of these non-linear systems, especially when they are subjected to time-varying external fields, makes a physical understanding of the structure formation process somehow elusive.', '0904.1160-1-3-1': 'To address this point, we present here a simple model based on an analytic solution for the homogeneous system for arbitrary magnetic fields [MATH] and magnetizations [MATH].', '0904.1160-1-3-2': 'This solution is then applied to the study of realistic, trapped spin-1 condensates by means of the Local Density Approximation (LDA).', '0904.1160-1-3-3': 'This approximation has already been applied successfully in a number of studies on scalar BECs, as well as cold Fermi gases.', '0904.1160-1-3-4': 'From the analysis of our results we are able to provide an intuitive picture of the process leading to the structure formation.', '0904.1160-1-3-5': 'Further, we argue that it should be possible to experimentally "tune" the spatial region where this process starts within the condensate.', '0904.1160-1-4-0': 'The paper is organized as follows.', '0904.1160-1-4-1': 'In Sect. [REF] we present the phase space of a homogeneous system under a magnetic field [MATH] and for arbitrary [MATH], and introduce the phase-space orbits that describe the dynamics of a conservative system.', '0904.1160-1-4-2': 'In Sect. [REF] we solve analytically the dynamical evolution of the homogeneous system.', '0904.1160-1-4-3': 'Then, in Sect. [REF] we describe our local-density approximation for a trapped system and present numerical results for its dynamics (Sect. [REF]), which we compare with simulations based on a mean-field treatment (Sect. [REF]).', '0904.1160-1-4-4': 'In Sect. [REF] we discuss the progressive dephasing of different spatial points of the condensate in a magnetic field, and relate this to the process of structure formation, with an indication of a possible experimental test.', '0904.1160-1-4-5': 'Finally, we conclude in Sect. [REF].', '0904.1160-1-5-0': '# Analytical results for the homogeneous system', '0904.1160-1-6-0': '## Energetics of the homogeneous system', '0904.1160-1-7-0': 'A homogeneous condensate of atoms with total spin [MATH] can be described by a vector order parameter [MATH] with [MATH] components, [EQUATION]', '0904.1160-1-7-1': 'The density of atoms in a given Zeeman component [MATH] is [MATH] and the total density is given by [MATH].', '0904.1160-1-7-2': 'Introducing the relative densities for the homogeneous system [MATH], one has [EQUATION]', '0904.1160-1-7-3': 'Given that [MATH] is a conserved quantity, Eq. ([REF]) will be fulfilled at all times during dynamical evolution.', '0904.1160-1-7-4': 'Moreover, the magnetization [EQUATION] is also a conserved quantity [CITATION].', '0904.1160-1-8-0': 'We now focus our analysis to the case of a [MATH] condensate.', '0904.1160-1-8-1': 'We write the various components of the order parameter as [MATH].', '0904.1160-1-8-2': 'This ansatz, together with conditions ([REF]) and ([REF]), leads to the following expression for the energy per particle of the homogeneous system in the mean-field approach [CITATION]: [EQUATION]', '0904.1160-1-8-3': 'Here [MATH], while [MATH] is given in terms of the [MATH]-wave scattering lengths [MATH] in the channels of total spin [MATH], by [MATH], with [MATH] the atomic mass.', '0904.1160-1-8-4': 'Finally, [MATH], where the energies of the atomic Zeeman states are given by the Breit-Rabi formula [CITATION] [MATH]), with [MATH] being the atomic hyperfine splitting and [MATH] is a function of the external magnetic field [MATH].', '0904.1160-1-8-5': 'A sketch of the surface [MATH] is given in Fig. [REF].', '0904.1160-1-9-0': 'As indicated above, [MATH] is a constant during dynamical evolution.', '0904.1160-1-9-1': 'Similarly, given initial conditions [MATH], [MATH] will also be conserved, thus defining an orbit on the surface [MATH] in [MATH] space.', '0904.1160-1-9-2': 'A sketch of one such orbit is presented in Fig. [REF].', '0904.1160-1-10-0': 'One should note that, depending on the initial conditions, the orbit defined by [MATH] can be closed or open.', '0904.1160-1-10-1': 'In the first case, [MATH] will be a periodic function of time, while in the latter case, [MATH] will grow indefinitely with time.', '0904.1160-1-10-2': 'In both cases, however, [MATH] will be a periodic function of time.', '0904.1160-1-11-0': '## Dynamics of the homogeneous system', '0904.1160-1-12-0': 'We are interested in the time evolution of the densities of the different Zeeman components, [MATH].', '0904.1160-1-12-1': 'From Eqs. ([REF]) and ([REF]) we have that [EQUATION]', '0904.1160-1-12-2': 'Therefore, we only need to follow the evolution of [MATH], which is given by [EQUATION]', '0904.1160-1-12-3': 'With Eq ([REF]), we rewrite this as [EQUATION]', '0904.1160-1-12-4': 'It can be shown that the term in [MATH] actually drops out and we are left with a cubic polynomial on [MATH], [EQUATION] with [EQUATION] and [MATH] are the roots of [MATH], [MATH].', '0904.1160-1-12-5': 'For ground state ([MATH]) alkalies [MATH].', '0904.1160-1-12-6': 'Therefore, [MATH] for [MATH] and [MATH] for [MATH] [CITATION].', '0904.1160-1-12-7': 'For concreteness, in the following we will assume [MATH], i.e., ferromagnetic interactions.', '0904.1160-1-13-0': 'We will now integrate the time evolution of [MATH].', '0904.1160-1-13-1': 'To do so, we introduce an auxiliary variable [MATH] through [MATH].', '0904.1160-1-13-2': 'This will satisfy the differential equation [EQUATION] where we defined [EQUATION]', '0904.1160-1-13-3': 'The first order differential equation ([REF]) can be solved analytically by separating the variables [MATH] and [MATH] and integrating: [EQUATION]', '0904.1160-1-13-4': 'The solution to the last integral can be expressed in terms of the elliptic integral of the first kind , [EQUATION]', '0904.1160-1-13-5': 'Taking as initial condition [MATH] and using the fact that [MATH], we can express [MATH] in a compact form by means of the Jacobi elliptic functions [CITATION], which are defined as the inverses of the elliptic integrals: [EQUATION] with [MATH], i.e., [MATH].', '0904.1160-1-13-6': 'Finally, we undo the change of variables to write down the time evolution of the population of the [MATH] state: [EQUATION]', '0904.1160-1-13-7': 'In accordance with the identity [CITATION] [EQUATION] and given that both [MATH] and [MATH] are periodic functions in [MATH] with period [MATH], [MATH] will be a periodic function of time with period [EQUATION]', '0904.1160-1-13-8': 'Here, [MATH] stands for the complete elliptic integral of first kind.', '0904.1160-1-13-9': 'We note that result ([REF]) agrees with that in Ref. [CITATION], where [MATH] was calculated directly by performing the integral [MATH] over a period of evolution.', '0904.1160-1-13-10': 'Further, let us note that the average value [MATH] does not necessarily coincide with the position of the minimum of [MATH], i.e., [MATH] may differ from the equilibrium value [MATH] (as given, e.g., in Ref. [CITATION] for the case [MATH]).', '0904.1160-1-13-11': 'This is illustrated in Figs. [REF](b) and [REF](b).', '0904.1160-1-14-0': '## Evolution in absence of a magnetic field', '0904.1160-1-15-0': 'We observe that the representation of [MATH] as a cubic polynomial on [MATH], Eq. ([REF]), cannot be performed when [MATH], i.e., when [MATH].', '0904.1160-1-15-1': 'In this case, the analytic expression ([REF]) is meaningless, as it would apparently result in no time evolution at all.', '0904.1160-1-15-2': 'Actually, in this situation, [MATH] can be written as a quadratic polynomial on [MATH]: [EQUATION]', '0904.1160-1-15-3': 'Here [MATH].', '0904.1160-1-15-4': 'Note that [MATH] for [MATH] as well as for [MATH], and in both cases we will have [MATH].', '0904.1160-1-15-5': 'Following a procedure analogous to that above, we arrive at [EQUATION] with [MATH].', '0904.1160-1-15-6': 'In this case, [MATH] follows a pure sinusoidal evolution as has been predicted before in a number of references, e.g. [CITATION].', '0904.1160-1-15-7': 'The average value is [MATH].', '0904.1160-1-15-8': 'Finally, the period reads (compare with [CITATION]) [EQUATION]', '0904.1160-1-15-9': 'We show in Fig. [REF] the time evolution of [MATH] for two representative cases.', '0904.1160-1-15-10': 'The different panels compare the analytic evolution -given by Eq. ([REF]) or ([REF])- with a numerical solution of the corresponding equation for [MATH].', '0904.1160-1-15-11': 'In all cases, we see that the amplitude as well as the period of the time evolution are well predicted by the analytic results.', '0904.1160-1-16-0': 'Finally, we show in Fig. [REF] a plot of [MATH] vs. [MATH] corresponding to the time evolution depicted in Fig. [REF].', '0904.1160-1-16-1': 'For the case with magnetic field and [MATH] we observe that the average value [MATH] (indicated by the dashed line) differs from the position of the minimum of [MATH], cf. Fig. [REF]) due to the deformation of the orbit.', '0904.1160-1-17-0': '# Dynamics of the trapped system', '0904.1160-1-18-0': 'We have established in the previous section the dynamic evolution of a homogeneous spin-1 condensate, in terms of orbits in the [MATH] plane constrained by (i) conservation of density, (ii), conservation of magnetization, and (iii) conservation of energy.', '0904.1160-1-18-1': 'The resulting dynamics of the population of the [MATH] Zeeman component has been shown to be a periodic function of time, with a period determined by the density [MATH] of the system, its magnetization [MATH], as well as the initial conditions of the evolution (implicit in [MATH] and, therefore, in [MATH] or [MATH]), cf. Eqs. ([REF]) and ([REF]).', '0904.1160-1-18-2': 'Now, we will transfer these results to a realistic case of a trapped, quasi-1D condensate.', '0904.1160-1-19-0': '## Local Density Approximation', '0904.1160-1-20-0': 'The initial conditions for the evolution of a trapped spinor condensate are the set of complex values [MATH] for all Zeeman components [MATH] and all positions [MATH] where the density is not zero.', '0904.1160-1-20-1': 'In typical experiments, the preparation of the initial state is such that [MATH] is a constant independent of position.', '0904.1160-1-20-2': 'This, together with the fact that [MATH] for the systems studied so far, has lead to some theoretical works based on the so-called Single Mode Approximation (SMA), which assumes that [MATH] for all times [MATH] of the evolution, i.e., that the spatial variation of the density of each Zeeman component is always given by the total density profile.', '0904.1160-1-20-3': 'However, numerical studies beyond the SMA (e.g. [CITATION]) predicted the formation of spin domains during the time evolution.', '0904.1160-1-20-4': 'These have been observed in a number of experiments, e.g. [CITATION].', '0904.1160-1-20-5': 'In order to be able to observe the formation of spin domains during time evolution in a trapped system, we will therefore not make use of the SMA, but apply the analytical results of Sect. [REF] via the Local Density Approximation (LDA), i.e., we will assume that the evolution of the [MATH] population at each point within the condensate, [MATH], is given by Eq. ([REF]) [or ([REF])] with the substitution [MATH].', '0904.1160-1-20-6': 'Here, the total density is normalized to the total number of atoms in the condensate, [MATH].', '0904.1160-1-20-7': 'Similarly, we introduce local densities of atoms in a given Zeeman state [MATH] normalized as [MATH].', '0904.1160-1-20-8': 'The conservation laws read now [MATH] and [MATH].', '0904.1160-1-20-9': 'We note that [MATH] does not change in time at low enough temperatures [CITATION].', '0904.1160-1-21-0': 'In terms of the phase space [MATH] introduced in Sect. [REF], a trapped system corresponds to an infinite-dimensional phase space [MATH], with a pair of variables [MATH] associated to each point [MATH].', '0904.1160-1-21-1': 'According to the LDA, we divide this whole phase space in sections corresponding to the different positions, [MATH], and assume that they are independent.', '0904.1160-1-21-2': 'The initial condition described above, [MATH], corresponds then to the dynamical system starting in all the different positions [MATH] at the same point of the corresponding phase space, [MATH].', '0904.1160-1-21-3': 'The dynamical evolution of the system corresponds, within LDA, to the population [MATH] at each point [MATH] following its own particular orbit in the corresponding [MATH] space, that is, [MATH] at position [MATH] follows the dynamic equation of the homogeneous system ([REF]) [or Eq. ([REF])] with the parameters [MATH] and [MATH] determined by the local density [MATH].', '0904.1160-1-21-4': 'In other words, we assume that the position dependence is only parametric, and comes through the values of the parameters [MATH] and [MATH].', '0904.1160-1-21-5': 'We will indicate this by [MATH].', '0904.1160-1-21-6': 'The density at position [MATH] of atoms in the Zeeman component [MATH] at time [MATH] will then be [EQUATION] with [MATH] given by Eq. ([REF]) with the substitution [MATH], and [MATH] is a conserved quantity [CITATION].', '0904.1160-1-22-0': 'Note that the orbits associated to different points [MATH] may differ from one another, as their shapes depend inter alia on the local density [MATH], cf. Eq. ([REF]).', '0904.1160-1-22-1': 'This fact, together with the position dependence of the parameters [MATH] and [MATH], is expected to lead to a dephasing of the evolution of the partial densities [MATH] at the different points, washing out the oscillations in the integrated populations, [MATH], in contrast to the stable oscillations that we have found for the homogeneous system, cf. Fig. [REF].', '0904.1160-1-23-0': 'In order to evaluate [MATH] it is necessary to know the density profile of the system, [MATH].', '0904.1160-1-23-1': 'A good estimate for [MATH] in trapped atomic gases is given by the Thomas-Fermi approximation: [EQUATION]', '0904.1160-1-23-2': 'For a quasi-1D system with total number of atoms [MATH] and central density [MATH], [MATH].', '0904.1160-1-23-3': 'The integrated population in [MATH] reads then [EQUATION]', '0904.1160-1-24-0': '## Analytic approximation with sinusoidal time dependence', '0904.1160-1-25-0': 'The time dependence of [MATH] has in principle to be calculated from Eq. ([REF]) for each position [MATH] at each time step, and then the integral ([REF]) performed numerically to determine [MATH].', '0904.1160-1-25-1': 'It is possible however to give an analytical estimation for [MATH] if we make a further assumption on the time evolution.', '0904.1160-1-25-2': 'From Fig. [REF], we see that the evolution of [MATH] for the homogeneous system is very close to a sinusoidal function even when [MATH]', '0904.1160-1-26-0': 'This is illustrated in Fig. [REF](b), where a function of the form [EQUATION] has been fitted to the numerical values obtained from Eq. ([REF]).', '0904.1160-1-26-1': 'The fit is very good, even for this case, where the orbit in phase space is strongly deformed [cf. Fig. [REF](b)].', '0904.1160-1-26-2': 'The advantage of approximating the time evolution of [MATH] by Eq. ([REF]) is that it allows for an analytic evaluation of the spatial integral ([REF]) taking into account the position dependence of [MATH].', '0904.1160-1-26-3': 'Indeed, from Eq. ([REF]) we expect [MATH].', '0904.1160-1-26-4': 'It is easy to show that [EQUATION]', '0904.1160-1-26-5': 'Here [MATH] and [MATH] are the Fresnel integrals [CITATION], and we introduced [MATH] and [MATH].', '0904.1160-1-27-0': 'We show in Fig. [REF] the time evolution of the integrated [MATH] population as given by Eqs. ([REF]) and ([REF]).', '0904.1160-1-28-0': 'This calculation has been done for a quasi-1D system of [MATH]Rb atoms in a trap such that the central density is [MATH] cm[MATH].', '0904.1160-1-28-1': 'The initial conditions are [MATH] and we have taken a magnetic field [MATH] mG (cf. Fig. [REF]).', '0904.1160-1-28-2': 'The solid line in the figure corresponds to the numerical integration of ([REF]) with [MATH] given by Eq. ([REF]).', '0904.1160-1-28-3': 'The circles stand for the analytic expression ([REF]) with the parameters [MATH] are taken so that [MATH] for a homogeneous system with density [MATH] reproduces the same behaviour as that given by Eq. ([REF]) at the same density: [MATH], [MATH].', '0904.1160-1-28-4': 'The agreement between the two calculations is very good at all times.', '0904.1160-1-28-5': 'Therefore, we conclude that the average value of [MATH] as well as the characteristic period of the oscillations is well determined by the values [MATH] and [MATH] calculated with the central density, while the time scale for the damping of the oscillations is determined by the spatial profile of the density.', '0904.1160-1-29-0': 'Regarding the dephasing of the evolution of [MATH] among different points, it is not very strong, in the sense that the damping of the oscillations is relatively slow.', '0904.1160-1-29-1': 'To be more precise, one can have a reasonable fit to the solid line in Fig. [REF] by a function of the form [EQUATION] with [MATH], [MATH], [MATH] and [MATH].', '0904.1160-1-30-0': '## Comparison with mean-field approach', '0904.1160-1-31-0': 'We proceed finally to compare the approximate calculation of [MATH] with a more complete approach in terms dynamical equations for the three components of the vector order parameter, [MATH], cf. Eq. ([REF]).', '0904.1160-1-31-1': 'In the mean-field approximation, such equations can be cast in the form of three coupled Gross-Pitaevskii equations [CITATION]: [EQUATION] where [MATH] and [MATH].', '0904.1160-1-32-0': 'The results of solving Eqs. ([REF]) with a Runge-Kutta algorithm are included in Fig. [REF] as a dashed line.', '0904.1160-1-32-1': 'The average value of the oscillating [MATH] is well estimated by the analytical model of Sect. [REF].', '0904.1160-1-32-2': 'Also, the characteristic timescale of the oscillations is well estimated by Eq. ([REF]).', '0904.1160-1-32-3': 'The overall agreement is good for times [MATH] ms. After this time, the analytical estimate keeps oscillating with a slowly decreasing amplitude, while the numerical solution of the coupled equations ([REF]) shows fluctuating oscillations.', '0904.1160-1-32-4': 'This behaviour has been observed before, and the transition at [MATH] ms has been related to a dynamical instability that leads to the formation of dynamical spin domains in the system [CITATION].', '0904.1160-1-32-5': 'It is thus not surprising that our simple model fails for [MATH].', '0904.1160-1-32-6': 'It is nevertheless remarkable that the time scale set by [MATH] ms is still a good estimate of the characteristic oscillation time even much later during the time evolution.', '0904.1160-1-33-0': '# Dephasing in a magnetic field and the process of structure formation in finite systems', '0904.1160-1-34-0': 'A qualitative difference between the homogeneous system and the confined one appears when a magnetic field is present and, therefore, [MATH].', '0904.1160-1-34-1': 'The dynamics of a spinor condensate in a magnetic field is known to show two limiting behaviours: the mean-field regime, where the interactions energy dominates the evolution, and the Zeeman regime, where the evolution is dominated by the Zeeman term of the Hamiltonian [CITATION].', '0904.1160-1-34-2': 'The crossover between the two regimes occurs when [MATH].', '0904.1160-1-34-3': 'This transition can be studied in real time by changing the homogeneous magnetic field on which the condensate is immersed [CITATION].', '0904.1160-1-35-0': 'This transition can also be observed between different spatial regions of an inhomogeneous system.', '0904.1160-1-35-1': 'Indeed, if we assume that the magnetic field, magnetization and central density are chosen so that [MATH] (so that at the center we are in the mean-field regime), then at the wings of the system, where [MATH] we will be in the Zeeman regime.', '0904.1160-1-35-2': 'Therefore, we expect to have a point in real space, where the behaviour with time changes qualitatively.', '0904.1160-1-35-3': 'For a profile as in Eq. ([REF]), this transition point is given by [EQUATION]', '0904.1160-1-35-4': 'Naturally, for [MATH], there is no transition (the density vanishes at [MATH]).', '0904.1160-1-35-5': 'On the other hand, for large enough magnetic field the whole system is in the Zeeman regime ([MATH]).', '0904.1160-1-36-0': 'These two regimes evolve with different characteristic times, [MATH] and [MATH] cf. Eqs. ([REF]) and ([REF]).', '0904.1160-1-36-1': 'Because of this, we can expect [MATH] and the phase in the inner part of the condensate ([MATH]) to evolve at a different rate than in the outer wings of the system ([MATH]), resulting in a particular spatial dependence of the phase.', '0904.1160-1-36-2': 'We note that the appearance of a spatial structure in the phase will lead to the creation of spin currents (cf. [CITATION]) and, thus, to spin textures as reported in [CITATION].', '0904.1160-1-36-3': 'Even though a smooth density profile will lead to a smooth variation of [MATH] with position, from our model we expect that these qualitatively different behaviours should be observable for times [MATH]', '0904.1160-1-37-0': 'Interestingly, in the light of the discussion in Sect. [REF], we observe that the time when the dynamical instability is expected to set in is close to the time when the divergence between mean-field and Zeeman regimes should be observable: [MATH].', '0904.1160-1-37-1': 'Because processes such as spin currents fall beyond LDA, their appearance implies a breakdown of our model, which is therefore not applicable to analyse the process of structure formation.', '0904.1160-1-37-2': 'This breakdown explains the lack of agreement between the results of our LDA model and those from Eqs. ([REF]) for [MATH] observed in Fig. [REF].', '0904.1160-1-38-0': 'The experiments reported in Ref. [CITATION] showed the appearance of spin domains to be simultaneous with that of topological defects (phase windings) and also spin currents.', '0904.1160-1-38-1': 'This observation is consistent with the model just sketched.', '0904.1160-1-38-2': 'The timescale for the appearance of spin domains is estimated in that reference to be [MATH]', '0904.1160-1-39-0': 'Similarly, Saito et al. [CITATION] estimate the timescale for the occurrence of a dynamical instability as [MATH] when the magnetic field is small; this estimate coincides with our [MATH].', '0904.1160-1-39-1': 'On the other hand, for larger magnetic fields ([MATH] with [MATH]), the relevant instability timescale is [MATH], which is similar to [MATH].', '0904.1160-1-40-0': 'From their simulations, Saito and Ueda indicated [CITATION] that the formation of spin domains starts at the center of the condensate, and then spreads out.', '0904.1160-1-40-1': 'In our model, however, the position where the phase slip appears is determined by [MATH], and therefore is in principle amenable to be modified experimentally.', '0904.1160-1-40-2': 'It seems interesting to investigate the prospect to control the spatial appearance of spin domains and phase structures as predicted by Eq. ([REF]).', '0904.1160-1-41-0': '# Summary and conclusions', '0904.1160-1-42-0': 'We have studied the dynamics of a trapped spin-1 condensate under a magnetic field.', '0904.1160-1-42-1': 'First, we have analyzed the homogeneous system, and seen that its dynamics can be understood in terms of orbits in the [MATH] space.', '0904.1160-1-42-2': 'We have then solved analytically for the dynamical evolution [MATH].', '0904.1160-1-42-3': 'We have used this information to study the trapped system by means of the Local Density Approximation (LDA).', '0904.1160-1-42-4': 'The results of this approach agree with those of a mean-field treatment for evolution times before the occurrence of a dynamical instability [CITATION].', '0904.1160-1-42-5': 'In particular, the expected average value of [MATH], as well as the characteristic time scale of its dynamics are well predicted by the formulas for the homogeneous system.', '0904.1160-1-43-0': 'Our analysis of the trapped system has shown that, in the presence of a magnetic field, both the mean-field and Zeeman regimes are realized in a single spinor condensate.', '0904.1160-1-43-1': 'The analysis of this model allows for some qualitative insight into the process of structure formation.', '0904.1160-1-43-2': 'In particular, our model identifies a transition point around which this structure is generated, and predicts that it should be tunable, cf. Eq. ([REF]), which could be tested in future experiments.', '0904.1160-1-44-0': 'We would like to acknowledge A. Polls, A. Sanpera and T. Petit for discussions and encouragement to develop this project, and M. Lee for comments on an early version of the manuscript.', '0904.1160-1-44-1': 'This work is supported by the UK EPSRC (grant no.', '0904.1160-1-44-2': 'EP/E025935).'}

{'0904.1160-2-0-0': 'We study the dynamics of a trapped spin-1 condensate in a magnetic field.', '0904.1160-2-0-1': 'First, we analyze the homogeneous system, for which the dynamics can be understood in terms of orbits in phase space.', '0904.1160-2-0-2': 'We analytically solve for the dynamical evolution of the populations of the various Zeeman components of the homogeneous system.', '0904.1160-2-0-3': 'This result is then applied via a local density approximation to trapped quasi-1D condensates.', '0904.1160-2-0-4': 'Our analysis of the trapped system in a magnetic field shows that both the mean-field and Zeeman regimes are simultaneously realized, and we argue that the border between these two regions is where spin domains and phase defects are generated.', '0904.1160-2-0-5': 'We propose a method to experimentally tune the position of this border.', '0904.1160-2-1-0': '# Introduction', '0904.1160-2-2-0': 'Bose-Einstein condensates (BECs) with a spin degree of freedom are an interesting field of research in many-body physics as they realize both superfluidity and magnetism in a well-controlled environment.', '0904.1160-2-2-1': 'First realized experimentally with [MATH]Na ten years ago [CITATION], their study has matured remarkably over the last few years, with several groups studying their dynamics [CITATION] and thermodynamics [CITATION].', '0904.1160-2-2-2': 'Of particular interest is the study of the process by which spin domains are formed during time evolution, a phenomenon observed experimentally [CITATION] and in numerical simulations based on a mean-field approach [CITATION].', '0904.1160-2-3-0': 'The complicated dynamics of these non-linear systems, especially when they are subjected to time-varying external fields, makes the physical understanding of the structure formation process somehow elusive.', '0904.1160-2-3-1': 'To address this point, we present here a simple model based on an analytic solution for the homogeneous system for arbitrary magnetic fields [MATH] and magnetizations [MATH].', '0904.1160-2-3-2': 'This solution is then applied to the study of realistic, trapped spin-1 condensates by means of the local density approximation (LDA).', '0904.1160-2-3-3': 'This approximation has already been applied successfully in a number of studies on scalar BECs, as well as cold Fermi gases.', '0904.1160-2-3-4': 'From the analysis of our results we are able to provide an intuitive picture of the process leading to the structure formation.', '0904.1160-2-3-5': 'Further, we argue that it should be possible to experimentally "tune" the spatial region where this process starts within the condensate.', '0904.1160-2-4-0': 'The paper is organized as follows.', '0904.1160-2-4-1': 'In Sect. [REF] we present the phase space of a homogeneous system under a magnetic field [MATH] and for arbitrary [MATH], and introduce the phase-space orbits that describe the dynamics of a conservative system.', '0904.1160-2-4-2': 'In Sect. [REF] we solve analytically the dynamical evolution of the homogeneous system.', '0904.1160-2-4-3': 'Then, in Sect. [REF] we describe our local-density approximation for a trapped system and present numerical results for its dynamics (Sect. [REF]), which we compare with simulations based on a mean-field treatment (Sect. [REF]).', '0904.1160-2-4-4': 'In Sect. [REF] we discuss the progressive dephasing of different spatial points of the condensate in a homogeneous magnetic field, and relate this to the process of structure formation, with an indication of a possible experimental test.', '0904.1160-2-4-5': 'Finally, we conclude in Sect. [REF].', '0904.1160-2-5-0': '# Analytical results for the homogeneous system', '0904.1160-2-6-0': '## Energetics of the homogeneous system', '0904.1160-2-7-0': 'A homogeneous condensate of atoms with total spin [MATH] can be described by a vector order parameter [MATH] with [MATH] components, [EQUATION]', '0904.1160-2-7-1': 'The density of atoms in a given Zeeman component [MATH] is [MATH] and the total density is given by [MATH].', '0904.1160-2-7-2': 'Introducing the relative densities for the homogeneous system [MATH], one has [EQUATION]', '0904.1160-2-7-3': 'Given that [MATH] is a conserved quantity, Eq. ([REF]) will be fulfilled at all times during the dynamical evolution.', '0904.1160-2-7-4': 'Moreover, the magnetization [EQUATION] is also a conserved quantity [CITATION].', '0904.1160-2-8-0': 'We now focus our analysis to the case of a [MATH] condensate.', '0904.1160-2-8-1': 'We write the various components of the order parameter as [MATH].', '0904.1160-2-8-2': 'This ansatz, together with conditions ([REF]) and ([REF]), leads to the following expression for the energy per particle of the homogeneous system in the mean-field approach [CITATION]: [EQUATION]', '0904.1160-2-8-3': 'Here [MATH], while [MATH] is given in terms of the [MATH]-wave scattering lengths [MATH] in the channels of total spin [MATH], by [MATH], with [MATH] as the atomic mass.', '0904.1160-2-8-4': 'Finally, [MATH], where the energies of the atomic Zeeman states are given by the Breit-Rabi formula [CITATION] [MATH]), with [MATH] being the atomic hyperfine splitting and [MATH] is a function of the external magnetic field [MATH].', '0904.1160-2-8-5': 'Here, [MATH] are the nuclear and electronic Lande factors, and [MATH] are the nuclear and Bohr magnetons, respectively.', '0904.1160-2-8-6': 'A sketch of the surface [MATH] is given in Fig. [REF].', '0904.1160-2-9-0': 'As indicated above, [MATH] is a constant during dynamical evolution.', '0904.1160-2-9-1': 'Similarly, given initial conditions [MATH], [MATH] will also be conserved, thus defining an orbit on the surface [MATH] in [MATH] space.', '0904.1160-2-9-2': 'A sketch of one such orbit is presented in Fig. [REF].', '0904.1160-2-10-0': 'One should note that, depending on the initial conditions, the orbit defined by [MATH] can be closed or open.', '0904.1160-2-10-1': 'In the first case, [MATH] will be a periodic function of time, while in the latter case, [MATH] will grow indefinitely with time.', '0904.1160-2-10-2': 'In both cases, however, [MATH] will be a periodic function of time.', '0904.1160-2-11-0': '## Dynamics of the homogeneous system', '0904.1160-2-12-0': 'We are interested in the time evolution of the densities of the different Zeeman components, [MATH].', '0904.1160-2-12-1': 'From Eqs. ([REF]) and ([REF]) we have that [EQUATION]', '0904.1160-2-12-2': 'Therefore, we only need to follow the evolution of [MATH], which is given by [EQUATION]', '0904.1160-2-12-3': 'With Eq ([REF]), we rewrite this as [EQUATION]', '0904.1160-2-12-4': 'It can be shown that the term in [MATH] actually drops out and we are left with a cubic polynomial on [MATH], [EQUATION] with [EQUATION] and [MATH] are the roots of [MATH], [MATH].', '0904.1160-2-12-5': 'For ground state ([MATH]) alkalies [MATH].', '0904.1160-2-12-6': 'Therefore, [MATH] for [MATH] and [MATH] for [MATH] [CITATION].', '0904.1160-2-12-7': 'For concreteness, in the following we will assume [MATH], i.e., ferromagnetic interactions.', '0904.1160-2-13-0': 'We will now integrate the time evolution of [MATH].', '0904.1160-2-13-1': 'To do so, we introduce an auxiliary variable [MATH] through [MATH].', '0904.1160-2-13-2': 'This will satisfy the differential equation [EQUATION] where we defined [EQUATION]', '0904.1160-2-13-3': 'The first order differential equation ([REF]) can be solved analytically by separating the variables [MATH] and [MATH], and integrating: [EQUATION]', '0904.1160-2-13-4': 'The solution to the last integral can be expressed in terms of the elliptic integral of the first kind , [EQUATION]', '0904.1160-2-13-5': 'Taking as initial condition [MATH] and using the fact that [MATH], we can express [MATH] in a compact form by means of the Jacobi elliptic functions [CITATION], which are defined as the inverses of the elliptic integrals, [EQUATION] with [MATH], i.e., [MATH].', '0904.1160-2-13-6': 'Finally, we undo the change in variables to write down the time evolution of the population of the [MATH] state, [EQUATION]', '0904.1160-2-13-7': 'In accordance with the identity [CITATION] [EQUATION] and given that both [MATH] and [MATH] are periodic functions in [MATH] with period [MATH], [MATH] will be a periodic function of time with period [EQUATION]', '0904.1160-2-13-8': 'Here, [MATH] stands for the complete elliptic integral of the first kind.', '0904.1160-2-13-9': 'We note that result ([REF]) agrees with that in Ref. [CITATION], where [MATH] was calculated directly by performing the integral [MATH] over a period of evolution.', '0904.1160-2-13-10': 'Further, let us point out that the average value [MATH] does not necessarily coincide with the position of the minimum of [MATH], i.e., [MATH] may differ from the equilibrium value [MATH] (as given, e.g., in Ref. [CITATION] for the case [MATH]).', '0904.1160-2-13-11': 'This is illustrated in Figs. [REF](b) and [REF](b).', '0904.1160-2-14-0': '## Evolution in the absence of a magnetic field', '0904.1160-2-15-0': 'We observe that the representation of [MATH] as a cubic polynomial on [MATH], Eq. ([REF]), cannot be performed when [MATH], i.e., when [MATH].', '0904.1160-2-15-1': 'In this case, the analytic expression ([REF]) is meaningless, as it would apparently result in no time evolution at all.', '0904.1160-2-15-2': 'Actually, in this situation, [MATH] can be written as a quadratic polynomial on [MATH]: [EQUATION]', '0904.1160-2-15-3': 'Here [MATH].', '0904.1160-2-15-4': 'Note that [MATH] for [MATH] as well as for [MATH], and in both cases we will have [MATH].', '0904.1160-2-15-5': 'Following a procedure analogous to that above, we arrive at [EQUATION] with [MATH].', '0904.1160-2-15-6': 'In this case, [MATH] follows a pure sinusoidal evolution as has been predicted before in a number of references, e.g., [CITATION].', '0904.1160-2-15-7': 'The average value is [MATH], and the period reads (compare with [CITATION]) [EQUATION]', '0904.1160-2-15-8': 'We show in Fig. [REF] the time evolution of [MATH] for two representative cases.', '0904.1160-2-15-9': 'The different panels compare the analytic evolution -given by Eq. ([REF]) or ([REF])- with a numerical solution of the corresponding equation for [MATH].', '0904.1160-2-15-10': 'In all cases, we see that the amplitude as well as the period of the time evolution are well predicted by the analytic results.', '0904.1160-2-16-0': 'Finally, we show in Fig. [REF] a plot of [MATH] vs. [MATH] corresponding to the time evolution depicted in Fig. [REF].', '0904.1160-2-16-1': 'For the case with magnetic field and [MATH] we observe that the average value [MATH] (indicated by the dashed line) differs from the position of the minimum of [MATH], cf. Fig. [REF]) due to the deformation of the orbit.', '0904.1160-2-17-0': '# Dynamics of the trapped system', '0904.1160-2-18-0': 'We have established in the previous section the dynamical evolution of a homogeneous spin-1 condensate, in terms of orbits in the [MATH] plane constrained by (i) conservation of density, (ii), conservation of magnetization, and (iii) conservation of energy.', '0904.1160-2-18-1': 'The resulting dynamics of the population of the [MATH] Zeeman component has been shown to be a periodic function of time, with a period determined by the density [MATH] of the system, its magnetization [MATH], as well as the initial conditions of the evolution (implicit in [MATH] and, therefore, in [MATH] or [MATH]), cf. Eqs. ([REF]) and ([REF]).', '0904.1160-2-18-2': 'Now, we will transfer these results to a realistic case of a trapped, quasi-one-dimensional (1D) condensate.', '0904.1160-2-19-0': '## Local-density approximation', '0904.1160-2-20-0': 'The initial conditions for the evolution of a trapped spinor condensate are the set of complex values [MATH] for all Zeeman components [MATH] and all positions [MATH] where the density is not zero.', '0904.1160-2-20-1': 'In typical experiments, the preparation of the initial state is such that [MATH] is a constant independent of position.', '0904.1160-2-20-2': 'This, together with the fact that [MATH] for the systems studied so far, has lead to some theoretical works based on the so-called single-mode approximation (SMA), which assumes that [MATH] for all times [MATH] of the evolution, i.e., that the spatial variation in the density of each Zeeman component is always given by the total density profile.', '0904.1160-2-20-3': 'However, numerical studies beyond the SMA (e.g., [CITATION]) predicted the formation of spin domains as time goes by.', '0904.1160-2-20-4': 'These have been observed in a number of experiments, e.g., [CITATION].', '0904.1160-2-20-5': 'In order to be able to observe the formation of spin domains during time evolution in a trapped system, we will therefore not make use of the SMA, but apply the analytical results of Sect. [REF] via the LDA, i.e., we will assume that the evolution of the [MATH] population at each point within the condensate, [MATH], is given by Eq. ([REF]) [or Eq. ([REF])] with the substitution [MATH].', '0904.1160-2-20-6': 'Here, the total density is normalized to the total number of atoms in the condensate, [MATH].', '0904.1160-2-20-7': 'Similarly, we introduce the local densities of atoms in a given Zeeman state [MATH] normalized as [MATH].', '0904.1160-2-20-8': 'The conservation laws read now [MATH] and [MATH].', '0904.1160-2-20-9': 'We note that [MATH] does not change in time at low enough temperatures [CITATION] unless momentum is imparted to the center of mass or to one or more of the Zeeman components [CITATION].', '0904.1160-2-21-0': 'In the language of the phase space introduced in Sect. [REF], a trapped system corresponds to an infinite-dimensional phase space, with a pair of variables [MATH] associated to each point [MATH].', '0904.1160-2-21-1': 'According to the LDA, we divide this whole phase space in sections corresponding to the different positions, and assume that they are independent.', '0904.1160-2-21-2': 'The initial condition described above, [MATH], corresponds then to the dynamical system starting in all the different positions [MATH] at the same point of the corresponding phase space, [MATH].', '0904.1160-2-21-3': 'The dynamical evolution of the system corresponds then to the population [MATH] at each point [MATH] following its own particular orbit in the corresponding [MATH] space, that is, [MATH] at position [MATH] follows the dynamical equation of the homogeneous system ([REF]) [or Eq. ([REF])] with the parameters [MATH] and [MATH] determined by the local density [MATH].', '0904.1160-2-21-4': 'In other words, we assume that the position dependence is only parametric, and comes through the values of the parameters [MATH] and [MATH].', '0904.1160-2-21-5': 'We will indicate this by [MATH].', '0904.1160-2-21-6': 'The density at position [MATH] of atoms in the Zeeman component [MATH] at time [MATH] will then be [EQUATION] with [MATH] given by Eq. ([REF]) with the substitution [MATH], and [MATH] is a conserved quantity [CITATION].', '0904.1160-2-22-0': 'Note that the orbits associated to different points [MATH] may differ from one another, as their shapes depend inter alia on the local density [MATH], cf. Eq. ([REF]).', '0904.1160-2-22-1': 'This fact, together with the position dependence of the parameters [MATH] and [MATH], is expected to lead to a dephasing of the evolution of the partial densities [MATH] at the different points, washing out the oscillations in the integrated populations, [MATH], in contrast to the stable oscillations that we have found for the homogeneous system, cf. Fig. [REF].', '0904.1160-2-23-0': 'In order to evaluate [MATH] it is necessary to know the density profile of the system.', '0904.1160-2-23-1': 'A good estimate for [MATH] in trapped atomic gases is given by the Thomas-Fermi approximation, [EQUATION]', '0904.1160-2-23-2': 'For a quasi-1D system with total number of atoms [MATH] and central density [MATH], [MATH].', '0904.1160-2-23-3': 'The integrated population in [MATH] then reads [EQUATION]', '0904.1160-2-24-0': '## Analytic approximation with sinusoidal time dependence', '0904.1160-2-25-0': 'The time dependence of [MATH] has in principle to be calculated from Eq. ([REF]) for each position [MATH] at each time step, and then the integral ([REF]) performed numerically to determine [MATH].', '0904.1160-2-25-1': 'It is possible however to give an analytical estimation for [MATH] if we make a further assumption on the time evolution.', '0904.1160-2-25-2': 'From Fig. [REF], we see that the evolution of [MATH] for the homogeneous system is very close to a sinusoidal function even when [MATH]', '0904.1160-2-26-0': 'This is illustrated in Fig. [REF](b), where a function of the form [EQUATION] has been fitted to the numerical values obtained from Eq. ([REF]).', '0904.1160-2-26-1': 'The fit is very good, even for this case, where the orbit in phase space is strongly deformed [cf. Fig. [REF](b)].', '0904.1160-2-26-2': 'The advantage of approximating the time evolution of [MATH] by Eq. ([REF]) is that it allows for an analytic evaluation of the spatial integral ([REF]), taking into account the position dependence of [MATH].', '0904.1160-2-26-3': 'Indeed, from Eq. ([REF]) we expect [MATH].', '0904.1160-2-26-4': 'It is easy to show that [EQUATION]', '0904.1160-2-26-5': 'Here [MATH] and [MATH] are the Fresnel integrals [CITATION], and we introduced [MATH] and [MATH].', '0904.1160-2-27-0': 'We show in Fig. [REF] the time evolution of the integrated [MATH] population as given by Eqs. ([REF]) and ([REF]).', '0904.1160-2-28-0': 'This calculation has been done for a quasi-1D system of [MATH]Rb atoms in a trap such that the central density is [MATH] cm[MATH].', '0904.1160-2-28-1': 'The initial conditions are [MATH] and [MATH] and we have taken a magnetic field [MATH] mG (cf. Fig. [REF]).', '0904.1160-2-28-2': 'The solid line in the figure corresponds to the numerical integration of Eq. ([REF]) with [MATH] given by Eq. ([REF]).', '0904.1160-2-28-3': 'The circles stand for the analytic expression ([REF]) with the parameters [MATH] taken so that [MATH] for a homogeneous system with density [MATH] reproduces the same behavior as that given by Eq. ([REF]) at the same density: [MATH], and [MATH].', '0904.1160-2-28-4': 'The agreement between the two calculations is very good at all times.', '0904.1160-2-28-5': 'Therefore, we conclude that the average value of [MATH] as well as the characteristic period of the oscillations is well determined by the values [MATH] and [MATH] calculated with the central density, while the time scale for the damping of the oscillations is determined by the spatial profile of the density.', '0904.1160-2-29-0': 'Regarding the dephasing of the evolution of [MATH] among different points, it is not very strong, in the sense that the damping of the oscillations is relatively slow.', '0904.1160-2-29-1': 'To be more precise, one can have a reasonable fit to the solid line in Fig. [REF] by a function of the form [EQUATION] with [MATH], [MATH], [MATH] and [MATH].', '0904.1160-2-30-0': '## Comparison with the mean-field approach', '0904.1160-2-31-0': 'We proceed finally to compare the approximate calculation of [MATH] with a more complete approach in terms of the dynamical equations for the three components of the vector order parameter, [MATH], cf. Eq. ([REF]).', '0904.1160-2-31-1': 'In the mean-field approximation, such equations can be cast in the form of three coupled Gross-Pitaevskii equations [CITATION], [EQUATION] where [MATH] and [MATH].', '0904.1160-2-32-0': 'The results of solving Eqs. ([REF]) with a Runge-Kutta algorithm are included in Fig. [REF] as a dashed line.', '0904.1160-2-32-1': 'The average value of the oscillating [MATH] is well estimated by the analytical model of Sect. [REF].', '0904.1160-2-32-2': 'Also, the characteristic time scale of the oscillations is well estimated by Eq. ([REF]).', '0904.1160-2-32-3': 'The overall agreement is good for times [MATH] ms. After this time, the analytical estimate keeps oscillating with a slowly decreasing amplitude, while the numerical solution of the coupled equations ([REF]) shows fluctuating oscillations.', '0904.1160-2-32-4': 'This behavior has been observed before, and the transition at [MATH] ms has been related to a dynamical instability that leads to the formation of dynamical spin domains in the system [CITATION].', '0904.1160-2-32-5': 'It is thus not surprising that our simple model fails for [MATH].', '0904.1160-2-32-6': 'It is nevertheless remarkable that the time scale set by [MATH] ms is still a good estimate of the characteristic oscillation time even much later during the time evolution.', '0904.1160-2-33-0': '# Dephasing in a magnetic field and the process of structure formation in finite systems', '0904.1160-2-34-0': 'A qualitative difference between the homogeneous system and the confined one appears when a magnetic field is present and, therefore, [MATH].', '0904.1160-2-34-1': 'The dynamics of a spinor condensate in a magnetic field is known to show two limiting behaviors: the mean-field regime, where the interaction energy dominates the evolution, and the Zeeman regime, where the evolution is driven by the Zeeman term of the Hamiltonian [CITATION].', '0904.1160-2-34-2': 'The crossover between the two regimes occurs when [MATH].', '0904.1160-2-34-3': 'This transition can be studied in real time by changing the (homogeneous) magnetic field on which the condensate is immersed [CITATION].', '0904.1160-2-35-0': 'This transition can also be observed between different spatial regions of an inhomogeneous system.', '0904.1160-2-35-1': 'Indeed, if we assume that the magnetic field, magnetization and central density are chosen so that [MATH] (so that at the center we are in the mean-field regime), then at the wings of the system, where [MATH], we will be in the Zeeman regime.', '0904.1160-2-35-2': 'Therefore, we expect to have a region in real space where the behavior with time changes qualitatively.', '0904.1160-2-35-3': 'For a profile as in Eq. ([REF]), this transition border is given by [EQUATION]', '0904.1160-2-35-4': 'Naturally, for [MATH], there is no transition (the density vanishes at [MATH]).', '0904.1160-2-35-5': 'On the other hand, for large enough magnetic field the whole system is in the Zeeman regime ([MATH]).', '0904.1160-2-36-0': 'These two regimes evolve with different characteristic times, [MATH] and [MATH], cf. Eqs. ([REF]) and ([REF]).', '0904.1160-2-36-1': 'Because of this, we can expect [MATH] and the phase in the inner part of the condensate ([MATH]) to evolve at a different rate than in the outer wings of the system ([MATH]), resulting in a particular spatial dependence of the phase.', '0904.1160-2-36-2': 'We note that the appearance of a spatial structure in the phase will lead to the creation of spin currents [CITATION] and, thus, to spin textures as reported in [CITATION].', '0904.1160-2-36-3': 'Even though a smooth density profile will lead to a smooth variation in [MATH] with position, from our model we expect that these qualitatively different behaviors should be observable for times [MATH]', '0904.1160-2-37-0': 'Interestingly, in light of the discussion in Sect. [REF], we observe that the time when the dynamical instability is expected to set in is close to the time when the divergence between mean-field and Zeeman regimes should be observable: [MATH].', '0904.1160-2-37-1': 'Because processes such as spin currents fall beyond LDA, their appearance implies a breakdown of our model, which is therefore not applicable to analyze the process of structure formation.', '0904.1160-2-37-2': 'This breakdown explains the lack of agreement between the results of our LDA model and those from Eqs. ([REF]) for [MATH] observed in Fig. [REF].', '0904.1160-2-38-0': 'The experiments reported in Ref. [CITATION] showed the appearance of spin domains to be simultaneous with that of topological defects (phase windings) and also spin currents.', '0904.1160-2-38-1': 'This observation is consistent with the model just sketched.', '0904.1160-2-38-2': 'The time scale for the appearance of spin domains is estimated in that reference to be [MATH]', '0904.1160-2-39-0': 'Similarly, Saito et al. [CITATION] determined the time scale for the occurrence of a dynamical instability to be [MATH] when the magnetic field is small; this estimate coincides with our [MATH].', '0904.1160-2-39-1': 'On the other hand, for larger magnetic fields [[MATH] with [MATH]], the relevant instability time scale is [MATH], which is similar to [MATH].', '0904.1160-2-40-0': 'From their simulations, Saito and Ueda indicated [CITATION] that the formation of spin domains starts at the center of the condensate, and then spreads out.', '0904.1160-2-40-1': 'In our model, however, the position where the phase slip appears is determined by [MATH], and therefore is in principle amenable to be modified experimentally.', '0904.1160-2-40-2': 'It seems interesting to investigate the prospect to control the spatial appearance of spin domains and phase structures as predicted by Eq. ([REF]).', '0904.1160-2-41-0': '# Summary and conclusions', '0904.1160-2-42-0': 'We have studied the dynamics of a trapped spin-1 condensate under a magnetic field.', '0904.1160-2-42-1': 'First, we have analyzed the homogeneous system and seen that its dynamics can be understood in terms of orbits in the [MATH] space.', '0904.1160-2-42-2': 'We have then solved analytically for the dynamical evolution [MATH].', '0904.1160-2-42-3': 'We have used this information to study the trapped system by means of the Local Density Approximation (LDA).', '0904.1160-2-42-4': 'The results of this approach agree with those of the mean-field treatment for evolution times before the occurrence of a dynamical instability [CITATION].', '0904.1160-2-42-5': 'In particular, the expected average value of [MATH], as well as the characteristic time scale of its dynamics are well predicted by the formulas for the homogeneous system.', '0904.1160-2-43-0': 'Our analysis of the trapped system has shown that, in the presence of a magnetic field, both the mean-field and Zeeman regimes are realized in a single spinor condensate.', '0904.1160-2-43-1': 'The analysis of this model allows for some qualitative insight into the process of structure formation.', '0904.1160-2-43-2': 'In particular, our model identifies a transition point [cf. Eq. ([REF])] around which this structure is generated, and predicts that it should be tunable, which could be tested in future experiments.', '0904.1160-2-44-0': 'I would like to acknowledge A. Polls, A. Sanpera, and T. Petit for discussions and encouragement to develop this project, and M. D. Lee for comments on an early version of the paper.', '0904.1160-2-44-1': 'This work was supported by the UK EPSRC (grant no.', '0904.1160-2-44-2': 'EP/E025935).'}

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[]

[['0904.1160-1-15-7', '0904.1160-2-15-7'], ['0904.1160-1-15-8', '0904.1160-2-15-7'], ['0904.1160-1-20-3', '0904.1160-2-20-3'], ['0904.1160-1-20-9', '0904.1160-2-20-9'], ['0904.1160-1-35-2', '0904.1160-2-35-2']]

[]

['0904.1160-1-15-3', '0904.1160-1-44-2', '0904.1160-2-15-3', '0904.1160-2-44-2']

{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/0904.1160

1708.09536

{'1708.09536-1-0-0': ': Explicit formulae are given for a type of Battle-Lemarie scaling functions and related wavelets.', '1708.09536-1-0-1': 'Compactly supported sums of their translations are established and applied to alternative norm characterization of sequence spaces isometrically isomorphic to Nikolskii-Besov spaces on [MATH].', '1708.09536-1-1-0': '# Introduction', '1708.09536-1-2-0': 'Battle-Lemarie scaling functions are polynomial splines with simple knots at [MATH] obtained by orthogonalisation process of the B-splines.', '1708.09536-1-2-1': 'For [MATH] the [MATH]th order B-spline is defined recursively by [EQUATION] with [MATH].', '1708.09536-1-2-2': 'It is known that [MATH] generates multiresolution analysis of [MATH] [CITATION].', '1708.09536-1-2-3': 'Moreover [CITATION],', '1708.09536-1-3-0': 'Battle-Lemarie scaling functions and related wavelets play an important role in approximation theory, numerical analysis (see e.g. [CITATION]), image, data and signal processing involving analysis of biological sequences and molecular biology-related signals, etc. [CITATION].', '1708.09536-1-3-1': 'On the strength of the differentiation property [EQUATION] this function class has appeared to be an effective tool for solving problems related to the theory of integration and differentiation operators in function spaces [CITATION].', '1708.09536-1-3-2': 'There is a number of papers devoted to Battle-Lemarie scaling functions and wavelets.', '1708.09536-1-3-3': 'Most of them deal with their implicit or approximate expressions.', '1708.09536-1-3-4': 'An idea of how to find explicit formulae for this function class was given in works by I.Ya.', '1708.09536-1-3-5': 'Novikov and S.B. Stechkin ([CITATION] and [CITATION], see also [CITATION] and [CITATION]).', '1708.09536-1-3-6': "The problem, we dealt in [CITATION], concerned the operators' compactness and approximation properties.", '1708.09536-1-3-7': 'The solution method required, first of all, explicit formulae for the chosen wavelet system.', '1708.09536-1-3-8': 'The other important points were in finding their proper transformations and sums in order to localise non-compactly supported scaling functions and wavelets of this type and, making use of [REF], connect their components (splines) with splines of lower or higher orders.', '1708.09536-1-3-9': 'All these questions were covered in [CITATION] for the scaling function and wavelets of the first order only.', '1708.09536-1-3-10': 'The results of the present work make us able to continue the study of compactness and approximation properties of integration operators in spaces of functions of higher smoothness than established in [CITATION].', '1708.09536-1-3-11': 'In the first part of this work we derive exact expressions for a family of Battle-Lemarie scaling functions and wavelets of all positive integer orders ( 2).', '1708.09536-1-3-12': 'The second part of the paper is devoted to the "localisation property" of the class ( 3).', '1708.09536-1-3-13': 'Namely, we establish compactly supported combinations [MATH] of wavelets of Battle-Lemarie type (see [REF] in Theorem [REF]), which contribute to simple connection between two B-splines of different orders by an integration (differential) operator as well as to relation between dilations of [MATH]splines (see [REF], [REF] and [REF]).', '1708.09536-1-3-14': 'Similar result is given for the scaling function (see [REF] in Theorem [REF] in combination with [REF]) and both are applied to equivalent norm characteristics in Besov type spaces (Proposition [REF]).', '1708.09536-1-4-0': 'Throughout the paper, we use [MATH], [MATH] and [MATH] for integers, natural and real numbers, respectively; symbol [MATH] for the complex plane.', '1708.09536-1-4-1': 'We make use of marks [MATH] and [MATH] for introducing new quantities.', '1708.09536-1-5-0': '# Construction of Battle-Lemarie scaling functions and related wavelets The Fourier transform of the normalised [MATH]-spline of the [MATH]th order has the form [EQUATION]', '1708.09536-1-5-1': 'In particular, if [MATH] then [EQUATION] and, therefore, [EQUATION]', '1708.09536-1-5-2': 'The general two-scale relation formula for [MATH], [MATH], has the following form [CITATION]: [EQUATION]', '1708.09536-1-5-3': 'Fixed [MATH] and any [MATH] put [MATH], [MATH].', '1708.09536-1-5-4': 'Each [MATH] let [MATH] denote the [MATH]closure of the linear span of the system [MATH].', '1708.09536-1-5-5': 'It is well-known [CITATION] that the spline spaces [MATH], [MATH], which are generated by the scaling function [MATH], constitute a multiresolution analysis of [MATH] in the sense that Further, there are the orthogonal complementary subspaces [MATH] such that Wavelet subspaces [MATH], [MATH], related to the spline [MATH], are also generated by some basic functions (wavelets) in the same manner as the spline spaces [MATH], [MATH], are generated by the spline [MATH].', '1708.09536-1-6-0': 'It is well-known that there exists another scaling function, whose integer translations form an orthonormal system within the same multiresolution analysis.', '1708.09536-1-6-1': 'The function [MATH] satisfying [EQUATION] is called the Battle-Lemarie scaling function [CITATION].', '1708.09536-1-6-2': 'Integer translations of [MATH] form an orthonormal basis in [MATH] of the multiresolution analysis generated by [MATH].', '1708.09536-1-7-0': 'The [MATH]th order Battle-Lemarie wavelet is the function [MATH] whose Fourier transform is [EQUATION]', '1708.09536-1-7-1': 'Integer translations of [MATH] form an orthonormal basis in [MATH] of the multiresolution analysis generated by [MATH].', '1708.09536-1-7-2': 'A scaling function [MATH] and one of its associated wavelets [MATH] form a wavelet system [MATH].', '1708.09536-1-8-0': 'In order to derive explicit expressions for spline wavelet systems of all orders [MATH] with properties analogous to [MATH] and [MATH] we follow the idea from [CITATION] (see also [CITATION], [CITATION] and [CITATION]).', '1708.09536-1-9-0': '## Auxiliary results', '1708.09536-1-10-0': 'Denote [EQUATION] and write [EQUATION] where the low-pass filter [MATH] (see [REF]) is given by [EQUATION].', '1708.09536-1-10-1': 'A wavelet function [MATH] related to [MATH] must be of the form [EQUATION] where the high-pass filter [MATH] is given by [EQUATION]', '1708.09536-1-11-0': 'It is well-known that [MATH] and [MATH], [MATH] form the Haar system.', '1708.09536-1-11-1': 'Write [EQUATION]', '1708.09536-1-11-2': 'The expressions [REF] were investigated in terms of rational polynomials in [CITATION].', '1708.09536-1-11-3': 'Let us shortly explain milestones of this well-known theory (see also [CITATION]).', '1708.09536-1-11-4': 'Differentiation of [REF] yields [EQUATION].', '1708.09536-1-11-5': 'As it was mentioned before, [MATH].', '1708.09536-1-11-6': 'Following [CITATION], we express [MATH] as polynomials in the variable [MATH] by means of [MATH] to obtain the recurrence relation for [EQUATION] with [MATH].', '1708.09536-1-11-7': 'Being an even polynomial for even [MATH], the [MATH] may be expressed as a polynomial [MATH] in the variable [MATH] of degree [MATH].', '1708.09536-1-11-8': 'By [CITATION], [MATH] has all simple and purely imaginary zeros [MATH].', '1708.09536-1-11-9': 'The change of variable transforms them into the zeros [MATH] of [MATH], which must all be positive and not less than [MATH].', '1708.09536-1-11-10': 'Thus, since [MATH], [EQUATION].', '1708.09536-1-11-11': 'We may and shall assume that [MATH] (see [CITATION]).', '1708.09536-1-11-12': 'To find zeros of [MATH], [MATH], with respect to [MATH] in this case, we write [MATH], that is [EQUATION].', '1708.09536-1-11-13': 'Thus, [MATH], where [MATH] and, therefore, [MATH].', '1708.09536-1-11-14': 'Notice that [EQUATION] and, therefore, [MATH].', '1708.09536-1-12-0': 'In view of [REF], [REF], [REF] and [REF] we construct a wavelet system of Battle-Lemarie type as follows.', '1708.09536-1-13-0': '## The construction Denote [EQUATION].', '1708.09536-1-13-1': 'Describe the case [MATH] before considering the general situation.', '1708.09536-1-13-2': 'If [MATH] then [MATH], [MATH] (see [REF] and [REF] in 2.1) and we have [REF] with [EQUATION]', '1708.09536-1-13-3': 'Taking into account [REF] and [REF] with [MATH], define [EQUATION]', '1708.09536-1-13-4': 'If we choose [MATH] in [REF] then [EQUATION] which is a shifted version of [MATH] of the form [REF].', '1708.09536-1-13-5': 'Further, since for [MATH] [EQUATION] then we can write [EQUATION]', '1708.09536-1-13-6': 'Choosing between [MATH] we obtain [MATH] and [MATH] with "left tails" of the forms [EQUATION] and, analogously, [MATH] and [MATH] with "right tails" of the forms [EQUATION]', '1708.09536-1-13-7': 'Since [MATH] and [MATH] differ by integer shifts only, they constitute the same multiresolution analysis of [MATH].', '1708.09536-1-14-0': 'To construct wavelets [MATH] related to [MATH] and [MATH] we write according to [REF], [REF] and [REF]: [EQUATION] where (see also [REF]) [EQUATION] that is [EQUATION].', '1708.09536-1-14-1': 'Thus, taking into account that for any [MATH] [EQUATION] we obtain [EQUATION] which means that we can construct two types of [MATH]: one having [MATH] in the numerator in [MATH] and the other one with [MATH].', '1708.09536-1-14-2': 'Symbol [MATH] will be used to emphasise our choice of [MATH] in [REF].', '1708.09536-1-15-0': 'With respect to [REF] and in view of [EQUATION] the [MATH] with [MATH] defined by [REF] for [MATH], have the following forms: Analogously, for the [MATH] with [MATH] defined by [REF] for [MATH], we get To obtain further results of the paper in the case [MATH] (see 3-4) we shall use wavelet systems [MATH], [MATH] and their shifted counterparts [MATH] and [MATH].', '1708.09536-1-16-0': 'For general [MATH] we define [MATH] as follows: [EQUATION] where (see [REF]) [EQUATION].', '1708.09536-1-16-1': 'The Fourier transform of a wavelet function [MATH] related to [MATH] must satisfy the condition [EQUATION] where, with [MATH], [MATH], [EQUATION]', '1708.09536-1-16-2': 'By [REF] [EQUATION]', '1708.09536-1-16-3': 'Considering [REF], we can vary [MATH] in different terms in the denumerator of [MATH].', '1708.09536-1-16-4': 'But, for simplicity, in this work we limit our attention to the cases with either all "[MATH]" or all "[MATH]" only.', '1708.09536-1-17-0': 'If [MATH] for all [MATH], we write, taking into account [REF], [EQUATION] and, in view of [MATH], where [MATH], we obtain [EQUATION]', '1708.09536-1-17-1': 'If [MATH] for at least one of [MATH] we write, similarly to the case [MATH]: [EQUATION]', '1708.09536-1-17-2': 'We say that [MATH] (or, alternatively, that [MATH]), where [MATH], if [MATH] (or [MATH]).', '1708.09536-1-17-3': 'Let [MATH] denote the cardinality of the set [MATH].', '1708.09536-1-17-4': 'Then, it follows from [REF] and [REF], that [EQUATION] where [MATH].', '1708.09536-1-17-5': 'By definition of [MATH] (see 2.1) and on the strength of [REF], the system [MATH], defined by [REF] or by [REF], is an orthonormal basis of [MATH] generated by [MATH].', '1708.09536-1-18-0': 'In order to construct [MATH], we write, taking into account [REF], [REF] and [REF], [EQUATION] where [EQUATION].', '1708.09536-1-18-1': 'Then, on the strength of [REF] and [REF], [EQUATION]', '1708.09536-1-18-2': 'Denote [MATH].', '1708.09536-1-18-3': 'Similarly to the case [MATH], we shall use symbols [MATH] to emphasise the choice of [MATH] and [MATH], [MATH], in the product [MATH] in [REF].', '1708.09536-1-19-0': 'Since [MATH], then the pre-image of [EQUATION] is a sum of [MATH] translations of [MATH] with coefficients depending on [MATH] and [MATH], [MATH].', '1708.09536-1-19-1': 'It holds: [EQUATION]', '1708.09536-1-19-2': 'The part [EQUATION] in [REF] is the most essential in the definition of [MATH].', '1708.09536-1-19-3': 'Its similarity with the right hand side of the two-scale relation formula [REF] plays an important role in connection between dilations of [MATH]splines (see [REF], [REF] and [REF]).', '1708.09536-1-19-4': 'Pre-image of [REF] has the form [EQUATION].', '1708.09536-1-19-5': 'Moreover, up to a constant, it is equal to [MATH]th order derivative of [MATH] (see [REF]): [EQUATION]', '1708.09536-1-20-0': 'Analogously one can construct [MATH] with all [MATH] possible combinations of [MATH], [MATH].', '1708.09536-1-20-1': 'For the sake of convenience it is reasonable to center them at [MATH].', '1708.09536-1-20-2': 'To obtain further results of the paper (see 3-4) we shall use wavelet systems [EQUATION] with [MATH], [MATH].', '1708.09536-1-20-3': 'By construction, translations of those [MATH] and [MATH] form orthonormal basis in subspaces [MATH] and [MATH] of [MATH] related to the multiresolution analysis generated by [MATH] and [MATH].', '1708.09536-1-20-4': 'Moreover, since [MATH], [MATH], then the system [EQUATION] where [MATH] and [MATH], consisting of [EQUATION] is an orthonormal basis in [MATH] with the following properties: The (iv) follows, in particular, from [CITATION] (see also [CITATION]).', '1708.09536-1-21-0': 'We complete the section by example of [MATH] and [MATH] of order [MATH].', '1708.09536-1-22-0': 'Let [MATH].', '1708.09536-1-22-1': 'Then [MATH] and [MATH].', '1708.09536-1-22-2': 'Therefore, [MATH] and [MATH].', '1708.09536-1-22-3': 'By [REF], we have with [MATH] (four options, depending on [MATH], in total): [EQUATION] where [EQUATION]', '1708.09536-1-22-4': 'By [REF] and [REF], say, for [MATH], [MATH], (that is in the case [MATH]) [EQUATION] with [MATH].', '1708.09536-1-22-5': 'Since [EQUATION] a wavelet [MATH], related to [MATH] may have, in particular, the following form: [EQUATION]', '1708.09536-1-23-0': '# Localisation property For a function [MATH] on [MATH] and [MATH] put [EQUATION] and define, recursively, (S_^mS_1^mF)(x):=(S_mS_1^mF)(x)+r_j_(S_mS_1^mF)(xm),1c_r, j_J_r;', '1708.09536-1-24-0': 'Let [MATH] be wavelet systems [REF] with [MATH] satisfying [REF] and [MATH] of the form [REF].', '1708.09536-1-24-1': 'For simplicity we assume that [MATH] are centred at [MATH] and [MATH] are centred at [MATH].', '1708.09536-1-24-2': 'Recall that [MATH] denotes cardinality of the set [MATH].', '1708.09536-1-24-3': 'Similarly, [MATH] stands for cardinality of the set [MATH].', '1708.09536-1-24-4': 'It holds [EQUATION] and where [MATH] stands for the [MATH]th order derivative of [MATH] and [EQUATION]', '1708.09536-1-24-5': 'Since [MATH] is centred at [MATH] we obtain by [REF], and based on [REF] is proven.', '1708.09536-1-25-0': 'On the strength of definition [REF] of [MATH], we obtain, similarly to [REF], that [EQUATION] where [MATH] defined by [REF].', '1708.09536-1-25-1': 'Recall that [MATH], [MATH].', '1708.09536-1-26-0': 'Given [MATH] and chosen "[MATH]" or "[MATH]" in [MATH] we consider the collection [MATH] of [MATH] functions [MATH], meaning [MATH] different combinations of [MATH], [MATH], in [MATH].', '1708.09536-1-26-1': 'We pair elements of [MATH] as follows: [MATH] and [MATH] from [MATH] are coupled if [MATH] for [MATH], while [MATH].', '1708.09536-1-26-2': 'Each couple we associate with the function [EQUATION] and call [MATH] the new collection of [MATH] functions [MATH].', '1708.09536-1-26-3': 'If [MATH], we finish the localisation process and obtain the function [MATH] satisfying [EQUATION] that is [EQUATION]', '1708.09536-1-26-4': 'Combination [REF] with substitution [MATH] brings, in particular, [EQUATION]', '1708.09536-1-26-5': 'If [MATH] then, similarly to the previous step, we pair elements of [MATH] by matching two functions [MATH] and [MATH] such that [MATH] for [MATH], but [MATH].', '1708.09536-1-26-6': 'Each couple we associate with the function [EQUATION] and call [MATH] the new collection of [MATH] functions [MATH].', '1708.09536-1-27-0': 'If [MATH] we continue the process.', '1708.09536-1-27-1': 'At [MATH]-th step we deal with functions [EQUATION] from [MATH] and form the new collection [MATH] of [MATH] elements [EQUATION]', '1708.09536-1-27-2': 'Overall, starting from [REF] requiring [MATH] steps, and making exactly [MATH] steps of the form [REF], that is [MATH] steps in total, we obtain the localised function [MATH] with [EQUATION].', '1708.09536-1-27-3': 'From here [REF] follows by [REF].', '1708.09536-1-28-0': 'We have [MATH].', '1708.09536-1-28-1': 'By [REF], it holds, where [MATH] stands for integer part of [MATH]: [EQUATION] [MATH] and [MATH] realize the localisation property of Battle-Lemarie wavelet systems.', '1708.09536-1-28-2': '[MATH] is constructed by integer shifts of [MATH], which generate the same multiresolution analysis in [MATH].', '1708.09536-1-28-3': 'For [MATH] we group proper shifts of wavelets [MATH] from the systems [REF].', '1708.09536-1-28-4': 'They constitute bases in subspaces [MATH] related to multiresolution analysis generated by [MATH] and [MATH].', '1708.09536-1-28-5': 'This localisation property is crucial, in particular, for the estimate [MATH] in the proof of Proposition [REF] in 4.', '1708.09536-1-29-0': '# Equivalent norms theorem', '1708.09536-1-30-0': '## Prerequisites', '1708.09536-1-31-0': '### Nikolskii-Besov spaces Let [MATH] and [MATH].', '1708.09536-1-31-1': 'For the definition of Nikolskii-Besov spaces [MATH] (or Besov type spaces, in other terminology, which is more commonly used) we refer to [CITATION].', '1708.09536-1-31-2': 'If, in addition, [MATH] then one can define [MATH] as follows.', '1708.09536-1-31-3': 'Let [MATH] with [MATH] be the set of all Lebesgue measurable functions [MATH] on [MATH], quasi-normed by [MATH] with the obvious modification for [MATH].', '1708.09536-1-31-4': 'For [MATH] and [MATH] put (see e.g. [CITATION]) [MATH], [MATH] and [EQUATION].', '1708.09536-1-31-5': 'If [MATH] and [MATH] then [MATH] if and only if [MATH] and [EQUATION].', '1708.09536-1-31-6': 'The theory and properties of Nikolskii-Besov spaces [MATH] may be found in [CITATION].', '1708.09536-1-32-0': '### Sequence spaces Let [MATH], [MATH] and [MATH].', '1708.09536-1-32-1': 'The sequence space [MATH] consists of all sequences [EQUATION] such that the quasi-norm [EQUATION] (wth the usual modification if [MATH] and/or [MATH]) is finite.', '1708.09536-1-32-2': 'Sequence spaces of the type [MATH] were introduced in [CITATION] in connection with atomic decomposition of the spaces [MATH].', '1708.09536-1-33-0': '### Spline bases in [MATH] Let [MATH] be the Schwartz space of all complex-valued rapidly decreasing, infinitely differentiable functions on [MATH], and let [MATH] denote its dual space of tempered distributions.', '1708.09536-1-33-1': '[CITATION] Let [MATH] and let [EQUATION] be an ([MATH]-normalised) orthogonal spline basis in [MATH] according to [REF].', '1708.09536-1-34-0': 'Let [MATH], [MATH] and [EQUATION].', '1708.09536-1-34-1': 'Let [MATH].', '1708.09536-1-34-2': 'Then [MATH] if and only if it can be represented as [EQUATION] unconditional convergence being in [MATH] and locally in any space [MATH] with [MATH].', '1708.09536-1-34-3': 'The representation [REF] is unique, [EQUATION] and [EQUATION] is an isomorphic map of [MATH] onto [MATH].', '1708.09536-1-34-4': 'If, in addition, [MATH], [MATH] then [EQUATION] is an unconditional (normalised) basis in [MATH].', '1708.09536-1-34-5': 'A proof of this proposition may be found in [CITATION].', '1708.09536-1-35-0': '## The result It follows from Proposition [REF] and [REF] that one can use [EQUATION] (with usual modifications for [MATH] and [MATH]) as an equivalent characterization of the norm in [MATH], when [MATH], [MATH] and [MATH].', '1708.09536-1-35-1': 'In this part of the paper we establish an equivalent characteristic for [REF].', '1708.09536-1-35-2': 'Let [MATH] and let [EQUATION] be an ([MATH]-normalised) orthogonal spline basis in [MATH] according to [REF].', '1708.09536-1-36-0': 'Let [MATH], [MATH] and [EQUATION].', '1708.09536-1-36-1': 'Then a distribution [MATH] belongs to [MATH] if and only if [EQUATION] (with usual modifications for [MATH] and [MATH]).', '1708.09536-1-36-2': 'Furthermore, [MATH] may be used as an equivalent norm on [MATH].', '1708.09536-1-37-0': 'Let [MATH] with [MATH].', '1708.09536-1-37-1': 'Argumentation for other cases of [MATH] is analogous.', '1708.09536-1-37-2': 'We write, by [REF], [EQUATION]', '1708.09536-1-37-3': 'Then, if [MATH], [EQUATION]', '1708.09536-1-37-4': "Analogously, for [MATH] we obtain using [MATH]times Holder's inequality with [MATH] and [MATH], and representing [MATH] for each [MATH], that [EQUATION]", '1708.09536-1-37-5': 'On the other side, according to the construction of [MATH] (see [REF]) with [MATH] and [MATH] in our case, [EQUATION]', '1708.09536-1-37-6': 'Thus, [EQUATION] where [MATH] for [MATH] and [MATH] if [MATH].', '1708.09536-1-38-0': 'For deriving an estimate similar to [REF] and [REF], but with, say, [MATH] this time, we write, by [REF], taking into account [REF], [REF] and [REF]: [EQUATION]', '1708.09536-1-38-1': 'Denote [MATH] the quantity in the square brackets.', '1708.09536-1-38-2': 'Similarly to [REF], Analogously, for [MATH] [EQUATION]', '1708.09536-1-38-3': 'Since for [MATH] and, similarly, for [MATH] we obtain, taking into account [REF] and [REF], For the reverse estimate we write, by [REF], [EQUATION]', '1708.09536-1-38-4': 'By the construction of [MATH], The required two-sided estimate for the second term in [MATH] follows from the inequalities above and the fact that [MATH] with [MATH] equal to [MATH] or [MATH] are equivalent.'}

{'1708.09536-2-0-0': ': Explicit formulae are given for a type of Battle-Lemarie scaling functions and related wavelets.', '1708.09536-2-0-1': 'Compactly supported sums of their translations are established and applied to alternative norm characterization of sequence spaces isometrically isomorphic to Nikolskii-Besov spaces on [MATH].', '1708.09536-2-1-0': '# Introduction', '1708.09536-2-2-0': 'Battle-Lemarie scaling functions are polynomial splines with simple knots at [MATH] obtained by orthogonalisation process of the B-splines.', '1708.09536-2-2-1': 'For [MATH] the [MATH]th order B-spline is defined recursively by [EQUATION] with [MATH].', '1708.09536-2-2-2': 'It is known that [MATH] generates multiresolution analysis of [MATH] [CITATION].', '1708.09536-2-2-3': 'Moreover [CITATION],', '1708.09536-2-3-0': 'Battle-Lemarie scaling functions and related wavelets play an important role in approximation theory, numerical analysis (see e.g. [CITATION]), image, data and signal processing involving analysis of biological sequences and molecular biology-related signals, etc. [CITATION].', '1708.09536-2-3-1': 'On the strength of the differentiation property [EQUATION] this function class has appeared to be an effective tool for solving problems related to the theory of integration and differentiation operators in function spaces [CITATION].', '1708.09536-2-3-2': 'There is a number of papers devoted to Battle-Lemarie scaling functions and wavelets.', '1708.09536-2-3-3': 'Most of them deal with their implicit or approximate expressions.', '1708.09536-2-3-4': 'An idea of how to find explicit formulae for this function class was given in works by I.Ya.', '1708.09536-2-3-5': 'Novikov and S.B. Stechkin ([CITATION] and [CITATION], see also [CITATION] and [CITATION]).', '1708.09536-2-3-6': "The problem, we dealt in [CITATION], concerned the operators' compactness and approximation properties.", '1708.09536-2-3-7': 'The solution method required, first of all, explicit formulae for the chosen wavelet system.', '1708.09536-2-3-8': 'The other important points were in finding their proper transformations and sums in order to localise non-compactly supported scaling functions and wavelets of this type and, making use of [REF], connect their components (splines) with splines of lower or higher orders.', '1708.09536-2-3-9': 'All these questions were covered in [CITATION] for the scaling function and wavelets of the first order only.', '1708.09536-2-3-10': 'The results of the present work make us able to continue the study of compactness and approximation properties of integration operators in spaces of functions of higher smoothness than established in [CITATION].', '1708.09536-2-3-11': 'In the first part of this work we derive exact expressions for a family of Battle-Lemarie scaling functions and wavelets of all positive integer orders ( 2).', '1708.09536-2-3-12': 'The second part of the paper is devoted to the "localisation property" of the class ( 3).', '1708.09536-2-3-13': 'Namely, we establish compactly supported combinations [MATH] of wavelets of Battle-Lemarie type (see [REF] in Theorem [REF]), which contribute to simple connection between two B-splines of different orders by an integration (differential) operator as well as to relation between dilations of [MATH]splines (see [REF], [REF] and [REF]).', '1708.09536-2-3-14': 'Similar result is given for the scaling function (see [REF] in Theorem [REF] in combination with [REF]) and both are applied to equivalent norm characteristics in Besov type spaces (Proposition [REF]).', '1708.09536-2-4-0': 'Throughout the paper, we use [MATH], [MATH] and [MATH] for integers, natural and real numbers, respectively; symbol [MATH] for the complex plane.', '1708.09536-2-4-1': 'We make use of marks [MATH] and [MATH] for introducing new quantities.', '1708.09536-2-5-0': '# Construction of Battle-Lemarie scaling functions and related wavelets The Fourier transform of the normalised [MATH]-spline of the [MATH]th order has the form [EQUATION]', '1708.09536-2-5-1': 'In particular, if [MATH] then [EQUATION] and, therefore, [EQUATION]', '1708.09536-2-5-2': 'The general two-scale relation formula for [MATH], [MATH], has the following form [CITATION]: [EQUATION]', '1708.09536-2-5-3': 'Fixed [MATH] and any [MATH] put [MATH], [MATH].', '1708.09536-2-5-4': 'Each [MATH] let [MATH] denote the [MATH]closure of the linear span of the system [MATH].', '1708.09536-2-5-5': 'It is well-known [CITATION] that the spline spaces [MATH], [MATH], which are generated by the scaling function [MATH], constitute a multiresolution analysis of [MATH] in the sense that Further, there are the orthogonal complementary subspaces [MATH] such that Wavelet subspaces [MATH], [MATH], related to the spline [MATH], are also generated by some basic functions (wavelets) in the same manner as the spline spaces [MATH], [MATH], are generated by the spline [MATH].', '1708.09536-2-6-0': 'It is well-known that there exists another scaling function, whose integer translations form an orthonormal system within the same multiresolution analysis.', '1708.09536-2-6-1': 'The function [MATH] satisfying [EQUATION] is called the Battle-Lemarie scaling function [CITATION].', '1708.09536-2-6-2': 'Integer translations of [MATH] form an orthonormal basis in [MATH] of the multiresolution analysis generated by [MATH].', '1708.09536-2-7-0': 'The [MATH]th order Battle-Lemarie wavelet is the function [MATH] whose Fourier transform is [EQUATION]', '1708.09536-2-7-1': 'Integer translations of [MATH] form an orthonormal basis in [MATH] of the multiresolution analysis generated by [MATH].', '1708.09536-2-7-2': 'A scaling function [MATH] and one of its associated wavelets [MATH] form a wavelet system [MATH].', '1708.09536-2-8-0': 'In order to derive explicit expressions for spline wavelet systems of all orders [MATH] with properties analogous to [MATH] and [MATH] we follow the idea from [CITATION] (see also [CITATION], [CITATION] and [CITATION]).', '1708.09536-2-9-0': '## Auxiliary results', '1708.09536-2-10-0': 'Denote [EQUATION] and write [EQUATION] where the low-pass filter [MATH] (see [REF]) is given by [EQUATION].', '1708.09536-2-10-1': 'A wavelet function [MATH] related to [MATH] must be of the form [EQUATION] where the high-pass filter [MATH] is given by [EQUATION]', '1708.09536-2-11-0': 'It is well-known that [MATH] and [MATH], [MATH] form the Haar system.', '1708.09536-2-11-1': 'Write [EQUATION]', '1708.09536-2-11-2': 'The expressions [REF] were investigated in terms of rational polynomials in [CITATION].', '1708.09536-2-11-3': 'Let us shortly explain milestones of this well-known theory (see also [CITATION]).', '1708.09536-2-11-4': 'Differentiation of [REF] yields [EQUATION].', '1708.09536-2-11-5': 'As it was mentioned before, [MATH].', '1708.09536-2-11-6': 'Following [CITATION], we express [MATH] as polynomials in the variable [MATH] by means of [MATH] to obtain the recurrence relation for [EQUATION] with [MATH].', '1708.09536-2-11-7': 'Being an even polynomial for even [MATH], the [MATH] may be expressed as a polynomial [MATH] in the variable [MATH] of degree [MATH].', '1708.09536-2-11-8': 'By [CITATION], [MATH] has all simple and purely imaginary zeros [MATH].', '1708.09536-2-11-9': 'The change of variable transforms them into the zeros [MATH] of [MATH], which must all be positive and not less than [MATH].', '1708.09536-2-11-10': 'Thus, since [MATH], [EQUATION].', '1708.09536-2-11-11': 'We may and shall assume that [MATH] (see [CITATION]).', '1708.09536-2-11-12': 'To find zeros of [MATH], [MATH], with respect to [MATH] in this case, we write [MATH], that is [EQUATION].', '1708.09536-2-11-13': 'Thus, [MATH], where [MATH] and, therefore, [MATH].', '1708.09536-2-11-14': 'Notice that [EQUATION] and, therefore, [MATH].', '1708.09536-2-12-0': 'In view of [REF], [REF], [REF] and [REF] we construct a wavelet system of Battle-Lemarie type as follows.', '1708.09536-2-13-0': '## The construction Denote [EQUATION].', '1708.09536-2-13-1': 'Describe the case [MATH] before considering the general situation.', '1708.09536-2-13-2': 'If [MATH] then [MATH], [MATH] (see [REF] and [REF] in 2.1) and we have [REF] with [EQUATION]', '1708.09536-2-13-3': 'Taking into account [REF] and [REF] with [MATH], define [EQUATION]', '1708.09536-2-13-4': 'If we choose [MATH] in [REF] then [EQUATION] which is a shifted version of [MATH] of the form [REF].', '1708.09536-2-13-5': 'Further, since for [MATH] [EQUATION] then we can write [EQUATION]', '1708.09536-2-13-6': 'Choosing between [MATH] we obtain [MATH] and [MATH] with "left tails" of the forms [EQUATION] and, analogously, [MATH] and [MATH] with "right tails" of the forms [EQUATION]', '1708.09536-2-13-7': 'Since [MATH] and [MATH] differ by integer shifts only, they constitute the same multiresolution analysis of [MATH].', '1708.09536-2-14-0': 'To construct wavelets [MATH] related to [MATH] and [MATH] we write according to [REF], [REF] and [REF]: [EQUATION] where (see also [REF]) [EQUATION] that is [EQUATION].', '1708.09536-2-14-1': 'Thus, taking into account that for any [MATH] [EQUATION] we obtain [EQUATION] which means that we can construct two types of [MATH]: one having [MATH] in the numerator in [MATH] and the other one with [MATH].', '1708.09536-2-14-2': 'Symbol [MATH] will be used to emphasise our choice of [MATH] in [REF].', '1708.09536-2-15-0': 'With respect to [REF] and in view of [EQUATION] the [MATH] with [MATH] defined by [REF] for [MATH], have the following forms: Analogously, for the [MATH] with [MATH] defined by [REF] for [MATH], we get To obtain further results of the paper in the case [MATH] (see 3-4) we shall use wavelet systems [MATH], [MATH] and their shifted counterparts [MATH] and [MATH].', '1708.09536-2-16-0': 'For general [MATH] we define [MATH] as follows: [EQUATION] where (see [REF]) [EQUATION].', '1708.09536-2-16-1': 'The Fourier transform of a wavelet function [MATH] related to [MATH] must satisfy the condition [EQUATION] where, with [MATH], [MATH], [EQUATION]', '1708.09536-2-16-2': 'By [REF] [EQUATION]', '1708.09536-2-16-3': 'Considering [REF], we can vary [MATH] in different terms in the denumerator of [MATH].', '1708.09536-2-16-4': 'But, for simplicity, in this work we limit our attention to the cases with either all "[MATH]" or all "[MATH]" only.', '1708.09536-2-17-0': 'If [MATH] for all [MATH], we write, taking into account [REF], [EQUATION] and, in view of [MATH], where [MATH], we obtain [EQUATION]', '1708.09536-2-17-1': 'If [MATH] for at least one of [MATH] we write, similarly to the case [MATH]: [EQUATION]', '1708.09536-2-17-2': 'We say that [MATH] (or, alternatively, that [MATH]), where [MATH], if [MATH] (or [MATH]).', '1708.09536-2-17-3': 'Let [MATH] denote the cardinality of the set [MATH].', '1708.09536-2-17-4': 'Then, it follows from [REF] and [REF], that [EQUATION] where [MATH].', '1708.09536-2-17-5': 'By definition of [MATH] (see 2.1) and on the strength of [REF], the system [MATH], defined by [REF] or by [REF], is an orthonormal basis of [MATH] generated by [MATH].', '1708.09536-2-18-0': 'In order to construct [MATH], we write, taking into account [REF], [REF] and [REF], [EQUATION] where [EQUATION].', '1708.09536-2-18-1': 'Then, on the strength of [REF] and [REF], [EQUATION]', '1708.09536-2-18-2': 'Denote [MATH].', '1708.09536-2-18-3': 'Similarly to the case [MATH], we shall use symbols [MATH] to emphasise the choice of [MATH] and [MATH], [MATH], in the product [MATH] in [REF].', '1708.09536-2-19-0': 'Since [MATH], then the pre-image of [EQUATION] is a sum of [MATH] translations of [MATH] with coefficients depending on [MATH] and [MATH], [MATH].', '1708.09536-2-19-1': 'It holds: [EQUATION]', '1708.09536-2-19-2': 'The part [EQUATION] in [REF] is the most essential in the definition of [MATH].', '1708.09536-2-19-3': 'Its similarity with the right hand side of the two-scale relation formula [REF] plays an important role in connection between dilations of [MATH]splines (see [REF], [REF] and [REF]).', '1708.09536-2-19-4': 'Pre-image of [REF] has the form [EQUATION].', '1708.09536-2-19-5': 'Moreover, up to a constant, it is equal to [MATH]th order derivative of [MATH] (see [REF]): [EQUATION]', '1708.09536-2-20-0': 'Analogously one can construct [MATH] with all [MATH] possible combinations of [MATH], [MATH].', '1708.09536-2-20-1': 'For the sake of convenience it is reasonable to center them at [MATH].', '1708.09536-2-20-2': 'To obtain further results of the paper (see 3-4) we shall use wavelet systems [EQUATION] with [MATH], [MATH].', '1708.09536-2-20-3': 'By construction, translations of those [MATH] and [MATH] form orthonormal basis in subspaces [MATH] and [MATH] of [MATH] related to the multiresolution analysis generated by [MATH] and [MATH].', '1708.09536-2-20-4': 'Moreover, since [MATH], [MATH], then the system [EQUATION] where [MATH] and [MATH], consisting of [EQUATION] is an orthonormal basis in [MATH] with the following properties: The (iv) follows, in particular, from [CITATION] (see also [CITATION]).', '1708.09536-2-21-0': 'We complete the section by example of [MATH] and [MATH] of order [MATH].', '1708.09536-2-22-0': 'Let [MATH].', '1708.09536-2-22-1': 'Then [MATH] and [MATH].', '1708.09536-2-22-2': 'Therefore, [MATH] and [MATH].', '1708.09536-2-22-3': 'By [REF], we have with [MATH] (four options, depending on [MATH], in total): [EQUATION] where [EQUATION]', '1708.09536-2-22-4': 'By [REF] and [REF], say, for [MATH], [MATH], (that is in the case [MATH]) [EQUATION] with [MATH].', '1708.09536-2-22-5': 'Since [EQUATION] a wavelet [MATH], related to [MATH] may have, in particular, the following form: [EQUATION]', '1708.09536-2-23-0': '# Localisation property For a function [MATH] on [MATH] and [MATH] put [EQUATION] and define, recursively, (S_^mS_1^mF)(x):=(S_mS_1^mF)(x)+r_j_(S_mS_1^mF)(xm),1c_r, j_J_r;', '1708.09536-2-24-0': 'Let [MATH] be wavelet systems [REF] with [MATH] satisfying [REF] and [MATH] of the form [REF].', '1708.09536-2-24-1': 'For simplicity we assume that [MATH] are centred at [MATH] and [MATH] are centred at [MATH].', '1708.09536-2-24-2': 'Recall that [MATH] denotes cardinality of the set [MATH].', '1708.09536-2-24-3': 'Similarly, [MATH] stands for cardinality of the set [MATH].', '1708.09536-2-24-4': 'It holds [EQUATION] and where [MATH] stands for the [MATH]th order derivative of [MATH] and [EQUATION]', '1708.09536-2-24-5': 'Since [MATH] is centred at [MATH] we obtain by [REF], and based on [REF] is proven.', '1708.09536-2-25-0': 'On the strength of definition [REF] of [MATH], we obtain, similarly to [REF], that [EQUATION] where [MATH] defined by [REF].', '1708.09536-2-25-1': 'Recall that [MATH], [MATH].', '1708.09536-2-26-0': 'Given [MATH] and chosen "[MATH]" or "[MATH]" in [MATH] we consider the collection [MATH] of [MATH] functions [MATH], meaning [MATH] different combinations of [MATH], [MATH], in [MATH].', '1708.09536-2-26-1': 'We pair elements of [MATH] as follows: [MATH] and [MATH] from [MATH] are coupled if [MATH] for [MATH], while [MATH].', '1708.09536-2-26-2': 'Each couple we associate with the function [EQUATION] and call [MATH] the new collection of [MATH] functions [MATH].', '1708.09536-2-26-3': 'If [MATH], we finish the localisation process and obtain the function [MATH] satisfying [EQUATION] that is [EQUATION]', '1708.09536-2-26-4': 'Combination [REF] with substitution [MATH] brings, in particular, [EQUATION]', '1708.09536-2-26-5': 'If [MATH] then, similarly to the previous step, we pair elements of [MATH] by matching two functions [MATH] and [MATH] such that [MATH] for [MATH], but [MATH].', '1708.09536-2-26-6': 'Each couple we associate with the function [EQUATION] and call [MATH] the new collection of [MATH] functions [MATH].', '1708.09536-2-27-0': 'If [MATH] we continue the process.', '1708.09536-2-27-1': 'At [MATH]-th step we deal with functions [EQUATION] from [MATH] and form the new collection [MATH] of [MATH] elements [EQUATION]', '1708.09536-2-27-2': 'Overall, starting from [REF] requiring [MATH] steps, and making exactly [MATH] steps of the form [REF], that is [MATH] steps in total, we obtain the localised function [MATH] with [EQUATION].', '1708.09536-2-27-3': 'From here [REF] follows by [REF].', '1708.09536-2-28-0': 'We have [MATH].', '1708.09536-2-28-1': 'By [REF], it holds, where [MATH] stands for integer part of [MATH]: [EQUATION] [MATH] and [MATH] realize the localisation property of Battle-Lemarie wavelet systems.', '1708.09536-2-28-2': '[MATH] is constructed by integer shifts of [MATH], which generate the same multiresolution analysis in [MATH].', '1708.09536-2-28-3': 'For [MATH] we group proper shifts of wavelets [MATH] from the systems [REF].', '1708.09536-2-28-4': 'They constitute bases in subspaces [MATH] related to multiresolution analysis generated by [MATH] and [MATH].', '1708.09536-2-28-5': 'This localisation property is crucial, in particular, for the estimate [MATH] in the proof of Proposition [REF] in 4.', '1708.09536-2-29-0': '# Equivalent norms theorem', '1708.09536-2-30-0': '## Prerequisites', '1708.09536-2-31-0': '### Nikolskii-Besov spaces Let [MATH] and [MATH].', '1708.09536-2-31-1': 'For the definition of Nikolskii-Besov spaces [MATH] (or Besov type spaces, in other terminology, which is more commonly used) we refer to [CITATION].', '1708.09536-2-31-2': 'If, in addition, [MATH] then one can define [MATH] as follows.', '1708.09536-2-31-3': 'Let [MATH] with [MATH] be the set of all Lebesgue measurable functions [MATH] on [MATH], quasi-normed by [MATH] with the obvious modification for [MATH].', '1708.09536-2-31-4': 'For [MATH] and [MATH] put (see e.g. [CITATION]) [MATH], [MATH] and [EQUATION].', '1708.09536-2-31-5': 'If [MATH] and [MATH] then [MATH] if and only if [MATH] and [EQUATION].', '1708.09536-2-31-6': 'The theory and properties of Nikolskii-Besov spaces [MATH] may be found in [CITATION].', '1708.09536-2-32-0': '### Sequence spaces Let [MATH], [MATH] and [MATH].', '1708.09536-2-32-1': 'The sequence space [MATH] consists of all sequences [EQUATION] such that the quasi-norm [EQUATION] (wth the usual modification if [MATH] and/or [MATH]) is finite.', '1708.09536-2-32-2': 'Sequence spaces of the type [MATH] were introduced in [CITATION] in connection with atomic decomposition of the spaces [MATH].', '1708.09536-2-33-0': '### Spline bases in [MATH] Let [MATH] be the Schwartz space of all complex-valued rapidly decreasing, infinitely differentiable functions on [MATH], and let [MATH] denote its dual space of tempered distributions.', '1708.09536-2-33-1': '[CITATION] Let [MATH] and let [EQUATION] be an ([MATH]-normalised) orthogonal spline basis in [MATH] according to [REF].', '1708.09536-2-34-0': 'Let [MATH], [MATH] and [EQUATION].', '1708.09536-2-34-1': 'Let [MATH].', '1708.09536-2-34-2': 'Then [MATH] if and only if it can be represented as [EQUATION] unconditional convergence being in [MATH] and locally in any space [MATH] with [MATH].', '1708.09536-2-34-3': 'The representation [REF] is unique, [EQUATION] and [EQUATION] is an isomorphic map of [MATH] onto [MATH].', '1708.09536-2-34-4': 'If, in addition, [MATH], [MATH] then [EQUATION] is an unconditional (normalised) basis in [MATH].', '1708.09536-2-34-5': 'A proof of this proposition may be found in [CITATION].', '1708.09536-2-35-0': '## The result It follows from Proposition [REF] and [REF] that one can use [EQUATION] (with usual modifications for [MATH] and [MATH]) as an equivalent characterization of the norm in [MATH], when [MATH], [MATH] and [MATH].', '1708.09536-2-35-1': 'In this part of the paper we establish an equivalent characteristic for [REF].', '1708.09536-2-35-2': 'Let [MATH] and let [EQUATION] be an ([MATH]-normalised) orthogonal spline basis in [MATH] according to [REF].', '1708.09536-2-36-0': 'Let [MATH], [MATH] and [EQUATION].', '1708.09536-2-36-1': 'Then a distribution [MATH] belongs to [MATH] if and only if [EQUATION] (with usual modifications for [MATH] and [MATH]).', '1708.09536-2-36-2': 'Furthermore, [MATH] may be used as an equivalent norm on [MATH].', '1708.09536-2-37-0': 'Let [MATH] with [MATH].', '1708.09536-2-37-1': 'Argumentation for other cases of [MATH] is analogous.', '1708.09536-2-37-2': 'We write, by [REF], [EQUATION]', '1708.09536-2-37-3': 'Then, if [MATH], [EQUATION]', '1708.09536-2-37-4': "Analogously, for [MATH] we obtain using [MATH]times Holder's inequality with [MATH] and [MATH], and representing [MATH] for each [MATH], that [EQUATION]", '1708.09536-2-37-5': 'On the other side, according to the construction of [MATH] (see [REF]) with [MATH] and [MATH] in our case, [EQUATION]', '1708.09536-2-37-6': 'Thus, [EQUATION] where [MATH] for [MATH] and [MATH] if [MATH].', '1708.09536-2-38-0': 'For deriving an estimate similar to [REF] and [REF], but with, say, [MATH] this time, we write, by [REF], taking into account [REF], [REF] and [REF]: [EQUATION]', '1708.09536-2-38-1': 'Denote [MATH] the quantity in the square brackets.', '1708.09536-2-38-2': 'Similarly to [REF], Analogously, for [MATH] [EQUATION]', '1708.09536-2-38-3': 'Since for [MATH] and, similarly, for [MATH] we obtain, taking into account [REF] and [REF], For the reverse estimate we write, by [REF], [EQUATION]', '1708.09536-2-38-4': 'By the construction of [MATH], The required two-sided estimate for the second term in [MATH] follows from the inequalities above and the fact that [MATH] with [MATH] equal to [MATH] or [MATH] are equivalent.'}

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'1708.09536-2-19-5'], ['1708.09536-1-8-0', '1708.09536-2-8-0'], ['1708.09536-1-34-2', '1708.09536-2-34-2'], ['1708.09536-1-34-3', '1708.09536-2-34-3'], ['1708.09536-1-34-4', '1708.09536-2-34-4'], ['1708.09536-1-34-5', '1708.09536-2-34-5'], ['1708.09536-1-33-1', '1708.09536-2-33-1'], ['1708.09536-1-3-0', '1708.09536-2-3-0'], ['1708.09536-1-3-1', '1708.09536-2-3-1'], ['1708.09536-1-3-2', '1708.09536-2-3-2'], ['1708.09536-1-3-3', '1708.09536-2-3-3'], ['1708.09536-1-3-4', '1708.09536-2-3-4'], ['1708.09536-1-3-5', '1708.09536-2-3-5'], ['1708.09536-1-3-6', '1708.09536-2-3-6'], ['1708.09536-1-3-7', '1708.09536-2-3-7'], ['1708.09536-1-3-8', '1708.09536-2-3-8'], ['1708.09536-1-3-9', '1708.09536-2-3-9'], ['1708.09536-1-3-10', '1708.09536-2-3-10'], ['1708.09536-1-3-11', '1708.09536-2-3-11'], ['1708.09536-1-3-12', '1708.09536-2-3-12'], ['1708.09536-1-3-13', '1708.09536-2-3-13'], ['1708.09536-1-3-14', '1708.09536-2-3-14'], ['1708.09536-1-4-0', '1708.09536-2-4-0'], ['1708.09536-1-4-1', '1708.09536-2-4-1'], ['1708.09536-1-36-1', '1708.09536-2-36-1'], ['1708.09536-1-36-2', '1708.09536-2-36-2'], ['1708.09536-1-5-1', '1708.09536-2-5-1'], ['1708.09536-1-5-2', '1708.09536-2-5-2'], ['1708.09536-1-5-4', '1708.09536-2-5-4'], ['1708.09536-1-5-5', '1708.09536-2-5-5'], ['1708.09536-1-10-0', '1708.09536-2-10-0'], ['1708.09536-1-10-1', '1708.09536-2-10-1'], ['1708.09536-1-7-0', '1708.09536-2-7-0'], ['1708.09536-1-7-1', '1708.09536-2-7-1'], ['1708.09536-1-7-2', '1708.09536-2-7-2'], ['1708.09536-1-6-0', '1708.09536-2-6-0'], ['1708.09536-1-6-1', '1708.09536-2-6-1'], ['1708.09536-1-6-2', '1708.09536-2-6-2'], ['1708.09536-1-11-0', '1708.09536-2-11-0'], ['1708.09536-1-11-2', '1708.09536-2-11-2'], ['1708.09536-1-11-3', '1708.09536-2-11-3'], ['1708.09536-1-11-4', '1708.09536-2-11-4'], ['1708.09536-1-11-5', '1708.09536-2-11-5'], ['1708.09536-1-11-6', '1708.09536-2-11-6'], ['1708.09536-1-11-7', '1708.09536-2-11-7'], ['1708.09536-1-11-8', '1708.09536-2-11-8'], ['1708.09536-1-11-9', '1708.09536-2-11-9'], ['1708.09536-1-11-11', '1708.09536-2-11-11'], ['1708.09536-1-11-12', '1708.09536-2-11-12'], ['1708.09536-1-11-13', '1708.09536-2-11-13'], ['1708.09536-1-11-14', '1708.09536-2-11-14'], ['1708.09536-1-20-0', '1708.09536-2-20-0'], ['1708.09536-1-20-1', '1708.09536-2-20-1'], ['1708.09536-1-20-2', '1708.09536-2-20-2'], ['1708.09536-1-20-3', '1708.09536-2-20-3'], ['1708.09536-1-20-4', '1708.09536-2-20-4'], ['1708.09536-1-0-0', '1708.09536-2-0-0'], ['1708.09536-1-0-1', '1708.09536-2-0-1'], ['1708.09536-1-12-0', '1708.09536-2-12-0'], ['1708.09536-1-15-0', '1708.09536-2-15-0'], ['1708.09536-1-32-1', '1708.09536-2-32-1'], ['1708.09536-1-32-2', '1708.09536-2-32-2'], ['1708.09536-1-27-0', '1708.09536-2-27-0'], ['1708.09536-1-27-1', '1708.09536-2-27-1'], ['1708.09536-1-27-2', '1708.09536-2-27-2'], ['1708.09536-1-27-3', '1708.09536-2-27-3'], ['1708.09536-1-35-1', '1708.09536-2-35-1'], ['1708.09536-1-35-2', '1708.09536-2-35-2'], ['1708.09536-1-18-3', '1708.09536-2-18-3'], ['1708.09536-1-31-1', '1708.09536-2-31-1'], ['1708.09536-1-31-2', '1708.09536-2-31-2'], ['1708.09536-1-31-3', '1708.09536-2-31-3'], ['1708.09536-1-31-5', '1708.09536-2-31-5'], ['1708.09536-1-31-6', '1708.09536-2-31-6'], ['1708.09536-1-38-1', '1708.09536-2-38-1'], ['1708.09536-1-38-2', '1708.09536-2-38-2'], ['1708.09536-1-38-3', '1708.09536-2-38-3'], ['1708.09536-1-38-4', '1708.09536-2-38-4'], ['1708.09536-1-14-0', '1708.09536-2-14-0'], ['1708.09536-1-14-1', '1708.09536-2-14-1'], ['1708.09536-1-14-2', '1708.09536-2-14-2'], ['1708.09536-1-28-1', '1708.09536-2-28-1'], ['1708.09536-1-28-2', '1708.09536-2-28-2'], ['1708.09536-1-28-3', '1708.09536-2-28-3'], ['1708.09536-1-28-4', '1708.09536-2-28-4'], ['1708.09536-1-28-5', '1708.09536-2-28-5'], ['1708.09536-1-24-0', '1708.09536-2-24-0'], ['1708.09536-1-24-1', '1708.09536-2-24-1'], ['1708.09536-1-24-2', '1708.09536-2-24-2'], ['1708.09536-1-24-3', '1708.09536-2-24-3'], ['1708.09536-1-24-4', '1708.09536-2-24-4'], ['1708.09536-1-24-5', '1708.09536-2-24-5'], ['1708.09536-1-16-0', '1708.09536-2-16-0'], ['1708.09536-1-16-1', '1708.09536-2-16-1'], ['1708.09536-1-16-3', '1708.09536-2-16-3'], ['1708.09536-1-16-4', '1708.09536-2-16-4'], ['1708.09536-2-19-2', '1708.09536-3-19-2'], ['1708.09536-2-19-3', '1708.09536-3-19-3'], ['1708.09536-2-19-4', '1708.09536-3-19-4'], ['1708.09536-2-19-5', '1708.09536-3-19-5'], ['1708.09536-2-14-0', '1708.09536-3-14-0'], ['1708.09536-2-14-1', '1708.09536-3-14-1'], ['1708.09536-2-14-2', '1708.09536-3-14-2'], ['1708.09536-2-38-1', '1708.09536-3-38-1'], ['1708.09536-2-38-2', '1708.09536-3-38-2'], ['1708.09536-2-38-3', '1708.09536-3-38-3'], ['1708.09536-2-38-4', '1708.09536-3-38-4'], ['1708.09536-2-15-0', '1708.09536-3-15-0'], ['1708.09536-2-28-1', '1708.09536-3-28-1'], ['1708.09536-2-28-2', '1708.09536-3-28-2'], ['1708.09536-2-28-3', '1708.09536-3-28-3'], ['1708.09536-2-28-4', '1708.09536-3-28-4'], ['1708.09536-2-28-5', '1708.09536-3-28-5'], ['1708.09536-2-26-4', '1708.09536-3-26-4'], ['1708.09536-2-26-5', '1708.09536-3-26-5'], ['1708.09536-2-26-6', '1708.09536-3-26-6'], ['1708.09536-2-11-0', '1708.09536-3-11-0'], ['1708.09536-2-11-2', '1708.09536-3-11-2'], ['1708.09536-2-11-3', '1708.09536-3-11-3'], ['1708.09536-2-11-4', '1708.09536-3-11-4'], ['1708.09536-2-11-5', '1708.09536-3-11-5'], ['1708.09536-2-11-6', '1708.09536-3-11-6'], ['1708.09536-2-11-7', '1708.09536-3-11-7'], ['1708.09536-2-11-8', '1708.09536-3-11-8'], ['1708.09536-2-11-9', '1708.09536-3-11-9'], ['1708.09536-2-11-11', '1708.09536-3-11-11'], ['1708.09536-2-11-12', '1708.09536-3-11-12'], ['1708.09536-2-11-13', '1708.09536-3-11-13'], ['1708.09536-2-11-14', '1708.09536-3-11-14'], ['1708.09536-2-36-1', '1708.09536-3-36-1'], ['1708.09536-2-36-2', '1708.09536-3-36-2'], ['1708.09536-2-13-1', '1708.09536-3-13-1'], ['1708.09536-2-13-3', '1708.09536-3-13-3'], ['1708.09536-2-13-4', '1708.09536-3-13-4'], ['1708.09536-2-13-5', '1708.09536-3-13-5'], ['1708.09536-2-34-2', '1708.09536-3-34-2'], ['1708.09536-2-34-3', '1708.09536-3-34-3'], ['1708.09536-2-34-4', '1708.09536-3-34-4'], ['1708.09536-2-34-5', '1708.09536-3-34-5'], ['1708.09536-2-35-1', '1708.09536-3-35-1'], ['1708.09536-2-35-2', '1708.09536-3-35-2'], ['1708.09536-2-12-0', '1708.09536-3-12-0'], ['1708.09536-2-6-0', '1708.09536-3-6-0'], ['1708.09536-2-6-1', '1708.09536-3-6-1'], ['1708.09536-2-6-2', '1708.09536-3-6-2'], ['1708.09536-2-31-1', '1708.09536-3-31-1'], ['1708.09536-2-31-2', '1708.09536-3-31-2'], ['1708.09536-2-31-3', '1708.09536-3-31-3'], ['1708.09536-2-31-5', '1708.09536-3-31-5'], ['1708.09536-2-31-6', '1708.09536-3-31-6'], ['1708.09536-2-16-0', '1708.09536-3-16-0'], ['1708.09536-2-16-1', '1708.09536-3-16-1'], ['1708.09536-2-16-3', '1708.09536-3-16-3'], ['1708.09536-2-16-4', '1708.09536-3-16-4'], ['1708.09536-2-32-1', '1708.09536-3-32-1'], ['1708.09536-2-32-2', '1708.09536-3-32-2'], ['1708.09536-2-33-1', '1708.09536-3-33-1'], ['1708.09536-2-27-0', '1708.09536-3-27-0'], ['1708.09536-2-27-1', '1708.09536-3-27-1'], ['1708.09536-2-27-3', '1708.09536-3-27-3'], ['1708.09536-2-5-1', '1708.09536-3-5-1'], ['1708.09536-2-5-2', '1708.09536-3-5-2'], ['1708.09536-2-5-4', '1708.09536-3-5-4'], ['1708.09536-2-5-5', '1708.09536-3-5-5'], ['1708.09536-2-10-0', '1708.09536-3-10-0'], ['1708.09536-2-10-1', '1708.09536-3-10-1'], ['1708.09536-2-21-0', '1708.09536-3-21-0'], ['1708.09536-2-24-0', '1708.09536-3-24-0'], ['1708.09536-2-24-1', '1708.09536-3-24-1'], ['1708.09536-2-24-2', '1708.09536-3-24-2'], ['1708.09536-2-24-3', '1708.09536-3-24-3'], ['1708.09536-2-24-4', '1708.09536-3-24-4'], ['1708.09536-2-24-5', '1708.09536-3-24-5'], ['1708.09536-2-18-3', '1708.09536-3-18-3'], ['1708.09536-2-7-0', '1708.09536-3-7-0'], ['1708.09536-2-7-1', '1708.09536-3-7-1'], ['1708.09536-2-7-2', '1708.09536-3-7-2'], ['1708.09536-2-0-0', '1708.09536-3-0-0'], ['1708.09536-2-0-1', '1708.09536-3-0-1'], ['1708.09536-2-3-0', '1708.09536-3-3-0'], ['1708.09536-2-3-1', '1708.09536-3-3-1'], ['1708.09536-2-3-2', '1708.09536-3-3-2'], ['1708.09536-2-3-3', '1708.09536-3-3-3'], ['1708.09536-2-3-4', '1708.09536-3-3-4'], ['1708.09536-2-3-5', '1708.09536-3-3-5'], ['1708.09536-2-3-6', '1708.09536-3-3-6'], ['1708.09536-2-3-7', '1708.09536-3-3-7'], ['1708.09536-2-3-8', '1708.09536-3-3-8'], ['1708.09536-2-3-9', '1708.09536-3-3-9'], ['1708.09536-2-3-10', '1708.09536-3-3-10'], ['1708.09536-2-3-11', '1708.09536-3-3-11'], ['1708.09536-2-3-12', '1708.09536-3-3-12'], ['1708.09536-2-3-13', '1708.09536-3-3-13'], ['1708.09536-2-3-14', '1708.09536-3-3-14'], ['1708.09536-2-4-0', '1708.09536-3-4-0'], ['1708.09536-2-4-1', '1708.09536-3-4-1'], ['1708.09536-2-20-0', '1708.09536-3-20-0'], ['1708.09536-2-20-1', '1708.09536-3-20-1'], ['1708.09536-2-20-2', '1708.09536-3-20-2'], ['1708.09536-2-20-3', '1708.09536-3-20-3'], ['1708.09536-2-20-4', '1708.09536-3-20-4'], ['1708.09536-2-17-0', '1708.09536-3-17-0'], ['1708.09536-2-17-1', '1708.09536-3-17-1'], ['1708.09536-2-17-2', '1708.09536-3-17-2'], ['1708.09536-2-17-3', '1708.09536-3-17-3'], ['1708.09536-2-17-4', '1708.09536-3-17-4'], ['1708.09536-2-17-5', '1708.09536-3-17-5'], ['1708.09536-2-8-0', '1708.09536-3-8-0'], ['1708.09536-2-2-0', '1708.09536-3-2-0'], ['1708.09536-2-2-1', '1708.09536-3-2-1'], ['1708.09536-2-2-2', '1708.09536-3-2-2'], ['1708.09536-2-27-2', '1708.09536-3-27-2'], ['1708.09536-2-13-6', '1708.09536-3-13-6'], ['1708.09536-2-13-7', '1708.09536-3-13-6']]

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['1708.09536-2-27-3', '1708.09536-3-27-3'], ['1708.09536-2-5-1', '1708.09536-3-5-1'], ['1708.09536-2-5-2', '1708.09536-3-5-2'], ['1708.09536-2-5-4', '1708.09536-3-5-4'], ['1708.09536-2-5-5', '1708.09536-3-5-5'], ['1708.09536-2-10-0', '1708.09536-3-10-0'], ['1708.09536-2-10-1', '1708.09536-3-10-1'], ['1708.09536-2-21-0', '1708.09536-3-21-0'], ['1708.09536-2-24-0', '1708.09536-3-24-0'], ['1708.09536-2-24-1', '1708.09536-3-24-1'], ['1708.09536-2-24-2', '1708.09536-3-24-2'], ['1708.09536-2-24-3', '1708.09536-3-24-3'], ['1708.09536-2-24-4', '1708.09536-3-24-4'], ['1708.09536-2-24-5', '1708.09536-3-24-5'], ['1708.09536-2-18-3', '1708.09536-3-18-3'], ['1708.09536-2-7-0', '1708.09536-3-7-0'], ['1708.09536-2-7-1', '1708.09536-3-7-1'], ['1708.09536-2-7-2', '1708.09536-3-7-2'], ['1708.09536-2-0-0', '1708.09536-3-0-0'], ['1708.09536-2-0-1', '1708.09536-3-0-1'], ['1708.09536-2-3-0', '1708.09536-3-3-0'], ['1708.09536-2-3-1', '1708.09536-3-3-1'], ['1708.09536-2-3-2', '1708.09536-3-3-2'], ['1708.09536-2-3-3', '1708.09536-3-3-3'], ['1708.09536-2-3-4', '1708.09536-3-3-4'], ['1708.09536-2-3-5', '1708.09536-3-3-5'], ['1708.09536-2-3-6', '1708.09536-3-3-6'], ['1708.09536-2-3-7', '1708.09536-3-3-7'], ['1708.09536-2-3-8', '1708.09536-3-3-8'], ['1708.09536-2-3-9', '1708.09536-3-3-9'], ['1708.09536-2-3-10', '1708.09536-3-3-10'], ['1708.09536-2-3-11', '1708.09536-3-3-11'], ['1708.09536-2-3-12', '1708.09536-3-3-12'], ['1708.09536-2-3-13', '1708.09536-3-3-13'], ['1708.09536-2-3-14', '1708.09536-3-3-14'], ['1708.09536-2-4-0', '1708.09536-3-4-0'], ['1708.09536-2-4-1', '1708.09536-3-4-1'], ['1708.09536-2-20-0', '1708.09536-3-20-0'], ['1708.09536-2-20-1', '1708.09536-3-20-1'], ['1708.09536-2-20-2', '1708.09536-3-20-2'], ['1708.09536-2-20-3', '1708.09536-3-20-3'], ['1708.09536-2-20-4', '1708.09536-3-20-4'], ['1708.09536-2-17-0', '1708.09536-3-17-0'], ['1708.09536-2-17-1', '1708.09536-3-17-1'], ['1708.09536-2-17-2', '1708.09536-3-17-2'], ['1708.09536-2-17-3', '1708.09536-3-17-3'], ['1708.09536-2-17-4', '1708.09536-3-17-4'], ['1708.09536-2-17-5', '1708.09536-3-17-5'], ['1708.09536-2-8-0', '1708.09536-3-8-0'], ['1708.09536-2-2-0', '1708.09536-3-2-0'], ['1708.09536-2-2-1', '1708.09536-3-2-1'], ['1708.09536-2-2-2', '1708.09536-3-2-2']]

[['1708.09536-2-27-2', '1708.09536-3-27-2']]

[]

[['1708.09536-2-13-6', '1708.09536-3-13-6'], ['1708.09536-2-13-7', '1708.09536-3-13-6']]

[]

['1708.09536-1-2-3', '1708.09536-1-5-3', '1708.09536-1-11-1', '1708.09536-1-11-10', '1708.09536-1-13-2', '1708.09536-1-16-2', '1708.09536-1-18-0', '1708.09536-1-18-1', '1708.09536-1-18-2', '1708.09536-1-19-0', '1708.09536-1-19-1', '1708.09536-1-22-0', '1708.09536-1-22-1', '1708.09536-1-22-2', '1708.09536-1-22-3', '1708.09536-1-22-4', '1708.09536-1-22-5', '1708.09536-1-25-0', '1708.09536-1-25-1', '1708.09536-1-26-0', '1708.09536-1-26-1', '1708.09536-1-26-2', '1708.09536-1-26-3', '1708.09536-1-28-0', '1708.09536-1-31-4', '1708.09536-1-34-0', '1708.09536-1-34-1', '1708.09536-1-36-0', '1708.09536-1-37-0', '1708.09536-1-37-1', '1708.09536-1-37-2', '1708.09536-1-37-3', '1708.09536-1-37-4', '1708.09536-1-37-5', '1708.09536-1-37-6', '1708.09536-1-38-0', '1708.09536-2-2-3', '1708.09536-2-5-3', '1708.09536-2-11-1', '1708.09536-2-11-10', '1708.09536-2-13-2', '1708.09536-2-16-2', '1708.09536-2-18-0', '1708.09536-2-18-1', '1708.09536-2-18-2', '1708.09536-2-19-0', '1708.09536-2-19-1', '1708.09536-2-22-0', '1708.09536-2-22-1', '1708.09536-2-22-2', '1708.09536-2-22-3', '1708.09536-2-22-4', '1708.09536-2-22-5', '1708.09536-2-25-0', '1708.09536-2-25-1', '1708.09536-2-26-0', '1708.09536-2-26-1', '1708.09536-2-26-2', '1708.09536-2-26-3', '1708.09536-2-28-0', '1708.09536-2-31-4', '1708.09536-2-34-0', '1708.09536-2-34-1', '1708.09536-2-36-0', '1708.09536-2-37-0', '1708.09536-2-37-1', '1708.09536-2-37-2', '1708.09536-2-37-3', '1708.09536-2-37-4', '1708.09536-2-37-5', '1708.09536-2-37-6', '1708.09536-2-38-0', '1708.09536-3-2-3', '1708.09536-3-5-3', '1708.09536-3-11-1', '1708.09536-3-11-10', '1708.09536-3-13-2', '1708.09536-3-16-2', '1708.09536-3-18-0', '1708.09536-3-18-1', '1708.09536-3-18-2', '1708.09536-3-19-0', '1708.09536-3-19-1', '1708.09536-3-22-0', '1708.09536-3-22-1', '1708.09536-3-22-2', '1708.09536-3-22-3', '1708.09536-3-22-4', '1708.09536-3-22-5', '1708.09536-3-25-0', '1708.09536-3-25-1', '1708.09536-3-26-0', '1708.09536-3-26-1', '1708.09536-3-26-2', '1708.09536-3-26-3', '1708.09536-3-28-0', '1708.09536-3-31-4', '1708.09536-3-34-0', '1708.09536-3-34-1', '1708.09536-3-36-0', '1708.09536-3-37-0', '1708.09536-3-37-1', '1708.09536-3-37-2', '1708.09536-3-37-3', '1708.09536-3-37-4', '1708.09536-3-37-5', '1708.09536-3-37-6', '1708.09536-3-38-0']

{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '3': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1708.09536

{'1708.09536-3-0-0': ': Explicit formulae are given for a type of Battle-Lemarie scaling functions and related wavelets.', '1708.09536-3-0-1': 'Compactly supported sums of their translations are established and applied to alternative norm characterization of sequence spaces isometrically isomorphic to Nikolskii-Besov spaces on [MATH].', '1708.09536-3-1-0': '# Introduction', '1708.09536-3-2-0': 'Battle-Lemarie scaling functions are polynomial splines with simple knots at [MATH] obtained by orthogonalisation process of the B-splines.', '1708.09536-3-2-1': 'For [MATH] the [MATH]th order B-spline is defined recursively by [EQUATION] with [MATH].', '1708.09536-3-2-2': 'It is known that [MATH] generates multiresolution analysis of [MATH] [CITATION].', '1708.09536-3-2-3': 'Moreover [CITATION],', '1708.09536-3-3-0': 'Battle-Lemarie scaling functions and related wavelets play an important role in approximation theory, numerical analysis (see e.g. [CITATION]), image, data and signal processing involving analysis of biological sequences and molecular biology-related signals, etc. [CITATION].', '1708.09536-3-3-1': 'On the strength of the differentiation property [EQUATION] this function class has appeared to be an effective tool for solving problems related to the theory of integration and differentiation operators in function spaces [CITATION].', '1708.09536-3-3-2': 'There is a number of papers devoted to Battle-Lemarie scaling functions and wavelets.', '1708.09536-3-3-3': 'Most of them deal with their implicit or approximate expressions.', '1708.09536-3-3-4': 'An idea of how to find explicit formulae for this function class was given in works by I.Ya.', '1708.09536-3-3-5': 'Novikov and S.B. Stechkin ([CITATION] and [CITATION], see also [CITATION] and [CITATION]).', '1708.09536-3-3-6': "The problem, we dealt in [CITATION], concerned the operators' compactness and approximation properties.", '1708.09536-3-3-7': 'The solution method required, first of all, explicit formulae for the chosen wavelet system.', '1708.09536-3-3-8': 'The other important points were in finding their proper transformations and sums in order to localise non-compactly supported scaling functions and wavelets of this type and, making use of [REF], connect their components (splines) with splines of lower or higher orders.', '1708.09536-3-3-9': 'All these questions were covered in [CITATION] for the scaling function and wavelets of the first order only.', '1708.09536-3-3-10': 'The results of the present work make us able to continue the study of compactness and approximation properties of integration operators in spaces of functions of higher smoothness than established in [CITATION].', '1708.09536-3-3-11': 'In the first part of this work we derive exact expressions for a family of Battle-Lemarie scaling functions and wavelets of all positive integer orders ( 2).', '1708.09536-3-3-12': 'The second part of the paper is devoted to the "localisation property" of the class ( 3).', '1708.09536-3-3-13': 'Namely, we establish compactly supported combinations [MATH] of wavelets of Battle-Lemarie type (see [REF] in Theorem [REF]), which contribute to simple connection between two B-splines of different orders by an integration (differential) operator as well as to relation between dilations of [MATH]splines (see [REF], [REF] and [REF]).', '1708.09536-3-3-14': 'Similar result is given for the scaling function (see [REF] in Theorem [REF] in combination with [REF]) and both are applied to equivalent norm characteristics in Besov type spaces (Proposition [REF]).', '1708.09536-3-4-0': 'Throughout the paper, we use [MATH], [MATH] and [MATH] for integers, natural and real numbers, respectively; symbol [MATH] for the complex plane.', '1708.09536-3-4-1': 'We make use of marks [MATH] and [MATH] for introducing new quantities.', '1708.09536-3-5-0': '# Construction of Battle-Lemarie scaling functions and related wavelets The Fourier transform of the normalised [MATH]-spline of the [MATH]th order has the form [EQUATION]', '1708.09536-3-5-1': 'In particular, if [MATH] then [EQUATION] and, therefore, [EQUATION]', '1708.09536-3-5-2': 'The general two-scale relation formula for [MATH], [MATH], has the following form [CITATION]: [EQUATION]', '1708.09536-3-5-3': 'Fixed [MATH] and any [MATH] put [MATH], [MATH].', '1708.09536-3-5-4': 'Each [MATH] let [MATH] denote the [MATH]closure of the linear span of the system [MATH].', '1708.09536-3-5-5': 'It is well-known [CITATION] that the spline spaces [MATH], [MATH], which are generated by the scaling function [MATH], constitute a multiresolution analysis of [MATH] in the sense that Further, there are the orthogonal complementary subspaces [MATH] such that Wavelet subspaces [MATH], [MATH], related to the spline [MATH], are also generated by some basic functions (wavelets) in the same manner as the spline spaces [MATH], [MATH], are generated by the spline [MATH].', '1708.09536-3-6-0': 'It is well-known that there exists another scaling function, whose integer translations form an orthonormal system within the same multiresolution analysis.', '1708.09536-3-6-1': 'The function [MATH] satisfying [EQUATION] is called the Battle-Lemarie scaling function [CITATION].', '1708.09536-3-6-2': 'Integer translations of [MATH] form an orthonormal basis in [MATH] of the multiresolution analysis generated by [MATH].', '1708.09536-3-7-0': 'The [MATH]th order Battle-Lemarie wavelet is the function [MATH] whose Fourier transform is [EQUATION]', '1708.09536-3-7-1': 'Integer translations of [MATH] form an orthonormal basis in [MATH] of the multiresolution analysis generated by [MATH].', '1708.09536-3-7-2': 'A scaling function [MATH] and one of its associated wavelets [MATH] form a wavelet system [MATH].', '1708.09536-3-8-0': 'In order to derive explicit expressions for spline wavelet systems of all orders [MATH] with properties analogous to [MATH] and [MATH] we follow the idea from [CITATION] (see also [CITATION], [CITATION] and [CITATION]).', '1708.09536-3-9-0': '## Auxiliary results', '1708.09536-3-10-0': 'Denote [EQUATION] and write [EQUATION] where the low-pass filter [MATH] (see [REF]) is given by [EQUATION].', '1708.09536-3-10-1': 'A wavelet function [MATH] related to [MATH] must be of the form [EQUATION] where the high-pass filter [MATH] is given by [EQUATION]', '1708.09536-3-11-0': 'It is well-known that [MATH] and [MATH], [MATH] form the Haar system.', '1708.09536-3-11-1': 'Write [EQUATION]', '1708.09536-3-11-2': 'The expressions [REF] were investigated in terms of rational polynomials in [CITATION].', '1708.09536-3-11-3': 'Let us shortly explain milestones of this well-known theory (see also [CITATION]).', '1708.09536-3-11-4': 'Differentiation of [REF] yields [EQUATION].', '1708.09536-3-11-5': 'As it was mentioned before, [MATH].', '1708.09536-3-11-6': 'Following [CITATION], we express [MATH] as polynomials in the variable [MATH] by means of [MATH] to obtain the recurrence relation for [EQUATION] with [MATH].', '1708.09536-3-11-7': 'Being an even polynomial for even [MATH], the [MATH] may be expressed as a polynomial [MATH] in the variable [MATH] of degree [MATH].', '1708.09536-3-11-8': 'By [CITATION], [MATH] has all simple and purely imaginary zeros [MATH].', '1708.09536-3-11-9': 'The change of variable transforms them into the zeros [MATH] of [MATH], which must all be positive and not less than [MATH].', '1708.09536-3-11-10': 'Thus, since [MATH], [EQUATION].', '1708.09536-3-11-11': 'We may and shall assume that [MATH] (see [CITATION]).', '1708.09536-3-11-12': 'To find zeros of [MATH], [MATH], with respect to [MATH] in this case, we write [MATH], that is [EQUATION].', '1708.09536-3-11-13': 'Thus, [MATH], where [MATH] and, therefore, [MATH].', '1708.09536-3-11-14': 'Notice that [EQUATION] and, therefore, [MATH].', '1708.09536-3-12-0': 'In view of [REF], [REF], [REF] and [REF] we construct a wavelet system of Battle-Lemarie type as follows.', '1708.09536-3-13-0': '## The construction Denote [EQUATION].', '1708.09536-3-13-1': 'Describe the case [MATH] before considering the general situation.', '1708.09536-3-13-2': 'If [MATH] then [MATH], [MATH] (see [REF] and [REF] in 2.1) and we have [REF] with [EQUATION]', '1708.09536-3-13-3': 'Taking into account [REF] and [REF] with [MATH], define [EQUATION]', '1708.09536-3-13-4': 'If we choose [MATH] in [REF] then [EQUATION] which is a shifted version of [MATH] of the form [REF].', '1708.09536-3-13-5': 'Further, since for [MATH] [EQUATION] then we can write [EQUATION]', '1708.09536-3-13-6': 'Choosing between [MATH] we obtain [MATH] and [MATH] with "left tails" of the forms [EQUATION] and, analogously, [MATH] and [MATH] with "right tails" of the forms [EQUATION] [MATH] and [MATH] differ by integer shifts only and constitute the same multiresolution analysis of [MATH].', '1708.09536-3-14-0': 'To construct wavelets [MATH] related to [MATH] and [MATH] we write according to [REF], [REF] and [REF]: [EQUATION] where (see also [REF]) [EQUATION] that is [EQUATION].', '1708.09536-3-14-1': 'Thus, taking into account that for any [MATH] [EQUATION] we obtain [EQUATION] which means that we can construct two types of [MATH]: one having [MATH] in the numerator in [MATH] and the other one with [MATH].', '1708.09536-3-14-2': 'Symbol [MATH] will be used to emphasise our choice of [MATH] in [REF].', '1708.09536-3-15-0': 'With respect to [REF] and in view of [EQUATION] the [MATH] with [MATH] defined by [REF] for [MATH], have the following forms: Analogously, for the [MATH] with [MATH] defined by [REF] for [MATH], we get To obtain further results of the paper in the case [MATH] (see 3-4) we shall use wavelet systems [MATH], [MATH] and their shifted counterparts [MATH] and [MATH].', '1708.09536-3-16-0': 'For general [MATH] we define [MATH] as follows: [EQUATION] where (see [REF]) [EQUATION].', '1708.09536-3-16-1': 'The Fourier transform of a wavelet function [MATH] related to [MATH] must satisfy the condition [EQUATION] where, with [MATH], [MATH], [EQUATION]', '1708.09536-3-16-2': 'By [REF] [EQUATION]', '1708.09536-3-16-3': 'Considering [REF], we can vary [MATH] in different terms in the denumerator of [MATH].', '1708.09536-3-16-4': 'But, for simplicity, in this work we limit our attention to the cases with either all "[MATH]" or all "[MATH]" only.', '1708.09536-3-17-0': 'If [MATH] for all [MATH], we write, taking into account [REF], [EQUATION] and, in view of [MATH], where [MATH], we obtain [EQUATION]', '1708.09536-3-17-1': 'If [MATH] for at least one of [MATH] we write, similarly to the case [MATH]: [EQUATION]', '1708.09536-3-17-2': 'We say that [MATH] (or, alternatively, that [MATH]), where [MATH], if [MATH] (or [MATH]).', '1708.09536-3-17-3': 'Let [MATH] denote the cardinality of the set [MATH].', '1708.09536-3-17-4': 'Then, it follows from [REF] and [REF], that [EQUATION] where [MATH].', '1708.09536-3-17-5': 'By definition of [MATH] (see 2.1) and on the strength of [REF], the system [MATH], defined by [REF] or by [REF], is an orthonormal basis of [MATH] generated by [MATH].', '1708.09536-3-18-0': 'In order to construct [MATH], we write, taking into account [REF], [REF] and [REF], [EQUATION] where [EQUATION].', '1708.09536-3-18-1': 'Then, on the strength of [REF] and [REF], [EQUATION]', '1708.09536-3-18-2': 'Denote [MATH].', '1708.09536-3-18-3': 'Similarly to the case [MATH], we shall use symbols [MATH] to emphasise the choice of [MATH] and [MATH], [MATH], in the product [MATH] in [REF].', '1708.09536-3-19-0': 'Since [MATH], then the pre-image of [EQUATION] is a sum of [MATH] translations of [MATH] with coefficients depending on [MATH] and [MATH], [MATH].', '1708.09536-3-19-1': 'It holds: [EQUATION]', '1708.09536-3-19-2': 'The part [EQUATION] in [REF] is the most essential in the definition of [MATH].', '1708.09536-3-19-3': 'Its similarity with the right hand side of the two-scale relation formula [REF] plays an important role in connection between dilations of [MATH]splines (see [REF], [REF] and [REF]).', '1708.09536-3-19-4': 'Pre-image of [REF] has the form [EQUATION].', '1708.09536-3-19-5': 'Moreover, up to a constant, it is equal to [MATH]th order derivative of [MATH] (see [REF]): [EQUATION]', '1708.09536-3-20-0': 'Analogously one can construct [MATH] with all [MATH] possible combinations of [MATH], [MATH].', '1708.09536-3-20-1': 'For the sake of convenience it is reasonable to center them at [MATH].', '1708.09536-3-20-2': 'To obtain further results of the paper (see 3-4) we shall use wavelet systems [EQUATION] with [MATH], [MATH].', '1708.09536-3-20-3': 'By construction, translations of those [MATH] and [MATH] form orthonormal basis in subspaces [MATH] and [MATH] of [MATH] related to the multiresolution analysis generated by [MATH] and [MATH].', '1708.09536-3-20-4': 'Moreover, since [MATH], [MATH], then the system [EQUATION] where [MATH] and [MATH], consisting of [EQUATION] is an orthonormal basis in [MATH] with the following properties: The (iv) follows, in particular, from [CITATION] (see also [CITATION]).', '1708.09536-3-21-0': 'We complete the section by example of [MATH] and [MATH] of order [MATH].', '1708.09536-3-22-0': 'Let [MATH].', '1708.09536-3-22-1': 'Then [MATH] and [MATH].', '1708.09536-3-22-2': 'Therefore, [MATH] and [MATH].', '1708.09536-3-22-3': 'By [REF], we have with [MATH] (four options, depending on [MATH], in total): [EQUATION] where [EQUATION]', '1708.09536-3-22-4': 'By [REF] and [REF], say, for [MATH], [MATH], (that is in the case [MATH]) [EQUATION] with [MATH].', '1708.09536-3-22-5': 'Since [EQUATION] a wavelet [MATH], related to [MATH] may have, in particular, the following form: [EQUATION]', '1708.09536-3-23-0': '# Localisation property For a function [MATH] on [MATH] and [MATH] put [EQUATION] and define, recursively, (S_^mS_1^mF)(x):=(S_mS_1^mF)(x)+r_j_(S_mS_1^mF)(xm),1c_r, j_J_r;', '1708.09536-3-24-0': 'Let [MATH] be wavelet systems [REF] with [MATH] satisfying [REF] and [MATH] of the form [REF].', '1708.09536-3-24-1': 'For simplicity we assume that [MATH] are centred at [MATH] and [MATH] are centred at [MATH].', '1708.09536-3-24-2': 'Recall that [MATH] denotes cardinality of the set [MATH].', '1708.09536-3-24-3': 'Similarly, [MATH] stands for cardinality of the set [MATH].', '1708.09536-3-24-4': 'It holds [EQUATION] and where [MATH] stands for the [MATH]th order derivative of [MATH] and [EQUATION]', '1708.09536-3-24-5': 'Since [MATH] is centred at [MATH] we obtain by [REF], and based on [REF] is proven.', '1708.09536-3-25-0': 'On the strength of definition [REF] of [MATH], we obtain, similarly to [REF], that [EQUATION] where [MATH] defined by[REF].', '1708.09536-3-25-1': 'Recall that [MATH], [MATH].', '1708.09536-3-26-0': 'Given [MATH] and chosen "[MATH]" or "[MATH]" in [MATH] we consider the collection [MATH] of [MATH] functions [MATH], meaning [MATH] different combinations of [MATH], [MATH], in [MATH].', '1708.09536-3-26-1': 'We pair elements of [MATH] as follows: [MATH] and [MATH] from [MATH] are coupled if [MATH] for [MATH], while [MATH].', '1708.09536-3-26-2': 'Each couple we associate with the function [EQUATION] and call [MATH] the new collection of [MATH] functions [MATH].', '1708.09536-3-26-3': 'If [MATH], we finish the localisation process and obtain the function [MATH] satisfying [EQUATION] that is [EQUATION]', '1708.09536-3-26-4': 'Combination [REF] with substitution [MATH] brings, in particular, [EQUATION]', '1708.09536-3-26-5': 'If [MATH] then, similarly to the previous step, we pair elements of [MATH] by matching two functions [MATH] and [MATH] such that [MATH] for [MATH], but [MATH].', '1708.09536-3-26-6': 'Each couple we associate with the function [EQUATION] and call [MATH] the new collection of [MATH] functions [MATH].', '1708.09536-3-26-7': 'Again, if [MATH], we stop the process with the function [MATH] such that [EQUATION] that is', '1708.09536-3-27-0': 'If [MATH] we continue the process.', '1708.09536-3-27-1': 'At [MATH]-th step we deal with functions [EQUATION] from [MATH] and form the new collection [MATH] of [MATH] elements [EQUATION]', '1708.09536-3-27-2': 'Overall, starting from [REF] requiring [MATH] steps, and making exactly [MATH] steps of the form [REF], that is [MATH] steps in total, we obtain the localised function [MATH] with [EQUATION]', '1708.09536-3-27-3': 'From here [REF] follows by [REF].', '1708.09536-3-28-0': 'We have [MATH].', '1708.09536-3-28-1': 'By [REF], it holds, where [MATH] stands for integer part of [MATH]: [EQUATION] [MATH] and [MATH] realize the localisation property of Battle-Lemarie wavelet systems.', '1708.09536-3-28-2': '[MATH] is constructed by integer shifts of [MATH], which generate the same multiresolution analysis in [MATH].', '1708.09536-3-28-3': 'For [MATH] we group proper shifts of wavelets [MATH] from the systems [REF].', '1708.09536-3-28-4': 'They constitute bases in subspaces [MATH] related to multiresolution analysis generated by [MATH] and [MATH].', '1708.09536-3-28-5': 'This localisation property is crucial, in particular, for the estimate [MATH] in the proof of Proposition [REF] in 4.', '1708.09536-3-29-0': '# Equivalent norms theorem', '1708.09536-3-30-0': '## Prerequisites', '1708.09536-3-31-0': '### Nikolskii-Besov spaces Let [MATH] and [MATH].', '1708.09536-3-31-1': 'For the definition of Nikolskii-Besov spaces [MATH] (or Besov type spaces, in other terminology, which is more commonly used) we refer to [CITATION].', '1708.09536-3-31-2': 'If, in addition, [MATH] then one can define [MATH] as follows.', '1708.09536-3-31-3': 'Let [MATH] with [MATH] be the set of all Lebesgue measurable functions [MATH] on [MATH], quasi-normed by [MATH] with the obvious modification for [MATH].', '1708.09536-3-31-4': 'For [MATH] and [MATH] put (see e.g. [CITATION]) [MATH], [MATH] and [EQUATION].', '1708.09536-3-31-5': 'If [MATH] and [MATH] then [MATH] if and only if [MATH] and [EQUATION].', '1708.09536-3-31-6': 'The theory and properties of Nikolskii-Besov spaces [MATH] may be found in [CITATION].', '1708.09536-3-32-0': '### Sequence spaces Let [MATH], [MATH] and [MATH].', '1708.09536-3-32-1': 'The sequence space [MATH] consists of all sequences [EQUATION] such that the quasi-norm [EQUATION] (wth the usual modification if [MATH] and/or [MATH]) is finite.', '1708.09536-3-32-2': 'Sequence spaces of the type [MATH] were introduced in [CITATION] in connection with atomic decomposition of the spaces [MATH].', '1708.09536-3-33-0': '### Spline bases in [MATH] Let [MATH] be the Schwartz space of all complex-valued rapidly decreasing, infinitely differentiable functions on [MATH], and let [MATH] denote its dual space of tempered distributions.', '1708.09536-3-33-1': '[CITATION] Let [MATH] and let [EQUATION] be an ([MATH]-normalised) orthogonal spline basis in [MATH] according to [REF].', '1708.09536-3-34-0': 'Let [MATH], [MATH] and [EQUATION].', '1708.09536-3-34-1': 'Let [MATH].', '1708.09536-3-34-2': 'Then [MATH] if and only if it can be represented as [EQUATION] unconditional convergence being in [MATH] and locally in any space [MATH] with [MATH].', '1708.09536-3-34-3': 'The representation [REF] is unique, [EQUATION] and [EQUATION] is an isomorphic map of [MATH] onto [MATH].', '1708.09536-3-34-4': 'If, in addition, [MATH], [MATH] then [EQUATION] is an unconditional (normalised) basis in [MATH].', '1708.09536-3-34-5': 'A proof of this proposition may be found in [CITATION].', '1708.09536-3-35-0': '## The result It follows from Proposition [REF] and [REF] that one can use [EQUATION] (with usual modifications for [MATH] and [MATH]) as an equivalent characterization of the norm in [MATH], when [MATH], [MATH] and [MATH].', '1708.09536-3-35-1': 'In this part of the paper we establish an equivalent characteristic for [REF].', '1708.09536-3-35-2': 'Let [MATH] and let [EQUATION] be an ([MATH]-normalised) orthogonal spline basis in [MATH] according to [REF].', '1708.09536-3-36-0': 'Let [MATH], [MATH] and [EQUATION].', '1708.09536-3-36-1': 'Then a distribution [MATH] belongs to [MATH] if and only if [EQUATION] (with usual modifications for [MATH] and [MATH]).', '1708.09536-3-36-2': 'Furthermore, [MATH] may be used as an equivalent norm on [MATH].', '1708.09536-3-37-0': 'Let [MATH] with [MATH].', '1708.09536-3-37-1': 'Argumentation for other cases of [MATH] is analogous.', '1708.09536-3-37-2': 'We write, by [REF], [EQUATION]', '1708.09536-3-37-3': 'Then, if [MATH], [EQUATION]', '1708.09536-3-37-4': "Analogously, for [MATH] we obtain using [MATH]times Holder's inequality with [MATH] and [MATH], and representing [MATH] for each [MATH], that [EQUATION]", '1708.09536-3-37-5': 'On the other side, according to the construction of [MATH] (see [REF]) with [MATH] and [MATH] in our case, [EQUATION]', '1708.09536-3-37-6': 'Thus, [EQUATION] where [MATH] for [MATH] and [MATH] if [MATH].', '1708.09536-3-38-0': 'For deriving an estimate similar to [REF] and [REF], but with, say, [MATH] this time, we write, by [REF], taking into account [REF], [REF] and [REF]: [EQUATION]', '1708.09536-3-38-1': 'Denote [MATH] the quantity in the square brackets.', '1708.09536-3-38-2': 'Similarly to [REF], Analogously, for [MATH] [EQUATION]', '1708.09536-3-38-3': 'Since for [MATH] and, similarly, for [MATH] we obtain, taking into account [REF] and [REF], For the reverse estimate we write, by [REF], [EQUATION]', '1708.09536-3-38-4': 'By the construction of [MATH], The required two-sided estimate for the second term in [MATH] follows from the inequalities above and the fact that [MATH] with [MATH] equal to [MATH] or [MATH] are equivalent.'}

0712.1413

{'0712.1413-1-0-0': 'We study the collapse of a spherically symmetric dust distribution in [MATH]-dimensional AdS spacetime.', '0712.1413-1-0-1': 'We investigate the role of dimensionality, and the presence of a negative cosmological constant, in determining the formation of trapped surfaces and the end state of gravitational collapse.', '0712.1413-1-0-2': 'We also show that, in the absence of a cosmological constant a self-similar solution exists only in 4+1 and 5+1 dimensional spacetimes, and that one cannot construct a self-similar solution when a cosmological constant is included.', '0712.1413-1-0-3': 'We calculate the surface gravity of an AdS black hole in arbitrary dimensions, and make some observations about the sign of the specific heat of the quantum black hole.', '0712.1413-1-1-0': '# Introduction', '0712.1413-1-2-0': 'There are many models of spherical gravitational collapse in classical general relativity which exhibit the formation of black holes as well as naked singularities, starting from regular initial data [CITATION], [CITATION].', '0712.1413-1-2-1': 'The study of quantum effects in the vicinity of the gravitational singularity then becomes significant.', '0712.1413-1-2-2': 'Such studies can be divided into two classes : (i) quantum field theory in curved space, and (ii) quantum general relativistic treatment of gravitational collapse.', '0712.1413-1-3-0': 'The earliest investigations of quantum field theory in the dynamical background of a collapsing spherical star were probably those due to Ford and Parker [CITATION] and Hiscock et al. [CITATION].', '0712.1413-1-3-1': 'These works introduced important techniques, such as the calculation of the quantum flux in the geometric optics approximation, and the regularization of the 2-d quantum stress tensor, which were used extensively in later studies.', '0712.1413-1-3-2': 'A systematic study of semiclassical effects in gravitational collapse was initiated by Vaz and Witten in [CITATION], [CITATION] and pursued in a series of papers [CITATION], [CITATION], [CITATION], [CITATION], [CITATION], [CITATION], [CITATION].', '0712.1413-1-3-3': 'Typically, these studies showed an important and interesting difference in the nature of quantum particle creation between the two cases - one in which collapse ends in a black hole, and another in which it ends in a naked singularity.', '0712.1413-1-3-4': 'The formation of a black hole is accompanied by the emission of Hawking radiation, as expected.', '0712.1413-1-3-5': 'However, when the collapse ends in a (shell-focusing) naked singularity, there is no evidence of some universal behaviour in the nature of quantum emission.', '0712.1413-1-3-6': 'It is typically found though, that the emitted quantum flux diverges in the approach to the Cauchy horizon.', '0712.1413-1-3-7': 'This divergence disappears when the calculation of the quantum flux is terminated about a Planck time before the formation of the Cauchy horizon, when the semiclassical approximation breaks down.', '0712.1413-1-3-8': 'Instead of the divergence, one finds that only about a Planck unit of energy is emitted during the semiclassical phase, and a full quantum gravitational treatment of the physics of the singularity and the Cauchy horizon becomes unavoidable.', '0712.1413-1-3-9': 'These developments have been reviewed in [CITATION].', '0712.1413-1-4-0': 'A full quantum gravitational treatment of collapse can be performed via a midisuperspace quantization within the framework of quantum general relativity.', '0712.1413-1-4-1': 'The aims of such a programme are manifold - to construct a quantum gravitational description of the black hole; to check if the gravitational singularity can be avoided in quantum gravity; to obtain a statistical derivation of the black-hole entropy from quantum gravitational microstates; and to determine the role of quantum gravity in ascertaining the nature of quantum emission from a naked singularity.', '0712.1413-1-4-2': 'The midisuperspace quantization programme has been carried out by us in a series of papers [CITATION], [CITATION], [CITATION], [CITATION], [CITATION], [CITATION], [CITATION], and work along these lines is still in progress.', '0712.1413-1-4-3': 'It is fair to say that while some progress has been made on aspects related to quantum black holes and black hole entropy, issues related to singularity avoidance and the nature of quantized naked singularities have thus far proved difficult to address, largely bcause of problems relating to finding a suitable regularization scheme for the quantized Hamiltonian constraint in canonical general relativity.', '0712.1413-1-4-4': 'Also, we still do not have a definitive answer as to the nature of quantum gravitational corrections to the semiclassical spectrum of Hawking radiation.', '0712.1413-1-4-5': 'By this we mean the following: starting from a candidate theory of quantum gravity such as quantum general relativity, one can derive Hawking radiation in the semiclassical approximation using a suitable midisuperspace model.', '0712.1413-1-4-6': 'Going beyond the semiclassical approximation, it is expected that quantum gravity will induce (possibly non-thermal) corrections to Hawking spectrum, but this still remains to be worked out in its full generality.', '0712.1413-1-4-7': 'It is hoped though that some progress will be possible on these unsolved problems if one makes contact with the methods of loop quantum gravity.', '0712.1413-1-5-0': 'All the classical and quantum studies mentioned so far have pertained to gravitational collapse in 3+1 dimensions.', '0712.1413-1-5-1': 'Motivated by the desire to overcome some of the obtacles faced in 3+1 physics, we turned attention to investigation of 2+1 gravitational collapse.', '0712.1413-1-5-2': 'This lower dimensional model, though simpler in some aspects, throws up new fascinating issues of its own, which have been studied in [CITATION], [CITATION], [CITATION].', '0712.1413-1-5-3': 'Classical 2+1 collapse admits a naked singularity for some initial data, but there is no coresponding quantum particle creation.', '0712.1413-1-5-4': 'A black hole solution (the well-known BTZ black hole [CITATION]) is possible in the presence of a negative cosmological constant, but the thermodynamics and statistics of the quantized BTZ black hole is completely different from that of the 4-d Schwarzschild black hole.', '0712.1413-1-6-0': 'These differences prompt us to the following question: in determining the nature of thermodynamics and statistics of the quantized black hole, and the nature of quantum emission from naked singularities, what is the role of the cosmological constant, and of the number of spatial dimensions?', '0712.1413-1-6-1': 'The present paper is the first in a series of three papers which addresses this question, by studying classical and quantum aspects of spherical dust collapse in an AdS spacetime with an arbitrary number of dimensions.', '0712.1413-1-6-2': 'In the current paper, we solve the Einstein equations for a collapsing dust ball in an asymptoticaly AdS spacetime, and examine the nature of the gravitational singularity.', '0712.1413-1-6-3': 'Quantization of this model will be taken up in two subsequent papers.', '0712.1413-1-7-0': 'The plan of the paper is as follows.', '0712.1413-1-7-1': 'In Section II we give results for spherical gravitational collapse of dust in an asymptotically flat [MATH]-dimensional spacetime.', '0712.1413-1-7-2': 'While this problem has been studied earlier by various authors [CITATION], [CITATION], [CITATION], [CITATION], we present here a simpler derivation of the occurrence of the naked singularity, and also obtain new results on the self-similar solution, besides giving an expression for the surface gravity of a [MATH]-dimensional Schwarzschild black hole, which will be used in the second paper of this series.', '0712.1413-1-7-3': 'More importantly, the results of this Section serve as a prelude to the corresponding analysis presented in Section III, for collapse in an AdS spacetime with arbitrary number of dimensions.', '0712.1413-1-7-4': 'While gravitational collapse of dust in four dimensional spacetime with a positive cosmlogical constant has been studied in [CITATION], (see also [CITATION]), to the best of our knowledge dust collapse in a [MATH]-dim AdS spacetime has not been studied before.', '0712.1413-1-8-0': 'One could question the introduction of a negative cosmological constant, as is done in this paper, when the observed Universe has a cosmological constant which is perhaps positive, or at best zero, but certainly not negative.', '0712.1413-1-8-1': 'Firstly, collapse physics in a deSitter spacetime is complicated by the presence of a deSitter event horizon, in addition to the black-hole event horizon.', '0712.1413-1-8-2': 'It thus seems natural to first address the AdS case before moving on to the more realistic, and more difficult, deSitter case.', '0712.1413-1-8-3': 'There are also reasons to believe that it would not make sense to directly construct a quantum black hole model in a higher dimensional space with a positive cosmological constant, because quantum gravity in such a spacetime may not exist nonperturbatively [CITATION], [CITATION].', '0712.1413-1-8-4': 'Pure quantum gravity with a positive cosmological constant may hence not exist as an exact theory, but only as a part of a larger system [CITATION].', '0712.1413-1-8-5': 'It is also a question of great interest as to whether studies of statistical properties of AdS black holes in canonical quantum general relativity can benefit from what is known about the AdS/CFT correspondence, as suggested recently in [CITATION] for the 4-d case.', '0712.1413-1-9-0': '# Higher Dimensional Spherically Symmetric Dust Collapse in the Absence of a Cosmological Constant', '0712.1413-1-10-0': 'Solution The metric for a spherically symmetric space-time can be written in the form [EQUATION] where [EQUATION]', '0712.1413-1-10-1': 'Here the number of space-time dimensions is (n+2) where [MATH] is the number of angular coordinates and the 2 designates one time dimension and one radial dimension.', '0712.1413-1-10-2': "For the case where the cosmological constant [MATH], Einstein equations are [EQUATION] where [MATH] is a constant related to Newton's constant of gravitation [MATH] (see section (2.4)) and [MATH] is the stress-energy tensor.", '0712.1413-1-10-3': 'For the case of non-rotating dust one can choose a synchronous and co-moving coordinate system in which the only non-zero component of the stress-energy tensor is [MATH], where [MATH] is the energy density of the dust.', '0712.1413-1-10-4': 'Further, in co-moving coordinates the [MATH] component of the metric can be chosen to be minus one.', '0712.1413-1-10-5': 'With this choice for the metric in ([REF]) we get the following independent set of Einstein equations [EQUATION]', '0712.1413-1-10-6': 'Components [MATH], [MATH] etc. are given by expressions similar to that for [MATH] except for overall sine squared factor(s).', '0712.1413-1-10-7': 'The Ricci scalar is given by[EQUATION]', '0712.1413-1-10-8': 'Solving the equation for [MATH] we obtain [EQUATION]', '0712.1413-1-10-9': 'In the above expression [MATH] is an arbitrary function called the energy function.', '0712.1413-1-10-10': 'Integration of the equation for [MATH] after using equation ([REF]) gives [EQUATION]', '0712.1413-1-10-11': 'Here [MATH] is another arbitrary function and is called the mass function.', '0712.1413-1-10-12': 'In what follows we will only consider the so called marginally bound case for which [MATH].', '0712.1413-1-10-13': 'In this case ([REF]) can be integrated easily and after choosing the negative sign for the square root corresponding to in-falling matter we get [EQUATION] where [MATH] is yet another arbitrary function which can be fixed by using the freedom in the choice of the [MATH]-coordinate.', '0712.1413-1-10-14': 'We relabel [MATH] such that at [MATH], [MATH].', '0712.1413-1-10-15': 'With this choice we have [EQUATION]', '0712.1413-1-10-16': 'From the above equations we see that at [MATH], [MATH] and this implies singularity formation for the shell labeled [MATH] as indicated by the blowing up of the Ricci scalar in ([REF]).', '0712.1413-1-10-17': 'Finally, substituting for [MATH] from ([REF]) in the equation for [MATH] we find that [EQUATION]', '0712.1413-1-10-18': 'From this one can obtain an expression for the mass function [EQUATION]', '0712.1413-1-11-0': '## A simple derivation of the naked singularity', '0712.1413-1-12-0': 'We now look at the nature of the [MATH] singularity formed at the center [MATH] of the dust cloud.', '0712.1413-1-12-1': 'For this we follow the method used in [CITATION] and start by assuming that the initial density profile [MATH] has the following series expansion near the center [MATH] of the dust cloud [EQUATION]', '0712.1413-1-12-2': 'Using this in ([REF]) we find that in this case the mass-function can be written as [EQUATION] where it is to be noted that [MATH] is not a free index but, as before, refers to the number of angular dimensions and [EQUATION] and [MATH] From ([REF]) we know that the singularity curve is given by [EQUATION]', '0712.1413-1-12-3': 'The central singularity at [MATH] forms at the time [EQUATION]', '0712.1413-1-12-4': 'Here, as a special case, we note that when [MATH], a constant (Oppenheimer-Snyder collapse), [MATH] and the singularity curve is given by [MATH] which is independent of [MATH] implying that all shells become singular at the same time as the central shell.', '0712.1413-1-12-5': 'Near [MATH] one can use the expansion for [MATH] as in ([REF]) and approximate the singularity curve as [EQUATION]', '0712.1413-1-12-6': 'In the above equation [MATH] is the first non-vanishing term beyond [MATH] in the expansion for [MATH].', '0712.1413-1-13-0': 'One would like to know whether the singularity at [MATH], [MATH] is naked or not, and for this we focus attention on radial null geodesics.', '0712.1413-1-13-1': 'We want to check if there are any outgoing radial null geodesics which terminate on the central singularity in the past.', '0712.1413-1-13-2': 'Assuming that there exist such geodesics we assume their form near [MATH] to be [EQUATION]', '0712.1413-1-13-3': 'Comparing this with ([REF]) we conclude that for the null geodesic to lie in the spacetime one must have [MATH] and if [MATH] then [EQUATION] (This is because [MATH] is negative, which will be the case if we demand that [MATH] be a decreasing function of [MATH]).', '0712.1413-1-14-0': 'Since one is interested in the region close to [MATH], we expand ([REF]) to leading order in [MATH] to obtain [EQUATION]', '0712.1413-1-14-1': 'From the metric one finds that for null geodesics [MATH].', '0712.1413-1-14-2': 'Differentiating ([REF]) w.r.t. [MATH] we get [EQUATION]', '0712.1413-1-14-3': 'Along the assumed geodesic, [MATH] is given by ([REF]).', '0712.1413-1-14-4': 'Substituting this in ([REF]) and equating it with the derivative of ([REF]), i.e. [MATH] gives [EQUATION]', '0712.1413-1-14-5': 'This is the main equation.', '0712.1413-1-14-6': 'If it admits a self-consistent solution then the singularity will be naked otherwise not.', '0712.1413-1-14-7': 'To simplify this we note that [MATH], as follows from ([REF]).', '0712.1413-1-15-0': 'We first consider the case [MATH].', '0712.1413-1-15-1': 'To leading order this gives [EQUATION]', '0712.1413-1-15-2': 'This equation implies [EQUATION]', '0712.1413-1-15-3': 'Since [MATH] is the first non-vanishing term beyond [MATH], we have the condition [MATH].', '0712.1413-1-15-4': 'Also, for consistency we require [MATH], which together with the previous condition on [MATH] implies [MATH].', '0712.1413-1-16-0': 'This implies that in 4-dimensions, where [MATH], we have [MATH], which means that [MATH] are the allowed values.', '0712.1413-1-16-1': 'That is, models for which either [MATH] (corresponding to [MATH]) or [MATH], [MATH] (corresponding to [MATH]) will have a naked singularity.', '0712.1413-1-17-0': 'Similarly in 5-dimensions, where [MATH], we find that [MATH] implying [MATH], i.e. only for [MATH] we get naked singularity.', '0712.1413-1-17-1': 'In 6-dimensions, [MATH] and we have [MATH], implying [MATH] as the only allowed value, i.e. the singularity is naked only if [MATH].', '0712.1413-1-17-2': 'One notes that for all higher dimensions [MATH] and therefore only [MATH], i.e. [MATH] gives naked singularity.', '0712.1413-1-18-0': 'As another special case we note that for [MATH], that is in (2+1)dimensions, [MATH] and therefore the condition [MATH] is always satisfied, implying that in this case we always have a naked singularity, which is in agreement with what has been observed in earlier work on (2+1) dimensional dust collapse [CITATION].', '0712.1413-1-19-0': 'We next consider the case where [MATH].', '0712.1413-1-19-1': 'Here ([REF]) gives [EQUATION] which implies [MATH].', '0712.1413-1-19-2': 'Now the conditions on [MATH] are that it be an integer greater than 1.', '0712.1413-1-19-3': 'These two conditions are met only for [MATH] and [MATH], that is, in (3+1) dimensions and in (4+1) dimensions respectively.', '0712.1413-1-19-4': 'For [MATH], [MATH] (which corresponds to [MATH]) and for [MATH], [MATH] (corresponding to [MATH]).', '0712.1413-1-19-5': 'Since the 4-dimensional case, corresponding to [MATH], is already reported in the literature [CITATION] we focus attention on the 5-dimensional case corresponding to [MATH].', '0712.1413-1-20-0': 'Substituting [MATH], [MATH] in ([REF]) we obtain [EQUATION]', '0712.1413-1-20-1': 'The above cubic for [MATH] has to be solved subject to the constraint [MATH] as mentioned earlier.', '0712.1413-1-20-2': 'By defining [MATH] and [MATH] the above equation is simplified to [EQUATION] and the constraint on [MATH] results in a constraint on [MATH] given by [MATH].', '0712.1413-1-20-3': 'By defining [MATH] and [MATH], the above cubic is further simplified to [EQUATION]', '0712.1413-1-20-4': 'For a naked singularity to form this equation for [MATH] should have a positive root subject to the constraint [MATH].', '0712.1413-1-21-0': 'Now for a general cubic [EQUATION] if we define [MATH] and [MATH], we have the following conditions on the roots of the cubic [CITATION] :', '0712.1413-1-22-0': '1.', '0712.1413-1-22-1': '[MATH], the roots of the cubic are all real.', '0712.1413-1-23-0': '2.', '0712.1413-1-23-1': '[MATH], the cubic has two imaginary roots.', '0712.1413-1-24-0': '3.', '0712.1413-1-24-1': '[MATH], two roots of the cubic are equal.', '0712.1413-1-25-0': '4.', '0712.1413-1-25-1': '[MATH] and [MATH], all three roots of the cubic are equal.', '0712.1413-1-26-0': 'Using these we can find the conditions on [MATH] for which the cubic in ([REF]) has at least one real root in the desired range.', '0712.1413-1-26-1': 'Here it should be noted that [MATH] as defined above has to be positive.', '0712.1413-1-26-2': 'It is found that for [MATH] all the three roots are real and at least one of these satisfies [MATH].', '0712.1413-1-26-3': 'For [MATH] the real root is negative.', '0712.1413-1-26-4': 'The range of [MATH] found above implies that for [MATH] one gets a naked singularity.', '0712.1413-1-27-0': 'We also note that for the Oppenheimer-Snyder collapse mentioned earlier, no naked singularity is formed since all shells become singular at the same time.', '0712.1413-1-28-0': '## Formation of Trapped Surfaces', '0712.1413-1-29-0': 'We now consider the formation of trapped surfaces.', '0712.1413-1-29-1': 'For this consider a congruence of outgoing radial null geodesics with tangent vector [MATH] where [MATH] is a parameter along the geodesic and [MATH] [CITATION].', '0712.1413-1-29-2': 'The expansion for these geodesics is given by [EQUATION]', '0712.1413-1-29-3': 'From this one finds that [EQUATION]', '0712.1413-1-29-4': 'Trapping occurs when [MATH] and the above equation with [MATH], [MATH] and [MATH] implies that this condition is met for [EQUATION]', '0712.1413-1-29-5': 'In 4-dimensions where [MATH] we get the well known result [EQUATION]', '0712.1413-1-29-6': 'For the general case one finds that the time at which trapping occurs [MATH] is given by [EQUATION] which means that the central shell is trapped at [MATH], that is, at the same time as the formation of the central singularity.', '0712.1413-1-29-7': 'For the outer shells trapping occurs before those shells become singular.', '0712.1413-1-30-0': '## Exterior Solution and Matching with the Interior', '0712.1413-1-31-0': 'We take the metric in the exterior to be independent of time and given by [EQUATION] where [MATH] are the coordinates in the spacetime exterior to the dust cloud.', '0712.1413-1-31-1': 'The components of the Einstein Tensor corresponding to the above metric are [EQUATION] and [MATH], [MATH] etc. are the related to [MATH] as in the interior.', '0712.1413-1-31-2': 'Solving the vacuum Einstein equations [MATH] one finds [EQUATION]', '0712.1413-1-31-3': 'Here [MATH] is a constant of integration.', '0712.1413-1-31-4': 'Thus the exterior metric is the Schwarzschild metric [EQUATION]', '0712.1413-1-31-5': 'For this to be a valid solution in the exterior we need to match the metric coefficients as well as their first derivatives (extrinsic curvature) in the exterior with the corresponding quantities in the interior at the boundary of the dust cloud [MATH] say, [CITATION], [CITATION].', '0712.1413-1-31-6': 'This will also determine the only unknown quantity [MATH] in the Schwarzchild solution.', '0712.1413-1-31-7': 'At the surface the exterior coordinates will be some functions [MATH] and [MATH] of the interior coordinates.', '0712.1413-1-31-8': 'These relations imply [MATH] and [MATH].', '0712.1413-1-31-9': 'Therefore at the surface (where [MATH]) [EQUATION]', '0712.1413-1-31-10': 'Matching the metric coefficients for [MATH] gives [MATH] and matching the metric coefficients for [MATH] then implies [EQUATION]', '0712.1413-1-31-11': 'To match the extrinsic curvature (second fundamental form) we need the normal to the surface.', '0712.1413-1-31-12': 'In the interior coordinates the components of the normal are found to be [MATH].', '0712.1413-1-31-13': 'Similarly in the exterior coordinates the normal is given by [MATH], where the relation [MATH] was used.', '0712.1413-1-31-14': 'The extrinsic curvature is given by [MATH], where [MATH] with [MATH] being the coordinates of the (n+2)-dimensional manifold and [MATH] being the coordinates on the boundary of the manifold.', '0712.1413-1-31-15': 'Since there is only one undetermined constant [MATH], we match only the [MATH] component of the extrinsic curvature.', '0712.1413-1-31-16': 'It can be easily checked that the other components do not give anything new.', '0712.1413-1-31-17': 'We find that at the surface the extrinsic curvature in the interior coordinates is given by [MATH].', '0712.1413-1-31-18': 'Similarly in the exterior coordinates we have [MATH].', '0712.1413-1-31-19': 'Equating these two expressions for [MATH] gives [EQUATION]', '0712.1413-1-31-20': 'Using ([REF]) and [MATH] (see ([REF])) the above equation gives [MATH], where from ([REF]) it is clear that [MATH] is proportional to the total mass of the dust cloud.', '0712.1413-1-31-21': 'Thus we find that for the metric coefficients and their first derivatives to be continuous across the boundary the metric in the exterior is given by [EQUATION]', '0712.1413-1-32-0': 'Now [MATH] and we know that mass of the dust cloud is given by [MATH] where [MATH] is the volume element of a spherical shell lying between [MATH] and [MATH] in [MATH] space dimensions.', '0712.1413-1-32-1': 'This volume element is given by [EQUATION] (The expression for the volume of an [MATH]-sphere is derived in an Appendix at the end of the paper).', '0712.1413-1-32-2': 'Therefore [EQUATION]', '0712.1413-1-32-3': 'This implies [EQUATION]', '0712.1413-1-32-4': 'Using this we find that the mass function can be written as [EQUATION]', '0712.1413-1-32-5': 'One can also find the constant [MATH] in the Schwarzchild solution using the weak field limit.', '0712.1413-1-32-6': "For this we assume that Newton's law for gravity holds for any number of dimensions i.e. [MATH] and [MATH].", '0712.1413-1-32-7': "Here [MATH] is the gravitational field strength and [MATH] is the gravitational potential (note: Newton's gravitational constant [MATH] being dimensionful will be different in different dimensions, however, this does not affect the form of the equations).", '0712.1413-1-32-8': 'Using this we find that in [MATH] spatial dimensions the gravitational potential is given by [EQUATION] where [MATH] (potential has a logarithmic dependence on [MATH] in [MATH] dimensions).', '0712.1413-1-32-9': 'In the weak field limit the Schwarzchild solution is [MATH] and [MATH].', '0712.1413-1-32-10': 'Also, using geodesic equation we find that generically, in the weak-static field limit [MATH] and [MATH].', '0712.1413-1-32-11': 'Comparing the two expressions for [MATH] (or for [MATH]) one finds that [MATH] and using the expression for [MATH] as found above one gets [EQUATION]', '0712.1413-1-32-12': "This expression for [MATH] will be the same as that found above from matching if the constant in Einstein's equations is chosen to be [MATH].", '0712.1413-1-32-13': 'For [MATH] this reduces to the value [MATH] as used in 4-dimensional theory and which when used in ([REF]) results in the familiar Schwarzchild solution [EQUATION]', '0712.1413-1-33-0': '## Surface Gravity', '0712.1413-1-34-0': 'For future reference, we compute the surface gravity at the event horizon of the [MATH]-dimensional black hole.', '0712.1413-1-34-1': "The exterior is given by [EQUATION] where [MATH], with [MATH], since as noted in the last section, the above formula for [MATH] is not valid for [MATH] as the correspondence between general relativity and Newton's theory breaks down in [MATH]-dimensions.", '0712.1413-1-34-2': 'From the form of the metric we have that at the event horizon [EQUATION]', '0712.1413-1-34-3': 'From this it is seen that the radius of the event horizon is given by [EQUATION]', '0712.1413-1-34-4': 'When [MATH] is even, [MATH] is odd and we have [EQUATION] which implies that the radius of the event horizon, when [MATH] is even, is [EQUATION]', '0712.1413-1-34-5': 'From above it is easy to see that for [MATH] we get the well known result [MATH].', '0712.1413-1-34-6': 'Similarly, when [MATH] is odd, [MATH] is an integer and the gamma function is related to the factorial [EQUATION] from which we find that the event horizon, in case when [MATH] is odd, is [EQUATION]', '0712.1413-1-34-7': 'Area of the event horizon is nothing but the volume of an [MATH]-sphere of radius [MATH] which is given by [EQUATION] where [MATH] is the volume of a unit [MATH]-sphere.', '0712.1413-1-34-8': 'Substituting for [MATH] we have [EQUATION]', '0712.1413-1-34-9': 'For [MATH] this gives [MATH].', '0712.1413-1-34-10': 'The surface gravity is given by [EQUATION]', '0712.1413-1-34-11': 'With [MATH], the surface gravity is given by [EQUATION]', '0712.1413-1-34-12': 'However, since [MATH] at the event horizon, the above equation gives the surface gravity as [EQUATION]', '0712.1413-1-34-13': 'From this we see that in [MATH]-dimensions [MATH].', '0712.1413-1-34-14': 'For completeness we also mention that in [MATH]-dimensions the surface gravity is zero since there is no event horizon.', '0712.1413-1-35-0': '## The Self-Similar Solution', '0712.1413-1-36-0': 'To see the effect of dimensions on the nature of quantum particle flux (which will be described in a work subsequent to this), we would like to have a globally naked singularity.', '0712.1413-1-36-1': 'It is known that a locally naked self-similar solution is also globally naked [CITATION], where self-similar spacetimes are defined by the existence of a homothetic Killing vector field.', '0712.1413-1-36-2': 'Therefore here we look at the dependence on dimensions of the self-similar dust model.', '0712.1413-1-36-3': 'It can be shown that for spherically symmetric, self-similar dust collapse, the mass function is given by [MATH], where [MATH] is a constant.', '0712.1413-1-36-4': 'With this ([REF]) becomes [EQUATION]', '0712.1413-1-36-5': 'Here [MATH] is a constant.', '0712.1413-1-36-6': 'This is because in self-similar collapse any dimensionless quantity has to be a function only of [MATH] whereas [MATH], being a function only of [MATH] (see ([REF])) has to be a constant.', '0712.1413-1-36-7': 'We are interested in finding the behavior of density [MATH] in the neighborhood of the centre [MATH].', '0712.1413-1-36-8': 'Using ([REF]) and [MATH] in the expression for density we find [EQUATION]', '0712.1413-1-36-9': 'For [MATH] we neglect the second order term in the above equation and obtain [EQUATION]', '0712.1413-1-36-10': 'Also for [MATH], [MATH], which implies [EQUATION]', '0712.1413-1-36-11': 'Substituting this in ([REF]) we get [EQUATION] where [MATH] and [MATH].', '0712.1413-1-36-12': 'This shows how the density profile should depend on the number of dimensions to obtain a self-similar solution.', '0712.1413-1-36-13': 'The above form for density profile implies that only in 4-dimensions ([MATH]) and in 5-dimensions ([MATH]) can there be a self-similar solution since in higher dimensions the density profile is no longer analytic.', '0712.1413-1-37-0': 'Here we also note that for [MATH], that is in [MATH] dimensions, [MATH] and is thus independent of [MATH] and therefore one requires that energy density [MATH] should be zero.', '0712.1413-1-37-1': 'Thus self-similarity in [MATH] dimensions is inconsistent with the presence of matter.', '0712.1413-1-38-0': '# Spherically symmetric inhomogeneous dust collapse in the presence of a negative cosmological constant', '0712.1413-1-39-0': '## Solution', '0712.1413-1-40-0': 'In the presence of a cosmological constant [MATH], Einstein equations are given by [MATH].', '0712.1413-1-40-1': 'For the case [MATH] we take [MATH] in which case the Einstein equations become [MATH], where now [MATH].', '0712.1413-1-40-2': 'The expressions for the components of Einstein tensor are still the same as in the [MATH] case.', '0712.1413-1-40-3': 'In particular since [MATH], therefore we again have [EQUATION]', '0712.1413-1-40-4': 'The solution of this equation is again given by ([REF]) and we again consider the marginally bound case so that [MATH].', '0712.1413-1-40-5': 'The 1-1 component of Einstein equations is [EQUATION]', '0712.1413-1-40-6': 'Integration of this equation gives [EQUATION] where as before [MATH] is the mass function.', '0712.1413-1-40-7': 'Integrating this equation after taking the negative sign for the square root (to account for in-falling matter) we get [EQUATION]', '0712.1413-1-40-8': 'Relabeling the [MATH] coordinate as in the previous case so that at [MATH], [MATH] we get [EQUATION]', '0712.1413-1-40-9': 'For [MATH] we again get [MATH] corresponding to the singularity formation for shell labeled [MATH].', '0712.1413-1-40-10': 'From [MATH] we obtain an expression for [MATH] which is again given by ([REF]).', '0712.1413-1-41-0': '## A simple derivation of the naked singularity', '0712.1413-1-42-0': 'As before we want to see if null geodesics can come out of the singularity.', '0712.1413-1-42-1': 'For this we proceed as before assuming that the density profile near the center is given by [EQUATION]', '0712.1413-1-42-2': 'From the form of the mass function [MATH] we have [EQUATION] where [MATH].', '0712.1413-1-42-3': 'From ([REF]) and ([REF]) we see that the singularity curve is given by [EQUATION]', '0712.1413-1-42-4': 'This implies that the central singularity at [MATH] forms at time [EQUATION]', '0712.1413-1-42-5': 'We again note that, as in the [MATH] case, when [MATH] is a constant all shells become singular at the same time as the central shell.', '0712.1413-1-43-0': 'We now rewrite the expression for the singularity curve as [EQUATION]', '0712.1413-1-43-1': 'It is reasonable to assume that for shells near [MATH] the time for singularity formation is close to the time for the central shell to become singular i.e. [MATH] and we can therefore write [MATH] where because of the assumption made [MATH].', '0712.1413-1-43-2': 'Using this we expand the left hand side of the above equation using the addition formula for sines and make use of [MATH] and [MATH] to get [EQUATION]', '0712.1413-1-43-3': 'Here [MATH] is the first non-zero term beyond [MATH] and is negative since we assume a decreasing density profile.', '0712.1413-1-43-4': 'Using [MATH] in the above equation we can finally write the expression for singularity curve for shells near the center as [EQUATION]', '0712.1413-1-43-5': 'To know whether the central singularity at [MATH], [MATH] is naked or not we focus attention on radial null geodesics and check if there are any outgoing radial null geodesics which terminate on the central singularity in the past.', '0712.1413-1-43-6': 'We proceed as in the earlier case, assuming that there exist such geodesics and take their form near [MATH] to be [EQUATION] where, comparing with ([REF]), we see that [MATH] and if [MATH] then [EQUATION] for the assumed geodesic to lie in the spacetime.', '0712.1413-1-43-7': 'We use ([REF]) and ([REF]) (retaining only the first two non-zero terms in the latter in the [MATH] approximation) in ([REF]) to get [EQUATION]', '0712.1413-1-43-8': 'Near [MATH], the time [MATH] appearing in the geodesic equation satisfies [MATH] and therefore the argument of the sine function in ([REF]) is close to zero and we use the approximation [MATH] obtaining [EQUATION]', '0712.1413-1-43-9': 'From the form of the metric we know that the radial null geodesics satisfy [MATH].', '0712.1413-1-43-10': 'We take the spatial derivative of the above equation, substitute for [MATH] from ([REF]) and equate the result to the derivative of ([REF]) [EQUATION]', '0712.1413-1-44-0': 'Consider first the case [MATH].', '0712.1413-1-44-1': 'Keeping terms only to lowest order in [MATH] we get [EQUATION]', '0712.1413-1-44-2': 'From this we have [EQUATION] where we have substituted for [MATH] in the argument of [MATH].', '0712.1413-1-44-3': 'Since the form of [MATH] is exactly the same as in the [MATH] case we find that the conditions for naked singularity formation are also the same as mentioned after ([REF]).', '0712.1413-1-44-4': 'That is, in [MATH]-dimensions [MATH], [MATH] implying [MATH] are the allowed values so that we get naked singularity for [MATH] or for [MATH], [MATH].', '0712.1413-1-44-5': 'Similarly in [MATH]-dimensions [MATH], [MATH] implying that only [MATH] is allowed so that we get naked singularity only for [MATH].', '0712.1413-1-44-6': 'In all higher dimensions we get naked singularity only if [MATH].', '0712.1413-1-45-0': 'Again [MATH], [MATH] and [MATH], [MATH] are critical cases satisfying [MATH].', '0712.1413-1-45-1': 'To analyze these we proceed as in the [MATH] case.', '0712.1413-1-45-2': 'For [MATH] ([REF]) becomes [EQUATION] which after substituting for [MATH] gives [EQUATION]', '0712.1413-1-45-3': 'Equating the power of [MATH] on the two sides gives [MATH] (as in the [MATH] case).', '0712.1413-1-45-4': 'Since [MATH] should be an integer greater than one we find that these conditions are satisfied only for [MATH]) and for [MATH]).', '0712.1413-1-46-0': 'Consider [MATH]; in this case the above equation can be written as [EQUATION] with the constraint that [EQUATION]', '0712.1413-1-46-1': 'If we define [MATH] and [MATH] the equation can be written in the simplified form [EQUATION] with the requirement that [MATH].', '0712.1413-1-46-2': 'If we further define [MATH] and [MATH] the equation becomes [EQUATION]', '0712.1413-1-46-3': 'For a naked singularity to form this equation for [MATH] should have a solution subject to the constraint [MATH] and [MATH].', '0712.1413-1-47-0': 'Using the conditions, as mentioned earlier, for the roots of a general cubic we can find the conditions on [MATH] for which the above cubic has at least one real root in the desired range.', '0712.1413-1-47-1': 'It is found that for [MATH] all the three roots are real and at least one of these satisfies [MATH].', '0712.1413-1-47-2': 'For [MATH] the real root is negative.', '0712.1413-1-47-3': 'The range of [MATH] found above implies that for [MATH] one gets a naked singularity.', '0712.1413-1-47-4': 'This shows that the critical case is also similar to the [MATH] case except that the allowed range for [MATH] has shifted.', '0712.1413-1-48-0': 'A similar analysis can be carried out for the case where [MATH] and [MATH].', '0712.1413-1-48-1': 'By defining [MATH] and [MATH] one gets a fourth order equation in [MATH].', '0712.1413-1-48-2': 'If one subsequently defines [MATH] and [MATH] one gets the equation [EQUATION] with the consistency conditions [MATH] and [MATH].', '0712.1413-1-48-3': 'It is found that the above conditions are satisfied for [MATH] or in terms of conditions on [MATH] we get [MATH].', '0712.1413-1-49-0': '## Formation of Trapped Surfaces', '0712.1413-1-50-0': 'As in the [MATH] case, we now look at the formation of trapped surfaces.', '0712.1413-1-50-1': 'Considering the expansion of outgoing radial null geodesics as in ([REF]) we find that [EQUATION]', '0712.1413-1-50-2': 'From this it is seen that the condition for trapping, [MATH], is met when [EQUATION] which for [MATH] (4-dimensions) reduces to the well known result [EQUATION]', '0712.1413-1-50-3': 'It is easy to see that, as in the [MATH] case, for the central shell, trapping coincides with singularity formation.', '0712.1413-1-51-0': '## Exterior Solution with a negative Cosmological constant', '0712.1413-1-52-0': 'As before we take the metric in the exterior to be [EQUATION]', '0712.1413-1-52-1': 'The components of Einstein tensor are the same as in ([REF])-([REF]).', '0712.1413-1-52-2': 'Solving the vacuum Einstein equations [MATH] we find [EQUATION]', '0712.1413-1-52-3': 'With this the metric in the exterior becomes [EQUATION]', '0712.1413-1-52-4': 'Here C is a constant of integration which is fixed by matching the exterior solution to the interior solution at the boundary in exactly the same way as for the [MATH] case and the result is [MATH] where [MATH] is the boundary of the dust cloud.', '0712.1413-1-52-5': 'Thus in 4-dimensions the exterior is [EQUATION]', '0712.1413-1-53-0': '## Surface Gravity', '0712.1413-1-54-0': 'As follows from ([REF]) the event horizon is defined by the condition [EQUATION]', '0712.1413-1-54-1': 'This implies that the radius of the event horizon is given as the solution to the following equation [EQUATION]', '0712.1413-1-54-2': 'For [MATH] we find that the event horizon is given by [EQUATION]', '0712.1413-1-54-3': 'Area of the event horizon is again given by ([REF]).', '0712.1413-1-54-4': 'To find the surface gravity we use ([REF]) obtaining [EQUATION]', '0712.1413-1-54-5': 'Using [MATH] we get [EQUATION]', '0712.1413-1-54-6': 'Anticipating that the Hawking temperature of the quantized black hole will be determined by surface gravity, we see that this expression for surface gravity has remarkable implications for black hole thermodynamics.', '0712.1413-1-54-7': 'The Hawking temperature will be controlled by three parameters: mass of the black hole, number of spatial dimensions and the cosmological constant.', '0712.1413-1-54-8': 'The presence of [MATH] could make the specific heat of the black hole positive (as it does for the BTZ black hole).', '0712.1413-1-54-9': 'To this effect we calculate the derivative [MATH] for fixed [MATH] and [MATH].', '0712.1413-1-54-10': 'From ([REF]) we get that [EQUATION] where to simplify notation we write [MATH] instead of [MATH].', '0712.1413-1-54-11': 'As is expected, and also evident from ([REF]), [MATH] increases monotonically with [MATH].', '0712.1413-1-54-12': 'Thus the condition for positivity of the specific heat is [EQUATION] which is a constraint on the dimensionless quantity constructed from the cosmological constant and the radius of the event horizon.', '0712.1413-1-54-13': 'After expressing [MATH] in terms of [MATH], [MATH] and [MATH], this becomes a condition on the mutual relation between the three free parameters.', '0712.1413-1-54-14': 'A few things can be read off immediately: (i) if [MATH] is zero, specific heat is necessarily negative, (ii) for the 4-d AdS case, i.e. [MATH], specific heat is positive if [MATH].', '0712.1413-1-54-15': 'Using the expression for [MATH] from ([REF]) this appears to translate into a complicated relation between [MATH] and [MATH], (iii) given any [MATH], and some value of [MATH], one can always choose an [MATH], and hence an [MATH], sufficiently large, so that the specific heat becomes positive.', '0712.1413-1-54-16': 'This important aspect will be further discussed in detail in the second paper of this series.', '0712.1413-1-55-0': '## The absence of a self-similar solution in the Presence of a Negative [MATH]', '0712.1413-1-56-0': 'It is interesting to note that it is not possible to have a self-similar solution in the presence of a cosmological constant.', '0712.1413-1-56-1': 'If we have a self-similar solution then we can write [MATH] with [MATH] being dimensionless.', '0712.1413-1-56-2': 'If we define [MATH] then the condition of self-similarity implies that [MATH] being dimensionless should be a function only of [MATH].', '0712.1413-1-56-3': 'With this if we now consider the equation [MATH] we get [EQUATION]', '0712.1413-1-56-4': 'The explicit presence of [MATH] in the above equation implies that [MATH] cannot be expressed as a function of [MATH] alone and thus we do not have a self-similar solution in the presence of [MATH].', '0712.1413-1-57-0': '# Conclusions', '0712.1413-1-58-0': 'We have studied the collapse of inhomogeneous spherically symmetric dust distribution in arbitrary number of space dimensions both in the absence and in the presence of a cosmological constant.', '0712.1413-1-58-1': 'From the analysis presented we see that even though naked singularity is allowed in all dimensions there is more freedom on initial conditions for obtaining naked singularity in 2+1, 3+1 and 4+1 dimensions, both in the absence as well as the presence of a negative cosmological constant.', '0712.1413-1-58-2': 'We have also seen that the formation of trapped surfaces is similar in all dimensions with the central shell getting trapped at the same time when it becomes singular.', '0712.1413-1-58-3': 'For outer shells trapping occurs before those shells become singular.', '0712.1413-1-58-4': 'We also saw explicitly that in the absence of a cosmological constant, globally naked self-similar models can be constructed only in 3+1 and 4+1 dimensions.', '0712.1413-1-58-5': 'For all other dimensions such a construction is ruled out whereas in the presence of a cosmological constant such a solution cannot be constructed at all.', '0712.1413-1-59-0': 'In the second paper in this series, we will study quantum field theory on the curved background provided by the classical solutions presented here, including the emission of Hawking radiation from an [MATH]-dim AdS black hole.', '0712.1413-1-59-1': 'In a third paper we will carry out a canonical quantization of this model, and also address the issue of black hole entropy, following the methods of [CITATION].', '0712.1413-1-60-0': 'Appendix Here we derive the formula for the volume of [MATH] sphere following the derivation given in [CITATION].', '0712.1413-1-60-1': 'For this consider [MATH]-dimensional eucilean space [MATH].', '0712.1413-1-60-2': 'We then have [EQUATION]', '0712.1413-1-60-3': 'We find the volume of the [MATH] sphere by evaluating the following integral in two different ways [EQUATION]', '0712.1413-1-60-4': 'Using ([REF]) the above integral can be written as [EQUATION]', '0712.1413-1-60-5': 'Each integral above is a Gaussian integral and we get [EQUATION]', '0712.1413-1-60-6': 'We now evaluate the same integral by considering the entire volume of [MATH] as made up of concentric spherical shells of radius [MATH] and thickness [MATH].', '0712.1413-1-60-7': 'The volume of each shell is then given by the product of the volume of the [MATH] sphere of radius [MATH] with the thickness [MATH].', '0712.1413-1-60-8': 'Thus we have [EQUATION] where [MATH] is the volume of a [MATH] sphere of radius [MATH].', '0712.1413-1-60-9': 'This is nothing but the product of the volume of a unit [MATH] sphere (written simply as [MATH] without any argument) with [MATH].', '0712.1413-1-60-10': 'Thus the above equation becomes [EQUATION]', '0712.1413-1-60-11': 'Making the change of variable [MATH] this becomes [EQUATION]', '0712.1413-1-60-12': 'Comparing this with ([REF]) we find that the volume of a unit [MATH] sphere is given by [EQUATION]'}

{'0712.1413-2-0-0': 'We study the collapse of a spherically symmetric dust distribution in [MATH]-dimensional AdS spacetime.', '0712.1413-2-0-1': 'We investigate the role of dimensionality, and the presence of a negative cosmological constant, in determining the formation of trapped surfaces and the end state of gravitational collapse.', '0712.1413-2-0-2': 'We obtain the self-similar solution for the case of zero cosmological constant, and show that one cannot construct a self-similar solution when a cosmological constant is included.', '0712.1413-2-1-0': '# Introduction', '0712.1413-2-2-0': 'There are many models of spherical gravitational collapse in classical general relativity which exhibit the formation of black holes as well as naked singularities, starting from regular initial data [CITATION], [CITATION].', '0712.1413-2-2-1': 'The study of quantum effects in the vicinity of the gravitational singularity then becomes significant.', '0712.1413-2-2-2': 'Such studies can be divided into two classes : (i) quantum field theory in curved space, and (ii) quantum general relativistic treatment of gravitational collapse.', '0712.1413-2-3-0': 'The earliest investigations of quantum field theory in the dynamical background of a collapsing spherical star were probably those due to Ford and Parker [CITATION] and Hiscock et al. [CITATION].', '0712.1413-2-3-1': 'These works introduced important techniques, such as the calculation of the quantum flux in the geometric optics approximation, and the regularization of the 2-d quantum stress tensor, which were used extensively in later studies.', '0712.1413-2-3-2': 'A systematic study of semiclassical effects in gravitational collapse was initiated by Vaz and Witten in [CITATION], [CITATION] and pursued in a series of papers [CITATION], [CITATION], [CITATION], [CITATION], [CITATION], [CITATION], [CITATION].', '0712.1413-2-3-3': 'Typically, these studies showed an important and interesting difference in the nature of quantum particle creation between the two cases - one in which collapse ends in a black hole, and another in which it ends in a naked singularity.', '0712.1413-2-3-4': 'The formation of a black hole is accompanied by the emission of Hawking radiation, as expected.', '0712.1413-2-3-5': 'However, when the collapse ends in a (shell-focusing) naked singularity, there is no evidence of some universal behaviour in the nature of quantum emission.', '0712.1413-2-3-6': 'It is typically found though, that the emitted quantum flux diverges in the approach to the Cauchy horizon.', '0712.1413-2-3-7': 'This divergence disappears when the calculation of the quantum flux is terminated about a Planck time before the formation of the Cauchy horizon, when the semiclassical approximation breaks down.', '0712.1413-2-3-8': 'Instead of the divergence, one finds that only about a Planck unit of energy is emitted during the semiclassical phase, and a full quantum gravitational treatment of the physics of the singularity and the Cauchy horizon becomes unavoidable.', '0712.1413-2-3-9': 'These developments have been reviewed in [CITATION].', '0712.1413-2-4-0': 'A full quantum gravitational treatment of collapse can be performed via a midisuperspace quantization within the framework of quantum general relativity.', '0712.1413-2-4-1': 'The aims of such a programme are manifold - to construct a quantum gravitational description of the black hole; to check if the gravitational singularity can be avoided in quantum gravity; to obtain a statistical derivation of the black-hole entropy from quantum gravitational microstates; and to determine the role of quantum gravity in ascertaining the nature of quantum emission from a naked singularity.', '0712.1413-2-4-2': 'The midisuperspace quantization programme has been carried out by us in a series of papers [CITATION], [CITATION], [CITATION], [CITATION], [CITATION], [CITATION], [CITATION], and work along these lines is still in progress.', '0712.1413-2-4-3': 'It is fair to say that while some progress has been made on aspects related to quantum black holes and black hole entropy, issues related to singularity avoidance and the nature of quantized naked singularities have thus far proved difficult to address, largely bcause of problems relating to finding a suitable regularization scheme for the quantized Hamiltonian constraint in canonical general relativity.', '0712.1413-2-4-4': 'Also, we still do not have a definitive answer as to the nature of quantum gravitational corrections to the semiclassical spectrum of Hawking radiation.', '0712.1413-2-4-5': 'By this we mean the following: starting from a candidate theory of quantum gravity such as quantum general relativity, one can derive Hawking radiation in the semiclassical approximation using a suitable midisuperspace model.', '0712.1413-2-4-6': 'Going beyond the semiclassical approximation, it is expected that quantum gravity will induce (possibly non-thermal) corrections to Hawking spectrum, but this still remains to be worked out in its full generality.', '0712.1413-2-4-7': 'It is hoped though that some progress will be possible on these unsolved problems if one makes contact with the methods of loop quantum gravity.', '0712.1413-2-5-0': 'All the classical and quantum studies mentioned so far have pertained to gravitational collapse in 3+1 dimensions.', '0712.1413-2-5-1': 'Motivated by the desire to overcome some of the obtacles faced in 3+1 physics, we turned attention to investigation of 2+1 gravitational collapse.', '0712.1413-2-5-2': 'Homogeneous dust collapse in 2+1 dimensions was first studied in [CITATION] and for the case of collapsing shells in [CITATION].', '0712.1413-2-5-3': 'This lower dimensional model, though simpler in some aspects, throws up new fascinating issues of its own, which have been studied in the context of inhomogeneous dust collapse in [CITATION], [CITATION], [CITATION].', '0712.1413-2-5-4': 'Classical 2+1 collapse admits a naked singularity for some initial data, but there is no coresponding quantum particle creation.', '0712.1413-2-5-5': 'A black hole solution (the well-known BTZ black hole [CITATION]) is possible in the presence of a negative cosmological constant, but the thermodynamics and statistics of the quantized BTZ black hole is completely different from that of the 4-d Schwarzschild black hole.', '0712.1413-2-6-0': 'These differences prompt us to the following question: in determining the nature of thermodynamics and statistics of the quantized black hole, and the nature of quantum emission from naked singularities, what is the role of the cosmological constant, and of the number of spatial dimensions?', '0712.1413-2-6-1': 'The present paper is the first in a series of three papers which addresses this question, by studying classical and quantum aspects of spherical dust collapse in an AdS spacetime with an arbitrary number of dimensions.', '0712.1413-2-6-2': 'In the current paper, we solve the Einstein equations for a collapsing dust ball in an asymptoticaly AdS spacetime, and examine the nature of the gravitational singularity.', '0712.1413-2-6-3': 'Quantization of this model will be taken up in two subsequent papers.', '0712.1413-2-7-0': 'The plan of the paper is as follows.', '0712.1413-2-7-1': 'In Section II we give results for spherical gravitational collapse of dust in an asymptotically flat [MATH]-dimensional spacetime.', '0712.1413-2-7-2': 'While this problem has been studied earlier by various authors [CITATION], [CITATION], [CITATION], [CITATION], we present here a simpler derivation of the occurrence of a locally naked singularity, and also obtain new results on the self-similar solution.', '0712.1413-2-7-3': 'More importantly, the results of this Section serve as a prelude to the corresponding analysis presented in Section III, for collapse in an AdS spacetime with arbitrary number of dimensions.', '0712.1413-2-7-4': 'While gravitational collapse of dust in four dimensional spacetime with a positive cosmlogical constant has been studied in [CITATION], (see also [CITATION]), and for a negative cosmological constant in [CITATION], to the best of our knowledge dust collapse in a [MATH]-dim AdS spacetime has not been studied before.', '0712.1413-2-8-0': 'One could question the introduction of a negative cosmological constant, as is done in this paper, when the observed Universe has a cosmological constant which is perhaps positive, or at best zero, but certainly not negative.', '0712.1413-2-8-1': 'Firstly, collapse physics in a deSitter spacetime is complicated by the presence of a deSitter event horizon, in addition to the black-hole event horizon.', '0712.1413-2-8-2': 'It thus seems natural to first address the AdS case before moving on to the more realistic, and more difficult, deSitter case.', '0712.1413-2-8-3': 'There are also reasons to believe that it would not make sense to directly construct a quantum black hole model in a higher dimensional space with a positive cosmological constant, because quantum gravity in such a spacetime may not exist nonperturbatively [CITATION], [CITATION].', '0712.1413-2-8-4': 'Pure quantum gravity with a positive cosmological constant may hence not exist as an exact theory, but only as a part of a larger system [CITATION].', '0712.1413-2-8-5': 'It is also a question of great interest as to whether studies of statistical properties of AdS black holes in canonical quantum general relativity can benefit from what is known about the AdS/CFT correspondence, as suggested recently in [CITATION] for the 4-d case.', '0712.1413-2-9-0': '# Higher Dimensional Spherically Symmetric Dust Collapse in the Absence of a Cosmological Constant', '0712.1413-2-10-0': 'Solution The metric for a spherically symmetric space-time can be written in the form [EQUATION] where [EQUATION]', '0712.1413-2-10-1': 'Here the number of space-time dimensions is (n+2) where [MATH] is the number of angular coordinates and the 2 designates one time dimension and one radial dimension.', '0712.1413-2-10-2': "For the case where the cosmological constant [MATH], Einstein equations are [EQUATION] where [MATH] is a constant related to Newton's constant of gravitation [MATH] (see section (2.4)) and [MATH] is the stress-energy tensor.", '0712.1413-2-10-3': 'For the case of non-rotating dust one can choose a synchronous and co-moving coordinate system in which the only non-zero component of the stress-energy tensor is [MATH], where [MATH] is the energy density of the dust.', '0712.1413-2-10-4': 'Further, in co-moving coordinates the [MATH] component of the metric can be chosen to be minus one.', '0712.1413-2-10-5': 'With this choice for the metric in ([REF]) we get the following independent set of Einstein equations [EQUATION]', '0712.1413-2-10-6': 'Components [MATH], [MATH] etc. are given by expressions similar to that for [MATH] except for overall sine squared factor(s).', '0712.1413-2-10-7': 'The Ricci scalar is given by[EQUATION]', '0712.1413-2-10-8': 'Solving the equation for [MATH] we obtain [EQUATION]', '0712.1413-2-10-9': 'In the above expression [MATH] is an arbitrary function called the energy function.', '0712.1413-2-10-10': 'Integration of the equation for [MATH] after using equation ([REF]) gives [EQUATION]', '0712.1413-2-10-11': 'Here [MATH] is another arbitrary function and is called the mass function.', '0712.1413-2-10-12': 'In what follows we will only consider the so called marginally bound case for which [MATH].', '0712.1413-2-10-13': 'In this case ([REF]) can be integrated easily and after choosing the negative sign for the square root corresponding to in-falling matter we get [EQUATION] where [MATH] is yet another arbitrary function which can be fixed by using the freedom in the choice of the [MATH]-coordinate.', '0712.1413-2-10-14': 'We relabel [MATH] such that at [MATH], [MATH].', '0712.1413-2-10-15': 'With this choice we have [EQUATION]', '0712.1413-2-10-16': 'From the above equations we see that at [MATH], [MATH] and this implies singularity formation for the shell labeled [MATH] as indicated by the blowing up of the Ricci scalar in ([REF]).', '0712.1413-2-10-17': 'Finally, substituting for [MATH] from ([REF]) in the equation for [MATH] we find that [EQUATION]', '0712.1413-2-10-18': 'From this one can obtain an expression for the mass function [EQUATION]', '0712.1413-2-11-0': '## A simple derivation of the naked singularity', '0712.1413-2-12-0': 'We now look at the nature of the [MATH] singularity formed at the center [MATH] of the dust cloud.', '0712.1413-2-12-1': 'For this we follow the method used in [CITATION] and start by assuming that the initial density profile [MATH] has the following series expansion near the center [MATH] of the dust cloud [EQUATION]', '0712.1413-2-12-2': 'Using this in ([REF]) we find that in this case the mass-function can be written as [EQUATION] where it is to be noted that [MATH] is not a free index but, as before, refers to the number of angular dimensions and [EQUATION] and [MATH] From ([REF]) we know that the singularity curve is given by [EQUATION]', '0712.1413-2-12-3': 'The central singularity at [MATH] forms at the time [EQUATION]', '0712.1413-2-12-4': 'Here, as a special case, we note that when [MATH], a constant (Oppenheimer-Snyder collapse), [MATH] and the singularity curve is given by [MATH] which is independent of [MATH] implying that all shells become singular at the same time as the central shell.', '0712.1413-2-12-5': 'Near [MATH] one can use the expansion for [MATH] as in ([REF]) and approximate the singularity curve as [EQUATION]', '0712.1413-2-12-6': 'In the above equation [MATH] is the first non-vanishing term beyond [MATH] in the expansion for [MATH].', '0712.1413-2-13-0': 'One would like to know whether the singularity at [MATH], [MATH] is naked or not, and for this we focus attention on radial null geodesics.', '0712.1413-2-13-1': 'We want to check if there are any outgoing radial null geodesics which terminate on the central singularity in the past.', '0712.1413-2-13-2': 'Assuming that there exist such geodesics we assume their form near [MATH] to be [EQUATION]', '0712.1413-2-13-3': 'Comparing this with ([REF]) we conclude that for the null geodesic to lie in the spacetime one must have [MATH] and if [MATH] then [EQUATION] (This is because [MATH] is negative, which will be the case if we demand that [MATH] be a decreasing function of [MATH]).', '0712.1413-2-14-0': 'Since one is interested in the region close to [MATH], we expand ([REF]) to leading order in [MATH] to obtain [EQUATION]', '0712.1413-2-14-1': 'From the metric one finds that for null geodesics [MATH].', '0712.1413-2-14-2': 'Differentiating ([REF]) w.r.t. [MATH] we get [EQUATION]', '0712.1413-2-14-3': 'Along the assumed geodesic, [MATH] is given by ([REF]).', '0712.1413-2-14-4': 'Substituting this in ([REF]) and equating it with the derivative of ([REF]), i.e. [MATH] gives [EQUATION]', '0712.1413-2-14-5': 'This is the main equation.', '0712.1413-2-14-6': 'If it admits a self-consistent solution then the singularity will be naked otherwise not.', '0712.1413-2-14-7': 'To simplify this we note that [MATH], as follows from ([REF]).', '0712.1413-2-15-0': 'We first consider the case [MATH].', '0712.1413-2-15-1': 'To leading order this gives [EQUATION]', '0712.1413-2-15-2': 'This equation implies [EQUATION]', '0712.1413-2-15-3': 'Since [MATH] is the first non-vanishing term beyond [MATH], we have the condition [MATH].', '0712.1413-2-15-4': 'Also, for consistency we require [MATH], which together with the previous condition on [MATH] implies [MATH].', '0712.1413-2-16-0': 'This implies that in 4-dimensions, where [MATH], we have [MATH], which means that [MATH] are the allowed values.', '0712.1413-2-16-1': 'That is, models for which either [MATH] (corresponding to [MATH]) or [MATH], [MATH] (corresponding to [MATH]) will have a naked singularity.', '0712.1413-2-17-0': 'Similarly in 5-dimensions, where [MATH], we find that [MATH] implying [MATH], i.e. only for [MATH] we get naked singularity.', '0712.1413-2-17-1': 'In 6-dimensions, [MATH] and we have [MATH], implying [MATH] as the only allowed value, i.e. the singularity is naked only if [MATH].', '0712.1413-2-17-2': 'One notes that for all higher dimensions [MATH] and therefore only [MATH], i.e. [MATH] gives naked singularity.', '0712.1413-2-18-0': 'As another special case we note that for [MATH], that is in (2+1)dimensions, [MATH] and therefore the condition [MATH] is always satisfied, implying that in this case we always have a naked singularity, which is in agreement with what has been observed in earlier work on (2+1) dimensional dust collapse [CITATION].', '0712.1413-2-19-0': 'We next consider the case where [MATH].', '0712.1413-2-19-1': 'Here ([REF]) gives [EQUATION] which implies [MATH].', '0712.1413-2-19-2': 'Now the conditions on [MATH] are that it be an integer greater than 1.', '0712.1413-2-19-3': 'These two conditions are met only for [MATH] and [MATH], that is, in (3+1) dimensions and in (4+1) dimensions respectively.', '0712.1413-2-19-4': 'For [MATH], [MATH] (which corresponds to [MATH]) and for [MATH], [MATH] (corresponding to [MATH]).', '0712.1413-2-19-5': 'Since the 4-dimensional case, corresponding to [MATH], is already reported in the literature [CITATION] we focus attention on the 5-dimensional case corresponding to [MATH].', '0712.1413-2-20-0': 'Substituting [MATH], [MATH] in ([REF]) we obtain [EQUATION]', '0712.1413-2-20-1': 'The above cubic for [MATH] has to be solved subject to the constraint [MATH] as mentioned earlier.', '0712.1413-2-20-2': 'By defining [MATH] and [MATH] the above equation is simplified to [EQUATION] and the constraint on [MATH] results in a constraint on [MATH] given by [MATH].', '0712.1413-2-20-3': 'By defining [MATH] and [MATH], the above cubic is further simplified to [EQUATION]', '0712.1413-2-20-4': 'For a naked singularity to form this equation for [MATH] should have a positive root subject to the constraint [MATH].', '0712.1413-2-21-0': 'Now for a general cubic [EQUATION] if we define [MATH] and [MATH], we have the following conditions on the roots of the cubic [CITATION] :', '0712.1413-2-22-0': '1.', '0712.1413-2-22-1': '[MATH], the roots of the cubic are all real.', '0712.1413-2-23-0': '2.', '0712.1413-2-23-1': '[MATH], the cubic has two imaginary roots.', '0712.1413-2-24-0': '3.', '0712.1413-2-24-1': '[MATH], two roots of the cubic are equal.', '0712.1413-2-25-0': '4.', '0712.1413-2-25-1': '[MATH] and [MATH], all three roots of the cubic are equal.', '0712.1413-2-26-0': 'Using these we can find the conditions on [MATH] for which the cubic in ([REF]) has at least one real root in the desired range.', '0712.1413-2-26-1': 'Here it should be noted that [MATH] as defined above has to be positive.', '0712.1413-2-26-2': 'It is found that for [MATH] all the three roots are real and at least one of these satisfies [MATH].', '0712.1413-2-26-3': 'For [MATH] the real root is negative.', '0712.1413-2-26-4': 'The range of [MATH] found above implies that for [MATH] one gets a naked singularity.', '0712.1413-2-27-0': 'We also note that for the Oppenheimer-Snyder collapse mentioned earlier, no naked singularity is formed since all shells become singular at the same time.', '0712.1413-2-28-0': '## Formation of Trapped Surfaces', '0712.1413-2-29-0': 'We now consider the formation of trapped surfaces.', '0712.1413-2-29-1': 'For this consider a congruence of outgoing radial null geodesics with tangent vector [MATH] where [MATH] is a parameter along the geodesic and [MATH] [CITATION].', '0712.1413-2-29-2': 'The expansion for these geodesics is given by [EQUATION]', '0712.1413-2-29-3': 'From this one finds that [EQUATION]', '0712.1413-2-29-4': 'Trapping occurs when [MATH] and the above equation with [MATH], [MATH] and [MATH] implies that this condition is met for [EQUATION]', '0712.1413-2-29-5': 'In 4-dimensions where [MATH] we get the well known result [EQUATION]', '0712.1413-2-29-6': 'For the general case one finds that the time at which trapping occurs [MATH] is given by [EQUATION] which means that the central shell is trapped at [MATH], that is, at the same time as the formation of the central singularity.', '0712.1413-2-29-7': 'For the outer shells trapping occurs before those shells become singular.', '0712.1413-2-30-0': '## Exterior Solution and Matching with the Interior', '0712.1413-2-31-0': 'We take the metric in the exterior to be independent of time and given by [EQUATION] where [MATH] are the coordinates in the spacetime exterior to the dust cloud.', '0712.1413-2-31-1': 'The components of the Einstein Tensor corresponding to the above metric are [EQUATION] and [MATH], [MATH] etc. are the related to [MATH] as in the interior.', '0712.1413-2-31-2': 'Solving the vacuum Einstein equations [MATH] one finds [EQUATION]', '0712.1413-2-31-3': 'Here [MATH] is a constant of integration.', '0712.1413-2-31-4': 'Thus the exterior metric is the Schwarzschild metric [EQUATION]', '0712.1413-2-31-5': 'For this to be a valid solution in the exterior we need to match the metric coefficients as well as their first derivatives (extrinsic curvature) in the exterior with the corresponding quantities in the interior at the boundary of the dust cloud [MATH] say, [CITATION], [CITATION].', '0712.1413-2-31-6': 'This will also determine the only unknown quantity [MATH] in the Schwarzchild solution.', '0712.1413-2-31-7': 'At the surface the exterior coordinates will be some functions [MATH] and [MATH] of the interior coordinates.', '0712.1413-2-31-8': 'These relations imply [MATH] and [MATH].', '0712.1413-2-31-9': 'Therefore at the surface (where [MATH]) [EQUATION]', '0712.1413-2-31-10': 'Matching the metric coefficients for [MATH] gives [MATH] and matching the metric coefficients for [MATH] then implies [EQUATION]', '0712.1413-2-31-11': 'To match the extrinsic curvature (second fundamental form) we need the normal to the surface.', '0712.1413-2-31-12': 'In the interior coordinates the components of the normal are found to be [MATH].', '0712.1413-2-31-13': 'Similarly in the exterior coordinates the normal is given by [MATH], where the relation [MATH] was used.', '0712.1413-2-31-14': 'The extrinsic curvature is given by [MATH], where [MATH] with [MATH] being the coordinates of the (n+2)-dimensional manifold and [MATH] being the coordinates on the boundary of the manifold.', '0712.1413-2-31-15': 'Since there is only one undetermined constant [MATH], we match only the [MATH] component of the extrinsic curvature.', '0712.1413-2-31-16': 'It can be easily checked that the other components do not give anything new.', '0712.1413-2-31-17': 'We find that at the surface the extrinsic curvature in the interior coordinates is given by [MATH].', '0712.1413-2-31-18': 'Similarly in the exterior coordinates we have [MATH].', '0712.1413-2-31-19': 'Equating these two expressions for [MATH] gives [EQUATION]', '0712.1413-2-31-20': 'Using ([REF]) and [MATH] (see ([REF])) the above equation gives [MATH], where from ([REF]) it is clear that [MATH] is proportional to the total mass of the dust cloud.', '0712.1413-2-31-21': 'Thus we find that for the metric coefficients and their first derivatives to be continuous across the boundary the metric in the exterior is given by [EQUATION]', '0712.1413-2-32-0': 'Now [MATH] and we know that mass of the dust cloud is given by [MATH] where [MATH] is the volume element of a spherical shell lying between [MATH] and [MATH] in [MATH] space dimensions.', '0712.1413-2-32-1': 'This volume element is given by [EQUATION]', '0712.1413-2-32-2': 'Therefore [EQUATION]', '0712.1413-2-32-3': 'This implies [EQUATION]', '0712.1413-2-32-4': 'Using this we find that the mass function can be written as [EQUATION]', '0712.1413-2-32-5': 'One can also find the constant [MATH] in the Schwarzchild solution using the weak field limit.', '0712.1413-2-32-6': "For this we assume that Newton's law for gravity holds for any number of dimensions i.e. [MATH] and [MATH].", '0712.1413-2-32-7': "Here [MATH] is the gravitational field strength and [MATH] is the gravitational potential (note: Newton's gravitational constant [MATH] being dimensionful will be different in different dimensions, however, this does not affect the form of the equations).", '0712.1413-2-32-8': 'Using this we find that in [MATH] spatial dimensions the gravitational potential is given by [EQUATION] where [MATH] (potential has a logarithmic dependence on [MATH] in [MATH] dimensions).', '0712.1413-2-32-9': 'In the weak field limit the Schwarzchild solution is [MATH] and [MATH].', '0712.1413-2-32-10': 'Also, using geodesic equation we find that generically, in the weak-static field limit [MATH] and [MATH].', '0712.1413-2-32-11': 'Comparing the two expressions for [MATH] (or for [MATH]) one finds that [MATH] and using the expression for [MATH] as found above one gets [EQUATION]', '0712.1413-2-32-12': "This expression for [MATH] will be the same as that found above from matching if the constant in Einstein's equations is chosen to be [MATH].", '0712.1413-2-32-13': 'For [MATH] this reduces to the value [MATH] as used in 4-dimensional theory and which when used in ([REF]) results in the familiar Schwarzchild solution [EQUATION]', '0712.1413-2-33-0': '## The Self-Similar Solution', '0712.1413-2-34-0': 'To see the effect of dimensions on the nature of quantum particle flux (which will be described in a work subsequent to this), we would like to have a globally naked singularity.', '0712.1413-2-34-1': 'It is known that a locally naked self-similar solution is also globally naked [CITATION], where self-similar spacetimes are defined by the existence of a homothetic Killing vector field.', '0712.1413-2-34-2': 'Therefore here we look at the dependence on dimensions of the self-similar dust model.', '0712.1413-2-35-0': 'In a self-similar collapse any dimensionless quantity made from the metric functions, has to be a function only of [MATH].', '0712.1413-2-35-1': 'This can be seen by starting from the definition of a homothetic Killing vector field [MATH], [EQUATION] (We emphasize here that we are dealing only with self-similarity of the first kind, which is defined by the above equation).', '0712.1413-2-35-2': 'This condition implies that [EQUATION] where [MATH] is the Einstein tensor for the spacetime [CITATION], [CITATION].', '0712.1413-2-35-3': 'It follows then that the the energy momentum tensor should also satisfy the equation [EQUATION]', '0712.1413-2-35-4': 'For a perfect fluid the energy-momentum tensor is [MATH].', '0712.1413-2-35-5': 'The condition ([REF]) implies the following, [EQUATION]', '0712.1413-2-35-6': 'Based on the above equations, the existence of a homothetic vector implies that the metric components in a comoving coordinate system will be of the form such that the dimensionless quantities become functions of [MATH], [CITATION].', '0712.1413-2-35-7': 'Following [CITATION] we assume a general spherically symmetric ansatz of the form [EQUATION] and also we have [EQUATION]', '0712.1413-2-35-8': 'If we assume the vector [MATH] has only [MATH] and [MATH] components given by [MATH] and expand ([REF]) we get PDEs for the vector components.', '0712.1413-2-35-9': 'The condition of comoving metric implies the condition that [MATH] and [MATH].', '0712.1413-2-35-10': 'After redefining the independent variables to [MATH] and [MATH] we now redefine the dependent variables by [EQUATION]', '0712.1413-2-35-11': 'Under the above change of variables the equations ([REF]) become, [EQUATION]', '0712.1413-2-35-12': 'So this shows that we can go to a coordinate system in which the metric functions are functions of [MATH] .', '0712.1413-2-35-13': 'So [MATH] and [MATH] are functions of [MATH] and [MATH] is [MATH].', '0712.1413-2-36-0': 'It can now be shown that for spherically symmetric, self-similar dust collapse, the mass function is given by [MATH], where [MATH] is a constant.', '0712.1413-2-36-1': 'For this we start with the [MATH] component of Einstein equation [EQUATION]', '0712.1413-2-36-2': 'Defining the self-similarity parameter [MATH] and writing [MATH], where [MATH] and [MATH] being dimensionless are functions only of [MATH], the above equation can be written as [EQUATION] where we have used [MATH] and [MATH].', '0712.1413-2-36-3': 'The above equation is solved easily to obtain [EQUATION] where [MATH] is a constant of integration and equals one for the marginally bound case (and will therefore be ignored in what follows).', '0712.1413-2-36-4': 'Similarly the [MATH] component of the Einstein equation gives [EQUATION]', '0712.1413-2-36-5': 'Solving this we obtain [EQUATION] where [MATH] is a constant of integration.', '0712.1413-2-37-0': 'Now the mass function is given by [MATH] and using [MATH] this can be written as [EQUATION]', '0712.1413-2-37-1': 'Using ([REF]) in the above equation we obtain [MATH] which is the desired result.', '0712.1413-2-38-0': 'With this result ([REF]) becomes [EQUATION]', '0712.1413-2-38-1': 'Here [MATH] is a constant.', '0712.1413-2-38-2': 'This is because in self-similar collapse any dimensionless quantity has to be a function only of [MATH] whereas [MATH], being a function only of [MATH] (see ([REF])) has to be a constant.', '0712.1413-2-38-3': 'We are interested in finding the behavior of density [MATH] in the neighborhood of the centre [MATH].', '0712.1413-2-38-4': 'Using ([REF]) and [MATH] in the expression for density we find [EQUATION]', '0712.1413-2-38-5': 'For [MATH] we neglect the second order term in the above equation and obtain [EQUATION]', '0712.1413-2-38-6': 'Also for [MATH], [MATH], which implies [EQUATION]', '0712.1413-2-38-7': 'Substituting this in ([REF]) we get [EQUATION] where [MATH] and [MATH].', '0712.1413-2-38-8': 'This shows how the density profile should depend on the number of dimensions to obtain a self-similar solution.', '0712.1413-2-38-9': 'The above form for density profile implies that in 4-dimensions ([MATH]) and in 5-dimensions ([MATH]) the self-similar solution corresponds to an analytic density profile whereas in higher dimensions the density profile is no longer analytic.', '0712.1413-2-39-0': 'Here we also note that for [MATH], that is in [MATH] dimensions, [MATH] and is thus independent of [MATH] and therefore one requires that energy density [MATH] should be zero.', '0712.1413-2-39-1': 'Thus self-similarity in [MATH] dimensions is inconsistent with the presence of matter.', '0712.1413-2-40-0': '# Spherically symmetric inhomogeneous dust collapse in the presence of a negative cosmological constant', '0712.1413-2-41-0': '## Solution', '0712.1413-2-42-0': 'In the presence of a cosmological constant [MATH], Einstein equations are given by [EQUATION]', '0712.1413-2-42-1': 'For the case [MATH] we take [MATH] in which case the Einstein equations become [MATH], where now [MATH].', '0712.1413-2-42-2': 'The expressions for the components of Einstein tensor are still the same as in the [MATH] case.', '0712.1413-2-42-3': 'In particular since [MATH], therefore we again have [EQUATION]', '0712.1413-2-42-4': 'The solution of this equation is again given by ([REF]) and we again consider the marginally bound case so that [MATH].', '0712.1413-2-42-5': 'The 1-1 component of Einstein equations is [EQUATION]', '0712.1413-2-42-6': 'Integration of this equation gives [EQUATION] where as before [MATH] is the mass function.', '0712.1413-2-42-7': 'Integrating this equation after taking the negative sign for the square root (to account for in-falling matter) we get [EQUATION]', '0712.1413-2-42-8': 'Relabeling the [MATH] coordinate as in the previous case so that at [MATH], [MATH] we get [EQUATION]', '0712.1413-2-42-9': 'For [MATH] we again get [MATH] corresponding to the singularity formation for shell labeled [MATH].', '0712.1413-2-42-10': 'From [MATH] we obtain an expression for [MATH] which is again given by ([REF]).', '0712.1413-2-43-0': '## A simple derivation of the naked singularity', '0712.1413-2-44-0': 'As before we want to see if null geodesics can come out of the singularity.', '0712.1413-2-44-1': 'For this we proceed as before assuming that the density profile near the center is given by [EQUATION]', '0712.1413-2-44-2': 'From the form of the mass function [MATH] we have [EQUATION] where [MATH].', '0712.1413-2-44-3': 'From ([REF]) and ([REF]) we see that the singularity curve is given by [EQUATION]', '0712.1413-2-44-4': 'This implies that the central singularity at [MATH] forms at time [EQUATION]', '0712.1413-2-44-5': 'We again note that, as in the [MATH] case, when [MATH] is a constant all shells become singular at the same time as the central shell.', '0712.1413-2-45-0': 'We now rewrite the expression for the singularity curve as [EQUATION]', '0712.1413-2-45-1': 'It is reasonable to assume that for shells near [MATH] the time for singularity formation is close to the time for the central shell to become singular i.e. [MATH] and we can therefore write [MATH] where because of the assumption made [MATH].', '0712.1413-2-45-2': 'Using this we expand the left hand side of the above equation using the addition formula for sines and make use of [MATH] and [MATH] to get [EQUATION]', '0712.1413-2-45-3': 'Here [MATH] is the first non-zero term beyond [MATH] and is negative since we assume a decreasing density profile.', '0712.1413-2-45-4': 'Using [MATH] in the above equation we can finally write the expression for singularity curve for shells near the center as [EQUATION]', '0712.1413-2-45-5': 'To know whether the central singularity at [MATH], [MATH] is naked or not we focus attention on radial null geodesics and check if there are any outgoing radial null geodesics which terminate on the central singularity in the past.', '0712.1413-2-45-6': 'We proceed as in the earlier case, assuming that there exist such geodesics and take their form near [MATH] to be [EQUATION] where, comparing with ([REF]), we see that [MATH] and if [MATH] then [EQUATION] for the assumed geodesic to lie in the spacetime.', '0712.1413-2-45-7': 'We use ([REF]) and ([REF]) (retaining only the first two non-zero terms in the latter in the [MATH] approximation) in ([REF]) to get [EQUATION]', '0712.1413-2-45-8': 'Near [MATH], the time [MATH] appearing in the geodesic equation satisfies [MATH] and therefore the argument of the sine function in ([REF]) is close to zero and we use the approximation [MATH] obtaining [EQUATION]', '0712.1413-2-45-9': 'From the form of the metric we know that the radial null geodesics satisfy [MATH].', '0712.1413-2-45-10': 'We take the spatial derivative of the above equation, substitute for [MATH] from ([REF]) and equate the result to the derivative of ([REF]) [EQUATION]', '0712.1413-2-46-0': 'Consider first the case [MATH].', '0712.1413-2-46-1': 'Keeping terms only to lowest order in [MATH] we get [EQUATION]', '0712.1413-2-46-2': 'From this we have [EQUATION] where we have substituted for [MATH] in the argument of [MATH].', '0712.1413-2-46-3': 'Since the form of [MATH] is exactly the same as in the [MATH] case we find that the conditions for naked singularity formation are also the same as mentioned after ([REF]).', '0712.1413-2-46-4': 'That is, in [MATH]-dimensions [MATH], [MATH] implying [MATH] are the allowed values so that we get naked singularity for [MATH] or for [MATH], [MATH].', '0712.1413-2-46-5': 'Similarly in [MATH]-dimensions [MATH], [MATH] implying that only [MATH] is allowed so that we get naked singularity only for [MATH].', '0712.1413-2-46-6': 'In all higher dimensions we get naked singularity only if [MATH].', '0712.1413-2-47-0': 'Again [MATH], [MATH] and [MATH], [MATH] are critical cases satisfying [MATH].', '0712.1413-2-47-1': 'To analyze these we proceed as in the [MATH] case.', '0712.1413-2-47-2': 'For [MATH] ([REF]) becomes [EQUATION] which after substituting for [MATH] gives [EQUATION]', '0712.1413-2-47-3': 'Equating the power of [MATH] on the two sides gives [MATH] (as in the [MATH] case).', '0712.1413-2-47-4': 'Since [MATH] should be an integer greater than one we find that these conditions are satisfied only for [MATH]) and for [MATH]).', '0712.1413-2-48-0': 'Consider [MATH]; in this case the above equation can be written as [EQUATION] with the constraint that [EQUATION]', '0712.1413-2-48-1': 'If we define [MATH] and [MATH] the equation can be written in the simplified form [EQUATION] with the requirement that [MATH].', '0712.1413-2-48-2': 'If we further define [MATH] and [MATH] the equation becomes [EQUATION]', '0712.1413-2-48-3': 'For a naked singularity to form this equation for [MATH] should have a solution subject to the constraint [MATH] and [MATH].', '0712.1413-2-49-0': 'Using the conditions, as mentioned earlier, for the roots of a general cubic we can find the conditions on [MATH] for which the above cubic has at least one real root in the desired range.', '0712.1413-2-49-1': 'It is found that for [MATH] all the three roots are real and at least one of these satisfies [MATH].', '0712.1413-2-49-2': 'For [MATH] the real root is negative.', '0712.1413-2-49-3': 'The range of [MATH] found above implies that for [MATH] one gets a naked singularity.', '0712.1413-2-49-4': 'This shows that the critical case is also similar to the [MATH] case except that the allowed range for [MATH] has shifted.', '0712.1413-2-50-0': 'A similar analysis can be carried out for the case where [MATH] and [MATH].', '0712.1413-2-50-1': 'By defining [MATH] and [MATH] one gets a fourth order equation in [MATH].', '0712.1413-2-50-2': 'If one subsequently defines [MATH] and [MATH] one gets the equation [EQUATION] with the consistency conditions [MATH] and [MATH].', '0712.1413-2-50-3': 'It is found that the above conditions are satisfied for [MATH] or in terms of conditions on [MATH] we get [MATH].', '0712.1413-2-51-0': '## Formation of Trapped Surfaces', '0712.1413-2-52-0': 'As in the [MATH] case, we now look at the formation of trapped surfaces.', '0712.1413-2-52-1': 'Considering the expansion of outgoing radial null geodesics as in ([REF]) we find that [EQUATION]', '0712.1413-2-52-2': 'From this it is seen that the condition for trapping, [MATH], is met when [EQUATION] which for [MATH] (4-dimensions) reduces to the well known result [EQUATION]', '0712.1413-2-52-3': 'It is easy to see that, as in the [MATH] case, for the central shell, trapping coincides with singularity formation.', '0712.1413-2-53-0': '## Exterior Solution with a negative Cosmological constant', '0712.1413-2-54-0': 'As before we take the metric in the exterior to be [EQUATION]', '0712.1413-2-54-1': 'The components of Einstein tensor are the same as in ([REF])-([REF]).', '0712.1413-2-54-2': 'Solving the vacuum Einstein equations [MATH] we find [EQUATION]', '0712.1413-2-54-3': 'With this the metric in the exterior becomes [EQUATION]', '0712.1413-2-54-4': 'Here C is a constant of integration which is fixed by matching the exterior solution to the interior solution at the boundary in exactly the same way as for the [MATH] case and the result is [MATH] where [MATH] is the boundary of the dust cloud.', '0712.1413-2-54-5': 'Thus in 4-dimensions the exterior is [EQUATION]', '0712.1413-2-55-0': '## The absence of a self-similar solution in the Presence of a [MATH]', '0712.1413-2-56-0': 'It is interesting to note that it is not possible to have a self-similar solution in the presence of a cosmological constant.', '0712.1413-2-56-1': 'To see this we begin by noting the condition that dimensionless functions made from the metric are functions only of [MATH] continues to hold.', '0712.1413-2-56-2': 'This follows because the cosmological constant term in the Einstein equations can be absorbed into the energy momentum tensor in the right hand side, by taking the [MATH]-term as a perfect fluid with equation of state [MATH].', '0712.1413-2-56-3': 'Eqn. ([REF]) then continues to hold, with the understanding that the contribution of the cosmological constant is included in the energy-momentum tensor.', '0712.1413-2-56-4': 'The remaining argument, leading to the conclusion that [MATH] is a function of [MATH] then follows.', '0712.1413-2-57-0': 'Now if we have a self-similar solution then we can write [MATH] with [MATH] being dimensionless.', '0712.1413-2-57-1': 'If we define [MATH] then the condition of self-similarity implies that [MATH] being dimensionless should be a function only of [MATH].', '0712.1413-2-57-2': 'With this if we now consider the equation [MATH] we get [EQUATION]', '0712.1413-2-57-3': 'The explicit presence of [MATH] in the above equation implies that [MATH] cannot be expressed as a function of [MATH] alone and thus we do not have a self-similar solution in the presence of [MATH].', '0712.1413-2-58-0': 'The same conclusion also follows from Eqn. ([REF]).', '0712.1413-2-58-1': 'With dust matter, the only contribution to pressure is coming from the cosmological constant, and this pressure is constant.', '0712.1413-2-58-2': 'The Lie derivative on the left hand side is thus zero, whereas on the right hand side the presure is non-zero, leading to a contradiction and showing that such a Kiling vector field cannot exist.', '0712.1413-2-58-3': 'Physically speaking, the presence of a cosmological constant introduces a length scale which prevents self-similarity.', '0712.1413-2-59-0': '# Conclusions', '0712.1413-2-60-0': 'We have studied the collapse of inhomogeneous spherically symmetric dust distribution in arbitrary number of space dimensions both in the absence and in the presence of a cosmological constant.', '0712.1413-2-60-1': 'From the analysis presented we see that even though naked singularity is allowed in all dimensions there is more freedom on initial conditions for obtaining naked singularity in 2+1, 3+1 and 4+1 dimensions, both in the absence as well as the presence of a negative cosmological constant.', '0712.1413-2-60-2': 'We have also seen that the formation of trapped surfaces is similar in all dimensions with the central shell getting trapped at the same time when it becomes singular.', '0712.1413-2-60-3': 'For outer shells trapping occurs before those shells become singular.', '0712.1413-2-60-4': 'We also saw explicitly that in the absence of a cosmological constant, globally naked self-similar models can be constructed all dimensions, whereas in the presence of a cosmological constant such a solution cannot be constructed.', '0712.1413-2-61-0': 'In the second paper in this series, we will study quantum field theory on the curved background provided by the classical solutions presented here, including the emission of Hawking radiation from an [MATH]-dim AdS black hole.', '0712.1413-2-61-1': 'In a third paper we will carry out a canonical quantization of this model, and also address the issue of black hole entropy, following the methods of [CITATION].'}

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'0712.1413-2-20-0'], ['0712.1413-1-20-1', '0712.1413-2-20-1'], ['0712.1413-1-20-2', '0712.1413-2-20-2'], ['0712.1413-1-20-3', '0712.1413-2-20-3'], ['0712.1413-1-20-4', '0712.1413-2-20-4'], ['0712.1413-1-25-1', '0712.1413-2-25-1'], ['0712.1413-1-6-0', '0712.1413-2-6-0'], ['0712.1413-1-6-1', '0712.1413-2-6-1'], ['0712.1413-1-6-2', '0712.1413-2-6-2'], ['0712.1413-1-6-3', '0712.1413-2-6-3'], ['0712.1413-1-3-0', '0712.1413-2-3-0'], ['0712.1413-1-3-1', '0712.1413-2-3-1'], ['0712.1413-1-3-2', '0712.1413-2-3-2'], ['0712.1413-1-3-3', '0712.1413-2-3-3'], ['0712.1413-1-3-4', '0712.1413-2-3-4'], ['0712.1413-1-3-5', '0712.1413-2-3-5'], ['0712.1413-1-3-6', '0712.1413-2-3-6'], ['0712.1413-1-3-7', '0712.1413-2-3-7'], ['0712.1413-1-3-8', '0712.1413-2-3-8'], ['0712.1413-1-3-9', '0712.1413-2-3-9'], ['0712.1413-1-13-0', '0712.1413-2-13-0'], ['0712.1413-1-13-1', '0712.1413-2-13-1'], ['0712.1413-1-13-2', '0712.1413-2-13-2'], ['0712.1413-1-13-3', '0712.1413-2-13-3'], ['0712.1413-1-47-0', '0712.1413-2-49-0'], ['0712.1413-1-47-1', '0712.1413-2-49-1'], ['0712.1413-1-47-2', '0712.1413-2-49-2'], ['0712.1413-1-47-3', '0712.1413-2-49-3'], ['0712.1413-1-47-4', '0712.1413-2-49-4'], ['0712.1413-1-50-0', '0712.1413-2-52-0'], ['0712.1413-1-50-1', '0712.1413-2-52-1'], ['0712.1413-1-50-2', '0712.1413-2-52-2'], ['0712.1413-1-50-3', '0712.1413-2-52-3'], ['0712.1413-1-4-0', '0712.1413-2-4-0'], ['0712.1413-1-4-1', '0712.1413-2-4-1'], ['0712.1413-1-4-2', '0712.1413-2-4-2'], ['0712.1413-1-4-3', '0712.1413-2-4-3'], ['0712.1413-1-4-4', '0712.1413-2-4-4'], ['0712.1413-1-4-5', '0712.1413-2-4-5'], ['0712.1413-1-4-6', '0712.1413-2-4-6'], ['0712.1413-1-4-7', '0712.1413-2-4-7'], ['0712.1413-1-18-0', '0712.1413-2-18-0'], ['0712.1413-1-59-0', '0712.1413-2-61-0'], ['0712.1413-1-59-1', '0712.1413-2-61-1'], ['0712.1413-1-37-0', '0712.1413-2-39-0'], ['0712.1413-1-37-1', '0712.1413-2-39-1'], ['0712.1413-1-29-0', '0712.1413-2-29-0'], ['0712.1413-1-29-1', '0712.1413-2-29-1'], ['0712.1413-1-29-2', '0712.1413-2-29-2'], ['0712.1413-1-29-3', '0712.1413-2-29-3'], ['0712.1413-1-29-4', '0712.1413-2-29-4'], ['0712.1413-1-29-5', '0712.1413-2-29-5'], ['0712.1413-1-29-6', '0712.1413-2-29-6'], ['0712.1413-1-29-7', '0712.1413-2-29-7'], ['0712.1413-1-40-1', '0712.1413-2-42-1'], ['0712.1413-1-40-2', '0712.1413-2-42-2'], ['0712.1413-1-40-3', '0712.1413-2-42-3'], ['0712.1413-1-40-4', '0712.1413-2-42-4'], ['0712.1413-1-40-5', '0712.1413-2-42-5'], ['0712.1413-1-40-6', '0712.1413-2-42-6'], ['0712.1413-1-40-7', '0712.1413-2-42-7'], ['0712.1413-1-40-8', '0712.1413-2-42-8'], ['0712.1413-1-40-9', '0712.1413-2-42-9'], ['0712.1413-1-40-10', '0712.1413-2-42-10'], ['0712.1413-1-27-0', '0712.1413-2-27-0'], ['0712.1413-1-0-0', '0712.1413-2-0-0'], ['0712.1413-1-0-1', '0712.1413-2-0-1'], ['0712.1413-1-5-0', '0712.1413-2-5-0'], ['0712.1413-1-5-1', '0712.1413-2-5-1'], ['0712.1413-1-5-3', '0712.1413-2-5-4'], ['0712.1413-1-5-4', '0712.1413-2-5-5'], ['0712.1413-1-46-0', '0712.1413-2-48-0'], ['0712.1413-1-46-1', '0712.1413-2-48-1'], ['0712.1413-1-46-2', '0712.1413-2-48-2'], ['0712.1413-1-46-3', '0712.1413-2-48-3'], ['0712.1413-1-43-0', '0712.1413-2-45-0'], ['0712.1413-1-43-1', '0712.1413-2-45-1'], ['0712.1413-1-43-2', '0712.1413-2-45-2'], ['0712.1413-1-43-3', '0712.1413-2-45-3'], ['0712.1413-1-43-4', '0712.1413-2-45-4'], ['0712.1413-1-43-5', '0712.1413-2-45-5'], ['0712.1413-1-43-6', '0712.1413-2-45-6'], ['0712.1413-1-43-7', '0712.1413-2-45-7'], ['0712.1413-1-43-8', '0712.1413-2-45-8'], ['0712.1413-1-43-9', '0712.1413-2-45-9'], ['0712.1413-1-43-10', '0712.1413-2-45-10'], ['0712.1413-1-44-0', '0712.1413-2-46-0'], ['0712.1413-1-44-1', '0712.1413-2-46-1'], ['0712.1413-1-44-2', '0712.1413-2-46-2'], ['0712.1413-1-44-3', '0712.1413-2-46-3'], ['0712.1413-1-44-4', '0712.1413-2-46-4'], ['0712.1413-1-44-5', '0712.1413-2-46-5'], ['0712.1413-1-44-6', '0712.1413-2-46-6'], ['0712.1413-1-7-0', '0712.1413-2-7-0'], ['0712.1413-1-7-1', '0712.1413-2-7-1'], ['0712.1413-1-7-3', '0712.1413-2-7-3'], ['0712.1413-1-26-0', 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'0712.1413-2-26-0'], ['0712.1413-1-26-1', '0712.1413-2-26-1'], ['0712.1413-1-26-2', '0712.1413-2-26-2'], ['0712.1413-1-26-3', '0712.1413-2-26-3'], ['0712.1413-1-26-4', '0712.1413-2-26-4'], ['0712.1413-1-45-0', '0712.1413-2-47-0'], ['0712.1413-1-45-1', '0712.1413-2-47-1'], ['0712.1413-1-45-2', '0712.1413-2-47-2'], ['0712.1413-1-45-3', '0712.1413-2-47-3'], ['0712.1413-1-45-4', '0712.1413-2-47-4'], ['0712.1413-1-2-0', '0712.1413-2-2-0'], ['0712.1413-1-2-1', '0712.1413-2-2-1'], ['0712.1413-1-2-2', '0712.1413-2-2-2'], ['0712.1413-1-48-0', '0712.1413-2-50-0'], ['0712.1413-1-48-1', '0712.1413-2-50-1'], ['0712.1413-1-48-2', '0712.1413-2-50-2'], ['0712.1413-1-48-3', '0712.1413-2-50-3'], ['0712.1413-1-16-0', '0712.1413-2-16-0'], ['0712.1413-1-16-1', '0712.1413-2-16-1'], ['0712.1413-1-31-0', '0712.1413-2-31-0'], ['0712.1413-1-31-1', '0712.1413-2-31-1'], ['0712.1413-1-31-2', '0712.1413-2-31-2'], ['0712.1413-1-31-3', '0712.1413-2-31-3'], ['0712.1413-1-31-4', '0712.1413-2-31-4'], ['0712.1413-1-31-5', '0712.1413-2-31-5'], ['0712.1413-1-31-6', '0712.1413-2-31-6'], ['0712.1413-1-31-7', '0712.1413-2-31-7'], ['0712.1413-1-31-8', '0712.1413-2-31-8'], ['0712.1413-1-31-9', '0712.1413-2-31-9'], ['0712.1413-1-31-10', '0712.1413-2-31-10'], ['0712.1413-1-31-11', '0712.1413-2-31-11'], ['0712.1413-1-31-12', '0712.1413-2-31-12'], ['0712.1413-1-31-13', '0712.1413-2-31-13'], ['0712.1413-1-31-14', '0712.1413-2-31-14'], ['0712.1413-1-31-15', '0712.1413-2-31-15'], ['0712.1413-1-31-16', '0712.1413-2-31-16'], ['0712.1413-1-31-17', '0712.1413-2-31-17'], ['0712.1413-1-31-18', '0712.1413-2-31-18'], ['0712.1413-1-31-19', '0712.1413-2-31-19'], ['0712.1413-1-31-20', '0712.1413-2-31-20'], ['0712.1413-1-31-21', '0712.1413-2-31-21'], ['0712.1413-1-56-0', '0712.1413-2-56-0'], ['0712.1413-1-56-2', '0712.1413-2-57-1'], ['0712.1413-1-56-3', '0712.1413-2-57-2'], ['0712.1413-1-56-4', '0712.1413-2-57-3'], ['0712.1413-1-36-0', '0712.1413-2-34-0'], ['0712.1413-1-36-1', '0712.1413-2-34-1'], ['0712.1413-1-36-2', '0712.1413-2-34-2'], ['0712.1413-1-36-5', '0712.1413-2-38-1'], ['0712.1413-1-36-6', '0712.1413-2-38-2'], ['0712.1413-1-36-7', '0712.1413-2-38-3'], ['0712.1413-1-36-8', '0712.1413-2-38-4'], ['0712.1413-1-36-9', '0712.1413-2-38-5'], ['0712.1413-1-36-10', '0712.1413-2-38-6'], ['0712.1413-1-36-11', '0712.1413-2-38-7'], ['0712.1413-1-36-12', '0712.1413-2-38-8'], ['0712.1413-1-32-0', '0712.1413-2-32-0'], ['0712.1413-1-32-4', '0712.1413-2-32-4'], ['0712.1413-1-32-5', '0712.1413-2-32-5'], ['0712.1413-1-32-6', '0712.1413-2-32-6'], ['0712.1413-1-32-7', '0712.1413-2-32-7'], ['0712.1413-1-32-8', '0712.1413-2-32-8'], ['0712.1413-1-32-9', '0712.1413-2-32-9'], ['0712.1413-1-32-10', '0712.1413-2-32-10'], ['0712.1413-1-32-11', '0712.1413-2-32-11'], ['0712.1413-1-32-12', '0712.1413-2-32-12'], ['0712.1413-1-32-13', '0712.1413-2-32-13']]

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[]

[['0712.1413-1-58-4', '0712.1413-2-60-4'], ['0712.1413-1-58-5', '0712.1413-2-60-4'], ['0712.1413-1-0-2', '0712.1413-2-0-2'], ['0712.1413-1-7-2', '0712.1413-2-7-2'], ['0712.1413-1-36-13', '0712.1413-2-38-9'], ['0712.1413-1-32-1', '0712.1413-2-32-1']]

[]

['0712.1413-1-21-0', '0712.1413-1-22-0', '0712.1413-1-22-1', '0712.1413-1-23-0', '0712.1413-1-23-1', '0712.1413-1-24-0', '0712.1413-1-24-1', '0712.1413-1-25-0', '0712.1413-1-32-2', '0712.1413-1-32-3', '0712.1413-2-21-0', '0712.1413-2-22-0', '0712.1413-2-22-1', '0712.1413-2-23-0', '0712.1413-2-23-1', '0712.1413-2-24-0', '0712.1413-2-24-1', '0712.1413-2-25-0', '0712.1413-2-32-2', '0712.1413-2-32-3']

{'1': 'http://arxiv.org/licenses/assumed-1991-2003/', '2': 'http://arxiv.org/licenses/assumed-1991-2003/'}

https://arxiv.org/abs/0712.1413

1401.0829

{'1401.0829-1-0-0': 'The Smoluchowski coagulation-diffusion PDE is a system of partial differential equations modelling the evolution in time of mass-bearing Brownian particles which are subject to short-range pairwise coagulation.', '1401.0829-1-0-1': 'This survey presents a fairly detailed exposition of the kinetic limit derivation of the Smoluchowski PDE from a microscopic model of many coagulating Brownian particles that was undertaken in [CITATION].', '1401.0829-1-0-2': 'It presents heuristic explanations of the form of the main theorem before discussing the proof, and presents key estimates in that proof using a novel probabilistic technique.', '1401.0829-1-0-3': "The survey's principal aim is an exposition of this kinetic limit derivation, but it also contains an overview of several topics which either motivate or are motivated by this derivation.", '1401.0829-1-1-0': '# Introduction', '1401.0829-1-2-0': '## Microscopic particles and macroscopic descriptions', '1401.0829-1-3-0': 'An important aim in statistical mechanics is to explain how the huge amount of information available in a microscopic description of a physical object, such as the positions and momenta of all the molecules comprising the air in a room, may be accurately summarised by first specifying a small number of physical parameters which are functions of macroscopic location, such as the density, temperature and pressure of this body of air at different points in the room, and then determining how these parameters evolve in space and time.', '1401.0829-1-4-0': '### The elastic billiards model and the heat equation', '1401.0829-1-5-0': 'The microscopic system may begin out of equilbrium: for example, a still body of warm air in one room may be separated by a partition from another still body of cooler air in another, and then the partition instantaneously removed, so that air molecules from one side and the other intermingle over time, and an equilibrium is eventually approached in which the body of air in the whole room is again close to still, at a temperature which is some average of those of the two isolated systems at the original time.', '1401.0829-1-5-1': 'In such a case as this, it is a natural task is to seek to summarise the evolution of a few suitable macroscopic physical quantities as the solution of partial differential equations.', '1401.0829-1-5-2': 'In the example, our object of study might be the temperature of the gas, and our aim to show that it is the heat equation, [MATH], which models the macroscopic evolution [MATH] (with [MATH] varying over the whole room, [MATH], say) of the temperature from the moment of the removal of the partition at time [MATH] until a late time, [MATH], at which a new equilibrium is approached.', '1401.0829-1-5-3': 'In an idealized and very classical choice of microscopic description of the gas, we might model the ensemble of air molecules as a system of tiny spheres of equal radius and mass, each moving according to some velocity, and each pair of which undergoes a perfectly elastic collision on contact, in the same manner that a pair of billiards would.', '1401.0829-1-5-4': 'On each of the walls that comprise the boundary [MATH] of the room, each sphere bounces elastically.', '1401.0829-1-5-5': 'The partition is modelled by the immobile sheet [MATH] on which spheres on either side also bounce elastically before time zero; the partition is removed instantaneously at that time.', '1401.0829-1-5-6': 'The initial instant of time may be taken to be zero, or some negative time.', '1401.0829-1-5-7': "At that moment, we may scatter the spheres in an independent Poissonian manner throughout the room [MATH] (the reader may notice that in fact some extra rule is needed here to ensure the spheres' disjointness); and on one and other side of the partition, choose their velocities independently, those to the right of partition according to a non-degenerate law of zero mean, and those on the left according to another such law of lower variance than the first; in this way, we model two bodies of still air, a warm one in the right chamber [MATH], and a cooler one in the left [MATH].", '1401.0829-1-5-8': 'In the microscopic model, there are huge numbers of tiny spheres in the system.', '1401.0829-1-5-9': 'Indeed, we may seek to understand the macroscopic evolution of temperature by in fact considering a whole sequence of microscopic models indexed by total particle number [MATH], in a limit of high [MATH].', '1401.0829-1-5-10': 'In the [MATH]-th model, spheres are initially scattered as we described, with a Poissonian intensity [MATH] thoughout [MATH].', '1401.0829-1-5-11': 'To carry out this task of understanding the large-scale evolution, we would wish to specify a microscopic definition of the notion of temperature, and then explain how it is in the high [MATH] limit that the microscopic temperature data may be meaningfully reduced to a macroscopic description, and that this latter description indeed evolves according to the heat equation.', '1401.0829-1-5-12': 'Microscopically, temperature is interpreted [CITATION] as the average kinetic energy of particles, where here the velocity of particles is measured relative to the average velocity of nearby particles.', '1401.0829-1-5-13': 'Since our particle systems are large microscopically, considering as we do a high [MATH] limit, we may specify in our [MATH]-th microscopic model a definition of temperature at any given location [MATH] as follows: first we may compute the mean velocity [MATH] of the set of spheres whose centres lie within some small distance [MATH] of a given location [MATH] in the room, and then we are able to define the microscopic temperature [MATH] to be the average of the square of the particle velocity minus [MATH], where the average is taken over the same set of spheres.', '1401.0829-1-5-14': 'Of course, the value [MATH] will change in time.', '1401.0829-1-5-15': 'As [MATH] approaches infinity with [MATH] being fixed but small, huge numbers of particles are involved in the empirical counts used for averaging.', '1401.0829-1-5-16': 'Our aim is to consider the space-time evolution of the microscopically specified temperatures after the high [MATH] high is taken, at which point, the weak of large numbers might suggest that the concerned empirical counts behave non-randomly to first order, so that our description becomes deterministic: the limit [MATH] will be some non-random function [MATH].', '1401.0829-1-5-17': 'In fact, since [MATH] is fixed, we should not yet expect our system to approximate the heat equation, since there is an effect of macroscopic smearing in our calculation of microscopic temperature.', '1401.0829-1-5-18': 'Rather, one might expect the heat equation description to emerge if we take a [MATH] limit of [MATH], after the first high [MATH] limit has been taken.', '1401.0829-1-5-19': 'Moreover, to hope to obtain this description, we will also need to scale time appropriately in the [MATH]-th microscopic model, as we take the first, high [MATH], limit.', '1401.0829-1-5-20': 'In the scaled time coordinates, the microscopic models should make their approach to the new thermal equilibrium at the same rate, as [MATH].', '1401.0829-1-5-21': 'What rate this is in fact depends on another important consideration concerning the microscopic models which our brief description left unspecified: the radius [MATH] of each sphere in the [MATH]-th model must certainly be chosen to satisfy [MATH] for some constant [MATH], if only to permit all of the spheres to inhabit the room disjointly; our choice of decay rate for [MATH] as a function of [MATH], subject to this constraint, will determine the factor by which we scale time in the [MATH]-th model in order to seek a heat equation description in the large.', '1401.0829-1-6-0': 'To implement the programme proposed in the preceding paragraph is an open problem, and in all likelihood, an extremely difficult one.', '1401.0829-1-6-1': 'There is no randomness in the model except in the initial selection of particle locations and velocities: from that time on, the deterministic laws of Newtonian mechanics govern the evolution of the microscopic models.', '1401.0829-1-6-2': 'Moreover, some choices for density admissible in the above description - such as when [MATH] converges in high [MATH], to a suitably small constant - lead to rather dense systems of particles.', '1401.0829-1-6-3': 'The derivation may be less ordinately hard were more dilute choices of limit considered, where [MATH] converges to zero more, and perhaps much more, quickly than does [MATH].', '1401.0829-1-7-0': 'It is important to note, however, that, if a choice of [MATH] as a function of [MATH] is made which is too rapidly decaying, we may leave the realm in which the heat equation is the appropriate macroscopic description.', '1401.0829-1-7-1': 'For example, if [MATH], it is a simple matter to check that a typical sphere after time zero will cross traverse the entire room on many occasions before meeting any other particle.', '1401.0829-1-7-2': 'The system will reach equilibrium after the removal of the partition simply by the free motion of the particles.', '1401.0829-1-7-3': 'The heat equation is only a suitable description when a typical particle experiences the thermal agitation caused by its collision with many other particles in short periods of macroscopic time.', '1401.0829-1-8-0': "### The elastic billiards model and Boltzmann's equation", '1401.0829-1-9-0': 'Moreover, the elastic billiards model crosses at least one interesting regime as it is diluted from the dense [MATH] phase towards the trivial free motion phase [MATH].', '1401.0829-1-9-1': 'Consider the choice [MATH].', '1401.0829-1-9-2': "A moment's thought shows that, in this regime, a typical sphere will travel (at unit-order velocity) for a duration before its first collision with another particle which on average neither tends to zero nor to infinity as [MATH].", '1401.0829-1-9-3': 'This is the regime of constant mean free path.', '1401.0829-1-9-4': 'The heat equation will not offer a suitable description for the evolution of temperature in this regime, because the mechanism providing for thermal agitation of particles occurs on a time scale which is marginally too slow.', '1401.0829-1-9-5': 'However, the programme of deriving a macroscopic description by means of a PDE does make sense, and in this case, offers a powerful model of gas dynamics.', '1401.0829-1-9-6': 'Suppose that, instead of using the microscopic data to form a description of temperature, we use it to describe the density of particles having a given velocity [MATH] nearby a given location [MATH].', '1401.0829-1-9-7': 'Particles may be scattered in a Poissonian fashion as before at the initial time, but with inhomogeneities in the intensity of this scattering permitted in both the space and velocity variables.', '1401.0829-1-9-8': 'With the macroscopic smearing parameter now being used to approximate velocity [MATH] as well as location [MATH], we may record a microscopic description [MATH] for the [MATH]-smeared density of spheres at space-velocity location [MATH].', '1401.0829-1-9-9': "Taking a high [MATH] and then low [MATH] limit as above, our macroscopic evolution is modelled by the fundamental system of equations in gas dynamics, Boltzmann's equation, valid for [MATH], [MATH] and [MATH]: [EQUATION]", '1401.0829-1-9-10': "Here, [MATH] is the free motion operator associated to particles of velocity [MATH], while [MATH] is a binary collision operator that reflects the microscopic elastic collision and whose form we will specify when we return to Boltzmann's equation in a brief discussion in Section [REF].", '1401.0829-1-9-11': 'For now, note that the time evolution of the macroscopic densities is governed both by the free motion and by the collision operator.', '1401.0829-1-9-12': 'This is what is to be expected in the regime of constant mean free path, where the typical particle experiences unit-order durations free of collision and other such periods where several collisions occur.', '1401.0829-1-10-0': 'Boltzmann carried out a derivation ([REF]) as a model of gas dynamics in 1872, based on several assumptions, including one of molecule chaos that he called the Stosszahlansatz and which we will later discuss.', '1401.0829-1-10-1': '(See [CITATION] for an English translation of his 1872 article.)', '1401.0829-1-10-2': "The validity of his derivation was a matter of controversy, not least due to Loschmidt's paradox concerning precollisional particle independence (see Subsection [REF]), and it was a fundamental advance made in 1975 by Lanford [CITATION] when the programme of rigorously deriving Boltzmann's equation from the elastic billiards model in the regime of mean free path was successfully implemented, for a short initial duration of time.", '1401.0829-1-10-3': 'By the latter condition, we mean that the validity of the description was established for some non-zero finite period, whose value depends on the form of the initial density profile of particles in space-velocity.', '1401.0829-1-11-0': "Lanford derived Boltzmann's equation by establishing that the correlation functions concerning several particles in the model satisfy a hierarchy of equations called the BBGKY hierarchy, where the index of an equation in the hierarchy is the number of particles whose correlation is being considered, and by showing that when the correlation functions adhere to the BBGKY hierarchy, the density profile follows Boltzmann's equation.", '1401.0829-1-11-1': "Illner and Pulvirenti implemented this approach in [CITATION] in order to derive Boltzmann's equation in a similar sense, but now globally in time, although with a comparable smallness condition, now on sparseness of the initial particle distribution; the cited derivation concerns a two dimensional gas, but this restriction on dimension was later lifted by the same authors.", '1401.0829-1-12-0': '### Our main goal: coagulating Brownian particles and the Smoluchowski PDE', '1401.0829-1-13-0': 'This survey is intended to offer a detailed overview of a programme for deriving the macroscopic description of a gas of particles in the same vein as the descriptions above propose.', '1401.0829-1-13-1': 'However, our microscopic particles will diffuse, each following a Brownian trajectory, and as such their evolution is random, not deterministic; the mechanism of interaction will be pairwise as above, but a coagulation in which only one particle survives rather than a collision in which both do.', '1401.0829-1-13-2': "On the other hand, in an effort to provide some generality in the microscopic description and richness in the macroscopic one, each of the particles will bear a mass, which the pairwise coagulation will conserve; and, moreover, we will permit the diffusivity of the Brownian trajectory of each particle to depend on the particle's mass.", '1401.0829-1-14-0': "The partial differential equation which the programme seeks to obtain in this case - the analogue of the heat equation or Boltzmann's equation in our opening examples - is, like Boltzmann's equation, in fact a system of PDE, in our case coupled in the mass parameter, known as the Smoluchowski coagulation-diffusion PDE.", '1401.0829-1-14-1': 'The choice made for diluteness in the high particle number limit will be that of the regime of constant mean free path.', '1401.0829-1-14-2': 'The programme of deriving the PDE in the case of constant mean free path is sometimes called a kinetic limit derivation.', '1401.0829-1-15-0': 'In the special case of mass-independent diffusion rates, the kinetic limit derivation was carried out in 1980 by Lang and Nguyen [CITATION], who followed the method of showing that the correlation functions between several particles are described by the BBGKY hierarchy which Lanford had introduced.', '1401.0829-1-16-0': "Introduced to the problem of generalizing Lang and Nguyen's derivation of the Smoluchowski PDE by James Norris, the author collaborated on it with Fraydoun Rezakhanlou.", '1401.0829-1-16-1': 'The principal aim of these notes is to give an informal but fairly detailed exposition of the kinetic limit derivation of the Smoluchowski PDE that was undertaken for dimension [MATH] in [CITATION].', '1401.0829-1-16-2': 'The treatment also first presents heuristic arguments with the aim that the reader may understand why the main theorem should be true before beginning a presentation of the proof of the theorem, and it also uses some novel probabilistic techniques to obtain key estimates used in the proof.', '1401.0829-1-16-3': 'The survey also touches on some related topics.', '1401.0829-1-17-0': '## The Smoluchowski coagulation-diffusion PDE', '1401.0829-1-18-0': 'We begin by recording the form of these equations and offering a brief explanation of the phenomenon that they may be expected to describe.', '1401.0829-1-19-0': 'Let the dimension [MATH] be given.', '1401.0829-1-19-1': 'A collection of functions [MATH], [MATH], is a strong solution of the discrete Smoluchowski coagulation-diffusion PDE with initial data [MATH], [MATH], if, for each [MATH] and [MATH], [MATH]; and, for each [MATH] and [MATH], [EQUATION] where the Laplacian acts on the spatial variable [MATH].', '1401.0829-1-19-2': 'The final two terms are interaction terms, a gain term given by [EQUATION] and a loss term by [EQUATION] (When [MATH], the partial time derivative on the left-hand side in ([REF]) is interpreted as a right derivative.)', '1401.0829-1-20-0': 'Note that the equations have two sets of parameters: the diffusion rates [MATH] and the coagulation propensities [MATH].', '1401.0829-1-20-1': 'The equations have a continuous counterpart, where the mass variable is now a positive real, and the above sums are replaced in an evident way by integrals, which we will not consider in this survey except in passing.', '1401.0829-1-21-0': 'To interpret the solution, consider a large number of minute particles in space [MATH], each carrying an integer mass.', '1401.0829-1-21-1': 'In a similar manner to our opening discussion, the quantity [MATH] is interpreted as the density of particles of mass [MATH] in the immediate vicinity of location [MATH] at time [MATH].', '1401.0829-1-21-2': 'The form of the right-hand side ([REF]) reflects the two dynamics for the particles: diffusive transport and binary coagulation.', '1401.0829-1-21-3': "Particles of mass [MATH] diffuse at rate [MATH], so that such a particle's displacement is given by [MATH], [MATH], where [MATH] is a standard Brownian motion.", '1401.0829-1-21-4': '(The factor of two appears because the infinitesimal generator of standard Brownian motion is a one-half multiple of the Laplacian; when we call [MATH] the diffusion rate, this is thus strictly speaking a misnomer.)', '1401.0829-1-21-5': "When a pair of particles are microscopically close, they may collide, disappearing from the model, to be replaced by a newcomer, whose mass is the sum of the two exiting particles'.", '1401.0829-1-21-6': 'The coagulation gain term ([REF]) expresses the possible means by which a new particle of mass [MATH] may appear in the immediate vicinity of location [MATH] at time [MATH]: by the coagulation of some pair of particles of masses [MATH], or [MATH] ... or [MATH].', '1401.0829-1-21-7': 'The product form [MATH] in the interaction term reflects an assumption that the particles in the immediate vicinity of [MATH] are well mixed, and the coefficient [MATH] models the tendency of particles at close range of pair-type [MATH] to coagulate in the immediate future.', '1401.0829-1-21-8': 'In the loss term ([REF]), we see the means by which the density [MATH] may fall due to coagulation: a particle of mass [MATH] may drop out of the count for this density due to coagulation with another particle, and that other particle may have any mass [MATH].', '1401.0829-1-22-0': 'Our aim in this survey to explain how the system ([REF]) may be derived in a kinetic limit from a collection of microscopic random models of diffusing mass-bearing particles that are liable to coagulate in pairs at close range.', '1401.0829-1-22-1': 'We now describe in precise terms the elements for this programme; for the case at hand, we are thus presenting an instance of the type of programme which we hazily sketched in our two opening examples.', '1401.0829-1-22-2': 'First, we specify the sequence of microscopic models: their initial particle distributions; and their dynamics: the free motion of individual particles, and the mechanism of pairwise coagulation at close range.', '1401.0829-1-22-3': 'In the main body of the article, we discuss only the derivation made in dimension [MATH], which was undertaken in [CITATION].', '1401.0829-1-22-4': 'Thus [MATH] may be assumed, except on one occasion when we make a short comment about the case when [MATH].', '1401.0829-1-23-0': '## The microscopic models', '1401.0829-1-24-0': 'The sequence of microscopic random models will be indexed by the total number [MATH] of particles intially present, at time zero.', '1401.0829-1-24-1': 'The [MATH]-indexed model will be specified by a probability measure [MATH].', '1401.0829-1-24-2': 'It is a measure not only on initial particle locations and masses but also on particle dynamics throughout [MATH].', '1401.0829-1-25-0': 'Initial particle distribution under [MATH].', '1401.0829-1-25-1': 'The quantity [MATH] may be interpreted as the density of particles of mass [MATH] in a tiny neighbourhood of [MATH] at time zero.', '1401.0829-1-25-2': 'Thus, [MATH] is interpreted as being proportional to the total number of particles of mass [MATH] and the constant [MATH], which we define by [MATH], as being proportional to the total number of initial particles.', '1401.0829-1-26-0': 'We will index the time-zero particle set under [MATH] by [MATH]; the initial mass and location of particle [MATH] will be denoted by [MATH].', '1401.0829-1-26-1': 'Reflecting the above density interpretation, we choose [MATH] independently, so that [MATH] has density [MATH] at [MATH].', '1401.0829-1-27-0': 'Notation for particle trajectories under [MATH].', '1401.0829-1-27-1': 'We wish to describe the subsequent evolution of each of the initial particles under [MATH].', '1401.0829-1-27-2': 'The trajectory of the [MATH]-th particle will be described by [MATH], where here [MATH] is an element called a cemetery state whose role, which we will shortly describe in precise terms, is to house particles that have disappeared from the model due to being on the wrong side of a pairwise collision.', '1401.0829-1-28-0': 'As such, at any given time [MATH], the particle configuration under [MATH] is described by a map [MATH], where [MATH] maps to [MATH] (or to [MATH]).', '1401.0829-1-29-0': 'To define the Markov process [MATH] precisely, we will specify its Markov generator, which acts on test functions [MATH].', '1401.0829-1-29-1': 'The action will be comprised of two parts: free motion of individual particles, and pairwise collision.', '1401.0829-1-29-2': 'We discuss our choice of each of these in words before providing the definition of the Markov generator.', '1401.0829-1-30-0': 'Free motion.', '1401.0829-1-30-1': 'A particle of mass [MATH] follows, independently of other particles, the trajectory [MATH] relative to its starting point, where [MATH] is a standard Brownian motion on [MATH].', '1401.0829-1-31-0': 'Pairwise collision.', '1401.0829-1-31-1': 'Any two particles will be liable to collide when their locations differ by order [MATH].', '1401.0829-1-31-2': 'Here, [MATH], the interaction range, is determined by [MATH] in a manner that we explain shortly.', '1401.0829-1-31-3': 'We introduce a compactly supported smooth interaction kernel [MATH] and a collection of microscopic interaction strengths [MATH], and declare that, at time [MATH], particles [MATH] and [MATH] collide at infinitesimal rate [MATH], where we adopt the convention that [MATH].', '1401.0829-1-31-4': 'The argument [MATH] for [MATH] indeed entails that collision may occur only between particles whose locations differ by order [MATH]; the prefactor of [MATH] is introduced because, in dimension [MATH], once a pair of particles have approached to distance of order [MATH], they are liable to remain at such a displacement for order [MATH] of time, since their relative displacement evolves as a Brownian motion of rate [MATH]; thus, the role of this prefactor is to ensure that the proportion of instances of particle pairs approaching into the interaction range that result in collision is of unit order, uniformly in [MATH].', '1401.0829-1-31-5': 'The role of the factor [MATH] is to control whether this proportion is close to one for a given particle mass pair (which would be ensured by choosing the value of [MATH] in question to be high) or closer to zero.', '1401.0829-1-32-0': 'The precise mechanism of collision.', '1401.0829-1-32-1': 'On collision of [MATH] and [MATH] at time [MATH], each of the pair of particles disappears, to be replaced by a new particle of mass [MATH] in the vicinity.', '1401.0829-1-32-2': 'As a matter of convenience for the ensuing proofs, the precise rule we pick for the appearance of the new particle is to choose its location to be [MATH] or [MATH], with probabilities [MATH] and [MATH].', '1401.0829-1-32-3': 'This rule permits the interpretation that, when two particles collide, one survives the collision and the other perishes; the probability of survival is proportional to incoming particle mass; the particle surviving collects the mass of the perishing particle, and the perishing particle vanishes from space.', '1401.0829-1-33-0': "In a formal device, the perishing particle's location and mass are each sent to the cemetery state [MATH], where they remain forever.", '1401.0829-1-33-1': "As such, for each [MATH], the [MATH]-th particle's trajectory is described by setting the vanishing time [MATH] equal to the first time at which particle [MATH] experiences a collision in which it perishes.", '1401.0829-1-33-2': 'The trajectory is then given by [MATH] on [MATH] and [MATH] on [MATH].', '1401.0829-1-34-0': 'The Markov generator of the dynamics.', '1401.0829-1-34-1': 'For any configuration [MATH], write [MATH], the surviving particle set, for those [MATH] such that [MATH] lies in [MATH] (rather than equalling [MATH]).', '1401.0829-1-34-2': 'Let [MATH] be smooth (in each hyperplane given by specifying the [MATH]-valued coordinates of the argument of [MATH]).', '1401.0829-1-34-3': 'Then the Markov generator [MATH] for [MATH] is given as follows.', '1401.0829-1-34-4': 'For each [MATH], [MATH], with the free-motion operator being given by [EQUATION] where [MATH] is the [MATH]-dimensional Laplacian acting on [MATH] viewed as a function of [MATH]; and, recalling that [MATH], with the collision operator being given by [EQUATION].', '1401.0829-1-34-5': 'Here, [MATH], the configuration adopted in the event that particle [MATH] survives collision with particle [MATH], is given by [EQUATION] while [MATH] is given by the same formula with the roles of [MATH] and [MATH] being reversed.', '1401.0829-1-35-0': '## The regime of bounded mean free path and the choice of interaction range', '1401.0829-1-36-0': 'It remains to specify how the interaction range [MATH] is determined by total initial particle number [MATH].', '1401.0829-1-36-1': 'This choice is made to be in the regime of constant mean free path: for dimension [MATH], [MATH] will be chosen to satisfy [EQUATION]', '1401.0829-1-36-2': 'To explain why this regime for the length of the free path is suitable, note that, since diffusion and coagulation terms are each present in the Smoluchowski PDE ([REF]), we expect that the evolution of a typical particle will be determined both by its free motion and its collision with other particles.', '1401.0829-1-36-3': 'It will neither diffuse without collision nor collide repeatedly before diffusing a macroscopic distance.', '1401.0829-1-37-0': 'The consideration that this regime be adopted forces the choice of scaling of [MATH] as a function of [MATH]: picking a uniformly random particle index [MATH] at the outset, [MATH] should be chosen so that the mean time to first collision of particle [MATH] converges as [MATH] to some strictly positive and finite constant.', '1401.0829-1-38-0': 'A heuristic argument explains why ([REF]) produces this outcome.', '1401.0829-1-38-1': 'We anticipate that, at any given time [MATH], a positive (although [MATH]-dependent) proportion of particles are surviving (rather than in the cemetery state).', '1401.0829-1-38-2': 'Assume that the surviving particles at time [MATH] are distributed so that the location and mass of each is chosen independently; the law of the location-mass statistic [MATH] of any given particle is equal to [MATH] (normalized to make the integral of this density equal to one).', '1401.0829-1-38-3': 'In other words, we are assuming a very strong sense that the density profile of particles under [MATH] mimics the solution of ([REF]).', '1401.0829-1-39-0': 'Pick a particle uniformly at random at the initial time and call the selected particle the tracer particle.', '1401.0829-1-39-1': 'We would like to estimate the mean number of collisions suffered by the tracer particle during [MATH] in terms of [MATH] and [MATH].', '1401.0829-1-39-2': 'As we briefly discussed in the paragraph under the heading "pairwise collision" in the preceding section, this quantity is expected to have the same order as the number of other particles which enter the [MATH]-neighbourhood of the given particle during [MATH].', '1401.0829-1-39-3': 'At any given time, our assumption on the distribution of other particles means that the probability that there is some other particle at distance less than [MATH] from the tracer particle is of order [MATH].', '1401.0829-1-39-4': 'Thus, the mean total amount of time during [MATH] that some other particle is at distance less than [MATH] from the tracer particle is also of order [MATH].', '1401.0829-1-39-5': 'Whenever another particle approaches the tracer particle to distance [MATH], it remains at the order of that distance for time of order [MATH] (since [MATH]).', '1401.0829-1-39-6': 'Thus, the mean number of different particles which during [MATH] approach to within distance [MATH] the tracer particle is of order [MATH].', '1401.0829-1-40-0': 'We thus see that imposing the relation ([REF]) may indeed be expected to ensure that the mean number of collisions suffered by the tracer particle in unit time is bounded away from zero and infinity uniformly in [MATH].', '1401.0829-1-41-0': '## The recipe for the macroscopic coagulation propensities', '1401.0829-1-42-0': 'The macroscopic coagulation propensities [MATH] that appear in the limiting system ([REF]) depend in a non-trivial fashion on the microscopic parameters [MATH], [MATH] and [MATH].', '1401.0829-1-42-1': 'Here is the recipe for obtaining [MATH] from these ingredients.', '1401.0829-1-42-2': 'As we will later explain, there exists a unique solution [MATH] of the equation [EQUATION] that satisfies [MATH] as [MATH].', '1401.0829-1-42-3': 'In fact, [MATH] for all [MATH], and [MATH] as [MATH].', '1401.0829-1-42-4': '(Here, as we will later, we write [MATH] for the Euclidean norm on [MATH].)', '1401.0829-1-43-0': 'The quantities [MATH] in ([REF]) are then specified by the formula [EQUATION]', '1401.0829-1-43-1': 'We mention that the minus sign appearing on the left-hand side of ([REF]) was not used in the original treatment in [CITATION].', '1401.0829-1-43-2': 'A positive choice for [MATH] permits an attractive probabilistic interpretation as we shall shortly discuss.', '1401.0829-1-44-0': '## The weak formulation of the Smoluchowski PDE', '1401.0829-1-45-0': 'We now recast the Smoluchowski PDE ([REF]) in weak form, since it is to this form of the equations that we will prove convergence.', '1401.0829-1-45-1': 'To do so, let [MATH] be the space of sequences [MATH] of smooth compactly supported functions [MATH].', '1401.0829-1-45-2': 'Then we say that [MATH], with [MATH] measurable for each [MATH], is a weak solution of ([REF]) if, for each [MATH], it satisfies the formula obtained from ([REF]) by multiplication by [MATH], integration in space-time, and integration by parts.', '1401.0829-1-45-3': 'Namely, such an [MATH] solves ([REF]) weakly if, for each [MATH] and [MATH], [EQUATION]', '1401.0829-1-46-0': '## Empirical densities', '1401.0829-1-47-0': 'In our opening discussion of the programme for deriving macroscopic limiting PDE, we suggested the use of [MATH]-macroscopically smeared particle counts as candidates to approximate the limiting evolution.', '1401.0829-1-47-1': 'Such counts play an important role in our derivation, and we will introduce them under the name microscopic candidate densities when we give an overview of the derivation of our main theorem, Theorem [REF], in Section [REF].', '1401.0829-1-48-0': 'However, to state this theorem, we will not use them.', '1401.0829-1-48-1': 'Rather, we will use a close cousin, empirical density measures defined under the microscopic models [MATH].', '1401.0829-1-48-2': 'We now define these.', '1401.0829-1-49-0': 'Under the law [MATH], let [MATH] denote the [MATH]-random variable, valued in measures on space-mass-time [MATH] such that, for each [MATH], its time-[MATH] marginal [MATH] is given by [EQUATION].', '1401.0829-1-50-0': 'For given [MATH], let [MATH] denote the [MATH]-random variable, valued in measures on space-time [MATH] such that, for each [MATH], its time-[MATH] marginal [MATH] is given by [EQUATION].', '1401.0829-1-51-0': 'Let [MATH] denote the space of measures [MATH] on [MATH] such that [MATH], and note that [MATH] is [MATH]-a.s. valued in [MATH].', '1401.0829-1-51-1': 'Equip [MATH] with the topology of vague convergence, under which this space is compact and metrizable.', '1401.0829-1-52-0': '## Hypotheses on microscopic parameters', '1401.0829-1-53-0': 'Our microscopic parameters are [MATH], [MATH] and [MATH].', '1401.0829-1-53-1': 'In the original paper [CITATION] and in the detailed overview of proof that we give in this survey, some hypotheses on these parameters must be imposed to enable the derivation to be made.', '1401.0829-1-53-2': 'We make some comments about the hypotheses made in [CITATION] and then specify and discuss those we make here.', '1401.0829-1-53-3': 'The two sets of assumptions will be called the original and the survey assumptions throughout.', '1401.0829-1-54-0': '### Original assumptions', '1401.0829-1-55-0': 'On the diffusion rate and the microscopic interaction strengths.', '1401.0829-1-55-1': 'Suppose that there exists a function [MATH] such that [MATH] for all [MATH], with [MATH] satisfying [EQUATION]', '1401.0829-1-55-2': 'On the initial condition.', '1401.0829-1-55-3': 'A slightly technical assumption is needed, of the membership in local [MATH] space of some sums over [MATH] of certain averages of [MATH]; the sums are weighted by certain powers of [MATH] and the above function [MATH].', '1401.0829-1-55-4': 'The reader may consult [CITATION] for the precise form.', '1401.0829-1-55-5': 'Although the assumption is a little technical to state, it is fairly weak: it is certainly satisfied if [MATH] is non-zero for only finitely many [MATH], and each [MATH] is compactly supported with bounded supremum; and in fact numerous blowups in [MATH] are also permitted.', '1401.0829-1-56-0': 'It is physically reasonable to think that the Brownian motion that is the free trajectory of the constituent particles in the models [MATH] arises due to thermal agitation caused by many collisions with the constituents of an ambient environment of much smaller air molecules.', '1401.0829-1-56-1': 'Viewed in these terms, it is very natural to suppose that the diffusion rate [MATH] will decrease as a function of the mass.', '1401.0829-1-56-2': 'Accepting this, the assumption ([REF]) is rather weak.', '1401.0829-1-56-3': 'If the diffusion rate is indeed decreasing, then ([REF]) is satisfied provided that there exists a function [MATH] for which [MATH] for all [MATH].', '1401.0829-1-56-4': 'Also, if the microscopic interaction strength [MATH] is identically constant, then the condition ([REF]) is equivalent to the function [MATH] being non-increasing; when [MATH], then, we are permitted choices of [MATH] that grow as quickly as [MATH].', '1401.0829-1-57-0': '### Survey assumptions', '1401.0829-1-58-0': 'These are:', '1401.0829-1-59-0': 'On [MATH].', '1401.0829-1-59-1': 'We will assume that [MATH] is a compactly supported smooth function.', '1401.0829-1-60-0': 'On [MATH].', '1401.0829-1-60-1': 'There is a uniform bound on the [MATH]-norm of each of these functions, and each is supported in a given compact region of [MATH].', '1401.0829-1-61-0': 'On [MATH].', '1401.0829-1-61-1': 'The function [MATH] is non-increasing and [MATH].', '1401.0829-1-62-0': 'On [MATH].', '1401.0829-1-62-1': 'We must suppose that [MATH].', '1401.0829-1-63-0': 'Among these, the assumptions are on the diffusion rates are genuinely restrictive: we must suppose that [MATH] grows less slowly than [MATH], which is not a particularly fast decay in any dimension [MATH].', '1401.0829-1-63-1': 'No such imposition was made in the original assumptions.', '1401.0829-1-63-2': 'The uniform bound demanded on [MATH], it must also be admitted, is also a significant restriction.', '1401.0829-1-63-3': 'The survey assumptions will permit us however to offer a method of proof of key estimates needed for the main result which is largely self-contained, as well as being novel and very probabilistic in nature; since it serves our expository purpose, we have decided to accept the more limited domain of validity demanded by these assumptions.', '1401.0829-1-64-0': '## Statement of main theorem', '1401.0829-1-65-0': 'Here is our main result.', '1401.0829-1-66-0': 'Let [MATH] and suppose that either of the above set of assumptions are in force.', '1401.0829-1-66-1': 'Let [MATH] denote the law on [MATH] given by the law of [MATH] under [MATH]; recall that [MATH] is related to [MATH] by means of the formula [MATH], with the constant [MATH] being given by the expression [MATH].', '1401.0829-1-67-0': 'The sequence [MATH] is tight in [MATH].', '1401.0829-1-67-1': 'Moreover, any limit point [MATH] of the sequence [MATH] is concentrated on the space of measures taking the form [MATH] where [MATH] ranges over weak solutions of ([REF]) that satisfy the initial condition [MATH]; recall that the collection of constants [MATH] is given by ([REF]).', '1401.0829-1-68-0': 'The reader may wonder what the meaning of this theorem is if it is not known that ([REF]) has a weak (global in time) solution for the relevant parameter choices of [MATH] and [MATH].', '1401.0829-1-68-1': 'In fact, the method of proof furnishes the existence of at least one weak solution.', '1401.0829-1-68-2': 'In any case, Laurencot and Mischler [CITATION] have established the existence of a global in time weak solution of ([REF]) whenever [MATH], and [MATH], for each [MATH], conditions which are significantly weaker than those demanded by the theorem.', '1401.0829-1-69-0': 'Theorem [REF] describes the evolution of the density profiles of particles of various masses in the limit of large particle number by means of the Smoluchowski PDE, and in this way it realizes the derivation programme that we began this article by outlining, for the diffusive coagulating system in question.', '1401.0829-1-69-1': 'The derivation has the merit of being global in time.', '1401.0829-1-69-2': 'However, note that, in general, there are limitations in the description offered of the large scale behaviour of the system.', '1401.0829-1-69-3': 'If the weak solution of this system of PDE is not known to be unique, we merely demonstrate convergence in a subsequential sense to the space of solutions.', '1401.0829-1-69-4': 'For example, admitting the possibility that the system ([REF]) has two distinct weak solutions [MATH] and [MATH] with initial condition [MATH], each of the following behaviours is consistent with Theorem [REF]:', '1401.0829-1-70-0': 'These peculiar scenarios are excluded if uniqueness of solutions to ([REF]) is known.', '1401.0829-1-70-1': 'Some conditions for uniqueness are furnished by [CITATION]; after deriving the kinetic limit of the PDE in [CITATION], Fraydoun Rezakhanlou and the author in [CITATION] provided uniqueness under rather weaker hypotheses.', '1401.0829-1-70-2': 'Indeed, as [CITATION] discusses, the next proposition is a consequence of Theorems 1.1, 1.2, 1.3 and 1.4 of [CITATION].', '1401.0829-1-71-0': 'Note that the survey assumptions in fact imply the hypotheses of Proposition [REF].', '1401.0829-1-71-1': 'This means that, in working with these assumptions, we automatically obtain the simpler statement of convergence available when uniqueness of the PDE system is known (and which we are about to state).', '1401.0829-1-72-0': 'It is a simple corollary of Theorem [REF] and Proposition [REF] that convergence to ([REF]) in fact holds in the following stronger sense.', '1401.0829-1-73-0': 'Let [MATH] and suppose that the original assumptions, and the assumptions of Proposition [REF], are in force.', '1401.0829-1-73-1': 'Let [MATH] be a bounded and continuous test function.', '1401.0829-1-73-2': 'Then, for each [MATH] and [MATH], [EQUATION] where again [MATH], with [MATH].', '1401.0829-1-73-3': 'In ([REF]), [MATH] denotes the unique weak solution to the system of partial differential equations ([REF]), with [MATH] again given by ([REF]).', '1401.0829-1-74-0': '## A simple computation about the collision of two particles', '1401.0829-1-75-0': 'The basic mechanism of interaction in our model concerns a pair of particles.', '1401.0829-1-75-1': 'Here, we explain a brief computation concerning such a pair, which offers a probabilistic interpretation of the function [MATH] used in the recipe ([REF]) for the macroscopic coagulation propensity [MATH].', '1401.0829-1-76-0': 'Suppose at a certain time, a particle of mass [MATH] is located at [MATH] and another, of mass [MATH], is located at [MATH], where [MATH].', '1401.0829-1-76-1': 'The pair are thus prone to interact shortly, in the next order [MATH] of time.', '1401.0829-1-76-2': 'Note also that, assuming uniform and independent placement of other particles in a compact region (in order to make an inference which we may find plausible for the actual model [MATH] at any given time), the typical distance from a particle to the set of other particles is of order [MATH], which is far greater than the [MATH] distance between the two particles in question.', '1401.0829-1-76-3': 'This means that in discussing the possible upcoming collision of this particle pair, we may harmlessly remove all other particles from the model.', '1401.0829-1-77-0': 'Left with a two particle problem, we set [MATH] equal to the probability of subsequent collision of the pair.', '1401.0829-1-77-1': 'We may now use Brownian scaling, zooming in by a factor of [MATH] and slowing down time by a factor of [MATH], to obtain a particle of mass [MATH] at the origin, one of mass [MATH] at [MATH], with the trajectories [MATH] being Brownian motions of speeds [MATH] and [MATH], and collision occurring at rate [MATH].', '1401.0829-1-77-2': 'That is, [MATH] is independent of [MATH], and we may take [MATH].', '1401.0829-1-78-0': 'As our notation suggests, [MATH] is nothing other than [MATH] from ([REF]):', '1401.0829-1-79-0': 'If [MATH], then [MATH] is the unique solution [MATH] of ([REF]).', '1401.0829-1-80-0': 'For occasional later use, we further define [MATH] for each [MATH] to be the probability that the two particles specified in the definition of [MATH] collide during [MATH].', '1401.0829-1-80-1': 'Thus, [MATH].', '1401.0829-1-80-2': 'In the expository discussion in Section [REF] (though not for the proof of Theorem [REF]), we will need the next result.', '1401.0829-1-81-0': 'Suppose that [MATH].', '1401.0829-1-81-1': 'Then [MATH] as [MATH].', '1401.0829-1-82-0': 'We present the proofs of these two lemmas by using a more general notation which we now present.', '1401.0829-1-83-0': '## Killed Brownian motion and the Feynman-Kac formula', '1401.0829-1-84-0': 'In our two particle problem after scaling, the displacement of the particles performs a Brownian motion at rate [MATH] until a random collision time.', '1401.0829-1-84-1': 'Slowing time by a factor of [MATH], this process is Brownian motion killed at rate [MATH] in a sense we now explain.', '1401.0829-1-85-0': 'Let [MATH] denote a smooth and compactly supported function.', '1401.0829-1-85-1': 'Let [MATH].', '1401.0829-1-85-2': 'Brownian motion in [MATH] begun at [MATH] and killed at rate [MATH] is the stochastic process [MATH] that we now specify.', '1401.0829-1-85-3': 'The process [MATH] maps [MATH] into [MATH] where, as before, [MATH] is a formal cemetery state.', '1401.0829-1-85-4': 'To define [MATH], let [MATH] denote rate one Brownian motion with [MATH].', '1401.0829-1-85-5': 'Define its interaction until time [MATH], [MATH], to be equal to [MATH].', '1401.0829-1-85-6': 'Let [MATH] denote an independent exponential random variable of rate one, and set the killing time [MATH] equal to [MATH], with the convention that [MATH].', '1401.0829-1-85-7': 'Then [EQUATION]', '1401.0829-1-85-8': 'We say that killing occurs if [MATH] and let [MATH] be such that [MATH] is the probability that killing occurs.', '1401.0829-1-86-0': 'For [MATH], [MATH] is a solution of the modified Poisson equation [EQUATION] satisfying [MATH] as [MATH].', '1401.0829-1-87-0': 'Remark.', '1401.0829-1-87-1': 'The solution is unique subject to [MATH] as [MATH].', '1401.0829-1-87-2': 'In a formal sense, this is verifed by observing that the difference [MATH] of two solutions solves [MATH] on [MATH] and then noting that [EQUATION] whose right-hand side is [MATH] and is thus at most zero.', '1401.0829-1-87-3': 'Hence, [MATH] and so [MATH] is identically zero on [MATH].', '1401.0829-1-87-4': 'We thus see that [MATH] is a constant function, and, since [MATH] as [MATH], [MATH] is identically equal to [MATH].', '1401.0829-1-87-5': "This would prove uniqueness, except that ([REF]) is a formal identity; if we integrate instead over the Euclidean ball [MATH] and take [MATH], then the boundary term in Green's theorem vanishes in the limit provided a decay condition on the solution [MATH] of ([REF]) such as [MATH] uniformly as [MATH] is made; thus, the solution is unique among those satisfying this decay condition.", '1401.0829-1-87-6': 'See [CITATION] for a proof of existence and uniqueness of the solution of ([REF]) (subject only to [MATH] as [MATH]) that uses Fredholm theory and compactness arguments.', '1401.0829-1-88-0': 'Proof.', '1401.0829-1-88-1': 'Let [MATH] given by [MATH], where the mean is taken over trajectories of standard Brownian motion [MATH].', '1401.0829-1-88-2': 'The Feynman-Kac formula [CITATION] shows that [MATH] satisfies the partial differential equation [EQUATION] for [MATH] and [MATH].', '1401.0829-1-88-3': 'Note that for any [MATH], [EQUATION] as [MATH], this probability tends to zero uniformly in [MATH], so that we find that [MATH] as [MATH], uniformly in [MATH].', '1401.0829-1-89-0': 'Note then that [MATH], which is the probability that Brownian motion [MATH] begun at [MATH] and killed at rate [MATH] is never killed, is equal to [MATH].', '1401.0829-1-89-1': 'That [MATH] solves [MATH] in a distributional sense follows by taking a high [MATH] limit of ([REF]), since [MATH] converges to [MATH] locally in [MATH].', '1401.0829-1-89-2': 'Since [MATH] is smooth, [MATH] being in local [MATH] implies that [MATH] is also in this space; thus, [MATH] is locally in [MATH].', '1401.0829-1-89-3': 'Iterating, we find that in fact [MATH], and so [MATH] solves ([REF]) in strong sense.', '1401.0829-1-89-4': '50 [MATH]', '1401.0829-1-90-0': 'Proof of Lemma [REF].', '1401.0829-1-90-1': 'Note that, by the spatial-temporal scaling satisfied by Brownian motion, [MATH] equals [MATH] where [MATH].', '1401.0829-1-90-2': 'Hence Lemma [REF] and the remark that follows it yield the result.', '1401.0829-1-90-3': '50 [MATH]', '1401.0829-1-91-0': 'Proof of Lemma [REF].', '1401.0829-1-91-1': 'Moreover, [MATH] equals [MATH], and [MATH] equals [MATH].', '1401.0829-1-91-2': 'Note that [MATH] is at most the probability that Brownian motion begun at [MATH] visits the support of [MATH] after time [MATH].', '1401.0829-1-91-3': 'With [MATH] denoting [MATH]-dimensional Lebesgue measure, this probability is at most a constant multiple of [MATH], independently of [MATH].', '1401.0829-1-91-4': '50 [MATH]', '1401.0829-1-92-0': '## A guide to the rest of the survey', '1401.0829-1-93-0': 'We have now set up the microscopic models [MATH] and laid out the programme for deriving their macroscopic evolution, including our main result, Theorem [REF].', '1401.0829-1-93-1': 'Our principal goal is to explain at a rather high, though not complete, level of detail, the proof of this theorem.', '1401.0829-1-93-2': 'We pause from pursuing this goal to explore two other directions first, however.', '1401.0829-1-93-3': "First, in Section [REF], we offer a glimpse of several topics which are tangentially related to this principal goal; this discussion is intended only to whet the reader's appetite for perhaps some of these topics and problems, and, for this reason as well as owing to limitations in the author's knowledge, it is brief and very inexhaustive.", '1401.0829-1-93-4': 'Second, in Section [REF], we offer a leisurely heuristic overview of our kinetic limit derivation in a very simplified special case, of annihilating constant diffusivity Brownian particles on a torus with a translation invariant initial condition.', '1401.0829-1-93-5': 'The argument is not rigorous at each step here, and in its method it does not provide a template for the derivation of Theorem [REF]; rather, its main goal is to provide an intuitive explanation for the form ([REF]) of the recipe for the macroscopic coagulation rates [MATH].', '1401.0829-1-93-6': 'The form of [MATH] is related to a repulsion effect present in the models [MATH] at positive times, whereby the microscopic neighbourhood of a given particle is less likely to contain other particles than typical points that lie beyond this region but in a macroscopic vicinity.', '1401.0829-1-93-7': 'We thus hope that, at the end of Section [REF], the reader will have a fuller understanding of why the statement of Theorem [REF] is true, if not yet of how it was proved in [CITATION].', '1401.0829-1-94-0': "We then return to the survey's principal goal.", '1401.0829-1-94-1': 'Section [REF] explains how Theorem [REF] will be proved, and the reader whose main interest is to see this proof explained may turn directly to this section.', '1401.0829-1-94-2': 'Therein, we introduce [MATH]-smeared approximations to the particle densities defined in the microscopic models [MATH], called microscopic candidate densities.', '1401.0829-1-94-3': 'We state a fundamental estimate, the Stosszahlansatz, which expresses total coagulation propensity in [MATH] approximately in terms of integrated products of microscopic candidate densities.', '1401.0829-1-95-0': 'The next Section [REF] describes the method of proof of the Stosszahlansatz.', '1401.0829-1-95-1': 'While so doing, it gives an alternative explanation to that of Section [REF] for the form ([REF]) of [MATH].', '1401.0829-1-96-0': 'The actual proof of the Stosszahlansatz is given in Section [REF].', '1401.0829-1-96-1': 'The proof relies on several estimates concerning various integrated sums of test functions over pairs and triples of particle indices.', '1401.0829-1-96-2': 'These bounds in turn are reduced to two key estimates.', '1401.0829-1-96-3': 'The first are particle concentration bounds, which state that if we have [MATH]-control at the initial time for the joint behaviour of [MATH]-tuples of particles in the models [MATH] (with [MATH] fixed, such as [MATH] equal to two or three), then this control propagates to all later times; it is here that the more restrictive aspect of the survey assumptions, on the decay rate of the diffusion rates [MATH], is invoked.', '1401.0829-1-96-4': 'The second key estimate are bounds on killing probabilities [MATH] (that we introduced in Section [REF]), which are uniform over [MATH] with given compact support.', '1401.0829-1-96-5': 'The proofs for this second key estimate also appear in this section.', '1401.0829-1-97-0': 'Section [REF] provides a proof of the first key estimate, the particle concentration bounds.', '1401.0829-1-97-1': 'The proof occupies several pages, but we hope that it is probabilistically interesting and intuitive.', '1401.0829-1-98-0': 'Finally, in Section [REF], we provide a summary of those points in our derivation where some steps were skipped, and mention where these omissions are treated in the original derivation in [CITATION].', '1401.0829-1-98-1': 'We also take this opportunity to explain how the proof in [CITATION] is obscure at a certain moment, and highlight how in the present paper we have endeavoured to structure the arguments to shed light on this obscurity.'}

{'1401.0829-2-0-0': 'The Smoluchowski coagulation-diffusion PDE is a system of partial differential equations modelling the evolution in time of mass-bearing Brownian particles which are subject to short-range pairwise coagulation.', '1401.0829-2-0-1': 'This survey presents a fairly detailed exposition of the kinetic limit derivation of the Smoluchowski PDE from a microscopic model of many coagulating Brownian particles that was undertaken in [CITATION].', '1401.0829-2-0-2': 'It presents heuristic explanations of the form of the main theorem before discussing the proof, and presents key estimates in that proof using a novel probabilistic technique.', '1401.0829-2-0-3': "The survey's principal aim is an exposition of this kinetic limit derivation, but it also contains an overview of several topics which either motivate or are motivated by this derivation.", '1401.0829-2-1-0': '# Introduction', '1401.0829-2-2-0': '## Microscopic particles and macroscopic descriptions', '1401.0829-2-3-0': 'An important aim in statistical mechanics is to explain how the huge amount of information available in a microscopic description of a physical object, such as the positions and momenta of all the molecules comprising the air in a room, may be accurately summarised by first specifying a small number of physical parameters which are functions of macroscopic location, such as the density, temperature and pressure of this body of air at different points in the room, and then determining how these parameters evolve in space and time.', '1401.0829-2-4-0': '### The elastic billiards model and the heat equation', '1401.0829-2-5-0': 'The microscopic system may begin out of equilbrium: for example, a still body of warm air in one room may be separated by a partition from another still body of cooler air in another, and then the partition instantaneously removed, so that air molecules from one side and the other intermingle over time, and an equilibrium is eventually approached in which the body of air in the whole room is again close to still, at a temperature which is some average of those of the two isolated systems at the original time.', '1401.0829-2-5-1': 'In such a case as this, it is a natural task to seek to summarise the evolution of a few suitable macroscopic physical quantities as the solution of partial differential equations.', '1401.0829-2-5-2': 'In the example, our object of study might be the temperature of the gas, and our aim to show that it is the heat equation, [MATH], which models the macroscopic evolution [MATH] (with [MATH] varying over the whole room, [MATH], say) of the temperature from the moment of the removal of the partition at time [MATH] until a late time, [MATH], at which a new equilibrium is approached.', '1401.0829-2-5-3': 'In an idealized and very classical choice of microscopic description of the gas, we might model the ensemble of air molecules as a system of tiny spheres of equal radius and mass, each moving according to some velocity, and each pair of which undergoes a perfectly elastic collision on contact, in the same manner that a pair of billiards would.', '1401.0829-2-5-4': 'On each of the walls that comprise the boundary [MATH] of the room, each sphere bounces elastically.', '1401.0829-2-5-5': 'The partition is modelled by the immobile sheet [MATH] on which spheres on either side also bounce elastically before time zero; the partition is removed instantaneously at that time.', '1401.0829-2-5-6': 'The initial instant of time may be taken to be zero, or some negative time.', '1401.0829-2-5-7': "At that moment, we may scatter the spheres in an independent Poissonian manner throughout the room [MATH] (the reader may notice that in fact some extra rule is needed here to ensure the spheres' disjointness); and on one side and the other of the partition, choose their velocities independently, those to the right of the partition according to a non-degenerate law of zero mean, and those on the left according to another such law of lower variance than the first; in this way, we model two bodies of still air, a warm one in the right chamber [MATH], and a cooler one in the left [MATH]: see Figure [REF].", '1401.0829-2-6-0': 'In the microscopic model, there are huge numbers of tiny spheres in the system.', '1401.0829-2-6-1': 'Indeed, we may seek to understand the macroscopic evolution of temperature by in fact considering a whole sequence of microscopic models indexed by total particle number [MATH], in a limit of high [MATH].', '1401.0829-2-6-2': 'In the [MATH] model, spheres are initially scattered as we described, with a Poissonian intensity [MATH] thoughout [MATH].', '1401.0829-2-6-3': 'To carry out this task of understanding the large-scale evolution, we would wish to specify a microscopic definition of the notion of temperature, and then explain how it is in the high [MATH] limit that the microscopic temperature data may be meaningfully reduced to a macroscopic description, and that this latter description indeed evolves according to the heat equation.', '1401.0829-2-6-4': 'Microscopically, temperature is interpreted [CITATION] as the average kinetic energy of particles, where here the velocity of particles is measured relative to the average velocity of nearby particles.', '1401.0829-2-6-5': 'Since our particle systems are large microscopically, considering as we do a high [MATH] limit, we may specify in our [MATH] microscopic model a definition of temperature at any given location [MATH] as follows: first we may compute the mean velocity [MATH] of the set of spheres whose centres lie within some small distance [MATH] of a given location [MATH] in the room, and then we are able to define the microscopic temperature [MATH] to be the average of the square of the particle velocity minus [MATH], where the average is taken over the same set of spheres.', '1401.0829-2-6-6': 'Of course, the value [MATH] will change in time.', '1401.0829-2-6-7': 'As [MATH] approaches infinity with [MATH] being fixed but small, huge numbers of particles are involved in the empirical counts used for averaging.', '1401.0829-2-6-8': 'Our aim is to consider the space-time evolution of the microscopically specified temperatures after the high [MATH] limit is taken, at which point, the weak law of large numbers might suggest that these empirical counts behave non-randomly to first order, so that our description becomes deterministic: the limit [MATH] will be some non-random function [MATH].', '1401.0829-2-6-9': 'In fact, since [MATH] is fixed, we should not yet expect our system to approximate the heat equation, since there is an effect of macroscopic smearing in our calculation of microscopic temperature.', '1401.0829-2-6-10': 'Rather, one might expect the heat equation description to emerge if we take a [MATH] limit of [MATH], after the first high [MATH] limit has been taken.', '1401.0829-2-6-11': 'Moreover, to hope to obtain this description, we will also need to scale time appropriately in the [MATH] microscopic model, as we take the first, high [MATH], limit.', '1401.0829-2-6-12': 'In the scaled time coordinates, the microscopic models should make their approach to the new thermal equilibrium at the same rate, as [MATH].', '1401.0829-2-6-13': 'What rate this is in fact depends on another important consideration concerning the microscopic models which our brief description left unspecified: the radius [MATH] of each sphere in the [MATH] model must certainly be chosen to satisfy [MATH] for some constant [MATH], if only to permit all of the spheres to inhabit the room disjointly; our choice of decay rate for [MATH] as a function of [MATH], subject to this constraint, will determine the factor by which we scale time in the [MATH] model in order to seek a heat equation description in the large.', '1401.0829-2-7-0': 'To implement the programme proposed in the preceding paragraph is an open problem, and in all likelihood, an extremely difficult one.', '1401.0829-2-7-1': 'There is no randomness in the model except in the initial selection of particle locations and velocities: from that time on, the deterministic laws of Newtonian mechanics govern the evolution of the microscopic models.', '1401.0829-2-7-2': 'Moreover, some choices for density admissible in the above description - such as when [MATH] converges in high [MATH], to a suitably small constant - lead to rather dense systems of particles.', '1401.0829-2-7-3': 'The derivation may be less inordinately hard were more dilute choices of limit considered, where [MATH] converges to zero more, and perhaps much more, quickly than does [MATH].', '1401.0829-2-8-0': 'It is important to note, however, that, if a choice of [MATH] as a function of [MATH] is made which is too rapidly decaying, we may leave the realm in which the heat equation is the appropriate macroscopic description.', '1401.0829-2-8-1': 'For example, if [MATH], it is a simple matter to check that a typical sphere after time zero will traverse the entire room on many occasions before meeting any other particle.', '1401.0829-2-8-2': 'The system will reach equilibrium after the removal of the partition simply by the free motion of the particles.', '1401.0829-2-8-3': 'The heat equation is only a suitable description when a typical particle experiences the thermal agitation caused by its collision with many other particles in short periods of macroscopic time.', '1401.0829-2-9-0': "### The elastic billiards model and Boltzmann's equation", '1401.0829-2-10-0': 'Moreover, the elastic billiards model crosses at least one interesting regime as it is diluted from the dense [MATH] phase towards the trivial free motion phase [MATH].', '1401.0829-2-10-1': 'Consider the choice [MATH].', '1401.0829-2-10-2': "A moment's thought shows that, in this regime, a typical sphere will travel (at unit-order velocity) for a duration before its first collision with another particle which on average neither tends to zero nor to infinity as [MATH].", '1401.0829-2-10-3': 'This is the regime of constant mean free path.', '1401.0829-2-10-4': 'The heat equation will not offer a suitable description for the evolution of temperature in this regime, because the mechanism providing for thermal agitation of particles - manifest only when a typical particle has suffered many collisions - occurs on a time scale which is marginally too slow.', '1401.0829-2-10-5': 'However, the programme of deriving a macroscopic description by means of a PDE does make sense, and in this case, offers a powerful model of gas dynamics.', '1401.0829-2-10-6': 'Suppose that, instead of using the microscopic data to form a description of temperature, we use it to describe the density of particles having a given velocity [MATH] nearby a given location [MATH].', '1401.0829-2-10-7': 'Particles may be scattered in a Poissonian fashion as before at the initial time, but with inhomogeneities in the intensity of this scattering permitted in both the space and velocity variables.', '1401.0829-2-10-8': 'With the macroscopic smearing parameter now being used to approximate velocity [MATH] as well as location [MATH], we may record a microscopic description [MATH] for the [MATH]-smeared density of spheres at space-velocity location [MATH].', '1401.0829-2-10-9': "Taking a high [MATH] and then low [MATH] limit as above, our macroscopic evolution is modelled by the fundamental system of equations in gas dynamics, Boltzmann's equation, valid for [MATH], [MATH] and [MATH]: [EQUATION]", '1401.0829-2-10-10': "Here, [MATH] is the free motion operator associated to particles of velocity [MATH], while [MATH] is a binary collision operator that reflects the microscopic elastic collision and whose form we will specify when we return to Boltzmann's equation in a brief discussion in Section [REF].", '1401.0829-2-10-11': 'For now, note that the time evolution of the macroscopic densities is governed both by the free motion and by the collision operator.', '1401.0829-2-10-12': 'This is what is to be expected in the regime of constant mean free path, where the typical particle experiences unit-order durations free of collision and other such periods where several collisions occur.', '1401.0829-2-11-0': 'Boltzmann carried out a derivation of ([REF]) as a model of gas dynamics in 1872, based on several assumptions, including one of molecule chaos that he called the Stosszahlansatz and which we will later discuss.', '1401.0829-2-11-1': '(See [CITATION] for an English translation of his 1872 article.)', '1401.0829-2-11-2': "The validity of his derivation was a matter of controversy, not least due to Loschmidt's paradox concerning precollisional particle independence (see Subsection [REF]), and it was a fundamental advance made in 1975 by Lanford [CITATION] when the programme of rigorously deriving Boltzmann's equation from the elastic billiards model in the regime of mean free path was successfully implemented, for a short initial duration of time.", '1401.0829-2-11-3': 'By the latter condition, we mean that the validity of the description was established for some non-zero finite period, whose value depends on the form of the initial density profile of particles in space-velocity.', '1401.0829-2-12-0': "Lanford derived Boltzmann's equation by establishing that the correlation functions concerning several particles in the model satisfy a hierarchy of equations called the BBGKY hierarchy, where the index of an equation in the hierarchy is the number of particles whose correlation is being considered, and by showing that when the correlation functions adhere to the BBGKY hierarchy, the density profile follows Boltzmann's equation.", '1401.0829-2-12-1': "Illner and Pulvirenti implemented this approach in [CITATION] in order to derive Boltzmann's equation in a similar sense, but now globally in time, although with a comparable smallness condition, now on sparseness of the initial particle distribution; the cited derivation concerns a two dimensional gas, but this restriction on dimension was later lifted by the same authors.", '1401.0829-2-13-0': '### Our main goal: coagulating Brownian particles and the Smoluchowski PDE', '1401.0829-2-14-0': 'This survey is intended to offer a detailed overview of a programme for deriving the macroscopic description of a gas of particles in the same vein as the descriptions above propose.', '1401.0829-2-14-1': 'However, our microscopic particles will diffuse, each following a Brownian trajectory, and as such their evolution is random, not deterministic; the mechanism of interaction will be pairwise as above, but a coagulation in which only one particle survives rather than a collision in which both do.', '1401.0829-2-14-2': "On the other hand, in an effort to provide some generality in the microscopic description and richness in the macroscopic one, each of the particles will bear a mass, which the pairwise coagulation will conserve; and, moreover, we will permit the diffusivity of the Brownian trajectory of each particle to depend on the particle's mass.", '1401.0829-2-15-0': "The partial differential equation which the programme seeks to obtain in this case - the analogue of the heat equation or Boltzmann's equation in our opening examples - is, like Boltzmann's equation, in fact a system of PDE, in our case coupled in the mass parameter, known as the Smoluchowski coagulation-diffusion PDE.", '1401.0829-2-15-1': 'The choice made for diluteness in the high particle number limit will be that of the regime of constant mean free path.', '1401.0829-2-15-2': 'The programme of deriving the PDE in the case of constant mean free path is sometimes called a kinetic limit derivation.', '1401.0829-2-16-0': 'In the special case of mass-independent diffusion rates, the kinetic limit derivation was carried out in 1980 by Lang and Nguyen [CITATION], who followed the method of showing that the correlation functions between several particles are described by the BBGKY hierarchy which Lanford had employed.', '1401.0829-2-17-0': "Introduced to the problem of generalizing Lang and Nguyen's derivation of the Smoluchowski PDE by James Norris, the author collaborated on it with Fraydoun Rezakhanlou.", '1401.0829-2-17-1': 'The principal aim of these notes is to give an informal but fairly detailed exposition of the kinetic limit derivation of the Smoluchowski PDE that was undertaken for dimension [MATH] in [CITATION].', '1401.0829-2-17-2': 'The treatment also first presents heuristic arguments with the aim that the reader may understand why the main theorem should be true before beginning a presentation of the proof of the theorem, and it also uses some novel probabilistic techniques to obtain key estimates used in the proof.', '1401.0829-2-17-3': 'The survey also touches on some related topics.'}

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[]

[]

[]

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{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1401.0829

0711.1432

{'0711.1432-1-0-0': 'Many extrasolar planets orbit closely to their parent star.', '0711.1432-1-0-1': 'Their existence raises the fundamental problem of loss and gain in their mass.', '0711.1432-1-0-2': 'For exoplanet HD209458b, reports on an unusually extended hydrogen corona and a hot layer in the lower atmosphere seem to support the scenario of atmospheric inflation by the strong stellar irradiation.', '0711.1432-1-0-3': 'However, difficulties in reconciling evaporation models with observations call for a reassessment of the problem.', '0711.1432-1-0-4': 'Here, we use HST archive data to report a new absorption rate of [MATH]% by atomic hydrogen during the HD209458b transit, and show that no sign of evaporation could be detected for the exoplanet.', '0711.1432-1-0-5': 'We also report evidence of time variability in the HD209458 flux, a variability that was not accounted for in previous studies, which corrupted their diagnostics.', '0711.1432-1-0-6': 'Mass loss rates thus far proposed in the literature in the range [MATH] must induce a spectral signature in the Lyman-[MATH] line profile of HD209458 that cannot be found in the present analysis.', '0711.1432-1-0-7': 'Either an unknown compensation effect is hiding the expected spectral feature or else the mass loss rate of neutrals from HD209458 is modest.', '0711.1432-1-1-0': '# Introduction', '0711.1432-1-2-0': 'Of all the planets discovered outside the realm of our solar system, some of the most dramatic new classes of objects are those in which the planet is a gas giant orbiting at merely a few stellar radii ([MATH]) from its parent star.', '0711.1432-1-2-1': 'These close-in extrasolar planets are Jupiter-like giants that are exposed to strong fluxes, magnetic fields, and plasma winds-a very harsh and active stellar environment.', '0711.1432-1-2-2': "Because of their stars' proximity, gravity, through tidal effects, distorts the shape of their atmosphere while the continuous extreme ultraviolet (UV) energy deposition inflates it ([CITATION]; [CITATION]; [CITATION]; [CITATION]; [CITATION]; [CITATION]; [CITATION]; [CITATION]).", '0711.1432-1-2-3': 'Unfortunately, little is known about those regions that separate extrasolar giant planets from their stars, particularly the immediate environment of the planet.', '0711.1432-1-3-0': 'One of the most extensively studied extrasolar systems is HD209458.', '0711.1432-1-3-1': 'For reference, HD209458b was first discovered transiting its parent star and covering [MATH]% of its disk ([CITATION]; [CITATION]).', '0711.1432-1-3-2': 'Some of the first attempts to learn about the immediate environment of the planet were ([MATH] nm) observations of the system using the Space Telescope Imaging Spectrometer (STIS) onboard the Hubble Space Telescope (HST).', '0711.1432-1-3-3': 'A first program of observation, obtained with the STIS/G140M grating and the 52"x0.1" slit, was implemented in [MATH] during HD209458 planetary transit but no conclusions were reported (see Table [REF]).', '0711.1432-1-3-4': 'Soon after, a second program visited the target during three transits ([CITATION]).', '0711.1432-1-3-5': 'An initial analysis of this data set concluded that a huge cloud of hydrogen is covering [MATH]% of the stellar disk ([CITATION]); it also claimed that spectral absorption during transit is deeper on the blue side of the stellar line.', '0711.1432-1-3-6': 'Accordingly, the hydrogen cloud was required to extend beyond the planetary Roche limit where an intense escape of [MATH] of hydrogen is a priori operating.', '0711.1432-1-3-7': 'These results, along with other far UV low-resolution observations of heavy constituents, led to the conclusion that the upper atmosphere of HD209458b should be in a hydrodynamic blow-off state ([CITATION]).', '0711.1432-1-4-0': 'Numerous studies then followed on different mechanisms for hydrogen loss from hot exoplanets closely orbiting their stars ([CITATION]; [CITATION]; [CITATION]; [CITATION]; [CITATION]; [CITATION]; [CITATION]).', '0711.1432-1-4-1': 'As noted by [CITATION], all loss rates thus far proposed by theoretical models in the range [MATH] exceed the lower limit provided by [CITATION].', '0711.1432-1-4-2': 'Unfortunately, most studies neglected to quantitatively translate their loss rate to a spectral absorption in the line profile that could be tested with the HST observation.', '0711.1432-1-4-3': 'Independently, pointing out that the observed mass function distribution of extrasolar giant planets (EGPs) follows a trend [MATH] for mass range [MATH], where [MATH] is the Jovian mass, [CITATION] derived the same mass function distribution for highly irradiated EGPs orbiting at distances smaller than [MATH] AU.', '0711.1432-1-4-4': "Accordingly, [CITATION] rejected substantial mass loss during EGPs' migration to smaller distances from their star, unless the loss mechanism is compensated by an unknown process.", '0711.1432-1-4-5': 'When combined with the unusual scales derived for the hydrogen extent and escape, all these studies then call for a careful reassessment of the HST observations thus far obtained on HD209458, at least in order to provide validated constraints on theoretical models.', '0711.1432-1-5-0': '# Observations and data analysis', '0711.1432-1-6-0': 'In the following, we report a new analysis of archive data obtained during the two HST/STIS programs described above.', '0711.1432-1-6-1': 'In total, we have four visits of the target corresponding to three exposures of roughly [MATH] duration each, resulting in [MATH] exposures of the systems around the transit period (Table [REF]).', '0711.1432-1-6-2': 'All observations were obtained in the time-tag mode, a technique that keeps track every [MATH] of photon events during each exposure.', '0711.1432-1-6-3': 'The question, then, is: why is this mode of observation important in the present case?', '0711.1432-1-6-4': 'First, we stress that the transit effect is a weak variation of the stellar signal.', '0711.1432-1-6-5': 'As such, its trend is best represented by a dense time series.', '0711.1432-1-6-6': 'Second, chromospheric and coronal variabilities of the star are unknown in the spectral window considered here, and this may seriously corrupt any diagnostic.', '0711.1432-1-7-0': 'To properly handle time tagged data, we partition each time-tag exposure into a set of shorter sub-exposures, taking into account the heliocentric and barycentric time correction procedure in which ephemerides are retrieved from the archives before the IRAF/STSDAS "odelaytime" procedure is applied ([CITATION]).', '0711.1432-1-7-1': 'After several trials, we found that [MATH] sampling of the time-tag data is a good compromise between acceptable signal-to-noise ratio (S/N) and time coverage.', '0711.1432-1-7-2': 'Next, each sub-exposure is calibrated through the STIS pipeline.', '0711.1432-1-7-3': 'The emitting source (the star plus sky background) is presumed extended, an option that allows efficient control of the subtraction of the sky background contamination.', '0711.1432-1-7-4': 'Time is then converted to fixed orbital phases measured from the transit central time (TCT), itself carefully taken from the most recent and accurate determination of the HD209458 system parameters ([CITATION]; [CITATION]).', '0711.1432-1-7-5': 'Sub-spectra of identical phase positions are accumulated from the initial twelve exposures, resulting in a unique 53 bins time series of the system versus the orbital phase angle (Fig [REF]).', '0711.1432-1-7-6': 'Unexpectedly, three gaps, lasting respectively [MATH], [MATH], and [MATH], appear in the time series, for which no observation is available.', '0711.1432-1-7-7': 'Because the three gaps are narrow and well-separated from each other, we determined that filling them has a negligible effect on our final conclusions ([CITATION]).', '0711.1432-1-7-8': 'Errors due to photon counting have been propagated, taking into account the correlation between the different phase positions relative to the initial sampling of sub-exposures time over the full observing time period.', '0711.1432-1-8-0': "We next define the wavelength domain of contamination by the sky background, including both the Earth's geocorona and the interplanetary medium emissions.", '0711.1432-1-8-1': "The difficulty comes from the uncertainty about the geocorona's strength when estimated from a detector sector, along the STIS slit, different from the one where the stellar signal was recorded.", '0711.1432-1-8-2': 'First, we subtracted the dark noise of the detector following ([CITATION] and [CITATION]) and then compared the sky background signal from different sectors along the slit and for different conditions of observation.', '0711.1432-1-8-3': 'Our conclusion is that the STIS MAMMA detector has an inherent non-uniformity corresponding to an incompressible uncertainty of [MATH]% on extended sources.', '0711.1432-1-8-4': 'Coincidentally, this uncertainty is comparable to photon statistical errors.', '0711.1432-1-8-5': 'To ensure that such error will not corrupt the stellar signal per wavelength pixel at the [MATH]% level, we deduce that a void window [MATH] nm should be disregarded in any spectral analysis that requires high accuracy, such as for a transit event or short-term stellar variability.', '0711.1432-1-9-0': '# Light curve trend and time variability', '0711.1432-1-10-0': 'Next, to obtain the trend of the planetary transit in , we integrated the time series spectra in the range [MATH] nm on the blue wing and [MATH] nm on the red wing.', '0711.1432-1-10-1': 'These ranges were selected so that the stellar signal per wavelength pixel remains above the statistical noise ([MATH]).', '0711.1432-1-10-2': 'The resulting light curve is noisy, but a trend is apparent and can be efficiently extracted (Fig. 2a).', '0711.1432-1-10-3': 'To dampen the signal noise while keeping a clear trend, the best compromise is to gather data by eight phase bins for a new bin of [MATH].', '0711.1432-1-10-4': 'Accumulating the signal from three new bins ([MATH] s) inside transit, we derive [MATH] drop-off of the stellar intensity during the planetary transit (Fig. 2a).', '0711.1432-1-10-5': 'Our absorption rate of [MATH] is much lower and accurate than reported in a previous study ([CITATION]), yet a marginal agreement could be found between our maximum rate (11) and their bottom value.', '0711.1432-1-10-6': 'If this obscuration is converted directly to a planetary occulting disk, then one would obtain a hydrogen cloud of [MATH] R[MATH] radius, much smaller than the Roche lobe limit of [MATH] R[MATH] ([CITATION]), where R[MATH] R[MATH] is the most recent estimate of the radius of HD209458b ([CITATION]; [CITATION]), and R[MATH] is the Jovian one.', '0711.1432-1-10-7': 'Now, to capture the trend of the transit curve, we used a sophisticated 2D model of planetary transit at that accurately accounts for the atmospheric radial structure of the planet ([CITATION]; [CITATION]) and properly estimates the atmospheric obscuration versus wavelength, including extinction by the interstellar gas intervening along the line of sight ([CITATION]).', '0711.1432-1-10-8': 'Our best least square fit is shown in Figure 2a.', '0711.1432-1-10-9': "For our purpose of time analysis, we remark that a functional fit could also be a good model to obtain the light curve's trend.", '0711.1432-1-11-0': 'We can now determine the stellar signal time evolution after we cancel the transit trend using our best fit to the observed light curve (Fig. 2a).', '0711.1432-1-11-1': 'The resulting ratio shows a variable behavior with an average amplitude [MATH] of the stellar integrated intensity (Fig. 2b).', '0711.1432-1-11-2': 'Using the Durbin-Watson statistical test ([CITATION]), we found no apparent serial correlation at the 1 confidence level in the corrected signal-a signal that also shows no evident periodicity.', '0711.1432-1-11-3': 'HD209458 was previously suspected to have a relatively moderate chromospheric activity from CaII H and K lines that were recorded over full orbits of the system ([CITATION]).', '0711.1432-1-11-4': "Our finding of a time variation of 8.6% on average in the stellar signal, with peaks that may reach [MATH]% ([MATH]), seems to support a relatively active corona of the star, presumably up to the planet's orbit.", '0711.1432-1-11-5': 'Such activity could be of common origin (flaring, non-uniformity of the stellar disc during transit, etc.) and/or related to an enhancement of magnetic activity on the star-planet line ([CITATION]).', '0711.1432-1-11-6': 'Also, one can speculate about the hydrogen cloud topology around the planet and its evolution with time.', '0711.1432-1-11-7': 'To that end, comparative studies with interacting binary stars may be useful in clarifying the different regimes of interaction between an exoplanet and its host star ([CITATION]).', '0711.1432-1-11-8': 'Unfortunately, the FUV observations thus far obtained do not cover a full orbit of the planet, thereby making it difficult to predict the exact configuration of the star-planet system.', '0711.1432-1-11-9': 'In any case, we believe that the unusual [MATH] obscuration previously reported ([CITATION]) was corrupted by this unaccounted-for variable component in the star-planet system signal.', '0711.1432-1-11-10': 'Here we can extract it because we are able to sample the transit period by a dense time series using the information gathered from the time tag mode of HST/STIS and [MATH] more observation time from the archives.', '0711.1432-1-12-0': '# Planetary mass loss or flux variability?', '0711.1432-1-13-0': 'In the following, we compare the in/out of transit stellar line profiles.', '0711.1432-1-13-1': 'The impetus of this study is the need to determine the relevance of a blueshifted absorption in the stellar line profile that may occur during transit, as claimed in earlier studies ([CITATION]).', '0711.1432-1-13-2': 'On the one hand, we derive an average unperturbed profile of the HD209458 emission line by merging all sub-spectra of the time series that we correct for the transit trend with the best fit shown in Figure 2a.', '0711.1432-1-13-3': 'The resulting profile is a good reference that best represents the out-transit stellar line and for which time variability has been reduced to the 1 signal level (Fig. 3a).', '0711.1432-1-13-4': 'On the other hand, the in-transit line profile, when corrected for the [MATH] drop-off during transit, properly recovers the unperturbed line profile (Fig. 3a), leaving no real possibility of extra absorption as claimed in prior studies ([CITATION]).', '0711.1432-1-14-0': 'To further investigate how time variability of the HD209458 emission line corrupts the diagnostic as it pertains to extra absorption or emission features that may appear in the stellar line during transit, we selected two phase windows inside the transit period for which we compared line profiles to the unperturbed stellar line.', '0711.1432-1-14-1': 'As shown in Figure 3b, a direct comparison would indicate that line peaks are equally absorbed for line profile B1, while for line profile B2, the red peak is the most absorbed.', '0711.1432-1-14-2': 'On the basis of line profile B2, the diagnostic would be just the opposite of that of ([CITATION]), leading to escaping hydrogen toward the star, while for line profile B1, the diagnostic would be no H escape.', '0711.1432-1-14-3': 'The problem is that these interpretations of preferred blueshifted or redshifted absorption do not account for the relatively strong modulation of the stellar signal evidenced in this study.', '0711.1432-1-14-4': 'Therefore, any claim of a preferred absorption during transit, either blue or redshifted, is not realistic, particularly at this relatively modest level of the signal to noise.', '0711.1432-1-14-5': 'It follows that the blueshifted absorption, advanced in previous studies ([CITATION]; [CITATION]) as a signature of atmospheric evaporation in a cometary-like tail of HD209458b, has, unfortunately, no foundation in the HST/STIS data set as it was only the effect of the stellar signal variability with time that corrupted the diagnostic.', '0711.1432-1-15-0': '# Conclusion', '0711.1432-1-16-0': 'We use HST archive observations of the emission of HD209458 to report an absorption rate of [MATH]% by atomic hydrogen during the transit of the planetary companion.', '0711.1432-1-16-1': 'If the planet is sketched as a compact blocking body, our analysis requires an H cloud effective extent that does not exceed [MATH] R[MATH]-a size that falls short of the Roche limit [MATH] R[MATH] of HD209458b.', '0711.1432-1-16-2': 'In addition, time variability of the stellar flux is evidenced, but no sign of extra or Doppler-shifted absorption could be detected during transit.', '0711.1432-1-16-3': 'This absence of extra absorption during transit and the relatively small size of the effective area of the hydrogen cloud around the exoplanet make it difficult to conceive of significant atmospheric evaporation from the planet.', '0711.1432-1-16-4': 'Of course, we cannot rule out that a complex atmospheric distribution, related to a particular planet-star interaction scenario, may hide or compensate the loss signature during the observing time.', '0711.1432-1-16-5': 'Future HST (when repaired) FUV observation of the system during a full planetary orbit should help to disentangle the different processes in play.', '0711.1432-1-17-0': 'The author acknowledges support from Université Pierre et Marie Curie (UPMC) and the Centre National de la Recherche Scientifique (CNRS) in France.', '0711.1432-1-17-1': 'This work is based on observations with the NASA/ESA Hubble Space Telescope, obtained at the Space Telescope Science Institute, which is operated by AURA, Inc.'}

{'0711.1432-2-0-0': 'Many extrasolar planets orbit closely to their parent star.', '0711.1432-2-0-1': 'Their existence raises the fundamental problem of loss and gain in their mass.', '0711.1432-2-0-2': 'For exoplanet HD209458b, reports on an unusually extended hydrogen corona and a hot layer in the lower atmosphere seem to support the scenario of atmospheric inflation by the strong stellar irradiation.', '0711.1432-2-0-3': 'However, difficulties in reconciling evaporation models with observations call for a reassessment of the problem.', '0711.1432-2-0-4': 'Here, we use HST archive data to report a new absorption rate of [MATH]% by atomic hydrogen during the HD209458b transit, and show that no sign of evaporation could be detected for the exoplanet.', '0711.1432-2-0-5': 'We also report evidence of time variability in the HD209458 flux, a variability that was not accounted for in previous studies, which corrupted their diagnostics.', '0711.1432-2-0-6': 'Mass loss rates thus far proposed in the literature in the range [MATH] must induce a spectral signature in the Lyman-[MATH] line profile of HD209458 that cannot be found in the present analysis.', '0711.1432-2-0-7': 'Either an unknown compensation effect is hiding the expected spectral feature or else the mass loss rate of neutrals from HD209458 is modest.', '0711.1432-2-1-0': '# Introduction', '0711.1432-2-2-0': 'Of all the planets discovered outside the realm of our solar system, some of the most dramatic new classes of objects are those in which the planet is a gas giant orbiting at merely a few stellar radii ([MATH]) from its parent star.', '0711.1432-2-2-1': 'These close-in extrasolar planets are Jupiter-like giants that are exposed to strong fluxes, magnetic fields, and plasma winds-a very harsh and active stellar environment.', '0711.1432-2-2-2': "Because of their stars' proximity, gravity, through tidal effects, distorts the shape of their atmosphere while the continuous extreme ultraviolet (UV) energy deposition inflates it ([CITATION]; [CITATION]; [CITATION]; [CITATION]; [CITATION]; [CITATION]; [CITATION]; [CITATION]).", '0711.1432-2-2-3': 'Unfortunately, little is known about those regions that separate extrasolar giant planets from their stars, particularly the immediate environment of the planet.', '0711.1432-2-3-0': 'One of the most extensively studied extrasolar systems is HD209458.', '0711.1432-2-3-1': 'For reference, HD209458b was first discovered transiting its parent star and covering [MATH]% of its disk ([CITATION]; [CITATION]).', '0711.1432-2-3-2': 'Some of the first attempts to learn about the immediate environment of the planet were ([MATH] nm) observations of the system using the Space Telescope Imaging Spectrometer (STIS) onboard the Hubble Space Telescope (HST).', '0711.1432-2-3-3': 'A first program of observation, obtained with the STIS/G140M grating and the 52"x0.1" slit, was implemented in [MATH] during HD209458 planetary transit but no conclusions were reported (see Table [REF]).', '0711.1432-2-3-4': 'Soon after, a second program visited the target during three transits ([CITATION]).', '0711.1432-2-3-5': 'An initial analysis of this data set concluded that a huge cloud of hydrogen is covering [MATH]% of the stellar disk ([CITATION]); it also claimed that spectral absorption during transit is deeper on the blue side of the stellar line.', '0711.1432-2-3-6': 'Accordingly, the hydrogen cloud was required to extend beyond the planetary Roche limit where an intense escape of [MATH] of hydrogen is a priori operating.', '0711.1432-2-3-7': 'These results, along with other far UV low-resolution observations of heavy constituents, led to the conclusion that the upper atmosphere of HD209458b should be in a hydrodynamic blow-off state ([CITATION]).', '0711.1432-2-4-0': 'Numerous studies then followed on different mechanisms for hydrogen loss from hot exoplanets closely orbiting their stars ([CITATION]; [CITATION]; [CITATION]; [CITATION]; [CITATION]; [CITATION]; [CITATION]).', '0711.1432-2-4-1': 'As noted by [CITATION], all loss rates thus far proposed by theoretical models in the range [MATH] exceed the lower limit provided by [CITATION].', '0711.1432-2-4-2': 'Unfortunately, most studies neglected to quantitatively translate their loss rate to a spectral absorption in the line profile that could be tested with the HST observation.', '0711.1432-2-4-3': 'Independently, pointing out that the observed mass function distribution of extrasolar giant planets (EGPs) follows a trend [MATH] for mass range [MATH], where [MATH] is the Jovian mass, [CITATION] derived the same mass function distribution for highly irradiated EGPs orbiting at distances smaller than [MATH] AU.', '0711.1432-2-4-4': "Accordingly, [CITATION] rejected substantial mass loss during EGPs' migration to smaller distances from their star, unless the loss mechanism is compensated by an unknown process.", '0711.1432-2-4-5': 'When combined with the unusual scales derived for the hydrogen extent and escape, all these studies then call for a careful reassessment of the HST observations thus far obtained on HD209458, at least in order to provide validated constraints on theoretical models.', '0711.1432-2-5-0': '# Observations and data analysis', '0711.1432-2-6-0': 'In the following, we report a new analysis of archive data obtained during the two HST/STIS programs described above.', '0711.1432-2-6-1': 'In total, we have four visits of the target corresponding to three exposures of roughly [MATH] duration each, resulting in [MATH] exposures of the systems around the transit period (Table [REF]).', '0711.1432-2-6-2': 'All observations were obtained in the time-tag mode, a technique that keeps track every [MATH] of photon events during each exposure.', '0711.1432-2-6-3': 'The question, then, is: why is this mode of observation important in the present case?', '0711.1432-2-6-4': 'First, we stress that the transit effect is a weak variation of the stellar signal.', '0711.1432-2-6-5': 'As such, its trend is best represented by a dense time series.', '0711.1432-2-6-6': 'Second, chromospheric and coronal variabilities of the star are unknown in the spectral window considered here, and this may seriously corrupt any diagnostic.', '0711.1432-2-7-0': 'To properly handle time tagged data, we partition each time-tag exposure into a set of shorter sub-exposures, taking into account the heliocentric and barycentric time correction procedure in which ephemerides are retrieved from the archives before the IRAF/STSDAS "odelaytime" procedure is applied ([CITATION]).', '0711.1432-2-7-1': 'After several trials, we found that [MATH] sampling of the time-tag data is a good compromise between acceptable signal-to-noise ratio (S/N) and time coverage.', '0711.1432-2-7-2': 'Next, each sub-exposure is calibrated through the STIS pipeline.', '0711.1432-2-7-3': 'The emitting source (the star plus sky background) is presumed extended, an option that allows efficient control of the subtraction of the sky background contamination.', '0711.1432-2-7-4': 'Time is then converted to fixed orbital phases measured from the transit central time (TCT), itself carefully taken from the most recent and accurate determination of the HD209458 system parameters ([CITATION]; [CITATION]).', '0711.1432-2-7-5': 'Sub-spectra of identical phase positions are accumulated from the initial twelve exposures, resulting in a unique 53 bins time series of the system versus the orbital phase angle (Fig [REF]).', '0711.1432-2-7-6': 'Unexpectedly, three gaps, lasting respectively [MATH], [MATH], and [MATH], appear in the time series, for which no observation is available.', '0711.1432-2-7-7': 'Because the three gaps are narrow and well-separated from each other, we determined that filling them has a negligible effect on our final conclusions ([CITATION]).', '0711.1432-2-7-8': 'Errors due to photon counting have been propagated, taking into account the correlation between the different phase positions relative to the initial sampling of sub-exposures time over the full observing time period.', '0711.1432-2-8-0': "We next define the wavelength domain of contamination by the sky background, including both the Earth's geocorona and the interplanetary medium emissions.", '0711.1432-2-8-1': "The difficulty comes from the uncertainty about the geocorona's strength when estimated from a detector sector, along the STIS slit, different from the one where the stellar signal was recorded.", '0711.1432-2-8-2': 'First, we subtracted the dark noise of the detector following ([CITATION] and [CITATION]) and then compared the sky background signal from different sectors along the slit and for different conditions of observation.', '0711.1432-2-8-3': 'Our conclusion is that the STIS MAMMA detector has an inherent non-uniformity corresponding to an incompressible uncertainty of [MATH]% on extended sources.', '0711.1432-2-8-4': 'Coincidentally, this uncertainty is comparable to photon statistical errors.', '0711.1432-2-8-5': 'To ensure that such error will not corrupt the stellar signal per wavelength pixel at the [MATH]% level, we deduce that a void window [MATH] nm should be disregarded in any spectral analysis that requires high accuracy, such as for a transit event or short-term stellar variability.', '0711.1432-2-9-0': '# Light curve trend and time variability', '0711.1432-2-10-0': 'Next, to obtain the trend of the planetary transit in , we integrated the time series spectra in the range [MATH] nm on the blue wing and [MATH] nm on the red wing.', '0711.1432-2-10-1': 'These ranges were selected so that the stellar signal per wavelength pixel remains above the statistical noise ([MATH]).', '0711.1432-2-10-2': 'The resulting light curve is noisy, but a trend is apparent and can be efficiently extracted (Fig. 2a).', '0711.1432-2-10-3': 'To dampen the signal noise while keeping a clear trend, the best compromise is to gather data by eight phase bins for a new bin of [MATH].', '0711.1432-2-10-4': 'Accumulating the signal from three new bins ([MATH] s) inside transit, we derive [MATH] drop-off of the stellar intensity during the planetary transit (Fig. 2a).', '0711.1432-2-10-5': 'Our absorption rate of [MATH] is much lower and accurate than reported in a previous study ([CITATION]), yet a marginal agreement could be found between our maximum rate (11) and their bottom value.', '0711.1432-2-10-6': 'If this obscuration is converted directly to a planetary occulting disk, then one would obtain a hydrogen cloud of [MATH] R[MATH] radius, much smaller than the Roche lobe limit of [MATH] R[MATH] ([CITATION]), where R[MATH] R[MATH] is the most recent estimate of the radius of HD209458b ([CITATION]; [CITATION]), and R[MATH] is the Jovian one.', '0711.1432-2-10-7': 'Now, to capture the trend of the transit curve, we used a sophisticated 2D model of planetary transit at that accurately accounts for the atmospheric radial structure of the planet ([CITATION]; [CITATION]) and properly estimates the atmospheric obscuration versus wavelength, including extinction by the interstellar gas intervening along the line of sight ([CITATION]).', '0711.1432-2-10-8': 'Our best least square fit is shown in Figure 2a.', '0711.1432-2-10-9': "For our purpose of time analysis, we remark that a functional fit could also be a good model to obtain the light curve's trend.", '0711.1432-2-11-0': 'We can now determine the stellar signal time evolution after we cancel the transit trend using our best fit to the observed light curve (Fig. 2a).', '0711.1432-2-11-1': 'The resulting ratio shows a variable behavior with an average amplitude [MATH] of the stellar integrated intensity (Fig. 2b).', '0711.1432-2-11-2': 'Using the Durbin-Watson statistical test ([CITATION]), we found no apparent serial correlation at the 1 confidence level in the corrected signal-a signal that also shows no evident periodicity.', '0711.1432-2-11-3': 'HD209458 was previously suspected to have a relatively moderate chromospheric activity from CaII H and K lines that were recorded over full orbits of the system ([CITATION]).', '0711.1432-2-11-4': "Our finding of a time variation of 8.6% on average in the stellar signal, with peaks that may reach [MATH]% ([MATH]), seems to support a relatively active corona of the star, presumably up to the planet's orbit.", '0711.1432-2-11-5': 'Such activity could be of common origin (flaring, non-uniformity of the stellar disc during transit, etc.) and/or related to an enhancement of magnetic activity on the star-planet line ([CITATION]).', '0711.1432-2-11-6': 'Also, one can speculate about the hydrogen cloud topology around the planet and its evolution with time.', '0711.1432-2-11-7': 'To that end, comparative studies with interacting binary stars may be useful in clarifying the different regimes of interaction between an exoplanet and its host star ([CITATION]).', '0711.1432-2-11-8': 'Unfortunately, the FUV observations thus far obtained do not cover a full orbit of the planet, thereby making it difficult to predict the exact configuration of the star-planet system.', '0711.1432-2-11-9': 'In any case, we believe that the unusual [MATH] obscuration previously reported ([CITATION]) was corrupted by this unaccounted-for variable component in the star-planet system signal.', '0711.1432-2-11-10': 'Here we can extract it because we are able to sample the transit period by a dense time series using the information gathered from the time tag mode of HST/STIS and [MATH] more observation time from the archives.', '0711.1432-2-12-0': '# Planetary mass loss or flux variability?', '0711.1432-2-13-0': 'In the following, we compare the in/out of transit stellar line profiles.', '0711.1432-2-13-1': 'The impetus of this study is the need to determine the relevance of a blueshifted absorption in the stellar line profile that may occur during transit, as claimed in earlier studies ([CITATION]).', '0711.1432-2-13-2': 'On the one hand, we derive an average unperturbed profile of the HD209458 emission line by merging all sub-spectra of the time series that we correct for the transit trend with the best fit shown in Figure 2a.', '0711.1432-2-13-3': 'The resulting profile is a good reference that best represents the out-transit stellar line and for which time variability has been reduced to the 1 signal level (Fig. 3a).', '0711.1432-2-13-4': 'On the other hand, the in-transit line profile, when corrected for the [MATH] drop-off during transit, properly recovers the unperturbed line profile (Fig. 3a), leaving no real possibility of extra absorption as claimed in prior studies ([CITATION]).', '0711.1432-2-14-0': 'To further investigate how time variability of the HD209458 emission line corrupts the diagnostic as it pertains to extra absorption or emission features that may appear in the stellar line during transit, we selected two phase windows inside the transit period for which we compared line profiles to the unperturbed stellar line.', '0711.1432-2-14-1': 'As shown in Figure 3b, a direct comparison would indicate that line peaks are equally absorbed for line profile B1, while for line profile B2, the red peak is the most absorbed.', '0711.1432-2-14-2': 'On the basis of line profile B2, the diagnostic would be just the opposite of that of ([CITATION]), leading to escaping hydrogen toward the star, while for line profile B1, the diagnostic would be no H escape.', '0711.1432-2-14-3': 'The problem is that these interpretations of preferred blueshifted or redshifted absorption do not account for the relatively strong modulation of the stellar signal evidenced in this study.', '0711.1432-2-14-4': 'Therefore, any claim of a preferred absorption during transit, either blue or redshifted, is not realistic, particularly at this relatively modest level of the signal to noise.', '0711.1432-2-14-5': 'It follows that the blueshifted absorption, advanced in previous studies ([CITATION]; [CITATION]) as a signature of atmospheric evaporation in a cometary-like tail of HD209458b, has, unfortunately, no foundation in the HST/STIS data set as it was only the effect of the stellar signal variability with time that corrupted the diagnostic.', '0711.1432-2-15-0': '# Conclusion', '0711.1432-2-16-0': 'We use HST archive observations of the emission of HD209458 to report an absorption rate of [MATH]% by atomic hydrogen during the transit of the planetary companion.', '0711.1432-2-16-1': 'If the planet is sketched as a compact blocking body, our analysis requires an H cloud effective extent that does not exceed [MATH] R[MATH]-a size that falls short of the Roche limit [MATH] R[MATH] of HD209458b.', '0711.1432-2-16-2': 'In addition, time variability of the stellar flux is evidenced, but no sign of extra or Doppler-shifted absorption could be detected during transit.', '0711.1432-2-16-3': 'This absence of extra absorption during transit and the relatively small size of the effective area of the hydrogen cloud around the exoplanet make it difficult to conceive of significant atmospheric evaporation from the planet.', '0711.1432-2-16-4': 'Of course, we cannot rule out that a complex atmospheric distribution, related to a particular planet-star interaction scenario, may hide or compensate the loss signature during the observing time.', '0711.1432-2-16-5': 'Future HST (when repaired) FUV observation of the system during a full planetary orbit should help to disentangle the different processes in play.', '0711.1432-2-17-0': 'The author acknowledges support from Université Pierre et Marie Curie (UPMC) and the Centre National de la Recherche Scientifique (CNRS) in France.', '0711.1432-2-17-1': 'This work is based on observations with the NASA/ESA Hubble Space Telescope, obtained at the Space Telescope Science Institute, which is operated by AURA, Inc.'}

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{'1': 'http://arxiv.org/licenses/assumed-1991-2003/', '2': 'http://arxiv.org/licenses/assumed-1991-2003/'}

https://arxiv.org/abs/0711.1432

1008.4414

{'1008.4414-1-0-0': 'We propose two efficient numerical methods of evaluating the luminosity distance in the spatially flat [MATH] universe.', '1008.4414-1-0-1': 'The first method is based on the Carlson symmetric form of elliptic integrals, which is highly accurate and can replace numerical quadratures.', '1008.4414-1-0-2': 'The second method, using a modified version of Hermite interpolation, is less accurate but involves only basic numerical operations and can be easily implemented.', '1008.4414-1-0-3': 'We compare our methods with other numerical approximation schemes and explore their respective features and limitations.', '1008.4414-1-0-4': 'Possible extensions of these methods to other cosmological models are also discussed.', '1008.4414-1-1-0': '# Introduction', '1008.4414-1-2-0': 'The computation of cosmological distances naturally arises in the study of cosmology, for example the luminosity distance [MATH] in the analysis of type Ia supernova (SNIa) data and the angular diameter distance [MATH] in the study of gravitational lensing.', '1008.4414-1-2-1': 'These distances depend on the underlying cosmological model and their parameters.', '1008.4414-1-2-2': 'Therefore they are useful as cosmological tests.', '1008.4414-1-2-3': 'As a result, accurate and efficient numerical algorithms of evaluating these distances become a necessity for the practitioners of cosmological research.', '1008.4414-1-3-0': 'The analytical form of the cosmological distances can be derived from the solution of the Friedmann equation, an ordinary differential equation involving the scale factor [MATH] as a function of cosmic time [MATH].', '1008.4414-1-3-1': 'Therefore, formulae for the distances usually involve an integral over the expansion history expressed in terms of the redshift [MATH] and cosmological parameters.', '1008.4414-1-3-2': 'In general, the integrations can be evaluated numerically by quadrature algorithms.', '1008.4414-1-3-3': 'However, numerical quadratures tend to be computationally heavy when high accuracy is desired.', '1008.4414-1-4-0': 'defcitealias1999ApJS.', '1008.4414-1-4-1': '.120...49PPen99 defcitealias2010MNRAS.406.', '1008.4414-1-4-2': '.548WWU10 In the presence of this performance issue, it is advantageous to develop algorithms that are restricted to specific cosmological models but are otherwise more efficient than general-purpose quadratures.', '1008.4414-1-4-3': 'For the spatially flat [MATH] model, efficient algorithms for the luminosity distnace have been proposed by [CITATION] and [CITATION].', '1008.4414-1-5-0': 'In this paper we propose two different numerical methods for the luminosity distance also in the context of the spatially flat [MATH] universe.', '1008.4414-1-5-1': 'The methods are presented in Sections [REF] and [REF] respectively.', '1008.4414-1-5-2': 'In Section [REF], the performances of these methods are discussed.', '1008.4414-1-5-3': 'Finally in Section [REF] we discuss some possible extensions to the methods presented in this paper.', '1008.4414-1-5-4': 'Throughout the paper we will focus on the luminosity distance only, but the results can be trivially extended to compute the angular diameter distance [MATH].', '1008.4414-1-6-0': '# Method I: Evaluation using Carlson symmetric forms', '1008.4414-1-7-0': 'The luminosity distance [MATH] in the spatially flat [MATH] universe is given by [EQUATION] where [MATH] and [MATH] are the energy densities corresponding to the matter and cosmological constant respectively: [MATH].', '1008.4414-1-7-1': 'Following the notation intalias1999ApJS.', '1008.4414-1-7-2': '.120...49P we introduce the parameter [MATH] and the change-of-variable [MATH], and re-write equation ([REF]) as [EQUATION] where [EQUATION]', '1008.4414-1-7-3': 'The integral in equations ([REF]) and ([REF]) are special cases of elliptic integrals.', '1008.4414-1-7-4': 'All elliptic integrals can be reduced to several basic forms, the best known of which are probably the three kinds of Legendre elliptic integrals , with reduction theorems and examples presented in .', '1008.4414-1-7-5': 'In our case it is clearer to express this integral by one of the Carlson symmetric forms [MATH], which is defined as [EQUATION]', '1008.4414-1-7-6': 'Using the reduction theorems, it is straightforward to verify that [EQUATION] where [EQUATION]', '1008.4414-1-7-7': 'It has been known that the Carlson forms can be computed numerically with high accuracy.', '1008.4414-1-7-8': '[CITATION] showed that the computation of [MATH] can be accomplished iteratively with the error decreasing by a factor of [MATH] after each iteration, therefore achieving fast convergence.', '1008.4414-1-7-9': 'Further analysis of the algorithms for [MATH] and other elliptic integrals can be found in , and computer implementation details have been discussed in and .', '1008.4414-1-8-0': '# Method II: Approximation by a Modified Hermite Interpolation', '1008.4414-1-9-0': 'The method presented in Section [REF] uses an iterative approach to the computation of [MATH].', '1008.4414-1-9-1': 'However, there are situations where a closed, approximate formula for the integral in equation ([REF]) is desired.', '1008.4414-1-9-2': 'Intalias1999ApJS.', '1008.4414-1-9-3': '.120...49P an approximation was obtained using polynomial fit for [MATH].', '1008.4414-1-9-4': 'Intalias2010MNRAS.406.', '1008.4414-1-9-5': '.548W another method with higher accuracy was proposed.', '1008.4414-1-9-6': 'In this section we show how a modified version of Hermite interpolation can lead to a class of approximations similar to that intalias1999ApJS.', '1008.4414-1-9-7': '.120...49P.', '1008.4414-1-10-0': 'We intend to approximate equation ([REF]) using only elementary operations, such as polynomial evaluation and [MATH]th root where [MATH] is a small integer.', '1008.4414-1-10-1': 'We note that the behavior of [MATH] has several deficiencies.', '1008.4414-1-10-2': 'First, the derivative of [MATH] becomes singular as [MATH].', '1008.4414-1-10-3': 'Second, the domain of [MATH] extends to infinity.', '1008.4414-1-10-4': 'Either one is detrimental to the approximation using polynomials.', '1008.4414-1-10-5': 'However, they can be removed by certain change-of-variables.', '1008.4414-1-10-6': 'For example, we can introduce a new function [EQUATION] that has smooth derivatives within the interval [MATH] and can be extended to the cases of [MATH] and [MATH].', '1008.4414-1-10-7': 'The limiting behaviors of [MATH] are shown below: [EQUATION] where [MATH] is a numerical constant.', '1008.4414-1-11-0': 'Using the end-point conditions in equation ([REF]) one can construct a 3rd-order polynomial, which is a linear combination of the four Hermite basis splines in [MATH], as a crude approximation with [MATH]20% relative error.', '1008.4414-1-11-1': 'This linear combination is unique, allowing no further improvements.', '1008.4414-1-11-2': 'However, we note that for realistic values of [MATH] it is not necessary to approximate [MATH] in the entire interval [MATH], because the subinterval [MATH] corresponds to the scenario of [MATH], i.e. "the future".', '1008.4414-1-11-3': 'Therefore, we can introduce a free parameter [MATH] as the alternative lower end-point, and only perform the approximation in the subinterval [MATH], if a constraint is put on [MATH] (or equivalently, [MATH]).', '1008.4414-1-12-0': 'To accommodate further refinements, a correction term [MATH] can be added to the Hermite approximation.', '1008.4414-1-12-1': 'We require the value and first derivative of [MATH] to vanish at either end-point, so that it can be added to the Hermite approximation without altering the coefficients on the basis splines.', '1008.4414-1-12-2': 'One choice of [MATH] is made possible by a family of functions [EQUATION] where [MATH] and [MATH] are adjustable parameters accounting for the deviation of the Hermite approximation from the true function.', '1008.4414-1-12-3': 'Other choices are possible, but we will begin with the simple case of equation ([REF]).', '1008.4414-1-13-0': 'By construction, the approximation described above has the property that the approximating function coincides with the true function at the end-points, [MATH] and [MATH], up to the first derivative.', '1008.4414-1-13-1': 'But we note that the goal is to approximate equation ([REF]) rather than equation ([REF]).', '1008.4414-1-13-2': 'This suggests that the implicit requirement of the coincidence of function values at end-points could be unnecessarily strong.', '1008.4414-1-13-3': 'Alternatively, we may refrain from requiring the approximating function values to match the true ones.', '1008.4414-1-13-4': 'Instead, we only require the matching of first derivatives at [MATH], and leave the end-point value at another free parameter.', '1008.4414-1-13-5': 'To summarize, we now have four free parameters that can be tuned: [MATH], [MATH] and [MATH], and the function value [MATH] at [MATH].', '1008.4414-1-13-6': 'The approximation to equation ([REF]) can be expressed as [EQUATION] where [EQUATION] and [MATH] are the Hermite basis splines, [EQUATION]', '1008.4414-1-13-7': 'Following the approach intalias1999ApJS.', '1008.4414-1-13-8': '.120...49P, we choose the objective function as the maximum relative error in [MATH] using the approximation ([Eq. [REF]]), with the restriction [MATH].', '1008.4414-1-13-9': 'Minimizing the objective function over the parameters, we obtain the best-fit [MATH], [MATH], [MATH], and [MATH].', '1008.4414-1-13-10': 'Substituting the numerical values into equation ([REF]), we therefore construct an approximation polynomial [EQUATION]', '1008.4414-1-13-11': 'Equation ([REF]) is the main result of this section.', '1008.4414-1-13-12': 'With the parameters determined, the approximation to [MATH] can be computed using this formula with equations ([REF]) and ([REF]).', '1008.4414-1-14-0': '# Performance of the Methods', '1008.4414-1-15-0': 'In this section we proceed to assess the performance of the methods in Sections [REF] and [REF].', '1008.4414-1-15-1': 'The assessment is mainly done in terms of the accuracy and efficiency.', '1008.4414-1-16-0': 'Accuracy', '1008.4414-1-17-0': 'The first method can be used to yield highly accurate numerical approximation of [MATH] for vast ranges of [MATH] and the parameter [MATH] if we adopt the algorithm for [MATH] by [CITATION].', '1008.4414-1-17-1': 'Unlike the methods based on the evaluation of a closed approximation formula, the desired cutoff error can be prescribed to determine when the iterative computation of [MATH] terminates.', '1008.4414-1-17-2': 'In practice, we found that the prescription of relative error [MATH] can be achieved without suffering significant loss in the computation speed.', '1008.4414-1-18-0': 'For the second method, we plot the distribution of the relative error of [MATH] in Figure [REF].', '1008.4414-1-18-1': 'As can be seen from the figure, the second method remains an approximation at best.', '1008.4414-1-18-2': 'Under our choice of fitting parameters and range of [MATH], the relative error in [MATH] is [MATH]0.5%.', '1008.4414-1-18-3': 'The major source of this error is contributed by [MATH].', '1008.4414-1-18-4': 'For [MATH] our method is comparable with that oftalias1999ApJS.', '1008.4414-1-18-5': '.120...49P, and ours slightly outperforms it when [MATH] is larger.', '1008.4414-1-19-0': 'Efficiency', '1008.4414-1-20-0': 'Theoretically, the best-, worst-, and average-case temporal efficiencies for each method can be calculated or estimated by tracking every operation taken during the course of the computing.', '1008.4414-1-20-1': 'However, such a thorough analysis is beyond the scope of this paper.', '1008.4414-1-20-2': 'Instead, we empirically compare the running time of the computer programs using the two methods with those of 1999ApJS.', '1008.4414-1-20-3': '.120...49P andtalias2010MNRAS.406.', '1008.4414-1-20-4': '.548W under a controlled environment.', '1008.4414-1-21-0': 'In Figure [REF] we display the benchmark results of our methods compared with that oftalias1999ApJS.', '1008.4414-1-21-1': '.120...49P and 2010MNRAS.406.', '1008.4414-1-21-2': '.548W. To simulate a "real-world" application of these methods, we creates a sample of SNIa redshifts using the Supernova/Acceleration Probe fiducial redshift distribution containing [MATH] redshift points distributed within [MATH] .', '1008.4414-1-21-3': 'Our sample satisfies the same distribution to the SNAP fiducial, but is 16 times as dense, i.e. with [MATH] points in total.', '1008.4414-1-21-4': 'We have made custom implementations of the methods fromtalias1999ApJS.', '1008.4414-1-21-5': '.120...49P,talias2010MNRAS.406.', '1008.4414-1-21-6': '.548W, and our Method II in the C programming language, and uses the GNU Scientific Library (GSL) implementation of the [MATH] algorithm in for Method I.', '1008.4414-1-21-7': 'In our benchmark routine, the computing of [MATH] values from our redshift sample is performed for [MATH] respectively, with each pass through the [MATH] sample repeated for 25 times (that sums up to a total of [MATH] evaluations of [MATH]).', '1008.4414-1-21-8': 'The benchmark itself is repeated for 2400 times.', '1008.4414-1-22-0': 'To interpret Figure [REF], we make two remarks.', '1008.4414-1-22-1': 'First, the execution time results were collected from the output of the gprof profiler and does not reflect the absolute time spent.', '1008.4414-1-22-2': 'It is only meaningful as a relative measure useful for comparing the speed of the codes with each other.', '1008.4414-1-22-3': 'Second, the results are dependent on our particular implementations as well as the computing environment.', '1008.4414-1-22-4': 'This is evident if our Figure [REF] is compared with Figure 4 intalias2010MNRAS.406..', '1008.4414-1-22-5': '.548W that shows a reversed result for the speeds of the two methods in 1999ApJS..', '1008.4414-1-23-0': '# Discussion', '1008.4414-1-24-0': 'As Figure [REF] suggests, both methods proposed in this paper is slower than thetalias1999ApJS.', '1008.4414-1-24-1': '.120...49P method.', '1008.4414-1-24-2': 'However, Method I is a very reasonable trade-off between an enormous gain in accuracy and small loss of efficiency.', '1008.4414-1-24-3': 'With Method I one does not need to resort to the numerical quadrature for the same level of accuracy.', '1008.4414-1-25-0': 'Method I can be extended to cover the [MATH] model with a curvature term [MATH], because in that case the equivalent of equation ([REF]) becomes [EQUATION] which is also an elliptic integral, and can be reduced to [MATH] accordingly.', '1008.4414-1-25-1': 'This is potentially useful for the analysis of future SNIa data, because it has been suggested that the spacetime curvature should not be ignored in the probe of dark energy using luminosity distance data .', '1008.4414-1-26-0': 'In contrast, Method II may not be as promising, because in its current form the accuracy does not outperform that oftalias1999ApJS.', '1008.4414-1-26-1': '.120...49P.', '1008.4414-1-26-2': 'However, the idea behind the method may be useful when extending to alternative cosmological models (for example, dynamical dark energy) which may not be reduced to the applicable scenarios of Method I.', '1008.4414-1-26-3': 'In the description of this method we have left some arbitrariness unjustified, notably the particular choice of the singularity-removing transformation (Eq. [[REF]]), the parameterization of the correction term (Eq. [[REF]]), and the very choice of Hermite basis splines.', '1008.4414-1-26-4': 'Alternative choices of them may be adopted to generate better approximations, for instance, the use of low-order Hermite-Birkhoff interpolation to selectively choose the point [MATH] near which the derivative information of the true function is to be best preserved.', '1008.4414-1-26-5': 'Moreover, our Method II uses only elementary numerical operations, while in 2010MNRAS.406.', '1008.4414-1-26-6': '.548W the numerical logarithm is extensively used.', '1008.4414-1-27-0': 'De-Zi Liu would like to thank Fang-Fang Zhu and Mao-Sheng Xiang for their kind help.', '1008.4414-1-27-1': 'This work was supported by the National Science Foundation of China (Grants No. 10473002), the Ministry of Science and Technology National Basic Science program (project 973) under grant No. 2009CB24901, Scientific Research Foundation of Beijing Normal University.'}

{'1008.4414-2-0-0': 'We propose two efficient numerical methods of evaluating the luminosity distance in the spatially flat [MATH] universe.', '1008.4414-2-0-1': 'The first method is based on the Carlson symmetric form of elliptic integrals, which is highly accurate and can replace numerical quadratures.', '1008.4414-2-0-2': 'The second method, using a modified version of Hermite interpolation, is less accurate but involves only basic numerical operations and can be easily implemented.', '1008.4414-2-0-3': 'We compare our methods with other numerical approximation schemes and explore their respective features and limitations.', '1008.4414-2-0-4': 'Possible extensions of these methods to other cosmological models are also discussed.', '1008.4414-2-1-0': '# Introduction', '1008.4414-2-2-0': 'The computation of cosmological distances naturally arises in the study of cosmology, for example the luminosity distance [MATH] in the analysis of type Ia supernova (SNIa) data and the angular diameter distance [MATH] in the study of gravitational lensing.', '1008.4414-2-2-1': 'These distances depend on the underlying cosmological model and their parameters.', '1008.4414-2-2-2': 'Therefore they are useful as cosmological tests.', '1008.4414-2-2-3': 'As a result, accurate and efficient numerical algorithms of evaluating these distances become a necessity for the practitioners of cosmological research.', '1008.4414-2-3-0': 'The analytical form of the cosmological distances can be derived from the solution of the Friedmann equation, an ordinary differential equation involving the scale factor [MATH] as a function of cosmic time [MATH].', '1008.4414-2-3-1': 'Therefore, formulae for the distances usually involve an integral over the expansion history expressed in terms of the redshift [MATH] and cosmological parameters.', '1008.4414-2-3-2': 'In general, the integrations can be evaluated numerically by quadrature algorithms.', '1008.4414-2-3-3': 'However, numerical quadratures tend to be computationally heavy when high accuracy is desired.', '1008.4414-2-4-0': 'defcitealias1999ApJS.', '1008.4414-2-4-1': '.120...49PPen99 defcitealias2010MNRAS.406.', '1008.4414-2-4-2': '.548WWU10 In the presence of this performance issue, it is advantageous to develop algorithms that are restricted to specific cosmological models but are otherwise more efficient than general-purpose quadratures.', '1008.4414-2-4-3': 'For the spatially flat [MATH] model, efficient algorithms for the luminosity distnace have been proposed by [CITATION] and [CITATION].', '1008.4414-2-5-0': 'In this paper we propose two different numerical methods for the luminosity distance also in the context of the spatially flat [MATH] universe.', '1008.4414-2-5-1': 'The methods are presented in Sections [REF] and [REF] respectively.', '1008.4414-2-5-2': 'In Section [REF], the performances of these methods are discussed.', '1008.4414-2-5-3': 'Finally in Section [REF] we discuss some possible extensions to the methods presented in this paper.', '1008.4414-2-5-4': 'Throughout the paper we will focus on the luminosity distance only, but the results can be trivially extended to compute the angular diameter distance [MATH].', '1008.4414-2-6-0': '# Method I: Evaluation using Carlson symmetric forms', '1008.4414-2-7-0': 'The luminosity distance [MATH] in the spatially flat [MATH] universe is given by [EQUATION] where [MATH] and [MATH] are the energy densities corresponding to the matter and cosmological constant respectively: [MATH].', '1008.4414-2-7-1': 'Following the notation intalias1999ApJS.', '1008.4414-2-7-2': '.120...49P we introduce the parameter [MATH] and the change-of-variable [MATH], and re-write equation ([REF]) as [EQUATION] where [EQUATION]', '1008.4414-2-7-3': 'The integral in equations ([REF]) and ([REF]) are special cases of elliptic integrals.', '1008.4414-2-7-4': 'All elliptic integrals can be reduced to several basic forms, the best known of which are probably the three kinds of Legendre elliptic integrals , with reduction theorems and examples presented in .', '1008.4414-2-7-5': 'In our case it is clearer to express this integral by one of the Carlson symmetric forms [MATH], which is defined as [EQUATION]', '1008.4414-2-7-6': 'Using the reduction theorems, it is straightforward to verify that [EQUATION] where [EQUATION]', '1008.4414-2-7-7': 'It has been known that the Carlson forms can be computed numerically with high accuracy.', '1008.4414-2-7-8': '[CITATION] showed that the computation of [MATH] can be accomplished iteratively with the error decreasing by a factor of [MATH] after each iteration, therefore achieving fast convergence.', '1008.4414-2-7-9': 'Further analysis of the algorithms for [MATH] and other elliptic integrals can be found in , and computer implementation details have been discussed in and .', '1008.4414-2-8-0': '# Method II: Approximation by a Modified Hermite Interpolation', '1008.4414-2-9-0': 'The method presented in Section [REF] uses an iterative approach to the computation of [MATH].', '1008.4414-2-9-1': 'However, there are situations where a closed, approximate formula for the integral in equation ([REF]) is desired.', '1008.4414-2-9-2': 'Intalias1999ApJS.', '1008.4414-2-9-3': '.120...49P an approximation was obtained using polynomial fit for [MATH].', '1008.4414-2-9-4': 'Intalias2010MNRAS.406.', '1008.4414-2-9-5': '.548W another method with higher accuracy was proposed.', '1008.4414-2-9-6': 'In this section we show how a modified version of Hermite interpolation can lead to a class of approximations similar to that intalias1999ApJS.', '1008.4414-2-9-7': '.120...49P.', '1008.4414-2-10-0': 'We intend to approximate equation ([REF]) using only elementary operations, such as polynomial evaluation and [MATH]th root where [MATH] is a small integer.', '1008.4414-2-10-1': 'We note that the behavior of [MATH] has several deficiencies.', '1008.4414-2-10-2': 'First, the derivative of [MATH] becomes singular as [MATH].', '1008.4414-2-10-3': 'Second, the domain of [MATH] extends to infinity.', '1008.4414-2-10-4': 'Either one is detrimental to the approximation using polynomials.', '1008.4414-2-10-5': 'However, they can be removed by certain change-of-variables.', '1008.4414-2-10-6': 'For example, we can introduce a new function [EQUATION] that has smooth derivatives within the interval [MATH] and can be extended to the cases of [MATH] and [MATH].', '1008.4414-2-10-7': 'The limiting behaviors of [MATH] are shown below: [EQUATION] where [MATH] is a numerical constant.', '1008.4414-2-11-0': 'Using the end-point conditions in equation ([REF]) one can construct a 3rd-order polynomial, which is a linear combination of the four Hermite basis splines in [MATH], as a crude approximation with [MATH]20% relative error.', '1008.4414-2-11-1': 'This linear combination is unique, allowing no further improvements.', '1008.4414-2-11-2': 'However, we note that for realistic values of [MATH] it is not necessary to approximate [MATH] in the entire interval [MATH], because the subinterval [MATH] corresponds to the scenario of [MATH], i.e. "the future".', '1008.4414-2-11-3': 'Therefore, we can introduce a free parameter [MATH] as the alternative lower end-point, and only perform the approximation in the subinterval [MATH], if a constraint is put on [MATH] (or equivalently, [MATH]).', '1008.4414-2-12-0': 'To accommodate further refinements, a correction term [MATH] can be added to the Hermite approximation.', '1008.4414-2-12-1': 'We require the value and first derivative of [MATH] to vanish at either end-point, so that it can be added to the Hermite approximation without altering the coefficients on the basis splines.', '1008.4414-2-12-2': 'One choice of [MATH] is made possible by a family of functions [EQUATION] where [MATH] and [MATH] are adjustable parameters accounting for the deviation of the Hermite approximation from the true function.', '1008.4414-2-12-3': 'Other choices are possible, but we will begin with the simple case of equation ([REF]).', '1008.4414-2-13-0': 'By construction, the approximation described above has the property that the approximating function coincides with the true function at the end-points, [MATH] and [MATH], up to the first derivative.', '1008.4414-2-13-1': 'But we note that the goal is to approximate equation ([REF]) rather than equation ([REF]).', '1008.4414-2-13-2': 'This suggests that the implicit requirement of the coincidence of function values at end-points could be unnecessarily strong.', '1008.4414-2-13-3': 'Alternatively, we may refrain from requiring the approximating function values to match the true ones.', '1008.4414-2-13-4': 'Instead, we only require the matching of first derivatives at [MATH], and leave the end-point value at another free parameter.', '1008.4414-2-13-5': 'To summarize, we now have four free parameters that can be tuned: [MATH], [MATH] and [MATH], and the function value [MATH] at [MATH].', '1008.4414-2-13-6': 'The approximation to equation ([REF]) can be expressed as [EQUATION] where [EQUATION] and [MATH] are the Hermite basis splines, [EQUATION]', '1008.4414-2-13-7': 'Following the approach intalias1999ApJS.', '1008.4414-2-13-8': '.120...49P, we choose the objective function as the maximum relative error in [MATH] using the approximation ([Eq. [REF]]), with the restriction [MATH].', '1008.4414-2-13-9': 'Minimizing the objective function over the parameters, we obtain the best-fit [MATH], [MATH], [MATH], and [MATH].', '1008.4414-2-13-10': 'Substituting the numerical values into equation ([REF]), we therefore construct an approximation polynomial [EQUATION]', '1008.4414-2-13-11': 'Equation ([REF]) is the main result of this section.', '1008.4414-2-13-12': 'With the parameters determined, the approximation to [MATH] can be computed using this formula with equations ([REF]) and ([REF]).', '1008.4414-2-14-0': '# Performance of the Methods', '1008.4414-2-15-0': 'In this section we proceed to assess the performance of the methods in Sections [REF] and [REF].', '1008.4414-2-15-1': 'The assessment is mainly done in terms of the accuracy and efficiency.', '1008.4414-2-16-0': 'Accuracy', '1008.4414-2-17-0': 'The first method can be used to yield highly accurate numerical approximation of [MATH] for vast ranges of [MATH] and the parameter [MATH] if we adopt the algorithm for [MATH] by [CITATION].', '1008.4414-2-17-1': 'Unlike the methods based on the evaluation of a closed approximation formula, the desired cutoff error can be prescribed to determine when the iterative computation of [MATH] terminates.', '1008.4414-2-17-2': 'In practice, we found that the prescription of relative error [MATH] can be achieved without suffering significant loss in the computation speed.', '1008.4414-2-18-0': 'For the second method, we plot the distribution of the relative error of [MATH] in Figure [REF].', '1008.4414-2-18-1': 'As can be seen from the figure, the second method remains an approximation at best.', '1008.4414-2-18-2': 'Under our choice of fitting parameters and range of [MATH], the relative error in [MATH] is [MATH]0.5%.', '1008.4414-2-18-3': 'The major source of this error is contributed by [MATH].', '1008.4414-2-18-4': 'For [MATH] our method is comparable with that oftalias1999ApJS.', '1008.4414-2-18-5': '.120...49P, and ours slightly outperforms it when [MATH] is larger.', '1008.4414-2-19-0': 'Efficiency', '1008.4414-2-20-0': 'Theoretically, the best-, worst-, and average-case temporal efficiencies for each method can be calculated or estimated by tracking every operation taken during the course of the computing.', '1008.4414-2-20-1': 'However, such a thorough analysis is beyond the scope of this paper.', '1008.4414-2-20-2': 'Instead, we empirically compare the running time of the computer programs using the two methods with those of 1999ApJS.', '1008.4414-2-20-3': '.120...49P andtalias2010MNRAS.406.', '1008.4414-2-20-4': '.548W under a controlled environment.', '1008.4414-2-21-0': 'In Figure [REF] we display the benchmark results of our methods compared with that oftalias1999ApJS.', '1008.4414-2-21-1': '.120...49P and 2010MNRAS.406.', '1008.4414-2-21-2': '.548W. To simulate a "real-world" application of these methods, we creates a sample of SNIa redshifts using the Supernova/Acceleration Probe fiducial redshift distribution containing [MATH] redshift points distributed within [MATH] .', '1008.4414-2-21-3': 'Our sample satisfies the same distribution to the SNAP fiducial, but is 16 times as dense, i.e. with [MATH] points in total.', '1008.4414-2-21-4': 'We have made custom implementations of the methods fromtalias1999ApJS.', '1008.4414-2-21-5': '.120...49P,talias2010MNRAS.406.', '1008.4414-2-21-6': '.548W, and our Method II in the C programming language, and uses the GNU Scientific Library (GSL) implementation of the [MATH] algorithm in for Method I.', '1008.4414-2-21-7': 'In our benchmark routine, the computing of [MATH] values from our redshift sample is performed for [MATH] respectively, with each pass through the [MATH] sample repeated for 25 times (that sums up to a total of [MATH] evaluations of [MATH]).', '1008.4414-2-21-8': 'The benchmark itself is repeated for 2400 times.', '1008.4414-2-22-0': 'To interpret Figure [REF], we make two remarks.', '1008.4414-2-22-1': 'First, the execution time results were collected from the output of the gprof profiler and does not reflect the absolute time spent.', '1008.4414-2-22-2': 'It is only meaningful as a relative measure useful for comparing the speed of the codes with each other.', '1008.4414-2-22-3': 'Second, the results are dependent on our particular implementations as well as the computing environment.', '1008.4414-2-22-4': 'This is evident if our Figure [REF] is compared with Figure 4 intalias2010MNRAS.406..', '1008.4414-2-22-5': '.548W that shows a reversed result for the speeds of the two methods in 1999ApJS..', '1008.4414-2-23-0': '# Discussion', '1008.4414-2-24-0': 'As Figure [REF] suggests, both methods proposed in this paper is slower than thetalias1999ApJS.', '1008.4414-2-24-1': '.120...49P method.', '1008.4414-2-24-2': 'However, Method I is a very reasonable trade-off between an enormous gain in accuracy and small loss of efficiency.', '1008.4414-2-24-3': 'With Method I one does not need to resort to the numerical quadrature for the same level of accuracy.', '1008.4414-2-25-0': 'Method I can be extended to cover the [MATH] model with a curvature term [MATH], because in that case the equivalent of equation ([REF]) assumes the form [EQUATION] where [EQUATION] is the expansion rate ([MATH]), and [EQUATION]', '1008.4414-2-25-1': 'The integral in equation ([REF]) is also an elliptic integral that can be reduced to [MATH] accordingly.', '1008.4414-2-25-2': 'This is potentially useful for the analysis of future SNIa data, because it has been suggested that the spacetime curvature should not be ignored in the probe of dark energy using luminosity distance data .', '1008.4414-2-26-0': 'In contrast, Method II may not be as promising, because in its current form the accuracy does not outperform that oftalias1999ApJS.', '1008.4414-2-26-1': '.120...49P.', '1008.4414-2-26-2': 'However, the idea behind the method may be useful when extending to alternative cosmological models (for example, dynamical dark energy) which may not be reduced to the applicable scenarios of Method I.', '1008.4414-2-26-3': 'In the description of this method we have left some arbitrariness unjustified, notably the particular choice of the singularity-removing transformation (Eq. [[REF]]), the parameterization of the correction term (Eq. [[REF]]), and the very choice of Hermite basis splines.', '1008.4414-2-26-4': 'Alternative choices of them may be adopted to generate better approximations, for instance, the use of low-order Hermite-Birkhoff interpolation to selectively choose the point [MATH] near which the derivative information of the true function is to be best preserved.', '1008.4414-2-26-5': 'Moreover, our Method II uses only elementary numerical operations, while in 2010MNRAS.406.', '1008.4414-2-26-6': '.548W the numerical logarithm is extensively used.', '1008.4414-2-27-0': 'De-Zi Liu would like to thank Fang-Fang Zhu and Mao-Sheng Xiang for their kind help.', '1008.4414-2-27-1': 'This work was supported by the National Science Foundation of China (Grants No. 10473002), the Ministry of Science and Technology National Basic Science program (project 973) under grant No. 2009CB24901, Scientific Research Foundation of Beijing Normal University.'}

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['1008.4414-2-18-5', '1008.4414-3-18-5'], ['1008.4414-2-25-0', '1008.4414-3-25-0'], ['1008.4414-2-25-2', '1008.4414-3-25-2'], ['1008.4414-2-9-1', '1008.4414-3-9-1'], ['1008.4414-2-9-3', '1008.4414-3-9-3'], ['1008.4414-2-9-5', '1008.4414-3-9-5'], ['1008.4414-2-12-0', '1008.4414-3-12-0'], ['1008.4414-2-12-1', '1008.4414-3-12-1'], ['1008.4414-2-12-2', '1008.4414-3-12-2'], ['1008.4414-2-12-3', '1008.4414-3-12-3'], ['1008.4414-2-7-0', '1008.4414-3-7-0'], ['1008.4414-2-7-1', '1008.4414-3-7-1'], ['1008.4414-2-7-2', '1008.4414-3-7-2'], ['1008.4414-2-7-3', '1008.4414-3-7-3'], ['1008.4414-2-7-4', '1008.4414-3-7-4'], ['1008.4414-2-7-5', '1008.4414-3-7-5'], ['1008.4414-2-7-6', '1008.4414-3-7-6'], ['1008.4414-2-7-7', '1008.4414-3-7-7'], ['1008.4414-2-7-8', '1008.4414-3-7-8'], ['1008.4414-2-7-9', '1008.4414-3-7-9'], ['1008.4414-2-3-0', '1008.4414-3-3-0'], ['1008.4414-2-3-1', '1008.4414-3-3-1'], ['1008.4414-2-3-2', '1008.4414-3-3-2'], ['1008.4414-2-3-3', '1008.4414-3-3-3'], ['1008.4414-2-26-0', '1008.4414-3-26-0'], ['1008.4414-2-26-2', '1008.4414-3-26-2'], ['1008.4414-2-26-3', '1008.4414-3-26-3'], ['1008.4414-2-26-4', '1008.4414-3-26-4'], ['1008.4414-2-26-5', '1008.4414-3-26-5'], ['1008.4414-2-26-6', '1008.4414-3-26-6'], ['1008.4414-2-0-0', '1008.4414-3-0-0'], ['1008.4414-2-0-1', '1008.4414-3-0-1'], ['1008.4414-2-0-2', '1008.4414-3-0-2'], ['1008.4414-2-0-3', '1008.4414-3-0-3'], ['1008.4414-2-0-4', '1008.4414-3-0-4'], ['1008.4414-2-10-0', '1008.4414-3-10-0'], ['1008.4414-2-10-1', '1008.4414-3-10-1'], ['1008.4414-2-10-2', '1008.4414-3-10-2'], ['1008.4414-2-10-3', '1008.4414-3-10-3'], ['1008.4414-2-10-4', '1008.4414-3-10-4'], ['1008.4414-2-10-5', '1008.4414-3-10-5'], ['1008.4414-2-10-6', '1008.4414-3-10-6'], ['1008.4414-2-10-7', '1008.4414-3-10-7'], ['1008.4414-2-4-2', '1008.4414-3-4-2'], ['1008.4414-2-4-3', '1008.4414-3-4-3'], ['1008.4414-2-13-0', '1008.4414-3-13-0'], ['1008.4414-2-13-1', '1008.4414-3-13-1'], ['1008.4414-2-13-2', '1008.4414-3-13-2'], ['1008.4414-2-13-3', '1008.4414-3-13-3'], ['1008.4414-2-13-4', '1008.4414-3-13-4'], ['1008.4414-2-13-5', '1008.4414-3-13-5'], ['1008.4414-2-13-6', '1008.4414-3-13-6'], ['1008.4414-2-13-7', '1008.4414-3-13-7'], ['1008.4414-2-13-8', '1008.4414-3-13-8'], ['1008.4414-2-13-9', '1008.4414-3-13-9'], ['1008.4414-2-13-10', '1008.4414-3-13-10'], ['1008.4414-2-13-11', '1008.4414-3-13-11'], ['1008.4414-2-13-12', '1008.4414-3-13-12'], ['1008.4414-2-20-0', '1008.4414-3-20-0'], ['1008.4414-2-20-1', '1008.4414-3-20-1'], ['1008.4414-2-20-2', '1008.4414-3-20-2'], ['1008.4414-2-20-4', '1008.4414-3-20-4'], ['1008.4414-2-24-0', '1008.4414-3-24-0'], ['1008.4414-2-24-2', '1008.4414-3-24-2'], ['1008.4414-2-24-3', '1008.4414-3-24-3'], ['1008.4414-2-17-0', '1008.4414-3-17-0'], ['1008.4414-2-17-1', '1008.4414-3-17-1'], ['1008.4414-2-17-2', '1008.4414-3-17-2'], ['1008.4414-2-21-0', '1008.4414-3-21-0'], ['1008.4414-2-21-2', '1008.4414-3-21-2'], ['1008.4414-2-21-3', '1008.4414-3-21-3'], ['1008.4414-2-21-4', '1008.4414-3-21-4'], ['1008.4414-2-21-6', '1008.4414-3-21-6'], ['1008.4414-2-21-7', '1008.4414-3-21-7'], ['1008.4414-2-21-8', '1008.4414-3-21-8'], ['1008.4414-2-22-0', '1008.4414-3-22-0'], ['1008.4414-2-22-1', '1008.4414-3-22-1'], ['1008.4414-2-22-2', '1008.4414-3-22-2'], ['1008.4414-2-22-3', '1008.4414-3-22-3'], ['1008.4414-2-22-4', '1008.4414-3-22-4'], ['1008.4414-2-22-5', '1008.4414-3-22-5']]

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[]

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[]

['1008.4414-1-4-0', '1008.4414-1-4-1', '1008.4414-1-9-2', '1008.4414-1-9-4', '1008.4414-1-9-7', '1008.4414-1-16-0', '1008.4414-1-19-0', '1008.4414-1-20-3', '1008.4414-1-21-1', '1008.4414-1-21-5', '1008.4414-1-24-1', '1008.4414-1-26-1', '1008.4414-2-4-0', '1008.4414-2-4-1', '1008.4414-2-9-2', '1008.4414-2-9-4', '1008.4414-2-9-7', '1008.4414-2-16-0', '1008.4414-2-19-0', '1008.4414-2-20-3', '1008.4414-2-21-1', '1008.4414-2-21-5', '1008.4414-2-24-1', '1008.4414-2-26-1', '1008.4414-3-4-0', '1008.4414-3-4-1', '1008.4414-3-9-2', '1008.4414-3-9-4', '1008.4414-3-9-7', '1008.4414-3-20-3', '1008.4414-3-21-1', '1008.4414-3-21-5', '1008.4414-3-24-1', '1008.4414-3-26-1']

{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '3': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1008.4414

{'1008.4414-3-0-0': 'We propose two efficient numerical methods of evaluating the luminosity distance in the spatially flat [MATH] universe.', '1008.4414-3-0-1': 'The first method is based on the Carlson symmetric form of elliptic integrals, which is highly accurate and can replace numerical quadratures.', '1008.4414-3-0-2': 'The second method, using a modified version of Hermite interpolation, is less accurate but involves only basic numerical operations and can be easily implemented.', '1008.4414-3-0-3': 'We compare our methods with other numerical approximation schemes and explore their respective features and limitations.', '1008.4414-3-0-4': 'Possible extensions of these methods to other cosmological models are also discussed.', '1008.4414-3-1-0': '# Introduction', '1008.4414-3-2-0': 'The computation of cosmological distances naturally arises in the study of cosmology, for example the luminosity distance [MATH] in the analysis of type Ia supernova (SNIa) data and the angular diameter distance [MATH] in the study of gravitational lensing.', '1008.4414-3-2-1': 'These distances depend on the underlying cosmological model and their parameters.', '1008.4414-3-2-2': 'Therefore they are useful as cosmological tests.', '1008.4414-3-2-3': 'As a result, accurate and efficient numerical algorithms of evaluating these distances become a necessity for the practitioners of cosmological research.', '1008.4414-3-3-0': 'The analytical form of the cosmological distances can be derived from the solution of the Friedmann equation, an ordinary differential equation involving the scale factor [MATH] as a function of cosmic time [MATH].', '1008.4414-3-3-1': 'Therefore, formulae for the distances usually involve an integral over the expansion history expressed in terms of the redshift [MATH] and cosmological parameters.', '1008.4414-3-3-2': 'In general, the integrations can be evaluated numerically by quadrature algorithms.', '1008.4414-3-3-3': 'However, numerical quadratures tend to be computationally heavy when high accuracy is desired.', '1008.4414-3-4-0': 'defcitealias1999ApJS.', '1008.4414-3-4-1': '.120...49PPen99 defcitealias2010MNRAS.406.', '1008.4414-3-4-2': '.548WWU10 In the presence of this performance issue, it is advantageous to develop algorithms that are restricted to specific cosmological models but are otherwise more efficient than general-purpose quadratures.', '1008.4414-3-4-3': 'For the spatially flat [MATH] model, efficient algorithms for the luminosity distnace have been proposed by [CITATION] and [CITATION].', '1008.4414-3-5-0': 'In this paper we propose two different numerical methods for the luminosity distance also in the context of the spatially flat [MATH] universe.', '1008.4414-3-5-1': 'The methods are presented in Sections 2 and 3 respectively.', '1008.4414-3-5-2': 'In Section 4, the performances of these methods are discussed.', '1008.4414-3-5-3': 'Finally in Section 5 we discuss some possible extensions to the methods presented in this paper.', '1008.4414-3-5-4': 'Throughout the paper we will focus on the luminosity distance only, but the results can be trivially extended to compute the angular diameter distance [MATH].', '1008.4414-3-6-0': '# Method I: Evaluation Using Carlson Symmetric Forms', '1008.4414-3-7-0': 'The luminosity distance [MATH] in the spatially flat [MATH] universe is given by [EQUATION] where [MATH] and [MATH] are the energy densities corresponding to the matter and cosmological constant respectively: [MATH].', '1008.4414-3-7-1': 'Following the notation intalias1999ApJS.', '1008.4414-3-7-2': '.120...49P we introduce the parameter [MATH] and the change-of-variable [MATH], and re-write equation ([REF]) as [EQUATION] where [EQUATION]', '1008.4414-3-7-3': 'The integral in equations ([REF]) and ([REF]) are special cases of elliptic integrals.', '1008.4414-3-7-4': 'All elliptic integrals can be reduced to several basic forms, the best known of which are probably the three kinds of Legendre elliptic integrals , with reduction theorems and examples presented in .', '1008.4414-3-7-5': 'In our case it is clearer to express this integral by one of the Carlson symmetric forms [MATH], which is defined as [EQUATION]', '1008.4414-3-7-6': 'Using the reduction theorems, it is straightforward to verify that [EQUATION] where [EQUATION]', '1008.4414-3-7-7': 'It has been known that the Carlson forms can be computed numerically with high accuracy.', '1008.4414-3-7-8': '[CITATION] showed that the computation of [MATH] can be accomplished iteratively with the error decreasing by a factor of [MATH] after each iteration, therefore achieving fast convergence.', '1008.4414-3-7-9': 'Further analysis of the algorithms for [MATH] and other elliptic integrals can be found in , and computer implementation details have been discussed in and .', '1008.4414-3-8-0': '# Method II: Approximation by a Modified Hermite Interpolation', '1008.4414-3-9-0': 'The method presented in Section 2 uses an iterative approach to the computation of [MATH].', '1008.4414-3-9-1': 'However, there are situations where a closed, approximate formula for the integral in equation ([REF]) is desired.', '1008.4414-3-9-2': 'In 1999ApJS.', '1008.4414-3-9-3': '.120...49P an approximation was obtained using polynomial fit for [MATH].', '1008.4414-3-9-4': 'Intalias2010MNRAS.406.', '1008.4414-3-9-5': '.548W another method with higher accuracy was proposed.', '1008.4414-3-9-6': 'In this section we show how a modified version of Hermite interpolation can lead to a class of approximations similar to that in 1999ApJS.', '1008.4414-3-9-7': '.120...49P.', '1008.4414-3-10-0': 'We intend to approximate equation ([REF]) using only elementary operations, such as polynomial evaluation and [MATH]th root where [MATH] is a small integer.', '1008.4414-3-10-1': 'We note that the behavior of [MATH] has several deficiencies.', '1008.4414-3-10-2': 'First, the derivative of [MATH] becomes singular as [MATH].', '1008.4414-3-10-3': 'Second, the domain of [MATH] extends to infinity.', '1008.4414-3-10-4': 'Either one is detrimental to the approximation using polynomials.', '1008.4414-3-10-5': 'However, they can be removed by certain change-of-variables.', '1008.4414-3-10-6': 'For example, we can introduce a new function [EQUATION] that has smooth derivatives within the interval [MATH] and can be extended to the cases of [MATH] and [MATH].', '1008.4414-3-10-7': 'The limiting behaviors of [MATH] are shown below: [EQUATION] where [MATH] is a numerical constant.', '1008.4414-3-11-0': 'Using the end-point conditions in equation ([REF]) one can construct a 3rd-order polynomial, which is a linear combination of the four Hermite basis splines in [MATH], as a crude approximation with [MATH]20% relative error.', '1008.4414-3-11-1': 'This linear combination is unique, allowing no further improvements.', '1008.4414-3-11-2': 'However, we note that for realistic values of [MATH] it is not necessary to approximate [MATH] in the entire interval [MATH], because the subinterval [MATH] corresponds to the scenario of [MATH], i.e. "the future".', '1008.4414-3-11-3': 'Therefore, we can introduce a free parameter [MATH] as the alternative lower end-point, and only perform the approximation in the subinterval [MATH], if a constraint is put on [MATH] (or equivalently, [MATH]).', '1008.4414-3-12-0': 'To accommodate further refinements, a correction term [MATH] can be added to the Hermite approximation.', '1008.4414-3-12-1': 'We require the value and first derivative of [MATH] to vanish at either end-point, so that it can be added to the Hermite approximation without altering the coefficients on the basis splines.', '1008.4414-3-12-2': 'One choice of [MATH] is made possible by a family of functions [EQUATION] where [MATH] and [MATH] are adjustable parameters accounting for the deviation of the Hermite approximation from the true function.', '1008.4414-3-12-3': 'Other choices are possible, but we will begin with the simple case of equation ([REF]).', '1008.4414-3-13-0': 'By construction, the approximation described above has the property that the approximating function coincides with the true function at the end-points, [MATH] and [MATH], up to the first derivative.', '1008.4414-3-13-1': 'But we note that the goal is to approximate equation ([REF]) rather than equation ([REF]).', '1008.4414-3-13-2': 'This suggests that the implicit requirement of the coincidence of function values at end-points could be unnecessarily strong.', '1008.4414-3-13-3': 'Alternatively, we may refrain from requiring the approximating function values to match the true ones.', '1008.4414-3-13-4': 'Instead, we only require the matching of first derivatives at [MATH], and leave the end-point value at another free parameter.', '1008.4414-3-13-5': 'To summarize, we now have four free parameters that can be tuned: [MATH], [MATH] and [MATH], and the function value [MATH] at [MATH].', '1008.4414-3-13-6': 'The approximation to equation ([REF]) can be expressed as [EQUATION] where [EQUATION] and [MATH] are the Hermite basis splines, [EQUATION]', '1008.4414-3-13-7': 'Following the approach intalias1999ApJS.', '1008.4414-3-13-8': '.120...49P, we choose the objective function as the maximum relative error in [MATH] using the approximation ([Eq. [REF]]), with the restriction [MATH].', '1008.4414-3-13-9': 'Minimizing the objective function over the parameters, we obtain the best-fit [MATH], [MATH], [MATH], and [MATH].', '1008.4414-3-13-10': 'Substituting the numerical values into equation ([REF]), we therefore construct an approximation polynomial [EQUATION]', '1008.4414-3-13-11': 'Equation ([REF]) is the main result of this section.', '1008.4414-3-13-12': 'With the parameters determined, the approximation to [MATH] can be computed using this formula with equations ([REF]) and ([REF]).', '1008.4414-3-14-0': '# Performance of the Methods', '1008.4414-3-15-0': 'In this section we proceed to assess the performance of the methods in Sections 2 and 3.', '1008.4414-3-15-1': 'The assessment is mainly done in terms of the accuracy and efficiency.', '1008.4414-3-16-0': '## Accuracy', '1008.4414-3-17-0': 'The first method can be used to yield highly accurate numerical approximation of [MATH] for vast ranges of [MATH] and the parameter [MATH] if we adopt the algorithm for [MATH] by [CITATION].', '1008.4414-3-17-1': 'Unlike the methods based on the evaluation of a closed approximation formula, the desired cutoff error can be prescribed to determine when the iterative computation of [MATH] terminates.', '1008.4414-3-17-2': 'In practice, we found that the prescription of relative error [MATH] can be achieved without suffering significant loss in the computation speed.', '1008.4414-3-18-0': 'For the second method, we plot the distribution of the relative error of [MATH] in Figure [REF].', '1008.4414-3-18-1': 'As can be seen from the figure, the second method remains an approximation at best.', '1008.4414-3-18-2': 'Under our choice of fitting parameters and range of [MATH], the relative error in [MATH] is [MATH]0.5%.', '1008.4414-3-18-3': 'The major source of this error is contributed by [MATH].', '1008.4414-3-18-4': 'For [MATH] our method is comparable with that oftalias1999ApJS.', '1008.4414-3-18-5': '.120...49P, and ours slightly outperforms it when [MATH] is larger.', '1008.4414-3-19-0': '## Efficiency', '1008.4414-3-20-0': 'Theoretically, the best-, worst-, and average-case temporal efficiencies for each method can be calculated or estimated by tracking every operation taken during the course of the computing.', '1008.4414-3-20-1': 'However, such a thorough analysis is beyond the scope of this paper.', '1008.4414-3-20-2': 'Instead, we empirically compare the running time of the computer programs using the two methods with those of 1999ApJS.', '1008.4414-3-20-3': '.120...49P andtalias2010MNRAS.406.', '1008.4414-3-20-4': '.548W under a controlled environment.', '1008.4414-3-21-0': 'In Figure [REF] we display the benchmark results of our methods compared with that oftalias1999ApJS.', '1008.4414-3-21-1': '.120...49P and 2010MNRAS.406.', '1008.4414-3-21-2': '.548W. To simulate a "real-world" application of these methods, we creates a sample of SNIa redshifts using the Supernova/Acceleration Probe fiducial redshift distribution containing [MATH] redshift points distributed within [MATH] .', '1008.4414-3-21-3': 'Our sample satisfies the same distribution to the SNAP fiducial, but is 16 times as dense, i.e. with [MATH] points in total.', '1008.4414-3-21-4': 'We have made custom implementations of the methods fromtalias1999ApJS.', '1008.4414-3-21-5': '.120...49P,talias2010MNRAS.406.', '1008.4414-3-21-6': '.548W, and our Method II in the C programming language, and uses the GNU Scientific Library (GSL) implementation of the [MATH] algorithm in for Method I.', '1008.4414-3-21-7': 'In our benchmark routine, the computing of [MATH] values from our redshift sample is performed for [MATH] respectively, with each pass through the [MATH] sample repeated for 25 times (that sums up to a total of [MATH] evaluations of [MATH]).', '1008.4414-3-21-8': 'The benchmark itself is repeated for 2400 times.', '1008.4414-3-22-0': 'To interpret Figure [REF], we make two remarks.', '1008.4414-3-22-1': 'First, the execution time results were collected from the output of the gprof profiler and does not reflect the absolute time spent.', '1008.4414-3-22-2': 'It is only meaningful as a relative measure useful for comparing the speed of the codes with each other.', '1008.4414-3-22-3': 'Second, the results are dependent on our particular implementations as well as the computing environment.', '1008.4414-3-22-4': 'This is evident if our Figure [REF] is compared with Figure 4 intalias2010MNRAS.406..', '1008.4414-3-22-5': '.548W that shows a reversed result for the speeds of the two methods in 1999ApJS..', '1008.4414-3-23-0': '# Discussion', '1008.4414-3-24-0': 'As Figure [REF] suggests, both methods proposed in this paper is slower than thetalias1999ApJS.', '1008.4414-3-24-1': '.120...49P method.', '1008.4414-3-24-2': 'However, Method I is a very reasonable trade-off between an enormous gain in accuracy and small loss of efficiency.', '1008.4414-3-24-3': 'With Method I one does not need to resort to the numerical quadrature for the same level of accuracy.', '1008.4414-3-25-0': 'Method I can be extended to cover the [MATH] model with a curvature term [MATH], because in that case the equivalent of equation ([REF]) assumes the form [EQUATION] where [EQUATION] is the expansion rate ([MATH]), and [EQUATION]', '1008.4414-3-25-1': 'The integral in equation ([REF]) is also an elliptic integral and can be reduced to [MATH] accordingly.', '1008.4414-3-25-2': 'This is potentially useful for the analysis of future SNIa data, because it has been suggested that the spacetime curvature should not be ignored in the probe of dark energy using luminosity distance data .', '1008.4414-3-26-0': 'In contrast, Method II may not be as promising, because in its current form the accuracy does not outperform that oftalias1999ApJS.', '1008.4414-3-26-1': '.120...49P.', '1008.4414-3-26-2': 'However, the idea behind the method may be useful when extending to alternative cosmological models (for example, dynamical dark energy) which may not be reduced to the applicable scenarios of Method I.', '1008.4414-3-26-3': 'In the description of this method we have left some arbitrariness unjustified, notably the particular choice of the singularity-removing transformation (Eq. [[REF]]), the parameterization of the correction term (Eq. [[REF]]), and the very choice of Hermite basis splines.', '1008.4414-3-26-4': 'Alternative choices of them may be adopted to generate better approximations, for instance, the use of low-order Hermite-Birkhoff interpolation to selectively choose the point [MATH] near which the derivative information of the true function is to be best preserved.', '1008.4414-3-26-5': 'Moreover, our Method II uses only elementary numerical operations, while in 2010MNRAS.406.', '1008.4414-3-26-6': '.548W the numerical logarithm is extensively used.'}

hep-ph-0312111

{'hep-ph-0312111-1-0-0': 'We set up a framework for a model-independent analysis of the time variation of [MATH], [MATH], and [MATH] indiviually.', 'hep-ph-0312111-1-0-1': 'It is shown that the time-evolution of each constant can be determined uniquely from the time evolution of the fine structure constant [MATH] provided that the choice of basic time-independent units (i.e., the clock and ruler) is fixed.', 'hep-ph-0312111-1-0-2': 'Realistic systems of units are considered as examples and implications for metrology are discussed.', 'hep-ph-0312111-1-1-0': 'Recently evidence has been reported for a possible time evolution of the fine structure constant on the cosmological time scale.', 'hep-ph-0312111-1-1-1': 'The study of absorbtion spectra of distant quasars yields [MATH] over the redshift range [MATH] [CITATION].', 'hep-ph-0312111-1-1-2': 'Preliminary results of a new study based on a larger sample support the earlier results [CITATION].', 'hep-ph-0312111-1-2-0': 'The constancy of the proton-to-electron mass ratio [MATH] is also being investigated [CITATION].', 'hep-ph-0312111-1-3-0': 'Motivation for possible variability of fundamental constants appears within different contexts such as Kaluza-Klein theories, superstring theories and other models.', 'hep-ph-0312111-1-3-1': 'We will not go into theoretical discussion of what would be a microscopic model that could explain why fundamental constants change ( see e.g. [CITATION] and references therein).', 'hep-ph-0312111-1-3-2': 'Apart from specific model building, a more general type of analysis is needed which would start with kinematical (i.e., metrological) considerations.', 'hep-ph-0312111-1-3-3': 'The role of the metrological approach is to complement and clarify the specific models (rather than to substitute them).', 'hep-ph-0312111-1-4-0': 'The question arises: if the fine structure constant does indeed evolve with time then which of the constants [MATH], [MATH], and [MATH] are evolving and how?', 'hep-ph-0312111-1-4-1': 'This question has been at the centre of recent debate [CITATION] and is the main topic of this paper.', 'hep-ph-0312111-1-4-2': 'Another issue of contention is the variation of dimensional quantities, where on one hand it is claimed that time variation of dimensional constants is a meaningless concept.', 'hep-ph-0312111-1-4-3': 'On the other hand, a number of theories have been proposed in which one (or more) dimensional constants (such as [MATH], [MATH] or [MATH]) are varying.', 'hep-ph-0312111-1-5-0': 'Our analysis will explicitely show in what sense and to what extent the time evolution of dimensional constants is meaningful and when it becomes misleading; hence bringing the two points of view closer to each other and eliminating some controversy in this area.', 'hep-ph-0312111-1-6-0': 'Fundamental constants and basic units form an interdependent system.', 'hep-ph-0312111-1-6-1': 'This is because the modern definitions of basic units (metre, second etc.) are given in terms of processes controlled by the magnitudes of the fundamental constants ([MATH], [MATH], [MATH] etc.).', 'hep-ph-0312111-1-7-0': 'If fundamental constants would change with time then, generally, so would the basic units.', 'hep-ph-0312111-1-8-0': 'Consequently, we face the following problem: how to define the basic units in such a way that they stay invariant even though the fundamental constants (which are involved in their definitions) may change?', 'hep-ph-0312111-1-8-1': 'This problem should arise in any specific theory of [MATH] variation at the stage when the contact with experiment is made; it is therefore desirable to perform a model-independent analysis first.', 'hep-ph-0312111-1-9-0': 'The purpose of the this paper is to propose a possible solution to this problem and to discuss some of its implications.', 'hep-ph-0312111-1-9-1': 'In particular, we will show that the time-dependence of [MATH], [MATH], and [MATH] can be reconstructed uniquely from the time dependence of [MATH] provided that the choice of basic units is fixed.', 'hep-ph-0312111-1-10-0': 'As a basis for our study we employ the standard methods of dimensional and metrological analysis supplemented by an additional postulate requiring that the fundamental units should be time-independent.', 'hep-ph-0312111-1-10-1': 'In other words, we require that the units should be fixed even though constants can vary.', 'hep-ph-0312111-1-10-2': 'Although it is possible, in principle, to use the system of time-dependent units that would greatly complicate both theory and experiment.', 'hep-ph-0312111-1-10-3': 'Theoretically, one would have to keep track of time derivatives of units in all equations making them much longer; the number of experimental parameters to be measured would also greatly increase.', 'hep-ph-0312111-1-11-0': 'For simplicity and correspondence with the previous literature, the centimetre-gram-second (CGS) system of units will be used.', 'hep-ph-0312111-1-11-1': '(This assumption, however, is not important for the essence of our argument.)', 'hep-ph-0312111-1-11-2': 'Also, as there is no strong observational or experimental evidence for the time-dependence of the fundamental constants apart from [MATH], such as the mass ratios ([MATH], ) and [MATH]-factors it will be assumed here that they do not vary with time.', 'hep-ph-0312111-1-12-0': 'To start with, our units of time, length and mass can be defined in the same way as the SI units (with proper scaling if necessary).', 'hep-ph-0312111-1-12-1': 'Thus [MATH] is the length travelled by light in a vacuum during a time interval of 1/299 792 45800 of a second, the second is the duration of 9 192 631 770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the cesium-133 atom and [MATH] is 1/1000 part of the mass of the international platinum-iridium prototype of the kilogram.', 'hep-ph-0312111-1-13-0': 'These definitions imply that the magnitudes of 1 second ([MATH]), 1 centimetre ([MATH]) and 1 gram ([MATH]) depend on the magnitudes of the fundamental constants as follows: [EQUATION]', 'hep-ph-0312111-1-13-1': 'It will be shown that the time evolution of [MATH], [MATH] and [MATH] crucially depend on the definitions of the centimetre and the second.', 'hep-ph-0312111-1-13-2': 'To illustrate this point, several alternative definitions of the centimetre and second (see e.g. [CITATION]) are introduced in the following way.', 'hep-ph-0312111-1-14-0': 'A "bar centimetre" is 1/100 of the length of the international platinum-iridium prototype of the metre.', 'hep-ph-0312111-1-14-1': 'Its dependence on the fundamental constants is controlled by the Bohr radius: [EQUATION]', 'hep-ph-0312111-1-14-2': 'The "krypton centimetre" is the length equal to the 16507.6373 wavelengths of the radiation corresponding to the transition between the levels [MATH] and [MATH] of krypton-86; this length depends on the fundamental constants as the inverse Rydberg constant: [EQUATION] (Both "bar" and "krypton" definitions of the metre were used by the SI system in the past.)', 'hep-ph-0312111-1-15-0': 'In addition, the second can be defined by choosing among different types of clocks other than cesium-133 clock adopted by the SI system; the ammonia clock (a representative of molecular clocks) measures time in terms of molecular vibration frequencies [CITATION]: [EQUATION]', 'hep-ph-0312111-1-15-1': 'An oscillator placed inside a superconducting cavity forms a high-stability device called a SCSO clock.', 'hep-ph-0312111-1-15-2': 'The frequency of such a device is inversely proportional to the size of the superconductive cavity and therefore to the Bohr radius [CITATION] [EQUATION]', 'hep-ph-0312111-1-15-3': 'Correspondingly, the "ammonia second" depends on the fundamental constants as [EQUATION] while the "SCSO second" has this dependence: [EQUATION]', 'hep-ph-0312111-1-15-4': 'Thus the three different definitions of the second are described by [EQUATION] where [MATH] corresponds to the SCSO, ammonia and cesium clock, respectively [CITATION].', 'hep-ph-0312111-1-16-0': 'Similarly, the three different definitions of the centimetre can be combined as [EQUATION] with [MATH] for the bar, krypton and light definitions of unit of length, correspondingly.', 'hep-ph-0312111-1-17-0': 'From dimensional considerations it follows that an arbitrary system of units, [MATH], can be characterized by Eq. ([REF]) and ([REF]) with a suitable choice of clock and ruler indices [MATH] and [MATH].', 'hep-ph-0312111-1-17-1': 'While theoretically all these [MATH] systems are equivalent, only a small number of them can be of interest for the past and future high-precision experiments.', 'hep-ph-0312111-1-17-2': 'The nine systems considered in this paper form a representative sample of such realistic systems of units.', 'hep-ph-0312111-1-18-0': 'Requiring that the centimetre, second and gram do not depend on time, we obtain [EQUATION]', 'hep-ph-0312111-1-18-1': 'These can be conveniently rewritten as [EQUATION]', 'hep-ph-0312111-1-18-2': 'Thus, if time dependence of [MATH] is experimentally measured and units are fixed, there is no further choice on how [MATH] depend on time and no extra measurements are required.', 'hep-ph-0312111-1-18-3': 'The results are presented in Table [REF].', 'hep-ph-0312111-1-19-0': 'How many dimensional constants vary?', 'hep-ph-0312111-1-19-1': 'In all cases except one, at least two constants must vary.', 'hep-ph-0312111-1-19-2': 'These varying constants cannot be set equal to 1 and treated as mere conversion factors.', 'hep-ph-0312111-1-19-3': 'The exceptional case is [MATH], i.e. the "bar" ruler plus the ammonia clock.', 'hep-ph-0312111-1-19-4': 'In this case [MATH] while [MATH] and [MATH] stay constant, which allows one to use the system [MATH], [MATH] (Hartree units).', 'hep-ph-0312111-1-19-5': 'In other words, in this case [MATH] and [MATH] can be treated as mere conversion factors.', 'hep-ph-0312111-1-20-0': 'Is the speed of light constant?', 'hep-ph-0312111-1-20-1': 'For the case of the "light centimetre" (i.e., SI-like definition), [MATH] remains constant automatically; however, in general, [MATH] varies.', 'hep-ph-0312111-1-20-2': 'Interestingly, for any choice of the centimetre there exists a "matching choice" of the second (i.e., [MATH]) so that the speed of light remains constant.', 'hep-ph-0312111-1-20-3': 'Furthermore, if we adopt a system with [MATH] (in such systems [MATH] is constant) then it follows that a) both [MATH] and [MATH] were smaller in the past, and b) the SCSO clock minimizes the time dependence of [MATH] and [MATH] (because this clock has lower [MATH] than other clocks).', 'hep-ph-0312111-1-21-0': 'In theoretical papers aimed at explaining the [MATH] variation in the context of quantum field theory the choice [MATH] is frequently made.', 'hep-ph-0312111-1-21-1': 'Consequently, it is implicitely assumed that both [MATH] and [MATH] do not depend on time.', 'hep-ph-0312111-1-21-2': 'From Eq. (15) and (16) it follows then that the only possible choice for [MATH] and [MATH] is [MATH] and [MATH], i.e. the system of units must be of [MATH]-type.', 'hep-ph-0312111-1-21-3': 'This system differs from the systems of units used in actual experiments (such as systems in Table 1 - e.g., the SI system is of [MATH]-type).', 'hep-ph-0312111-1-21-4': "In other words, the theorist's units would appear time-dependent from the point of view of the experimenter (and the other way round).", 'hep-ph-0312111-1-21-5': 'Therefore the translation from one system to the other should be done explicitely in order to make the contact between the theory and observations.', 'hep-ph-0312111-1-22-0': 'An interesting question is whether or not one can design a high-precision clock with [MATH] or [MATH].', 'hep-ph-0312111-1-22-1': 'Such clocks would be useful in removing the time dependence of [MATH] and [MATH], or [MATH] and [MATH] respectively.', 'hep-ph-0312111-1-23-0': 'These results should be taken into account while comparing the predictions of [MATH] varying theories and the forthcoming experiments on high-precision clock comparison.', 'hep-ph-0312111-1-24-0': 'Particular attention should be taken to ensure that the theoretical and experimental results are expressed in the same system of units.', 'hep-ph-0312111-1-24-1': 'In the case that experiment finds a discrepancy between the readings of two different clocks, and thus confirms the variability of [MATH], the problem will arise as to which clock shows the "true" time.', 'hep-ph-0312111-1-24-2': 'Our discussion would help in making this choice.', 'hep-ph-0312111-1-25-0': 'A topic of active debate in recent literature has been whether or not it is meaningful to talk about the time evolution of dimensional fundamental constants.', 'hep-ph-0312111-1-25-1': 'The discussion presented in this paper goes some way in resolving this problem by demonstrating a framework in which time dependence of dimensional constants becomes a well defined concept.', 'hep-ph-0312111-1-25-2': 'Such frameworks require a set of basic assumptions to be fixed (in this particular case, that the units of length, time and mass remain constant.)', 'hep-ph-0312111-1-25-3': 'This can be compared to gauge-dependent quantities and gauge fixing in quantum field theory.', 'hep-ph-0312111-1-26-0': 'In summary, the time evolution of the three fundamental constants [MATH], [MATH] , and [MATH] has been studied from a metrological perspective.', 'hep-ph-0312111-1-26-1': 'Assuming that [MATH] evolution is known from observation, and the choice of time-independent units is fixed, it has been shown that the three separate evolution laws of [MATH], [MATH], and [MATH] can be found explicitely for a generic system of units.', 'hep-ph-0312111-1-26-2': 'As an application of these findings, nine specific unit systems of interest for experiment and theory, based on different realistic choices of clocks and rulers, have been considered.', 'hep-ph-0312111-1-26-3': 'Time variations of [MATH], [MATH], and [MATH] in these nine cases are found and compared, and their significance to the forthcoming experimental tests of [MATH] variation are discussed.', 'hep-ph-0312111-1-27-0': 'We are grateful to G.C.Joshi, W.McBride, B.H.J.McKellar and R.R.Volkas for stimulating discussions.'}

{'hep-ph-0312111-2-0-0': 'We set up a framework for a model-independent analysis of the time variation of [MATH], [MATH], and [MATH] individually.', 'hep-ph-0312111-2-0-1': 'It is shown that the time-evolution of each constant can be determined uniquely from the time evolution of the fine structure constant [MATH] provided that the choice of basic time-independent units (i.e., the clock and ruler) is fixed.', 'hep-ph-0312111-2-0-2': 'Realistic systems of units are considered as examples and implications for metrology are discussed.', 'hep-ph-0312111-2-1-0': 'Recently evidence has been reported for a possible time evolution of the fine structure constant on the cosmological time scale.', 'hep-ph-0312111-2-1-1': 'The study of absorbtion spectra of distant quasars yields [MATH] over the redshift range [MATH] [CITATION] .', 'hep-ph-0312111-2-1-2': 'Preliminary results of a new study based on a larger sample support the earlier results [CITATION].', 'hep-ph-0312111-2-1-3': 'However, a similar analysis performed by a different group finds no evidence for the evolution of [MATH]: [MATH] over the redshift range [MATH] [CITATION].', 'hep-ph-0312111-2-1-4': 'Also, the variation of the fine structure constant in [MATH] decay over the age of the solar system (corresponding to [MATH]) has been found consistent with zero at the level of [MATH] [CITATION].', 'hep-ph-0312111-2-1-5': 'The source for these discrepancies is yet to be understood.', 'hep-ph-0312111-2-2-0': 'The constancy of the proton-to-electron mass ratio [MATH] is also being investigated [CITATION].', 'hep-ph-0312111-2-3-0': 'Motivation for possible variability of fundamental constants appears within different contexts such as Kaluza-Klein theories, superstring theories and other models.', 'hep-ph-0312111-2-3-1': 'We will not go into theoretical discussion of what would be a microscopic model that could explain why fundamental constants change ( see e.g. [CITATION] and references therein ).', 'hep-ph-0312111-2-3-2': 'Apart from specific model building, a more general type of analysis is needed which would start with kinematical (i.e., metrological) considerations.', 'hep-ph-0312111-2-3-3': 'The role of the metrological approach is to complement and clarify the specific models (rather than to substitute them).', 'hep-ph-0312111-2-4-0': 'The question arises: if the fine structure constant does indeed evolve with time then which of the constants [MATH], [MATH], and [MATH] are evolving and how?', 'hep-ph-0312111-2-4-1': 'This question has been at the centre of recent debate [CITATION] and is the main topic of this paper.', 'hep-ph-0312111-2-4-2': 'Another issue of contention is the variation of dimensional quantities, where on one hand it is claimed that time variation of dimensional constants is a meaningless concept.', 'hep-ph-0312111-2-4-3': 'On the other hand, a number of theories have been proposed in which one (or more) dimensional constants (such as [MATH], [MATH] or [MATH]) are varying.', 'hep-ph-0312111-2-5-0': 'Our analysis will explicitely show in what sense and to what extent the time evolution of dimensional constants is meaningful and when it becomes misleading; hence bringing the two points of view closer to each other and eliminating some controversy in this area.', 'hep-ph-0312111-2-6-0': 'Fundamental constants and basic units form an interdependent system.', 'hep-ph-0312111-2-6-1': 'This is because the modern definitions of basic units (metre, second etc.) are given in terms of processes controlled by the magnitudes of the fundamental constants ([MATH], [MATH], [MATH] etc.).', 'hep-ph-0312111-2-7-0': 'If fundamental constants would change with time then, generally, so would the basic units.', 'hep-ph-0312111-2-8-0': 'Consequently, we face the following problem: how to define the basic units in such a way that they stay invariant even though the fundamental constants (which are involved in their definitions) may change?', 'hep-ph-0312111-2-8-1': 'This problem should arise in any specific theory of [MATH] variation at the stage when the contact with experiment is made; it is therefore desirable to perform a model-independent analysis first.', 'hep-ph-0312111-2-9-0': 'The purpose of the this paper is to propose a possible solution to this problem and to discuss some of its implications.', 'hep-ph-0312111-2-9-1': 'In particular, we will show that the time-dependence of [MATH], [MATH], and [MATH] can be reconstructed uniquely from the time dependence of [MATH] provided that the choice of basic units is fixed.', 'hep-ph-0312111-2-10-0': 'As a basis for our study we employ the standard methods of dimensional and metrological analysis supplemented by an additional postulate requiring that the fundamental units should be time-independent.', 'hep-ph-0312111-2-10-1': 'In other words, we require that the units should be fixed even though constants can vary.', 'hep-ph-0312111-2-10-2': 'Although it is possible, in principle, to use the system of time-dependent units that would greatly complicate both theory and experiment.', 'hep-ph-0312111-2-10-3': 'Theoretically, one would have to keep track of time derivatives of units in all equations making them much longer; the number of experimental parameters to be measured would also greatly increase.', 'hep-ph-0312111-2-11-0': 'For simplicity and correspondence with the previous literature, the centimetre-gram-second (CGS) system of units will be used.', 'hep-ph-0312111-2-11-1': '(This assumption, however, is not important for the essence of our argument.)', 'hep-ph-0312111-2-11-2': 'Also, as there is no strong observational or experimental evidence for the time-dependence of the fundamental constants apart from [MATH], such as the mass ratios ([MATH], [MATH], ) and [MATH]-factors it will be assumed here that they do not vary with time (for theoretical discussions see,e.g., [CITATION]).', 'hep-ph-0312111-2-12-0': 'To start with, our units of time, length and mass can be defined in the same way as the SI units (with proper scaling if necessary).', 'hep-ph-0312111-2-12-1': 'Thus [MATH] is the length travelled by light in a vacuum during a time interval of 1/299 792 45800 of a second, the second is the duration of 9 192 631 770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the cesium-133 atom and [MATH] is 1/1000 part of the mass of the international platinum-iridium prototype of the kilogram.', 'hep-ph-0312111-2-13-0': 'These definitions imply that the magnitudes of 1 second ([MATH]), 1 centimetre ([MATH]) and 1 gram ([MATH]) depend on the magnitudes of the fundamental constants as follows: [EQUATION]', 'hep-ph-0312111-2-13-1': 'It will be shown that the time evolution of [MATH], [MATH] and [MATH] crucially depend on the definitions of the centimetre and the second.', 'hep-ph-0312111-2-13-2': 'To illustrate this point, several alternative definitions of the centimetre and second (see e.g. [CITATION]) are introduced in the following way.', 'hep-ph-0312111-2-14-0': 'A "bar centimetre" is 1/100 of the length of the international platinum-iridium prototype of the metre.', 'hep-ph-0312111-2-14-1': 'Its dependence on the fundamental constants is controlled by the Bohr radius: [EQUATION]', 'hep-ph-0312111-2-14-2': 'The "krypton centimetre" is the length equal to the 16507.6373 wavelengths of the radiation corresponding to the transition between the levels [MATH] and [MATH] of krypton-86; this length depends on the fundamental constants as the inverse Rydberg constant: [EQUATION] (Both "bar" and "krypton" definitions of the metre were used by the SI system in the past.)', 'hep-ph-0312111-2-15-0': 'In addition, the second can be defined by choosing among different types of clocks other than cesium-133 clock adopted by the SI system; the ammonia clock (a representative of molecular clocks) measures time in terms of molecular vibration frequencies [CITATION]: [EQUATION]', 'hep-ph-0312111-2-15-1': 'An oscillator placed inside a superconducting cavity forms a high-stability device called a SCSO clock.', 'hep-ph-0312111-2-15-2': 'The frequency of such a device is inversely proportional to the size of the superconductive cavity and therefore to the Bohr radius [CITATION] [EQUATION]', 'hep-ph-0312111-2-15-3': 'Correspondingly, the "ammonia second" depends on the fundamental constants as [EQUATION] while the "SCSO second" has this dependence: [EQUATION]', 'hep-ph-0312111-2-15-4': 'Thus the three different definitions of the second are described by [EQUATION] where [MATH] corresponds to the SCSO, ammonia and cesium clock, respectively [CITATION].', 'hep-ph-0312111-2-16-0': 'Similarly, the three different definitions of the centimetre can be combined as [EQUATION] with [MATH] for the bar, krypton and light definitions of unit of length, correspondingly.', 'hep-ph-0312111-2-17-0': 'From dimensional considerations it follows that an arbitrary system of units, [MATH], can be characterized by Eq. ([REF]) and ([REF]) with a suitable choice of clock and ruler indices [MATH] and [MATH].', 'hep-ph-0312111-2-17-1': 'While theoretically all these [MATH] systems are equivalent, only a small number of them can be of interest for the past and future high-precision experiments.', 'hep-ph-0312111-2-17-2': 'The nine systems considered in this paper form a representative sample of such realistic systems of units.', 'hep-ph-0312111-2-18-0': 'Requiring that the centimetre, second and gram do not depend on time, we obtain [EQUATION]', 'hep-ph-0312111-2-18-1': 'These can be conveniently rewritten as [EQUATION]', 'hep-ph-0312111-2-18-2': 'Thus, if time dependence of [MATH] is experimentally measured and units are fixed, there is no further choice on how [MATH] depend on time and no extra measurements are required.', 'hep-ph-0312111-2-18-3': 'The results are presented in Table [REF].', 'hep-ph-0312111-2-19-0': 'How many dimensional constants vary?', 'hep-ph-0312111-2-19-1': 'In all cases except one, at least two constants must vary.', 'hep-ph-0312111-2-19-2': 'These varying constants cannot be set equal to 1 and treated as mere conversion factors.', 'hep-ph-0312111-2-19-3': 'The exceptional case is [MATH], i.e. the "bar" ruler plus the ammonia clock.', 'hep-ph-0312111-2-19-4': 'In this case [MATH] while [MATH] and [MATH] stay constant, which allows one to use the system [MATH], [MATH] (Hartree units).', 'hep-ph-0312111-2-19-5': 'In other words, in this case [MATH] and [MATH] can be treated as mere conversion factors.', 'hep-ph-0312111-2-20-0': 'Is the speed of light constant?', 'hep-ph-0312111-2-20-1': 'For the case of the "light centimetre" (i.e., SI-like definition), [MATH] remains constant automatically; however, in general, [MATH] varies.', 'hep-ph-0312111-2-20-2': 'Interestingly, for any choice of the centimetre there exists a "matching choice" of the second (i.e., [MATH]) so that the speed of light remains constant.', 'hep-ph-0312111-2-20-3': 'Furthermore, if we adopt a system with [MATH] (in such systems [MATH] is constant) then it follows that a) both [MATH] and [MATH] were smaller in the past, and b) the SCSO clock minimizes the time dependence of [MATH] and [MATH] (because this clock has lower [MATH] than other clocks).', 'hep-ph-0312111-2-21-0': 'In theoretical papers aimed at explaining the [MATH] variation in the context of quantum field theory the choice [MATH] is frequently made.', 'hep-ph-0312111-2-21-1': 'Consequently, it is implicitely assumed that both [MATH] and [MATH] do not depend on time.', 'hep-ph-0312111-2-21-2': 'From Eq. (15) and (16) it follows then that the only possible choice for [MATH] and [MATH] is [MATH] and [MATH], i.e. the system of units must be of [MATH]-type.', 'hep-ph-0312111-2-21-3': 'This system differs from the systems of units used in actual experiments (such as systems in Table 1 - e.g., the SI system is of [MATH]-type).', 'hep-ph-0312111-2-21-4': "In other words, the theorist's units would appear time-dependent from the point of view of the experimenter (and the other way round).", 'hep-ph-0312111-2-21-5': 'Therefore the translation from one system to the other should be done explicitely in order to make the contact between the theory and observations.', 'hep-ph-0312111-2-22-0': 'An interesting question is whether or not one can design a high-precision clock with [MATH] or [MATH].', 'hep-ph-0312111-2-22-1': 'Such clocks would be useful in removing the time dependence of [MATH] and [MATH], or [MATH] and [MATH] respectively.', 'hep-ph-0312111-2-23-0': 'These results should be taken into account while comparing the predictions of [MATH] varying theories and the forthcoming experiments on high-precision clock comparison.', 'hep-ph-0312111-2-24-0': 'Particular attention should be taken to ensure that the theoretical and experimental results are expressed in the same system of units.', 'hep-ph-0312111-2-24-1': 'In the case that experiment finds a discrepancy between the readings of two different clocks, and thus confirms the variability of [MATH], the problem will arise as to which clock shows the "true" time.', 'hep-ph-0312111-2-24-2': 'Our discussion would help in making this choice.', 'hep-ph-0312111-2-25-0': 'A topic of active debate in recent literature has been whether or not it is meaningful to talk about the time evolution of dimensional fundamental constants.', 'hep-ph-0312111-2-25-1': 'The discussion presented in this paper goes some way in resolving this problem by demonstrating a framework in which time dependence of dimensional constants becomes a well defined concept.', 'hep-ph-0312111-2-25-2': 'Such frameworks require a set of basic assumptions to be fixed (in this particular case, that the units of length, time and mass remain constant.)', 'hep-ph-0312111-2-25-3': 'This can be compared to gauge-dependent quantities and gauge fixing in quantum field theory.', 'hep-ph-0312111-2-26-0': 'In summary, the time evolution of the three fundamental constants [MATH], [MATH] , and [MATH] has been studied from a metrological perspective.', 'hep-ph-0312111-2-26-1': 'Assuming that [MATH] evolution is known from observation, and the choice of time-independent units is fixed, it has been shown that the three separate evolution laws of [MATH], [MATH], and [MATH] can be found explicitely for a generic system of units.', 'hep-ph-0312111-2-26-2': 'As an application of these findings, nine specific unit systems of interest for experiment and theory, based on different realistic choices of clocks and rulers, have been considered.', 'hep-ph-0312111-2-26-3': 'Time variations of [MATH], [MATH], and [MATH] in these nine cases are found and compared, and their significance to the forthcoming experimental tests of [MATH] variation are discussed.', 'hep-ph-0312111-2-27-0': 'We are grateful to G.C.Joshi, W.McBride, B.H.J.McKellar and R.R.Volkas for stimulating discussions and to T.Dent and D.F.Mota for valuable comments.'}

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[['hep-ph-0312111-1-0-0', 'hep-ph-0312111-2-0-0'], ['hep-ph-0312111-1-3-1', 'hep-ph-0312111-2-3-1'], ['hep-ph-0312111-1-11-2', 'hep-ph-0312111-2-11-2'], ['hep-ph-0312111-1-1-1', 'hep-ph-0312111-2-1-1']]

[]

[['hep-ph-0312111-1-27-0', 'hep-ph-0312111-2-27-0']]

[]

[]

{'1': 'http://arxiv.org/licenses/assumed-1991-2003/', '2': 'http://arxiv.org/licenses/assumed-1991-2003/'}

https://arxiv.org/abs/hep-ph/0312111

0811.3275

{'0811.3275-1-0-0': 'We formulate the expectation value of the Bell-Zukowski operator acting on qubit states of a two-particle Bell experiment.', '0811.3275-1-0-1': 'By using the equivalence between a set of [MATH] copies of a two-qubit experiment and a standard two-setting Bell experiment in an entangled [MATH]-particle state, we obtain an inequality, which we may call the Bell-Zukowski inequality.', '0811.3275-1-0-2': 'It determines whether the measured correlation functions of two-particle states can be modeled locally and realistically.', '0811.3275-1-0-3': 'In this Bell experiment of two particles, the conflict between local realism and quantum mechanics is discussed in conjunction with the violation of the Bell-Zukowski inequality.', '0811.3275-1-0-4': 'The main point of the result is that the Bell-Zukowski operator can be represented by the Bell-Mermin operator.', '0811.3275-1-0-5': 'The threshold visibility of two-particle interference analyzed in this scheme shows good agreement with the value to cause a violation of the Bell-Zukowski inequality.', '0811.3275-1-1-0': '# Introduction', '0811.3275-1-2-0': 'Bell inequalities that correlation functions satisfying local realistic theories must obey can be violated by certain quantum predictions, as Bell reported in 1964 [CITATION].', '0811.3275-1-2-1': 'Bell used the singlet state, or EPR pairs [CITATION], to show that the correlation functions measured in such singlet states cannot be modeled by local realistic models.', '0811.3275-1-2-2': 'Likewise, a certain set of correlation functions produced by quantum measurements of a quantum state contradicts certain predictions of local realistic theories.', '0811.3275-1-2-3': 'Those states also cannot be modeled by local realistic models.', '0811.3275-1-2-4': 'Up to now, local realistic theories have been studied extensively [CITATION].', '0811.3275-1-2-5': 'Many experiments have shown that Bell inequalities and local realistic theories are violated [CITATION].', '0811.3275-1-2-6': 'Later, in a work by Fine [CITATION], a set of correlation functions can be described with the property that they are reproducible by local realistic theories for a system in two-partite states if and only if the set of correlation functions satisfies the complete set of (two-setting) Bell inequalities.', '0811.3275-1-2-7': 'This result is generalized [CITATION] to a system described by multipartite states in the case where two dichotomic observables are measured per site.', '0811.3275-1-3-0': 'In this paper, we present a method using two Bell operators [CITATION] to refute local realistic models of a quantum state.', '0811.3275-1-3-1': 'In order to do so, we need only a two-setting and two-particle Bell experiment reproducible by local realistic theories.', '0811.3275-1-3-2': 'Such a Bell experiment also reveals the conflict between local realism and quantum mechanics in the sense that the Bell-Zukowski inequality [CITATION] is violated.', '0811.3275-1-4-0': 'Let us consider two-qubit states that, under specific settings, give correlation functions reproducible by local realistic theories.', '0811.3275-1-4-1': 'Imagine that [MATH] copies of the states can be distributed among [MATH] parties in such a way that each pair of parties shares one copy of the state.', '0811.3275-1-4-2': 'The parties perform a Bell-Greenberger-Horne-Zeilinger (GHZ) [MATH]-particle experiment [CITATION] on their qubits.', '0811.3275-1-4-3': 'Each of the pairs of parties uses the measurement settings noted above.', '0811.3275-1-4-4': 'The Bell-Mermin operator [CITATION], [MATH], for their experiment does not show any violation of local realism.', '0811.3275-1-4-5': 'Nevertheless, one can find another Bell operator, which differs from [MATH] by a numerical factor, that does show such a violation.', '0811.3275-1-4-6': 'That is, the original two-qubit states cannot be modeled by local realistic models.', '0811.3275-1-5-0': 'More specifically, the situation is as follows: A given two-setting and two-particle Bell experiment is reproducible by local realistic theories.', '0811.3275-1-5-1': 'However, the experimental correlation functions can compute a violation of the Bell-Zukowski inequality.', '0811.3275-1-5-2': 'Therefore, actually measured data reveal that the measured state cannot be modeled by local realistic models.', '0811.3275-1-5-3': 'Thus, a conflict between local realism and quantum mechanics is revealed.', '0811.3275-1-5-4': 'We can see this phenomenon by the simple algebra presented below.', '0811.3275-1-6-0': 'This phenomenon can occur when the system is in a mixed two-qubit state.', '0811.3275-1-6-1': 'We analyze the threshold visibility for two-particle interference to reveal the conflict mentioned above.', '0811.3275-1-6-2': 'It is found that the threshold visibility agrees with the value to obtain a violation of the Bell-Zukowski inequality.', '0811.3275-1-7-0': '# Bell-Mermin operator and Bell-Zukowski operator', '0811.3275-1-8-0': 'Let [MATH] be [MATH].', '0811.3275-1-8-1': 'We consider the following specific Bell-Mermin operator (see Eq. ([REF])): [EQUATION]', '0811.3275-1-8-2': 'Here, the states [MATH] are GHZ states [CITATION], i.e., [EQUATION]', '0811.3275-1-8-3': 'An average of the Bell-Mermin operator is evaluated by using a standard two-setting Bell experiment.', '0811.3275-1-8-4': 'See Fig. [REF].', '0811.3275-1-9-0': 'One can introduce a [MATH]-partite Bell operator, which one may call the Bell-Zukowski operator [MATH], which differs from [MATH] only by a numerical factor.', '0811.3275-1-9-1': 'The Bell-Zukowski operator [MATH] [CITATION] is [EQUATION]', '0811.3275-1-9-2': 'An average of the Bell-Zukowski operator is evaluated by using an all-setting Bell experiment.', '0811.3275-1-9-3': 'See Fig. [REF].', '0811.3275-1-10-0': 'Clearly, we see that the Bell-Mermin operator given in Eq. ([REF]) is connected to the Bell-Zukowski operator [MATH] in the following relation: [EQUATION]', '0811.3275-1-10-1': 'One can see that the specific two-setting Bell [MATH]-particle experiment in question computes an average value of the Bell-Zukowski operator [MATH] when an average value of [MATH] is evaluated.', '0811.3275-1-10-2': 'Of course, this argument is due to the validity of quantum mechanics.', '0811.3275-1-10-3': 'See Fig. [REF].', '0811.3275-1-11-0': 'From the Bell-Zukowski inequality [EQUATION] we have a condition on the average of the Bell-Mermin operator [MATH], which is written by [EQUATION]', '0811.3275-1-11-1': "Please notice that the Bell-Zukowski inequality [MATH] is derived under the assumption that there are predetermined 'hidden' results of the measurement for all directions in the rotation plane for the system in a state.", '0811.3275-1-11-2': "On the other hand, the Bell-Mermin inequality is derived under the assumption that there are predetermined 'hidden' results of the measurement for two directions for the system in a state.", '0811.3275-1-11-3': 'We see that a violation of the condition in Eq. ([REF]) implies a violation of the Bell-Zukowski inequality.', '0811.3275-1-11-4': 'Our aim is to compute an expectation value of the Bell-Mermin operator given in Eq. ([REF]) by using a two-particle Bell experiment reproducible by local realistic theories.', '0811.3275-1-11-5': 'The Bell-Zukowski inequality is stronger than the standard Bell inequalities for [MATH].', '0811.3275-1-11-6': 'This is why a standard Bell experiment reproducible by local realistic theories reveals the conflict between local realism and quantum mechanics.', '0811.3275-1-12-0': '# Experimental situation', '0811.3275-1-13-0': 'We consider the following two-qubit states: [EQUATION] where [MATH] is a Bell state defined as [EQUATION] [MATH] is the random noise admixture.', '0811.3275-1-13-1': 'The value of [MATH] can be interpreted as the reduction factor of the interferometric contrast observed in the two-particle correlation experiment.', '0811.3275-1-13-2': 'The states [MATH] are eigenstates of the [MATH]-component of the Pauli observable, [MATH], for the [MATH]th observer.', '0811.3275-1-13-3': 'Here, [MATH] and [MATH] are the labels of the parties (say Alice and Bob).', '0811.3275-1-13-4': 'Then, we have [MATH] and [MATH].', '0811.3275-1-14-0': 'Here, [MATH] and [MATH] are Pauli-spin operators for the [MATH]-component and for the [MATH]-component, respectively.', '0811.3275-1-14-1': 'This set of experimental correlation functions is described with the property that they are reproducible by local realistic theories.', '0811.3275-1-14-2': 'See the following relations along with the arguments in Ref. 10: [EQUATION]', '0811.3275-1-14-3': 'In the following section, we will use this kind of experimental situation.', '0811.3275-1-14-4': 'Those experimental correlation functions can compute a violation of the Bell-Zukowski inequality.', '0811.3275-1-14-5': 'In order to do so, we shall compute an expectation value of the Bell-Mermin operator in the next section.', '0811.3275-1-15-0': '# Conflict between local realism and quantum mechanics', '0811.3275-1-16-0': 'Imagine that [MATH] copies of the states introduced in the preceding section can be distributed among [MATH] parties in such a way that each pair of parties shares one copy of the state [EQUATION]', '0811.3275-1-16-1': 'Suppose that spatially separated [MATH] observers perform measurements on each of [MATH] particles.', '0811.3275-1-16-2': 'The decision processes for choosing measurement observables are space-like separated.', '0811.3275-1-16-3': 'It can be regarded as a standard two-setting Bell experiment in an entangled state in [MATH] particles.', '0811.3275-1-16-4': 'See Fig. [REF].', '0811.3275-1-17-0': 'We assume that a two-orthogonal-setting Bell-GHZ [MATH]-particle correlation experiment is performed.', '0811.3275-1-17-1': 'We choose measurement observables such that [EQUATION]', '0811.3275-1-17-2': 'Namely, each of the pairs of parties uses measurement settings such that they can check the condition in Eq. ([REF]).', '0811.3275-1-17-3': 'Therefore, it should be that given [MATH] correlation functions are described with the property that they are reproducible by local realistic theories.', '0811.3275-1-17-4': 'The Bell-Mermin operators [MATH] and [MATH] do not show any violation of local realism as shown below.', '0811.3275-1-18-0': 'Let [MATH] denote the function [EQUATION] [MATH] is invertible as [EQUATION]', '0811.3275-1-18-1': 'The Bell-Mermin operators [MATH] and [MATH] are defined by [CITATION] [EQUATION]', '0811.3275-1-18-2': 'The Bell-Mermin inequality can be expressed as [CITATION] [EQUATION]', '0811.3275-1-18-3': 'We also define [MATH] for any subset [MATH] by [EQUATION]', '0811.3275-1-18-4': 'It is easy to see that, when [MATH] are disjoint, [EQUATION] which leads to following equations: [EQUATION]', '0811.3275-1-18-5': 'In specific operators [MATH] and [MATH] given in Eq. ([REF]), where [EQUATION] and [EQUATION] we have (cf. [CITATION]) [EQUATION] and [EQUATION]', '0811.3275-1-18-6': 'Hence, we obtain [EQUATION] where [EQUATION]', '0811.3275-1-18-7': 'Measurements on each of [MATH] particles enable them to obtain [MATH] correlation functions.', '0811.3275-1-18-8': 'Thus, they get an average value of the specific Bell-Mermin operator given in Eq. ([REF]).', '0811.3275-1-18-9': 'According to Eq. ([REF]), we obtain [EQUATION]', '0811.3275-1-18-10': 'Clearly, the Bell-Mermin operators [MATH] and [MATH] for their experiment do not show any violation of local realism as we have mentioned above.', '0811.3275-1-19-0': 'Nevertheless, when [MATH] and [MATH] is given by [EQUATION] we have a violation of the condition in Eq. ([REF]), i.e., one can compute a violation of the Bell-Zukowski inequality [MATH] that is, the measured two-qubit state cannot be modeled by local realistic models.', '0811.3275-1-19-1': 'The condition in Eq. ([REF]) says that the threshold visibility decreases when the number of copies, [MATH], increases.', '0811.3275-1-19-2': 'In an extreme situation, when [MATH], we have the desired condition [EQUATION] to show the conflict in question.', '0811.3275-1-19-3': 'This agrees with the value to get a violation of the Bell-Zukowski inequality.', '0811.3275-1-19-4': 'It is worth noting that the condition in Eq. ([REF]) gives [MATH], which does not seem to conflict with the condition in Eq. ([REF]).', '0811.3275-1-20-0': 'The given example using two-qubit states reveals the violation of the Bell-Zukowski inequality.', '0811.3275-1-20-1': 'The interesting point is that all the information to get the violation of the Bell-Zukowski inequality can be obtained only by a two-setting and two-particle Bell experiment reproducible by local realistic theories.', '0811.3275-1-21-0': '# SUMMARY', '0811.3275-1-22-0': 'In summary, we have shown that the Bell-Zukowski operator can be represented by the Bell-Mermin operator.', '0811.3275-1-22-1': 'This fact provides a means to check whether a quantum state can be modeled by local realistic models, i.e., if the conflict between local realism and quantum mechanics occurs.', '0811.3275-1-22-2': 'Our argument relies only on a two-setting and two-particle Bell experiment reproducible by local realistic theories.', '0811.3275-1-22-3': 'Given a two-setting and two-particle Bell experiment reproducible by local realistic theories, one can compute a violation of the Bell-Zukowski inequality.', '0811.3275-1-22-4': 'Measured data, thus, indicate that the measured state cannot be modeled by local realistic models.', '0811.3275-1-22-5': 'Thus, the conflict between local realism and quantum mechanics is revealed.', '0811.3275-1-22-6': 'This phenomenon can occur when the system is in a mixed state.', '0811.3275-1-22-7': 'We also analyzed the threshold visibility for two-particle interference in order to bring about the phenomenon.', '0811.3275-1-22-8': 'The threshold visibility agrees well with the value to obtain a violation of the Bell-Zukowski inequality.', '0811.3275-1-23-0': '# ACKNOWLEDGEMENTS', '0811.3275-1-24-0': 'This work was supported by the Frontier Basic Research Programs at Korea Advanced Institute of Science and Technology.', '0811.3275-1-24-1': 'K. N. is supported by a BK21 research grant.'}

{'0811.3275-2-0-0': 'We formulate the expectation value of the Bell-Zukowski operator acting on qubit states of a two-particle Bell experiment.', '0811.3275-2-0-1': 'By using the equivalence between a set of [MATH] copies of a two-qubit experiment and a standard two-setting Bell experiment in an entangled [MATH]-particle state, we obtain an inequality, which we may call the Bell-Zukowski inequality.', '0811.3275-2-0-2': 'It determines whether the measured correlation functions of two-particle states can be modeled locally and realistically.', '0811.3275-2-0-3': 'In this Bell experiment of two particles, the conflict between local realism and quantum mechanics is discussed in conjunction with the violation of the Bell-Zukowski inequality.', '0811.3275-2-0-4': 'The main point of the result is that the Bell-Zukowski operator can be represented by the Bell-Mermin operator.', '0811.3275-2-0-5': 'The threshold visibility of two-particle interference analyzed in this scheme shows good agreement with the value to cause a violation of the Bell-Zukowski inequality.', '0811.3275-2-1-0': '# Introduction', '0811.3275-2-2-0': 'Bell inequalities that correlation functions satisfying local realistic theories must obey can be violated by certain quantum predictions, as Bell reported in 1964 [CITATION].', '0811.3275-2-2-1': 'Bell used the singlet state, or EPR pairs [CITATION], to show that the correlation functions measured in such singlet states cannot be modeled by local realistic models.', '0811.3275-2-2-2': 'Likewise, a certain set of correlation functions produced by quantum measurements of a quantum state contradicts certain predictions of local realistic theories.', '0811.3275-2-2-3': 'Those states also cannot be modeled by local realistic models.', '0811.3275-2-2-4': 'Up to now, local realistic theories have been studied extensively [CITATION].', '0811.3275-2-2-5': 'Many experiments have shown that Bell inequalities and local realistic theories are violated [CITATION].', '0811.3275-2-2-6': 'Later, in a work by Fine [CITATION], a set of correlation functions can be described with the property that they are reproducible by local realistic theories for a system in two-partite states if and only if the set of correlation functions satisfies the complete set of (two-setting) Bell inequalities.', '0811.3275-2-2-7': 'This result is generalized [CITATION] to a system described by multipartite states in the case where two dichotomic observables are measured per site.', '0811.3275-2-3-0': 'In this paper, we present a method using two Bell operators [CITATION] to refute local realistic models of a quantum state.', '0811.3275-2-3-1': 'In order to do so, we need only a two-setting and two-particle Bell experiment reproducible by local realistic theories.', '0811.3275-2-3-2': 'Such a Bell experiment also reveals the conflict between local realism and quantum mechanics in the sense that the Bell-Zukowski inequality [CITATION] is violated.', '0811.3275-2-4-0': 'Let us consider two-qubit states that, under specific settings, give correlation functions reproducible by local realistic theories.', '0811.3275-2-4-1': 'Imagine that [MATH] copies of the states can be distributed among [MATH] parties in such a way that each pair of parties shares one copy of the state.', '0811.3275-2-4-2': 'The parties perform a Bell-Greenberger-Horne-Zeilinger (GHZ) [MATH]-particle experiment [CITATION] on their qubits.', '0811.3275-2-4-3': 'Each of the pairs of parties uses the measurement settings noted above.', '0811.3275-2-4-4': 'The Bell-Mermin operator [CITATION], [MATH], for their experiment does not show any violation of local realism.', '0811.3275-2-4-5': 'Nevertheless, one can find another Bell operator, which differs from [MATH] by a numerical factor, that does show such a violation.', '0811.3275-2-4-6': 'That is, the original two-qubit states cannot be modeled by local realistic models.', '0811.3275-2-5-0': 'More specifically, the situation is as follows: A given two-setting and two-particle Bell experiment is reproducible by local realistic theories.', '0811.3275-2-5-1': 'However, the experimental correlation functions can compute a violation of the Bell-Zukowski inequality.', '0811.3275-2-5-2': 'Therefore, actually measured data reveal that the measured state cannot be modeled by local realistic models.', '0811.3275-2-5-3': 'Thus, a conflict between local realism and quantum mechanics is revealed.', '0811.3275-2-5-4': 'We can see this phenomenon by the simple algebra presented below.', '0811.3275-2-6-0': 'This phenomenon can occur when the system is in a mixed two-qubit state.', '0811.3275-2-6-1': 'We analyze the threshold visibility for two-particle interference to reveal the conflict mentioned above.', '0811.3275-2-6-2': 'It is found that the threshold visibility agrees with the value to obtain a violation of the Bell-Zukowski inequality.', '0811.3275-2-7-0': '# Bell-Mermin operator and Bell-Zukowski operator', '0811.3275-2-8-0': 'Let [MATH] be [MATH].', '0811.3275-2-8-1': 'We consider the following specific Bell-Mermin operator (see Eq. ([REF])): [EQUATION]', '0811.3275-2-8-2': 'Here, the states [MATH] are GHZ states [CITATION], i.e., [EQUATION]', '0811.3275-2-8-3': 'An average of the Bell-Mermin operator is evaluated by using a standard two-setting Bell experiment.', '0811.3275-2-8-4': 'See Fig. [REF].', '0811.3275-2-9-0': 'One can introduce a [MATH]-partite Bell operator, which one may call the Bell-Zukowski operator [MATH], which differs from [MATH] only by a numerical factor.', '0811.3275-2-9-1': 'The Bell-Zukowski operator [MATH] [CITATION] is [EQUATION]', '0811.3275-2-9-2': 'An average of the Bell-Zukowski operator is evaluated by using an all-setting Bell experiment.', '0811.3275-2-9-3': 'See Fig. [REF].', '0811.3275-2-10-0': 'Clearly, we see that the Bell-Mermin operator given in Eq. ([REF]) is connected to the Bell-Zukowski operator [MATH] in the following relation: [EQUATION]', '0811.3275-2-10-1': 'One can see that the specific two-setting Bell [MATH]-particle experiment in question computes an average value of the Bell-Zukowski operator [MATH] when an average value of [MATH] is evaluated.', '0811.3275-2-10-2': 'Of course, this argument is due to the validity of quantum mechanics.', '0811.3275-2-10-3': 'See Fig. [REF].', '0811.3275-2-11-0': 'From the Bell-Zukowski inequality [EQUATION] we have a condition on the average of the Bell-Mermin operator [MATH], which is written by [EQUATION]', '0811.3275-2-11-1': "Please notice that the Bell-Zukowski inequality [MATH] is derived under the assumption that there are predetermined 'hidden' results of the measurement for all directions in the rotation plane for the system in a state.", '0811.3275-2-11-2': "On the other hand, the Bell-Mermin inequality is derived under the assumption that there are predetermined 'hidden' results of the measurement for two directions for the system in a state.", '0811.3275-2-11-3': 'We see that a violation of the condition in Eq. ([REF]) implies a violation of the Bell-Zukowski inequality.', '0811.3275-2-11-4': 'Our aim is to compute an expectation value of the Bell-Mermin operator given in Eq. ([REF]) by using a two-particle Bell experiment reproducible by local realistic theories.', '0811.3275-2-11-5': 'The Bell-Zukowski inequality is stronger than the standard Bell inequalities for [MATH].', '0811.3275-2-11-6': 'This is why a standard Bell experiment reproducible by local realistic theories reveals the conflict between local realism and quantum mechanics.', '0811.3275-2-12-0': '# Experimental situation', '0811.3275-2-13-0': 'We consider the following two-qubit states: [EQUATION] where [MATH] is a Bell state defined as [EQUATION] [MATH] is the random noise admixture.', '0811.3275-2-13-1': 'The value of [MATH] can be interpreted as the reduction factor of the interferometric contrast observed in the two-particle correlation experiment.', '0811.3275-2-13-2': 'The states [MATH] are eigenstates of the [MATH]-component of the Pauli observable, [MATH], for the [MATH]th observer.', '0811.3275-2-13-3': 'Here, [MATH] and [MATH] are the labels of the parties (say Alice and Bob).', '0811.3275-2-13-4': 'Then, we have [MATH] and [MATH].', '0811.3275-2-14-0': 'Here, [MATH] and [MATH] are Pauli-spin operators for the [MATH]-component and for the [MATH]-component, respectively.', '0811.3275-2-14-1': 'This set of experimental correlation functions is described with the property that they are reproducible by local realistic theories.', '0811.3275-2-14-2': 'See the following relations along with the arguments in Ref. 10: [EQUATION]', '0811.3275-2-14-3': 'In the following section, we will use this kind of experimental situation.', '0811.3275-2-14-4': 'Those experimental correlation functions can compute a violation of the Bell-Zukowski inequality.', '0811.3275-2-14-5': 'In order to do so, we shall compute an expectation value of the Bell-Mermin operator in the next section.', '0811.3275-2-15-0': '# Conflict between local realism and quantum mechanics', '0811.3275-2-16-0': 'Imagine that [MATH] copies of the states introduced in the preceding section can be distributed among [MATH] parties in such a way that each pair of parties shares one copy of the state [EQUATION]', '0811.3275-2-16-1': 'Suppose that spatially separated [MATH] observers perform measurements on each of [MATH] particles.', '0811.3275-2-16-2': 'The decision processes for choosing measurement observables are space-like separated.', '0811.3275-2-16-3': 'It can be regarded as a standard two-setting Bell experiment in an entangled state in [MATH] particles.', '0811.3275-2-16-4': 'See Fig. [REF].', '0811.3275-2-17-0': 'We assume that a two-orthogonal-setting Bell-GHZ [MATH]-particle correlation experiment is performed.', '0811.3275-2-17-1': 'We choose measurement observables such that [EQUATION]', '0811.3275-2-17-2': 'Namely, each of the pairs of parties uses measurement settings such that they can check the condition in Eq. ([REF]).', '0811.3275-2-17-3': 'Therefore, it should be that given [MATH] correlation functions are described with the property that they are reproducible by local realistic theories.', '0811.3275-2-17-4': 'The Bell-Mermin operators [MATH] and [MATH] do not show any violation of local realism as shown below.', '0811.3275-2-18-0': 'Let [MATH] denote the function [EQUATION] [MATH] is invertible as [EQUATION]', '0811.3275-2-18-1': 'The Bell-Mermin operators [MATH] and [MATH] are defined by [CITATION] [EQUATION]', '0811.3275-2-18-2': 'The Bell-Mermin inequality can be expressed as [CITATION] [EQUATION]', '0811.3275-2-18-3': 'We also define [MATH] for any subset [MATH] by [EQUATION]', '0811.3275-2-18-4': 'It is easy to see that, when [MATH] are disjoint, [EQUATION] which leads to following equations: [EQUATION]', '0811.3275-2-18-5': 'In specific operators [MATH] and [MATH] given in Eq. ([REF]), where [EQUATION] and [EQUATION] we have (cf. [CITATION]) [EQUATION] and [EQUATION]', '0811.3275-2-18-6': 'Hence, we obtain [EQUATION] where [EQUATION]', '0811.3275-2-18-7': 'Measurements on each of [MATH] particles enable them to obtain [MATH] correlation functions.', '0811.3275-2-18-8': 'Thus, they get an average value of the specific Bell-Mermin operator given in Eq. ([REF]).', '0811.3275-2-18-9': 'According to Eq. ([REF]), we obtain [EQUATION]', '0811.3275-2-18-10': 'Clearly, the Bell-Mermin operators [MATH] and [MATH] for their experiment do not show any violation of local realism as we have mentioned above.', '0811.3275-2-19-0': 'Nevertheless, when [MATH] and [MATH] is given by [EQUATION] we have a violation of the condition in Eq. ([REF]), i.e., one can compute a violation of the Bell-Zukowski inequality [MATH] that is, the measured two-qubit state cannot be modeled by local realistic models.', '0811.3275-2-19-1': 'The condition in Eq. ([REF]) says that the threshold visibility decreases when the number of copies, [MATH], increases.', '0811.3275-2-19-2': 'In an extreme situation, when [MATH], we have the desired condition [EQUATION] to show the conflict in question.', '0811.3275-2-19-3': 'This agrees with the value to get a violation of the Bell-Zukowski inequality.', '0811.3275-2-19-4': 'It is worth noting that the condition in Eq. ([REF]) gives [MATH], which does not seem to conflict with the condition in Eq. ([REF]).', '0811.3275-2-20-0': 'The given example using two-qubit states reveals the violation of the Bell-Zukowski inequality.', '0811.3275-2-20-1': 'The interesting point is that all the information to get the violation of the Bell-Zukowski inequality can be obtained only by a two-setting and two-particle Bell experiment reproducible by local realistic theories.', '0811.3275-2-21-0': '# SUMMARY', '0811.3275-2-22-0': 'In summary, we have shown that the Bell-Zukowski operator can be represented by the Bell-Mermin operator.', '0811.3275-2-22-1': 'This fact provides a means to check whether a quantum state can be modeled by local realistic models, i.e., if the conflict between local realism and quantum mechanics occurs.', '0811.3275-2-22-2': 'Our argument relies only on a two-setting and two-particle Bell experiment reproducible by local realistic theories.', '0811.3275-2-22-3': 'Given a two-setting and two-particle Bell experiment reproducible by local realistic theories, one can compute a violation of the Bell-Zukowski inequality.', '0811.3275-2-22-4': 'Measured data, thus, indicate that the measured state cannot be modeled by local realistic models.', '0811.3275-2-22-5': 'Thus, the conflict between local realism and quantum mechanics is revealed.', '0811.3275-2-22-6': 'This phenomenon can occur when the system is in a mixed state.', '0811.3275-2-22-7': 'We also analyzed the threshold visibility for two-particle interference in order to bring about the phenomenon.', '0811.3275-2-22-8': 'The threshold visibility agrees well with the value to obtain a violation of the Bell-Zukowski inequality.', '0811.3275-2-23-0': '# ACKNOWLEDGEMENTS', '0811.3275-2-24-0': 'This work was supported by the Frontier Basic Research Programs at Korea Advanced Institute of Science and Technology.', '0811.3275-2-24-1': 'K. N. is supported by a BK21 research grant.'}

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[]

[]

[]

[]

['0811.3275-1-8-0', '0811.3275-1-8-4', '0811.3275-1-9-3', '0811.3275-1-10-3', '0811.3275-1-16-4', '0811.3275-2-8-0', '0811.3275-2-8-4', '0811.3275-2-9-3', '0811.3275-2-10-3', '0811.3275-2-16-4']

{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/0811.3275

1812.01362

{'1812.01362-1-0-0': 'We analyse the complexity of constructing centralisers of strong involutions in unitary groups over finite fields of odd order.', '1812.01362-1-0-1': 'Let [MATH] be an involution in [MATH], with fixed point space [MATH] of dimension between [MATH] and [MATH] (a strong involution).', '1812.01362-1-0-2': 'For each [MATH] such that [MATH] has even order, there is a unique power [MATH] of [MATH] that is an involution, and [MATH].', '1812.01362-1-0-3': 'We prove that there exists a constant [MATH] such that with probability at least [MATH], the restriction [MATH] is a strong involution, and similarly for [MATH].', '1812.01362-1-0-4': 'This enables us to prove logarithmic bounds on the number of random conjugates of [MATH] required to generate a subgroup of [MATH] containing the last term of its derived series.', '1812.01362-1-0-5': 'This has consequences for the complexity of recognition algorithms for unitary groups in odd characteristic.', '1812.01362-1-1-0': '# Introduction', '1812.01362-1-2-0': 'Parker and Wilson showed in 2010 (see [CITATION]) that involution-centraliser methods could be used to solve several computationally difficult problems, and gave complexity analyses for these algorithms in simple Lie type groups in odd characteristic.', '1812.01362-1-2-1': 'Central to these approaches are conjugate pairs [MATH] of involutions.', '1812.01362-1-2-2': "If [MATH] is a uniformly distributed element of a group [MATH], and [MATH] has odd order [MATH], then [MATH] is an element of [MATH] (this observation dates back to Bray's paper [CITATION]).", '1812.01362-1-2-3': 'Furthermore, if [MATH] is random and uniformly distributed in [MATH], then [MATH] is uniformly distributed in [MATH].', '1812.01362-1-2-4': 'Parker and Wilson showed that if [MATH] is a simple classical group of dimension [MATH], then the proportion of [MATH] such that [MATH] has odd order is bounded below by [MATH], for some constant [MATH], so that with high probability [MATH] random elements [MATH] suffice to construct such a random element [MATH].', '1812.01362-1-2-5': 'Moreover, for infinitely many odd field orders, if [MATH] is linear or unitary then the lower bound [MATH] cannot be improved (see [CITATION] and [CITATION]).', '1812.01362-1-3-0': 'If [MATH] has even order, then [MATH] can also be used to construct an element [MATH] of [MATH]; here [MATH] is the power of [MATH] that is an involution.', '1812.01362-1-3-1': 'However, these elements [MATH] are not uniformly distributed in [MATH]; instead [MATH] is uniformly distributed only within its [MATH]-conjugacy class (Remark [REF]).', '1812.01362-1-3-2': 'In this paper we analyse the centralisers of strong involutions in unitary groups (Definition [REF]).', '1812.01362-1-3-3': 'We first analyse the fundamental case of conjugate involution pairs [MATH] whose product [MATH] is regular semisimple on the underlying vector space.', '1812.01362-1-3-4': 'We go on to show that there exists an absolute constant [MATH] such that [MATH] random elements [MATH] suffice to construct a set of involutions that generate a group containing the last term in the derived series of [MATH].', '1812.01362-1-3-5': 'Our methods build on those of [CITATION] and [CITATION], but we encounter fundamental new difficulties, as the structure of regular semisimple elements in [MATH] that are "almost irreducible" (in a sense that we shall make precise in Definition [REF]) and conjugate to their inverses is very different from those in [MATH].', '1812.01362-1-3-6': 'In future work, we plan to address the symplectic and orthogonal groups.', '1812.01362-1-3-7': 'For these families of groups completely different arguments will be required: for example, one may readily compute that in [MATH] and [MATH] there are no regular semisimple elements that are inverted by involutions.', '1812.01362-1-4-0': 'For an involution [MATH], we write [MATH] and [MATH] to denote its eigenspaces for eigenvalues [MATH] and [MATH].', '1812.01362-1-4-1': 'We say that such a [MATH] is strong if [MATH].', '1812.01362-1-5-0': 'For [MATH] a group, and [MATH], we write [MATH] for [MATH] when [MATH] is even, and for [MATH] when [MATH] is odd.', '1812.01362-1-6-0': 'A random variable [MATH] on a finite group [MATH] is nearly uniformly distributed if for all [MATH] the probability [MATH] that [MATH] takes the value [MATH] satisfies [EQUATION].', '1812.01362-1-7-0': 'Our first main technical theorem is as follows.', '1812.01362-1-8-0': 'There exist positive constants [MATH] such that the following is true.', '1812.01362-1-8-1': 'Suppose that [MATH], that [MATH] is a strong involution in [MATH] with [MATH] odd, and that [MATH] is a nearly uniformly distributed random element of [MATH].', '1812.01362-1-8-2': 'Let [MATH], and let [MATH] be the restriction of [MATH] to the eigenspace [MATH] or [MATH].', '1812.01362-1-8-3': 'Then', '1812.01362-1-9-0': '[(i)] [MATH] is a strong involution with probability at least [MATH]; and [MATH] is a strong involution with probability at least [MATH].', '1812.01362-1-10-0': 'Our proof shows that the values [MATH] and [MATH] suffice.', '1812.01362-1-10-1': "The comparable values in [CITATION] for the case of special linear groups with nearly uniform random elements would be [MATH], and echoing the view expressed there, 'we believe that these constants are far from best possible'.", '1812.01362-1-10-2': 'From Theorem [REF], we are able to deduce the following result (see Section [REF]).', '1812.01362-1-11-0': 'There exist constants [MATH] such that the following is true.', '1812.01362-1-11-1': 'Let [MATH], let [MATH] with [MATH] odd, let [MATH] be a strong involution, let [MATH] and let [MATH].', '1812.01362-1-11-2': 'Let [MATH] be a sequence of at least [MATH] random elements of [MATH], chosen independently and nearly uniformly, and let [MATH].', '1812.01362-1-11-3': 'Then [EQUATION].', '1812.01362-1-12-0': 'One of our motivations for proving the preceding two theorems was an application to computational group theory.', '1812.01362-1-13-0': 'There exists a constant [MATH] such that for all [MATH] and for all odd [MATH], the following holds.', '1812.01362-1-13-1': 'There is a Las Vegas algorithm which on input a strong involution [MATH], constructs generators for a subgroup of [MATH] containing [MATH].', '1812.01362-1-13-2': 'It requires [MATH] nearly uniformly distributed random elements and succeeds with probability at least [MATH].', '1812.01362-1-14-0': "We focus now on the algorithm of Leedham-Green and O'Brien [CITATION] for constructive recognition of special unitary groups.", '1812.01362-1-14-1': 'Let [MATH] denote an upper bound on the number of field operations needed to construct an independent nearly uniformly distributed random element of [MATH], and let [MATH] be an upper bound on the number of field operations equivalent to a call to a discrete logarithm oracle for [MATH].', '1812.01362-1-14-2': 'Reasoning in exactly the same way as [CITATION], the following theorem can be deduced from [CITATION] and Theorem [REF].', '1812.01362-1-15-0': 'There is a Las Vegas algorithm that takes as input a subset [MATH] of bounded cardinality in [MATH], with [MATH] odd, and returns standard generators for [MATH] as straight line programmes of length [MATH] in [MATH].', '1812.01362-1-15-1': 'The algorithm has complexity [MATH], measured in field operations.', '1812.01362-1-16-0': 'To prove Theorem [REF], we carry out an extensive analysis of the products of conjugate involutions in [MATH].', '1812.01362-1-16-1': 'Some of our results may be of independent interest, so in the remainder of this section we describe them.', '1812.01362-1-17-0': 'An [MATH] matrix [MATH] is regular semisimple if its characteristic polynomial [MATH] is multiplicity-free.', '1812.01362-1-18-0': 'An involution [MATH] has type [MATH] if [MATH] and [MATH].', '1812.01362-1-18-1': 'Given [MATH], an involution [MATH] of type [MATH] is [MATH]-balanced if [MATH].', '1812.01362-1-18-2': 'An involution [MATH] of type [MATH] is perfectly balanced.', '1812.01362-1-19-0': 'Let [MATH], equipped with a unitary form with Gram matrix [MATH].', '1812.01362-1-19-1': 'Following [CITATION], we define [MATH] to be the class of perfectly balanced involutions in [MATH], with [MATH] odd, and we define [MATH] to be [MATH].', '1812.01362-1-20-0': 'We let [EQUATION] and let [MATH] be the probability that a random element of [MATH] lies in [MATH].', '1812.01362-1-21-0': 'Let [MATH] be as above, with [MATH] odd.', '1812.01362-1-21-1': 'If [MATH], then [MATH], whilst [MATH].', '1812.01362-1-22-0': 'We prove that for [MATH] even, [MATH].', '1812.01362-1-22-1': 'We show in Lemma [REF] that we can derive the probability [MATH] for [MATH] odd from the even-dimensional case, and hence show that for [MATH] odd, [MATH].', '1812.01362-1-22-2': 'We prove in Corollary [REF] that the limits as [MATH] of [MATH] and [MATH] exist, and determine each limit.', '1812.01362-1-23-0': 'The structure of this paper is as follows.', '1812.01362-1-23-1': 'In Section [REF] we begin our exploration of the conjugacy classes in [MATH], and of the characteristic polynomials of elements of [MATH].', '1812.01362-1-23-2': 'In Section [REF] we define a set of ordered pairs of conjugate involutions [MATH] such that [MATH] is guaranteed to be a strong involution.', '1812.01362-1-23-3': 'Thus to prove Theorem [REF] it suffices to show that this set is sufficiently large.', '1812.01362-1-23-4': 'In Sections [REF], [REF] and [REF] we classify the [MATH]-irreducible regular semisimple elements of [MATH] (that is, such elements that are as close to irreducible as possible, see Definition [REF]), determine their centralisers, and count the number of involutions inverting them.', '1812.01362-1-23-5': 'In Section [REF] we calculate various upper and lower bounds on the number of monic polynomials that correspond to irreducible factors of the characteristic polynomials of these [MATH]-irreducbile regular semisimple elements.', '1812.01362-1-23-6': 'In Section [REF] we define and analyse our key generating function, [MATH].', '1812.01362-1-23-7': 'In Section [REF] we factorise [MATH], in order control the dimension of the [MATH]-eigenspace of [MATH], and prove bounds on the coefficients of certain generating functions that refine the information in [MATH].', '1812.01362-1-23-8': 'In Section [REF] we prove Theorem [REF], and finally in Section [REF] we prove Theorems [REF], [REF] and [REF].'}

{'1812.01362-2-0-0': 'We analyse the complexity of constructing involution centralisers in unitary groups over fields of odd order.', '1812.01362-2-0-1': 'In particular, we prove logarithmic bounds on the number of random elements required to generate a subgroup of the centraliser of a strong involution that contains the last term of its derived series.', '1812.01362-2-0-2': 'We use this to strengthen previous bounds on the complexity of recognition algorithms for unitary groups in odd characteristic.', '1812.01362-2-0-3': 'Our approach generalises and extends two previous papers by the second author and collaborators on strong involutions and regular semisimple elements of linear groups.', '1812.01362-2-1-0': '# Introduction', '1812.01362-2-2-0': 'Parker and Wilson [CITATION] showed in 2010 that involution-centraliser methods could be used to solve several computationally difficult problems, and gave complexity analyses for these algorithms in simple Lie type groups in odd characteristic.', '1812.01362-2-2-1': 'Central to these approaches are conjugate pairs [MATH] of involutions.', '1812.01362-2-2-2': 'If [MATH] is a uniformly distributed random element of a group [MATH], and [MATH] has odd order [MATH], then [MATH] is a uniformly distributed random element of [MATH].', '1812.01362-2-2-3': 'This observation is due to Richard Parker, see [CITATION].', '1812.01362-2-2-4': 'Parker and Wilson in [CITATION] showed that if [MATH] is a simple classical group of dimension [MATH], then the proportion of elements [MATH] of [MATH] such that [MATH] has odd order is bounded below by [MATH], for some constant [MATH], so that with high probability [MATH] random elements [MATH] suffice to construct such a random element [MATH].', '1812.01362-2-2-5': 'Moreover, for infinitely many odd field orders, if [MATH] is linear or unitary then the lower bound [MATH] cannot be improved (see [CITATION] and [CITATION]).', '1812.01362-2-2-6': 'If [MATH] has even order [MATH], then [MATH] is an involution in the centraliser [MATH] of [MATH].', '1812.01362-2-2-7': 'However, these elements [MATH] are not uniformly distributed in [MATH]; instead [MATH] is uniformly distributed only within its [MATH]-conjugacy class.', '1812.01362-2-3-0': 'In this paper we analyse the centralisers [MATH] of strong involutions [MATH] (see Definition [REF]) in unitary groups [MATH] in odd characteristic.', '1812.01362-2-4-0': 'We show that there exists an absolute constant [MATH] such that given a strong involution [MATH], a set of [MATH] random elements [MATH] suffices to construct a set of involutions that generates a group containing the last term in the derived series of [MATH].', '1812.01362-2-4-1': 'A careful analysis of the highest power of 2 dividing [MATH] is required here.', '1812.01362-2-4-2': 'Our methods build on the work of Praeger and Seress [CITATION], and of Dixon, Praeger and Seress [CITATION], but we encounter fundamental new difficulties: the structure of regular semisimple elements in [MATH] that are "almost irreducible" (in a sense that we shall make precise in Definition [REF]) and conjugate to their inverses is very different from those in [MATH].', '1812.01362-2-4-3': 'In future work, we plan to address the symplectic and orthogonal groups.', '1812.01362-2-4-4': 'For these families of groups completely different arguments will be required: for example, one may readily compute that in [MATH] and [MATH] there are no regular semisimple elements that are inverted by involutions.', '1812.01362-2-5-0': 'For an involution [MATH], we write [MATH] and [MATH] to denote its eigenspaces for eigenvalues [MATH] and [MATH].', '1812.01362-2-5-1': 'Such a [MATH] is strong if [MATH].', '1812.01362-2-5-2': 'For an element [MATH] of a group [MATH], let [MATH] denote [MATH] when [MATH] is even, and [MATH] otherwise.', '1812.01362-2-6-0': 'A random variable [MATH] on a finite group [MATH] is nearly uniformly distributed if for all [MATH] the probability [MATH] that [MATH] takes the value [MATH] satisfies [EQUATION].', '1812.01362-2-7-0': 'Our first main technical theorem is as follows.', '1812.01362-2-8-0': 'There exist positive constants [MATH] such that the following is true.', '1812.01362-2-8-1': 'Suppose that [MATH], that [MATH] is a strong involution in [MATH] with [MATH] odd, and that [MATH] is a nearly uniformly distributed random element of [MATH].', '1812.01362-2-8-2': 'Let [MATH], and let [MATH] be the restriction of [MATH] to the eigenspace [MATH] (where [MATH]).', '1812.01362-2-8-3': 'Then', '1812.01362-2-9-0': '[(i)] [MATH] is a strong involution with probability at least [MATH]; and [MATH] is a strong involution with probability at least [MATH].', '1812.01362-2-10-0': 'Our proof shows that the values [MATH] and [MATH] suffice.', '1812.01362-2-10-1': "The comparable values in [CITATION] for the case of special linear groups with nearly uniform random elements are [MATH], and echoing the view expressed there, 'we believe that these constants are far from best possible'.", '1812.01362-2-11-0': 'From Theorem [REF], and [CITATION], we are able to deduce the following result (see [REF]).', '1812.01362-2-12-0': 'There exist constants [MATH] such that the following is true.', '1812.01362-2-12-1': 'Let [MATH], let [MATH] or [MATH] with [MATH] odd, and let [MATH] be a strong involution.', '1812.01362-2-12-2': 'For [MATH], let [MATH] if [MATH], or [MATH] if [MATH].', '1812.01362-2-12-3': 'Let [MATH] be a sequence of at least [MATH] random elements of [MATH], chosen independently and nearly uniformly, and let [MATH].', '1812.01362-2-12-4': 'Then [EQUATION].', '1812.01362-2-13-0': 'One of our motivations for proving the preceding two theorems was an application to computational group theory.', '1812.01362-2-13-1': 'Two key steps in many algorithms (for example, those insLOB, LO, ParkerWilson) are first to construct an involution [MATH] in a group [MATH] of Lie type, and then to construct a subgroup of the centraliser of [MATH] that contains the last term, [MATH], in the derived series of [MATH].', '1812.01362-2-13-2': 'For some of these algorithms, including the constructive recognition algorithms in [CITATION], the involution [MATH] is required to be strong.', '1812.01362-2-14-0': 'Let [MATH] be a group.', '1812.01362-2-14-1': 'For an involution [MATH] and an element [MATH] of [MATH], we let [MATH] be [MATH] when [MATH] has even order, and [MATH] when [MATH] is odd.', '1812.01362-2-14-2': 'It follows that [MATH].', '1812.01362-2-15-0': 'Building on work of Lubeck, Niemeyer and Praeger [CITATION], we can remove the degree restriction in Theorem [REF], and include the step of finding a strong involution, whilst only slightly worsening the probability of success (see [REF]).', '1812.01362-2-16-0': 'There exists a positive constant [MATH] such that for all [MATH], for all odd [MATH], and for [MATH] or [MATH], the following holds with probability at least [MATH].', '1812.01362-2-16-1': 'A sequence [MATH] of [MATH] independent nearly uniformly distributed random elements of [MATH] contains an element [MATH] such that [MATH] is a strong involution, and moreover [MATH].', '1812.01362-2-17-0': "Leedham-Green and O'Brien in [CITATION] define certain generating sets for the quasisimple classical groups in odd characteristic, called standard generators, and use a recursive approach, via repeated involution centralisers, to find these standard generators in the given group.", '1812.01362-2-17-1': 'Our improved analysis in Theorem [REF] of the number of random elements required to construct an involution centraliser enables us to replace a factor of [MATH] in their complexity analysis with a factor of [MATH].', '1812.01362-2-17-2': 'Let [MATH] denote an upper bound on the number of field operations needed to construct an independent nearly uniformly distributed random element of [MATH], and let [MATH] be an upper bound on the number of field operations equivalent to a call to a discrete logarithm oracle for [MATH].', '1812.01362-2-17-3': 'Reasoning in the same way as [CITATION], the following can be deduced from [CITATION] and Theorem [REF].', '1812.01362-2-18-0': 'Let [MATH] be odd, and let [MATH].', '1812.01362-2-18-1': 'There is a Las Vegas algorithm that takes as input a set [MATH] of generators for [MATH] of bounded cardinality, and returns standard generators for [MATH] as straight line programmes of length [MATH] in [MATH].', '1812.01362-2-18-2': 'The algorithm has complexity [MATH], measured in field operations.', '1812.01362-2-19-0': 'To prove Theorem [REF], we carry out an extensive analysis of the products of conjugate involutions in [MATH].', '1812.01362-2-19-1': 'Some of our results may be of independent interest, so in the remainder of this section we describe them.', '1812.01362-2-20-0': 'Denote the characteristic polynomial of a square matrix [MATH] by [MATH].', '1812.01362-2-20-1': 'Such a matrix [MATH] is regular semisimple if [MATH] is multiplicity-free.', '1812.01362-2-20-2': 'Let [MATH], with [MATH] odd, equipped with a unitary form having Gram matrix the identity matrix [MATH].', '1812.01362-2-20-3': 'We say that an involution [MATH] is perfectly balanced if [MATH].', '1812.01362-2-20-4': 'Following [CITATION], we define [MATH] to be the class of perfectly balanced involutions in [MATH], and we define [MATH] to be [MATH].', '1812.01362-2-21-0': 'We let [EQUATION]', '1812.01362-2-21-1': 'For [MATH] odd, let [MATH] be the probability that a random element [MATH] lies in [MATH].', '1812.01362-2-21-2': 'If [MATH] then [MATH], and [MATH].', '1812.01362-2-22-0': 'We prove that [MATH], that [MATH] for [MATH] even,', '1812.01362-2-23-0': 'and that [MATH] for [MATH] odd.', '1812.01362-2-23-1': 'We shall also prove in Corollary [REF] that the limits as [MATH] of [MATH] and [MATH] exist, and determine each limit.', '1812.01362-2-24-0': 'The structure of this paper is as follows.', '1812.01362-2-24-1': 'In [REF] we begin our exploration of the conjugacy classes of [MATH], and of the characteristic polynomials of elements of [MATH].', '1812.01362-2-24-2': 'In [REF] we define a set of ordered pairs of conjugate involutions [MATH] such that [MATH] is guaranteed to be a strong involution.', '1812.01362-2-24-3': 'Thus to prove Theorem [REF] it suffices to show that this set is sufficiently large.', '1812.01362-2-24-4': 'In [REF], [REF] and [REF] we classify the [MATH]-irreducible regular semisimple elements of [MATH] (that is, such elements that are as close to irreducible as possible, see Definition [REF]), determine their centralisers, and count the number of involutions inverting them.', '1812.01362-2-24-5': 'In [REF] we calculate various upper and lower bounds on the number of monic polynomials that correspond to irreducible factors of the characteristic polynomials of these [MATH]-irreducible regular semisimple elements.', '1812.01362-2-24-6': 'In [REF] we define and analyse our key generating function, [MATH].', '1812.01362-2-24-7': 'In [REF] we factorise [MATH], and prove bounds on the coefficients of certain generating functions that refine the information in [MATH].', '1812.01362-2-24-8': 'This additional information allows us to control the powers of [MATH] dividing the orders of the roots of the characteristic polynomial of [MATH], and hence to bound the dimension of the [MATH]-eigenspace of [MATH].', '1812.01362-2-24-9': 'In [REF] we prove Theorem [REF], and finally in [REF] we prove Theorems [REF], [REF] and [REF].'}

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[]

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[]

['1812.01362-1-5-0', '1812.01362-1-7-0', '1812.01362-1-8-3', '1812.01362-1-11-3', '1812.01362-1-18-0', '1812.01362-1-18-1', '1812.01362-1-18-2', '1812.01362-1-21-0', '1812.01362-1-21-1', '1812.01362-2-7-0', '1812.01362-2-8-3', '1812.01362-2-12-2', '1812.01362-2-12-4', '1812.01362-2-21-0', '1812.01362-2-21-2', '1812.01362-2-22-0']

{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1812.01362

1703.10576

{'1703.10576-1-0-0': 'We present a uniform framework for the treatment of a large class of toy models of quantum theory.', '1703.10576-1-0-1': 'Specifically, we will be interested in theories of wavefunctions valued in commutative involutive semirings, and which give rise to some semiring-based notion of classical non-determinism via the Born rule.', '1703.10576-1-0-2': 'The models obtained with our construction possess many of the familiar structures used in Categorical Quantum Mechanics.', '1703.10576-1-0-3': 'We also provide a bestiary of increasingly exotic examples: some well known, such as real quantum theory and relational quantum theory; some less known, such as hyperbolic quantum theory, p-adic quantum theory and "parity quantum theory"; and some entirely new, such as "finite-field quantum theory" and "tropical quantum theory".', '1703.10576-1-0-4': 'As a further bonus, the measurement scenarios arising within these theories can be studied using the sheaf-theoretic framework for non-locality and contextuality.', '1703.10576-1-1-0': '# Introduction', '1703.10576-1-2-0': 'The construction of toy models plays a key role in many foundational efforts across mathematics, physics and computer science.', '1703.10576-1-2-1': 'In the foundations of quantum theory, toy models help to understand which abstract structural features of quantum systems-and their interface to the classical world-are involved in providing different kinds of non-classical behaviour.', '1703.10576-1-2-2': 'In turn, this informs practical research into quantum computation and communication technology, helping to cut down the noise and focus on those features that truly contribute to quantum advantage.', '1703.10576-1-3-0': "The categorical and diagrammatic methods from Categorical Quantum Mechanics (CQM) [CITATION] have proven particularly well suited to the construction and study of toy models, with the majority of efforts focussed on Spekkens's toy model [CITATION] and the more general relational quantum theory [CITATION].", '1703.10576-1-3-1': 'In the same years, models have been developed within a variety of other frameworks: examples include real quantum theory, of special interest in the context of generalised/operational probabilistic theories and the study of Jordan algebras [CITATION], hyperbolic quantum theory [CITATION], [MATH]-adic quantum theory [CITATION], and modal quantum theory [CITATION].', '1703.10576-1-4-0': 'When constructing a toy model, it is essential to consider both the quantum side and the corresponding quantum-classical interface: indeed, many toy models result in notions of classical non-determinism which are different from the conventional probabilistic one, and special care needs to be taken in order to achieve a consistent treatment of classical systems.', '1703.10576-1-4-1': "Examples of this phenomenon include the possibilistic non-determinism arising from Spekkens's toy model and relational quantum theory [CITATION], the [MATH]-adic non-determinism arising from [MATH]-adic quantum theory [CITATION], and the signed probabilities arising from hyperbolic quantum theory [CITATION].", '1703.10576-1-5-0': 'Because of this issue, we adopt a recently developed framework, that of Categorical Probabilistic Theories [CITATION], which can simultaneously treat quantum-like systems and classical systems endowed with generic semiring-based notion of non-determinism.', '1703.10576-1-5-1': 'Categorical Probabilistic Theories have been introduced with the intent of bridging the gap between CQM and Operational Probabilistic Theories (OPTs) [CITATION]: they aim to provide categorical and diagrammatic methods in the style of CQM to talk about the problems that OPTs are concerned with.', '1703.10576-1-5-2': 'As a side-product of their abstract categorical formulation, these theories natively admit a general, semiring-based notion of classical non-determinism, and are therefore perfect to construct and study exotic toy models of quantum theory.', '1703.10576-1-6-0': 'In this work, we focus our attention on a very large class of finite-dimensional quantum-like theories, where wavefunctions of complex amplitude are replaced with wavefunctions valued in some arbitrary commutative semiring with involution [MATH].', '1703.10576-1-6-1': 'In the quantum-classical transition, probabilities are still deemed to arise via the Born rule, and as a consequence classical non-determinism is naturally and necessarily modelled by the semiring [MATH] of positive elements for [MATH], generalising the traditional probabilistic semiring [MATH] (which is the semiring of positive elements for [MATH]).', '1703.10576-1-7-0': 'As an underlying model for [MATH]-valued wavefunctions we consider the category [MATH], with objects in the form [MATH] for all finite sets [MATH], and morphisms [MATH] given by [MATH], the [MATH]-semimodule of [MATH]-by-[MATH] matrices with values in [MATH], equipped with matrix composition and identities.', '1703.10576-1-7-1': 'The category [MATH] generalises [MATH], and is a dagger compact category with Kronecker product as tensor product and conjugate transpose as dagger.', '1703.10576-1-7-2': 'We model mixed-state quantum theory using the CPM construction [CITATION], and the full quantum-classical theory using the dagger-idempotent variant of the CP* construction [CITATION] presented in Ref. [CITATION], which results in quantum-classical transitions governed by a generalised categorical flavour of the Born rule.', '1703.10576-1-7-3': 'Hence our full theory will be described by [MATH], which we will show to be an [MATH]-probabilistic theory in the sense of Ref. [CITATION], and hence amenable for treatment in the framework of Categorical Probabilistic Theories.', '1703.10576-1-8-0': 'Synopsis.', '1703.10576-1-8-1': 'We begin in Sections [REF] and [REF] by summarising the structure of Categorical Probabilistic Theories and by introducing our general framework for the construction of theories of wavefunctions valued in commutative involutive semirings.', '1703.10576-1-8-2': 'The remainder of the work is then dedicated to showing that our new framework is expressive enough to cover all existing toy models mentioned above.', '1703.10576-1-9-0': 'In Sections [REF] and [REF] we briefly cover real quantum theory and relational quantum theory, two models already dear to the CQM and OPT communities.', '1703.10576-1-9-1': 'In Sections [REF], [REF] and [REF] we cover hyperbolic quantum theory, parity quantum theory and [MATH]-adic quantum theory, already known in some specialised circles but not yet part of the categorical bestiary.', '1703.10576-1-9-2': "Hyperbolic quantum theory is particularly interesting: it is a quasi-probabilistic theory which can support at least as many protocols as real quantum theory (e.g. Simon's problem can be solved efficiently), but at the same time it is entirely local thanks to negative probabilities.", '1703.10576-1-9-3': 'Furthermore, parity quantum theory was already known as the [MATH] case of modal quantum theory, although a different interpretation is given here to its classical non-determinism.', '1703.10576-1-10-0': 'In Sections [REF] and [REF] we present two new families of toy models: finite-field quantum theory and tropical quantum theory, based on finite fields and tropical semirings respectively.', '1703.10576-1-10-1': 'Finite-field quantum theory proves to be especially interesting: it possesses a rich phase group, and can support a number of non-trivial quantum protocols and arguments, but at the same time it is entirely local.', '1703.10576-1-10-2': 'Furthermore, its pure quantum states are points of finite projective spaces, and can be studies using tools from finite geometry.', '1703.10576-1-10-3': 'Tropical quantum theory, on the other hand, turns out to be an extension of boolean quantum theory alternative to relational quantum theory, with which it shares many structural and operational traits (despite the lack of multiplicative idempotence in its semirings).', '1703.10576-1-11-0': '# Categorical Probabilistic Theories', '1703.10576-1-12-0': 'The main intuition behind a generalised, semiring-based notion of classical non-deterministic systems is borrowed from computer science, where the use of (commutative) semirings to model resources used by automata is commonplace.', '1703.10576-1-12-1': 'We look at probabilities in physics as a resource modelling non-determinism of classical systems, with properties captured by those of the commutative semiring [MATH]: from this perspective, it makes sense to study what classical non-determinism looks like when [MATH] is replaced by some other commutative semiring [MATH].', '1703.10576-1-12-2': 'Interesting alternative choices for [MATH] which already appeared in the literature include the boolean semiring [MATH] and other locales (in relational quantum theory), the quasi-probabilistic semiring [MATH] (a field, in hyperbolic quantum theory), the [MATH]-adic semiring [MATH] (another field, in [MATH]-adic quantum theory), and finite fields (in modal quantum theory).', '1703.10576-1-12-3': 'We will refer to classical systems with non-determinism governed by a commutative semiring [MATH] as classical [MATH]-probabilistic systems.', '1703.10576-1-13-0': 'One of the reason for the wide adoption of semirings in mathematics is that they capture the bare minimum algebraic structure required by matrix multiplication, with commutativity being a necessary addendum when a symmetric tensor product of matrices is of interest (as is the case in many physical applications).', '1703.10576-1-13-1': 'As our category of classical [MATH]-probabilistic systems we take the category [MATH] of free finite-dimensional [MATH]-semimodules and [MATH]-linear maps between them: objects are in the form [MATH] with [MATH] finite sets, and morphisms [MATH] form the free finite-dimensional [MATH]-semimodule of [MATH]-by-[MATH] matrices with values in [MATH].', '1703.10576-1-13-2': 'The category [MATH] is a compact closed symmetric monoidal category, with Kronecker product of matrices as tensor product.', '1703.10576-1-13-3': 'It is enriched in itself, and hence in commutative monoids ([MATH]-enriched), so that each homset [MATH] comes with a mixing operation [MATH] and an impossible process [MATH].', '1703.10576-1-13-4': 'The category [MATH] contains the category [MATH] of finite sets and functions (the category of classical deterministic systems) as a subcategory, and from [MATH] it inherits an environment structure [MATH] [CITATION] given by the discarding maps [MATH].', '1703.10576-1-14-0': 'In Ref. [CITATION], it is argued that the minimal requirements for a categorical probabilistic theory should include: (i) the explicit existence of classical systems; (ii) the extendibility of probabilistic mixing to all systems; (iii) the possibility of defining a meaningful notion of local state and discarding of systems.', '1703.10576-1-15-0': '[[MATH]-probabilistic Theory]', '1703.10576-1-16-0': 'An [MATH]-probabilistic theory is a symmetric monoidal category [MATH] which satisfies the following requirements.', '1703.10576-1-17-0': '[(i)] There is a full sub-SMC of [MATH], denoted by [MATH], which is equivalent to [MATH].', '1703.10576-1-17-1': '[(ii)] The SMC [MATH] is enriched in commutative monoids, and the enrichment on the subcategory [MATH] coincides with the one given by the linear structure of [MATH].', '1703.10576-1-17-2': '[(iii)] The SMC [MATH] comes with an environment structure, i.e. with a family [MATH] of morphisms which satisfy the following requirements: [EQUATION]', '1703.10576-1-17-3': 'On the subcategory [MATH], this environment structure must coincide with the canonical one of [MATH].', '1703.10576-1-18-0': 'We refer to [MATH] as classical theory, and to its objects and morphisms as classical systems and processes.', '1703.10576-1-18-1': 'As diagrammatic convention, we use dashed wires for classical systems, and solid wires for generic ones.', '1703.10576-1-19-0': '[MATH]-probabilistic theories come with a number of native features that are commonplace in the modelling of quantum protocols: it is possible to exert classical control, to define tests with classical outcomes, to marginalise over classical outcomes, to work with preparations and measurements, and to apply any kind of classical pre- and post-processing.', '1703.10576-1-19-1': 'Amongst the many mixed quantum-classical processes, we can consider Bell-type measurement scenarios.', '1703.10576-1-19-2': 'An [MATH]-party Bell-type measurement scenario in an [MATH]-probabilistic theory is a process in the following form, where the processes [MATH] and the state [MATH] are all normalised (recall that a process [MATH] is said to be normalised if [MATH]): [EQUATION]', '1703.10576-1-19-3': 'In the context of a Bell-type measurement scenario, the processes [MATH] are often referred to as measurements, their inputs as measurement choices and their outputs as (measurement) outcomes.', '1703.10576-1-19-4': 'The following result from Ref. [CITATION] ensures that non-locality in [MATH]-probabilistic theories can always be studied using the well-established sheaf-theoretic framework for non-locality and contextuality [CITATION].', '1703.10576-1-19-5': '[Bell-type measurement scenarios [CITATION]]', '1703.10576-1-20-0': 'A Bell-type measurement scenario in an [MATH]-probabilistic theory always corresponds to a no-signalling empirical model in the sheaf-theoretic framework for non-locality and contextuality [CITATION].', '1703.10576-1-21-0': 'An immediate consequence of the connection with the sheaf-theoretic framework is that we can straightforwardly adapt a proof of Ref. [CITATION] to rule out non-locality in a large class of toy models.', '1703.10576-1-21-1': '[Locality of [MATH]-probabilistic theories over fields]', '1703.10576-1-22-0': 'If [MATH] is a field, then all [MATH]-probabilistic theories are local.', '1703.10576-1-23-0': '# Theories of wavefunctions valued in commutative involutive semirings', '1703.10576-1-24-0': 'Note that two different linear structures intervene in the definition of quantum theory: the [MATH]-linear structure of wavefunctions, modelling superposition, interference and phases, and the [MATH]-linear structure of probability distributions over classical systems.', '1703.10576-1-24-1': 'We have already seen that the framework of [MATH]-probabilistic theories replaces the probability semiring [MATH] with a more general commutative semiring [MATH] as a model of classical non-determinism.', '1703.10576-1-24-2': 'In this Section, we construct a large class of toy models of quantum theory by considering theories of wavefunctions with amplitudes valued in some commutative semiring with involution [MATH], generalising the field with involution [MATH] traditionally used in quantum mechanics.', '1703.10576-1-24-3': 'To do so, we consider the dagger compact category [MATH] (dagger and compact closed structure will be defined using the involution), and we require classical non-determinism to arise via the Born rule, as embodied by the CP* construction.', '1703.10576-1-24-4': 'The corresponding quantum-classical theory will therefore be modelled by [MATH], and the main result of this Section (Theorem [REF]) will show that this is an [MATH]-probabilistic theory (where [MATH] the sub-semiring of positive elements of [MATH], see Definition [REF] below).', '1703.10576-1-25-0': 'The category [MATH] for a commutative semiring with involution [MATH] is defined as in the previous Section, but it comes with additional structure.', '1703.10576-1-25-1': 'Indeed, we can defined dagger and compact closed structures on [MATH] exactly as done in [MATH] (which is [MATH]), with conjugation taken using the involution [MATH] of [MATH] in place of complex conjugation.', '1703.10576-1-25-2': 'Each object [MATH] in [MATH] comes with at least one orthonormal basis [MATH], as well as an associated special commutative [MATH]-Frobenius algebra [MATH]: [EQUATION]', '1703.10576-1-25-3': 'For any group structure [MATH] on any finite set [MATH], one also obtains an associated [MATH]-Frobenius algebra [MATH] on [MATH] by linearly extending the group multiplication and unit: [EQUATION]', '1703.10576-1-25-4': 'The [MATH]-Frobenius algebra is commutative if and only if the group is, and it always satisfies the following: [EQUATION]', '1703.10576-1-25-5': 'Unfortunately, [MATH] is not quasi-special (a.k.a. normalisable) unless the scalar [MATH] takes the form [MATH] for some [MATH] which is multiplicatively invertible: when this is the case, however, we have a legitimate strongly complementary pair [MATH] in [MATH] corresponding to the finite group [MATH].', '1703.10576-1-25-6': 'When [MATH] is abelian these strongly complementary pairs can be used (under additional conditions) to implement quantum protocols such as the algorithm to solve the abelian Hidden Subgroup Problem [CITATION] or generalised Mermin-type arguments [CITATION].', '1703.10576-1-25-7': 'This also means that certain objects in [MATH] support fragments of the ZX calculus [CITATION], opening the way to the application of well-established diagrammatic techniques.', '1703.10576-1-26-0': 'In quantum theory, the probabilistic semiring [MATH] arises as a sub-semiring of [MATH] fixed by complex conjugation, namely the sub-semiring of those elements [MATH] taking the form [MATH]: this is, essentially, a hallmark of the Born rule.', '1703.10576-1-26-1': 'In general commutative semirings with involution, elements in the form [MATH] need not be closed under addition, but it is true their closure under addition always form a semiring.', '1703.10576-1-27-0': 'Let [MATH] be a commutative semiring with involution.', '1703.10576-1-27-1': 'Then we define its sub-semiring of positive elements [MATH] to be the closure under addition in [MATH] of the set [MATH].', '1703.10576-1-28-0': 'When classical non-determinism is introduced via the Born rule, quantum theory naturally gives rise to a probabilistic theory.', '1703.10576-1-28-1': 'Similarly, it is possible to prove that any theory of wavefunctions valued in a commutative semirings with involution [MATH] gives rise to an [MATH]-probabilistic theory, where [MATH] is the corresponding sub-semiring of positive elements.', '1703.10576-1-29-0': 'Let [MATH] be a commutative semiring with involution, and let [MATH] be its sub-semiring of positive elements.', '1703.10576-1-29-1': 'Then [MATH] is [MATH]-probabilistic under the [MATH]-enrichment inherited from [MATH].', '1703.10576-1-30-0': 'Note that the scalars of [MATH] are the elements of [MATH], and that the pure scalars are those in the form [MATH] for some [MATH]: as a consequence, not all scalars of [MATH] need be pure (in contrast to what happens with ordinary quantum theory).', '1703.10576-1-30-1': 'In what follows, we will try as much as possible to construct theories where all scalars are pure, but there are examples (such as the case of [MATH]-adic quantum theory) where this cannot be achieved.', '1703.10576-1-30-2': 'When all scalars are pure, the requirement that [MATH] is always automatically satisfied for all finite groups [MATH], and we only need to care about [MATH] being invertible as a scalar in [MATH] (a fact which always holds true whenever [MATH] is a semi-field/field and [MATH] is non-zero in [MATH]).', '1703.10576-1-31-0': 'We will now proceed to construct a number of toy models within this framework.', '1703.10576-1-31-1': 'For each toy model we will: (i) study the phase group; (ii) discuss which Hidden Subgroup Problems can be efficiently solved, and which generalised Mermin-type arguments can be implemented, using the phase group and the strongly complementary pairs; (iii) assess the overall locality of the theory, via the connection with the sheaf-theoretic framework for non-locality and contextuality.', '1703.10576-1-32-0': '# Real quantum theory', '1703.10576-1-33-0': 'The simplest non-conventional example is given by the ring [MATH] of signed reals (with the trivial involution), which yields the probability semiring [MATH] as its sub-semiring of positive elements; in particular, all positive elements are pure scalars.', '1703.10576-1-33-1': 'The corresponding probabilistic theory [MATH] is known as real quantum theory [CITATION]: it is arguably the most well-studied of the quantum-like theories, and the closest to ordinary quantum theory.', '1703.10576-1-33-2': 'Thus said, real quantum theory can be distinguished from ordinary quantum theory because it fails to be locally tomographic [CITATION], i.e. bipartite (mixed) states in real quantum theory cannot in general be distinguished by product measurements alone.', '1703.10576-1-33-3': 'Equivalently, one can check that the CP maps [MATH] and [MATH] on [MATH] in [MATH] cannot be distinguished by applications to mixed states of [MATH] alone, because the latter are described by density matrices which are always real symmetric.', '1703.10576-1-34-0': 'The group of phases in [MATH] is [MATH], and non-trivial interference is possible in real quantum theory.', '1703.10576-1-34-1': 'For example, each of the Pauli [MATH] eigenstates [MATH] of the qubit [MATH] in real quantum theory yields the uniform distribution when measured in the Pauli [MATH] basis [MATH], but their superposition [MATH] yields the outcome corresponding to [MATH] with certainty.', '1703.10576-1-34-2': "Simon's problem, and other Hidden Subgroup Problems on [MATH], can be solved efficiently in real quantum theory.", '1703.10576-1-34-3': 'More generally, consider a classical structure [MATH] on [MATH] with enough classical states, which corresponds to an orthonormal basis of [MATH] (because [MATH] is multiplicatively cancellative [CITATION]).', '1703.10576-1-34-4': 'Then the group of [MATH]-phase gates is isomorphic to the group [MATH] of [MATH]-bit strings under bitwise xor.', '1703.10576-1-34-5': 'Because of the structure of phase groups, generalised Mermin-type arguments only yield local empirical models [CITATION].', '1703.10576-1-34-6': "Nevertheless, Bell's theorem goes through in real quantum theory (as it only involves states and measurements on the [MATH] great circle of the Bloch sphere), which is therefore a non-local probabilistic theory.", '1703.10576-1-35-0': '# Relational quantum theory', '1703.10576-1-36-0': 'Examples of an entirely different nature are given by considering distributive lattices [MATH] (with the trivial involution), which yield themselves back as their sub-semirings of positive elements (because of multiplicative idempotence); in particular, all positive elements are pure scalars.', '1703.10576-1-36-1': 'Distributive lattices seem to be almost as far as one can get from the probabilistic semiring [MATH], but the category [MATH] has been studied extensively as a toy model for quantum theory (especially in the boolean case [MATH]) [CITATION], and the corresponding CPM category has also received some attention on its own [CITATION].', '1703.10576-1-36-2': 'We refer to the corresponding [MATH]-probabilistic (or possibilistic) theory as relational quantum theory.', '1703.10576-1-37-0': 'The group of phases in [MATH] is the singleton [MATH], and no interference is possible in relational quantum theory.', '1703.10576-1-37-1': 'Relational quantum theories also feature very few quantum-to-classical transitions: there is a unique basis on each system, namely the one given by the elements of the underlying set.', '1703.10576-1-37-2': 'They are local tomographic on pure states, but they fail to be tomographic altogether on mixed states: for example, the pure state [MATH] for [MATH] and the mixed state [MATH] are distinct, but cannot be distinguished by measurement.', '1703.10576-1-37-3': 'In fact, a characteristic trait of relational quantum theories is exactly that superposition and mixing are essentially indistinguishable (because of idempotence) [CITATION].', '1703.10576-1-37-4': 'Classical structures [MATH] in relational quantum theory over the booleans are known to correspond to abelian groupoids [MATH] [CITATION], and the corresponding group of [MATH]-phase gates is isomorphic to [MATH].', '1703.10576-1-37-5': 'It can be shown that generalised Mermin-type arguments only yield local empirical models [CITATION].', '1703.10576-1-37-6': 'In fact, it can be shown that that relational quantum theories are entirely local [CITATION].', '1703.10576-1-38-0': '# Hyperbolic quantum theory', '1703.10576-1-39-0': 'Turning our attention back to real algebras, we can consider the commutative ring of split complex numbers [MATH], a two-dimensional real algebra.', '1703.10576-1-39-1': 'Split complex numbers take the form [MATH], where [MATH] and [MATH]; in particular, they have non-trivial zero-divisors in the form [MATH], because [MATH].', '1703.10576-1-39-2': 'They come with the involution [MATH], which yields the signed-probability ring [MATH] as sub-semiring of positive elements; in particular, all positive elements are pure scalars.', '1703.10576-1-39-3': 'We refer to the corresponding [MATH]-probabilistic theory [MATH] as hyperbolic quantum theory [CITATION].', '1703.10576-1-40-0': 'Hyperbolic quantum theory is an extremely interesting theory.', '1703.10576-1-40-1': "On the one hand, it contains real quantum theory as a sub-theory, and as a consequence every scenario and protocol which can be implemented in real quantum theory (such as the algorithm to efficiently solve Simon's problem [CITATION]) can also be implemented in hyperbolic quantum theory.", '1703.10576-1-40-2': 'On the other hand, hyperbolic quantum theory is a local theory, in the sense that every empirical model arising in hyperbolic quantum theory admits a local hidden variable model in terms of signed probabilities (the notion of classical non-determinism for hyperbolic quantum theory) [CITATION].', '1703.10576-1-40-3': 'While signed probabilities might at first sound unphysical, an operational interpretation exists in terms of unsigned probabilities on signed events [CITATION].', '1703.10576-1-41-0': 'The group of phases in [MATH] consists of the elements with square norm [MATH], i.e. the elements in the form [MATH] which lie on the following unit hyperbola of the real plane: [EQUATION]', '1703.10576-1-41-1': 'In fact, the natural geometry for the split complex numbers is that of the real plane endowed with the Lorentzian metric [MATH], i.e. that of the Minkowski plane.', '1703.10576-1-41-2': 'Just like multiplication by phases in [MATH] forms the circle group [MATH] of rotations around the origin for the Euclidean plane, multiplication by phases in [MATH] forms the group [MATH] of orthochronous homogeneous Lorentz transformations for the Minkowski plane.', '1703.10576-1-41-3': "We have the isomorphism of Lie groups [MATH] given by [MATH]: as a consequence, the [MATH]-valued multiplicative characters for finite groups are exactly the same as the [MATH]-valued multiplicative characters, and the only finite groups with enough multiplicative characters to form a Fourier basis are the ones in the form [MATH]; Simon's problem, and other Hidden Subgroup Problems for [MATH], can be efficiently solved in hyperbolic quantum theory, despite the latter being local.", '1703.10576-1-41-4': 'Things are different for infinite groups such as [MATH], which have enough [MATH]-valued multiplicative characters but not enough [MATH]-valued multiplicative characters.', '1703.10576-1-42-0': 'Now consider a classical structure [MATH] corresponding to an orthonormal basis of [MATH] (we have to ask explicitly for orthogonality because the result of [CITATION] does not apply to hyperbolic quantum theory: [MATH] has non-trivial zero-divisors, and hence it is not multiplicatively cancellative).', '1703.10576-1-42-1': 'The group of [MATH]-phase gates is isomorphic to [MATH], and has [MATH] as a maximal finite subgroup: as a consequence, generalised Mermin-type arguments (which involve finite groups) only yield local empirical models, just as in real quantum theory.', '1703.10576-1-42-2': 'However, future extensions of Mermin-type arguments to infinite groups might yield different results: this is because subgroups like [MATH] would become available, and there are equations (such as [MATH]) which have no solutions in the subgroup [MATH] but have solutions (e.g. [MATH] and [MATH]) in the larger group [MATH].', '1703.10576-1-43-0': '# Parity quantum theory', '1703.10576-1-44-0': 'A simple variation on relational quantum theory (over the booleans) is given by using symmetric difference of sets, instead of union, as the superposition operation.', '1703.10576-1-44-1': 'This leads us to consider the finite field with two elements [MATH], with trivial involution, in place of the booleans [MATH], also with trivial involution.', '1703.10576-1-44-2': 'The multiplication is the same, but addition is now non-idempotent, and superposition is no longer the same as mixing.', '1703.10576-1-44-3': 'The parity semiring [MATH] yields itself back as its sub-semiring of positive elements (in particular, all positive elements are pure scalars), and we refer to the corresponding [MATH]-probabilistic theory [MATH] as parity quantum theory.', '1703.10576-1-45-0': 'Because the involution is trivial, parity quantum theory as defined here pretty much coincides with the [MATH] case of modal quantum theory [CITATION], but it should be noted that the philosophical interpretation of [MATH]-valued probabilities is significantly different.', '1703.10576-1-45-1': 'In modal quantum theory, the interest is in generating possibilistic tables by using finite fields, subsequently interpreting all zero values as the boolean [MATH] and all non-zero values as the boolean [MATH].', '1703.10576-1-45-2': 'In parity quantum theory, the non-determinism itself is interpreted to be natively [MATH]-valued, and no attempt is made to translate the resulting empirical models into possibilistic ones.', '1703.10576-1-45-3': 'Indeed, such an interpretation would not be natural within our semiring-oriented framework, as no semiring homomorphism can exists from any finite field into the booleans.', '1703.10576-1-46-0': 'The group of phases in [MATH] is the singleton [MATH], but interference is still possible in parity quantum theory: this somewhat counter-intuitive situation is made possible by the fact that [MATH] is its own additive inverse in [MATH], so that triviality of the group of phases is slightly deceptive.', '1703.10576-1-46-1': 'Indeed, consider the four two-qubit states below, which form an orthonormal basis for [MATH]: [EQUATION]', '1703.10576-1-46-2': 'For example, we have [MATH].', '1703.10576-1-46-3': 'When measured in the computational basis [MATH], the normalised states [MATH], [MATH] and [MATH] all have non-zero [MATH]-probability of yielding an outcome in the set [MATH], but their superposition [MATH] (also a normalised state) has zero [MATH]-probability of yielding an outcome in that set.', '1703.10576-1-47-0': 'Because the group of phases is trivial, so are all the groups of phase gates, as well as all the [MATH]-valued multiplicative characters of all groups; as a consequence, parity quantum theory admits no non-trivial generalised Mermin-type arguments, and no implementation of the algorithm to solve the HSP.', '1703.10576-1-47-1': 'Furthermore, Theorem [REF] shows that parity quantum theory is local, because [MATH] is a field.', '1703.10576-1-48-0': '[MATH]-probabilistic theories can be similarly constructed for modal quantum theory over any other finite field [MATH] [CITATION], by taking [MATH] with the trivial involution.', '1703.10576-1-48-1': 'However, these theories have a lot of non-pure scalars-namely the [MATH] non-squares in [MATH]-and their phases are close to trivial-namely they are [MATH] if [MATH] and [MATH] if [MATH].', '1703.10576-1-48-2': 'Instead, we will consider a more sophisticated construction based on quadratic extensions of finite fields, which we will refer to as finite-field quantum theory.', '1703.10576-1-49-0': 'What will make finite-field quantum theory extremely attractive for Categorical Quantum Mechanics is the fact that it is a local theory (by Theorem [REF]), in which it is nonetheless possible to formulate non-trivial quantum algorithms (such as the one solving the abelian Hidden Subgroup Problem), as well as non-trivial non-locality arguments (such as generalised Mermin-type arguments).', '1703.10576-1-49-1': "This is in stark contrast with the more traditional toy models such as Spekkens's toy model [CITATION] and relational quantum theory, in which the quantum Fourier transform cannot be performed for non-trivial groups [CITATION] (precluding the implementation of algorithms based on it), and in which all Mermin-type arguments are necessarily trivial [CITATION].", '1703.10576-1-50-0': '# Finite-field quantum theory', '1703.10576-1-51-0': 'Consider a finite field [MATH] (with [MATH] odd), and let [MATH] be a generator for the cyclic group [MATH] of invertible (aka non-zero) elements in [MATH] (i.e. a primitive element for [MATH]).', '1703.10576-1-51-1': 'We consider the ring [MATH], equipped with the involution [MATH]: because [MATH] is a primitive element, [MATH] is a field.', '1703.10576-1-51-2': 'We are in fact working with the quadratic extension of fields [MATH], equipped with the usual involution and (squared) norm: [EQUATION]', '1703.10576-1-51-3': 'The sub-field [MATH] (given by the elements in the form [MATH]) is the sub-semiring of positive elements (and we will shortly see that all positive elements are pure scalars).', '1703.10576-1-52-0': 'The phases in [MATH] are the points [MATH] of the [MATH] plane lying on the unit hyperbola [MATH], which does not factor as a product of two lines because [MATH] is a primitive element.', '1703.10576-1-52-1': 'The following iconic result of Galois theory due to Hilbert can be used to characterise them (see e.g. [CITATION] for a proof).', '1703.10576-1-52-2': "[Hilbert's Theorem 90]", '1703.10576-1-53-0': 'Let [MATH] be a finite cyclic field extension, and let [MATH] be a generator for its cyclic Galois group.', '1703.10576-1-53-1': 'Then the multiplicative group of elements [MATH] of unit relative norm [MATH] is isomorphic to the quotient group [MATH].', '1703.10576-1-54-0': 'The phases in [MATH] form the cyclic group [MATH].', '1703.10576-1-55-0': "Another interesting consequence of Hilbert's Theorem 90 is the fact that the positive elements in finite-field quantum theory are all pure scalars.", '1703.10576-1-56-0': 'All scalars in [MATH] are pure.', '1703.10576-1-57-0': 'We have seen that finite-field quantum theory comes with a non-trivial phase group, which in turn allow for non-trivial implementations of certain quantum protocols.', '1703.10576-1-57-1': 'We open with a result about the Quantum Fourier Transform, which combined with the main result of Ref. [CITATION] implies that the Hidden Subgroup Problem can be solved efficiently in finite-field quantum theory for arbitrarily large families of finite abelian groups (as [MATH] grows larger).', '1703.10576-1-58-0': 'Let [MATH] be a finite abelian group.', '1703.10576-1-58-1': 'Then [MATH] has enough [MATH]-valued unitary multiplicative characters if and only if [MATH] with [MATH] for all [MATH].', '1703.10576-1-58-2': 'When this is the case, the Hidden Subgroup Problem for [MATH] can be solved efficiently in finite-field quantum theory.', '1703.10576-1-59-0': 'Now consider a classical structure [MATH] with enough classical states on a [MATH]-dimensional quantum system in finite-field quantum theory, which corresponds to an orthonormal basis of the vector space [MATH] (because [MATH] is multiplicatively cancellative [CITATION]).', '1703.10576-1-59-1': 'Then the group of [MATH]-phase gates in [MATH] is isomorphic to the group [MATH].', '1703.10576-1-60-0': 'It is possible to formulate non-trivial generalised Mermin-type arguments in finite-field quantum theory if and only if [MATH] is not a square-free natural number.', '1703.10576-1-61-0': 'While finite-field quantum theory and parity quantum theory might not have as direct a physical interpretation as hyperbolic quantum theory and relational quantum theory, they come with the major advantage of having wavefunction valued over a field, so that objects are finite-dimensional vector spaces (equipped with a non-standard inner product, in the case of finite-field quantum theory).', '1703.10576-1-61-1': 'This opens the door for a systematic study of quantum systems in these theories using standard tools from finite geometry.', '1703.10576-1-61-2': 'Further investigation in this direction is left to future work.', '1703.10576-1-62-0': '# [MATH] p-adic quantum theory', '1703.10576-1-63-0': 'We now look at the construction of [MATH]-adic quantum mechanics [CITATION], where [MATH] is the field of [MATH]-adic numbers, and [MATH] is some quadratic extension.', '1703.10576-1-63-1': 'In this Section, we will use the notation [MATH] to denote the [MATH]-adic numbers, and [MATH] to denote the [MATH]-adic integers, to distinguish them from the finite field [MATH] of integers modulo [MATH]; note that this convention is different from the one used in many texts on [MATH]-adic arithmetic, where [MATH] is used for the [MATH]-adic integers (and [MATH] for the [MATH]-adic numbers).', '1703.10576-1-64-0': 'When [MATH], the [MATH]-adic numbers [MATH] fall within four distinct quadratic classes-jointly labelled by the parity of the order [MATH] and by the quadratic class of the first non-zero digit [MATH]-and there are three corresponding inequivalent quadratic extensions.', '1703.10576-1-64-1': 'This means that there is no way to obtain all positive elements as pure scalars by a single quadratic extension.', '1703.10576-1-64-2': 'This would seem to indicate that mixed states play a necessary role in the emergence of [MATH]-adic probabilities, which cannot all be obtained from pure states alone: the potential physical significance of this observation might become the topic of future work on [MATH]-adic quantum theory within CQM.', '1703.10576-1-65-0': 'We consider the quadratic extension [MATH], where [MATH] and [MATH] is a primitive element in the field [MATH] of integers modulo [MATH], and we follow the presentation of Ref. [CITATION].', '1703.10576-1-65-1': 'A generic element of [MATH] takes the form [MATH], for [MATH], and its square norm is [MATH].', '1703.10576-1-65-2': 'Whether an element [MATH] can be written in this form, i.e. whether is is a pure scalar in [MATH], is determined by the sign function [MATH], which takes the value [MATH] if [MATH] for some [MATH], and the value [MATH] otherwise.', '1703.10576-1-65-3': 'An explicit form for the sign function (in the [MATH] case) is given by Equation (2.34) of Ref. [CITATION], which specialised to our case ([MATH] and [MATH]) reads [MATH].', '1703.10576-1-65-4': 'Hence the pure scalars in [MATH] are exactly the [MATH]-adic numbers [MATH] with even order [MATH]; closure of this set under addition yields [MATH] as sub-semiring (field, in fact) of positive elements in [MATH].', '1703.10576-1-66-0': 'The phases in [MATH]-adic quantum theory are those [MATH] such that [MATH].', '1703.10576-1-66-1': 'In Ref. [CITATION] (Equation (4.35) of Section IV.C, and Equation (C12b) of Appendix C.3) it is shown that phases form a multiplicative group [MATH] isomorphic to the additive group [MATH]-here [MATH] are the integers modulo [MATH], while [MATH] is the additive subgroup of [MATH] formed by those [MATH]-adic integers which are divisible by [MATH].', '1703.10576-1-66-2': 'In particular, [MATH] is an infinite group with the cardinality of the continuum, and each sheet [MATH] is a profinite torsion-free group, which is best understood by looking at the descending normal series [MATH] and considering the finite cyclic quotients [MATH].', '1703.10576-1-67-0': 'The scalar [MATH] is always invertible, and it is in the form [MATH] if and only if the largest power of [MATH] which divides [MATH] is even.', '1703.10576-1-67-1': "Furthermore, [MATH] has enough [MATH]-valued multiplicative characters if and only if [MATH] with [MATH] for all [MATH] (in the light of Hensel's Lemma, this parallelism between [MATH]-adic quantum theory and finite-field quantum theory on [MATH] should not come as a big surprise): finite abelian groups [MATH] satisfying this condition admit efficient solutions for Hidden Subgroup Problems in [MATH]-adic quantum theory (because we necessarily have that [MATH] cannot divide [MATH]).", '1703.10576-1-67-2': 'Similarly, it is possible to formulate non-trivial generalised Memrin-type arguments in [MATH]-adic quantum theory if and only if [MATH] is not square-free.', '1703.10576-1-67-3': 'Thus said, [MATH]-adic quantum theory is a local theory by virtue of Theorem [REF].', '1703.10576-1-68-0': 'Similar considerations apply to the the construction of [MATH]-adic quantum theory for the other two quadratic extensions [MATH] and [MATH] available in the case of [MATH] (although the cases [MATH] and [MATH] have to be treated separately), as well as the seven quadratic extensions available in the case of [MATH].', '1703.10576-1-68-1': 'The phase groups take a similar (but not identical) form to the one presented here, and the full details can be found in Ref. [CITATION] (Section IV.C and Appendices C.3, C.4).', '1703.10576-1-69-0': '# Tropical quantum theory', '1703.10576-1-70-0': 'Relational quantum theory involves semirings which are both additively and multiplicatively idempotent, parity quantum theory involves a semiring which is only multiplicatively idempotent, and ordinary quantum theory involves a semiring which is neither additively nor multiplicatively idempotent.', '1703.10576-1-70-1': 'We now give examples of theories with wavefunctions based in semirings which are additively idempotent but not multiplicatively idempotent, namely the tropical semirings [CITATION].', '1703.10576-1-71-0': 'A tropical semiring is the commutative semiring [MATH] obtained from a totally ordered commutative monoid [MATH] having an absorbing element [MATH] which is larger than all elements in the monoid.', '1703.10576-1-71-1': 'In the tropical semiring, [MATH] is the addition, [MATH] is the additive unit, [MATH] is the multiplication and [MATH] is the multiplicative unit.', '1703.10576-1-71-2': 'The nomenclature is extended to semirings isomorphic to the explicitly min-plus semirings used above (e.g. max-plus formulations, or the Viterbi semiring).', '1703.10576-1-72-0': 'Examples of tropical semirings appearing in the literature include the tropical reals [MATH], the tropical integers [MATH], the tropical naturals [MATH], and the Viterbi semiring [MATH] (which is a tropical semiring because it is isomorphic to the explicitly min-plus semiring [MATH] via the semiring homomorphism [MATH]).', '1703.10576-1-72-1': 'In fact, there is an easy characterisation of which commutative semirings arise as tropical semirings (the proof is omitted as it is a straightforward check).', '1703.10576-1-73-0': 'A commutative semiring [MATH] is a tropical semiring if and only if for all [MATH] we have [MATH] or [MATH] (in which case we can set [MATH]).', '1703.10576-1-74-0': 'From now on, we will revert back to usual semiring notation, and we will rely on the result above to connect with the min-plus notation typical of tropical geometry [CITATION].', '1703.10576-1-74-1': 'We will, however, remember that tropical semirings come with a total order respected by the multiplication, and we will occasionally use [MATH], [MATH] and [MATH] in addition to the addition/multiplication.', '1703.10576-1-75-0': 'The only involution possible on a tropical semiring [MATH] is the trivial one, and the positive elements form the sub-semiring of squares [MATH].', '1703.10576-1-76-0': 'If [MATH] is a tropical semiring and [MATH] is its sub-semiring of positive elements, we refer to the [MATH]-probabilistic theory [MATH] as tropical quantum theory.', '1703.10576-1-77-0': 'Just as in the case of relational quantum theory, the group of phases in a tropical semiring [MATH] is always trivial (because [MATH] implies [MATH] in any totally ordered monoid [MATH]), and there is no interference.', '1703.10576-1-77-1': 'Similarly, there is a unique orthonormal basis on each system, the only unitaries/invertible maps are permutations, and superposition cannot be distinguished from mixing by measurements alone.', '1703.10576-1-77-2': 'Tropical quantum theory does not admit any implementation of the algorithm for the abelian Hidden Subgroup Problem, nor does it admit any generalised Mermin-type non-locality arguments.', '1703.10576-1-78-0': 'The parallelisms with relational quantum theory become less surprising when one realises that tropical quantum theory is another generalisation of quantum theory over the booleans, which form a totally ordered distributive lattice, and hence are a particular case of tropical semiring.', '1703.10576-1-78-1': '(Proof of the following result is omitted, as it is a straightforward check.)', '1703.10576-1-79-0': 'Any totally ordered distributive lattice [MATH] is a tropical semiring [MATH]; conversely, every tropical semiring [MATH] which has [MATH] as least element and such that [MATH] for all [MATH] is a totally ordered distributive lattice [MATH].', '1703.10576-1-80-0': 'In the light of the result above, we expect tropical quantum theory to be local, exactly like relational quantum theory, but further investigation of this question is left to future work.', '1703.10576-1-81-0': '# Conclusions and Future Work', '1703.10576-1-82-0': 'In the first two Sections of this work, we have provided a general framework, based on enrichment of CP* categories, for the construction of toy models of quantum theory.', '1703.10576-1-82-1': 'Specifically, we have focussed our efforts on theories of wavefunctions valued in some commutative semirings with involution [MATH], replacing the field with involution [MATH] used in conventional quantum theory: in the process, the dagger compact category [MATH] of finite-dimensional complex Hilbert spaces was replaced by the symmetric monoidal category [MATH] of free finite-dimensional [MATH]-semimodules, equipped with the dagger compact structure given by the involution of [MATH] (which generalises complex conjugation).', '1703.10576-1-82-2': 'We have also postulated classical non-determinism to arise via a generalisation of the Born rule, as embodied by the CP* construction, and we have shown that our construction yields special cases [MATH]-probabilistic theories, as defined in the recently introduced framework of Categorical Probabilistic Theories (here [MATH] is the sub-semiring of [MATH] given by the positive elements, generalising the probabilistic semiring [MATH] modelling classical non-determinism in conventional quantum theory).', '1703.10576-1-83-0': 'In subsequent Sections, we have shown our framework to be expressive enough to capture many toy models which have appeared in the literature in (more or less) recent years.', '1703.10576-1-83-1': 'Aside from real quantum theory and relational quantum theory, which have already found their special place in the heart of categorical quantum mechanicians and operational probabilistic theorists, we considered hyperbolic quantum theory, [MATH]-adic quantum theory and parity quantum theory (the [MATH] case of modal quantum theory), all interesting enough to deserve their own place in our growing zoo of categorical toy models.', '1703.10576-1-84-0': 'We have also introduced two new families of toy models, one based on quadratic extensions of finite field (finite-field quantum theory), and the other based on tropical semirings (tropical quantum theory).', '1703.10576-1-84-1': 'While tropical quantum theory proves to be a variant on relational quantum theory, finite-field quantum theory is of independent interest: it boasts a rich phase group which allows non-trivial quantum protocols to be implemented, while at the same time remaining fully local, as well as amenable to treatment with tools from finite geometry.', '1703.10576-1-85-0': 'Future work.', '1703.10576-1-85-1': 'This work leaves a number of directions open to future investigation.', '1703.10576-1-85-2': 'Firstly, some of the toy theories presented in this work have barely had their surface scratched from the point of view of Categorical Quantum Mechanics: an in-depth study of the categorical features they possess (e.g. unitaries, measurements/preparations, [MATH]-Frobenius algebras, complementary and strongly complementary observables), as well as a more thorough understanding of which quantum protocols can be implemented in them, will be a priority in further developments.', '1703.10576-1-86-0': 'Secondly, this work mainly focussed on existing toy models, or variations thereof, to show that the framework we presented truly is expressive enough for its intended purpose.', '1703.10576-1-86-1': 'However, there are many other examples of semirings, rings and fields that could potentially produce interesting and unexpected features, and we expect our zoo of toy models to continue growing in the coming years.', '1703.10576-1-87-0': 'Finally, it was not possible, for reasons of space, to explore the applications of finite geometry to finite-field quantum theory, despite the promise of interesting connections between toy quantum systems and finite projective spaces.', '1703.10576-1-87-1': 'Similarly, it was not possible to establish whether tropical quantum theory is always local.', '1703.10576-1-87-2': 'A thorough exploration of these matters is left to future work.'}

{'1703.10576-2-0-0': 'We present a uniform framework for the treatment of a large class of toy models of quantum theory.', '1703.10576-2-0-1': 'Specifically, we will be interested in theories of wavefunctions valued in commutative involutive semirings, and which give rise to some semiring-based notion of classical non-determinism via the Born rule.', '1703.10576-2-0-2': 'The models obtained with our construction possess many of the familiar structures used in Categorical Quantum Mechanics.', '1703.10576-2-0-3': 'We also provide a bestiary of increasingly exotic examples: some well known, such as real quantum theory and relational quantum theory; some less known, such as hyperbolic quantum theory, p-adic quantum theory and "parity quantum theory"; and some entirely new, such as "finite-field quantum theory" and "tropical quantum theory".', '1703.10576-2-0-4': 'As a further bonus, the measurement scenarios arising within these theories can be studied using the sheaf-theoretic framework for non-locality and contextuality.', '1703.10576-2-0-5': 'Their computational complexity can similarly be studied within existing frameworks for affine and unitary circuits over commutative semirings.', '1703.10576-2-1-0': '# Introduction', '1703.10576-2-2-0': 'The construction of toy models plays a key role in many foundational efforts across mathematics, physics and computer science.', '1703.10576-2-2-1': 'In the foundations of quantum theory, toy models help to understand which abstract structural features of quantum systems-and their interface to the classical world-are involved in providing different kinds of non-classical behaviour.', '1703.10576-2-2-2': 'In turn, this informs practical research into quantum computation and communication technology, helping to cut down the noise and focus on those features that truly contribute to quantum advantage.', '1703.10576-2-3-0': "The categorical and diagrammatic methods from Categorical Quantum Mechanics (CQM) [CITATION] have proven particularly well suited to the construction and study of toy models, with the majority of efforts focussed on Spekkens's toy model [CITATION] and the more general relational quantum theory [CITATION].", '1703.10576-2-3-1': 'In the same years, models have been developed within a variety of other frameworks: examples include real quantum theory, of special interest in the context of generalised/operational probabilistic theories and the study of Jordan algebras [CITATION], hyperbolic quantum theory [CITATION], [MATH]-adic quantum theory [CITATION], and modal quantum theory [CITATION].', '1703.10576-2-4-0': 'When constructing a toy model, it is essential to consider both the quantum side and the corresponding quantum-classical interface: indeed, many toy models result in notions of classical non-determinism which are different from the conventional probabilistic one, and special care needs to be taken in order to achieve a consistent treatment of classical systems.', '1703.10576-2-4-1': "Examples of this phenomenon include the possibilistic non-determinism arising from Spekkens's toy model and relational quantum theory [CITATION], the [MATH]-adic non-determinism arising from [MATH]-adic quantum theory [CITATION], and the signed probabilities arising from hyperbolic quantum theory [CITATION].", '1703.10576-2-5-0': 'Because of this issue, we adopt a recently developed framework, that of Categorical Probabilistic Theories [CITATION], which can simultaneously treat quantum-like systems and classical systems endowed with generic semiring-based notion of non-determinism.', '1703.10576-2-5-1': 'Categorical Probabilistic Theories have been introduced with the intent of bridging the gap between CQM and Operational Probabilistic Theories (OPTs) [CITATION]: they aim to provide categorical and diagrammatic methods in the style of CQM to talk about the problems that OPTs are concerned with.', '1703.10576-2-5-2': 'As a side-product of their abstract categorical formulation, these theories natively admit a general, semiring-based notion of classical non-determinism, and are therefore perfect to construct and study exotic toy models of quantum theory.', '1703.10576-2-6-0': 'In this work, we focus our attention on a very large class of finite-dimensional quantum-like theories, where wavefunctions of complex amplitude are replaced with wavefunctions valued in some arbitrary commutative semiring [MATH] with involution.', '1703.10576-2-6-1': 'In the quantum-classical transition, probabilities are still deemed to arise via the Born rule, and as a consequence classical non-determinism is naturally and necessarily modelled by the semiring [MATH] of positive elements for [MATH], generalising the traditional probabilistic semiring [MATH] (which is the semiring of positive elements for [MATH], with complex conjugation as involution).', '1703.10576-2-7-0': 'As an underlying model for [MATH]-valued wavefunctions we consider the category [MATH], with objects in the form [MATH] for all finite sets [MATH], and morphisms [MATH] given by [MATH], the [MATH]-semimodule of [MATH]-by-[MATH] matrices with values in [MATH] (equipped with matrix composition and identities).', '1703.10576-2-7-1': 'The category [MATH] generalises [MATH], and is a dagger compact category with Kronecker product as tensor product and conjugate transpose as dagger.', '1703.10576-2-7-2': 'We model mixed-state quantum theory using the CPM construction [CITATION], and the full quantum-classical theory using the dagger-idempotent variant of the CP* construction presented in Ref. [CITATION] (instead of the one from Refs. [CITATION]), which results in quantum-classical transitions governed by a categorical flavour of the Born rule.', '1703.10576-2-7-3': 'Hence our full theory will be described by [MATH], which we will show to be an [MATH]-probabilistic theory in the sense of Ref. [CITATION], and hence amenable for treatment in the framework of Categorical Probabilistic Theories.', '1703.10576-2-7-4': 'Furthermore, this formulation is compatible with the framework developed in Ref. [CITATION], which can therefore be used to investigate natural notions of computational complexity for the toy models we present.', '1703.10576-2-8-0': 'Synopsis.', '1703.10576-2-8-1': 'We begin in Sections [REF] and [REF] by summarising the structure of Categorical Probabilistic Theories, and by introducing our general framework for the construction of theories of wavefunctions valued in commutative involutive semirings.', '1703.10576-2-8-2': 'The remainder of the work is then dedicated to showing that our new framework is expressive enough to cover all the interesting toy models which we mentioned.', '1703.10576-2-9-0': 'In Sections [REF] and [REF] we briefly cover real quantum theory and relational quantum theory, two models already dear to the CQM and OPT communities.', '1703.10576-2-9-1': 'In Sections [REF], [REF] and [REF] we cover hyperbolic quantum theory, parity quantum theory and [MATH]-adic quantum theory, already known in some specialised circles but not yet part of the categorical bestiary.', '1703.10576-2-9-2': "Hyperbolic quantum theory is particularly interesting: it is a quasi-probabilistic theory which can support at least as many protocols as real quantum theory (e.g. Simon's problem can be solved efficiently), but at the same time it is entirely local thanks to negative probabilities.", '1703.10576-2-9-3': 'Furthermore, parity quantum theory was already known as the [MATH] case of modal quantum theory, although a different interpretation is given here to its classical non-determinism.', '1703.10576-2-10-0': 'In Sections [REF] and [REF] we present two new families of toy models: finite-field quantum theory and tropical quantum theory, based on finite fields and tropical semirings respectively.', '1703.10576-2-10-1': 'Finite-field quantum theory proves to be especially interesting: it possesses a rich phase group, and can support a number of non-trivial quantum protocols and arguments, but at the same time it is entirely local.', '1703.10576-2-10-2': 'Furthermore, its pure quantum states are points of finite projective spaces, and can be studies using tools from finite geometry.', '1703.10576-2-10-3': 'Tropical quantum theory, on the other hand, turns out to be an extension of boolean quantum theory alternative to relational quantum theory, with which it shares many structural and operational traits.', '1703.10576-2-11-0': '# Categorical Probabilistic Theories', '1703.10576-2-12-0': 'The main intuition behind a generalised, semiring-based notion of classical non-deterministic systems is borrowed from computer science, where the use of (commutative) semirings to model resources used by automata is commonplace.', '1703.10576-2-12-1': 'We look at probabilities in physics as a resource modelling non-determinism of classical systems, with properties captured by those of the commutative semiring [MATH]: from this perspective, it makes sense to study what classical non-determinism looks like when [MATH] is replaced by some other commutative semiring [MATH].', '1703.10576-2-12-2': 'Interesting alternative choices for [MATH] which already appeared in the literature include the boolean semiring [MATH] and other locales (in relational quantum theory), the quasi-probabilistic semiring [MATH] (a field, in hyperbolic quantum theory), the [MATH]-adic semiring [MATH] (another field, in [MATH]-adic quantum theory), and finite fields (in modal quantum theory).', '1703.10576-2-12-3': 'We will refer to classical systems with non-determinism governed by a commutative semiring [MATH] as classical [MATH]-probabilistic systems.', '1703.10576-2-13-0': 'One of the reason for the wide adoption of semirings in mathematics is that they capture the bare minimum algebraic structure required by matrix multiplication, with commutativity being a necessary addendum when a symmetric tensor product of matrices is of interest (as is the case in many physical applications).', '1703.10576-2-13-1': 'As our category of classical [MATH]-probabilistic systems we take the category [MATH] of free finite-dimensional [MATH]-semimodules and [MATH]-linear maps between them: objects are in the form [MATH] with [MATH] finite sets, and morphisms [MATH] form the free finite-dimensional [MATH]-semimodule of [MATH]-by-[MATH] matrices with values in [MATH].', '1703.10576-2-13-2': 'The category [MATH] is a compact closed symmetric monoidal category, with Kronecker product of matrices as tensor product.', '1703.10576-2-13-3': 'It is enriched in itself, and hence in commutative monoids ([MATH]-enriched), so that each homset [MATH] comes with a mixing operation [MATH] and an impossible process [MATH].', '1703.10576-2-13-4': 'The category [MATH] contains the category [MATH] of finite sets and functions (the category of classical deterministic systems) as a subcategory, and from [MATH] it inherits an environment structure [MATH] [CITATION] given by the discarding maps [MATH].', '1703.10576-2-14-0': 'In Ref. [CITATION], it is argued that the minimal requirements for a categorical probabilistic theory should include: (i) the explicit existence of classical systems; (ii) the extendibility of probabilistic mixing to all systems; (iii) the possibility of defining a meaningful notion of local state and discarding of systems.', '1703.10576-2-15-0': '[[MATH]-probabilistic Theory]', '1703.10576-2-16-0': 'An [MATH]-probabilistic theory is a symmetric monoidal category [MATH] which satisfies the following requirements.', '1703.10576-2-17-0': '[(i)] There is a full sub-SMC of [MATH], denoted by [MATH], which is equivalent to [MATH].', '1703.10576-2-17-1': '[(ii)] The SMC [MATH] is enriched in commutative monoids, and the enrichment on the subcategory [MATH] coincides with the one given by the linear structure of [MATH].', '1703.10576-2-17-2': '[(iii)] The SMC [MATH] comes with an environment structure, i.e. with a family [MATH] of morphisms which satisfy the following requirements: [EQUATION]', '1703.10576-2-17-3': 'On the subcategory [MATH], this environment structure must coincide with the canonical one of [MATH].', '1703.10576-2-18-0': 'We refer to [MATH] as classical theory, and to its objects and morphisms as classical systems and processes.', '1703.10576-2-18-1': 'As diagrammatic convention, we use dashed wires for classical systems, and solid wires for generic ones.', '1703.10576-2-19-0': '[MATH]-probabilistic theories come with a number of native features that are commonplace in the modelling of quantum protocols: it is possible to exert classical control, to define tests with classical outcomes, to marginalise over classical outcomes, to work with preparations and measurements, and to apply any kind of classical pre- and post-processing.', '1703.10576-2-19-1': 'Amongst the many mixed quantum-classical processes, we can consider Bell-type measurement scenarios.', '1703.10576-2-19-2': 'An [MATH]-party Bell-type measurement scenario in an [MATH]-probabilistic theory is a process in the following form, where the processes [MATH] and the state [MATH] are all normalised (recall that a process [MATH] is said to be normalised if [MATH]): [EQUATION]', '1703.10576-2-19-3': 'In the context of a Bell-type measurement scenario, the processes [MATH] are often referred to as measurements, their inputs as measurement choices and their outputs as (measurement) outcomes.', '1703.10576-2-19-4': 'The following result from Ref. [CITATION] ensures that non-locality in [MATH]-probabilistic theories can always be studied using the well-established sheaf-theoretic framework for non-locality and contextuality [CITATION].', '1703.10576-2-19-5': '[Bell-type measurement scenarios [CITATION]]', '1703.10576-2-20-0': 'A Bell-type measurement scenario in an [MATH]-probabilistic theory always corresponds to a no-signalling empirical model in the sheaf-theoretic framework for non-locality and contextuality [CITATION].', '1703.10576-2-21-0': 'An immediate consequence of the connection with the sheaf-theoretic framework is that we can straightforwardly adapt a proof of Ref. [CITATION] to rule out non-locality in a large class of toy models.', '1703.10576-2-21-1': '[Locality of [MATH]-probabilistic theories over fields]', '1703.10576-2-22-0': 'If [MATH] is a field, then all [MATH]-probabilistic theories are local.', '1703.10576-2-23-0': '# Theories of wavefunctions valued in commutative involutive semirings', '1703.10576-2-24-0': 'Note that two different linear structures intervene in the definition of quantum theory: the [MATH]-linear structure of wavefunctions, modelling superposition, interference and phases, and the [MATH]-linear structure of probability distributions over classical systems.', '1703.10576-2-24-1': 'We have already seen that the framework of [MATH]-probabilistic theories replaces the probability semiring [MATH] with a more general commutative semiring [MATH] as a model of classical non-determinism.', '1703.10576-2-24-2': 'In this Section, we construct a large class of toy models of quantum theory by considering theories of wavefunctions with amplitudes valued in some commutative semiring [MATH] with involution, generalising the field with involution [MATH] traditionally used in quantum mechanics.', '1703.10576-2-24-3': 'To do so, we consider the dagger compact category [MATH] (dagger and compact closed structure will be defined using the involution), and we require classical non-determinism to arise via the Born rule, as embodied by the CP* construction.', '1703.10576-2-24-4': 'The corresponding quantum-classical theory will therefore be modelled by [MATH], and the main result of this Section (Theorem [REF]) will show that this is an [MATH]-probabilistic theory (where [MATH] is the sub-semiring of positive elements of [MATH]; see Definition [REF] below).', '1703.10576-2-25-0': 'The category [MATH] for a commutative semiring [MATH] with involution is defined as in the previous Section, but it comes with additional structure.', '1703.10576-2-25-1': 'Indeed, we can defined dagger and compact closed structures on [MATH] exactly as done in [MATH] (which is [MATH]), with conjugation taken using the involution [MATH] of [MATH] in place of complex conjugation.', '1703.10576-2-25-2': 'Each object [MATH] in [MATH] comes with at least one orthonormal basis [MATH], as well as an associated special commutative [MATH]-Frobenius algebra [MATH]: [EQUATION]', '1703.10576-2-25-3': 'For any group structure [MATH] on any finite set [MATH], one also obtains an associated [MATH]-Frobenius algebra [MATH] on [MATH] by linearly extending the group multiplication and unit: [EQUATION]', '1703.10576-2-25-4': 'The [MATH]-Frobenius algebra is commutative if and only if the group is, and it always satisfies the following: [EQUATION]', '1703.10576-2-25-5': 'Unfortunately, [MATH] is not quasi-special (a.k.a. normalisable) unless the scalar [MATH] takes the form [MATH] for some [MATH] which is multiplicatively invertible: when this is the case, however, we have a legitimate strongly complementary pair [MATH] in [MATH] corresponding to the finite group [MATH].', '1703.10576-2-25-6': 'When [MATH] is abelian these strongly complementary pairs can be used (under additional constraints) to implement quantum protocols such as the algorithm to solve the abelian Hidden Subgroup Problem [CITATION] or generalised Mermin-type arguments [CITATION].', '1703.10576-2-25-7': 'This also means that certain objects in [MATH] support fragments of the ZX calculus [CITATION], opening the way to the application of well-established diagrammatic techniques.', '1703.10576-2-26-0': 'In quantum theory, the probabilistic semiring [MATH] arises as a sub-semiring of [MATH] fixed by complex conjugation, namely the sub-semiring of those elements [MATH] taking the form [MATH]: this is, essentially, a hallmark of the Born rule.', '1703.10576-2-26-1': 'In general commutative semirings with involution, elements in the form [MATH] need not be closed under addition, but it is true their closure under addition always form a semiring.', '1703.10576-2-27-0': 'Let [MATH] be a commutative semiring with involution.', '1703.10576-2-27-1': 'Then we define its sub-semiring of positive elements [MATH] to be the closure under addition in [MATH] of the set [MATH].', '1703.10576-2-28-0': 'When classical non-determinism is introduced via the Born rule, quantum theory naturally gives rise to a probabilistic theory.', '1703.10576-2-28-1': 'Similarly, it is possible to prove that any theory of wavefunctions valued in a commutative semiring [MATH] with involution gives rise to an [MATH]-probabilistic theory, where [MATH] is the corresponding sub-semiring of positive elements.', '1703.10576-2-29-0': 'Let [MATH] be a commutative semiring with involution, and let [MATH] be its sub-semiring of positive elements.', '1703.10576-2-29-1': 'Then [MATH] is [MATH]-probabilistic under the [MATH]-enrichment inherited from [MATH].', '1703.10576-2-30-0': 'Note that the scalars of [MATH] are the elements of [MATH], and that the pure scalars are those in the form [MATH] for some [MATH]: as a consequence, not all scalars of [MATH] need be pure (in contrast to what happens with ordinary quantum theory).', '1703.10576-2-30-1': 'In what follows, we will try as much as possible to construct theories where all scalars are pure, but there are examples (such as the case of [MATH]-adic quantum theory) where this cannot be achieved.', '1703.10576-2-30-2': 'When all scalars are pure, the requirement that [MATH] is always automatically satisfied for all finite groups [MATH], and we only need to care about [MATH] being invertible as a scalar in [MATH] (a fact which always holds true whenever [MATH] is a semi-field/field and [MATH] is non-zero in [MATH]).', '1703.10576-2-31-0': 'We will now proceed to construct a number of toy models within this framework.', '1703.10576-2-31-1': 'For each toy model we will try to: (i) study the phase group; (ii) discuss which Hidden Subgroup Problems can be efficiently solved, and which generalised Mermin-type arguments can be implemented, using the phase group and the strongly complementary pairs; (iii) assess the overall locality of the theory, via the connection with the sheaf-theoretic framework for non-locality and contextuality.', '1703.10576-2-32-0': '# Real quantum theory', '1703.10576-2-33-0': 'The simplest non-conventional example is given by the ring [MATH] of signed reals (with the trivial involution), which yields the probability semiring [MATH] as its sub-semiring of positive elements; in particular, all positive elements are pure scalars.', '1703.10576-2-33-1': 'The corresponding probabilistic theory [MATH] is known as real quantum theory [CITATION]: it is arguably the most well-studied of the quantum-like theories, and the closest to ordinary quantum theory.', '1703.10576-2-33-2': 'Thus said, real quantum theory can be distinguished from ordinary quantum theory because it fails to be locally tomographic [CITATION], i.e. bipartite (mixed) states in real quantum theory cannot in general be distinguished by product measurements alone.', '1703.10576-2-33-3': 'Equivalently, one can check that the CP maps [MATH] and [MATH] on [MATH] in [MATH] cannot be distinguished by applications to mixed states of [MATH] alone, because the latter are described by density matrices which are always real symmetric.', '1703.10576-2-34-0': 'The group of phases in [MATH] is [MATH], and non-trivial interference is possible in real quantum theory.', '1703.10576-2-34-1': 'For example, each of the Pauli [MATH] eigenstates [MATH] of the qubit [MATH] in real quantum theory yields the uniform distribution when measured in the Pauli [MATH] basis [MATH], but their superposition [MATH] yields the outcome corresponding to [MATH] with certainty.', '1703.10576-2-34-2': "Simon's problem, and other Hidden Subgroup Problems on [MATH], can be solved efficiently in real quantum theory.", '1703.10576-2-34-3': 'More generally, consider a classical structure [MATH] on [MATH] with enough classical states, which corresponds to an orthonormal basis of [MATH] (because [MATH] is multiplicatively cancellative [CITATION]).', '1703.10576-2-34-4': 'Then the group of [MATH]-phase gates is isomorphic to the group [MATH] of [MATH]-bit strings under bitwise xor.', '1703.10576-2-34-5': 'Because of the structure of phase groups, generalised Mermin-type arguments only yield local empirical models [CITATION].', '1703.10576-2-34-6': "Nevertheless, Bell's theorem goes through in real quantum theory (as it only involves states and measurements on the [MATH] great circle of the Bloch sphere), which is therefore a non-local probabilistic theory.", '1703.10576-2-35-0': '# Relational quantum theory', '1703.10576-2-36-0': 'Examples of an entirely different nature are given by considering distributive lattices [MATH] (with the trivial involution), which yield themselves back as their sub-semirings of positive elements (because of multiplicative idempotence); in particular, all positive elements are pure scalars.', '1703.10576-2-36-1': 'Distributive lattices seem to be almost as far as one can get from the probabilistic semiring [MATH], but the category [MATH] has been studied extensively as a toy model for quantum theory (especially in the boolean case [MATH]) [CITATION], and the corresponding CPM category has also received some attention on its own [CITATION].', '1703.10576-2-36-2': 'We refer to the corresponding [MATH]-probabilistic (or possibilistic) theory as relational quantum theory.', '1703.10576-2-37-0': 'The group of phases in [MATH] is the singleton [MATH], and no interference is possible in relational quantum theory.', '1703.10576-2-37-1': 'Relational quantum theories also feature very few quantum-to-classical transitions: there is a unique basis on each system, namely the one given by the elements of the underlying set.', '1703.10576-2-37-2': 'They are local tomographic on pure states, but they fail to be tomographic altogether on mixed states: for example, the pure state [MATH] for [MATH] and the mixed state [MATH] are distinct, but cannot be distinguished by measurement.', '1703.10576-2-37-3': 'In fact, a characteristic trait of relational quantum theories is exactly that superposition and mixing are essentially indistinguishable (because of idempotence) [CITATION].', '1703.10576-2-37-4': 'Classical structures [MATH] in relational quantum theory over the booleans are known to correspond to abelian groupoids [MATH] [CITATION], and the corresponding group of [MATH]-phase gates is isomorphic to [MATH].', '1703.10576-2-37-5': 'It can be shown that generalised Mermin-type arguments only yield local empirical models [CITATION].', '1703.10576-2-37-6': 'In fact, it can be shown that that relational quantum theories are entirely local [CITATION].', '1703.10576-2-38-0': '# Hyperbolic quantum theory', '1703.10576-2-39-0': 'Turning our attention back to real algebras, we can consider the commutative ring of split complex numbers [MATH], a two-dimensional real algebra.', '1703.10576-2-39-1': 'Split complex numbers take the form [MATH], where [MATH] and [MATH]; in particular, they have non-trivial zero-divisors in the form [MATH], because [MATH].', '1703.10576-2-39-2': 'They come with the involution [MATH], which yields the signed-probability ring [MATH] as sub-semiring of positive elements; in particular, all positive elements are pure scalars.', '1703.10576-2-39-3': 'We refer to the corresponding [MATH]-probabilistic theory [MATH] as hyperbolic quantum theory [CITATION].', '1703.10576-2-40-0': 'Hyperbolic quantum theory is an extremely interesting theory.', '1703.10576-2-40-1': "On the one hand, it contains real quantum theory as a sub-theory, and as a consequence every scenario and protocol which can be implemented in real quantum theory (such as the algorithm to efficiently solve Simon's problem [CITATION]) can also be implemented in hyperbolic quantum theory.", '1703.10576-2-40-2': 'On the other hand, hyperbolic quantum theory is a local theory, in the sense that every empirical model arising in hyperbolic quantum theory admits a local hidden variable model in terms of signed probabilities (the notion of classical non-determinism for hyperbolic quantum theory) [CITATION].', '1703.10576-2-40-3': 'While signed probabilities might at first sound unphysical, an operational interpretation exists in terms of unsigned probabilities on signed events [CITATION].', '1703.10576-2-41-0': 'The group of phases in [MATH] consists of the elements with square norm [MATH], i.e. the elements in the form [MATH] which lie on the following unit hyperbola of the real plane: [EQUATION]', '1703.10576-2-41-1': 'In fact, the natural geometry for the split complex numbers is that of the real plane endowed with the Lorentzian metric [MATH], i.e. that of the Minkowski plane.', '1703.10576-2-41-2': 'Just like multiplication by phases in [MATH] forms the circle group [MATH] of rotations around the origin for the Euclidean plane, multiplication by phases in [MATH] forms the group [MATH] of orthochronous homogeneous Lorentz transformations for the Minkowski plane.', '1703.10576-2-41-3': "We have the isomorphism of Lie groups [MATH] given by [MATH]: as a consequence, the [MATH]-valued multiplicative characters for finite groups are exactly the same as the [MATH]-valued multiplicative characters, and the only finite groups with enough multiplicative characters to form a Fourier basis are the ones in the form [MATH]; Simon's problem, and other Hidden Subgroup Problems for [MATH], can be efficiently solved in hyperbolic quantum theory, despite the latter being local.", '1703.10576-2-41-4': 'Things are different for infinite groups such as [MATH], which have enough [MATH]-valued multiplicative characters but not enough [MATH]-valued multiplicative characters.', '1703.10576-2-42-0': 'Now consider a classical structure [MATH] corresponding to an orthonormal basis of [MATH] (we have to ask explicitly for orthogonality, because the result of [CITATION] does not apply to hyperbolic quantum theory: [MATH] has non-trivial zero-divisors, and hence it is not multiplicatively cancellative).', '1703.10576-2-42-1': 'The group of [MATH]-phase gates is isomorphic to [MATH], and has [MATH] as a maximal finite subgroup: as a consequence, generalised Mermin-type arguments (which involve finite groups) only yield local empirical models, just as in real quantum theory.', '1703.10576-2-42-2': 'However, future extensions of Mermin-type arguments to infinite groups might yield different results: this is because subgroups like [MATH] would become available, and there are equations (such as [MATH]) which have no solutions in the subgroup [MATH] but have solutions (e.g. [MATH] and [MATH]) in the larger group [MATH].', '1703.10576-2-43-0': '# Parity quantum theory', '1703.10576-2-44-0': 'A simple variation on relational quantum theory (over the booleans) is given by using symmetric difference of sets, instead of union, as the superposition operation.', '1703.10576-2-44-1': 'This leads us to consider the finite field with two elements [MATH], with trivial involution, in place of the booleans [MATH], also with trivial involution.', '1703.10576-2-44-2': 'The multiplication is the same, but addition is now non-idempotent, and superposition is no longer the same as mixing.', '1703.10576-2-44-3': 'The parity semiring [MATH] yields itself back as its sub-semiring of positive elements (in particular, all positive elements are pure scalars), and we refer to the corresponding [MATH]-probabilistic theory [MATH] as parity quantum theory.', '1703.10576-2-45-0': 'Parity quantum theory as defined here (the same as in Ref. [CITATION]) pretty much coincides with the [MATH] case of modal quantum theory [CITATION], but it should be noted that the philosophical interpretation of [MATH]-valued probabilities is significantly different.', '1703.10576-2-45-1': 'In modal quantum theory, the interest is in generating possibilistic tables by using finite fields, subsequently interpreting all zero values as the boolean [MATH] and all non-zero values as the boolean [MATH].', '1703.10576-2-45-2': 'In parity quantum theory, the non-determinism itself is interpreted to be natively [MATH]-valued, and no attempt is made to translate the resulting empirical models into possibilistic ones.', '1703.10576-2-45-3': 'Indeed, such an interpretation would not be natural within our semiring-oriented framework, as no semiring homomorphism can exists from any finite field into the booleans.', '1703.10576-2-46-0': 'The group of phases in [MATH] is the singleton [MATH], but interference is still possible in parity quantum theory: this somewhat counter-intuitive situation is made possible by the fact that [MATH] is its own additive inverse in [MATH], so that triviality of the group of phases is slightly deceptive.', '1703.10576-2-46-1': 'Indeed, consider the four two-qubit states below, which form an orthonormal basis for [MATH]: [EQUATION]', '1703.10576-2-46-2': 'For example, we have [MATH].', '1703.10576-2-46-3': 'When measured in the computational basis [MATH], the normalised states [MATH], [MATH] and [MATH] all have non-zero [MATH]-probability of yielding an outcome in the set [MATH], but their superposition [MATH] (also a normalised state) has zero [MATH]-probability of yielding an outcome in that set.', '1703.10576-2-47-0': 'Because the group of phases is trivial, so are all the groups of phase gates, as well as all the [MATH]-valued multiplicative characters of all groups; as a consequence, parity quantum theory admits no non-trivial generalised Mermin-type arguments, and no implementation of the algorithm to solve the HSP.', '1703.10576-2-47-1': 'Furthermore, Theorem [REF] shows that parity quantum theory is local, because [MATH] is a field.', '1703.10576-2-48-0': '[MATH]-probabilistic theories can be similarly constructed for modal quantum theory over any other finite field [MATH] [CITATION], by taking [MATH] with the trivial involution.', '1703.10576-2-48-1': 'However, these theories have a lot of non-pure scalars-namely the [MATH] non-squares in [MATH]-and their phases are close to trivial-namely they are [MATH] if [MATH] and [MATH] if [MATH].', '1703.10576-2-48-2': 'Instead, we will consider a more sophisticated construction based on quadratic extensions of finite fields, which we will refer to as finite-field quantum theory.', '1703.10576-2-48-3': 'Finite-field quantum theory is a local theory (by Theorem [REF]), in which it is nonetheless possible to formulate non-trivial quantum algorithms, as well as non-trivial Mermin-type non-locality arguments.', '1703.10576-2-48-4': "This is in stark contrast with more traditional toy models such as Spekkens's toy model [CITATION] and relational quantum theory, in which the quantum Fourier transform cannot be performed for non-trivial groups [CITATION] (precluding the implementation of algorithms based on it), and in which all Mermin-type arguments are necessarily trivial [CITATION].", '1703.10576-2-49-0': '# Finite-field quantum theory', '1703.10576-2-50-0': 'Consider a finite field [MATH] (with [MATH] odd), and let [MATH] be a generator for the cyclic group [MATH] of invertible (aka non-zero) elements in [MATH] (i.e. a primitive element for [MATH]).', '1703.10576-2-50-1': 'We consider the ring [MATH], equipped with the involution [MATH]: because [MATH] is a primitive element, [MATH] is a field.', '1703.10576-2-50-2': 'We are in fact working with the quadratic extension of fields [MATH], equipped with the usual involution and (squared) norm from Galois theory: [EQUATION]', '1703.10576-2-50-3': 'The sub-field [MATH] (given by the elements in the form [MATH]) is the sub-semiring of positive elements (and we will shortly see that all positive elements are pure scalars).', '1703.10576-2-51-0': 'The phases in [MATH] are the points [MATH] of the [MATH] plane lying on the unit hyperbola [MATH], which does not factor as a product of two lines because [MATH] is a primitive element.', '1703.10576-2-51-1': 'The following iconic result of Galois theory, due to Hilbert, can be used to characterise them (see e.g. Ref. [CITATION] for a proof).', '1703.10576-2-51-2': "[Hilbert's Theorem 90]", '1703.10576-2-52-0': 'Let [MATH] be a finite cyclic field extension, and let [MATH] be a generator for its cyclic Galois group.', '1703.10576-2-52-1': 'Then the multiplicative group of elements [MATH] of unit relative norm [MATH] is isomorphic to the quotient group [MATH].', '1703.10576-2-53-0': 'The phases in [MATH] form the cyclic group [MATH].', '1703.10576-2-54-0': "Another interesting consequence of Hilbert's Theorem 90 is the fact that the positive elements in finite-field quantum theory are all pure scalars.", '1703.10576-2-55-0': 'All scalars in [MATH] are pure.', '1703.10576-2-56-0': 'We have seen that finite-field quantum theory comes with a non-trivial phase group, which in turn allows for non-trivial implementations of certain quantum protocols.', '1703.10576-2-56-1': 'We open with a result about the Quantum Fourier Transform, which combined with the main result of Ref. [CITATION] implies that the Hidden Subgroup Problem can be solved efficiently in finite-field quantum theory for arbitrarily large families of finite abelian groups (as [MATH] grows larger).', '1703.10576-2-57-0': 'Let [MATH] be a finite abelian group.', '1703.10576-2-57-1': 'Then [MATH] has enough [MATH]-valued unitary multiplicative characters if and only if [MATH] with [MATH] for all [MATH].', '1703.10576-2-57-2': 'When this is the case, the Hidden Subgroup Problem for [MATH] can be solved efficiently in finite-field quantum theory.', '1703.10576-2-58-0': 'Now consider a classical structure [MATH] with enough classical states on a [MATH]-dimensional quantum system in finite-field quantum theory, which corresponds to an orthonormal basis of the vector space [MATH] (because [MATH] is multiplicatively cancellative [CITATION]).', '1703.10576-2-58-1': 'Then the group of [MATH]-phase gates in [MATH] is isomorphic to the group [MATH].', '1703.10576-2-59-0': 'It is possible to formulate non-trivial generalised Mermin-type arguments in finite-field quantum theory if and only if [MATH] is not a square-free natural number.', '1703.10576-2-60-0': 'While finite-field quantum theory and parity quantum theory might not have as direct a physical interpretation as hyperbolic quantum theory and relational quantum theory, they come with the major advantage of having wavefunction valued over a field, so that objects are finite-dimensional vector spaces (equipped with a non-standard inner product, in the case of finite-field quantum theory).', '1703.10576-2-60-1': 'This opens the door for a systematic study of quantum systems in these theories using standard tools from finite geometry.', '1703.10576-2-60-2': 'Further investigation in this direction is left to future work.', '1703.10576-2-61-0': '# [MATH] p-adic quantum theory', '1703.10576-2-62-0': 'We now look at the construction of [MATH]-adic quantum mechanics [CITATION], where [MATH] is the field of [MATH]-adic numbers, and [MATH] is some quadratic extension.', '1703.10576-2-62-1': 'In this Section, we will use the notation [MATH] to denote the [MATH]-adic numbers, and [MATH] to denote the [MATH]-adic integers, to distinguish them from the finite field [MATH] of integers modulo [MATH]; note that this convention is different from the one used in many texts on [MATH]-adic arithmetic, where [MATH] is used for the [MATH]-adic integers (and [MATH] for the [MATH]-adic numbers).', '1703.10576-2-63-0': 'When [MATH], the [MATH]-adic numbers [MATH] fall within four distinct quadratic classes-jointly labelled by the parity of the order [MATH] and by the quadratic class of the first non-zero digit [MATH]-corresponding to three inequivalent quadratic extensions.', '1703.10576-2-63-1': 'This means that there is no way to obtain all positive elements as pure scalars by a single quadratic extension.', '1703.10576-2-63-2': 'This would seem to indicate that mixed states play a necessary role in the emergence of [MATH]-adic probabilities, which cannot all be obtained from pure states alone: the potential physical significance of this observation might become the topic of future work on [MATH]-adic quantum theory within CQM.', '1703.10576-2-64-0': 'We consider the quadratic extension [MATH], where [MATH] and [MATH] is a primitive element in the field [MATH] of integers modulo [MATH], and we follow the presentation of Ref. [CITATION].', '1703.10576-2-64-1': 'A generic element of [MATH] takes the form [MATH], for [MATH], and its square norm is [MATH].', '1703.10576-2-64-2': 'Whether an element [MATH] can be written in this form, i.e. whether is is a pure scalar in [MATH], is determined by the sign function [MATH], which takes the value [MATH] if [MATH] for some [MATH], and the value [MATH] otherwise.', '1703.10576-2-64-3': 'An explicit form for the sign function (in the [MATH] case) is given by Equation (2.34) of Ref. [CITATION], which specialised to our case ([MATH] and [MATH]) reads [MATH].', '1703.10576-2-64-4': 'Hence the pure scalars in [MATH] are exactly the [MATH]-adic numbers [MATH] with even order [MATH]; closure of this set under addition yields [MATH] as sub-semiring (field, in fact) of positive elements in [MATH].', '1703.10576-2-65-0': 'The phases in [MATH]-adic quantum theory are those [MATH] such that [MATH].', '1703.10576-2-65-1': 'In Ref. [CITATION] (Equation (4.35) of Section IV.C, and Equation (C12b) of Appendix C.3) it is shown that phases form a multiplicative group [MATH] isomorphic to the additive group [MATH]-here [MATH] are the integers modulo [MATH], while [MATH] is the additive subgroup of [MATH] formed by those [MATH]-adic integers which are divisible by [MATH].', '1703.10576-2-65-2': 'In particular, [MATH] is an infinite group with the cardinality of the continuum, and each sheet [MATH] is a profinite torsion-free group, which is best understood by looking at the descending normal series [MATH] and considering the finite cyclic quotients [MATH].', '1703.10576-2-66-0': 'The scalar [MATH] is always invertible, and it is in the form [MATH] if and only if the largest power of [MATH] which divides [MATH] is even.', '1703.10576-2-66-1': "Furthermore, [MATH] has enough [MATH]-valued multiplicative characters if and only if [MATH] with [MATH] for all [MATH] (in the light of Hensel's Lemma, this parallelism between [MATH]-adic quantum theory and finite-field quantum theory on [MATH] should not come as a big surprise): finite abelian groups [MATH] satisfying this condition admit efficient solutions for Hidden Subgroup Problems in [MATH]-adic quantum theory (because we necessarily have that [MATH] cannot divide [MATH]).", '1703.10576-2-66-2': 'Similarly, it is possible to formulate non-trivial generalised Memrin-type arguments in [MATH]-adic quantum theory if and only if [MATH] is not square-free.', '1703.10576-2-66-3': 'Thus said, [MATH]-adic quantum theory is a local theory by virtue of Theorem [REF].', '1703.10576-2-67-0': 'Similar considerations apply to the the construction of [MATH]-adic quantum theory for the other two quadratic extensions [MATH] and [MATH] available in the case of [MATH] (although the cases [MATH] and [MATH] have to be treated separately), as well as the seven quadratic extensions available in the case of [MATH].', '1703.10576-2-67-1': 'The phase groups take a similar (but not identical) form to the one presented here, and the full details can be found in Ref. [CITATION] (Section IV.C and Appendices C.3, C.4).', '1703.10576-2-68-0': '# Tropical quantum theory', '1703.10576-2-69-0': 'Relational quantum theory involves semirings which are both additively and multiplicatively idempotent, parity quantum theory involves a semiring which is only multiplicatively idempotent, and ordinary quantum theory involves a semiring which is neither additively nor multiplicatively idempotent.', '1703.10576-2-69-1': 'We now give examples of theories with wavefunctions based in semirings which are additively idempotent but not multiplicatively idempotent, namely the tropical semirings [CITATION].', '1703.10576-2-70-0': 'A tropical semiring is the commutative semiring [MATH] obtained from a totally ordered commutative monoid [MATH] having an absorbing element [MATH] which is larger than all elements in the monoid.', '1703.10576-2-70-1': 'In the tropical semiring, [MATH] is the addition, [MATH] is the additive unit, [MATH] is the multiplication and [MATH] is the multiplicative unit.', '1703.10576-2-70-2': 'The nomenclature is extended to semirings isomorphic to the explicitly min-plus semirings used above (e.g. max-plus formulations, or the Viterbi semiring).', '1703.10576-2-71-0': 'Examples of tropical semirings appearing in the literature include the tropical reals [MATH], the tropical integers [MATH], the tropical naturals [MATH], and the Viterbi semiring [MATH] (which is a tropical semiring because it is isomorphic to the explicitly min-plus semiring [MATH] via the semiring homomorphism [MATH]).', '1703.10576-2-71-1': 'In fact, there is an easy characterisation of which commutative semirings arise as tropical semirings (the proof is omitted as it is a straightforward check).', '1703.10576-2-72-0': 'A commutative semiring [MATH] is a tropical semiring if and only if for all [MATH] we have [MATH] or [MATH] (in which case we can set [MATH]).', '1703.10576-2-73-0': 'From now on, we will revert back to usual semiring notation, and we will rely on the result above to connect with the min-plus notation typical of tropical geometry [CITATION].', '1703.10576-2-73-1': 'We will, however, remember that tropical semirings come with a total order respected by the multiplication, and we will occasionally use [MATH], [MATH] and [MATH] in addition to the addition/multiplication.', '1703.10576-2-74-0': 'The only involution possible on a tropical semiring [MATH] is the trivial one, and the positive elements form the sub-semiring of squares [MATH].', '1703.10576-2-75-0': 'If [MATH] is a tropical semiring and [MATH] is its sub-semiring of positive elements, we refer to the [MATH]-probabilistic theory [MATH] as tropical quantum theory.', '1703.10576-2-76-0': 'Just as in the case of relational quantum theory, the group of phases in a tropical semiring [MATH] is always trivial (because [MATH] implies [MATH] in any totally ordered monoid [MATH]), and there is no interference.', '1703.10576-2-76-1': 'Similarly, there is a unique orthonormal basis on each system, the only unitaries/invertible maps are permutations, and superposition cannot be distinguished from mixing by measurements alone.', '1703.10576-2-76-2': 'Tropical quantum theory does not admit any implementation of the algorithm for the abelian Hidden Subgroup Problem, nor does it admit any generalised Mermin-type non-locality arguments.', '1703.10576-2-77-0': 'The parallelisms with relational quantum theory become less surprising when one realises that tropical quantum theory is another generalisation of quantum theory over the booleans: the latter form a totally ordered distributive lattice, and hence are a particular case of tropical semiring.', '1703.10576-2-77-1': '(Proof of the following result is omitted, as it is a straightforward check.)', '1703.10576-2-78-0': 'Any totally ordered distributive lattice [MATH] is a tropical semiring [MATH]; conversely, every tropical semiring [MATH] which has [MATH] as least element and such that [MATH] for all [MATH] is a totally ordered distributive lattice [MATH].', '1703.10576-2-79-0': 'In the light of the result above, we expect tropical quantum theory to be local, exactly like relational quantum theory, but further investigation of this question is left to future work.', '1703.10576-2-80-0': '# Conclusions and Future Work', '1703.10576-2-81-0': 'In the first two Sections of this work, we have provided a general framework, based on enrichment of CP* categories, for the construction of toy models of quantum theory.', '1703.10576-2-81-1': 'Specifically, we have focussed our efforts on theories of wavefunctions valued in some commutative semiring [MATH] with involution, replacing the field with involution [MATH] used in conventional quantum theory.', '1703.10576-2-81-2': 'In the process, the dagger compact category [MATH] of finite-dimensional complex Hilbert spaces was replaced by the symmetric monoidal category [MATH] of free finite-dimensional [MATH]-semimodules, equipped with the dagger compact structure given by the involution of [MATH] (which generalises complex conjugation).', '1703.10576-2-81-3': 'We have also postulated classical non-determinism to arise via a generalisation of the Born rule, as embodied by the CP* construction, and we have shown that our construction yields special cases of [MATH]-probabilistic theories, as defined in the recently introduced framework of Categorical Probabilistic Theories (here [MATH] is the sub-semiring of [MATH] given by the positive elements, generalising the probabilistic semiring [MATH] modelling classical non-determinism in conventional quantum theory).', '1703.10576-2-82-0': 'In subsequent Sections, we have shown our framework to be expressive enough to capture many toy models which have appeared in the literature in (more or less) recent years.', '1703.10576-2-82-1': 'Aside from real quantum theory and relational quantum theory, which have already found their special place in the heart of categorical quantum mechanicians and operational probabilistic theorists, we considered hyperbolic quantum theory, [MATH]-adic quantum theory and parity quantum theory (the [MATH] case of modal quantum theory), all interesting enough to deserve their own place in our growing zoo of categorical toy models.', '1703.10576-2-83-0': 'We have also introduced two new families of toy models, one based on quadratic extensions of finite field (finite-field quantum theory), and the other based on tropical semirings (tropical quantum theory).', '1703.10576-2-83-1': 'While tropical quantum theory proves to be a variant on relational quantum theory, finite-field quantum theory is of independent interest: it boasts a rich phase group which allows non-trivial quantum protocols to be implemented, while at the same time remaining fully local, as well as amenable to treatment with tools from finite geometry.', '1703.10576-2-84-0': 'Future work.', '1703.10576-2-84-1': 'This work leaves a number of directions open to future investigation.', '1703.10576-2-84-2': 'Firstly, some of the toy theories presented in this work have barely had their surface scratched from the point of view of Categorical Quantum Mechanics: an in-depth study of the categorical features they possess (e.g. unitaries, measurements/preparations, [MATH]-Frobenius algebras, complementary and strongly complementary observables) will be a priority in further developments, together with a more thorough understanding of which quantum protocols can be implemented in them.', '1703.10576-2-85-0': 'Secondly, this work mainly focussed on existing toy models, or variations thereof, to show that the framework we presented truly is expressive enough for its intended purpose.', '1703.10576-2-85-1': 'However, there are many other examples of semirings, rings and fields that could potentially produce interesting and unexpected features, and we expect our zoo to continue growing in the coming years.', '1703.10576-2-86-0': 'Finally, it was not possible, for reasons of space, to explore the applications of finite geometry to finite-field quantum theory, despite the promise of interesting connections between toy quantum systems and finite projective spaces.', '1703.10576-2-86-1': 'Similarly, it was not possible to establish whether tropical quantum theory is always local.', '1703.10576-2-86-2': 'A thorough exploration of these matters is left to future work.'}

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[]

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{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1703.10576

1405.3317

{'1405.3317-1-0-0': 'We study the radial migration of stars driven by recurring multi-arm spiral features in an exponential disk embedded in a dark matter halo.', '1405.3317-1-0-1': 'The spiral perturbations redistribute angular momentum within the disk and lead to substantial radial displacements of individual stars, in a manner that largely preserves the circularity of their orbits and that results, after [MATH] Gyr ([MATH] full rotations at the disk scalelength), in little radial heating and no appreciable changes to the vertical or radial structure of the disk.', '1405.3317-1-0-2': "Our results suggest that the radial displacements are driven by Sellwood Binney's corotation scattering process and clarify a number of issues related to the spatial distribution and kinematics of migrators.", '1405.3317-1-0-3': 'In particular, we find that migrators are a heavily biased subset of stars with preferentially low vertical velocity dispersions.', '1405.3317-1-0-4': 'This "provenance bias" for migrators is not surprising in hindsight, for stars with small vertical excursions spend more time near the disk plane and thus respond more readily to non-axisymmetric perturbations.', '1405.3317-1-0-5': 'We also find that the vertical velocity dispersion of outward migrators always decreases, whereas the opposite holds for inward migrators.', '1405.3317-1-0-6': 'To first order, newly arrived migrators simply replace stars that have migrated off to other radii, thus inheriting the vertical bias of the latter.', '1405.3317-1-0-7': 'Extreme migrators might therefore be recognized, if present, by the unexpectedly small amplitude of their vertical excursions.', '1405.3317-1-0-8': 'Our results show that migration, understood as changes in angular momentum that preserve circularity, can affect strongly the thin disk, but cast doubts on models that envision the Galactic thick disk as a relic of radial migration.', '1405.3317-1-1-0': '1Department of Astronomy, University of Wisconsin, 2535 Sterling Hall, 475 N. Charter Street, Madison, WI 53076, USA.', '1405.3317-1-1-1': 'e-mail:ciro@astro.wisc.edu', '1405.3317-1-2-0': '2Alfred P. Sloan Fellow', '1405.3317-1-3-0': '3Senior CIfAR Fellow, Department of Physics and Astronomy, University of Victoria, Victoria, BC, Canada V8P 5C2', '1405.3317-1-4-0': '4Observatorio Astronomico, Universidad Nacional de Cordoba, Cordoba X5000BGR, Argentina', '1405.3317-1-5-0': '# Introduction', '1405.3317-1-6-0': 'Cold, thin stellar disks are dynamically fragile entities that react strongly to changes in the gravitational potential, such as those that result from changes in the mass of the galaxy itself, from external perturbations such as the accretion of a satellite or the tidal effects of substructure, or from internal processes such as bar instability or the development of spiral structure.', '1405.3317-1-6-1': 'In the absence of dissipation, these perturbations generally heat the disk, moving stars away from their birth radii at the expense of decreasing the circularity of their orbits and increasing the amplitude of their vertical oscillations.', '1405.3317-1-6-2': 'Stars on nearly circular orbits must have therefore been born at about the same radius where they are found today.', '1405.3317-1-7-0': 'One difficulty with this simple scenario arises in our Galaxy from the weak correlation between age and metallicity for thin disk stars in the solar neighbourhood , which is essentially flat for ages between [MATH] and [MATH] Gyr .', '1405.3317-1-7-1': 'Although this may be taken to suggest a simple (but somewhat contrived) scenario where the average metallicity of the interstellar medium (ISM) at the solar circle has not changed since [MATH], the sizeable scatter in the correlation would also require the ISM to have been rather inhomogenous in the past, a possibility that seems at face value unattractive .', '1405.3317-1-8-0': 'An alternative view holds that thin disk stars do not necessarily remain close to their birth radii .', '1405.3317-1-8-1': 'If true, then long-lived stars in the solar neighbourhood might actually sample the chemical properties of the Galaxy over a wide range of radii at the time of their formation, possibly blurring any correlation between the age and metallicity of stars selected in local samples.', '1405.3317-1-8-2': 'This scenario gained support after Sellwood Binney, in an influential paper, identified a mechanism that allows stars to migrate in radius whilst remaining on nearly circular orbits .', '1405.3317-1-8-3': 'The mechanism relies on angular momentum exchanges between stars near corotation of transient spiral patterns, and allows stars to switch radii without substantially heating the disk.', '1405.3317-1-8-4': 'Given that spiral patterns seem ubiquitous in disk galaxies, this mechanism must operate to some degree in most galaxy disks.', '1405.3317-1-9-0': 'Since its inception, the corotation resonance (CR) mechanism of Sellwood Binney has been the subject of a number of studies in the literature.', '1405.3317-1-9-1': 'These studies have generally confirmed the validity of the arguments laid out in their original paper, but have also raised a number of questions about the initial and final properties of migrating stars and about the consequences of radial migration for the vertical structure of the disk.', '1405.3317-1-9-2': 'It is unclear, for example, what biases, if any, might affect the kinematics or spatial distribution of migrators, what strategies might be used to identify migrators observationally, or whether radial migration plays a role in the origin of the thick disk of the Galaxy.', '1405.3317-1-10-0': 'The latter issue, in particular, has been debated vigorously, especially since [CITATION] proposed that the chemical differentiation between thin and thick disks could be explained solely by radial migration, which might have brought stars with high vertical kinetic energy from the inner Galaxy out to the solar neighbourhood.', '1405.3317-1-10-1': '[CITATION], on the other hand, have argued that migration does little for disk thickening because outward migrators should cool vertically as they move out, a result that has been further debated by [CITATION] and [CITATION].', '1405.3317-1-11-0': 'This unsettled state of affairs might be traced, in part, to the disparate numerical contexts used to study the effects of migration.', '1405.3317-1-11-1': 'Although most published work has relied on direct numerical simulation, some studies are based on the response of equilibrium disk models to transient spiral modes or to the tidal perturbation of a massive satellite , whereas others have focused on hydrodynamical simulations of disks of gas and stars that evolve in isolation or that result from the dissipative collapse of a rotating sphere of gas in the gravitational potential of a dark matter halo .', '1405.3317-1-12-0': 'The first approach benefits from the controlled nature of the initial disk, which allows for the identification of numerical artifact, but suffers from the arbitrariness involved in choosing the orbit and mass of the perturbers, or from differences in the origin of the spiral perturbations, which either develop freely or result from the forcing of particular modes with predetermined amplitude and duration.', '1405.3317-1-12-1': 'The second approach allows the disk to evolve more freely, but its analysis is obscured by the complexities that arise from the continuously evolving disk potential and from the presence of rapidly-evolving multiple modes and instabilities.', '1405.3317-1-12-2': 'Furthermore, the numerical resolution of hydrodynamical simulations is inevitably much lower than that attainable with [MATH]-body models, making it much harder to discern and isolate the effects of noise associated with the finite number of particles used.', '1405.3317-1-13-0': 'Finally, there are also semantic issues, since some authors use the term "radial migration" as a catch-all phrase to denote any radial displacement of a star, regardless of its cause or effect on its orbit, whereas others reserve the term "migration" for changes in the guiding centre radius (i.e., the orbital angular momentum) of orbits that remain nearly circular through the process.', '1405.3317-1-13-1': 'We shall adopt the latter meaning in this contribution, since it is the defining feature of the corotation scattering mechanism of [CITATION].', '1405.3317-1-14-0': 'We address some of these issues here using numerical simulations that evolve a kinematically-cold self-gravitating disk of stars in the fixed potential of a cosmologically-motivated dark matter halo.', '1405.3317-1-14-1': 'Spiral patterns are seeded by perturbations modeled on the gravitational influence of molecular clouds and persist almost unchanged for over [MATH] Gyr, the duration of the simulation (more than [MATH] full rotations at the exponential disk scalelength).', '1405.3317-1-14-2': 'Our simulations differ from earlier work mainly on numerical resolution: the disk is made up of [MATH] particles, an improvement of nearly two orders of magnitude over earlier work.', '1405.3317-1-14-3': 'The large number of particles allows us to study the orbital evolution of individual stars without having to worry about the discreteness noise associated with the particle representation of the disk.', '1405.3317-1-15-0': 'Our simulations also differ from earlier attempts because the disk is not subject to the rapid development of instabilities that may change its radial or vertical structure; a control run without perturbers shows that the disk does not develop strong non-axisymmetric features over several rotation periods.', '1405.3317-1-15-1': 'Finally, our simulations also stand out because the spiral features seeded by the "molecular clouds" do not heat appreciably the disk and persist for over [MATH] Gyr.', '1405.3317-1-15-2': 'Both the vertical and azimuthally-averaged structure of the disk remain basically unchanged throughout.', '1405.3317-1-16-0': 'The plan for this paper is as follows.', '1405.3317-1-16-1': 'We describe the numerical simulations used in this work in Section [REF], present our main results in Section [REF] and conclude with a brief summary in Section [REF].', '1405.3317-1-17-0': '# Numerical Methods', '1405.3317-1-18-0': '## The galaxy model', '1405.3317-1-19-0': 'We use here [MATH]-body simulations of the same galaxy model presented by [CITATION] .', '1405.3317-1-19-1': 'For the sake of completeness we give here a brief description of the simulations and refer the interested reader to the original reference for further details.', '1405.3317-1-19-2': 'The models evolve a self-gravitating disk of particles embedded in a rigid halo potential described by a Hernquist profile with total mass [MATH] and scale length [MATH] kpc.', '1405.3317-1-20-0': 'The disk is a [MATH]-particle realization of an exponential disk of constant scaleheight: [EQUATION] whose positions and velocities were generated using a modified version of the method presented by [CITATION], as described by [CITATION].', '1405.3317-1-20-1': 'We adopt a radial scalelength [MATH] kpc, scaleheight [MATH] kpc and total disk mass [MATH].', '1405.3317-1-21-0': 'With these choices, the disk is submaximal, and makes up about 27% of the total mass enclosed within two radial scalelengths.', '1405.3317-1-21-1': 'A particle in a circular orbit completes a full rotation at [MATH] in [MATH] Myr.', '1405.3317-1-21-2': 'We show in Fig. [REF] the circular velocity profile of the galaxy model (top panel), and the circular orbit timescale, [MATH], in Myr (bottom panel).', '1405.3317-1-21-3': 'The middle panel shows the Toomre stability parameter [MATH], chosen to ensure the disk is marginally stable at all radii.', '1405.3317-1-21-4': '(Here [MATH] is the epicycle frequency and [MATH] the velocity dispersion in radial motions.)', '1405.3317-1-22-0': '## The control simulation', '1405.3317-1-23-0': 'We have evolved the galaxy model described in the previous section for from [MATH] Myr to [MATH] Myr, corresponding to [MATH] circular orbit timescales at [MATH], or [MATH] full rotations at [MATH] kpc.', '1405.3317-1-23-1': 'The system was evolved using an updated version of Gadget , using a Plummer-equivalent softening length for the stellar particles of [MATH] pc and default integration parameters.', '1405.3317-1-24-0': 'After [MATH] Myr the "control" disk has evolved little from its initial configuration, as may be seen in the top two panels of Fig. [REF], and has developed no obvious non-axisymmetric features.', '1405.3317-1-24-1': 'It does develop, however, "ring-like" features that travel like radial waves through the disk, and which are easily appreciate in the top right-panel of Fig. [REF].', '1405.3317-1-25-0': 'These waves do not seem to be caused by discreteness effects, since they appear spontaneously and are coherent over the whole disk.', '1405.3317-1-25-1': 'The radial waves seem to travel back and forth through the disk, without causing any discernable change in the radial distribution of disk particles.', '1405.3317-1-25-2': 'They do not perturb appreciably the vertical structure of the disk, as shown in the top two panels of Fig. [REF], nor change the surface density profile of the disk, as shown in the left panel of Fig. [REF].', '1405.3317-1-26-0': 'The left panels of Fig. [REF] show that the radial waves have little effect on the kinematics of disk particles: there is little evolution in the velocity dispersion profiles of the disk over the [MATH] Myr that the disk runs unperturbed.', '1405.3317-1-26-1': 'We conclude that our galaxy disk model is not subject to the strong non-axisymmetric instabilities that have plagued [MATH]-body studies of disk galaxies over the years.', '1405.3317-1-26-2': 'Although our disks are fully self-gravitating, it appears as if the number of particles used is sufficient to suppress discreteness noise and to prevent, for several rotation periods, the development of strong non-axisymmetric features.', '1405.3317-1-27-0': '## The "spiral" simulation', '1405.3317-1-28-0': 'As shown by [CITATION], non-axisymmetric features do develop when the disk is perturbed in a manner consistent with the presence of [MATH] "molecular clouds".', '1405.3317-1-28-1': 'We refer the reader to [CITATION] for details on the numerical implementation but, in brief, the perturbations are modeled by the addition of softened, [MATH] particles constrained to travel in circular orbits and scattered about the disk.', '1405.3317-1-28-2': 'The perturbation is transient, and the molecular-cloud perturbers are removed [MATH] Myr after their inclusion.', '1405.3317-1-29-0': 'The disk reacts strongly to these perturbations, and develops a long-lived multi-arm spiral pattern, as shown in the bottom four panels of Fig. [REF].', '1405.3317-1-29-1': 'In this sequence, [MATH] shows the disk just after the perturbers are removed; [MATH] corresponds, as in the control simulation, to [MATH] Myr later; whereas [MATH] and [MATH] denote later stages of the evolution, [MATH] Myr and [MATH] Myr after the removal of the perturbers, respectively.', '1405.3317-1-30-0': 'Visual inspection of Fig. [REF] shows that the multi-arm spirals seeded by the molecular cloud perturbers are long lived and that they change little in nature over [MATH] Gyr of evolution.', '1405.3317-1-30-1': 'Despite their longevity, the spiral patterns seem to have little effect on the vertical structure of the disk, as shown by the bottom four panels of Fig. [REF].', '1405.3317-1-30-2': 'The radial structure of the disk also remains almost unchanged, as indicated by the azimuthally-averaged surface density profiles at [MATH], [MATH], [MATH], and [MATH] shown in the right-hand panel of Fig. [REF].', '1405.3317-1-31-0': 'Only the in-plane kinematics of the disk seems to be noticeably affected, as shown by the slight, but monotonic, heating in the radial and azimuthal directions seen in the right-hand panels of Fig. [REF].', '1405.3317-1-31-1': 'The vertical structure of the disk, however, is unaffected throughout: the vertical velocity dispersion profile, [MATH], at [MATH] is indistinguishable from that at [MATH] (see bottom right panel of Fig. [REF]).', '1405.3317-1-32-0': '## Azimuthal structure', '1405.3317-1-33-0': 'We examine the azimuthal structure of the disks in Fig. [REF], where we show the fluctuations in the surface density and in the gravitational potential at various times as a function of the azimuthal angle [MATH] and of the cylindrical radius, [MATH].', '1405.3317-1-34-0': 'The leftmost panels of Fig. [REF] shows that a barely discernible multi-arm spiral pattern develops in the control simulation after [MATH] Myr of evolution.', '1405.3317-1-34-1': 'Given its small amplitude (the rms fluctuations in the surface density are just [MATH] at [MATH] kpc) this pattern is only evident in the residuals and, as we will see below, leads to basically no secular evolution in the disk over the [MATH] Myr it was evolved for.', '1405.3317-1-34-2': 'This confirms our earlier statement that the model galaxy is free from the violent instabilities or rapidly-growing non-axisymmetric disturbances characteristic of models where either the galaxy is initially out of equilibrium, or where discreteness noise seeds perturbations that transform the galaxy over a short period of time.', '1405.3317-1-35-0': 'The multi-arm spiral structure of the "spiral" simulation stands out clearly in Fig. [REF], with interarm surface density fluctuations of roughly [MATH].', '1405.3317-1-35-1': 'The actual gravitational potential fluctuations are much smaller, however, due to the presence of the dark matter halo.', '1405.3317-1-35-2': 'The spiral structure in the "spiral" simulation remains roughly constant from [MATH] to [MATH], and is characterized by a radially increasing number of arms that bifurcate outwards: at [MATH] kpc, near the center, only about [MATH] arms are clearly defined, whereas more than [MATH] arms can be counted [MATH] kpc away from the center.', '1405.3317-1-35-3': 'We ascribe the very slight evolution in the spiral structure from [MATH] to [MATH] to the weak, but monotonic, radial heating of the disk seen in the increase of the radial velocity dispersion shown in Fig. [REF].', '1405.3317-1-35-4': 'We emphasize again that the overall heating is quite modest; in [MATH] Gyr of evolution the radial velocity dispersion at [MATH] kpc increases from [MATH] km/s to just over [MATH] km/s.', '1405.3317-1-35-5': 'The increase is even smaller at other radii.', '1405.3317-1-36-0': 'The spirals shown in Fig. [REF] have pattern speeds that approximately match the angular speed of the disk at the radii where they are most prominent.', '1405.3317-1-36-1': 'This is seen in the spectrograms of Fig. [REF], where we show the power spectra for the [MATH], [MATH], [MATH], and [MATH] harmonic modes, measured from snapshots sampled every [MATH] Myr for the spiral disk in the [MATH]Myr[MATH] interval.', '1405.3317-1-36-2': 'Note that each mode spans a limited radial range and travels with a pattern speed similar to that of circular orbits at that radius.', '1405.3317-1-37-0': 'As discussed by [CITATION], arms in the pattern fade away and recur while traveling at about the circular angular frequency, in a seemingly "self-perpetuating" cycle.', '1405.3317-1-37-1': 'This implies that stars on nearly circular orbits will be consistently near corotation of the spiral mode that dominates at each radius, favouring the corotation scattering mechanism of [CITATION].', '1405.3317-1-37-2': 'We examine its effect over many rotation periods next.', '1405.3317-1-38-0': '# Results', '1405.3317-1-39-0': '## Radial Migration', '1405.3317-1-40-0': 'In a stationary potential and in the absence of non-axisymmetric perturbations disk stars should conserve both their energy, [MATH], and the [MATH]-component of their specific angular momentum, [MATH].', '1405.3317-1-40-1': 'The presence of the transient spiral patterns discussed in the last section, however, leads to exchanges of angular momentum amongst disk particles and to variations in their orbital energies.', '1405.3317-1-41-0': 'We examine these changes by selecting, at [MATH], particles within two [MATH] pc-wide cylindrical shells centered at [MATH] and [MATH] kpc, respectively.', '1405.3317-1-41-1': 'We track these particles at later times and compare their current energies and angular momenta with those at [MATH].', '1405.3317-1-41-2': 'The comparison is shown in Fig. [REF] for the control simulation (at [MATH]) and for the spiral simulation at [MATH], [MATH] and [MATH].', '1405.3317-1-42-0': 'Particles in the control simulation exhibit very slight changes in their values of [MATH] and [MATH], as expected given the smooth radial and azimuthal structure of the control disk at all times during its evolution.', '1405.3317-1-43-0': 'Particles in the spiral simulation, on the other hand, undergo correlated changes in [MATH] and [MATH] that can reach (and exceed, in some cases) a factor of [MATH] in [MATH].', '1405.3317-1-43-1': 'Recalling that the guiding center radius, [MATH], of a star in a nearly circular orbit is directly proportional to [MATH]; [EQUATION] (where [MATH] and [MATH] is the gravitational potential on the disk plane) this implies changes in the guiding center that can exceed a factor of [MATH].', '1405.3317-1-44-0': 'These variations, however, do not alter the nearly circular nature of the orbits.', '1405.3317-1-44-1': 'Indeed, the correlated changes in [MATH] and [MATH] keep the stars very close to the minimum energy consistent with its angular momentum (indicated by the solid curves in Fig. [REF]), which defines circular orbits.', '1405.3317-1-44-2': 'Thus, the spiral perturbations shown in Figs. [REF] and [REF] cause substantial migration in the spiral disk with minimal heating, as envisioned in the CR mechanism of [CITATION].', '1405.3317-1-45-0': 'Fig. [REF] also hints that, as time goes by, the radial displacements of disk particles increase monotonically.', '1405.3317-1-45-1': 'We show this more explicitly in Fig. [REF], where we plot the distribution of guiding center radii at [MATH] of particles that migrate at later times to [MATH] kpc (left panels) or [MATH] kpc (right).', '1405.3317-1-45-2': 'The width of these distributions increase with time roughly as [MATH], suggestive of a simple diffusion mechanism.', '1405.3317-1-46-0': 'After more than [MATH] Gyr of evolution (i.e., at [MATH]) more than [MATH] of particles with [MATH] kpc have migrated to their current orbits from radii that were initially either [MATH] smaller or larger.', '1405.3317-1-46-1': 'At [MATH] kpc the fraction of such extreme migrators reduces to [MATH] in the same time, presumably because of the longer orbital times characteristic of larger radii.', '1405.3317-1-46-2': 'The radial distribution of migrators at [MATH] kpc is also clearly asymmetric, with a larger fraction of extreme migrators that come from the inner regions.', '1405.3317-1-46-3': 'This is again not surprising, since there are few particles beyond [MATH] kpc in the disk.', '1405.3317-1-47-0': 'We show in Fig. [REF] the evolution in radius of a few "extreme" migrators that have moved roughly [MATH] inwards or outwards to reach their final radius.', '1405.3317-1-47-1': 'The top left and right panels correspond to migrators that reach [MATH] or [MATH] kpc at [MATH], respectively.', '1405.3317-1-47-2': "In spite of the large changes in guiding center, the migrators' orbital circularities barely change.", '1405.3317-1-47-3': 'This is shown in the bottom panels of Fig. [REF], where we plot the evolution of the circularity parameter, [MATH], defined as the specific angular momentum in units of that of a circular orbit of the same energy.', '1405.3317-1-47-4': 'As these panels show, even extreme migrators stay on nearly circular orbits throughout the evolution.', '1405.3317-1-48-0': '## Migrator Properties', '1405.3317-1-49-0': 'We explore the properties of migrators in Fig. [REF], where we show, in the left-hand panels, the vertical velocity dispersion of particles with guiding center radii [MATH] kpc (shaded area), as a function of their original guiding center, [MATH].', '1405.3317-1-49-1': 'Particles to the left of the shaded area have migrated outwards to reach [MATH] kpc; those to the right have migrated inwards.', '1405.3317-1-49-2': 'For reference, the thick dashed curves in each panel of this figure indicate the vertical velocity dispersion profile of all particles in the disk, [MATH].', '1405.3317-1-50-0': 'The blue curves in Fig. [REF] show the current velocity dispersion of subsets of particles binned according to [MATH].', '1405.3317-1-50-1': 'The inverted "U" shape indicates that migrators are a kinematically-biased population with lower velocity dispersions than non-migrators, a bias that increases the farther away a particle has migrated from in order to reach its current radius.', '1405.3317-1-50-2': 'The bias affects both inward and outward migrators, and holds at all times.', '1405.3317-1-50-3': 'The bias is not only kinematical; as expected given their low vertical velocity dispersion, the vertical distribution of migrators is also thinner than that of non-migrators, as shown by their current vertical scaleheight (see the blue curves in the right-hand panels of Fig. [REF]).', '1405.3317-1-50-4': 'Again, the farther particles have migrated from the thinner their current vertical distribution.', '1405.3317-1-51-0': 'The cause of this vertical bias is twofold.', '1405.3317-1-51-1': 'Firstly, migrating stars are a heavily-biased subset of particles at their original radii, a "provenance bias" that increases the farther a particle migrates in a given time.', '1405.3317-1-51-2': 'This may be seen by comparing the red curves in Fig. [REF], which show the velocity dispersion and scaleheight of migrators at their initial radius, with the dashed black line, which shows the same but averaged for all stars at that radius.', '1405.3317-1-51-3': 'The same vertical bias is clearly present at their radius of origin: radial migration mainly affects thinner-than-average populations at both their destination and provenance radii.', '1405.3317-1-52-0': "A second reason for the migrators' vertical bias may be seen by comparing the initial and final velocity dispersions of migrating stars.", '1405.3317-1-52-1': 'This is given by the difference, at given [MATH], between the red and blue curves in Fig. [REF].', '1405.3317-1-52-2': 'Outward migrators (to the left of the gray zone) clearly cool as they move out (i.e., the blue curve is below the red), whereas the opposite holds for inward migrators.', '1405.3317-1-52-3': 'We note that this in itself would not be enough to explain the vertical bias at the final radius, since outward (inward) migrators originate in regions of higher (lower) velocity dispersion of the disk.', '1405.3317-1-52-4': 'However, Fig. [REF] shows that such cooling or heating is but a small correction to the "provenance bias" affecting all migrators.', '1405.3317-1-53-0': 'We have checked that a similar provenance bias and heating/cooling process affect migrators at other radii, and that similar results are obtained when analyzing the other two components of the velocity dispersion, [MATH] and [MATH].', '1405.3317-1-53-1': 'Therefore, radial migration does not only operate most effectively on particles closest to the disk, but also on those closest to circular orbits.', '1405.3317-1-53-2': 'These biases have been hinted at in earlier work but their importance has not been properly appreciated, obscured perhaps by the concurrence of other heating mechanisms (i.e., satellite perturbations, disk growth, major disk instabilities) whose effectiveness in displacing stars radially can exceed that of the corotation scattering mechanism we focus on here.', '1405.3317-1-54-0': 'With the benefit of hindsight, our results are not surprising.', '1405.3317-1-54-1': 'Radial migration is caused by non-axisymmetric patterns in the disk that are more readily felt by stars that remain closer to the disk during their orbits; thus, the smaller the vertical amplitude of their oscillations the farther stars can migrate.', '1405.3317-1-54-2': 'This is, of course, true at all radii affected by the spiral patterns.', '1405.3317-1-54-3': 'If radial migration is not to cause vertical heating or a change in the surface density profile (as is the case in our simulation) then migrating stars at their final location must simply "substitute" other stars that have migrated off, which were themselves biased in the same way.', '1405.3317-1-54-4': 'In other words, to first order radial migration leads stars to merely "exchange" places in the disk, and affects first and foremost stars in the very thin disk.', '1405.3317-1-55-0': '## Effects of radial migration', '1405.3317-1-56-0': 'Our results extend and clarify earlier work on the possible effects of radial migration.', '1405.3317-1-56-1': 'For example, some authors have argued that outward migration causes the disk to thicken because it would shift stars with high vertical kinetic energy to regions where the disk gravity is lessened, an mechanism that [CITATION] have cited as a plausible origin for the Galactic thick disk.', '1405.3317-1-56-2': 'This assumption has been qualified and contested in later work, which have argued that migrating stars should cool vertically if they migrate out and heat up if they migrate inward.', '1405.3317-1-56-3': 'This was first proposed by [CITATION], who argued that such kinematic changes might signal the conservation of the vertical action, a result that has been further elaborated in more recent work .', '1405.3317-1-56-4': 'It is still unclear from that work, however, whether disk thickening can actually result from radial migration.', '1405.3317-1-57-0': 'Our results help clarify this issue.', '1405.3317-1-57-1': 'Although it is true that migrators heat or cool as they move in or out of the disk, these changes play only a small role in determining the final vertical distribution of migrators.', '1405.3317-1-57-2': 'This is mainly set by the provenance bias we describe above, which has not been fully accounted for in earlier work.', '1405.3317-1-57-3': 'Because the bias adds to the cooling associated with outward migration, our results suggest that "thick disks" cannot result from radial migration caused by the corotation scattering mechanism suggested by [CITATION].', '1405.3317-1-58-0': 'Of course, this does not rule out the possibility that thick disk stars in the solar neighbourhood might have formed elsewhere in the Galaxy: an external origin or a pre-existing thin disk heated by external perturbations , early mergers , or the deepening potential of the forming Galaxy, remain viable options for the origin of the Galactic thick disk.', '1405.3317-1-59-0': 'Our results also suggest that radial migration could have had a substantial effect on the thin disk and that, in local volumes such as the solar neighbourhood, the most extreme migrators might actually be found amongst stars closest to the disk.', '1405.3317-1-59-1': 'This is consistent with observations that suggest that some of the clearest signs of migration are found amongst thin disk stars.', '1405.3317-1-60-0': 'One compelling example involves the origin of the large spread in metallicity of the thin disk, which spans the range [MATH][Fe/H][MATH] .', '1405.3317-1-60-1': 'If the spread is caused by radial migration of stars from the inner and outer Galaxy, then one would expect that those most chemically distinct should correspond, on average, to the most extreme migrators.', '1405.3317-1-61-0': 'As discussed, for example, by [CITATION], this might explain the puzzling anti-correlation between metallicity and rotation velocity found for stars in the thin disk.', '1405.3317-1-61-1': 'In this scenario, the anti-correlation merely reflects the difference in angular momentum expected for inward and outward migrators that have reached the solar neighbourhood.', '1405.3317-1-61-2': 'Arrivals from the outer disk (presumably those populating the metal-poor tail of the thin disk) should have higher-than-average angular momentum, while the opposite should hold for those coming from the inner disk.', '1405.3317-1-61-3': 'The net effect is a negative gradient in the [MATH] vs [Fe/H] relation (or, equivalently, in [MATH] vs [Fe/H]), which is indeed observed .', '1405.3317-1-62-0': 'Our results confirm this expectation.', '1405.3317-1-62-1': 'Fig. [REF] shows, for particles identified in a narrow cylindrical shell of radius [MATH] kpc at the final time, [MATH], the azimuthal velocity, [MATH], as a function of the initial guiding centre radius, [MATH].', '1405.3317-1-62-2': 'This figure shows that stars that have migrated inward from large radii tend to have larger [MATH] (or, equivalently, larger [MATH]) than those that have migrated outward from well within the inner disk, leading to a well defined correlation between rotation velocity and provenance radius.', '1405.3317-1-62-3': 'If radius correlates with metallicity, then a negative metallicity gradient in the Galaxy would give rise to an anti-correlation between [MATH] and metal content such as observed.', '1405.3317-1-63-0': 'On the other hand, we note that stars chemically identified to be part of the Galactic thick disk by their [MATH] enhancement show a gradient in the [MATH] vs [Fe/H] relation of opposite sign , a result quite difficult to reconcile in a migration-driven scenario for the origin of the thick disk.', '1405.3317-1-63-1': 'The different signs of the gradient suggest a qualitative distinction in the origin of [MATH]-poor (thin-disk) stars and [MATH]-rich (thick-disk) stars.', '1405.3317-1-63-2': 'Such kinematic distinction might also present an interesting challenge to models that view the Galactic disk as a single entity made by the overlap of several mono-abundance populations .', '1405.3317-1-64-0': 'Finally, our results suggest an extra consistency check of the migration interpretation of the thin-disk [MATH] vs [Fe/H] anti-correlation: those that migrate the most should also have lower-than-average vertical velocity dispersions.', '1405.3317-1-64-1': 'In this interpretation, both unusually metal-poor or unusually metal-rich stars should have lower scaleheights, although the trend might be more readily noticeable in the metal-poor population (see Fig. [REF]).', '1405.3317-1-64-2': 'No such effect has been reported in the literature, but this may be due to difficulties in separating thin disk from thick disk stars at the metal-poor end.', '1405.3317-1-64-3': 'Indeed, we note that a kindred effect has been argued by [CITATION] to explain the drop in [MATH] observed for the most [MATH]-rich stars in their sample.', '1405.3317-1-65-0': '# Summary and Conclusions', '1405.3317-1-66-0': 'We have analyzed the effects of a persistent, multi-arm spiral pattern on the radial distribution of particles in an exponential thin disk.', '1405.3317-1-66-1': 'Our disk is modeled with [MATH] self-gravitating particles and a rigid halo potential, thus minimizing the effects of noise.', '1405.3317-1-66-2': 'Evolved in isolation, the disk develops no obvious non-axisymmetric patterns nor is subject to the violent instabilities that have often plagued [MATH]-body models of cold stellar disks.', '1405.3317-1-67-0': 'Spiral patterns are seeded in the disk by the temporary addition of massive perturbers intended to simulate the presence of molecular clouds.', '1405.3317-1-67-1': 'The disk responds to the molecular clouds by developing an intricate multi-arm spiral pattern that persists, after the perturbers are removed, for over [MATH] Gyr.', '1405.3317-1-67-2': 'Interestingly, the perturbations cause only minor heating and have no discernible effect on the vertical structure of the disk.', '1405.3317-1-67-3': 'Thus, our simulations are especially well suited to analyze the effect of long-lived spiral patterns on the orbits of disk stars.', '1405.3317-1-68-0': 'Stars in the disk respond to the spiral perturbations by migrating widely through the disk whilst largely preserving the nearly circular nature of their orbits, in a manner consistent with the corotation scattering mechanism proposed by [CITATION].', '1405.3317-1-68-1': 'Adopting this as a definition of radial migration, we focus our analysis on the properties of migrators and on the effects of radial migration on the vertical structure of the disk.', '1405.3317-1-68-2': 'Our main conclusions may be summarized as follows.', '1405.3317-1-69-0': 'Our results endorse the view that migration might play a substantial role in the radial redistribution of stars in galaxies that have experienced sustained spiral structure.', '1405.3317-1-69-1': 'Although we have tried to emphasize the results that seem of general applicability, we also recognize that our conclusions are based on a single simulation that, in many ways, fails to capture the fascinating complexity of galaxies like our own Milky Way.', '1405.3317-1-69-2': 'The evolving potential of the forming galaxy; the accretion of gas and the ongoing formation of stars; the effects of satellites and substructure; the presence of a bar and of evolving spirals; all must have left an imprint on the structure of the Galaxy that our simulations cannot attempt to match.', '1405.3317-1-70-0': 'Further, our simulation adopts a disk that is lighter than that of the Milky Way and whose spiral patterns, therefore, might differ from that of the Galaxy.', '1405.3317-1-70-1': 'A more massive disk would probably sport spirals with fewer arms that might lead to faster heating.', '1405.3317-1-70-2': 'Although we do not believe that this would affect qualitatively our conclusions, a more definitive, quantitative comparison with observation should be attempted with models that reproduce better the known structural parameters of the Galaxy.', '1405.3317-1-70-3': 'Still, we believe that simplified models like the one we discuss here are useful, since they allow us to isolate and understand the effects of radial migration so as to inform the analysis of more ambitious modelling.', '1405.3317-1-71-0': '2em', '1405.3317-1-72-0': 'This work is funded by NSF Grant No. 1211258 and NASA Grant No. 13-ATP13-0053.', '1405.3317-1-72-1': 'ED gratefully acknowledges the support of the Alfred P. Sloan Foundation.', '1405.3317-1-72-2': 'ED and JN express their appreciation towards the Aspen Center for Physics for their hospitality.', '1405.3317-1-72-3': 'MA and JN acknowledge the support of ANPCyT, Argentina by Grant No. PICT 2012-1137.', '1405.3317-1-72-4': 'This work was supported in part by the National Science Foundation under Grant No. PHYS-1066293.', '1405.3317-1-72-5': 'This simulation has been run on Odyssey at Harvard and on the High Performance Computing cluster provided by the Advanced Computing Infrastructure (ACI) and Center for High Throughput Computing (CHTC) at the University of Wisconsin.', '1405.3317-1-73-0': '2em'}

{'1405.3317-2-0-0': 'We study the radial migration of stars driven by recurring multi-arm spiral features in an exponential disk embedded in a dark matter halo.', '1405.3317-2-0-1': 'The spiral perturbations redistribute angular momentum within the disk and lead to substantial radial displacements of individual stars, in a manner that largely preserves the circularity of their orbits and that results, after [MATH] Gyr ([MATH] full rotations at the disk scalelength), in little radial heating and no appreciable changes to the vertical or radial structure of the disk.', '1405.3317-2-0-2': 'Our results clarify a number of issues related to the spatial distribution and kinematics of migrators.', '1405.3317-2-0-3': 'In particular, we find that migrators are a heavily biased subset of stars with preferentially low vertical velocity dispersions.', '1405.3317-2-0-4': 'This "provenance bias" for migrators is not surprising in hindsight, for stars with small vertical excursions spend more time near the disk plane and thus respond more readily to non-axisymmetric perturbations.', '1405.3317-2-0-5': 'We also find that the vertical velocity dispersion of outward migrators always decreases, whereas the opposite holds for inward migrators.', '1405.3317-2-0-6': 'To first order, newly arrived migrators simply replace stars that have migrated off to other radii, thus inheriting the vertical bias of the latter.', '1405.3317-2-0-7': 'Extreme migrators might therefore be recognized, if present, by the unexpectedly small amplitude of their vertical excursions.', '1405.3317-2-0-8': 'Our results show that migration, understood as changes in angular momentum that preserve circularity, can affect strongly the thin disk, but cast doubts on models that envision the Galactic thick disk as a relic of radial migration.', '1405.3317-2-1-0': '1Department of Astronomy, University of Wisconsin, 2535 Sterling Hall, 475 N. Charter Street, Madison, WI 53076, USA.', '1405.3317-2-1-1': 'mailto:ciro@astro.wisc.edue-mail:ciro@astro.wisc.edu', '1405.3317-2-2-0': '2Department of Physics and Astronomy, University of Victoria, Victoria, BC, V8P 5C2, Canada', '1405.3317-2-3-0': '3Observatorio Astronomico, Universidad Nacional de Cordoba, Cordoba X5000BGR, Argentina', '1405.3317-2-4-0': '4Alfred P. Sloan Fellow', '1405.3317-2-5-0': '5Senior CIfAR Fellow', '1405.3317-2-6-0': '# Introduction', '1405.3317-2-7-0': 'Cold, thin stellar disks are dynamically fragile entities that react strongly to changes in the gravitational potential, such as those that result from changes in the mass of the galaxy itself, from external perturbations such as the accretion of a satellite or the tidal effects of substructure, or from internal processes such as bar instability or the development of spiral structure.', '1405.3317-2-7-1': 'In the absence of dissipation, these perturbations generally heat the disk, moving stars away from their birth radii at the expense of decreasing the circularity of their orbits and increasing the amplitude of their vertical oscillations.', '1405.3317-2-7-2': 'Stars on nearly circular orbits must have therefore been born at about the same radius where they are found today.', '1405.3317-2-8-0': 'One difficulty with this simple scenario arises in our Galaxy from the weak correlation between age and metallicity for thin disk stars in the solar neighborhood , which is essentially flat for ages between [MATH] and [MATH] Gyr .', '1405.3317-2-8-1': 'Although this may be taken to suggest a simple (but somewhat contrived) scenario where the average metallicity of the interstellar medium (ISM) at the solar circle has not changed since [MATH], the sizable scatter in the correlation would also require the ISM to have been rather in-homogeneous in the past, a possibility that seems at face value unattractive .', '1405.3317-2-9-0': 'An alternative view holds that thin disk stars do not necessarily remain close to their birth radii .', '1405.3317-2-9-1': 'If true, then long-lived stars in the solar neighborhood might actually sample the chemical properties of the Galaxy over a wide range of radii at the time of their formation, possibly blurring any correlation between the age and metallicity of stars selected in local samples.', '1405.3317-2-9-2': 'This scenario gained support after Sellwood and Binney, in an influential paper, identified a mechanism that allows stars to migrate in radius whilst remaining on nearly circular orbits .', '1405.3317-2-9-3': 'The mechanism relies on angular momentum exchanges between stars near corotation of transient spiral patterns, and allows stars to switch radii without substantially heating the disk.', '1405.3317-2-9-4': 'Given that spiral patterns seem ubiquitous in disk galaxies, this mechanism must operate to some degree in most galaxy disks.', '1405.3317-2-10-0': 'Since its inception, the corotation resonance (CR) mechanism of [CITATION] has been the subject of a number of studies in the literature.', '1405.3317-2-10-1': 'These studies have generally confirmed the validity of the arguments laid out in their original paper and extended their application to multiple, interacting spiral patterns , but have also raised a number of questions about the initial and final properties of migrating stars and about the consequences of radial migration for the vertical structure of the disk.', '1405.3317-2-10-2': 'It is unclear, for example, what biases, if any, might affect the kinematics or spatial distribution of migrators, what strategies might be used to identify migrators observationally, or whether radial migration plays a role in the origin of the thick disk of the Galaxy.', '1405.3317-2-11-0': 'The latter issue, in particular, has been debated vigorously, especially since [CITATION] proposed that the chemical differentiation between thin and thick disks could be explained solely by radial migration, which might have brought stars with high vertical kinetic energy from the inner Galaxy out to the solar neighborhood.', '1405.3317-2-11-1': '[CITATION], on the other hand, have argued that migration does little for disk thickening because outward migrators should cool vertically as they move out, a result that has been further debated by [CITATION] and [CITATION].', '1405.3317-2-12-0': 'This unsettled state of affairs might be traced, in part, to the disparate numerical contexts used to study the effects of migration.', '1405.3317-2-12-1': 'Although most published work has relied on direct numerical simulation, some studies are based on the response of equilibrium disk models to transient spiral modes or to the tidal perturbation of a massive satellite , whereas others have focused on hydrodynamical simulations of disks of gas and stars that evolve in isolation or that result from the dissipative collapse of a rotating sphere of gas in the gravitational potential of a dark matter halo .', '1405.3317-2-13-0': 'The first approach benefits from the controlled nature of the initial disk, which allows for the identification of numerical artifact, but suffers from the arbitrariness involved in choosing the orbit and mass of the perturbers, or from differences in the origin of the spiral perturbations, which either develop freely or result from the forcing of particular modes with predetermined amplitude and duration.', '1405.3317-2-13-1': 'The second approach allows the disk to evolve more freely, but its analysis is obscured by the complexities that arise from the continuously evolving disk potential and from the presence of rapidly evolving multiple modes and instabilities.', '1405.3317-2-13-2': 'Furthermore, the numerical resolution of hydrodynamical simulations is inevitably much lower than that attainable with [MATH]-body models, making it much harder to discern and isolate the effects of noise associated with the finite number of particles used.', '1405.3317-2-14-0': 'Finally, there are also semantic issues, since some authors use the term "radial migration" as a catch-all phrase to denote any radial displacement of a star, regardless of its cause or effect on its orbit, whereas others reserve the term "migration" for changes in the guiding center radius (i.e., the orbital angular momentum) of orbits that remain nearly circular through the process.', '1405.3317-2-14-1': 'We shall adopt the latter meaning in this contribution.', '1405.3317-2-15-0': 'We address some of these issues here using numerical simulations that evolve a kinematically cold self-gravitating disk of stars in the fixed potential of a cosmologically motivated dark matter halo.', '1405.3317-2-15-1': 'Spiral patterns are seeded by perturbations modeled on the gravitational influence of molecular clouds and persist almost unchanged for over [MATH] Gyr, the duration of the simulation (more than [MATH] full rotations at the exponential disk scalelength).', '1405.3317-2-15-2': 'Our simulations differ from earlier work mainly on numerical resolution: the disk is made up of [MATH] particles, an improvement of nearly two orders of magnitude over earlier work.', '1405.3317-2-15-3': 'The large number of particles allows us to study the orbital evolution of individual stars without having to worry about the discreteness noise associated with the particle representation of the disk.', '1405.3317-2-16-0': 'Our simulations also differ from earlier attempts because the disk is not subject to the rapid development of instabilities that may change its radial or vertical structure; a control run without perturbers shows that the disk does not develop strong non-axisymmetric features over several rotation periods.', '1405.3317-2-16-1': 'Finally, our simulations also stand out because the spiral features seeded by the "molecular clouds" do not heat appreciably the disk and persist for over [MATH] Gyr.', '1405.3317-2-16-2': 'Both the vertical and azimuthally averaged structure of the disk remain basically unchanged throughout.', '1405.3317-2-17-0': 'The plan for this paper is as follows.', '1405.3317-2-17-1': 'We describe the numerical simulations used in this work in Section [REF], present our main results in Section [REF], and conclude with a brief summary in Section [REF].', '1405.3317-2-18-0': '# Numerical Methods', '1405.3317-2-19-0': '## The Galaxy Model', '1405.3317-2-20-0': 'We use here [MATH]-body simulations of the same galaxy model presented by [CITATION] .', '1405.3317-2-20-1': 'For the sake of completeness we give here a brief description of the simulations and refer the interested reader to the original reference for further details.', '1405.3317-2-20-2': 'The models evolve a self-gravitating disk of particles embedded in a rigid halo potential described by a Hernquist profile with total mass [MATH] and scale length [MATH] kpc.', '1405.3317-2-21-0': 'The disk is a [MATH] particle realization of an exponential disk of constant scaleheight: [EQUATION] whose positions and velocities were generated using a modified version of the method presented by [CITATION], as described by [CITATION].', '1405.3317-2-21-1': 'We adopt a radial scalelength [MATH] kpc, scaleheight [MATH] kpc and total disk mass [MATH].', '1405.3317-2-22-0': 'With these choices, the disk is submaximal, and makes up about 27% of the total mass enclosed within two radial scalelengths.', '1405.3317-2-22-1': 'A particle in a circular orbit completes a full rotation at [MATH] in [MATH] Myr.', '1405.3317-2-22-2': 'We show in Figure [REF] the circular velocity profile of the galaxy model (top panel), and the circular orbit timescale, [MATH], in Myr (bottom panel).', '1405.3317-2-22-3': 'The middle panel shows the Toomre stability parameter [MATH], chosen to ensure the disk is marginally stable at all radii.', '1405.3317-2-22-4': '(Here [MATH] is the epicycle frequency and [MATH] the velocity dispersion in radial motions.)', '1405.3317-2-23-0': '## The Control Simulation', '1405.3317-2-24-0': 'We have evolved the galaxy model described in the previous section for from [MATH] Myr to [MATH] Myr, corresponding to six circular orbit timescales at [MATH], or three full rotations at [MATH] kpc.', '1405.3317-2-24-1': 'The system was evolved using an updated version of Gadget , using a Plummer-equivalent softening length for the stellar particles of [MATH] pc and default integration parameters.', '1405.3317-2-25-0': 'After [MATH] Myr the "control" disk has evolved little from its initial configuration, as may be seen in the top two panels of Figure [REF], and has developed no obvious non-axisymmetric features.', '1405.3317-2-25-1': 'It does develop, however, "ring-like" features that travel like radial waves through the disk, and which are easily appreciate in the top right panel of Figure [REF].', '1405.3317-2-26-0': 'These waves do not seem to be caused by discreteness effects, since they appear spontaneously and are coherent over the whole disk.', '1405.3317-2-26-1': 'The radial waves seem to travel back and forth through the disk, without causing any discernable change in the radial distribution of disk particles.', '1405.3317-2-26-2': 'They do not perturb appreciably the vertical structure of the disk, as shown in the top two panels of Figure [REF], nor change the surface density profile of the disk, as shown in the left panel of Figure [REF].', '1405.3317-2-27-0': 'The left panels of Figure [REF] show that the radial waves have little effect on the kinematics of disk particles: there is little evolution in the velocity dispersion profiles of the disk over the [MATH] Myr that the disk runs unperturbed.', '1405.3317-2-27-1': 'We conclude that our galaxy disk model is not subject to the strong non-axisymmetric instabilities that have plagued [MATH]-body studies of disk galaxies over the years.', '1405.3317-2-27-2': 'Although our disks are fully self-gravitating, it appears as if the number of particles used is sufficient to suppress discreteness noise and to prevent, for several rotation periods, the development of strong non-axisymmetric features.', '1405.3317-2-28-0': '## The "Spiral" Simulation', '1405.3317-2-29-0': 'As shown by [CITATION], non-axisymmetric features do develop when the disk is perturbed in a manner consistent with the presence of [MATH] "molecular clouds".', '1405.3317-2-29-1': 'We refer the reader to [CITATION] for details on the numerical implementation but, in brief, the perturbations are modeled by the addition of softened, [MATH] particles constrained to travel in circular orbits and scattered about the disk.', '1405.3317-2-29-2': 'The perturbation is transient, and the molecular-cloud perturbers are removed [MATH] Myr after their inclusion.', '1405.3317-2-30-0': 'The disk reacts strongly to these perturbations, and develops a long-lived multi-arm spiral pattern, as shown in the bottom four panels of Figure [REF].', '1405.3317-2-30-1': 'In this sequence, [MATH] shows the disk just after the perturbers are removed; [MATH] corresponds, as in the control simulation, to [MATH] Myr later; whereas [MATH] and [MATH] denote later stages of the evolution, [MATH] Myr and [MATH] Myr after the removal of the perturbers, respectively.', '1405.3317-2-31-0': 'Visual inspection of Figure [REF] shows that the multi-arm spirals seeded by the molecular cloud perturbers are long lived and that they change little in nature over [MATH] Gyr of evolution.', '1405.3317-2-31-1': 'Despite their longevity, the spiral patterns seem to have little effect on the vertical structure of the disk, as shown by the bottom four panels of Figure [REF].', '1405.3317-2-31-2': 'The radial structure of the disk also remains almost unchanged, as indicated by the azimuthally averaged surface density profiles at [MATH], [MATH], [MATH], and [MATH] shown in the right-hand panel of Figure [REF].', '1405.3317-2-32-0': 'Only the in-plane kinematics of the disk seems to be noticeably affected, as shown by the slight, but monotonic, heating in the radial and azimuthal directions seen in the right-hand panels of Figure [REF].', '1405.3317-2-32-1': 'The vertical structure of the disk, however, is unaffected throughout: the vertical velocity dispersion profile, [MATH], at [MATH] is indistinguishable from that at [MATH] (see bottom right panel of Figure [REF]).', '1405.3317-2-33-0': '## Azimuthal Structure', '1405.3317-2-34-0': 'We examine the azimuthal structure of the disks in Figure [REF], where we show the fluctuations in the surface density and in the gravitational potential at various times as a function of the azimuthal angle [MATH] and of the cylindrical radius, [MATH].', '1405.3317-2-35-0': 'The leftmost panels of Figure [REF] shows that a barely discernible multi-arm spiral pattern develops in the control simulation after [MATH] Myr of evolution.', '1405.3317-2-35-1': 'Given its small amplitude (the rms fluctuations in the surface density are just [MATH] at [MATH] kpc) this pattern is only evident in the residuals and, as we will see below, leads to basically no secular evolution in the disk over the [MATH] Myr it was evolved for.', '1405.3317-2-35-2': 'This confirms our earlier statement that the model galaxy is free from the violent instabilities or rapidly-growing non-axisymmetric disturbances characteristic of models where either the galaxy is initially out of equilibrium, or where discreteness noise seeds perturbations that transform the galaxy over a short period of time.', '1405.3317-2-36-0': 'The multi-arm spiral structure of the "spiral" simulation stands out clearly in Figure [REF], with interarm surface density fluctuations of roughly [MATH].', '1405.3317-2-36-1': 'The actual gravitational potential fluctuations are much smaller, however, due to the presence of the dark matter halo.', '1405.3317-2-36-2': 'The spiral structure in the "spiral" simulation remains roughly constant from [MATH] to [MATH], and is characterized by a radially increasing number of arms that bifurcate outward: at [MATH] kpc, near the center, only about [MATH] arms are clearly defined, whereas more than [MATH] arms can be counted [MATH] kpc away from the center.', '1405.3317-2-36-3': 'We ascribe the very slight evolution in the spiral structure from [MATH] to [MATH] to the weak, but monotonic, radial heating of the disk seen in the increase of the radial velocity dispersion shown in Figure [REF].', '1405.3317-2-36-4': 'We emphasize again that the overall heating is quite modest; in [MATH] Gyr of evolution the radial velocity dispersion at [MATH] kpc increases from [MATH] km s-1 to just over [MATH] km s-1.', '1405.3317-2-36-5': 'The increase is even smaller at other radii.', '1405.3317-2-37-0': 'The spirals shown in Figure [REF] have pattern speeds that approximately match the angular speed of the disk at the radii where they are most prominent.', '1405.3317-2-37-1': 'This is seen in the spectrograms of Figure [REF], where we show the power spectra for the [MATH], [MATH], [MATH], and [MATH] harmonic modes, measured from snapshots sampled every [MATH] Myr for the spiral disk in the [MATH]Myr[MATH] interval.', '1405.3317-2-37-2': 'Note that each mode spans a limited radial range and that the higher modes travel with a pattern speed similar to that of circular orbits at that radius.', '1405.3317-2-38-0': 'As discussed by [CITATION], arms in the pattern fade away and recur while traveling at about the circular angular frequency, in a seemingly "self-perpetuating" cycle.', '1405.3317-2-38-1': 'This implies that stars on nearly circular orbits will be consistently near CR of the spiral mode that dominates at each radius, favoring the corotation scattering mechanism of [CITATION].', '1405.3317-2-38-2': 'We examine its effect over many rotation periods next.', '1405.3317-2-39-0': '# Results', '1405.3317-2-40-0': '## Radial Migration', '1405.3317-2-41-0': 'In a stationary potential and in the absence of non-axisymmetric perturbations disk stars should conserve both their energy, [MATH], and the [MATH]-component of their specific angular momentum, [MATH].', '1405.3317-2-41-1': 'The presence of the transient spiral patterns discussed in the last section, however, leads to exchanges of angular momentum amongst disk particles and to variations in their orbital energies.', '1405.3317-2-42-0': 'We examine these changes by selecting, at [MATH], particles within two [MATH] pc wide cylindrical shells centered at [MATH] and [MATH] kpc, respectively.', '1405.3317-2-42-1': 'We track these particles at later times and compare their current energies and angular momenta with those at [MATH].', '1405.3317-2-42-2': 'The comparison is shown in Figure [REF] for the control simulation (at [MATH]) and for the spiral simulation at [MATH], [MATH] and [MATH].', '1405.3317-2-43-0': 'Particles in the control simulation exhibit very slight changes in their values of [MATH] and [MATH], as expected given the smooth radial and azimuthal structure of the control disk at all times during its evolution.', '1405.3317-2-44-0': 'Particles in the spiral simulation, on the other hand, undergo correlated changes in [MATH] and [MATH] that can reach (and exceed, in some cases) a factor of [MATH] in [MATH].', '1405.3317-2-44-1': 'Recalling that the guiding center radius, [MATH], of a star in a nearly circular orbit is directly proportional to [MATH]; [EQUATION] (where [MATH] and [MATH] is the gravitational potential on the disk plane) this implies changes in the guiding center that can exceed a factor of [MATH].', '1405.3317-2-45-0': 'These variations, however, do not alter the nearly circular nature of the orbits.', '1405.3317-2-45-1': 'Indeed, the correlated changes in [MATH] and [MATH] keep the stars very close to the minimum energy consistent with its angular momentum (indicated by the solid curves in Figure [REF]), which defines circular orbits.', '1405.3317-2-45-2': 'Thus, the spiral perturbations shown in Figs. [REF] and [REF] cause substantial migration in the spiral disk with minimal heating, as envisioned in the CR mechanism of [CITATION].', '1405.3317-2-46-0': 'Figure [REF] also hints that, as time goes by, the radial displacements of disk particles increase monotonically.', '1405.3317-2-46-1': 'We show this more explicitly in Figure [REF], where we plot the distribution of guiding center radii at [MATH] of particles that migrate at later times to [MATH] kpc (left panels) or [MATH] kpc (right panels).', '1405.3317-2-46-2': 'The width of these distributions increase with time roughly as [MATH], suggestive of a simple diffusion mechanism.', '1405.3317-2-47-0': 'After more than [MATH] Gyr of evolution (i.e., at [MATH]) more than [MATH] of particles with [MATH] kpc have migrated to their current orbits from radii that were initially either [MATH] smaller or larger.', '1405.3317-2-47-1': 'At [MATH] kpc the fraction of such extreme migrators reduces to [MATH] in the same time, presumably because of the longer orbital times characteristic of larger radii.', '1405.3317-2-47-2': 'The radial distribution of migrators at [MATH] kpc is also clearly asymmetric, with a larger fraction of extreme migrators that come from the inner regions.', '1405.3317-2-47-3': 'This is again not surprising, since there are few particles beyond [MATH] kpc in the disk.', '1405.3317-2-48-0': 'We show in Figure [REF] the evolution in radius of a few "extreme" migrators that have moved roughly [MATH] inward or outward to reach their final radius.', '1405.3317-2-48-1': 'The top left and right panels correspond to migrators that reach [MATH] or [MATH] kpc at [MATH], respectively.', '1405.3317-2-48-2': "In spite of the large changes in guiding center, the migrators' orbital circularities barely change.", '1405.3317-2-48-3': 'This is shown in the bottom panels of Figure [REF], where we plot the evolution of the circularity parameter, [MATH], defined as the specific angular momentum in units of that of a circular orbit of the same energy.', '1405.3317-2-48-4': 'As these panels show, even extreme migrators stay on nearly circular orbits throughout the evolution.', '1405.3317-2-49-0': '## Migrator Properties', '1405.3317-2-50-0': 'We explore the properties of migrators in Figure [REF], where we show, in the left-hand panels, the vertical velocity dispersion of particles with guiding center radii [MATH] kpc (shaded area), as a function of their original guiding center, [MATH].', '1405.3317-2-50-1': 'Particles to the left of the shaded area have migrated outwards to reach [MATH] kpc; those to the right have migrated inward.', '1405.3317-2-50-2': 'For reference, the thick dashed curves in each panel of this figure indicate the vertical velocity dispersion profile of all particles in the disk, [MATH].', '1405.3317-2-51-0': 'The blue curves in Figure [REF] show the current velocity dispersion of subsets of particles binned according to [MATH].', '1405.3317-2-51-1': 'The inverted "U" shape indicates that migrators are a kinematically biased population with lower velocity dispersions than non-migrators, a bias that increases the farther away a particle has migrated from in order to reach its current radius.', '1405.3317-2-51-2': 'The bias affects both inward and outward migrators, and holds at all times.', '1405.3317-2-51-3': 'The bias is not only kinematical; as expected given their low vertical velocity dispersion, the vertical distribution of migrators is also thinner than that of non-migrators, as shown by their current vertical scaleheight (see the blue curves in the right-hand panels of Figure [REF]).', '1405.3317-2-51-4': 'Again, the farther particles have migrated from the thinner their current vertical distribution.', '1405.3317-2-52-0': 'The cause of this vertical bias is twofold.', '1405.3317-2-52-1': 'Firstly, migrating stars are a heavily-biased subset of particles at their original radii, a "provenance bias" that increases the farther a particle migrates in a given time.', '1405.3317-2-52-2': 'This may be seen by comparing the red curves in Figure [REF], which show the velocity dispersion and scaleheight of migrators at their initial radius, with the dashed black line, which shows the same but averaged for all stars at that radius.', '1405.3317-2-52-3': 'The same vertical bias is clearly present at their radius of origin: radial migration mainly affects thinner-than-average populations at both their destination and provenance radii.', '1405.3317-2-53-0': "A second reason for the migrators' vertical bias may be seen by comparing the initial and final velocity dispersions of migrating stars.", '1405.3317-2-53-1': 'This is given by the difference, at given [MATH], between the red and blue curves in Figure [REF].', '1405.3317-2-53-2': 'Outward migrators (to the left of the gray zone) clearly cool as they move out (i.e., the blue curve is below the red), whereas the opposite holds for inward migrators.', '1405.3317-2-53-3': 'We note that this in itself would not be enough to explain the vertical bias at the final radius, since outward (inward) migrators originate in regions of higher (lower) velocity dispersion of the disk.', '1405.3317-2-53-4': 'However, Figure [REF] shows that such cooling or heating is but a small correction to the "provenance bias" affecting all migrators.', '1405.3317-2-54-0': 'We have checked that a similar provenance bias and heating/cooling process affect migrators at other radii, and that similar results are obtained when analyzing the other two components of the velocity dispersion, [MATH] and [MATH].', '1405.3317-2-54-1': 'Therefore, radial migration does not only operate most effectively on particles closest to the disk, but also on those closest to circular orbits.', '1405.3317-2-54-2': 'These biases have been hinted at in earlier work but their importance has not been properly appreciated, obscured perhaps by the concurrence of other heating mechanisms (i.e., satellite perturbations, disk growth, major disk instabilities) whose effectiveness in displacing stars radially can exceed that of the corotation scattering mechanism we focus on here .', '1405.3317-2-55-0': 'With the benefit of hindsight, our results are not surprising.', '1405.3317-2-55-1': 'Radial migration is caused by non-axisymmetric patterns in the disk that are more readily felt by stars that remain closer to the disk during their orbits; thus, the smaller the vertical amplitude of their oscillations the farther stars can migrate.', '1405.3317-2-55-2': 'This is, of course, true at all radii affected by the spiral patterns.', '1405.3317-2-55-3': 'If radial migration is not to cause vertical heating or a change in the surface density profile (as is the case in our simulation) then migrating stars at their final location must simply "substitute" other stars that have migrated off, which were themselves biased in the same way.', '1405.3317-2-55-4': 'In other words, to first order radial migration leads stars to merely "exchange" places in the disk, and affects first and foremost stars in the very thin disk.', '1405.3317-2-56-0': '## Effects of Radial Migration', '1405.3317-2-57-0': 'Our results extend and clarify earlier work on the possible effects of radial migration.', '1405.3317-2-57-1': 'For example, some authors have argued that outward migration causes the disk to thicken because it would shift stars with high vertical kinetic energy to regions where the disk gravity is lessened, an mechanism that [CITATION] have cited as a plausible origin for the Galactic thick disk.', '1405.3317-2-57-2': 'This assumption has been qualified and contested in later work, which have argued that migrating stars should cool vertically if they migrate out and heat up if they migrate inward.', '1405.3317-2-57-3': 'This was first proposed by [CITATION], who argued that such kinematic changes might signal the conservation of the vertical action, a result that has been further elaborated in more recent work .', '1405.3317-2-58-0': 'Our results help clarify this issue.', '1405.3317-2-58-1': 'Although it is true that migrators heat or cool as they move in or out of the disk, these changes play only a small role in determining the final vertical distribution of migrators.', '1405.3317-2-58-2': 'This is mainly set by the provenance bias we describe above, which has not been fully accounted for in earlier work.', '1405.3317-2-58-3': 'Because the bias adds to the cooling associated with outward migration, our results suggest that "hick disks" might not result from the only effect of radial migration, at least in multi-armed spiral galaxies.', '1405.3317-2-59-0': 'Of course, this does not rule out the possibility that thick disk-stars in the solar neighborhood might have formed elsewhere in the Galaxy: an external origin or a pre-existing thin disk heated by external perturbations , early mergers , or the deepening potential of the forming Galaxy, remain viable options for the origin of the Galactic thick disk.', '1405.3317-2-60-0': 'Our results also suggest that radial migration could have had a substantial effect on the thin disk and that, in local volumes such as the solar neighborhood, the most extreme migrators might actually be found amongst stars closest to the disk.', '1405.3317-2-60-1': 'This is consistent with observations that suggest that some of the clearest signs of migration are found amongst thin disk stars.', '1405.3317-2-61-0': 'One compelling example involves the origin of the large spread in metallicity of the thin disk, which spans the range [MATH][Fe/H][MATH] .', '1405.3317-2-61-1': 'If the spread is caused by radial migration of stars from the inner and outer Galaxy, then one would expect that those most chemically distinct should correspond, on average, to the most extreme migrators.', '1405.3317-2-62-0': 'As discussed, for example, by [CITATION], this might explain the puzzling anti-correlation between metallicity and rotation velocity found for stars in the thin disk.', '1405.3317-2-62-1': 'In this scenario, the anti-correlation merely reflects the difference in angular momentum expected for inward and outward migrators that have reached the solar neighborhood.', '1405.3317-2-62-2': 'Arrivals from the outer disk (presumably those populating the metal-poor tail of the thin disk) should have higher-than-average angular momentum, while the opposite should hold for those coming from the inner disk.', '1405.3317-2-62-3': 'The net effect is a negative gradient in the [MATH] versus [Fe/H] relation (or, equivalently, in [MATH] versus [Fe/H]), which is indeed observed .', '1405.3317-2-63-0': 'Our results confirm this expectation.', '1405.3317-2-63-1': 'Figure [REF] shows, for particles identified in a narrow cylindrical shell of radius [MATH] kpc at the final time, [MATH], the azimuthal velocity, [MATH], as a function of the initial guiding center radius, [MATH].', '1405.3317-2-63-2': 'This figure shows that stars that have migrated inward from large radii tend to have larger [MATH] (or, equivalently, larger [MATH]) than those that have migrated outward from well within the inner disk, leading to a well defined correlation between rotation velocity and provenance radius.', '1405.3317-2-63-3': 'If radius correlates with metallicity, then a negative metallicity gradient in the Galaxy would give rise to an anti-correlation between [MATH] and metal content such as observed.', '1405.3317-2-64-0': 'On the other hand, we note that stars chemically identified to be part of the Galactic thick disk by their [MATH] enhancement show a gradient in the [MATH] versus [Fe/H] relation of opposite sign , a result that is difficult to reconcile in our models assuming a migration-driven scenario for the origin of the thick disk.', '1405.3317-2-64-1': 'However we notice that in this scenario the anti-correlation has been reproduced as reported in Figure 15 of [CITATION].', '1405.3317-2-64-2': 'The reason of the differences between our results and previous results might be in the different models of galaxies adopted and will be investigated in a forthcoming work.', '1405.3317-2-64-3': 'Note that the different signs of the gradient suggest a qualitative distinction in the origin of [MATH]-poor (thin-disk) stars and [MATH]-rich (thick-disk) stars.', '1405.3317-2-64-4': 'Such kinematic distinction might also present an interesting challenge to models that view the Galactic disk as a single entity made by the overlap of several mono-abundance populations .', '1405.3317-2-65-0': 'Finally, our results suggest an extra consistency check of the migration interpretation of the thin disk [MATH] versus [Fe/H] anti-correlation: those that migrate the most should also have lower-than-average vertical velocity dispersions.', '1405.3317-2-65-1': 'In this interpretation, both unusually metal-poor or unusually metal-rich stars should have lower scaleheights, although the trend might be more readily noticeable in the metal-poor population (see Figure [REF]).', '1405.3317-2-65-2': 'No such effect has been reported in the literature, but this may be due to difficulties in separating thin-disk from thick-disk stars at the metal-poor end.', '1405.3317-2-65-3': 'Indeed, we note that a kindred effect has been argued by [CITATION] to explain the drop in [MATH] observed for the most [MATH]-rich stars in their sample.', '1405.3317-2-66-0': '# Summary and Conclusions', '1405.3317-2-67-0': 'We have analyzed the effects of a persistent, multi-arm spiral pattern on the radial distribution of particles in an exponential thin disk.', '1405.3317-2-67-1': 'Our disk is modeled with [MATH] self-gravitating particles and a rigid halo potential, thus minimizing the effects of noise.', '1405.3317-2-67-2': 'Evolved in isolation, the disk develops no obvious non-axisymmetric patterns nor is subject to the violent instabilities that have often plagued [MATH]-body models of cold stellar disks.', '1405.3317-2-68-0': 'Spiral patterns are seeded in the disk by the temporary addition of massive perturbers intended to simulate the presence of molecular clouds.', '1405.3317-2-68-1': 'The disk responds to the molecular clouds by developing an intricate multi-arm spiral pattern that persists , after the perturbers are removed, for over [MATH] Gyr.', '1405.3317-2-68-2': 'Interestingly, the perturbations cause only minor heating and have no discernible effect on the vertical structure of the disk.', '1405.3317-2-68-3': 'Thus, our simulations are especially well suited to analyze the effect of long-lived spiral patterns on the orbits of disk stars.', '1405.3317-2-69-0': 'Stars in the disk respond to the spiral perturbations by migrating widely through the disk whilst largely preserving the nearly circular nature of their orbits.', '1405.3317-2-69-1': 'Adopting this as a definition of radial migration, we focus our analysis on the properties of migrators and on the effects of radial migration on the vertical structure of the disk.', '1405.3317-2-69-2': 'Our main conclusions may be summarized as follows.', '1405.3317-2-70-0': 'Our results endorse the view that migration might play a substantial role in the radial redistribution of stars in galaxies that have experienced sustained spiral structure.', '1405.3317-2-70-1': 'Although we have tried to emphasize the results that seem of general applicability, we also recognize that our conclusions are based on a single simulation that, in many ways, fails to capture the fascinating complexity of galaxies like our own Milky Way.', '1405.3317-2-70-2': 'The evolving potential of the forming galaxy; the accretion of gas and the ongoing formation of stars; the effects of satellites and substructure; the presence of a bar and of evolving spirals; all must have left an imprint on the structure of the Galaxy that our simulations cannot attempt to match.', '1405.3317-2-71-0': 'Further, our simulation adopts a disk that is lighter than that of the Milky Way and whose spiral patterns, therefore, might differ from that of the Galaxy .', '1405.3317-2-71-1': 'We cannot therefore rule out that other disturbances may give rise to a thick disk, as argued by [CITATION].', '1405.3317-2-71-2': 'A more massive disk would probably sport spirals with fewer arms that might lead to faster heating.', '1405.3317-2-71-3': 'Although we do not believe that this would affect qualitatively our conclusions, a more definitive, quantitative comparison with observation should be attempted with models that reproduce better the known structural parameters of the Galaxy.', '1405.3317-2-71-4': 'Still, we believe that simplified models like the one we discuss here are useful, since they allow us to isolate and understand the effects of radial migration so as to inform the analysis of more ambitious modeling.', '1405.3317-2-72-0': '2em', '1405.3317-2-73-0': 'This work is funded by NSF grant No. 1211258 and NASA grant No. 13-ATP13-0053.', '1405.3317-2-73-1': 'ED gratefully acknowledges the support of the Alfred P. Sloan Foundation.', '1405.3317-2-73-2': 'CV-C, ED and JN express their appreciation towards the Aspen Center for Physics for their hospitality.', '1405.3317-2-73-3': 'MA and JN acknowledge the support of ANPCyT, Argentina by grant No. PICT 2012-1137.', '1405.3317-2-73-4': 'This work was supported in part by the National Science Foundation under grant No. PHYS-1066293.', '1405.3317-2-73-5': 'This simulation has been run on Odyssey at Harvard and on the High Performance Computing cluster provided by the Advanced Computing Infrastructure (ACI) and Center for High Throughput Computing (CHTC) at the University of Wisconsin.', '1405.3317-2-73-6': 'This work was supported in part by the National Science Foundation under Grant No. PHYS-1066293.', '1405.3317-2-74-0': '2em'}

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[]

[['1405.3317-1-57-3', '1405.3317-2-58-3'], ['1405.3317-1-68-0', '1405.3317-2-69-0'], ['1405.3317-1-0-2', '1405.3317-2-0-2'], ['1405.3317-1-13-1', '1405.3317-2-14-1']]

[]

['1405.3317-1-1-1', '1405.3317-1-2-0', '1405.3317-1-4-0', '1405.3317-1-71-0', '1405.3317-1-72-0', '1405.3317-1-72-3', '1405.3317-1-72-4', '1405.3317-1-73-0', '1405.3317-2-1-1', '1405.3317-2-3-0', '1405.3317-2-4-0', '1405.3317-2-5-0', '1405.3317-2-72-0', '1405.3317-2-73-0', '1405.3317-2-73-3', '1405.3317-2-73-4', '1405.3317-2-73-6', '1405.3317-2-74-0']

{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1405.3317